anal - massachusetts institute of technologyhohmann.mit.edu/papers/lim_phd.pdf · anal ysis and...

ANALYSIS AND CONTROL OF LINEAR

PARAMETER-VARYING SYSTEMS

a dissertation

submitted to the department of aeronautics & astronautics

and the committee on graduate studies

of stanford university

in partial fulfillment of the requirements

for the degree of

doctor of philosophy

By

Sungyung Lim

September 1998

Copyright c 1998 by Sungyung Lim

All Rights Reserved.

ii

I certify that I have read this dissertation and that in my opinion it is fully adequate,

in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

Jonathan P. HowDepartment of Aeronautics and Astronautics

(Principal Adviser)



Stephen P. Boyd

Department of Electrical Engineering



Stephen M. Rock

Department of Aeronautics and Astronautics



Andrew PackardDepartment of Mechanical Engineering

University of California, Berkeley

Approved for the University Committee on Graduate Studies:

iii

To my wife, Yunhee and my daughter, Ellen

iv

Abstract

The area of analysis and control of linear parameter-varying (LPV) systems has received

much recent attention because of its importance in developing systematic techniques for

gain-scheduling. An LPV system resembles a linear system that nonlinearly depends on

one or more time-varying parameters. Nonlinear systems are often modeled in the LPV

system via the parameterized Jacobian linearization.

Typical approaches for the analysis and control of LPV systems are the scaled small-

gain approach and the dissipative systems framework using smooth parameter-dependent

Lyapunov functions (PDLFs). The dissipative systems framework is the more desirable of

the two techniques because it can directly treat time-varying parameters and yield an LPV-

type controller. Furthermore, the dissipative systems framework attractively formulates

analysis and synthesis problems as convex optimization problems involving linear matrix

inequalities (LMIs), which are now very e�ciently solved by computer. However, the current

dissipative systems framework has two major potential drawbacks: (1) di�culty in selecting

an optimal PDLF in order to reduce conservatism of the dissipative systems approach; (2)

di�culty in solving exactly convex optimization problems involving an in�nite number of

LMIs.

The thesis presents new analysis and control design techniques to avoid these poten-

tial drawbacks of the smooth dissipative systems framework. The thesis focuses on a

piecewise-a�ne parameter-dependent linear parameter-varying (PALPV) system which is

a new class of LPV systems. Associated with the PALPV system is a piecewise-a�ne

parameter-dependent Lyapunov function (PAL). To address the non-di�erential nature of

both the PALPV system and the PAL, the thesis develops a nonsmooth dissipative systems

framework. Then, the thesis fully characterizes several interesting analysis and synthe-

sis problems, such as L2-gain, L1-gain, H2-norm, passivity, and robust counterparts, ofPALPV system with the developed nonsmooth dissipative systems framework.

v

The new approach is shown to yield a less conservative, reliable result than previously

published LPV approaches. The improvement is a direct result of (1) using a more accurate

model in the analysis and control, and (2) using a very general class of PDLFs. The

derived analysis and synthesis formulations are also �nite-dimensional convex optimization

problems which can be solved extremely e�ciently by computer. Furthermore, the new

approach enables us a trade-o� between conservatism and computational e�ort of the design

technique.

Several benchmark problems including a missile autopilot design problem are used to

demonstrate the usefulness, reliability, and feasibility of the proposed new approach.

vi

Acknowledgments

Thanks to all.

vii

Contents

Abstract v

Acknowledgments vii

List of Tables xi

List of Figures xii

List of Symbols xiv

List of Acronyms xvi

1 Introduction 1

1.1 Previous Research : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3

1.1.1 Linearization Methods : : : : : : : : : : : : : : : : : : : : : : : : : 3

1.1.2 Analysis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4

1.1.3 Synthesis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7

1.2 Thesis Objectives and Overview : : : : : : : : : : : : : : : : : : : : : : : : 10

2 Mathematical Preliminary 13

2.1 Norms and Normed Spaces of Signals : : : : : : : : : : : : : : : : : : : : : 13

2.2 Stability : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14

2.2.1 Lyapunov Stability : : : : : : : : : : : : : : : : : : : : : : : : : : : 15

2.2.2 Input-Output Stability : : : : : : : : : : : : : : : : : : : : : : : : : 16

2.3 Small-Gain Theorem and Passivity Theorem : : : : : : : : : : : : : : : : : 17

2.4 Lebesgue Integral Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : 18

2.5 Ordinary Di�erential Equations : : : : : : : : : : : : : : : : : : : : : : : : 21

viii

2.6 Linear Matrix Inequality : : : : : : : : : : : : : : : : : : : : : : : : : : : : 24

2.6.1 Linear Matrix Inequality Problems : : : : : : : : : : : : : : : : : : 24

2.6.2 Numerical Algorithms for LMI Problems : : : : : : : : : : : : : : : 25

2.6.3 Miscellaneous Results on LMIs : : : : : : : : : : : : : : : : : : : : : 28

2.7 \Convexifying" Techniques : : : : : : : : : : : : : : : : : : : : : : : : : : : 29

2.8 Bilinear Matrix Inequality : : : : : : : : : : : : : : : : : : : : : : : : : : : 32

3 Dissipative Systems Framework 34

3.1 De�nition of Dissipative Systems : : : : : : : : : : : : : : : : : : : : : : : : 34

3.2 \Dini-Di�erential" Dissipation Inequality : : : : : : : : : : : : : : : : : : : 37

3.3 \Dissipation" implies : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 38


3.3.2 L2-Gain : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 413.3.3 L1-Gain : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 433.3.4 H2-Norm : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 453.3.5 Passivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 46

3.4 Dissipation of Feedback Systems : : : : : : : : : : : : : : : : : : : : : : : : 48

3.5 System with Structured Dynamics Uncertainties : : : : : : : : : : : : : : : 50

4 Analysis 56

4.1 PALPV System : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 56

4.2 PAL : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 58

4.3 Analysis Formulations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 60



4.3.6 Robust L2-Gain and Others : : : : : : : : : : : : : : : : : : : : : : 714.4 Generalized Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 75

4.4.1 PALPV system : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 75

4.4.2 PAL with s � 3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 774.4.3 Analysis formulations : : : : : : : : : : : : : : : : : : : : : : : : : : 79

ix

5 Synthesis 81

5.1 QPALPV System : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 82

5.1.1 PALPV System : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 82

5.1.2 LPV Controller : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 83

5.2 QPAL : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 85

5.3 Synthesis Formulations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 87


5.3.5 Robust L2-Gain : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1205.4 Generalized Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 133

5.4.1 QPALPV system : : : : : : : : : : : : : : : : : : : : : : : : : : : : 133

5.4.2 QPAL with s � 3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1345.4.3 Synthesis formulations : : : : : : : : : : : : : : : : : : : : : : : : : 135

6 Numerical Studies 138

6.1 Stability Margin and L2-Gain Problems : : : : : : : : : : : : : : : : : : : : 1386.2 L2-Gain Synthesis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1436.3 Autopilot Design : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 149

6.3.1 Missile Model and Performance Objective : : : : : : : : : : : : : : : 149

6.3.2 PALPV Modeling : : : : : : : : : : : : : : : : : : : : : : : : : : : : 152

6.3.3 Comparison of LPV control techniques : : : : : : : : : : : : : : : : 155

6.3.4 Autopilot Design and Simulations : : : : : : : : : : : : : : : : : : : 158

7 Conclusions 168

7.1 Summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 168

7.2 Conclusions and Contributions : : : : : : : : : : : : : : : : : : : : : : : : : 170

7.3 Recommendations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 171

Bibliography 173

x

List of Tables

6.1 Relative RMS of Ek(M;�) in (%) : : : : : : : : : : : : : : : : : : : : : : : 153

6.2 Features of LPV control techniques : : : : : : : : : : : : : : : : : : : : : : 156

6.3 Maximum eigenvalues from the synthesis and the post-analysis : : : : : : : 157

6.4 Characteristics of design techniques and their results. Note that the NGS

technique does not provide any guaranteed L2-gain. : : : : : : : : : : : : 159

xi

List of Figures

2.1 Feedback system : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17

3.1 Feedback system : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 48

3.2 Dynamic system with two uncertainty blocks in a feedback loop : : : : : : 50

4.1 Partitioned parameter subspaces (s = 2) with m1 = m2 = 2 : : : : : : : : 57

4.2 Example of a continuous, piecewise-a�ne P (�) with s = 2. For simplicity,

the parameter space is only split into 2� 2 regions. : : : : : : : : : : : : : 595.1 Example of a continuous, piecewise-a�ne X(�) with s = 2. Y (�) is also

similarly de�ned with Yij 's. For simplicity, each parameter space is only

split into 2 regions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 84

6.1 �max vs. _�max from the gridding technique, multi-convexity, S-procedure,and our PAL with a number of di�erent partitions (N). The multi-convexity

approach is equivalent to our PAL with N = 1. : : : : : : : : : : : : : : : 140

6.2 P (�) from our PAL with N = 5 at di�erent values of _�max's. : : : : : : : : 141

6.3 L2-gain vs. _�max from the gridding technique, multi-convexity, S-procedureand our PAL with a number of di�erent partitions (N). The multi-convexity

approach is equivalent to our PAL with N = 1. : : : : : : : : : : : : : : 143

6.4 Block diagram of a benchmark problem : : : : : : : : : : : : : : : : : : : 144

6.5 L2-gain vs. _�max from the gridding technique, S-procedure, multi-convexityapproach and our QPAL. The multi-convexity approach is equivalent to our

QPAL with N = 1. PCS means the Popov controller for _�max = 0. The dot-

line indicates the minimum L2-gain that can be obtained by the pointwiseH1 control for _�max = 0. : : : : : : : : : : : : : : : : : : : : : : : : : : : 145

