realrobustnessradii and performance limitations of ......to thank professor b. ross barmish for...

Real Robustness Radii and Performance Limitations

of LTI Control Systems

by

Simon Sai-Ming Lam

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Electrical and Computer Engineering

University of Toronto

Copyright c© 2011 by Simon Sai-Ming Lam

Abstract

Real Robustness Radii and Performance Limitations

of LTI Control Systems

Simon Sai-Ming Lam

Doctor of Philosophy

Graduate Department of Electrical and Computer Engineering

University of Toronto

2011

In the study of linear time-invariant systems, a number of definitions, such as control-

lability, observability, not having decentralized fixed modes, minimum phase, etc., have

been made. These definitions are highly useful in obtaining existence results for solving

various types of control problems, but a drawback to these definitions is that they are

binary, which simply determines whether a system is, for instance, either controllable

or uncontrollable. In practical situations, however, there are many uncertainties in a

system’s parameters caused by linearization, modelling errors, discretizations, and other

numerical approximations and/or errors. So knowing that a system is controllable can

sometimes be misleading if the controllable system is actually “almost” uncontrollable as

a result of such uncertainties. Since an “almost” uncontrollable system poses significant

difficulty in designing a quality controller, a continuous measure of controllability, called

a controllability radius, is more desirable to use and has been widely studied in the past.

The main focus of this thesis is to extend the development behind the controllability

radius, with an emphasis on real parametric perturbations, to other definitions, replacing

the traditional binary ‘yes/no’ metrics with continuous measures. We study four topics

related to this development. First, we generalize the concept of real perturbation values

of a matrix to the cases of matrix pairs and matrix triplets. By doing so, we are able

to deal with more general perturbation structures and subsequently study, in addition

ii

to standard LTI systems, other types of systems such as LTI descriptor and time-delay

systems. Second, we introduce the real decentralized fixed mode (DFM) radius, the real

transmission zero at s radius, and the real minimum phase radius, which respectively

measure how “close” i) a decentralized LTI system is to having a DFM, ii) a centralized

system is to having a transmission zero at a particular point s in the complex plane,

and iii) a minimum phase system is to being a nonminimum phase system. These radii

are defined in terms of real parametric perturbations, and computable formulas for these

radii are derived using a characterization based on real perturbation values and the

aforementioned generalizations. Third, we present two efficient algorithms to i) solve the

general real perturbation value problem, and ii) evaluate the various real LTI robustness

radii introduced in this thesis. Finally as the last topic, we study the ability of a LTI

system to achieve high performance control, and characterize the difficulty of achieving

high performance control using a new continuous measure called the Toughness Index.

A number of examples involving the various measures are studied in this thesis.

iii

Dedication

To my dad, mom, sister, and Betty.

iv

Acknowledgements

First and foremost, I would like to thank Professor Edward J. Davison (Ted), for his

extraordinary guidance and patience as an exceptional supervisor and mentor. I have

learned a great deal from him, and will forever be inspired by his enthusiastic approach

to research. I will always cherish the time we spent working together, from the start of

my master’s to the end of my Ph.D.

I want to thank the members of my Ph.D. supervisory and defense committee: Pro-

fessor Bruce A. Francis, Professor Raymond H. Kwong, Professor Aleksandar Prodic,

Professor Manfredi Maggiore, and Professor Lacra Pavel. Their feedback and construc-

tive criticism really helped improve the quality of this thesis. In addition, I would like

to thank Professor B. Ross Barmish for being the external appraiser of this thesis and

providing valuable suggestions to this work.

Special thanks go to the students in the Systems Control Group for their friendship

and help over these many years.

I also want to acknowledge the financial support provided by the various scholarships

from the ECE department at University of Toronto, the research assistantships from

Professor Davison, and the Ontario Graduate Scholarship.

