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Nonlinear Controlof

Dynamic Networks

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AUTOMATION AND CONTROL ENGINEERINGA Series of Reference Books and Textbooks

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Nonlinear Control of Dynamic Networks, Tengfei Liu; Zhong-Ping Jiang; David J. Hill

Modeling and Control for Micro/Nano Devices and Systems, Ning Xi; Mingjun Zhang; Guangyong Li

Linear Control System Analysis and Design with MATLAB®, Sixth Edition, Constantine H. Houpis; Stuart N. Sheldon

Real-Time Rendering: Computer Graphics with Control Engineering, Gabriyel Wong; Jianliang Wang

Anti-Disturbance Control for Systems with Multiple Disturbances, Lei Guo; Songyin Cao

Tensor Product Model Transformation in Polytopic Model-Based Control, Péter Baranyi; Yeung Yam; Péter Várlaki

Fundamentals in Modeling and Control of Mobile Manipulators, Zhijun Li; Shuzhi Sam Ge

Optimal and Robust Scheduling for Networked Control Systems, Stefano Longo; Tingli Su; Guido Herrmann; Phil Barber

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Quantitative Process Control Theory, Weidong Zhang

Classical Feedback Control: With MATLAB® and Simulink®, Second Edition, Boris Lurie; Paul Enright

Intelligent Diagnosis and Prognosis of Industrial Networked Systems, Chee Khiang Pang; Frank L. Lewis; Tong Heng Lee; Zhao Yang Dong

Synchronization and Control of Multiagent Systems, Dong Sun

Subspace Learning of Neural Networks, Jian Cheng; Zhang Yi; Jiliu Zhou

Reliable Control and Filtering of Linear Systems with Adaptive Mechanisms, Guang-Hong Yang; Dan Ye

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Sliding Mode Control in Electro-Mechanical Systems, Second Edition, Vadim Utkin; Juergen Guldner; Jingxin Shi

Autonomous Mobile Robots: Sensing, Control, Decision Making and Applications, Shuzhi Sam Ge; Frank L. Lewis

Linear Control Theory: Structure, Robustness, and Optimization, Shankar P. Bhattacharyya; Aniruddha Datta; Lee H.Keel

Optimal Control: Weakly Coupled Systems and Applications, Zoran Gajic

Deterministic Learning Theory for Identification, Recognition, and Control, Cong Wang; David J. Hill

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Nonlinear Controlof

Dynamic Networks Tengfe i L iu

P o l y t e c h n i c I n s t i t u t e o f N e w Yo r k U n i v e r s i t yD e p a r t m e n t o f E l e c t r i c a l & C o m p u t e r E n g i n e e r i n g

Zhong-P ing J iangP o l y t e c h n i c I n s t i t u t e o f N e w Yo r k U n i v e r s i t y

D e p a r t m e n t o f E l e c t r i c a l & C o m p u t e r E n g i n e e r i n g

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D e p a r t m e n t o f E l e c t r i c a l & E l e c t r o n i c E n g i n e e r i n g

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Dedication

This work is dedicated to

Lina and Debbie (TFL)Xiaoming, Jenny, and Jack (ZPJ)

Gloria (DJH)

Contents

Chapter 1 Introduction ...........................................................................1

1.1 Control Problems with Dynamic Networks....................11.2 Lyapunov Stability.........................................................41.3 Input-to-State Stability..................................................81.4 Input-to-Output Stability ............................................151.5 Input-to-State Stabilization and an Overview of the

Book.............................................................................16

Chapter 2 Interconnected Nonlinear Systems .......................................19

2.1 Trajectory-Based Small-Gain Theorem........................212.2 Lyapunov-Based Small-Gain Theorem.........................262.3 Small-Gain Control Design ..........................................302.4 Notes ............................................................................36

Chapter 3 Large-Scale Dynamic Networks............................................39

3.1 Continuous-Time Dynamic Networks...........................423.2 Discrete-Time Dynamic Networks................................543.3 Hybrid Dynamic Networks...........................................633.4 Notes ............................................................................75

Chapter 4 Control under Sensor Noise .................................................79

4.1 Static State Measurement Feedback Control ...............804.2 Dynamic State Measurement Feedback Control ..........934.3 Decentralized Output Measurement Feedback

Control ....................................................................... 1014.4 Event-Triggered and Self-Triggered Control .............. 1164.5 Synchronization under Sensor Noise .......................... 1314.6 Application: Robust Adaptive Control ...................... 1374.7 Notes .......................................................................... 139

Chapter 5 Quantized Nonlinear Control ............................................. 143

5.1 Static Quantization: A Sector Bound Approach ........ 1445.2 Dynamic Quantization ............................................... 1575.3 Quantized Output-Feedback Control ......................... 1805.4 Notes .......................................................................... 190

ix

x Contents

Chapter 6 Distributed Nonlinear Control ........................................... 193

6.1 A Cyclic-Small-Gain Result in Digraphs.................... 1966.2 Distributed Output-Feedback Control ....................... 1986.3 Formation Control of Nonholonomic Mobile Robots . 2076.4 Distributed Control with Flexible Topologies ............ 2246.5 Notes .......................................................................... 250

Chapter 7 Conclusions and Future Challenges ................................... 255

Appendix A Related Notions in Graph Theory ..................................... 261

Appendix B Systems with Discontinuous Dynamics .............................. 263

B.1 Basic Definitions ........................................................ 263B.2 Extended Filippov Solution ....................................... 264B.3 Input-to-State Stability.............................................. 265B.4 Large-Scale Dynamic Networks of Discontinuous

Subsystems................................................................. 266

Appendix C Technical Lemmas Related to Comparison Functions........ 269

Appendix D Proofs of the Small-Gain Theorems 2.1, 3.2 and 3.6.......... 273

D.1 A Useful Technical Lemma ........................................ 273D.2 Proof of Theorem 2.1: The Asymptotic Gain

Approach.................................................................... 273D.3 Sketch of Proof of Theorem 3.2 ................................. 275D.4 Proof of Theorem 3.6 ................................................. 279

Appendix E Proofs of Technical Lemmas in Chapter 4 ......................... 285

E.1 Proof of Lemma 4.2 ................................................... 285E.2 Proof of Lemma 4.3 ................................................... 286E.3 Proof of Lemma 4.5 ................................................... 287E.4 Proof of Lemma 4.6 ................................................... 289

Appendix F Proofs of Technical Lemmas in Chapter 5 ......................... 293

F.1 Proof of Lemma 5.1 ................................................... 293F.2 Proof of Lemma 5.3 ................................................... 295F.3 Proof of Lemma 5.4 ................................................... 297F.4 Proof of Lemma 5.5 ................................................... 298F.5 Proof of Lemma 5.8 ................................................... 303

References.................................................................................. 305

Contents xi

Index.......................................................................................... 321

List of Figures

1.1 Feedback control system........................................................................11.2 Robust control configuration. ................................................................21.3 An interconnected system. ....................................................................31.4 A high-order nonlinear system as a dynamic network...........................41.5 Asymptotic gain property......................................................................91.6 Equivalence between ISS and the existence of an ISS-Lyapunov

function. ..............................................................................................14

2.1 An interconnected system with no external input...............................192.2 An interconnected system with external inputs. .................................222.3 Definitions of sets A, B and O. ...........................................................272.4 Small-gain-based recursive control design. ..........................................35

3.1 The block diagram of the closed-loop system in Example 3.1.............403.2 The gain digraph of the dynamic network (3.8). .................................413.3 The replacement of the ISS gains. .......................................................443.4 The j subsystems on a specified simple path ma ending at Via1 . .........473.5 A disconnected gain digraph. ..............................................................513.6 A time-delay component......................................................................543.7 An example of the gain margin property of discrete-time systems......563.8 The gain digraph of the dynamic network in Example 3.5..................623.9 The evolutions of VΠ and uΠ of the dynamic network in Example 3.5.633.10 The equivalence between cyclic-small-gain and gains less than Id. .....66

4.1 The block diagram of a measurement feedback control system...........794.2 Boundaries of set-valued map S1 and the definition of e2. ..................884.3 The gain digraph of the e-system........................................................914.4 The estimator for ei: the ei-subsystem................................................954.5 The interconnection with each ei-system. ...........................................964.6 The gain digraph of the closed-loop system. .......................................984.7 The sensor noise and the system states............................................. 1024.8 The estimator states and the control input....................................... 1024.9 The block diagram of the large-scale system (4.141)–(4.146). ........... 1034.10 State trajectories of Example 4.5. ..................................................... 1174.11 Control signals and disturbances of Example 4.5. ............................. 1174.12 The block diagram of an event-triggered control system................... 1184.13 Event-triggered control problem as a robust control problem. .......... 1184.14 An illustration of Θ1(x(tk)) ⊆ Θ2(x(tk))........................................... 121

xiii

xiv List of Figures

5.1 The block diagram of a quantized control system. ............................ 1435.2 Two examples of quantizers. ............................................................. 1435.3 A truncated logarithmic quantizer. ................................................... 1455.4 Quantized control as a robust control problem. ................................ 1465.5 Set-valued map S1 and the definition of e2. ...................................... 1495.6 The gain interconnection graph of the closed-loop quantized system.1555.7 State trajectories of the example in Subsection 5.1.4. ....................... 1585.8 Control input of the example in Section 5.1.4. .................................. 1585.9 A uniform quantizer q with a finite number of levels. ....................... 1595.10 Basic idea of dynamic quantization................................................... 1605.11 The quantized control structure for high-order nonlinear systems. ... 1645.12 Three-level uniform quantizer with M = 3........................................ 1655.13 The gain digraph of the e-system...................................................... 1705.14 Motions of Θ(t) and W (t) =W (X(t),Θ(t)) in the zooming-in stage.1795.15 Quantized output-feedback control.................................................... 190

6.1 A multi-vehicle system. ..................................................................... 1946.2 An example of information exchange digraph. .................................. 1956.3 The block diagram of each controlled agent i. .................................. 2016.4 Kinematics of the unicycle robot....................................................... 2086.5 An example for φxi and φyi............................................................... 2186.6 The position sensing graph of the formation control system............. 2236.7 The velocities of the robots. .............................................................. 2246.8 The stages of the distributed controllers. .......................................... 2246.9 The trajectories of the robots............................................................ 2256.10 Property 1 of Proposition 6.2. ........................................................... 2286.11 The motion of the point (η, ζ) and the rigid body ζ = ψ(η − µ). ..... 2336.12 Digraphs representing the switching information exchange topology.2496.13 The switching sequence of the information exchange topology. ........ 2506.14 The linear velocities and angular velocities of the robots.................. 2506.15 The stages of the distributed controllers. .......................................... 2516.16 The trajectories of the robots............................................................ 251

Preface

The rapid development of computing, communications, and sensing technolo-gies has been enabling new potential applications of advanced control of com-plex systems like smart power grids, biological processes, distributed com-puting networks, transportation systems, and robotic networks. Significantproblems are to integrally deal with the fundamental system characteristicssuch as nonlinearity, dimensionality, uncertainty, and information constraints,and diverse kinds of networked behaviors, which may arise from quantization,data sampling, and impulsive events.

Physical systems are inherently nonlinear and interconnected in nature.Significant progress has been made on nonlinear control systems in the pastthree decades. However, new system analysis and design tools that are capa-ble of addressing more communication and networking issues are still highlydesired to handle the emerging theoretical challenges underlying the new en-gineering problems. As an example, small quantization errors may cause theperformance of a “well-designed” nonlinear control system to deteriorate. Theneed for new tools motivates this book, the purpose of which is to present aset of novel analysis and design tools to address the newly arising theoreticalproblems from the viewpoint of dynamic networks. The results are intended tohelp solve real-world nonlinear control problems, including quantized controland distributed control aspects.

In this book, dynamic networks are regarded as systems composed of struc-turally interconnected subsystems. Such systems often display complex dy-namic behaviors. The control problem of such a complex system could besimplified with the notion of a dynamic network if the subsystems have somecommon characteristic which, together with the structural feature of the dy-namic network, can guarantee the achievement of the control objective. Forthe research in this book, one such characteristic is Sontag’s input-to-statestability (ISS), based on which, refined small-gain theorems are extremelyuseful in solving control problems of complex systems by taking advantage ofthe structural feature.

By bridging the gap between the stability concepts defined in the input–output and the state–space contexts, the notion of ISS has proved to be ex-tremely useful in analysis and control design of nonlinear systems with theinfluence of external inputs represented by nonlinear gains. Its essential rela-tionship with robust stability provides an effective approach to robust controlby means of input-to-state stabilization. For a dynamic network of ISS subsys-tems, the small-gain theorem is capable of testing the overall ISS by directlychecking compositions of the ISS gains of the subsystems. Based on the ISSsmall-gain theorem, complex systems can be input-to-state stabilized by ap-propriately designing the subsystems.

xv

xvi Preface

This book is based on the authors’ recent research results on nonlinear con-trol of dynamic networks. In particular, it contains refined small-gain resultsfor dynamic networks and their applications in solving the control problemsof nonlinear uncertain systems subject to sensor noise, quantization error, andinformation exchange constraints. The widely known Lyapunov functions ap-proach is mainly used for proofs and discussions. The relationship betweenthe new tools and the existing nonlinear control methods is highlighted. Inthis way, not only control researchers but also students interested in relatedtopics may understand and use the tools for control designs.

The organization of the book is as follows. To make the book self-contained,Chapter 1 provides some prerequisite knowledge on useful characteristics ofLyapunov stability and ISS. Chapter 2 presents ISS small-gain results forinterconnected systems composed of two subsystems. Both trajectory-basedand Lyapunov-based formulations of the ISS small-gain theorem are reviewedwith proofs. For dynamic networks that may contain more than two subsys-tems, Chapter 3 introduces more readily usable cyclic-small-gain methods toreduce the complexity of analysis and control design problems for more gen-eral dynamic networks. Detailed proofs of some of the background theoremsin Chapters 1–3, which need a higher level of mathematical sophistication andare available in the literature, are not provided. However, the basic ideas arehighlighted.

The applications of the cyclic-small-gain theorem to nonlinear control de-signs are studied in Chapters 4–6. Specifically, Chapter 4 investigates the im-portant measurement feedback control problem for uncertain nonlinear sys-tems with disturbed measurements. In Chapter 5, the quantized nonlinearcontrol problem is studied. Chapter 6 discusses the distributed control prob-lem for coordination of groups of nonlinear systems under information ex-change constraints. The control problems are transformed into input-to-statestabilization problems of dynamic networks, and the influence of the uncertainsources, i.e., sensor noise, quantization, and information exchange constraints,are explicitly evaluated and attenuated by new cyclic-small-gain designs.

Certainly, most of the results presented in this book can be extended formore general systems. Some of the easier extensions mentioned in the book arenot thoroughly discussed and may be used as exercises for interested readers.Several future challenges in this research direction are outlined in Chapter7. The Appendix gives supplementary materials on graph theory and dis-continuous systems, and the proofs of the technical lemmas which seem toomathematical to be placed in the main chapters. Finally, historical discussionwill be confined to brief notes at appropriate points in the text.

TFL wishes to express his sincere gratitude to his coauthors, ProfessorZhong-Ping Jiang and Professor David Hill, who are also TFL’s postdoctoraladviser and PhD supervisor, respectively. They introduced TFL to nonlinearcontrol, drew his attention to ISS and small-gain, and offered him preciousopportunities for working on the frontier research subjects in the field. Their

Preface xvii

persistent support, expert guidance, and willingness to share wisdom havebeen invaluable for TFL’s academic career. TFL would also like to give thanksto his previous and current labmates in Canberra and New York. He hasbenefited a lot from the discussions/debates with them. TFL would needmore than a lifetime to thank his wife, Lina Zhang, for her understanding andpatience and their daughter, Debbie Liu, for a lot of happiness.

ZPJ would like to thank, from the bottom of his heart, all his coauthors andfriends for sharing their passion for nonlinear control. The ideas and methodspresented in this book truly reflect their wisdom and vision for the nonlinearcontrol of dynamic networks. Special thanks go to Iven Mareels, Laurent Praly(his former PhD adviser), Andy Teel, and Yuan Wang for collaborations onthe very first, nonlinear ISS small-gain theorems, and to Hiroshi Ito, IassonKarafyllis, Pierdomenico Pepe, and again Yuan Wang for recent joint work onvarious extensions of the small-gain theorem for dynamic networks. Finally,it is only under the strong and constant support and love of his family thatZPJ can discover the beauty of nonlinear feedback and control, while havingfun doing research.

DJH firstly thanks his coauthors for their hard work and collaborationthroughout the research leading to this book. It has been a pleasure to see theideas of state-space-based small-gain theorems progress through all our PhDtheses, as well as work with colleagues—with special mention of Iven Mareels,and now into this book. (As a memorial note, his thesis was over 30 years ago,following the seminal work on dissipative systems by Jan Willems, who sadlypassed away during our writing.) Personally, DJH would like to thank his wife,Gloria Sunnie Wright, whose positive supportive approach and excitement forlife are a perfect match for an academic who (as she often hears) has “got torun” for deadlines.

The authors are grateful to the series editors, Frank Lewis and Sam Ge, forthe opportunity to publish the book. The authors would also like to thank theeditorial staff, in particular, Nora Konopka, Michele Smith, Amber Donley,Michael Davidson, John Gandour, and Shashi Kumar, of Taylor & Francis fortheir efforts in publishing the book.

The research presented in this book was supported partly by the NYU-Poly Faculty Fellowship provided to the first author during his visit at thePolytechnic Institute of New York University, partly by the U.S. NationalScience Foundation and by the Australian Research Council.

Tengfei Liu New York, USAZhong-Ping Jiang New York, USADavid J. Hill Hong Kong, China

Author Biographies

Dr. Tengfei Liu received a B.E. degree in automation and a M.E. degree incontrol theory and control engineering from South China University of Tech-nology, in 2005 and 2007, respectively. He received a Ph.D. in engineering fromthe Australian National University in 2011. Tengfei Liu is a visiting assistantprofessor at the Polytechnic Institute of New York University. His researchinterests include stability theory, robust nonlinear control, quantized control,distributed control, and their applications in mechanical systems, power sys-tems, and transportation systems.

Dr. Liu, with Z. P. Jiang and D. J. Hill, received the Guan Zhao-Zhi BestPaper Award at the 2011 Chinese Control Conference.

Professor Zhong-Ping Jiang received a B.Sc. degree in mathematics fromthe University of Wuhan, Wuhan, China, in 1988, a M.Sc. degree in statisticsfrom the University of Paris XI, France, in 1989, and a Ph.D. degree in auto-matic control and mathematics from the Ecole des Mines de Paris, France, in1993.

Currently, he is a professor of electrical and computer engineering at thePolytechnic School of Engineering of New York University. His main researchinterests include stability theory, robust and adaptive nonlinear control, adap-tive dynamic programming, and their applications to underactuated mechan-ical systems, communication networks, multi-agent systems, smart grid, andsystems neuroscience. He is coauthor of the book Stability and Stabilizationof Nonlinear Systems (with Dr. I. Karafyllis, Springer 2011).

A Fellow of both the IEEE and IFAC, Dr. Jiang is an editor for the Interna-tional Journal of Robust and Nonlinear Control and has served as an associateeditor for several journals including Mathematics of Control, Signals and Sys-tems (MCSS), Systems and Control Letters, IEEE Transactions on AutomaticControl, European Journal of Control, and Science China: Information Sci-ences. Dr. Jiang is a recipient of the prestigious Queen Elizabeth II FellowshipAward from the Australian Research Council, the CAREER Award from theU.S. National Science Foundation, and the Young Investigator Award fromthe NSF of China. He received the Best Theory Paper Award (with Y. Wang)at the 2008 WCICA, and with T. Liu and D.J. Hill, the Guan Zhao ZhiBest Paper Award at the 2011 CCC. The paper with his PhD student Y.Jiang entitled “Robust Adaptive Dynamic Programming for Optimal Nonlin-ear Control Design” received the Shimemura Young Author Prize at the 2013

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xx Author Biographies

Asian Control Conference in Istanbul, Turkey.

Professor David J. Hill received BE (electrical engineering) and BSc (math-ematics) degrees from the University of Queensland, Australia, in 1972 and1974, respectively. He received a PhD degree in electrical engineering from theUniversity of Newcastle, Australia, in 1977.

He holds the Chair of Electrical Engineering in the Department of Electricaland Electronic Engineering at the University of Hong Kong. He is also a part-time professor in the Centre for Future Energy Networks at the University ofSydney, Australia. During 2005–2010, he was an Australian Research CouncilFederation Fellow at the Australian National University and, from 2006, also achief investigator and theme leader (complex networks) in the ARC Centre ofExcellence for Mathematics and Statistics of Complex Systems. Since 1994, hehas held various positions at the University of Sydney, Australia, including theChair of Electrical Engineering until 2002 and again during 2010–2013 alongwith an ARC Professorial Fellowship. He has also held academic and substan-tial visiting positions at the universities of Melbourne, California (Berkeley),Newcastle (Australia), Lund (Sweden), Munich, and Hong Kong (City andPolytechnic). During 1996–1999 and 2001–2004, he served as head of the re-spective departments in Sydney and Hong Kong. He currently holds honoraryprofessorships at City University of Hong Kong, South China University ofTechnology, Wuhan University, and Northeastern University, China.

His general research interests are in control systems, complex networks,power systems, and stability analysis. His work is now mainly on control andplanning of future energy networks and basic stability questions for dynamicnetworks.

Professor Hill is a Fellow of the Institute of Electrical and Electronics En-gineers, USA, the Society for Industrial and Applied Mathematics, USA, theAustralian Academy of Science, and the Australian Academy of TechnologicalSciences and Engineering. He is also a foreign member of the Royal SwedishAcademy of Engineering Sciences.

Notations

C The set of complex numbersR The set of real numbersR+ The set of nonnegative real numbersRn The n-dimensional Euclidean spaceZ The set of integersZ+ The set of nonnegative integersN The set of natural numbersxT The transpose of vector x|x| Euclidean norm of vector x|A| Induced Euclidean norm of matrix Asgn(x) The sign of x ∈ R: sgn(x) = 1 if x > 0; sgn(x) = 0 if

x = 0; sgn(x) = −1 if x < 0a mod b Remainder of the Euclidean division of a by b for a ∈

R, b ∈ R\0‖u‖∆ ess supt∈∆ |u(t)| with ∆ ⊆ R+ for u : R+ → Rn

‖u‖∞ ‖u‖∆ with ∆ = [0,∞)

:= ordef= Equal by definition

≡ Identically equalf g Composition of functions f and gλmax (λmin) Largest (smallest) eigenvaluet+ (t−) Time right after (right before) t∂ Partial derivative∇V (x) Gradient vector of function V at xId The identity functionBn The unit ball centered at the origin in Rn

cl(S) The closure of set Sint(S) The interior of set Sco(S) The convex hull of set Sco(S) The closed convex hull of set Sdom(F ) The domain of map Fgraph(F ) The graph of map Frange(F ) The range of map F

ABBREVIATIONS

AG Asymptotic GainAS Asymptotic StabilityGAS Global Asymptotic StabilityGS Global StabilityIOpS Input-to-Output Practical Stability

xxi

xxii Notations

IOS Input-to-Output StabilityISpS Input-to-State Practical StabilityISS Input-to-State StabilityOAG Output Asymptotic GainRS Robust StabilityUBIBS Uniform Bounded-Input Bounded-State StabilityUO Unboundedness ObservabilityWRS Weakly Robust Stability

1 Introduction

1.1 CONTROL PROBLEMS WITH DYNAMIC NETWORKS

The basic idea for control of dynamic networks is to consider complex sys-tems as structural interconnections of subsystems with specific properties,and solve their control problems using the subsystem and structural features.Such ideas can be traced back to the original development of circuit theory.The rapid development of computing, communication, and sensing technologyhas enabled new potential applications of advanced control to complex sys-tems. Significant problems are to integrally deal with the fundamental systemcharacteristics, such as nonlinearity, dimensionality, uncertainty and informa-tion constraints, and diverse kinds of networked behaviors like quantization,data sampling, and impulsive events. With the development of new tools, thisbook studies the analysis and control problems of complex systems from theviewpoint of dynamic networks.

Even the single-loop control system may be considered as a dynamic net-work if detailed behaviors of the sensor and the actuator are taken into ac-count. In a typical single-loop state-feedback control system, as shown inFigure 1.1, the state of the plant is measured by the sensor and sent to thecontroller, which computes the needed control actions. These are implementedby the actuator for a desired behavior of the plant. A key issue with controlsystems is stability. By designing an asymptotically stable control system,the error between the actual state signal and a desired signal is expected toconverge to zero ultimately.

controller

plant

sensoractuator

xm

u

ud

x

FIGURE 1.1 State-feedback control system: x is the state of the plant, u is the

control input, xm is the measurement of x, and ud is the desired control input

computed by the controller.

Practical control systems are inevitably subject to uncertainties, whichmay be caused by the sensing and actuation components, and the unmodeleddynamics of the plant. By considering a control system as an interconnection ofthe perturbation-free nominal system and the perturbation terms, the basic

1

2 Nonlinear Control of Dynamic Networks

idea of robust control is to design the nominal system to be robust to theperturbations.

Based on this idea, a linear state-feedback control system

x = Ax +Bu (1.1)

ud = −Kxm (1.2)

can be rewritten as the closed-loop nominal system with the perturbationterms:

x = Ax+B(−K(x+ x)− u)

= (A−BK)x−BKx−Bu, (1.3)

where x = xm−x and u = ud−u. Suppose that the control objective is to makethe system practically stable at the origin, i.e., to steer the state x to withina specific bounded neighborhood of the origin. If x, u are bounded, then suchan objective can be achieved if (A−BK) is Hurwitz, i.e., all the eigenvaluesof (A−BK) are on the open left-half of the complex plane. For such a linearsystem, we can study the influence of u and x separately, due to the well-knownSuperposition Principle. If the eigenvalues of (A − BK) can be arbitrarilyassigned by an appropriate choice of K (with complex eigenvalues occurringin conjugate pairs), then the influence of u can be attenuated to within anarbitrarily small level. But this may not be the case for the perturbation termBKx (because it depends on K).

controller

plant

Λu Λx

xm

u

ud

x

nominal system

FIGURE 1.2 Robust control configuration, where Λu,Λx represent perturbation

terms.

The problem can still be handled even if the perturbation terms may notbe bounded. For example, they can satisfy the following properties:

|x| ≤ δx|x|+ cx (1.4)

|u| ≤ δu|ud|+ cu, (1.5)

where δx, cx, δu, cu are nonnegative constants. Such perturbations are said tohave the sector bound property.

Introduction 3

One may denote x = δx(t)x + cx(t) and u = δu(t)K(1 + δx(t))x +δu(t)Kcx(t) + cu(t) with |δx(t)| ≤ δx, |δu(t)| ≤ δu, |cx(t)| ≤ cx, |cu(t)| ≤ cufor t ≥ 0. Then, system (1.3) can be represented by x = (A−BK)x+w with

w(t) = −BK(δx(t) + δu(t)(1 + δx(t)))x(t)

−B(Kcx(t) +Kδu(t)cx(t) + cu(t))

:= φ(x(t), cx(t), cu(t), t). (1.6)

It can be directly checked that |φ(x, cx, cu, t)| ≤ a1|x|+ a2|cx|+ a3|cu| for allt ≥ 0 with constants a1, a2, a3 ≥ 0.

As shown in Figure 1.3, the system is transformed into the interconnectionof the nominal system and the perturbation term. There have been standardmethods to solve this kind of problem in robust linear control theory [288].One of them is the classical small-gain theorem, due to Sandberg and Zames.Interested readers may consult [48, Chapter 5] and [54, Chapter 4] for thedetails. See also [207, Section V] for a small-gain result of large-scale systems.

w = φ(x, cx, cu, t)

x = (A−BK)x+ w

cxcuw

x

FIGURE 1.3 An interconnected system.

For genuinely nonlinear control systems, the problems discussed above willtypically be more complicated. Consider the popular strict-feedback nonlinearsystem:

xi = ∆i(xi, w) + xi+1, i = 1, . . . , n− 1 (1.7)

xn = ∆n(xn, w) + u, (1.8)

where [x1, . . . , xn]T := x ∈ Rn is the state, xi = [x1, . . . , xi]

T , u ∈ R is thecontrol input, w ∈ Rnw represents the external disturbances, and ∆i : R

i → R

for i = 1, . . . , n are locally Lipschitz functions. For this system, we considerx1 as the output. Recursive designs have proved to be useful for the controlof such system; see e.g., [153, 235, 151]. By representing the system as adynamic network composed of xi-subsystems for i = 1, . . . , n, the basic ideais to recursively design control laws for the xi-subsystems by considering xi+1

as the control inputs until the true control input u occurs. For such system,the influence of the disturbance w might be amplified through the numerousinterconnections between the subsystems as shown in Figure 1.4. The problemwould be more complicated if the system is subject to sensor noise. As shown


in Example 4.1, even for a first-order nonlinear system, small sensor noise maydrive the system state to infinity, although the state of the noise-free systemasymptotically converges to the origin. Quantized control provides anotherinteresting example for robust control of nonlinear systems with measurementerrors satisfying the sector bound property; see Section 5.1 for details.

x1 xi xi+1 xn· · · · · ·

FIGURE 1.4 A high-order nonlinear system as a dynamic network.

Dynamic networks also occur in distributed control of interconnected sys-tems, for which each i-th subsystem takes the following form:

xi = fi(xi, ui) (1.9)

yi = hi(xi), (1.10)

where yi ∈ Rpi is the output, xi ∈ Rni is the state, ui ∈ Rmi is the controlinput, and fi : R

mi+ni → Rni , hi : Rni → Rpi are properly defined functions.

In a distributed control structure, each subsystem may be equipped with acontroller. Through information exchange, the controllers for the subsystemscoordinate with each other, and the outputs of the subsystems achieve somedesired group behavior, e.g., limt→∞(yi(t)−yj(t)) = 0. In this case, the inter-connections in the dynamic network are formed by the information exchangebetween the controllers, and in some cases, e.g., power systems and telephonenetworks, by direct physical interconnections.

This book develops new design tools for nonlinear control of dynamic net-works, which are applicable to measurement feedback control, quantized con-trol, and distributed control. With the new tools, the related control problemscan be transformed into solvable stability problems of dynamic networks com-posed of subsystems admitting the input-to-state stability (ISS) property orthe more general input-to-output stability (IOS) property. To introduce thesebasic notions, we begin with Lyapunov stability for systems without externalinputs.

1.2 LYAPUNOV STABILITY

The stabilization problem is one of the most important problems in controltheory. In general terms, a control system is stabilizable if one can find a

Introduction 5

control law which makes the closed-loop system stable at an equilibrium point.This section reviews some basic concepts of Lyapunov stability [195, 78, 144]for systems with no external inputs.

The comparison functions defined below are used to characterize Lyapunovstability and the notions of ISS and IOS.

Definition 1.1 A function α : R+ → R+ is said to be positive definite ifα(0) = 0 and α(s) > 0 for s > 0.

Definition 1.2 A continuous function α : R+ → R+ is said to be a class Kfunction, denoted by α ∈ K, if it is strictly increasing and α(0) = 0; it is saidto be a class K∞ function, denoted by α ∈ K∞, if it is a class K function andsatisfies α(s) → ∞ as s→ ∞.

Definition 1.3 A continuous function β : R+ × R+ → R+ is said to be aclass KL function, denoted by β ∈ KL, if, for each fixed t ∈ R+, functionβ(·, t) is a class K function and, for each fixed s ∈ R+, function β(s, ·) isdecreasing and satisfies limt→∞ β(s, t) = 0.

For convenience of the further discussions, we also give the following defi-nitions on Lipschitz continuity.

Definition 1.4 A function h : X → Y with X ⊆ Rn and Y ⊆ R

m is said tobe Lipschitz continuous, or simply Lipschitz, on X , if there exists a constantLh ≥ 0, such that for any x1, x2 ∈ X ,

|h(x1)− h(x2)| ≤ Lh|x1 − x2|. (1.11)

Definition 1.5 A function h : X → Y with X ⊆ Rn being open and con-nected, and Y ⊆ Rm is said to be locally Lipschitz on X , if each x ∈ X has aneighborhood X0 ⊆ X such that h is Lipschitz on X0.

Definition 1.6 A function h : X → Y with X ⊆ Rn and Y ⊆ Rm is said tobe Lipschitz on compact sets, if h is Lipschitz on every compact set D ⊆ X .

Consider the system

x = f(x), (1.12)

where f : Rn → Rn is a locally Lipschitz function. Assume that the origin isan equilibrium of the nonlinear system, i.e., f(0) = 0. Note that if an equi-librium other than the origin, say xe, is of interest, one may use a coordinatetransformation x′ = x− xe to move the equilibrium to the origin. Therefore,the assumption of the equilibrium at the origin is with no loss of generality.Denote x(t, x0) or simply x(t) as the solution of system (1.12) with initial con-dition x(0) = x0, and let [0, Tmax) with 0 < Tmax ≤ ∞ be the right maximalinterval for the definition of x(t, x0).


The standard definition of Lyapunov stability is usually given by using “ǫ-δ”terms, which can be found in the standard textbooks on nonlinear systems;see, e.g., [78] and [144, Chapter 4]. Definition 1.7 employs the comparisonfunctions α ∈ K and β ∈ KL for convenience of the comparison betweenLyapunov stability and ISS. A proof of the equivalence between the standarddefinition and Definition 1.7 can be found in [144, Appendix C.6]. See alsothe discussions in [78, Definitions 2.9 and 24.2].

Definition 1.7 System (1.12) is

• stable at the origin if there exist an α ∈ K and a constant c > 0 suchthat for any |x0| ≤ c,

|x(t, x0)| ≤ α(|x0|) (1.13)

for all t ≥ 0;• globally stable (GS) at the origin if property (1.13) holds for all initial

states x0 ∈ Rn;

• asymptotically stable (AS) at the origin if there exist a β ∈ KL anda constant c > 0 such that for any |x0| ≤ c,

|x(t, x0)| ≤ β(|x0|, t) (1.14)

for all t ≥ 0;• globally asymptotically stable (GAS) at the origin if condition (1.14)

holds for any initial state x0 ∈ Rn.

With the standard definition, GAS at the origin can be defined based on GSby adding the global convergence property at the origin: limt→∞ x(t, x0) = 0for all x0 ∈ Rn; see [144, Definition 4.1]. It can be observed that GAS is morethan global convergence.

Theorem 1.1, which is known as Lyapunov’s Second Theorem (or the Lya-punov Direct Method), gives sufficient conditions for stability and AS.

Theorem 1.1 Let the origin be an equilibrium of system (1.12) and Ω ⊂Rn be a domain containing the origin. Let V : Ω → R+ be a continuouslydifferentiable function such that

V (0) = 0, (1.15)

V (x) > 0 for x ∈ Ω\0, (1.16)

∇V (x)f(x) ≤ 0 for x ∈ Ω. (1.17)

Then, system (1.12) is stable at the origin. Moreover, if

∇V (x)f(x) < 0 for x ∈ Ω\0, (1.18)

then system (1.12) is AS at the origin.

Introduction 7

A function V that satisfies (1.15)–(1.17) is called a Lyapunov function. Ifmoreover, V satisfies (1.18), then it is called a strict Lyapunov function [16].

It is natural to ask whether the condition for AS in Theorem 1.1 can guar-antee GAS by directly replacing the Ω with Rn. Example 1.1, which was givenin [78, p. 109], answers this question.

Example 1.1 Consider system

x1 =−6x1

(1 + x21)2+ 2x2, (1.19)

x2 =−2(x1 + x2)

(1 + x21)2. (1.20)

Let

V (x) =x21

1 + x21+ x22. (1.21)

It can be directly verified that V satisfies all the conditions for AS at the origingiven by Theorem 1.1 with n = 2 and Ω = R2. By testing the vector field onthe boundary of hyperbola x2 = 2/(x1 −

√2), the trajectories to the right of

the branch in the first quadrant cannot cross that branch. This means that thesystem is not GAS at the origin.

Theorem 1.2 gives extra conditions on the Lyapunov function V for GAS.

Theorem 1.2 Let the origin be an equilibrium of system (1.12). Let V :Rn → R+ be a continuously differentiable function such that

V (0) = 0, (1.22)

V (x) > 0 for x ∈ Rn\0, (1.23)

|x| → ∞ ⇒ V (x) → ∞, (1.24)

∇V (x)f(x) < 0 for x ∈ Rn\0. (1.25)

Then, system (1.12) is globally asymptotically stable at the origin.

According to Theorem 1.2, it is not sufficient to guarantee GAS by simplyreplacing the Ω in the condition for AS in Theorem 1.1 with Rn. Condition(1.24) is also needed for GAS.

Condition (1.22)–(1.24) is equivalent to the statement that V is positivedefinite and radially unbounded, which can be represented with comparisonfunctions α, α ∈ K∞ as

α(|x|) ≤ V (x) ≤ α(|x|) (1.26)


for all x ∈ Rn. Moreover, condition (1.25) is equivalent to the existence of acontinuous and positive definite function α such that

∇V (x)f(x) ≤ −α(V (x)) (1.27)

holds for all x ∈ Rn. See [144, Lemma 4.3] for the details.Theorems 1.1 and 1.2 give sufficient conditions for stability, AS and GAS.

A proof of the converse Lyapunov theorem for the necessity of the conditionscan be found in [144].

1.3 INPUTTOSTATE STABILITY

For systems with external inputs, the notion of input-to-state stability (ISS),invented by Sontag, has proved to be powerful for evaluating the influence ofthe external inputs.

1.3.1 DEFINITION

Consider the system

x = f(x, u), (1.28)

where x ∈ Rn is the state, u ∈ Rm represents the input, and f : Rn × Rm →Rn is a locally Lipschitz function and satisfies f(0, 0) = 0. By consideringthe input u as a function of time, assume that u is measurable and locallyessentially bounded. Recall that u is locally essentially bounded if for anyt ≥ 0, ‖u‖[0,t] exists. Denote x(t, x0, u), or simply x(t), as the solution ofsystem (1.12) with initial condition x(0) = x0 and input u.

In [241], the original definition of ISS is given in the “plus” form; see (1.31).For convenience of discussions, we mainly use the definition in the equivalent“max” form. The equivalence is discussed later.

Definition 1.8 System (1.28) is said to be input-to-state stable (ISS) if thereexist β ∈ KL and γ ∈ K such that for any initial state x(0) = x0 and anymeasurable and locally essentially bounded input u, the solution x(t) satisfies

|x(t)| ≤ maxβ(|x0|, t), γ(‖u‖∞) (1.29)

for all t ≥ 0.

Here, γ is called the ISS gain of the system. Notice that, if u ≡ 0, thenDefinition 1.8 is reduced to Definition 1.7 for GAS at the origin. Due tocausality, x(t) depends on x0 and the past inputs u(τ) : 0 ≤ τ ≤ t, andthus, the ‖u‖∞ in (1.29) can be replaced with ‖u‖[0,t].

Since

maxa, b ≤ a+ b ≤ max(1 + 1/δ)a, (1 + δ)b (1.30)

Introduction 9

for any a, b ≥ 0 and any δ > 0, property (1.29) in the “max” form is equivalentto

|x(t)| ≤ β′(|x0|, t) + γ′(‖u‖∞), (1.31)

where β′ ∈ KL and γ′ ∈ K. It should be noted that although the transfor-mation from (1.29) to (1.31) can be done by directly replacing the “max”operation with the “+” operation without changing functions β and γ, thetransformation from (1.31) to (1.29) may result in a pair of β and γ differentfrom the pair of β′ and γ′. To get a γ very close to γ′, one may choose a verysmall δ for the transformation, but this could result in a very large β.

With property (1.31), x(t) asymptotically converges to within the regiondefined by |x| ≤ γ′(‖u‖∞), i.e.,

limt→∞

|x(t)| ≤ γ′(‖u‖∞). (1.32)

As shown in Figure 1.5, γ′ describes the “steady-state” performance of thesystem, and is usually called the asymptotic gain (AG), while the “transientperformance” is described by β′.

0 t

x(t)

x0

γ′(‖u‖∞)

β′(|x0|, t) + γ′(‖u‖∞)

FIGURE 1.5 Asymptotic gain property.

Intuitively, since only large values of t determine the value limt→∞ |x(t)|,one may replace the γ′(‖u‖∞) in (1.32) with γ′(limt→∞ |u(t)|) orlimt→∞ γ′(|u(t)|). See [250, 247] for more detailed discussions.

When system (1.28) is reduced to a linear system, a necessary and sufficientcondition for the ISS property can be derived.

Theorem 1.3 A linear time-invariant system

x = Ax+Bu (1.33)

is ISS if and only if A is Hurwitz.


Proof. With initial condition x(0) = x0 and input u, the solution of system(1.33) is

x(t) = eAtx0 +

∫ t

0

eA(t−τ)Bu(τ)dτ, (1.34)

which implies

|x(t)| ≤ |eAt||x0|+(∫ ∞

0

|eAτ |dτ)

|B|‖u‖∞. (1.35)

If A is Hurwitz, i.e., every eigenvalue of A has negative real part, then∫∞0

|eAs|ds < ∞. Define β′(s, t) = |eAt|s and γ′(s) =(∫∞

0|eAτ |dτ

)

|B|s fors, t ∈ R+. Clearly, β

′ ∈ KL and γ′ ∈ K∞. Then, the linear system is ISS inthe sense of (1.31). The sufficiency part is proved.

For the necessity, one may consider the case of u ≡ 0. In this case, the ISSof system (1.33) implies GAS of system

x = Ax (1.36)

at the origin. According to linear systems theory [28], system (1.36) is GASat the origin if and only if A is Hurwitz. ♦

Based on the proof of Theorem 1.3, one may consider the ISS property(1.31) as a nonlinear modification of property (1.35) of linear systems. Lemma1.1 shows that any KL function β(s, t) can be considered as a nonlinear mod-ification of function se−t.

Lemma 1.1 For any β ∈ KL, there exist α1, α2 ∈ K∞ such that

β(s, t) ≤ α2(α1(s)e−t) (1.37)

for all s, t ≥ 0.

See [243, Proposition 7] and its proof therein.According to Lemma 1.1, if property (1.31) holds, then there exist α′

1, α′2 ∈

K∞ such that

|x(t)| ≤ α′2(α

′1(|x0|)e−t) + γ′(‖u‖∞), (1.38)

which shows a close analogy of ISS to the solution property (1.35) of linearsystem (1.33) with A being Hurwitz.

Also, with Lemma 1.1, property (1.29) implies

|x(t)| ≤ maxα2(α1(|x0|)e−t), γ(‖u‖∞), (1.39)

where α1, α2 are appropriate class K∞ functions. This means, for any x0and ‖u‖∞ satisfying α2 α1(|x0|) > γ(‖u‖∞), there exists a finite time t∗ =log(α1(|x0|)) − log(α−1

2 γ(‖u‖∞)), after which solution x(t) is within the

Introduction 11

range defined by |x| ≤ γ(‖u‖∞). This shows the difference between the ISSgain γ defined in (1.29) and the asymptotic gain γ′ defined in (1.31).

Theorem 1.3 means that a linear system is ISS if the corresponding input-free system is GAS at the origin. But this may not be true for nonlinearsystems. Consider Example 1.2 given by [243].

Example 1.2 Consider the nonlinear system

x = −x+ ux (1.40)

with x, u ∈ R. If u ≡ 0, then the resulting system x = −x is GAS at the origin.But system (1.40) is not ISS. Just consider the class of constant inputs u > 1.

However, it has been proved that AS at the origin of system (1.28) withu ≡ 0 is equivalent to a local ISS property of system (1.28) [250]. The definitionof local ISS is given by Definition 1.9.

Definition 1.9 System (1.28) is said to be locally input-to-state stable if thereexist β ∈ KL, γ ∈ K, and constants ρx, ρu > 0 such that for any initial statex(0) = x0 satisfying |x0| ≤ ρx and any measurable and locally essentiallybounded input u satisfying ‖u‖∞ ≤ ρu, the solution x(t) satisfies

|x(t)| ≤ maxβ(|x0|, t), γ(‖u‖∞) (1.41)

for all t ≥ 0.

Theorem 1.4 presents the equivalence between AS and local ISS.

Theorem 1.4 System (1.28) is locally ISS if and only if the zero-input system

x = f(x, 0) (1.42)

is AS at the origin.

Proof. The proof of Theorem 1.4 is motivated by the proof of [78, Theorems56.3 and 56.4] on the equivalence between total stability and AS at the origin,and the proof of [250, Lemma I.2] on the sufficiency of GAS for local ISS.

The necessity part is obvious. We prove the sufficiency part. By using theconverse Lyapunov theorem (see e.g., [144]), the AS of system (1.42) at theorigin implies the existence of a Lyapunov function V : Ω → R+ with Ω ⊆ Rn

being a domain containing the origin such that properties (1.15)–(1.18) hold.For such V , one can find an Ω′ ⊆ Ω still containing the origin such that forall x ∈ Ω′,

α(|x|) ≤ V (x) ≤ α(|x|) (1.43)

∇V (x)f(x, 0) ≤ −α(V (x)), (1.44)


where α, α ∈ K∞ and α is a continuous and positive definite function.By using the continuity of ∇V and f , for any x ∈ Ω′\0, one can find a

δ > 0 such that

|∇V (x)f(x, ǫ) −∇V (x)f(x, 0)| ≤ 1

2α(V (x)) (1.45)

for all |ǫ| ≤ δ. Thus, there is a positive definite function χ0 such that for anyx ∈ Ω′, property (1.45) holds for all |ǫ| ≤ χ0(|x|).

Then, we choose Ω0 as a compact set containing the origin and belongingto Ω′, and choose χ ∈ K such that

χ(s) ≤ χ0(s) (1.46)

for all 0 ≤ s ≤ max|x| : x ∈ Ω0. It can be directly proved that if x ∈ Ω0

and χ(|u|) ≤ |x|, then

∇V (x)f(x, u) ≤ −1

2α(V (x)). (1.47)

Thus, with (1.43), property (1.47) holds if

V (x) ≤ max α(|x|) : x ∈ Ω0 , (1.48)

V (x) ≥ α χ(‖u‖∞) := γ(‖u‖∞). (1.49)

Then, the sufficiency part can be proved following the same line as (1.54)–(1.56) given later for ISS-Lyapunov functions. The interested reader may alsoconsult the proof of [250, Lemma I.2]. ♦

From Definition 1.8, an ISS system is always forward complete, i.e., for anyinitial state x(0) = x0 and any measurable and locally essentially bound inputu, the solution x(t) is defined for all t ≥ 0. Moreover, it has the uniformlybounded-input bounded-state (UBIBS) property.

Definition 1.10 System (1.28) is said to have the UBIBS property if thereexists σ1, σ2 ∈ K such that for any initial state x(0) = x0 and any measurableand locally essentially bounded input u,

|x(t)| ≤ maxσ1(|x0|), σ2(‖u‖∞) (1.50)

for all t ≥ 0.

Recall Definition 1.3 for classKL functions. If system (1.28) is ISS satisfying(1.29), then it admits property (1.50) by defining σ1(s) = β(s, 0) and σ2(s) =γ(s) for s ∈ R+.

More importantly, ISS is equivalent to the conjunction of UBIBS and AG[250]. This result can be used for the proof of the ISS small-gain theorem forinterconnected nonlinear systems; see detailed discussions in Chapter 2.

Theorem 1.5 System (1.28) is ISS if and only if it has the properties ofUBIBS and AG in the sense of (1.50) and (1.32), respectively.

Introduction 13

1.3.2 ISSLYAPUNOV FUNCTION

Similar to stability, we can employ ISS-Lyapunov functions to formulate thenotion of ISS. For system (1.28), the equivalence between ISS and the existenceof ISS-Lyapunov functions is originally presented in [249].

Theorem 1.6 System (1.28) is ISS if and only if it admits a continuouslydifferentiable function V : Rn → R+, for which

1. there exist α, α ∈ K∞ such that

α(|x|) ≤ V (x) ≤ α(|x|), ∀x, (1.51)

2. there exist a γ ∈ K and a continuous, positive definite α such that

V (x) ≥ γ(|u|) ⇒ ∇V (x)f(x, u) ≤ −α(V (x)), ∀x, u. (1.52)

A function V satisfying (1.51) and (1.52) is called an ISS-Lyapunov func-tion and γ is called the Lyapunov-based ISS gain. ISS-Lyapunov functionsdefined with (1.52) are said to be in the gain margin form. It can be observedthat, under condition (1.52), the state x ultimately converges to within theregion such that V (x) ≤ γ(‖u‖∞). If input u ≡ 0, then the sufficiency part ofTheorem 1.6 is reduced to Theorem 1.2 for GAS.

An equivalent formulation to (1.52) is in the dissipation form:

∇V (x)f(x, u) ≤ −α′(V (x)) + γ′(|u|), (1.53)

where α′ ∈ K∞ and γ′ ∈ K.The proof of the sufficiency part of Theorem 1.6 can be found in the original

ISS paper [241], while the necessity part is proved for the first time in [249].Here, according to [241, 249], we give a sketch of the proof, which could behelpful in understanding ISS-Lyapunov functions.

With property (1.52), it can be proved that there exists a β ∈ KL satisfyingβ(s, 0) = s for all s ∈ R+ such that

V (x(t)) ≤ β(V (x(0)), t), (1.54)

as long as V (x(t)) ≥ γ(‖u‖∞). This means

V (x(t)) ≤ maxβ(V (x(0)), t), γ(‖u‖∞) (1.55)

for all t ≥ 0. Define β(s, t) = α−1(β(α(s), t)) and γ(s) = α−1 γ(s) fors, t ∈ R+. Then, β ∈ KL, γ ∈ K, and

|x(t)| ≤ maxβ(|x(0)|, t), γ(‖u‖∞) (1.56)

holds for all t ≥ 0. The sufficiency part of Theorem 1.6 is proved.


It should be noted that an ISS-Lyapunov function is not necessarily con-tinuously differentiable. Sometimes, it is more convenient to construct locallyLipschitz ISS-Lyapunov functions, which are still sufficient for ISS. Accordingto Rademacher’s theorem [59, p. 216], a locally Lipschitz function is continu-ously differentiable almost everywhere. For a locally Lipschitz ISS-Lyapunovfunction V , condition (1.52) holds for almost all x. In this case, the argumentsused in the original ISS paper [241] are still valid to show that the existenceof such a V implies ISS.

The necessity part of Theorem 1.6 can be proved by constructing ISS-Lyapunov functions. The proof given in [249] employs the notion of weaklyrobust stability (WRS), and the basic idea is shown in Figure 1.6.

ISS-Lyapunov function ISS

WRS

FIGURE 1.6 Equivalence between ISS and the existence of an ISS-Lyapunov func-

tion.

The WRS property describes the capability of a system to handle state-dependent perturbations. System (1.28) is said to be WRS if it admits astability margin ρ ∈ K∞ such that system

x = f(x, d(t)ρ(|x|)) (1.57)

is GAS at the origin uniformly with respect to time, for all possible d : R+ →Bm. Recall that Bm represents the unit ball with center at the origin in R

m. Itis proved in [249] that ISS implies WRS for system (1.28), and the Lyapunovfunction of system (1.57) can be used as the ISS-Lyapunov function for system(1.28).

The proof of the existence of a Lyapunov function for a WRS system isrelated to the converse Lyapunov theorem. Reference [170] presents a resulton the construction of smooth Lyapunov functions for weakly robustly stablesystems. In [137, Chapter 3], converse Lyapunov results for general nonlinearsystems were proved by using the relationship between exponential conver-gence and KL convergence. According to the proof given in [249], a Lyapunovfunction for system (1.57) can be used as an ISS-Lyapunov function for system(1.28). The interested reader may consult [249] for a detailed proof.

A property stronger than WRS is the robust stability (RS) property [249],which considers systems with state-dependent perturbations in the more gen-eral form:

x = f(x,∆(t, x)), (1.58)

where the perturbation term ∆(t, x) might be caused by uncertainty of thesystem dynamics. System (1.28) is said to be RS with a gain margin ρ ∈ K

Introduction 15

if the perturbed system (1.58) is uniformly GAS at the origin as long as|∆(t, x)| ≤ ρ(|x|). The equivalence between RS and ISS has also been provedin [249]. In Chapter 2, we consider the RS property of ISS as a special case ofthe ISS small-gain theorem.

In the context of dissipativity, the α′ and γ′ functions in (1.53) are knownas the supply functions. Reference [248] presents a result on the freedom ofmodifying the supply functions and its application to stability analysis and theconstruction of Lyapunov functions for cascade systems. Connections betweenISS and dissipativity [277, 89, 92] were made in [248, 249], and have persistedthroughout the ISS approach, but generally there are many differences be-tween the dissipative systems and ISS approaches which would be interestingto explore as an extension of [248, 243, 53, 7, 108] on variants of both. Further,more effort is needed for systems with supply functions in more general forms.As seen in the basic ideas [277, 87], the supply functions can take very gen-eral forms of which quadratics and monotone gains are just examples. Also,some generalization of passivity concepts away from the quadratic form canbe found in [235, 62, 63].

1.4 INPUTTOOUTPUT STABILITY

Consider the system

x = f(x, u) (1.59)

y = h(x), (1.60)

where x ∈ Rn is the state, u ∈ Rm is the input, y ∈ Rl is the output, andf : Rn × Rm → Rn and h : Rn → Rl are locally Lipschitz functions. It isassumed that f(0, 0) = 0 and h(0) = 0.

The notion of input-to-output stability (IOS) can be derived by directlyreplacing the state x on the left-hand side of (1.29) with the output y.

Definition 1.11 System (1.59)–(1.60) is said to be IOS if there exist a β ∈KL and a γ ∈ K such that for any initial state x(0) = x0, any measurable andlocally essentially bounded u, and any t where x(t) is defined, it holds that

|y(t)| ≤ maxβ(|x0|, t), γ(‖u‖∞). (1.61)

Here, γ is called the IOS gain of system (1.59)–(1.60).As for ISS, property (1.61) can be equivalently represented by

|y(t)| ≤ β′(|x0|, t) + γ′(‖u‖∞) (1.62)

with β′ ∈ KL and γ′ ∈ K.Corresponding to the AG property of ISS, an IOS system has the output

asymptotic gain (OAG) property: for any initial state x(0) = x0, any measur-able and locally essentially bounded input u, and any t where x(t) is defined,

limt→∞

|y(t)| ≤ γ′(‖u‖∞), (1.63)


where γ′ is called the output asymptotic gain.The IOS studied in this book has connections with the classical input-

output stability, but the two concepts are not entirely equivalent. To avoidconfusion, we use I/O stability as the abbreviation of input-output stability.The study of the stability problem for the systems with input-output (I/O)representations in the operator setting goes back to functional analysis andother approaches developed in the 1960s; see [233, 283, 48]. The work of Hill[88] and Mareels and Hill [199] proposed a generalized notion of I/O stabilityby introducing monotone gains. This is a nonlinear extension of the classicalfinite-gain stability. References [92, 93, 87] made the role of initial conditionin I/O stability explicit to find connections to Lyapunov stability/instability,but did not assume asymptotic stability a priori. By introducing the KLfunction (see the β function in (1.61)), IOS provides an explicit description ofthe converging effect of the initial condition and a link to partial stability ofinternal states [269].

For systems with outputs, observability describes the capability to estimatethe internal state by using the input and output data. In the literature, severalobservability notions have been used for guaranteeing asymptotic stability innonlinear systems; see e.g., [276, 91, 128, 130, 85] and the references therein.Now, we recall the notion of unboundedness observability (UO) from [130]which, together with IOS, will be used in the following chapters.

Definition 1.12 System (1.59)–(1.60) is said to be unboundedness observableif there exist αO ∈ K∞ and constant DO ≥ 0 such that for each measurableand locally essentially bounded input u and for any initial condition x(0) = x0,the solution x(t) of the system satisfies

|x(t)| ≤ αO(|x(0)| + ‖u‖[0,t] + ‖y‖[0,t]) +DO (1.64)

for all t where x(t) is defined.

If system (1.59)–(1.60) has the UO property in the form of (1.64) withDO = 0,then it is said to be UO with zero offset.

This section only briefly reviews the notions that are used in the fol-lowing chapters. The interested reader may consult the original papers[104, 130, 131, 141, 142, 223, 241, 242, 243, 247, 249, 250, 251, 252] for morecharacterizations and properties of ISS and IOS. Specifically, discussions re-lated to ISS and robust stability of systems describing difference equations ordifference inclusions can be found in [131, 142, 143, 22].

1.5 INPUTTOSTATE STABILIZATION AND AN OVERVIEW OF THEBOOK

By characterizing model uncertainty as well as external inputs as a perturba-tion of a nominal model, the goal of robust control is to design feedback control

Introduction 17

laws such that the closed-loop system is robust with respect to a certain levelof perturbations. Due to its relationship to robust stability, ISS is a natu-ral tool for robust control. One approach is input-to-state stabilization, i.e.,designing ISS systems via feedback, with the influence of the perturbationsrepresented by ISS gains. A one-to-one correspondence between input-to-statestability and input-to-state stabilization has been discussed in [137]. Noticethat another line of research to cope with uncertainty leads to adaptive controldesigns, for which ISS has also been used as a powerful tool [115].

The above-mentioned systems with uncertainties and disturbances can beconsidered as an interconnected system. Other interconnected systems occurin emerging control applications ranging from conventional and smart electricgrids, robotic networks and transportation networks to communication andbiological networks. The ISS small-gain theorem is capable of testing the ISSof an interconnected system by directly checking the composition of ISS gainsfor the subsystems. It may significantly reduce the complexity of analyzingand designing interconnected systems. For interconnected systems, with theISS small-gain theorem, the input-to-state stabilization problem can be solvedby designing control laws to make the subsystems ISS with appropriate gains.The basic ideas of the ISS small-gain and the more general IOS small-gaintheorems together with an introduction to their use in control designs arereviewed in Chapter 2.

Inspired by ISS small-gain methods, the basic idea of this book is to trans-form several robust nonlinear control problems into input-to-state stabiliza-tion problems of large-scale dynamic networks, which may contain more thantwo subsystems. For this purpose, a cyclic-small-gain theorem is developed inChapter 3 for large-scale dynamic networks composed of ISS and the moregeneral IOS subsystems. With the cyclic-small-gain theorem, the problem oftesting the ISS property of a dynamic network is reduced to checking thespecific compositions of the ISS gains along the simple cycles in the network.Moreover, as shown in the following chapters, this technique is extremely effi-cient for analyzing the influence of perturbations through the numerous linksand loops in a complex dynamic network.

The robust control problem for nonlinear uncertain systems subject to sen-sor noise is challenging, yet important. Chapter 4 contributes new cyclic-small-gain design methods to solve the problems caused by sensor noise. Measure-ment feedback control issues in the settings of static state-feedback, dynamicstate-feedback and output-feedback, and the applications to event-based con-trol, synchronization, and robust adaptive control are thoroughly studied.With small-gain control designs, the closed-loop systems can be transformedinto large-scale dynamic networks of ISS subsystems, for which the influenceof sensor noise can be explicitly described by ISS gains and attenuated to thelevel of sensor noise.

In modern automatic control systems, signals are usually quantized beforetransmission via communication channels. A quantizer can be mathematically


modeled as a discontinuous map from a continuous region to a discrete set ofnumbers, which leads to a special class of system uncertainties. In Chapter 5,quantized control of nonlinear systems with static quantization is first solvedthrough a cyclic-small-gain design. Due to the finite word-length of digitaldevices, a practical quantizer has a finite number of quantization levels. InChapter 5, dynamic quantization is developed for high-order nonlinear systemssuch that the quantization levels can be dynamically scaled during the controlprocedure for semiglobal quantized stabilization. Quantized output-feedbackcontrol is also studied.

The trend of controlling complex systems composed of spatially distributedsubsystems motivates the idea of distributed control. In a distributed controlsystem, the subsystems are controlled by local controllers with information ex-change between these controllers for coordination purposes. Formation controlof mobile robots is an example. Chapter 6 develops cyclic-small-gain methodsfor distributed control of nonlinear systems. By representing the informationexchange in distributed control systems with directed graphs (digraphs), acyclic-small-gain result in digraphs is first proposed. Then, we use this resultto solve the problems with distributed output-feedback control and distributedformation control.

Based on the results in this book, nontrivial efforts are required for furtherdevelopment of nonlinear control of more general dynamic networks. Somechallenging problems in this research direction are listed in Chapter 7.

2 Interconnected NonlinearSystems

The small-gain theorem is an extremely useful tool for the analysis and controldesign of interconnected systems. This chapter introduces the first, fundamen-tally nonlinear variant of the small-gain theorem, known as the ISS small-gaintheorem, as well as the related methods. With the ISS small-gain theorem,the ISS property of an interconnected system composed of two ISS subsystemscan be tested by checking the composition of the ISS gains.

We start with a simple case in which the interconnected system is composedof one dynamic subsystem and one static subsystem and the interconnectedsystem does not have any external input. The block diagram of the system isshown in Figure 2.1. Suppose that subsystem Ξ1 is in the form of (1.28) andsubsystem Ξ2 is defined as

u = ∆(x, t). (2.1)

Ξ2

Ξ1

x

u

FIGURE 2.1 An interconnected system with no external input.

With the robust stability property of ISS, the interconnected system isGAS at the origin if subsystem Ξ1 is ISS in the sense of (1.29) and thereexists an appropriate ρ ∈ K∞ such that subsystem Ξ2 satisfies

|∆(x, t)| ≤ ρ(|x|) (2.2)

for all x and all t ≥ 0. A sufficient condition for ρ to guarantee GAS is

ρ(γ(s)) < s (2.3)

for all s > 0, i.e.,

ρ γ < Id. (2.4)

19


As shown later, the robust stability property of ISS can be consideredas a special case of the ISS small-gain theorem. Here, we give a proof of thesufficiency of condition (2.4) for GAS, which is a reduced version of the originalproof of the ISS small-gain theorem given by Jiang, Teel, and Praly [130]. Theproof is carried out in two steps. We first show the GS of the interconnectedsystem at the origin, and then prove the GAS with the help of Lemma D.1.Lemma D.1 is a slight modification of [130, Lemma A.1].

Suppose that the solution x(t) of the interconnected system is right maxi-mally defined on [0, T ) with 0 < T ≤ ∞. By applying |u| = |∆(x, t)| ≤ ρ(|x|)to (1.29), we have

|x(t)| ≤ maxβ(|x(0)|, 0), γ(‖u‖[0,T ))≤ maxβ(|x(0)|, 0), γ ρ(‖x‖[0,T )) (2.5)

for all 0 ≤ t < T . Then, taking the supremum of |x| over [0, T ) implies

‖x‖[0,T ) ≤ maxβ(|x(0)|, 0), γ ρ(‖x‖[0,T )). (2.6)

With condition (2.4) satisfied, γ ρ(‖x‖[0,T )) < ‖x‖[0,T ), and thus

‖x‖[0,T ) ≤ β(|x(0)|, 0) := σ(|x(0)|). (2.7)

It can be directly verified that if (2.7) is not true, then (2.6) cannot be satisfied.This means that x(t) is defined on [0,∞), and the T in (2.7) can be directly

replaced by ∞, which implies

|x(t)| ≤ σ(|x(0)|) (2.8)

for all t ≥ 0. GS at the origin, in the sense of Definition 1.7, is proved.For system Ξ1, with the time-invariance property, (1.29) implies

|x(t)| ≤ maxβ(|x(t0)|, t− t0), γ(‖u‖[t0,∞)) (2.9)

for any 0 ≤ t0 ≤ t. Then, by choosing t0 = t/2 and using |u| ≤ ρ(|x|), we have

|x(t)| ≤ max

β

(∣

∣

∣

∣

x

(

t

2

)∣

∣

∣

∣

,t

2

)

, γ(

‖u‖[t/2,∞)

)

≤ max

β

(∣

∣

∣

∣

x

(

t

2

)∣

∣

∣

∣

,t

2

)

, γ ρ(

‖x‖[t/2,∞)

)

. (2.10)

Note that property (2.8) implies |x(t/2)| ≤ σ(|x(0)|) for all t ≥ 0. Thus,

|x(t)| ≤ max

β

(

σ (|x(0)|) , t2

)

, γ ρ(

‖x‖[t/2,∞)

)

:= max

β (|x(0)| , t) , γ ρ(

‖x‖[t/2,∞)

)

. (2.11)

Interconnected Nonlinear Systems 21

With Lemma D.1, there exists a β ∈ KL such that

|x(t)| ≤ β(|x(0)|, t). (2.12)

The GAS property is thus proved.Recall that GAS is equivalent to the conjunction of GS and global con-

vergence; see the discussions below Definition 1.7. An alternative proof thatdoes not use Lemma D.1 can be carried out by proving the global convergenceproperty. This is not provided in this book, but one may consider it as a spe-cial case of the proof of Theorem 2.1 for interconnected ISS systems by usingthe AG arguments in Appendix D.2. A drawback of this approach is that onemay not get an explicit β ∈ KL for the GAS property (2.12).

Property (2.2) of subsystem Ξ2 implies that

|u(t)| ≤ ρ(‖x‖∞) (2.13)

for all t ≥ 0, which can be considered as a special case of (1.29) with ρconsidered as the gain. Then, condition (2.4) means that the composition ofthe gains of the two subsystems is less than the identity function. Such acondition is called the nonlinear small-gain condition.

This chapter reviews the small-gain results for interconnected systems com-posed of ISS or more general IOS subsystems. Specifically, Section 2.1 extendsthe above robustness analysis to the trajectory-based small-gain results devel-oped in [130]. Due to the importance of Lyapunov functions, the Lyapunov-based ISS small-gain theorem originally developed in [126] is reviewed in Sec-tion 2.2. Section 2.3 introduces the basic idea of the important gain assign-ment technique [130, 223, 123, 125] for small-gain designs and provides a casestudy of applying the small-gain theorem to robust control of nonlinear un-certain systems. Note that small-gain results for more general systems, e.g.,systems modeled by retarded functional differential equations, have also beendeveloped in the literature; see e.g., [137]. Some related topics on large-scaledynamic networks are discussed in Chapter 3 in the cyclic-small-gain frame-work.

2.1 TRAJECTORYBASED SMALLGAIN THEOREM

Consider an interconnected system composed of two subsystems

x1 = f1(x, u1) (2.14)

x2 = f2(x, u2), (2.15)

where x = [xT1 , xT2 ]T with x1 ∈ Rn1 and x2 ∈ Rn2 is the state, u1 ∈ Rm1

and u2 ∈ Rm2 are external inputs, and f1 : Rn1+n2 × Rm1 → Rn1 and f2 :Rn1+n2×Rm2 → Rn2 are locally Lipschitz functions satisfying f1(0, 0) = 0 andf2(0, 0) = 0. For convenience of notation, define u = [uT1 , u

T2 ]T . By considering


x2

x1u1

u2

FIGURE 2.2 An interconnected system with external inputs.

u as a function of time, assume that it is measurable and locally essentiallybounded.

For i = 1, 2, assume that each xi-subsystem is ISS with x3−i and ui as theinputs. Specifically, for each i = 1, 2, there exist βi ∈ KL and γi(3−i), γ

ui ∈ K

such that for any initial state xi(0) = xi0 and any measurable and locallyessentially bounded inputs x3−i, ui, it holds that

|xi(t)| ≤ maxβi(|xi0|, t), γi(3−i)(‖x3−i‖∞), γui (‖ui‖∞) (2.16)

for all t ≥ 0. Here, the ISS property of the subsystems are in the “max” form,which is equivalent to the “plus” form used in [130].

With the discussions below, we show that the interconnected system is ISSwith u as the input if

γ12 γ21 < Id. (2.17)

It should be noted that for any γ12, γ21 ∈ K,

γ12 γ21 < Id ⇔ γ21 γ12 < Id. (2.18)

Indeed, for the implication “⇒”, assume that γ21γ12 < Id does not hold. Thatis, there exists a positive s such that γ21(γ12(s)) ≥ s. Then, γ12γ21(γ12(s)) ≥γ12(s), which leads to a contradiction with γ12 γ21 < Id. By symmetry, theother implication “⇐” holds.

Theorem 2.1 presents a trajectory-based ISS small-gain result.

Theorem 2.1 Consider the interconnected system composed of two subsys-tems in the form of (2.14)–(2.15) satisfying (2.16). The interconnected systemis ISS with u as the input if the small-gain condition (2.17) is satisfied.

Proof. The proof is basically a reduced version of the proof for the IOS small-gain theorem given in [130]. We only make slight modifications to handle thedifference between the two forms of the ISS property. Pick any specific initialstate x(0) and any measurable and locally essentially bounded input u.


Step 1–UBIBS: Suppose that the solution x(t) of the interconnected systemis defined on [0, T ) with T > 0. Define σi(s) = βi(s, 0) for s ∈ R+. For i = 1, 2,by using the ISS property (2.16), one has

|xi(t)| ≤ maxσi(|xi(0)|), γi(3−i)(‖x3−i‖[0,T )), γui (‖ui‖∞) (2.19)

for 0 ≤ t < T , and thus, by taking the supremum of |xi(t)| over [0, T ), wehave

‖xi‖[0,T ) ≤ maxσi(|xi(0)|), γi(3−i)(‖x3−i‖[0,T )), γui (‖ui‖∞). (2.20)

By substituting (2.20) with i replaced by 3 − i in the right-hand side of(2.19), it is achieved that

|xi(t)| ≤ maxσi(|xi(0)|), γi(3−i) σ3−i(|x3−i(0)|),γi(3−i) γ(3−i)i(‖xi‖[0,T )),

γi(3−i) γu3−i(‖u3−i‖∞), γui (‖ui‖∞)≤ maxσi(|xi(0)|), γi(3−i) σ3−i(|x3−i(0)|),

γi(3−i) γ(3−i)i(‖xi‖[0,T )),

γi(3−i) γu3−i(‖u3−i‖∞), γui (‖ui‖∞). (2.21)

Define

σi1(s) = maxσi(s), γi(3−i) σ3−i(s), (2.22)

σi2(s) = maxγui (s), γi(3−i) γu3−i(s) (2.23)

for s ∈ R+. By taking the supremum of xi(t) over [0, T ) and using (2.21), onehas

‖xi‖[0,T ) ≤ maxσi1(|x(0)|), σi2(‖u‖∞), γi(3−i) γ(3−i)i(‖xi‖[0,T ))≤ maxσi1(|x(0)|), σi2(‖u‖∞), (2.24)

where the small-gain condition (2.17) is used for the last inequality. Thismeans that |xi(t)| is defined on [0,∞). With the T in (2.24) replaced by ∞,it is achieved that

|xi(t)| ≤ maxσi1(|x(0)|), σi2(‖u‖∞) (2.25)

for all t ≥ 0. UBIBS property is proved as property (2.25) holds for any initialstate x(0) and any measurable and locally essentially bounded input u.

Step 2–ISS: Denote x∗i = maxσi1(|x(0)|), σi2(‖u‖∞) for i = 1, 2.By means of time invariance and causality, (2.16) implies

|xi(t)| ≤ max

βi(|xi(t0)|, t− t0), γi(3−i)(

‖x3−i‖[t0,t])

, γui (‖ui‖∞)

(2.26)


for any 0 ≤ t0 ≤ t, and thus, by choosing t0 = t/2, we have

|xi(t)| ≤ max

βi

(∣

∣

∣

∣

xi

(

t

2

)∣

∣

∣

∣

,t

2

)

, γi(3−i)(

‖x3−i‖[t/2,t])

, γui (‖ui‖∞)

≤ max

βi

(

x∗i ,t

2

)

, γi(3−i)(

‖x3−i‖[t/2,t])

, γui (‖ui‖∞)

(2.27)

for i = 1, 2.By taking the maximum of xi(t) over [t/2, t], it is achieved that

‖xi‖[t/2,t]≤ maxt/2≤τ≤t

βi

(∣

∣

∣xi

(τ

2

)∣

∣

∣ ,τ

2

)

, γi(3−i)(

‖x3−i‖[τ/2,τ ])

, γui (‖ui‖∞)

≤max

βi

(

x∗i ,t

4

)

, γi(3−i)(

‖x3−i‖[t/4,t])

, γui (‖ui‖∞)

(2.28)

for i = 1, 2.Then, by substituting (2.28) with i replaced by 3− i into (2.27), one has

|xi(t)| ≤ max

βi

(

x∗i ,t

2

)

, γi(3−i) β3−i(

x∗3−i,t

4

)

,

γi(3−i) γ(3−i)i(

‖xi‖[t/4,t])

, γi(3−i) γu3−i (‖u3−i‖∞) ,

γui (‖ui‖∞)

. (2.29)

Recall the x∗i = maxσi1(|x(0)|), σi2(‖u‖∞) for i = 1, 2. Property (2.29)implies that

|xi(t)| ≤ max

βi(|x(0)|, t), γi(3−i) γ(3−i)i(

‖xi‖[t/4,t])

, γui (‖u‖∞)

(2.30)

for all t ≥ 0, where

βi(s, t) = max

βi

(

σi1(s),t

2

)

, γi(3−i) β3−i(

σ(3−i)1(s),t

4

)

, (2.31)

γui (s) = max

γui (s), βi (σi2(s), 0) , γi(3−i) γu3−i(s),

γi(3−i) β3−i(

σ(3−i)2(s), 0)

. (2.32)

Clearly, βi ∈ KL, γui ∈ K.

Then, by using Lemma D.1, there exists a βi ∈ KL such that

|xi(t)| ≤ max

βi(|x(0)|, t), γui (‖u‖∞)

(2.33)


for all t ≥ 0, for i = 1, 2. Note that property (2.33) holds for any initial statex(0) and any measurable and locally essentially bounded u. The ISS of theinterconnected system is proved. ♦

It should be noted that by using Theorem 1.5, Theorem 2.1 can also beproved by showing the UBIBS and AG properties of the interconnected sys-tem. The proof of AG is provided in Appendix D.2 for interested readers.

If γ12 or γ21 is zero, then the interconnected system is reduced to a cas-cade system, for which the small-gain condition is satisfied automatically. Ifmoreover, u1 = u2 = 0, then Theorem 2.1 is reduced to [144, Lemma 4.7]for GAS. Also note that if γu1 = γu2 = 0 and one of the subsystems, say thex2-subsystem, has β2 = 0, then the result of Theorem 2.1 is reduced to therobust stability result given at the beginning of this chapter.

The small-gain result developed in [130] can cover the more general case inwhich the subsystems are interconnected with each other by outputs insteadof states. Consider the following interconnected system:

xi = fi(xi, y3−i, ui) (2.34)

yi = hi(xi) (2.35)

where, for i = 1, 2, xi ∈ Rni is the state, ui ∈ Rmi is the input, yi ∈ Rli is theoutput, and fi, hi are locally Lipschitz functions satisfying fi(0, 0, 0) = 0 andhi(0) = 0.

Assume that each i-th subsystem is UO with zero offset and IOS withy3−i, ui as the inputs and yi as the output. Specifically, there exist αOi ∈ K∞,βi ∈ KL, γi(3−i) ∈ K, and γui ∈ K such that

|xi(t)| ≤ αOi(

|xi(0)|+ ‖y3−i‖[0,t] + ‖ui‖[0,t])

(2.36)

|yi(t)| ≤ maxβi(|xi(0)|, t), γi(3−i)(‖y3−i‖[0,t]), γui (‖ui‖[0,t]) (2.37)

for all t ∈ [0, Tmax), where [0, Tmax) with 0 < Tmax ≤ ∞ is the right maximalinterval for the definition of (x1(t), x2(t)).

Theorem 2.2 gives a small-gain result for the interconnected IOS system.

Theorem 2.2 Consider the interconnected system (2.34)–(2.35) satisfying(2.36) and (2.37) for i = 1, 2. Then the interconnected system is UO andIOS if

γ12 γ21 < Id. (2.38)

Theorem 2.2 does not assume the forward completeness of the subsystems.Following the discussions in [130], IOS together with UO implies the forwardcompleteness of the subsystems. If the small-gain condition is satisfied, thenthe forward completeness of the interconnected system is guaranteed by theIOS and UO properties of the subsystems. In [134], Theorem 2.2 is generalizedfor large-scale dynamic networks composed of more than two subsystems. Thisresult is reviewed in Chapter 3.


Reference [130] also takes into account the practical convergence issues byintroducing the notion of input-to-output practical stability (IOpS) property,and the small-gain theorem therein is more general than Theorem 2.2. Furtherextensions of [130] can be found in [133]. References [135, 167, 169] as well asthe book [137] show the extensions of the small-gain theorem to more generalcomplex systems such as hybrid systems and systems modeled by retardedfunctional differential equations.

2.2 LYAPUNOVBASED SMALLGAIN THEOREM

Lyapunov functions play an irreplaceable role in the analysis and controlof nonlinear systems. With the Lyapunov-based formulation of ISS, the ISSproperty of nonlinear systems is often tested by constructing ISS-Lyapunovfunctions. This section reviews the Lyapunov-based ISS small-gain theoremdeveloped in [126] for feedback systems. In particular, it is shown that if aninterconnected system satisfies the Lyapunov-based ISS small-gain condition,then ISS-Lyapunov functions can be constructed for the system by using theISS-Lyapunov functions of the subsystems.

For the interconnected system (2.14)–(2.15), assume that each xi-subsystem for i = 1, 2 admits a continuously differentiable ISS-Lyapunovfunction Vi : R

ni → R+ satisfying the following:

1. there exist αi, αi ∈ K∞ such that

αi(|xi|) ≤ Vi(xi) ≤ αi(|xi|), ∀xi; (2.39)

2. there exist χi(3−i), χui ∈ K and a continuous, positive definite αi such that

Vi(xi) ≥ maxχi(3−i)(V3−i(x3−i)), χui (|ui|)⇒∇Vi(xi)fi(x, ui) ≤ −αi(Vi(xi)), ∀x, ui. (2.40)

Theorem 2.3 gives a Lyapunov formulation of the ISS small-gain theorem.

Theorem 2.3 Interconnected system (2.14)–(2.15) with each xi-subsystemadmitting an ISS-Lyapunov function Vi satisfying (2.39)–(2.40) is ISS if thefollowing small-gain condition is satisfied:

χ12 χ21 < Id. (2.41)

Proof. Theorem 2.3 is proved by constructing an ISS-Lyapunov function Vfor the interconnected system.

For χ12, χ21 ∈ K satisfying the small-gain condition (2.41), we find a σ ∈K∞ such that it is continuously differentiable on (0,∞) and satisfies

σ > χ21, σ−1 > χ12. (2.42)


This can be achieved because for χ12, χ21 ∈ K satisfying condition (2.41),there exists a χ12 ∈ K∞ such that χ12 > χ12 and χ12 χ21 < Id. One canalways find a σ ∈ K∞ such that it is continuously differentiable on (0,∞) andsatisfies

χ21 < σ < χ−112 , (2.43)

which guarantees the satisfaction of (2.42). See [126] for the detailed proof ofthe existence of such σ.

An ISS-Lyapunov function candidate for the interconnected system is de-fined as

V (x) = max σ(V1(x1)), V2(x2) . (2.44)

Clearly, V is positive definite and radially unbounded. Also, V is continuouslydifferentiable almost everywhere.

Let f(x, u) = [fT1 (x, u1), fT2 (x, u2)]

T . In the following procedure, we provethat there exist a χ ∈ K and a continuous, positive definite α such that

V (x) ≥ χ(|u|) ⇒ ∇V (x)f(x, u) ≤ −α(V (x)) (2.45)

for almost all x and all u.For this purpose, define the following sets, as shown in Figure 2.3:

A = (x1, x2) : V2(x2) < σ(V1(x1)), (2.46)

B = (x1, x2) : V2(x2) > σ(V1(x1)), (2.47)

O = (x1, x2) : V2(x2) = σ(V1(x1)). (2.48)

0 V1

V2

A

B

O

χ−112

χ21

σ

FIGURE 2.3 Definitions of sets A, B and O.

For any fixed point p = (p1, p2) 6= (0, 0) and a control value v = (v1, v2),consider the following three cases.


Case 1: p ∈ A

In this case, V (x) = σ(V1(x1)) in a neighborhood of p, and consequently

∇V (p)f(p, v) =∂σ(V1(p1))

∂V1(p1)∇V1(p1)f1(p, v1). (2.49)

For p ∈ A, it holds that V2(p2) < σ(V1(p1)), and based on the definition of σ,V1(p1) > χ12(V2(p2)). With (2.40), this implies

∇V1(p1)f1(p, v1) ≤ −α1(v1(p1)) (2.50)

whenever V1(p1) ≥ σ χu1 (|v1|). It follows that, for p ∈ A,

∇V (p)f(p, v) ≤ −α1(V (p)) (2.51)

whenever V (p) ≥ χu1 (|v1|), where

χu1 (s) = σ χu1 (s) (2.52)

for s ∈ R+, and α1 is a continuous and positive definite function such that

α1(s) ≤ σd(σ−1(s))α1(σ−1(s)) (2.53)

for s > 0, with σd(s) = dσ(s)/ds.

Case 2: p ∈ B

In this case, by using similar arguments as in Case 1, it can be proved that

∇V (p)f(p, v) ≤ −α2(V (p)) (2.54)

whenever V (p) ≥ χu2 (|v2|), where α2 = α2 and χu2 = χu2 .

Case 3: p ∈ O

First note that it holds for the locally Lipschitz function V that

∇V (p)f(p, v) =d

dt

∣

∣

∣

∣

t=0

V (ϕ(t)) (2.55)

for almost all p and all v, with ϕ(t) = [ϕT1 (t), ϕT2 (t)]

T being the solution ofthe initial-value problem

ϕ(t) = f(ϕ(t), v), ϕ(0) = p. (2.56)

In this case, assume p = (p1, p2) 6= (0, 0) and

V1(p1) ≥ χu1 (|v1|), (2.57)

V2(p2) ≥ χu2 (|v2|). (2.58)


Then, by using similar arguments as for Cases 1 and 2, one has

∇σ(V1(p1))f1(p, v1) ≤ −α1(V (p)) (2.59)

∇V2(p2)f2(p, v2) ≤ −α2(V (p)) (2.60)

where α1 and α2 are continuous and positive definite functions.Note that in this case p1 6= 0 and p2 6= 0. Then, because of the contin-

uous differentiability of σ, V1, and V2, and the continuity of f , there existneighborhoods X1 of p1 and X2 of p2 such that

∇σ(V1(x1))f1(x, v1) ≤ −α1(V (p)), (2.61)

∇V2(x2)f2(x, v2) ≤ −α2(V (p)) (2.62)

for all x ∈ X1×X2. Note also that there exists δ > 0 such that ϕ(t) ∈ X1×X2

for all 0 ≤ t < δ.Now pick ∆t ∈ (0, δ). If ϕ(∆t) ∈ A ∪O, then

V (ϕ(∆t)) − V (p) = σ(V1(ϕ1(∆t))) − σ(V1(p1))

≤ −1

2α1(V (p))∆t. (2.63)

Similarly, if ϕ(∆t) ∈ B ∪O, thenV (ϕ(∆t)) − V (p) = V2(ϕ2(∆t))− V2(p2)

≤ −1

2α2(V (p))∆t. (2.64)

Hence, if V is differentiable at p, then

∇V (p)f(p, v) ≤ −α(V (p)) (2.65)

where α(s) = minα1(s)/2, α1(s)/2 for s ∈ R+. Note that conditions (2.57)and (2.58) can be guaranteed by

V (p) ≥ maxχu1 (|v1|), χu2 (|v2|). (2.66)

By combining the three cases, it can be concluded that

V (p) ≥ maxχu1 (|v1|), χu2 (|v2|) ⇒ ∇V (p)f(p, v) ≤ −α(V (p)). (2.67)

Since V is continuously differentiable almost everywhere, (2.67) holds foralmost all p and all v. Property (2.45) is then proved by defining χ(s) =maxχu1(s), χu2 (s) for s ∈ R+. Thus, V is an ISS-Lyapunov function of theinterconnected system. Theorem 2.3 is proved. ♦

Actually, the selected σ is not necessarily continuously differentiable on(0,∞), as the constructed V is not required to be continuously differentiable.A similar proof can be carried out with a σ ∈ K∞ which is locally Lipschitz on(0,∞). More generally, the small-gain condition and the construction of theISS-Lyapunov function are still valid even if V1 and V2 are locally Lipschitz onRn1\0 and R

n2\0, respectively. With the techniques presented in [170],a smooth ISS-Lyapunov function can be constructed based on the V definedabove.


2.3 SMALLGAIN CONTROL DESIGN

If a system can be transformed into an interconnection of ISS subsystemsthrough control design, then one may employ the ISS small-gain theoremto analyze the stability property of the closed-loop system. This section in-troduces a small-gain control design method for nonlinear uncertain systemsbased on the gain assignment technique [130, 223, 123, 125].

2.3.1 GAIN ASSIGNMENT

Assigning an appropriate ISS gain to a system by means of feedback is akey step in applying the ISS small-gain theorem to nonlinear control design.This subsection introduces the basic idea of the gain assignment technique,whereby a system is transformed by feedback into one with a given Lyapunovfunction and specific ISS gains. We illustrate with the following first-ordersystem:

η = φ(η, w1, w2) + κ (2.68)

where η ∈ R is the state, κ ∈ R is the control input, w1 ∈ Rm1 , w2 ∈Rm2 represent external disturbance inputs, and the nonlinear function φ :

Rm1+m2+1 → R satisfies

|φ(η, w1, w2)| ≤ ψηφ(|η|) +∑

i=1,2

ψwi

φ (|wi|), ∀η, w1, w2, (2.69)

with ψηφ, ψw1

φ , ψw2

φ ∈ K∞. Define

V (η) = αV (|η|) (2.70)

with αV (s) = s2/2 for s ∈ R+. We look for a feedback control law in the formof

κ = κ(η) (2.71)

such that the closed-loop system composed of (2.68) and (2.71) with w1, w2

as the external inputs is ISS with V defined in (2.70) as an ISS-Lyapunovfunction. Moreover, the closed-loop system will be designed to have specificISS gains χw1

η , χw2η ∈ K∞ corresponding to the external inputs. To further

realize small-gain-based recursive control design, the control law κ is expectedto be continuously differentiable.

For any constants ǫ, ℓ > 0, we find a ν : R+ → R+ which is positive,nondecreasing, and continuously differentiable on (0,∞) such that

ν(s)s ≥ ψηφ(s) +∑

i=1,2

ψwi

φ (

χwiη

)−1 αV (s) +ℓ

2s (2.72)


for all s ≥√2ǫ. This is achievable by using Lemma C.8 because the right-hand

side of (2.72) is a classK∞ function of s. Inequality (2.72) enables constructionof a feedback law to ensure the specified ISS properties for the closed loop.

Define

κ(r) = −ν(|r|)r (2.73)

for r ∈ R. Clearly, κ is odd, strictly decreasing, radially unbounded, andcontinuously differentiable on (−∞, 0) ∪ (0,∞). Direct calculation yields:

limr→0+

dκ(r)

dr= lim

r→0+

(

−ν(r)− dν(r)

drr

)

, (2.74)

limr→0−

dκ(r)

dr= lim

r→0−

(

−ν(−r) + dν(−r)d(−r) r

)

= limr′→0+

(

−ν(r′)− dν(r′)

d(r′)r′)

, (2.75)

which implies limr→0+dκ(r)dr = limr→0−

dκ(r)dr . Thus, κ is continuously differ-

entiable on R.Recall the definition of V in (2.70). Consider the case of

V (η) ≥ maxi=1,2

χwiη (|wi|), ǫ. (2.76)

In this case, |wi| ≤(

χwiη

)−1 αV (|η|) for i = 1, 2, and |η| ≥√2ǫ. Direct

calculation yields:

∇V (η)(φ(η, w1, w2) + κ(η)) = η(φ(η, w1, w2) + κ)

= η(φ(η, w1, w2)− ν(|η|)η)≤ |η||φ(η, w1, w2)| − ν(|η|)|η|2

≤ |η|

ψηφ(|η|) +∑

i=1,2

ψwi

φ (|wi|)− ν(|η|)|η|

≤ − ℓ

2|η|2 = −ℓV (η). (2.77)

As a result, for any specific χw1η , χw2

η ∈ K∞ and constants ǫ, ℓ > 0, onecan find a continuously differentiable, odd, strictly decreasing, and radiallyunbounded κ in the form of (2.73), such that V satisfies

V (η) ≥ maxi=1,2

χwiη (|wi|), ǫ

⇒∇V (η)(φ(η, w1 , w2) + κ(η)) ≤ −ℓV (η), ∀η, w1, w2. (2.78)

It should be noted that even if w1 = w2 = 0, property (2.78) can only guar-antee that V (η(t)) ultimately converges to within the region V (η) ≤ ǫ. This


means practical convergence. If we also consider ǫ as an external input, thenthe system is ISS with V as an ISS-Lyapunov function. More precisely, thesystem is said to be input-to-state practically stable (ISpS); see [130] for thedefinition of ISpS. With the gain assignment technique, the Lyapunov-basedISS gains χw1

η , χw2η can be chosen to be any class K∞ functions.

By using Lemma C.8, if ψηη and ψwi

φ (

χwi

φ

)−1

αV for i = 1, 2 are Lipschitz

on compact sets, then we can choose ǫ = 0 for (2.72). If ψηφ and ψwi

φ are

Lipschitz on compact sets, then we may choose χwiη = αV

(

ϑwiη

)−1or χwi

η =(

ϑwiη

)−1 αV with ϑwiη ∈ K∞ being Lipschitz on compact sets.

In the following chapters, nontrivial modifications of the gain assignmenttechnique will be made to solve the specific problems.

2.3.2 SMALLGAIN CONTROL DESIGN: A CASE STUDY

Reference [123] successfully applied the IOS small-gain theorem to recursivecontrol design of general cascade nonlinear systems with dynamic uncertain-ties. In this subsection, we consider a much simpler case to show the basicapproach.

Consider the following nonlinear system in the strict-feedback form [153]:

xi = xi+1 +∆i(xi), i = 1, . . . , n− 1 (2.79)

xn = u+∆n(xn) (2.80)

where [x1, . . . , xn]T ∈ Rn is the state, xi = [x1, . . . , xi]

T and u ∈ R is thecontrol input. It is assumed that, for each i = 1, . . . , n, there exists a ψ∆i

∈ K∞such that

|∆i(xi)| ≤ ψ∆i(|xi|). (2.81)

Through small-gain design, the [x1, . . . , xn]T -system is recursively trans-

formed into a new [e1, . . . , en]T -system with ISS ei-subsystems by defining

coordinate transformation

e1e2...en

=

x1x2 − κ1(e1)...xn − κn−1(en−1)

(2.82)

and control law

u = κn(en) (2.83)

where κ1, . . . , κn : R → R are appropriately chosen functions.Define an ISS-Lyapunov function candidate for each ei-subsystem as

Vei (ei) = αV (|ei|) (2.84)


with αV (s) = s2/2 for s ∈ R+. For convenience of notation, denote ei =[e1, . . . , ei]

T and en+1 = xn+1 − κn(en) with xn+1 = u. The ei-subsystems fori = 1, . . . , n are designed to be ISS one-by-one.