6.6 L2-gain vs. _�max from the same synthesis techniques as in Fig. 6.5 exceptthat Y (�) is constrained to be constant. All the labels are same as Fig. 6.5. 146

xii

6.7 H1-norm vs. � of the closed-loop system for _�max = 0, the controller of whichis designed by pointwise-H1, PCS, and QPAL approach with N = f1; 2; 3g,respectively. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 148

6.8 Weighted open-loop interconnection of the missile plant : : : : : : : : : : 151

6.9 f2(M;�) and its approximation error (E2(M;�)) with a number of di�erent

partitions (N = 1; 3; 5) : : : : : : : : : : : : : : : : : : : : : : : : : : : : 154

6.10 L2-gain v.s. N (number of partitions) from GRID, GRID1, QPAL andQPAL1. Black 2 indicates the result is veri�ed by the post-analysis, while

white � means the result is not. : : : : : : : : : : : : : : : : : : : : : : : : 1576.11 Mach number pro�le for Case I : : : : : : : : : : : : : : : : : : : : : : : : 163

6.12 Normal acceleration �(t) from NGS, C-�, GRID, and QPAL for Case I : 163

6.13 Mach number pro�le for Case II without noise : : : : : : : : : : : : : : : 164

6.14 Normal acceleration �(t) from C-�, GRID, and QPAL for Case II without

noise : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 164

6.15 Mach number pro�le for Case II with and without noise : : : : : : : : : : 165

6.16 Normal acceleration �(t) from the QPAL approach for Case II with and

without noise : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 165

6.17 Response of tail deection rate ( _�(t)) from the QPAL approach for Case II

with v(t) = 0 and v(t) 2 [�0:05; 0:05] : : : : : : : : : : : : : : : : : : : : : 1666.18 Response of tail deection rate ( _�(t)) from the QPAL approach for Case II

with v(t) 2 [�0:15; 0:15] and v(t) 2 [�0:25; 0:25] : : : : : : : : : : : : : : 1666.19 Angle-of-attack (�) and its estimated value (~�) from the QPAL approach

for Case II with v(t) = 0 and v(t) 2 [�0:25; 0:25] : : : : : : : : : : : : : : 167

xiii

List of Symbols

R set of real numbers

R+ set of nonnegative numbers

Rn set of real n-vectors

I [t0;1)D set of statesDcl set of closed-loop system statesW set of performance inputsU set of control inputsZ set of performance outputsY set of outputsFs

n� 2 C1(I;Rs) : �(t) 2 P; _�(t) 2 ; 8t 2 I

o; where

P = [�1; �1]� � � � � [�s; �s] and = [��1; �1]� � � � � [��s; �s]C1 class of continuously di�erentiable functions

K class of continuous, strictly increasing functions with f(0) = 0

kxk"

nXi=1

jxij2#1=2

and the corresponding normed space l2

kxkp�Z

1

t0

kx(t)kpdt�1=p

and the corresponding normed space Lph�; �i inner productB�(c) fx 2 D : kx� ck � �gI identity matrix

diag(X1; � � � ;Xn) diagni=1(Xi) i.e., block-diagonal matrix with X1; � � � ;XnX? orthogonal complement of X, i.e., X

TX? = 0 and

[X X?] is of maximum rank

xiv

TrX trace of X

XT transpose of a matrix X

X�1 inverse of a matrix X

X > 0 positive de�nite, i.e., X = XT and xTXx > 0 8x 2 RnX(�) > 0 positive de�nite for all � 2 PCo[X1; � � � ;Xr] convex hull of the set fX1; � � � ; Xrg� uncertainty block

S� set of uncertaintiesD

+V (x(t); �(t)) lim sup

h!0+

1

h

[V (x(t+ h); �(t+ h))� V (x(t); �(t))]

D+V (x; �)(u; 0) lim sup

h!0+

1

h

[V (x+ hu; �)� V (x; �)]

D+V (x; �)(u; 0) lim infh!0+

1

h

[V (x+ hu; �)� V (x; �)]Ex The expected value of the random variable x

l ii � � � is1(s) 1 � � � 1| {z }

s

with 1(0) = ;

[A(�)]ij Aij(�ij)

2 \belongs to"8 \for all"2 end of the proof

xv

List of Acronyms

ALF A�ne parameter-dependent Lyapunov Function

ALPV A�ne parameter-dependent Linear Parameter-Varying

BMI Bilinear Matrix Inequality

C-� Complex � synthesis

GRID Gridding approach in the dissipative systems framework

IQC Integral Quadratic Constraint

LDI Linear Di�erential Inclusion

LFT Linear Fraction Transformation

LMI Linear Matrix Inequality

LPV Linear Parameter-Varying

LTI Linear Time-Invariant

MIMO Multi-Input Multi-Output

NGS Naive Gain-Scheduling

PAL Piecewise A�ne parameter-dependent Lyapunov function

PALPV Piecewise A�ne parameter-dependent Linear Parameter-Varying

PDLF Parameter-Dependent Lyapunov Function

PJL Parameterized Jacobian Linearization

QAL Quasi-A�ne parameter-dependent Lyapunov function

QLPV Quasi Linear Parameter-Varying

QPAL Quasi-Piecewise-A�ne parameter-dependent Lyapunov function

QPALPV Quasi-Piecewise-A�ne parameter-dependent Linear Parameter-Varying

xvi

Chapter 1

Introduction

All physical systems are virtually nonlinear and time-varying in nature. Examples of non-

linearities we are often confronted with range from simple nonlinearity, such as saturation,

rate limiters, and backlash, to the inherently nonlinear behavior of physical systems such

as robotic manipulators, aircraft, and chemical process plants. Nevertheless, it is often

possible to describe the operation of the physical system by a linear system, under the cir-

cumstances that the real operation of the physical system does not deviate too much from

the \nominal" operating (or equilibrium) condition. Therefore, the analysis and synthesis

of linear systems have occupied an important place in systems theory. Consequently, many

(computationally tractable) analysis and synthesis techniques have been developed.

We often encounter situations where the linearized model around a nominal operating

condition is inadequate or inaccurate. For example, the missile dynamics involves a wide

range of variation in system dynamics over the operation range so that it can not be rep-

resented by a linearized model. In this case, a linear controller from linear systems theory

may perform well on the linearized model but may not even be stable when implemented

on the real physical system. To play with \inaccuracies," linear systems theory has been

extended to linear robust systems theory that takes into account these inherent inaccuracies

as uncertainties and then provides systematic analysis and design techniques in the face of

these \uncertainties." However, introducing a uncertainty in the design process leads to

degradation in the performance of a designed controller (Note that the performance degra-

dation is often alternatively expressed by conservatism in the literature). Therefore, one

1

CHAPTER 1. INTRODUCTION 2

of the important issues in linear robust systems theory is how to e�ciently play with un-

certainties, i.e., to reduce conservatism of analysis and synthesis techniques exploiting the

nature of uncertainties, such as \structured," \real" and \time-invariant."

The \inaccuracy" can also be reduced via sophisticated linearization methods, such as

the parameterized Jacobian linearization method (PJL). PJL linearizes a nonlinear physical

system around parameterized operating conditions rather than a single operating condition.

The control of the linearized model from PJL is traditionally done by the gain-scheduling

technique. The gain-scheduling technique designs linear controllers at several operating

conditions and then interpolates these designed controllers along the user-de�ned param-

eter trajectory. The gain scheduling method does enjoy widespread usage in a variety of

applications, such as aircraft control, missile autopilot, jet-engine control, and process con-

trol. However, it remains an ad hoc methodology because robustness, performance, or even

nominal stability of a gain-scheduling controller are not addressed explicitly in the design

process. Rather, such properties are inferred from extensive simulations.

The demand for systematic, theoretically rigorous techniques for the gain-scheduling

method has stimulated a great deal of research on linear parameter-varying (LPV) systems,

where system matrices are matrix functions of time-varying parameters. Associated with

LPV systems are roughly two main analysis and design approaches: linear robust systems

theory and dissipative systems theory using various parameter-dependent Lyapunov func-

tions (PDLFs). The dissipative systems framework is the more desirable of the two tech-

niques because it can directly treat real time-varying parameters and also yield an LPV-type

controller. Furthermore, the dissipative systems framework attractively formulates analysis

and synthesis problems as convex optimization problems involving linear matrix inequalities

(LMIs) which are now very e�ciently solved by computer. However, two major important

issues still remain unsolved: (1) di�culty in selecting an optimal PDLF in order to reduce

conservatism of the dissipative systems approach; (2) di�culty in solving exactly convex op-

timization problems involving an in�nite number of LMIs when using the PDLF. Note that

these issues imply a zero-sum game between conservatism and computational complexity of

the dissipative systems framework.

In this thesis, we investigate a new approach to avoid these potential issues associated

with the dissipative systems framework. In other words, the devised new approach should

automatically select an optimal PDLF during the analysis and synthesis optimization pro-

cess and then yield �nite-dimensional convex optimization problems. Furthermore, the new


approach should enable us an explicit trade-o� between conservatism and computational

complexity of the dissipative systems framework.

1.1 Previous Research

This section is intended to review some of the key approaches most relevant to this research

in order to place the present work in context. This section mainly reviews some linearization

methods for a nonlinear system, linear robust systems theory and LPV systems theory.

We should clarify the di�erence between a parametric uncertainty in linear robust sys-

tems theory and a parameter in LPV systems theory. Both are assumed to be unknown but

constrained a priori to lie in some known, bounded real set. However, the parameter is even

further assumed to be measurable in real time. Therefore, both are not distinguishable in

the analysis, while they are distinguishable in the synthesis. In other words, we can exploit

a parameter-dependent controller in the LPV framework.