Finally, this work would not be possible without the love and support from my parents

David and Amy, my sister Christine, and my girlfriend Betty. I am forever indebted to

them, for their constant encouragement and belief in me, for all the sacrifices they had

to make, and for the great care they have given me.....not to mention the cooking.

v

Contents

List of Tables xiv

List of Figures xvi

List of Algorithms xvii

List of Symbols xviii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Real LTI Robustness Radii . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Decentralized fixed mode radius . . . . . . . . . . . . . . . . . . . 4

1.2.2 Transmission zero at s radius . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Minimum phase radius . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.4 Controllability radius of LTI descriptor systems . . . . . . . . . . 5

1.2.5 Spectral controllability radius of LTI time-delay systems . . . . . 6

1.3 Real Perturbation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Generalized real perturbation values . . . . . . . . . . . . . . . . 8

1.3.2 Restricted real perturbation values . . . . . . . . . . . . . . . . . 8

1.4 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 A New Continuous Measure of Performance Limitation . . . . . . . . . . 9

1.6 Thesis Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . 10

vi

1.7 Preliminaries and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7.1 Matrix notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7.2 Eigenvalues, singular values, real perturbation values notation . . 13

1.7.3 Real representation of complex matrices . . . . . . . . . . . . . . 14

1.7.4 Review of the robust servomechanism problem (RSP) . . . . . . . 15

2 Determination of Structural Real Perturbation Values 18

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Review of Real Perturbation Values . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2 Construction of minimum-norm perturbation ∆ . . . . . . . . . . 22

2.3 Generalized Real Perturbation Values . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.3 Generalized real perturbation values of the first kind . . . . . . . 25

2.4 Restricted Real Perturbation Values . . . . . . . . . . . . . . . . . . . . . 26

2.4.1 Lower bounds for restricted real perturbation values . . . . . . . . 26

2.4.2 Approximation of restricted real perturbation values . . . . . . . . 27

2.4.3 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 LTI Robustness Radii I 34

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Review of Controllability Radius . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Decentralized Fixed Mode Radius . . . . . . . . . . . . . . . . . . . . . . 40

3.3.1 Review of decentralized LTI systems and DFMs . . . . . . . . . . 40

3.3.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.3 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

vii

3.3.4 Extension to general information flow constraints . . . . . . . . . 45

3.4 Transmission Zero at s Radius and Minimum Phase Radius . . . . . . . . 48

3.4.1 Review of transmission zeros and minimum phase . . . . . . . . . 48

3.4.2 Transmission zero at s radius - definition . . . . . . . . . . . . . . 49

3.4.3 Minimum phase radius - definition . . . . . . . . . . . . . . . . . 49

3.4.4 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5.1 Properties of the DFM radius . . . . . . . . . . . . . . . . . . . . 52

3.5.2 Properties of the TZ and MP radius . . . . . . . . . . . . . . . . 52

3.5.3 Bounds on the global minimizers . . . . . . . . . . . . . . . . . . 55

3.6 Construction of Minimum-Norm Parametric Perturbations . . . . . . . . 56

3.6.1 Minimum-norm parametric perturbations - TZ at s radius . . . . 57

3.6.2 Minimum-norm parametric perturbations - MP radius . . . . . . . 58

3.6.3 Minimum-norm parametric perturbations - DFM radius . . . . . . 59

3.7 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.7.1 Example 1: real DFM radius . . . . . . . . . . . . . . . . . . . . . 61

3.7.2 Example 2: real TZ at s and MP radius . . . . . . . . . . . . . . 62

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 LTI Robustness Radii II 65

4.1 Structured LTI Robustness Radii . . . . . . . . . . . . . . . . . . . . . . 66

4.1.1 Structured real controllability radius . . . . . . . . . . . . . . . . 66

4.1.2 Structured real TZ at s radius and minimum phase radius . . . . 68

4.1.3 Structured real DFM radius . . . . . . . . . . . . . . . . . . . . . 70

4.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Extension to LTI Descriptor Systems . . . . . . . . . . . . . . . . . . . . 74