Initial Step: The e1subsystem

The e1-subsystem is in the following form:

e1 = x2 +∆1(e1). (2.85)

Recall that e2 = x2 − κ1(e1). Then, the e1-subsystem can be rewritten as

e1 = κ1(e1) + e2 +∆1(e1)

:= κ1(e1) + ∆∗1(e2). (2.86)

With (2.81) satisfied, from the definition of ∆∗1, we can find ψe1∆∗

i, ψe2∆∗

i∈

K∞ such that |∆∗1(e2)| ≤ ψe1∆∗

i(|e1|) + ψe2∆∗

i(|e2|). With the gain assignment

technique introduced in Subsection 2.3.1, for any specified constants ǫe1 , ℓe1 >0 and γe2e1 αV ∈ K∞, we can find a continuously differentiable, odd, strictlydecreasing, and radially unbounded function κ1 such that

Ve1(e1) ≥ maxγe2e1 αV (|e2|), ǫe1⇒∇Ve1(e1)(κ1(e1) + ∆∗

1(e2)) ≤ −ℓe1Ve1 (e1), (2.87)

and thus,

Ve1(e1) ≥ maxγe2e1 (Ve2 (e2)), ǫe1⇒∇Ve1(e1)(κ1(e1) + ∆∗

1(e2)) ≤ −ℓe1Ve1 (e1). (2.88)

Recursive Step: The eisubsystem (i = 2, . . . , n)

Suppose that the ei−1-subsystem has been designed to be in the followingform:

e1 = κ1(e1) + ∆∗1(e2) (2.89)

...

ei−1 = κi−1(ei−1) + ∆∗i−1(ei) (2.90)

where κ1, . . . , κi−1 are appropriately chosen continuously differentiable, odd,strictly decreasing, and radially unbounded functions.

For convenience of notation, denote ˙ei−1 = Fi−1(ei).Also suppose that the ei−1-subsystem is ISS with an ISS-Lyapunov function

Vei−1 satisfying

αei−1(|ei−1|) ≤ Vei−1(ei−1) ≤ αei−1(|ei−1|), (2.91)


and

Vei−1(ei−1) ≥ maxγeiei−1(Vei (ei)), ǫei−1

⇒∇Vei−1 (ei−1)Fi−1(ei) ≤ −αei−1(Vei−1 (ei−1)) a.e., (2.92)

where αei−1, αei−1 , γ

eiei−1

∈ K∞, ǫei−1 > 0 is a constant and αei−1 is a contin-uous and positive definite function.

By taking the derivative of ei, we have

ei = xi −∂κi−1(ei−1)

∂ei−1ei−1

= xi+1 +∆i(xi)−∂κi−1(ei−1)

∂ei−1(κi−1(ei−1) + ∆∗

i−1(ei)). (2.93)

With the recursive definition (2.82), we can represent xi with ei. Also notethat ei+1 = xi+1 − κi(ei). Then, the ei-subsystem can be rewritten as

ei = κi(ei) + ei+1 +∆i(xi)−∂κi−1(ei−1)

∂ei−1(κi−1(ei−1) + ∆∗

i−1(ei))

:= κi(ei) + ∆∗i (ei+1). (2.94)

It can be proved that there exist ψei−1

∆∗

i, ψei∆∗

i, ψ

ei+1

∆∗

i∈ K∞ such that

|∆∗i (ei+1)| ≤ ψ

ei−1

∆∗

i(|ei−1|)+ψei∆∗

i(|ei|)+ψei+1

∆∗

i(|ei+1|). With the gain assignment

technique, for any specified constants ǫei , ℓei > 0 and γei−1ei αei−1 , γ

ei+1ei αV ∈

K∞, we can find a continuously differentiable, odd, strictly decreasing, andradially unbounded function κi such that

Vei(ei) ≥ maxγ ei−1ei αei−1(|ei−1|), γei+1

ei αV (|ei+1|), ǫei⇒∇Vei (ei)(κi(ei) + ∆∗

i (ei+1)) ≤ −ℓeiVi(ei), (2.95)

and thus,

Vei (ei) ≥ maxγ ei−1ei (Vei−1 (ei−1)), γ

ei+1ei (Vei+1 (ei+1)), ǫei

⇒∇Vei (ei)(κi(ei) + ∆∗i (ei+1)) ≤ −ℓeiVi(ei). (2.96)

The ei-subsystem is an interconnection of the ei−1-subsystem and the ei-subsystem, as shown in Figure 2.4. For convenience of notation, denote ˙ei =Fi(ei+1). With the Lyapunov-based ISS small-gain theorem given in Section2.2, the ei-subsystem is ISS if we choose γ

ei−1ei such that

γ ei−1ei γeiei−1

< Id. (2.97)

Moreover, we can construct an ISS-Lyapunov function Vei for the ei-subsystem as

Vei(ei) = maxσei−1(Vei−1 (ei−1)), Vei (ei) (2.98)


where σei−1 ∈ K∞ is continuously differentiable on (0,∞) and satisfies σei−1 >

γei−1ei and σei−1 γeiei−1

< Id.

Then, there exist αei , αei , γei+1

ei ∈ K∞, constants ǫei > 0 and αei that arecontinuous and positive definite such that

αei(|ei|) ≤ Vei (ei) ≤ αei(|ei|), (2.99)

Vei (ei) ≥ maxγei+1

ei (Vei+1(ei+1)), ǫei⇒∇Vei (ei)Fi(ei+1) ≤ −αei(Vei (ei)) a.e. (2.100)

ei-subsystem

ei−1-subsystemei−1

ei

ei+1ei−1 ei ...

ei-subsystem

FIGURE 2.4 Small-gain-based recursive control design.

In the case of i = n, the control input u = xn+1 occurs and en+1 = 0.We can construct an ISS-Lyapunov function Ven for the en-system, i.e., thee-system, which satisfies (2.100) with i = n and ei+1 = 0. In this way, theclosed-loop system is designed to be ISS with the ǫei ’s as the inputs, or moreprecisely, practically stable [78, 156]. With the gain assignment technique, theǫei ’s can be chosen arbitrarily small, and the closed-loop signals ei’s ultimatelyconverge to within an arbitrarily small neighborhood of the origin.

Actually, for systems in the form of (2.79)–(2.80), if each ∆i satisfies (2.81)with ψ∆i

∈ K∞ being Lipschitz on compact sets, then asymptotic stabiliza-tion can be achieved with ǫen = 0. For this purpose, one may choose the

γ(·)(·) ’s and the σ(·)’s such that their inverse functions are Lipschitz on com-

pact sets. Then, each ISS-Lyapunov function Vei , recursively constructed withVe1 , . . . , Vei , satisfies property (2.99) with αei in the form of

αei = (ϑei)−1 αV (2.101)

where ϑei ∈ K∞ is Lipschitz on compact sets.Then, the γ

ei−1ei αei−1 used for the design of each ei-subsystem is in the

form of(

ϑ′ei−1

)−1

αV with ϑ′ei−1= ϑei−1

(

γei−1ei

)−1

being Lipschitz on


compact sets. Note that γei−1ei αei−1 and γ

ei+1ei αV correspond to the χw1

φ

and χw2

φ in (2.72) for gain assignment. This fulfills the condition given at theend of Subsection 2.3.1 for zero ǫ.

2.4 NOTES

The small-gain condition, i.e., a loop-gain of less than unity, is one way to en-sure stability of interconnected systems. In the past twenty years, tremendousefforts have been made in stability analysis and control design of intercon-nected nonlinear systems. The idea of the small-gain theorem was originallystudied with the gain property taking a linear or affine form; see, e.g., [48, 283]for input-output feedback systems, as well as the recent works [20, 76]. Thesmall-gain theorem for nonlinear feedback systems with non-affine gains waspresented in [88, 199] within the input-output context.

Taking explicit advantage of Sontag’s seminal work on ISS [249, 241, 242],the first generalized, nonlinear ISS small-gain theorem was proposed in [130].The IOS counterpart of the small-gain theorem is also available in [130]. Asa fundamental difference with respect to the earlier small-gain theorems, inthe ISS or IOS framework, the role of the initial conditions is made explicit toensure asymptotic stability in the Lyapunov sense as well as bounded-inputbounded-output stability. A new small-gain design tool was presented for thefirst time in [123, 130] for robust global stabilization of nonlinear systemswith dynamic uncertainties. In parallel, Teel presents a small-gain tool for theanalysis and synthesis of control systems with saturation in [260]. A Lyapunovreformulation of the ISS small-gain theorem can be found in [126]. Necessaryand sufficient small-gain conditions for interconnected integral input-to-statestable (iISS) systems can be found in [107, 108, 110]. Further extensions of thesmall-gain theorem have also been made for general nonlinear systems possiblywith time delays using the concept of vector Lyapunov functions [138]. Asa powerful tool, the ISS small-gain theorem has been included in standardtextbooks on nonlinear systems; see, e.g., [106, 144]. See also the book [137]and the references cited therein for other more recent developments along theline of ISS small-gain.

This chapter mainly focuses on continuous-time interconnected systemsdescribed by differential equations, while the counterparts of the results fordiscrete-time systems [131, 154] and hybrid systems [208, 167, 211, 169] havealso been developed based on the corresponding extensions of ISS. The in-terconnected hybrid systems studied in [169] may involve both stable andunstable dynamics.

There have also been numerous successful applications of the small-gaintheorem to nonlinear control designs. The applications of the small-gain the-orem to output regulation and global stabilization of nonlinear feedforwardsystems can be found in [100, 31, 32, 30, 99]. References [208, 25, 169] em-ploy the small-gain theorem for networked and quantized control designs. In[220], the authors employ a modified small-gain theorem to solve the stabil-


ity problem arising from observer-based control designs. Another interestingapplication of the small-gain theorem lies in robust adaptive dynamic pro-gramming; see e.g., [120].

Chapter 3 introduces an extension of the small-gain theorem to large-scaledynamic networks which contain more than two subsystems. More relateddiscussions are given in Chapter 3.

3 LargeScale DynamicNetworks

The small-gain theorem introduced in Chapter 2 has found wide application instability analysis, stabilization, robust adaptive control, observer design, andoutput regulation for interconnected nonlinear systems. Although one mayuse the small-gain theorem recursively for interconnected systems involvingmore than one cycle, refined small-gain criteria are highly desired to handlelarge-scale dynamic networks more efficiently.

Example 3.1 shows a control system which is transformable into an inter-connected system of three ISS subsystems and contains more than one cyclein the system structure.

Example 3.1 Consider the single-mode approximation of the PDE model ofan axial compressor introduced in [206]:

R = σR(−2φ− φ2 −R), R(t) ≥ 0 (3.1)

φ = −ψ − 3

2φ+

1

2− 1

2(φ+ 1)3 − 3(φ+ 1)R (3.2)

ψ =1

β2(φ+ 1− v), (3.3)

where φ and ψ are the deviations of the mass flow and the pressure rise fromtheir set points, R is the nonnegative magnitude of the first stall mode, thecontrol input v is the flow through the throttle, and σ, β are positive constants.For this system, ψ and R are not measurable. The control objective is tostabilize the system and make φ asymptotically converge to the origin.

In [153, Section 2.4], a state-feedback control law is designed for globalasymptotic stabilization. Reference [152] improves the design by only using φand ψ for feedback. By only using the measurement of φ, [196] realize semi-global stabilization with a high-gain observer. Global asymptotic stabilizationis achieved in [10] through an ISS-induced output-feedback design.

For convenience of discussions, we denote z = R, x1 = φ, x2 = −ψ,y = x1, and u = v/β2, and rewrite the system as

z = g(z, x1) (3.4)

x1 = f1(x1, z) + x2 (3.5)

x2 = f2(x1) + u (3.6)

y = x1. (3.7)

39


In [10], by using y and u, a reduced-order observer with state x2 is designedto estimate the unmeasurable x2 such that the estimation error system withstate x2−x2 is ISS with z as the input. A control law u = u(y, x2) is designedsuch that the (x1, x2)-subsystem is ISS with both x2 − x2 and z as the inputs.The z-subsystem is also proved to be ISS with x1 as the input. Thus, theclosed-loop system is a dynamic network composed of three ISS subsystems.The system structure is shown in Figure 3.1.

(x1, x2) z

x2 − x2

FIGURE 3.1 The block diagram of the closed-loop system in Example 3.1.

The axial compressor model considered in this example is in the widelyrecognized output-feedback form. With the tools developed in this book, we cansolve more complicated control problems, e.g., quantized control and distributedcontrol, with this class of nonlinear systems.

To illustrate the need of small-gain results for network analysis in more gen-eral terms than the example, consider a nonlinear dynamic network composedof three subsystems:

xi = fi(x), i = 1, 2, 3, (3.8)

where xi ∈ Rni is the state of the i-th subsystem, x = [xT1 , xT2 , x

T3 ]T , and

fi : Rn1+n2+n3 → Rni is a locally Lipschitz function satisfying fi(0) = 0.

Suppose that each xi-subsystem has an ISS-Lyapunov function Vi, whichis positive definite and radially unbounded, and satisfies

Vi(xi) ≥ maxj 6=i

γij(Vj(xj)) ⇒ ∇Vi(xi)fi(x) ≤ −αi(Vi(xi)), ∀x, (3.9)

where γij ∈ K∪0 represents the ISS gains and αi is a continuous and positivedefinite function. We consider the case in which only γ12, γ13, γ21, γ32, γ31 arenonzero ISS gains.

The gain interconnection structure of the dynamic network can be repre-sented with a digraph, called the gain digraph, by considering the subsystemsas vertices and the nonzero gain interconnections as directed links. Since thegain digraph describes the relation between the Lyapunov functions, each xi-subsystem is represented with its ISS-Lyapunov function Vi. The gain digraphof the dynamic network defined above is shown in Figure 3.2.

LargeScale Dynamic Networks 41

V1

V2

V3

FIGURE 3.2 The gain digraph of the dynamic network (3.8).

We consider using the small-gain theorem introduced in Chapter 2 twiceto analyze the stability property of the dynamic network. First, we divide thedynamic network into two parts: the (x1, x2)-subsystem with x3 as the inputand the x3-subsystem with (x1, x2) as the input.

The (x1, x2)-subsystem is ISS because the small-gain condition is satisfied:

γ12 γ21 < Id. (3.10)

We construct an ISS-Lyapunov function for the (x1, x2)-subsystem as

V(1,2)(x1, x2) = max V1(x1), σ(V2(x2)) , (3.11)

where σ is a class K∞ function that is continuously differentiable on (0,∞)and satisfies

σ > γ12, σ−1 > γ21. (3.12)

Then, it holds that

V(1,2)(x1, x2) ≥ γ13(V3(x3))

⇒∇V(1,2)(x1, x2)f(1,2)(x) ≤ −α(1,2)(V(1,2)(x1, x2)) a.e., (3.13)

where f(1,2)(x) := [fT1 (x), fT2 (x)]T and α(1,2) is a continuous and positivedefinite function.

From (3.9), the influence of V(1,2)(x1, x2) to V3(x3) can be represented by

V3(x3) ≥ γ3(1,2)(V(1,2)(x1, x2))

⇒∇V3(x3)f3(x) ≤ −α3(V3(x3)), (3.14)

where γ3(1,2)(s) := maxγ31(s), γ32 σ−1(s) for s ≥ 0.Then, we consider the interconnection of the (x1, x2)-subsystem and the

x3-subsystem. The dynamic network is asymptotically stable at the origin ifit satisfies the small-gain condition γ13 γ3(1,2) < Id, or equivalently,

γ13 γ31 < Id, (3.15)

γ13 γ32 σ−1 < Id. (3.16)


The satisfaction of condition (3.16) depends on the choice of σ, which is sub-ject to constraints σ > γ12 and σ−1 > γ21. Note that (3.10) guarantees theexistence of σ to satisfy the constraints. By choosing σ such that σ−1 > γ21and σ−1 is very close to γ21, (3.16) can be guaranteed by

γ13 γ32 γ21 < Id. (3.17)

Thus, the dynamic network is asymptotically stable at the origin if (3.10),(3.15), and (3.17) are satisfied. This means that the composition of the ISSgains along every simple cycle in the gain digraph should be less than Id. Thiscondition is referred to as the cyclic-small-gain condition.

Considering the wide interest in studying large-scale dynamic networks, itis natural to ask:

1. Is the cyclic-small-gain condition valid for general large-scale dynamic net-works composed of ISS subsystems?

2. How do we construct an ISS-Lyapunov function for a dynamic network ifit satisfies the cyclic-small-gain condition?

In this chapter, we develop cyclic-small-gain results to solve the problemsfor continuous-time, discrete-time, and more general hybrid dynamic net-works. To make the results more accessible, we mainly consider ISS systemsin this chapter, while some extensions to IOS systems are also provided.

The first three sections of this chapter study continuous-time, discrete-time,and hybrid dynamic networks, respectively. The fourth section discusses thedevelopment of the relevant literature.

3.1 CONTINUOUSTIME DYNAMIC NETWORKS

Consider the following large-scale dynamic network containing N subsystems:

xi = fi (x, ui) , i = 1, . . . , N, (3.18)

where x =[

xT1 , . . . , xTN

]Twith xi ∈ Rni is the state, ui ∈ Rmi represents

the external inputs, and each fi : Rn+mi → Rni with n =∑N

j=1 nj is alocally Lipschitz function satisfying fi(0, 0) = 0. The external input u =[

uT1 , . . . , uTN

]Tis a measurable and locally essentially bounded function from

R+ to Rm with m =∑N

i=1mi. Denote f(x, u) = [fT1 (x, u1), . . . , fTN (x, uN )]T .

Assume that for i = 1, . . . , N , each xi-subsystem admits a continuouslydifferentiable ISS-Lyapunov function Vi : R

ni → R+ satisfying the following


αi(|xi|) ≤ Vi(xi) ≤ αi(|xi|), ∀xi; (3.19)


2. there exist γij ∈ K∪ 0 (j = 1, . . . , N , j 6= i) and γui ∈ K∪ 0 such that

Vi(xi) ≥ maxj 6=i

γij(Vj(xj)), γui(|ui|)

⇒∇Vi(xi)fi(x, ui) ≤ −αi(Vi(xi)), ∀x, ui, (3.20)

where αi is a continuous and positive definite function.

For systems that are formulated in the dissipation form, property 2 aboveshould be replaced by

2′. there exist α′i ∈ K∞, σ′

ij ∈ K∪0 (j = 1, . . . , N , j 6= i) and σ′ui ∈ K∪0

such that

∇Vi(xi)fi(x, ui) ≤ −α′i(Vi(xi)) + max

σ′ij(Vj(xj)), σ

′ui(|ui|)

. (3.21)

Due to the equivalence of the two forms for continuous-time systems, we onlyconsider the gain margin form in the following discussions.

By considering the subsystems as vertices and the nonzero gain intercon-nections as directed links, the gain interconnection structure of the dynamicnetwork can be represented by a digraph, called the gain digraph. Then, con-cepts from graph theory, such as path, reachability, and simple cycle, canbe used to describe the gain interconnections in the dynamic network. Sincethe gains are defined with Lyapunov functions, for i = 1, . . . , N , each xi-subsystem is represented by its Lyapunov function Vi. Appendix A gives thedefinitions of the related notions in graph theory.

Theorem 3.1 answers the question on the validity of the cyclic-small-gaincondition for continuous-time large-scale dynamic networks with subsystemsadmitting ISS-Lyapunov functions.

Theorem 3.1 Consider the continuous-time dynamic network (3.18) witheach xi-subsystem admitting a continuously differentiable ISS-Lyapunov func-tion Vi satisfying (3.19)–(3.20). Then, it is ISS with x as the state and u asthe input if for every simple cycle (Vi1 , Vi2 , . . . , Vir , Vi1) in the gain digraph,

γi1i2 γi2i3 · · · γiri1 < Id, (3.22)

where r = 2, . . . , N and 1 ≤ ij ≤ N , ij 6= ij′ if j 6= j′.

Condition (3.22) means that the composition of the ISS gains along everysimple cycle in the gain digraph is less than the identity function Id. Weprove Theorem 3.1 by constructing ISS-Lyapunov functions. In this way, theproblem of constructing ISS-Lyapunov functions for the large-scale dynamicnetworks is solved at the same time.


3.1.1 BASIC IDEA OF CONSTRUCTING ISSLYAPUNOV FUNCTIONS

The small-gain theorem introduced in Chapter 2 considers the case in whichdynamic network (3.18) contains two subsystems, i.e., N = 2. In this case, ifγ12γ21 < Id, then the dynamic network is ISS and an ISS-Lyapunov functioncan be constructed as:

V (x) = maxV1(x1), σ(V2(x2)), (3.23)

where σ ∈ K∞ is continuously differentiable on (0,∞) and satisfies

σ > γ12, σ−1 > γ21. (3.24)

Recall the fact that γ12 γ21 < Id ⇔ γ21 γ12 < Id. By using LemmaC.1 twice, there exist γ12, γ21 ∈ K∞ which are continuously differentiable on(0,∞) and satisfy γ12 > γ12, γ21 > γ21 and γ12 γ21 < Id. Thus, with γ12, γ21replaced by γ12, γ21 (as shown Figure 3.3), the small-gain condition is stillsatisfied.

V1 V2 V1 V2

γ12

γ21

γ12

γ21

FIGURE 3.3 The replacement of the ISS gains.

If we choose σ = γ12, then condition (3.24) is satisfied and the resultingISS-Lyapunov function is

V (x) = max V1(x1), γ12(V2(x2)) . (3.25)

Since γ12 is a modification of the ISS gain γ12, the term γ12(V2(x2)) can beconsidered as the “potential influence” of V2 acting on V1 with modified gainγ12.

3.1.2 A CLASS OF ISSLYAPUNOV FUNCTIONS FOR DYNAMIC NETWORKS

Based on the idea of potential influence, a class of ISS-Lyapunov functions areconstructed for large-scale dynamic networks satisfying the cyclic-small-gaincondition.

Recall the fact that for any χ1, χ2 ∈ K∪ 0, χ1 χ2 < Id ⇔ χ2 χ1 < Id.Consider a dynamic network in the form of (3.18) with the cyclic-small-gaincondition (3.22) satisfied. For each i∗ = 1, . . . , N , it holds that

γi∗i2 γi2i3 · · · γiri∗ < Id (3.26)


for r = 2, . . . , N , 1 ≤ ij ≤ N , ij 6= i∗, ij 6= ij′ if j 6= j′. With Lemma C.1, ifγi∗i2 6= 0, then one can find a γi∗i2 ∈ K∞ which is continuously differentiableon (0,∞) and satisfies γi∗i2 > γi∗i2 such that (3.26) still holds with γi∗i2replaced by γi∗i2 .

By repeating this procedure for all the γi∗i2 with i∗ = 1, . . . , N , i2 6= i∗,there exist γ(·)’s such that

1. γ(·) ∈ K∞ and γ(·) > γ(·) if γ(·) ∈ K; γ(·) = 0 if γ(·) = 0.2. γ(·)’s are continuously differentiable on (0,∞).3. for each r = 2, . . . , N ,

γi1i2 · · · γiri1 < Id (3.27)

holds for all 1 ≤ ij ≤ N and ij 6= ij′ if j 6= j′.

Through the approach above, all the nonzero gains in the dynamic networkare replaced by the γ(·)’s, which are of classK∞ and continuously differentiableon (0,∞) such that the cyclic-small-gain condition is still satisfied. Note thatthe replacement of the nonzero gains does not influence the gain digraph.

In the large-scale dynamic network, the potential influence acting on thep-th subsystem from all the subsystems can be described as

V[p] =

⋃

j=1,...,N

V[p]j (x) (3.28)

with

V[p]j (x) =

γi[p]1 i

[p]2

· · · γi[p]j−1i

[p]j

(

Vi[p]j

(

xi[p]j

))

,

where i[p]1 = p, i

[p]k ∈ 1, . . . , N, k ∈ 1, . . . , j, i[p]k 6= i

[p]k′ if k 6= k′, for

j = 1, . . . , N . Clearly, each element in V[p]j (x) corresponds to a simple path

ending at Vp in the gain digraph.Note that γ(·) ∈ K∞ ∪ 0. It is easy to verify that maxV[p] is positive

definite and radially unbounded with respect to the Lyapunov functions ofthe subsystems with indices belonging to RS(p).

Correspondingly, the potential influence of the external input u =[

uT1 , . . . , uTN

]Tacting on the p-th subsystem can be described as

U[p] =

⋃

j=1,...,N

U[p]j (3.29)

with

U[p]j =

γi[p]1 i

[p]2

· · · γi[p]j−1i

[p]j

γui

[p]j

(|ui[p]j−1

|)

(3.30)

for j = 1, . . . , N .


Define

VΠ(x) = maxVΠ(x) = max

⋃

p∈Π

V[p](x)

(3.31)

where set Π ⊆ 1, . . . , N satisfies⋃

p∈Π(RS(p)) = 1, . . . , N.It can be directly verified that max

(

⋃

p∈Π V[p])

is positive definite and

radially unbounded with respect to max V1, . . . , VN and thus with respectto x, i.e., there exist α, α ∈ K∞ such that α(|x|) ≤ VΠ(x) ≤ α(|x|) for allx. It can also be observed that VΠ is locally Lipschitz on Rn\ 0. Thanksto Rademacher’s theorem (see, e.g., [59, p. 216]), VΠ is differentiable almosteverywhere.

Correspondingly, denote

uΠ = maxUΠ = max

⋃

p∈Π

U[p]

. (3.32)

It can be verified that there exists a γu ∈ K∞ such that uΠ ≤ γu(|u|) for allu.

In Subsection 3.1.3, we show that VΠ (x) is an ISS-Lyapunov function (notnecessarily continuously differentiable) of the dynamic network with uΠ as thenew input; see (3.54). Then, it is directly proved that the dynamic network isISS with u as the input; see (3.55).

3.1.3 PROOF OF THE CYCLICSMALLGAIN THEOREM FOR CONTINUOUSTIME DYNAMIC NETWORKS

Throughout the proof, we consider the case where VΠ(x) ≥ uΠ and x 6= 0.Intuitively, if VΠ is strictly decreasing on the timeline, then it is an ISS-Lyapunov function of the dynamic network. The decreasing property of VΠ =maxVΠ is determined by the decrease in all the elements in VΠ that take thevalue of VΠ.

Note that the elements in VΠ are defined with the compositions of ISSgains along specific simple paths and each element corresponds to the ISS-Lyapunov function of one subsystem. For convenience of notation, we use ato mark the simple path corresponding to an element taking the value of VΠand use A to denote the set of all such a’s. Consider a specific simple pathma := (Via

j, . . . , Via2 , Via1 ) in the gain digraph. The simple path is highlighted

by the thick arrows in Figure 3.4. According to the definition, for a fixed x,it holds that

γia1 ia2 · · · γiak−1

iak · · · γia

j−1iaj

(

Viaj

(

xiaj

))

= VΠ (x) , (3.33)

where iak ∈ 1, . . . , N, k ∈ 1, . . . , j, and iak 6= iak′ if k 6= k′, for all a ∈ A.


Via1 Via2 Viaj−1

Viaj

· · · ...

FIGURE 3.4 The j subsystems on a specified simple path ma ending at Via1.

With the cyclic-small-gain condition (3.22) satisfied, by using property(3.33), we study the relation between Viaj and the Lyapunov functions of theother subsystems.

1. Relation between Viaj and Vl with l ∈ ia1, . . . , iaj−1If j ≥ 2, for all k = 1, . . . , j − 1, property (3.33) implies

γia1 ia2 · · · γiak−1i

ak

(

Viak(xia

k))

≤γia1 ia2 · · · γiak−1i

ak · · · γiaj−1i

aj

(

Viaj (xiaj ))

, (3.34)

and thus,

Viak(xia

k) ≤ γia

kiak+1

· · · γiaj−1i

aj

(

Viaj(xia

j))

(3.35)

by canceling out γia1 ia2 · · · γiak−1

iak.

Note that Viakwith k = 1, . . . , j − 1 represents the Lyapunov functions

of all the subsystems on simple path ma. Property (3.35) describes therelation between Via

kand Via

j. If j = 1, then there is only the subsystem

corresponding to Via1 on simple path ma.Condition (3.27) means that the composition of the modified gains γ(·)’salong every simple cycle in the gain digraph is less than Id. Specifically, forthe subsystems along simple path ma, it holds that

γiaj iak γiakiak+1

· · · γiaj−1iaj< Id. (3.36)

Then, property (3.35) can be simplified as:

γiaj iak(

Viak(xia

k))

≤ γiaj iak γiakiak+1

· · · γiaj−1iaj

(

Viaj (xiaj ))

< Viaj (xiaj ) (3.37)

for all k = 1 . . . , j − 1. Equivalently,

Viaj(xia

j) > γia

jl (Vl(xl)) (3.38)

holds for all l ∈ ia1, . . . , iaj−1.


2. Relation between Viajand Vl with l ∈ 1, . . . , N\ia1, . . . , iaj

We first consider the case of j ≤ N−1. For each l ∈ 1, . . . , N\ia1, . . . , iaj,if γia

j−1l6= 0, then γia1 ia2 · · · γia

j−2iaj−1

γiaj−1l

(Vl(xl)) belongs to VΠ(x);

otherwise γia1 ia2 · · · γiaj−2iaj−1

γiaj−1l

(Vl(xl)) = 0. Thus, if j ≤ N − 1, then

γia1 ia2 · · · γiaj−2iaj−1

(

Viaj−1(xiaj−1

))

≥ γia1 ia2 · · · γiaj−2iaj−1

γiaj−1l(Vl(xl)) , (3.39)

holds for all l ∈ 1, . . . , N\ia1, . . . , iaj, which can be simplified by cancel-ing out the common terms on both sides as

Viaj(xia

j) ≥ γia

jl (Vl(xl)) . (3.40)

If j = N , then all the subsystems of the dynamic network are on simplepath ma.

Properties (3.38) and (3.40) together imply

Viaj (xiaj ) ≥ maxl∈1,...,N\iaj

γiaj l (Vl(xl))

≥ maxl∈1,...,N\iaj

γiajl (Vl(xl))

. (3.41)

From the definition of the new input uΠ, it can also be guaranteed that

VΠ(x) ≥ uΠ ⇒ Viaj (x) ≥ γuiaj (|uiaj |). (3.42)

For each a ∈ A, if conditions (3.41) and (3.42) hold, then the property ofthe ISS-Lyapunov functions, given in (3.20), yields:

∇Viaj(xia

j)fia

j(x, uia

j) ≤ −αia

j

(

Viaj(xia

j))

≤ −αiaj(VΠ(x)) (3.43)

where αiaj := αiaj γ−1iaj−1i

aj · · · γ−1

ia1 ia2and the second inequality holds due to

(3.33).Define

γma = γia1 ia2 · · · γiaj−1i

aj, (3.44)

Vma(xiaj ) = γma

(

Viaj (xiaj ))

. (3.45)

For each a ∈ A, since γia1 ia2 · · · γiaj−1iaj∈ K∞ and the γ(·)’s are continuously

differentiable on (0,∞), with (3.43), there exists a continuous and positivedefinite function αma such that

∇Vma(xma)fiaj (x, uiaj ) = γ′ma

(

Viaj (xiaj ))

∇Viaj (xiaj )fiaj (x, uiaj )≤ −

(

γ′ma γ−1ma (VΠ(x))

)

αiaj(VΠ(x))

≤ −αma (VΠ(x)) (3.46)


if VΠ (x) 6= 0 and VΠ (x) ≥ uΠ. Property (3.46) means that all the elementsthat take the value of VΠ(x) are decreasing in the case of VΠ (x) ≥ uΠ. In thefollowing procedure, we study the decreasing property of VΠ(x).

Denote the size of A as NA. We consider two cases: NA = 1 and NA ≥ 2.Case 1: NA = 1. The decreasing property of VΠ is determined by Vma (and

thus Viaj ) in a neighborhood of x, which can be guaranteed by (3.46) as

∇VΠ(x)f(x, u) = ∇Vma(xma)fiaj (x, uiaj−1) ≤ −αma (VΠ(x)) (3.47)

whenever VΠ(x) ≥ uΠ.Case 2: NA ≥ 2. Recall that x 6= 0. Then, for all a ∈ A, xiaj 6= 0. Using the

continuous differentiability of γ(·)’s and Viaj and the continuity of fiaj , one sees

that ∇Vma(xma)fiaj (x, uiaj ) is continuous with respect to x for a specified uiaj ,and there exists a neighborhood X = X1 × · · · × XN of x such that

∇Vma(xma)fiaj (ξ, uiaj ) ≤ −1

2αma (VΠ(x)) (3.48)

holds for all ξ ∈ X and all a ∈ A.For the locally Lipschitz function VΠ, it holds for almost all pairs of (x, u)

that

∇VΠ(x)f(x, u) =d

dt

∣

∣

∣

∣

t=0

VΠ(φ(t)), (3.49)

where φ(t) = [φT1 (t), . . . , φTN (t)]T is the solution of the initial-value problem

φ(t) = f(φ(t), u), φ(0) = x. (3.50)

Because of the continuity of φ(t) with respect to t, there exists a δ > 0such that φ(t) ∈ X and any element in VΠ corresponding to a simple path

marked by b with b /∈ A satisfies Vmb(φibj (t)) < max

Vma(φiaj(t)) : a ∈ A

for

t ∈ [0, δ).

For any t ∈ (0, δ), irrespective of which element in

Vma(φiaj (t)) : a ∈ A

,

VΠ(φ(t)) takes the value of, there exists a continuous and positive definitefunction αA such that

VΠ(φ(t)) − VΠ(x)

t≤ −αA (VΠ(x)) . (3.51)

For instance, we can take αA(s) = mina∈A αma(s)/3 for all s ≥ 0.Hence, if VΠ is differentiable at x, then

∇VΠ(x)f(x, u) ≤ −αA (VΠ(x)) . (3.52)

By combining (3.47) and (3.52), it follows that if VΠ is differentiable at x,then

VΠ(x) ≥ uΠ ⇒ ∇VΠ(x)f(x, u) ≤ −αA (VΠ(x)) . (3.53)


Note that for different x, different elements may take the value of VΠ(x)and set A may be different. Define αΠ(s) as the minimum of all the possibleαA(s)’s for s ≥ 0. Then, αΠ is a continuous and positive definite function. Forany x, if VΠ is differentiable at x, then

VΠ(x) ≥ uΠ ⇒ ∇VΠ(x)f(x, u) ≤ −αΠ (VΠ(x)) . (3.54)

Since VΠ is differentiable almost everywhere, (3.54) holds almost everywhere.Recall that the definition of uΠ in (3.32) implies uΠ ≤ γu(|u|) for all u withγu ∈ K∞. Thus, as a direct consequence of (3.54), we have

VΠ(x) ≥ γu(|u|) ⇒ ∇VΠ(x)f(x, u) ≤ −αΠ (VΠ(x)) . (3.55)

This proves that VΠ is an ISS-Lyapunov function of the dynamic network withu as the input, and at the same time, proves the cyclic-small-gain theorem forcontinuous-time dynamic networks.

The ISS-Lyapunov function VΠ proposed above is not continuously differ-entiable. With the technique given in [170], one can further construct smoothISS-Lyapunov functions based on VΠ.

For simplicity of discussions, the ISS-Lyapunov functions of the subsys-tems are assumed to be continuously differentiable. The construction methodis still valid for systems with Lyapunov functions that are continuously dif-ferentiable almost everywhere and satisfy (3.20) almost everywhere. In thatcase, it can still be proved that the constructed VΠ satisfies (3.54) almosteverywhere. Moreover, the modified gains γ(·)’s are not required to be con-tinuously differentiable on (0,∞) either. The ISS-Lyapunov function VΠ canstill be constructed by using γ(·)’s which are continuously differentiable almosteverywhere.

Example 3.2 Consider the N = 3 dynamic network (3.8). With no externalinput, it satisfies the cyclic-small-gain condition and is asymptotically stableat the origin according to Theorem 3.1.

For the gain digraph shown in Figure 3.2, RS(i) = 1, 2, 3 for i = 1, 2, 3.Different ISS-Lyapunov functions VΠ’s can be constructed by choosing differ-ent Π’s. For example,

V1(x) = max V1(x1), γ12(V2(x2)), γ13 γ32(V2(x2)), γ13(V3(x3)) , (3.56)

V2(x) = max V2(x2), γ21(V1(x1)), γ21 γ13(V3(x3)) . (3.57)

There are two terms depending on V2(x2) in the definition of V1(x), becausethere are two simple paths leading from V2 to V1 in the gain digraph.

Example 3.3 If the gain digraph of a dynamic network is disconnected, thenit is impossible to find one single subsystem which is reachable from all theother subsystems, and the Π should contain more than one element to con-struct a positive definite and radially unbounded VΠ. Consider a dynamic net-work with the gain digraph shown in Figure 3.5.


V1

V2

V3

V4

V5

FIGURE 3.5 A disconnected gain digraph.

Since RS(1)∪RS(5) = 1, 2, 3, 4, 5, we choose Π = 1, 5, with which anISS-Lyapunov function is constructed as

VΠ(x) = max V1(x1), γ13(V3(x3)), γ13 γ32(V2(x2)), V5(x5), γ54(V4(x4))(3.58)

where γ(·)’s are the appropriately modified ISS gains. It can be observed thatmaxV1(x1), γ13(V3(x3)), γ13 γ32(V2(x2)) and maxV5(x5), γ54(V4(x4)) arethe Lyapunov functions of the (x1, x2, x3)-subsystem (the part on the left-handside in Figure 3.5) and the (x4, x5)-subsystem (the part on the right-handside), respectively. In fact, the Lyapunov function for a disconnected dynamicnetwork can be directly defined as the maximum of the Lyapunov functions ofall the disconnected parts.

With Π = 1, 5, define

uΠ = max u1, γ13 γu3(u3), γ13 γ32 γu2(u2), u5, γ54 γu4(u4) . (3.59)

Then, there exists a continuous and positive definite function αΠ such that

VΠ(x) ≥ uΠ ⇒ ∇VΠ(x)f(x, u) ≤ −αΠ(VΠ(x)), a.e. (3.60)

To analyze the influence of the external inputs to each subsystem, one mayfirst transform the Lyapunov-based ISS property into the trajectory-based ISSproperty: for any initial state x(0) = x0,

VΠ(x(t)) ≤ max

β(VΠ(x0), t), ‖uΠ‖[0,t]

(3.61)

where β ∈ KL. Consider the x2-subsystem, for example. From the definitionof VΠ, V2(x2) ≤ γ−1

32 γ−113 (VΠ(x)) for all x, and the influence of the external

inputs on the x2-subsystem can be easily estimated through the following IOSproperty by considering V2(x2) as the output:

V2(x2(t)) ≤ max

γ−132 γ−1

13 (β(VΠ(x0), t)) , γ−132 γ−1

13

(

‖uΠ‖[0,t])

(3.62)

for any initial state x(0) = x0. According to the definition of uΠ in (3.59),property (3.62) implies that the x2-subsystem is influenced by u4 and u5. How-ever, due to the disconnected system structure, u4 and u5 do not influence the


x2-subsystem. For a more accurate estimation, one may just use the Lyapunovfunction of the (x1, x2, x3)-subsystem to estimate the influence of u1, u2, andu3 on the x2-subsystem.

3.1.4 DISCONTINUOUS DYNAMIC NETWORKS

Reference [84] extends the concepts of ISS and the ISS-Lyapunov functionto discontinuous systems and also proposes an extended Filippov solutionfor interconnected discontinuous systems by using differential inclusions. Ap-pendix B provides some related concepts on discontinuous systems. Basedon the concept of extended Filippov solution, an ISS small-gain theorem hasbeen developed for discontinuous systems. Based on the results in [84], wedevelop a cyclic-small-gain theorem for discontinuous dynamic networks withthe subsystems represented by differential inclusions:

xi ∈ Fi(x, ui), i = 1, . . . , N, (3.63)

where Fi : Rn+mi

Rni is a convex, compact, and upper semi-continuousset-valued map satisfying 0 ∈ Fi(0, 0), and the variables are defined in thesame way as for (3.18).

Assume that each xi-subsystem in (3.63) admits an ISS-Lyapunov functionVi satisfying (3.19) and

Vi (xi) ≥ maxj=1,...,N ;j 6=i

γij (Vj (xj)) , γui (|ui|)

⇒ maxfi∈Fi(x,ui)

∇Vi(xi)fi ≤ −αi (Vi (xi)) (3.64)

wherever ∇Vi exists. Clearly, (3.64) is a direct modification of (3.20).We have such a cyclic-small-gain result for discontinuous dynamic networks:

if the cyclic-small-gain condition (3.22) is satisfied, then the discontinuousdynamic network is ISS and an ISS-Lyapunov function VΠ can be constructedas in (3.31) such that

VΠ(x) ≥ uΠ ⇒ maxf∈F (x,u)

∇VΠ(x)f ≤ −αΠ (VΠ(x)) (3.65)

wherever ∇V exists, with F (x, u) = [FT1 (x, u1), . . . , FTN (x, uN )]T . Note that

property (3.65) is an extension of property (3.54).

3.1.5 DYNAMIC NETWORKS OF IOS SUBSYSTEMS

Corresponding to Theorem 2.2, this subsection presents cyclic-small-gain re-sults for large-scale dynamic networks composed of IOS subsystems. Time-delay issues are also discussed.


Consider a large-scale dynamic network in the form of

x1 = f1(x1, y2, y3, . . . , yn, u1) (3.66)

x2 = f2(x2, y1, y3, . . . , yn, u2) (3.67)

...

xn = fn(xn, y1, y2, . . . , yn−1, un) (3.68)

with output maps

yi = hi(xi), i = 1, . . . , n. (3.69)

For each i-th subsystem, xi ∈ Rni is the state, ui ∈ R

mi is the input,yi ∈ Rli is the output, and fi, hi are locally Lipschitz functions. Denotex = [xT1 , . . . , x

Tn ]T , y = [yT1 , . . . , y

Tn ]T , and u = [uT1 , . . . , u

Tn ]T . By considering

u as a function of time, assume that u is measurable and locally essentiallybounded.

Suppose that each i-th subsystem is UO with zero offset and IOS with yjfor j 6= i and ui as the inputs and yi as the output. Specifically, there existαOi ∈ K∞, βi ∈ KL, γij ∈ K and γui ∈ K such that

|xi(t)| ≤ αOi

|xi(0)|+∑

j 6=i‖yj‖[0,t] + ‖ui‖[0,t]

(3.70)

|yi(t)| ≤ maxj 6=i

βi(|xi(0)|, t), γij(‖yj‖[0,t]), γui (‖ui‖∞) (3.71)

for all t ∈ [0, Tmax), where [0, Tmax) with 0 < Tmax ≤ ∞ is the right maximalinterval for the definition of (x1(t), . . . , xn(t)).

A cyclic-small-gain theorem for large-scale dynamic networks composed ofIOS subsystems is given in Theorem 3.2.

Theorem 3.2 Consider dynamic network (3.66)–(3.69) satisfying (3.70)–(3.71) for i = 1, . . . , n. Then the dynamic network is UO and IOS if thecyclic-small-gain condition (3.22) is satisfied.

Reference [134] presents a cyclic-small-gain theorem for large-scale dynamicnetworks composed of output-Lagrange input-to-output stable (OLIOS) sub-systems with an induction-based proof. For the OLIOS systems, the cyclic-small-gain theorem can be proved by using the equivalence between OLIOSand the conjunction of UBIBS and the output asymptotic gain property. Butthis method seems not directly applicable to the systems with only UO andIOS properties. For Theorem 3.2, Appendix D.3 presents a sketch of a proof,which can be considered as a combination of the methods in [130] and [134].

The cyclic-small-gain condition is also valid for the large-scale dynamicnetworks with interconnection time delays. This topic has been studied in


[137, 265]. Consider a dynamic network in the following form:

x1(t) = f1(x1(t), y2(t− τ12), y3(t− τ13), . . . , yn(t− τ1n), u1(t)) (3.72)

x2(t) = f2(x2(t), y1(t− τ21), y3(t− τ23), . . . , yn(t− τ2n), u2(t)) (3.73)

...

xn(t) = fn(xn(t), y1(t− τn1), y2(t− τn2), . . . , yn−1(t− τn(n−1)), un(t))(3.74)

with output maps defined in (3.69), where τij : R+ → [0, θ] for i 6= j representsthe time delay of the interconnection from the j-th subsystem to the i-thsubsystem with constant θ ≥ 0 being the largest time delay. The analogousdefinitions of UO and IOS for systems with delays can be found in [265].

Intuitively, (but maybe not mathematically rigorously), since

|yi(t− τji)| ≤ ‖yi‖[−θ,∞), (3.75)

one may consider the time-delay components (shown in Figure 3.6) as sub-systems with the identity gain, so they should not cause violation of thecyclic-small-gain condition for a system when introduced.

yi(t)time delay

τjiyi(t− τji)

FIGURE 3.6 A time-delay component.

Theorem 3.3 gives a cyclic-small-gain result for large-scale dynamic net-works with time-delays.

Theorem 3.3 Consider dynamic network (3.72)–(3.74) with output maps de-fined by (3.69). Suppose that if the time delays do not exist, i.e., θ = 0, eachi-th subsystem with i = 1, . . . , n satisfies (3.70)–(3.71). Then the dynamicnetwork with θ ≥ 0 is UO and IOS if the cyclic-small-gain condition (3.22) issatisfied.

3.2 DISCRETETIME DYNAMIC NETWORKS

In view of the critical importance of discrete-time system theory in computer-aided control engineering applications, in this section, we generalize the ISScyclic-small-gain theorem introduced in Section 3.1 to discrete-time dynamicnetworks. Due to phenomena which are particular to discrete-time systems(see Example 3.4 below), such a generalization is nontrivial.

Analogous to the continuous-time dynamic network studied in Section 3.1,the discrete-time dynamic network addressed in this section is composed of


N discrete-time subsystems in the following form:

xi(T + 1) = fi(x(T ), ui(T )), i = 1, . . . , N, (3.76)

where x = [xT1 , . . . , xTN ]T with xi ∈ Rni is the state, ui ∈ Rnui represent the

external inputs, and fi : Rn+nui → Rni with n :=

∑Ni=1 ni is continuous. T

takes values in Z+. It is assumed that fi (0, 0) = 0, and the external inputu = [uT1 , . . . , u

TN ]T is bounded. Denote f(x, u) = [fT1 (x, u1), . . . , f

TN (x, uN )]T .

There are two kinds of Lyapunov formulations for discrete-time ISS sys-tems: the dissipation form and the gain margin form. We first give the dissi-pation form.

For i = 1, . . . , N , each xi-subsystem admits a continuous ISS-Lyapunovfunction Vi : R

ni → R+ satisfying the following:


αi(|xi|) ≤ Vi(xi) ≤ αi(|xi|), ∀xi; (3.77)

2. there exist αi ∈ K∞, σij ∈ K ∪ 0 and σui ∈ K ∪ 0 such that

Vi(fi(x, ui)) − Vi(xi)

≤ − αi(Vi(xi)) + maxj 6=i

σij(Vj(xj)), σui(|ui|) , ∀x, ui. (3.78)

Without loss of generality, we assume (Id−αi) ∈ K. Note that if (Id−αi) /∈K, one can always find an α′

i < αi such that (Id−α′i) ∈ K and property (3.78)

holds with αi replaced by α′i. Then, we consider

γij = α−1i (Id− ρi)

−1 σij (3.79)

as the ISS gain from Vj to Vi, with ρi being a continuous and positive definitefunction and satisfying (Id− ρi) ∈ K∞.

Correspondingly, the ISS gain from the external input ui to Vi is definedas

γui = α−1i (Id− ρi)

−1 σui. (3.80)

The gain margin formulation of the ISS-Lyapunov functions can be ob-tained by modifying property 2 as: there exist a continuous and positive def-inite function α′

i and γ′ij , γ

′ui ∈ K ∪ 0 such that

Vi(xi) ≥ maxj 6=i

γ′ij(Vj(xj)), γ′ui(|ui|)

⇒Vi(fi(x, ui))− Vi(xi) ≤ −α′i(Vi(xi)), ∀x, ui. (3.81)

In contrast to the continuous-time systems discussed in Section 3.1, thetrajectories of a discrete-time systemmay “jump out” of the region determinedby the gain margin in (3.81), which means that the γ′ij and the γ′ui in (3.81)may not be the true ISS gains. Consider Example 3.4.


Example 3.4 Consider a discrete-time system

z(T + 1) = g(z(T ), |w(T )|), (3.82)

where z ∈ R is the state, w ∈ Rm is the external input, and g : Rm+1 → R

defined in Figure 3.7 is continuous. Define Vz(z) = |z|. Then, one can find asmall δ > 0 such that

Vz(z) ≥ (1 + δ)|w| ⇒ Vz(g(z, |w|))− Vz(z) ≤ −αz(Vz(z)), (3.83)

where αz is a continuous and positive definite function. Then, (1 + δ) is the“ISS gain” defined by the gain margin formulation (3.81).

z|w| w45−|w|

g(z, |w|)

0

FIGURE 3.7 An example of the gain margin property of discrete-time systems, where

w = (1 + δ)|w|.

However, from Figure 3.7, it is possible that Vz(g(z, |w|)) > (1+δ)|w|, evenif Vz(z) ≤ (1+δ)|w|, which means that the state of the discrete-time nonlinearsystem may “jump out” of the region defined by the gain margin. To solve thisproblem, one may find an αg ∈ K such that |g(z, |w|)| ≤ αg(|w|) whenever|z| ≤ αg(|w|), and define γw(s) = max(1 + δ)s, αg(s) for s ≥ 0. Then, thephenomenon of “jump out” can be avoided as

Vz(z) ≥ γw(|w|) ⇒ Vz(g(z, |w|))− Vz(z) ≤ −αz(Vz(z)) (3.84)

Vz(z) ≤ γw(|w|) ⇒ Vz(g(z, |w|)) ≤ γw(|w|), (3.85)

and γw can be used as the ISS gain for the discrete-time system.


Following the idea in Example 3.4, to take into account the “jump out”issue, the gain margin formulation (3.81) is consolidated with

Vi(xi) ≤ maxj 6=i


⇒Vi(fi(x, ui)) ≤ (Id− δ′i)(

max


)

, (3.86)

where δ′i is a continuous and positive definite function satisfying (Id− δ′i) ∈K∞.

By combining (3.81) and (3.86), the refined gain margin formulation isdescribed with property 1 above and

2′. there exist γij ∈ K ∪ 0 and γui ∈ K ∪ 0 such that

Vi(fi(x, ui)) ≤ (Id− δi)

(

maxj 6=i

γij(Vj(xj)), Vi(xi), γui(|ui|))

, ∀x, ui,(3.87)

where δi is a continuous and positive definite function satisfying (Id− δi) ∈K∞.

We present the cyclic-small-gain results for discrete-time dynamic networksformulated in the dissipation form and the gain margin form in Theorems 3.4and 3.5, respectively.

Theorem 3.4 Consider the discrete-time dynamic network (3.76) with eachxi-subsystem having a continuous ISS-Lyapunov function Vi satisfying (3.77)and (3.78). Then, it is ISS with x as the state and u as the input, if thereexist continuous and positive definite functions ρi satisfying (Id − ρi) ∈ K∞such that for every simple cycle (Vi1 , Vi2 , . . . , Vir , Vi1) in the gain digraph,

γi1i2 γi2i3 . . . γiri1 < Id, (3.88)

where r = 2, . . . , N and 1 ≤ ij ≤ N , ij 6= ij′ if j 6= j′.

Theorem 3.5 Consider the discrete-time dynamic network (3.76) with eachxi-subsystem having a continuous ISS-Lyapunov function Vi satisfying (3.77)and (3.87). Then, the dynamic network is ISS with x as the state and u asthe input if for every simple cycle (Vi1 , Vi2 , . . . , Vir , Vi1) in the gain digraph,

γi1i2 γi2i3 . . . γiri1 < Id, (3.89)

where r = 2, . . . , N , 1 ≤ ij ≤ N , ij 6= ij′ if j 6= j′.

As for continuous-time dynamic networks, Theorems 3.4 and 3.5 are provedby constructing ISS-Lyapunov functions. The ISS-Lyapunov function candi-dates for discrete-time dynamic networks are constructed like continuous-time


dynamic networks in Subsection 3.1.2. One difference is that the ISS-Lyapunovfunctions constructed for discrete-time systems are only required to be con-tinuous.

The proofs of Theorems 3.4 and 3.5 are given in Subsections 3.2.1 and 3.2.2,respectively.

3.2.1 PROOF OF THE CYCLICSMALLGAIN THEOREM FOR DISCRETETIMEDYNAMIC NETWORKS IN DISSIPATION FORM

In the proof for continuous-time dynamic networks, we only consider the be-havior of the largest elements in VΠ. However, for discrete-time dynamic net-works, we should study the motion of all the elements in VΠ. Denote thelargest element of VΠ at time T as V ∗

Π(x∗(T )). Then, VΠ(x(T +1))−VΠ(x(T ))

is determined by all the elements in VΠ, not only by the largest elements. Thisleads to another difference between discrete-time systems and continuous-timesystems.

According to the definition of VΠ, each element of VΠ(x) corresponds to asimple path in the gain interconnection digraph. Consider any specific elementin VΠ(x) that corresponds to a simple path na = (Viaj , . . . , Via2 , Via1 ), as shownin Figure 3.4. The definitions of VΠ and uΠ imply


iak · · · γia

j−1iaj(Via

j(xia

j)) ≤ VΠ(x), (3.90)

γia1 ia2 · · · γiaj−1iaj γuiaj (|uiaj |) ≤ uΠ. (3.91)

We first study the relation between VΠ and the ISS-Lyapunov functions ofthe subsystems. We consider the following two cases.

1. Relation between VΠ and Vl with l ∈ ia1, ia2 , . . . , iaj−1If j = 1, then the simple path na contains only the ia1-th subsystem. Ifj ≥ 2, then for all k ∈ 1, . . . , j − 1, we have

γia1 ia2 · · · γiak−1i

ak(Via

k(xia

k)) ≤ VΠ(x). (3.92)

With the satisfaction of the cyclic-small-gain condition, it holds that

γiakiak+1

· · · γiaj−1i

aj γia

jiak< Id (3.93)

for all k ∈ 1, . . . , j − 1.Then, (3.92) and (3.93) together imply


iak γia

kiak+1

· · · γiaj−1iaj γiaj iak(Viak (xiak )) ≤ VΠ(x) (3.94)

for all k ∈ 1, . . . , j − 1, and equivalently,

γia1 ia2 · · · γiaj−1iaj γiaj l(Vl(xl)) ≤ VΠ(x) (3.95)

for all l ∈ ia1 , ia2, . . . , iaj−1.


2. Relation between VΠ and Vl with l ∈ 1, . . . , N\ia1, ia2 , . . . , iaj−1If j = N , then the simple path na contains all the subsystems in thedynamic network. If j ≤ N − 1, then for all l ∈ 1, . . . , N\ia1, ia2 , . . . , iaj ,we directly have

γia1 ia2 · · · γiaj−1i

aj γia

jl(Vl(xl)) ≤ VΠ(x), (3.96)

because γia1 ia2 · · · γiaj−1i

aj γia

jl(Vl(xl)) is an element of VΠ.

By using the definition of γiaj l, properties (3.95) and (3.96) can be equiva-lently represented by

(Id− ρiaj ) αiaj γ−1iaj−1i

aj · · · γ−1

ia1 ia2(VΠ(x)) ≥ max

l6=iaj

σiaj l(Vl(xl))

. (3.97)

Based on the relation between VΠ and the Lyapunov functions of the sub-systems given in (3.97), we prove that VΠ is an ISS-Lyapunov function of thediscrete-time dynamic network and satisfies the refined gain margin propertydefined by (3.87).

1. Case 1: VΠ(x) ≥ uΠ.Using (3.91), we have

γuiaj (|uiaj |) ≤ γ−1iaj−1i

aj · · · γ−1

ia1 ia2(VΠ(x)) (3.98)

i.e.,

σuiaj (|uiaj |) ≤ (Id− ρiaj ) αiaj γ−1iaj−1i

aj · · · γ−1

ia1 ia2(VΠ(x)). (3.99)

By combining (3.78), (3.90), (3.97), and (3.99), we have

Viaj (fiaj (x, uiaj )) ≤ (Id− αiaj )(Viaj (xiaj )) + maxl6=iaj

σiaj l(Vl(xl)), σuiaj (|uiaj |)

≤ (Id− αiaj) γ−1

iaj−1iaj · · · γ−1

ia1 ia2(VΠ(x))

+ (Id− ρiaj ) αiaj γ−1iaj−1i

aj · · · γ−1

ia1 ia2(VΠ(x))

= γ−1iaj−1i

aj · · · γ−1

ia1 ia2(VΠ(x))

− ρiaj αia

j γ−1

iaj−1iaj · · · γ−1

ia1 ia2(VΠ(x)). (3.100)

With Lemma C.3, by considering Viaj (xiaj ) as s, γ−1iaj−1i

aj · · · γ−1

ia1 ia2(VΠ(x))

as s′, ρiaj αiaj as α, and γia1 ia2 · · · γiaj−1iajas χ, there exists a continuous

and positive definite function αna such that


aj(Via

j(fia

j(x, uia

j))) − VΠ(x)

≤− αna γ−1iaj−1i

aj · · · γ−1

ia1 ia2(VΠ(x))

≤− αna(VΠ(x)), (3.101)


i.e.,

γia1 ia2 · · · γiaj−1iaj(Viaj (fiaj (x, uiaj ))) ≤ (Id− αna)(VΠ(x)), (3.102)

where αna is positive definite and satisfies (Id− αna) ∈ K∞.2. Case 2: VΠ(x) < uΠ.

Property (3.91) can be rewritten as

σuiaj(|uia

j|) ≤ (Id− ρia

j) αia

j γ−1

iaj−1iaj · · · γ−1

ia1 ia2(uΠ). (3.103)

From property (3.97), one can observe

(Id− ρiaj) αia

j γ−1

iaj−1iaj · · · γ−1

ia1 ia2(uΠ) ≥ max

l6=iaj

σiajl(Vl(xl))

. (3.104)

Combining (3.78), (3.90), (3.103), and (3.104), we obtain

Viaj (fiaj (x, uiaj )) ≤ (Id− αiaj )(Viaj (xiaj ))

+ maxl6=iaj

σiaj l(Vl(xl)), σuiaj (|uiaj |)

≤ (Id− αiaj) γ−1

iaj−1iaj · · · γ−1

ia1 ia2(uΠ)

+ (Id− ρiaj ) αiaj γ−1iaj−1i

aj · · · γ−1

ia1 ia2(uΠ)

= γ−1iaj−1i

aj · · · γ−1

ia1 ia2(uΠ)

− ρiaj αia

j γ−1

iaj−1iaj · · · γ−1

ia1 ia2(uΠ). (3.105)

With Lemma C.3, as in Case 1, one can achieve

γia1 ia2 · · · γiaj−1iaj(Viaj (fiaj (x, uiaj )))− uΠ ≤ −αna(uΠ), (3.106)

i.e.,


aj(Via

j(Via

j(fia

j(x, uia

j)))) ≤ (Id− αna)(uΠ). (3.107)

Note that γia1 ia2 · · · γiaj−1iaj(Viaj (xiaj )) is an arbitrary element of VΠ(x).

Choose αΠ as the minimum of the αna ’s corresponding to all the elements inVΠ(x). Then, αΠ is a continuous and positive definite function and satisfies(Id− αΠ) ∈ K∞. Considering both Case 1 and Case 2, we have

VΠ(f(x, u)) ≤ (Id− αΠ)(maxVΠ(x), uΠ), (3.108)

which is in the refined gain margin form defined by (3.87).In the proof of Theorem 3.4, VΠ is shown to satisfy the gain margin for-

mulation. An ISS-Lyapunov function in the dissipation form can be furtherconstructed based on VΠ with a similar idea as in [131, Remark 3.3] and theproof of [23, Proposition 2.6]. Notice that αΠ in (3.108) should be of class K∞to apply these methods. This problem can be solved with [132, Lemma 2.8].


3.2.2 PROOF OF CYCLICSMALLGAIN THEOREM FOR DISCRETETIMEDYNAMIC NETWORKS IN GAIN MARGIN FORM

Properties (3.90) and (3.91) still hold for VΠ in this case.As with the proof of Theorem 3.4, consider an arbitrary element in VΠ(x)

that corresponds to a simple path na = (Viaj, . . . , Via2 , Via1 ). With approaches

similar to properties (3.95) and (3.96), we can ultimately obtain

γiaj l(Vl(xl)) ≤ γ−1iaj−1i

aj · · · γ−1

ia1 ia2(VΠ(x)) (3.109)

for all l ∈ 1, . . . , N\iaj.With (3.95), (3.96), and (3.109) satisfied, property (3.87) implies

Viaj (fiaj (x, uiaj )) ≤ (Id− δiaj ) γ−1iaj−1i

aj · · · γ−1

ia1 ia2(max VΠ(x), uΠ) . (3.110)

With Lemma C.4, by considering γ−1iaj−1i

aj · · · γ−1

ia1 ia2as the χ and δiaj as the

ε, there exists a continuous, positive definite δiajsatisfying (Id − δia

j) ∈ K∞

such that

Viaj(fia

j(x, uia

j)) ≤ γ−1

iaj−1iaj · · · γ−1

ia1 ia2 (Id− δia

j) (max VΠ(x), uΠ) , (3.111)

i.e.,

γia1 ia2 · · · γiaj−1iaj(Viaj (fiaj (x, uiaj ))) ≤ (Id− δiaj ) (max VΠ(x), uΠ) . (3.112)

Define δ(s) = mini∈1,...,N

δi(s)

for s ≥ 0. It is clear that δ is a contin-uous and positive definite function, and satisfies (Id− δ) ∈ K∞. Note that na

corresponds to an arbitrary element in VΠ(x). It can be concluded that

VΠ(f(x, u)) ≤ (Id− δ) (max VΠ(x), uΠ) . (3.113)

Theorem 3.5 is proved.We employ an example to show the construction of an ISS-Lyapunov func-

tion for a discrete-time dynamic network.

Example 3.5 Consider a discrete-time dynamic network in the form of(3.76) with N = 3. The dynamics of the subsystems are defined as:

f1(x, u1) = 0.6x1 +max0.36x32, 3.2x33, u1, (3.114)

f2(x, u2) = 0.4x2 +max0.6x1/31 , 1.2x3, u2, (3.115)

f3(x, u3) = 0.2x3 +max0.36x1/31 , 0.36x2, u3. (3.116)

Each subsystem is ISS with Vi(xi) = |xi| as an ISS-Lyapunov functionsatisfying the dissipation formulation:

Vi(fi(x, ui))− Vi(xi) = −αi(Vi(xi)) + maxj 6=i

σij(Vj(xj)), σui(ui), (3.117)


where

α1(s) = 0.4s, σ12(s) = 0.36s3, σ13(s) = 3.2s3, σu1(s) = s,

α2(s) = 0.6s, σ21(s) = 0.6s1/3, σ23(s) = 1.2s, σu2(s) = s,

α3(s) = 0.8s, σ31(s) = 0.36s1/3, σ32(s) = 0.36s, σu1(s) = s,(3.118)

for s ∈ R+.Choose ρ1(s) = ρ2(s) = ρ3(s) = 0.02s and define

γ12(s) = 0.9184s3, γ13(s) = 8.1633s3,

γ21(s) = 1.0204s1/3, γ23(s) = 2.0408s,

γ31(s) = 0.4592s1/3, γ32(s) = 0.4592s(3.119)

for s ∈ R+. The γ(·)’s are considered as the ISS gains of the subsystems. Thegain digraph of the dynamic network is shown in Figure 3.8.

V1

V2

V3

FIGURE 3.8 The gain digraph of the dynamic network in Example 3.5.

It can be directly checked that the dynamic network satisfies the cyclic-small-gain condition:

γ12 γ21 < Id, γ23 γ32 < Id, γ31 γ13 < Id,

γ12 γ23 γ31 < Id, γ13 γ32 γ21 < Id, (3.120)

and thus it is ISS.In the gain digraph, RS(1) = 1, 2, 3. By choosing Π = 1, we construct

the following ISS-Lyapunov function for the dynamic network:

VΠ(x) = max

V1(x1), γ12(V2(x2)), γ13 γ32(V2(x2)),γ13(V3(x3)), γ12 γ23(V3(x3))

= maxV1(x1), 0.9184V 32 (x2), 8.1633V

33 (x3). (3.121)

Correspondingly,

uΠ = max

σu1(u1), γ12 σu2(u2), γ13 γ32 σu2(u2),γ13 σu3(u3), γ12 γ23 σu3(u3)

= max2.5|u1|, 4.2516|u2|3, 15.9439|u3|3. (3.122)

Figure 3.9 shows the evolutions of VΠ and uΠ with initial condition x(0) =[0.6, 0.87, 0.42]T and inputs u(T ) = [0.1 sin(9T ), 0.1 sin(11T ), 0.1 sin(17T )]T .From Figure 3.9, VΠ ultimately converges to the region determined by themagnitude of |uΠ|. This is in accordance with the theoretical result (3.108).


FIGURE 3.9 The evolutions of VΠ and uΠ of the dynamic network in Example 3.5.

3.3 HYBRID DYNAMIC NETWORKS

Based on the results for continuous-time dynamic networks and discrete-timedynamic networks, it is possible to develop a cyclic-small-gain result for hy-brid dynamic networks, which involve both continuous-time and discrete-timedynamics. The hybrid dynamic network studied in this section is composedof N subsystems whose trajectories may be continuous, piecewise constant,or impulsive on the timeline. Define N = 1, . . . , N as the set of indices ofthe subsystems. For i ∈ N , each i-th subsystem of the dynamic network ismodeled by

xi(t) = fi(x(t), ui(t)), t ∈ R+\πi (3.123)

xi(t) = gi(x(t−), ui(t

−)), t ∈ πi, (3.124)

where xi ∈ Rni is the state of the i-th subsystem, x = [xT1 , . . . , xTN ]T ∈ Rn

with n :=∑N

i=1 ni is the state of the dynamic network, ui : R+ → Rmi is theinput of the i-th subsystem, and πi ⊂ R+ is the set of impulsive time instantsof the i-th subsystem. For each i ∈ N , it is assumed that fi : R

n+mi → Rni

is locally Lipschitz and fi(0, 0) = 0; gi : Rn+mi → Rni is continuous andgi(0, 0) = 0. Denote u = [uT1 , . . . , u

TN ]T as the input vector of the hybrid

dynamic network. Assume that each ui is piecewise continuous and bounded.Note that differential equation (3.123) represents continuous-time dynam-

ics, and difference equation (3.124) captures discrete-time dynamics. We con-sider three kinds of subsystems: the first kind is described by only continuous-time models (3.123) with πi = ∅, the second kind is purely described bydiscrete-time models (3.124), and the third kind is described by a mix of(3.123) and (3.124). Assumption 3.1 is made on the ISS properties of thesubsystems.


Assumption 3.1 Each xi-subsystem for i ∈ N has Lyapunov-based ISSproperties. Specifically, for i ∈ N , there exists a function Vi : R

ni → R+ whichis locally Lipschitz on Rni\0, positive definite and radially unbounded, andsatisfies the following:

1. for each i ∈ NC , πi = ∅, and there exist γij , γui∈ K ∪ 0 with j 6= i such

that

Vi(xi) ≥ maxj 6=i

γij(Vj(xj)), γui(|ui|)

⇒∇Vi(xi)fi(x, ui) ≤ −αi(Vi(xi)) a.e., (3.125)

where αi is a continuous and positive definite function;2. for each i ∈ ND, fi ≡ 0, and there exist γij , γui

∈ K ∪ 0 with j 6= i suchthat

Vi(gi(x, ui)) ≤ (Id− ρi)

(

maxj 6=i

γij(Vj(xj)), Vi(xi), γui(|ui|)

)

, (3.126)

where ρi is a continuous and positive definite function and satisfies (Id −ρi) ∈ K∞;

3. for each i ∈ NH , πi 6= ∅, and there exist γij , γui∈ K ∪ 0 with j 6= i such

that both properties (3.125) and (3.126) are satisfied.

It should be noted that, if the πi’s are different, then the impulsive timeinstants of different subsystems are different. From this point of view, thehybrid dynamic network (3.123)–(3.124) is more general than the discrete-time dynamic network (3.76), even if NC ∪NH = ∅.

A mild assumption is made on the intervals between the impulsive timeinstants. For each i ∈ ND, πi is of the form πi = tiw > 0 : w ∈ Z+, andthere exist constants δt, δt > 0 such that for all i ∈ ND,

δt ≤ ti(w+1) − tiw ≤ δt (3.127)

holds for all w ∈ Z+.If for each i-th subsystem with i ∈ ND, there is an upper bound and a

lower bound of the intervals between the impulsive time instants, then onecan always find common bounds for all the discrete-time subsystems.

Under the conditions above, the existence and uniqueness of solutions ofthe dynamic network with subsystems in the form of (3.123)–(3.124) canbe guaranteed in the sense of Caratheology [60]. We use x(t, t0, ξ, u) =[xT1 (t, t0, ξ, u), . . . , x

TN (t, t0, ξ, u)]

T or simply x(t) to denote the state trajec-tory of the dynamic network with initial condition ξ ∈ Rn at time t0 andinput u. For each i ∈ ND ∩NH , xi(t, t0, ξ, u) is right-continuous on the time-line. It should be noted that the semi-group property is satisfied for the hybriddynamic network because the impulsive time sets πi’s are fixed and do not


depend on the initial condition [135]. With the state trajectories defined oncontinuous time, we still use Definition 1.8 for hybrid dynamic networks.

The main result of this section is that a hybrid dynamic network com-posed of subsystems (3.123)–(3.124) is ISS if it satisfies the cyclic-small-gaincondition.

3.3.1 EQUIVALENCE BETWEEN CYCLICSMALLGAIN AND GAINS LESSTHAN THE IDENTITY

The proofs of the cyclic-small-gain theorems for continuous-time and discrete-time dynamic networks mainly deal with the simple cycles in the gain di-graphs. For hybrid dynamic networks, the analysis of the cycles involving bothcontinuous-time and discrete-time dynamics could be much more complicated.In this subsection, a result on the equivalence between cyclic-small-gain andgains less than Id is developed. Based on this observation, in the following sub-section, the cyclic-small-gain theorem for hybrid dynamic networks is provedby showing that hybrid dynamic networks with interconnection gains less thanId are ISS.

The proof of the equivalence is based on the fact that, for a continuous-timeor discrete-time system, if V is an ISS-Lyapunov function, then for any σ ∈K∞ being locally Lipschitz on (0,∞), σ(V ) is also an ISS-Lyapunov function.Consider a continuous-time system, for example. Assume that system x =f(x, u) with state x ∈ Rn and external input u ∈ Rm is ISS with V : Rn → R+

as an ISS-Lyapunov function satisfying

V (x) ≥ γu(|u|) ⇒ ∇V (x)f(x, u) ≤ −α(V (x)), a.e., (3.128)

where γ ∈ K and α is a continuous and positive definite function. Then, forany σ ∈ K∞ being locally Lipschitz on (0,∞), V := σ(V ) is also continuouslydifferentiable almost everywhere, and there exists a continuous and positivedefinite function α such that

V (x) ≥ σ γ(|u|) ⇒ ∇V (x)f(x, u) ≤ −α(V (x)), a.e. (3.129)

Such transformation is also valid for discrete-time systems.

Example 3.6 Consider the continuous-time dynamic network (3.8). DefineVi = σi(Vi) for i = 1, 2, 3 with σi ∈ K∞ being locally Lipschitz on (0,∞).Then, Vi is still an ISS-Lyapunov function of the xi-subsystem and satisfies

Vi(xi) ≥ maxj 6=i

γij(Vi(xi) ⇒ ∇Vi(xi)fi(x) ≤ −αi(Vi(xi)), ∀x, (3.130)

where αi is a continuous and positive definite function, and γij = σiγij σ−1j .

Now we consider the special case where γ12, γ13, γ21, γ32, γ31 are of classK∞ and are continuously differentiable on (0,∞). Also assume that the cyclic-


small-gain condition is satisfied, i.e.,

γ13 γ31 < Id, (3.131)

γ13 γ32 γ21 < Id. (3.132)

Then, by using Lemma C.1, there exists a continuous and positive definite δsuch that

(γ13 + δ) γ31 < Id, (3.133)

(γ13 + δ) γ32 (γ21 + δ) < Id. (3.134)

By choosing σ1 = γ21+ δ, σ2 = Id and σ3 = (γ21 + δ) (γ13 + δ), we can showthat all the interconnection gains γij’s are less than Id.

V1

V2

V3γ13

γ12γ21 γ32

γ31V1

V2

V3γ13

γ12γ21 γ32

γ31

FIGURE 3.10 The equivalence between cyclic-small-gain and gains less than Id:

γij = σi γij σ−1

j .

Under Assumption 3.1, the rest of this subsection shows that, if a hybriddynamic network satisfies the cyclic-small-gain condition, then we can finda transformation σi(Vi) for each Vi as done in Example 3.6, such that theinterconnection gains in the dynamic network are less than Id.

The first step is to replace the ISS gains γ(·)’s with γ(·)’s such that

1. γ(·) ∈ K∞ and γ(·) > γ(·) if γ(·) ∈ K; γ(·) = 0 if γ(·) = 0,2. γ(·)’s are locally Lipschitz on (0,∞), and3. for each r = 2, . . . , N ,

γi1i2 · · · γiri1 < Id (3.135)

holds for all 1 ≤ ij ≤ N and ij 6= ij′ if j 6= j′.

The existence of such γ(·)’s can be guaranteed for hybrid dynamic networks, byreasoning similar to continuous-time dynamic networks; see Subsection 3.1.2.The only difference is that, here we consider a more general case in whichγ(·)’s are chosen to be locally Lipschitz on (0,∞).


Recall that RS(i) represents the reaching set of the i-th subsystem. For aset Ξ satisfying

⋃

i∈Ξ RS(i) = N , define

Γ[q→Ξ](s) =⋃

p∈Ξ

⋃

j∈NΓ[q→p]j (s), (3.136)

where

Γ[q→p]j (s) =

γ0i

[q→p]1

γi[q→p]1 i

[q→p]2

· · · γi[q→p]j−1 i

[q→p]j

(s)

(3.137)

for s ≥ 0, with γ0i

[q→p]1

= Id, i[q→p]1 = q, i

[q→p]k ∈ N , k ∈ 1, . . . , j, i[q→p]

k 6=i[q→p]k′ if k 6= k′, for j ∈ N . Clearly, each Γ

[q→p]j corresponds to a simple path

from the xq-subsystem to the xp-subsystem, and Γq→Ξ corresponds to thesimple paths from the xq-subsystem to all the xp-subsystems with p ∈ Ξ. Aspecial case is q = p and j = 1.

Define

VΞq (xq) = Γ[q→Ξ](Vq(xq))

V Ξq (xq) = γ[q→Ξ](Vq(xq)) (3.138)

with γ[q→Ξ](s) = maxΓ[q→Ξ](s) for s ≥ 0. Clearly, V Ξq (xq) = max VΞ

q (xq).As there exists at least one subsystem in Ξ that is reachable from the xq-

subsystem and γ(·) ∈ K∞∪0, it can be observed that for all q ∈ N , γ[q→Ξ] ∈K∞. Moreover, V Ξ

q is positive definite and radially unbounded with respectto xq, and is continuously differentiable almost everywhere. Correspondingly,to simplify the discussions, we define

uΞq = γ[q→Ξ] γuq(|uq|) (3.139)

as the new input of the xq-subsystem.Proposition 3.1 presents the main result on the equivalence between cyclic-

small-gain and interconnection gains less than Id.

Proposition 3.1 A hybrid dynamic network composed of subsystems(3.123)–(3.124) which satisfy Assumption 3.1 and the cyclic-small-gain condi-tion (3.22) can be reformulated as one with interconnection gains less than Idby considering V Ξ

q defined in (3.138) as the new ISS-Lyapunov function and

uΞq defined in (3.139) as the new input for each xq-subsystem with q ∈ N .

Proposition 3.1 is proved below by studying both the continuous-time dy-namics and the discrete-time dynamics.

With Lemma C.5, there exists a continuous and positive definite functionδ satisfying (Id− δ) ∈ K∞ such that γij (Id− δ) ≥ γij for all γij 6= 0. Thus,for all i, j ∈ N , i 6= j, γij (Id− δ) ≥ γij .


Furthermore, there exists a continuous and positive definite function δ′

satisfying (Id− δ′) ∈ K∞ such that

(Id− δ′) γi[q→p]1 i

[q→p]2

· · · γi[q→p]j−1 i

[q→p]j

≥ γi[q→p]1 i

[q→p]2

· · · γi[q→p]j−1 i

[q→p]j

(Id− δ) (3.140)

for all q ∈ N , all p ∈ Ξ, and all j ∈ N , where i[q→p]k ∈ N for k ∈ 1, . . . , j,

and i[q→p]k 6= i

[q→p]k′ if k 6= k′.

We now study the continuous-time dynamics and the discrete-time dynam-ics separately.

ContinuousTime Dynamics

For any q∗ ∈ NC ∪ NH , consider the case where

V Ξq∗(xq∗) ≥ max

j 6=q∗

(Id− δ′)(V Ξj (xj)), u

Ξq∗

. (3.141)

Denote any one of the elements in VΞq∗(xq∗) taking the value of V Ξ

q∗(xq∗) as

γi∗1i∗2 · · · γi∗j∗−1

i∗j∗(Vi∗

j∗(xi∗

j∗)), (3.142)

where i∗1 = p∗ ∈ Ξ and i∗j∗ = q∗.For each i∗k ∈ i∗1, . . . , i∗j∗−1, Vi∗k satisfies

γi∗1i∗2 · · · γi∗k−1

i∗k (Id− δ)(Vi∗

k(xi∗

k))

≤ (Id− δ′) γi∗1i∗2 · · · γi∗k−1i

∗

k(Vi∗

k(xi∗

k))

≤ max VΞi∗k(xi∗

k)

≤ V Ξq∗(xq∗)

≤ γi∗1i∗2 · · · γi∗j∗−1

i∗j∗(Vi∗

j∗(xi∗

j∗)) (3.143)

and thus,

(Id− δ)(Vi∗k(xi∗

k)) ≤ γi∗

ki∗k+1

· · · γi∗j∗−1

i∗j∗(Vi∗

j∗(xi∗

j∗)), (3.144)

which implies

γi∗j∗i∗k (Id− δ)(Vi∗

k(xi∗

k)) ≤ γi∗

j∗i∗k γi∗

ki∗k+1

· · · γi∗j∗−1

i∗j∗(Vi∗

j∗(xi∗

j∗)).

(3.145)

With the cyclic-small-gain condition satisfied, one has

γi∗j∗i∗k γi∗

ki∗k+1

· · · γi∗j∗−1

i∗j∗< Id. (3.146)


Then, (3.145) implies

Vi∗j∗(xi∗

j∗) ≥ γi∗

j∗i∗k (Id− δ)(Vi∗

k(xi∗

k)) ≥ γi∗

j∗i∗k(Vi∗

k(xi∗

k)) (3.147)

for all i∗k ∈ i∗1, i∗2, . . . , i∗j∗−1.For each Vi∗

kwith i∗k ∈ S\i∗1, i∗2, . . . , i∗j, it can be observed that

γi∗1i∗2 · · · γi∗j∗−1

i∗j∗


k(xi∗

k))

≤ (Id− δ′) γi∗1i∗2 · · · γi∗j∗−1

i∗j∗

γi∗j∗i∗k(Vi∗

k(xi∗

k))

≤ max VΞi∗k(xi∗

k)

≤ V Ξq∗(xq∗)

≤ γi∗1i∗2 · · · γi∗j∗−1

i∗j∗(Vi∗

j∗(xi∗

j∗)). (3.148)

Thus, it is achieved that

Vi∗j∗(xi∗

j∗) ≥ γi∗

j∗i∗k (Id− δ)(Vi∗

k(xi∗

k)) ≥ γi∗

j∗i∗k(Vi∗

k(xi∗

k)) (3.149)

for all i∗k ∈ S\i∗1, i∗2, . . . , i∗j.Note that i∗j∗ = q∗. By combining (3.147) and (3.149) and considering

V Ξq∗ ≥ uΞq∗ , one gets

Vq∗(xq∗ ) ≥ maxl6=q∗

γq∗l(Vl(xl)), γuq∗(|uq∗ |)

, (3.150)

which implies that

∇Vq∗(xq∗)fq∗(x, uq∗) ≤ −αq∗(Vq∗(xq∗)) (3.151)

holds almost everywhere.Recall that V Ξ

q∗(xq∗) = γ[q∗→Ξ](Vq∗(xq∗)). The continuous differentiability

of the γ(·)’s implies that the K∞ function γ[q∗→Ξ] is continuously differen-

tiable almost everywhere. Then, there exist continuous and positive definitefunctions αΞ

q∗ and αΞq∗ such that

∇V Ξq∗(xq∗)fq∗(x, uq∗) ≤ −αΞ

q∗(Vq∗(xq∗))

= −αΞq∗ (γ[q∗→Ξ])−1(V Ξ

q∗(xq∗))

≤ −αΞq∗(V

Ξq∗(xq∗)) (3.152)

holds almost everywhere.


DiscreteTime Dynamics

For any q∗ ∈ ND ∪ NH , using Lemma C.5, one can find a continuous andpositive definite function ρ′q∗ satisfying (Id− ρ′q∗) ∈ K∞ such that

V Ξq∗(gq∗(x, uq∗))

≤ γ[q∗→Ξ] (Id− ρq∗)(max

j 6=q∗γq∗j(Vj(xj)), Vq∗(xq∗ ), γuq∗

(|uq∗ |))

≤ (Id− ρ′q∗)

(

maxj 6=q∗

γ[q∗→Ξ] γq∗j(Vj(xj)),

γ[q∗→Ξ](Vq∗(xq∗)), γ

[q∗→Ξ] γuq∗(|uq∗ |)

)

= (Id− ρ′q∗)(maxj 6=q∗

γ[q∗→Ξ] γq∗j(Vj(xj)), V Ξq∗(xq∗ ), u

Ξq∗). (3.153)

Denote any element in Γ[q∗→Ξ](s) as γi∗1i∗2 · · · γi∗j∗−1

i∗j∗(s) for s ∈ R+,

with i∗1 = p∗ ∈ Ξ and i∗j∗ = q∗. For any j ∈ N\q∗, consider γi∗1i∗2 · · · γi∗

j∗−1i∗j∗

γi∗j∗j(Vj(xj)).

If j ∈ i∗1, i∗1, . . . , i∗j∗−1, then denote j = i∗k with i∗k ∈ i∗1, . . . , i∗j∗−1. Inthis case, the following property holds

γi∗1 i∗2 · · · γi∗j∗−1i∗j∗

γi∗j∗j(Vj(xj))

= γi∗1 i∗2 · · · γi∗ki∗k+1 · · · γi∗

j∗−1i∗j∗

γi∗j∗i∗k(Vi∗

k(xi∗

k))

≤ γi∗1 i∗2 · · · γi∗ki∗k+1 · · · γi∗

j∗−1i∗j∗


k(xi∗

k))

≤ γi∗1 i∗2 · · · γi∗k−1i∗

k (Id− δ)(Vi∗

k(xi∗

k))

≤ (Id− δ′) γ[j→Ξ](Vj(xj))

= (Id− δ′)(V Ξj (xj)), (3.154)

where the cyclic-small-gain condition γi∗ki∗k+1

· · · γi∗j∗−1

i∗j∗

γi∗j∗i∗k< Id is

used for the second inequality.If j ∈ N\i∗1, i∗2, . . . , i∗j∗, then it can be directly derived that

γi∗1 i∗2 · · · γi∗j∗−1i∗j∗

γi∗j∗j (Vj (xj))

≤ γi∗1 i∗2 · · · γi∗j∗−1i∗j∗

γi∗j∗j (Id− δ)(Vj(xj))

≤ (Id− δ′) γ[j→Ξ](Vj(xj))

= (Id− δ′) (V Ξj (xj)). (3.155)

Combining (3.154) and (3.155), for any j ∈ S\q∗ = N\i∗j∗ and for any

γi∗1 i∗2 · · · γi∗j∗−1i∗j∗(s) in Γ[q∗→Ξ](s), one has

γi∗1i∗2 · · · γi∗j∗−1

i∗j∗

γi∗j∗j(Vj(xj)) ≤ (Id− δ′) (V Ξ

j (xj)). (3.156)


Recall that γ[q∗→Ξ](s) = maxΓ[q∗→Ξ](s) for s ∈ R+. Then, for any j ∈

N\q∗ = N\i∗j∗, one has

γ[q∗→Ξ] γi∗

j∗j(Vj(xj)) ≤ (Id− δ′) (V Ξ

j (xj)). (3.157)

Then, from (3.153), it can be achieved that

V Ξq∗(gq∗(x, uq∗)) ≤ (Id− ρ′q∗)

(

maxj 6=q∗

(Id− δ′)(V Ξj (xj)), V

Ξq∗(xq∗ ), u

Ξi

)

≤ (Id− ρ′q∗)

(

maxj 6=q∗

V Ξj (xj), V

Ξq∗(xq∗), u

Ξi

)

(3.158)

for any q∗ ∈ ND ∪ NH .Properties (3.152) and (3.158) imply that a hybrid dynamic network of

(3.123)–(3.124) satisfying the cyclic-small-gain condition (3.22) can be trans-formed into a network with interconnection gains less than Id by appropriatelyscaling the ISS-Lyapunov functions of the subsystems. Based on this observa-tion, the cyclic-small-gain theorem for hybrid dynamic networks can be provedby checking the ISS of hybrid dynamic networks with interconnection gainsless than Id.

3.3.2 CYCLICSMALLGAIN THEOREM FOR HYBRID DYNAMIC NETWORKS

In this subsection, we first consider the dynamic networks composed ofsubsystems in the form of (3.123)–(3.124) with interconnection gains γij ’s(i, j ∈ N , i 6= j) defined in (3.125) and (3.126) less than Id. Based on theequivalence result developed in Subsection 3.3.1, the Lyapunov function con-structed for such systems is further used to validate the Lyapunov functionsin the form of (3.31) for general dynamic networks.

For a continuous-time dynamic network composed of two subsystems, ifthe interconnection gains are less than Id, then one may choose σ = Id in(3.23), and construct an ISS-Lyapunov function as the maximum of the Lya-punov functions of the subsystems. Similarly, for the hybrid dynamic networkswith interconnection gains less than the identity, we construct the followingLyapunov function candidate

V (x) = maxV(x) (3.159)

with

V(x) = V1(x1), . . . , VN (xN ). (3.160)

Since for each i ∈ N , Vi(xi) is positive definite and radially unbounded withrespect to xi, it can be verified that V (x) is positive definite and radiallyunbounded with respect to x. Moreover, V is locally Lipschitz on Rn\0.


Correspondingly, we define

u = maxU (3.161)

with

U = γu1(|u1|), . . . , γuN(|uN |). (3.162)

Denote π =⋃

i∈N πi as the set of the impulsive time instants of the dynamicnetwork. The following theorem shows that the hybrid dynamic network withgains less than Id is ISS with V (x) defined in (3.159) as a weak Lyapunovfunction in the sense that V (x(t)) is, not necessarily strictly, decreasing alongthe solutions x(t).

Theorem 3.6 Consider the hybrid dynamic network composed of subsystems(3.123)–(3.124). Under Assumption 3.1, if all the interconnection gains γij ’s(i, j ∈ N , i 6= j) are less than Id, i.e., γij < Id, then V (x) defined in (3.159)is a weak ISS-Lyapunov function and admits the following properties:

1. for any ξ, u and t0 ≥ 0,

V (x(t, t0, ξ, u)) ≥ u(t) ⇒ V (x(t, t0, ξ, u)) ≤ 0 (3.163)

holds for almost all t ∈ [t0,∞)\π;2. for any ξ, u and t0 ≥ 0,

V (x(t, t0, ξ, u)) ≤ maxV (x(t−, t0, ξ, u)), u(t−) (3.164)

holds for all t ∈ (t0,∞) ∩ π;3. there exist a δtD > 0 and a positive definite function ρ∗ satisfying (Id −

ρ∗) ∈ K∞, such that for any ξ and any u,

V (x(t, t0, ξ, u)) ≤ max(Id− ρ∗)(V (ξ)), ‖u‖[t0,t] (3.165)

holds for any pair of nonnegative numbers (t, t0) satisfying t − t0 ≥ δtD,and the hybrid dynamic network is ISS.

The proof of Theorem 3.6 is given in Appendix D.4.Based on the observation that a hybrid dynamic network satisfying the

cyclic-small-gain condition can be reformulated as one with interconnectiongains less than the identity function, we develop a cyclic-small-gain theoremfor hybrid dynamic networks.

If a hybrid dynamic network composed of subsystems (3.123)–(3.124) satis-fies the cyclic-small-gain condition (3.22), based on Proposition 3.1 and The-orem 3.6, we can construct an ISS-Lyapunov function as

V Ξ(x) = maxq∈N

V Ξq (xq). (3.166)


Corresponding to V Ξ, we define

uΞ = maxq∈N

uΞq (3.167)

as the new input of the hybrid dynamic network. Then, properties (3.163)–(3.165) hold for the hybrid dynamic network with V replaced by V Ξ and ureplaced by uΞ. Our main theorem is as follows.

Theorem 3.7 A hybrid dynamic network composed of subsystems (3.123)–(3.124) satisfying the cyclic-small-gain condition (3.22) is ISS with V Ξ definedin (3.166) as a weak ISS-Lyapunov function.

3.3.3 AN EXAMPLE

Consider a hybrid dynamic network in the form of (3.123)–(3.124) with N = 3and each xi ∈ R.

The x1-subsystem involves only continuous-time dynamics with

f1(x, u1) = −|x1|x1 + 0.3x2 + 0.3x23 + u21. (3.168)

The x2-subsystem is defined on discrete time with

g2(x, u2) = 0.4x2 + 0.25x21 + 0.25x23 + u2 (3.169)

and π2 = Z+\0.The x3-subsystem is an impulsive system with

f3(x, u3) = −2|x3|x3 + 0.5x21 + 0.5x2 + 3u23, (3.170)

g3(x, u3) = 0.4x3 + 0.4x1 (3.171)

and π3 = k + 0.2(k mod 2) : k ∈ Z+\0.Define Vi(xi) = |xi| for i = 1, 2, 3. Then, each Vi is positive definite, radially

unbounded, and continuously differentiable on R\0.For the continuous-time dynamics, it can be verified that

V1(x1) ≥ max0.707V2(x2)1/2, V3(x3), 5|u1|⇒∇V1(x1)f1(x, u1) ≤ −0.06V1(x1)

2, (3.172)

V3(x3) ≥ max0.9V1(x1), 0.707V2(x2)1/2, 3|u3|⇒∇V3(x3)f3(x, u3) ≤ −0.0494V3(x3)

2, (3.173)

and for the discrete-time dynamics, it can be verified that

V2(g2(x, u2)) ≤ 0.8maxV2(x2), 1.875V1(x1)2, 1.875V3(x3)2, 7.5|u2|,(3.174)

V3(g3(x, u3)) ≤ 0.8889maxV3(x3), 0.9V1(x1). (3.175)


Define γ12(s) = 0.707s1/2, γ13(s) = s, γ21(s) = 1.875s2, γ23(s) = 1.875s2,γ31(s) = 0.9s, γ32(s) = 0.707s1/2, γu1(s) = 5s, γu2(s) = 7.5s, and γu3(s) = 3sfor s ∈ R+. Then,

γ12 γ21 < Id (3.176)

γ13 γ31 < Id (3.177)

γ23 γ32 < Id (3.178)

γ13 γ32 γ21 < Id (3.179)

γ12 γ23 γ31 < Id. (3.180)

Thus, the hybrid dynamic network composed of x1-, x2-, and x3-subsystemssatisfies the cyclic-small-gain condition, and from the cyclic-small-gain theo-rem, it is ISS with [u1, u2, u3]

T as input.We employ the technique in Subsection 3.3.2 to construct an ISS-Lyapunov

function for the hybrid dynamic network. Firstly, define γ12(s) = 0.71s1/2,γ13(s) = 1.01s, γ21(s) = 1.88s2, γ23(s) = 1.88s2, γ31(s) = 0.91s, and γ32(s) =0.71s1/2 for s ∈ R+.