1.1.1 Linearization Methods

Some direct linearization methods for nonlinear systems could be roughly categorized into

three types: (I) linearization about an equilibrium, (II) linearization about a (parame-

terized) state trajectory, and (III) global linearization. Method I is used to represent a

nonlinear system with a linear time-invariant (LTI) system around an equilibrium con-

dition [Vid92]. The system representation in Method I is the simplest of three di�erent

approaches and so are the related analysis and synthesis techniques. However, this ap-

proach is limited to characterizing only the local properties of a nonlinear system around

an equilibrium condition [Vid92].

Method II is used where the nonlinear system follows prescribed trajectories from re-

peated maneuvers and the outcome of some trajectory optimization [BH75]. Method II is

also used where the nonlinear system can be approximated by a family of linearizations or

the parameterized linearization [Rug91, SA90]. In particular, this case motivates an LPV

system representation which will be intensively studied here. Since the model from Method

II is valid around a state trajectory rather than a single equilibrium, Method II can represent

a nonlinear system in a wider range of operating conditions than Method I.

The last approach is used to represent a nonlinear system with a set of linear time-

varying (LTV) systems [Liu68, Lsl69], which is often called linear di�erential inclusions


(LDIs) in the literature [BGFB94]. Since Method III approximates the set of trajectories of

a nonlinear system with trajectories of LDIs, it can represent a nonlinear system in the entire

operating range. However, Method III can be very conservative because there may be many

trajectories of the LDI that are not actual trajectories of the nonlinear system [BGFB94].

Method II lies between Method I and Method III in every category of comparison. The

main di�erence between Method II and Method III is whether or not a nonlinear system

can be parameterized by index or parameters or whether a nonlinear system is represented

by an LTI system with parametric uncertainties or an LPV system.

Note that the above linearization methods are a direct way to derive a linear system,

while the feedback linearization method [Isi89] in nonlinear controls is an indirect way in

the sense that a nonlinear system is linearized by a feedback loop for further analysis.

1.1.2 Analysis

Analysis techniques for LPV systems fall roughly into one of two categories: the scaled small-

gain framework and the dissipative systems framework. These approaches overlap with

counterparts for uncertain systems because an LPV system can be treated as a uncertain

system with parametric uncertainties. Note that as shown in the literature [How93,Vid92],

these two approaches for uncertain systems are closely related to each other.

The small-gain framework originates from the work done by Zames [Zam66], which

provided an exact robust stability test for an LTI system with unstructured dynamic un-

certainty. This approach led to the use of singular values as an important tool in robust

control [DFT92]. However, the small-gain framework provides only su�cient conditions for

an LTI system with structured parametric or dynamic uncertainty and thus may be very

conservative. Hereby, the small-gain framework have been modi�ed to exploit the structure

and type of a uncertainty. The most remarkable results are the structured singular value

(�) [Doy82] and the multivariable stability margin (Km) [Saf82] for linear fractional trans-

formation (LFT) systems [Red60]. However, these quantitative measures are very di�cult

to calculate exactly, so that they are often estimated by both the (computationally feasible)

upper and lower bounds. Therefore, one of the main issues associated with these approaches

is how to derive a tight upper and lower bound. The problem of deriving a tight upper

bound is focused on in particular, because an upper bound condition is often convex and

provides a su�cient condition for robust analysis.


A number of researchers have considerably derived tighter upper and lower bounds

with scaling techniques [FTD91,Hel95,PD93,Pag96] and developing e�cient computation

schemes for these bounds [BDGPS91,Hel95,Yo93]. The scaling matrix for the upper bound

is originally frequency-dependent for static parametric uncertainties [FTD91,Hel95] or LTI

uncertainties [PD93]. The scaling matrix is constrained to be constant for quickly vary-

ing parametric uncertainties or structured nonlinear (dynamic) uncertainties [AG95,Hel95,

Dan96,Pac94,SG,Sha94]. Furthermore, the constant scaling is extended to the Popov mul-

tiplier which e�ectively takes into account the \real" uncertainty. However, the constant

scaling matrix can be very conservative for the analysis of LPV systems with slowly varying

parameters.

In a parallel approach, tight upper bounds have been also derived by the passivity

framework [BHPD94, Hel95] and integral quadratic constraints (IQC) approach [Jon96,

Meg93] using various multipliers. As the IQC approach describes the uncertainty in terms

of integral quadratic constraints, it generalizes the standard passivity approach [DV75].

One of the attractive properties of the IQC approach is its ability to build a new multiplier

as a convex combination of known basic multipliers. These multiplier-based approaches can

analyze less conservatively the uncertain system with slowly varying uncertainties using the

swapping lemma [Mor80]. An explicit result is shown in [Jon96] using the Popov multiplier,

which is the simplest type of multiplier. However, further study is required to investigate

the desired type of multiplier which leads to a tight upper bound. Furthermore, the use of a

general (frequency-dependent) multiplier makes the synthesis problem very computationally

di�cult.

The dissipative systems framework was originally developed to de�ne \dissipativeness"

(a generalized concept of passivity) in terms of an inequality involving the storage func-

tion (or Lyapunov function)and the supply rate [Wil72]. This framework has been ex-

tended to formulate many su�cient conditions for performance analysis problems of LTI

systems [BGFB94, Iwa93, SGC97], LTV systems [HM76], and nonlinear systems [Scha96].

One of the attractive characteristics of the dissipative systems framework is its ability to

formulate analysis problems as convex optimization problems involving linear matrix in-

equalities (LMIs) [BGFB94], which are now very e�ciently solved by computer.

The appearance of LMIs in the control community started with the Lyapunov stability

theory. The important role of LMIs in control theory was already recognized in the early

1960's, especially by Yakubovich [Yak67]. Since the late 1980's, the LMI approach in control


theory have been revived because of the development of computationally e�cient interior-

point algorithms for LMIs [NG94,NN94,VB96]. Several software packages now exist which

allow users to represent LMI problems with a high-level language and to interface with

MATLAB (LMILAB [GNLC95] and SDPSOL [WB96] are examples of this software).

In the dissipative systems framework, a supply rate is uniquely related to a performance

analysis problem. Therefore, one of the important issues associated with the dissipative

systems framework is to determine for a given uncertain system an optimal Lyapunov

function which leads to a less conservative analysis result (Note that the dissipative sys-

tems framework often provides only su�cient conditions). So far, various types of Lya-

punov functions have been proposed, from the typical (smooth) quadratic Lyapunov func-

tion [BGFB94, Iwa93, SGC97] to nonsmooth Lyapunov functions [AC84, SS95]. However,

one of these Lyapunov functions is selected such that the Lyapunov function should have

the same uncertainty-dependence as the uncertain system. A famous example is the Lur�e-

Postnikov Lyapunov function for the analysis of Lur�e systems [NT73]. This key idea has

been also applied to the selection of parameter-dependent Lyapunov functions (PDLFs) for

the analysis of LPV systems [AA97,AWU97,Beck95,FAG95,GAC96,YS95]. For example, as-

sociated with an a�ne parameter-dependent LPV system is an a�ne parameter-dependent

Lyapunov function.

The use of a PDLF allows the dissipative systems framework to directly analyze an LPV

system with slowly varying parameters [Wu95]. Hereby, using the PDLF implies that the

dissipative systems framework provides a less conservative analysis tool for the analysis of

LPV systems than the scaled small-gain framework. However, using the PDLF introduces

two major potential issues: di�culty in selecting an optimal PDLF vs. given time variation

of parameters and solving exactly an in�nite number of LMIs. The �rst issue can be

addressed by an example. Consider an a�ne LPV system, where the parameter (p) has

broad range of time variation, _p 2 [0;1]. Based on the result of the Lur�e system [NT73], ana�ne PDLF is associated with the a�ne LPV system with _p = 0. However, a (parameter-

independent) quadratic Lyapunov function is associated with the a�ne LPV system with

_p = 1 (See [GAC96]). Thus, the current rule for selecting PDLFs may not be e�ectivewhen applied to an LPV system with time-varying parameters. This potential problem

spurs a new study on the selection of an optimal PDLF.

The second issue is one of the main di�culties in the analysis of uncertain and LPV

systems. Due to the nature of these systems, the analysis technique should play with the


(convex) compact set of uncertainties and parameters. Therefore, the derived formula-

tions actually represent convex problems involving an in�nite number of LMIs. As these

systems are constrained to a couple of special cases, such as LDIs or polynomial (parameter-

dependent) LPV systems, the number of LMIs can, however, be reduced to a �nite number

of LMIs using \convexity" [AA97, AWU97, FAG95, GAC96, YS95]. Note that associated

with these systems are the PDLFs with the same parameter-dependency. However, LPV

systems and PDLFs that are more general than these special cases are desired to develop

a less conservative, guaranteed analysis technique for a broad class of LPV systems. Un-

fortunately, the corresponding analysis formulations are an in�nite number of LMIs, so the

guaranteed result cannot be found by computer. In this case, �nite basis methods and

heuristic gridding techniques [Wu95] are typically used. However, these approaches do not

guarantee the analysis result and furthermore need further study on the selection of the

best basis functions.

The dissipative systems approach using a nonsmooth Lyapunov function can be found

in the analysis problems for the switched, hybrid and other nonlinear controls [Bran94,

HelZ97, MBA96, JR96, RJ97, WP94]. The stability analysis of switched or hybrid sys-

tems [Bran94,MBA96,WP94] is attempted by a piecewise continuous Lyapunov function.