4.3 Extension to LTI Time-Delay Systems . . . . . . . . . . . . . . . . . . . 77

4.4 Eigenvalue Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

viii

4.4.1 Controllability radius and mobility of λ(A +BK) . . . . . . . . . 79

4.4.2 Transmission zero at s radius and eigenvalue mobility of RSP . . 81

4.4.3 DFM radius and eigenvalue mobility . . . . . . . . . . . . . . . . 82

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Algorithms 85

5.1 Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2 Basic Idea of Algorithm I and II . . . . . . . . . . . . . . . . . . . . . . . 88

5.2.1 Current approximation of maximum . . . . . . . . . . . . . . . . 89

5.2.2 Maximizing set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2.3 Updating the current approximation yk . . . . . . . . . . . . . . . 93

5.2.4 Stopping criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3 Algorithm I: Computing Real Perturbation Values . . . . . . . . . . . . . 94

5.3.1 Obtaining the maximizing set Sk . . . . . . . . . . . . . . . . . . 94

5.3.2 Updating the current maximum rk . . . . . . . . . . . . . . . . . 100


5.3.4 Computational requirements . . . . . . . . . . . . . . . . . . . . . 102


5.4 Algorithm II: Computing LTI Robustness Radii . . . . . . . . . . . . . . 103

5.4.1 The minimizing set Sk . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4.2 Updating the current minimum rk . . . . . . . . . . . . . . . . . . 110


5.4.4 Algorithm for solving Problem (2a) . . . . . . . . . . . . . . . . . 111

5.4.5 Algorithm for computing the real stabilizability radius . . . . . . 113

5.4.6 Computational requirements . . . . . . . . . . . . . . . . . . . . . 113


5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

ix

6 Toughness Index for High Performance Control 119

6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.2 Review: Asymptotic Locations of the Optimal Closed-Loop Poles . . . . 121

6.3 Toughness Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.3.1 Reduced model based on H(s) approximation . . . . . . . . . . . 124

6.3.2 Toughness Index - definition . . . . . . . . . . . . . . . . . . . . . 125

6.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.4.1 Example 1: longitudinal control of an airplane . . . . . . . . . . . 128

6.4.2 Example 2: a simple example . . . . . . . . . . . . . . . . . . . . 130

6.4.3 Example 3: distillation column . . . . . . . . . . . . . . . . . . . 131

6.5 Application to the Robust Servomechanism Problem . . . . . . . . . . . 132

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7 Applications 134

7.1 DFM Radius of Industrial Plants . . . . . . . . . . . . . . . . . . . . . . 135

7.1.1 DFM radius vs. CFM radius . . . . . . . . . . . . . . . . . . . . . 135

7.1.2 Unstable DFM radius vs. unstable CFM radius . . . . . . . . . . 136

7.1.3 Real DFM radius wrt s vs. real CFM radius wrt s . . . . . . . . . 137

7.2 Unstructured Decentralized Fixed Modes . . . . . . . . . . . . . . . . . . 137

7.3 Pairing of Inputs/Outputs: Real DFM Radius vs. Relative Gain Array . 140

7.3.1 Review: relative gain array (RGA) . . . . . . . . . . . . . . . . . 141

7.4 Structured Real Controllability Radius of the Pendulum . . . . . . . . . 144

7.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.5 Example - Transmission Zero at s Radius . . . . . . . . . . . . . . . . . . 148

7.5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.6 Toughness Index: Robust Servomechanism Problem . . . . . . . . . . . . 152

7.6.1 Example 1: RSP for a mass-spring system . . . . . . . . . . . . . 152

7.6.2 Example 2: control of a commercial hard disc drive . . . . . . . . 156

x

8 Conclusions 160

8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8.2.1 Restricted real perturbation values formula . . . . . . . . . . . . . 162

8.2.2 Extensions to other radii . . . . . . . . . . . . . . . . . . . . . . . 162

8.2.3 Coordinate dependence of LTI robustness radii . . . . . . . . . . . 163

A Review of Singular Values 165

A.1 Singular Values of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 165

A.2 Generalized Singular Values of a Matrix Pair . . . . . . . . . . . . . . . . 167