Define Ξ = 2. Then, ⋃i∈Ξ RS(i) = 1, 2, 3. Define

γ[1→Ξ] = maxΓ[1→Ξ](s) = maxγ21(s), γ23 γ31(s) = 1.88s2 (3.181)

γ[2→Ξ] = maxΓ[2→Ξ](s) = s (3.182)

γ[3→Ξ] = maxΓ[3→Ξ](s) = maxγ23(s), γ21 γ13(s) = 1.92s2. (3.183)

Define

V Ξ1 (x1) = γ[1→Ξ](V1(x1)) = 1.88V1(x1)

2 (3.184)

V Ξ2 (x2) = γ[2→Ξ](V2(x2)) = V2(x2) (3.185)

V Ξ3 (x3) = γ[3→Ξ](V3(x3)) = 1.92V3(x3)

2 (3.186)

uΞ1 = γ[1→Ξ] γu1(|u1|) = 47|u1|2 (3.187)

uΞ2 = γ[2→Ξ] γu2(|u2|) = 7.5|u2| (3.188)

uΞ3 = γ[3→Ξ] γu3(|u3|) = 17.3|u3|2. (3.189)

Then, for the continuous-time dynamics, it holds that

V Ξ1 (x1) ≥ max0.94V Ξ

2 (x2), 0.98VΞ3 (x3), u

Ξ1

⇒∇V Ξ1 (x1)f1(x, u1) ≤ −0.089V Ξ

1 (x1)3/2 (3.190)

V Ξ3 (x3) ≥ max0.827V Ξ

1 (x1), 0.96VΞ2 (x2), u

Ξ3

⇒∇V Ξ3 (x3)f3(x, u3) ≤ −0.0713V Ξ

3 (x3)3/2, (3.191)

and for the discrete-time dynamics, it holds that

V Ξ2 (g2(x, u2)) ≤ 0.8maxV Ξ

2 (x2), 0.998VΞ1 (x1), 0.98V

Ξ3 (x3), u

Ξ2 (3.192)

V Ξ3 (g3(x, u3)) ≤ 0.79maxV Ξ

3 (x3), 0.83VΞ1 (x1). (3.193)


In this way, the subsystems of the hybrid dynamic network are reformulatedwith new ISS-Lyapunov functions, with which the interconnection gains areless than the identity. Based on this achievement, we can construct the ISS-Lyapunov function of the hybrid dynamic network as:

V Ξ(x) = maxV Ξ1 (x1), V

Ξ2 (x2), V

Ξ3 (x3). (3.194)

3.4 NOTES

Some recent extensions of the ISS small-gain theorem can be found in[221, 43, 231, 134, 121, 138, 137]. To the best of the authors’ knowledge,Teel [259] stated an extension of the nonlinear small-gain theorem for thefirst time, for networks of discrete-time ISS systems. Shortly, the authors of[43, 44, 231] developed a matrix-small-gain criterion for networks with plus-type interconnections. In [134, 121], a more general cyclic-small-gain theoremfor networks of IOS systems was developed. The corresponding Lyapunovformulations have been developed in [178, 179]. It should be noted that thematrix-small-gain condition is given by matrix inequalities of nonlinear func-tions, which is usually not easily checkable. As shown in this chapter, thecyclic-small-gain condition can be easily verified by directly testing specificcompositions of ISS gains of the subsystems.

The small-gain methods have also been introduced in hybrid systems,which involve both continuous-time and discrete-time dynamics; see e.g.,[167, 168, 135, 86, 211, 23, 42, 138]. In [86], the impulses are time-triggeredand a (converse) dwell-time-based strategy is developed to evaluate the ISSproperty of impulsive systems. In [23], the discrete evolution is state triggeredand both the continuous evolution and the discrete evolution are required topossess some stability property to guarantee the ISS of a hybrid system. ISSsmall-gain criteria for hybrid feedback systems and their corresponding Lya-punov formulations have also been developed by [167, 135, 211]. The interestin these results for quantized control, impulsive control, and networked controlcan be found in recent papers; see e.g., [86, 168]. Reference [84] considers non-linear systems with discontinuous right-hand sides. References [138, 42] gener-alize the small-gain results to large-scale hybrid dynamic networks, based onvector Lyapunov functions and the matrix-small-gain theorem, respectively.One recent result on global stabilization of nonlinear systems based on vector-control Lyapunov functions can be found in [139]. It should be pointed outthat, in [167, 211, 42], the impulses of the subsystems are supposed to betriggered at the same time. A cyclic-small-gain theorem for hybrid dynamicnetworks with the impulses of the subsystems triggered asynchronously is de-veloped in [188]. A time-delay version of the cyclic-small-gain theorem can befound in [265].

This chapter has presented cyclic-small-gain results for continuous-time,discrete-time, and hybrid dynamic networks with the ISS property of thesubsystems formulated by ISS-Lyapunov functions based on recent results in


[134, 121, 178, 179, 188].The continuous-time dynamic networks considered in this chapter are mod-

eled by differential equations. For discontinuous systems, i.e., systems withdiscontinuous dynamics, differential inclusions are used. In the discontinuouscase, the cyclic-small-gain condition is still valid as long as the subsystems areISS. See [84] for the extension of the original ISS small-gain theorem for dis-continuous systems. A similar approach also applies to discrete-time dynamicnetworks.

The hybrid dynamic networks considered in this chapter are composedof ISS subsystems, whose motions may be continuous, piecewise constant,or impulsive on the timeline. In particular, the impulses of the subsystemsare time-triggered and the impulsive time instants of different subsystemsare allowed to be different. For hybrid dynamic networks, this chapter hasonly studied time-triggered impulsive events. Small-gain theorems for hybridsystems with state-triggered impulses were studied in [211, 42] for the KLLstability and input-to-state stability based on hybrid inclusions [71] and hy-brid input-to-state stability in [23]. In these results, the impulses of differentsubsystems are triggered by the same state conditions. However, the impulsesof different subsystems may be triggered under different state conditions inpractical systems. Based on the achievements in this chapter, further effortmay be devoted to a small-gain result for hybrid dynamic networks with theimpulses of different subsystems triggered by different state conditions.

At a fundamental level, passivity and dissipativity concepts would appearto add extra flexibility to a reliance on just gain. However, there are other ques-tions to explore. Firstly, throughout stability theory, the input–output andLyapunov approaches have been linked by versions of the famous Kalman-Yakubovich (or Positive-Real) Lemma, from linear systems [5] to abstractsystems [92]. Further, this lemma established the Lyapunov function in anadditive form, which became the basic idea of dissipative systems, i.e., theLyapunov functions are constructed by adding up the “storage functions”of all the subsystems. This was consistent with the study of large-scale sys-tem stability [205, 207] where Lyapunov functions took a weighted additiveor vector form with discussions of their relative merits being a major pointof interest. Some step towards exploring connections with ISS and the moregeneral iISS has been taken recently in [41, 109]. Other features of the earlierstability theory were equivalences between passivity and small-gain theoremsvia transformations [4], the capability to study stability with different norms[48] (passivity goes naturally with Hilbert spaces where finite energy signalscan be usefully studied) as noted for gain properties in [248, 249], the distinc-tions between the input–output and state stability concepts [91] (of interestin adaptive control where internal chaos was consistent with robust stabi-lization [198]), and instability theorem counterparts to the stability results[48, 93]. The last area includes ways to describe unstable systems within again/dissipativity framework. Some applications of passivity methods to non-


linear control can be found in [268].This chapter has only considered the dynamic networks with stable subsys-

tem dynamics. It is well known that appropriately switching between unstabledynamics may still lead to stable behaviors. This is often formulated by the“dwell-time” condition [86]. This issue was addressed in [169] for a Lyapunov-based small-gain theorem for hybrid systems. More effort is desired for thetheoretical development of the cyclic-small-gain theorem in this research di-rection. Some extensions of passivity theory can be found in recent works[62, 63].

4 Control underSensor Noise

The robust control problem for nonlinear uncertain systems with measurementfeedback (i.e., in the presence of sensor noise) is challenging, yet important.The purpose of this chapter is to show that several nonlinear control problemscan be studied in a unified framework of measurement feedback control. Figure4.1 shows the block diagram of a measurement feedback control system.

controller

plantxu xm

w

FIGURE 4.1 The block diagram of a measurement feedback control system: u is the

control input, x is the state of the plant, and xm = x+w is the measurement of the

state with w representing the additive sensor noise.

First, by means of an elementary example, we show that stabilization inthe absence of sensor noise may not imply external stability or robustness inthe presence of sensor noise.

Example 4.1 Consider a first-order nonlinear system

x = x2 + u, (4.1)

where x ∈ R is the state and u ∈ R is the control input. If there is no sensornoise, we can design a feedback linearizing control law u = −x2 − 0.1x suchthat the closed-loop system is x = −0.1x, which is asymptotically, and evenexponentially in this case, stable at the origin. If the measurement of the stateis subject to additive sensor noise, denoted by w, then the realizable feedbackcontrol law is u = −(x+w)2− 0.1(x+w) and the resulting closed-loop systemis

x = −(0.1 + 2w)x − w2 − 0.1w, (4.2)

which clearly does not satisfy the Bounded-Input Bounded-State (BIBS) sta-bility property when w is considered as the input.

79


This chapter contributes new cyclic-small-gain design methods to cope withthe problems caused by sensor noise. In Section 4.1, we propose a new mea-surement feedback control design for nonlinear uncertain systems in the strict-feedback form. As an alternative to static state feedback, a dynamic state feed-back control strategy is developed in Section 4.2 for measurement feedbackcontrol. A further extension of the design to decentralized control of nonlinearsystems with output measurements is given in Section 4.3. The applications ofthe designs to event-triggered and self-triggered control, synchronization, androbust adaptive control are given in Sections 4.4, 4.5, and 4.6, respectively.With nontrivial modifications, the tools introduced in this chapter are alsovery useful in solving the problems in the following chapters.

4.1 STATIC STATE MEASUREMENT FEEDBACK CONTROL

In this section, we propose a small-gain design approach to robust control ofnonlinear uncertain systems with disturbed measurement. As a design ingre-dient, a modified gain assignment lemma for measurement feedback control offirst-order nonlinear systems is first proposed. Then, the measurement feed-back control problem for nonlinear uncertain systems in the strict-feedbackform is solved by recursively applying the modified gain assignment technique.

4.1.1 A MODIFIED GAIN ASSIGNMENT LEMMA

Consider the system

η = φ(η, w1, . . . , wn−2) + κ (4.3)

ηm = η + wn−1 + sgn(η)|wn|, (4.4)

where η ∈ R is the state, κ ∈ R is the control input, w1, . . . , wn ∈ R representexternal disturbance inputs, and the function φ(η, w1, . . . , wn−2) is locallyLipschitz and satisfies

|φ(η, w1, . . . , wn−2)| ≤ ψφ(|[η, w1, . . . , wn−2]T |), ∀η, w1, . . . , wn−2 (4.5)

with known ψφ ∈ K∞. The ηm defined by (4.4) is considered as the mea-surement of η. The case of wn = 0 was considered in the past literature; seee.g., [130, 223, 123, 125]. The reason why we introduce this additional termsgn(η)|wn| in (4.4) is that we need to develop a tool as stated in Lemma 4.1for the development of robust small-based measurement feedback controllersfor higher-dimensional nonlinear systems in Subsection 4.1.3. As will be clearlater, the ISS-gain from wn to η can always be made zero. Additionally, thegain from wn−1 to η has a specific form.

Lemma 4.1 shows that one can design a measurement feedback control lawso that the closed-loop system is ISS with the disturbances as external inputs.Define αV (s) = s2/2 for s ∈ R+.

Control under Sensor Noise 81

Lemma 4.1 Consider system (4.3)–(4.4). For any specified 0 < c < 1, ǫ > 0,ℓ > 0, and χw1

η , . . . , χwn−2η ∈ K∞, one can find a continuously differentiable,

odd, strictly decreasing, and radially unbounded function κ : R → R such thatwith control law

κ = κ(ηm), (4.6)

Vη(η) = αV (|η|) is an ISS-Lyapunov function of the closed-loop system andsatisfies

Vη(η) ≥ maxk=1,...,n−2

χwkη (|wk|), αV

( |wn−1|c

)

, ǫ

⇒ ∇Vη(η)(φ(η, w1, . . . , wn−2) + κ(ηm)) ≤ −ℓVη(η), ∀η, w1, . . . , wn. (4.7)

If moreover, ψφ is Lipschitz on compact sets and each χwkη for k = 1, . . . , n−

2 is chosen such that(

χwkη

)−1 αV is Lipschitz on compact sets, then anappropriate κ can be found such that (4.7) holds with ǫ = 0.

Proof. With (4.5) satisfied, one can find ψηφ, ψw1

φ , . . . , ψwn−2

φ ∈ K∞ such that

|φ(η, w1, . . . , wn−2)| ≤ ψηφ(|η|) +n−2∑

k=1

ψwk

φ (|wk|). (4.8)

Since ψηφ(s) +∑n−2k=1 ψ

wk

φ (

χwkη

)−1 αV (s) + ℓs/2 is a class K∞ functionof s, from Lemma C.8 in the Appendix, for any constants 0 < c < 1 andǫ > 0, one can find a ν : R+ → R+ which is positive, nondecreasing, andcontinuously differentiable on (0,∞) such that

(1− c)ν((1 − c)s)s ≥ ψηφ(s) +

n−2∑

k=1

ψwk

φ (

χwkη

)−1 αV (s) +ℓ

2s (4.9)

for all s ≥√2ǫ.

With the ν satisfying (4.9), define

κ(r) = −ν(|r|)r (4.10)

for r ∈ R. Then, κ is continuously differentiable, odd, strictly decreasing, andradially unbounded.

With Vη(η) = αV (|η|) = |η|2/2, we consider the case of

Vη(η) ≥ maxk=1,...,n−2

χwkη (|wk|), αV

( |wn−1|c

)

, ǫ

. (4.11)

In this case, we have

|wk| ≤(

χwkη

)−1 αV (|η|), k = 1, . . . , n− 2, (4.12)

|wn−1| ≤ cα−1V (Vη(η)) = c|η|, (4.13)

|η| ≥√2ǫ. (4.14)


Recall the definition of ηm in (4.4). With 0 < c < 1 and property (4.13),when η 6= 0, we have

sgn(ηm) = sgn(η), (4.15)

|ηm| ≥ (1− c)|η|. (4.16)

In the case of (4.11), using (4.8)–(4.10) and (4.12)–(4.16), we have

∇Vη(η)(φ(η, w1 , . . . , wn−2) + κ(ηm))

= η(φ(η, w1, . . . , wn−2)− ν(|ηm|)ηm)

≤ |η||φ(η, w1, . . . , wn−2)| − |η|ν(|ηm|)|ηm|

≤ |η|(

ψηφ(|η|) +n−2∑

k=1

ψwk

φ (|wk|)− (1− c)ν((1 − c)|η|)|η|)

≤ |η|(

ψηφ(|η|) +n−2∑

k=1

ψwk

φ (

χwkη

)−1 αV (|η|) − (1− c)ν((1 − c)|η|)|η|)

≤ − ℓ

2|η|2 = −ℓVη(η). (4.17)

According to Lemma 4.1, if ψφ is Lipschitz on compact sets, thenψηφ, ψ

w1

φ , . . . , ψwn−2

φ can be chosen to be Lipschitz on compact sets. With(

χwkη

)−1 αV being Lipschitz on compact sets, one can guarantee that theright-hand side of (4.9) is Lipschitz on compact sets. In this case, by usingLemma C.8 in the Appendix, one can find an appropriate ν, and thus κ, forǫ = 0. This ends the proof. ♦

If wn−1 is bounded, then by using a set-valued map to cover the influenceof wn−1, the closed-loop system composed of (4.3), (4.4), and (4.6) can berepresented with a differential inclusion:

η ∈ φ(η, w1, . . . , wn−2) + κ(η + awn−1 + sgn(η)|wn|) : |a| ≤ 1:= F (η, w1, . . . , wn−2, wn−1, wn), (4.18)

where wn−1 is an upper bound of |wn−1|. Clearly, 0 ∈ F (0, . . . , 0). With thedifferential inclusion formulation, property (4.7) can be equivalently repre-sented by

Vη(η) ≥ maxk=1,...,n−2

χwkη (|wk|), αV

( |wn−1|c

)

, ǫ

⇒ maxf∈F (η,w1,...,wn−2,wn−1,wn)

∇Vη(η)f ≤ −ℓVη(η). (4.19)

Compared with the gain assignment result in Subsection 2.3.1, Lemma 4.1takes into account sensor noise caused by wn−1 and wn. Here, wn is taken intoaccount for the later recursive control design. The gain assignment techniquesgiven in [223, 125] do not take into account the influence of wn 6= 0.


It should be noted that with the proposed design, the ISS gain from wn toη is zero. However, property (4.7) implies that the ISS gain from wn−1 to ηis 1/c, where the constant c should be chosen to satisfy 0 < c < 1. Thus, thegain from wn−1 to η is larger than one. This means that the influence of thesensor noise cannot be attenuated to an arbitrarily small level. In fact, thisis also the case if the system (4.3) is reduced to a linear system; see Example4.2.

Example 4.2 Consider a first-order linear time-invariant system in the formof (4.3)–(4.4) with w1, . . . , wn−2, wn = 0, and

φ(η, w1, . . . , wn−2) = aη, (4.20)

where a is an unknown constant satisfying 0 < a ≤ a with known constant a >0. With a linear measurement feedback control law u = −kηm with constantk > a, the closed-loop system is

η = (a− k)η − kwn−1, (4.21)

which can be equivalently represented by transfer function

(Lη)(s)(Lwn−1)(s)

=−k

s+ (k − a):= G(s), (4.22)

where L represents the Laplace transform, and s ∈ C. For the linear sys-tem, the gain from wn−1 to η can be calculated in the frequency domain assupωf≥0 |G(jωf )| = k/(k−a), which is larger than one and can be designed tobe arbitrarily close to one by choosing k large enough. This is in accordancewith the modified gain assignment lemma.

With the gain assignment technique in Lemma 4.1, measurement feedbackcontrol can be solved for system (4.3) appended with ISS dynamic uncertain-ties [125]. Example 4.3 considers a simplified case to show the basic idea ofthis ISS small-gain design.

Example 4.3 Consider the nonlinear system

z = q(z, x) (4.23)

x = f(x, z) + u (4.24)

xm = x+ w, (4.25)

where [z, x] ∈ R2 is the state, u ∈ R is the control input, x is considered asthe output, xm is the measurement of x with w ∈ R representing sensor noise,and q, f : R

2 → R are locally Lipschitz functions. Only the measurementxm is available for feedback control design. For this system, the z-subsystemrepresents dynamic uncertainties.


Assume that the z-subsystem is ISS with x as the input and admits anISS-Lyapunov function Vz : R → R+ such that

αz(|z|) ≤ Vz(z) ≤ αz(|z|), ∀z, (4.26)

Vz(z) ≥ χxz (|x|) ⇒ ∇Vz(z)q(z, x) ≤ −αz(Vz(z)), ∀z, x, (4.27)

where αz, αz ∈ K∞, χxz ∈ K, and αz is a continuous and positive definitefunction. Also assume that there exists a ψf ∈ K∞ such that for all x, z,

|f(x, z)| ≤ ψf (|[x, z]T |). (4.28)

Based on Lemma 4.1, we can design a control law u = u(xm) such thatthe x-subsystem is ISS with Vx(x) = αV (|x|) = x2/2 as an ISS-Lyapunovfunction. In particular, for any specific χzx ∈ K∞, continuous and positivedefinite function αx, and constant ǫ > 0, the control law can be designed suchthat Vx(x) = αV (|x|) = x2/2 satisfies

Vx(x) ≥ max

χzx(|z|), αV( |w|c

)

, ǫ

⇒∇Vx(x)(f(x, z) + u(xm)) ≤ −αx(Vx(x)) ∀x, z, w, (4.29)

and thus

Vx(x) ≥ max

χzx α−1z (Vz(z)), αV

( |w|c

)

, ǫ

⇒∇Vx(x)(f(x, z) + u(xm)) ≤ −αx(Vx(x)) ∀x, z, w. (4.30)

.Also note that property (4.27) implies

Vz(z) ≥ χxz α−1V (Vx(x)) ⇒ ∇Vz(z)q(z, x) ≤ −αz(Vz(z)). (4.31)

With the design above, the closed-loop system is transformed into an in-terconnection of two ISS subsystems. The closed-loop system is ISS if χzx ischosen to satisfy the small-gain condition:

χzx α−1z χxz α−1

V < Id. (4.32)

By using the Lyapunov-based ISS small-gain theorem, we can also constructan ISS-Lyapunov function to analyze the influence of the sensor noise on theconvergence of x.

If ψf is Lipschitz on compact sets, and there exists a χzx ∈ K∞ such that(χzx)

−1 αV is Lipschitz on compact sets and the small-gain condition (4.32)is satisfied, then the ǫ in (4.30) can be chosen to be zero.


4.1.2 PROBLEMS WITH HIGHORDER NONLINEAR SYSTEMS

By a repeated application of the gain assignment Lemma 4.1, this subsectiondevelops a class of measurement feedback controllers for nonlinear uncertainsystems in the strict-feedback form:

xi = xi+1 +∆i(xi, d), i = 1, . . . , n− 1 (4.33)

xn = u+∆n(xn, d) (4.34)

xmi = xi + wi, i = 1, . . . , n, (4.35)

where [x1, . . . , xn]T := x ∈ R

n is the state, xi = [x1, . . . , xi]T , u ∈ R is

the control input, d ∈ Rnd represents external disturbance inputs, xmi is themeasurement of xi with wi being the corresponding sensor noise, and ∆i’s(i = 1, . . . , n) are uncertain, locally Lipschitz functions.

The following assumptions are made on system (4.33)–(4.35).

Assumption 4.1 For each i = 1, . . . , n, there exists a known ψ∆i∈ K∞ such

that for all xi, d,

|∆i(xi, d)| ≤ ψ∆i(|[xTi , dT ]T |). (4.36)

Assumption 4.2 There exists a constant d ≥ 0 such that

|d(t)| ≤ d (4.37)

for t ≥ 0.

Assumption 4.3 For each i = 1, . . . , n, there exists a constant wi > 0 suchthat

|wi(t)| ≤ wi (4.38)

for t ≥ 0.

A small-gain design has been developed for the stabilization of strict-feedback systems in the form of (4.33)–(4.34) and more general cascade sys-tems with dynamic uncertainties [123]; see also the discussions for a simplifiedcase in Subsection 2.3.2. If system (4.33)–(4.34) is free of sensor noise, i.e.,wi = 0, then it can be stabilized with a nonlinear controller in the form of

x∗1 = κ1(x1) (4.39)

x∗i+1 = κi(xi − x∗i ), i = 2, . . . , n− 1 (4.40)

u = κn(xn − x∗n), (4.41)


where the κi’s for i = 1, . . . , n are appropriately designed nonlinear functions,and u is the implementable control law. In this case, to analyze the stabilityproperty of the closed-loop system, we can define new state variables as

e1 = x1, (4.42)

ei = xi − κi−1(ei−1), i = 2, . . . , n. (4.43)

For continuous differentiability of the new state variables, the functions κi fori = 1, . . . , n− 1 are required to be continuously differentiable.

To take into account the influence of the sensor noise, we may considerreplacing each xi in (4.39)–(4.41) with xmi . We choose the new measurementfeedback control law in the following form:

x∗1 = κ1(xm1 ) (4.44)

x∗i+1 = κi(xmi − x∗i ), i = 2, . . . , n− 1 (4.45)

u = κn(xmn − x∗n), (4.46)

where the κi’s are not necessarily the same with the κi’s in (4.39)–(4.41). Forsuch a control law, the state transformation in the form of (4.42)–(4.43) maybe modified as

e1 = xm1 , (4.47)

ei = xmi − κi−1(ei−1), i = 2, . . . , n. (4.48)

However, with such treatment, if the sensor noise is not differentiable, then thenew ei’s for i = 2, . . . , n are not differentiable and one cannot use differentialequations to represent the dynamics of the ei-subsystems. Practically, it mightbe too restrictive to assume the continuous differentiability of sensor noise.

The main objective of this section is to develop a new small-gain designmethod which leads to nonlinear controllers that are:

1. robust to nondifferentiable and even discontinuous sensor noise, and more-over,

2. capable of attenuating the influence of the sensor noise on the control sys-tem to the largest extent possible.

4.1.3 RECURSIVE CONTROL DESIGN

This subsection employs set-valued maps to handle the problem caused bysensor noise. With the new design, the closed-loop system is transformed intoan interconnection of ISS subsystems represented by differential inclusions.Specifically, the [x1, . . . , xn]

T -system is transformed into a new [e1, . . . , en]T -

system through the following transformation:

e1 = x1 (4.49)

ei = ~d(xi, Si−1(xi−1)), i = 2, . . . , n, (4.50)


where Si : Ri R is an appropriately chosen set-valued map, and

~d(z,Ω) := z − argminz′∈Ω

|z − z′| (4.51)

for any z ∈ R and any compact Ω ⊂ R. Basically, the set-valued maps areemployed to cover the influence of the sensor noise and to represent the possi-ble control laws in the control design procedure. A control law in the form of(4.44)–(4.46) is at last found as a selection of the set-valued map Sn : Rn R.

For convenience of notation, denote ei = [e1, . . . , ei]T and Wi =

[w1, . . . , wi]T for i = 1, . . . , n. Also denote xn+1 = u.


By taking the derivative of e1, we have

e1 = x2 +∆1(x1, d)

= x2 − e2 +∆1(x1, d) + e2. (4.52)

Recall that e2 = ~d(x2, S1(x1)). Then,

x2 − e2 ∈ S1(x1). (4.53)

Define the set-valued map S1 as

S1(x1) = κ1(x1 + a1w1) : |a1| ≤ 1, (4.54)

where κ1 : R → R is a continuously differentiable, odd, strictly decreasing,and radially unbounded function.

Since κ1 is strictly decreasing, maxS1(x1) = κ1(x1− w1) and minS1(x1) =κ1(x1 + w1). Set-valued map S1 and the definition of e2 are shown in Figure4.2.

Intuitively, set-valued map S1 is defined such that if x2 is the con-trol input of the e1-subsystem, then the measurement feedback control lawx2 = κ1(x1 + w1) = κ1(e1 + w1) with |w1| ≤ w1 is a selection of S1(x1).Moreover, by choosing a continuously differentiable κ1, the boundaries of S1

are continuously differentiable, and as shown below, the derivative of e2 existsalmost everywhere. As a result, the problem caused by the nondifferentiablesensor noise is solved.

Recursive Step: The eisubsystems for i = 2, . . . , n

For convenience, denote S0(x0) = 0. For each k = 1, . . . , i − 1, define set-valued map Sk as

Sk(xk) = κk(xk − pk−1 + akwk) : pk−1 ∈ Sk−1(xk−1), |ak| ≤ 1 , (4.55)

where κk : R → R is a continuously differentiable, odd, strictly decreasing,and radially unbounded function.


0 x1

x2

(x1, x2)

e2

w1 w1

x2 = κ1(x1)

FIGURE 4.2 Boundaries of set-valued map S1 and the definition of e2.

Lemma 4.2 Consider the [x1, . . . , xn]T -system defined by (4.33)–(4.35) with

Assumptions 4.1–4.3 satisfied. If each Sk is defined in (4.55) for k = 1, . . . , i−1, then when ei 6= 0, the ei-subsystem can be written in the form of

ei = xi+1 + φ∗i (xi, d), (4.56)

where

|φ∗i (xi, d)| ≤ ψφ∗

i(|[eTi , dT ,WT

i−1]T |) (4.57)

with known ψφ∗

i∈ K∞. If, moreover, the ψ∆k

’s for k = 1, . . . , i are Lipschitzon compact sets, then one can find a ψφ∗

i∈ K∞ which is Lipschitz on compact

sets.

The proof of Lemma 4.2 is provided in Appendix E.1.Define set-valued map Si as

Si(xi) =

κi(xi − pi−1 + aiwi) : pi−1 ∈ Si−1(xi−1), |ai| ≤ 1

, (4.58)

where κi is a continuously differentiable, odd, strictly decreasing, and radiallyunbounded function.

Then, the ei-subsystem can be rewritten as

ei = xi+1 − ei+1 + φ∗i (xi, d) + ei+1, (4.59)


where

xi+1 − ei+1 ∈ Si(xi) (4.60)

according to the definition ei+1 = ~d(xi+1, Si(xi)).It can be observed that the S1(x1) defined in (4.54) is also in the form of

(4.58) with S0(x0) = 0, and the e1-subsystem defined in (4.52) is in theform of (4.59).

State Measurement Feedback Control Law and the ClosedLoop System

We design the measurement feedback control law as

p∗1 = κ1(xm1 ), (4.61)

p∗i = κi(xmi − p∗i−1), i = 2, . . . , n− 1 (4.62)

u = κn(xmn − p∗n−1). (4.63)

Recall that xmi = xi + wi for i = 1, . . . , n. It is directly checked that

p∗1 ∈ S1(x1) ⇒ · · · ⇒ p∗i ∈ Si(xi) ⇒ · · · ⇒ u ∈ Sn(xn), (4.64)

which means en+1 = 0.Considering xi+1 − ei+1 ∈ Si(xi) for i = 1, . . . , n and en+1 = 0, when

ei 6= 0, we can represent each ei-subsystem for i = 1, . . . , n with a differentialinclusion:

ei ∈ pi + φ∗i (xi, d) + ei+1 : pi ∈ Si(xi):= Fi(xi, ei+1, d). (4.65)

Thus, the closed-loop system with control law (4.61)–(4.63) is transformedinto a network composed of the ei-subsystems, each of which is representedby a differential inclusion.

Also, when ei 6= 0, for each pi−1 ∈ Si−1(xi−1), it holds that |xi−pi−1| > |ei|and sgn(xi−pi−1) = sgn(ei), which imply sgn(xi−ei−pi−1) = sgn(ei). Thus,for i = 1, . . . , n, each Si(xi) in the form of (4.58) can be rewritten as

Si(xi) =

κi(ei + sgn(ei)|wi0|+ aiwi) : |ai| ≤ 1

, (4.66)

where wi0 = xi − ei − pi−1 with pi−1 ∈ Si−1(xi−1).By combining (4.65) and (4.66), we can recognize that each ei-subsystem

is in the form of (4.18). In the next subsection, we employ the gain assign-ment technique introduced in Subsection 4.1.1 to choose the κi’s such that allthe ei-subsystems are ISS with the ISS gains satisfying the cyclic-small-gaincondition.


Clearly, if there is no sensor noise, then the problem is reduced to the oneconsidered in Section 2.3. In this case, the state transformation (4.49)–(4.50)is reduced to

e1 = x1 (4.67)

ei = xi − κi−1(ei−1), i = 2, . . . , n, (4.68)

and the control law (4.61)–(4.63) is reduced to

p∗1 = κ1(x1) (4.69)

p∗i = κi(xi − p∗i−1), i = 2, . . . , n− 1 (4.70)

u = κn(xn − p∗n−1). (4.71)

4.1.4 CYCLICSMALLGAIN SYNTHESIS

Define

Vi(ei) = αV (|ei|) :=1

2|ei|2 (4.72)

as the ISS-Lyapunov function candidate for each ei-subsystem. For conve-nience of discussions, denote Vn+1(en+1) = αV (|en+1|).

Consider each ei-subsystem (4.65) with Si in the form of (4.66). By usingLemma 4.1, for any specified ǫi > 0, ℓi > 0, 0 < ci < 1, γekei , γ

wkei ∈ K∞ (k =

1, . . . , i − 1), and γei+1ei , γdei ∈ K∞, we can find a continuously differentiable,

odd, strictly decreasing, and radially unbounded function κi for Si such that

Vi(ei) ≥ maxk=1,...,i−1

γekei αV (|ek|), γei+1ei αV (|ei+1|),

γwkei (wk), γ

wiei (wi), γ

dei(d), ǫi

⇒ maxfi∈Fi(xi,ei+1,d)

∇Vi(ei)fi ≤ −ℓiVi(ei) (4.73)

and thus,

Vi(ei) ≥ maxk=1,...,i−1

γekei (Vk(ek)), γei+1ei (Vi+1(ei+1)),

γwkei (wk), γ

wiei (wi), γ

dei(d), ǫi

⇒ maxfi∈Fi(xi,ei+1,d)

∇Vi(ei)fi ≤ −ℓiVi(ei), (4.74)

where

γwiei (s) = αV

(

s

ci

)

(4.75)

for s ∈ R+.Recall that ei = [e1, . . . , ei]

T . Denote e = en. The gain digraph of thee-system is shown in Figure 4.3.


V1 Vi Vi+1 Vn· · · · · ·

FIGURE 4.3 The gain digraph of the e-system.

According to the recursive design, given the ei−1-subsystem, by appropri-ately choosing set-valued map Si for the ei-subsystem, we can design the ISSgains γekei for 1 ≤ k ≤ i− 1 such that

γe2e1 γe3e2 γe4e3 · · · γeiei−1 γe1ei < Id

γe3e2 γe4e3 · · · γeiei−1 γe2ei < Id

...γeiei−1

γei−1ei < Id

. (4.76)

By applying this reasoning repeatedly, we can guarantee (4.76) for all 2 ≤ i ≤n. In this way, the e-system satisfies the cyclic-small-gain condition.

With the Lyapunov-based ISS cyclic-small-gain theorem in Chapter 3, anISS-Lyapunov function can be constructed for the e-system to evaluate theinfluence of the sensor noise:

V (e) = maxi=1,...,n

σi(Vi(ei)) (4.77)

where σ1(s) = s, σi(s) = γe2e1 · · · γeiei−1(s) for i = 2, . . . , n for s ∈ R+. The

γ(·)(·) ’s are class K∞ functions that are continuously differentiable on (0,∞),

slightly larger than the corresponding γ(·)(·) ’s, and still satisfy the cyclic-small-

gain condition. (Recall the construction in Subsection 3.1.2.)The influence of wi’s, ǫi’s, and d can be represented by

θ = maxi=1,...,n

σi

(

maxk=1,...,i

γwkei (wk), γ

dei(d), ǫi

)

. (4.78)

Then, it holds that

V (e) ≥ θ ⇒ maxf∈F (x,e,d)

∇V (e)f ≤ −α(V (e)) a.e., (4.79)

where F (x, e, d) = [F1(x1, e2, d), . . . , Fn(xn, en+1, d)]T and α is a continuous

and positive definite function.


By choosing γwkei (i = 2, . . . , n, k = 1, . . . , i − 1), γdei (i = 1, . . . , n), ǫi

(i = 1, . . . , n), and γei+1ei (i = 1, . . . , n− 1) small enough, we can make σi for

i = 2, . . . , n small enough and get

θ = γw1e1 (w1) = αV

(

w1

c1

)

. (4.80)

Then, from (4.79), it is achieved that

V (e) ≥ αV

(

w1

c1

)

⇒ maxf∈F (x,e,d)

∇V (e)f ≤ −α(V (e)) (4.81)

holds wherever ∇V exists.Property (4.81) implies that V (e) ultimately converges to within the region

V (e) ≤ αV (w/c1). Using the definitions of e1, V1(e1), V (e) (see (4.72) and(4.77)), we have |x1| = |e1| = α−1

V (V1(e1)) ≤ α−1V (V (e)), which implies that x1

ultimately converges to within the region |x1| ≤ w1/c1. Note that constant c1can be arbitrarily chosen as long as 0 < c1 < 1. By choosing c1 to be arbitrarilyclose to one, x1 can be steered arbitrarily close to the region |x1| ≤ w1.

According to the design above, because of the nonzero ǫi terms, asymptoticstability of the closed-loop system cannot be guaranteed even if wi ≡ 0 fori = 1, . . . , n and d ≡ 0. This problem can be solved if the ψ∆i

’s for i =1, . . . , n in Assumption 4.1 are Lipschitz on compact sets. In this case, foreach ei-subsystem, we can find a ψφ∗

ithat is Lipschitz on compact sets such

that (4.57) holds. Then, according to Lemma 4.1, by choosing the γekei for

k = 1, . . . , i − 1 and γdei such that(

γekei αV)−1 αV and

(

γdei)−1 αV are

Lipschitz on compact sets, (4.73) and thus (4.74) can be realized with ǫi = 0.Asymptotic stabilization is achieved if wi ≡ 0 for i = 1, . . . , n and d ≡ 0.

The main result on state measurement feedback control of nonlinear un-certain systems is given in Theorem 4.1.

Theorem 4.1 Consider system (4.33)–(4.35). Under Assumptions 4.1, 4.2and 4.3, the closed-loop signals are bounded and, in particular, state x1 can besteered arbitrarily close to the region |x1| ≤ w1 with the measurement feedbackcontrol law (4.61)–(4.63). If the system is disturbance-free, i.e., wi ≡ 0 fori = 1, . . . , n and d ≡ 0, and the ψ∆i

’s for i = 1, . . . , n in Assumption 4.1 areLipschitz on compact sets, then one can design the control law in the form of(4.61)–(4.63) such that x1 asymptotically converges to the origin.

It should be noted that the design proposed in this section can also beapplied to strict-feedback systems with ISS dynamic uncertainties:

z = g(z, x1, d) (4.82)

xi = xi+1 +∆i(xi, z, d), i = 1, . . . , n− 1 (4.83)

xn = u+∆n(xn, z, d) (4.84)

xmi = xi + wi, i = 1, . . . , n, (4.85)


where the z-subsystem with z ∈ Rm represents the dynamic uncertainties.If the z-subsystem is ISS with x1, d as the inputs, then following the designprocedure in this section, the closed-loop system can still be transformed intoa network of ISS subsystems, and the control problem can then be solved byusing the cyclic-small-gain theorem.

4.2 DYNAMIC STATE MEASUREMENT FEEDBACK CONTROL

In practical industrial applications, low-pass filters are often employed to at-tenuate high-frequency noise and to estimate the measured signals. Motivatedby low-pass filters, in this section, we develop a dynamic state measurementfeedback control structure for input-to-state stabilization of nonlinear systemsunder sensor noise.

We still consider nonlinear systems in the strict-feedback form:

xi = xi+1 +∆i(xi), i = 1, . . . , n− 1 (4.86)

xn = u+∆n(xn) (4.87)

xmi = xi + wi, i = 1, . . . , n, (4.88)

where [x1, . . . , xn]T := x ∈ Rn is the state, u ∈ R is the control input, xi =

[x1, . . . , xi]T , xmi ∈ R is the disturbed measurement of xi with sensor noise

wi ∈ R, and ∆i’s for i = 1, . . . , n are unknown locally Lipschitz functions.

Assumption 4.4 For each ∆i with i = 1, . . . , n in (4.86)–(4.87), there existsa known ψ∆i

∈ K∞ such that for all xi,

|∆i(xi)| ≤ ψ∆i(|xi|). (4.89)

The objective is to design a dynamic state measurement feedback controllerof the form

ζ = ϕ(ζ, xm) (4.90)

u = λ(ζ) (4.91)

such that system (4.86)–(4.88) is made ISS with the wi’s as the inputs, andthus is IOS with the wi’s as the inputs and x1 as the output. Moreover, itis desired that the IOS gain from w1 to x1 can be designed to be arbitrarilyclose to the identity function, and the IOS gains from w2, . . . , wn to x1 canbe designed to be arbitrarily small.

The gain assignment technique is still an ingredient for the design in thissection. The basic idea is to transform the closed-loop system into an inter-connection of first-order nonlinear systems in the following form:

η = φ(η, w1, . . . , wm) + κ (4.92)

ηm = η + wm+1, (4.93)


where η ∈ R is the state, κ ∈ R is the control input, w1, . . . , wm+1 ∈ R

represent external inputs, ηm ∈ R is the measurement of η, the nonlinearfunction φ(η, w1, . . . , wm) is locally Lipschitz and satisfies

|φ(η, w1, . . . , wm)| ≤ ψηφ(|η|) +m∑

k=1

ψwk

φ (|wk|) (4.94)

with known ψηφ, ψw1

φ , . . . , ψwm

φ ∈ K∞.

Now, a new system with state [e1, e1, . . . , en, en]T is constructed based on

the x-system. Moreover, all the ei-subsystems and the ei-subsystems are de-signed to be ISS, and the cyclic-small-gain theorem is used to check the sta-bility of the closed-loop system.

4.2.1 DYNAMIC STATE MEASUREMENT FEEDBACK CONTROL DESIGN

In this subsection, we design the dynamic state measurement feedback con-troller through a recursive approach.

If there is no sensor noise, then we may design a controller to transform theclosed-loop system into ISS ei-subsystems with ei defined by (4.67)–(4.68). Inthe presence of sensor noise, we use an estimate ei−1 to replace the ei−1 onthe right-hand side of (4.68), and the new state transformation is

e1 = x1 (4.95)

ei = xi − κ(i−1)2(ei−1), i = 2, . . . , n, (4.96)

where κ(i−1)2 : R → R is a continuously differentiable, odd, strictly decreasingand radially unbounded function.

Denote xn+1 = xmn+1 = u. For i = 1, . . . , n, each ei is generated by thefollowing estimator:

˙ei = κi1(ei − emi ) + xmi+1, (4.97)

where κi1 : R → R is an odd and strictly decreasing function, and

em1 = xm1 (4.98)

emi = xmi − κ(i−1)2(ei−1), i = 2, . . . , n. (4.99)

The structure of the ei-subsystem is shown in Figure 4.4.For i = 1, . . . , n, define

ei = ei − ei (4.100)

as the estimation error for ei. By taking the derivative of ei and using xmi =


xi

wi

xmi

κ(i−1)2

ei−1

κi1

xmi+1

∫ ei++

+− −+

++

FIGURE 4.4 The estimator for ei: the ei-subsystem.

xi + wi and xmi+1 = xi+1 + wi+1, we have

˙ei = ˙ei − ei

= ˙ei − xi +∂κ(i−1)2(ei−1)

∂ei−1

˙ei−1

= κi1(ei − wi) + wi+1 + xi+1 −∆i(xi)− xi+1

+∂κ(i−1)2(ei−1)

∂ei−1

˙ei−1, (4.101)

which can be represented in the form of

˙ei = ∆∗i1(e1, e1, . . . , ei, ei, wi−1, wi, wi+1) + κi1(ei − wi)

:= fei(e1, e1, . . . , ei, ei, wi−1, wi, wi+1) (4.102)

with

∆∗i1(e1, e1, . . . , ei, ei, wi−1, wi, wi+1)

= wi+1 −∆i(xi) +∂κ(i−1)2(ei−1)

∂ei−1

˙ei−1. (4.103)

With Assumption 4.4 satisfied, we can find ψek∆∗

i1, ψek∆∗

i1∈ K∞ for k = 1, . . . , i

and ψwi−1

∆∗

i1, ψwi

∆∗

i1, ψ

wi+1

∆∗

i1∈ K∞ such that

|∆∗i1(e1, e1, . . . , ei, ei, wi−1, wi, wi+1)|

≤i∑

k=1

(

ψek∆∗

i1(|ek|) + ψek∆∗

i1(|ek|)

)

+ ψwi−1

∆∗

i1(|wi−1|) + ψwi

∆∗

i1(|wi|) + ψ

wi+1

∆∗

i1(|wi+1|). (4.104)

By using xmi = xi + wi and xmi+1 = xi+1 + wi+1, we can rewrite the ei-subsystem as

˙ei = κi1(ei − wi) + wi+1 + xi+1. (4.105)


With ei+1 = xi+1 − κi2(ei) according to (4.96), we have xi+1 = ei+1 +κi2(ei), and thus, the ei-subsystem (4.105) can be rewritten as

˙ei = ∆∗i2(ei, ei+1, ei+1, wi, wi+1) + κi2(ei)

:= fei(ei, ei+1, ei, ei+1, wi, wi+1), (4.106)

where

∆∗i2(ei, ei+1, ei+1, wi, wi+1) = κi1(ei − wi) + wi+1 + ei+1 − ei+1. (4.107)

Since κi1 is odd and strictly decreasing, we can find ψei∆i2, ψ

ei+1

∆i2, ψ

ei+1

∆i2,

ψwi

∆i2, ψ

wi+1

∆i2∈ K∞ such that

|∆∗i2(ei, ei+1, ei+1, wi, wi+1)|

= ψei∆i2(|ei|) + ψ

ei+1

∆i2(|ei+1|) + ψ

ei+1

∆i2(|ei+1|) + ψwi

∆i2(|wi|)

+ ψwi+1

∆i2(|wi+1|). (4.108)

Denote ei = [ei, ei]T for i = 1, . . . , n. The interconnection in the ei-

subsystem is shown in Figure 4.5.

ei ei

ek

ek

ei+1

ei+1

k = 1, . . . , i− 1

FIGURE 4.5 The interconnection with each ei-system (i = 1, . . . , n).

Define αV (s) = s2/2 for s ∈ R+. For each ei-subsystem and each ei-subsystem, we define the following ISS-Lyapunov function candidates, respec-tively:

Vei (ei) = αV (|ei|), i = 1, . . . , n, (4.109)

Vei (ei) = αV (|ei|), i = 1, . . . , n. (4.110)

For convenience of notation, denote e0 = e0 = 0 and w0 = 0.Denote ¯ei = [eT1 , . . . , e

Ti ]T for i = 1, . . . , n. Given the ¯ei-subsystem, we

choose γ ekei , γekei

∈ K∞ with k = 1, . . . , i such that the compositions of thegain functions along all the simple loops through the ei-subsystem in the[¯eTi−1, ei]

T -subsystem are less than the identity function.Consider the ei-subsystem defined in (4.102). Using Lemma 4.1, for any

specific constants 0 < ci1 < 1, ǫi1 > 0 and ℓi1 > 0, any specific


γ eiei , γwi−1

ei, γwi

ei, γwi+1

ei∈ K∞, and the γ ekei , γ

ekei

∈ K∞ for k = 1, . . . , i chosenabove, we design κi1 in the form of (4.6) such that Vei satisfies

Vei ≥ maxk=1,...,i−1

γ ekei (Vek ), γekei(Vek), γ

eiei(Vei ),

γwi−1

ei(|wi−1|), γwi

ei(|wi|),

γwi+1

ei(|wi+1|), ǫi1

⇒∇Vei(ei)fei(e1, e1, . . . , ei, ei, wi−1, wi, wi+1) ≤ −ℓi1Vei , (4.111)

where

γwi

ei= αV

(

s

ci1

)

(4.112)

for s ∈ R+.Given the [¯ei−1, ei]

T -subsystem, we choose γ eiei ∈ K∞ such that the com-positions of the gain functions along all the simple loops through the ei-subsystem in the ¯ei-subsystem are less than the identity function.

Consider the ei-subsystem defined in (4.106). Using Lemma 4.1, for any

specified constants ǫi2 > 0 and ℓi2 > 0, any specified γei+1

ei, γei+1

ei∈ K∞ and

the γ eiei chosen above, we design κi2 in the form of (4.6) such that Vei satisfies

Vei(ei) ≥ max

γ eiei (Vei(ei)), γei+1

ei(Vei+1(ei+1)), γ

ei+1

ei(Vei+1 (ei+1)),

γwi

ei(|wi|), γwi+1

ei(|wi+1|), ǫi2

⇒∇Vei (ei)fei(ei, ei+1, ei, ei+1, wi, wi+1) ≤ −ℓi2Vei (ei). (4.113)

In the case of i = n, xi+1 = xmi+1 = u, ei+1 = ei+1 = 0, and wi+1 = 0. Thedynamic state measurement feedback control law is designed as

u = κn2(en). (4.114)

4.2.2 ISS OF THE CLOSEDLOOP SYSTEM

Define e = [e1, . . . , en]T . The e-system can be represented by

e = fe(e, w1, . . . , wn). (4.115)

The gain digraph of the e-system is shown in Figure 4.6.According to the recursive design, given the ei−1-system, by designing κi1

for the ei-subsystem, we can assign the ISS gains γ ekei , γekei’s for k = 1, . . . , i−1

such that all the simple loops in the [eT

i−1, ei]T -system through the ei-

subsystem satisfy the cyclic-small-gain condition. By designing κi2 for theei-subsystem, we can assign the ISS gain γ eiei such that the simple loop in

the ei-system through the ei-subsystem satisfies the cyclic-small-gain condi-tion. Through the recursive control design procedure, the e-system satisfiesthe cyclic-small-gain condition and is ISS.


e1 ei ei+1 en· · · · · ·

FIGURE 4.6 The gain digraph of the [e1, . . . , en]T -system.

For the e-system, we construct the ISS-Lyapunov function candidate

Ve(e) = maxi=1,...,n

σi1(Vei (ei)), σi2(Vei (ei)) , (4.116)

where σ12 = Id, and σi1 with i = 1, . . . , n and σi2 with i = 2, . . . , n are com-

positions of γ(·)(·) ’s which are continuously differentiable on (0,∞) and slightly

larger than the corresponding γ(·)(·) ’s, and still satisfy the cyclic-small-gain con-

dition. Here, it is not necessary to give an explicit representation of the σi1and σi2 to analyze the effect of the sensor noise.

Correspondingly, the influence from wi and ǫi for i = 1, . . . , n can be rep-resented as:

θ = maxi=1,...,n

σi1 γwi−1

ei(|wi−1|), σi1 γwi

ei(|wi|),

σi1 γwi+1

ei(|wi+1|), σi1(ǫi1),

σi2 γwi

ei(|wi|), σi2 γwi+1

ei(|wi+1|), σi2(ǫi2)

. (4.117)

Then, it holds that

Ve(e) ≥ θ ⇒ ∇Ve(e)fe(e, w1, . . . , wn) ≤ −αe(Ve(e)) (4.118)

wherever ∇Ve exists, where αe is a continuous and positive definite function.By default, γwn

en+1:= 0, γw1

e0:= 0, γw1

e0:= 0, σ(n+1)1 := 0, and σ01 := 0.

Define

γwie (s) = max

σ(i+1)1 γwi

ei+1(s), σi1 γwi

ei(s),

σ(i−1)1 γwi

ei−1(s), σi2 γwi

ei(s),

σ(i−1)2 γwi

ei−1(s)

(4.119)

ǫ = maxi=1,...,n

σi1(ǫi1), σi2(ǫi2) . (4.120)

Then, the θ defined in (4.117) can be equivalently represented by

θ = maxi=1,...,n

γwie (|wi|), ǫ . (4.121)


By choosing the γ(·)(·) ’s small enough, we can make σi1 for i = 1, . . . , n and

σi2 for i = 2, . . . , n small enough such that

σi1 γwi

ei(s) ≥ max

σ(i+1)1 γwi

ei+1(s), σ(i−1)1 γwi

ei−1(s),

σi2 γwi

ei(s), σ(i−1)2 γwi

ei−1(s)

. (4.122)

Then, it is achieved that

θ = maxi=1,...,n

σi1 γwi

ei(|wi|), ǫ

. (4.123)

Property (4.118) implies that there exists a βe ∈ KL such that

Ve(e(t)) ≤ max

βe(Ve(e(t0)), t− t0), supt0≤τ≤t

(θ(τ))

, (4.124)

where

θ(τ) = maxi=1,...,n

σi1 γwi

ei(|wi(t)|), ǫ

. (4.125)

From the definition of Ve in (4.118), using σ12 = Id, we have

|x1| = |e1| = |e1 − e1| ≤ |e1|+ |e1|= α−1

V (Ve1 (e1)) + α−1V (Ve1(e1))

≤ α−1V σ−1

12 (Ve(e)) + α−1V σ−1

11 (Ve(e))

= (α−1V + α−1

V σ−111 )(Ve(e)). (4.126)

Define

γwix1

= (α−1V + α−1

V σ−111 ) σi1 γwi

ei, i = 1, . . . , n, (4.127)

βx1 = (α−1V + α−1

V σ−111 ) βe, (4.128)

ǫx1 = (α−1V + α−1

V σ−111 )(ǫ). (4.129)

Then, from (4.124) and (4.125), we obtain

|x1(t)| ≤(α−1V + α−1

V σ−111 )(Ve(e(t)))

≤max

βx1(Ve(e(t0)), t− t0), supt0≤τ≤t

(

maxi=1,...,n

γwix1(|wi(τ)|)

)

, ǫ

.

(4.130)

Thus, the closed-loop system is IOS with x1 as the output and the IOSgain from wi to x1 is γwi

x1.

Recall that γw1

e1(s) = αV (s/c11) for s ∈ R+. From the definition of γw1

x1in

(4.127) with i = 1, we have

γw1x1

= (α−1V + α−1

V σ−111 ) σ11 γw1

e1

= (Id + α−1V σ11 αV )

(

s

c11

)

. (4.131)


Note that for i = 1, . . . , n, each σi1 is a composition of the γ(·)(·) ’s, which can

be chosen to be arbitrarily small. Thus, the IOS gains γwix1

for i = 2, . . . , ncan be designed to be arbitrarily small. If we choose c11 to be arbitrarily closeto one, and σ11 to be arbitrarily small, then γw1

x1is arbitrarily close to the

identity function.The main result of this section is summarized in the following theorem.

Theorem 4.2 Under Assumption 4.4, system (4.86)–(4.88) can be input-to-state stabilized with the dynamic state measurement feedback control law de-fined in (4.96), (4.99), (4.97), and (4.114). Moreover, the closed-loop systemis IOS with the sensor noise w1, . . . , wn as the inputs and x1 as the output,the IOS gain from w1 to x1 can be designed to be arbitrarily close to the iden-tity function, and the IOS gains from w2, . . . , wn to x1 can be designed to bearbitrarily small.

4.2.3 A DESIGN EXAMPLE

To verify the main result of this section, consider the following second-ordernonlinear system:

x1 = x2 (4.132)

x2 = 0.2x22 + u (4.133)

xm1 = x1 + w1 (4.134)

xm2 = x2 + w2. (4.135)

For the sake of simplicity, we consider the case of w2 = 0.Define e1 = x1. Following the design procedure in Subsection 4.2.1, we have

˙e1 = κ11(e1 − w1) + e2 − e2 + κ12(e1) (4.136)

˙e1 = κ11(e1 − w1), (4.137)

where e1 is the estimate of e1, e1 = e1 − e1, and e2 and e2 are defined later.Consider the e1-subsystem. Clearly, ∆∗

11 = 0. We choose c11 = 0.8 andℓ11 = 0.02. Then, the κ11 is designed in the form of κ11(r) = −ν11(|r|)r withν11 satisfying

(1 − c11)ν11((1− c11)s)s ≥ 0.01s. (4.138)

Then, we choose ν11(s) = 0.05 for s ∈ R+ and κ11(r) = −0.05r for r ∈ R.With κ11 designed, we have ∆

∗12(e1, e2, e2, w1) = −0.05e1+0.05w1+ e2− e2.

Thus, ψe1∆∗

12(s) = 0.05s, ψw1

∆∗

12(s) = 0.05s, ψe2∆∗

12(s) = s, and ψe2∆∗

12(s) = s.

Choose ℓ12 = 0.02, γ e1e1 (s) = s, γ e2e1 (s) = 0.99s, γw1

e1(s) = 0.5s2, and γ e2e1 (s) = s.

Then, the κ12 is designed in the form of κ12(r) = −ν12(|r|)r with ν12(s) = 2.11for s ∈ R+.


Define e2 = x2 − κ12(e1). The estimator for e2 is designed in the followingform:

˙e2 = κ21(e2) + u, (4.139)

where e2 is the estimate of e2. Define e2 = e2 − e2. By directly taking thederivative of e2, we have

˙e2 = ∆∗21(e1, e1, e2, e2, w1) + κ21(e2), (4.140)

where |∆∗21(e1, e1, e2, e2, w1)| satisfies |∆∗

21(e1, e1, e2, e2, w1)| ≤ 0.1055|e1| +1.7344|e1|2+4.4521|e1|+0.822|e2|2+2.11|e2|+0.822|e2|2+2.11|e2|+0.1055|w1|.Thus, we have ψe2∆∗

21(s) = 0.822s2+2.11s, ψe2∆∗

21(s) = 0.822s2+2.11s, ψe1∆∗

21(s) =

1.7344s2 + 4.4521s, ψe1∆∗

21(s) = 0.1055s, and ψw1

∆∗

21(s) = 0.1055s. We choose

ℓ21 = 0.02, γ e1e2 (s) = 0.99s, γ e2e2 (s) = 0.99s, γ e1e2 (s) = 0.99s, and γw1

e2(s) =

0.5s2. Then, the γ21 is designed in the form of κ21(r) = −ν21(|r|)r withν21(s) = 3.3784s+ 8.7876 for s ∈ R+.

With κ21 designed, we have ∆∗22(e2) = −3.3784|e2|e2 − 8.7876e2. Thus,

ψe2∆∗

22(s) = 3.3784s2 + 8.7876s. We choose ℓ22 = 0.02 and γ e2e2 (s) = 0.99s.

Then, the κ22 is designed in the form of κ22(r) = −ν22(|r|)r with ν22(s) =3.3784s+ 8.7976 for s ∈ R+.

In the construction of the ISS-Lyapunov function for the closed-loop sys-tem, we can choose all σ(·) = Id. With direct calculation following the proce-dure in Subsection 4.2.2, we have γw1

x1(s) = 2.5s for s ∈ R+.

Simulation results shown in Figures 4.7–4.8 are in accordance with thetheoretical design.

4.3 DECENTRALIZED OUTPUT MEASUREMENT FEEDBACKCONTROL

Decentralized control problems arise from various engineering applications,such as power systems, transportation networks, water systems, chemical engi-neering, and telecommunication networks [239, 118]. Among the main charac-teristics of decentralized control are the dramatic reduction of computationalcomplexity and the enhancement of robustness against uncertain interactions.This section studies decentralized output measurement feedback control oflarge-scale nonlinear systems with nonlinear dynamical interactions. It shouldbe noted that, when reduced to single (centralized) systems, the observer-based design is still useful in handling the sensor noise. The discussions inthis section neglect the possible influence of external disturbances, and focuson the impact of sensor noise to decentralized output-feedback control. Theproblems caused by external disturbances can be solved as in Section 4.1.

Consider the large-scale system with subsystems in the output-feedback


FIGURE 4.7 The sensor noise and the system states.

FIGURE 4.8 The estimator states and the control input.

form:

zi = ∆i0(zi, yi, wi) (4.141)

xij = xi(j+1) +∆ij(yi, zi, wi), j = 1, . . . , ni − 1 (4.142)

xini= ui +∆ini

(yi, zi, wi) (4.143)

wi = [y1, . . . , yi−1, yi+1, yN ]T (4.144)

yi = xi1 (4.145)

ymi = yi + di, (4.146)


where, for each i = 1, . . . , N , [zTi , xi1, . . . , xini]T with zi ∈ Rnzi and xij ∈ R

(j = 1, . . . , ni) is the state, ui ∈ R is the control input, yi ∈ R is the output,zi and [xi2, . . . , xini

]T are the unmeasured portions of the state, ymi ∈ R isthe measurement of the output with di ∈ R being sensor noise, and ∆ij ’s(j = 1, . . . , ni) are unknown locally Lipschitz functions.

Figure 4.9 shows the block diagram including the i-th and the i′-th sub-systems (1 ≤ i, i′ ≤ N, i 6= i′) of the large-scale system.

wi zi xi

ui

yi

di

ymi

wi′ zi′ xi′

ui′

yi′

di′

ymi′

......

...

...

FIGURE 4.9 The block diagram of the large-scale system (4.141)–(4.146).

Assumptions 4.5, 4.6, and 4.7 are made on system (4.141)–(4.146).

Assumption 4.5 For i = 1, . . . , N , j = 1, . . . , ni, each ∆ij satisfies

|∆ij(yi, wi)| ≤ ψ∆ij(|[yi, wTi ]T |) (4.147)

for all [yi, wTi ]T ∈ RN , with a known ψ∆ij

∈ K∞.

Assumption 4.6 For each i = 1, . . . , N , there exists a constant di ≥ 0, suchthat

|di(t)| ≤ di (4.148)

for t ≥ 0.

Assumption 4.7 For i = 1, . . . , N , each zi-subsystem (4.141) with yi andwi as inputs admits a continuously differentiable ISS-Lyapunov function Vzi ,satisfying the following:


1. there exist αzi , αzi ∈ K∞ such that for all zi,

αzi(|zi|) ≤ Vzi(zi) ≤ αzi(|zi|); (4.149)

2. there exist χyizi , χwizi ∈ K and a continuous and positive definite function αzi

such that for all zi, yi, wi,

Vzi(zi) ≥ maxχyizi (|yi|), χwizi (|wi|)

⇒∇Vzi(zi)∆i0(zi, yi, wi) ≤ −αzi(Vzi(zi)). (4.150)

In the presence of the sensor noise, the objective of this section is to design adecentralized controller for the large-scale system composed of (4.141)–(4.146)by using the measurements ymi for i = 1, . . . , N , such that the outputs yi’s(i = 1, . . . , N) are steered to within some small neighborhoods of the origin.

4.3.1 DECENTRALIZED REDUCEDORDER OBSERVER

For each i-th subsystem, a decentralized reduced-order observer is designedto estimate the unmeasurable internal states by using the measurement ymi ofthe output:

ξij = ξi(j+1) + Li(j+1)ymi − Lij(ξi2 + Li2y

mi ), j = 2, . . . , ni − 1 (4.151)

ξini= ui − Lini

(ξi2 + Li2ymi ), (4.152)

where ξij is to be used as an estimate of xij − Lijyi for each j = 2, . . . , ni.With ymi = yi + di, observer (4.151)–(4.152) can be equivalently representedby

ξij = ξi(j+1) + Li(j+1)yi − Lij(ξi2 + Li2yi) + (Li(j+1) − LijLi2)di,

j = 2, . . . , ni − 1 (4.153)

ξini= ui − Lini

(ξi2 + Li2yi)− LiniLi2di. (4.154)

Define ζi = [xi2 − Li2yi − ξi2, . . . , xini− Lini

yi − ξini]T as the observation

error. Then, direct calculation yields:

ζi = Aiζi + φi0(yi, zi, wi, di)

:= fζi(ζi, yi, zi, wi, di), (4.155)


where

Ai =

−Li2...

−Li(ni−1)

Ini−2

−Lini0 · · · 0

,

φi0(yi, zi, wi, di) =

−Li2...

−Lini

Ini−1

∆i1(yi, zi, wi)...

∆ini(yi, zi, wi)

+

L2i2 − Li3

...Li(ni−1)Li2 − Lini

LiniLi2

di.

Under Assumption 4.5, by using the definition of φi0, we can findψyiφi0

, ψziφi0, ψwi

φi0, ψdiφi0

∈ K∞ such that |φi0(yi, zi, wi, di)|2 ≤ ψyiφi0(|yi|) +

ψziφi0(|zi|) + ψwi

φi0(|wi|) + ψdiφi0

(|di|) for all yi, zi, wi, di.The real constants Lij ’s (j = 2, . . . , ni) are chosen so that Ai is a Hurwitz

matrix, i.e., its eigenvalues have negative real parts. As a result, there exists apositive definite matrix Pi = PTi ∈ R

(ni−1)×(ni−1) satisfying PiAi + ATi Pi =−2Ini−1. Define Vζi(ζi) = ζTi Piζi. Then, there exist αζi , αζi ∈ K∞ such thatαζi(|ζi|) ≤ Vζi (ζi) ≤ αζi(|ζi|) holds for all ζi. With direct calculation, we have

∇Vζi(ζi)fζi(ζi, yi, zi, wi, di)=− 2ζTi ζi + 2ζTi Piφi0(yi, zi, wi, di)

≤− ζTi ζi + |Pi|2|φi0(yi, zi, wi, di)|2

≤− 1

λmax(Pi)Vζi(ζi)

+ |Pi|2(

ψyiφi0(|yi|) + ψziφi0

(|zi|) + ψwi

φi0(|wi|) + ψdiφi0

(|di|))

. (4.156)

Define χyiζi = 4λmax(Pi)|Pi|2ψyiφi0, χziζi = 4λmax(Pi)|Pi|2ψziφi0

, χwi

ζi=

4λmax(Pi)|Pi|2ψwi

φi0, and χdiζi = 4λmax(Pi)|Pi|2ψdiφi0

. Then, direct calculationyields:

Vζi(ζi) ≥ maxχyiζi (|yi|), χziζi(|zi|), χwi

ζi(|wi|), χdiζi (|di|)

⇒∇Vζi(ζi)fζi(ζi, yi, zi, wi, di) ≤ −αζi(Vζi (ζi)), (4.157)

where αi0(s) = s/4λmax(Pi) for s ∈ R+.The reduced-order observer in this subsection is motivated by the design

in [116]. Due to the unavailability of the accurate output yi, the measurementymi is used instead. The influence of the sensor noise di is represented by anISS gain.



In this subsection, a new [eTi0, ei1, . . . , eini]T -system composed of ISS subsys-

tems is recursively constructed based on the [ζTi , zTi , yi, ξi2, . . . , ξini

]T -system:

ζi = Aiζi + φi0(yi, zi, wi, di) (4.158)

zi = ∆i0(zi, yi, wi) (4.159)

yi = ξi2 + φi1(ζi, zi, yi, wi) (4.160)

ξij = ξi(j+1) + φij(yi, ξi2, di), j = 2, . . . , ni − 1 (4.161)

ξini= ui + φini

(yi, ξi2, di), (4.162)

where

φi1(ζi, zi, yi, wi) = Li2yi + (xi2 − Li2ei1 − ξi2) + ∆i1(yi, zi, wi),

φij(yi, ξi2, di) = Li(j+1)yi − Lij(ξi2 + Li2yi) + (Li(j+1) − LijLi2)di,

φini(yi, ξi2, di) = −Lini

(ξi2 + Li2yi)− LiniLi2di.

The φi1 is denoted as a function of ζi, zi, yi, wi because (xi2 − Li2yi − ξi2)is the first element of vector ζi. Based on the observer design in Subsection4.3.1, the large-scale system (4.141)–(4.146) is stabilized if (4.158)–(4.162)is stabilized. Note that the part composed of (4.160), (4.161) and (4.162) isin the strict-feedback form, and the part composed of (4.158) and (4.159)can be considered as dynamic uncertainty. The difference from the standardstabilization problem is that yi is not available and y

mi should be used instead.

Define ei0 = [ζTi , zTi ]T and ei1 = yi. The eij for j = 2, . . . , ni are defined

one by one in the following design procedure. The ISS-Lyapunov functioncandidates of the eij-subsystems (j = 1, . . . , ni) are defined as

Vij(eij) = αV (|eij |) =1

2|eij |2. (4.163)

In the following discussions in this subsection, we sometimes use Vij insteadof Vij(eij) to simplify the notations. For convenience of notation, denote eij =[eTi0, ei1, . . . , eij ]

T and ξij = [ξi2, . . . , ξij ]T .

The ei0subsystem

Recall that the ζi-subsystem and the zi-subsystem are ISS. The ei0-subsystemis a cascade connection of the ζi-subsystem and the zi-subsystem, which satis-fies the small-gain condition automatically. Define Lyapunov-based ISS gainsγei1ζi = χyiζi α

−1V , γziζi = χziζi α−1

zi , and γei1zi = χyizi α−1

V . Then, from (4.157)and (4.150), we have

Vζi ≥ maxγei1ζi (Vi1), γziζi(Vzi), χ

wi

ζi(|wi|), χdiζi (|di|)

⇒∇Vζi(ζi)fζi(ζi, yi, zi, wi, di) ≤ −αζi(Vζi), (4.164)

Vzi ≥ maxγei1zi (Vi1), χwizi (|wi|)

⇒∇Vzi(zi)∆i0(zi, yi, wi) ≤ −αzi(Vzi). (4.165)


With the Lyapunov-based ISS cyclic-small-gain theorem, the ISS-Lyapunovfunction of the ei0-subsystem is constructed as:

Vi0(ei0) = maxVζi(ζi), γziζi (Vzi(zi)), (4.166)

where γziζi ∈ K∞ is slightly larger than γziζi and continuously differentiableon (0,∞). Then, there exist αi0, αi0 ∈ K∞ such that αi0(|ei0|) ≤ Vi0(ei0) ≤αi0(|ei0|).

Define

γei1ei0 (s) = maxγei1ζi (s), γziζi γei1zi (s) (4.167)

χwiei0(s) = maxχwi

ζi(s), γziζi χ

wizi (s) (4.168)

χdiei0(s) = χdiζi (s) (4.169)

for s ∈ R+. Then, γei1ei0 is the ISS gain from Vi1 to Vi0, χ

wiei0 is the ISS gain

from wi to Vi0, and χdiei0 is the ISS gain from di to Vi0.

With the cyclic-small-gain theorem, there exists a continuous and positivedefinite function αi0 such that

Vi0 ≥ maxγei1ei0 (Vi1), χwiei0 (|wi|), χdiei0(|di|)

⇒∇Vi0(ei0)fei0(ei0, ei1, wi, di) ≤ −αi0(Vi0) (4.170)

wherever Vi0 is differentiable. Here, fei0 represents the dynamics of the ei0-subsystem, i.e., ei0 = fei0(ei0, ei1, wi, di). It should be noted that Vi0 is locallyLipschitz, and thus continuously differentiable almost everywhere.

The ei1subsystem

The design for the ei1-subsystem (i.e., the yi-subsystem) is quite similar tothe e1-subsystem in Subsection 4.1.3.

To deal with the sensor noise, we employ a set-valued map

Si1(yi) = κi1(yi + δdi) : |δ| ≤ 1 (4.171)

with κi1 continuously differentiable, odd, strictly decreasing, and radiallyunbounded, to be determined later. Then, maxSi1(yi) = κi1(yi − di) andminSi1(yi) = κi1(yi + di).

Recall the definition of ~d in (4.51). Define ei2 as

ei2 = ~d(ξi2, Si1(yi)). (4.172)

Then, we have

ξi2 − ei2 ∈ Si1(yi), (4.173)


and the ei1-subsystem can be represented by

ei1 =ξi2 − ei2 + ei2 + φi1(ζi, zi, yi, wi)

:=ξi2 − ei2 + φ∗i1(ζi, zi, yi, ei2, wi)

∈ξi2 − ei2 + φ∗i1(ζi, zi, yi, ei2, wi) : ξi2 − ei2 ∈ Si1(yi):=Fei1 (ζi, zi, yi, ei2, wi). (4.174)

From Assumption 4.5 and the definition of φi1, we can find a ψφ∗

i1∈ K∞ such

that |φ∗i1(ζi, zi, yi, ei2, wi)| ≤ ψφ∗

i1(|[eTi2, wTi ]T |).

Note that ei1 = yi. With Lemma 4.1, for any 0 < ci1 < 1, ǫi1 > 0, ℓi1 > 0,γei0ei1 , γ

ei2ei1 , χ

wiei1 ∈ K∞, we can find a continuously differentiable, odd, strictly

decreasing and radially unbounded κi1 such that the ei1-subsystem (4.174)with ξi2 − ei2 satisfying (4.173) is ISS with Vi1 satisfying

Vi1(ei1) ≥ max

γei0ei1 αi0(|ei0|), γei2ei1 αV (|ei2|), χwiei1 (|wi|), χdiei1 (di), ǫi1

⇒ maxfei1∈Fei1

(ζi,zi,yi,ei2,wi)∇Vi1(ei1)fei1 ≤ −ℓi1Vi1(ei1) (4.175)

where

χdiei1(s) = αV

(

1

ci1s

)

(4.176)

for s ∈ R+. Note that Vi0 ≥ αi0(|ei0|) and Vi2 = αV (|ei2|). Thus, with theappropriately designed κi1, we can achieve

Vi1(ei1) ≥ max

γei0ei1 (Vi0), γei2ei1 (Vi2), χ

wiei1 (|wi|), χdiei1(di), ǫi1

⇒ maxfei1∈Fei1

(ζi,zi,yi,ei2,wi)∇Vi1(ei1)fei1 ≤ −ℓi1Vi1(ei1). (4.177)

The eijsubsystem (j = 2, . . . , ni)

By convention, Si1(yi, ξi1) := Si1(yi). When j = 3, . . . , ni, for each k =2, . . . , j − 1, a set-valued map Sik is defined as

Sik(yi, ξik) =

κik(ξik − pik) : pik ∈ Si(k−1)(yi, ξi(k−1))

, (4.178)

where κik is a continuously differentiable, odd, strictly decreasing and radiallyunbounded function, and ei(k+1) is defined as

ei(k+1) = ~d(ξi(k+1), Sik(yi, ξik)). (4.179)

Lemma 4.3 Consider the [ζTi , zTi , yi, ξi2, . . . , ξini

]T -system defined by (4.158)–(4.162). If for k = 1, . . . , j−1, Sik and ei(k+1) are defined as (4.171), (4.172),(4.178) and (4.179), then when eij 6= 0, the eij-subsystem can be representedwith

eij = ξi(j+1) + φ′ij(zi, yi, ξij , wi, di), (4.180)


where

|φ′ij(yi, ξij , wi, di)| ≤ ψφ′

ij(|[eTij , wi, di]T |) (4.181)

with ψφ′

ij∈ K∞ known. Specifically, ξi(ni+1) = ui.

The proof of Lemma 4.3 is given in Appendix E.2.Define a set-valued map Sij as

Sij(yi, ξij) = κij(ξij − pij) : pij ∈ Si(j−1)(yi, ξi(j−1)) (4.182)

with κij continuously differentiable, odd, strictly decreasing, and radially un-bounded, to be determined later. Define

ei(j+1) = ~d(ξi(j+1), Sij(yi, ξij)). (4.183)

Then, ξi(j+1) − ei(j+1) ∈ Sij(yi, ξij), and when eij 6= 0, the eij-subsystem canbe rewritten as:

eij = ξi(j+1) − ei(j+1) + ei(j+1) + φ′ij(zi, yi, ξij , wi, di)

:= ξi(j+1) − ei(j+1) + φ∗ij(zi, yi, ξij , ei(j+1), wi, di)

∈ ξi(j+1) − ei(j+1) + φ∗ij(zi, yi, ξij , ei(j+1), wi, di) :

ξi(j+1) − ei(j+1) ∈ Sij(yi, ξij):= Feij (zi, yi, ξij , ei(j+1), wi, di). (4.184)

Clearly, under Assumption 4.5, there exists a ψφ∗

ij∈ K∞ such that

φ∗ij(zi, yi, ξij , ei(j+1), wi, di) ≤ ψφ∗

ij(|[eTi(j+1), w

Ti , di]

T |). (4.185)

From the definition of eij in (4.179), in the case of eij 6= 0, for all pij ∈Si(j−1)(yi, ξi(j−1)), it holds that |ξij−pij | > |eij | and sgn(ξij−pij) = sgn(eij),which implies sgn(ξij − pij − eij) = sgn(eij), and thus ξij − pij = eij + (ξij −pij − eij) = eij + sgn(eij)|ξij − pij − eij |. Then, we can rewrite the set-valuedmap Sij as

Sij(yi, ξij) =

κij(eij + sgn(eij)|ξij − pij − eij |) :pij ∈ Si(j−1)(yi, ξi(j−1))

. (4.186)

With Lemma 4.1, for any ǫij > 0, ℓij > 0, γei0eij , . . . , γei(j−1)eij ,

γei(j+1)eij , χwi

eij , χdieij ∈ K∞, we can find a continuously differentiable, odd, strictly

decreasing, and radially unbounded κij such that the eij-subsystem (4.184)with ξi(j+1) − ei(j+1) ∈ Sij(yi, ξij) is ISS with Vij satisfying

Vij(eij) ≥ maxk=1,...,j−1,j+1

γei0eij αi0(|ei0|), γeikeij αV (|eik|),χwieij (|wi|), χdieij (di), ǫij

⇒ maxfeij∈Feij

(zi,yi,ξij ,ei(j+1),wi,di)∇Vij(eij)feij ≤ −ℓijVij(eij). (4.187)


Notice that Vi0 ≥ αi0(|ei0|) and Vik = αV (|eik|) for k = 1, . . . , j − 1, j + 1.With the appropriately designed κij , we can achieve

Vij(eij) ≥ maxk=0,...,j−1,j+1

γeikeij (Vik), χwieij (|wi|), χdieij (di), ǫij

⇒ maxfeij∈Feij

(zi,yi,ξij ,ei(j+1),wi,di)∇Vij(eeij )feij ≤ −ℓijVij(eij). (4.188)

Decentralized Control Law

In the case of j = ni, the true control input ui occurs and thus we can setei(ni+1) = 0 and Vi(ni+1) = 0. Indeed, our decentralized control law can bechosen as

p∗i2 = κi1(yi + di) (4.189)

p∗ij = κi(j−1)(ξi(j−1) − p∗i(j−1)), j = 3, . . . , ni (4.190)

ui = κini(ξini

− p∗ini). (4.191)

It is directly checked that

|di| ≤ di ⇒ p∗i2 ∈ Si1(yi) ⇒ · · · ⇒ p∗ini∈ Si(ni−1)(yi, ξi(ni−1))

⇒ ui ∈ Sini(yi, ξini

).

4.3.3 CYCLICSMALLGAIN SYNTHESIS OF THE SUBSYSTEMS

Denote ei = eini. With the recursive control design, each ei-subsystem is an

interconnection of ISS subsystems. With the cyclic-small-gain theorem, in thissubsection, the decentralized controller designed above is fine-tuned, to yieldthe ISS property of each ei-subsystem.

According to the recursive design, given the ei(j−1)-subsystem, by appro-priately choosing set-valued map Sij for the eij-subsystem, we can design theISS gains γeikeij ’s (k = 0, . . . , j − 1) such that

γei(k+1)eik · · · γeijei(j−1)

γeikeij < Id, k = 0, . . . , j − 1. (4.192)

By applying this reasoning repeatedly, we can guarantee (4.192) for all j =1, . . . , ni. In this way, the ei-system satisfies the cyclic-small-gain condition inChapter 3.

The gain interconnections between the subsystems can be representedby the gain digraph. In the gain digraph of the ei-subsystem, the ei1-subsystem is reachable from the subsystems of ei0, ei2, . . . , eini

. With theLyapunov-based cyclic-small-gain theorem in Chapter 3, we construct anISS-Lyapunov function for the ei-system as the “potential influence” fromVi0(ei0), . . . , . . . , Vini

(eini) to Vi1(ei1):

Vi(ei) = maxj=0,...,ni

σij(Vij(eij)) (4.193)


with σi1(s) = s, σij(s) = γei2ei1 · · · γeijei(j−1)(s) (j = 2, . . . , ni), and σi0(s) =

maxj=1,...,niσij γei0eij (s) for s ∈ R+, where the γ

(·)(·) ’s are K∞ functions

continuously differentiable on (0,∞), slightly larger than the corresponding

γ(·)(·) ’s, and still satisfy (4.192) for all j = 1, . . . , ni.

For convenience, denote ǫi0 = 0. We represent “potential influence” fromwi, di, and ǫij ’s to Vi1 as

θi = maxj=0,...,ni

σij χwieij (|wi|), σij χdieij (di), σij(ǫij)

. (4.194)

Using the cyclic-small-gain theorem, we have that

Vi(ei) ≥ θi

⇒ maxfei∈Fei

(zi,yi,ξini,ei,wi,di)

∇Vi(ei)fei ≤ −αi(Vi(ei)) (4.195)

holds wherever ∇Vi(ei) exists, where αi is a continuous and positive definitefunction, and

Fei(zi, yi, ξini, ei, wi, di) :=

fei0(ei0, ei1, wi, di)Fei1(ζi, zi, yi, ei2, wi)...Feini

(zi, yi, ξini, ei(ni+1), wi, di)

(4.196)

with eni+1 = 0.From the definitions of wi, el1’s, Vl1’s, and Vl’s (1 ≤ l ≤ N) and Assumption

4.6, we have

|wi| ≤√

∑

1≤l≤N, l6=i(α−1V (Vl))2 + de2i

≤√

max1≤l≤N, l6=i

N · (α−1V (Vl))2, N · de2i

= max1≤l≤N, l6=i

√N · (α−1

V (Vl)),√N · dei. (4.197)

Define γ elei (s) = maxj=0,...,niσij χwi

eij (√N · ail · α−1

V (s)), χdeiei (s) =

maxj=0,...,niσij χwi

eij (√Ns), χdiei (s) = maxj=0,...,ni

σij χdieij (s) and ǫi =maxj=0,...,ni

σij(ǫij) for s ∈ R+. Then, by substituting (4.197) into (4.194)and substituting (4.194) into (4.195), we get

Vi(ei) ≥ max1≤l≤N, l6=i

γ elei (Vl(el)), χdeiei (d

ei ), χ

diei (di), ǫi

⇒ maxfei∈Fei

(zi,yi,ξini,ei,wi,di)

∇Vi(ei)fei ≤ −αi(Vi(ei)) (4.198)


wherever∇Vi(ei) exists, with αi being a continuous and positive definite func-tion.

According to the recursive design approach, we can design the γ(·)(·) ’s (and

thus the γ(·)(·) ’s) to be arbitrarily small to get arbitrarily small σij ’s (j =

0, 2, . . . , ni). We can also design the χwieij ’s (j = 1, . . . , ni), the ǫij ’s (j =

1, . . . , ni), and the χdieij ’s (j = 2, . . . , ni) to be arbitrarily small. Thus, from

the definitions of γ elei and χdeiei , we can design the γ elei ’s (1 ≤ l ≤ N, l 6= i),

the χdeiei (d

ei ), and the ǫi in (4.198) to be arbitrarily small. By designing the

σij χdieij ’s (j = 0, 2, . . . , ni) small enough, from the definitions of χdiei and χdiei1(see (4.176)), it can be achieved that

χdiei (s) = σi1 χdiei1(s) = χdiei1(s) = αV

(

s

ci1

)

, (4.199)

where ci1 may be chosen to be any value satisfying 0 < ci1 < 1.

4.3.4 ANALYSIS OF THE CLOSEDLOOP DECENTRALIZED SYSTEM

The closed-loop decentralized system is an interconnection of ISS ei-subsystems (i = 1, . . . , N) with ISS-Lyapunov functions satisfying (4.198).As the discussion in Subsection 4.3.3 shows, we can design all the ISSgains γ elei ’s (1 ≤ i, l ≤ N, i 6= l) to be arbitrarily small. Thus, the γ elei ’s(1 ≤ i, l ≤ N, i 6= l) can be tuned such that all the simple loops in the closed-loop decentralized system satisfy the cyclic-small-gain condition. In this way,the ISS of the closed-loop decentralized system is achieved.

Consider the gain digraph of the [eT1 , . . . , eTN ]T -system. Recall the definition

of reaching set, denoted by RS, in Definition A.5 in Appendix A.To analyze the effect of the disturbances on each ei-subsystem (i =

1, . . . , N), we construct an ISS-Lyapunov function of the interconnected sys-tem composed of the er-subsystems with r ∈ RS(i) ⊆ 1, . . . , N as:

Vi(ei) = maxr∈RS(i)

ρr(Vr(er)), (4.200)

where state ei consists of all the er’s (r ∈ RS(i)), ρi = Id, and the ρr’s(r ∈ RS(i)\i) are compositions of γ

er′er ’s (r, r′ ∈ RS(i), r 6= r′) which are

continuously differentiable on (0,∞) and slightly larger than the correspond-ing γ

er′er ’s. Note that the dynamics of the ei-system can be described by a

differential inclusion ˙ei ∈ Fi(·).Correspondingly, we can represent the influence of the disturbances to

Vi(ei) as

θi = maxr∈RS(i)

ρr χderer (d

er), ρr χ

dmrer (dr), ρr(ǫr). (4.201)


Again, by using the ISS cyclic-small-gain theorem, we have

Vi(ei) ≥ θi ⇒ maxfi∈Fi(·)

∇Vi(ei)fi ≤ −αi(Vi(ei)) (4.202)

with αi positive definite, wherever ∇Vi(ei) exists.Note that, in (4.201), ρr (r ∈ RS(i)\i) can be designed to be arbitrarily

small by designing γer′er ’s (and of course the γ

er′er ’s) to be arbitrarily small, χ

derer

and ǫr can be designed to be arbitrarily small, and χdmrer (s) can be designed to

be αV

(

scr1

)

. Thus, through an appropriate design, we can get

θi = ρi αV(

dici1

)

= αV

(

dici1

)

. (4.203)

From (4.202) and (4.203), we can see that Vi(ei) ultimately converges towithin the region Vi(ei) ≤ αV

(

di/ci1)

. Using the definitions of ei1, Vi1(ei1),Vi(ei), and Vi(ei) (see (4.163), (4.193), and (4.200)), we have αV (|ei1|) =Vi1(ei1) ≤ Vi(ei) ≤ Vi(ei), which implies that yi = ei1 ultimately converges towithin the region |yi| ≤ di/ci1. By choosing ci1 to be arbitrarily close to one,the output yi can be driven arbitrarily close to the region |yi| ≤ di.

The main result of the section is summarized in Theorem 4.3.

Theorem 4.3 Consider the large-scale system (4.141)–(4.146). Under As-sumptions 4.5, 4.6, and 4.7, the closed-loop signals are bounded, and in par-ticular, each output yi (i = 1, . . . , N) can be steered arbitrarily close to theregion |yi| ≤ di with the decentralized controller composed of the decentral-ized reduced-order observer (4.151)–(4.152) and the decentralized control law(4.189)–(4.191).

In the case of N = 1, system (4.141)–(4.146) is reduced to

z = ∆0(z, y) (4.204)

xi = xi+1 +∆i(y, z), i = 1, . . . , n− 1 (4.205)

xn = u+∆n(y, z) (4.206)

y = x1 (4.207)

ym = y + d, (4.208)

where [zT , x1, . . . , xn]T with z ∈ Rm and xi ∈ R (i = 1, . . . , n) is the state,

u ∈ R is the control input, y ∈ R is the output, z and [x2, . . . , xn]T are

the unmeasured portions of the state, ym ∈ R is the measurement of theoutput with d ∈ R being sensor noise, and ∆i’s (i = 1, . . . , ni) are unknownlocally Lipschitz functions. In this case, the proposed decentralized controlleris reduced to a centralized controller, and the design is still valid.


Example 4.4 Consider the axial compressor model in Example 3.1. By defin-ing the transformations z = R, x1 = φ, x2 = −ψ, y = x1, and u = (v− 1)/β2

(which transform the control problem into a stabilization problem with respectto the origin), one can rewrite system (3.1)–(3.3) in the output-feedback formwith y as the output and u as the control input:

z = g(z, x1) (4.209)

x1 = x2 +∆1(x1, z) (4.210)

x2 = u+∆2(x1) (4.211)

y = x1, (4.212)

where

∆1(x1, z) = −3

2x1 +

1

2− 1

2(x1 + 1)3 − 3(x1 + 1)z, (4.213)

∆2(x1) = − 1

β2x1. (4.214)

The proposed measurement output-feedback design can be readily applied evenif ∆1 and ∆2 contain uncertainties.

Example 4.5 We employ an interconnected system composed of two identicalsubsystems to demonstrate the control design procedure. Each i-th subsystem(i = 1, 2) is in the following form:

zi = −2zi + wixi1 = xi2 +

14x

2i1

xi2 = ui +√28 xi1 +

14x

2i1 +

√28 xi1zi

wi = y(3−i)yi = xi1ymi = yi + di,

(4.215)

where [zi, xi1, xi2]T ∈ R

3 is the state, ui ∈ R is the control input, yi ∈ R

is the output, zi and xi2 are unmeasured portions of the state, ymi ∈ R isthe measurement of output yi, and di represents the sensor noise satisfying|di(t)| ≤ di = d = 0.1 for t ≥ 0. Define αV (s) = 0.5s2 for s ∈ R+.

Define Vzi(zi) = 0.5z2i . Then, we have

Vzi(zi) ≥ χwizi (|wi|) ⇒ ∇Vzi(zi)zi ≤ −Vzi(zi), (4.216)

where χwizi (s) = 0.25s2 for s ∈ R+.

For each i-th subsystem, the reduced-order observer is constructed as

ξi2 = u− ξi2 − ymi . (4.217)

Define ζi = ζi2 = xi2 − yi − ξi2. Then,

ζi = −ζi + di +

√2

8xi1 +

√2

8zi. (4.218)


Define Vζi(ζi) = ζ2i . Then, we have

Vζi(ζi) ≥ maxγziζi (Vzi(zi)), γei1ζi

(Vi1(ei1))⇒ ∇Vζi(ζi)ζi ≤ −0.25Vζi(ζi), (4.219)

where ei1 = yi, Vi1(ei1) = αV (|ei1|), and γziζi (s) = s and γei1ζi (s) = s fors ∈ R+.

Define ei0 = [ζi, zi]T . Then, the ISS-Lyapunov function for the ei0-

subsystem can be constructed as

Vi0(ei0) = maxVζi(ζi), γziζi (Vzi(zi)), (4.220)

where γziζi (s) = 1.1s for s ∈ R+. Moreover, Vi0(ei0) satisfies

Vi0(ei0) ≥ maxγei1ei0 (Vi1(ei1)), χwiei0(|wi|), χdiei0(di)

⇒ ∇Vi0(ei0)ei0 ≤ −0.25Vi0(ei0), (4.221)

where γei1ei0 (s) = s, χwiei0(s) = 0.275s2 and χdiei0(s) = 8s2.

The ei1-subsystem can be rewritten as

ei1 = ξi2 − ei2 + (ζi + ei1 + 0.25e2i1 + ei2). (4.222)

Choose ci1 = 0.5 and γei2ei1 (s) = 0.9s for s ∈ R+. Select γei0ei1 (s) = 0.9s fors ∈ R+ such that γei0ei1 γei1ei0 < Id. With the gain assignment lemma, designκi1(r) = −νi1(|r|)r with νi1(s) = s + 6.5989 for s ∈ R+. In this way, ifκi1(ei1 + di) ≤ ξi2 − ei2 ≤ κi1(ei1 − di), then the implication

Vi1(ei1) ≥ maxγei0ei1 (Vi0(ei0)), γei2ei1 (Vi2(ei2)), χdiei1 (di)⇒ ∇Vi1(ei1)ei1 ≤ −Vi1(ei1) (4.223)

holds, where χdiei1(s) = αV (s/ci1) = 2s2.The ei2-subsystem can be rewritten as

ei2 = u+ φ∗i2(ζi, ei1, ei2, ξi2, di), (4.224)

where |φ∗i2(ζi, ei1, ei2, ξi2, di)| ≤ ψei0φ∗

i2(|ei0|) + ψei1φ∗

i2(|ei1|)ψei2φ∗

i2(|ei2|) + ψdiφ∗

i2(di)

with ψei0φ∗

i2(s) = 6.5989s + 2s2, ψei1φ∗

i2(s) = 44.5455s + 12.8489s2 + 9.75s3,

ψei2φ∗

i2(s) = 7.5989s + 2s2, and ψdiφ∗

i2(s) = 36.9466s + 41.5934s2 + 9.25s3 for

s ∈ R+.Choose χdiei2(s) = 2s2. Select γei0ei2 (s) = 0.9s and γei1ei2 (s) = s for s ∈ R+

such that γei1ei0 γei2ei1 γei0ei2 < Id and γei2ei1 γei1ei2 < Id. For the ei2-subsystem,design κi2(r) = −νi2(|r|)r with νi2(s) = 76.0.62 + 58.7526s + 10.9063s2 fors ∈ R+. Then, with minSi2(yi, ξi2) ≤ ui ≤ maxSi2(yi, ξi2), Vi2(ei2) satisfies

Vi2(ei2) ≥ maxγei0ei2 (Vi0(ei0)), γei1ei2 (Vi1(ei1)), χdiei2 (di)⇒ ∇Vi2(ei2)ei2 ≤ −Vi2(ei2). (4.225)


Denote ei = [ei0, ei1, ei2]T . Define γei0ei1 (s) = 0.95s, γei2ei1 (s) = 0.95s and

γei0ei2 (s) = 0.95s for s ∈ R+. For each i-th subsystem, we can construct an ISS-Lyapunov function Vi(ei) = maxσi0(Vi0(ei0)), σi1(Vi1(ei1)), σi2(Vi2(ei2))with σi0(s) = maxγei0ei1 (s), γei2ei1 γei0ei2 (s) = 0.95s, σi1(s) = s and σi2(s) =

γei2ei1 (s) = 0.95s for s ∈ R+. We can also calculate χdiei (s) = 2s2 for s ∈ R+.