While this Lyapunov function reduces conservatism of the stability analysis, it is limited

to the stability analysis problem. As using continuous, piecewise quadratic Lyapunov func-

tions, LMI formulations have been developed for other performance analysis problems of

the switched system [JR96,RJ97]. However, these Lyapunov functions may be conservative

for the analysis of LPV systems because they cannot directly treat time-varying parameters

in LPV systems. Furthermore, extensions of these approaches to performance synthesis

problems are not often straightforward.

1.1.3 Synthesis

The synthesis is a more complicated problem than the analysis: while an analysis prob-

lem �nds only the optimal scaling matrix or PDLF, a synthesis problem �nds an optimal

scaling matrix or PDLF and unknown controllers. Thus, it involves more technical steps

and assumptions than the analysis. However, most synthesis techniques stem from the

analysis techniques so synthesis techniques can be categorized as analysis techniques: the

scaled small-gain framework and the dissipative systems framework. Therefore, most of the

discussions and references made in the analysis are still e�ective here.


It should be mentioned that the LPV control is obviously di�erent from the robust

control. The di�erence lies in the assumption that while a parametric uncertainty in an

uncertain system is not measurable, a parameter in an LPV system is measurable in real

time. Therefore, the LPV control can design an LPV-type controller rather than an LTI

controller.

Before we discuss synthesis techniques for LPV systems, we briey address the typ-

ical gain-scheduling approach. The gain-scheduling technique is based on the matured

linear systems theory and user-de�ned scheduling scheme [AW89]. In detail, local linear

controllers are designed for linearized models of a nonlinear system at several di�erent

operating conditions. A global nonlinear controller for the nonlinear system is then ob-

tained by interpolating, or scheduling, the gains of the local linear controllers. Due to

simplicity, the gain-scheduling approach has been successfully applied to many interesting

problems [AW89,NRR93, Ste80,WP94]. However, it remains an ad hoc methodology. For

example, the robustness, performance, or even nominal stability properties of the global

gain-scheduled controller are not addressed explicitly in the design process [SA92].

The lack of guaranteed properties stems from the fact that there is no theoretic tool to

verify the interpolation process. The theoretic lack for the interpolation has been compen-

sated with a couple of useful guidelines, such as \the scheduling variable should capture the

plant's nonlinearities" and \the scheduling variable should vary slowly" [SA92]. However,

these guidelines may not be applicable when applied to advanced missile autopilot designs.

For example, some parameters, such as the angle-of-attack, are arbitrarily quickly varying.

Furthermore, these guidelines may lead to a very complicated scheduling scheme because

missile dynamics is a highly nonlinear MIMO system [NRR93]. A theoretic improvement

of gain-scheduling is obviously to include the scheduling process in the controller design

problem. This idea spurs the studies on LPV systems.

The scaled small-gain approach for the LPV control is a special case of the counterpart

for the robust control except that the scaling matrix in the LPV control is constrained to

be constant and a block full matrix associated with a block-repeated uncertainty structure.

The �rst property is obvious because parameters of LPV systems are time-varying. Note

that the constant scaling matrix includes the Popov multiplier which e�ectively takes into

account the \real" parameter. The second property is due to the fact that the controller is

assumed to be the same as parameter-dependency as the LPV system, i.e., parameters and

their copies for the controller appear in the uncertainty block. This property contributes


LMI formulations for the synthesis [AG95, Hel95, Pac94, SG] rather than bilinear matrix

inequality (BMI) formulations [Hel95,PD93,Yo93], which are typical in the robust control

synthesis.

Due to the nature of the scaled small-gain approach, most researches have focused

on the (robust) L2-gain synthesis problem. However, a systematic approach using fullblock scaling, which is a simpli�ed IQC approach with constant multiplier, was recently

devised [Sche96, SW96]. This approach can be applied to other performance synthesis

problems and also improve the results from the scaled small-gain approach because of the

richness of scaling matrices [Sche96, SW96]. Note that the integral quadratic constraint

is originally shown in the dissipative systems framework [BGFB94], so versatility of the

IQC approach is as much as the dissipative systems approach. The advantage of the scaled

small-gain approach makes full use of the matured scaled small-gain approach in the robust

control. In other words, any result in the robust control could be directly extended to the

LPV control. Furthermore, it could be even simpli�ed due to the structure of the uncertainty

block. However, the bene�t is obviously limited by conservatism of the synthesis tool

because the constant scaling matrix cannot e�ciently account for the slowly time-varying

parameter.

The synthesis procedure of the dissipative systems framework originates from the stan-

dard three steps for the H1 synthesis of an LTI system [GA94]: (1) to derive the analysisformulation for the closed-loop system; (2) to eliminate the unknown controller dynamics

from the analysis formulation and then solve the remaining formulation; (3) to construct the

unknown controller dynamics from the results of (2). This procedure has been re�ned and

applied to many other performance synthesis problems of an LTI system [Iwa93, SGC97].

Including the \convexifying" step at (2), this standard step has also been applied to the

L2-gain synthesis problem of an LPV system [AA98, BP94, KJS96, Wu95, Woo95, YS95].However, the nature of the dissipative systems framework enables us to derive many other

performance synthesis problems for the LPV system with the same format of the L2-gainsynthesis. As discussed in the analysis, the crucial factor associated with the dissipative

systems framework is selecting an optimal PDLF and \convexifying" an in�nite number of

LMIs.

Note that we observe results similar to the scaled small-gain approach. Consider the

parametric robust control for a Lur'e system subject to parametric uncertainties [Ban97,


FAG95]. The corresponding Lyapunov function, V = xTclPcl(p)xcl, is

Pcl(p) =

24 X(p) Z(p)Z(p)T E(p)

35

0@P�1cl (p) =

24 Y (p) �

� �

351A;

where X(p) is as same uncertainty dependency as the Lur'e system and Z(p) and E(p) are

constant. The reason for Z(p) and E(p) being constant is that the feedback controller should

be uncertainty-independent (Note that these Z(p) and E(p) are related to the controller

dynamics [Ban97]). As shown in [Ban97, FAG95], the parametric robust control problem

then yields a nonconvex optimization involving BMIs. Next, consider the LPV control for

an a�ne LPV system. Since the controller can be parameter-dependent, consider the PDLF

such that X(p) and Y (p) is as same parameter-dependency as the a�ne LPV system, i.e.,

a�ne in p [AA98]. In this case, the LPV control leads a convex optimization problem

involving LMIs of X(p) and Y (p). Furthermore, it yields a less conservative result than the

robust control because of the richness of PDLFs.

1.2 Thesis Objectives and Overview

The primary goal of this thesis is to develop analysis and synthesis tools to avoid two poten-

tial issues associated with the dissipative systems framework: (1) di�culty in selecting an

optimal PDLF in order to reduce conservatism of the dissipative systems approach; (2) di�-

culty in solving exactly convex optimization problems involving an in�nite number of LMIs.

In other words, the devised new approach should automatically select an optimal PDLF

during the analysis and synthesis optimization process and then yield �nite-dimensional

convex optimization problems. The approach is also desired to provide an explicit trade-o�

between conservatism and computational complexity of the design technique. Furthermore,

the approach can derive several performance analysis and synthesis formulations for LPV

systems without any di�culty.

This thesis attempts to consider both a piecewise-a�ne parameter-dependent linear

parameter-varying (PALPV) system and a continuous, (quasi-) piecewise-a�ne parameter-

dependent Lyapunov function (PAL) for analysis and synthesis. From the survey of pre-

vious works, both an a�ne LPV system and a�ne PDLF turns out to be the simplest

pair that leads to a �nite convex optimization problem. Therefore, a generalization for

reducing conservatism should be made without destroying the attractive property of \a�ne


parameter-dependency." It intuitively leads to the idea that \piecewise-a�ne parameter-

dependency" may provide one solution for the potential issues associated with the dissipative

systems framework (Similar approaches can be found in many other �elds, such as hybrid

control [RJ97]). However, the concept of \piecewise-a�ne parameter-dependency" needs

a \nonsmooth" dissipative systems framework rather than the typical smooth dissipative

systems framework. As a result, this thesis focuses on the development of a nonsmooth

dissipative systems framework and applications on interesting performance analysis and

synthesis problems of PALPV systems.

The results of this thesis address the usefulness of the concept of \piecewise-a�ne

parameter-dependency." The new approach allows a trade-o� between conservatism and

computational e�ort: tuning the number of piecewise terms, the new approach produces

various results ranging from the existing result (based on a�ne parameter-dependency) to

improved new results. As a supporting tool, the developed nonsmooth dissipative systems

framework turns out to be very similar to the current smooth dissipative systems frame-

work. Furthermore, this nonsmooth dissipative systems framework contributes to deriving

many analysis and synthesis formulations for PALPV systems.

Chapter 2 of the thesis outlines some of the mathematical preliminaries for this work. It

presents the overview of several important topics, such as stability of systems, Lebesgue in-

tegral theorem, ordinary di�erential equations (or inequalities) and linear matrix inequality.

In particular, the overview of Lebesgue integral theorem and ordinary di�erential inequal-

ities are presented because of the essential roles that they play in the later developments.

Through this chapter, some extensions of the existing mathematical results have been also

presented for the later developments.

Chapter 3 builds a nonsmooth dissipative systems framework using results. This new

framework is developed with the Lebesgue integral theorem. The derived results are exten-

sions of the results for LTI systems [BGFB94, Iwa93,SGC97] to LPV systems. This frame-

work is a simpli�cation of general nonsmooth dissipative systems frameworks [AC84,SS95]

to e�ectively support our idea in mathematics. This chapter explicitly demonstrates that

many interesting analysis problems for LPV systems, such as Lyapunov stability, L2-gain,L1-gain, H2-norm, passivity and robust counterparts, can be formulated in terms of aLipschitz Lyapunov function and the supply rate.