A.3 Restricted Singular Values of a Matrix Triplet . . . . . . . . . . . . . . . 168

B Construction of a Minimum-Norm Perturbation 172

C Proofs of Theorems and Lemmas 176

C.1 Proofs of Theorems of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . 176

C.1.1 Proof of Theorem 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . 176



C.2 Proofs of Theorems and Lemmas of Chapter 3 . . . . . . . . . . . . . . . 180


C.2.2 Proof of Lemma 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 182


C.2.4 Proof of Lemma 3.5.5 . . . . . . . . . . . . . . . . . . . . . . . . . 183

C.3 Proofs of Theorems of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . 184





xi




C.4 Proofs of Theorems and Lemmas of Chapter 5 . . . . . . . . . . . . . . . 189



C.4.3 Proof of Lemma 5.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 191

C.4.4 Proof of Lemma 5.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . 192


C.5 Proofs of Lemmas of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . 193

C.5.1 Proof of Lemma 6.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . 193

Bibliography 195

xii

List of Tables

2.1 Approximation and lower bounds of τi(Mj, Lj , Nj) for j = 0, 1, 2 . . . . . 32

2.2 Summary of various singular value and real perturbation value problems

for (M,L,N) and i ∈ 1, 2, . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1 Mobility bounds for closed-loop eigenvalues of example system . . . . . . 81

5.1 Summary of computational requirements of Algorithm I . . . . . . . . . . 102

5.2 Estimates of the global maximum (rk), the maximizer (γk), and the max-

imizing set (Sk) at each iteration k . . . . . . . . . . . . . . . . . . . . . 105

5.3 Summary of computational requirements of Algorithm II . . . . . . . . . 114

5.4 Estimates of the global minimum radius (rk), the minimizer (sk), and the

minimizing set (Sk) at each iteration k . . . . . . . . . . . . . . . . . . . 116

7.1 Comparison of the real DFM radius and the real CFM radius of various

industrial system models . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.2 Comparison of the real unstable DFM radius and the real unstable CFM

radius of various industrial system models . . . . . . . . . . . . . . . . . 137

7.3 Comparison of the modal real DFM radius and the modal real CFM radius

with respect to the given unstable pole s . . . . . . . . . . . . . . . . . . 138

7.4 Input/Output pairing using real DFM radius vs. RGA approach . . . . . 143

7.5 The structured real controllability radius of a multi-link inverted pendulum

system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

xiii

7.6 Transmission zero at s = 0 radius for various industrial examples . . . . . 150

7.7 Mobility bounds of the RSP’s closed-loop eigenvalues for various industrial

examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.8 Summary of the mass-spring example for tracking/disturbance poles =

[0,±j1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.9 Summary of the mass-spring example for tracking/disturbance poles =

[0,±j1,±j2,±j4,±j8,±j10] . . . . . . . . . . . . . . . . . . . . . . . . . 156

xiv

List of Figures

5.1 Plot of (a) y = 1 − (1− x)2, (b) the current approximation of the max-

imum, yk−1, and the corresponding maximizer xk−1, (c) all points that

achieve yk−1 and the maximizing set Sk (shaded area), and (d) the new

approximation yk and xk. . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Computing the maximizing set Sk: (a) current approximation rk−1 which

is achieved at γk−1; and (b) other points that achieve rk−1 and Sk =

(0, s1) ∪ (γk−1, s2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3 Example of using the derivative information of g ∈ Γ(rk−1) to determine

the maximizing set, Sk−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4 Intermediate results at Iteration 1 (top) and Iteration 2 (bottom). . . . 104

5.5 Illustration of computing Rθk: (a) endpoints that achieve a radius equal to

rk−1; and (b) interval(s) that achieve a radius less than rk−1; i.e., Rθk. . . 109