Note that |wi| = |y(3−i)| ≤ α−1V (V(3−i)(e(3−i))). Then, γ

e(3−i)

ei (s) =

σi0 χwiei0 α−1

V (s) = 0.5225. Thus, γe(3−i)

ei γ eie(3−i)< Id. Define Vi(ei) =

maxVi(ei), γ e(3−i)

ei (V(3−i)(e(3−i))) with γe(3−i)

ei (s) = 0.55s for s ∈ R+. Then,

we can calculate θi = maxχdiei (di), γe(3−i)

ei χdm(3−i)

e(3−i)(d(3−i)) = χdiei (di) = 2s2.

Based on the theory, the measurement feedback controller with κi1 and κi2designed above could drive the output yi to the region |yi| ≤ di/ci1 = 2di = 0.2.Simulation results with disturbances dm1 (t) = 0.09 sin(5t) + 0.01sign(sin(30t))and dm2 (t) = 0.09 cos(5t) + 0.01sign(cos(30t)), and initial conditions z1(0) =0.5, x11(0) = 0, x12(0) = 0, ξ12(0) = 0, z2(0) = −0.5, x21(0) = 0, x22(0) = 0,and ξ22(0) = 0, shown in Figures 4.10 and 4.11, are in accordance with ourtheoretic results.

4.4 EVENTTRIGGERED AND SELFTRIGGERED CONTROL

An event-triggered control system is a sampled-data system in which thesampling time instants are determined by events generated by the real-timesystem state. By taking the advantage of the inter-sample behavior, event-triggered sampling may realize improved control performance over periodicsampling [160, 83].

An event-triggered state-feedback control system is generally in the follow-ing form:

x(t) = f(x(t), u(t)) (4.226)

u(t) = v(x(tk)), t ∈ [tk, tk+1), k ∈ S, (4.227)

where x ∈ Rn is the state, u ∈ Rm is the control input, f : Rn × Rm → Rn

is a locally Lipschitz function representing system dynamics, v : Rn → Rm

is a locally Lipschitz function representing the control law, tk represents thesampling time instants, and S ⊆ Z+ is the set of the indices of all the samplingtime instants. It is assumed that f(0, v(0)) = 0. The sequence tkk∈S isdetermined online based on the measurement of the real-time system state. Ifthere is an infinite number of sampling time instants, then S = Z+; otherwise,S is in the form of 0, . . . , k∗ with k∗ ∈ Z+ being the last sampling timeinstant. For convenience of notation, we denote tk∗+1 = ∞. Figure 4.12 showsthe block diagram of an event-triggered control system.

Define

w(t) = x(tk)− x(t), t ∈ [tk, tk+1), k ∈ S (4.228)


FIGURE 4.10 State trajectories of Example 4.5.

FIGURE 4.11 Control signals and disturbances of Example 4.5.

as the measurement error caused by data sampling, and rewrite

u(t) = v(x(t) + w(t)). (4.229)


controller

plant

event trigger &samplerhold

x(tk)

u(t)

u(tk)

x(t)

FIGURE 4.12 The block diagram of an event-triggered control system.

Then, by substituting (4.229) into (4.226), we have

x(t) = f(x(t), v(x(t) + w(t)))

:= f(x(t), x(t) + w(t)). (4.230)

Clearly, the event-triggered control problem is closely related to the mea-surement feedback control problem. However, through event-based triggering,the measurement error w caused by data sampling is adjustable, and the ob-jective of event-triggered control is to adjust w online to asymptotically steerthe system state x(t) to the origin, if possible.

Based on the idea of robust control, a widely recognized approach to event-triggered control contains two steps:

1. Designing a continuous-time controller which guarantees the robustness ofthe closed-loop system with respect to the measurement error caused bydata sampling;

2. Designing an appropriate event trigger to restrict the measurement errorcaused by data sampling to be within the margin of robust stability.

The block diagram of the system is shown in Figure 4.13.

event trigger & sampler

x(t) = f(x(t), v(x(t) + w(t)))

+

−

x(t)w(t)

x(tk)

FIGURE 4.13 Event-triggered control problem as a robust control problem.

Due to its equivalence to robust stability, ISS has been used for event-triggered control of nonlinear systems. In [255, 160, 272], it is assumed thatsystem (4.230) is ISS with w as the input and has an ISS-Lyapunov function


V : Rn → R+ satisfying

∇V (x)f (x, x + w) ≤ −α(V (x)) + γ(|w|) (4.231)

for all x,w, where α ∈ K∞ and γ ∈ K. Then, by designing the event triggersuch that

|w(t)| ≤ ρ(V (x(t))) (4.232)

always holds with ρ ∈ K satisfying

α−1 γ ρ < Id, (4.233)

asymptotic stability of the closed-loop system is achieved. In this case, V is aLyapunov function of the closed-loop system.

Theoretically, a system is ISS if and only if it has an ISS-Lyapunov func-tion. However, even if a nonlinear system has been designed to be ISS, theconstruction of an ISS-Lyapunov function may not be straightforward. Notethat, given an ISS-Lyapunov function, one can easily determine the ISS char-acteristics of a system. By using the relationship between ISS and robuststability, the study in this section shows that a known ISS-Lyapunov functionmay not be necessary for the design of event-triggered control.

For physical realization of event-triggered sampling, a positive lower boundof the inter-sample periods should be guaranteed throughout the process ofevent-triggered control, i.e., infk∈S(tk+1 − tk) > 0, to avoid infinitely fastsampling. A special case is the Zeno behavior, with which, limk→∞ tk < ∞[70].

In this section, we first present a trajectory-based ISS condition for asymp-totic stabilization of general event-triggered control systems without infinitelyfast sampling. Then, in Subsection 4.4.2, we discuss the event-triggered con-trol problem in the presence of external disturbances. Subsection 4.4.3 givesa design for strict-feedback systems to fulfill the condition for event-triggeredcontrol.

4.4.1 AN ISS GAIN CONDITION FOR EVENTTRIGGERED CONTROL

In this subsection, we assume that a measurement feedback controller existsfor system (4.226) such that system (4.230) is ISS with the measurement erroras the input.

Assumption 4.8 System (4.230) is ISS with w as the input, that is, thereexist β ∈ KL and γ ∈ K such that for any initial state x(0) and any piecewisecontinuous, bounded w, it holds that

|x(t)| ≤ max β(|x(0)|, t), γ(‖w‖∞) (4.234)

for all t ≥ 0.


Under Assumption 4.8, with the robust stability property of ISS, if theevent trigger is designed such that |w(t)| ≤ ρ(|x(t)|) for all t ≥ 0 with ρ ∈ Ksatisfying

ρ γ < Id, (4.235)

then x(t) asymptotically converges to the origin. Based on this idea, the eventtrigger considered in this section is defined as: if x(tk) 6= 0, then

tk+1 = inf t > tk : H(x(t), x(tk)) = 0 , (4.236)

where H : Rn × Rn → R is defined by

H(x, x′) = ρ(|x|)− |x− x′|. (4.237)

If x(tk) = 0 or t > tk : H(x(t), x(tk)) = 0 = ∅, then the data samplingevent is not triggered and in this case, tk+1 = ∞. Note that, under the as-sumption of f(0, v(0)) = 0, if x(tk) = 0, then u(t) = v(x(tk)) = 0 keeps thestate at the origin for all t ∈ [tk,∞).

With the event trigger proposed above, given tk and x(tk) 6= 0, tk+1 is thefirst time instant after tk such that

ρ(|x(tk+1)|)− |x(tk+1)− x(tk)| = 0. (4.238)

Since ρ(|x(tk)|)−|x(tk)−x(tk)| > 0 for any x(tk) 6= 0 and x(t) is continuouson the timeline, the proposed event trigger guarantees that

ρ(|x(t)|) − |x(t) − x(tk)| ≥ 0 (4.239)

for t ∈ [tk, tk+1). Recall the definition of w(t) in (4.228). Property (4.239)implies that

|w(t)| ≤ ρ(|x(t)|) (4.240)

holds for t ∈ [tk, tk+1). At this stage, it cannot be readily guaranteed that(4.240) holds for all t ≥ 0, as

⋃

k∈S[tk, tk+1) may not cover the whole time-

line, i.e., R+\⋃

k∈S[tk, tk+1) 6= ∅.

As mentioned above, for physical realization of (4.240) with event-triggeredsampling, a positive lower bound of the inter-sample periods should be guaran-teed throughout the event-triggered control procedure, i.e., infk∈Stk+1−tk >0, to avoid infinitely fast sampling.

Theorem 4.4 presents a condition on the ISS gain γ to find a ρ such thatinfk∈Stk+1 − tk > 0, and the closed-loop event-triggered control system isasymptotically stable at the origin.

Theorem 4.4 Consider the event-triggered control system (4.230) with lo-cally Lipschitz f satisfying f(0, 0) = 0 and w defined in (4.228). If Assump-tion 4.8 is satisfied with a γ which is Lipschitz on compact sets, then one canfind a ρ ∈ K∞ such that


• ρ satisfies (4.235), and• ρ−1 is Lipschitz on compact sets.

Moreover, with the sampling time instants triggered by (4.236) with H definedin (4.237), for any specific initial state x(0), system state x(t) satisfies

|x(t)| ≤ β(|x(0)|, t) (4.241)

for all t ≥ 0, with β ∈ KL, and the inter-sample periods are lower bounded bya positive constant.

Proof. With a γ ∈ K being Lipschitz on compact sets, one can always finda γ ∈ K∞ being Lipschitz on compact sets such that γ > γ. By choosingρ = γ−1, we have ρ γ = γ γ < γ γ−1 < Id, and ρ−1 = γ is Lipschitz oncompact sets.

Along each trajectory of the closed-loop system, for each k ∈ S with statex(tk) at time instant tk, define

Θ1(x(tk)) =

x ∈ Rn : |x− x(tk)| ≤ ρ (Id + ρ)−1(|x(tk)|)

, (4.242)

Θ2(x(tk)) = x ∈ Rn : |x− x(tk)| ≤ ρ(|x|) . (4.243)

By directly using Lemma C.6, it can be proved that Θ1(x(tk)) ⊆ Θ2(x(tk)).An illustration with x = [x1, x2]

T ∈ R2 is given in Figure 4.14.

0 x1

x2

ρ

1

x(tk)

Θ1

Θ2

FIGURE 4.14 An illustration of Θ1(x(tk)) ⊆ Θ2(x(tk)).

Given a ρ ∈ K∞ such that ρ−1 is Lipschitz on compact sets, it can beproved that (ρ (Id + ρ)−1)−1 = (Id + ρ) ρ−1 = ρ−1 + Id is Lipschitz oncompact sets and there exists a continuous, positive function ρ : R+ → R+

such that (ρ−1 + Id)(s) ≤ ρ(s)s := ρ(s) for s ∈ R+. By using the definition ofρ, one has s =

(

ρ ρ−1(s))

ρ−1(s) and thus ρ−1(s) = s/(

ρ ρ−1(s))

:= ρ(s)s.Here, it can be directly checked that ρ : R+ → R+ is continuous and positive.Thus,

ρ (Id + ρ)−1(s) = (ρ−1 + Id)−1(s) ≥ ρ−1(s) = ρ(s)s. (4.244)


Property (4.244) implies that, if

|x− x(tk)| ≤ ρ(|x(tk)|)|x(tk)|, (4.245)

then x ∈ Θ1(x(tk)).Also, for any x ∈ Θ1(x(tk)), by using the locally Lipschitz property of f ,

it holds that

|f(x, v(x(tk)))| = |f(x, x(tk))|= |f(x− x(tk) + x(tk), x(tk))|≤ Lf

(

|[xT − xT (tk), xT (tk)]

T |)

|[xT − xT (tk), xT (tk)]

T |≤ L(|x(tk)|)|x(tk)|, (4.246)

where Lf , L are continuous, positive functions defined on R+. Property (4.245)is used for the last inequality.

Then, the minimum time Tmink needed for the state of the closed-loop sys-

tem starting at x(tk) to go outside the region Θ1(x(tk)) can be estimatedby

Tmink ≥ ρ(|x(tk)|)|x(tk)|

L(|x(tk)|)|x(tk)|=ρ(|x(tk)|)L(|x(tk)|)

, (4.247)

which is well defined and strictly larger than zero for any x(tk). SinceΘ1(x(tk)) ⊆ Θ2(x(tk)) and x(t) is continuous on the timeline, the minimuminterval needed for the state starting at x(tk) to go outside Θ2(x(tk)) is notless than Tmin

k .By directly using (4.247), one has

Tmin0 ≥ ρ(|x(0)|)

L(|x(0)|) . (4.248)

If S = 0, then w(t) is continuous and (4.239) holds for t ∈ [0,∞). Wenow consider the case of S 6= 0. Suppose that for a specific k ∈ Z+\0, theevent trigger (4.236) with H defined by (4.237) guarantees that for t ∈ [0, tk),w(t) is piecewise continuous and (4.240) holds. Under condition (4.234), byusing the robust stability property of ISS, one has

|x(t)| ≤ β(|x(0)|, t) (4.249)

for all t ∈ [0, tk), with β ∈ KL. Due to the continuity of x(t) with respect

to t, x(tk) = limt→t−kx(t). Thus, |x(tk)| ≤ β(|x(0)|, 0). This, together with

property (4.247) implies that

Tmink ≥ min

ρ(|x|)L(|x|) : |x| ≤ β(|x(0)|, 0)

. (4.250)


This means that for t ∈ [0, tk+1), w(t) is piecewise continuous and (4.240)holds. By induction, w(t) is piecewise continuous and (4.239) holds for t ∈[0, tk+1) for any k ∈ S. If S is an infinite set, then limk→∞ tk+1 = ∞ by using(4.248); if S is a finite set, say 0, . . . , k∗, then tk∗+1 = ∞. In both cases,w(t) is piecewise continuous and (4.239) holds for t ∈ [0,∞).

With the robust stability property of ISS, property (4.249) holds for t ∈[0,∞). This ends the proof. ♦

The proof of Theorem 4.4 also naturally leads to a self-triggered samplingstrategy, which computes tk+1 by using tk and x(tk), and thus does not contin-uously monitor the trajectory of x(t). Suppose that Assumption 4.8 is satisfiedfor the closed-loop system composed of (4.226) and (4.229) with locally Lip-schitz f . With property (4.247), given tk and x(tk), tk+1 can be computedas

tk+1 =ρ(|x(tk)|)L(|x(tk)|)

+ tk (4.251)

for k ∈ Z+. Based on the proof of Theorem 4.4, it can be directly verifiedthat ρ(|x(t)|) − |x(t) − x(tk)| ≥ 0 holds for all t ∈ [tk, tk+1), k ∈ Z+, andall the inter-sample periods are lower bounded by a positive constant, givenany specific initial state x(0). With Assumption 4.8 satisfied, the state x(t)ultimately converges to the origin.

Example 4.6 The condition for event-triggered control without infinitely fastsampling can be readily fulfilled by the linear system

x = Ax+Bu (4.252)

with x ∈ Rn as the state and u ∈ Rm as the control input, if the systemis controllable. One can find a K such that A − BK is Hurwitz and designu = −K(x+w) with w being the measurement error caused by data sampling.Then,

x = Ax−BK(x+ w) = (A−BK)x−BKw. (4.253)

With initial state x(0) and input w, the solution of the closed-loop system is

x(t) = e(A−BK)tx(0)−∫ t

0

e(A−BK)(t−τ)BKw(τ)dτ. (4.254)

for t ≥ 0. It can be verified that x(t) satisfies property (4.234) with β(s, t) =(1 + 1/δ)|e(A−BK)t|s and γ(s) = (1 + δ)

(∫∞0 |e(A−BK)τ |dτ

)

s, where δ can beselected as any positive constant. Clearly, γ is Lipschitz on compact sets.


4.4.2 EVENTTRIGGERED CONTROL AND SELFTRIGGERED CONTROL INTHE PRESENCE OF EXTERNAL DISTURBANCES

Theorem 4.4 does not take into account the influence of external disturbances.To study the influence of the disturbances, we consider the following system

x(t) = f(x(t), u(t), d(t)), (4.255)

where d(t) ∈ Rnd represents the external disturbances, and the other variablesare defined as for (4.226). It is assumed that d is piecewise continuous andbounded. The control law and the event trigger are still in the form of (4.227)and (4.236), respectively. In this case, we still discuss the ISS condition forthe realization of event-triggered control with guaranteed positive inter-sampleperiods.

With w defined in (4.228) as the measurement error caused by data sam-pling, the control law (4.227) can be rewritten as (4.229). By substituting(4.229) into (4.255), we have

x(t) = f(x(t), v(x(t) + w(t)), d(t))

:= f(x(t), x(t) + w(t), d(t)). (4.256)

Corresponding to Assumption 4.8 for the disturbance-free case, we makethe following assumption on system (4.256).

Assumption 4.9 System (4.256) is ISS with w and d as the inputs, that is,there exist β ∈ KL and γ, γd ∈ K such that for any initial state x(0) and anypiecewise continuous, bounded w and d, it holds that

|x(t)| ≤ max

β(|x(0)|, t), γ(‖w‖∞), γd(‖d‖∞)

(4.257)

for all t ≥ 0.

Under Assumption 4.9, if the event trigger is still capable of guaranteeing(4.239) with ρ ∈ K such that ρ γ < Id, then by using the robust stabilityproperty of ISS, we can prove that

|x(t)| ≤ max

β(|x(0)|, t), γd(‖d‖∞)

(4.258)

with β ∈ KL and γd ∈ K. As x converges to the origin, the upper boundof |w(t)| = |x(tk) − x(t)| converges to zero according to (4.239). However,due to the presence of the external disturbance d, the system dynamicsf(x(t), v(x(t) + w(t)), d(t)) may not converge to zero as x converges to theorigin. This means that the inter-sample period tk+1 − tk could be arbitrarilysmall.


EventTriggered Sampling with ǫ Modification

Inspired by the recent result [52], we modify the event trigger (4.236) byreplacing H defined in (4.237) with

H(x, x′) = maxρ(|x|), ǫ − |x− x′|, (4.259)

where ρ is a class K∞ function satisfying ρ γ < Id and constant ǫ > 0. Inthis case, corresponding to (4.239), we have

|x(t) − x(tk)| < maxρ(|x(t)|), ǫ (4.260)

for t ∈ [tk, tk+1), k ∈ Z+. With robust stability property of ISS, it holds that

|x(t)| ≤ max

β(|x(0)|, t), γ(ǫ), γd(‖d‖∞)

(4.261)

with β ∈ KL and γ, γd ∈ K. It should be noted that, with ǫ > 0, tk+1− tk > 0is guaranteed for all k ∈ S and function ρ−1 is no longer required to beLipschitz on compact sets. This result is summarized by Theorem 4.5 withoutproof.

Theorem 4.5 Consider the event-triggered control system (4.256) with lo-cally Lipschitz f and w defined in (4.228). If Assumption 4.9 is satisfied, withthe sampling time instants triggered by (4.236) with H defined in (4.259), forany specific initial state x(0), the system state x(t) satisfies (4.261) for all

t ≥ 0, with β ∈ KL and γ, γd ∈ K, and the inter-sample periods are lowerbounded by a positive constant.

For such an event-triggered control system, even if d ≡ 0, only practicalconvergence can be guaranteed, that is, x(t) can only be guaranteed to con-verge to within a neighborhood of the origin defined by |x| ≤ γ(ǫ). In the nextsubsection, we present a self-triggered sampling mechanism to overcome thisobstacle, under the assumption of an a priori known upper bound of ‖d‖∞.

SelfTriggered Sampling

We show that if an upper bound of ‖d‖∞ is known a priori, then we candesign a self-triggered sampling mechanism such that x(t) is practically steeredto within a neighborhood of the origin with size depending solely on ‖d‖∞.Moreover, if d(t) converges to zero, then x(t) asymptotically converges to theorigin.

Assumption 4.10 There is a known constant Bd ≥ 0 such that

‖d‖∞ ≤ Bd. (4.262)

Lemma 4.4 on a property of locally Lipschitz functions is used in the fol-lowing self-triggered control design.


Lemma 4.4 For any locally Lipschitz function h : Rn1×Rn2×· · ·×Rnm → Rp

satisfying h(0, . . . , 0) = 0 and any ϕ1, . . . , ϕm ∈ K∞ with ϕ−11 , . . . , ϕ−1

m beingLipschitz on compact sets, there exists a continuous, positive, and nondecreas-ing function Lh : R+ → R+ such that

|h(z1, . . . , zm)| ≤ Lh

(

maxi=1,...,m

|zi|)

maxi=1,...,m

ϕi(|zi|) (4.263)

for all z, where z = [zT1 , . . . , zTm]T .

Proof. For a locally Lipschitz h satisfying h(0, . . . , 0) = 0, one can always finda continuous, positive, and nondecreasing function Lh0 : R+ → R+ such that

|h(z1, . . . , zm)| ≤ Lh0

(

maxi=1,...,m

|zi|)

maxi=1,...,m

|zi| (4.264)

for all z.Define

ϕ(s) = maxi=1,...,m

ϕ−1i (s) (4.265)

for s ∈ R+. Then, ϕ ∈ K∞. Since ϕ−11 , . . . , ϕ−1

m are Lipschitz on compact sets,ϕ is Lipschitz on compact sets.

From the definition, one has

ϕ

(

maxi=1,...,m

ϕi(|zi|))

= maxi=1,...,m

ϕ ϕi(|zi|)

≥ maxi=1,...,m

ϕ−1i ϕi(|zi|)

= maxi=1,...,m

|zi|. (4.266)

With the ϕ which is Lipschitz on compact sets, there exists a continuous,positive, and nondecreasing function Lϕ : R+ → R+ such that

ϕ

(

maxi=1,...,m

ϕi(|zi|))

≤ Lϕ

(

maxi=1,...,m

ϕi(|zi|))

maxi=1,...,m

ϕi(|zi|). (4.267)

Lemma 4.4 is proved by substituting (4.266) and (4.267) into (4.264), anddefining a continuous, positive, and nondecreasing Lh such that

Lh

(

maxi=1,...,m

|zi|)

≥ Lh0

(

maxi=1,...,m

|zi|)

Lϕ

(

maxi=1,...,m

ϕi(|zi|))

(4.268)

for all z. ♦Assume that f is locally Lipschitz and f(0, 0, 0) = 0. Then, with Lemma

4.4, for any specific χ, χd ∈ K∞ with χ−1,(

χd)−1

being Lipschitz on compactsets, one can find a continuous, positive and nondecreasing Lf such that

|f(x + w, x, d)| ≤ Lf (max |x|, |w|, |d|)max

χ(|x|), |w|, χd(|d|)

(4.269)


for all x,w, d.

By choosing χ, χd ∈ K∞ with χ−1,(

χd)−1

being locally Lipchitz, the self-triggered sampling mechanism is designed as

tk+1 = tk +1

Lf (max χ(|x(tk)|), χd(Bd)), (4.270)

where χ(s) = maxχ(s), s and χd(s) = maxχd(s), s for s ∈ R+.Theorem 4.6 provides the main result of this subsection.

Theorem 4.6 Consider the event-triggered control system (4.256) with lo-cally Lipschitz f satisfying f(0, 0, 0) = 0 and w defined in (4.228). If As-sumption 4.9 holds with a γ being Lipschitz on compact sets, then one canfind a ρ ∈ K∞ such that

• ρ satisfies

ρ γ < Id, (4.271)

and• ρ−1 is Lipschitz on compact sets.

Moreover, under Assumption 4.10, by choosing χ = ρ(Id+ρ)−1 and χd ∈ K∞with

(

χd)−1

being Lipschitz on compact sets for the self-triggered samplingmechanism (4.270), for any specific initial state x(0), the system state x(t)satisfies

|x(t)| ≤ maxβ(|x(0)|, t), γ χd(‖d‖∞), γd(‖d‖∞) (4.272)

for all t ≥ 0, with β ∈ KL and γ, γd ∈ K, and the inter-sample periods arelower bounded by a positive constant.

Proof. Note that χ = ρ (Id+ρ)−1 implies χ−1 = Id+ρ−1. If ρ−1 is Lipschitz

on compact sets, then χ−1 is Lipschitz on compact sets. Also note that(

χd)−1

is chosen to be Lipschitz on compact sets.For the locally Lipschitz f satisfying f(0, 0, 0) = 0, by using Lemma 4.4,

one can find a continuous, positive, and nondecreasing Lf such that (4.269)holds.

We first prove that the self-triggered sampling mechanism achieves that

|x(t) − x(tk)| ≤ maxχ(|x(tk)|), χd(‖d‖∞) (4.273)

for t ∈ [tk, tk+1).By taking the integration of both the sides of (4.256), one has

x(t) − x(tk) =

∫ t

tk

f(x(tk) + w(τ), x(tk), d(τ))dτ, (4.274)


and thus,

|x(t)− x(tk)| ≤∫ t

tk

|f(x(tk) + w(τ), x(tk), d(τ))|dτ. (4.275)

Denote Ω(x(tk), ‖d‖∞) as the region of x such that |x − x(tk)| ≤maxχ(|x(tk)|), χd(‖d‖∞). Then, the minimum time needed for x(t) to gooutside the region Ω(x(tk), ‖d‖∞) can be estimated by

maxχ(|x(tk)|), χd(‖d‖∞)C(x(tk), ‖d‖∞)

≥ maxχ(|x(tk)|), χd(‖d‖∞)Lf (maxχ(|x(tk)|), χd(‖d‖∞))maxχ(|x(tk)|), χd(‖d‖∞)

=1

Lf (maxχ(|x(tk)|), χd(‖d‖∞))

≥ 1

Lf (maxχ(|x(tk)|), χd(Bd)), (4.276)

where χ(s) = maxχ(s), s and χd(s) = maxχd(s), s for s ∈ R+, and

C(x(tk), ‖d‖∞) = max

|f(x(tk) + w, x(tk), d)| :|w| ≤ maxχ(|x(tk)|), χd(‖d‖∞),|d| ≤ ‖d‖∞

. (4.277)

Thus, the proposed self-triggered sampling mechanism (4.270) guarantees(4.273).

With Lemma C.6, (4.273) implies

|w(t)| = |x(t) − x(tk)| ≤ maxρ(|x(t)|), χd(‖d‖∞) (4.278)

for t ∈ [tk, tk+1). Note that ρ γ < Id. Using the robust stability property ofISS and employing a similar induction procedure as for the proof of Theorem4.4, one can prove that (4.272) holds for all t ≥ 0. ♦

With the asymptotic gain property of ISS, if d(t) converges to zero, thenx(t) asymptotically converges to the origin.

4.4.3 EVENTTRIGGERED CONTROL DESIGN FOR NONLINEAR UNCERTAINSYSTEMS

The measurement feedback control design in Section 4.1 provides a solutionto robust control of nonlinear systems in the presence of measurement errors.Since the key of event-triggered control is to deal with the measurement er-ror caused by data sampling, this subsection presents a design to fulfill therequirements for event-triggered control for nonlinear uncertain systems by re-fining the design in Section 4.1. In this subsection, we consider the case where


the systems are free of external disturbances. The proposed design can be di-rectly extended to fulfill the requirement for self-triggered control of systemsunder external disturbances as discussed in Subsection 4.4.2.

Consider the following nonlinear system in the strict-feedback form:

xi(t) = xi+1(t) + ∆i(xi(t)), i = 1, . . . , n− 1 (4.279)

xn(t) = u(t) + ∆n(xn(t)), (4.280)

where [x1, . . . , xn]T := x ∈ Rn is the state, u ∈ R is the control input, and

∆i’s for i = 1, . . . , n with xi = [x1, . . . , xi]T are unknown, locally Lipschitz

functions.

Assumption 4.11 For each i = 1, . . . , n, there exists a known ψ∆i∈ K∞

which is Lipschitz on compact sets such that for all xi,

|∆i(xi)| ≤ ψ∆i(|xi|). (4.281)

Define w(t) as in (4.228) as the measurement error caused by data sampling.For convenience of notation, denote w = [w1, . . . , wn]

T . In the design, wefirst assume the boundedness of w, i.e., the existence of ‖w‖∞, denoted byw∞. Equivalently, ‖wi‖∞, denoted by w∞

i , exists for i = 1, . . . , n. Denotew∞i = [w∞

1 , . . . , w∞i ]T .

Following the approach in Section 4.1, one may design a control law suchthat the closed-loop system is robust to measurement error. To clarify theinfluence of data sampling on the closed-loop system, we slightly modify thedesign procedure in Section 4.1 as follows.

The basic idea of the control design in this subsection is still to transformthe closed-loop system into an interconnection of ISS subsystems, and usethe cyclic-small-gain theorem to guarantee the ISS of the closed-loop system.Specifically, the state variables of the ISS subsystems are defined as

e1 = x1 (4.282)

ei = ~d(xi, Si−1(xi−1, w∞i )), i = 2, . . . , n, (4.283)

and the control law is designed such that

u ∈ Sn(xn, w∞n ), (4.284)

where ~d(z,Ω) := z− argminz′∈Ω

|z− z′| for any z ∈ R and any compact Ω ⊂ R,

and for each i = 1, . . . , n, Si : Ri × Ri R is an appropriately designedset-valued map to cover the influence of the measurement errors.

Moreover, the set-valued maps are recursively defined as

S1(x1, w∞1 ) = κ1(x1 + a1w

∞1 ) : |a1| ≤ 1 (4.285)

Si(xi, w∞i ) = κi(xi + aiw

∞i − pi−1) : |ai| ≤ 1,

pi−1 ∈ Si−1(xi−1, w∞i−1)

i = 2, . . . , n, (4.286)


where the κi’s for i = 1, . . . , n are continuously differentiable, odd, and strictlydecreasing functions.

It can be proved that for i = 1, . . . , n, each ei-subsystem can be representedby a differential inclusion as

ei ∈ Si(xi, w∞i ) + Φi(xi, w

∞i , ei+1), (4.287)

where Φi satisfies

|Φi(xi, w∞i , ei+1)| ≤ ψΦi

(|[xTi , w∞Ti , ei+1]

T |) (4.288)

for all xi, w∞i , ei+1, with ψΦi

∈ K∞ being Lipschitz on compact sets.As shown in Section 4.1, by designing the κi’s, the ei-subsystems can be

designed to be ISS with ISS gains satisfying the cyclic-small-gain conditionfor the ISS of the closed-loop system with e as the state. Based on this design,we show that the closed-loop system with x as the state and w as the inputis ISS with an ISS gain being Lipschitz on compact sets.

With Assumption 4.12 satisfied, as discussed in Section 4.1, by choosingthe γekei ’s for i = 1, . . . , n, k = 1, . . . , i − 1, i + 1 to be Lipschitz on compactsets and choosing γwk

ei for k = 1, . . . , i− 1 to be in the form of αV γwkei with

γwkei being Lipschitz on compact sets, (4.74) can be realized with ǫi = 0 for

each ei-subsystem.Then, one may construct an ISS-Lyapunov function in the form of (4.77)

with σi’s for i = 1, . . . , n being Lipschitz on compact sets, and represent theinfluence of the measurement error caused by data sampling with

θ = maxi=1,...,n

σi

(

maxk=1,...,i

γwkei (wk)

)

. (4.289)

With such treatment, property (4.79) still holds, which means

V (e(t)) ≤ maxβ(V (e(0)), t), γ(w∞), (4.290)

where β is a class KL function and

γ(s) := maxi=1,...,n

σi

(

maxk=1,...,i

γwkei (s)

)

(4.291)

for s ∈ R+. Thus,

|e(t)| ≤ max

α−1V β (αV (|e(0)|), t) , α−1

V γ(w∞)

. (4.292)

According to the definitions of e1, . . . , en, it can be observed that the in-crease of w∞

i ’s for i = 1, . . . , n leads to the decrease of |ei|’s for i = 2, . . . , n.Notice that, in the case of w∞

i = 0 for i = 1, . . . , n, ei = xi − κi−1(ei−1) fori = 2, . . . , n. Thus, if w∞

i ≥ 0 for i = 1, . . . , n, then

|ei| ≤ |xi − κi−1(ei−1)| ≤ |xi|+ |κi−1(ei−1)|. (4.293)


Then, one can find an αx ∈ K∞ such that

|e| ≤ αx(|x|). (4.294)

Also, from the definitions of e1, . . . , en, we have

|x1| = |e1|, (4.295)

|xi| ≤ max |maxSi−1(xi−1, w∞i ) + ei|, |minSi−1(xi−1, w

∞i )− ei| ,

i = 2, . . . , n. (4.296)

Due to the continuous differentiability of the κi’s used for the definitionof the set-valued maps Si’s, there exist functions αe, αw ∈ K∞ which areLipschitz on compact sets such that

|x| ≤ maxαe(|e|), αw(|w∞|). (4.297)

By substituting (4.294) and (4.297) into (4.292), one achieves

|x(t)| ≤ max

αe α−1V β (αV αx(|x(0)|), t) , αe α−1

V γ(w∞), αw(w∞)

:= maxβ(|x(0)|, t), γ(w∞). (4.298)

It can be verified that β ∈ KL and γ ∈ K.Note that the design of the control law does not depend on w∞

1 , . . . , w∞n .

The control law guarantees (4.298) for all w∞. This proves the ISS of theclosed-loop system with x as the state and w as the input.

Since αe and αw are Lipschitz on compact sets, one can prove that γ isLipschitz on compact sets by showing that α−1

v γ is Lipschitz on compactsets. This, according to the definition of γ, can be proved by proving thatα−1V σi γwk

ei is Lipschitz on compact sets. Recall that each γwkei for k =

1, . . . , i− 1 is chosen to be in the form of αV γwkei with γwk

ei being Lipschitzon compact sets and each γwi

ei is in the form of αV (s/ci); check (4.75). Then,

α−1V σi γwk

ei is Lipschitz on compact sets as α−1V σi αV is Lipschitz on

compact sets.The design result in this subsection is summarized in Theorem 4.7.

Theorem 4.7 Consider nonlinear uncertain system (4.279)–(4.280) with As-sumption 4.12 satisfied. Then, one can design an event-triggered controllerwith the event trigger in the form of (4.236) with H defined by (4.237) andcontrol law in the form of (4.61)–(4.63) such that infinitely fast sampling isavoided and the state x of the system is bounded and converges to the originasymptotically.

4.5 SYNCHRONIZATION UNDER SENSOR NOISE

The measurement feedback control design presented in Section 4.1 can alsobe extended for synchronization control of nonlinear uncertain systems in thepresence of sensor noise.


We consider nonlinear systems in the strict-feedback form:

xij = xi(j+1) + fj(Xij), j = 1, . . . , n− 1 (4.299)

xin = ui + fn(Xin) (4.300)

xmij = xij + dij , (4.301)

where, for i = 1, 2, [xi1, . . . , xin]T := xi ∈ Rn and ui ∈ R are the state

and the control input, respectively, xmij is the measurement of xij , dij is the

measurement disturbance, Xij = [xi1, . . . , xij ]T , fj is an unknown locally

Lipschitz function, and xi1 is referred to as the output of the xi-system.In absence of disturbances, accurate synchronization, i.e., limt→∞(x1(t)−

x2(t)) = 0 for any initial state x1(0), x2(0), is usually expected. However,due to the sensor noise, perfect synchronization would not be realizable, andpartial synchronization in the sense of ISS is practically meaningful.

Assumptions 4.12 and 4.13 are made on system (4.299)–(4.301).

Assumption 4.12 For j = 1, . . . , n, each fj satisfies

|fj(X1j)− fj(X2j)| ≤ ϕj (|X1j |, |X2j |)ψj(|X1j −X2j |) (4.302)

for X1j, X2j ∈ Rj, where ψj ∈ K∞ and ϕj : R+ × R+ → R+ is positive and

nondecreasing with respect to |X1j | and |X2j |.

Assumption 4.13 For i = 1, 2, j = 1, . . . , n, each dij satisfies

|dij(t)| ≤ dij (4.303)

for t ≥ 0, with dij ≥ 0 being a constant.

It should be noted that no global Lipschitz condition is assumed on thedynamics of the nonlinear systems.

For convenience of notation, denote Xmij = [xmi1 , . . . , x

mij ]T for i = 1, 2,

j = 1, . . . , n.


For j = 1, . . . , n, define zj = x1j − x2j as the synchronization error betweenthe two systems, and define φj(Xj) = fj(X1j) − fj(X2j) as the difference ofthe system dynamics. Then, each zj-subsystem can be represented by

zj = zj+1 + φj(X1j , X2j). (4.304)

Specifically, zn+1 = x1(n+1) − x2(n+1) = u1 − u2. Define dj = d1j + d2j forj = 1, . . . , n. Then, Assumption 4.13 implies |d1j − d2j | ≤ dj . Due to thesensor noise, zmj = xm1j − xm2j instead of zj is available for feedback. Then,

|zmj − zj | ≤ dj .


Define Zj = [z1, . . . , zj]T . Assumption 4.12 implies, for each j = 1, . . . , n,

|φj(X1j , X2j)| ≤ ϕj (|X1j |, |X2j |)ψj(|Zj |). (4.305)

The zj-subsystems (j = 1, . . . , n) describe the dynamical behavior of thesynchronization error system. By considering X1j and X2j as external inputs,the Zn-system is in the lower-triangular form. Following a similar idea as inSection 4.1, we construct a new [e1, . . . , en]

T -system composed of ISS subsys-tems by recursively designing a nonlinear control law for the Zn-system.

For convenience of notation, define Dij = [di1, . . . , dij ]T and Dj =

[d1, . . . , dj ]T for i = 1, 2, j = 1, . . . , n, and denote Ej = [e1, . . . , ej ]

T forj = 1, . . . , n.


Define e1 = z1. Then, the e1-subsystem can be written as

e1 = z2 + φ1(X11, X21). (4.306)

Define a set-valued map

S1(X11, X21) =

κ1(X11 + δ11, X21 + δ21, z1 + δ1) :

− D11 ≤ δ11 ≤ D11,−D21 ≤ δ21 ≤ D21, |δ1| ≤ d1

,

(4.307)

where κ1 is in the form of

κ1(a1, a2, a3) = µ1(|a1|, |a2|)θ1(a3) (4.308)

with µ1 : R+×R+ → R+ being continuously differentiable on (0,∞)× (0,∞),positive and nondecreasing with respect to the two variables, and θ1 : R → R

being continuously differentiable, odd, strictly decreasing, and radially un-bounded. Both µ1 and θ1 are defined later.

Recall the definition of ~d in (4.51). Define

e2 = ~d(z2, S1(X11, X21)), (4.309)

and rewrite

e1 = z2 − e2 + φ1(X11, X21) + e2, (4.310)

where z2 − e2 ∈ S1(X11, X21) based on (4.309).


Recursive Step: The ejsubsystems

For convenience, S0(X0) := 0. For each k = 1, . . . , j − 1, a set-valued mapSk is defined as

Sk(X1k, X2k) =

κk(X1k + δ1k, X2k + δ2k, zk − pk−1 + δk) :

− D1k ≤ δ1k ≤ D1k,−D2k ≤ δ2k ≤ D2k,

|δk| ≤ dk, pk−1 ∈ Sk−1(X1(k−1), X2(k−1))

, (4.311)

where κk is in the form of

κk(a1, a2, a3) = µk(|a1|, |a2|)θk(a3) (4.312)

with µk : R+×R+ → R+ being continuously differentiable on (0,∞)× (0,∞),positive and nondecreasing with respect to the two variables, and θk : R → R

being continuously differentiable, odd, strictly decreasing, and radially un-bounded; and ek+1 is defined as

ek+1 = ~d(zk+1, Sk(X1k, X2k)). (4.313)

Lemma 4.5 Consider the zj-subsystems defined in (4.304). If Sk and ek+1

are defined as in (4.311) and (4.313) for each k = 1, . . . , j − 1, then, whenej 6= 0, the ej-subsystem can be represented by

ej ∈

zj+1 + φ∗j : φ∗j ∈ Φ∗

j (X1j , X2j)

, (4.314)

where Φ∗j is a convex, compact, and upper semi-continuous set-valued map,

and for any φ∗j ∈ Φ∗j (X1j , X2j), it holds that

|φ∗j | ≤ ϕ∗j (|X1j |, |X2j |)ψ∗

j (|[ETj , DTj−1]

T |) (4.315)

with ϕ∗j : R+×R+ → R+ is positive and nondecreasing with respect to the two

variables, and ψ∗j ∈ K∞.

The proof of Lemma 4.5 is in Appendix E.3.Define a set-valued map

Sj(X1j , X2j) =

κj(X1j + δ1j , X2j + δ2j , zj − pj−1 + δj) :

− D1j ≤ δ1j ≤ D1j,−D2j ≤ δ2j ≤ D2j ,

|δj | ≤ dj , pj−1 ∈ Sj−1(X1(j−1), X2(j−1))

, (4.316)

where κk is in the form of

κj(a1, a2, a3) = µj(|a1|, |a2|)θj(a3) (4.317)


with µj : R+×R+ → R+ being continuously differentiable on (0,∞)× (0,∞),positive and nondecreasing with respect to the two variables, and θj : R → R

being continuously differentiable, odd, strictly decreasing, and radially un-bounded. Both µj and θj are defined later.

Define

ej+1 = ~d(zj+1, Sj(X1j , X2j)). (4.318)

Clearly, zj+1−ei+1 ∈ Sj(X1j , X2j). Then, the ej-subsystem can be representedby the differential inclusion:

ej ∈ zj+1 − ej+1 + φ∗j + ej+1 : φ∗j ∈ Φ∗j (X1j , X2j)

zj+1 − ej+1 ∈ Sj(X1j , X2j):= Fj(X1j , X2j , ej+1). (4.319)

It can be observed that S1(X11, X21) defined in (4.307) is in the form of(4.316) with S0(X10, X20) = 0, and the e1-subsystem defined in (4.310) isin the form of (4.319) with Φ∗

1(X11, X21) = φ1(X11, X21).With Lemma 4.5, the [z1, . . . , zn]

T -system has been transformed into the[e1, . . . , en]

T -system with each ej-subsystem (j = 1, . . . , n) in the form of(4.319).

4.5.2 ISS OF THE TRANSFORMED SUBSYSTEMS

Define αV (s) = s2/2 for s ∈ R+. For j = 1, . . . , n, each ej-subsystem isdesigned to be ISS with ISS-Lyapunov function

Vj(ej) = αV (|ej |). (4.320)

For convenience of notation, define Vn+1(en+1) = αV (|en+1|). In the followingdiscussions, we sometimes simply use Vj instead of Vj(ej).

Lemma 4.6 Consider the ej-subsystem defined in (4.319) and the set-valuedmap Sj defined in (4.316). For any specified ǫj > 0, ℓj > 0, 0 < cj < 1,γekej , γ

dkej ∈ K∞ (k = 1, . . . , j − 1), and γ

ej+1ej ∈ K∞, one can find a µj : R+ ×

R+ → R+ being continuously differentiable on (0,∞) × (0,∞), positive andnondecreasing, and a θj : R → R being continuously differentiable, odd, strictlydecreasing, and radially unbounded such that with zj+1−ej+1 ∈ Sj(X1j , X2j),it holds that

Vj ≥ maxk=1,...,j−1

γekej (Vk), γej+1ej (Vj+1), γ

dkej (dk), γ

djej (dj), ǫj

⇒ maxfj∈Fj(X1j ,X2j ,ej+1)

∇Vj(ej)fj ≤ −ℓjVj(ej), (4.321)

where

γdjej (s) = αV

(

s

cj

)

(4.322)

for s ∈ R+.


The proof of Lemma 4.6 is in Appendix E.4.

4.5.3 REALIZABLE CONTROL LAWS

The true control inputs u1 and u2 occur in the form of u1 − u2 in the en-subsystem. We set en+1 = 0 and Vn+1 = 0, and choose the following controllaw such that u1 − u2 ∈ Sn(X1n, X2n):

p∗1 = µ1(|Xm11|, |Xm

21|)θ1(zm1 ) (4.323)

p∗j = µj(|Xm1j |, |Xm

2j |)θj(zmj − p∗j−1) (4.324)

u1 − u2 = µn(|Xm1n|, |Xm

2n|)θn(zmn − p∗n−1). (4.325)


p∗1 ∈ S1(X11, X21) ⇒ · · · ⇒ p∗j ∈ Sj(X1j , X2j) ⇒ · · ·⇒ u1 − u2 ∈ Sn(X1n, X2n). (4.326)


Define e = [e1, . . . , en]T . With Lemma 4.6, the closed-loop system has been

transformed into a network of ISS subsystems. Moreover, the gains can bedesigned to satisfy the cyclic-small-gain condition. Now consult the similardesign in Section 4.1 for the cyclic-small-gain condition for the synchroniza-tion control problem. An ISS-Lyapunov function for the e-system can be con-structed as

V (e) = maxj=1,...,n

σj(Vj(ej)), (4.327)

where σ1(s) = s, σj(s) = γe2e1 · · · γejej−1 (s) (j = 2, . . . , n) for s ∈ R+, where

the γ(·)(·) ’s areK∞ functions being continuously differentiable on (0,∞), slightly

larger than the corresponding γ(·)(·) ’s, and still satisfying the cyclic-small-gain

condition.The influence of dj ’s and ǫj ’s can be represented as

θ = maxj=1,...,n

σj

(

maxk=1,...,j

γdkej (dk), ǫj

)

. (4.328)

Using the Lyapunov-based ISS cyclic-small-gain theorem, we have that

V (e) ≥ θ ⇒ maxf∈F (X1n,X2n,e)

∇V (e)f ≤ −α(V (e)) (4.329)

holds wherever ∇V exists, where α is a continuous and positive definite func-tion, and

F (X1n, X2n, e) := [F1(X11, X21, e2), . . . , Fn(X1n, X2n, en+1)]T (4.330)


with en+1 = 0.By choosing γdkej (j = 2, . . . , n, k = 1, . . . , j−1), ǫj (j = 1, . . . , n) and γ

ej+1ej

(j = 1, . . . , n− 1) (and thus σj (j = 2, . . . , n)) small enough, we can achieve

θ = γd1e1 (d1) = αV

(

d1c1

)

. (4.331)

Then, from (4.329), we have

V (e) ≥ αV

(

d1c1

)

⇒ maxf∈F (X1n,X2n,e)

∇V (e)f ≤ −α(V (e)) (4.332)

holds wherever ∇V exists.Property (4.332) implies that V (e) ultimately converges to within the re-

gion V (e) ≤ αV (d/c1). Using the definitions of e1, V1(e1), V (e) (see (4.320)and (4.327)), we have |z1| = |e1| = α−1

V (V1(e1)) ≤ α−1V (V (e)), which implies

that z1 ultimately converges to within the region |z1| ≤ d1/c1. Notice thatc1 can be chosen arbitrarily from the interval (0, 1) in the recursive designprocedure. Recall that z1 = x11 − x21. By choosing c1 to be arbitrarily closeto one, x11−x21 can be steered arbitrarily close to the region |x11−x21| ≤ d1.

The main result of this section is summarized in the following theorem.

Theorem 4.8 Consider the two uncertain nonlinear systems defined in(4.299)–(4.301). Under Assumptions 4.12 and 4.13, one can design synchro-nization controllers in the form of (4.323)–(4.325) such that synchronizationerrors zj = x1j − x2j for j = 1, . . . , n are bounded and z1 = x11 − x21 can besteered arbitrarily close to the region |x11 − x21| ≤ d1.

In this section, we only consider the synchronization problem of two non-linear systems. It is certainly of interest and deserves more effort to generalizethe design based on the cyclic-small-gain theorem to the synchronization prob-lem of more than two systems. Notice that the agreement problem, which isclosely related to the synchronization problem, is studied in Chapter 6.

4.6 APPLICATION: ROBUST ADAPTIVE CONTROL UNDER SENSORNOISE

This chapter has mainly considered the measurement feedback control prob-lem of two classes of nonlinear systems: the strict-feedback system and theoutput-feedback system. In these designs, the dynamics of the systems areassumed to be bounded by class K∞ functions; see Assumptions 4.1, 4.4, 4.5,and 4.12. This means that the origin is an equilibrium of the considered sys-tems if the inputs, including external disturbances and control inputs, arezero.

This section shows that if the origin is not an equilibrium of the input-free systems, the measurement feedback control problem can still be solved


by a robust adaptive controller. A practical example is the fan speed controlproblem; see [66, 124].

Instead of Assumption 4.1, for strict-feedback system (4.33)–(4.35), wemake the following assumption.

Assumption 4.14 For each i = 1, . . . , n, there exists a known ψ∆i∈ K∞

such that for all xi, d,

|∆i(xi, d)−∆i(0, 0)| ≤ ψ∆i(|[xTi , dT ]T |). (4.333)

It is not restrictive to assume a known ψ∆i. But the uncertainty of ∆i

means uncertainty of ∆i(0, 0). By finding a known constant ∆0i > 0 such

that |∆i(0, 0)| ≤ ∆0i and considering ∆i(0, 0) as an external disturbance, the

design proposed in Section 4.1 still works. However, since ∆i(0, 0) 6= 0, perfectconvergence cannot be guaranteed even if the system is disturbance-free.

It is intuitive to introduce some adaptive mechanism to deal with the con-stant uncertainty ∆i(0, 0). We propose an ISS-induced design for robust adap-tive control of system (4.33)–(4.35) in the presence of sensor noise. For con-venience of notation, denote ci = ∆i(0, 0).

First consider the x1-subsystem. Introduce new variables x∗2 and v1 suchthat x∗2 = v1. Define e1 = x1, e2 = x2 − x∗2, and x

∗2 = x∗2 + c1.

Then, we have

e1 = x∗2 +∆1(e1, d) + e2 (4.334)

˙x∗2 = v1. (4.335)

It should be noted that x∗2 is unknown. By considering d and e2 as externalinputs, and e1 as the output, the (e1, x

∗2)-system is in the output-feedback

form. According to the method in Section 4.3, we employ a reduced-orderobserver

ξ1 = v1 − L1(ξ1 + L1(e1 + w1)) (4.336)

to estimate x∗2 − L1e1. Here, the available e1 + w1 = xm1 is used. Defineζ1 = x∗2 − L1e1 − ξ1 as the estimation error. Then, direct calculation yields:

ζ1 = −L1ζ1 − L1(e2 +∆1(e1, d)− L1w1), (4.337)

which is ISS with e1, e2, d, w1 as the inputs.According to the definitions above, x∗2 = ζ1 + L1e1 + ξ1. Consider the

(e1, ξ1)-system

e1 = ξ1 + ζ1 + L1e1 +∆1(e1, d) + e2 (4.338)

ξ1 = v1 − L1(ξ1 + L1(e1 + w1)), (4.339)

which is in the strict-feedback form. By using the method in Section 4.1, wedesign a control law v1 := v1(e1 + w1, ξ1) such that the (e1, ξ1)-system is ISS


with ζ1, d, e2, w1 as the inputs. Moreover, the ISS gains from ζ1, d, e2 can bedesigned to be arbitrarily small.

Define e2 = x2 − x∗2. Then,

e2 = x2 − x∗2= x3 +∆2(x2, d) + c2 − v1(x

m1 , ξ1)

:= x3 +∆′2(e1, e2, ζ1, d, w1) + c′2, (4.340)

which is in the form of the e1-subsystem (4.334).Recursive design can be performed until the true control input u occurs.

Then, the closed-loop system is transformed into a network of ISS subsystems.The ISS gains can be appropriately designed to satisfy the cyclic-small-gaincondition. A detailed proof is left to interested readers. It should be noted thatthis design is still valid for strict-feedback systems with ISS inverse dynamics(4.82)–(4.85).

Compared with strict-feedback systems, output-feedback systems in theform of (4.204)–(4.208) with unknown ∆i(0, 0, 0) can be handled more easily.

Define

x′1 = x1 (4.341)

x′i+1 = xi+1 +∆i(0, 0, 0), for i = 1, . . . , n− 1 (4.342)

x′n+1 = u (4.343)

and introduce a dynamic compensator

u = v. (4.344)

Then, system (4.204)–(4.208) can be transformed into

z = ∆0(z, y, w) (4.345)

x′i = x′i+1 +∆i(y, z, w)−∆i(0, 0, 0), i = 1, . . . , n (4.346)

x′n+1 = v (4.347)

y = x′1 (4.348)

ym = y + d, (4.349)

the measurement feedback control problem of which can be readily solvedwith the design in Section 4.3.

4.7 NOTES

Despite its importance, the robust control of nonlinear systems in the pres-ence of sensor noise has not received considerable attention in the presentliterature. Examples showing the difficulty of the problem can be found in[66, Chapter 6] and [64]. In [66], a controller is designed with set-valued mapsand “flattened” Lyapunov functions following the backstepping methodology


such that the control system is ISS with respect to the measurement distur-bances. However, in that result, the influence of the sensor noise grows withthe order of the system. Reference [125] studies nonlinear systems composedof two subsystems; one is ISS and the other one is input-to-state stabiliz-able with respect to the measurement disturbance. In [125], the ISS of thecontrol system is guaranteed by the gain assignment technique introduced in[130, 223, 123] and the nonlinear small-gain theorem proposed in [130, 126]. In[159], it is shown that, for general nonlinear control systems under persistentlyacting disturbances, the existence of smooth Lyapunov functions is equivalentto the existence of (possibly discontinuous) feedback stabilizers which are ro-bust with respect to small measurement errors and small additive externaldisturbances. Discontinuous controllers are developed in [159] for nonlinearsystems such that the closed-loop system is insensitive to small measurementerrors. With a refined gain assignment technique, this chapter has studied themeasurement feedback control problem for nonlinear uncertain systems.

Gain assignment is a vital tool for small-gain-based nonlinear control de-signs [130, 223, 123]. This chapter has employed modified gain assignmentmethods for robust control of nonlinear systems under sensor noise. Refer-ence [125, Proposition 4.1] presents a gain assignment technique to guaranteethe ISS of the control system with respect to the measurement disturbanceand the gain from the measurement disturbance to the corresponding outputis assigned to be of class K∞. Lemmas 4.1 and 4.6 in this chapter considermore general cases in which the control laws are considered as selections ofappropriately designed set-valued maps.

Significant contributions have been made to the development of decentral-ized control theory (see e.g., [239, 129, 237, 274, 279, 147, 118] and refer-ences therein). For large-scale systems, with small-gain methods, the basicidea is to design decentralized controllers to make the subsystems have de-sired gain properties. For example, in [94], the large-scale small-gain criteriain the finite-gain setting [207] were used in stability analysis of decentralizedadaptive control. ISS small-gain methods have played an important role indecentralized control of large-scale nonlinear systems [118]. In this chapter,we have proposed an ISS cyclic-small-gain design for decentralized nonlin-ear control in the presence of sensor noise. The disturbed output-feedbacknonlinear system defined in (4.141)–(4.146) is a measurement-disturbed ver-sion of the system considered in [129, 148] and exists in mechanical systems,e.g., the interconnected system of cart-inverted double pendulum [237]. Differ-ent from centralized systems, the input–output feedback linearization methodcannot be implemented in the decentralized system due to the dependenceof the ∆ij ’s on wi. In particular, the zi-subsystem is referred to as the non-linear zero-dynamics systems, which forms nonlinear dynamical interactionsbetween the subsystems of the decentralized system. The nonlinearities in(4.141)–(4.146) are simply assumed to be bounded by K∞ functions, whichrelaxes the polynomial-type growth conditions imposed in [237, 274, 279].


The decentralized reduced-order observer design in this chapter is motivatedby the designs in [222, 129]. The only difference is that the measurements ofthe outputs are subject to sensor noise, the impact of which on the controlperformance has been well handled with the new cyclic-small-gain design inthis chapter.

The impact of the observer-based design also extends to the estimator de-sign for dynamic state-feedback control and the case study of robust adaptivecontrol subject to sensor noise in this chapter. Notice that ISS has been usedas a powerful tool [115] for robust adaptive control of nonlinear systems withno sensor noise. By employing neural network approximation, small-gain over-comes a circularity issue in adaptive control for non-affine pure-feedback sys-tems in [270]. More discussions on robust control of nonlinear systems undersensor noise can be found in [187, 186, 177].

5 Quantized NonlinearControl

In modern automatic control systems, signals are usually quantized beforebeing transmitted via communication channels. Figure 5.1 shows the blockdiagram of a typical quantized control system.

controller

plant

quantizerquantizer

xq

uq

u

x

FIGURE 5.1 The block diagram of a quantized control system: u is the control

input computed by the controller, x is the state of the plant, and uq and xq are the

quantized signals of u and x, respectively.

A quantizer can be mathematically modeled as a discontinuous map froma continuous region to a discrete set of numbers. Two examples of commonlyconsidered quantizers are shown in Figure 5.2.

x

xq

0 x

xq

0

FIGURE 5.2 Two examples of quantizers.

The difference between the input and the output of a quantizer is called thequantization error. If the quantization errors are bounded, then by consideringthem as sensor noise, one may directly use the methods proposed in Chapter4 to design quantized controllers. Usually, however, the quantization errorscannot be guaranteed to be bounded. For the quantizers shown in Figure

143


5.2, the quantization errors go to infinity as the inputs of the quantizers goto infinity. With nontrivial modifications of the methods in Chapter 4, thischapter resolves the quantized control problems for nonlinear systems.

Based on ISS cyclic-small-gain methods, in Section 5.1, a sector boundapproach is developed for quantized control of nonlinear systems with staticquantization. Due to the finite word-length of digital devices, practical quan-tizers have finite quantization levels. Section 5.2 introduces a dynamic quanti-zation strategy such that the quantization levels can be dynamically adjustedduring the control procedure for semiglobal quantized stabilization. In Section5.3, the quantized output-feedback control problem is studied for a class ofnonlinear systems taking the generalized output-feedback form. Section 5.4gives some notes and references on quantized nonlinear control.

5.1 STATIC QUANTIZATION: A SECTOR BOUND APPROACH

This section considers the strict-feedback system as the plant:

xi = xi+1 +∆i(xi), 1 ≤ i ≤ n− 1 (5.1)

xn = u+∆n(xn) (5.2)

xqi = qi(xi), 1 ≤ i ≤ n, (5.3)

where x = [x1, . . . , xn]T ∈ Rn is the state, u ∈ R is the control input, xi =

[x1, . . . , xi]T , xqi is the quantization of xi, qi’s are state quantizers, each of

which is a map from R to some discrete set Ωqi , and ∆i’s are unknown,locally Lipschitz functions.

Assumption 5.1 For 1 ≤ i ≤ n, the map qi : R → Ωi of the i-th quantizeris piecewise constant, and there exist known constants 0 ≤ bi < 1 and ai ≥ 0such that for all xi ∈ R,

|qi(xi)− xi| ≤ bi|xi|+ (1 − bi)ai. (5.4)

One example of a quantizer which satisfies condition (5.4) is the truncatedlogarithmic quantizer defined as

qi(xi) =

(1+bi)k+1ai

(1−bi)k , if (1+bi)kai

(1−bi)k < xi ≤ (1+bi)k+1ai

(1−bi)k+1 , k ∈ Z+;

0, if 0 ≤ xi ≤ ai;

−qi(−xi), if xi < 0.

(5.5)

See also Figure 5.3. It should be noted that condition (5.4) can be satisfiedby more general quantizers as long as their maps are bounded by sectors withan offset.

We make the following assumption on the system dynamics.

Quantized Nonlinear Control 145

xi

qi(xi)

0

ai

1 + bi

1− bi

FIGURE 5.3 A truncated logarithmic quantizer: |qi(xi)−xi| ≤ bi|xi|+(1− bi)ai for

all xi ∈ R with ai ≥ 0 and 0 ≤ bi < 1.

Assumption 5.2 For each ∆i (1 ≤ i ≤ n), there exists a known ψ∆i∈ K∞

such that for all xi ∈ Ri,

|∆i(xi)| ≤ ψ∆i(|xi|). (5.6)

The objective of this section is to find, if possible, a static quantized state-feedback controller in the form of

u = u(xq) (5.7)

with xq := [xq1, . . . , xqn]T , for system (5.1)–(5.3) such that the closed-loop

signals are bounded and the state x1(t) converges to some neighborhood ofthe origin whose size depends on the quantization error near the origin.

To convey the basic approach to quantized nonlinear control, this sectiondoes not consider the influence of the external disturbances. However, thanksto the natural robustness of the cyclic-small-gain design, quantized control canstill be realized for the systems subject to dynamic uncertainty and externaldisturbance:

z = g(z, x1) (5.8)

xi = xi+1 +∆i(xi, d), 1 ≤ i ≤ n− 1 (5.9)

xn = u+∆n(xn, d) (5.10)

xqi = qi(xi), 1 ≤ i ≤ n, (5.11)

where d ∈ Rnd represents external disturbance inputs, z ∈ Rnz representsthe state of the inverse dynamics representing dynamic uncertainties and is


not measurable, and the other variables are defined the same as for system(5.1)–(5.3).

This section only considers the case where the state measurement is quan-tized. Actuator quantization (i.e., the control signal u is quantized beforeit is applied to the plant) is surely of interest. Based on the proposed de-sign, interested readers may study quantized nonlinear control with both statequantization and actuator quantization. It should be noted that the actua-tor quantization error depends on the magnitude of the control signal, and itcannot be trivially treated as an external disturbance.

By considering the influence of the quantization error as an uncertain term,the closed-loop quantized system can be represented with the block diagramshown in Figure 5.4, where the operator Λ satisfies condition (5.4). For linearsystems, it is standard to employ robust control designs to solve the problem.If the system is linear, or more generally, globally Lipschitz, then one mayconjecture that there exists some appropriate bi such that the closed-loopquantized system can be made robust to the quantization error. In the follow-ing discussions, global Lipschitz continuity is not assumed on the dynamics ofsystem (5.1)–(5.3).

Λ

controller plantu xxq

xq − x

FIGURE 5.4 Quantized control as a robust control problem.

Example 5.1 shows that the quantized control problem can be solved forfirst-order systems based on a modification of the gain assignment technique.

Example 5.1 Consider the system

η = κ+ φ(η) (5.12)

ηq = q(η), (5.13)

where η ∈ R is the state, κ ∈ R is the control input, q : R → R is the statequantizer, and φ : R → R is an uncertain, locally Lipschitz function. Assumethat there exists a known locally Lipschitz ψφ ∈ K∞ such that |φ(x)| ≤ ψφ(|x|)for all x ∈ R. Also assume the sector bound property on the quantizer, i.e.,|q(η) − η| ≤ b|η|+ (1− b)a with a ≥ 0 and 0 ≤ b < 1 being constants.


With Lemma C.8, for the locally Lipschitz ψφ ∈ K∞ and any specified b ≥ 0and 0 < c < 1, one can always find a continuous, positive, and nondecreasingν : R+ → R+ such that

(1− b)(1− c)ν((1 − b)(1− c)s)s ≥ 1

2s+ ψφ(s) (5.14)

for all s ∈ R+. The quantized control law is designed as

κ = κ(ηq) := −ν(|ηq|)ηq . (5.15)

For the closed-loop quantized system, we define a Lyapunov function can-didate as V (η) = η2/2. Consider the case of V (η) ≥ a2/2c2. In this case,|η| ≥ a/c. By also using |q(η) − η| ≤ b|η|+ (1− b)a, it can be verified that

sgn(ηq) = sgn(η), (5.16)

|ηq| ≥ (1− b)(1− c)|η|. (5.17)

Then,

∇V (η)(κ(ηq) + φ(η)) = η(−ν(|ηq|)ηq + φ(η))

≤ −ν(|ηq|)|ηq||η|+ |η||φ(η)|= −|η|(−ν(|ηq|)|ηq|+ |φ(η)|)

≤ −|η|(

−ν((1− b)(1− c)|η|)(1 − b)(1− c)|η|

+ ψφ(|η|))

≤ −1

2|η|2. (5.18)

That is,

V (η) ≥ a2

2c2⇒ ∇V (η)(κ+ φ(η)) ≤ −1

2|η|2, (5.19)

which means that η can be steered to within the region |η| ≤ a/c, where con-stant a represents the quantization error around the origin and parameter ccan be chosen to be arbitrarily close to one. Theoretically, if a = 0, then asymp-totic convergence can be achieved. Under mild conditions, such a design canalso guarantee robustness to external disturbances.

This example shows that the influence of the quantization error can beattenuated through a modified gain assignment design.

This section extends the design in Example 5.1 in a recursive manner tosolve the quantized control problem for system (5.1)–(5.3).

The main result of this section is summarized in Theorem 5.1.


Theorem 5.1 Consider system (5.1)–(5.3) under Assumptions 5.1 and 5.2.A quantized control law in the form of (5.7) can be designed such that theclosed-loop signals are bounded. Moreover, if a1 6= 0, then given any 0 < c1 <1, the state x1 can be steered to within the region |x1| ≤ a1/c1; if a1 = 0,then given an arbitrarily small δ > 0, the state x1 can be steered to within theregion |x1| ≤ δ.

The basic idea of designing the quantized control law is to transform the[x1, . . . , xn]

T -system into a new [e1, . . . , en]T -system through a recursive con-

trol design procedure. We employ set-valued maps to cover the discontinuitycaused by the quantizers. By appropriately choosing the set-valued maps, eachei-subsystem is designed to be ISS with the other states ej ’s (j 6= i) as theinputs. Then, the cyclic-small-gain theorem is used to guarantee the ISS of the[e1, . . . , en]

T -system. For convenience of discussions, denote ei = [e1, . . . , ei]T

and ai = [a1, . . . , ai]T .



Let e1 = x1. Rewrite the e1-subsystem as

e1 = x2 − e2 + φ∗1(x1, e2), (5.20)

where e2 is a new state variable to be defined later and φ∗1(x1, e2) := ∆1(e1)+e2. Under condition (5.6), one can find a ψφ∗

1∈ K∞ such that

|φ∗1(x1, e2)| ≤ ψφ∗

1(|e2|). (5.21)

Inspired by the set-valued map design for measurement feedback control,we use set-valued maps to cover the discontinuity of quantization. With (5.4)satisfied, define set-valued maps S1 and S1 as

S1(x1) =

κ1(x1 + d11) : |d11| ≤ b1|x1|+ (1− b1)a1

(5.22)

S1(x1) =

d12p1 : p1 ∈ S1(x1),1

1 + b2≤ d12 ≤ 1

1− b2

, (5.23)

where κ1 is a continuously differentiable, odd, strictly decreasing, and radiallyunbounded function, to be determined later.

Recall that ~d(z,Ω) := z− argminz′∈Ω

|z− z′| for any z ∈ R and any compact

Ω ⊂ R. Define

e2 = ~d(x2, S1(x1)). (5.24)

Then, we have x2 − e2 ∈ S1(x1).In the definition of S1, the term d11 represents the impact of the quan-

tization error q1(x1) − x1, which satisfies (5.4). If x2 is accurately available,


following the idea of measurement feedback control in Chapter 4, one maydesign S1 such that if x2 ∈ S1(x1), then the e1-subsystem is ISS. However,q2(x2) instead of x2 is available. In this case, the new set-valued map S1(x1)is employed to deal with the quantization error q2(x2)− x2.

With a continuously differentiable κ1, the boundaries of S1(x1), i.e.,maxS1(x1) and minS1(x1), are continuously differentiable almost everywhereand the derivative of e2 exists almost everywhere. Then, one may use a differ-ential inclusion to represent the dynamics of the e2-subsystem. An example ofS1(x1) is shown in Figure 5.5. Here, κ1 is chosen to be continuously differen-tiable for simplicity of discussions. The design procedure is still valid if κ1 isonly locally Lipschitz, which means, κ1 is continuously differentiable almosteverywhere.

0 x1

x2

e2

(x1, x2)

x2 = κ1(q1(x1))

x2 = minS1(x1)

x2 = maxS1(x1)

FIGURE 5.5 Set-valued map S1 and the definition of e2.

If a1 = b1 = b2 = 0, then q1(x1) = x1, S1(x1) = S1(x1) = κ1(x1) ande2 = x2 − κ1(x1), and the set-valued map design is reduced to the function-based design in Section 2.3.

In the following procedure, the new ei-subsystems (2 ≤ i ≤ n) are derivedin a recursive manner and are represented by differential inclusions.


Recursive Step: The eisubsystem (2 ≤ i ≤ n)

By default, S0(x0) := 0. With condition (5.4) satisfied, for each k =1, . . . , i− 1, define set-valued maps Sk and Sk as

Sk(xk) =

κk(xk − pk−1 + dk1) : pk−1 ∈ Sk−1(xk−1),

|dk1| ≤ bk|xk|+ (1− bk)ak

(5.25)

Sk(xk) =

dk2pk : pk ∈ Sk(xk),1

1 + bk+1≤ dk2 ≤ 1

1− bk+1

, (5.26)

where κk is continuously differentiable, odd, strictly decreasing, and radiallyunbounded.

For each k = 1, . . . , i− 1, define ek+1 as

ek+1 = ~d(xk+1, Sk(xk)). (5.27)

It can be observed that ek is defined as the directed distance from xk toSk−1(xk−1). However, ek cannot be directly used for feedback. Alternatively,xk+1 − pk with pk ∈ Sk(xk) is employed to define set-valued map Sk and thusSk. With this treatment, if pk ∈ Sk(xk), then xk+1 can be steered to withinSk(xk) by using the control design below.

Lemma 5.1 shows that with the set-valued maps defined above, each ei-subsystem can be transformed into an appropriate form for gain assignmentdesign.

Lemma 5.1 Consider system (5.1)–(5.3) satisfying Assumptions 5.1 and 5.2.With the definitions in (5.25)–(5.27) for 1 ≤ k ≤ i− 1, for any variable ei+1,when ei 6= 0, the dynamics of each ei-subsystem can be represented by thedifferential inclusion

ei ∈ xi+1 − ei+1 + φ∗i : φ∗i ∈ Φ∗

i (xi, ei+1), (5.28)

where Φ∗i (xi, ei+1) is a convex, compact, and upper semi-continuous set-valued

map, and there exists a ψΦi∈ K∞ such that for any xi, ei+1 and any φ∗i ∈

Φ∗i (xi, ei+1),

|φ∗i | ≤ ψΦ∗

i(|[eTi+1, a

Ti−1]

T |). (5.29)

The proof of Lemma 5.1 is in Appendix F.1. In the case without quantiza-tion, i.e., ai = 0 and bi = 0 for 1 ≤ i ≤ n, we have maxSi−1 = minSi−1 =κi−1(ei−1). In this case, the differential inclusion (5.28) can be equivalentlyrepresented by a differential equation.

Define set-valued maps Si and Si as in (5.25) and (5.26) with k = i,respectively. Specifically, in the case of i = n, bi+1 = bn+1 = 0 and


Si(xi) = Sn(xn) = Sn(xn). Define ei+1 as in (5.27) with k = i. Then, italways holds that

xi+1 − ei+1 ∈ Si(xi). (5.30)

By default, denote xn+1 = u.Then, when ei 6= 0, we can further represent the ei-subsystem as

ei ∈ Si(xi) + Φ∗i (xi, ei+1). (5.31)

Define Φ∗1(x1, e2) = φ∗1(x1, e2) and ψΦ∗ = ψφ∗ . Then, the e1-subsystem

is also in the form of (5.31) with a0 = 0 and Φ∗1 satisfying (5.29). With

Lemma 5.1, through the recursive design, the [x1, . . . , xn]T -system has been

transformed into the [e1, . . . , en]T -system with each ei-subsystem (1 ≤ i ≤ n)

in the form of (5.31).The extended Filippov solution of each ei-subsystem can be defined with

differential inclusion (5.31) with the convex, compact, and upper semi-continuous set-valued map Si(xi) + Φ∗

i (xi, ei+1).

5.1.2 QUANTIZED CONTROLLER

At Step i = n, the true control input u occurs, and thus we can set en+1 = 0.Indeed, the desired quantized controller u can be chosen as follows:

p∗1 = κ1(q1(x1)) (5.32)

p∗i = κi(qi(xi)− p∗i−1), 2 ≤ i ≤ n− 1 (5.33)

u = κn(qn(xn)− p∗n−1). (5.34)


p∗1 ∈ S1(x1) ⇒ · · · ⇒ p∗i ∈ Si(xi) ⇒ · · · ⇒ u ∈ Sn(xn), (5.35)

which implies en+1 = 0.

5.1.3 ISS OF THE TRANSFORMED SUBSYSTEMS AND CYCLICSMALLGAINTHEOREMBASED SYNTHESIS

Define αV (s) = s2/2 for s ∈ R+. In this subsection, we refine the gain assign-ment technique and show that each ei-subsystem (1 ≤ i ≤ n) can be renderedISS with the ISS-Lyapunov function

Vi(ei) = αV (|ei|) (5.36)

by appropriately choosing the κi’s for the Si’s and the Si’s.


Lemma 5.2 Consider the ei-subsystem (1 ≤ i ≤ n) in (5.31) with Φ∗i sat-

isfying (5.29). Under condition (5.4), for any specified ǫi > 0, 0 < ci < 1,ℓi > 0, γe1ei , . . . , γ

ei−1ei , γ

ei+1ei , χa1ei , . . . , χ

ai−1ei ∈ K∞, one can find a continuously

differentiable, odd, strictly decreasing, and radially unbounded function κi forSi(xi) such that Vi(ei) satisfies

Vi(ei) ≥ maxk=1,...,i−1

γekei (Vk(ek)), γei+1ei (Vi+1(ei+1)), χ

akei (ak), αV

(

aici

)

, ǫi

⇒ maxfi∈(Si(xi)+Φ∗

i (xi,ei+1))∇Vi(ei)fi ≤ −ℓiVi(ei), (5.37)

where Vn+1(en+1) = 0.

Proof. With (5.29) satisfied, there exist ψe1Φ∗

i, . . . , ψ

ei+1

Φ∗

i, ψa1Φ∗

i, . . . , ψ

ai−1

Φ∗

i∈ K∞

such that for any φ∗i ∈ Φ∗i (xi, ei+1),

|φ∗i | ≤i+1∑

k=1

ψekΦ∗

i(|ek|) +

i−1∑

k=1

ψakΦ∗

i(ak). (5.38)

Under Assumption 5.1, 0 ≤ bi, bi+1 < 1. From Lemma C.8, for any ǫi >0 and any 0 < ci < 1, one can find a νi : R+ → R+ which is positive,nondecreasing and continuously differentiable on (0,∞), and satisfies

(1 − bi)(1− ci)

1 + bi+1νi((1− bi)(1− ci)s)s

≥ℓi2s+ ψeiΦ∗

i(s) +

∑

k=1,...,i−1,i+1

ψekΦ∗

i α−1

V (γekei )−1 αV (s)

+∑

k=1,...,i−1

ψakΦ∗

i (χakei )−1 αV (s) (5.39)

for s ≥ √2ǫi.

With this kind of νi, define κi(r) = −νi(|r|)r for r ∈ R. Then, κi is odd,strictly decreasing, radially unbounded, and continuously differentiable.

Recall that Vi(ei) = αV (|ei|) = 12 |ei|2. Consider the case of

Vi(ei) ≥ maxk=1,...,i−1,i+1

γekei (Vk(ek)), χakei (ak), αV

(

aici

)

, ǫi

. (5.40)


|ek| ≤ α−1V (γekei )−1 αV (|ei|), k = 1, . . . , i− 1, i+ 1 (5.41)

|ei| ≥√2ǫi (5.42)

ei 6= 0 (5.43)

ai ≤ ci|ei| (5.44)

ak ≤ (χakei )−1 αV (|ei|). (5.45)


We simply use Sk and Sk to denote Sk(xk) and Sk(xk) for 1 ≤ k ≤ n. Fromthe definition of Si−1, we have

maxSi−1 ≥ max

max

(

1

1 + biSi−1

)

,max

(

1

1− biSi−1

)

minSi−1 ≤ min

min

(

1

1 + biSi−1

)

,min

(

1

1− biSi−1

)

.

Consider the following cases for xi − pi−1 + di1 with pi−1 ∈ Si−1 anddi1 ≤ bi|xi|+ (1− bi)ai:

a) xi > maxSi−1 (i.e., ei > 0) and xi ≥ 0:

xi − pi−1 + di1 ≥ (1− bi)xi −max Si−1 − (1 − bi)ai

= (1− bi)

(

xi −max

(

1

1− biSi−1

)

− ai

)

≥ (1− bi) (xi −maxSi−1 − ai) = (1− bi)(ei − ai).

b) xi < minSi−1 (i.e., ei < 0) and xi ≥ 0:

xi − pi−1 + di1 ≤ (1 + bi)xi −min Si−1 + (1− bi)ai

= (1 + bi)

(

xi −min

(

1

1 + biSi−1

))

+ (1− bi)ai

≤ (1− bi) (xi −minSi−1 + ai) = (1− bi)(ei + ai).

c) xi < minSi−1 (i.e., ei < 0) and xi ≤ 0:

xi − pi−1 + di1 ≤ (1− bi)xi −min Si−1 + (1− bi)ai

= (1− bi)

(

xi −min

(

1

1− biSi−1

)

+ ai

)

≤ (1− bi) (xi −minSi−1 + ai) = (1− bi)(ei + ai).

d) xi > maxSi−1 (i.e., ei > 0) and xi ≤ 0:

xi − pi−1 + di1 ≥ (1 + bi)xi −max Si−1 − (1 − bi)ai

= (1 + bi)

(

xi −max

(

1

1 + biSi−1

))

− (1− bi)ai

≥ (1− bi) (xi −maxSi−1 − ai) = (1− bi)(ei − ai).

Then, using (5.44), we have

|xi − pi−1 + di1| ≥ (1− bi)(1 − ci)|ei|, (5.46)

sign(xi − pi−1 + di1) = sign(ei). (5.47)


In the case of (5.40), for any |di1| ≤ bi|xi| + (1 − bi)ai, pi−1 ∈ Si−1,1/(1 + bi+1) ≤ di2 ≤ 1/(1− bi+1), and φ

∗i ∈ Φ∗

i (xi, ei+1), using (5.39)–(5.47),we successfully achieve

∇Vi(ei)(di2κi(xi − pi−1 + di1) + φ∗i )

=ei(

−di2νi(|xi − pi−1 + di1|)(xi − pi−1 + di1) + φ∗i)

≤− di2νi(|xi − pi−1 + di1|)|xi − pi−1 + di1||ei|+ |ei||φ∗i |

≤|ei|(

− 1

1 + bi+1νi((1− bi)(1 − ci)|ei|)(1 − bi)(1 − ci)|ei|

+

i+1∑

k=1

ψekΦ∗

i(|ek|) +

i−1∑

k=1

ψakΦ∗

i(ak)

)

≤|ei|(

− (1− bi)(1 − ci)

1 + bi+1νi((1 − bi)(1 − ci)|ei|)|ei|

+∑

k=1,...,i−1,i+1

ψekΦ∗

i α−1

V (γekei )−1 αV (|ei|)

+ ψeiΦ∗

i(|ei|) +

i−1∑

k=1

ψakΦ∗

i (χakei )−1 αV (|ei|)

)

≤− ℓi2|ei|2 = −ℓiVi(ei). (5.48)

As a result, we obtain that

max∇Vi(ei)(Si(xi) + Φ∗i (xi, ei+1)) ≤ −ℓiVi(ei) (5.49)

holds almost everywhere. ♦Thus, the ei-subsystem is ISS with e1, . . . , ei−1, ei+1, ak (1 ≤ k ≤ i), and

ǫi as the inputs. With Lemma 5.2, the ei-subsystems can be rendered to beISS one-by-one in the recursive design procedure. Furthermore, the ISS gains

γ(·)(·) ’s and χ

(·)(·)’s can be designed to be arbitrarily small or small enough to

satisfy the cyclic-small-gain condition.With the help of the cyclic-small-gain theorem, the quantized controller

designed above can be fine-tuned to yield the ISS property of the closed-loopquantized system.

Denote e = [e1, . . . , en]T . For convenience, denote e = F (e, x). With the re-

cursive control design procedure, the e-system is an interconnection of ISS sub-systems with a1, . . . , an, ǫ1, . . . , ǫn as inputs. The gain interconnection graphof the closed-loop quantized system is shown in 5.6.

In the recursive design procedure, given the ei−1-subsystem, by designingthe set-valued maps Si and Si for the ei-subsystem, we can assign the ISSgains γekei for 1 ≤ k ≤ i−1 to satisfy the cyclic-small-gain condition (4.76). Byapplying this reasoning repeatedly, we can guarantee (4.76) for all 2 ≤ i ≤ n.In this way, the e-system satisfies the cyclic-small-gain condition.


V1 Vi Vi+1 Vn· · · · · ·

FIGURE 5.6 The gain interconnection graph of the closed-loop quantized system.

With the Lyapunov-based ISS cyclic-small-gain result in Chapter 3, anISS-Lyapunov function can be constructed for the e-system as:

V (e) = max1≤i≤n

σi(Vi(ei)) (5.50)

with σ1(s) = s and σi(s) = γe2e1 · · · γeiei−1(s) (2 ≤ i ≤ n) for s ∈ R+,

where the γ(·)(·) ’s are K∞ functions being continuously differentiable on (0,∞),

slightly larger than the corresponding γ(·)(·) ’s and still satisfying (4.76) for all

2 ≤ i ≤ n. Clearly, V is differentiable almost everywhere.The influence of a1, . . . , an, ǫ1, . . . , ǫn can be represented by

ϑ = max1≤i≤n

σi αV(

aici

)

, maxk=1,...,i−1

σi χakei (ak), σi(ǫi)

:= max

ϑ0, σ1 αV(

a1c1

)

= max

ϑ0, αV

(

a1c1

)

. (5.51)

By using the Lyapunov-based cyclic-small-gain theorem (Theorem 3.1), itholds that

V (e) ≥ ϑ⇒ max∇V (e)F (e, x) ≤ −α(V (e)) (5.52)

wherever ∇V (e) exists, where α is a continuous and positive definite function.Property (5.52) implies that V (e) ultimately converges to within the region

V (e) ≤ ϑ. Recall the definitions of V1 and V (see (5.36) and (5.50)). Onecan see V (e) ≥ σ1(V1(e1)) = V1(e1) = αV (|e1|). Thus, x1 = e1 ultimatelyconverges to within the region |x1| ≤ α−1

V (ϑ).

In the recursive design procedure, we can design the γ(·)(·) ’s (and thus the

γ(·)(·) ’s) to be arbitrarily small to get arbitrarily small σi’s (2 ≤ i ≤ n). We can

also design the χakei ’s (1 ≤ k ≤ i − 1, 1 ≤ i ≤ n) and the ǫi’s (1 ≤ i ≤ n)to be arbitrarily small. In this way, if a1 6= 0, then we can design ϑ0 to bearbitrarily small such that ϑ = αV (a1/c1). By choosing c1 arbitrarily close toone, x1 can be driven arbitrarily close to the region |x1| ≤ a1. If a1 = 0, thenx1 ultimately converges to within the region |x1| ≤ α−1

V (ϑ0), where ϑ0 can bedesigned to be arbitrarily small.


5.1.4 A NUMERICAL EXAMPLE

We employ a simple example to demonstrate the control design approach.Consider the system

x1 = x2 + 0.5x1 (5.53)

x2 = u+ 0.5x22 (5.54)

xq1 = q1(x1) (5.55)

xq2 = q2(x2), (5.56)

where [x1, x2]T ∈ R

2 is the state, u ∈ R is the control input, q1 and q2 arestate quantizers satisfying (5.4) with b1 = b2 = 0.1 and a1 = a2 = 0.2.

Define e1 = x1. Then, the e1-subsystem can be written as

e1 = x2 − e2 + (0.5e1 + e2), (5.57)

where e2 is defined as (5.24) with S1 defined as (5.23). Define Φ∗1(e1, e2) =

0.5e1 + e2. Then, ψe1Φ∗

1(s) = 0.5s and ψe2Φ∗

1(s) = s. Choose γe2e1 (s) = 0.95s,

c1 = 0.2, and ℓ1 = 0.89. Choose ν1(s) = 3.06 according to (5.39). Then,κ1(r) = −ν1(|r|)r = −3.06r.

Following the design procedure provided in the proof of Lemma 5.1 inAppendix F.1, the e2-subsystem is in the following form:

e2 ∈ u+ φ∗2 : φ∗2 ∈ Φ∗2(x1, x2). (5.58)

Here, we only present the calculation of Φ∗2(x1, x2) in the case of x2 >

maxS1(x1). The calculation in the case of x2 < maxS1(x1) is quite similar.Firstly, max S1(x1) can be calculated as:

max S1(x1) = −3.06(x1 − 0.1|x1| − 0.18)

=

−3.06(0.9x1 − 0.18), if x1 ≥ 0;

−3.06(1.1x1 − 0.18), if x1 < 0.(5.59)

Then, it can be observed that max S1(x1) > 0 if x1 < 0.2 and max S1(x1) ≤0 if x1 ≥ 0.2. Then, we have

maxS1(x1) =

10.9 max S1(x1), if x1 < 0.2;11.1 max S1(x1), if x1 ≥ 0.2.

(5.60)

Combining (5.59) and (5.60), direct calculation yields:

maxS1(x1) =

−2.5036x1 + 0.5007, if x1 ≥ 0.2;

−3.06x1 + 0.612, if 0 ≤ x1 < 0.2;

−3.74x1 + 0.612, if x1 < 0.

(5.61)


Thus, we get

∂maxS1(x1) =

2.5036, if x1 > 0.2;

[2.5036, 3.06], if x1 = 0.2;

3.06, if 0 < x1 < 0.2;

[3.06, 3.74], if x1 = 0;

3.74, if x1 < 0.

(5.62)

Define φ21(x1, x2) = −1.2518x1−2.5036x2+0.5x22, φ22(x1, x2) = −1.53x1−3.06x2 +0.5x22 and φ23(x1, x2) = −1.87x1 − 3.74x2 +0.5x22. Then, in the caseof x2 > maxS1(x1), we can calculate

Φ∗2(x1, x2) =

φ21(x1, x2), if x1 > 0.2;

coφ21(x1, x2), φ22(x1, x2), if x1 = 0.2;

φ22(x1, x2), if 0 < x1 < 0.2;

coφ22(x1, x2), φ23(x1, x2), if x1 = 0;

φ23(x1, x2), if x1 < 0.

(5.63)

Then, it can be verified that for any φ∗2 ∈ Φ∗2(x1, x2), |φ∗2| ≤ ψe1Φ∗

2(|e1|) +

ψe2Φ∗

2(|e2|)+ψa1Φ∗

2(a1) with ψ

e1Φ∗

2(s) = 15.8576s, ψe2Φ∗

2(s) = 3.74s+s2 and ψa1Φ∗

2(s) =

11.44s for s ∈ R+.To satisfy the cyclic-small-gain condition, choose γe1e2 (s) = s. Choose

χa1e2 (s) = αV (s/0.3) and c2 = 0.3. Choose ν2(s) = 23.5 + s according to(5.39). Then, κ2(r) = −ν2(|r|)r.

The quantized controller is designed as

p∗1 = −3.06q1(x1) (5.64)

u = −(q2(x2)− p∗1)(23.5 + |q2(x2)− p∗1|). (5.65)

With the design method in Subsection 5.1.3, x1 would ultimately convergeto within the region |x1| ≤ a1/c1 = 1. Simulation results with initial conditionsx1(0) = 1 and x2(0) = −3, shown in Figures 5.7 and 5.8, are in accordancewith the theoretical results.

5.2 DYNAMIC QUANTIZATION

Practically, due to the finite word-length of digital devices, quantizers haveonly finite numbers of quantization levels. An example is the finite-level uni-form quantizer, as shown in Figure 5.9. If the input of the quantizer is withinthe quantization range Mµ, then the quantization error is less than µ; oth-erwise, the output of the quantizer is saturated. Clearly, the sector boundcondition assumed in Section 5.1 cannot be satisfied by the finite-level quan-tizers.


FIGURE 5.7 State trajectories of the example in Subsection 5.1.4.

FIGURE 5.8 Control input of the example in Section 5.1.4.


µ

Mµ

r

q(r, µ)

0

FIGURE 5.9 A uniform quantizer q with a finite number of levels: µ represents the

quantization error within the quantization rangeMµ, i.e., |r| ≤ Mµ⇒ |q(r, µ)−r| ≤

µ, with M being a positive integer.

By considering µ as a variable of the quantizer, the basic idea of dynamicquantization is to dynamically update µ during the quantized control proce-dure for improved quantized control performance, e.g., semiglobal asymptoticstabilization. Example 5.2 shows the basic idea.

Example 5.2 Consider a closed-loop quantized system

x = f(x, κ(q(x, µ))) (5.66)

where x ∈ R is the state, f : R2 → R is a locally Lipschitz function, κ : R → R

is the control law, q : R × R+ → R is the quantizer as shown in Figure 5.9with parameter M > 0, and µ ∈ R+ is the variable of the quantizer.

By defining quantization error d(x, µ) = q(x, µ) − x, the closed-loop quan-tized system can be rewritten as

x = f(x, κ(x + d(x, µ))). (5.67)

Assume that system (5.67) is ISS with d as the input and admits an ISS-Lyapunov function V : R → R+ satisfying

α(|x|) ≤ V (x) ≤ α(|x|) (5.68)

V (x) ≥ γ(|d|) ⇒ ∇V (x)f(x, κ(x + d)) ≤ −α(V (x)), (5.69)

where α, α ∈ K∞, γ ∈ K, and α is a continuous and positive definite function.Also assume that α, γ, and M satisfy

α−1 γ(µ) ≤Mµ (5.70)


for all µ ∈ R+.Consider the case of α(Mµ) ≥ V (x) ≥ γ(µ). Direct calculation yields:

V (x) ≤ α(Mµ) ⇒ |x| ≤Mµ⇒ |d| ≤ µV (x) ≥ γ(µ)

⇒ V (x) ≥ γ(|d|). (5.71)

Then, by using (5.69), we have

α(Mµ) ≥ V (x) ≥ γ(µ) ⇒ ∇V (x)f(x, κ(x + d)) ≤ −α(V (x)). (5.72)

Suppose that an upper bound of V (x(0)) is known a priori. By choosingµ(0) such that α(Mµ(0)) ≥ V (x(0)) and reducing µ on the timeline slowly, asshown in Figure 5.10, asymptotic stabilization can be achieved. In fact, prop-erty (5.72) defines nested invariant sets of the quantized control system, whichplay a central role in dynamically quantized control of nonlinear systems. Theprocess of reducing µ is usually known as the “zooming-in” stage of dynamicquantization.

tk tk+10

α(Mµ(t))

γ(µ(t))

V (x(t))

FIGURE 5.10 Basic idea of dynamic quantization.

In the case where the upper bound of V (x(0)) is unknown, semiglobalstabilization can be achieved for forward complete systems by employing a“zooming-out” stage, i.e., increasing µ and thus α(Mµ) fast enough thatα(Mµ(t∗)) ≥ V (x(t∗)) at some finite time t∗. Very detailed discussions of thisidea can be found in [164]. The dynamically quantized control design for high-order nonlinear systems in the following sections are based on these “zooming”ideas.