Chapter 4 derives several interesting analysis formulations for PALPV systems using the

developed dissipative systems framework with the continuous, Lipschitz PAL. It includes


Lyapunov stability, L2-gain, L1-gain, H2-norm, passivity analysis problems and robustcounterparts for PALPV systems. A construction method for the PAL is also explicitly

presented. The promising property of the dissipative systems framework allows us to sys-

tematically derive these analysis problems. The derived formulations are �nite-dimensional

convex optimization problems involving LMIs.

Chapter 5 continues the work of Chapter 4 to address the synthesis. The PAL is extended

to be a continuous, quasi-piecewise-a�ne parameter-dependent Lyapunov function (QPAL)

for the synthesis problem. This chapter then presents several control design problems of

PALPV systems. It includes L2-gain, L1-gain, H2-norm, passivity and robust L2-gainsynthesis problems for PALPV system. The derived formulations are �nite-dimensional

convex optimization problems involving LMIs.

Chapter 6 demonstrates the e�ectiveness of the proposed approach in the analysis and

synthesis. Several benchmark problems are used to demonstrate conservatism of the analysis

and synthesis techniques. A realistic benchmark problem of a missile autopilot design is

also used to address some important issues: the impact of the model used in the controller

design on performance and reliability of the designed controller; and the richness and its

conservatism of PDLFs used for the LPV control.

Chapter 2

Mathematical Preliminary

This chapter is comprised of some basic de�nitions and elementary results in linear alge-

bra, system theory, Lebesgue integral theorem, ordinary di�erential equation and convex

optimization. While the treatment of this material is by no means exhaustive, it should be

su�cient as a reference for this work.

2.1 Norms and Normed Spaces of Signals

We de�ne some standard norms and normed spaces for signals. Various norms for engineer-

ing problems are introduced in [BGFB94,DV75,Vid92] and references therein.

De�nition 2.1 Norms on the linear vector space Rn are as follows:

� kxk1 =nXi=1

jxij and the corresponding normed space l1.

� kxkp ="

nXi=1

jxijp#1=p

and the corresponding normed space lp.

� kxk1 = max1�i�n

jxij and the corresponding normed space l1.

Note that for simplicity, we de�ne k � k as the l2-norm on Rn.

De�nition 2.2 Let I = [t0;1) and E = ff : I ! Rn j f locally Lebesgue integrable g.Norms on appropriate subsets of E are as follows:

� kxk1 =Z1

t0

kx(t)kdt and the corresponding normed space L1(I;Rn).

13

CHAPTER 2. MATHEMATICAL PRELIMINARY 14

� kxkp =�Z

1

t0

kx(t)kpdt�1=p

and the corresponding normed space Lp(I;Rn).

� kxk1 = ess supt2I

kx(t)k and the corresponding normed space L1(I;Rn).

De�nition 2.3 Suppose f : I ! Rn. Then for each T 2 I, the function fT : I ! Rn isde�ned by

fT (t) =

8<: f(t) t0 � t � T0 t > T

and is called the truncation of f to the interval [t0; T ].

De�nition 2.4 The normed space Lpe consists of all Lebesgue integrable functions f :I ! Rn with property that fT 2 Lp for all �nite T , and is called the extension of Lp or theextended Lp-space.

De�nition 2.5 Let H : Lpe ! Lpe be causal (non-anticipative) if

(Hf)T = (HfT )T ; 8T 2 I and f 2 Lpe:

Another popular norm is the inner product, denoted by < �; � >.

De�nition 2.6 Let E � L2(I;Rn). The inner product h�; �i : E � E ! R+ is de�ned asfollow:

� hx; yi =Z1

t0

x(t)T y(t)dt for x; y 2 E

� hx; xi = kxk22 for x 2 E

2.2 Stability

We briey discuss some de�nitions of stability: Lyapunov stability, input-output stability

and input-state stability. The �rst two de�nitions [Kh96,Vid92] have been widely used in

the control community. However, the last de�nition has been recently recognized as a useful

tool for the Lyapunov nonlinear controls such as backstepping [Son89, KKK95]. We will

review only the �rst two de�nitions here.


2.2.1 Lyapunov Stability

We introduce uniform (asymptotic) stability and exponential stability of a nonlinear time-

varying system. The \uniformity" is necessary to characterize time-varying systems whose

behavior has a certain consistency for di�erent values of initial time t0 (Refer to [Kh96,

Vid92] for details).

We de�ne a nonlinear time-varying system:

_x = f(x; t); x(t0) = x0:

This system is assumed to have at least a solution x(�) on I. It is also assumed that theorigin x = 0 is an equilibrium point for this system, i.e.,

f(0; t) = 0; 8 t 2 I:

Note that if the equilibrium under study is not the origin, we can always rede�ne the coordi-

nates on Rn in such a way that the equilibrium of interest becomes the new origin [Vid92].

De�nition 2.7 The equilibrium point x = 0 is locally uniformly stable if for any R > 0,

there exists a positive scalar r = r(R) such that

kx(t0)k < r =) kx(t)k < R 8 t � t0:

De�nition 2.8 The equilibrium point x = 0 is locally uniformly asymptotically stable if for

any R0 > 0, there exist positive scalars R1; R2 and T (R1; R2) > 0 such that 0 < R2 < R1 <

R0 and 8 t � 0,

kx(t0)k < R1 =) kx(t)k < R2 8 t � t0 + T (R1; R2):

De�nition 2.9 The equilibrium point x = 0 is locally exponentially stable if there exist two

positive numbers, � and �, such that for su�ciently small x(t0),

kx(t)k � �kx(t0)ke��(t�t0) 8 t � t0:


Note that exponential stability implies uniformly asymptotic stability but the converse

is not necessarily true.

2.2.2 Input-Output Stability

We briey state the basic de�nitions of input-output stability (Refer to [Kh96, Vid92] for

details).

Suppose H : X ! X is a mapping. H de�nes a binary relation R on X, i.e.,

R = f(x;Hx) : x 2 Xg:

De�nition 2.10 Suppose R is a binary relation on Lpe. Then R is said to be Lp-stable if

(x; y) 2 R; x 2 Lp =) y 2 Lp:

R is Lp-stable with �nite gain if it is Lp-stable, and in addition there exist �nite constant�p and �p such that

(x; y) 2 R; x 2 Lp =) kykp � �pkxkp + �p: (2.1)

Furthermore, when �p = 0, R is Lp-stable with �nite gain and zero bias.

De�nition 2.11 Suppose R is a binary relation on Lpe. If R is Lp-stable with �nite gain,then the Lp-gain of R is de�ned as p = inff�p : 9�p � 0 such that Eq. 2.1 holds g:

De�nition 2.12 Suppose H : Lpe ! Lpe. Then the map H is said to be Lp-stable if andonly if the corresponding binary relation R on Lpe is Lp-stable.

An alternative approach to the L2-stability is the passivity approach using the innerproduct.

De�nition 2.13 Suppose H : L2e ! L2e. Then the map H is said to be passive if thereexists constant � such that

hHx; xiT � �:


-H1

v1

v2

w1 z1

w2z2H2

Fig. 2.1: Feedback system

Furthermore, the map H is said to be strictly passive if there exists constant � such that

hHx; xiT > �: (2.2)

where hHx; xiT = h(Hx)T ; xT i :Note that there exist several di�erent formulations on Eq. 2.2 [TGPS96].

Eq. 2.2 () h(Hx)T ; xT i � �kxT k22 + � input strictly passive() h(Hx)T ; xT i � �k(Hx)T k22 + � output strictly passive() h(Hx)T ; xT i � �(kxT k22 + k(Hx)T k22) + � input/output strictly passive

2.3 Small-Gain Theorem and Passivity Theorem

The general frameworks to study input-output stability of complex systems such as a feed-

back connected system are the small-gain theorem and passivity theorem for the L2-stability.

Theorem 2.1 [DV75] Consider the feedback system in Fig. 2.1. Let H1;H2 : Lpe !Lpe; p 2 [1;1], be causal and Lp-stable operators with �nite gains 1; 2 and associatedconstants �1; �2. If

12 < 1;

then the feedback system is Lp-stable, i.e., w1T , w2T , z1T and z2T have bounded Lp-norm'sfor v1; v2 2 Lpe.


Theorem 2.2 [DV75] Consider the feedback system in Fig. 2.1. Suppose there exist con-

stants �i; �i; i = 1; 2, such that

hHix; xiT � �ikxT k22 + �ik(Hix)T k22; 8T � 0; 8x 2 L2e; i = 1; 2:

Then the feedback system is L2-stable with �nite gain and zero bias if �1+�2 > 0; �2+�1 > 0:

Theorem 2.2 implies the following corollary.

Corollary 2.1 Consider the feedback system in Fig. 2.1. Then the feedback system is L2-stable with �nite gain and zero bias if

� H1 and H2 are input strictly passive or

� H1 and H2 are output strictly passive or

� H1 is passive and H2 is input and output strictly passive or reversely.

2.4 Lebesgue Integral Theorem

The reconstruction of a function from its derivative plays an important role in the dissipative

systems framework discussed later. This problem, which shows the connection between

di�erentiation and integration, is also a fundamental problem in real analysis [KF70].

Theorem 2.3 [KF70] Suppose that f : I �! Rn is absolutely continuous on the intervalI; that is, given any � > 0, there is a � > 0 such that

nXk=1

kf(bk)� f(ak)k < �

for every �nite system of pairwise disjoint subintervals, (ak; bk) � I, of total lengthnX

k=1

(bk � ak) < �:

Then, the derivative _f is summable or integrable on I and

f(t2) = f(t1) +

Z t2t1

_f(�)d�


for any t1; t2 2 I:

Note that _f exists almost everywhere ( _f exists except for the measure zero set). There-

fore, _f is not de�ned on the measure zero set. However, a Lipschitz function has a nice

property that the integration can be constructed by the Dini-derivative which is well-de�ned

everywhere.