5.6 Illustration of computing Rk: (a) the set Rkθ is obtained for a particular θ;

and (b) the set Rθ is obtained by varying θ from 0→ π. . . . . . . . . . 110

5.7 Grid plot of the real controllability radius with respect to s ∈ C. . . . . . 117

5.8 Plot of Rk at (a) iteration 1; (b) iteration 3; (c) iteration 4; and (d)

iteration 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.1 Real DFM radius of the sampled system of (7.3) with respect to the mode

e2T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.2 Model of a multi-link inverted pendulum with v links [51]. . . . . . . . . 144

xv

7.3 Output response of the mass-spring system using cheap servomechanism

control with tracking/disturbance poles [0,±j1], and (a) ǫ = 10−4, (b)

ǫ = 10−8, and (c) ǫ = 10−12. . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.4 Input response of the mass-spring system using cheap servomechanism

control with tracking/disturbance poles [0,±j1], and (a) ǫ = 10−4, (b)

ǫ = 10−8, and (c) ǫ = 10−12. . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.5 Output response of the mass-spring system using cheap servomechanism

control with tracking/disturbance poles [0,±j1,±j2,±j4,±j8,±j10], and

(a) ǫ = 10−12, (b) ǫ = 10−18, and (c) ǫ = 10−24. . . . . . . . . . . . . . . . 157

7.6 Input response of the mass-spring system using cheap servomechanism

control with tracking/disturbance poles [0,±j1,±j2,±j4,±j8,±j10], and

(a) ǫ = 10−12, (b) ǫ = 10−18, and (c) ǫ = 10−24. . . . . . . . . . . . . . . . 158

xvi

List of Algorithms

5.1 Basic structure of Algorithm I and II. . . . . . . . . . . . . . . . . . . . . 90

5.2 Algorithm I for computing real perturbation values. . . . . . . . . . . . . . 101

5.3 Algorithm II for computing LTI robustness radii. . . . . . . . . . . . . . . 112

xvii

List of Symbols

Basic Algebra/Matrix Notation

R field of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

C field of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

C+ closed right-half of the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

CU closed upper-half of the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

F denotes either R or C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

M complex conjugate of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

MT transpose of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

MHcomplex conjugate transpose of M ; i.e., MH = M

T. . . . . . . . . . . . . . . . . . . 12

M−H =(MH

)−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

M+ Moore-Penrose pseudoinverse of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

rank(M) rank of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

nullity(M) nullity of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

ReM real components of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

ImM imaginary components of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

det(M) determinant of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

trace(M) trace of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

‖M‖2 spectral norm of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

λ(M) the set of eigenvalues of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

λi(M) i-th (ordinary) eigenvalue of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

xviii

λi(M,N) i-th generalized eigenvalue of the matrix pair (M,N) . . . . . . . . . . . . . . . . . 13

σi(M) i-th (ordinary) singular value of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

σi(M,N) i-th generalized singular value of the matrix pair (M,N) . . . . . . . . . . . . . 14

σi(M,L,N) i-th restricted singular value of the matrix triplet (M,L,N) . . . . . . . . . . 14

τi(M) i-th (ordinary) real perturbation value of M . . . . . . . . . . . . . . . . . . . . . . . . . . 14

τi(M,N) i-th generalized real perturbation value of the matrix pair (M,N) . . . . 14

τi(M,L,N) i-th restricted real perturbation value of the matrix triplet (M,L,N) . 14

Other Notation

n = 1, 2, . . . , n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

deg(p(s)) degree of the polynomial p(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74

MR =[ReM − ImMImM ReM

]; the real representation of M . . . . . . . . . . . . . . . . . . . . . . . . . 14

Tq = 1√2

[Iq jIq

−jIq −Iq

]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

MR

γ =[

ReM −γ ImMγ−1 ImM ReM

]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

xix

Chapter 1

Introduction

1.1 Motivation

In this thesis, we focus on linear-time invariant (LTI) systems described by

x = Ax+Bu (1.1)

y = Cx+Du,

where x ∈ Rn, u ∈ Rm, and y ∈ Rr are respectively the system states, inputs, and

outputs, and A, B, C, and D are constant matrices with the appropriate dimensions for

n ≥ 1, m ≥ 1, r ≥ 1, and max(m, r) ≤ n. In the study of such LTI systems, a num-

ber of definitions have been made that are highly useful in obtaining existence results