5.2.1 PROBLEM FORMULATION

The objective of this section is to design a new class of quantized controllers forstabilization of high-order nonlinear uncertain systems with dynamic quan-tization. The influence of dynamic uncertainty is also taken into account.Specifically, we consider the strict-feedback system with dynamic uncertain-ties:

z = g(z, x1) (5.73)

xi = xi+1 +∆i(xi, z), i = 1, . . . , n− 1 (5.74)

xn = u+∆n(xn, z), (5.75)


where [x1, . . . , xn]T := x ∈ Rn is the measurable state, z ∈ Rnz represents

the state of the inverse dynamics representing dynamic uncertainties and isnot measurable, u ∈ R is the control input, xi = [x1, . . . , xi]

T , and ∆i’s(i = 1, . . . , n) are unknown locally Lipschitz continuous functions. We considerthe general case in which both the measurement x and the control input uare quantized.

To realize quantized control with partial-state feedback, we assume thatsystem (5.73)–(5.75) is unboundedness observable (UO) and small-time final-state norm-observable with x as the output. The definition of UO is givenin Definition 1.12. The notion of small-time final-state norm-observability isrecalled from [85] and is closely linked to the notion of UO [128]. It is of interestto note that a dynamic system which is small-time final-state norm-observableis automatically UO, but the converse statement is not true. See also [247]for detailed discussions of observability notions for nonlinear systems in theframework of ISS.

Definition 5.1 Consider a dynamic system x = f(x), y = h(x) where x ∈Rn is the state, y ∈ R

m is the output, f : Rn → Rn is a locally Lipschitz

function with f(0) = 0, and h : Rn → Rm is a continuous function withh(0) = 0. The system is said to be small-time final-state norm-observable iffor any τ > 0, there exists γ ∈ K∞ such that

|x(τ)| ≤ γ(‖y‖[0,τ ]), ∀x(0) ∈ Rn. (5.76)

Throughout the section, the following assumptions are made on system(5.73)–(5.75).

Assumption 5.3 System (5.73)–(5.75) with u = 0 is forward complete andsmall-time final-state norm-observable with x as the output, i.e., for u = 0,

∀td > 0 ∃ϕ ∈ K∞ such that

|X(td)| ≤ ϕ(‖x‖[0,td]), ∀X(0) ∈ Rn+nz , (5.77)

where X := [zT , xT ]T .

Assumption 5.3 is needed for semiglobal quantized stabilization. However,it is important to mention that Assumption 5.3 is not needed if the bounds ofthe initial state of system (5.73)–(5.75) are known a priori. See the discussionson dynamic quantization in Subsection 5.2.8.

Assumptions 5.4 and 5.5 are made on the system dynamics.

Assumption 5.4 For each ∆i with i = 1, . . . , n, there exists a known λ∆i∈

K∞ such that for all xi, z,

|∆i(xi, z)| ≤ λ∆i(|(xi, z)|). (5.78)


Assumption 5.5 The z-subsystem (5.73) with x1 as the input admits anISS-Lyapunov function V0 : Rnz → R+ which is locally Lipschitz on Rnz\0and satisfies the following:

1. there exist α0, α0 ∈ K∞ such that

α0(|z|) ≤ V0(z) ≤ α0(|z|), ∀z ∈ Rnz ; (5.79)

2. there exist a χx1z ∈ K and a continuous and positive definite α0 such that

V0(z) ≥ χx1z (|x1|) ⇒ ∇V0(z)g(z, x1) ≤ −α0(V0(z)) (5.80)

wherever ∇V0 exists.

Under Assumption 5.5, system (5.73)–(5.75) represents an important classof minimum-phase nonlinear systems, which have been studied extensivelyby many researchers in the context of (non-quantized) robust and adaptivenonlinear control [153].

5.2.2 QUANTIZATION

This subsection provides a more detailed description of the quantizer shownin Figure 5.9. A quantizer q(r, µ) is defined as q(r, µ) = µqo(r/µ), wherer ∈ R is the input of the quantizer, µ > 0 is a variable to be explained later,and qo : R → R is a piecewise constant function. Specifically, there exists aconstant M > 0 such that

|qo(a)−M | ≤ 1, if a > M ; (5.81)

|qo(a)− a| ≤ 1, if |a| ≤M ; (5.82)

|qo(a) +M | ≤ 1, if a < −M ; (5.83)

qo(0) = 0. (5.84)

Then, quantizer q(r, µ) satisfies:

|q(r, µ)−Mµ| ≤ µ, if r > Mµ; (5.85)

|q(r, µ)− r| ≤ µ, if |r| ≤Mµ; (5.86)

|q(r, µ) +Mµ| ≤ µ, if r < −Mµ; (5.87)

q(0, µ) = 0. (5.88)

Mµ is the quantization range of quantizer q(r, µ), and µ represents the largestquantization error when |r| ≤Mµ. Clearly, the quantizer shown in Figure 5.9satisfies properties (5.85)–(5.88).

In several existing quantized control results (see e.g., [164]), two positiveparameters, say M ′, δ′, are used to formulate a quantizer q′ as:

|q′(r, µ′)− r| ≤ δ′µ′, if |r| ≤M ′µ′; (5.89)

|q′(r, µ′)| > (M ′ − δ′)µ′, if |r| > M ′µ′. (5.90)


Actually, a quantizer satisfying (5.85)–(5.87) has such properties if the vari-ables are appropriately defined. Indeed, by defining M = M ′/δ′, µ = δ′µ′

and a new quantizer q(r, µ) = q′(r, µ/δ′), properties (5.85)–(5.87) hold for thenew quantizer q. Moreover, properties (5.85) and (5.87) explicitly representthe saturation property of the quantizer, and are quite useful in realizing therecursive design in this section.

As shown in Example 5.2, given fixed M , the basic idea of dynamic quan-tization is to dynamically update µ (and thus Mµ) to improve the controlperformance. Increasing µ, referred to as zooming-out, enlarges µ and thusthe quantization range Mµ. Decreasing µ, referred to as zooming-in, reducesµ andMµ. According to the literature, µ is called the zooming variable. Withthe design in this section, the zooming variables are updated in discrete time.

5.2.3 QUANTIZED CONTROLLER STRUCTURE AND CONTROL OBJECTIVE

We introduce a new quantized control structure, which is a natural extensionof the ISS small-gain design without quantization. With the ISS small-gaindesign method in Chapter 2, we can recursively design a non-quantized con-troller for system (5.73)–(5.75) as

vi = κi(xi − vi−1), i = 1, . . . , n− 1 (5.91)

u = κn(xn − vn−1), (5.92)

where v0 = 0 and κi’s for i = 1, . . . , n are appropriately chosen continuousfunctions. The maps defined in (5.91) are usually called virtual control laws,and (5.92) defines the true control law.

Our solution to the quantized control problem for system (5.73)–(5.75) isto add quantizers before and after each (virtual) control law defined in (5.91)–(5.92). Based on this idea, the quantized controller is in the form of

vi = qi2(κi(qi1(xi − vi−1, µi1)), µi2), i = 1, . . . , n− 1 (5.93)

u = qn2(κn(qn1(xn − vn−1, µn1)), µn2), (5.94)

where v0 = 0, the qij ’s are quantizers with zooming variables µij ’s for i =1, . . . , n, j = 1, 2, and the κi’s for i = 1, . . . , n are nonlinear functions. Eachκi in (5.93)–(5.94) is not necessarily the same as the corresponding κi in(5.91)–(5.92) due to the implementation of the quantizers. The block diagramof the proposed quantized control system is shown in Figure 5.11.

In Assumption 5.6, each quantizer qij is assumed to have properties in theform of (5.85)–(5.88).

Assumption 5.6 For i = 1, . . . , n, j = 1, 2, each quantizer qij with zooming


xn = u+∆n(xn, z)

...

xi = xi+1 +∆i(xi, z)

...

x1 = x2 +∆1(x1, z)

z = g(z, x1)

κn

qn2

qn1

...

+−

κi

qi2

qi1

...

+−

κ1

q12

q11

FIGURE 5.11 The quantized control structure for high-order nonlinear systems.

variable µij satisfies

|qij(r, µij)−Mijµij | ≤ µij , if r > Mijµij ; (5.95)

|qij(r, µij)− r| ≤ µij , if |r| ≤Mijµij ; (5.96)

|qij(r, µij) +Mijµij | ≤ µij , if r < −Mijµij ; (5.97)

qij(0, µij) = 0, (5.98)

where Mi1 > 2 and Mi2 > 1.

The assumption on the parameters Mi1 and Mi2 is not restrictive. FromFigure 5.12, it can be observed that the simplest three-level quantizer satisfies(5.85)–(5.88) with M = 3.

In the dynamic quantization design in this section, the zooming variablesµij for i = 1, . . . , n, j = 1, 2 are piecewise constant and updated in discrete-time. Without loss of generality, they are assumed to be right-continuous onthe timeline. The dynamic quantization is composed of two stages: zooming-out and zooming-in. To simplify the discussions, the time sequences for theupdates of all the zooming variables are designed to be the same and denotedby tkk∈Z+ , in which tk+1 − tk = td with constant td > 0.

The update law of each µij is expected to be in the following form:

µij(tk+1) = Qij(µij(tk)), k ∈ Z+. (5.99)


µ

Mµ

r

q(r, µ)

0

FIGURE 5.12 Three-level uniform quantizer with M = 3.

In the zooming-out stage, Qij = Qoutij ; in the zooming-in stage, Qij = Qin

ij .The goal is to design a quantized controller in the form of (5.93)–(5.94) with

dynamic quantization (5.99) to semiglobally stabilize system (5.73)–(5.75)such that all the signals including the state x in the closed-loop quantizedsystem are bounded, and moreover, to steer x1 to an arbitrarily small neigh-borhood of the origin.

5.2.4 RECURSIVE CONTROL DESIGN WITH SETVALUED MAPS FOR STATICQUANTIZATION

A fundamental technical obstacle for quantized feedback control design is thatthe quantized control system in question must be made robust with respect tothe quantization errors. The nonlinearity and dimensionality of system (5.73)–(5.75) and the saturation and discontinuity of quantization together cause themajor difficulties.

This subsection develops a recursive design procedure for κi in (5.93)–(5.94)by taking into account the effects of static quantization, such that the closed-loop quantized system admits nested invariant sets for further dynamic quan-tization designs. Set-valued maps are still used to handle the discontinuity ofquantization. With appropriately designed set-valued maps, the closed-loopquantized system is transformed into a network of ei-subsystems representedby differential inclusions. Moreover, each ei-subsystem is designed to be ISSwith gains satisfying the cyclic-small-gain condition. More importantly, it isshown in Subsection 5.2.6 that the design guarantees the existence of nestedinvariant sets for the closed-loop quantized system to realize dynamic quan-tization.


This subsection focuses on the influence of quantization error, and assumesthat the zooming variables are constant.

Assumption 5.7 For i = 1, . . . , n, j = 1, 2, each zooming variable µij isconstant on the timeline.

Note that Assumption 5.7 is removed for dynamic quantization design inSubsection 5.2.8.


Let e1 = x1. The e1-subsystem is in the following form:

e1 = x2 +∆1(x1, z). (5.100)

Define a set-valued map S1 as

S1(x1, µ11, µ12) =

κ1(x1 + b11) + b12 : |b11| ≤ maxc11|e1|, µ11, |b12| ≤ µ12

,

(5.101)

where κ1 is a continuously differentiable, odd, strictly decreasing, and radi-ally unbounded function and 0 < c11 < 1 is a constant, both of which aredetermined later. It should be noted that b11, b12 defined in (5.101) are usedas auxiliary variables to define set-valued map S1.

Recall that ~d(ξ,Ω) := ξ − argminξ′∈Ω

|ξ − ξ′| for any ξ ∈ R and any Ω ⊂ R.

Define

e2 = ~d(x2, S1(x1, µ11, µ12)). (5.102)

Rewrite the e1-subsystem (5.100) as

e1 = x2 − e2 +∆1(x1, z) + e2, (5.103)

where x2 − e2 ∈ S1(x1, µ11, µ12) due to (5.102).It is necessary to give a detailed description of the set-valued map S1.

Consider the first-order nonlinear system e1 = x2 + ∆1(x1, z). Recall thedefinition e1 = x1. With the gain assignment technique in Subsection 2.3,one can design a control law x2 = κ1(x1) to stabilize the e1-system. In theexistence of quantization errors, control law x2 = κ1(x1) should be modifiedas x2 = q12(κ1(q11(x1, µ11)), µ12). The set-valued map S1 takes into accountthe quantization errors of both the quantizers q11 and q12. As shown below,the motion of the new variable e2 defined based on S1 can be representedby a differential inclusion, and the problem caused by the discontinuity ofquantization is solved. In the control design procedure below, we still useset-valued maps to deal with the discontinuity of quantization.


Recursive Step: The eisubsystems

Denote µi1 = [µ11, . . . , µi1]T and µi2 = [µ12, . . . , µi2]

T for i = 1, . . . , n. Foreach i = 2, . . . , n, define a set-valued map Si as

Si(xi, µi1, µi2) =

κi(xi − ςi−1 + bi1) + bi2 : ςi−1 ∈ Si−1(xi−1, µ(i−1)1, µ(i−1)2),

|bi1| ≤ maxci1|ei|, µi1, |bi2| ≤ µi2

,

(5.104)

where κi is a continuously differentiable, odd, strictly decreasing, and radiallyunbounded function and 0 < ci1 < 1 is a constant. Both κi and ci1 aredetermined later. The definition of Si guarantees its convexity, compactness,and upper semi-continuity of the set-valued map Si. Here, bi1, bi2 are auxiliaryvariables used to define the set-valued map Si.

It can be observed that S1(x1, µ11, µ12), defined in (5.101), is also in theform of (5.104) with S0(x0, µ01, µ02) := 0.

For each i = 2, . . . , n, define ei+1 as

ei+1 = ~d(xi+1, Si(xi, µi1, µi2)). (5.105)

Lemma 5.3 shows that with the recursive definitions of set-valued maps in(5.104) and new state variables in (5.105), the ei-subsystems with i = 1, . . . , ncan be represented by differential inclusions with specific properties.

Lemma 5.3 Consider the (x1, . . . , xn)-system in (5.74)–(5.75). Under As-sumptions 5.4 and 5.7, with the definitions in (5.101), (5.102), (5.104), and(5.105), each ei-subsystem for 1 ≤ i ≤ n can be represented with the differen-tial inclusion:

ei ∈ Si(xi, µi1, µi2) + Φ∗i (ei+1, xi, µ(i−1)1, µ(i−1)2, z), (5.106)

where Φ∗i is a convex, compact and upper semi-continuous set-valued map, and

there exists a λΦ∗

i∈ K∞ such that for all (ei+1, xi, µ(i−1)1, µ(i−1)2, z), any

φ∗i ∈ Φ∗i (ei+1, xi, µ(i−1)1, µ(i−1)2, z) satisfies

|φ∗i | ≤ λΦ∗

i(|(ei+1, µ(i−1)1, µ(i−1)2, z)|), (5.107)

where ei := [e1, . . . , ei]T .

The proof of Lemma 5.3 is in Appendix F.With Lemma 5.3, through the recursive design approach, the (x1, . . . , xn)-

system has been transformed into the new (e1, . . . , en)-system with each ei-subsystem (i = 1, . . . , n) in the form of (5.106). The extended Filippov solu-tion of each ei-subsystem can be defined with differential inclusion (5.106),because set-valued maps Si and Φ∗

i are convex, compact, and upper semi-continuous.


ISS of the Subsystems

Denote e0 = z. Then,

e0 = g(e0, e1). (5.108)

Define γe1e0 (s) = χx1z α−1

V (s) for s ∈ R+. Under Assumption 5.5, it holdsthat

V0(e0) ≥ γe1e0 (V1(e1)) ⇒ ∇V0(e0)g(e0, e1) ≤ −α0(V0(e0)) (5.109)

wherever ∇V0 exists.For each ei-subsystem with i = 1, . . . , n, define the following ISS-Lyapunov

function candidate

Vi(ei) = αV (|ei|), (5.110)

where αV (s) = s2/2 for s ∈ R+. For convenience of notation, defineVn+1(en+1) = αV (|en+1|).

Lemma 5.4 states that, for i = 1, . . . , n, by appropriately choosing κi, eachei-subsystem can be rendered to be ISS with Vi defined in (5.110) as an ISS-Lyapunov function. Furthermore, the ISS gains from µi1 and µi2 to ei satisfyspecific conditions to guarantee the existence of nested invariant sets.

Lemma 5.4 Consider the ei-subsystem (i = 1, . . . , n) in the form of (5.106)with Si defined in (5.101) and (5.104). Under Assumptions 5.4 and 5.7, forany specified constants ǫi > 0, ιi > 0, 0 < ci1, ci2 < 1, γekei ∈ K∞ for k =0, . . . , i − 1, i + 1, and γµk1

ei , γµk2ei ∈ K∞ for k = 1, . . . , i − 1, one can find

a continuously differentiable, odd, strictly decreasing, and radially unboundedκi for the set-valued map Si such that the ei-subsystem is ISS with Vi(ei) =αV (|ei|) as an ISS-Lyapunov function satisfying

Vi(ei) ≥ maxk=1,...,i−1

γe0ei (V0(e0)), γekei (Vk(ek)), γ

ei+1ei (Vi+1(ei+1)),

γµk1ei (µk1), γ

µk2ei (µk2), γ

µi1ei (µi1), γ

µi2ei (µi2), ǫi

⇒ maxψi∈Ψi(ei+1,xi,µi1,µi2,z)

∇Vi(ei)ψi ≤ −ιiVi(ei), (5.111)

where

γµi1ei (s) = αV

(

1

ci1s

)

(5.112)

γµi2ei (s) = αV

(

1

1− ci1κ−1i

(

1

ci2s

))

(5.113)

Ψi(ei+1, xi, µi1, µi2, z) := Si(xi, µi1, µi2) + Φ∗i (ei+1, xi, µ(i−1)1, µ(i−1)2, z)

(5.114)

with κi(s) = |κi(s)| for s ∈ R+.


The proof of Lemma 5.4 is in Appendix F.Recall the definition Vi(ei) = αV (|ei|). Consider the γµi1

ei and γµi2ei defined

in (5.112) and (5.113), respectively. It can be observed that the ISS gainfrom the quantization error µi1 through the quantized control system to thesignal ei is s/ci1. Direct calculation yields that the ISS gain from the quan-tization error µi2 through the quantized control system to the signal κi(ei)is κi

(

κ−1i (s/ci2)/(1− ci1)

)

, which may not be linear, but is closely related tothe linear function s/ci2. As clarified later, properties (5.112) and (5.113) playa crucial role for the implementation of dynamic quantization in the quantizedcontrol system.

5.2.5 QUANTIZED CONTROLLER

In Subsection 5.2.4, set-valued maps are used to transform the closed-loopquantized system into a network of ISS subsystems described by differentialinclusions. In this subsection, it is shown that the quantized control law u inthe form of (5.93)–(5.94) with the κi’s defined above belongs to the set-valuedmap Sn under realizable conditions. In this way, the closed-loop quantizedsystem with the proposed quantized control law u can be represented as adynamic network composed of ISS subsystems.

Recall that κi(s) = |κi(s)| for s ∈ R+. Lemma 5.5 provides conditionsunder which the quantized control law u in the form of (5.93)–(5.94) belongsto the set-valued map Sn.

Lemma 5.5 Under Assumption 5.6, if

1

Mi1< ci1 ≤ 0.5,

1

Mi2< ci2 < 1 (5.115)

for all i = 1, . . . , n and if

|ei| ≤Mi1µi1, (5.116)

κi((1− ci1)|ei|) ≤Mi2µi2 (5.117)

for all i = 1, . . . , n, then vi for i = 1, . . . , n− 1 and u defined in (5.93)–(5.94)satisfy

vi ∈ Si(xi, µi1, µi2), i = 1, . . . , n− 1, (5.118)

u ∈ Sn(xn, µn1, µn2). (5.119)

The proof of Lemma 5.5 is given in Appendix F by fully using the propertiesof the quantizers and the set-valued maps.

Conditions (5.116) and (5.117) imply that the signals |ei| and |κi((1 −ci)|ei|)| should be covered by the quantization ranges Mi1µi1 and Mi2µi2,respectively, such that the quantized control law u belongs to the set-valuedmap Sn. This is because of the saturation property (see (5.95) and (5.97)) ofthe quantizers.


5.2.6 SMALLGAINBASED SYNTHESIS AND NESTED INVARIANT SETS OFTHE CLOSEDLOOP QUANTIZED SYSTEM

Recall that e = [eT0 , e1, . . . , en]T . The gain digraph of the e-system is shown

in Figure 5.13. The purpose of this subsection is to design the ISS gains toyield the ISS property of the closed-loop quantized system with e as the stateby using the cyclic-small-gain theorem.

V0 V1 Vi Vn· · · · · ·

FIGURE 5.13 The gain digraph of the e-system.

Recall that ei = [e1, . . . , ei]T . For each (e0, ei)-subsystem (i = 1, . . . , n),

given the (e0, ei−1)-subsystem, by designing the set-valued map Si for the ei-subsystem, the ISS gains from states e0, . . . , ei−1 to state ei can be assigned.With the recursive design, the ISS gains γekei for k = 0, . . . , i−1 can be designedsuch that

γe1e0 γe2e1 γe3e2 · · · γeiei−1 γe0ei < Id

γe2e1 γe3e2 · · · γeiei−1 γe1ei < Id

...γeiei−1

γei−1ei < Id

. (5.120)

By applying this reasoning repeatedly, (5.120) can be guaranteed for all i =1, . . . , n. In this way, the e-system satisfies the cyclic-small-gain condition.

In the gain digraph of the e-system shown in Figure 5.13, the e1-subsystemis reachable from the subsystems of e0, e2, . . . , en, i.e., there are sequences ofdirected arcs from the subsystems of e0, e2, . . . , en to the e1-subsystem.

By using the Lyapunov-based cyclic-small-gain theorem in Chapter 3, anISS-Lyapunov function can be constructed for the e-system as

V (e) = maxi=0,...,n

σi(Vi(ei)) (5.121)

with σ1(s) = s, σi(s) = γe2e1 · · · γeiei−1(s) (i = 2, . . . , n), and σ0(s) =

maxi=1,...,nσi γe0ei (s) for s ∈ R+, where the γ(·)(·) ’s are K∞ functions con-

tinuously differentiable on (0,∞) and slightly larger than the corresponding

γ(·)(·) ’s, and still satisfy the cyclic-small-gain condition.


The following lemma states that by appropriately choosing the κi’s forthe set-valued maps Si’s, the cyclic-small-gain condition (5.120) can be sat-isfied and the closed-loop quantized system with state e admits specific ISSproperties.

Lemma 5.6 Consider the e-system composed of the ei-subsystems in the formof (5.108) and (5.106) satisfying (5.109) and (5.111), respectively. If the ISSgains defined in (5.109) and (5.111) satisfy (5.120) for all i = 1, . . . , n and ifu ∈ Sn(xn, µn1, µn2), then the ISS-Lyapunov function candidate V defined in(5.121) for the e-system satisfies

V (e) ≥ θ(µn1, µn2, ǫn) ⇒ maxψ∈Ψ(e,x,µn1,µn2)

∇V (e)ψ ≤ −α(V (e)) (5.122)

wherever ∇V exists, where α is a continuous and positive definite function,and

θ(µn1, µn2, ǫn) := maxi=1,...,n

σi

(

maxk=1,...,i

γµk1ei (µk1), γ

µk2ei (µk2), ǫi

)

(5.123)

Ψ(e, x, µn1, µn2) := [gT (e0, e1),Ψ1(e2, x1, µ11, µ12), . . . ,Ψn(0, xn, µn1, µn2)]T

(5.124)

with ǫn := [ǫ1, . . . , ǫn]T .

Proof. In the case of u ∈ Sn(xn, µn1, µn2), it holds that en+1 = 0 and thusVn+1(en+1) = 0. With the cyclic-small-gain condition (5.120) satisfied for alli = 1, . . . , n, (5.122) can be proved. ♦

For specified σi for i = 1, . . . , n, by designing the γµk1ei ’s (k = 1, . . . , i− 1)

and the γµk2ei ’s (k = 1, . . . , i− 1) small enough, we can achieve

θ(µn1, µn2, ǫn) = maxi=1,...,n

σi γµi1ei (µi1), σi γµi2

ei (µi2), σi(ǫi)

(5.125)

for all µi1, µi2, ǫi > 0 for i = 1, . . . , n.Motivated by [164], the dynamic quantization design in this section is based

on nested invariant sets with sizes depending on zooming variables µn1 andµn2.

Define

B1(µn1, µn2) = maxi=1,...,n

σi αV (Mi1µi1),

σi αV(

11−ci1 κ

−1i (Mi2µi2)

)

, (5.126)

B2(µn1, µn2) = maxi=1,...,n

σi αV(

1ci1µi1

)

,

σi αV(

11−ci1 κ

−1i

(

1ci2µi2

))

. (5.127)

Lemma 5.7 summarizes this section by showing the existence of the nestedinvariant sets, defined by B1 and B2, for the closed-loop quantized systemdesigned based on Lemmas 5.3–5.6.


Lemma 5.7 Consider the quantized control system consisting of the plant(5.73)–(5.75) and the quantized control law (5.93)–(5.94). Under Assumptions5.4, 5.5, 5.6 and 5.7, the closed-loop quantized system can be transformed intoa large-scale system composed of ei-subsystems in the form of (5.108) and(5.106), and for specified constants ci1, ci2 satisfying (5.115) for i = 1, . . . , n,specified ISS gains γ

ek′

ek (k 6= k′) satisfying the cyclic-small-gain condition(5.120) for all i = 1, . . . , n, specified ISS gains γµk1

ei , γµk2ei for i = 1, . . . , n, k =

1, . . . , i− 1 satisfying (5.125), and specified arbitrarily small constants ǫi fori = 1, . . . , n, we can find continuously differentiable, odd, strictly decreasing,and radially unbounded functions κi for i = 1, . . . , n such that (5.111) holdsfor i = 1, . . . , n. Moreover, if

σi αV (Mi1µi1) = σi αV(

1

1− ci1κ−1i (Mi2µi2)

)

=σj αV (Mj1µj1) = σj αV(

1

1− cj1κ−1j (Mj2µj2)

)

(5.128)

for all i, j = 1, . . . , n and if

B1(µn1, µn2) ≥ θ0 (5.129)

with θ0 = maxi=1,...,nσi(ǫi), then the ISS-Lyapunov function candidate Vdefined in (5.121) satisfies

B1(µn1, µn2) ≥ V (e) ≥ maxB2(µn1, µn2), θ0 (5.130)

⇒ maxψ∈Ψ(e,x,µn1,µn2)

∇V (e)ψ ≤ −α(V (e)), (5.131)

where Ψ is defined in (5.124).

Proof. Under Assumptions 5.4 and 5.7, with Lemma 5.3, we can transform theclosed-loop quantized system into a large-scale system with state e composedof ei-subsystems in the form of (5.108) and (5.106).

Under Assumptions 5.4, 5.5, 5.6, and 5.7, by directly using Lemma 5.4,for any specified constants ci1, ci2 satisfying (5.115) for i = 1, . . . , n, any ISSgains γ

ek′

ek (k 6= k′) satisfying the cyclic-small-gain condition (5.120) for alli = 1, . . . , n, any specified ISS gains γµk1

ei , γµk2ei for i = 1, . . . , n, k = 1, . . . , i−1

satisfying (5.125) and specified arbitrarily small constants ǫi for i = 1, . . . , n,we can find continuously differentiable, odd, strictly decreasing and radi-ally unbounded functions κi for i = 1, . . . , n such that (5.111) holds fori = 1, . . . , n.

The satisfaction of (5.115) by appropriately choosing the κi’s for i =1, . . . , n guarantees that B1(µn1, µn2) > B2(µn1, µn2) for all positivezooming variables µi1, µi2. By using (5.129), we have B1(µn1, µn2) ≥maxB2(µn1, µn2), θ0. Recall the definitions of Vi(ei) in (5.110) and V (e)


in (5.121). The equalities in (5.128) and the left inequality in (5.130) guar-antee (5.116)–(5.117). Under Assumption 5.6, with (5.115) satisfied, by usingLemma 5.5, we have u ∈ Sn(xn, µn1, µn2).

Note that (5.125) is satisfied by appropriately choosing κi for i =1, . . . , n. By virtue of (5.112), (5.113), (5.125) and (5.127), θ(µn1, µn2, ǫn) =maxB2(µn1, µn2), θ0. With the cyclic-small-gain condition (5.120) satisfiedby appropriately choosing κi for i = 1, . . . , n and u ∈ Sn(xn, µn1, µn2), Lemma5.6 guarantees the implication in (5.130) and (5.131). ♦

In the following subsection, based on Lemma 5.7, the invariant sets areused to design dynamic quantization.

5.2.7 A GUIDELINE FOR QUANTIZED CONTROL LAW DESIGN

To clarify the design procedure, we provide a guideline to choosing the func-tions κi for the quantized control law (5.93)–(5.94) such that the closed-loopquantized system satisfies property (5.130)–(5.131). The guideline includestwo major steps:

1. Choose the ISS parameters of the ei-subsystems.a. Choose constants ci1, ci2 to satisfy (5.115) for i = 1, . . . , n.b. Choose ISS gains γ

ejei ∈ K∞ (j 6= i) and the corresponding functions

γejei > γ

ejei to satisfy the cyclic-small-gain condition (5.120) for all

i = 1, . . . , n, and calculate σi for i = 1, . . . , n in (5.121).c. Choose ISS gains γµk1

ei , γµk2ei for i = 1, . . . , n, k = 1, . . . , i− 1 such that

(5.125) holds for all µi1, µi2, ǫi > 0 for i = 1, . . . , n.d. Choose specified ǫi, ιi > 0 for i = 1, . . . , n.

2. Choose κi’s based on Lemma 5.4 with the ISS parameters chosen in Step1.

In Step 1, it is only required that Step (c) is after Step (b), because con-dition (5.125) in Step (c) depends on the σi calculated in Step (b). UnderAssumptions 5.4–5.7, if the ISS parameters and the κi are chosen accordingto the guideline and if conditions (5.128) and (5.129) are satisfied, then fromLemma 5.7, the nested invariant sets exist.

5.2.8 DYNAMIC QUANTIZATION

Because of the saturation property of the quantizers, the quantized controllaw designed in Subsection 5.2.4 can only guarantee local stabilization; see(5.130)–(5.131). In this subsection, based on the nested invariant sets givenin Lemma 5.7, we design a dynamic quantization logic in the form of (5.99),composed of a zooming-in stage and a zooming-out stage, to dynamicallyadjust the zooming variables µij (i = 1, . . . , n, j = 1, 2) such that the closed-loop quantized system is semiglobally stabilized. In this design, the zoomingvariables µij(t) are piecewise constant signals, and are adjusted on a discretetime sequence tkk∈Z+ , where tk+1 − tk = td with constant td > 0.


To satisfy condition (5.128) in Lemma 5.7, we design dynamic quantizationsuch that for all t ∈ R+,

σi αV (Mi1µi1(t)) = σi αV(

1

1− ci1κ−1i (Mi2µi2(t))

)

:= Θ(t) (5.132)

for i = 1, . . . , n. Equivalently, it is required that

µi1(t) =1

Mi1α−1V σ−1

i (Θ(t)) := Υi1(Θ(t)), (5.133)

µi2(t) =1

Mi2κi(

(1− ci1)α−1V σ−1

i (Θ(t)))

:= Υi2(Θ(t)) (5.134)

for i = 1, . . . , n. According to the definitions, Υi1 and Υi2 are invertible fori = 1, . . . , n. Thus, the dynamic quantization logic (5.99) can be designedby choosing an appropriate update law for Θ, which may reduce the designcomplexity for all the zooming variables µij (i = 1, . . . , n, j = 1, 2). Theupdate law for Θ is expected to be in the following form:

Θ(tk+1) = Q(Θ(tk)), k ∈ Z+. (5.135)

In the zooming-out stage, Q = Qout; in the zooming-in stage, Q = Qin. WithQout and Qin designed, we can design the dynamic quantization logic (5.99)for µij by choosing

Qoutij = Υij Qout Υ−1

ij , (5.136)

Qinij = Υij Qin Υ−1

ij . (5.137)

Using the definition of B1 in (5.126), we also have

Θ(t) = B1(µn1(t), µn2(t)). (5.138)

Before designing dynamic quantization, the relation between zooming vari-ables µn1, µn2 and control error e should be clarified. For i = 1, . . . , n, usingthe definitions of Si in (5.104), the strictly decreasing property of κi implies

maxSi(xi, µi1, µi2)

= κi(xi −maxSi−1(xi−1, µ(i−1)1, µ(i−1)2)−maxci1|ei|, µi1) + µi2,(5.139)

and

minSi(xi, µi1, µi2)

= κi(xi −minSi−1(xi−1, µ(i−1)1, µ(i−1)2) + maxci1|ei|, µi1)− µi2. (5.140)


Recall that e = [eT0 , e1, . . . , en]T . Given the definitions of ei for i = 2, . . . , n

in (5.105), denote

e = e(X, µn1, µn2) (5.141)

with X = [zT , xT ]T ∈ Rn+nz . It can be observed that e is a continuous

function of X, µn1, µn2. Clearly, the piecewise updates of µn1, µn2 cause jumpsof e on the timeline. This should be carefully handled in the design.

ZoomingOut Stage

The purpose of the zooming-out stage in this subsection is to increase thezooming variables µij such that at some finite time tk∗ , the state of theclosed-loop quantized system is restricted to be in the larger invariant set cor-responding to B1 in (5.130). In this stage, the components κi’s for i = 1, . . . , nof the controller are set to be zero, i.e., u = 0.

The small-time norm-observability assumed in Assumption 5.3 guaranteesthat for td > 0, there exists a ϕ ∈ K∞ such that

|X(tk + td)| ≤ ϕ(‖x‖[tk,tk+td]) (5.142)

for any k ∈ Z+. Considering the definitions of V and e in (5.121) and (5.141),for td > 0, property (5.142) can be represented with the Lyapunov functionV as

|V (e(X(tk + td), 0, 0))| ≤ ϕ(‖x‖[tk,tk+td]) (5.143)

for any k ∈ Z+, where ϕ ∈ K∞.With the forward completeness property assumed in Assumption 5.3, we

design a zooming-out logic Qout : R+ → R+ to increase Θ fast enough todominate the growth rate of ϕ(|x|) such that at some finite time tk∗ > 0 withk∗ ∈ Z+, it holds that

Mi1µi1(tk∗) ≥ |xi(tk∗)|, i = 1, . . . , n, (5.144)

Θ(tk∗) ≥ ϕ(‖x‖[tk∗−td,tk∗ ]). (5.145)

Due to the saturation of the quantizer, if the input signal of a quantizeris outside the range of the quantizer, then one cannot estimate the bound ofthe signal without using additional information. In the zooming-out stage, theκi’s are set to be zero, and the input of the quantizer qi1 is xi; see control law(5.93)–(5.94). Inequality (5.144) means that at some finite time tk∗ , xi is inthe quantization range of qi1. Then, we can estimate the bound of |xi(tk∗)|.

Using (5.143) and (5.145), we have

Θ(tk∗) ≥ maxV (e(X(tk∗), 0, 0)), θ0. (5.146)

From the definitions of maxSi(xi, µi1, µi2), minSi(xi, µi1, µi2), and ei+1,one observes that increase of µn1, µn2 leads to increase of maxSi(xi, µi1, µi2),


decrease of minSi(xi, µi1, µi2) and thus decrease or hold of |ei+1| for i =1, . . . , n− 1. Thus, with the zooming-out logic Qout, we achieve that, at timetk∗ > 0 with k∗ ∈ Z+, it holds that

Θ(tk∗) ≥ maxV (e(X(tk∗), µn1(tk∗), µn2(tk∗))), θ0. (5.147)

With Qout designed, we can design the zooming-out logic Qoutij for i =

1, . . . , n, j = 1, 2 according to (5.136)–(5.137).It should be noted that, if a bound of the initial state X(0) is known a

priori, then we can directly set Θ(tk∗) to satisfy (5.146) with tk∗ = 0. Inthis case, the zooming-out stage is not necessary and Assumption 5.3 is notrequired.

ZoomingIn Stage

The zooming-out stage achieves (5.147) at time tk∗ with k∗ ∈ Z+. Supposethat at some tk > 0 with k ≥ k∗, it is achieved that

Θ(tk) ≥ maxV (e(X(tk), µn1(tk), µn2(tk))), θ0. (5.148)

We first design a Qin : R+ → R+ for the zooming-in stage such that

Θ(tk+1) = Qin(Θ(tk))

≥ maxV (e(X(tk+1), µn1(tk+1), µn2(tk+1))), θ0. (5.149)

This objective is achievable by using Lemmas 5.8 and 5.9. Then, we show theconvergence property of the update law (5.135) for Θ in the zooming-in stageby Lemma 5.10.

Recall that if (5.149) is achieved based on (5.148), then one can recursivelyguarantee that the state e of the closed-loop quantized system is always in thelarger invariant set represented by B1 in spite of the discontinuous update ofΘ; see (5.130) and (5.138).

Lemma 5.8 describes the decreasing property of V during the time interval[tk, tk+1), based on which we design the zooming-in update law Qin for Θ.

Lemma 5.8 Consider the closed-loop quantized system with V satisfyingproperty (5.130)–(5.131). If (5.148) holds at time tk with k ∈ Z+, then thereexists a continuous and positive definite function ρ such that

(Id− ρ) ∈ K∞, (5.150)

V (e(X(tk+1), µn1(tk), µn2(tk))) ≤ max(Id− ρ)(Θ(tk)), θ0. (5.151)

The proof of Lemma 5.8 is in Appendix F.From the definition in (5.141), the piecewise constant update of the zooming

variables µn1, µn2 causes jumps of e and thus jumps of V . Based on (5.151),we design the zooming-in logic Qin to achieve (5.149) by taking into accountthe jumps.


For convenience of notation, define

W (ξ, s) = V (e(ξ, Υn1(s), Υn2(s))) (5.152)

for ξ ∈ Rn+nz and s ∈ R+, where

Υn1(s) = [Υ11(s), . . . ,Υn1(s)]T , (5.153)

Υn1(s) = [Υ12(s), . . . ,Υn2(s)]T . (5.154)

Then, W (ξ, s) is a continuous function of (ξ, s).Consider (ξ, s) satisfying

0 ≤ s ≤ Θ(tk∗) (5.155)

W (ξ, s) ≤ Θ(tk∗). (5.156)

From the definitions of V andW in (5.121) and (5.152), we can find a compactset Ωo ⊂ Rn+nz ×R+ such that all the (ξ, s) satisfying (5.155)–(5.156) belongto Ωo. By using the property of continuous functions, we can find a continuousand positive definite function ρo < Id such that for all (ξ, s) ∈ Ωo and allh ≥ 0, it holds that

|W (ξ, s− ρo(h)) −W (ξ, s)| ≤ h. (5.157)

We propose the following update law for Θ in the zooming-in stage:

Qin(Θ) = Θ− ρo(

Θ−maxΞ(Θ), θ02

)

, (5.158)

where Ξ = (Id−ρ). In the following procedure, we use Lemma 5.9 to guaranteethe achievement of objective (5.149) and employ Lemma 5.10 to show theconvergence of Θ with the update law defined in (5.158).

Lemma 5.9 shows that property (5.149) can be achieved with the zooming-in update law (5.158) for Θ, given that (5.147) and (5.148) are satisfied.

Lemma 5.9 Consider the closed-loop quantized system with V satisfyingproperty (5.130)–(5.131). Suppose that condition (5.147) holds at some finitetime tk∗ and condition (5.148) holds at some time tk with k ≥ k∗. Then, prop-erty (5.149) is satisfied at time tk+1 with the update law Θ(tk+1) = Qin(Θ(tk))with Qin defined in (5.158).

Proof. With 0 < ρo < Id, it can be guaranteed that

Θ(tk+1) ≤ Θ(tk) (5.159)

and

Θ(tk+1) = Θ(tk)− ρo(

Θ(tk)−maxΞ(Θ(tk)), θ02

)

≥ Θ(tk) + maxΞ(Θ(tk)), θ02

(5.160)


for k ≥ k∗. Thus, 0 < Θ(tk) ≤ Θ(tk∗) for k ≥ k∗.From Lemma 5.8, (5.151) holds. Using (5.147), (5.151), and (5.152), we

have

W (X(tk+1),Θ(tk)) ≤ maxΞ(Θ(tk)), θ0 ≤ Θ(tk∗) (5.161)

for k ≥ k∗. Hence, (X(tk+1),Θ(tk)) ∈ Ωo for k ≥ k∗. Given (X(tk+1),Θ(tk)) ∈Ωo, from (5.158) and (5.161), we obtain

W (X(tk+1),Θ(tk+1))

≤W (X(tk+1),Θ(tk)) + |W (X(tk+1),Θ(tk+1))−W (X(tk+1),Θ(tk))|≤W (X(tk+1),Θ(tk)) + |W (X(tk+1), Q

in(Θ(tk))−W (X(tk+1),Θ(tk))|

≤ maxΞ(Θ(tk)), θ0+Θ(tk)−maxΞ(Θ(tk)), θ0

2

=Θ(tk) + maxΞ(Θ(tk)), θ0

2. (5.162)

From (5.148), we have θ0 ≤ Θ(tk), which implies

θ0 ≤ Θ(tk) + maxΞ(Θ(tk)), θ02

. (5.163)

Properties (5.160), (5.162), and (5.163) together with the definition of Win (5.152) guarantee (5.149). ♦

Lemma 5.10 shows the convergence property of the update law (5.135) forΘ with Q = Qin defined in (5.158).

Lemma 5.10 Suppose that at some tk∗ > 0 with k∗ ∈ Z+, Θ(tk∗) ≥ θ0. Thenwith Qin defined in (5.158), update law Θ(tk+1) = Qin(Θ(tk)) achieves

limk→∞

Θ(tk) = θ0. (5.164)

Proof. Consider the following two cases.

• Ξ(Θ(tk)) ≥ θ0. From the definition of Ξ, one can find a continuous andpositive definite function ρ∗1 such that Ξ ≤ Id−ρ∗1. Then, one can find

a continuous and positive definite function ρ∗2 such that ρo(

s−Ξ(s)2

)

≥ρ∗2(s) for s ∈ R+. In the case of Ξ(Θ(tk)) ≥ θ0, we have

Θ(tk+1) = Θ(tk)− ρo(

Θ(tk)− Ξ(Θ(tk))

2

)

≤ Θ(tk)− ρ∗2(Θ(tk)), (5.165)

which guarantees that there exists a tko > tk with ko ∈ Z+ such thatΞ(Θ(tko)) < θ0.


• 0 ≤ Ξ(Θ(tk)) < θ0. Define Θ′(tk) = Θ(tk)− θ0 for k ∈ Z+. Then, weobtain

Θ′(tk+1) = Θ′(tk)− ρo(

Θ′(tk)2

)

, (5.166)

which is an asymptotically stable first-order discrete-time system[131].

Recall the definition of Ξ. We can see Ξ−1 > Id, Ξ−1(θ0) is larger than θ0 andΘ < Ξ−1(θ0) is an invariant set of system (5.166). Thus, limk→∞ Θ′(tk) = 0and equivalently limk→∞ Θ(tk) = θ0. ♦

The motions of Θ(t) and W (X(t),Θ(t)) are illustrated in Figure 5.14.

tk tk+1 tk+20

θ0

Θ(t)W (t)

FIGURE 5.14 Motions of Θ(t) and W (t) =W (X(t),Θ(t)) in the zooming-in stage.

The zooming-in update law for Θ can be designed by finding the ρ withLemma 5.8 and ρo by using the continuity ofW . Lemmas 5.9 and 5.10 are usedto prove the effectiveness of the zooming-in update law Qin defined in (5.158).With Qin designed, we can design the zooming-in logic Qin

ij for i = 1, . . . , n,j = 1, 2 according to (5.136)–(5.137).

Main Result

Based on the design above, the main result of quantized control is summarizedin Theorem 5.2.

Theorem 5.2 For system (5.73)–(5.75), under Assumptions 5.3–5.6, bychoosing constants ci1, ci2 satisfying (5.115) for i = 1, . . . , n, ISS gains γ

ek′

ek

(k 6= k′) satisfying the cyclic-small-gain condition (5.120) for all i = 1, . . . , n,ISS gains γµk1

ei , γµk2ei for i = 1, . . . , n, k = 1, . . . , i − 1 satisfying (5.125)

and constants ǫi > 0 for i = 1, . . . , n, one can design the functions κi fori = 1, . . . , n in (5.93)–(5.94) and the dynamic quantization logic Qij fori = 1, . . . , n, j = 1, 2 in (5.99) such that the closed-loop solutions z and xare bounded. Moreover, by choosing the constants ǫi > 0 for i = 1, . . . , n to bearbitrarily small, the output x1(t) can be steered to within an arbitrarily smallneighborhood of origin.


Proof. With Assumption 5.3 satisfied, at some time tk∗ > 0 with k∗ ∈ Z+,(5.147) can be achieved by the zooming-out logic Qout

ij .With Assumptions 5.4–5.6 satisfied, using Lemma 5.7, by appropriately

designing the functions κi for i = 1, . . . , n such that the ISS parameters satisfythe conditions (5.115), (5.120), and (5.125), the closed-loop quantized systemhas the nested invariant sets defined in (5.131).

Using Lemmas 5.8 and 5.9, (5.149) can be guaranteed with the designedzooming-in logic, and it holds that V (e(X(tk), µn1(tk), µn2(tk))) ≤ Θ(tk) fork ≥ k∗. Moreover, limk→∞ V (e(X(tk), µn1(tk), µn2(tk))) ≤ θ0 according toLemma 5.10. Recall the definition of V in (5.121). The closed-loop signal x1is driven to within the region |x1| ≤ α−1

V (θ0). Recall the definition of θ0 inLemma 5.7. By designing ǫi (i = 1, . . . , n) to be arbitrarily small, the state x1can be steered to within an arbitrarily small neighborhood of the origin. ♦

If there are no inverse dynamics (i.e., the z-subsystem does not exist) insystem (5.73)–(5.75), the assumption on small-time norm-observability in As-sumption 5.3 is not needed.

5.3 QUANTIZED OUTPUTFEEDBACK CONTROL

By designing a quantized observer, this section studies quantized output-feedback control of nonlinear systems. The main result shows that the outputof the quantized control system can be steered to within an arbitrarily smallneighborhood of the origin even with a three-level uniform quantizer.

Consider the disturbed output-feedback nonlinear system with quantizedoutput:

xi = xi+1 + fi(y, d), i = 1, . . . , n− 1 (5.167)

xn = u+ fn(y, d) (5.168)

y = x1 (5.169)

yq = q(y, µ), (5.170)

where [x1, . . . , xn]T ∈ Rn is the state, u ∈ R is the control input, d ∈ Rnd rep-

resents external disturbance inputs, y ∈ R is the output, q(y, µ) is the outputquantizer with variable µ > 0, yq ∈ R is the quantized output, [x2, . . . , xn]

T

is the unmeasured portion of the state, and fi’s (i = 1, . . . , n) are uncertainlocally Lipschitz continuous functions.

The output quantizer is assumed to satisfy

|y| ≤Mµ⇒ |q(y, µ)− y| ≤ µ, (5.171)

where M > 0, Mµ is the quantization range and µ > 0 is the maximumquantization error when |y| ≤Mµ.

In this section, the basic idea of dynamic quantization is still to appropri-ately update the zooming variable µ during the quantized control procedure


for improved control performance. It is assumed that µ is right-continuouswith respect to time and is updated in discrete-time as:

µ(tk+1) = Q(µ(tk)), k ∈ Z+, (5.172)

where Q : R+ → R+ represents the dynamic quantization logic, and tk ≥ 0with k ∈ Z+ are updating time instants satisfying tk+1 − tk = dt with dt > 0.

Assumptions 5.8–5.11 are made throughout this section.

Assumption 5.8 System (5.167)–(5.169) with u = 0 is forward complete andsmall-time norm-observable with y as the output.

Assumption 5.9 For each fi(y, d) (i = 1, . . . , n) in (5.167)–(5.168), thereexists a known ψfi ∈ K∞ such that for all y, d,

|fi(y, d)| ≤ ψfi(|[y, dT ]T |). (5.173)

Assumption 5.10 There exists a d ≥ 0 such that

|d(t)| ≤ d (5.174)

for t ≥ 0.

Assumption 5.11 Quantizer q(y, µ) satisfies (5.171) with M > 1.

5.3.1 REDUCEDORDER OBSERVER DESIGN

For convenience of notation, denote w = yq − y.Inspired by the reduced-order observer used for measurement feedback con-

trol in Section 4.3, we design the following reduced-order observer which usesthe quantized output yq:

ξi = ξi+1 + Li+1yq − Li(ξ2 + L2y

q), i = 2, . . . , n− 1 (5.175)

ξn = u− Ln(ξ2 + L2yq), (5.176)

where ξi is an estimate for the unmeasured state xi−Liy for each i = 2, . . . , n.Define e0 = [x2−L2y−ξ2, . . . , xn−Lny−ξn]T as the observation error. Then,from (5.167)–(5.170) and (5.175)–(5.176), the observation error system is

e0 = Ae0 + φ0(y, d, w)

:= fe0(e0, y, d, w), (5.177)


where

A =

−L2

...−Ln−1

In−2

−Ln 0 · · · 0

, (5.178)

φ0(y, d, w) =

−L2

...−Ln

In−1

f1(y, d)...

fn(y, d)

+

L22 − L3

...Ln−1L2 − LnLnL2

w.

(5.179)

The real constants Li’s in (5.178) are chosen so that A is Hurwitz, andthus there exists a matrix P = PT > 0 satisfying PA+ ATP = −2In−1. Forφ0 defined in (5.179), using Assumption 5.9, we can find ψyφ0

, ψdφ0, ψwφ0

∈ K∞such that |φ0(y, d, w)|2 ≤ ψyφ0

(|y|) + ψdφ0(|d|) + ψwφ0

(|w|) holds for all y, d, w.Define V0(e0) = eT0 Pe0. Define α0(s) = λmin(P )s

2 and α0(s) = λmax(P )s2

for s ∈ R+. Then, α0(|e0|) ≤ V0(e0) ≤ α0(|e0|) holds for all e0. Direct compu-tation yields:

∇V0(e0)fe0(e0, y, d, w) =− 2eT0 e0 + 2eT0 Pφ0(y, d, w)

≤− eT0 e0 + |P |2|φ0(y, d, w)|2

≤− 1

λmax(P )V0(e0) + |P |2

(

ψyφ0(|y|)

+ ψdφ0(|d|) + ψwφ0

(|w|))

. (5.180)

Define χy0 = 4λmax(P )|P |2ψyφ0, χd0 = 4λmax(P )|P |2ψdφ0

, and χµ0 =

4λmax(P )|P |2ψwφ0. Then, we have

V0(e0) ≥ maxχy0(|y|), χd0(|d|), χµ0 (|w|)⇒∇V0(e0)fe0(e0, y, d, w) ≤ −α0(V0(e0)), (5.181)

where α0(s) = s/4λmax(P ) for s ∈ R+.

5.3.2 QUANTIZED CONTROL DESIGN

The gain assignment technique still plays a central role in quantized output-feedback control design. Consider the following first-order system:

η = φ(η, ω1, . . . , ωn−2) + κ, (5.182)

where η ∈ R is the state, κ ∈ R is the control input, ω1, . . . , ωn−2 ∈ R representexternal disturbance inputs, and the nonlinear function φ(η, ω1, . . . , ωn−2) is


locally Lipschitz and satisfies

|φ(η, ω1, . . . , ωn−2)| ≤ ψφ(|[η, ω1, . . . , ωn−2]T |), (5.183)

with ψφ ∈ K∞ known. Define αV (s) = 12s

2 for s ∈ R+. Notice that sgndenotes the standard sign function.

Lemma 5.11 Consider system (5.182). For any specified 0 < c < 1, ǫ > 0,ℓ > 0, and χω1

η , . . . , χωn−2η ∈ K∞, one can find a continuously differentiable,

odd, strictly decreasing, and radially unbounded κ such that if κ in (5.182)satisfies

κ ∈ κ(η + sgn(η)|ωn|) + δ|ωn−1| : |δ| ≤ 1, (5.184)

where ωn−1, ωn ∈ R represent measurement disturbances, then it holds that

Vη(η) ≥ maxk=1,...,n−2

χωkη (|ωk|), αV

( |ωn−1|c

)

, ǫ

⇒∇Vη(η)(φ(η, ω1, . . . , ωn−2) + κ) ≤ −ℓVη(η). (5.185)

Lemma 5.11 is a set-valued map version of Lemma 4.1 and can be provedin the same way.

Define e1 = y. Consider the [eT0 , e1, ξ2, . . . , ξn]T -system:

e0 = Ae0 + φ0(e1, d, w) (5.186)

e1 = ξ2 + φ1(e0, e1, d) (5.187)

ξi = ξi+1 + φi(e1, ξ2, w), i = 2, . . . , n− 1 (5.188)

ξn = u+ φn(e1, ξ2, w), (5.189)

where

φ1(e0, e1, d) = L2y + (x2 − L2y − ξ2) + f1(y, d)

φi(e1, ξ2, w) = Li+1yq − Li(ξ2 + L2y

q), i = 2, . . . , n− 1

φn(e1, ξ2, w) = −Ln(ξ2 + L2yq).

We get (5.187) from the x1-subsystem (5.167) using the fact that (x2−L2e1−ξ2) is the first element of vector e0. We get (5.188) and (5.189) from (5.175)and (5.176) by using yq = y + w = e1 + w.

We construct a new [eT0 , e1, . . . , en]T -system consisting of ISS subsystems

obtained through a recursive design of the [eT0 , e1, ξ2, . . . , ξn]T -system. The

ISS-Lyapunov function V0 for the e0-subsystem is defined in Subsection 5.3.1.For i = 1, . . . , n, each ei-subsystem is designed with an ISS-Lyapunov functioncandidate

Vi(ei) = αV (|ei|), (5.190)


where αV (s) = s2/2 for s ∈ R+. Denote ei = [eT0 , e1, . . . , ei]T and ξi =

[ξ2, . . . , ξi]T .

In this subsection, we suppose that µ is constant and consider only the caseof |e1| = |y| ≤Mµ. From (5.171), this means |w| = |yq − y| ≤ µ.

The e0subsystem

Define γ10 = χy0 α−1V and χµ0 = χw0 . Then, from (5.181), we have

V0(e0) ≥ maxγ10(V1(e1)), χd0(|d|), χµ0 (µ)⇒∇V0(e0)fe0(e0, y, d, w) ≤ −α0(V0(e0)). (5.191)

The e1subsystem

The e1-subsystem can be rewritten as

e1 = ξ2 − e2 + (φ1(e0, e1, d) + e2)

:= ξ2 − e2 + φ∗1(e2, d)

:= fe1(e2, ξ2, d) (5.192)

with the new state variable e2 to be defined below. From Assumption 5.9and the definition of φ1, we can find a ψφ∗

1∈ K∞ such that |φ∗1(e2, d)| ≤

ψφ∗

1(|[eT2 , dT ]T |).Define a set-valued map S1 as

S1(e1, µ) = κ1(e1 + aµ) : |a| ≤ 1 (5.193)

with κ1 continuously differentiable, odd, strictly decreasing, and radially un-bounded, to be determined later. State variable e2 is defined as

e2 = ~d(ξ2, S1(e1, µ)). (5.194)

Then, we have ξ2 − e2 ∈ S1(e1, µ).For any γ01 , γ

21 ∈ K∞, choose χ0

1 = γ01 α0 and χ21 = γ21 αV . Then,

γ01(V0) = χ01 α−1

0 (V0) ≥ χ01(|e0|) and γ21(V2) = χ2

1 α−1V (V2) = χ2

1(|e2|). WithLemma 5.11, for any specified 0 < c1 < 1, ǫ1 > 0, ℓ1 > 0, γ01 , γ

21 , χ

d1 ∈ K∞, we

can find a continuously differentiable, odd, strictly decreasing, and radiallyunbounded κ1 such that the e1-subsystem with ξ2− e2 ∈ S1(e1, µ) is ISS withV1 satisfying

V1 ≥ max

γ01(V0), γ21(V2), χ

d1(|d|), χµ1 (µ), ǫ1

⇒∇V1(e1)fe1(e2, ξ2, d) ≤ −ℓ1V1, (5.195)

where χµ1 (s) = αV (s/c1) for s ∈ R+.The definition of set-valued map S1 is quite similar with the definition

of the S1 in Section 4.1 and can be represented by Figure 4.2 with the w1

replaced by µ.


The eisubsystem (i = 2, . . . , n)

When i = 3, . . . , n, for each k = 2, . . . , i− 1, a set-valued map Sk is defined as

Sk(e1, ξk, µ) = κk(ξk − pk) : pk ∈ Sk−1(e1, ξk−1, µ), (5.196)

where κk is continuously differentiable, odd, strictly decreasing, and radiallyunbounded; and the new state variable ek+1 is defined as

ek+1 = ~d(ξk+1, Sk(e1, ξk, µ)). (5.197)

It is worth noting that, since κk is strictly decreasing, it holds that

maxSk(e1, ξk, µ) = κk(ξk −maxSk−1(e1, ξk−1, µ)), (5.198)

minSk(e1, ξk, µ) = κk(ξk −minSk−1(e1, ξk−1, µ)). (5.199)

Lemma 5.12 Consider the [eT0 , e1, ξ2, . . . , ξn]T -system in (5.186)–(5.189)

with |e1| ≤Mµ. With Sk(ek, µ) and ek+1 defined in (5.193), (5.194), (5.196),and (5.197) for k = 1, . . . , i − 1, for any variable ei+1, when ei 6= 0, theei-subsystem can be represented as

ei = ξi+1 − ei+1 + φ∗i (ei+1, d, µ, w, ξi), (5.200)

where

|φ∗i (ei+1, d, µ, w, ξi)| ≤ ψφ∗

i(|[eTi+1, d

T , µ]T |) (5.201)

with ψφ∗

i∈ K∞. Specifically, ξn+1 = u.

With the quantized observer designed above, the system in the output-feedback form has been transformed into the strict-feedback form and Lemma5.12 can be proved in the same way as for the strict-feedback system in Section5.2.

Define a set-valued map Si as

Si(e1, ξi, µ) = κi(ξi − pi) : pi ∈ Si−1(e1, ξi−1, µ) (5.202)

with κi continuously differentiable, odd, strictly decreasing, and radially un-bounded, to be defined later. Define ei+1 as

ei+1 = ~d(ξi+1, Si(e1, ξi, µ)). (5.203)

Then, we have ξi+1 − ei+1 ∈ Si(e1, ξi, µ).From the definition of ei (i.e., ek+1 with k = i − 1) in (5.197), in the

case of ei 6= 0, for all pi ∈ Si−1(e1, ξi−1, µ), it holds that |ξi − pi| ≥ |ei|and sgn(ξi − pi) = sgn(ei), which means sgn(ξi − pi − ei) = sgn(ei), andthus ξi − pi = ei + (ξi − pi − ei) = ei + sgn(ei)|ξi − pi − ei|. Note that


ξi+1 − ei+1 ∈ Si(e1, ξi, µ). There always exists a pi ∈ Si−1(e1, ξi−1, µ) suchthat ξi+1 − ei+1 = κi(ξi − pi) = κi(ei + sgn(ei)|ξi − pi − ei|).

With Lemma 5.11, for any ǫi > 0, ℓi > 0, γ0i , . . . , γi−1i , γi+1

i , χdi , χµi ∈ K∞,

we can find a continuously differentiable, odd, strictly decreasing, and radiallyunbounded κi such that the ei-subsystem with ξi+1 − ei+1 ∈ Si(e1, ξi, µ) isISS with Vi satisfying

Vi ≥ maxk=0,...,i−1,i+1

γki (Vk), χdi (|d|), χµi (µ), ǫi

⇒∇Vi(ei)fei(ei+1, ξi+1, d, µ, w) ≤ −ℓiVi. (5.204)

Here, fei represents the dynamics of the ei-subsystem, i.e., ei =fei(ei+1, ξi+1, d, µ, w). By default, Vn+1 := αV (|en+1|). The true control inputu = ξn+1 occurs with the en-subsystem, and we set en+1 = 0.

Realizable Quantized Controller

From (5.204) with i = n, our desired quantized controller u can be chosen inthe following form:

p∗2 = κ1(yq) (5.205)

p∗i = κi−1(ξi−1 − p∗i−1), i = 3, . . . , n (5.206)

u = κn(ξn − p∗n). (5.207)

In the case of |y| ≤ Mµ, we have |w| = |yq − y| ≤ µ and thus κ1(yq) =

κ1(y + w) = κ1(e1 + w) ∈ S1(e1, µ). It is then directly checked that

p∗2 ∈ S1(e1, µ) ⇒· · · ⇒ p∗i ∈ Si−1(e1, ξi−1, µ)

⇒· · · ⇒ u = ξn+1 − en+1 ∈ Sn(e1, ξn, µ),

where en+1 = 0. Thus, if |y| ≤ Mµ, then the quantized control law (5.205)–(5.207) guarantees (5.195) and (5.204).


Denote e = en and ξ = ξn. For i = 0, . . . , n, each ei-subsystem has beenmade ISS (or more precisely, practically ISS). In this subsection, we choosethe ISS-gains such that the e-system satisfies the cyclic-small-gain condition.The gain digraph of the e-system is still in the form shown in Figure 5.13.

According to the recursive design, given the ei−1-subsystem, by designingthe set-valued map Si for the ei-subsystem, we assign the ISS gains γki (k =1, . . . , i− 1) such that

γk+1k γk+2

k+1 · · · γi−1i−2 γii−1 γki < Id. (5.208)

Applying this reasoning repeatedly, the e-system satisfies the cyclic-small-gaincondition.


An ISS-Lyapunov function is constructed as:

V (e) = maxi=0,...,n

σi(Vi(ei)) (5.209)

with σ1(s) = s, σi(s) = γ21 · · · γii−1(s) (i = 2, . . . , n), and σ0(s) =

maxi=1,...,nσi γ0i (s) for s ∈ R+, where the γ(·)(·) ’s are K∞ functions con-

tinuously differentiable on (0,∞) and slightly larger than the corresponding

γ(·)(·) ’s and still satisfy the cyclic-small-gain condition.

Recall that |d| ≤ d. Denote ǫ0 = 0. We represent the maximal influence ofd, µ, and ǫi (i = 1, . . . , n) as

θ = maxi=0,...,n

σi χdi (d), σi χµi (µ), σi(ǫi)

. (5.210)

Using the Lyapunov-based cyclic-small-gain theorem, we achieve that if|y| ≤ Mµ, then the e-system with quantized control law (5.205)–(5.207) sat-isfies

V (e) ≥ θ ⇒ ∇V (e)fe(e, ξ, d, µ, w) ≤ −α(V (e)) (5.211)

holds wherever ∇V (e) exists, with α positive definite. Note that ∇V (e) existsalmost everywhere. Here, fe represents the dynamics of the e-system, i.e.,e = fe(e, ξ, d, µ, w).

In the recursive design approach, we can make the γ(·)(·) ’s (and thus the

γ(·)(·) ’s) arbitrarily small to get arbitrarily small σi’s (i = 0, 2, . . . , n). We can

also select the χdi ’s (i = 0, . . . , n), the ǫi’s (i = 1, . . . , n), and the χµi ’s (i =0, 2, . . . , n) to be arbitrarily small. In this way, for arbitrarily small θ0 > 0,we can design the gains such that maxi=1,...,n

σi χdi (d), σi(ǫi)

≤ θ0 andmaxi=0,2,...,n σi χµi (µ) ≤ θ0.

Recall that χµ1 (s) = αV (s/c1) for s ∈ R+ defined in (5.195). If |y| ≤ Mµ,then quantized control law (5.205)–(5.207) guarantees

V (e) ≥ max αV (µ/c1), θ0 ⇒ ∇V (e)fe(e, ξ, d, µ, w) ≤ −α(V (e)) (5.212)

wherever ∇V (e) exists.

5.3.4 DYNAMIC QUANTIZATION AND MAIN RESULT

Define Θ = αV (Mµ). Then, µ = α−1V (Θ)/M . The update law for µ can

be determined by designing an update law for Θ. Denote x = [x1, . . . , xn]T

and ξ = [ξ2, . . . , ξn]T . Recall that e = [e0, . . . , en]

T and the definition ofei for i = 0, . . . , n. The transformed state variable e can be considered asa continuous function of x, ξ, µ. By using µ = α−1

V (Θ)/M , we can denotee = e(x, ξ,Θ). In dynamic quantization, Θ is piecewise updated on the timelineand denoted as Θ(t). Clearly, the piecewise update of Θ leads to jumps of e.

Some of the results in this section can be considered as special cases ofSubsection 5.2.8 and are presented without detailed proofs.


ZoomingOut Stage

In this stage, the control input u and the state ξ of the observer are setto be zero. The small-time norm-observability assumed in Assumption 5.8guarantees that for dt > 0, there exists a ϕ ∈ K∞ such that

|x(tk + dt)| ≤ ϕ(‖y‖[tk,tk+dt]) (5.213)

for all k ∈ Z+. Considering the definitions of V and e, for dt > 0, there existsa ϕ ∈ K∞ such that

|V (e(x(tk + dt), 0, 0))| ≤ ϕ(‖y‖[tk,tk+dt]) (5.214)

for all k ∈ Z+.The forward completeness assumed in Assumption 5.8 guarantees that we

can increase Θ fast enough to dominate the growth rate of ϕ(|y|). Thus, wecan design the zooming-out logic to increase Θ (and thus µ) fast enough suchthat at some time tk∗ > 0 with k∗ ∈ Z+, it holds that

Θ(tk∗) ≥ ϕ(‖y‖[tk∗−dt,tk∗ ]) ≥ maxV (e(x(tk∗), 0, 0)), θ0. (5.215)

From the definition of Si in (5.193) and (5.202), it can be observed thatan increase of µ (and thus Θ) leads to an increase of maxSi and a decreaseof minSi. Using the definition of ei+1, an increase of Θ leads to a decreaseor hold of |ei+1| (and thus a decrease or hold of V (e)). Note that ξ(tk∗) = 0.From (5.215), we achieve

Θ(tk∗) ≥ maxV (e(x(tk∗ ), ξ(tk∗),Θ(tk∗))), θ0. (5.216)

ZoomingIn Stage

With the help of Assumption 5.11, in the recursive control design procedure,we can choose c1 satisfying 1/M < c1 < 1. Then, one can find a positivedefinite ρz1 such that

αV (µ/c1) ≤ (Id− ρz1)(Θ). (5.217)

Suppose that at some time tk > 0 with k ∈ Z+, it holds that

Θ(tk) ≥ maxV (e(x(tk), ξ(tk),Θ(tk))), θ0. (5.218)

We want to find a QΘin : R+ → R+ such that Θ(tk+1) = QΘ

in(Θ(tk)) satisfies

Θ(tk+1) ≥ maxV (e(x(tk+1), ξ(tk+1),Θ(tk+1))), θ0, (5.219)

where tk+1 − tk = dt.One can find a positive definite ρz2 such that (Id − ρz2) ∈ K∞ and (Id −

ρz2)(s) ≥ max(Id− ρz1)(s), s− dt ·min(Id−ρz1)(s)≤v≤s α(V ) for s ∈ R+. Define

Ξ = Id− ρz2. (5.220)


Condition (5.218) implies that V (e(x(tk), ξ(tk),Θ(tk))) ≤ αV (Mµ(tk)).From (5.212) and (5.217), if (5.218) holds, then

V (e(x(tk+1), ξ(tk+1),Θ(tk))) ≤ maxΞ(Θ(tk)), θ0. (5.221)

Using the property of continuous functions, we can find a positive definiteρz3 < Id such that for all x ∈ Rn, ξ ∈ Rn−1, Θ > 0, and h ≥ 0, it holds that

|V (e(x, ξ,Θ − ρz3(h)))− V (e(x, ξ,Θ))| ≤ h. (5.222)

Define

ΘΘin(Θ) = Θ− ρz3

(

Θ−maxΞ(Θ), θ02

)

. (5.223)

Then, (5.221), (5.222), and (5.223) imply

V (e(x(tk+1), ξ(tk+1),Θ(tk+1))) ≤Θ(tk) + maxΞ(Θ(tk)), θ0

2, (5.224)

and (5.218) and (5.223) imply

Θ(tk+1) ≥Θ(tk) + maxΞ(Θ(tk)), θ0

2≥ θ0. (5.225)

Properties (5.224) and (5.225) together guarantee (5.219).

Lemma 5.13 Suppose that Θ(tk∗) ≥ θ0 with k∗ ∈ Z+. Then, with zooming-in logic Θ(tk+1) = QΘ

in(Θ(tk)) for k ∈ Z+ with QΘin defined in (5.223), it holds

that

limk→∞

Θ(tk) = θ0. (5.226)

Lemma 5.13 can be proved in the same way as Lemma 5.10.With the appropriately designed zooming-in logic in (5.223), it always holds

that V (e(x(t), ξ(t),Θ(t))) ≤ Θ(t). Thus, the closed-loop signals are bounded.By using (5.226), we have

limk→∞

V (e(x(t), ξ(t),Θ(t))) = θ0. (5.227)

Recall the definition of V in (5.209). It can be observed that y = x1 = e1ultimately converges to within the region |y| ≤ α−1

V (θ0). By choosing θ0 tobe arbitrarily small, output y can be steered to within an arbitrarily smallneighborhood of the origin.

Recall that Θ = αV (Mµ). With QΘin defined in (5.223), the zooming-in

logic for µ is designed as

Q(µ) = Qin(µ) =1

Mα−1V QΘ

in αV (Mµ). (5.228)

The main result of this section is summarized in Theorem 5.3.


Theorem 5.3 Consider system (5.167)–(5.170) with output quantization sat-isfying (5.171). Under Assumptions 5.8–5.11, the closed-loop signals arebounded, and in particular, the output y can be steered to within an arbitrarilysmall neighborhood of the origin with the quantized output-feedback controllercomposed of reduced-order observer (5.175)–(5.176), control law (5.205)–(5.207), and dynamic quantization in the form of (5.172) with zooming-indynamics Q = Qin defined in (5.228).

The block diagram of the quantized output-feedback control system de-signed in this section is shown in Figure 5.15. Interested readers may trydesigning quantized controllers with actuator quantization by combining thedesigns in this section and Section 5.2. Notice that the recent paper [191]presents a result on output-feedback control of nonlinear systems with actu-ator quantization.

observer

control law plant

quantizer

yu

yq

ξ2, . . . , ξn

FIGURE 5.15 Quantized output-feedback control.

5.4 NOTES

Recent years have seen considerable efforts devoted to quantized control oflinear and nonlinear systems. Reference [55] studied quantized stabilization ofsingle-input single-output (SISO) linear systems with the coarsest quantizers,and showed that the coarsest quantizer should follow a logarithmic law forquadratic stabilization. By characterizing the coarsest quantizer as a sectorbounded uncertainty, the authors of [67] considered the quantized control ofmulti-input multi-output (MIMO) linear systems and analyzed the robust-ness of the quantized control systems. An early result on quantized controlof nonlinear systems with logarithmic quantizers appeared in [175], in whichthe general idea of using (robust) control Lyapunov functions to design (ro-bust) quantized controllers is employed. Reference [26] studied the conditionsunder which a logarithmic quantizer does not cancel the stabilizing effect ofa continuous feedback control law, for quantized control of dissipative nonlin-ear systems. In [26], set-valued maps are employed to overcome the problemscaused by the discontinuity of the quantizers.

Based on the idea of scaling quantization levels, the authors of [19, 163,168, 164] studied quantized control of linear and nonlinear systems with dy-


namic quantization. If the quantizer admits a finite number of levels, thenthe quantization error is large if the original signal is outside the range of thequantizer. To deal with this problem, a growth condition should be satisfiedby the quantization error and the quantized signal [209, 163]. Reference [46]presents a semiglobal stabilization result for nonlinear systems in the feedfor-ward form. Reference [163] clarifies the relation between ISS and quantizedcontrol. ISS with respect to quantization error appears to be fundamentalin several results on quantized control; see e.g., Liberzon and Nesic’s results[163, 165, 166, 168]. Reference [209] established a unified framework for con-trol design of nonlinear systems with quantization and time scheduling viaan emulation-like approach, with the ISS small-gain theorem [130] (see alsoChapter 2) as a tool. For the systems with partial state information, a quan-tized output-feedback control strategy was developed [166].

Because of the discontinuity of the quantizers, a closed-loop quantized sys-tem is basically a discontinuous system and can be modeled by differentialinclusions. Then, the extended Filippov solution introduced in [84] can beused to represent the motion of such systems. Based on the extended Filippovsolution, the cyclic-small-gain theorem proposed in Chapter 3 can be directlygeneralized to dynamic networks described by differential inclusions. Detaileddiscussions on discontinuous systems and the Filippov solution can be foundin [60, 35]. Appendix B gives a brief introduction to the related notions andthe basic results.

The sector bound approach presented in this chapter for static quantizationis motivated by [67, 26], which directly assume the sector bound property ofthe quantizers instead of discussing the nonlinearity and the discontinuity ofthe quantizers in details. In this chapter, set-valued maps have been employedto cover the sector bound of the quantizers such that the closed-loop quantizedsystem is transformed into a network of ISS subsystems. Then, the quantizedcontrol design is finalized with the cyclic-small-gain theorem, and the influenceof the quantization errors are explicitly represented by ISS gains.

System (5.73)–(5.75) represents an important class of minimum-phase non-linear systems, which have been studied extensively by many authors inthe context of (non-quantized) robust and adaptive nonlinear control. Thereader may consult [153, 222, 262] and references therein for the details. Forsemiglobal quantized stabilization of system (5.73)–(5.75), it is assumed thatthe uncontrolled system is final-state norm-observable such that the quantizedcontroller can estimate an upper bound of the internal state in some finite timeat the zooming-out stage. Reference [85] discussed the equivalent characteri-zations of initial-state norm-observability, final-state norm-observability, andKL norm-observability for forward complete (or unboundedness observable)nonlinear systems with external inputs. By means of dynamic quantization,an n-dimensional strict-feedback nonlinear system with measurement and ac-tuator quantization can be semiglobally stabilized (with global convergenceof the closed-loop state signals) by a quantized controller with 2n three-level


dynamic quantizers. When the strict-feedback system is reduced to the output-feedback form, quantized output-feedback control is solved by introducing aquantized observer. This result is introduced in Section 5.3.

In spite of the obtained results, several related problems should be ad-dressed in the future research:

• Quantized control is closely related to other network control problemssuch as sampled-data control and control with time-delays. How todeal with more complicated network behaviors in a systematic way,in particular those hybrid/switching systems satisfying only a weaksemigroup property (see [137]), should be studied in greater detail.

• The cyclic-small-gain theorem was originally developed for large-scalesystems. It is thus very natural to ask whether decentralized quan-tized controllers can be developed for a class of large-scale nonlinearsystems.

• Controllers are expected to possess adaptive capabilities to cope with“large” system uncertainties. A further extension of the presentedmethodology to quantized adaptive control is of practical interest forengineering applications.

More discussions on quantized control of nonlinear uncertain systems can befound in [122, 191, 189, 190] as well as Chapter 7.

6 Distributed NonlinearControl

The spatially distributed structure of complex systems motivates the ideaof distributed control. In a distributed control system, the subsystems arecontrolled by local controllers through information exchange with neighbor-ing agents for coordination purposes. Formation control of mobile robots is anexample. The major difficulties of distributed control are due to complex char-acteristics such as nonlinearity, dimensionality, uncertainty, and informationconstraints. This chapter develops small-gain methods for distributed controlof nonlinear systems.

The discussion in this chapter starts with an example of a multi-vehicleformation control system in which each vehicle is modeled by an integra-tor. In the case of leader-following with fixed topology, it is shown that theproblem can be transformed into the stability problem of a specific dynamicnetwork composed of ISS subsystems. This motivates a cyclic-small-gain re-sult in digraphs, which is given in Section 6.1. It is shown that the new resultis extremely useful for distributed control of nonlinear systems. Specifically,Section 6.2 presents a cyclic-small-gain design for distributed output-feedbackcontrol of nonlinear systems. In Section 6.3, we study the distributed forma-tion control problem of nonholonomic mobile robots with a fixed informationexchange topology. An extension to the case of flexible topology is developedin Section 6.4.

Example 6.1 Consider a group of N + 1 vehicles (multi-vehicle system) asshown in Figure 6.1, with each vehicle modeled by an integrator:

xi = vi, i = 0, . . . , N, (6.1)

where xi ∈ R is the position and vi ∈ R is the velocity of the i-th vehicle. Thevehicle with index 0 is the leader while the other vehicles are the followers.The objective is to control the follower vehicles to specific positions relative tothe leader by adjusting the velocities vi for i = 1, . . . , N . More specifically, itis required that

limt→∞

(xi(t)− xj(t)) = dij , i, j = 0, . . . , N, (6.2)

where constants dij represent the desired relative positions. Clearly, to definethe problem well, dij = dik + dkj for any i, j, k = 0, . . . , N and dij = −djifor any i, j = 0, . . . , N . Also, by default, dii = 0 for any i = 0, . . . , N . In theliterature of distributed control, the vehicles are usually considered as agentsand the multi-vehicle system is studied as a multi-agent system.

193


· · · · · ·

x0 xi xN

v0 vi vN

FIGURE 6.1 A multi-vehicle system.

Compared with global positions, relative positions between the vehicles areoften easily measurable in practice, and are used for feedback in this example.Considering the position information exchange, agent j is called a neighborof agent i if (xi − xj) is available to agent i, and Ni ⊆ 0, . . . , N is used todenote the set of agent i’s neighbors. We consider the case where each vehicleonly uses the position differences with the vehicles right before and after it,i.e., Ni = i− 1, i+ 1 for i = 1, . . . , N − 1 and NN = N − 1.

Define xi = xi − x0 − di0 and vi = vi − v0. By taking the derivative of xi,we have

˙xi = vi, i = 1, . . . , N. (6.3)

According to the definition of xi, xi − xj = xi − xj − dij . Thus, the controlobjective is achieved if limt→∞(xi− x0) = 0. Also, (xi− xj) is available to thecontrol of the xi-subsystem if (xi − xj) is available to agent i. This problemis normally known as the consensus problem. If the position information ex-change topology has a spanning tree with agent 0 as the root, then the followingdistributed control law is effective:

vi = ki∑

j∈Ni

(xj − xi), (6.4)

where ki is a positive constant. Moreover, if the velocities vi are required tobe bounded, one may modify (6.4) as

vi = ϕi

∑

j∈Ni

(xj − xi)

, (6.5)

where ϕi : R → [vi, vi] with constants vi < 0 < vi is a continuous, strictlyincreasing function satisfying ϕi(0) = 0. With control law (6.5), vi ∈ [v0 +vi, v0 + vi]. The validity of the control laws defined by (6.4) and (6.5) can bedirectly verified by using the state agreement result in [172].

With control law (6.5), each xi-subsystem can be rewritten as

˙xi = ϕi

∑

j∈Ni

xj −Nixi

:= fi(x), (6.6)

Distributed Nonlinear Control 195

where Ni is the size of Ni and x = [x0, . . . , xN ]T . Define Vi(xi) = |xi| as anISS-Lyapunov function candidate for the xi-subsystem for i = 1, . . . , N . It canbe verified that for any δ > 0, there exists a continuous, positive definite αsuch that

Vi(xi) ≥1

(1− δi)Ni

∑

j∈Ni

Vj(xj) ⇒ ∇Vi(xi)fi(x) ≤ −αi(Vi(xi)) a.e., (6.7)

where, for convenience of notation, V0(x0) = 0. This shows the ISS of eachxi-subsystem with i = 1, . . . , N . If the network of ISS subsystems is asymp-totically stable, then the control objective is achieved.

We employ a digraph Gf to represent the underlying interconnection struc-ture of the dynamic network. The vertices of the digraph correspond to agents1, . . . , N , and for i, j = 1, . . . , N , directed edge (j, i) exists in the graph if andonly if xj is an input of the xi-subsystem. We use N i to represent the setof neighbors of agent i in Gf . Then, it is directly verified that N i = Ni\0.Recall that V0(x0) = 0. Then, the Ni in (6.7) can be directly replaced by N i.Figure 6.2 shows the digraph Gf for the case in which each follower vehicleuses the position differences with the vehicles right before and after it.

1 2 3 · · · N

FIGURE 6.2 An example of information exchange digraph Gf , for which each vehicle

uses the position differences with the vehicles right before and after it. In this figure,

N i = i− 1, i+ 1 for i = 2, . . . , N − 1, N 1 = 2 and NN = N − 1.

Notice that for any positive constants a1, . . . , an satisfying∑n

i=1 1/ai ≤n, it holds that

∑ni=1 di =

∑ni=1(1/ai)aidi ≤ nmaxi=1,...,naidi for all

d1, . . . , dn ≥ 0. Then, property (6.7) implies

Vi(xi) ≥N i

(1 − δi)Nimaxj∈N i

aijVj(xj) ⇒ ∇Vi(xi)fi(x) ≤ −αi(Vi(xi)), (6.8)

where N i is the size of N i and aij are positive constants satisfying∑

j∈N i1/aij ≤ N i. It can be observed that Ni = N i + 1 if 0 ∈ Ni and

Ni = N i if 0 /∈ Ni.Given specific aij > 0, one can test the stability property of the closed-loop

system by directly checking whether the cyclic-small-gain condition is satisfied.But, for a specific Gf , can we find appropriate coefficients aij to satisfy thecyclic-small-gain condition, and how?

It should be noted that the effectiveness of control law (6.5) can be provedby using the result in [172]. Here, our objective is to transform the probleminto a stability problem of dynamic networks, and develop a result which ishopefully useful for more general distributed control problems.

To answer the question in Example 6.1, a cyclic-small-gain result in di-graphs is developed in Section 6.1.


6.1 A CYCLICSMALLGAIN RESULT IN DIGRAPHS

Consider a digraph Gf which has N vertices. For i = 1, . . . , N , define N i suchthat if there is a directed edge (j, i) from the j-th vertex to the i-th vertex,then j ∈ N i. Each edge (j, i) is assigned a positive variable aij . For a simplecycle O of Gf , denote AO as the product of the positive values assigned to theedges of the cycle. For i = 1, . . . , N , denote C(i) as the set of simple cycles ofGf through the i-th vertex.

Lemma 6.1 If the digraph Gf has a spanning tree Tf with vertices i∗1, . . . , i∗q

as the roots, then for any ǫ > 0, there exist aij > 0 for i = 1, . . . , N , j ∈ N i,such that

∑

j∈N i

1

aij≤ N i, i = 1, . . . , N (6.9)

AO < 1 + ǫ, O ∈ C(i∗1) ∪ · · · ∪ C(i∗q) (6.10)

AO < 1, O ∈

⋃

i=1,...,N

C(i)

\(

C(i∗1) ∪ · · · ∪ C(i∗q))

, (6.11)

where N i is the size of N i.

Proof. We only consider the case of q = 1. The case of q ≥ 2 can be provedsimilarly. Denote i∗ as the root of the tree.

Define a0ij = 1 for 1 ≤ i ≤ N , j ∈ N i. If aij = a0ij for 1 ≤ i ≤ N , j ∈ N i,then

∑

j∈N i

1

a0ij≤ N i, i = 1, . . . , N (6.12)

AO = 1, O ∈⋃

i=1,...,N

C(i). (6.13)

Consider one of the paths leading from root i∗ in the spanning tree Tf .Denote the path as (p1, . . . , pm) with p1 = i∗.

One can find a1p2p1 = a0p2p1 + ǫ0p2p1 > 0 with ǫ0p2p1 > 0 and a1p2j = a0p2j −ǫp2j > 0 with ǫp2j > 0 for j ∈ N p2\p1 such that if aij = a1ij for i = p2 and

aij = a0ij for i 6= p2, then (6.12) is satisfied, and also

AO < 1 + ǫ′ for O ∈ C(p1), (6.14)

AO < 1 for O ∈ C(p2)\C(p1) (6.15)

with 0 < ǫ′ < ǫ.Then, one can find a1p3p2 = a0p3p2 + ǫ0p3p2 > 0 with ǫ0p3p2 > 0 and a1p3j =

a0p3j − ǫ0p3j > 0 with ǫ0p3j > 0 for j ∈ N p3\p2 such that if aij = a1ij for


i ∈ p2, p3, and aij = a0ij for i /∈ p2, p3, then (6.12) is satisfied, and also

AO < 1 + ǫ′′ for O ∈ C(p1), (6.16)

AO < 1 for O ∈ (C(p2) ∪ C(p3)) \C(p1) (6.17)

with 0 < ǫ′ ≤ ǫ′′ < ǫ.By doing this for i = p2, . . . , pm, we can find a1ij > 0 for i ∈ p2, . . . , pm,

j ∈ N i, such that

AO < 1 + ǫ1 for O ∈ C(p1), (6.18)

AO < 1 for O ∈ (C(p2) ∪ · · · ∪ C(pm)) \C(p1) (6.19)

with 0 < ǫ0 < ǫ.By considering each path leading from the root i∗ in the spanning tree one-

by-one, we can find a1ij > 0 for i ∈ 1, . . . , N, j ∈ N i, such that if aij = a1ijfor i ∈ 1, . . . , N, j ∈ N i, then (6.12) and (6.11) are satisfied and

AO < 1 + ǫ1 for O ∈ C(i∗1) ∪ · · · ∪ C(i∗q), (6.20)

where 0 < ǫ1 < ǫ.Note that the left-hand sides of inequalities (6.9), (6.10), and (6.11) con-

tinuously depend on aij for i ∈ 1, . . . , N, j ∈ N i. One can find a2ij > 0 for

i ∈ 1, . . . , N, j ∈ N i, such that if aij = a2ij for i ∈ 1, . . . , N, j ∈ N i, thenconditions (6.9), (6.10), and (6.11) are satisfied. ♦

Example 6.2 Continue Example 6.1. Define L = i ∈ 1, . . . , N : 0 ∈ Ni.Considering the relation between Ni and N i, and N i ≤ N , the cyclic-small-gain condition can be satisfied by the network of ISS subsystems with property(6.8) if

AO <(1− δ)N (N + 1)

N, O ∈

⋃

i∈LC(i), (6.21)

AO < (1− δ)N , O ∈

⋃

i∈1,...,NC(i)

\(

⋃

i∈LC(i)

)

, (6.22)

where δ = maxi=1,...,Nδi.By using Lemma 6.1, if graph Gf has a spanning tree with the agents be-

longing to L as the roots, one can find a constant δ > 0 and constants aij > 0satisfying

∑

j∈N i1/aij ≤ N i such that conditions (6.21) and (6.22) are sat-

isfied. The graph shown in Figure 6.2 satisfies this condition.

Lemma 6.1 proves very useful in constructing distributed controllers fornonlinear agents to achieve convergence of their outputs to an agreementvalue. It provides for a form of gain assignment in the network coupling.


6.2 DISTRIBUTED OUTPUTFEEDBACK CONTROL

In this section, the basic idea of cyclic-small-gain design for distributed con-trol is generalized to high-order nonlinear systems. Consider a group of Nnonlinear agents, of which each agent i (1 ≤ i ≤ N) is in the output-feedbackform:

xij = xi(j+1) +∆ij(yi, wi), 1 ≤ j ≤ ni − 1 (6.23)

xini= ui +∆ini

(yi, wi) (6.24)

yi = xi1, (6.25)

where [xi1, . . . , xini]T := xi ∈ R

ni with xij ∈ R (1 ≤ j ≤ ni) is the state,ui ∈ R is the control input, yi ∈ R is the output, [xi2, . . . , xini

]T is theunmeasured portion of the state, wi ∈ R

nwi represents external disturbances,and ∆ij ’s (1 ≤ j ≤ ni) are unknown locally Lipschitz functions.

The objective of this section is to develop a new class of distributed con-trollers for the multi-agent system based on available information such thatthe outputs yi for 1 ≤ i ≤ N converge to the same desired agreement valuey0. This problem is called the output agreement problem in this book.

In Section 4.3, decentralized control was developed such that a group ofnonlinear systems can be stabilized despite the nonlinear interconnections be-tween them. Different from decentralized control, the major objective of dis-tributed control is to control the agents in a coordinated way for some desiredgroup behavior. For the output agreement problem, the objective is to controlthe agents so that the outputs converge to a desired common value. Informa-tion exchange between the agents is required for coordination purposes. Inpractice, the information exchange is subject to constraints. As considered inExample 6.1, the position x0 of the leader vehicle is only available to some ofthe follower vehicles, and the formation control objective is achieved throughinformation exchange between the neighboring vehicles.

For distributed control of the multi-agent nonlinear system (6.23)–(6.25),we employ a digraph Gc to represent the information exchange topology be-tween the agents. Digraph Gc contains N vertices corresponding to the Nagents andM directed edges corresponding to the information exchange links.Specifically, if yi − yk is available to the local controller design of agent i,then there is a directed link from agent k to agent i and agent k is called aneighbor of agent i; otherwise, there is no link from agent k to agent i. SetNi ⊆ 1, . . . , N is used to represent agent i’s neighbors. In this section, anagent is not considered as a neighbor of itself and thus i /∈ Ni for 1 ≤ i ≤ N .Agent i is called an informed agent if it has access to the knowledge of theagreement value y0 for its local controller design. Let L ⊆ 1, . . . , N representthe set of all the informed agents.

The following assumption is made on the agreement value and system(6.23)–(6.25).

Assumption 6.1 There exists a nonempty set Ω ⊆ R such that


1. y0 ∈ Ω;2. for each 1 ≤ i ≤ N , 1 ≤ j ≤ ni,

|∆ij(yi, wi)−∆ij(zi, 0)| ≤ ψ∆ij(|[yi − zi, w

Ti ]T |) (6.26)

for all [yi, wTi ]T ∈ R

1+nwi and all zi ∈ Ω, where ψ∆ij∈ K∞ is Lipschitz on

compact sets and known.

It should be noted that a priori information on the bounds of y0 (and thusΩ) is usually known in practice. In this case, condition 2 in Assumption 6.1can be guaranteed if for each zi, there exists a ψzi∆ij

∈ K∞ that is Lipschitzon compact sets such that

|∆ij(yi, wi)−∆ij(zi, 0)| = |∆ij((yi − zi) + zi, wi)−∆ij(zi, 0)|≤ ψzi∆ij

(|[yi − zi, wTi ]T |). (6.27)

Then, ψ∆ijcan be defined as ψ∆ij

(s) = supzi∈Ω ψzi∆ij

(s) for s ∈ R+. In fact,

there always exists a ψzi∆ij∈ K∞ that is Lipschitz on compact sets to fulfill

condition (6.27) if ∆ij is locally Lipschitz.It is also assumed that the external disturbances are bounded.

Assumption 6.2 For each i = 1, . . . , N , there exists a wi ≥ 0 such that

|wi(t)| ≤ wi (6.28)

for all t ≥ 0.

The basic idea is to design observer-based local controllers for the agentssuch that each controlled agent i is IOS, and moreover, has the UO property.Then, the cyclic-small-gain theorem in digraphs can be used to guarantee theIOS of the closed-loop multi-agent system and then the achievement of outputagreement.