Corollary 2.2 If f : I �! Rn is Lipschitz on the interval I; that is, there exists a � suchthat for any t1; t2 2 I, kf(t2)� f(t1)k � �kt2 � t1k:

Then the Dini-derivative D+f(t) exists everywhere on I and

f(t2) = f(t1) +

Z t2t1

D+f(�)d�

for any t1; t2 2 I: Here, D+f(t) = lim suph!0+

1

h

[f(t+ h)� f(t)]:

Proof: The proof is based on the facts [KF70,RHL77]; the Dini-derivative is equal to the

derivative if the derivative exists; the Dini-derivative of a Lipschitz function f(t) is de�ned

everywhere on I; and the behavior of _f(t) on the measure zero set does not a�ect theintegral. Since a Lipschitz function is also absolutely continuous, the following equation is

then immediately obtained:

f(t2) = f(t1) +

Z t2t1

_f(�)d� = f(t1) +

Z t2t1

D+f(�)d�

for any t1; t2 2 I. 2

When f is a multi-variable function, the following chain-rule of the Dini-derivative is

useful. Note that this lemma is an extension of the result [AC84,Yos66].

Lemma 2.1 Let x 2 D � C1(I;Rn) and � 2 F � C1(I;Rs); that is, x and � are continu-ously di�erentiable on the interval I. Suppose that a continuous f : D�F �! Rn satis�esthe following conditions:

� f is continuous, Lipschitz on x for each �xed �; that is, there is a � such that

kf(x2; �)� f(x1; �)k � �kx2 � x1k

for each � 2 F and x1; x2 2 D.


� f is continuous, Lipschitz on � for each �xed x; that is, there is a � such that

kf(x; �2)� f(x; �1)k � �k�2 � �1k

for each x 2 D and �1; �2 2 F .

Then

D+f(x; �)( _x; 0) +D+f(x; �)(0; _�) � D+f(x(t); �(t)) � D+f(x; �)( _x; 0) +D+f(x; �)(0; _�):

(2.3)

Here, the \partial" Dini-derivative is de�ned:

D+f(x; �)(u; 0) = lim sup

h!0+

1

h

[f(x+ hu; �)� f(x; �)];

D+f(x; �)(u; 0) = lim infh!0+

1

h

[f(x+ hu; �)� f(x; �)]:

Proof: The proof is based on the result [AC84, Yos66] and the well known properties of

`lim sup' and `lim inf.' According to the property of `lim sup,' D+f is bounded above:

D+f(x(t); �(t)) = lim sup

h!o+

1

h

[f(x(t+ h); �(t+ h))� f(x(t); �(t))]

� lim suph!o+

1

h

[f(x(t+ h); �(t+ h))� f(x(t); �(t+ h))] (2.4)

+ lim suph!o+

1

h

[f(x(t); �(t+ h))� f(x(t); �(t))]: (2.5)

We consider the �rst term (Eq. 2.4). Since x is smooth and f is continuous, Eq. 2.4 becomes

lim suph!o+

1

h

[f(x(t) + _x(t)h+ �(t; x; h)h; �(t)) � f(x(t); �(t))]: (2.6)

Furthermore, f(x; �) is Lipschitz on x so Eq. 2.6 becomes

lim suph!o+

1

h

[f(x(t) + _x(t)h; �(t))� f(x(t); �(t))]: (2.7)

Therefore,

Eq. 2.4 = D+f(x; �)( _x; 0):


Similarly,

Eq. 2.5 = D+f(x; �)(0; _�):

As a result,

D+f(x(t); �(t)) � D+f(x; �)( _x; 0) +D+f(x; �)(0; _�): (2.8)

According to the property of `lim sup' and `lim inf,' D+f is bounded below:

D+f(x(t); �(t)) = lim sup

h!o+

1

h

[f(x(t+ h); �(t+ h))� f(x(t); �(t))]

� lim suph!o+

1

h

[f(x(t+ h); �(t+ h))� f(x(t); �(t+ h))] (2.9)

+ lim infh!o+

1

h

[f(x(t); �(t+ h))� f(x(t); �(t))]: (2.10)

As in the derivation of upper bound, we can derive

D+f(x(t); �(t)) � D+f(x; �)( _x; 0) +D+f(x; �)(0; _�): (2.11)

Eqs. 2.8 and 2.11 implies Eq. 2.3. 2

2.5 Ordinary Di�erential Equations

When dealing with continuous-time systems, it is necessary to have a good understanding

of the basic facts regarding initial-value problems of di�erential equations. Consider

_x = f(t; x); with x(t0) = x0; (2.12)

for x 2 D � Rn. Such equations result when control and parameter trajectories aresubstituted in the right-hand side of a nonlinear parameter-varying system:

_x = f(x; �(t); w(t)):

We state the main result on existence and uniqueness associated with the initial-value

problem.

Theorem 2.4 [Son90] Assume that f : I � D �! Y, where D � Rn is open, Y � Rn isopen and I is an interval, satis�es the following conditions:


� f is continuous, locally Lipschitz on x for each �xed t; that is, there are for eachx0 2 D a real number � > 0 and a � such that the ball B�(x0) of radius � centered atx0 is contained in D and

kf(t; x)� f(t; y)k � �kx� yk

for each t 2 I and x; y 2 B�(x0).

� f is measurable, locally integrable on t for each �xed x0; that is for each �xed x0 thereis a � such that

kf(t; x0)k � �

for all t.

Then, for each pair (t0; x0) 2 I � D there is some nonempty subinterval J � I andthere exists the unique continuous, Lipschitz solution x of Eq. 2.12 on J .

Note that the interval J could be arbitrary small. However, the interval J can beextended to J = [t0;+1) in the following two cases.

Corollary 2.3 [Son90] Let D = Rn. Suppose that f satis�es the assumptions of Theo-rem 2.4 except that the function � can be chosen independently of x0 with � = 1. ThenJ = [t0;+1).

Corollary 2.4 [Kh96] Let W be a compact subset of D � Rn. Suppose that f satis�es theassumptions of Theorem 2.4 and for x0 2W it is known that every solution of Eq. 2.12 liesentirely in W . Then, J = [t0;+1).

We discuss some properties of the �rst-order \di�erential" inequality which is frequently

shown in the input-output stability or input-to-state stability [KKK95,Son89].

Lemma 2.2 [Son90] Assume given an interval I, a constant c � 0, and two functions,�; � : I �! R+ such that � is locally integrable and � is continuous. Suppose further thatfor some t0 2 I it holds that

�(t) � c+Z tt0

�(�)�(�)d�


for all t � t0. Then, it must hold that

�(t) � ceR tt0�(�)d�

:

Note that this lemma is called the Bellman-Gronwall.

Lemma 2.3 Assume constants b; c � 0, and two functions v; w : I ! R+ such that

D+v(t) � �cv(t) + bw(t)2; with v(t0) � 0 (2.13)

� thenv(t) � v(t0)e�c(t�t0) +

Z tt0

be�c(t��)

w(�)2d�:

� If, in addition, w 2 L2(I;R+), then v 2 L1(I;R+) and

kvk1 �1

c

(v(t0) + bkwk22):


v(t) � v(t0)e�c(t�t0) + bkwk22:


v(t) � v(t0)e�c(t�t0) +b

c

kwk21:

Proof: We show that Eq. 2.13 implies the �rst result. Since v(t) is Lipschitz and thus

absolutely continuous, Eq. 2.13 implies

_v(t) � �cv(t) + bw(t)2 for almost all t:

Multiplying this inequality by ect, it becomes

d

dt

(v(t)ect) � bw(t)2ect for almost all t:


Since v(t)ect is absolutely continuous, Theorem 2.3 implies that

v(t)ect � v(t0)ect0 �Z tt0

bw(�)2ec�d�:

Dividing both sides by e�ct, we derive

v(t) � v(t0)e�c(t�t0) +Z tt0

be�c(t��)

w(�)2d�:

The other results can be derived by the same approach as [KKK95]. 2

2.6 Linear Matrix Inequality

2.6.1 Linear Matrix Inequality Problems

Many problems in the LPV systems theory as well as robust linear systems theory can be

formulated as convex optimization problems involving linear matrix inequalities (LMIs).

Detailed references on this topic can be found in [BGFB94,GNLC95,VB96]. An LMI has

the form

F (x) = F0 +mXi=1

xiFi > 0 (2.14)

where the symmetric matrices Fi = FTi 2 Rn�n, i = 0; 1; � � � ;m are given and x 2 Rm is

variable. The inequality in Eq. 2.14 means that F (x) is positive- de�nite, i.e., uTF (x)u > 0

for all u 2 Rn, u 6= 0 or the smallest eigenvalue of F (x) is positive. However, most LMIproblems include matrices as variables. For example, the Lyapunov stability problem is

formulated: 9P > 0 such that F (P ) = ��ATP + PA

�> 0: In this case, we must check

whether or not F (P ) can be converted to an LMI. A couple of known facts are useful

to investigate the property of F (P ): if matrix variables (P ) a�nely enter into the matrix

inequality and are not coupled with other matrix variables, the matrix inequality is an LMI;

even a matrix inequality including coupled matrix variables can be an (enlarged) LMI, when

these coupled terms are related to the schur complement.


Lemma 2.4 [BGFB94] Let F (P ) =

24 F11(P ) F12(P )F21(P ) F22(P )

35 and where F11(P ) is square.