for solving various types of control problem. The following definitions, for instance, are

widely used: controllability and stabilizability of (A,B); observability and detectability

of (C,A); not having decentralized fixed modes (DFM); and minimum phase. In these

cases, the definitions are all binary metrics; i.e., a system is either controllable or not, a

decentralized system either does or does not have a DFM, a system is either minimum

phase or nonminimum phase, etc. In real life, however, there are many uncertainties

in the system parameters such as those arising from modelling errors, linearization, dis-

cretization, and other numerical and/or approximation errors. As a result, a controllable

1

Chapter 1. Introduction 2

system may in fact be uncontrollable when these uncertainties are accounted for; in other

words, the system is “almost” uncontrollable. Similarly, a minimum phase system may

almost be nonminimum phase, a decentralized system may almost have a DFM, and so

on. As a motivating example, consider the following two systems:

Plant #1:

x =

1 2

0 3

x+

2

1

u

y =

[

1 2

]

x

Plant #2:

x =

1 2

0 3

x+

2

10−12

u

y =

[

10−12 2

]

x,

where both systems are controllable and observable. Note, however, that if the two rel-

atively small 10−12 terms in the model parameters of Plant #2 are replaced with zeros,

then Plant #2 becomes uncontrollable and unobservable, and in fact, the system becomes

unstabilizable and undetectable. So in the case of Plant #2, it is rather misleading to

know that the system is controllable and observable, since a relatively very small para-

metric perturbation can cause the system to no longer be so. In comparison, Plant #1

is in some sense “more” controllable and observable than Plant #2, since it can tolerate

much larger uncertainties. From this example, one can see that a continuous measure of

controllability,1 called a controllability radius, is more informative and desirable than the

traditional ‘yes/no’ controllability metric, which simply determines whether a system is

controllable or not. The controllability radius has been widely studied in the literature,

and it is desirable to have similar continuous measures to characterize various other LTI

definitions and properties found in the control literature, which is the main focus of this

thesis.

Two types of continuous measures are studied in this thesis. In the first, we extend

the notion of the controllability radius to other LTI properties, with a focus on real

1Likewise in terms of observability, stabilizability, detectability, and the other binary definitionsmentioned above.


parametric perturbations. In particular, we consider parametric perturbations of the

following form:

A → A+∆A, B → B +∆B, (1.2)

C → C +∆C , D → D +∆D,

where (∆C ,∆A,∆B,∆D) are real perturbations, and define various real LTI robustness

radii, which like the controllability radius, measure a system’s ability to maintain the

various LTI properties when the system is subject to uncertainties described by the

perturbation structure in (1.2). We also extend the controllability radius of LTI systems

given in (1.1) to LTI descriptor and time-delay systems. Associated with the study of

LTI robustness radii, we investigate two related topics: (i) the study of real perturbation

values; and (ii) the development of efficient algorithms for computing these robustness

radii. In the second type of measure, we define a Toughness index as a measure of

a system’s limitation in achieving high performance, which may occur even when the

system is minimum phase.

1.2 Real LTI Robustness Radii

In this thesis, we extend the notion of a controllability radius to three other important

LTI definitions: i) decentralized fixed modes; ii) transmission zeros; and iii) minimum

phase property of a system. We also extend the controllability radius to LTI descriptor

and time-delay systems. Note that some of these radii were previously studied in the

literature, but often in terms of complex parametric perturbations. A contribution of

this thesis is then the study of these radii in terms of real perturbations, which is more

realistic since the system matrices are real matrices.


1.2.1 Decentralized fixed mode radius

Decentralized fixed modes (DFM) are generalizations of the centralized notion of uncon-

trollable and unobservable modes to decentralized systems. Recall that if one is to design

a centralized LTI controller for the LTI system (1.1), then the closed-loop spectrum will

always contain the plant’s uncontrollable and unobservable modes. In a similar manner,

the spectrum of a closed-loop decentralized LTI system always contains the set of de-

centralized fixed modes [87]. Subsequently, a LTI system can be decentrally stabilized

by means of a LTI controller if and only if it has no unstable DFMs. Thus, following

the same motivation as for the controllability radius, we propose a continuous measure,

called the real decentralized fixed mode (DFM) radius, to determine the following:

How “close” is a decentralized LTI control system to having a decentralized

fixed mode?