By introducing a dynamic compensator

ui = vi (6.29)

and defining x′i1 = yi − y0 and x′i(j+1) = xi(j+1) +∆ij(y0, 0) for 1 ≤ j ≤ ni,

we can transform each agent i defined by (6.23)–(6.25) into the form of

x′ij = x′i(j+1) +∆ij(yi, wi)−∆ij(y0, 0), 1 ≤ j ≤ ni + 1 (6.30)

x′ini= vi +∆ini

(yi, wi)−∆ini(y0, 0) (6.31)

y′i = x′i1 (6.32)

with the output tracking error y′i = yi − y0 as the new output and vi as thenew control input.


Moreover, the dynamic compensator (6.29) guarantees that the origin is anequilibrium of the transformed agent system (6.30)–(6.32) if it is disturbance-free, and the distributed control objective can be achieved if the equilibriumat the origin of each transformed agent system is stabilized.

The local controller for each agent i is designed by directly using the avail-able ymi , defined as follows:

ymi =1

Ni + 1

(

∑

k∈Ni

(yi − yk) + (yi − y0)

)

, i ∈ L (6.33)

ymi =1

Ni

∑

k∈Ni

(yi − yk), i ∈ 1, . . . , N\L, (6.34)

where Ni is the size of Ni. For convenience of discussions, we represent ymiwith the new outputs as

ymi = y′i − µi (6.35)

with

µi =1

Ni + 1

∑

k∈Ni

y′k, i ∈ L (6.36)

µi =1

Ni

∑

k∈Ni

y′k, i ∈ 1, . . . , N\L. (6.37)

6.2.1 DISTRIBUTED OUTPUTFEEDBACK CONTROLLER

Owing to the output-feedback structure, we design a local observer for eachtransformed agent system (6.30)–(6.32):

ξi1 = ξi2 + Li2ξi1 + ρi1(ξi1 − ymi ) (6.38)

ξij = ξi(j+1) + Li(j+1)ξi1 − Lij(ξi2 + Li2ξi1), 2 ≤ j ≤ ni (6.39)

ξi(ni+1) = vi − Li(ni+1)(ξi2 + Li2ξi1), (6.40)

where ρi1 : R → R is an odd and strictly decreasing function, and Li2, . . . , Lini

are positive constants. In the observer, ξi1 is an estimate of y′i, and ξij is anestimate of x′ij − Lijy

′i for 2 ≤ j ≤ ni + 1.

Here, equation (6.38) is constructed to estimate y′i by using ymi which isinfluenced by the outputs y′k (k ∈ Ni) of the neighbor agents (see (6.35)). Thenonlinear function ρi1 in (6.38) is used to assign an appropriate nonlineargain to the observation error system. As shown later, it is the key to makingeach controlled agent IOS with specific gains satisfying the cyclic-small-gaincondition. Equations (6.39)–(6.40) of the observer are in the same spirit of thereduced-order observer in Section 4.3. Slightly differently, we use ξi1 insteadof the unavailable y′i in (6.39)–(6.40).


With the estimates, a nonlinear local control law is designed as

ei1 = ξi1, (6.41)

eij = ξij − κi(j−1)(ei(j−1)), 2 ≤ j ≤ ni + 1 (6.42)

vi = κi(ni+1)(ei(ni+1)), (6.43)

where κi1, . . . , κi(ni+1) are continuously differentiable, odd, strictly decreasing,and radially unbounded functions.

Consider Zi = [x′i1, . . . , x′i(ni+1), ξi1, . . . , ξi(ni+1)]

T as the internal state of

each controlled agent composed of the transformed agent system (6.30)–(6.32)and the local observer-based controller (6.38)–(6.43). The block diagram ofcontrolled agent i with µi as the input and y

′i as the output is shown in Figure

6.3.

−+ymilocal observerlocal control law

vi ∫ uiagent i

yi +−y0

y′i

µi

FIGURE 6.3 The block diagram of each controlled agent i.

The following proposition presents the UO and IOS properties of each con-trolled agent i.

Proposition 6.1 Each controlled agent i composed of (6.30)–(6.32) and(6.38)–(6.43) has the following UO and IOS properties with µi as the inputand y′i as the output: for all t ≥ 0,

|Zi(t)| ≤ αUO

i (|Zi0|+ ‖µi‖[0,t]) (6.44)

|y′i(t)| ≤ max

βi(|Zi0|, t), χi(‖µi‖[0,t]), γi(‖wi‖[0,t])

, (6.45)

for any initial state Zi(0) = Zi0 and any µi, wi, where βi ∈ KL and χi, γi, αi ∈K∞. Moreover, γi can be designed to be arbitrarily small, and for any specifiedconstant bi > 1, χi can be designed such that χi(s) ≤ bis for all s ≥ 0.

The proof of Proposition 6.1 is given in Subsection 6.2.4.


With the proposed distributed output-feedback controller, the closed-loopmulti-agent system has been transformed into a network of IOS subsystems.This subsection presents the main result of output agreement and provides aproof based on the cyclic-small-gain result in digraphs.


Theorem 6.1 Consider the multi-agent system in the form of (6.23)–(6.25)satisfying Assumptions 6.1 and 6.2. If there is at least one informed agent,i.e., L 6= ∅, and the communication digraph Gc has a spanning tree with theinformed agents as the roots, then we can design distributed observers (6.38)–(6.40) and distributed control laws (6.29), (6.41)–(6.43) such that all the sig-nals in the closed-loop multi-agent system are bounded, and the output yi ofeach agent i can be steered to within an arbitrarily small neighborhood of thedesired agreement value y0. Moreover, if wi = 0 for i = 1, . . . , N , then eachoutput yi asymptotically converges to y0.

Proof. Notice that for any constants a1, . . . , an > 0 satisfying∑n

i=1(1/ai) ≤ n,it holds that

n∑

i=1

di =

n∑

i=1

1

aiaidi ≤ n max

1≤i≤naidi (6.46)

for all d1, . . . , dn ≥ 0.Recall the definition of µi in (6.36) and (6.37). We have

|µi| ≤ δi maxk∈Ni

aik|y′k|, (6.47)

where δi = Ni

Ni+1 if i ∈ L, δi = 1 if i /∈ L, and aik are positive constantssatisfying

∑

k∈Ni

1

aik≤ Ni. (6.48)

Then, using the fact that the Ni in (6.47) is time-invariant, property (6.45)implies

|y′i(t)| ≤ max

βi(|Zi0|, t), biδi maxk∈Ni

aik‖y′k‖[0,t], γi(‖wi‖[0,t])

(6.49)

for any initial state Zi0 and any wi, for all t ≥ 0.It can be observed that the interconnection topology of the controlled

agents is in accordance with the information exchange topology, representedby digraph Gc. For i ∈ N , k ∈ Ni, we assign the positive value aik to theedge (k, i) in Gc. Denote C as the set of all simple cycles in Gc and CL as theset of all simple cycles through the vertices belonging to L. Denote AO as theproduct of the positive values assigned to the edges of the cycle O ∈ C.

Note that bi can be designed to be arbitrarily close to one. By using thecyclic-small-gain theorem for networks of IOS systems, the closed-loop multi-agent system is IOS if

AON

N + 1< 1, O ∈ CL (6.50)

AO < 1, O ∈ C\CL. (6.51)


If Gc has a spanning tree with vertices belonging to L as the roots, thenaccording to Lemma 6.1, there exist positive constants aik satisfying (6.48),(6.50) and (6.51). Then, the closed-loop distributed system is UO and IOSwith wi as the inputs and y

′i as the outputs. With Assumption 6.2, the external

disturbances wi are bounded. The boundedness of the signals of the closed-loop distributed system can be directly verified under Assumption 6.2.

By designing the IOS gains γi arbitrarily small (this can be done accord-ing to Proposition 6.1), the influence of the external disturbances wi is madearbitrarily small, and y′i can be driven to within an arbitrarily small neigh-borhood of the origin. Equivalently, yi can be driven to within an arbitrarilysmall neighborhood of y0. In the case of wi = 0 for i = 1, . . . , N , each outputyi asymptotically converges to y0. This ends the proof of Theorem 6.1. ♦

6.2.3 ROBUSTNESS TO TIME DELAYS OF INFORMATION EXCHANGE

If there are communication delays, then ymi as defined in (6.33) and (6.34)should be modified as

ymi (t) =1

Ni + 1

(

∑

k∈Ni

(yi(t)− yk(t− τik(t))) + (yi(t)− y0)

)

, i ∈ L

(6.52)

ymi (t) =1

Ni

∑

k∈Ni

(yi(t)− yk(t− τik(t))), i ∈ 1, . . . , N\L, (6.53)

where τik : R+ → R+ represents non-constant time delays of exchanged infor-mation.

In this case, ymi (t) can still be written in the form of ymi (t) = y′i(t)− µi(t)with

µi(t) =1

Ni + 1

∑

k∈Ni

y′k(t− τik(t)), i ∈ L (6.54)

µi(t) =1

Ni

∑

k∈Ni

y′k(t− τik(t)), i ∈ 1, . . . , N\L. (6.55)

We assume that there exists a τ ≥ 0 such that, for i = 1, . . . , N , k ∈ Ni,0 ≤ τik(t) ≤ τ holds for all t ≥ 0. By considering µi and wi as the externalinputs, each controlled agent i composed of (6.30)–(6.32) and (6.38)–(6.43) isstill UO and property (6.49) should be modified as

|y′i(t)| ≤ max

βi(|Zi0|, t), biδi maxk∈Ni

aik‖y′k‖[−τ ,∞), γi(‖wi‖[0,∞))

(6.56)

for any initial state Zi0 and any wi, for all t ≥ 0.By using the time-delay version of the cyclic-small-gain theorem, Theorem

3.3, we can still guarantee the IOS of the closed-loop multi-agent system withy′i as the outputs and wi as the inputs, following analysis similar to that forthe proof of Theorem 6.1.


6.2.4 PROOF OF UO AND IOS OF EACH CONTROLLED AGENT

This subsection gives the proof of Proposition 6.1.

The Observation Error System

Define ζi1 = y′i − ξi1 and ζij = x′ij − Lijy′i − ξij for 2 ≤ j ≤ ni + 1 as

the observation errors. Denote ζi2 = [ζi2, . . . , ζi(ni+1)]T and ∆ij(y

′i, y0, wi) =

∆ij(y′i + y0, wi)−∆ij(y0, 0) for 1 ≤ j ≤ ni.

By taking the derivatives of ζi1, ζi2, we obtain

ζi1 = ρi1(ζi1 − µi) + φi1(y0, ζi1, ζi2, ξi1, wi) (6.57)

˙ζi2 = Aiζi2 + φi2(y0, ζi1, ξi1, wi), (6.58)

where

Ai =

−Li2...

−Li(ni−1)

Ini−2

−Lini0 · · · 0

, (6.59)

φi2(y0, ζi1, ξi1, wi) =

φi2(y0, ζi1, ξi1, wi)...φi(ni+1)(y0, ζi1, ξi1, wi)

, (6.60)

with

φi1 = ∆i1 + ζi2 + Li2ζi1 (6.61)

φij = ∆ij − Lij∆i1 + (Li(j+1) − LijLi2)ζi1, 2 ≤ j ≤ ni (6.62)

φi(ni+1) = −Li(ni+1)∆i1 − LiniLi2ζi1. (6.63)

With Assumption 6.1 satisfied, one can find ψφij∈ K∞ such that

|φi1(y0, ζi1, ζi2, ξi1, wi)| ≤ ψφij(|[ζi1, ζi2, ξi1, wi]T |), (6.64)

|φij(y0, ζi1, ξi1, wi)| ≤ ψφij(|[ζi1, ξi1, wi]T |). (6.65)

The positive constants Li2, . . . , Liniare chosen so that Ai is a Hurwitz matrix,

i.e., its eigenvalues have negative real parts.With Lemma 4.1, we can find a continuously differentiable ρi1 such that

for any constants 0 < ci < 1, ℓζi1 > 0 and any χζi2ζi1 , χξi1ζi1, χwi

ζi1∈ K∞ being

Lipschitz on compact sets, the ζi1-subsystem is ISS with Vζi1(ζi1) = |ζi1| asan ISS-Lyapunov function, which satisfies

Vζi1 (ζi1) ≥ max

χµi

ζi1(|µi|), χζi2ζi1(|ζi2|), χ

ξi1ζi1

(|ξi1|), χwi

ζi1(|wi|)

⇒∇Vζi1 (ζi1)ζi1 ≤ −ℓζi1Vζi1 , a.e., (6.66)


where χµi

ζi1(s) = s/ci for s ∈ R+.

Noticing that Ai is Hurwitz, there exists a positive definite matrix Pi =PTi ∈ R(ni−1)×(ni−1) satisfying PiAi + ATi Pi = −2Ini−1. Define Vζi2(ζi2) =

ζTi2Piζi2. Then, there exist αζi2 , αζi2 ∈ K∞ such that αζi2(|ζi2|) ≤ Vζi2 (ζi2) ≤αζi2(|ζi2|). With direct calculation, we have

∇Vζi2 (ζi2) ˙ζi2 =− 2ζTi2ζi2 + 2ζTi2Piφi2(ζi1, ξi1, wi)

≤− ζTi2ζi2 + |Pi|2|φi2(ζi1, ξi1, wi)|2

≤− 1

λmax(Pi)Vζi2(ζi2)

+ |Pi|2(

ψζi1φi2

(|ζi1|) + ψξi1φi2

(|ξi1|) + ψwi

φi2(|wi|)

)

. (6.67)

This means that the ζi2-subsystem is ISS with Vζi2 as an ISS-Lyapunov func-

tion. The ISS gains can be chosen as follows. Define χζi1ζi2

= 4λmax(Pi)|P 2i |ψζi1φi2

,

χξi1ζi2

= 4λmax(Pi)|P 2i |ψξi1φi2

and χwi

ζi2= 4λmax(Pi)|P 2

i |ψwi

φi2. Then,

Vζi2(ζi2) ≥ max

χζi1ζi2

(|ζi1|), χξi1ζi2(|ξi1|), χwi

ζi2(|wi|)

⇒∇Vζi2 (ζi2) ˙ζi2 ≤ −ℓζi2Vζi2(ζi2), (6.68)

where ℓζi2 = 14λmax(Pi)

.

The Control Error System

By taking the derivatives of ei1, . . . , ei(ni+1), direct calculation yields:

ei1 = κi1(ei1) + ϕi1(ei1, ei2, µi, ζi1), (6.69)

eij = κij(eij) + ϕij(ei1, . . . , ei(j+1), µi, ζi1), 2 ≤ j ≤ ni + 1, (6.70)

where

ϕi1 = ei2 + Li2ξi1 + ρi1(ξi1 − ymi ) (6.71)

ϕij = ei(j+1) + Li(j+1)ξi1 − Lij(ξi2 + Li2ξi1)

− ∂κi(j−1)(ei(j−1))

∂ei(j−1)ei(j−1). (6.72)

By default, ei(ni+2) = 0. Clearly, ϕi1 and ϕij are locally Lipschitz functions.Also, we can find ψϕij

∈ K∞ such that

|ϕij | ≤ ψϕij(|[ei1, . . . , ei(j+1), µi, ζi1]

T |) (6.73)

for 1 ≤ j ≤ ni + 1.


With Lemma 4.1, we can find continuously differentiable functions κij for1 ≤ j ≤ ni+1 such that each eij-subsystem is ISS with Veij (eij) = |eij | as anISS-Lyapunov function:

Veij (eij) ≥ maxk=1,...,j−1,j+1

χeikeij (|eik|), χµieij (|µi|), χζi1eij (|ζi1|)

⇒∇Veij (eij)eij ≤ −ℓeijVeij (eij), a.e., (6.74)

where ℓ(·) can be any specified positive constants, χei0ei1 , χei(ni+2)

ei(ni+1) = 0 and the

other χ(·)(·)’s can be any specified K∞ functions that are Lipschitz on compact

sets.

UO and IOS

Define Zi = [ζi1, ζTi2, ei1, . . . , ei(ni+1)]

T . Each controlled agent with state Zi hasbeen transformed into a network of ISS subsystems. With the Lyapunov-basedcyclic-small-gain theorem, controlled agent i is ISS with wi and µi as inputsif the composition of the ISS gains along every simple cycle in the systemdigraph is less than the identity function. The cyclic-small-gain condition canbe satisfied by choosing the ISS gains of the ζi1, ei1, . . . , ei(ni+1)-subsystemsto be small enough.

With the cyclic-small-gain condition satisfied, to find the IOS gains, weconstruct an ISS-Lyapunov function in the following form:

Vi(Zi) = max1≤j≤ni+1

σζi1 (Vζi1(ζi1)), σζi2(Vζi2 (ζi2)), σeij (Veij (eij))

, (6.75)

where σei1 = Id, and the other σ(·)’s are compositions of χ(·)(·)’s which are of

class K∞, smooth on (0,∞), slightly larger than the corresponding χ(·)(·)’s, and

still satisfy the cyclic-small-gain condition. Thus, Vi(Zi) is positive definiteand radially unbounded with respect to Zi. Here, it is not necessary to givean explicit representation of the σ(·)’s.

Accordingly, we define

χi(|µi|) = max1≤j≤ni+1

σζi1 χµi

ζi1(|µi|), σeij χµi

eij (|µi|)

, (6.76)

γi(|wi|) = max1≤j≤ni+1

σζi1 χwi

ζi1(|wi|), σζi2 χ

wi

ζi2(|wi|)

. (6.77)

By choosing the ISS gains χµieij and χζi1ei1 small enough, it can be achieved that

χi = σζi1 χµi

ζi1, (6.78)

where σζi1 can be designed to be arbitrarily small. Similarly, γi can also bedesigned to be arbitrarily small.

Then, there exists a continuous and positive definite function αi such that

Vi(Zi) ≥ maxχi(|µi|), γi(|wi|) ⇒ ∇Vi(Zi)Zi ≤ −αi(Vi(Zi)) (6.79)


holds almost everywhere. Thus, there exists a βi ∈ KL such that for all t ≥ 0,

Vi(Zi(t)) ≤ max

βi(Vi(Zi0), t), χi(‖µi‖[0,t]), γi(‖wi‖[0,t])

(6.80)

holds for any initial state Zi0.Recall the definitions of ζi1, ζi2, ei1, . . . , ei(ni+1). One can find αi, αi ∈ K∞

such that αi(|Zi|) ≤ Vi(Zi) ≤ αi(|Zi|) holds for all Zi. Based on (6.80), onecan find an αUO

i ∈ K∞ such that the UO property (6.44) holds.From the definition of Vi(Zi), using σζi1 = Id, we have |y′i| ≤ |ei1|+ |ζi1| ≤

σ−1ei1 (Vi(Zi)) + σ−1

ζi1(Vi(Zi)) = (Id + σ−1

ζi1)(Vi(Zi)). By defining

χi(s) = (Id + σ−1ζi1

) σζi1 χµi

ζi1= (Id + σζi1) χµi

ζi1(s) (6.81)

γi(s) = (Id + σ−1ζi1

) γi(s) (6.82)

βi(s, t) = (Id + σ−1ζi1

) βi(αi(s), t) (6.83)

for s, t ≥ 0, we can prove (6.45).Recall that χµi

ζi1(s) = s/ci for s ∈ R+. For any specified constant bi > 1,

by choosing ci to be close enough to one and σζi1 to be small enough, χi canbe designed such that χi(s) ≤ bis for all s ≥ 0. With fixed σζi1 , by choosingγi to be arbitrarily small, γi can be designed to be arbitrarily small.

6.3 FORMATION CONTROL OF NONHOLONOMIC MOBILE ROBOTS

Formation control of autonomous mobile agents is aimed at forcing agents toconverge toward, and maintain, specific relative positions. Distributed forma-tion control of multi-agent systems based on available local information, e.g.,relative position measurements, has attracted tremendous attention from therobotics and controls communities.

Motivated by the cyclic-small-gain design for distributed output-feedbackcontrol of nonlinear systems in Section 6.2, this section proposes a class ofdistributed controllers for leader-following formation control of unicycle robotsusing the practically available relative position measurements. The kinematicsof the unicycle robot are demonstrated by Figure 6.4.

For this purpose, the formation control problem is first transformed into astate agreement problem of double-integrators through dynamic feedback lin-earization. The nonholonomic constraint causes singularity for dynamic feed-back linearization when the linear velocity of the robot is zero. This issueshould be well taken into consideration for the validity of the transformeddouble-integrator models. Then, distributed formation control laws are de-veloped. To avoid the singularity problem caused by the nonholonomic con-straint, saturation functions are introduced to the control design to restrict thelinear velocities of the robots to be larger than zero. It should be noted thatlinear analysis methods may not be directly applicable due to the employmentof the saturation functions. Then, the closed-loop system is transformed intoa dynamic network of IOS systems. The cyclic-small-gain result in digraphs is


used to guarantee the IOS of the dynamic network and thus the achievementof formation control.

θ

ωv

(x, y)

X

Y

0

FIGURE 6.4 Kinematics of the unicycle robot, where (x, y) represents the Cartesian

coordinates of the center of mass of the robot, v is the linear velocity, θ is the heading

angle, and ω is the angular velocity.

With the effort mentioned above, the proposed design has three advantages:

1. The proposed distributed formation control law does not use global positionmeasurements or assume tree position sensing structures.

2. The formation control objective can be practically achieved in the presenceof position measurement errors.

3. The linear velocities of the robots can be designed to be less than certaindesired values, as practically required.

This section considers the formation control problem of a group of N +1 mobile robots. For i = 0, 1, . . . , N , the kinematics of each i-th robot aredescribed by the unicycle model:

xi = vi cos θi (6.84)

yi = vi sin θi (6.85)

θi = ωi, (6.86)

where [xi, yi]T ∈ R

2 represent the Cartesian coordinates of the center of massof the i-th robot, vi ∈ R is the linear velocity, θi ∈ R is the heading angle,and ωi ∈ R is the angular velocity.

The robot with index 0 is the leader robot, and the robots with indices1, . . . , N are follower robots. The linear velocity vi and the angular velocityωi are considered as the control inputs of the i-th robot for i = 1, . . . , N . For


this system, the formation control objective is to control each i-th followerrobot such that

limt→∞

(xi(t)− xj(t)) = dxij (6.87)

limt→∞

(yi(t)− yj(t)) = dyij (6.88)

with dxij , dyij being appropriate constants representing the desired relativepositions, and

limt→∞

((θi(t)− θj(t)) mod 2π) = 0 (6.89)

for any i, j = 0, . . . , N , where mod represents the modulo operation. Forconvenience of notation, let dxii = dyii = 0 for any i = 0, . . . , N . We assumethat dxij = dxik − dxkj and dyij = dyik − dykj for any i, j, k = 0, . . . , N .

Assumption 6.3 on v0 is made throughout this section.

Assumption 6.3 The linear velocity v0 of the leader robot is differentiablewith bounded derivative, i.e., v0(t) exists and is bounded on [0,∞), and hasupper and lower constant bounds v0, v0 > 0 such that v0 ≤ v0(t) ≤ v0 for allt ≥ 0.

One technical problem of controlling groups of mobile robots is that ac-curate global positions of the robots are usually not available for feedback,and relative position measurements should be used instead. The hardware forrelative position measurements for each follower robot may include a rangesensor (e.g., sonar, laser range finder, and light detection and ranging (LI-DAR) component) and a gyroscope for orientation measurement [57]. A di-graph can be employed to represent the relative position sensing structurebetween the robots. The position sensing digraph G has N + 1 vertices withindices 0, 1, . . . , N corresponding to the robots. If the relative position be-tween robot i and robot j is available to robot j, then G has a directed edgefrom vertex i to vertex j; otherwise G does not have such an edge.

The goal of this section is to present a class of distributed formation con-trollers for mobile robots by using local relative position measurements as wellas the velocity and acceleration information of the leader. The basic idea ofthe design is to first transform the unicycle model into the double integratorthrough dynamic feedback linearization under constraints, and at the sametime, to reformulate the formation control problem as a stabilization problem.Then, distributed control laws are designed to make each controlled mobilerobot IOS. Finally, the cyclic-small-gain theorem is used to guarantee theachievement of the formation control objective.

6.3.1 DYNAMIC FEEDBACK LINEARIZATION

In this subsection, the distributed formation control problem is reformulatedwith the dynamic feedback linearization technique. For details of dynamic


feedback linearization and its applications to nonholonomic systems, pleaseconsult [40, 61].

For each i = 0, . . . , N , introduce a new input ri ∈ R such that

vi = ri. (6.90)

Define vxi = vi cos θi and vyi = vi sin θi. Then, xi = vxi and yi = vyi. Takethe derivatives of vxi and vyi, respectively. Then,

(

vxivyi

)

=

(

cos θi −vi sin θisin θi vi cos θi

)(

riωi

)

. (6.91)

In the case of vi 6= 0, by designing(

riωi

)

=

(

cos θi sin θi− sin θi

vicos θivi

)(

uxiuyi

)

, (6.92)

the unicycle model (6.84)–(6.86) can be transformed into two double-integrators with new inputs uxi and uyi:

xi = vxi, vxi = uxi, (6.93)

yi = vyi, vyi = uyi. (6.94)

Define xi = xi−x0−dxi, yi = yi−y0−dyi, vxi = vxi−vx0, vyi = vyi−vy0,uxi = uxi − ux0, and uyi = uyi − uy0. Then,

˙xi = vxi, ˙vxi = uxi, (6.95)

˙yi = vyi, ˙vyi = uyi. (6.96)

The formation control problem is solvable if we can design control laws forsystem (6.95)–(6.96) with uxi and uyi as the control inputs, so that vi 6= 0 isguaranteed, and at the same time,

limt→∞

xi(t) = 0, (6.97)

limt→∞

yi(t) = 0. (6.98)

It should be noted that the validity of (6.93)–(6.94) (and thus (6.95)–(6.96))for the unicycle model is under the condition that vi 6= 0. Such requirementis basically caused by the nonholonomic constraint of the mobile robot. Thisleads to the major difference between this problem and the distributed controlproblem for double-integrators.

To use (6.95)–(6.96) for control design, each follower robot should haveaccess to ux0, uy0, which represent the acceleration of the leader robot. Thisrequirement can be fulfilled if the leader robot can calculate ux0, uy0 by usingr0, ω0, θ0, v0 according to (6.92) and transmit them to the follower robots.Note that ω0, θ0, v0 are usually measurable, and r0 is normally available as itis the control input of the leader robot.


6.3.2 A CLASS OF IOS CONTROL LAWS

As an ingredient for the distributed control design, this subsection presents aclass of nonlinear control laws for the following double-integrator system withan external input, such that the closed-loop system is UO and IOS:

η = ζ (6.99)

ζ = µ (6.100)

η = η + w, (6.101)

where [η, ζ]T ∈ R2 is the state, µ ∈ R is the control input, w ∈ R representsan external input, η can be considered as a measurement of η subject to w,and only (η, ζ) are used for feedback. As shown later, each controlled robotcan be transformed into the form of (6.99)–(6.101) with w representing theinteraction between the robots.

Lemma 6.2 For system (6.99)–(6.101), consider a control law taking theform:

µ = −kµ(ζ − φ(η)). (6.102)

For any constant φ > 0, one can find an odd, strictly decreasing, continuouslydifferentiable function φ : R → [−φ, φ] and a positive constant kµ satisfying

−kµ4<dφ(r)

dr< 0 (6.103)

for all r ∈ R, such that the closed-loop system (6.99)–(6.102) is UO with zerooffset, and is IOS with the identity function as the gain, i.e., the followingproperties hold:

|η(t)| ≤ β(|[η(0), ζ(0)]T |, t) + ‖w‖t (6.104)

|ζ(t)| ≤ |ζ(0)|+ αUO(‖η‖t + ‖w‖t) (6.105)

for some β ∈ KL, αUO ∈ K∞, and all t ≥ 0.

It is necessary to note that condition (6.103) is easily checkable for practicalimplementation of the control law (6.102).

Proof. Denote w = ‖w‖T with T ≥ 0. Recall that ~d(r, S) represents the directdistance from r ∈ R to S ⊂ R. We first introduce the following transformationto state ζ:

ζ = ~d(ζ, Sζ(η, w)) (6.106)

with

Sζ(η, w) = cφ(η + w) : |w| ≤ w, c ∈ [c2, c1] , (6.107)

where 0 < c2 < c1 are constants to be defined later. Then, we have ζ − ζ ∈Sζ(η, w).


The ηsubsystem

With ζ − ζ ∈ Sζ(η, w), the η-subsystem can be represented by the followingdifferential inclusion:

η ∈

ζ∗ + ζ : ζ∗ ∈ Sζ(η, w)

. (6.108)

Due to the influence of w, we study the convergence of η to the set Sζ(w) =w : |w| ≤ w when 0 ≤ t ≤ T . Define

η = ~d(η, Sη(w)). (6.109)

Then, from (6.108), we have

˙η ∈

ζ∗ + ζ : ζ∗ ∈ Sζ(η, w)

:= Fη(η, ζ, w). (6.110)

The properties of φ and the definition of η guarantee

|cφ(η + w)| ≥ c2|φ(η)| (6.111)

sgn(cφ(η + w)) = sgn(c2φ(η)) = −sgn(η) (6.112)

for c ∈ [c2, c1] and |w| ≤ w, when η 6= 0.We consider the case of (1 − δ)c2|φ(η)| ≥ |ζ| with δ < 1 and η 6= 0. In this

case, it holds that

|cφ(η + w) + ζ| ≥ |cφ(η + w)| − |ζ|≥ c2|φ(η)| − (1− δ)c2|φ(η)|= δc2|φ(η)| (6.113)

and

sgn(cφ(η + w) + ζ) = sgn(c2φ(η)) = −sgn(η) (6.114)

for c ∈ [c2, c1] and |w| ≤ w.Define Vη(η) = η2/2 as a Lyapunov function candidate for the η-subsystem.

Then, in the case of (1 − δ)c2|φ(η)| ≥ |ζ|, direct calculation yields:

maxfη∈Fη(η,ζ,w)

∇Vη(η)fη ≤ −δc2|η||φ(η)| = −δc2ηφ(η). (6.115)

The ζsubsystem

To simplify the discussions, we only study the case of ζ > 0. The case of ζ < 0can be studied in the same way. The definition of Sζ in(6.107) implies

maxSζ(η, w) =

c2φ(η − w), when η ≥ w;

c1φ(η − w), when η < w.(6.116)


Since φ is continuously differentiable, maxSζ(η, w) is continuously differen-

tiable almost everywhere. Denote dφ(r)dr = φd(r) for r ∈ R. When 0 ≤ t ≤ T ,

by taking the derivative of ζ and using the control law (6.102), we can use adifferential inclusion to represent the ζ-subsystem:

˙ζ ∈

µ− cφd(η − w)η : c ∈ Sc(η)

=

−kµ(ζ − φ(η + w)) − cφd(η − w)ζ : c ∈ Sc(η)

⊆

−(

kµ + cφd(η − w))

(

ζ − kµkµ + cφd(η − w)

φ(η + w)

)

:

c ∈ [c2, c1], |w| ≤ w

:= Fζ(η, ζ, w), (6.117)

where Sc(η) = c2 when η > w, Sc(η) = c1 when η < w, and Sc(η) =[c2, c1] when η = w.

Define kφ = − infr∈Rφd(r) = − infr∈Rdφ(r)/dr. Condition (6.103) im-plies 0 < 4kφ < kµ.

Choose c1 = kµ/2kφ. Then, given 0 < 4kφ < kµ, i.e., (4kφkµ−k2µ)/4kφ < 0,we have kφc

21− kµc1+ kµ < 0, i.e., kµ/(kµ− c1kφ) < c1. Choose c2 < 1. Then,

it can be proved that

c2 <kµ

kµ + cφd(η − w)< c1 (6.118)

for c ∈ [c2, c1] and −kφ ≤ φd(η − w) < 0.Choose 0 < kµ ≤ 1

2kµ. Then,

kµ + cφd(η − w) ≥ kµ (6.119)

for c ∈ [c2, c1] and −kφ ≤ φd(η − w) < 0.Denote

kµkµ + cφd(η − w)

φ(η + w) = ∆(η, w,w). (6.120)

When 0 ≤ t ≤ T , by using (6.118) and |w| ≤ w, we have

∆(η, w,w) ∈ cφ(η + w) : c ∈ [c2, c1], |w| ≤ w (6.121)

for c ∈ [c2, c1] and −kφ ≤ φd(η − w) < 0, and thus

|ζ −∆(η, w,w)| ≥ |ζ| (6.122)

sgn(ζ −∆(η, w,w)) = sgn(ζ) (6.123)


when ζ 6= 0.Define Vζ(ζ) =

12 ζ

2 as a Lyapunov function candidate for the ζ-subsystem.Then, by using (6.119)–(6.123), we have

maxfζ∈Fζ(η,ζ,w)

∇Vζ(ζ)fζ ≤ −kµζ2 = −2kµVζ(ζ). (6.124)

IOS and UO

From (6.124), we have

Vζ(ζ(t)) ≤ e−2kµtVζ(ζ(0)) (6.125)

and thus

|ζ(t)| ≤ e−kµt|ζ(0)|. (6.126)

It can be calculated that it takes T0 = max 0, T ′0 for |ζ(t)| to converge to

the region |ζ| ≤ (1− δ)c2φ, where T′0 = 1

kµ

(

ln(|ζ(0)|)− ln((1 − δ)c2φ))

.

Considering properties (6.111) and (6.112) of set-valued map Sζ(η, w), wehave

maxFη(η, ζ, w) ≤ |ζ| (6.127)

minFη(η, ζ, w) ≥ −|ζ| (6.128)

and thus

|η(t)| ≤ |η(0)|+∫ t

0

|ζ(τ)|dτ

≤ |η(0)|+∫ t

0

e−kµτdτ |ζ(0)|

≤ |η(0)|+ 1

kµ|ζ(0)|, (6.129)

where we used property (6.126) for the second inequality.When 0 ≤ t ≤ T , we first consider the case of ζ(t) ≥ (1 − δ)c2φ. In this

case, t ≤ T ′0 = T0.

Choose αβ0(s) = (2 + kµ)s2/(2(1 − δ)c2φkµ) for s ∈ R+ and β0(s, t) =

e−kµtαβ0(s) for s, t ∈ R+. Then, β0 ∈ KL. It can be proved that

β0(|[η(0), ζ(0)]T |, T ′0) =

1 + kµ

2kµ|ζ(0)|(η2(0) + ζ2(0))

≥ |η(0)|+ 1

kµ|ζ(0)|, (6.130)


where Young’s inequality—see [153]—is used for the second inequality.When 0 ≤ t ≤ T , if ζ(t) ≥ (1−δ)c2φ, by using (6.129), (6.130), and t ≤ T ′

0,we have

|η(t)| ≤ β0(|[η(0), ζ(0)]T |, T ′0) ≤ β0(|[η(0), ζ(0)]T |, t). (6.131)

When 0 ≤ t ≤ T , in the case of |ζ(t)| ≤ (1 − δ)c2φ, we can consider the(η, ζ)-system as a cascade connection of the input-to-stable η-subsystem andthe asymptotically stable ζ-subsystem, and use the small-gain theorem in [130]to directly prove the existence of β1 ∈ KL such that

|η(t)| ≤ β1(|[η(T0), ζ(T0)]T |, t− T0). (6.132)

Define β ∈ KL as

β(s, t) = maxβ0(s, t), β1(s, t) (6.133)

for s, t ∈ R+. Then, for any η0, ζ0 ∈ R, with initial condition η(0) = η0,ζ(0) = ζ0, it holds that

|η(t)| ≤ β(|[η0, ζ0]T |, t) (6.134)

for 0 ≤ t ≤ T .From the definition of η in (6.109), we have

|η(t)| ≤ |η(t)| ≤ |η(t)|+ ‖w‖T . (6.135)

Recall the definition of ζ in (6.106). From the fact that

φ(η(t)) ∈ cφ(η(t) + w) : |w| ≤ w, c ∈ [c2, c1] (6.136)

with 0 < c2 < 1 < c1, we get

|ζ(t)| ≤ |ζ(t)− φ(η(t))| ≤ |ζ(t)| + |φ(η(t))| (6.137)

for 0 ≤ t ≤ T . From the properties of φ, we can find |φ(r)| ≤ kφ|r| for r ∈ R.Thus, there exists an α0 ∈ K∞ such that

|[η(0), ζ(0)]T | ≤ α0(|[η(0), ζ(0)]T |). (6.138)

Property (6.134) together with (6.135) and (6.138) implies that

|η(T )| ≤ β(|[η(0), ζ(0)]T |, t) + ‖w‖T , (6.139)

where β(s, t) := β(α0(s), t) for s, t ∈ R+.From (6.126) and (6.137), we have

|ζ(t)| ≤ |ζ(0)| ≤ |ζ(0)|+ |φ(η(0))|. (6.140)


Using the definition of ζ, we have

|ζ(t)| ≤ |ζ(t)|+ maxζ∗∈Sζ(η(t),w)

|ζ∗|

≤ |ζ(0)|+ |φ(η(0))| + c1|φ(|η(t)| + ‖w‖T )|≤ |ζ(0)|+ kφ|η(0)|+ c1kφ(|η(t)|+ ‖w‖T )≤ |ζ(0)|+ kφ|η(0)|+ c1kφ(‖η‖T + ‖w‖T ) (6.141)

for 0 ≤ t ≤ T . Then, one can find an αUO ∈ K∞ such that

|ζ(T )| ≤ |ζ(0)|+ αUO(‖η‖T , ‖w‖T ). (6.142)

It can be observed that the definitions of β in (6.139) and αUO in (6.142)do not depend on the signals η, ζ, w and time T , and it can be concluded thatfor any η0, ζ0 ∈ R, with initial states η(0) = η0 and ζ(0) = ζ0, and inputw : R+ → R being piecewise continuous and bounded, it holds that

|η(T )| ≤ β(|[η0, ζ0]T |, T ) + ‖w‖T (6.143)

|ζ(T )| ≤ |ζ0|+ αUO(‖η‖T + ‖w‖T ) (6.144)

for all T ≥ 0. Note that we used T instead of t in the proof to avoid confusion.This ends the proof of Lemma 6.2. ♦

6.3.3 DISTRIBUTED FORMATION CONTROLLER DESIGN AND SMALLGAINANALYSIS

As discussed in Subsection 6.3.1, for the validity of (6.95)–(6.96) of the for-mation control design, vi should be guaranteed to be nonzero. For a specifiedλ∗ satisfying 0 < λ∗ < v0, by designing a control law for the i-th robot suchthat

max |vxi|, |vyi| ≤√2

2(v0 − λ∗) ≤

√2

2(v0 − λ∗), (6.145)

it can be guaranteed that |vi| =√

v2xi + v2yi =√

(vx0 + vxi)2 + (vy0 + vyi)2 ≥λ∗ > 0 and thus vi 6= 0. In this way, singularity is avoided.

Practically, the linear velocity of each robot is usually required to be lessthan a desired value. For any given λ∗ > v0, we can also guarantee |vi| ≤ λ∗

by designing a control law such that


2(λ∗ − v0). (6.146)

For specified constants λ∗, λ∗, v0, v0 satisfying 0 < λ∗ < v0 < v0 < λ∗, wedefine

λ = min

√2

2(v0 − λ∗),

√2

2(λ∗ − v0)

. (6.147)


Then, conditions (6.145) and (6.146) can be satisfied if

max |vxi|, |vyi| ≤ λ. (6.148)

The proposed distributed control law is composed of two stages: (a) initial-ization and (b) formation control. The initialization stage is employed becausethe formation control stage cannot solely guarantee vi 6= 0 if (6.145) is not sat-isfied at the beginning of the control procedure. With the initialization stage,the linear velocity and the heading direction of each follower robot can be con-trolled to satisfy (6.148) within some finite time. Then, the formation controlstage is triggered, and thereafter, the satisfaction of (6.148) is guaranteed,and at the same time, the formation control objective is achieved.

Initialization Stage

For this stage, we design the following control law

ωi = φθi(θi − θ0) + ω0 (6.149)

ri = φvi(vi − v0) + v0 (6.150)

for each i-th follower robot, where φθi, φvi : R → R are nonlinear functions.Define θi = θi − θ0 and vi = vi − v0. Taking the derivatives of θi and vi,

respectively, and using (6.149) and (6.150), we have

˙θi = φθi(θi), (6.151)

˙vi = φvi(vi). (6.152)

By designing φθi, φvi such that −φθi(s), φθi(−s),−φvi(s), φvi(−s) are positivedefinite for s ∈ R+, we can guarantee the asymptotic stability of systems(6.151) and (6.152). Moreover, there exist βθ, βv ∈ KL such that |θ(t)| ≤βθ(|θ(0)|, t) and |v(t)| ≤ βv(|v(0)|, t).

By directly using the property of continuous functions, there exist δv0 > 0and δθ0 > 0 such that, for all v0 ∈ [v0, v0], θ0 ∈ R, |δv0| ≤ δv0 and |δθ0| ≤ δθ0,

|(v0 + δv0) cos(θ0 + δθ0)− v0 cos θ0| ≤ λ, (6.153)

|(v0 + δv0) sin(θ0 + δθ0)− v0 sin θ0| ≤ λ. (6.154)

Recall that for any β ∈ KL, there exist α1, α2 ∈ K∞ such that β(s, t) ≤α1(s)α2(e

−t) for all s, t ∈ R+ according to Lemma 1.1; see also [243, Lemma8]. With control law (6.149)–(6.150), there exists a finite time TOi for the i-throbot such that |θi(TOi) − θ0(TOi)| ≤ δθ0 and |vi(TOi) − v0(TOi)| ≤ δv0, andthus condition (6.148) is satisfied at time TOi.

It should be noted that if vi(0) ≤ λ∗, then control law (6.150) guaranteesvi(t) ≤ λ∗ for t ∈ [0, TOi] because of v0(t) ≤ v0 < λ∗.


Formation Control Stage

At time TOi, the distributed control law for the i-th follower robot is switchedto the formation control stage.

In this stage, we design

uxi = −kxi(vxi − φxi(zxi)) (6.155)

uyi = −kyi(vyi − φyi(zyi)), (6.156)

where φxi, φyi : R → [−λ, λ] are odd, strictly decreasing, and continuouslydifferentiable functions and kxi, kyi are positive constants satisfying

− kxi/4 < dφxi(r)/dr < 0 (6.157)

− kyi/4 < dφyi(r)/dr < 0 (6.158)

for all r ∈ R. An example for φxi and φyi is shown in Figure 6.5.

0 r

φ(r)

λ

−λ

FIGURE 6.5 An example for φxi and φyi.

The variables zxi and zyi are defined as

zxi =1

Ni

∑

j∈Ni

(xi − xj − (dxi − dxj)) (6.159)

zyi =1

Ni

∑

j∈Ni

(yi − yj − (dyi − dyj)), (6.160)

where Ni is the size of Ni with Ni representing the position sensing structure.If j ∈ Ni, then the position sensing digraph G has a directed edge (j, i) fromvertex j to vertex i. Note that dxi − dxj , dyi − dyj in (6.159) and (6.160)represent the desired relative position between the i-th robot and the j-throbot. By default, dx0 = dy0 = 0.

In the formation control stage, the control inputs ri and ωi are defined as(6.92) with uxi = uxi + ux0 and uyi = uyi + uy0.


Consider the (vxi, vyi)-system defined in (6.95)–(6.96). With condition(6.148) satisfied at time TOi, the boundedness of φxi and φyi together with thecontrol law (6.155)–(6.156) guarantees the satisfaction of (6.148) after TOi. Forthe proof of this statement, we can consider (vxi, vyi) : max|vxi|, |vyi| ≤ λas an invariant set of the (vxi, vyi)-system.

The main result of distributed formation control is summarized below.

Theorem 6.2 Consider the multi-robot model (6.84)–(6.86) and the dis-tributed control laws defined by (6.90), (6.92), (6.149), (6.150), (6.155), and(6.156) with parameters kxi, kyi satisfying (6.157)–(6.158). Under Assumption6.3, if the position sensing digraph G has a spanning tree with vertex 0 as theroot, then for any constants dxi, dyi ∈ R with i = 1, . . . , N , the coordinates(xi(t), yi(t)) and the angle θi(t) of each i-th robot asymptotically converge to(x0(t) + dxi, y0(t) + dyi) and θ0(t) + 2kπ with k ∈ Z, respectively. Moreover,given any λ∗ > v0, if vi(0) ≤ λ∗ for i = 1, . . . , N , then vi(t) ≤ λ∗ for allt ≥ 0.

The two-stage distributed control law results in a switching incident of thecontrol signal for each follower robot during the control procedure. The tra-jectories of such systems can be well defined in the spirit of Rademacher (seee.g., [60]), and the performance of the system can be analyzed by consideringthe two stages one-by-one.

6.3.4 SMALLGAIN ANALYSIS AND PROOF OF THEOREM 6.2

Recall the definition of λ in (6.147). With condition (6.148) satisfied afterTOi, we have vi 6= 0 and thus the validity of (6.92) for all t ≥ TOi. Under thecondition of vi(0) ≤ λ∗, the boundedness of vi(t), i.e., vi(t) ≤ λ∗, can also bedirectly proved based on the discussions in Subsection 6.3.3.

Denote x0 = 0 and y0 = 0. We equivalently represent zxi and zyi as

zxi =1

Ni

∑

j∈Ni

(xi − dxi − x0 − (xj − dxj − x0))

=1

Ni

∑

j∈Ni

(xi − xj) = xi −1

Ni

∑

j∈Ni

xj (6.161)

and similarly,

zyi = yi −1

Ni

∑

j∈Ni

yj. (6.162)


Denote

ωxi =1

Ni

∑

j∈Ni

xj , (6.163)

ωyi =1

Ni

∑

j∈Ni

yj . (6.164)

Then, control laws (6.155) and (6.156) are in the form of (6.102).In the following proof, we only consider the (xi, vxi)-system (6.95). The

(yi, vyi)-system (6.96) can be studied in the same way.Define TO = maxi=1,...,NTOi. By using Lemma 6.2, for each i = 1, . . . , N ,

the closed-loop system composed of (6.95) and (6.155) has the following prop-erties: for any xi0, vxi0 ∈ R, with xi(TO) = xi0 and vxi(TO) = vxi0,

|xi(t)| ≤ βxi(|[xi0, vxi0]T |, t− TO) + ‖ωxi‖[TO,t] (6.165)

|vxi(t)| ≤ |vxi0|+ αxi(‖xi‖[TO,t] + ‖ωxi‖[TO,t]), (6.166)

where βxi ∈ KL and αxi ∈ K∞.Notice that for any constants a1, . . . , an > 0 satisfying

∑ni=1(1/ai) ≤ n, it

holds that∑n

i=1 di =∑ni=1(1/ai)aidi ≤ nmax1≤i≤naidi for all d1, . . . , dn ≥

0. We have

|ωxi| ≤ δi maxj∈N i

aij |xj |, (6.167)

where δi = (Ni− 1)/Ni, N i = Ni\0, and∑

j∈N i(1/aij) ≤ Ni− 1 if 0 ∈ Ni;

δi = 1, N i = Ni and∑

j∈N i(1/aij) ≤ Ni if 0 /∈ Ni.

Then, properties (6.165) and (6.166) imply

|xi(t)| ≤ βxi(|[xi0, vxi0]T |, t− TO) + δi maxj∈N i

aij‖xj‖[TO,t], (6.168)

|vxi(t)| ≤ |vxi0|+ αxi(‖xi‖[TO,t] + δi maxj∈N i

aij‖xj‖[TO,t]). (6.169)

Define the follower sensing digraph Gf as a subgraph of G. Digraph Gf hasN vertices with indices 1, . . . , N corresponding to the vertices with indices1, . . . , N of G and representing the follower robots. From the definitions of N i

and Gf , it can be observed that, for i = 1, . . . , N , if j ∈ N i, then there isa directed edge (j, i) from the j-th vertex to the i-th vertex in Gf . Clearly,Gf represents the interconnection topology of the network composed of the(xi, vxi)-systems (6.95).

Define F0 = i ∈ 1, . . . , N : 0 ∈ Ni. Denote Cf as the set of all simplecycles of Gf , and denote C0 ⊆ Cf as the set of all simple cycles through thevertices with indices belonging to F0.

For i = 1, . . . , N , j ∈ N i, we assign the positive value aij to edge (j, i)in Gf . For a simple cycle O ∈ Cf , denote AO as the product of the positivevalues assigned to the edges of the cycle.


Consider xi with i = 1, . . . , N as the outputs of the network composed ofthe (xi, vxi)-systems (6.95). By using the IOS small-gain theorem for generalnonlinear systems in [134, 137], xi(t) with i = 1, . . . , N converge to the originif

AON − 1

N< 1 for O ∈ C0, (6.170)

AO < 1 for O ∈ Cf\C0. (6.171)

Note that AO(N − 1)/N < 1 is equivalent to AO < N/(N − 1) = 1 +1/(N − 1).

If G has a spanning tree with vertex 0 as the root, then Gf has a spanningtree with the indices of the root vertices belonging to F0. According to Lemma6.1, there exist positive constants aij such that both conditions (6.170) and(6.171) are satisfied. For system (6.95), with the convergence of each xi tothe origin and the boundedness of uxi, we can guarantee the convergence ofvxi to the origin by using Barbalat’s lemma [144]. Similarly, we can prove theconvergence of vyi to the origin. By using the definitions of vxi and vyi, theconvergence of θi to θ0 + 2kπ with k ∈ Z can be concluded. This ends theproof of Theorem 6.2.

6.3.5 ROBUSTNESS TO RELATIVE POSITION MEASUREMENT ERRORS

Measurement errors can decrease the performance of a nonlinear control sys-tem. In this section, we discuss the robustness of our distributed formationcontroller in the presence of relative position measurement errors.

It can be observed that the initialization stage of the distributed controllaw defined in (6.149)–(6.150) is not affected by the position measurementerrors. Also, condition (6.148) still holds for t ≥ TOi for i = 1, . . . , N .

For the formation control stage, in the presence of relative position mea-surement errors, the zxi and zyi defined for the distributed control law (6.155)–(6.156) should be modified as

zxi =1

Ni

∑

j∈Ni

(

xi − xj − (dxi − dxj) + ωxij)

(6.172)

zyi =1

Ni

∑

j∈Ni

(

yi − yj − (dyi − dyj) + ωyij)

, (6.173)

where Ni is the size of Ni and ωxij , ωyij ∈ R represent the relative position

measurement errors corresponding to (xi−xj) and (yi−yj), respectively. Dueto the boundedness of the designed φxi and φyi in (6.155)–(6.156), condition(6.148) is still satisfied in the existence of position measurement errors, whichguarantees the validity of (6.92).


Here, we only consider each xi-subsystem. The yi-subsystems can be stud-ied in the same way. By defining

ωxi =1

Ni

∑

j∈Ni

(

xj + ωxij)

, (6.174)

we have zxi = xi − ωxi. With such definition, if the measurement errors ωxijare piecewise continuous and bounded, then each xi-subsystem still has theIOS and UO properties given by (6.165) and (6.166), respectively.

As in the discussion above (6.167), we have

|ωxi| ≤ max

ρiNi

∑

j∈Ni

(|xj |),ρ′iNi

∑

j∈Ni

(|ωxij |)

:= max

ρiNi

∑

j∈Ni

(|xj |), ωexi

≤ maxj∈N i

δiaij |xj |, ωexi , (6.175)

where ρi, ρ′i > 0 satisfying 1/ρi + 1/ρ′i ≤ 1, and δi = ρi(Ni − 1)/Ni,

N i = Ni\0 and∑

j∈N i(1/aij) ≤ Ni − 1 if 0 ∈ Ni; δi = ρi, N i = Ni

and∑

j∈N i(1/aij) ≤ Ni if 0 /∈ Ni.

In the existence of the relative position measurement errors, we can stillguarantee the IOS of the closed-loop distributed system by using the cyclic-small-gain theorem. In this case, the cyclic-small-gain condition is as follows:

AOρ(N − 1)

N< 1 for O ∈ C0, (6.176)

AOρ < 1 for O ∈ Cf\C0, (6.177)

where ρ := maxi∈1,...,Nρi is larger than one according to 1ρi

+ 1ρ′i

≤ 1, and

can be chosen to be very close to one. Lemma 6.1 can guarantee (6.176) and(6.177) if G has a spanning tree with vertex 0 as the root. Thus, the proposeddistributed control law is robust with respect to relative position measurementerrors.

6.3.6 A NUMERICAL EXAMPLE

Consider a group of 6 robots with indices 0, 1, . . . , 5. Notice that the robotwith index 0 is the leader. The neighbor sets of the robots are defined asfollows: N1 = 0, 5, N2 = 1, 3, N3 = 2, 5, N4 = 3, N5 = 4.

By default, the values of all the variables in this simulation are in SIunits. For convenience, we omit the units. The desired relative position ofthe follower robots are defined by dx1 = −

√3d/2, dx2 = −

√3d/2, dx3 =

0, dx4 =√3d/2, dx5 =

√3d/2, dy1 = −d/2, dy2 = −3d/2, dy3 = −2d, dy4 =


−3d/2, dy5 = −d/2 with d = 30. Figure 6.6 shows the position sensing graphof the formation control system. Clearly, the position sensing graph has aspanning tree with vertex 0 as the root.

2 3 4

1 5

0

FIGURE 6.6 The position sensing graph of the formation control system.

It should be noted that the control law for each follower robot also uses thevelocity and acceleration information of the leader robot, the communicationtopology of which is not shown in Figure 6.6.

The control inputs of the leader robot are r0(t) = 0.1 sin(0.4t) and ω0(t) =0.1 cos(0.2t). With such control inputs, the linear velocity v0 with v0(0) = 3satisfies v0 ≤ v0(t) ≤ v0 with v0 = 3 and v0 = 3.5.

Choose λ∗ = 0.45 and λ∗ = 6.05. The distributed control laws for theinitialization stage are in the form of (6.149)–(6.150) with φθi(r) = φvi(r) =−0.5(1−exp(−0.5r))/(1+exp(−0.5r)) for i = 1, . . . , 5. The distributed controllaws for the formation control stage are in the form of (6.155)–(6.156) withkxi = kyi = 2 and φxi(r) = φyi(r) = −1.8(1− exp(−0.5r))/(1 + exp(−0.5r))for i = 1, . . . , 5. With direct calculation, it can be verified that the designedkxi, kyi, φxi, φyi satisfy (6.157) and (6.158). Also, φxi(r), φyi(r) ∈ [−1.8, 1.8]for all r ∈ R. With v0 = 3 and v0 = 3.5, the control laws can restrict the linearvelocities of the follower robots to be in the range of [3−1.8

√2, 3.5+1.8

√2] =

[0.454, 6.046]⊂ [λ∗, λ∗].The initial states of the robots are chosen as

i (xi(0), yi(0)) vi(0) θi(0)

0 (0, 0) 3 π/61 (−40, 10) 4 π2 (−20,−40) 3.5 5π/63 (5,−40) 2.5 04 (50,−10) 2 −2π/35 (50, 10) 3 0

The measurement errors are: ωxij(t) = 0.3(cos(t+ iπ/6)+ cos(t/3+ iπ/6)+cos(t/5 + iπ/6) + cos(t/7 + iπ/6)) and ωyij(t) = 0.3(sin(t + iπ/6) + sin(t/3 +iπ/6) + sin(t/5 + iπ/6) + sin(t/7 + iπ/6)) for i = 1, . . . , N , j ∈ Ni.

The linear velocities and the angular velocities of the robots are shown inFigure 6.7. The stage changes of the distributed controllers are shown in Figure


6.8 with “0” representing initialization stage and “1” representing formationcontrol stage. Figure 6.9 shows the trajectories of the robots. The simulationverifies the theoretical results.

FIGURE 6.7 The linear velocities and the angular velocities of the robots.

FIGURE 6.8 The stages of the distributed controllers.

6.4 DISTRIBUTED CONTROL WITH FLEXIBLE TOPOLOGIES

In Section 6.3, the formation control problem of the nonholonomic mobilerobots is transformed into the distributed nonlinear control problem of double-


FIGURE 6.9 The trajectories of the robots. The dashed curve represents the trajec-

tory of the leader.

integrators. With fixed information exchange topologies under a connectivitycondition, the control objective can be achieved. This section takes a step for-ward toward solving the distributed nonlinear control problem for multi-agentsystems modeled by double integrators under switching information exchangetopologies. The goal is to develop a new class of distributed nonlinear con-trol laws to solve a strong output agreement problem, that is, the outputs ofthe agents converge to each other and the internal states converge to the ori-gin. From a practical point of view, it is assumed that the double-integratorsinteract with each other through output interconnections (more specifically,the differences of the outputs), for coordination. In this section, an invariantset method is developed such that the strong output agreement problem issolvable if the information exchange graph satisfies a mild joint connectivitycondition. Moreover, the proposed design is also valid for systems under phys-ical constraints such as velocity limitation. The result proposed in this sectionis also used for distributed formation control of nonholonomic mobile robots.

Strong Output Agreement Problem

Consider a group of N double-integrator agents with switching topologies bydistributed control:

ηi = ζi (6.178)

ζi = µi, (6.179)

where [ηi, ζi]T ∈ R2 is the state and µi ∈ R is the control input.


The basic idea of the strong output agreement problem is to design a classof distributed nonlinear control laws in the form of

µi = ϕi(ζi, ξi) (6.180)

ξi = φσ(t)

i (η1, . . . , ηN ), (6.181)

where σ : [0,+∞) → P is a piecewise constant signal representing switchinginformation exchange topology with P ⊂ N being a finite set representing allthe possible information exchange topologies, ϕi : R

2 → R and φp

i : RN → R

for each p ∈ P , such that the following properties hold:

limt→∞

(ηi(t)− ηj(t)) = 0, for any i, j = 1, . . . , N, (6.182)

limt→∞

ζi(t) = 0, for i = 1, . . . , N. (6.183)

Notice that the strong output agreement as defined above is a stronger prop-erty of state agreement. For state agreement, the internal states ξi are onlyrequired to converge to each other.

6.4.1 PROPERTIES OF A CLASS OF NONLINEAR SYSTEMS

Our strong output agreement result is developed based on several propertiesof the following second-order nonlinear system:

η = ζ (6.184)

ζ = ϕ(ζ − φ(η − ω)), (6.185)

where [η, ζ]T ∈ R2 is the state, ω ∈ R is an external disturbance input, andϕ, φ : R → R are nonincreasing and locally Lipschitz functions.

For convenience of notation, we define two new classes of functions. Afunction β : R+ × R+ → R+ is called an I+L function, denoted by β ∈ I+L,if β ∈ KL, β(s, 0) = s for s ∈ R+, and for any specified T > 0, there existcontinuous, positive definite, and nondecreasing α1, α2 < Id such that for alls ∈ R+, β(s, t) ≥ α1(s) for t ∈ [0, T ] and β(s, t) ≤ α2(s) for t ∈ [T,∞). Afunction β : R × R+ → R is called an IL function, denoted by β ∈ IL, ifthere exist β′, β′′ ∈ I+L such that for t ≥ 0, β(r, t) = β′(r, t) for r ≥ 0, andβ(r, t) = −β′′(−r, t) for r < 0. The new notations are necessary when wewant to avoid the finite-time convergence in a network with a time-variabletopology, which may lead to oscillation. A similar problem arises in the stateagreement problem of coupled nonlinear systems. See [172, Lemma 5.2] forsome details.

Proposition 6.2 If ω ∈ [ω, ω] with ω ≤ ω being constants, and if functions


ϕ and φ satisfy

ϕ(0) = φ(0) = 0, (6.186)

ϕ(r)r < 0, φ(r)r < 0 for r 6= 0, (6.187)

supr∈R

max ∂ϕ(r) < 4 infr∈R

min ∂φ(r), (6.188)

then system (6.184)–(6.185) has the following properties:

1. There exist strictly decreasing and locally Lipschitz functions ψ, ψ : R → R

satisfying ψ(0) = ψ(0) = 0 such that

S(ω, ω) =

(η, ζ) : ψ(η − ω) ≤ ζ ≤ ψ(η − ω)

(6.189)

is an invariant set of system (6.184)–(6.185).2. For any specified initial state (η(0), ζ(0)), there exist a finite time t1 and

constants µ, µ ∈ R such that

ψ(η(t1)− µ) ≤ ζ(t1) ≤ ψ(η(t1)− µ). (6.190)

3. For any specified σ, σ ∈ R, if (η(t), ζ(t)) ∈ S(σ, σ) for t ∈ [0, T ], then thereexist β

1, β1 ∈ IL such that

−β1(σ − η(0), t) + σ ≤ η(t) ≤ β1(η(0)− σ, t) + σ (6.191)

for all t ∈ [0, T ].4. For any specified compact M ⊂ R, there exist β

2, β2 ∈ IL such that if

(η(0), ζ(0)) ∈ S(µ0, µ0) with µ

0≤ µ0 belonging to M , then one can find

µ(t), µ(t) satisfying

−β2(ω − µ

0, t) + ω ≤ µ(t) ≤ µ(t) ≤ β2(µ0 − ω, t) + ω (6.192)

such that

(η(t), ζ(t)) ∈ S(µ(t), µ(t)) (6.193)

for all t ≥ 0.

The proof of Proposition 6.2 is given in Subsection 6.4.2.Figure 6.10 shows Property 1 of system (6.184)–(6.185) given in Proposition

6.2. The region between ζ = ψ(η−ω) and ζ = ψ(η−ω) forms an invariant setof system (6.184)–(6.185). Properties 2, 3, and 4 are based on the definitionsof ψ and ψ.


η0

ζ

ωω

ζ = ψ(η − ω)

ζ = ψ(η − ω)

FIGURE 6.10 The boundaries of the invariant set in Property 1 of Proposition 6.2.

6.4.2 PROOF OF PROPOSITION 6.2

We first present a technical lemma on a class of first-order nonlinear systems.

Lemma 6.3 Consider the following first-order system

ς = α(ς), (6.194)

where ς ∈ R is the state and α is a nonincreasing and locally Lipschitz functionsatisfying α(0) = 0 and rα(r) < 0 for all r 6= 0. There exists β ∈ IL suchthat for any ς0 ∈ R, with initial condition ς(0) = ς0, it holds that

ς(t) ≤ β(ς0, t) (6.195)

for all t ≥ 0.

Proof. Denote ς∗(ς0, t) as the solution of system (6.194) with initial conditionς(0) = ς0. Define β′(s, t) = ς∗(s, t) and β′′(s, t) = −ς∗(−s, t) for s, t ∈ R+.

Consider the case of ς(0) ≥ 0.Since α is locally Lipschitz, for any specified ς > 0, there exists a constant

kα > 0 such that α(s) ≥ −kαs for s ≤ ς and thus

ς(t) ≥ −kας(t) (6.196)

for 0 ≤ ς(t) ≤ ς.For any specified T > 0, define

α1(s) = e−kαT min s, ς (6.197)

for s ∈ R+. Then, α1 is continuous and positive definite.If ς(0) ≤ ς, then

ς(T ) ≥ ς(0)e−kαT ≥ α1(ς(0)). (6.198)


Consider the case of ς(0) > ς. If there is a time 0 < t′ ≤ T such thatς(t′) = ς , then

ς(T ) ≥ ς(t′)e−kα(T−t′) ≥ ςe−kαT > α1(ς(0)); (6.199)

otherwise,

ς(T ) > ς > ςe−kαT > α1(ς(0)). (6.200)

According to the definition of β′, for the specified T > 0, it holds thatβ′(s, T ) ≥ α1(s). Because of the nonincreasing property of ς(t) with ς(0) ≥ 0,we have

β′(s, t) ≥ α1(s) (6.201)

for t ∈ [0, T ].For any specified T > 0, define

α2(s) = max

1

2s, s+ T max

12 s≤τ≤s

α(τ)

(6.202)

for s ∈ R+. It can be verified that α2 is continuous, positive definite, and lessthan Id.

If ς(T ) ≥ 12 ς(0), then

12 ς(0) ≤ ς(t) ≤ ς(0) for 0 ≤ t ≤ T , and

ς(t) ≤ max12 ς(0)≤τ≤ς(0)

α(τ). (6.203)

Then, we have

ς(T ) ≤ ς(0) +

∫ T

0

max12 ς(0)≤τ≤ς(0)

α(τ)dt

= ς(0) + T max12 ς(0)≤τ≤ς(0)

α(τ)

≤ α2(ς(0)). (6.204)

If ς(T ) < ς(0)/2, then ς(T ) < α2(ς(0)) automatically. According to the def-inition of β′, for the specified T > 0, it holds that β′(s, T ) ≤ α2(s). Becauseof the nonincreasing property of ς(t) with ς(0) ≥ 0, we have

β′(s, t) ≤ α2(s) (6.205)

for t ∈ [T,∞).It can be directly verified that β′ ∈ KL and β′(s, 0) = s for s ≥ 0. By also

using (6.201) and (6.205), we can prove β′ ∈ I+L. Due to symmetry, we canalso prove β′′ ∈ I+L. Thus, β ∈ IL.

Define β(r, t) = β′(r, t) for r ≥ 0, t ≥ 0 and β(r, t) = −β′′(−r, t) for r ≤ 0,t ≥ 0. Then, (6.195) holds and β ∈ IL. This ends the proof. ♦


Property 1

Denote Sa(ω) = (η, ζ) : ζ ≤ ψ(η − ω) and Sb(ω) = (η, ζ) : ζ ≥ ψ(η − ω).Then, S(ω, ω) = Sa(ω) ∩ Sb(ω). If both Sa(ω) and Sb(ω) are invariant setsof system (6.184)–(6.185) and ψ(r) ≤ ψ(r) for all r ∈ R, then S(ω, ω) is an

invariant set. In the following procedure, we find appropriate ψ such thatSa(ω) is an invariant set. Function ψ can be found in the same way.

For a nonincreasing and locally Lipschitz function φ satisfying (6.186) and(6.187), there exists a function φ : R → R which is strictly decreasing andcontinuously differentiable on (−∞, 0)∪(0,∞) such that φ(0) = 0, φ(r) ≥ φ(r)for r ∈ R and

infr∈R

min∂φ(r) ≥ infr∈R

min ∂φ(r) − ǫ (6.206)

for any specified arbitrarily small ǫ > 0.Define

ψ(r) = max

cφ(r) : c ∈ [c1, c2]

, (6.207)

where c1 and c2 are constants satisfying 0 < c2 < 1 < c1 to be deter-mined later. Then, ψ is strictly decreasing and continuously differentiableon (−∞, 0) ∪ (0,∞) and ψ(0) = 0.

Define ζ = ζ−ψ(η−ω). When ζ ≥ ψ(η−ω), directly taking the derivativeof ζ yields:

˙ζ ∈

ζ − ψdη : ψ

d ∈ ∂ψ(η − ω)

=

ϕ(ζ − φ(η − ω))− ψdζ : ψ

d ∈ ∂ψ(η − ω)

⊆

ϕ(ζ − φ(η − ω))− ψdζ : ψ

d ∈ ∂ψ(η − ω), ω ≤ ω ≤ ω

=

−(kϕ + ψd)

(

ζ − kϕφ(η − ω)

kϕ + ψd

)

+ ϕ(ζ − φ(η − ω)) :

ψd ∈ ∂ψ(η − ω), ω ≤ ω ≤ ω

:= Fζ(η, ζ, ω, ω), (6.208)

where kϕ := − sup∂ϕ(r) : r ∈ R and ϕ(r) := ϕ(r)+kϕ(r) for r ∈ R. Clearly,

ϕ(r)r ≤ 0 (6.209)

for r ∈ R.Denote kφ = − infr∈Rmin∂φ(r). With condition (6.188) and (6.206) sat-

isfied, we can choose φ such that 0 < 4kφ ≤ kϕ. Choose c1 = kϕ/2kφ. Then,

kφc21−kϕc1+kϕ ≤ 0, i.e., kϕ/(kϕ−c1kφ) ≤ c1. Clearly, c1 ≥ 2. Choose c2 ≤ 1.


The definition of ψ in (6.207) implies ∂ψ(r) ⊆

cφd: c ∈ [c2, c1], φ

d ∈ ∂φ(r)

,

and thus infr∈Rmin∂ψ(r) ≥ −c1kφ. By also using supr∈Rmax ∂ψ(r) ≤ 0

(due to the strict decreasing of ψ), we can prove

c2 ≤ kϕ

kϕ + ψd≤ c1 (6.210)

for ψd ∈ ∂ψ(η − ω).

Using ω ≤ ω and the nonincreasing of φ and φ, from (6.210) we have

kϕφ(η − ω)

kϕ + ψd

≤ max

cφ(η − ω) : c ∈ [c1, c2]

= ψ(η − ω), (6.211)

which implies

ζ − kϕφ(η − ω)

kϕ + ψd

≥ ζ (6.212)


Based on the definitions of ψdand c1, we also have

kϕ + ψd ≥ kϕ + inf

r∈R

min ∂ψ(r) = kϕ − c1kφ =1

2kϕ (6.213)


Based on (6.208), (6.209), (6.212), and (6.213), it can be proved that

maxfζ∈Fζ(η,ζ,ω,ω)

fζ ≤ −1

2kϕζ (6.214)

when ζ ≥ ψ(η − ω), i.e., ζ ≥ 0. This guarantees the invariance of set Sa(ω).Following a similar approach, we can also find ψ : R → R such that it is

strictly decreasing and continuously differentiable on (−∞, 0) ∪ (0,∞) andsatisfies ψ(0) = 0 and ψ ≤ ψ(r) for all r ∈ R, and prove that Sb(ω) is aninvariant set.

Property 2

We present only the proof of the second inequality in (6.190). The first in-equality in (6.190) can be proved in the same way.

We first consider the case in which φ(r) → ∞ as r → −∞. In this case,ψ(r) → ∞ as r → −∞ according to the definition of ψ in (6.207). In this case,for any (η(0), ζ(0)), one can always find a µ such that the second inequalityin (6.190) holds with t1 = 0.


If the condition for the first case is not satisfied, then there exist constantsφu > 0 and 2/3 < φδ < 1 such that φ(r) ≤ φu for all r ∈ R and one can findan r∗ satisfying φ(r∗) ≥ φδφu. According to the definition of ψ in (6.207), itholds that ψ(r∗) ≥ c1φ

δφu, where c1 ≥ 2, and thus ψ(r∗) ≥ 4φu/3.Define ζ = ζ − φu. When ζ ≥ φu, taking the derivative of ζ yields

˙ζ = ζ = ϕ(ζ − φ(η − ω)) ≤ φ(ζ − φu) = ϕ(ζ), (6.215)

where ϕ satisfies (6.186), (6.187), and (6.188). Then, there exists a β ∈ KLsuch that for any ζ(0) ≥ φu,

ζ(t) ≤ β(ζ(0), t) (6.216)

for all t ≥ 0. According to [243, Lemma 8], there exist αβ1, αβ2 ∈ K∞ suchthat β(s, t) ≤ αβ1(s)αβ2(e

−t) for all s, t ∈ R+, and thus there exists a finite

time t1 such that β(ζ(0), t1) ≤ φu/3, which guarantees ζ(t1) ≤ φu/3, i.e.,ζ(t1) ≤ 4φu/3. During finite time interval [0, t1], the boundedness of ζ(t)implies the boundedness of η(t). Then, one can find a µ such that the secondinequality in (6.190) holds.

Property 3

For t ∈ [0, T ], it holds that

η(t) = ζ(t) ≤ ψ(η(t) − σ). (6.217)

Define ς(t) as the solution of the initial-value problem

ς(t) = ψ(ς(t)− σ) (6.218)

with ς(0) = η(0). By using the comparison principle (see e.g., [144]), it can beproved that

η(t) ≤ ς(t) (6.219)

for t ∈ [0, T ].Note that ψ is locally Lipschitz and satisfies rψ(r) < 0 for r 6= 0. Define

ς = ς − σ. Then, (6.218) implies ˙ς(t) = ψ(ς(t)). By using Lemma 6.3, thereexists a β1 ∈ IL such that ς(t) ≤ β1(ς(0), t) for t ∈ [0, T ], i.e.,

ς(t) ≤ β1(ς(0)− σ, t) + σ. (6.220)

The second inequality in (6.191) is proved by using ς(0) = η(0) and η(t) ≤ς(t) for t ∈ [0, T ]. The first inequality in (6.191) can be proved in the sameway.


Property 4

Define ω∗ = minµ0, η(0), ω and ω∗ = maxµ0, η(0), ω. Then, (η(0), ζ(0)) ∈

S(ω∗, ω∗) ∩ (η, ζ) : ω∗ ≤ η ≤ ω∗ := S(ω∗, ω∗).Because ω ∈ [ω, ω] ⊆ [ω∗, ω∗], S(ω∗, ω∗) is an invariant set, and S(ω∗, ω∗) is

also an invariant set. Given (η(0), ζ(0)) ∈ S(ω∗, ω∗), it holds that (η(t), ζ(t)) ∈S(ω∗, ω∗) for all t ≥ 0.

In the following proof, we adopt some idea from kinematics of the planetranslational motion of a rigid body; see e.g., [96]. Define

ηd = ζ, (6.221)

ζd = ϕ(ζ − φ(η − ω)), (6.222)

v = [ηd, ζd]T , (6.223)

v1 =

[

minψ

d∈∂ψ(η)

ζd

ψd, ζd

]T

, (6.224)

v2 =

[

ηd − minψ

d∈∂ψ(η)

ζd

ψd, 0

]T

. (6.225)

Clearly, v2(t) = v(t) − v1(t).

η0

ζ

µ1

ζ = ψ(η − µ1)

(η, ζ)v2

ζd vv1

ηd

FIGURE 6.11 The motion of the point (η, ζ) and the rigid body ζ = ψ(η − µ): v

is the velocity of the point, which is the composition of ηd and ζd and also the

composition of v1 and v2; v1 represents the relative velocity of the point along the

rigid body and v2 represents the translational motion velocity of the rigid body.

For any specified µ, if ω ∈ [ω∗, µ], then S(ω∗, µ) is an invariant set. From(6.214), for any (η, ζ) satisfying ζ = ψ(η − µ), it holds that

ηd − minψ

d∈∂ψ(η)

ζd

ψd≤ 0, (6.226)


i.e.,

ζ − minψ

d∈∂ψ(η)

ϕ(ζ − φ(η − ω))

ψd

≤ 0 (6.227)

for all ω ∈ [ω∗, µ]. Thus, it can be concluded that

ζ − minψ

d∈∂ψ(η)

ϕ(ζ − φ(η − µ))

ψd

≤ 0 (6.228)

for any (η, ζ) satisfying ζ = ψ(η − µ).Then, for any (η, ζ) satisfying ζ = ψ(η − µ), it holds that

ηd − minψ

d∈∂ψ(η)

ζd

ψd

= ζ − minψ

d∈∂ψ(η)

ϕ(ζ − φ(η − ω))

ψd

= ζ − minψ

d∈∂ψ(η)

ϕ(ζ − φ(η − µ))

ψd

− minψ

d∈∂ψ(η)

ϕ(ζ − φ(η − ω))− ϕ(ζ − φ(η − µ))

ψd

≤ − minψ

d∈∂ψ(η)

ϕ(ζ − φ(η − ω))− ϕ(ζ − φ(η − µ))

ψd

, (6.229)

where we used (6.228) for the inequality.In the case of µ ≥ ω, by using the continuous and strictly decreasing

properties of ϕ and φ, one can always find a positive definite, nondecreasing,locally Lipschitz αa2 such that

minψ

d∈∂ψ(η)

ϕ(ζ − φ(η − ω))− ϕ(ζ − φ(η − µ))

ψd

≥ αa2(µ− ω) (6.230)

for (η, ζ) ∈ S(ω∗, ω∗) and ω ∈ [ω, ω].In the case of µ < ω, because ϕ and φ are locally Lipschitz and strictly

decreasing, one can find a positive definite, nondecreasing, locally Lipschitzαb2 such that

minψ

d∈∂ψ(η)

ϕ(ζ − φ(η − ω))− ϕ(ζ − φ(η − µ))

ψd

≥ −αb2(ω − µ) (6.231)

for (η, ζ) ∈ S(ω∗, ω∗) and ω ∈ [ω, ω].Define

α2(r) =

−αa2(r) for r ≥ 0;

αb2(−r) for r < 0.(6.232)


Then, α2(0) = 0, rα2(r) < 0 for all r 6= 0, and α2 is nonincreasing and locallyLipschitz.

Define ς(t) as the solution of the initial-value problem

ς(t) = α2(ς(t)− ω) (6.233)

with initial condition ς(0) = µ0. Then, ζ(t) ≤ ψ(η(t) − ς(t)) for t ≥ 0. Ifµ(t) ≥ σ(t) for t ≥ 0, then ζ(t) ≤ ψ(η(t)− µ(t)) for t ≥ 0.

Similarly, one can find a nonincreasing, locally Lipschitz α2 which satisfiesα2(0) = 0, rα2(r) < 0 for all r 6= 0, such that ζ(t) ≥ ψ(η(t) − µ(t)) for t ≥ 0,if µ(t) ≤ σ(t) for t ≥ 0, where ς(t) is the solution of the initial-value problem

ς(t) = α2(ς(t)− ω) (6.234)

with initial condition ς(0) = µ0.

Define α′2 = maxα2(r), α2(r) and α′

2 = minα2(r), α2(r) for r ∈ R.Define µ(t) and µ(t) as the solutions of the initial-value problems

µ(t) = α′2(µ(t)− ω) (6.235)

µ(t) = α′2(µ(t)− ω) (6.236)

with initial conditions µ(0) = µ0 and µ(0) = µ0. Then, the comparison prin-

ciple can guarantee µ(t) ≤ σ(t) and µ(t) ≤ σ(t). Moreover, µ(t) ≥ µ(t) fort ≥ 0.

By using Lemma 6.3, one can find β2, β2 ∈ IL such that (6.192) holds.

6.4.3 MAIN RESULTS OF STRONG OUTPUT AGREEMENT WITH FLEXIBLETOPOLOGIES

Consider the multi-agent system (6.178)–(6.179). We propose a class of dis-tributed control laws in the following form:

µi = ϕi(ζi − φi(ξi)) (6.237)

ξi =1

∑

j∈Ni(σ(t))aij

∑

j∈Ni(σ(t))

aij(ηi − ηj), (6.238)

where σ : [0,∞) → P is a piecewise constant signal, which describes theinformation exchange between the systems, with P ⊂ N being a finite setrepresenting all the possible information exchange topologies, and Ni(p) ⊆1, . . . , N denotes the neighbor set of agent i for each i = 1, . . . , N and eachp ∈ P . In (6.237)–(6.238), constant aij > 0 if i 6= j and aij ≥ 0 if i = j. Thefunctions ϕi, φi : R → R are nonincreasing, locally Lipschitz, and satisfy

ϕi(0) = φi(0) = 0, (6.239)

ϕi(r)r < 0, φi(r)r < 0 for r 6= 0, (6.240)

supr∈R

max ∂ϕi(r) : r ∈ R < 4 infr∈R

min ∂φi(r), (6.241)


for i = 1, . . . , N .By defining ωi =

∑

j∈Ni(σ(t))aijηj/

∑

j∈Ni(σ(t))aij , it can be observed that

each controlled agent (6.178)–(6.179) with control law (6.237)–(6.238) is in theform of (6.184)–(6.185), and conditions (6.239)–(6.241) are in accordance withconditions (6.186)–(6.188).

Before proposing our main result on strong output agreement, we firstuse a switching digraph G(σ(t)) = (N , E(σ(t))) to represent the informationexchange topology between the agents, where N is the set of N vertices cor-responding to the agents, and for each p ∈ P , if j ∈ Ni(p), then there is adirected edge (j, i) belonging to E(p). By default, (i, i) for i ∈ N belong toE(p) for all p ∈ P .

A digraph G = (N , E) is quasi-strongly connected (QSC) if there exists ac ∈ N such that there is a directed path from c to i for each i ∈ N ; vertex cis called the center of G. For a switching digraph G(σ(t)) = (N , E(σ(t))), wedefine the union digraph over [t1, t2] as G(σ([t1, t2])) = (N ,

⋃

t∈[t1,t2]E(σ(t))).

A switching digraph G(σ(t)) with σ : [0,∞) → P is said to be uniformly quasi-strongly connected (UQSC) with time constant T > 0 if G(σ([t, t+T ])) is QSCfor all t ≥ 0. A switching digraph G(σ(t)) with σ : [0,∞) → P has an edgedwell time τD > 0 if for each t ∈ [0,∞), for any directed edge (i1, i2) ∈ E(σ(t)),there exists a t∗ ≥ 0 such that t ∈ [t∗, t∗ + τD] and (i1, i2) ∈ E(σ(τ)) forτ ∈ [t∗, t∗ + τD].

Lemma 6.4 Consider a switching digraph G(σ(t)) = (N , E(σ(t))) with σ :[0,∞) → P, which is UQSC with time constant T > 0 and has an edge dwelltime τD > 0. If c ∈ N is a center of G(σ([t, t + T ])), then for any N1 suchthat c ∈ N1, there exist i1 ∈ N1, i2 ∈ N\N1, and t

′ ∈ [t− τD, t+T ] such that(i1, i2) ∈ E(σ(τ)) for τ ∈ [t′, t′ + τD].

Lemma 6.4 can be proved by directly using the definitions of UQSC and edgedwell-time.

The following theorem presents our main result on the strong output agree-ment problem.

Theorem 6.3 Consider the double-integrators (6.178)–(6.179) with controllaws in the form of (6.237)–(6.238). Assume conditions (6.239)–(6.241) aresatisfied. If G(σ(t)) = (N , E(σ(t))) with σ : [0,∞) → P is UQSC and has anedge dwell-time τD > 0, then the strong output agreement problem is solvable.

The proof of Theorem 6.3, which is based on Proposition 6.2, is in Section6.4.4.

In previously published papers on coordinated control of continuous-timesystems with switching topologies, e.g., [111, 172, 236], it is usually assumedthat the switching graph has a dwell-time τ ′D, which means each specific topol-ogy remains unchanged for a period larger than τ ′D. As edge dwell-time is


directly used in our analysis, we assume edge dwell-time instead of the cus-tomary graph dwell-time. It should be noted that the assumption of edgedwell-time does not cause restrictions to the main results. In fact, the edgedwell-time is normally larger than the graph dwell-time for a specific switchinggraph.

In a leader–follower structure, the motion of the leader does not depend onthe outputs of the followers and the output of the leader is accessible to someof the followers. For a group of systems with a leader i∗, to achieve strongoutput agreement, it is required that there exists a finite constant T > 0 suchthat for all t ≥ 0, the union digraph G(σ([t, t+T ])) is QSC with i∗ as a center,according to Theorem 6.3.

Conditions (6.239)–(6.241) allow us to choose bounded and nonsmooth φifor i = 1, . . . , N . One example of φi is

φi(r) =

−1, when r > 1;

−r, when − 1 ≤ r ≤ 1;

1, when r < −1.

(6.242)

Correspondingly, we can choose ϕi(r) = −kr with constant k > 4. This couldbe of practical interest when the velocities ζi are required to be bounded inthe process of controlling the positions ηi to achieve agreement. Consider theζi-system (6.179) with µi defined by (6.237). With bounded φi, the velocityζi can be restricted to within a specific bounded range depending on theinitial state ζi(0) and the bounds of φi. With bounded velocities, we can alsoguarantee the boundedness of the control signals µi, which may be requiredto be bounded due to actuator saturation.

6.4.4 PROOF OF THEOREM 6.3

Define

ωi =

∑

j∈Ni(σ(t))aijηj

∑

j∈Ni(σ(t))aij

. (6.243)

Then, ξi = ηi − ωi, and each (ηi, ζi)-system (6.178)–(6.179) with µi definedin (6.237)–(6.238) is in the form of (6.184)–(6.185). With conditions (6.239)–(6.241) satisfied, each closed-loop (ηi, ζi)-system has the properties given inProposition 6.2.

According to Property 1 in Proposition 6.2, for i = 1, . . . , N , one can findψi, ψi such that

Si(ωi, ωi) =

(ηi, ζi) : ψi(ηi − ωi) ≤ ζi ≤ ψi(ηi − ω)

(6.244)

is an invariant set of the (ηi, ζi)-system if ωi ∈ [ωi, ωi].


Given the chosen ψi, ψi, we suppose that there exist µ

i(0), µi(0) such that

(ηi(0), ζi(0)) ∈ Si(µi(0), µi(0)) (6.245)

µ(0) ≤ ηi(0) ≤ µi(0) (6.246)

for i ∈ N . Otherwise, there exists a finite time t∗, at which property (6.245)holds, according to Property 2 in Proposition 6.2.

Denote η = [η1, . . . , ηN ]T and ζ = [ζ1, . . . , ζN ]T . Then, (η(0), ζ(0)) ∈S(µ(0), µ(0)) with

S(µ(0), µ(0)) = (η, ζ) : (ηi, ζi) ∈ Si(µ(0), µ(0)) for i ∈ N, (6.247)

where µ(0) = mini∈N µi(0) and µ(0) = maxi∈N µi(0) with µi(0), µi(0) satis-

fying (6.245) and (6.246).For each i ∈ N , (6.243) implies µ(0) ≤ ωi ≤ µ(0) if µ(0) ≤ ηi ≤ µ(0).

Based on Proposition 6.2, it can be proved that S(µ(0), µ(0)) is an invariantset of the interconnected system with (η, ζ) as the state. Thus, (η(t), ζ(t)) ∈S(µ(0), µ(0)) and µ(0) ≤ ωi(t) ≤ µ(0) for all t ≥ 0.

The basic idea of the proof is to find appropriate µi(t), µi(t) such that

ψi(ηi(t)− µi(t)) ≤ ζi(t) ≤ ψi(ηi(t)− µi(t)). (6.248)

We define two sets, Q1 and Q2, which satisfy Q1 ∪ Q2 = N and have thefollowing properties: if i ∈ Q1, then the µ

i(0) defined in (6.245) satisfies

µi(0) ≥ (µ(0) + µ(0))/2; (6.249)

if i ∈ Q2, then the µi(0) defined in (6.245) satisfies

µi(0) ≤ (µ(0) + µ(0))/2. (6.250)

Note that either Q1 or Q2 can be an empty set. Also, the existence of the pair(Q1,Q2) may not be unique.

Define T ∗ = ∆T + N(T + 2τD + ∆T ) with ∆T > 0. For each i ∈ Q1, byusing Property 4 in Proposition 6.2, there exists a µ

i(t) satisfying

µi(t)− µ(0) ≥ αi(µi(0)− µ(0)) (6.251)

such that the first inequality of (6.248) holds for 0 ≤ t ≤ T ∗, where αi iscontinuous, positive definite, and less than Id.

For each i ∈ Q1, by also using (6.249), we have

µi(t)− µ(0)

≥minαi(r − µ(0)) : (µ(0) + µ(0))/2 < r ≤ µ(0)=minαi(r′) : (µ(0)− µ(0))/2 < r′ ≤ µ(0)− µ(0)=αli(µ(0)− µ(0)) (6.252)


for 0 ≤ t ≤ T ∗, where αli(s) := minαi(s′) : s/2 ≤ s′ ≤ s for s ∈ R+. Clearly,αli is continuous, positive definite, and less than Id. For each i ∈ Q1, usingProperty 3 in Proposition 6.2, for specific ∆T > 0, one can find a continuousand positive definite function αl0i < Id such that

ηi(t)− µ(0) ≥ αl0i ( min0≤t≤T∗

µi(t)− ηi(0))

≥ αl0i ( min0≤t≤T∗

µi(t)− µ(0))

≥ αl0i αli(µ(0)− µ(0))

:= αli(µ(0)− µ(0)) (6.253)

for t ∈ [∆T , T∗]. Clearly, αli is continuous, positive definite, and less than Id.

Similarly, for each i ∈ Q2, one can find µi(t) satisfying

µ(0)− µi(t) ≥ αui (µ(0)− µ(0)) (6.254)

such that the second inequality of (6.248) holds for 0 ≤ t ≤ T ∗, where αui iscontinuous, positive definite, and less than Id. Also, one can find continuousand positive definite αui < Id such that

µ(0)− ηi(t) ≥ αui (µ(0)− µ(0)) (6.255)

for t ∈ [∆T , T∗].

Initial Step: Because G(σ(t)) is UQSC, G(σ([∆T + τD,∆T + T + τD]))has a center, denoted by l1. Suppose l1 ∈ Q2. (If l1 ∈ Q1, then the theoremcan be proved in the same way.) Then, according to Lemma 6.4, there existl′1 ∈ Q2, f1 ∈ Q1 and t′ ∈ [∆T ,∆T + T + τD] such that (l′1, f1) ∈ E(σ(t)) fort ∈ [t′, t′ + τD] ⊆ [∆T ,∆T + T + 2τD].

By using the fact that µ(0)− ηl′1(t) ≥ αul′1(µ(0)− µ(0)), we have

ωf1(t) =

∑

j∈Ni(σ(t))aijηj(t)

∑

j∈Ni(σ(t))aij

≤ µ(0)−af1l′1 α

ul′1(µ(0)− µ(0))

∑

j∈Ni(σ(t))aij

. (6.256)

Also, we have ωf1(t) ≥ µ(0) for t ∈ [t′, t′ + τD]. Property 4 in Proposition 6.2guarantees that one can find µ

f1(t) and µf1(t) satisfying µf1

(t) ≥ µ(0) and

µf1(t) ≤ µ(0)− αu0f1 (µ(0)− maxt∈[t′,t′+τD]

ωf1(t))

≤ µ(0)− αu0f1

(

af1l′1αul′1(µ(0)− µ(0))

∑

j∈Ni(σ(t))aij

)

(6.257)

such that (6.248) with i = f1 holds at t = t′ + τD, where αu0f1

is continuous,positive definite, nondecreasing, and less than Id.


For t ∈ [t′ + τD, T∗], it can be guaranteed that µ(0) ≤ ωf1(t) ≤ µ(0). By

using Property 4 in Proposition 6.2 again, one can find µf1(t) ≥ µ(0) and

µf1(t) ≤ µ(0)− αuf1(µ(0)− µ(0)) (6.258)

such that (6.248) with i = f1 holds for t ∈ [∆T +T +2τD, T∗] ⊆ [t′ + τD, T

∗],where αuf1 < Id is continuous, positive definite, and less than Id.

By using Property 3 in Proposition 6.2, for specific ∆T > 0, there exists acontinuous and positive definite αu0f1 < Id such that

µ(0)− ηf1(t) ≥ αu0f1 (µ(0)− max0≤t≤T∗

µf1(t))

≥ αu0f1 αuf1(µ(0)− µ(0))

:= αuf1(µ(0)− µ(0)), (6.259)

i.e.,

ηf1(t) ≤ µ(0)− αuf1(µ(0)− µ(0)) (6.260)

for t ∈ [∆T +(T +2τD+∆T ), T∗]. According to the definitions, αuf1 is contin-

uous, positive definite, and less than Id. Since f1 ∈ Q1, according to (6.253),there also exists a continuous and positive definite αlf1 < Id such that

ηf1(t) ≥ µ(0) + αlf1(µ(0)− µ(0)) (6.261)

for t ∈ [∆T + (T + 2τD +∆T ), T∗].

Recursive Step: Denote Fk = f1, . . . , fk ⊂ N . Suppose that for eachi ∈ Fk, there exist continuous and positive definite functions αli, α

ui < Id such

that

ηi(t) ≥ µ(0) + αli(µ(0)− µ(0)), (6.262)

ηi(t) ≤ µ(0)− αui (µ(0)− µ(0)) (6.263)

for t ∈ [∆T + k(T + 2τD +∆T ), T∗].