Then F (P ) > 0 if and only if

F11(P ) > 0 and F22(P )� F12(P )F�111 (P )F21(P ) > 0:

Here, F22(P )� F12(P )F�111 (P )F21(P ), is called schur complement of F (P ).

Note that an LMI (Eq. 2.14) can be nonlinear and nonsmooth on x but still a convex

constraint on x. A set of LMIs can be also combined into an (enlarged) LMI using methods

such as the diagonalization or S-procedure discussed below.

Lemma 2.5 [BGFB94] Let F0; F1; � � � ; Fp be quadratic functions of the variable � 2 Rn:

Fi(�) = �TTi� + 2u

Ti � + vi; i = 0; 1; � � � ; p;

where Ti = TTi . Suppose that there exist �1; � � � ; �p � 0 such that for all �,

F0(�)�pX

i=1

�iFi(�) � 0:

Then F0(�) � 0 for all � such that Fi(�) � 0, i = 1; � � � ; p.

Since an LMI de�nes a convex constraint on the variable x, optimization problems

involving the minimization (or maximization) of a convex performance function f : F ! Rwith F = fx jF (x) > 0g belong to the class of convex optimization problems and thus canemploy the full power of convex optimization theory. Associated with the study of LMIs

are three generic problems:

� Feasibility problem: whether or not F is an empty set

� Optimization problem: infx

ncTx jF (x) > 0

o

� Generalized eigenvalue problem: infxf� j �F (x)�G(x) > 0; F (x) > 0; G(x) > 0g

2.6.2 Numerical Algorithms for LMI Problems

The three typical LMI problems can be solved in a numerically e�cient way, i.e., poly-

nomial time algorithms such as the cutting plane, the ellipsoid method and very e�cient


interior-point methods (Refer to [BGFB94] and references therein for details). In prac-

tice, the interior-point algorithms are much more e�cient than the �rst two methods.

Among interior-point methods, the most e�cient methods today appear to be primal-dual

methods and projective methods [NN94]. Furthermore, these primal-dual and projective

methods have been extended to exploit the special (Lyapunov) structure of LMI prob-

lems [NG94,VB95].

We will now discuss some basic ideas of interior-point methods (Detailed references on

this topic can be found in [BGFB94,GNLC95,VB96]). Consider an optimization problem:

opt = infx

ncTx jF (x) > 0

o(2.15)

A simple approach, called the Method of Centers, converts this problem to a -feasibility

problem:

min such that F =nxjF (x) > 0; ( � cTx) > 0

o6= ;:

It de�nes a barrier function � such that

� � is smooth and strictly convex on the interior of the feasibility set F .

� � approaches in�nity along each sequence of points xn in the interior of F thatconverge to a boundary point of F .

One candidate of � is

� =

8<: log detF (x)

�1 + log( � cTx)�1 if x 2 F1 otherwise:

The Method of Centers then solves the problem with the following algorithm:

Repeat x 2 F , > cTx, 0 < � < 1

Inner Loop

� �nd the path of center x�(),

x�() = argmin

x

�log detF (x)�1 + log( � cTx)�1

�

by the (iterative) Newton method starting at x


Until (New) = (1� �)cTx� + � converges to opt

Note that this algorithm is comprised of the inner and outer loop so that it is not e�cient

to solve.

A sophisticated approach is the primal-dual optimization method that minimizes to zero

the duality gap, which is de�ned the di�erence between an upper bound and a lower bound

of the optimal value (opt). Therefore, this approach has some nice properties: the optimal

value of the primal-dual optimization framework is always known as `0'; the algorithm is

supported by the simpli�ed theory; and the interval of the optimal solution (opt) is always

known so the stopping criteria of the algorithm is directly related to the accuracy of the

optimal solution. A dual problem of Eq. 2.15 is

supZ=ZT�0

f�TrF0Z : TrFiZ = cig :

The primal-dual optimization formulation is then

infx;Z=ZT�0

ncTx+TrF0Z jF (x) > 0; TrFiZ = ci; i = 1; � � � ;m

o:

The performance function is the \duality gap" (� = cTx + TrF0Z), which is always non-

negative and specially zero at the optimal condition. Therefore, a barrier function (�(x;Z) =

� log det (F (x)Z)) to �nd the path of center (x�; Z�) (for example, see the algorithmof the Method of Center) has a nice property that (x;Z) = �(x;Z) � �(x�; Z�) =n log (�=n) + �(x;Z) is always non-negative and specially zero at the path of center. Fur-

thermore, an augmented primal-dual potential function '(x;Z) = �pn log(�)+ (x;Z) can

combine the inner and outer loop of the Method of Centers into one loop.

Repeat given strictly feasible x and Z

� �nd feasible search direction �x and �Z by solving a least-square problem� plane search: argmin

p;q' (x+ p �x; Z + q �Z)

� update: x = x+ p �x and Z = Z + q �Z

Until � = cTx+TrF0Z � �


The most time-consuming step is to solve the least-square problem at each iteration.

This least-square problem can be more e�ciently solved exploiting the problem struc-

ture [VB95]. Note that there exist several software packages which allow users to represent

LMI problems with a high-level language and to interface with MATLAB. Example are

SDPPACK [AHN97], LMITOOL [GNLC95] and SDPSOL [WB96].

2.6.3 Miscellaneous Results on LMIs

We state some results on matrix function elimination and completion. These lemmas form

the backbone of the synthesis in this study. Note that the known properties of a matrix

associated with the matrix elimination and completion are directly extended for a matrix

function, as long as the matrix function is continuous over a compact set.

Lemma 2.6 [ND77] Let � 2 P, hyper-rectangle (� Rs). Suppose a symmetric, continuousmatrix function A(�) > 0 and B(�) = S(�)TA(�)S(�), where S(�) is a nonsingular continu-

ous matrix function; that is, A(�) and B(�) are congruent. Then B(�) > 0. In other words,

the congruent transformation does not change inequality.

Lemma 2.7 [BGFB94] Let � 2 P, hyper-rectangle (� Rs). Given a symmetric, continuousmatrix function G : P ! Rn�n and continuous matrix functions U; V : P ! Rn�m, letU?(�) and V?(�) be continuous matrices functions whose columns form bases for the kernels

of U(�) and V (�), respectively. There exists a continuous matrix function K : P ! Rm�msatisfying

G(�) + U(�)K(�)V (�)T + V (�)K(�)TU(�)T < 0; 8 � 2 P

if and only if U?(�)TG(�)U?(�) < 0 and V?(�)

TG(�)V?(�) < 0 for all � 2 P.

Lemma 2.8 [Pac94] Let � 2 P, hyper-rectangle (� Rs). Given a pair of continuouspositive-de�nite matrix functions X;Y : P ! Rn�n+ , there exist continuous matrix func-tions M;N : P ! Rn�r and S; T : P ! Rr�r+ such that

P (�) =

24 X(�) M(�)M(�)T S(�)

35> 0 and Q(�) = P (�)�1 =

24 Y (�) N(�)N(�)T T (�)

35> 0


if and only if

24 X(�) I

I Y (�)

35 � 0 and rank

0@24 X(�) I

I Y (�)

351A � n+ r:

Note that given X(�); Y (�) 2 Rn�n+ , there exists an integer r such that the dilation canbe completed if and only if X(�) � Y (�)�1 � 0. If this semi-de�nite condition holds, thenthe rank of X(�) � Y (�)�1 determines the dimension necessary for the dilation. It can beeasily drawn that

I �X(�)Y (�) =M(�)N(�)T : (2.16)

Lemma 2.9 [PZPB91] Suppose r = n (implies X(�)�Y (�)�1 > 0). Then, P (�) and Q(�)can be parameterized as follows:

P (�) =

24 X(�) M(�)M(�)T M(�)T

�X(�)� Y (�)�1��1M(�)

35

and

Q(�) =

24 Y (�) N(�)N(�)T N(�)T

�Y (�)�X(�)�1��1N(�)

35:

2.7 \Convexifying" Techniques

We will frequently encounter in�nite-dimensional LMI problems in an LPV systems frame-

work. For example, the stability problem of an LPV system can be formulated: for all

� 2 [�1; 1] and _� = 0,A(�)TP (�) + P (�)A(�) < 0: (2.17)

Eq. 2.17 actually represents an in�nite number of LMIs because it should be checked over

all (in�nite) �'s inside the interval [�1; 1]. Therefore, Eq. 2.17 may be very di�cult toexactly solve. However, for some special cases, Eq. 2.17 can be reduced to a �nite number

of LMI conditions using \convexifying" techniques [AA97,FAG95,GAC96,WUFS94,YS95].

We consider two simple \convexifying" techniques for our study.


Let � 2 P, hyper-rectangle (� Rs), and � be 2s vertices or corners of this hyper-rectangle. Consider a matrix quadratic function F : P ! Rn�n such that

F (�1; � � � ; �s) = C0 +sX

k=1

�kCk +sX

k=1

k�1Xp=1

�k�pCkp +sX

k=1

�2kDk: (2.18)

The following lemma summarizes the results of [GAC96] for our study. The �rst \con-

vexifying" technique is stated.

Lemma 2.10 [GAC96] Suppose that F (�) is bounded by a multi-convex function, Fub(�) =

F (�) +sX

k=1

�2kMk for Mk > 0, such that

@2Fub(�)

@�2k

= 2(Dk +Mk) � 0; for k = 1; � � � ; s: (2.19)

Then F (�) (Eq. 2.18) is negative-de�nite on all � 2 P if Fub(w) is negative-de�nite at allthe corner points w 2 �.

Note that Eq. 2.19 is non-strict inequality but still strictly feasible with appropriateMk > 0.