1.2.2 Transmission zero at s radius

Transmission zeros of a LTI system are the invariants of a system that play an important

role in various areas of control theory, such as regulation problems, decoupling theory,

control performance, etc. In the robust servomechanism problem (RSP), for instance,

one of the necessary and sufficient conditions for the existence of a solution is that the

plant’s set of transmission zeros do not coincide with any of the tracking and disturbance

poles of the servocompensator [15]; e.g., there exists a solution to the RSP for tracking

constant references and rejecting constant disturbance signals if and only if the plant has

no transmission zeros at the origin. Hence, one may be interested in the real transmission

zero at s radius, which measures the following:

How “close” is a LTI system to having a transmission zero at a particular

point s (e.g., the origin s = 0) in the complex plane?


1.2.3 Minimum phase radius

Minimum phase systems are systems that have no transmission zeros in the closed right-

half of the complex plane. It is well known that minimum phase systems have certain

advantages over nonminimum phase systems. For example, minimum phase systems

can achieve perfect regulation [79], and perfect tracking/disturbance rejection [20]. So

similarly, we propose a continuous measure, called the real minimum phase radius, to

characterize the following:

How “close” is a minimum phase system to becoming a nonminimum phase

system?

1.2.4 Controllability radius of LTI descriptor systems

A LTI descriptor system, also known as a singular system or a generalized state-space

system, is described by

Ex = Ax+Bu (1.3)

y = Cx+Du,

where x ∈ Rn, u ∈ Rm, and y ∈ Rr are the system states, inputs, and outputs, respec-

tively, and E ∈ Rn×n is a singular matrix. Such systems often arise when modelling

interconnected systems and constrained mechanical systems (e.g., many engineering sys-

tems, such as power systems and electrical networks), where they cannot be described

by the regular state-space model (1.1). Definitions of controllability similar to that of

(1.1) have been defined, and a controllability radius for descriptor systems was studied

by Byers [9]. In this thesis, we extend the work of Byers [9] to the study of the real con-

trollability radius of descriptor systems, where one is interested in only real parametric

perturbations.


1.2.5 Spectral controllability radius of LTI time-delay systems

In this thesis, we consider the class of LTI systems with time-delays described by

x(t) = A0x(t) + A1x(t− τ) +Bu(t− τ) (1.4)

y = Cx(t− τ),

where x ∈ Rn, u ∈ R

m, and y ∈ Rr are the system states, inputs, and outputs, re-

spectively, and τ > 0 represents a time-delay. Spectral controllability is a property of

time-delay systems somewhat analogous to the controllability property of LTI systems

(1.1). In particular, spectral controllability is a necessary and sufficient condition for solv-

ing the finite spectrum assignment problem for the time-delay system (1.4) (see [63, 88]

and the references therein). So, we introduce the real spectral controllability radius, which

measures

How “close” is a spectrally controllable time-delay system to becoming spec-

trally uncontrollable?

1.3 Real Perturbation Values

The so-called real perturbation values of a matrix introduced in [4] is a key tool used

to compute the various real LTI robustness radii described above. As we see in more

detail in Chapter 3, one of the key steps in obtaining a formula for computing the real

controllability radius is solving the following subproblem: Given s ∈ C and system (1.1),

compute the “smallest” real parametric perturbations (∆A,∆B) such that the perturbed

system – perturbed according to (1.2) – is uncontrollable at s. This problem is formulated

as follows (see Section 3.2):

inf

∆A ∈ Rn×n

∆B ∈ Rn×m

∥∥∥∥

[

∆A ∆B

]∥∥∥∥2

∣∣∣∣(A+∆A, B +∆B) is uncontrollable at s

, (1.5a)


or equivalently,

inf

∆A ∈ Rn×n

∆B ∈ Rn×m

∥∥∥∥

[

∆A ∆B

]∥∥∥∥2

∣∣∣∣rank

([

A− sI B

]

+

[

∆A ∆B

])

< n

.