Note that, according to (6.253) and (6.255), for each i ∈ Q1∪Fk\Fk, thereexists a continuous and positive definite αli < Id such that (6.262) holds forall t ∈ [∆T + k(T + 2τD + ∆T ), T

∗], and for each i ∈ Q2 ∪ Fk\Fk, thereexists a continuous and positive definite αui < Id such that (6.263) holds fort ∈ [∆T + k(T + 2τD +∆T ), T

∗].Due to the UQSC of G(σ(t)), the union digraph G(σ([∆T + k(T + 2τD +

∆T ) + τD,∆T + k(T + 2τD +∆T ) + T + τD)) has a center, denoted by lk+1.There are two possible cases: lk+1 ∈ Q1 and lk+1 ∈ Q2. We only considerthe first case, while the second case can be studied following a quite similarapproach.

If lk+1 ∈ Q1, then lk+1 ∈ Q1 ∪ Fk and according to Lemma 6.4, thereexist l′k+1 ∈ Q1 ∪ Fk, fk+1 ∈ Q2 ∪ Fk\Fk, and t′ ∈ [∆T + k(T + 2τD +


∆T ),∆T + k(T + 2τD + ∆T ) + T + τD] such that (l′k+1, fk+1) ∈ E(σ(t)) fort ∈ [t′, t′ + τD] ⊆ [∆T + k(T +2τD +∆T ),∆T + k(T +2τD +∆T ) +T +2τD].

By using (6.262) with i = l′k+1, we have

ωfk+1(t) =

∑

j∈Nfk+1(σ(t)) afk+1jηj(t)

∑

j∈Nfk+1(σ(t)) afk+1j

≥ µ(0) +afk+1l′k+1

αll′k+1

(µ(0)− µ(0))∑


(6.264)

for t ∈ [t′, t′ + τD]. Also, we have ωfk+1(t) ≤ µ(0) for t ∈ [t′, t′ + τD]. Property

4 in Proposition 6.2 guarantees that one can find µfk+1(t) and µfk+1

(t) suchthat µfk+1

(t) ≤ µ(0) and

µfk+1

(t) ≥ µ(0) + αl0fk+1( mint∈[t′,t′+τD]

ωfk+1(t)− µ(0))

≥ µ(0) + αl0fk+1

(

afk+1l′k+1αll′

k+1(µ(0)− µ(0))

∑


)

(6.265)

such that (6.248) with i = fk+1 holds at t = t′+τD, where αl0fk+1is continuous,

positive definite, nondecreasing, and less than Id.For t ∈ [t′ + τD, T

∗], we have µ(0) ≤ ωfk+1(t) ≤ µ(0). By using Property 4

in Proposition 6.2 again, we can find µfk+1(t) ≤ µ(0) and

µfk+1

(t) ≥ µ(0) + αlfk+1(µ(0)− µ(0)) (6.266)

such that (6.248) with i = fk+1 holds for t ∈ [∆T + k(T + 2τD +∆T ) + T +2τD, T

∗], where αlfk+1is continuous, positive definite, and less than Id.

By using Property 3 in Proposition 6.2, for specific ∆T > 0, there exists acontinuous, positive definite αl0fk+1

< Id such that

ηfk+1(t)− µ(0) ≥ αl0fk+1

(mint∈T

µfk+1

(t)− µ(0))

≥ αl0fk+1 αlfk+1

(µ(0)− µ(0))

:= αlfk+1(µ(0)− µ(0)), (6.267)

i.e.,

ηfk+1(t) ≥ µ(0) + αlfk+1

(µ(0)− µ(0)) (6.268)

for t ∈ [∆T + (k + 1)(T + 2τD + ∆T ), T∗], where T = [∆T + k(T + 2τD +

∆T ) + T + 2τD, T∗]. Since fk+1 ∈ Q2, according to (6.255), there also exists

a continuous and positive definite αufk+1< Id such that

ηfk+1(t) ≤ µ(0)− αufk+1

(µ(0)− µ(0)) (6.269)


for t ∈ [∆T + (k + 1)(T + 2τD +∆T ), T∗].

Denote Fk+1 = f1, . . . , fk+1. For each i ∈ Fk+1, there exist continuousand positive definite functions αli, α

ui < Id such that (6.262) and (6.263) hold

for all t ∈ [∆T + (k + 1)(T + 2τD +∆T ), T∗].

Final Step: Repeat the procedure in Step k+1 until k+1 = k∗ such that

Q1 ⊆ Fk∗ , (6.270)

or

Q2 ⊆ Fk∗ . (6.271)

Note that for i = 1, . . . , k∗ − 1, fi+1 ∈ Q1 ∪ Fi\Fi or fi+1 ∈ Q2 ∪ Fi\Fi. Itcan be concluded that fi+1 /∈ Fi and thus k∗ ≤ N . Otherwise, the size of Fk∗

is larger than the size of N .Also note that it is a special case that Q1 = ∅ or Q2 = ∅.Convergence: Recall that Q1 ∪Q2 = N . Condition (6.270) implies Fk∗ ∪

Q2 = N ; condition (6.271) implies Fk∗ ∪ Q1 = N .If (6.270) holds, then for i ∈ Fk∗ ∪ Q2 = N , define

µi(T∗) = µ(0)− αui (µ(0)− µ(0)) (6.272)

and define

µ(T ∗) = maxi∈N

µi(T∗). (6.273)

Then, there exists a continuous and positive definite αu < Id such thatµ(T ∗) ≤ µ(0)− αu(µ(0)−µ(0)). By defining µ(T ∗) = µ(0) and µ = µ−µ, wecan achieve

µ(T ∗) = µ(T ∗)− µ(T ∗)

≤ µ(0)− αu(µ(0)− µ(0))− µ(0)

= µ(0)− αu(µ(0)). (6.274)

If (6.271) holds, then for i ∈ Fk∗ ∪ Q1 = N , define

µi(T ∗) = µ(0) + αli(µ(0)− µ(0)) (6.275)

and define

µ(T ∗) = mini∈N

µi(T ∗). (6.276)

By defining µ(T ∗) = µ(0) and µ = µ− µ, we can achieve

µ(T ∗) ≤ µ(0)− αl(µ(0)), (6.277)

where αl is continuous, positive definite, and less than Id.


Define α(s) = min

αl(s), αu(s)

for s ∈ R+. Then, α is continuous, pos-itive definite, and less than Id. By recursively analyzing the system, we canachieve

µ((k + 1)T ∗) ≤ µ(kT ∗)− α(µ(kT ∗)) (6.278)

for k ∈ Z+. By using the asymptotic stability result for discrete-time nonlinearsystems in [132], we can conclude that µ(kT ∗) → 0 as k → ∞.

Define µ(t) = µ(kT ∗) and µ(t) = µ(kT ∗) if t ∈ [kT ∗, (k+1)T ∗). Accordingto the analysis above, during the control procedure, it always holds that

(ηi(t), ζi(t)) ∈ Si(µ(t), µ(t)) (6.279)

and

µ(t) ≤ ηi(t) ≤ µ(t). (6.280)

Properties (6.182) and (6.183) can be proved as µ = µ − µ asymptoticallyconverges to the origin. This ends the proof of Theorem 6.3.

6.4.5 DISTRIBUTED FORMATION CONTROL OF MOBILE ROBOTS

As a practical engineering application, we study the distributed formationcontrol of a group of N + 1 nonholonomic mobile robots under switchingposition measurement topology. For i = 0, . . . , N , each i-th robot is modeledby the unicycle model (6.84)–(6.86).

The robot with index 0 is the leader robot, and the robots with indices1, . . . , N are follower robots. We consider vi and ωi as the control inputs ofthe i-th robot for i = 1, . . . , N . For system (6.84)–(6.86), the position of theleader robot is assumed to be accessible to (some of) the follower robots, andthe control objective is still to achieve (6.87)–(6.89). That is, (xi(t), yi(t))converges to (x0(t) + dxi, y0(t) + dyi) with dxi, dyi being constants and θi(t)converges to θ0(t) + 2kπ with k ∈ Z.

The following assumption is made throughout this section.

Assumption 6.4 The linear velocity v0 of the leader robot is differentiablewith bounded derivative, i.e., v0(t) exists and is bounded on [0,∞), and thereexists constants v0 > v0 > 0 such that v0 ≤ v0(t) ≤ v0 for all t ≥ 0.

Global position measurements of the robots are usually unavailable and thesensing topology may be switching in practical formation control systems. Weemploy the strong output agreement result proposed in the previous sectionto develop a new class of coordinated controllers, which are capable of over-coming the problems caused by the nonholonomic constraint and achievingthe formation control objective by using local relative position measurementsunder switching position sensing topologies as well as the velocity information


of the leader. With our new coordinated controller, the velocity vi of each i-thfollower robot can also be guaranteed to be upper bounded by a constantλ∗i > v0 if required.

The distributed controller for each follower robot is composed of two stages:(a) initialization and (b) formation control. With the initialization stage, theheading direction of each follower robot can be controlled to converge to withindesired ranges in some finite time. Then, the formation control stage is trig-gered and the formation control objective is achieved during the formationcontrol stage.

Initialization Stage

The objective of the initialization stage is to control the angles θi(t) for i =1, . . . , N to within a specific small neighborhood of θ0(t).

For each i-th mobile robot (6.84)–(6.86), we propose the following initial-ization control law:

ωi = φθi(θi − θ0) + ω0 (6.281)

vi = v0, (6.282)

where φθi : R → R is a nonlinear function such that φθi(r)r < 0 for r 6= 0 andφθi(0) = 3.

Define θi = θi − θ0. By taking the derivative of θi and using (6.281) and(6.86), we have

˙θi = φθi(θi). (6.283)

With the appropriately designed φθi, we can guarantee the asymptotic sta-bility of system (6.283). Moreover, there exists βθ ∈ KL such that |θ(t)| ≤βθ(|θ(0)|, t) for t ≥ 0.

For specified 0 < λ∗ < v0 < v0 < λ∗, define

λ = min

√2

2(v0 − λ∗),

√2

2(λ∗ − v0)

. (6.284)

By directly using the property of continuous functions, there exists a δθ0 > 0such that

|v0 cos(θ0 + δθ0)− v0 cos θ0| ≤ λ, (6.285)

|v0 sin(θ0 + δθ0)− v0 sin θ0| ≤ λ (6.286)

for all v0 ∈ [v0, v0], θ0 ∈ R and |δθ0| ≤ δθ0. Recall that for any β ∈ KL, thereexist α1, α2 ∈ K∞ such that β(s, t) ≤ α1(s)α2(e

−t) for all s, t ∈ R+ [243,Lemma 8]. With control law (6.281)–(6.282), there exists a finite time TOi for


the i-th robot such that |θi(TOi)− θ0(TOi)| ≤ δθ0, and thus,

|vi(TOi) cos θi(TOi)− v0(TOi) cos θ0(TOi)| ≤ λ, (6.287)

|vi(TOi) sin θi(TOi)− v0(TOi) sin θ0(TOi)| ≤ λ. (6.288)

At time TOi, the distributed control law for the i-th follower robot isswitched to the formation control stage.

Formation Control Stage

With the dynamic feedback linearization technique recalled in Subsection6.3.1, the unicycle model (6.84)–(6.86) can be transformed into two double-integrators in the form of (6.95)–(6.96) by introducing a new input ri for(6.90), if vi 6= 0 is satisfied. The formation control objective is achieved ifcontrol laws can be designed for the robots to guarantee vi 6= 0, and at thesame time, stabilize (6.95)–(6.96) at the origin.

In this way, the formation control problem is transformed into the issue ofdesigning control laws for system (6.95)–(6.96) with uxi and uyi as the controlinputs, so that vi 6= 0 is guaranteed, and at the same time, the formationcontrol objective is achieved.

The condition vi 6= 0 for the validity of (6.93)–(6.94) can be equivalently

represented by√

v2xi + v2yi > 0 based on the definition of vxi and vyi. To

implement the transformation in (6.92), we need to design the control law forthe i-th robot such that


2(v0 − λ∗) (6.289)

for a specified 0 < λ∗ < v0. In doing so, we can guarantee that |vi| =√

v2xi + v2yi =√

(vx0 + vxi)2 + (vy0 + vyi)2 ≥ λ∗ > 0 and thus vi 6= 0.

Similarly, to guarantee that |vi| ≤ λ∗ for any given λ∗ > v0, we design acontrol law such that


2(λ∗ − v0). (6.290)

The formation control problem is now transformed into the issue of design-ing control laws for system (6.95)–(6.96) with uxi and uyi as the control inputs,so that (6.289) and (6.290) are guaranteed during the control procedure, andat the same time, the formation control objective is achieved.

Recall the definition of λ in (6.284). Both (6.289) and (6.290) can be sat-isfied if

max|vxi|, |vyi| ≤ λ. (6.291)

After the initialization stage, at time TOi, the satisfaction of (6.287) and(6.288) implies that (6.291) is satisfied at time TOi.


Considering the requirement of relative position measurement, we proposea distributed control law in the following form:

uxi = −ϕxi(vxi − φxi(zxi)) (6.292)

uyi = −ϕyi(vyi − φyi(zyi)), (6.293)

where ϕxi, ϕyi, φxi, φyi are strictly decreasing and locally Lipschitz, and satisfyϕxi(0) = ϕyi(0) = φxi(0) = φyi(0) = 0, ϕxi(r)r < 0, ϕyi(r)r < 0, φxi(r)r < 0and φyi(r)r < 0 for r 6= 0, and

supr∈R

max ∂ϕxi(r) : r ∈ R < 4 infr∈R

min ∂φxi(r), (6.294)

supr∈R

max ∂ϕyi(r) : r ∈ R < 4 infr∈R

min∂φyi(r), (6.295)

for i = 1, . . . , N . The functions φxi, φyi are also designed to satisfy

−λ ≤ φxi(r), φyi(r) ≤ λ (6.296)

for r ∈ R.The variables zxi and zyi are defined as

zxi =

∑

j∈Ni(σ(t))aij(xi − xj − dxij)

∑

j∈Ni(σ(t))aij

(6.297)

zyi =

∑

j∈Ni(σ(t))bij(yi − yj − dyij)

∑

j∈Ni(σ(t))bij

, (6.298)

where σ : [0,∞) → P is a piecewise constant switching signal describing theposition sensing topology with P being the set of all the possible positionsensing topologies, Ni(p) ⊆ 0, . . . , N for each i = 1, . . . , N and each p ∈ P ,constant aij > 0 if i 6= j and aij ≥ 0 if i = j. Note that dxij , dyij in (6.297)and (6.298) represent the desired relative position between the i-th robot andthe j-th robot. By default, dxii = dyii = 0.

Consider the (vxi, vyi)-system defined in (6.93)–(6.94). With (6.287) and(6.288) achieved, the boundedness of φxi and φyi in (6.296) together with thecontrol law (6.292)–(6.293) guarantees that

max |vxi(t)|, |vyi(t)| ≤ λ (6.299)

for t ≥ TOi. By considering (vxi, vyi) : max|vxi|, |vyi| ≤ λ as an in-variant set of the (vxi, vyi)-system, (6.299) can be proved. With (6.299)achieved, we have maxi=1,...,N |vxi(t)|, |vyi(t)| ≤ λ for all t ≥ TO withTO := maxi=1,...,NTOi. This guarantees the validity of the transformedmodel (6.93)–(6.94).

Consider the multi-robot model (6.84)–(6.86) and the distributed controllaws defined by (6.281), (6.282), (6.90), (6.92), (6.292), and (6.293) with non-linear functions ϕxi, ϕyi, φxi, φyi satisfying (6.294), (6.295), and (6.296).


We represent the switching position sensing topology by a switching di-graph G(σ(t)) = (N , E(σ(t))), where N = 0, . . . , N and E(σ(t)) is de-fined based on Ni(σ(t)) given in (6.297) and (6.298) for i = 1, . . . , N andN0(σ(t)) ≡ 0. Theorem 6.4 presents our main result on formation controlof unicycle mobile robots.

Theorem 6.4 Under Assumption 6.4, if G(σ(t)) = (N , E(σ(t))) with σ :[0,∞) → P is UQSC and has an edge dwell-time τD > 0, then (6.87)–(6.89)can be achieved for any i, j = 0, . . . , N . Moreover, given any λ∗ > v0, ifvi(0) ≤ λ∗ for i = 1, . . . , N , then vi(t) ≤ λ∗ for all t ≥ 0.

Proof. The states of the mobile robots remain bounded during the finite timeinterval [0, TO]. We study the motion of the robots during [TO,∞). Note thatthe model (6.95)–(6.96) is valid during [TO,∞).

Note that vx0 = vy0 = ux0 = uy0 = 0. One can find appropriateϕx0, ϕy0, φx0, φy0, a00, b00 to represent ux0 and uy0 by (6.292) and (6.293)with zx0 and xy0 in the form of (6.297) and (6.298), respectively.

For i = 0, . . . , N , rewrite

zxi =

∑

j∈Ni(σ(t))aij(xi − xj)

∑

j∈Ni(σ(t))aij

, (6.300)

zyi =

∑

j∈Ni(σ(t))bij(yi − yj)

∑

j∈Ni(σ(t))bij

. (6.301)

For i = 0, . . . , N , all the uxi and uyi are in the form of µi defined in (6.237)and all the zxi and zyi are in the form of ξi defined in (6.238).

Theorem 6.3 guarantees that

limt→∞

(xi(t)− xj(t)) = 0, (6.302)

limt→∞

(yi(t)− yj(t)) = 0, (6.303)

for any i, j = 0, . . . , N . Then, we can prove (6.87) and (6.88) by using thedefinitions of xi and yi and the fact that x0(t) = y0(t) ≡ 0. The result ofvi(t) ≤ λ∗ can be proved based on the discussions below (6.299).

By using Theorem 6.3, we can also prove the convergence of vxi, vyi to theorigin. By using the definitions of vxi, vyi, the convergence such that (6.89)can be proved. This ends the proof. ♦

If there is no leader in the mobile robot system, the velocities of the robotsare usually hard to control. To overcome this problem, we may employ avirtual leader, which generates a reference velocity for the follower robots.Note that the global and relative positions of the virtual leader are usually notavailable. In this case, our proposed formation control strategy is still validsuch that the follower robots are controlled to converge to specific relative


positions of each other, i.e., (6.87) and (6.88) are achieved for any i, j =1, . . . , N .

We represent the switching position sensing topology of the follower robotswith a switching digraph Gf (σ(t)) = (N f , Ef (σ(t))), where N f = 1, . . . , Nand Ef (σ(t)) is defined based on Ni(σ(t)) given in (6.297) and (6.298) fori = 1, . . . , N .

Theorem 6.5 presents such an extension of our main formation controlresult.

Theorem 6.5 Under Assumption 6.4, if Gf (σ(t)) = (N f , Ef (σ(t))) with σ :

[0,∞) → P is UQSC and has an edge dwell-time τfD > 0, then (6.87)–(6.89)can be achieved for any i, j = 1, . . . , N . Moreover, given any λ∗ > v0, ifvi(0) ≤ λ∗ for i = 1, . . . , N , then vi(t) ≤ λ∗ for all t ≥ 0.

Proof. The proof of Theorem 6.5 is still based on Theorem 6.3 and quitesimilar as the proof of Theorem 6.4. In the case of Theorem 6.5, we need notconsider x0 and y0, and properties (6.302) and (6.302) can be achieved forany i, j = 1, . . . , N . Given the definitions of xi and yi for i = 1, . . . , N , we canprove (6.87) and (6.88) for any i, j = 1, . . . , N . ♦

6.4.6 SIMULATION RESULTS

Consider a group of 6 robots with indices 0, 1, . . . , 5 with robot 0 being theleader. By default, the values of all the variables in this simulation are in SIunits.

The control inputs of the leader robot are r0(t) = 0.1 sin(0.4t) andω0(t) = 0.1 cos(0.2t). With such control inputs, the linear velocity v0 sat-isfies v0 ≤ v0(t) ≤ v0 with v0 = 3 and v0 = 3.5, given v0(0) = 0.The desired relative position of the follower robots are defined by dx01 =−√3d/2, dx02 = −

√3d/2, dx03 = 0, dx04 =

√3d/2, dx05 =

√3d/2, dy01 =

−d/2, dy02 = −3d/2, dy03 = −2d, dy04 = −3d/2, dy05 = −d/2 with d = 80.Choose λ∗ = 0.45 and λ∗ = 6.05. The distributed control laws for the

initialization stage are designed in the form of (6.281)–(6.282) with

φθi(r) = φvi(r) = −0.5(1− exp(−0.5r))/(1 + exp(−0.5r)) (6.304)

for i = 1, . . . , 5. The distributed control laws for the formation control stageare in the form of (6.292)–(6.293) with kxi = kyi = 6 and

φxi(r) = φyi(r) = −1.8(1− exp(−r))/(1 + exp(−r)) (6.305)

for i = 1, . . . , 5. It can be verified that kxi, kyi, φxi, φyi satisfy (6.294) and(6.295). Also, φxi(r), φyi(r) ∈ [−1.8, 1.8] for all r ∈ R. With v0 = 3 andv0 = 3.5, the linear velocities of the robots are restricted to be within therange of [3− 1.8

√2, 3.5 + 1.8

√2] = [0.454, 6.046]⊂ [λ∗, λ∗].


The initial states of the robots are chosen as

i (xi(0), yi(0)) vi(0) θi(0)

0 (0, 0) 3 π/61 (−20, 50) 4 π2 (30,−40) 3.5 5π/63 (50,−100) 2.5 04 (200,−100) 2 −2π/35 (100,−120) 3 0

The information exchange topology switches between the digraphs in Fig-ure 6.12, and the switching sequence is shown in Figure 6.13.

2 3 4

1 0 5

(5)

2 3 4

1 0 5

(6)

2 3 4

1 0 5

(3)

2 3 4

1 0 5

(4)

2 3 4

1 0 5

(1)

2 3 4

1 0 5

(2)

FIGURE 6.12 Digraphs representing the switching information exchange topology.

The linear velocities and the angular velocities of the robots are shownin Figure 6.14. The stage changes of the distributed controllers are shown inFigure 6.15 with “0” representing the initialization stage and “1” representingthe formation control stage. Figure 6.16 shows the trajectories of the robots.


FIGURE 6.13 The switching sequence of the information exchange topology.

The simulation verifies the theoretical result of the paper.

FIGURE 6.14 The linear velocities and angular velocities of the robots.

6.5 NOTES

Distributed control of multi-agent systems for group coordination has recentlyattracted significant attention within the controls and robotics communities:see, for example, [212, 214, 172, 236, 176] based on Lyapunov methods; [8, 17]using passivity methods; [58, 215, 39, 240, 225, 253, 162] based on linear al-


FIGURE 6.15 The stages of the distributed controllers.

FIGURE 6.16 The trajectories of the robots. The trajectory of the leader robot is

represented by the dashed curve.

gebra and graph theory; and [271, 275, 254] using output regulation theory.The main objective of distributed control is to achieve some desired groupbehavior for multi-agent systems by taking advantage of local system infor-mation and information exchanges among neighboring systems. Distributedcontrol may find applications in sensor networks [213], vehicle coordinationand formation [256, 229, 111, 68, 282], and smart power grids [280]. Refer-ence [192] studies the synchronization problem of multi-agent systems withoutconnectivity assumptions. In [287], the synchronization of dynamic networks


with nonidentical nodes is achieved by reorganizing the connection topologies.One group behavior of wide interest is the agreement property, for which therelated variables of multi-agent systems are steered to a common value. Itshould be noted that most of the previously published papers focus on linearmodels.

This chapter has presented cyclic-small-gain tools for distributed control ofnonlinear multi-agent systems. In Section 6.2, with the proposed distributedobservers and control laws, the outputs of the agents can be steered to withinan arbitrarily small neighborhood of the desired agreement value under exter-nal disturbances. Asymptotic output agreement can be achieved if the systemis disturbance-free. The robustness to bounded time delays of exchanged infor-mation can also be guaranteed. In the problem setting, each agent can use theoutput of itself and the outputs of its neighbors, while only the informed agentshave access to the desired agreement value. This makes the distributed controlproblem considered in this chapter significantly different from the decentral-ized control problem, in which each decentralized controller often assumes thea priori knowledge of the reference signal and does not take advantage of theavailable information of neighboring agents; see e.g., [239, 129]. It should benoted that the term of distributed control is also used for decentralized con-trol under arbitrary information structure constraints in some recent works[285, 286].

Section 6.2 only considers the case with time-invariant agreement value y0.It is practically interesting to further study the distributed nonlinear controlfor agreement with a time-varying agreement value. Recent developments onthe output-feedback tracking control of nonlinear systems (see, e.g., [116])should be helpful for the research in this direction.

Section 6.3 studies the formation control of nonholonomic mobile robots.For the formation control of mobile robots, assuming a tree sensing struc-ture generally leads to cascade interconnection structures of the closed-loopsystems [47]. Along this line of research, [258] employs nonlinear gains toestimate the influence of leader behavior on the formation by using the con-cept of ISS. Several researchers have attempted to relax the assumption oftree sensing structures, at the price of using global position measurements. In[212], a control Lyapunov function approach was introduced to multi-agentcoordination. The authors of [50] proposed a constructive design method forthe formation tracking control and collision avoidance of unicycle robots. Ref-erence [51] transforms the cooperative control problem into a decentralizedbackstepping design problem. In [161], the artificial potential function-basedapproach was studied for the flocking control of nonholonomic mobile robots.The wiggling controller proposed in [171] does not use global position mea-surements and is capable of controlling unicycles to stationary points, butthe controller seems unable to overcome the nonholonomic constraint of theunicycle to achieve moving formations. It should be noted that if each robothas access to its desired path, as in the coordinated path following problems


studied in [2, 103, 157], then the requirement of global position measurementscan be relaxed.

The distributed formation control law proposed in Section 6.3 uses relativeposition measurements without assuming a tree structure. For this purpose,the formation control problem is first transformed into a state agreement prob-lem of double-integrators with dynamic feedback linearization [40]. The non-holonomic constraint leads to singularity for dynamic feedback linearizationwhen the linear velocity of the robot is zero. Then, a class of distributed non-linear control laws is designed. Saturation functions are employed to restrictthe linear velocities of the robots to be larger than zero to avoid the singularity.With the proposed distributed nonlinear control law, the closed-loop systemcan be transformed into a network of IOS systems, and the achievement of theformation control objective can be guaranteed by using the cyclic-small-gaintheorem. The special case in which there are only two robots and the desiredrelative positions are zero has been studied extensively in the past literature;see [127, 49] and the references therein.

To further relax the requirement on the fixed topologies, Section 6.4 haspresented a modified distributed nonlinear control design for strong outputagreement of multi-agent systems modeled by double-integrators. With thenew design, the information exchange topology of the large-scale intercon-nected multi-agent system is allowed to be directed and switching, as longas a mild connectivity condition is satisfied. By appropriately designing thedistributed control law, the internal states of the agents can be restricted tobe within an arbitrarily small neighborhood of the origin, which is of practi-cal interest. As an application, a distributed formation control algorithm hasalso been developed for groups of unicycle mobile robots with flexible topolo-gies and relative position measurements. The singularity problem caused bythe nonholonomic constraint is solved by properly designing the distributedcontrol law.

By showing that a distributed control problem can be transformed intothe stability/convergence problem of a dynamic network composed of IOSsubsystems, Sections 6.1–6.3 provide some partial answers to the questionasked by Open Problem #5 in [137]: “Application of small-gain results fordistributed feedback design of large-scale nonlinear systems.” More discussionson the application of the cyclic-small-gain theorem to distributed control canbe found in [184, 185, 180, 183].

7 Conclusions and FutureChallenges

This book has presented recent results on the stability and control of intercon-nected systems which are modeled as dynamic networks. Such networks arisenaturally in engineering applications, such as robotic networks and power sys-tems, as well as in other areas such as biology, physics, and economics. Thetreatment is all based on the concept of input-to-state stability (ISS) and theidea of small-gain in loops of the network as a means to achieve stability whensystems are interconnected. The ISS property for each subsystem includesnonlinear gain functions and corresponds to the existence of an ISS-Lyapunovfunction. The stability conditions are intuitively and conveniently expressedin terms of compositions of the gains associated with cycles in the systemgraph being less than one, generalizing the well-known small-gain theoremfor feedback systems. After establishing the small-gain theorems for classesof dynamic networks (continuous-time, discrete-time, and hybrid), the bookproposed a set of tools for input-to-state stabilization and robust control ofcomplex nonlinear systems from the viewpoint of dynamic networks. Amongthese tools and applications are:

• Lyapunov-based cyclic-small-gain theorems for continuous-time,discrete-time, and hybrid dynamic networks composed of multipleISS subsystems;

• novel small-gain-based static feedback and dynamic feedback designsfor robust control of nonlinear uncertain systems with disturbed mea-surements;

• quantized stabilization designs for nonlinear uncertain systems withstatic quantization and dynamic quantization; and

• distributed coordination control of nonlinear multi-agent systems un-der information exchange constraints.

The idea of small-gain is about 40 years old, and thinking of the Nyquistcriterion interpretation, is one of two fundamental ideas for preserving stabilityin a feedback loop. Nevertheless, after all the work of numerous researchers,taking this idea to ever more complex systems, the topics considered in thisbook show that many interesting research problems, in particular at the levelof control synthesis, remain to be considered. These new challenges relateto theoretical advances as well as further application of the results. Somesuggestions for such work will now be discussed.

The hybrid dynamic network considered in Chapter 3 involves only stabledynamics. However, it is well known that appropriately switching between un-

255


stable dynamics may still lead to stability. In fact, this can be a way to achievehigher performance. This behavior is often formulated with the “dwell-time”condition. There are two major difficulties in introducing this approach tohybrid dynamic networks: (a) The subsystems may have different impulsivetime instants, which may lead to restrictiveness if we consider the dynamicnetwork as one single system and directly apply the “dwell-time” method;(b) Some necessary transformation is often needed to transform the decreas-ing/increasing rates of the subsystems from non-exponential to exponential,without influencing the validity of the cyclic-small-gain condition. Recently,researchers have started to consider the possibility of unstable subsystems.One recent contribution in this direction provides revised small-gain theoremsfor hybrid feedback systems [169]. ISS stability criteria for hybrid systemswhich include unstable subsystems (discrete-time or continuous-time) havebeen given in [174, 169]. Generally, there seem to be possibilities to advancethe complexity to include switching, impulses, delays, and interconnections[173].

A major problem in analyzing complex systems and networks can be tofind ways to manage the computations involved as the system scale becomeslarger. This can arise from the complexity of the dynamics and/or the net-work structure, and so affect finding ISS-Lyapunov functions and testing thesmall-gain criteria. The latter involves functional tests which grow with thediameter and connectivity of the network. It would appear that much workcan yet be done to find computational tools and results which deal with specialclasses of systems. One such tool is the sum-of-squares technique, which hasrecently been applied to networked systems [79]. These issues of scale havebeen around since the effort on so-called large-scale systems in the 1970s.For stability analysis, the paradigm was generally to establish strongly stablesubsystems and weak interconnections, which reduced the stability test to amatrix condition [205, 207]. In its most accessible form, this expressed thedominance of the subsystem stability over the strength of the interconnec-tions. It has been of interest ever since to find stability results which allowmore diverse interplay between subsystem and interconnection properties. Thesynchronization theory for dynamical networks demonstrated stability criteriawith strong coupling. However, again, as systems get more complicated, thereis motivation to find ways to “divide and conquer” the system scale. One ideais to think in terms of stability certificates where a sequence of stability testsare applied to sub-regions of the network, perhaps each subsystem and itsnearest neighbors [216]. It follows that there is no need for an overall stabilitytest. Another idea exploits detailed system structure [226]. These ideas arein their infancy and it remains to see how far they can be taken. Computa-tional aspects need to be sensitive to the properties of solutions to dissipationinequalities [112].

The control designs proposed in this book are restricted to systems inspecific forms: strict-feedback form and output-feedback form. While popular

Conclusions and Future Challenges 257

in research to demonstrate designs with supporting theory, they are basedmore in mathematical convenience than any physical forms. However, in givenpractical case studies, such results are a useful guide if not directly applicable.Nevertheless, despite the popularity of these forms, extensions of the designsto more general nonlinear systems would contribute a lot to nonlinear controltheory. Again, these structural issues have an earlier form in discussions ofhow to extend control designs based on backstepping and forwarding to moregeneral structures [235].

In modern networked control systems, data transmission through commu-nication channels inevitably causes time delays and thus late response of thecontrol system to control commands. Similar problems also arise from datasampling. Reference [261] discovered the connections between ISS small-gaintheorems and the Razumikhin theorem, the latter being dedicated to time-delay nonlinear systems. Recently, [136] presented a sufficient and necessarycondition for stabilizability of nonlinear, time-varying systems with delayedstate measurements. However, robust control designs for nonlinear systems(in the popular forms) with both uncertain dynamics and time delay are stillto be developed. Related problems have also been noted in [137]. It shouldbe mentioned that the recent work of Krstic [149, 150] has proposed newsolutions to stabilization with input delays.

Uncertain actuator dynamics may cause the performance of a nonlinearcontrol system to deteriorate. Quite a few references (see, e.g., [123, 119] andthe references therein) studied the controller design problem for nonlinearsystems with uncertain actuator dynamics. Another source of destabilizingfactors is the uncertain measurement dynamics (due to inertia of the sensor,for example). However, there has not been much research on robust design ofnonlinear control systems in the presence of uncertain measurement dynamics.The tools developed in this book may provide some potential solution to thisproblem. As we can design input-to-state stabilizing controllers for nonlinearsystems with measurement errors, one possible solution to the problem is tofind some way to formulate the uncertain measurement dynamics with ISSand use the ISS cyclic-small-gain theorem to guarantee stabilization.

An alternative approach to the control of uncertain systems is adaptivecontrol, which is extremely useful for systems with “large” parameterized un-certainties. Passivity has played a central role in adaptive control designs sinceits earliest days [12]. Robust adaptive control methods have also been devel-oped for systems with both parameterized uncertainties and disturbances.Design techniques for nonlinear systems have been obtained via backstep-ping and passivity in [153]. However, the influence of sensor noise on adaptivecontrol performance of nonlinear uncertain systems has not been systemat-ically investigated, and refined robust adaptive control methods remain tobe achieved. Considering the usefulness of passivity to adaptive control andthe importance of ISS and small-gain theorems to robust control (and theirmore minor roles, each in the other area), one may think about the combina-


tion of the methods. Preliminary results were obtained in [115]. The notionof dissipativity [277] has also been used to realize such a combination forinterconnected systems, and apply these to giving general stability criteriawhich combine passivity and finite gain aspects in [90, 207, 93]. Another ap-proach has suggested using the passivity and finite gain properties side byside [72]. In these results, the supply functions are in the quadratic form. Todeal with complex nonlinear dynamics, more efforts are needed for the combi-nation of passivity and ISS small-gain methods for systems with more generalsupply functions. With the expected generalization, the problem mentionedabove and more general robust adaptive control problems may hopefully besolved for nonlinear uncertain systems. Similar problems also arise from thedistributed adaptive control.

Another promising research area, still in its infancy, is the quantized non-linear control which, as shown in this book, is strongly connected to the robustnonlinear control. Advanced robust control design methods, such as the ISSsmall-gain approach, are powerful in handling the new problems caused byquantization in nonlinear control. Expectedly, the preliminary results pre-sented in the book can be further generalized in several directions, in view ofthe rich literature of nonlinear control over the last three decades. It should bementioned that there are more open problems in this field than the availableresults. Some open problems of great interest are stated here:

• Geometric nonlinear control with quantized signals. The classical yetimportant topic of controllability and observability for nonlinear sys-tems needs to be revisited [105, 244], when only quantized signals areallowed. In addition, the relationships between controllability andstabilizability [37], and feedback linearization theory [105], need tobe revisited as well in the context of quantized feedback control.

• Tracking via quantized feedback. While this book focuses on quan-tized stabilization, the problem of quantized feedback tracking is ofmore practical interest and covers stabilization as a special case. In-stead of forcing the state or the output to the origin or a set point ofinterest, the quantized feedback tracking problem seeks a quantizedfeedback controller so that the output follows a desirable referencesignal or the state follows the desired state of a reference model. Thisproblem has received practically no attention in the present litera-ture. Closely related to this problem is the output regulation theory[99, 21] that consists of searching for (unquantized) feedback controllaws to achieve asymptotic tracking with disturbance rejection, whenthe disturbance and reference signals are generated by an exo-system.The well-known internal model principle serves as a bridge to con-vert the output regulation problem into a stabilization problem for atransformed system. To what extent will the internal model principleremain valid and applicable when only quantized signals are used?

Conclusions and Future Challenges 259

• Decentralized quantized control. In the decentralized control setting,local controllers are used to control the subsystems of a large-scalesystem [239]. Among the main characteristics of decentralized con-trol are the dramatic reduction of computational complexity andthe enhancement of robustness against uncertain interactions or lossof interaction. The ISS small-gain designs for decentralized control[129, 117] makes it possible to further take into account the effect ofquantization; see also the survey [118]. The decentralized measure-ment feedback control problem, which is closely related to decentral-ized quantized control, has been studied in this book. For dynamicquantization, the zooming variables of the quantizers of different sub-systems should coordinate with each other. This is still the case whendecentralized control is reduced to centralized control, as shown inSection 5.2. Another problem with decentralized dynamic quantiza-tion is that the updates of the zooming variables of the quantizersmay not be synchronized. For such problem, the small-gain resultsfor hybrid systems [137, 138, 188] should be helpful.

• Quantized adaptive control. Controllers are expected to possess adap-tive capabilities to cope with “large” uncertainties. A further exten-sion of the previously developed methodology to quantized adaptivecontrol is of practical interest for engineering applications. The re-cent achievements [81, 82] provide a basis for future research in thisdirection. Reference [81] proposed a Lyapunov-based framework foradaptive quantized control of linear uncertain systems modeled indiscrete-time. In [82], a direct adaptive control strategy was devel-oped for nonlinear uncertain systems with input quantizers underthe assumption that the system is robustly stabilizable with respectto sector bounded uncertainties.

• Quantized control systems with time delays. As discussed above, datatransmission through communication channels inevitably results intime delays, a severe cause of poor performance and even instabilityof the system in question. Recently, a necessary and sufficient condi-tion for robust stabilizability of nonlinear, time-varying systems withdelayed state measurements was presented in [136]. A new frameworkcapable of dealing with quantization and time delay at the same timeis of paramount importance for the transition of advanced nonlinearcontrol theory to practice.

The potential for application to physical systems certainly needs furtherexploration. Power systems are the most complex nonlinear dynamic networkswithin engineering systems. They connect generation and load devices acrosswhole continents with dynamics on time scales from microseconds to hours,and with such distances the effect of time delays has to be considered. In termsof power flows, the device dynamics and interconnections are highly nonlin-ear. Further, the control loops routinely involve switching leading to large


hybrid networks. There are many sources of modeling uncertainty. So for sta-bility analysis and control, techniques associated with the whole theoreticaltoolkit, especially nonlinearity, uncertainty, hybrid models, large scale, robust-ness, discrete events, and adaptivity are all potentially useful. Certainly, themajor stability and control questions can be formulated as nonlinear and/ornetwork control problems. Trends in the industry, including new approachesto reliability (away from worst-case scenarios) and increasing use of renew-ables mean there will be much more uncertainty and systems will be drivenharder. This translates into a need for nonlinear systems and methods, whichreduce the effects of uncertainty. There has been a lot of work on stabilityanalysis [34] and nonlinear control [27, 194] using modern nonlinear systemstechniques, including gain-related ideas to handle disturbances [75]. The newconcern is how stability and control become robust to major changes such asplacement, size, and variability of renewable generation [29]. There appearsto be considerable work to be done to use methods based on ISS to exploresuch issues. The recently developed robust adaptive dynamic programmingmethods [113, 114] provide new solutions; see also [120] for a recent review.

Similarly, in other areas involving interconnections of systems in networkstructures, there are ideas to pursue. The distributed control problems ofrobotic networks have been partially studied in this book. Another area wheresmall-gain and dissipativity concepts appear useful, which is outside the ex-perience of the authors, is biology. A recent book has studied biomolecularsystems using ISS and related concepts [145], including contractive systems[193] which are closely related to the small-gain idea. Also see the recent workof Sontag and coworkers in systems biology [246, 45, 56]. The recent reviewarticle [120] gives an application of the ISS small-gain theorem to biologicalmotor control, an important problem in systems neuroscience.

These topics—and no doubt there are more that can be seen in the work ofothers—promise that the fundamental concepts of ISS and small-gain can bedeveloped even further to provide important tools for stability analysis andcontrol of nonlinear interconnected systems.

A Related Notions inGraph Theory

This section gives the standard definitions of the notions in graph theory thatare used in this book.

Definition A.1 A graph G is a collection of points V1, . . . ,Vn and a collectionof lines a1, . . . , am joining all or some of these points. The points are calledvertices, and the lines, denoted by the pairs of points they connect, are calledlinks.

Definition A.2 A graph G is called a directed graph or simply a digraph ifthe lines in it have a direction. The lines are called directed links or arcs.

Definition A.3 A path in a digraph G is any sequence of arcs where the finalvertex of one is the initial vertex of the next one, denoted as the sequence ofthe vertices it contains. If a path has no repeated vertices, then it is called asimple path.

Definition A.4 In a digraph G, if there exists a path leading from vertex ito vertex j, then vertex j is reachable from vertex i. Specifically, any vertex iis reachable from itself.

Definition A.5 In a digraph G, the reaching set of a vertex j, denoted byRS(j), is the set of the vertices from which vertex j is reachable.

Definition A.6 In a digraph G, a path such that the starting vertex and theending vertex are the same is called a cycle. If a cycle has no repeated verticesother than the starting and ending vertices, then it is called simple cycle.

Definition A.7 A directed tree is a digraph which has no cycle and thereexists a vertex from which all the other vertices are reachable.

Definition A.8 A spanning tree T of a digraph G is a directed tree formedby all the vertices and some or all of the edges of G.

261

© 2014 by Taylor & Francis Group, LLC

B Systems withDiscontinuous Dynamics

This appendix provides some basic concepts and preliminary results of systemswith discontinuous dynamics. Detailed studies on this topic can be found in[36, 60, 13, 84].

B.1 BASIC DEFINITIONS

Definition B.1 For a set M ∈ Rn, a point x ∈ Rn is an interior point ofM if there exists an open ball centered at x which is contained in M. Theinterior of M, denoted by int(M), contains all the interior points of M.

Definition B.2 A set M ⊆ Rn is called convex if for every x, y ∈ M andevery λ ∈ [0, 1], λx+ (1− λ)y ∈ M.

Definition B.3 The convex hull of a set M ⊆ Rn, denoted by co(M), is theintersection of all the convex sets containing M.

Definition B.4 The closed convex hull of a set M ⊆ Rn, denoted by co(M),is the intersection of all the closed convex sets containing M.

Let X and Y be two sets in Euclidian spaces.

Definition B.5 A set-valued map F : X Y is a map that associates withany x ∈ X a subset F (x) of Y, and the subsets F (x) are called the images ofF .

Definition B.6 The domain of a set-valued map F : X Y is defined as

dom(F )def= x ∈ X : F (x) 6= ∅. (B.1)

Definition B.7 A set-valued map F : X Y is called strict if dom(F ) = X .

Definition B.8 A function f : X → Y is a selection of a strict set-valuedmap F : X Y if f(x) ∈ F (x) for all x ∈ X .

Definition B.9 The graph of a set-valued map F : X Y is the subset ofpairs (x, y) where y ∈ F (x), that is,

graph(F )def= (x, y) ∈ X × Y : y ∈ F (x). (B.2)

263


Definition B.10 The range of a set-valued map F : X Y, denoted byrange(F ), is defined as

range(F )def=⋃

x∈XF (x). (B.3)

Definition B.11 A set-valued map F : X Y is upper semi-continuous(USC) at x0 ∈ X if for any open N containing F (x0) there exists a neighbor-hood M of x0 such that F (M) ⊂ N .

Let B(y, d) with y ∈ Rm and d ∈ R+ represent the unit ball with center yand radius d in R

m. With F : X Y, define

B(F (x), d) = y : B(y, d) ∩ F (x) 6= ∅ (B.4)

for x ∈ X and d ∈ R+.

Definition B.12 A set-valued map F : X Y is locally Lipschitz if for anyx0 ∈ X , there exist a neighborhood N (x0) ⊂ X and a constant L ≥ 0 suchthat for any x, x′ ∈ N (x0),

F (x) ⊆ B(F (x′), L|x− x′|). (B.5)

Definition B.13 A set-valued map F : X Y is Lipschitz if there exists aconstant L ≥ 0 such that for any x, x′ ∈ X ,

F (x) ⊆ B(F (x′), L|x− x′|). (B.6)

B.2 EXTENDED FILIPPOV SOLUTION

Consider the continuous-time system

x = f(x, u), (B.7)

where x ∈ Rn is the state and u ∈ Rm is the control input.The vector field f is assumed to be a piecewise continuous function from

Rn × Rm to Rn in the sense that

f(x, u) = f i(x, u) (B.8)

when [xT , uT ]T ∈ Ωi, i ∈ M , where M = 1, . . . ,M, Ω1, . . . ,ΩM are closedsubsets of Rn × Rm satisfying

⋃

i∈M Ωi = Rn × Rm and int(Ωi) ∩ intΩj = ∅for i 6= j. The dynamics of system (B.7) are discontinuous and the system iscalled a discontinuous system for convenience.

It is assumed that f i : Ωi → Rn is locally Lipschitz on the domain Ωi, andcl(int(Ωi)) = Ωi for all i ∈ M .

Systems with Discontinuous Dynamics 265

The following differential inclusion is used to define the extended Filippovsolution:

x ∈ F (x, u) (B.9)

with

F (x, u) := cof i(x, u) : i ∈ I(x, u), (B.10)

I(x, u) := i ∈ M : [xT , uT ]T ∈ Ωi. (B.11)

The definition of F implies that

1. F is strict, that is, Dom(F ) = Rn × R

m;2. F (x, u) is a compact convex subset of Rn for every pair (x, u) in Rn×Rm;3. F is upper semi-continuous.

Definition B.14 A function x : [a, b] → Rn is an extended Filippov solutionto the discontinuous system (B.7) with u : R+ → Rm measurable and essen-tially ultimately bounded, if x is locally absolutely continuous and satisfies

x(t) ∈ F (x(t), u(t)) (B.12)

for almost all t ∈ [a, b].

For a general discontinuous f(x, u), the set-valued map F (x, u) can bedefined as

F (x, u) =⋂

ǫ>0

⋂

µ(M=0)

cof(Bǫ(x, u)\M), (B.13)

where Bǫ(x, u) is an open ball of radius ǫ around (x, u), and M represents allsets of zero measure (i.e., µ(M) = 0).

Compared with the standard Filippov solution [60], the definition of theextended Filippov solution takes into account both state x and input u. Thistreatment is helpful for the study of interconnected discontinuous systems.It should be noted that if the system does not have external input, then thedefinition of extended Filippov solution is reduced to the definition of Filippovsolution. See also books [281, 33] and the tutorial [38] for more basic conceptsof discontinuous systems. For related concepts in set-valued maps, see [13].

B.3 INPUTTOSTATE STABILITY

Definition B.15 System (B.7) is said to be ISS if there exist β ∈ KL andχ ∈ K such that for any initial state x(0) = x0 and any measurable and locallyessentially bounded u, the extended Filippov solution exits and satisfies

|x(t)| ≤ max β(|x0|, t), χ(‖u‖) (B.14)

for t ≥ 0.


An ISS-Lyapunov function candidate of a discontinuous system can bepiecewisely defined as:

V (x) = V i(x) (B.15)

when x ∈ Γi for i ∈ MV , where MV = 1, . . . ,MV , and Γ1, . . . ,ΓMV areclosed subsets of Rn satisfying

⋃

i∈MVΓi = Rn and int(Γi) ∩ int(Γj) = ∅ for

i 6= j.Each Vi is supposed to be continuous differentiable on some open domain

containing Γi for i ∈ MV . Moreover, it is assumed that

Vi(x) = Vj(x) (B.16)

when x ∈ Γi ∩ Γj 6= ∅. Define

J(x) = j ∈ MV : x ∈ Γj. (B.17)

Definition B.16 The ISS-Lyapunov function candidate V of form (B.15) issaid to be an ISS-Lyapunov function for system (B.7) if

1. V is locally Lipschitz;2. V is positive definite and radially unbounded, that is, there exist α, α ∈ K∞

such that for all x ∈ Rn,

α(|x|) ≤ V (x) ≤ α(|x|), ∀x; (B.18)

3. there exist a γ ∈ K and a continuous, positive definite α such that foralmost all x, u,

V (x) ≥ γ(|u|) ⇒ ∇V (x)f i(x, u) ≤ −α(V (x)), ∀i ∈ I(x, u). (B.19)

Theorem B.1 System (B.7) is ISS if it admits an ISS-Lyapunov function.

The proof of Theorem B.1 can be found in [84].For a discontinuous system (B.7) with its Filippov solution defined by dif-

ferential inclusion (B.9), condition (B.19) can be replaced with

V (x) ≥ γ(|u|) ⇒ maxf∈F (x,u)

∇V (x)f ≤ −α(V (x)). (B.20)

B.4 LARGESCALE DYNAMIC NETWORKS OF DISCONTINUOUSSUBSYSTEMS

Consider the network of discontinuous subsystems

xi = fi(x, ui) = f jii (x, ui), i = 1, . . . , N (B.21)

Systems with Discontinuous Dynamics 267

when [xT , uTi ]T ∈ Ωjii for ji ∈ Mi, where xi ∈ Rni is the state of the i-th

subsystem, x = [xT1 , . . . , xTN ]T is the state of the dynamic network, ui ∈ Rmi

is the input of the xi-subsystem. By considering ui as the input of the i-thsubsystem, assume that ui is measurable and locally essentially bounded.

The dynamic network with state x = [xT1 , . . . , xTN ]T and external input

u = [uT1 , . . . , uTN ]T can be rewritten as

x = [f j1T1 (x, u1), . . . , fjNTN (x, uN )]T

:= f(x, u) (B.22)

when [xT , uT ]T ∈ Ω(j1,...,jN ) with

Ω(j1,...,jN ) :=

[xT , uT ]T : [xT , uTi ]T ∈ Ωjii , i = 1, . . . , N

(B.23)

for each combination of (j1, . . . , jN ) ∈ M1 × · · · MN .Define

Fi(x, ui) = cof jii : ji ∈ Ii(x, ui) (B.24)

Ii(x, ui) = ji ∈ M i : [xT , uTi ]T ∈ Ωjii (B.25)

F (x, u) = F1(x, u1)× · · · × FN (x, uN ). (B.26)

Then, it can be proved that

1. F is strict, that is, dom(F ) = Rn × Rnu ;2. F (x, u) is a compact convex subset of Rn for every pair (x, u) in R

n×Rnu;

3. F is upper semi-continuous.

If cl(int(Ωjii )) = Ωjii , then the extended Filippov solution x : [a, b] →R

∑i=1,...,N ni of the discontinuous dynamic network can be defined by differ-

ential inclusion

x(t) ∈ F (x, u) (B.27)

for almost all t ∈ [a, b].

C Technical Lemmas Relatedto Comparison Functions

Lemma C.1 Consider χ ∈ K and χi ∈ K∪0 for i = 1, . . . , n. If χχi < Idfor i = 1, . . . , n, then there exists a χ ∈ K∞ such that χ > χ, χ is continuouslydifferentiable on (0,∞), and χ χi < Id for i = 1, . . . , n.

Proof. Define χ (s) = maxi=1,...,n χi (s) for all s ≥ 0. Then, χ ∈ K∪0 andχ χ < Id. Following the proofs of Theorem 3.1 and Lemma A.1 in [126], onecan find a χ ∈ K∞ such that χ > χ, χ is continuously differentiable on (0,∞)and χ χ < Id. It is easy to verify that χ χi < Id for i = 1, . . . , n. ♦

Lemma C.2 Consider χi1, χi2 ∈ K ∪ 0 for i = 1, . . . , n. If χi1 χi2 < Idfor i = 1, . . . , n, then there exists a positive definite function η such that(Id− η) ∈ K∞ and χi1 (Id− η)−1 χi2 < Id for i = 1, . . . , n.

Proof. Recall the fact that for any χ1, χ2 ∈ K∪0, χ1χ2 < Id ⇔ χ2χ1 < Id.Property χi1 (Id− η)−1 χi2 < Id is equivalent to (Id− η)−1 χi2 χi1 < Id.

Define χ0(s) = min 12 (χ

−1i1 χ−1

i2 (s) + s) for s ≥ 0. Obviously, χ0 ∈ K∞.

For all i = 1, . . . , n, since χi2 χi1 < Id, we have χ−1i1 χ−1

i2 > Id. Thus,χ0 > Id. We also have χ0 χi2 χi1 ≤ 1

2 (Id + χi2 χi1) < Id for all i =1, . . . , n. Define η = χ0 − Id. Then, η is positive definite, (Id + η) ∈ K∞, and(Id + η) χi2 χi1 < Id for i = 1, . . . , n. The proof follows readily by definingη = Id− (Id + η)−1, or equivalently η = η (Id + η)−1. ♦

Lemma C.3 For any positive definite function α, and any class K∞ functionχ, there exists a positive definite function α such that χ(s′)−χ(s) ≥ α(s′) forany pair of nonnegative numbers (s, s′) satisfying s′ − s ≥ α(s′).

Proof. s′− s ≥ α(s′) can be written as (Id−α)(s′) ≥ s. Assume (Id−α) ∈ K.(Otherwise, one can find an smaller α′ to replace α such that (Id− α′) ∈ K.)

Note that χ−1χ(Id−α) = Id−α < Id implies χ(Id−α)χ−1 < Id. WithLemma C.2, we can find a positive definite function α satisfying (Id−α) ∈ K∞,such that

(Id− α)−1 χ (Id− α) χ−1 < Id. (C.1)

Consequently,

χ (Id− α) < (Id− α) χ. (C.2)

269


Define α0 = αχ. Then, α0 is positive definite and for any one positive definitefunction α′ ≤ α0,

(χ− α′)(s′) = (Id− α) χ(s′) ≥ χ (Id− α)(s′) ≥ χ(s) (C.3)

holds for any pair of nonnegative numbers (s, s′) satisfying s′ − s ≥ α(s′). ♦

Lemma C.4 For any K∞ function χ and any continuous, positive definitefunction ε satisfying (Id−ε) ∈ K∞, there exists a continuous, positive definitefunction µ satisfying (Id− µ) ∈ K∞ such that χ (Id− µ) = (Id− ε) χ.

Proof. (Id − ε) χ χ−1 = Id − ε < Id implies χ−1 (Id − ε) χ < Id. Theresult is proved by defining µ = Id− χ−1 (Id− ε) χ. ♦

Lemma C.5 For any χi ∈ K∞ and χi ∈ K satisfying χi > χi for i =1, . . . , n, there exist continuous and positive definite κ satisfying (Id− κ) ∈K∞ and continuous and positive definite κ′ satisfying (Id− κ′) ∈ K∞ suchthat χi (Id− κ) > χi and (Id− κ′) χi > χi for i = 1, . . . , n.

Proof. From the proof of the Lemma A.1 in [126], there exists a K∞ func-tion χi such that χi > χi > χi. The proof is concluded by definingκ(s) = mini=1,...,ns− χ−1

i χi(s) and κ′(s) = mini=1,...,ns− χi χ−1i (s)

for s ≥ 0. ♦

Lemma C.6 For any a, b ∈ R, if there exists a θ ∈ K and a constant c ≥ 0such that

|a− b| ≤ maxθ (Id + θ)−1(|a|), c, (C.4)

then

|a− b| ≤ maxθ(|b|), c. (C.5)

Proof. We first consider the case of θ (Id+θ)−1(|a|) ≥ c, which together with(C.4) implies

|a− b| ≤ θ (Id + θ)−1(|a|). (C.6)

In this case,

|a| − |b| ≤ θ (Id + θ)−1(|a|), (C.7)

and thus,

(Id− θ (Id + θ)−1)(|a|) ≤ |b|. (C.8)

Technical Lemmas Related to Comparison Functions 271

Notice that Id−θ(Id+θ)−1 = (Id+θ)(Id+θ)−1−θ(Id+θ)−1 = (Id+θ)−1.Then, we have

|a| ≤ (Id + θ)(|b|). (C.9)

By using (C.6) again, it can be achieved that

|a− b| ≤ θ(|b|). (C.10)

Property (C.5) is then proved by also considering the case of θ (Id +θ)−1(|a|) < c, i.e., |a− b| ≤ c. ♦

Lemma C.7 Consider a signal µ : [t0,∞) → R+ which is right-continuousand differentiable almost everywhere on [t0,∞)\ with = τk : k ∈ Z+ ⊂[t0,∞) being a strictly increasing sequence. Suppose that there exists a con-stant ω ≥ 0 such that

µ(t) ≥ ω ⇒ µ(t) ≤ −ϕ(µ(t)) (C.11)

for almost all t ∈ [t0,∞), with ϕ being positive semi-definite and locally Lip-schitz, and

µ(t) ≤ maxµ(t−), ω (C.12)

when t ∈ . Then,

µ(t) ≤ maxη(t), ω (C.13)

for all t ∈ [t0,∞), where η(t) is the unique solution of η = −ϕ(η) with η(t0) ≥µ(t0).

Proof. If µ(t1) ≤ ω for some t1 ≥ t0, then µ(t) ≤ ω for all t ∈ [t1,∞). Ifµ(t2) > ω for some t2 ≥ t0, then there exists a t3 > t2 such that µ(t) > ω forall t ∈ [t0, t3).

We study the case where µ(t0) > ω. For any t3 such that µ(t) > ω for allt ∈ [t0, t3), it holds that

µ(t) ≤ maxµ(t−), ω ≤ µ(t−) (C.14)

for all t ∈ [t0, t3) ∩.By using [155, Theorem 1.10.2] or [264, Lemma 1], we have

µ(t) ≤ η(t) (C.15)

for all t ∈ [t0, t3)∩ [τk, τk+1) for any k ∈ Z+ satisfying [t0, t3)∩ [τk, τk+1) 6= ∅.Inequalities (C.14) and (C.15) together imply µ(t) ≤ η(t) for all t ∈ [t0, t3)

with any t3 satisfying µ(t) > ω for all t ∈ [t0, t3).If there exists a t∗µ such that µ(t) > ω for all t ∈ [t0, t

∗µ) and µ(t) ≤ ω for all

t ∈ [t∗µ,∞), then µ(t) ≤ η(t) for all t ∈ [t0, t∗µ) and µ(t) ≤ ω for all t ∈ [t∗µ,∞).

Thus, µ(t) ≤ maxη(t), ω for t ∈ [t0,∞).If µ(t) > ω for all t ∈ [t0,∞), then µ(t) ≤ η(t) for all t ∈ [t0,∞). ♦


Lemma C.8 For any function γ ∈ K and any δ > 0, there exist constantsk, δ′ > 0 and a continuously differentiable γ ∈ K∞ such that

γ(s) ≤ γ(s) for s ≥ δ, (C.16)

γ(s) = ks for s ∈ [0, δ′). (C.17)

Moreover, if γ is linearly bounded near zero, then (C.16) can be satisfied withδ = 0; if γ is globally bounded by a linear function, then (C.16) and (C.17)hold with δ = 0 and δ′ = ∞. In particular, such functions can be taken convex.

See [123, Lemma 1] for the proof of Lemma C.8.

D Proofs of the SmallGainTheorems 2.1, 3.2 and 3.6

D.1 A USEFUL TECHNICAL LEMMA

Lemma D.1 is used for the proofs of the trajectory-based small-gain theorems.See [265, Lemma 5.4] for the original version.

Lemma D.1 Let β ∈ KL, let ρ ∈ K such that ρ < Id, and let µ be a realnumber in (0, 1]. There exists a β ∈ KL such that for any nonnegative realnumbers s and d, and any nonnegative real function z defined on [0,∞) andsatisfying

z(t) ≤ maxβ(s, t), ρ(‖z‖[µt,∞)), d (D.1)

for all t ∈ [0,∞), it holds that

z(t) ≤ maxβ(s, t), d (D.2)

for all t ∈ [0,∞).

The employment of this kind of technical lemma is motivated by the originalsmall-gain result developed by [130]; see [130, Lemma A.1]. It should be notedthat [130] mainly considers “plus”-type interconnections, while Lemma D.1is used for the systems with “max”-type interconnections in this book. Themajor difference is that the signal z(t) in [130, Lemma A.1] satisfies z(t) ≤β(s, t) + ρ(‖z‖[µt,∞)) + d instead of (D.1), and the corresponding result is in

the form of z(t) ≤ β(s, t) + d′ instead of (D.2).

D.2 PROOF OF THEOREM 2.1: THE ASYMPTOTIC GAIN APPROACH

Consider the interconnected system composed of two subsystems in the formof (2.14)–(2.15) satisfying (2.16). Assume that the small-gain condition (2.17)is satisfied.

Consider any specific initial state x(0) and any piecewise continuous,bounded input u. Denote x∗i = maxσi1(|x(0)|), σi2(‖u‖∞) for i = 1, 2.

Due to time invariance and causality properties, (2.16) implies

|xi(t)| ≤ maxβi(|xi(t0)|, t− t0), γi(2−i)(‖x2−i‖[t0,t]), γui (‖ui‖∞). (D.3)

273


Then, direct calculation yields:

limt→∞

‖xi‖[4t,8t] = limt→∞

max0≤τ≤4t

|x(4t+ τ)|

≤ limt→∞

max0≤τ≤4t

βi(|xi(2t)|, 2t+ τ),

γi(2−i)(‖x2−i‖[2t,4t+τ ]), γui (‖ui‖∞)≤ limt→∞

maxβi(x∗i , 2t), γi(2−i)(‖x2−i‖[2t,8t]), γui (‖ui‖∞),(D.4)

where

‖x2−i‖[2t,8t] = max0≤τ ′≤6t

|x(2t+ τ ′)|

≤ max0≤τ ′≤6t

β2−i(|x2−i(t)|, t+ τ ′),

γ(2−i)i(‖xi‖[t,2t+τ ′]), γu2−i(‖u2−i‖∞)

≤ maxβ2−i(x∗2−i, t), γ(2−i)i(‖xi‖[t,8t]), γu2−i(‖u2−i‖∞). (D.5)

By substituting (D.5) into (D.4), one has

limt→∞

‖xi‖[4t,8t] = limt→∞

maxβi(x∗i , 2t), γi(2−i) β2−i(x∗2−i, t),

γi(2−i) γ(2−i)i(‖xi‖[t,8t]),γi(2−i) γu2−i(‖u2−i‖∞), γui (‖ui‖∞). (D.6)

Note that

limt→∞

‖xi‖[t,8t] = limt→∞

max‖xi‖[t,2t], ‖xi‖[2t,4t], ‖xi‖[4t,8t]

= limt→∞

‖xi‖[4t,8t] (D.7)

since limt→∞ ‖xi‖[t,2t] = limt→∞ ‖xi‖[2t,4t] = limt→∞ ‖xi‖[4t,8t].Then, from (D.6), with the small-gain condition (2.17) satisfied, it holds

that

limt→∞

‖xi‖[t,8t] ≤ max limt→∞

βi(x∗i , 2t),

limt→∞

γi(2−i) β2−i(x∗2−i, t),

limt→∞

γi(2−i) γ(2−i)i(‖xi‖[t,8t]),

γi(2−i) γu2−i(‖u2−i‖∞), γui (‖ui‖∞)≤ max lim

t→∞βi(x

∗i , 2t),

limt→∞

γi(2−i) β2−i(x∗2−i, t),

γi(2−i) γu2−i(‖u2−i‖∞), γui (‖ui‖∞)= maxγi(2−i) γu2−i(‖u2−i‖∞), γui (‖ui‖∞). (D.8)

Proofs of the SmallGain Theorems 2.1, 3.2 and 3.6 275

The AG property is proved as

limt→∞

|xi(t)| ≤ limt→∞

‖xi‖[t,8t]≤ maxγi(2−i) γu2−i(‖u2−i‖∞), γui (‖ui‖∞). (D.9)

This ends the proof of Theorem 2.1.

D.3 SKETCH OF PROOF OF THEOREM 3.2

Inspired by [130], the cyclic-small-gain theorem for large-scale dynamic net-works composed of IOS subsystems can be proved in two steps:

1. Forward completeness of the system and boundedness of solutions for allt ∈ [0,∞);

2. IOS of the large-scale dynamic network.

For large-scale dynamic networks, the results are proved by induction. This ismotivated by the proof of the cyclic-small-gain theorem for output-Lagrangeinput-to-output stable (OLIOS) systems in [134].

D.3.1 FORWARD COMPLETENESS OF THE SYSTEM AND BOUNDEDNESSOF SOLUTIONS

Pick any initial state x(0) and any measurable and locally essentially boundedu. Suppose that x(t) is right maximally defined on [0, T ) with T possiblyinfinite.

By (3.71), for i = 1, . . . , n, it holds that

|yi(t)| ≤ maxj=1,...,n;j 6=i

σi(|xi(0)|), γij(

‖yj‖[0,T )

)

, γui(

‖uj‖[0,T )

)

(D.10)

for t ∈ [0, T ), where σi(s) = βi(s, 0) for s ∈ R+. Clearly, σi ∈ K.We first consider the case of n = 2. In this case, γi(3−i) γ(3−i)i < Id for

i = 1, 2.By (D.10), one has for i = 1, 2,

|yi(t)| ≤ max

σi(|xi(0)|), γi(3−i)(

‖y3−i‖[0,T )

)

, γui(

‖uj‖[0,T )

)

(D.11)

for t ∈ [0, T ). By taking the supremum of |yi| over [0, T ) and defining

σi(s) = maxσi(s), γi(3−i) σ3−i(s), (D.12)

one has

‖yi‖[0,T ) ≤ max

σi(|x(0)|), γui(

‖ui‖[0,T )

)

, γi(3−i) γ(3−i)i(

‖yi‖[0,T )

)

,

γi(3−i) γu3−i(

‖u3−i‖[0,T )

)

. (D.13)


Since γi(3−i) γ(3−i)i < Id, one achieves

‖yi‖[0,T ) ≤ max

σi(|x(0)|), γui(

‖ui‖[0,T )

)

, γi(3−i) γu3−i(

‖u3−i‖[0,T )

)

.(D.14)

Hence, for any initial state x(0), and any measurable and locally essentiallybounded u, y is bounded over the interval [0, T ). By using the UO property(3.70), the state x is bounded over [0, T ). This means that the maximuminterval for the definition of x is [0,∞). It again follows (D.14) that thereexists σ, γ ∈ K such that for any initial state x(0), and any measurable andlocally essentially bounded u,

|y(t)| ≤ max σ(|x(0)|), γ(‖u‖∞) (D.15)

for all t ≥ 0.Suppose that for any dynamic network with n = n∗, the existence and

boundedness of the solutions on [0,∞) can be proved and property (D.15)holds for all t ≥ 0, if the subsystems for i = 1, . . . , n∗ are UO in the sense of(3.70), have the property (D.10), and satisfy the cyclic-small-gain condition.

We consider a dynamic network with n = n∗ + 1, with the subsystemsbeing UO in the sense of (3.70), having property (D.10) and satisfying thecyclic-small-gain condition.

By (D.10), one has for i = 1, . . . , n∗ + 1,

|yi(t)| ≤ maxj=1,...,n∗+1;j 6=i

σi(|xi(0)|), γij(

‖yj‖[0,T )

)

, γui(

‖ui‖[0,T )

)

,

γi(n∗+1)

(

‖yn∗+1‖[0,T )

)

, (D.16)

and thus, for i = 1, . . . , n∗,

|yi(t)| ≤ maxj=1,...,n∗;j 6=i;l=1,...,n∗

σi(|xi(0)|), γij(

‖yj‖[0,T )

)

, γui(

‖ui‖[0,T )

)

,

γi(n∗+1) σn∗+1(|xn∗+1(0)|), γi(n∗+1) γ(n∗+1)l

(

‖yl‖[0,T )

)

,

γi(n∗+1) γun∗+1

(

‖un∗+1‖[0,T )

)

≤ maxj=1,...,n∗;j 6=i

σi(|x(0)|), γij(

‖yj‖[0,T )

)

, γui(

‖u‖[0,T )

)

, (D.17)

where

σi(s) = max

σi(s), γi(n∗+1) σn∗+1(s)

γij(s) = maxj=1,...,n∗;j 6=i

γij(s), γi(n∗+1) γ(n∗+1)j(s)

γui (s) = max

γui (s), γi(n∗+1) γun∗+1(s)

(D.18)

for s ∈ R+. We used the cyclic-small-gain condition γi(n∗+1) γ(n∗+1)i < Idto get the last inequality of (D.17).


Consider the dynamic network with the n∗ subsystems being UO with zerooffset and having property (D.17). Note that the new IOS gains γij still satisfythe cyclic-small-gain condition according to [134, Lemma 5.3].

According to the hypothesis, for any initial state x(0) and any measurableand locally essentially bounded u, the solution (x1(t), . . . , xn∗(t)) exists and isbounded for t ≥ 0, which implies the existence and boundedness of xn∗+1(t)for t ≥ 0 by using the UO and IOS properties of the (n∗ + 1)-th subsystem.Then, for the dynamic network with n = n∗ + 1, property (D.15) holds fort ≥ 0.

D.3.2 INPUTTOOUTPUT STABILITY

The UO of the subsystems in the sense of (3.70) implies the UO of the dynamicnetwork, i.e., there exist αO ∈ K∞ such that

|x(t)| ≤ αO (|x(0)|+ ‖u‖∞ + ‖y‖∞) (D.19)

for t ≥ 0.For any specific initial state x(0) and any measurable and locally essentially

bounded u, with (D.15) proved, we define

c = αO (|x(0)|+ ‖u‖∞ +max σ(|x(0)|), γ (‖u‖∞)) . (D.20)

Then,

x(t) ≤ c (D.21)

for t ≥ 0.By using the time-invariance property, property (3.71) implies


βi(|xi(t0)|, t− t0), γij(

‖yj‖[t0,∞)]

)

, γui (‖ui‖∞) (D.22)

for all 0 ≤ t0 ≤ t. By choosing t0 = µt with µ ≤ 0.5 and using (D.21), one has


βi(c, (1− µ)t), γij(

‖yj‖[µt,∞)]

)

, γui (‖ui‖∞)

≤ maxj 6=i

βi(c, µt), γij(

‖yj‖[µt,∞)]

)

, γui (‖ui‖∞) (D.23)

for t ≥ 0.With the satisfaction of the cyclic-small-gain condition, by proving the

existence of βi ∈ KL, χi ∈ K satisfying χi < Id, γui ∈ K, and 0 < µi ≤ 1 suchthat

|yi(t)| ≤ max

βi(c, t), χi(

‖yi‖[µit,∞)

)

, γui (‖u‖∞)

(D.24)

for i = 1, . . . , n, we can use Lemma D.1 to prove the IOS of the dynamicnetwork.


Consider the case of n = 2. In this case, for i = 1, 2,

|yi(t)| ≤ max

βi(c, µt), γi(3−i)(

‖y3−i‖[µt,∞)

)

, γui (‖ui‖)

, (D.25)

and

‖yi‖[µt,∞) ≤ maxµt≤τ<∞

βi(c, µτ), γi(3−i)(

‖y3−i‖[µτ,∞)

)

, γui (‖ui‖)

≤ max

βi(c, µ2t), γi(3−i)

(

‖y3−i‖[µ2t,∞)

)

, γui (‖ui‖)

. (D.26)

By substituting (D.26) with i replaced by 3− i into (D.25), one has

|yi(t)| ≤ max

βi(c, µt), γi(3−i) β3−i(c, µ2t), γi(3−i) γ(3−i)i(

‖yi‖[µ2t,∞)

)

,

γi(3−i) γu3−i (‖u3−i‖∞) , γui (‖ui‖∞)

. (D.27)

Thus, property (D.24) is proved for n = 2.Suppose that property (D.24) can be proved for n = n∗. Consider the case

of n = n∗+1. In this case, by still using the time-invariance property, one hasfor i = 1, . . . , n∗, Then,

|yi(t)| ≤ maxj=1,...,n∗,j 6=i

βi(c, µt), γij(

‖yj‖[µt,∞)

)

, γi(n∗+1)

(

‖yn∗+1‖[µt,∞)

)

,

γui (‖ui‖∞)

(D.28)

for i = 1, . . . , n∗, and

‖yn∗+1‖[µt,∞) = maxµt≤τ<∞

|yn∗+1(τ)|

≤ maxj=1,...,n∗

βn∗+1(c, µ2t), γ(n∗+1)j

(

‖yj‖[µ2t,∞)

)

,

γun∗+1 (‖un∗+1‖∞)

. (D.29)

By substituting (D.29) into (D.28), one has

|yi(t)| ≤ maxj=1,...,n∗,j 6=i

βi(c, t), γij(

‖yj‖[µ2t,∞)

)

, γui (‖u‖∞)

(D.30)

for i = 1, . . . , n∗, where

βi(s, t) = max

βi(s, µt), γi(n∗+1) βn∗+1(s, µ2t)

(D.31)

γij(s) = max

γij(s), γi(n∗+1) γ(n∗+1)j(s)

(D.32)

γui (s) = max

γui (s), γi(n∗+1) γun∗+1(s)

(D.33)

for s, t ≥ 0. Note that the new IOS gains γij still satisfy the cyclic-small-gaincondition according to [134, Lemma 5.3]. For the dynamic network composedof n∗ subsystems satisfying (D.30), there exist βi ∈ KL, χi ∈ K satisfying


χi < Id, γui ∈ K, and 0 < µi ≤ 1 so that (D.24) holds for i = 1, . . . , n∗. Notethat one can pick the n∗ subsystems arbitrarily from the n∗ + 1 subsystems.Property (D.24) can be proved for i = 1, . . . , n∗ + 1.

By induction, property (D.24) is proved for each i-th subsystem with i =1, . . . , n of the dynamic network under the cyclic-small-gain condition.

Then, with Lemma D.1, for i = 1, . . . , n, there exist βi ∈ KL and γui ∈ Ksuch that

|yi(t)| ≤ max

βi(c, t), γui (‖u‖∞)

(D.34)

for t ≥ 0, and thus, there exist β ∈ KL and γu ∈ K such that

|y(t)| ≤ max

β(c, t), γu (‖u‖∞)

(D.35)

for t ≥ 0.Recall the definition of c in (D.20). One can find β ∈ KL and γ ∈ K so that

for any initial state x(0) and any measurable and locally essentially boundedu,

|y(t)| ≤ max β(|x(0)|, t), γu (‖u‖∞) (D.36)

for t ≥ 0. This ends the proof of Theorem 3.2.

D.4 PROOF OF THEOREM 3.6

For convenience of notation, we denote VCH(x) = maxi∈NCHVi(xi)

with NCH = NC ∪ NH and VD(x) = maxi∈NDVi(xi). Then, V (x) =

maxVCH(x), VD(x). Denote f(x, u) = [fT1 (x, u1), . . . , fTN(x, uN )]T .

Proof of Property 1

Under the conditions of Property 1, consider t /∈ π.Simply denote x(t, t0, ξ, u) as x(t). We study the decreasing property of

V (x(t)) at time t in the case of V (x(t)) ≥ u(t) when t /∈ π.At time t, define

A = j ∈ N : Vj(xj(t)) = V (x(t)). (D.37)

Recall that V (x(t)) = maxV(x(t)) = maxi∈N Vi(xi(t)). Since all the γij ’s(i, j ∈ N , i 6= j) are less than Id, it holds that

Vj(xj(t)) ≥ maxl6=j

Vl(xl(t)) > maxl6=j

γjl(Vl(xl(t))) (D.38)

for all j ∈ A. Furthermore, in the case of V (x(t) ≥ u(t), with the definitionof u in (3.161), it holds that Vj(xj(t)) = V (x(t)) ≥ u(t) ≥ γuj

(|uj(t)|) for allj ∈ A. It follows that

Vj(xj(t)) ≥ maxl6=j

γjl(Vl(xl(t))), γuj(|uj(t)|). (D.39)


If ∇V is defined at x(t), then

∇V (x(t))f(x(t), u(t)) =d

dr

∣

∣

∣

∣

r=0

V (φ(r)), (D.40)

where φ(r) = [φ1(r), . . . , φN (r)]T is the continuous solution of the initial-valueproblem

φ(r) = f(φ(r), u(t)), φ(0) = x(t). (D.41)

In the case of NCH ∩ A 6= ∅, for j ∈ NCH ∩ A, if ∇Vj is well defined atxj(t), then from (3.125) and (D.39), it holds that

∇Vj(xj(t))fj(x(t), uj(t)) ≤ −αj(Vj(xj(t))). (D.42)

Considering the continuity of fj and the continuity of ∇Vj at xj(t), thereexists a neighborhood X = X1 × · · · × XN of x(t) such that for j ∈ NCH ∩A,

∇Vj(ζj)fj(ζ, uj(t)) ≤ −1

2αj(Vj(xj(t))) (D.43)

holds for all ζ = [ζT1 , . . . , ζTN ]T ∈ X .

Because of the continuity of φ(r), there exists a δ > 0 such that φ(r) ∈ Xand

maxj∈A

Vj(φj(r)) = maxj∈N

Vj(φj(r)) = V (φ(r)) (D.44)

for r ∈ [0, δ).For j ∈ NCH ∩A, from (D.43),

Vj(φj(r)) − Vj(xj(t))

r≤ −1

2αj(Vj(xj(t))) (D.45)

holds for r ∈ (0, δ).For j ∈ ND, since fj ≡ 0,

Vj(φj(r)) − Vj(xj(t))

r= 0 (D.46)

holds for r ∈ (0, δ).From (D.45) and (D.46), we have

V (φ(r)) − V (x(t))

r=

maxj∈AVj(φj(r)) −maxj∈AVj(xj(t))r

≤ 0 (D.47)

for r ∈ (0, δ). Hence, if ∇V is well defined at x(t), then

V (x(t)) ≥ u(t) ⇒ ∇V (x(t))f(x(t), u(t)) ≤ 0. (D.48)


Note that V (x) is smooth almost everywhere. By using the results of [86,Lemma 1] or [264, Section 2]), from (D.48), we conclude V (x(t)) is differen-tiable almost everywhere on the timeline and we achieve for any ξ, u, t0 ≥ 0,

V (x(t, t0, ξ, u)) ≥ ‖u‖[t0,t] ⇒ V (x(t, t0, ξ, u)) ≤ 0 (D.49)

holds for almost all t ∈ [t0,∞)\π. Property 1 in Theorem 3.6 is proved.

Proof of Property 2

Under the conditions of Property 2, consider t ∈ π.For any ξ, u, and t0 ≥ 0, if t > t0 and t /∈ πi, then xi(t, t0, ξ, u) is continuous

at time t and

Vi(xi(t, t0, ξ, u)) = Vi(xi(t−, t0, ξ, u)) ≤ V (x(t−, t0, ξ, u)); (D.50)

else if t > t0 and t ∈ πi (obviously, i ∈ ND ∪ NH), then

Vi(xi(t, t0, ξ, u))

=Vi(gi((xi(t−, t0, ξ, u), ui(t

−)))

≤(Id− ρi)(maxl6=i

γil(Vl(xl(t−, t0, ξ, u))), Vi(xi(t−, t0, ξ, u)), γui(|ui(t−)|))

≤maxV (x(t−, t0, ξ, u)), u(t−). (D.51)

From (D.50) and (D.51), when t > t0 and t ∈ π, we obtain

V (x(t, t0, ξ, u)) = maxi∈N

Vi(xi(t, t0, ξ, u))

≤ maxV (x(t−, t0, ξ, u)), u(t−). (D.52)

Property 2 in Theorem 3.6 is proved.

Uniformly BoundedInput BoundedState (UBIBS) Property

Note that u(t−) ≤ ‖u‖[t0,t] and u(t) ≤ ‖u‖[t0,t] hold for any t > t0 ≥ 0. WithLemma C.7, by considering V (x(t, t0, ξ, u)) as µ(t) and ‖u‖[t0,t] as ω, from theproperties (D.49) and (D.52), for any ξ, u, and t0 ≥ 0, we have

V (x(t, t0, ξ, u)) ≤ maxV (ξ), ‖u‖[t0,t] (D.53)

for all t > t0. Note that V (x(t0, t0, ξ, u)) = V (ξ). Thus, for any ξ, u, andt0 ≥ 0, property (D.53) holds for all t ≥ t0.

Proof of Property 3

Define

tD0 = minti0 : i ∈ ND − δ (D.54)

tD(w+1) − tDw = δt+ δ, w ∈ Z+\0 (D.55)

π = tD(2w) : w ∈ Z+, (D.56)


where 0 < δ < minti0 : i ∈ ND can be arbitrarily close to zero. Thedefinition of tDw means that for each i ∈ ND, (tD(2w), tD(2w+1)] ∩ πi 6= ∅ andof course (tD(2w), tD(2w+2)] ∩ πi 6= ∅.

Denote ∆ = [tD(2w), tD(2w+2)]. Property (D.53) implies that for any ξ andu,

V (x(t, tD(2w), ξ, u)) ≤ maxV (ξ), ‖u‖∆ (D.57)

for t ∈ ∆.If V (ξ) ≤ ‖u‖∆, then

V (x(t, tD(2w), ξ, u)) ≤ ‖u‖∆. (D.58)

For each i ∈ ND, in the case of V (ξ) > ‖u‖∆, we have

Vi(xi(t, tD(2w), ξ, u))

=Vi(gi(x(t−, tD(2w), ξ, u), ui(t

−)))

≤(Id− ρi)(maxV (x(t−, tD(2w), ξ, u)), γui(|ui(t−)|))

≤(Id− ρi)(maxV (ξ), ‖u‖∆)=(Id− ρi)(V (ξ)) (D.59)

for all t ∈ (tD(2w), tD(2w+2)) ∩ πi.Note that the state of each xi-subsystem (i ∈ ND) keeps constant when

t /∈ πi. For i ∈ ND, because (tD(2w), tD(2w+1)] ∩ πi 6= ∅, from (D.59), we get

Vi(xi(t, tD(2w), ξ, u)) ≤ max(Id− ρi)(V (ξ)), ‖u‖∆, i ∈ ND (D.60)

for t ∈ [tD(2w+1), tD(2w+2)].Define ρ(s) = mini∈ND

ρi(s) for s ≥ 0. Then,

VD(x(t, tD(2w), ξ, u)) ≤ max(Id− ρ)(V (ξ)), ‖u‖∆ (D.61)

for t ∈ [tD(2w+1), tD(2w+2)]. Clearly, ρ is continuous and positive definite, and(Id− ρ) ∈ K∞.

For i ∈ NCH , when t ∈ [tD(2w+1), tD(2w+2)]\(⋃

i∈NHπi), only the

continuous-time dynamics work. Consider VCH(x) as the ISS-Lyapunov func-tion and VD and u as the inputs of the interconnection of the continuous-timesubsystems (i ∈ NC) and the hybrid subsystems (i ∈ NH). Using the cyclic-small-gain theorem for continuous-time dynamic networks in Section 3.1 andProperty (D.61), we can find a continuous and positive definite function αCHsuch that if ∇VCH is defined at x(t, tD(2w), ξ, u), then

VCH(x(t, tD(2w), ξ, u)) ≥ maxVD(x(t, tD(2w), ξ, u)), u(t)⇒ ∇VCH(x(t, tD(2w), ξ, u))fCH(x(t, tD(2w), ξ, u), u(t))

≤− αCH(VCH(x(t, tD(2w), ξ, u))) (D.62)


holds for t ∈ [tD(2w+1), tD(2w+2)]\(⋃

i∈NHπi), where fCH is the vector of the

continuous-time dynamics of the continuous-time subsystems and the hybridsubsystems.

Using (D.61) and u(t) ≤ ‖u‖∆, we have

VCH(x(t, tD(2w), ξ, u)) ≥ max(Id− ρ)(V (ξ)), ‖u‖∆⇒ ∇VCH(x(t, tD(2w), ξ, u))fCH(x(t, tD(2w), ξ, u), u(t))

≤− αCH(VCH(x(t, tD(2w), ξ, u))) (D.63)

for t ∈ [tD(2w+1), tD(2w+2)]\(⋃

i∈NHπi).

By using [86, Lemma 1] or [264, Section 2], we have VCH(x(t, tD(2w), ξ, u))is differentiable almost everywhere on the timeline and

VCH(x(t, tD(2w), ξ, u)) ≥ max(Id− ρ)(V (ξ)), ‖u‖∆⇒VCH(x(t, tD(2w), ξ, u)) ≤ −αCH(VCH(x(t, tD(2w), ξ, u))) (D.64)

for almost all t ∈ [tD(2w+1), tD(2w+2)]\(⋃

i∈NHπi).

For t ∈ [tD(2w+1), tD(2w+2)] ∩ (⋃

i∈NHπi), from (D.52) and (D.61), we

have

VCH(x(t, tD(2w), ξ, u))

≤V (x(t, tD(2w), ξ, u))

≤maxV (x(t−, tD(2w), ξ, u)), u(t−)

=maxVCH(x(t−, tD(2w), ξ, u)), VD(x(t−, tD(2w), ξ, u)), u(t

−)≤max(Id− ρ)(V (ξ)), VCH(x(t−, tD(2w), ξ, u)), u(t

−), ‖u‖∆≤max(Id− ρ)(V (ξ)), VCH(x(t−, tD(2w), ξ, u)), ‖u‖∆. (D.65)

For the last inequality above, We used the fact that u(t−) ≤ ‖u‖∆ for t ∈[tD(2w+1), tD(2w+2)].

By considering VCH(x(t, tD(2w), ξ, u)) as µ(t), αCH as ϕ and max(Id −ρ)(V (ξ)), ‖u‖∆ as ω, with Lemma C.7, from (D.64) and (D.65), it can beproved that, for all t ∈ [tD(2w+1), tD(2w+2)],

VCH(x(t, tD(2w), ξ, u) ≤ maxν(t), (Id− ρ)(V (ξ)), ‖u‖∆, (D.66)

where ν(t) is the solution of ν = −αCH(ν) with ν(tD(2w+1)) = V (ξ) ≥VCH(x(t, tD(2w+1), ξ, u)).

From Proposition 2.5 and Theorem 2.8 in [170], the uniform asymptoticstability of ν = −αCH(ν) implies the existence of βCH ∈ KL satisfyingβCH(s, 0) = s for all s ≥ 0 such that ν(t) ≤ βCH(V (ξ), t − tD(2w+1)) forall t ∈ [tD(2w+1), tD(2w+2)], and one can find a continuous and positive defi-nite function ρ′ satisfying (Id− ρ′) ∈ K∞ such that

ν(tD(2w+2)) ≤ βCH(V (ξ), tD(2w+2) − tD(2w+1))

≤ βCH(V (ξ), maxi∈ND

δti)

≤ (Id− ρ′)(V (ξ)). (D.67)


Define ρ∗(s) = minρ(s), ρ′(s) for s ∈ R+. Then, from (D.66) and (D.67),

VCH(x(tD(2w+2), tD(2w), ξ, u)) ≤ max(Id− ρ∗)(V (ξ)), ‖u‖∆, (D.68)

which together with (D.61) implies

V (x(tD(2w+2), tD(2w), ξ, u)) ≤ max(Id− ρ∗)(V (ξ)), ‖u‖∆. (D.69)

Define δtD = 2maxw∈Z+tD(2w+2)− tD(2w). Then, for any pair of nonneg-

ative numbers (t, t0) satisfying t − t0 ≥ δtD, there exists some w ∈ Z+ suchthat [tD(2w), tD(2w+2)] ∈ [t0, t], and thus

V (x(t, t0, ξ, u))

≤maxV (x(tD(2w+2), t0, ξ, u)), ‖u‖[tD(2w+2),t]≤max(Id− ρ∗)(V (x(tD(2w), t0, ξ, u))), ‖u‖[tD(2w),t]≤max(Id− ρ∗)(V (ξ)), ‖u‖[t0,t]. (D.70)

The ISS of the dynamic network can be proved based on (D.53) and (D.70)following a similar approach as in the proof of [86, Theorem 1].

E Proofs of Technical Lemmasin Chapter 4

E.1 PROOF OF LEMMA 4.2

For simplicity, we use Sk instead of Sk(xk) for k = 1, . . . , i − 1. We onlyconsider the case of ei > 0. The proof for the case of ei < 0 is similar.

Consider the recursive definition of Sk’s in (4.55). For k = 1, . . . , i− 1, thestrictly decreasing property of the κk’s implies

maxSk = κk(xk −maxSk−1 − wk), (E.1)

minSk = κk(xk −minSk−1 + wk). (E.2)

The continuous differentiability of the κk’s implies the continuous differ-entiability of maxSk with respect to xk and maxSk−1 for k = 1, . . . , i − 1.Using the property of composition of continuously differentiable functions, wecan see maxSi−1 is continuously differentiable with respect to xi−1 and thus∇maxSi−1 is continuous with respect to xi−1.

In the case of ei > 0, the dynamics of ei can be rewritten as

ei = xi −∇maxSi−1 ˙xi−1

= xi+1 +∆i(xi, d)−∇maxSi−1 ˙xi−1

:= xi+1 + φ∗i (xi, d). (E.3)

Note that ˙xi−1 = [x2 +∆1(x1, d), . . . , xi +∆i−1(xi−1, d)]T . With Assump-

tion 4.1, one can find a ψ ˙xi−1∈ K∞ such that | ˙xi−1| ≤ ψ ˙xi−1

(|[xTi , dT ]T |).Since ∇maxSi−1 is continuous with respect to xi−1, one can find a ψ0

φ∗

i∈ K∞

such that

|φ∗i (xi, d)| ≤ ψ0φ∗

i(|[xTi , dT ]T |). (E.4)

To prove (4.57), for each k = 1, . . . , i− 1, we look for a ψxk+1∈ K∞ such

that |xk+1| ≤ ψxk+1(|[eTk+1,W

Tk ]T |).

For k = 1, . . . , i− 1, from the definitions of ek+1 in (4.50), we can observeminSk ≤ xk+1 − ek+1 ≤ maxSk and thus

|xk+1| ≤ max|maxSk|, |minSk|+ |ek+1|. (E.5)

For each k = 1, . . . , i − 1, define κ0k(s) = |κk(s)| for s ∈ R+. Because κkis odd, strictly decreasing, and radially unbounded, we have κ0k ∈ K∞. From

285


(E.1), we have

|maxSk| ≤ κ0k(|xk −maxSk−1 − wk|)≤ κ0k(|xk −maxSk−1|+ |wk|)≤ κ0k(|maxSk−1|+ |minSk−1|+ |ek|+ wk). (E.6)

We used |xk−maxSk−1| ≤ |maxSk−1|+ |minSk−1|+ |ek|, which was derivedfrom minSk ≤ xk+1 − ek+1 ≤ maxSk. Similarly, we can also get

|minSk| ≤ κ0k(|maxSk−1|+ |minSk−1|+ |ek|+ wk). (E.7)

For each xk+1 (k = 1, . . . , i−1), using (E.5) and repeatedly using (E.6) and(E.7), one can find a ψxk+1

∈ K∞ such that |xk+1| ≤ ψxk+1(|[eTk+1,W

Tk ]T |).

This, together with (E.4), leads to the satisfaction of (4.57).If the ψ∆k

’s for k = 1, . . . , i are Lipschitz on compact sets, then all theclass K∞ functions determining ψφ∗

iare Lipschitz on compact sets, and one

can find a ψφ∗

i∈ K∞ which is Lipschitz on compact sets.


By convention, Si1(yi, ξi1) := Si1(yi). We simply use Sik instead of Sik(yi, ξik)for k = 1, . . . , j − 1. We only consider the case of eij > 0. The proof for thecase of eij < 0 is similar.