In fact, to introduce Mk > 0 always makes a non-strict inequality strictly feasible, admit-

tedly with some conservatism. In numerical studies, we use the strictly feasible condition

of Eq. 2.19, i.e., @2Fub(�)

@�2k

= 2(Dk +Mk) > 0; for k = 1; � � � ; s, rather than the non-strictinequality.

Another approach is based on the quadratic relation shown in the following lemma and

S-procedure.

Lemma 2.11 [FAG95] Let x; q 2 Rn. There exists � 2 R such that q = �x and j�j � � ifand only if there exist symmetric matrix S and skew matrix T such that

qTSq � �2xTSx and qTTx = 0: (2.20)

Note that the skew matrix T is used to account for the \real" quantity of x and q. For

convenience, we de�ne some matrices:


C = [C1 � � �Cs] T = [T1 � � � Ts] JT = [In�n � � � In�n]D = diagsk=1Dk S = diag

sk=1Sk � = diag

sk=1�

2kIn�n

and

� =

26666664

0 0 � � � 0C21 0 � � � 0...

......

...

Cs1 Cs2 Cs(s�1) 0

37777775:

Lemma 2.12 Let j�kj � �k for k = 1; � � � ; s. F (�) (Eq. 2.18) is negative-de�nite on all� 2 P if there exist Sk = STk > 0 and Tk = �T Tk , k = 1; � � � ; s such that

24 C0 + JT�SJ 12C � T

12CT + T D � S + �

35< 0: (2.21)

Proof: Pre-multiply and post-multiply Eq. 2.18 with x. Then,

xTF (�)x = xT

24C0 +

sXk=1

�kCk +sX

k=1

k�1Xp=1

�k�pCkp +sX

k=1

�2kDk

35x < 0: (2.22)

De�ne a new variable qk = �kx and then

Eq. 2.22 = xTC0xT +

sXk=1

qTk Ckx+

sXk=1

k�1Xp=1

qTk Ckpqp +

sXk=1

qTkDkqk < 0: (2.23)

According to Lemma 2.11, the relation between qk and x can be formulated as follows:

qTk Skqk � �2kxTSkx for Sk = STk > 0 (2.24)

qTk Tkx = 0 for Tk = �T Tk : (2.25)


Eqs. 2.24 and 2.25 can be combined into Eq. 2.23 using S-procedure (Lemma 2.5), admit-tedly with some conservatism. Then,

xTC0x

T +sX

k=1

0@qTk Ckx+

k�1Xp=1

qTk Ckpqp + q

TkDkqk � qTk Skqk + �2kxTSkx+ 2q2kTkx

1A< 0:

This inequality obviously implies Eq. 2.21. 2

Remark 2.1 We should properly use these \convexifying" techniques. The technique based

on Lemma 2.10 is less computationally intensive than that based on Lemma 2.12 for a large

problem. For example, the LMI solver based on the projection method [GNLC95] has the

relationship between the number N(�) of ops needed to compute an �-accurate solution

and the problem size:N(�) � M N3 log(V=�); where M is the total row size of the LMIsystem, N is the total number of scalar decision variables, and V is a data-dependent scaling

factor. This inequality implies that the LMI problem involving many number of unknown

variables is likely to be more computationally intensive than that involving many number

of LMIs. While the S-procedure approach increases a number of unknown variables, themulti-convexity approach increases a number of LMIs. Therefore, the technique based on

Lemma 2.10 is less computationally intensive than that based on Lemma 2.12 for a large

problem. For our study, we will use Lemma 2.10 as a \convexifying" technique. The studies

using Lemma 2.12 have been investigated in [LimH97,LimH98].

2.8 Bilinear Matrix Inequality

An optimization problem involving bilinear matrix inequalities (BMIs) is often regarded as

an extension of the LMI problem in the literature [Ban97]. A BMI has the form

F (x; y) = F00 +mXi=1

xiFi0 +nX

j=1

yjF0j +mXi=1

nXj=1

xiyjFij > 0: (2.26)

where Fij = FTij 2 Rp�p are given and variables are x 2 Rm and y 2 Rn. For example,

F (Q;T; Y;�) = B+QCTT + (AQ+BY )T CT� > 0: Note that the coupling terms such as

QCTT cannot be eliminated by the schur complement because CT is not a positive-de�nite


matrix. The optimization involving BMI constraints is illustrated as

infx;y

ncTx+ dT y jF (x; y) � 0

o: (2.27)

This optimization often arises in a vast number of robust control synthesis problems such as

the parametric robust controller design, �xed order and decentralized controller design. This

optimization is known as an NP-hard problem so that it requires very intensive computation.

However, these many heuristic methods have been developed which can �nd only local

solutions. One of these heuristic methods is the alternative algorithm, called D � K orV �K iteration, that utilizes various e�cient LMI solvers in order to solve the optimizationinvolving BMIs: For �xed x, �nd y to minimize Eq. 2.27 using LMI solvers because Eq. 2.27

is an LMI on y; Similarly, �nd x to minimize Eq. 2.27 with �xed y. These approaches do

not guarantee the convergence of the algorithm but they seem to converge the local minima

in many practical applications [Ban97].

Chapter 3

Dissipative Systems Framework

We investigate various analysis problems { such as the stability, L2-gain, passivity and otherperformance measures { of a nonlinear parameter-varying system within a nonsmooth dis-

sipative systems framework. In [BGFB94, SW96, Wil72], it is shown that many analysis

problems can be formulated in terms of storage functions and supply rates within the dissi-

pative systems framework. This chapter extends these results to address analysis problems

for a special class of LPV systems which will be discussed later.

3.1 De�nition of Dissipative Systems

We now consider a nonlinear parameter-varying system �:

_x = f(x; �; w); x(t0) = x0 (3.1)

z = g(x; �; w); (3.2)

where x : I �! D � Rn is the state, w : I �! W � Rw is the input, z : I �! Z � Rzis the output, and � : I �! Fs is the parameter of the system. Here,

Fs 4=n� 2 C1(I;Rs) : �(t) 2 P; _�(t) 2 ; 8t 2 I

o;

where P = [�1; �1]� � � � � [�s; �s] and = [��1; �1]� � � � � [��s; �s].Note that with a speci�c trajectory �(�), Eq. 3.1 is a nonlinear time-varying system.

Thus, we can use some useful facts of nonlinear time-varying systems to explore the prop-

erties of nonlinear parameter-varying systems. However, it should be noted that � is a set

34

CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK 35

of nonlinear time-varying systems because � is not known in advance except that � and _�

are bounded by compact sets. Throughout this chapter, we assume

� f and g are unbiased in the sense that

f(0; �; 0) = 0; g(0; �; 0) = 0; 8 � 2 Fs:

� f and g are continuous, locally Lipschitz on x and w jointly for each �xed �; there area real number � > 0, � and � such that the ball B�(0; 0) of radius � centered at (0; 0)is contained in D �W and given � 2 Fs,

kf(x; �; w) � f(y; �; v)k � �(kx� yk+ kw � vk)kg(x; �; w) � g(y; �; v)k � �(kx� yk+ kw � vk)

for (x;w); (y; v) 2 B�(0; 0).

� f is locally integrable on � for each �xed x and w; there is a such that given(x;w) 2 B�(0; 0),

kf(x; �; w)k �

for � 2 Fs.

� every solution of Eq. 3.1 lies entirely in a compact set Dv � D that includes x0.

These assumptions imply some properties: the origin x(t) = 0 is an equilibrium for �

such that f(0; �; 0) = 0; 8 � 2 Fs; Eq. 3.1 has the unique continuous, Lipschitz solution x(�)over I with any x0 2 B�(0), � 2 Fs, and input w 2 B�(0) by Theorem 2.4 and Corollary 2.4.Note that the last assumption simpli�es the derivation of the dissipative systems framework.

However, this assumption will be shown to be satis�ed by the dissipative systems framework

discussed later; for any locally square integrable w(t), the resulting functions x(t) and z(t)

can be locally square integrable.

Let

r : W �Z �! R

be a mapping and assume for all t0; t1 2 R and for all input-output pairs (w; z), the com-position function r(w; z) is locally integrable, i.e.,

Z t1t0

jr(w(t); z(t))jdt

CHAPTER 3. DISSIPATIVE SYSTEMS FRAMEWORK 36

r will be referred to as the supply rate. Note that since z itself can be a state of �, r(w; z)

may also be a function of x.

De�nition 3.1 The class K is a set of functions, f : R+ ! R+, that are continuous,strictly increasing functions with f(0) = 0.

De�nition 3.2 � is said to be dissipative with respect to the supply rate r for all � 2 Fs ifthere exists a continuous function V : D �P ! R+ such that for some functions a; b 2 K,all t1 2 I, all x0 2 D and all (�; w) 2 Fs �W,

(I) a(kxk) � V (x; �) � b(kxk)

(II) V (x(t1); �(t1)) � V (x(t0); �(t0)) +R t1t0r(w(�); z(�))d�

where x(t1) is the state of � at time t1 resulting from the initial condition x0, �(�) and w(�).

The supply rate r should be interpreted as the supply delivered to the system. This

means that during time interval [t0; t1] work has been done on the system whenever

Z t1t0

rd�

is positive, while work is done by the system if this integral is negative. The function V ,

called a storage function or Lyapunov function, generalizes the notion of an energy function

for a dissipative system. (II) of De�nition 3.2, called the dissipation inequality, formalizes

the intuitive idea that a dissipative system is characterized by the property that the change

of the storage in any time interval [t0; t1] will never exceed the amount of supply that

ows into the system. Hence, there can be no int

anal - massachusetts institute of technologyhohmann.mit.edu/papers/lim_phd.pdf · anal ysis and...

Documents