Consider the definition of Si1 in (4.171) and the iteration-type definitionsof Sik’s in (4.178). The strictly decreasing properties of the κ(·)’s imply

maxSi1 = κi1(yi − dmi ), (E.8)

maxSik = κik(ξik −maxSi(k−1)), k = 2, . . . , j − 1, (E.9)

minSi1 = κi1(yi + dmi ), (E.10)

minSik = κik(ξik −minSi(k−1)), k = 2, . . . , j − 1. (E.11)

The continuous differentiability of the κ(·)’s implies the continuous differen-tiability of maxSi1 with respect to yi and the continuous differentiability ofmaxSik with respect to ξik and maxSi(k−1) for k = 2, . . . , j − 1. Using theproperty of the composition of continuously differentiable functions, we cansee maxSi(j−1) is continuously differentiable with respect to [yi, ξ

Ti(j−1)]

T and

thus ∇maxSi(j−1) in (E.12) is continuous with respect to [yi, ξTi(j−1)]

T .In the case of eij > 0, the dynamics of eij can be derived as

eij = ξij −∇maxSi(j−1)[yi,˙ξTi(j−1)]

T

= ξi(j+1) + φij(yi, ξi2, di)−∇maxSi(j−1)[yi,˙ξTi(j−1)]

T

:= ξi(j+1) + φ′ij(zi, yi, ξij , wi, di). (E.12)


Denote Xk = [XT1k, X

T2k]

T . Define

∂maxSj−1 =⋂

ǫ>0

⋂

µ(M)=0

co∇maxSj−1(Bǫ(Xj−1)\M), (E.19)

where Bǫ(Xj−1) is a ball of radius ǫ around Xj−1. Then, ∂maxSj−1 is convex,compact, and upper semi-continuous.

In the case of ej > 0, the ej-subsystem can be represented with a differentialinclusion as

ej ∈

zj+1 + φj(X1j , X2j)− φ0j : φ0j ∈ ∂maxSj−1Xj−1

:=

zj+1 + φ∗j : φ∗j ∈ Φ∗

j (X1j , X2j)

, (E.20)

where

Φ∗j (X1j , X2j) =

φj(X1j , X2j)− φ0j : φ0j ∈ ∂maxSj−1Xj−1

. (E.21)

Because φj(X1j , X2j) and Xj−1 are locally Lipschitz, and ∂maxSj−1 isconvex, compact, and upper semi-continuous, Φ∗

j is convex, compact, andupper semi-continuous.

maxSk can be considered as a discontinuous function of X1k, X2k, zk,maxSk−1, and dk. From the definitions of µk and θk, there exist ϕ0

Skpositive

and nondecreasing, and ψ0Sk

∈ K∞ such that

|maxSk| ≤ ϕ0Sk

(|X1k|, |X2k|)ψ0Sk

(|[zk,maxSk−1, dk]T |). (E.22)

We can also calculate

∇maxSkXk =∂µk

∂[XT1k, X

T2k]

T[XT

1k, XT2k]

T θk(zk −maxSk−1 − dk)

+∂θk

∂[zk,maxSk−1]T[zk,∇maxSk−1Xk−1]

T , (E.23)

where (δ1k, δ2k) may take the value of (D1k, D2k) or (−D1k,−D2k), dependingon the sign of θk(zk −maxSk−1 − dk).

From the definition of zk, there exist ϕ0∇Sk

positive and nondecreasing, and

ψ0∇Sk

∈ K∞ such that

|∇maxSkXk| ≤ϕ0∇Sk

(|X1(k+1)|, |X2(k+1)|,∇maxSk−1)×ψ0∇Sk

(|[zk, zk+1,maxSk−1,∇maxSk−1, dk]T |). (E.24)

By recursively using (E.22) and (E.24), one can find ϕ1∇Sj−1

positive and

nondecreasing and ψ1∇Sj−1

∈ K∞ such that

|∇maxSj−1Xj−1| ≤ ϕ1∇Sj−1

(|X1j |, |X2j |)ψ1∇Sj−1

(|[ZTj , DTj−1]

T |). (E.25)


As for (E.22), there exist ϕ0Sk

positive and nondecreasing, and ψ0Sk

∈ K∞such that

|minSk| ≤ ϕ0Sk

(|X1k|, |X2k|)ψ0Sk

(|[zk,maxSk−1, dk]T |). (E.26)

The definitions of ek+1 (k = 1, . . . , j − 1) in (4.313) implies

|zk+1| ≤ max|maxSk|, |minSk| + |ek+1|. (E.27)

With (E.27), and iteratively using (E.22) and (E.26), for each k = 1, . . . , j−1, one can find ϕzk positive and nondecreasing and ψzk ∈ K∞ such that

|zk+1| ≤ ϕzk(|X1k|, |X2k|)ψzk(|[ETk+1, DTk ]T |). (E.28)

Thus, from (E.25) and (E.28), we can find that ϕ2∇Sj−1

positive and non-

decreasing and ψ2∇Sj−1

∈ K∞ such that

|∇maxSj−1Xj−1| ≤ ϕ2∇Sj−1

(|X1j |, |X2j |)ψ2∇Sj−1

(|[ETj , DTj−1]

T |). (E.29)

From the definition of Φ∗j in (E.21) and the definition of ∂maxSj−1Xj−1

in (E.19), we can find ϕ∗j : R+ × R+ → R+ is positive and nondecreasing

with respect to the two variables, and ψ∗j ∈ K∞, such that for any φ∗j ∈

Φ∗j (X1j , X2j), (4.305) holds.


With (4.305) satisfied, one can find ψekΦ∗

j(k = 1, . . . , j + 1) and ψdkΦ∗

j∈ K∞

(k = 1, . . . , j − 1), such that for any φ∗j ∈ Φ∗j (X1j , X2j), it holds that

|φ∗j | ≤ ϕ∗j (|X1j |, |X2j |)

(

j∑

k=1

ψekΦ∗

j(|ek|) +

j−1∑

k=1

ψdkΦ∗

j(dk)

)

. (E.30)

By convenience, define γejej = Id. Define

Πj(s) =

j∑

k=1

ψekΦ∗

j α−1

V (

γekej

)−1

αV (s)

+

j−1∑

k=1

ψdkΦ∗

j(

γdkej

)−1

αV (s) (E.31)

for s ∈ R+. Then, Πj ∈ K∞.For any 0 < cj < 1, ǫj > 0, one can find a νj : R+ → R+ positive,

nondecreasing and continuously differentiable on (0,∞) and satisfying

(1− cj)νj((1 − cj)s)s ≥ Πj(s) +ℓj2s+ α−1

V (

γej+1ej

)−1

αV (s) (E.32)


for s ∈ R+.One can also find a µj : R+×R+ → R+, which is continuously differentiable

on (0,∞)× (0,∞), positive, and nondecreasing, such that

min

µj(|X1j + δ1j |, |X2j + δ2j |) : −D1j ≤ δ1j ≤ D1j ,−D2j ≤ δ2j ≤ D2j

≥ max

ϕ∗j (|X1j |, |X2j|), 1

(E.33)

for all X1j , X2j ∈ Rj .Define κj(a1, a2, a3) = µj(|a1|, |a2|)θj(a3) with θj(a3) = −νj(|a3|)a3 for

a1, a2, a3 ∈ R.Recall that Vk(ek) = αV (|ek|) = 1

2e2k for k = 1, . . . , n+1. We use Vk instead

of Vk(ek) for convenience. Consider the case of

Vj ≥ maxk=1,...,j−1

γekej (Vk), γej+1ej (Vj+1), γ

dkej (dk), γ

djej (dj), ǫj

(E.34)

with γdjej (s) = αV

(

scj

)

for s ∈ R+.

In this case, for any φ∗j ∈ Φ∗j (X1j , X2j), it holds that

|φ∗j + ej+1| ≤ µj(|X1j , X2j |)(

Πj(|ej |) + α−1V

(

γej+1ej

)−1

αV (|ej |))

,

(E.35)

we can also get

dj ≤ cj |ej| (E.36)

dk ≤(

γdkej

)−1

αV (|ej |), k = 1, . . . , j − 1 (E.37)

|ej | ≥√

2ǫj. (E.38)

With 0 < cj < 1, for pj−1 ∈ Sj−1 and |δ| ≤ dj , we have

|zj − pj−1 + δj | ≥ (1− cj)|ej | (E.39)

sgn(zj − pj−1 + δj) = sgn(ej) (E.40)

and thus

νj(|zj − pj−1 + δj |)|zj − pj−1 + δj |≥ (1 − cj)νj((1− cj)|ej |)|ej |. (E.41)

In the case of (E.34), for pj−1 ∈ Sj−1 and |δj | ≤ dj , for any φ∗j ∈


Φ∗j (X1j , X2j), we have

∇Vj(zj+1 − ej+1 + φ∗j + ej+1)

= ej

(

−µj(|X1j |, |X2j |)νj(|zj − pj−1 + δj |)(zj − pj−1 + δj) + φ∗j + ej+1

)

≤ − µj(|X1j |, X2j |)νj(|zj − pj−1 + δj |)|zj − pj−1 + δj ||ej|+ |φ∗j + ej+1||ej |

≤ − µj(|X1j |, X2j |)νj((1− cj)|ej |)(1 − cj)|ej |2

+ µj(|X1j |, |X2j |)(

Πj(|ej |) + α−1V

(

γej+1ej

)−1

αV (|ej |))

|ej|

≤ − ℓj2µj(|X1j |, |X2j |)|ej |2

≤ − ℓjVj . (E.42)

F Proofs of Technical Lemmasin Chapter 5

F.1 PROOF OF LEMMA 5.1

We simply use Sk and Sk to denote Sk(xk) and Sk(xk) for 1 ≤ k ≤ i− 1. Weonly consider the case of ei > 0. The proof for the case of ei < 0 is similar.

Consider the recursive definitions of Sk and Sk in (5.25)–(5.26). With con-dition 5.4 satisfied, we have 0 ≤ bk < 1 and ak ≥ 0 for 1 ≤ k ≤ n. For1 ≤ k ≤ i− 1, the strictly decreasing properties of the κk’s imply

maxSk = max

dk2 max Sk :1

1 + bk+1≤ dk2 ≤ 1

1− bk+1

(F.1)

max Sk = κk(xk −max Sk−1 − bk|xk| − (1− bk)ak). (F.2)

By iteratively using (F.2), we can see that max Si−1 is continuously differen-tiable almost everywhere with respect to xi−1. From (F.1), maxSi−1 is con-tinuously differentiable almost everywhere with respect to max Si−1. Thus,maxSi−1 is continuously differentiable almost everywhere with respect toxi−1.

Considering the definition of ei in (5.27) with k = i − 1, when ei > 0, wecan represent the ei-subsystem with a differential equation

ei = xi+1 +∆i(xi)−∇maxSi−1 ˙xi−1 (F.3)

wherever maxSi−1 is continuously differentiable, or equivalently, wherever∇maxSi−1 exists. Because maxSi−1 is continuously differentiable almost ev-erywhere, ∇maxSi−1 is discontinuous and thus the ei-subsystem is a discon-tinuous system. We represent the ei-subsystem with a differential inclusion byembedding the discontinuous ∇maxSi−1 into a set-valued map

∂maxSi−1 =⋂

ǫ>0

⋂

µ(M)=0

co∇maxSi−1(Bǫ(xi−1)\M), (F.4)

where Bǫ(xi−1) is a ball of radius ǫ around xi−1 and M represents all setsof zero measure (i.e., µ(M) = 0). Then, ∂maxSi−1 is convex, compact, andupper semi-continuous (see [84] for recent results on such properties for dis-continuous systems).

Then, in the case of ei > 0, the ei-subsystem can be represented with adifferential inclusion as

ei ∈ xi+1 +∆i(xi)− φi : φi ∈ ∂maxSi−1 ˙xi−1:= xi+1 − ei+1 + φ∗i : φ

∗i ∈ Φ∗

i (xi, ei+1), (F.5)

293


where

Φ∗i (xi, ei+1) = ei+1 +∆i(xi)− φi : φi ∈ ∂maxSi−1 ˙xi−1. (F.6)

Since ∆i(xi) and ˙xi−1 are locally Lipschitz, and ∂maxSi−1 is convex, com-pact, and upper semi-continuous, Φ∗

i (xi, ei+1) is convex, compact, and uppersemi-continuous.

For system (5.1)–(5.2), with condition (5.6), |∆i(xi)| is bounded by a K∞function of |xi| and | ˙xi−1| = |[x1, . . . , xi]T | is bounded by a K∞ function of|xi|. Hence, there exists a ψΦ∗

i1 ∈ K∞ such that for any φ∗i ∈ Φ∗

i (xi, ei+1), itholds that

|φ∗i | ≤ ψΦ∗

i1(|[xTi , ei+1]

T |). (F.7)

We show that |xi| is bounded by a K∞ function of |ei| and ai−1. Thedefinitions of ek+1 (1 ≤ k ≤ i− 1) in (5.27) imply

|xk+1| ≤ max|maxSk|, |minSk|+ |ek+1|. (F.8)

Define κok(s) = |κk(s)| for s ∈ R+. Then, κok ∈ K∞. From (F.1) and (F.2),

for 1 ≤ k ≤ i− 1,

|maxSk| ≤∣

∣

∣

∣

1

1− bk+1

∣

∣

∣

∣

|max Sk| (F.9)

|max Sk| ≤ κok((1 + bk)|xk|+ |max Sk−1|+ (1 − bk)ak))

≤ κok((1 + bk)(|xk − ek|+ |ek|) + |max Sk−1|+ (1− bk)ak)

≤ κok

(

(1 + bk)(|max Sk−1|+ |min Sk−1|+ |ek|)

+ |max Sk−1|+ (1− bk)ak

)

. (F.10)

In the same way, for 1 ≤ k ≤ i− 1, we can also get

|minSk| ≤∣

∣

∣

∣

1

1− bk+1

∣

∣

∣

∣

|min Sk| (F.11)

|min Sk| ≤ κok

(

(1 + bk)(|max Sk−1|+ |min Sk−1|+ |ek|)

+ |min Sk−1|+ (1 − bk)ak

)

. (F.12)

Note that x1 = e1. With (F.8), and iteratively using (F.9)–(F.12), we canprove that for each 1 ≤ k ≤ i − 1, |xk+1| is bounded by a K∞ function of|ek+1| and ak. This, together with (F.7), implies that there exists a ΨΦ∗

i∈ K∞

such that for any φ∗i ∈ Φ∗i (xi, ei+1), it holds that

|φ∗i | ≤ ψΦ∗

i(|[eTi+1, a

Ti−1]

T |). (F.13)

This ends the proof.



We simply use Sk instead of Sk(xk, µk1, µk2) for k = 1, . . . , i − 1. We onlyconsider the case of ei > 0.

Consider the recursive definition of Sk’s in (5.104). For k = 1, . . . , i−1, thestrictly decreasing property of the κk’s implies

maxSk = κk(xk −maxSk−1 −maxck1|ek|, µk1) + µk2, (F.14)

minSk = κk(xk −minSk−1 +maxck1|ek|, µk1)− µk2. (F.15)

From the iteration type definition of ek’s for k = 1, . . . , i−1, ei−1 is continu-ous and differentiable almost everywhere with respect to xi−1, µ(i−2)1, µ(i−2)2.

Since κk’s are continuously differentiable for k = 1, . . . , i− 1, using (F.14),we can see maxSi−1 is continuously differentiable almost everywhere withrespect to xi−1, µ(i−1)1, µ(i−1)2.

Considering the definition of ei in (5.105) with k = i − 1, when ei > 0, wecan represent the ei-subsystem with a differential equation

ei = xi+1 +∆i(xi, z)−∇maxSi−1[ ˙xi−1, 0(i−1), 0(i−1)]T (F.16)

with 0(i−1) being the vector composed of i − 1 zero elements, wherevermaxSi−1 is continuously differentiable, or equivalently, ∇maxSi−1 exists. Be-cause maxSi−1 is continuously differentiable almost everywhere, ∇maxSi−1

is discontinuous and thus the ei-subsystem is a discontinuous system. Werepresent the ei-subsystem with a differential inclusion by embedding the dis-continuous ∇maxSi−1 into a set-valued map

∂maxSi−1 =⋂

ǫ>0

⋂

τ(M)=0

co∇maxSi−1(Bε(ζi−1)\M), (F.17)

where Bε(ζi−1) is an open ball of radius ε around ζi−1 := [xTi−1, µT(i−1)1, µ

T(i−1)2]

T ,

and M represents all sets of zero measure (i.e., τ(M) = 0).Then, in the case of ei > 0, the ei-subsystem can be represented with a

differential inclusion as

ei ∈ xi+1 +∆i(xi, z)− ϕi : ϕi ∈ ∂maxSi−1[ ˙xTi−1, 0(i−1), 0(i−1)]

T := xi+1 + φi : φi ∈ Φi(xi, µ(i−1)1, µ(i−1)2, z), (F.18)

where

Φi(xi, µ(i−1)1, µ(i−1)2, z)

= ∆i(xi, z)− ϕi : ϕi ∈ ∂maxSi−1[ ˙xTi−1, 0(i−1), 0(i−1)]

T . (F.19)

Because ∆i and ˙xi are locally Lipschitz and ∂maxSi−1 is convex, com-pact, and upper semi-continuous, Φi is convex, compact, and upper semi-continuous. Considering the definition of ∂maxSi−1, one can find a continuous


function si−1 such that for all xi−1, µ(i−1)1, µ(i−1)2, any si−1 ∈ ∂maxSi−1 sat-isfies |si−1| ≤ si−1(xi−1, µ(i−1)1, µ(i−1)2). Thus, for all xi−1, µ(i−1)1, µ(i−1)2, z,any φi ∈ Φi(xi, µ(i−1)1, µ(i−1)2, z) satisfies

|φi| ≤ |∆i(xi, z)|+ si−1(xi−1, µ(i−1)1, µ(i−1)2)| ˙xi−1|. (F.20)

From (5.74)–(5.75) and Assumption 5.4, ∆i(xi, z) is bounded by a K∞function of (xi, z) and ˙xi−1 is bounded by a K∞ function of (xi, z). Thus,there exists a λ0Φi

∈ K∞ such that for any φi ∈ Φi(xi, µ(i−1)1, µ(i−1)2, z), itholds that

|φi| ≤ λ0Φi(|(xi, µ(i−1)1, µ(i−1)2, z)|). (F.21)

For the purpose of (5.107), for each k = 1, . . . , i− 1, we find a λxk+1∈ K∞

such that |xk+1| ≤ λxk+1(|(ek+1, µk1, µk2)|). For k = 1, . . . , i − 1, from the

definitions of xk+1 in (5.105), we have minSk ≤ xk+1 − ek+1 ≤ maxSk andthus

|xk+1| ≤ max|maxSk|, |minSk|+ |ek+1|. (F.22)

For each k = 1, . . . , i− 1, define κok(s) = κk(|s|) for s ∈ R+. Because κk isodd, strictly decreasing, and radially unbounded, κok ∈ K∞. From (F.14), wehave

|maxSk| ≤ κok(|xk −maxSk−1 −maxck1|ek|, µk1|) + µk2

≤ κok(|xk −maxSk−1|+maxck1|ek|, µk1) + µk2

≤ κok(|maxSk−1|+ |minSk−1|+ |ek|+maxck1|ek|, µk1)+ µk2. (F.23)

In (F.23), we used the fact that minSk−1 ≤ xk − ek ≤ maxSk−1 and thusminSk−1−maxSk−1+ek ≤ xk−maxSk−1 ≤ ek, to arrive at |xk−maxSk−1| ≤|maxSk−1|+ |minSk−1|+ |ek|. Similarly, we obtain

|minSk| ≤ κok(|maxSk−1|+ |minSk−1|+ |ek|+maxck1|ek|, µk1)+ µk2. (F.24)

For each xk+1 (k = 1, . . . , i − 1), using (F.22), (F.23), and (F.24), onecan find a λxk+1

∈ K∞ such that |xk+1| ≤ λxk+1(|(ek+1, µk1, µk2)|). This,

together with (F.21), guarantees that there exists a λΦi∈ K∞ such that for

all (xi, µ(i−1)1, µ(i−1)2, z), any φi ∈ Φi(xi, µ(i−1)1, µ(i−1)2, z) satisfies

|φi| ≤ λΦi(|(ei, z, µ(i−1)1, µ(i−1)2)|), (F.25)

where ei := [e1, . . . , ei]T .

Define

Φ∗i (ei+1, xi, µ(i−1)1, µ(i−1)2, z)

= φi + ei+1 : φi ∈ Φi(xi, µ(i−1)1, µ(i−1)2, z). (F.26)


From (5.105), xi+1 − ei+1 ∈ Si(xi, µi1, µi2). Then, equation (F.18) can berewritten as (5.106).

From (F.25), we can find a λΦ∗

i∈ K∞ such that any φ∗i ∈

Φ∗i (ei+1, xi, µ(i−1)1, µ(i−1)2, z) satisfies (5.107) for all (ei+1, xi, µ(i−1)1, µ(i−1)2, z).By also considering the cases of ei = 1 and ei < 0, Lemma 5.3 can be

proved.


Note that e0 = z. With (5.107) satisfied, one can find λekΦ∗

i∈ K∞ for k =

0, . . . , i + 1 and λµk1

Φ∗

i, λµk2

Φ∗

i∈ K∞ for k = 1, . . . , i − 1 such that for any φ∗i ∈

Φ∗i (ei+1, xi, µ(i−1)1, µ(i−1)2, z), it holds that

|φ∗i | ≤ Σi+1k=1λ

ekΦ∗

i(|ek|) + Σi−1

k=1

(

λµk1

Φ∗

i(µk1) + λµk2

Φ∗

i(µk2)

)

. (F.27)

By convenience, let γeiei = Id. Define

Πi(s) = λe0Φ∗

i α−1

0 (

γe0ei)−1 αV (s) + Σi+1

k=1λekΦ∗

i α−1

V (

γekei)−1 αV (s)

+ Σi−1k=1λ

µk1

Φ∗

i(

γµk1ei

)−1 αV (s) + Σi−1k=1λ

µk2

Φ∗

i(

γµk2ei

)−1 αV (s)

+ιi2s (F.28)

for s ∈ R+. Then, Πi ∈ K∞.From Lemma 1 in [123], for any 0 < ci1, ci2 < 1, ǫi > 0, one can find a

νi : R+ → R+ that is positive, nondecreasing and continuously differentiableon (0,∞), and satisfies

(1− ci2)(1 − ci1)νi ((1 − ci1)s) s ≥ Πi(s) (F.29)

for s ≥ √2ǫi. With the νi satisfying (F.29), define κi(r) = −νi(|r|)r for

r ∈ R. Noticing that limt→0+dκi(r)dr = limt→0−

dκi(r)dr , κi is continuously dif-

ferentiable, odd, strictly decreasing, and radially unbounded.Recall that Vk(ek) = αV (|ek|) = 1

2e2k for k = 1, . . . , n. We use Vk instead of

Vk(ek) for k = 1, . . . , n. Consider the case of

Vi ≥ maxk=1,...,i−1

γe0ei (V0), γekei (Vk), γ

ei+1ei (Vi+1),

γµk1ei (µk1), γ

µk2ei (µk2), γ

µi1ei (µi1), γ

µi2ei (µi2), ǫi

. (F.30)


Πi(|ei|)−ιi2|ei| ≥ φ∗i (F.31)

for all φ∗i ∈ Φ∗i (ei+1, xi, µ(i−1)1, µ(i−1)2, z). And it also holds that

µi1 ≤ ci1|ei| (F.32)

µi2 ≤ ci2κi ((1− ci1)|ei|) |ei| (F.33)

|ei| ≥√2ǫi. (F.34)


When ei 6= 0, with 0 < ci1 < 1, for ςi−1 ∈ Si−1 and |bi1| ≤maxci1|ei|, µi1 = ci1|ei|, we have

|xi − ςi−1 + bi1| ≥ (1− ci1)|ei| (F.35)

sgn(xi − ςi−1 + bi1) = sgn(ei) (F.36)

and thus

νi(|xi − ςi−1 + bi1|)|xi − ςi−1 + bi1| ≥ (1− ci1)νi((1 − ci1)|ei|)|ei|. (F.37)

In the case of (F.30), for any φ∗i ∈ Φ∗i (ei+1, xi, µ(i−1)1, µ(i−1)2, z), with

ςi−1 ∈ Si−1, |bi1| ≤ maxci1|ei|, µi1 and |bi2| ≤ µi2, using (F.31)–(F.37), wehave

∇Vi (κi(xi − ςi−1 + bi1) + bi2 + φ∗i )

= ei

(

−νi(|xi − ςi−1 + bi1|)(xi − ςi−1 + bi1) + bi2 + φ∗i)

≤ − νi(|xi − ςi−1 + bi1|)|xi − ςi−1 + bi1||ei|+ |bi2||ei|+ |φ∗i ||ei|≤ − (1 − ci2)(1 − ci1)νi((1 − ci1)|ei|)|ei|2 +Πi(|ei|)|ei| −

ιi2|ei|2

≤ − ιi2|ei|2 = −ιiαV (|ei|), (F.38)

which implies (5.111).


For convenience of notation, define vn = u. Note that S1(x1, µ11, µ12) de-fined in (5.101) is in the form of (5.104) with S0(x0, µ01, µ02) := 0.Then, v0 ∈ S0(x0, µ01, µ02). Suppose that vi−1 ∈ Si−1(xi−1, µ(i−1)1, µ(i−1)2).We will find ςi−1, bi1 and bi2 satisfying ςi−1 ∈ Si−1(xi−1, µ(i−1)1, µ(i−1)2),|bi1| ≤ maxci1|ei|, µi1, and |bi2| ≤ µi2, respectively, such that

vi = κi(xi − ςi−1 + bi1) + bi2 ∈ Si(xi, µi1, µi2). (F.39)

By applying this reasoning repeatedly, property (5.118) can be proved. Weconsider only the case of ei ≥ 0. The proof for the case of ei < 0 is similar.We study the following cases (A) and (B).

(A) |κi(qi1(xi − vi−1, µi1))| ≤Mi2µi2.With Assumption 5.6 satisfied, one can find a |bi2| ≤ µi2 such that

qi2(κi(qi1(xi − vi−1, µi1)), µi2) = κi(qi1(xi − vi−1, µi1)) + bi2.(F.40)

(A1) |xi − vi−1| ≤Mi1µi1.


In this case, Assumption 5.6 implies that there exists a |bi1| ≤ µi1such that

qi1(xi − vi−1, µi1) = xi − vi−1 + bi1. (F.41)

Choose ςi−1 = vi−1. Then, ςi−1 ∈ Si−1(xi−1, µ(i−1)1, µ(i−1)2) and

qi1(xi − vi−1, µi1) = xi − ςi−1 + bi1. (F.42)

(A2) |xi − vi−1| > Mi1µi1.In this case, Assumption 5.6 implies that there exists a |bi1| ≤ µi1such that

qi1(xi − vi−1, µi1) = sgn(xi − vi−1)Mi1µi1 + bi1. (F.43)

We study the following two cases:– ei > 0. Recall (5.105) and (5.116). In this case, we have xi >

vi−1 and

xi − vi−1 > Mi1µi1 ≥ ei

= xi −maxSi−1(xi−1, µ(i−1)1, µ(i−1)2). (F.44)

One can find a ςi−1 ∈ [vi−1,maxSi−1(xi−1, µ(i−1)1, µ(i−1)2)]such that xi − ςi−1 =Mi1µi1 and thus


– ei = 0. In this case, by using (5.105), we have xi ∈Si−1(xi−1, µ(i−1)1, µ(i−1)2) and can directly find a ςi−1 ∈Si−1(xi−1, µ(i−1)1, µ(i−1)2), which is closer to xi than vi−1 suchthat xi − ςi−1 = sgn(xi − vi−1)Mi1µi1 and thus


From (F.42) and (F.46), in the case of |κi(qi1(xi − vi−1))| ≤Mi2µi2, we can find ςi−1 ∈ Si−1(xi−1), |bi1| ≤ µi1 and |bi2| ≤ µi2such that

vi = qi2(κi(qi1(xi − vi−1))) = κi(xi − ςi−1 + bi1) + bi2. (F.47)

(B) |κi(qi1(xi − vi−1, µi1))| > Mi2µi2.Before the discussions, we give the following lemma.

Lemma F.1 Under the conditions of Lemma 5.5, if |κi(qi1(xi −vi−1, µi1))| > Mi2µi2, then

sgn(xi − vi−1) = sgn(qi1(xi − vi−1, µi1)). (F.48)


The proof of Lemma F.1 is in Appendix F.4.1Note that κi is an odd and strictly decreasing function. Then, wehave

sgn(κi(qi1(xi − vi−1, µi1))) = −sgn(qi1(xi − vi−1, µi1))

= −sgn(xi − vi−1). (F.49)

Under Assumption 5.6, using Lemma F.1, one can find a |bi2| ≤ µi2such that

qi2(κi(qi1(xi − vi−1, µi1)), µi2)

= sgn(κi(qi1(xi − vi−1, µi1)))Mi2µi2 + bi2

= − sgn(xi − vi−1)Mi2µi2 + bi2. (F.50)

(B1) ei > 0.In this case, using (5.105), we have xi > vi−1 and thus

κi(qi1(xi − vi−1, µi1)) < 0, (F.51)

qi2(κi(qi1(xi − vi−1, µi1))) = −Mi2µi2 + bi2. (F.52)

With |κi(qi1(xi − vi−1, µi1))| > Mi2µi2, property (F.51) implies

κi(qi1(xi − vi−1, µi1)) < −Mi2µi2. (F.53)

Consider the following two cases:– xi − vi−1 ≤ Mi1µi1. In this case, under Assumption 5.6, one

can find a |b′i1| ≤ µi1 such that

κi(xi − vi−1 + b′i1) = κi(qi1(xi − vi−1, µi1))

< −Mi2µi2. (F.54)

– x1 − vi−1 > Mi1µi1. In this case, under Assumption 5.6, onecan find a |b′i1| ≤ µi1 such that

κi(Mi1µi1 + b′i1) = κi(qi1(xi − vi−1, µi1))

< −Mi2µi2. (F.55)

By using the strictly decreasing property of κi, we have

κi(xi − vi−1 + b′i1) < κi(Mi1µi1 + b′i1) < −Mi2µi2. (F.56)

Thus, in both the cases above, one can find a |b′i1| ≤ µi1 such that

κi(xi − vi−1 + b′i1) < κi(Mi1µi1 + b′i1) < −Mi2µi2. (F.57)

From (5.117), we have

κi((1 − ci1)|ei|) < Mi2µi2. (F.58)


From the definition of ei, using ei > 0, we have

κi(xi −maxSi−1(xi−1, µ(i−1)1, µ(i−1)2)− ci1ei)

>−Mi2µi2. (F.59)

From (F.57) and (F.59), by using the continuity of κi, one canfind aςi−1 ∈ [vi−1,maxSi−1(xi−1, µ(i−1)1, µ(i−1)2)] and a bi1 ∈[−ci1ei, b′i1] such that

κi(xi − ςi−1 + bi1) = −Mi2µi2. (F.60)

Recall (F.52). We have

vi = qi2(κi(qi1(xi − vi−1, µi1)))

= κi(xi − ςi−1 + bi1) + bi2. (F.61)

(B2) ei = 0.In this case, xi ∈ Si−1(xi−1, µ(i−1)1, µ(i−1)2). From Lemma F.1,we have xi − vi−1 6= 0. Consider the following two cases:– |xi − vi−1| ≤ Mi1µi1. In this case, define ς ′i−1 = vi−1. With

Assumption 5.6, one can find a |b′i1| ≤ µi1 such that

κi(xi − ς ′i−1 + b′i1) = κi(qi1(xi − vi−1, µi1))

> Mi2µi2, if xi < ς ′i−1

< −Mi2µi2, if xi > ς ′i−1.(F.62)

We used |κi(qi1(xi − vi−1, µi1))| > Mi2µi2 and (F.49) for thelast part of (F.62).

– |xi − vi−1| > Mi1µi1. In this case, under Assumption 5.6, onecan find a |b′i1| ≤ µi1 such that

κi(sgn(xi − vi−1)Mi1µi1 + b′i1)

= κi(qi1(xi − vi−1, µi1))

> Mi2µi2, if xi < vi−1

< −Mi2µi2, if xi > vi−1.(F.63)

In the case of |xi − vi−1| > Mi1µi1, one can find a ς ′i−1 ∈[xi, vi−1] satisfying sgn(xi−ς ′i−1) = sgn(xi−vi−1) and sgn(xi−vi−1)Mi1µi1 = xi − ς ′i−1. In this way, we achieve

κi(xi − ς ′i−1 + b′i1)

> Mi2µi2, if xi < ς ′i−1

< −Mi2µi2, if xi > ς ′i−1.(F.64)


Note that κi(xi − xi + 0) = κi(0) = 0. By using the continuity ofκi, one can find a ςi−1 ∈ [xi, ς

′i−1] and a bi1 ∈ [0, b′i1] such that

sgn(xi − ςi−1) = sgn(xi − ς ′i−1) = sgn(xi − vi−1) (F.65)

κi(xi − ςi−1 + bi1) = −sgn(xi − vi−1)Mi2µi2. (F.66)

Clearly, ςi−1 ∈ Si−1(xi−1, µ(i−1)1, µ(i−1)2) and |bi1| ≤ µi1. Recall(F.50). We have

vi = qi2(κi(qi1(xi − vi−1, µi1)))

= κi(xi − ςi−1 + bi1) + bi2. (F.67)

Considering both cases (A) and (B), the proof of Lemma 5.5 is concluded.

F.4.1 PROOF OF LEMMA F.1

Consider the following two cases:

• |xi−vi−1| > Mi1µi1. In this case, under Assumption 5.6, one can finda |bi1| ≤ µi1 such that

qi1(xi − vi−1, µi1) = sgn(xi − vi−1)Mi1µi1 + bi1 (F.68)

Note that Mi1 > 2. Thus,

sgn(xi − vi−1) = sgn(qi1(xi − vi−1, µi1)). (F.69)

• |xi−vi−1| ≤Mi1µi1. In this case, under Assumption 5.6, one can finda |bi1| ≤ µi1 such that

qi1(xi − vi−1, µi1) = xi − vi−1 + bi1. (F.70)

Condition |κi(qi1(xi − vi−1, µi1))| > Mi2µi2 implies qi1(xi −vi−1, µi1) 6= 0.If sgn(xi − vi−1) 6= sgn(qi1(xi − vi−1, µi1)), then sgn(bi1) =sgn(qi1(xi−vi−1, µi1)) and |bi1| > |xi−vi−1|. Thus, |xi−vi−1+bi1| ≤|bi1| ≤ µi1. Note that 1

Mi1< ci1 < 0.5. Then, we can derive

|κi(qi1(xi − vi−1, µi1))| ≤ κi(µi1)

≤ κi

(

1− ci1ci1

µi1

)

< κi((1 − ci1)Mi1µi1)

=Mi2µi2, (F.71)

which leads to a contradiction with |κi(qi1(xi− vi−1))| > Mi2µi2. Weused (5.128) for the last equality in (F.71).



Recall that if χ1, χ2 ∈ K∞ satisfies χ1(s) > χ2(s) for s ∈ R+, then (Id −χ) χ1(s) ≥ χ2(s) for s ∈ R+ with χ := Id − χ2 χ−1

1 being continuous andpositive definite. For each i = 1, . . . , n, with (5.115) satisfied, we can find acontinuous and positive definite ρi such that

σi αV(

1

ci1s

)

≤ (Id− ρi) σi αV (Mi1s) (F.72)

σi αV(

1

1− ci1κ−1i

(

1

ci2s

))

≤ (Id− ρi) σi αV(

1

1− ci1κ−1i (Mi2s)

)

(F.73)

for all s ∈ R+. Define ρ(s) = mini=1,...,nρi(s) for s ∈ R+. Then, ρ iscontinuous and positive definite. Using (5.126), (5.127), (5.132), and (5.138),we have

B2(µn1(t), µn2(t)) ≤ (Id− ρ)(Θ(t)) (F.74)

for any t ∈ R+.Note that the zooming variables µn1(t) and µn2(t) are constant on [tk, tk+1),

that is, µn1(t) = µn1(tk) and µn2(t) = µn2(tk) for t ∈ [tk, tk+1). Suppose(5.148) holds. We study the following two cases:

(a) V (e(X(tk+1), µn1(tk), µn2(tk))) < max(Id− ρ)(Θ(tk)), θ0.(b) V (e(X(tk+1), µn1(tk), µn2(tk))) ≥ max(Id − ρ)(Θ(tk)), θ0. In

this case, from (5.131), (5.148), and (F.74), it follows thatV (e(X(t), µn1(t), µn2(t))) is strictly decreasing for t ∈ [tk, tk+1) and

max(Id− ρ)(Θ(tk)), θ0 ≤ V (e(X(t), µn1(t), µn2(t)))

≤ Θ(tk) (F.75)

for all t ∈ [tk, tk+1). By using (5.131), we have

V (e(X(tk+1), µn1(tk), µn2(tk)))

≤ V (e(X(tk), µn1(tk), µn2(tk)))

−∫ tk+1

tk

α(V (e(X(τ), µn1(τ), µn2(τ))))dτ

≤ Θ(tk)− td · minmax(Id−ρ)(Θ(tk)),θ0≤v≤Θ(tk)

α(v)

≤ Θ(tk)− td · min(Id−ρ)(Θ(tk))≤v≤Θ(tk)

α(v), (F.76)

where td = tk+1 − tk. Define ρ′(s) = td · min(Id−ρ)(s)≤v≤s α(v) fors ∈ R+. Then, it can be directly verified that ρ′ is continuous andpositive definite and that

V (e(X(tk+1), µn1(tk), µn2(tk))) ≤ (Id− ρ′)(Θ(tk)). (F.77)


Lemma 5.8 is proved by finding a continuous and positive definite functionρ such that (Id− ρ) ∈ K∞ and (Id− ρ)(s) ≥ max(Id− ρ)(s), (Id− ρ′)(s) fors ∈ R+.

References

1. D. Aeyels and J. Peuteman. A new asymptotic stability criterion for nonlineartime-variant differential equations. IEEE Transactions on Automatic Control,43:968–971, 1998.

2. A. P. Aguiar and A. M. Pascoal. Coordinated path-following control fornonlinear systems with logic-based communication. In Proceedings of the

46th Conference on Decision and Control, pages 1473–1479, 2007.3. M. A. Aizerman and F. R. Gantmacher. Absolute Stability of Regulator Sys-

tems. Holden Day, San Francisco, 1963.4. B. D. O. Anderson. The small gain theorem, the passivity theorem and their

equivalence. Journal of Franklin Institute, 293:105–115, 1972.5. B. D. O. Anderson and S. Vongpanitlerd. Network Analysis and Synthesis: A

Modern Systems Approach. Prentice-Hall, New Jersey, 1973.6. D. Angeli and E. D. Sontag. Forward completeness, unbounded observability

and Their Lyapunov characterizations. Systems & Control Letters, 38:209–217, 1999.

7. D. Angeli, E. D. Sontag, and Y. Wang. A characterization of integral input-to-state stability. IEEE Transactions on Automatic Control, 45:1082–1097,2000.

8. M. Arcak. Passivity as a design tool for group coordination. IEEE Transac-

tions on Automatic Control, 52:1380–1390, 2007.9. M. Arcak. Passivity approach to network stability analysis and distributed

control synthesis. In W. Levine, editor, The Control Handbook, pages 1–18.CRC Press, 2010.

10. M. Arcak and P. V. Kokotovic. Nonlinear observers: A circle criterion designand robustness analysis. Automatica, 37:1923–1930, 2001.

11. A. Astolfi, D. Karagiannis, and R. Ortega. Nonlinear and Adaptive Control

with Applications. Springer, 2008.12. K. J. Astrom and B. Wittenmark. Adaptive Control. Pearson Education,

second edition, 1995.13. J.-P. Aubin and H. Frankowska. Set-Valued Analysis. Birkhauser, Boston,

1990.14. A. Bacciotti. Local Stabilizability of Nonlinear Control Systems. World Sci-

entific, 1992.15. A. Bacciotti and L. Rosier. Lyapunov stability and Lagrange stability: Inverse

theorems for discontinuous systems. Mathematics of Control, Signals and

Systems, 11:101–125, 1998.16. A. Bacciotti and L. Rosier. Liapunov Functions and Stability in Control

Theory. Springer-Verlag, Berlin, second edition, 2005.17. H. Bai, M. Arcak, and J. Wen. Cooperative Control Design: A Systematic,

Passivity-Based Approach. Springer, New York, 2011.18. D. P. Bertsekas and J. N. Tsitsiklis. Parallel and Distributed Computation:

Numerical Methods. Athena Scientific, 1997.19. R. W. Brockett and D. Liberzon. Quantized feedback stabilization of linear

305

306 References

systems. IEEE Transactions on Automatic Control, 45:1279–1289, 2000.20. B. Brogliato, R. Lozano, B. Maschke, and O. Egeland. Dissipative Systems

Analysis and Control: Theory and Applications. Springer-Verlag, London,second edition, 2007.

21. C. I. Byrnes, F. Delli Priscoli, and A. Isidori. Output Regulation of Uncertain

Nonlinear Systems. Birkhauser, 1997.22. C. Cai and A. R. Teel. Input-output-to-state stability for discrete-time sys-

tems. Automatica, 44:326–336, 2008.23. C. Cai and A. R. Teel. Characterizations of input-to-state stability for hybrid

systems. Systems & Control Letters, 58:47–53, 2009.24. Y. Cao and W. Ren. Distributed coordinated tracking with reduced inter-

action via a variable structure approach. IEEE Transactions on Automatic

Control, 57:33–48, 2012.25. D. Carnevale, A. R. Teel, and D. Nesic. A Lyapunov proof of an improved

maximum allowable transfer interval for networked control systems. IEEE

Transactions on Automatic Control, 52:892–897, 2007.26. F. Ceragioli and C. De Persis. Discontinuous stabilization of nonlinear sys-

tems: Quantized and switching controls. Systems & Control Letters, 56:461–473, 2007.

27. J. W. Chapman, M. D. Ilic, C. A. King, L. Eng, and H. Kaufman. Stabilizinga multimachine power system via decentralized feedback linearizing excitationcontrol. IEEE Transactions on Power Systems, 8:830–839, 1993.

28. C. T. Chen. Linear System Theory and Design. Oxford University Press,1999.

29. P.-C. Chen, R. Salcedo, Q. Zhu, F. de Leon, D. Czarkowski, Z. P. Jiang,V. Spitsa, Z. Zabar, and R. E. Uosef. Analysis of voltage profile problemsdue to the penetration of distributed generation in low-voltage secondarydistribution networks. IEEE transactions on Power Delivery, 27:2020–2028,2012.

30. T. Chen and J. Huang. A small-gain approach to global stabilization ofnonlinear feedforward systems with input unmodeled dynamics. Automatica,46:1028–1034, 2010.

31. Z. Chen and J. Huang. A general formulation and solvability of the globalrobust output regulation problem. IEEE Transactions on Automatic Control,50:448–462, 2005.

32. Z. Chen and J. Huang. Global robust output regulation problem for outputfeedback systems. IEEE Transactions on Automatic Control, 50:117–121,2005.

33. D. Cheng, X. Hu, and T. Shen. Analysis and Design of Nonlinear Control

Systems. Springer, 2011.34. H.-D. Chiang. Direct Methods for Stability Analysis of Electric Power Sys-

tems: Theoretical Foundation, BCU Methodologies, and Applications. Wiley,2011.

35. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski. Asymptoticstability and smooth Lyapunov functions. Journal of Differential Equations,149:69–114, 1998.

36. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski. Nonsmooth

Analysis and Control Theory. Springer, New York, 1998.

References 307

37. J.-M. Coron. Control and Nonlinearity. American Mathematical Society,2007.

38. J. Cortes. Discontinuous dynamical systems: A tutorial on solutions, non-smooth analysis, and stability. IEEE Control Systems Magazine, 28:36–73,2008.

39. J. Cortes, S. Martnez, and F. Bullo. Robust rendezvous for mobile au-tonomous agents via proximity graphs in arbitrary dimensions. IEEE Trans-

actions on Automatic Control, 51:1289–1298, 2006.40. B. d’Andrea-Novel, G. Bastin, and G. Campion. Dynamic feedback lineariza-

tion of nonholonomic wheeled mobile robots. In Proceedings of the 1992

IEEE International Conference on Robotics and Automation, pages 2527–2532, 1992.

41. S. Dashkovskiy, H. Ito, and F. Wirth. On a small gain theorem for ISSnetworks in dissipative Lyapunov form. European Journal of Control, 17:357–365, 2011.

42. S. Dashkovskiy and M. Kosmykov. Stability of networks of hybrid ISS sys-tems. In Proceedings of Joint 48th IEEE Conference on Decision and Control

and 28th Chinese Control Conference, pages 3870–3875, 2009.43. S. Dashkovskiy, B. S. Ruffer, and F. R. Wirth. An ISS small-gain theorem for

general networks. Mathematics of Control, Signals and Systems, 19:93–122,2007.

44. S. Dashkovskiy, B. S. Ruffer, and F. R. Wirth. Small gain theorems for largescale systems and construction of ISS Lyapunov functions. SIAM Journal on

Control and Optimization, 48:4089–4118, 2010.45. P. De Leenheer, D. Angeli, and E. D. Sontag. On predator-prey systems and

small-gain theorems. Mathematical Biosciences and Engineering, 2:25–42,2005.

46. C. De Persis. n-bit stabilization of n-dimensional nonlinear systems in feed-forward form. IEEE Transactions on Automatic Control, 50:299–311, 2005.

47. J. P. Desai, J. P. Ostrowski, and V. Kumar. Modeling and control of for-mations of nonholonomic mobile robots. IEEE Transactions on Robotics and

Automation, 17:905–908, 2001.48. C. A. Desoer and M. Vidyasagar. Feedback Systems: Input-Output Properties.

Academic Press, New York, 1975.49. W. E. Dixon, D. M. Dawson, E. Zergeroglu, and A. Behal. Nonlinear Control

of Wheeled Mobile Robots. Springer, London, 2001.50. K. D. Do. Formation tracking control of unicycle-type mobile robots with

limited sensing ranges. IEEE Transactions on Control Systems Technology,16:527–538, 2008.

51. W. Dong and J. A. Farrell. Decentralized cooperative control of multiplenonholonomic dynamic systems with uncertainty. Automatica, 45:706–710,2009.

52. T. Donkers and M. Heemels. Output-based event-triggered control withguaranteed L∞-gain and improved and decentralized event-triggering. IEEE

Transactions on Automatic Control, 57:1362–1376, 2012.53. P. M. Dower and M. R. James. Dissipativity and nonlinear systems with

finite power gain. International Journal of Robust and Nonlinear Control,8:699–724, 1998.

308 References

54. J. Doyle, B. Francis, and A. Tannenbaum. Feedback Control Systems. MacMil-lan Publishing Co., 1992.

55. N. Elia and S. K. Mitter. Stabilization of linear systems with limited infor-mation. IEEE Transactions on Automatic Control, 46:1384–1400, 2001.

56. G. A. Enciso and E. D. Sontag. Global attractivity, I/O monotone small-gaintheorems, and biological delay systems. Discrete and Continuous Dynamical

Systems, 14:549–578, 2006.57. H. R. Everett. Sensors for Mobile Robots: Theory and Application. A. K.

Peters, 1995.58. J. A. Fax and R. M. Murray. Information flow and cooperative control of

vehicle formation. IEEE Transactions on Automatic Control, 49:1465–1476,2004.

59. H. Federer. Geometric Measure Theory. Springer-Verlag, New York, 1969.60. A. F. Filippov. Differential Equations with Discontinuous Righthand Sides.

Kluwer Academic Publishers, 1988.61. M. Fliess, J. L. Levine, P. Martin, and P. Rouchon. Flatness and defect of

non-linear systems: Introductory theory and examples. International Journalof Control, 61:1327–1361, 1995.

62. F. Forni, R. Sepulchre, and A. van der Schaft. On differential passivity ofphysical systems. CoRR, abs/1309.2558, 2013.

63. F. Forni, R. Sepulchre, and A. van der Schaft. On differentially dissipativedynamical systems. In Proceedings of the 9th IFAC Symposium on Nonlinear

Control Systems, pages 15–20, 2013.64. R. A. Freeman. Global internal stabilization does not imply global external

stabilizability for small sensor disturbances. IEEE Transactions on Automatic

Control, 40:2119–2122, 1995.65. R. A. Freeman and P. V. Kokotovic. Global robustness of nonlinear systems to

state measurement disturbances. In Proceedings of the 32nd IEEE Conference

on Decision and Control, pages 1507–1512, 1993.66. R. A. Freeman and P. V. Kokotovic. Robust Nonlinear Control Design: State-

space and Lyapunov Techniques. Birkhauser, Boston, 1996.67. M. Fu and L. Xie. The sector bound approach to quantized feedback control.

IEEE Transactions on Automatic Control, 50:1698–1711, 2005.68. R. Ghabcheloo, A. P. Aguiar, A. Pascoal, C. Silvestre, I. Kaminer, and J. Hes-

panha. Coordinated path-following in the presence of communication lossesand time delays. SIAM Journal on Control and Optimization, 48:234–265,2009.

69. R. Goebel. Set-valued Lyapunov functions for difference inclusions. Automat-

ica, 47:127–132, 2011.70. R. Goebel, R. G. Sanfelice, and A. R. Teel. Hybrid Dynamical Systems:

Modeling, Stability, and Robustness. Princeton University Press, 2012.71. R. Goebel and A. R. Teel. Solution to hybrid inclusions via set and graphical

convergence with stability theory applications. Automatica, 42:573–587, 2006.72. W. M. Griggs, B. D. O. Anderson, A. Lanzon, and M. Rotkowitz. A stability

result for interconnections of nonlinear systems with “mixed” small gain andpassivity properties. In Proceedings of the 46th IEEE Conference on Decision

and Control, pages 4489–4494, 2011.73. L. Grune. Asymptotic Behavior of Dynamical and Control Systems under

References 309

Perturbation and Discretization. Springer, Berlin, 2002.74. L. Grune. Input-to-state dynamical stability and its Lyapunov function char-

acterization. IEEE Transactions on Automatic Control, 47:1499–1504, 2002.75. Y. Guo, D. J. Hill, and Y. Wang. Nonlinear decentralized control of large-scale

power systems. Automatica, 36:1275–1289, 2000.76. W. M. Haddad and V. Chellaboina. Nonlinear Dynamical Systems and Con-

trol: A Lyapunov-Based Approach. Princeton University Press, 2008.77. W. M. Haddad and S. G. Nersesov. Stability and Control of Large-Scale

Dynamical Systems: A Vector Dissipative Systems Approach. Princeton Uni-versity Press, 2011.

78. W. Hahn. Stability of Motion. Springer-Verlag, Berlin, 1967.79. E. J. Hancock and A. Papachristodoulou. Structured sum of squares for

networked systems analysis. In Proceedings of the 50th IEEE Conference on

Decision and Control and European Control Conference, pages 7236–7241,2011.

80. J. Harmard, A. R. Rapaport, and F. Mazenc. Output tracking of continuousbioreactors through recirculation and by-pass. Automatica, 42:1025–1032,2006.

81. T. Hayakawa, H. Ishii, and K. Tsumura. Adaptive quantized control for linearuncertain discrete-time systems. Automatica, 45:692–700, 2009.

82. T. Hayakawa, H. Ishii, and K. Tsumura. Adaptive quantized control fornonlinear uncertain systems. Systems & Control Letters, 58:625–632, 2009.

83. W. P. M. H. Heemels, K. H. Johansson, and P. Tabuada. An introductionto event-triggered and self-triggered control. In Proceedings of the 51st IEEE

Conference on Decision and Control, pages 3270–3285, 2012.84. W. P. M. H. Heemels and S. Weiland. Input-to-state stability and inter-

connections of discontinuous dynamical systems. Automatica, 44:3079–3086,2008.

85. J. P. Hespanha, D. Liberzon, D. Angeli, and E. D. Sontag. Nonlinear norm-observability notions and stability of switched systems. IEEE Transactions

on Automatic Control, 50:154–168, 2005.86. J. P. Hespanha, D. Liberzon, and A. R. Teel. Lyapunov conditions for input-

to-state stability of impulsive systems. Automatica, 44:2735–2744, 2008.87. D. J. Hill. Dissipativeness, stability theory and some remaining problems. In

C. I. Byrnes, C. F. Martin, and R. E. Saeks, editors, Analysis and Control of

Nonlinear Systems, Amsterdam, 1988. North-Holland.88. D. J. Hill. A generalization of the small-gain theorem for nonlinear feedback

systems. Automatica, 27:1043–1045, 1991.89. D. J. Hill and P. J. Moylan. The stability of nonlinear dissipative systems.

IEEE Transactions on Automatic Control, 21:708–711, 1976.90. D. J. Hill and P. J. Moylan. Stability results for nonlinear feedback systems.

Automatica, 13:377–382, 1977.91. D. J. Hill and P. J. Moylan. Connections between finite-gain and asymptotic

stability. IEEE Transactions on Automatic Control, 25:931–936, 1980.92. D. J. Hill and P. J. Moylan. Dissipative dynamical systems: Basic input-

output and state properties. Journal of Franklin Institute, 5:327–357, 1980.93. D. J. Hill and P. J. Moylan. General instability results for interconnected

systems. SIAM Journal on Control and Optimization, 21:256–279, 1983.

310 References

94. D. J. Hill, C. Wen, and G. C. Goodwin. Stability analysis of decentralisedrobust adaptive control. Systems & Control Letters, 11:277–284, 1988.

95. S. Hirche, T. Matiakis, and M. Buss. A distributed controller approachfor delay-independent stability of networked control systems. Automatica,45:1828–1836, 2009.

96. J. Hirschhorn. Kinematics and Dynamics of Plane Mechanisms. McGraw-HillBook Company, 1962.

97. Y. Hong, G. Chen, and L. Bushnell. Distributed observers design for leader-following control of multi-agent networks. Automatica, 44:846–850, 2008.

98. Y. Hong, L. Gao, D. Cheng, and J. Hu. Lyapunov-based approach to multia-gent systems with switching jointly connected interconnection. IEEE Trans-

actions on Automatic Control, 52:943–948, 2007.99. J. Huang. Nonlinear Output Regulation: Theory and Applications. SIAM,

Philadelphia, 2004.100. J. Huang and Z. Chen. A general framework for tackling the output regulation

problem. IEEE Transactions on Automatic Control, 49:2203–2218, 2004.101. S. Huang, M. R. James, D. Nesic, and P. M. Dower. Analysis of input-to-

state stability for discrete time nonlinear systems via dynamic programming.Automatica, 41:2055–2065, 2005.

102. S. Huang, M. R. James, D. Nesic, and P. M. Dower. A unified approach tocontroller design for achieving ISS and related properties. IEEE Transactions

on Automatic Control, 50:1681–1697, 2005.103. I.-A. F. Ihle, M. Arcak, and T. I. Fossen. Passivity-based designs for synchro-

nized path-following. Automatica, 43:1508–1518, 2007.104. B. Ingalls, E. D. Sontag, and Y. Wang. Generalizations of asymptotic gain

characterizations of ISS to input-to-output stability. In Proceedings of the

2001 American Control Conference, pages 704–708, 2001.105. A. Isidori. Nonlinear Control Systems. Springer, London, third edition, 1995.106. A. Isidori. Nonlinear Control Systems II. Springer-Verlag, London, 1999.107. H. Ito. State-dependent scaling problems and stability of interconnected iISS

and ISS systems. IEEE Transactions on Automatic Control, 51:1626–1643,2006.

108. H. Ito and Z. P. Jiang. Necessary and sufficient small gain conditions forintegral input-to-state stable systems: A Lyapunov perspective. IEEE Trans-

actions on Automatic Control, 54:2389–2404, 2009.109. H. Ito, Z. P. Jiang, and P. Pepe. Construction of Lyapunov-Krasovskii func-

tionals for networks of iISS retarded systems in small-gain formulation. Au-

tomatica, 49:3246–3257, 2013.110. H. Ito, P. Pepe, and Z. P. Jiang. A small-gain condition for iISS of inter-

connected retarded systems based on Lyapunov-Krasovskii functionals. Au-

tomatica, 46:1646–1656, 2010.111. A. Jadbabaie, J. Lin, and A. Morse. Coordination of groups of mobile au-

tonomous agents using nearest neighbor rules. IEEE Transactions on Auto-

matic Control, 48:988–1001, 2003.112. M. R. James. A partial differential inequality for dissipative nonlinear sys-

tems. Automatica, 21:315–320, 1993.113. Y. Jiang and Z. P. Jiang. Robust adaptive dynamic programming for large-

scale systems with an application to multimachine power systems. IEEE

References 311

Transactions on Circuits and Systems–II: Express Briefs, 59:693–697, 2012.114. Y. Jiang and Z. P. Jiang. Robust adaptive dynamic programming with an

application to power systems. IEEE Transactions on Neural Networks and

Learning Systems, 24:1150–1156, 2013.115. Z. P. Jiang. A combined backstepping and small-gain approach to adaptive

output-feedback control. Automatica, 35:1131–1139, 1999.116. Z. P. Jiang. Decentralized and adaptive nonlinear tracking of large-scale sys-

tems via output feedback. IEEE Transactions on Automatic Control, 45:2122–2128, 2000.

117. Z. P. Jiang. Decentralized disturbance attenuating output-feedback trackersfor large-scale nonlinear systems. Automatica, 38:1407–1415, 2002.

118. Z. P. Jiang. Decentralized control for large-scale nonlinear systems: A reviewof recent results. Dynamics of Continuous, Discrete and Impulsive Systems

Series B: Algorithms and Applications, 11:537–552, 2004.119. Z. P. Jiang and M. Arcak. Robust global stabilization with ignored input dy-

namics: An input-to-state stability (ISS) small-gain approach. IEEE Trans-

actions on Automatic Control, 46:1411–1415, 2001.120. Z. P. Jiang and Y. Jiang. Robust adaptive dynamic programming for linear

and nonlinear systems: An overview. European Journal of Control, 19:417–425, 2013.

121. Z. P. Jiang, Y. Lin, and Y. Wang. Nonlinear small-gain theorems for discrete-time large-scale systems. In Proceedings of the 27th Chinese Control Confer-

ence, pages 704–708, 2008.122. Z. P. Jiang and T. Liu. Quantized nonlinear control–A survey. Acta Auto-

matica Sinica, 2013. In press.123. Z. P. Jiang and I. M. Y. Mareels. A small-gain control method for non-

linear cascade systems with dynamic uncertainties. IEEE Transactions on

Automatic Control, 42:292–308, 1997.124. Z. P. Jiang and I. M. Y. Mareels. Robust nonlinear integral control. IEEE

Transactions on Automatic Control, 46:1336–1342, 2001.125. Z. P. Jiang, I. M. Y. Mareels, and D. J. Hill. Robust control of uncertain non-

linear systems via measurement feedback. IEEE Transactions on Automatic

Control, 44:807–812, 1999.126. Z. P. Jiang, I. M. Y. Mareels, and Y. Wang. A Lyapunov formulation of

the nonlinear small-gain theorem for interconnected systems. Automatica,32:1211–1215, 1996.

127. Z. P. Jiang and H. Nijmeijer. Tracking control of mobile robots: A case studyin backstepping. Automatica, 33:1393–1399, 1997.

128. Z. P. Jiang and L. Praly. Preliminary results about robust Lagrange stabilityin adaptive nonlinear regulation. International Journal of Adaptive Control

and Signal Processing, 6:285–307, 1992.129. Z. P. Jiang, D. W. Repperger, and D. J. Hill. Decentralized nonlinear output-

feedback stabilization with disturbance attenuation. IEEE Transactions on

Automatic Control, 46:1623–1629, 2001.130. Z. P. Jiang, A. R. Teel, and L. Praly. Small-gain theorem for ISS systems and

applications. Mathematics of Control, Signals and Systems, 7:95–120, 1994.131. Z. P. Jiang and Y. Wang. Input-to-state stability for discrete-time nonlinear

systems. Automatica, 37:857–869, 2001.

312 References

132. Z. P. Jiang and Y. Wang. A converse Lyapunov theorem for discrete-timesystems with disturbances. Systems & Control Letters, 45:49–58, 2002.

133. Z. P. Jiang and Y. Wang. Small-gain theorems on input-to-output stability.In Proceedings of the 3rd International DCDIS Conference, pages 220–224,2003.

134. Z. P. Jiang and Y. Wang. A generalization of the nonlinear small-gain theoremfor large-scale complex systems. In Proceedings of the 7th World Congress on

Intelligent Control and Automation, pages 1188–1193, 2008.135. I. Karafyllis and Z. P. Jiang. A small-gain theorem for a wide class of feedback

systems with control applications. SIAM Journal on Control and Optimiza-

tion, 46:1483–1517, 2007.136. I. Karafyllis and Z. P. Jiang. Necessary and sufficient Lyapunov-like condi-

tions for robust nonlinear stabilization. ESAIM: Control, Optimisation and

Calculus of Variations, 16:887–928, 2009.137. I. Karafyllis and Z. P. Jiang. Stability and Stabilization of Nonlinear Systems.

Springer, London, 2011.138. I. Karafyllis and Z. P. Jiang. A vector small-gain theorem for general nonlinear

control systems. IMA Journal of Mathematical Control and Information,28:309–344, 2011.

139. I. Karafyllis and Z. P. Jiang. Global stabilization of nonlinear systems basedon vector control Lyapunov functions. IEEE Transactions on Automatic Con-

trol, 58:2550–2562, 2013.140. I. Karafyllis, C. Kravaris, L. Syrou, and G. Lyberatos. A vector Lyapunov

function characterization of input-to-state stability with application to robustglobal stabilization of the chemostat. European Journal of Control, 14:47–61,2008.

141. I. Karafyllis and J. Tsinias. Non-uniform in time input-to-state stability andthe small-gain theorem. IEEE Transactions on Automatic Control, 49:196–216, 2004.

142. C. Kellett and A. R. Teel. Smooth Lyapunov functions and robustness ofstability for difference inclusions. Systems & Control Letters, 52:395–405,2004.

143. C. Kellett and A. R. Teel. On the robustness of KL-stability for differenceinclusions: Smooth discrete-time Lyapunov functions. SIAM Journal on Con-

trol and Optimization, 44:777–800, 2005.144. H. K. Khalil. Nonlinear Systems. Prentice-Hall, New Jersey, third edition,

2002.145. H. Koeppl, D. Densmore, G. Setti, and M. di Bernardo, editors. Design and

Analysis of Biomolecular Circuits. Springer, 2011.146. N. N. Krasovskii. Stability of Motion. Stanford University Press, 1963.147. P. Krishnamurthy and F. Khorrami. Decentralized control and disturbance

attenuation for large-scale nonlinear systems in generalized output-feedbackcanonical form. Automatica, 39:1923–1933, 2003.

148. P. Krishnamurthy, F. Khorrami, and Z. P. Jiang. Global output feedbacktracking for nonlinear systems in generalized output-feedback canonical form.IEEE Transactions on Automatic Control, 47:814–819, 2002.

149. M. Krstic. Delay Compensation for Nonlinear, Adaptive, and PDE Systems.Birkhauser, Boston, 2009.

References 313

150. M. Krstic. Input delay compensation for forward complete and feedforwardnonlinear systems. IEEE Transactions on Automatic Control, 55:287–303,2010.

151. M. Krstic and H. Deng. Stabilization of Nonlinear Uncertain Systems.Springer-Verlag, London, 1998.

152. M. Krstic, D. Fontaine, P. V. Kokotovic, and J. Paduano. Useful nonlinearitiesand global bifurcation control of jet engine stall and surge. IEEE Transactions

on Automatic Control, 43:1739–1745, 1998.153. M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic. Nonlinear and Adaptive

Control Design. John Wiley & Sons, New York, 1995.154. D. S. Laila and D. Nesic. Lyapunov based small-gain theorem for param-

eterized discrete-time interconnected ISS systems. IEEE Transactions on

Automatic Control, 48:1783–1788, 2003.155. V. Lakshmikantham and S. Leela. Differential and Integral Inequalities: The-

ory and Applications. Volume I–Ordinary Differential Equations. AcademicPress, New York, 1969.

156. V. Lakshmikantham, S. Leela, and A. A. Martyuk. Practical Stability of

Nonlinear Systems. World Scientific, New Jersey, 1990.157. Y. Lan, G. Yan, and Z. Lin. Synthesis of distributed control of coordinated

path following. IEEE Transactions on Automatic Control, 56:1170–1175,2011.

158. J. LaSalle and S. Lefschetz. Stability by Liapunov’s Direct Method with Ap-

plications. Academic Press, New York, 1961.159. Y. S. Ledyaev and E. D. Sontag. A Lyapunov characterization of robust

stabilization. Nonlinear Analysis, 37:813–840, 1999.160. M. D. Lemmon. Event-triggered feedback in control, estimation, and opti-

mization. In A. Bemporad, M. Heemels, and M. Johansson, editors, Networked

Control Systems, Lecture Notes in Control and Information Sciences, pages293–358, Berlin, 2010. Springer-Verlag.

161. Q. Li and Z. P. Jiang. Flocking control of multi-agent systems with applicationto nonholonomic multi-robots. Kybernetika, 45:84–100, 2009.

162. T. Li, M. Fu, L. Xie, and J. F. Zhang. Distributed consensus with limitedcommunication data rate. IEEE Transactions on Automatic Control, 56:279–292, 2011.

163. D. Liberzon. Hybrid feedback stabilization of systems with quantized signals.Automatica, 39:1543–1554, 2003.

164. D. Liberzon. Switching in Systems and Control. Birkhauser, Boston, 2003.165. D. Liberzon. Quantization, time delays, and nonlinear stabilization. IEEE

Transactions on Automatic Control, 51:1190–1195, 2006.166. D. Liberzon. Observer-based quantized output feedback control of nonlinear

systems. In Proceedings of the 17th IFAC World Congress, pages 8039–8043,2008.

167. D. Liberzon and D. Nesic. Stability analysis of hybrid systems via small-gain theorems. In J. P. Hespanha and A. Tiwari, editors, Proceedings of the

9th International Workshop on Hybrid Systems: Computation and Control,

Lecture Notes in Computer Science, pages 421–435, Berlin, 2006. Springer.168. D. Liberzon and D. Nesic. Input-to-state stabilization of linear systems with

quantized state measurements. IEEE Transactions on Automatic Control,

314 References

52:767–781, 2007.169. D. Liberzon, D. Nesic, and A. R. Teel. Lyapunov-based small-gain theorems

for hybrid systems. Submitted to IEEE Transactions on Automatic Control.170. Y. Lin, E. D. Sontag, and Y. Wang. A smooth converse Lyapunov theorem

for robust stability. SIAM Journal on Control and Optimization, 34:124–160,1996.

171. Z. Lin, B. Francis, and M. Maggiore. Necessary and sufficient graphical con-ditions for formation control of unicycles. IEEE Transactions on Automatic

Control, 50:121–127, 2005.172. Z. Lin, B. Francis, and M. Maggiore. State agreement for continuous-time cou-

pled nonlinear systems. SIAM Journal on Control and Optimization, 46:288–307, 2007.

173. B. Liu and D. J. Hill. Stability via hybrid-event-time Lyapunov functionand impulsive stabilization for discrete-time delayed switched systems. SIAM

Journal on Control and Optimization, 2013. In press.174. B. Liu, D. J. Hill, and K. L. Teo. Input-to-state stability for a class of hybrid

dynamical systems via hybrid time approach. In Proceedings of the 48th IEEE

Conference on Decision and Control and 28th Chinese Control Conference,pages 3926–3931, 2009.

175. J. Liu and N. Elia. Quantized feedback stabilization of non-linear affine sys-tems. International Journal of Control, 77:239–249, 2004.

176. S. Liu, L. Xie, and H. Zhang. Distributed consensus for multi-agent systemswith delays and noises in transmission channels. Automatica, 47:920–934,2011.

177. T. Liu. Input-to-State Stability Methods for Nonlinear and Network Systems.PhD thesis, The Australian National University, 2011.

178. T. Liu, D. J. Hill, and Z. P. Jiang. Lyapunov formulation of ISS cyclic-small-gain in continuous-time dynamical networks. Automatica, 47:2088–2093, 2011.

179. T. Liu, D. J. Hill, and Z. P. Jiang. Lyapunov formulation of the large-scale,ISS cyclic-small-gain theorem: The discrete-time case. Systems & Control

Letters, 61:266–272, 2012.180. T. Liu and Z. P. Jiang. Distributed nonlinear control of mobile autonomous

multi-agents. Submitted to Automatica.181. T. Liu and Z. P. Jiang. Event-triggered control of nonlinear systems with

partial state feedback. Submitted to Automatica.182. T. Liu and Z. P. Jiang. A small-gain approach to robust event-triggered

control of nonlinear systems. Submitted to IEEE Transactions on Automatic

Control.183. T. Liu and Z. P. Jiang. Distributed control of nonlinear uncertain systems:

A cyclic-small-gain approach. Acta Automatica Sinica, 2013. In press.184. T. Liu and Z. P. Jiang. Distributed formation control of nonholonomic mobile

robots without global position measurements. Automatica, 49:592–600, 2013.185. T. Liu and Z. P. Jiang. Distributed output-feedback control of nonlinear

multi-agent systems. IEEE Transactions on Automatic Control, 58:2912–2917, 2013.

186. T. Liu, Z. P. Jiang, and D. J. Hill. Robust control of nonlinear strict-feedbacksystems with measurement errors. In Proceedings of the 50th IEEE Conference

on Decision and Control and European Control Conference, pages 2034–2039,

References 315

2011.187. T. Liu, Z. P. Jiang, and D. J. Hill. Decentralized output-feedback control of

large-scale nonlinear systems with sensor noise. Automatica, 48:2560–2568,2012.

188. T. Liu, Z. P. Jiang, and D. J. Hill. Lyapunov formulation of the ISS cyclic-small-gain theorem for hybrid dynamical networks. Nonlinear Analysis: Hy-

brid Systems, 6:988–1001, 2012.189. T. Liu, Z. P. Jiang, and D. J. Hill. Quantized stabilization of strict-feedback

nonlinear systems based on ISS cyclic-small-gain theorem. Mathematics of

Control, Signals, and Systems, 24:75–110, 2012.190. T. Liu, Z. P. Jiang, and D. J. Hill. A sector bound approach to feedback

control of nonlinear systems with state quantization. Automatica, 48:145–152, 2012.

191. T. Liu, Z. P. Jiang, and D. J. Hill. Small-gain based output-feedback controllerdesign for a class of nonlinear systems with actuator dynamic quantization.IEEE Transactions on Automatic Control, 57:1326–1332, 2012.

192. Z. Liu and L. Guo. Synchronization of multi-agent systems without connec-tivity assumptions. Automatica, 45:2744–2753, 2009.

193. W. Lohmiller and J.-J. E. Slotine. On contraction analysis for non-linearsystems. Automatica, 34:683–696, 1998.

194. Q. Lu, Y. Sun, and S. Mei. Nonlinear Control Systems and Power System

Dynamics. Kluwer Academic, 2001.195. A. M. Lyapunov. The General Problem of the Stability of Motion. Taylor &

Francis, London, 1992.196. M. Maggiore and K. Passino. Output feedback control for stabilizable and

incompletely observable nonlinear systems: Jet engine stall and surge control.In Proceedings of the 2000 American Control Conference, pages 3626–3630,2000.

197. M. Malisoff and F. Mazenc. Constructions of Strict Lyapunov Functions.Springer-Verlag, London, 2009.

198. I. M. Y. Mareels and R. R. Bitmead. Non-linear dynamics in adaptive control:Chaotic and periodic stabilization. Automatica, 22:641–655, 1986.

199. I. M. Y. Mareels and D. J. Hill. Monotone stability of nonlinear feedbacksystems. Journal of Mathematical Systems, Estimation, and Control, 2:275–291, 1992.

200. S. J. Mason. Feedback theory–Some properties of signal flow graphs. Pro-

ceedings of the IRE, 41:1144–1156, 1953.201. S. J. Mason. Feedback theory–Further properties of signal flow graphs. Pro-

ceedings of the IRE, 44:920–926, 1956.202. F. Mazenc and Z. P. Jiang. Global output feedback stabilization of a chemo-

stat with an arbitrary number of species. IEEE Transactions on Automatic

Control, 55:2570–2575, 2010.203. F. Mazenc, M. Malisoff, and J. Harmand. Further results on stabilization of

periodic trajectories for a chemostat with two species. IEEE Transactions on

Automatic Control, 53:66–74, 2008.204. M. Mesbahi and M. Egerstedt. Graph Theoretic Methods in Multiagent Net-

works. Princeton University Press, New Jersey, 2010.205. A. N. Michel and R. K. Miller. Qualitative Analysis of Large Scale Dynamical

316 References

Systems. Academic Press, New York, 1977.206. F. K. Moore and E. M. Greitzer. A theory of post-stall transients in axial com-

pression systems-Part I: Development of equations. Journal of Engineering

for Gas Turbines and Power, 108:68–76, 1986.207. P. J. Moylan and D. J. Hill. Stability criteria for large-scale systems. IEEE

Transactions on Automatic Control, 23:143–149, 1978.208. D. Nesic and D. Liberzon. A small-gain approach to stability analysis of

hybrid systems. In Proceedings of the 44th IEEE Conference on Decision and

Control, pages 5409–5414, 2005.209. D. Nesic and D. Liberzon. A unified framework for design and analysis of

networked and quantized control systems. IEEE Transactions on Automatic

Control, 54:732–747, 2009.210. D. Nesic and A. R. Teel. Changing supply functions in input to state stable

systems: The discrete-time case. IEEE Transactions on Automatic Control,46:960–962, 2001.

211. D. Nesic and A. R. Teel. A Lyapunov-based small-gain theorem for hybridISS systems. In Proceedings of the 47th IEEE Conference on Decision and

Control, pages 3380–3385, 2008.212. P. Ogren, M. Egerstedt, and X. Hu. A control Lyapunov function approach

to multiagent coordination. IEEE Transactions on Robotics and Automation,18:847–851, 2002.

213. P. Ogren, E. Fiorelli, and N. Leonard. Cooperative control of mobile sen-sor networks: Adaptive gradient climbing in a distributed network. IEEE

Transactions on Automatic Control, 49:1292–1302, 2004.214. R. Olfati-Saber. Flocking for multi-agent dynamic systems: Algorithms and

theory. IEEE Transactions on Automatic Control, 51:401–420, 2006.215. R. Olfati-Saber and R. M. Murray. Consensus problems in networks of agents

with switching topology and time-delays. IEEE Transactions on Automatic

Control, 49:1520–1533, 2004.216. R. Pates and G. Vinnicombe. Stability certificates for networks of heteroge-

neous linear systems. In Proceedings of the 51st IEEE Conference on Decision

and Control, pages 6915–6920, 2012.217. P. Pepe. On the asymptotic stability of coupled delay differential and contin-

uous time difference equations. Automatica, 41:107–112, 2005.218. P. Pepe and H. Ito. On saturation, discontinuities, and delays, in iISS and

ISS feedback control redesign. IEEE Transactions on Automatic Control,57:1125–1140, 2012.

219. P. Pepe, I. Karafyllis, and Z. P. Jiang. On the Liapunov-Krasovskii method-ology for the ISS of systems described by coupled delay differential and dif-ference equations. Automatica, 44:2266–2273, 2008.

220. R. Postoyan and D. Nesic. Trajectory based small gain theorems for parame-terized systems. In Proceedings of 2010 American Control Conference, pages184–189, 2010.

221. H. G. Potrykus, F. Allgower, and S. J. Qin. The character of an idempotent-analytic nonlinear small gain theorem. In L. Benvenuti, A. De Santis, andL. Farina, editors, Positive Systems, volume 294 of Lecture Notes in Control

and Information Science, pages 361–368, Berlin, 2003. Springer.222. L. Praly and Z. P. Jiang. Stabilization by output feedback for systems with

References 317

ISS inverse dynamics. Systems & Control Letters, 21:19–33, 1993.223. L. Praly and Y. Wang. Stabilization in spite of matched unmodeled dynam-

ics and an equivalent definition of input-to-state stability. Mathematics of

Control, Signals and Systems, 9:1–33, 1996.224. Z. Qu. Cooperative Control of Dynamical Systems: Applications to Au-

tonomous Vehicles. Springer-Verlag, 2009.225. Z. Qu, J. Wang, and R. A. Hull. Cooperative control of dynamical systems

with application to autonomous vehicles. IEEE Transactions on Automatic

Control, 53:894–911, 2008.226. A. Rantzer. Distributed performance analysis of heterogeneous systems. In

Proceedings of the 49th IEEE Conference on Decision and Control, pages2682–2685, 2010.

227. W. Ren. On consensus algorithms for double-integrator dynamics. IEEE

Transactions on Automatic Control, 53:1503–1509, 2008.228. W. Ren and R. Beard. Distributed Consensus in Multi-vehicle Cooperative

Control. Springer-Verlag, London, 2008.229. W. Ren, R. Beard, and E. Atkins. Information consensus in multivehicle

cooperative control. IEEE Control Systems Magazine, 27:71–82, 2007.230. R. Rifford. Existence of Lipschitz and semiconcave control-Lyapunov func-

tions. SIAM Journal on Control and Optimization, 39:1043–1064, 2000.231. B. S. Ruffer. Monotone Systems, Graphs, and Stability of Large-Scale Inter-

connected Systems. PhD thesis, University of Bremen, Germany, 2007.232. I. W. Sandberg. On the L2-boundedness of solutions of nonlinear functional

equations. Bell System Technical Journal, 43:1581–1599, 1964.233. I. W. Sandberg. An observation concerning the application of the contraction

mapping fixed-point theorem and a result concerning the norm-boundednessof solutions of nonlinear functional equations. Bell System Technical Journal,44:1809–1812, 1965.

234. A. V. Savkin and R. J. Evans. Hybrid Dynamical Systems: Controller and

Sensor Switching Problems. Birkhauser, Boston, 2002.235. R. Sepulchre, M. Jankovic, and P. V. Kokotovic. Constructive Nonlinear

Control. Springer-Verlag, New York, 1997.236. G. Shi and Y. Hong. Global target aggregation and state agreement of nonlin-

ear multi-agent systems with switching topologies. Automatica, 45:1165–1175,2009.

237. L. Shi and S. K. Singh. Decentralized control for interconnected uncertain sys-tems: Extensions to high-order uncertainties. International Journal of Con-

trol, 57:1453–1468, 1993.238. H. Shim and A. R. Teel. Asymptotic controllability and observability im-

ply semiglobal practical asymptotic stabilizability by sampled-data outputfeedback. Automatica, 39:441–454, 2003.

239. D. D. Siljak. Decentralized Control of Complex Systems. Academic Press,Boston, 1991.

240. D. D. Siljak. Dynamic graphs. Nonlinear Analysis: Hybrid Systems, 2:544–567, 2008.

241. E. D. Sontag. Smooth stabilization implies coprime factorization. IEEE

Transactions on Automatic Control, 34:435–443, 1989.242. E. D. Sontag. Further facts about input to state stabilization. IEEE Trans-

318 References

actions on Automatic Control, 35:473–476, 1990.243. E. D. Sontag. Comments on integral variants of ISS. Systems & Control

Letters, 34:93–100, 1998.244. E. D. Sontag. Mathematical Control Theory: Deterministic Finite-

Dimensional Systems. Springer, New York, second edition, 1998.245. E. D. Sontag. Clocks and insensitivity to small measurement errors. ESAIM:

Control, Optimisation and Calculus of Variations, 4:537–557, 1999.246. E. D. Sontag. Asymptotic amplitudes and Cauchy gains: A small-gain prin-

ciple and an application to inhibitory biological feedback. Systems & Control

Letters, 47:167–179, 2002.247. E. D. Sontag. Input to state stability: Basic concepts and results. In P. Nistri

and G. Stefani, editors, Nonlinear and Optimal Control Theory, pages 163–220, Berlin, 2007. Springer-Verlag.

248. E. D. Sontag and A. R. Teel. Changing supply functions in input/state stablesystems. IEEE Transactions on Automatic Control, 40:1476–1478, 1995.

249. E. D. Sontag and Y. Wang. On characterizations of the input-to-state stabilityproperty. Systems & Control Letters, 24:351–359, 1995.

250. E. D. Sontag and Y. Wang. New characterizations of input-to-state stability.IEEE Transactions on Automatic Control, 41:1283–1294, 1996.

251. E. D. Sontag and Y. Wang. Notions of input to output stability. Systems &

Control Letters, 38:351–359, 1999.252. E. D. Sontag and Y. Wang. Lyapunov characterization of input to output

stability. SIAM Journal on Control and Optimization, 39:226–249, 2001.253. H. Su, X. Wang, and Z. Lin. Flocking of multi-agents with a virtual leader.

IEEE Transactions on Automatic Control, 54:293–307, 2009.254. Y. Su and J. Huang. Cooperative output regulation of linear multi-agent

systems. IEEE Transactions on Automatic Control, 57:1062–1066, 2012.255. P. Tabuada. Event-triggered real-time scheduling of stabilizing control tasks.

IEEE Transactions on Automatic Control, 52:1680–1685, 2007.256. H. Tanner, A. Jadbabaie, and G. Pappas. Stable flocking of mobile agents,

Part I: Fixed topology. In Proceedings of the 42nd IEEE Conference on De-

cision and Control, pages 2010–2015, 2003.257. H. Tanner, A. Jadbabaie, and G. Pappas. Stable flocking of mobile agents,

Part II: Dynamic topology. In Proceedings of the 42nd IEEE Conference on

Decision and Control, pages 2016–2021, 2003.258. H. G. Tanner, G. J. Pappas, and V. Kummar. Leader-to-formation stability.

IEEE Transactions on Robotics and Automation, 20:443–455, 2004.259. A. R. Teel. Input-to-state stability and the nonlinear small-gain theorem.

Private communications.260. A. R. Teel. A nonlinear small gain theorem for the analysis of control systems

with saturation. IEEE Transactions on Automatic Control, 41:1256–1270,1996.

261. A. R. Teel. Connections between Razumikhin-type theorems and the ISSnonlinear small gain theorem. IEEE Transactions on Automatic Control,43:960–964, 1998.

262. A. R. Teel and L. Praly. Tools for semiglobal stabilization by partial state andoutput feedback. SIAM Journal on Control and Optimization, 33:1443–1488,1995.

References 319

263. A. R. Teel and L. Praly. On assigning the derivative of a disturbance at-tenuation control Lyapunov function. Mathematics of Control, Signals, and

Systems, 13:95–124, 2000.264. A. R. Teel and L. Praly. A smooth Lyapunov function from a class-KL

estimate involving two positive functions. ESAIM Control Optimisation and

Calculus of Variations, 5:313–367, 2000.265. S. Tiwari, Y. Wang, and Z. P. Jiang. Nonlinear small-gain theorems for large-

scale time-delay systems. Dynamics of Continuous, Discrete and Impulsive

Systems Series A: Mathematical Analysis, 19:27–63, 2012.266. J. Tsinias. Sufficient Lyapunov-like conditions for stabilization. Mathematics

of Control, Signals, and Systems, 2:343–357, 1989.267. J. Tsinias and N. Kalouptsidis. Output feedback stabilization. IEEE Trans-

actions on Automatic Control, 35:951–954, 1990.268. A. J. van der Schaft. L2-Gain and Passivity Techniques in Nonlinear Control.

Lecture Notes in Control and Information Sciences. Springer-Verlag, London,1996.

269. V. I. Vorotnikov. Partial Stability and Control. Birkhauser, Boston, 1998.270. C. Wang, S. S. Ge, D. J. Hill, and G. Chen. An ISS-modular approach for

adaptive neural control of pure-feedback systems. Automatica, 42:723–731,2006.

271. X. Wang, Y. Hong, J. Huang, and Z. P. Jiang. A distributed control ap-proach to a robust output regulation problem for multi-agent systems. IEEETransactions on Automatic Control, 55:2891–2895, 2010.

272. X. Wang and M. D. Lemmon. On event design in event-triggered feedbacksystems. Automatica, 47:2319–2322, 2011.

273. C. Wen and J. Zhou. Decentralized adaptive stabilization in the presence ofunknown backlash-like hysteresis. Automatica, 43:426–440, 2007.

274. C. Wen, J. Zhou, and W. Wang. Decentralized adaptive backstepping stabi-lization of interconnected systems with dynamic input and output interaction.Automatica, 45:55–67, 2009.

275. P. Wieland, R. Sepulchre, and F. Allgower. An internal model principleis necessary and sufficient for linear output synchronization. Automatica,47:1068–1074, 2011.

276. J. C. Willems. The generation of Lyapunov functions for input-output stablesystems. SIAM Journal on Control, 9:105–134, 1971.

277. J. C. Willems. Dissipative dynamical systems, Part I: General theory; Part II:Linear systems with quadratic supply rates. Archive for Rationale Mechanics

Analysis, 45:321–393, 1972.278. J. C. Willems. The behavioral approach to open and interconnected systems.

IEEE Control Systems Magazine, 27:46–99, 2007.279. S. Xie and L. Xie. Decentralized global robust stabilization of a class of large-

scale interconnected minimum-phase nonlinear systems. In Proceedings of the

37th IEEE Conference on Decision and Control, pages 1482–1487, 1998.280. H. Xin, Z. Qu, J. Seuss, and A. Maknouninejad. A self organizing strategy

for power flow control of photovoltaic generators in a distribution network.IEEE Transactions on Power Systems, 26:1462–1473, 2011.

281. V. A. Yakubovich, G. A. Leonov, and A. Kh. Gelig. Stability of Stationary

Sets in Control Systems with Discontinuous Nonlinearities. World Scientific

320 References

Press, 2004.282. C. Yu, B. D. O. Anderson, S. Dasgupta, and B. Fidan. Control of minimally

persistent formations in the plane. SIAM Journal on Control and Optimiza-

tion, 48:206–233, 2009.283. G. Zames. On the input-output stability of time-varying nonlinear feedback

systems–Part I: Conditions derived using concepts of loop gain, conicity, andpositivity. IEEE Transactions on Automatic Control, 11:228–238, 1966.

284. G. Zames. On the input-output stability of time-varying nonlinear feedbacksystems–Part II: Conditions involving circles in the frequency plane and sectornonlinearities. IEEE Transactions on Automatic Control, 11:465–476, 1966.

285. A. Zecevic and D. D. Siljak. Control design with arbitrary information struc-ture constraints. Automatica, 44:2642–2647, 2008.

286. A. Zecevic and D. D. Siljak. Control of Complex Systems: Structural Con-

straints and Uncertainty. Springer, 2010.287. J. Zhao, D. J. Hill, and T. Liu. Synchronization of dynamical networks with

nonidentical nodes: Criteria and control. IEEE Transactions on Circuits and

Systems–I: Regular Papers, 58:584–594, 2011.288. K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control. Prentice

Hall, 1995.

Index

ǫ modification, 125

Adaptive control, 17, 257Distributed, 258Robust, see Robust adaptive

controlAG, 9, 12, 21, 25Asymptotic gain, see AG

Output, 15Axial compressor, 39

Comparison functionsClass KL function, 5Class K function, 5Class K∞ function, 5Positive definite function, 5

Control under sensor noise, 79Dynamic state measurement feed-

back, 93Static state measurement feed-

back, 80Coordinate transformation, 5, 32Cyclic-small-gain

In digraphs, 196

Decentralized control, 101, 140, 252Output measurement feedback,

101Digraph, 18, 196, 261

Path, 261Reachability, 261Reaching set, 261Simple cycle, 261Spanning tree, 261Tree, 261

Discontinuous system, 52, 264Dissipativity, 15, 76, 258Distributed control, 4, 18, 193

Formation, 207, 224Output-feedback, 198

Dynamic feedback linearization, 207Dynamic network, 1

Continuous-time, 42Discontinuous, 52, 266Discrete-time, 54Hybrid, 63With interconnection time de-

lays, 54Dynamic uncertainty, 32, 83, 92, 106,

145, 160

Equilibrium, 5, 200Event-triggered control, 116

Filippov solution, 52, 264Extended, 52, 264

Forward completeness, 12, 25, 275

Gain assignment, 30, 33, 93, 140, 146,150, 151, 166, 182

Gain digraph, 40, 43, 90, 97, 110, 170,186, 195

Graph, 261Directed, see Digraph

I/O stability, 16iISS, 36, 76Infinitely fast sampling, 119Information exchange digraph, 195,

198Input-output stability, see I/O stabil-

ityInput-to-output stability, see IOSInput-to-state stability, see ISSInput-to-state stabilization, 16, 80,

93, 101Integral input-to-state stability, see

iISSInterconnected system, 3, 4, 19

Large-scale, see Dynamic net-work

Invariant set, 165, 219Nested, 165

IOS, 15, 201, 211

321

322 INDEX

IOS gain, 15ISS, 8, 265

Local, 11Lyapunov formulation, see ISS-

Lyapunov functionISS cyclic-small-gain theorem

Continuous-time, 42Control design, 90, 97, 110, 128,

136, 137, 151, 170, 186, 201,219

Discrete-time, 54Hybrid, 63

ISS gain, 8Lyapunov-based, 13, 55

ISS small-gain theoremControl design, 30Cyclic, see ISS cyclic-small-gain

theoremLyapunov-based, 26Trajectory-based, 21

ISS-Lyapunov function, 13, 266Construction, 27, 44Dissipation form, 13, 43, 55Gain margin form, 13, 26, 42, 57Supply functions, 15

Linear time-invariant system, 9, 83Lipschitz continuity, 5

Local, 5On compact sets, 5

Local essential boundedness, 8Logarithmic quantizer, 144Lyapunov function, 7

Strict, 7

Multi-agent system, 193, 198, 207,224

Nominal system, 2Nonholonomic mobile robot, 208, 243

ObserverDecentralized, 104Distributed, 200ISS-induced, 104, 138, 181, 200

Nonlinear, 200Quantized, 181Reduced-order, 104, 138, 181

Output agreement, 198Strong, 225

Output-feedback form, 102, 138, 180,198

Passivity, 15, 257Perturbation, 2, 14, 17Power system, 259

QuantizationDynamic, 157Static, 144

Quantized control, 143Output-feedback, 180

Quantizer, 143Finite-level, 157Uniform, 157

Recursive design, 3, 86, 94, 106, 132,137, 148, 165, 182, 200

Robust adaptive control, 17, 39, 137Robust control, 2, 16

Nonlinear, see Robust nonlinearcontrol

Robust nonlinear control, 79, 143, 193Robust stability, 14

Weakly, 14Robustness, 14, 101, 118, 221, 259

To sensor noise, 79To time delays, 203

Sector bound property, 2, 144Self-triggered control, 123, 125Sensing digraph, 218Set-valued map based design, 80, 107,

129, 133, 148, 165, 182Small-gain theorem, 3

Classical, 3, 255Cyclic, see ISS cyclic-small-gain

theoremISS, see ISS small-gain theoremNonlinear, 21, 36, see also ISS

small-gain theorem

INDEX 323

StabilityAsymptotic, 6Global, 6Global asymptotic, 6Lyapunov, 4Practical, 2, 35

Stabilization, 4Input-to-state, see Input-to-state

stabilizationQuantized, 157, 180, 186

Strict-feedback form, 3, 85, 93, 129,132, 144, 160

Superposition Principle, 2Synchronization under sensor noise,

132

Time delay, 54Of information exchange, 203

UBIBS, 12, 23, 25, 53, 281Unboundedness observability, see UOUniform bounded-input bounded-state,

see UBIBSUO, 16

With zero offset, 16, 201, 211

Zeno behavior, 119Zooming

Variable, 163, 180Zooming-in, 160, 163, 164, 176,

188Zooming-out, 160, 163, 164, 175,

188

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