(1.5b)

So from (1.5b), we see that to obtained a formula for computing the real controllability

radius, one must first solve the following type of problem: Given a complex matrix M

and an integer i > 0, determine

inf∆ real

‖∆‖2 | rank(M −∆) < i . (1.6)

In fact, computing the various real LTI robustness radii described above all rely heavily

on solving (1.6), where the solution to (1.6) is obtained in [4] and is given by the real

perturbation values of the matrix M .

However, we see from (1.6) that the concept of real perturbation values considers

only unstructured perturbations, which is sufficient for computing the real controllability

radius with respect to the unstructured perturbation form in (1.2). Now suppose we

consider structured parametric perturbations of the following form:

A → A+ E∆AF , B → B + E∆BG, (1.7)

C → C +H∆CF , D → D +H∆DG,

where E , F , G, and H are given complex matrices. The formulation in (1.5b) now

becomes

inf

∆A ∈ Rn×n

∆B ∈ Rn×m

∥∥∥∥

[

∆A ∆B

]∥∥∥∥2

∣∣∣∣∣∣∣

rank

[

A− sI B

]

+E[

∆A ∆B

]

F 0

0 G

< n

,

(1.8)

which cannot be solved using ordinary real perturbation values as described above. So in

order to account for more general structured perturbations, which is important in various


applications of the LTI robustness radii and also in studying the real controllability radius

of LTI descriptor and time-delay systems, the following two generalizations of (1.6) are

obtained in this thesis: (i) the generalized real perturbation values; and (ii) the restricted

real perturbation values.

1.3.1 Generalized real perturbation values

In the first generalization, we extend (1.6) to the following problem: Given a complex

matrix pair (M,N) and an integer i > 0, determine

inf∆ real

‖∆‖2 | rank(M −∆N) < i . (1.9)

We call the set of solutions to (1.9) (i.e., for various integer i) the generalized real per-

turbation values of (M,N), and obtain formulas for computing such values.

1.3.2 Restricted real perturbation values

In the second generalization of (1.6), we consider the following problem: Given a complex

matrix triplet (M,L,N) and an integer i > 0, determine

inf∆ real

‖∆‖2 | rank(M − L∆N) < i . (1.10)

We call the set of solutions to (1.10) the restricted real perturbation values of (M,L,N),

and provide lower bounds for these values.

1.4 Algorithms

A formula for computing the i-th real perturbation value of a complex matrixM ∈ Cn×m,

denoted by τi(M), for i ∈ 1, . . . ,min(n,m) is given by ([4])

τi(M) = supγ∈(0,1]

σ2i−1

ReM −γ ImM

γ−1 ImM ReM

, (1.11)


while a formula for the real controllability radius rcRof a system (A,B) is given by ([42])

rcR(A,B) = inf

s∈Csup

γ∈(0,1]σ2n−1

ReW −γ ImW

γ−1 ImW ReW

, (1.12)

where W =

[

A− sI B

]

. Therefore, to compute real perturbation values and the

controllability radius, one has to solve a 1-D and 2-D optimization problem, respectively,

where both, in general, are nonlinear optimization problems with possibly multiple local

extrema.

In this thesis, we propose two algorithms to solve the general forms of problems (1.11)

and (1.12). These general forms correspond to solving various types of real perturba-

tion values (including the generalized and restricted real perturbation values mentioned

above), and various LTI robustness radii introduced in this thesis. One of the key ad-

vantages of the proposed algorithms is that they are insensitive to initial search points,

and so are more efficient compared to using generic nonlinear search techniques, which

generally need to be executed multiple times using many different initial search points.

1.5 A New ContinuousMeasure of Performance Lim-

itation

In this thesis, we also introduce a continuous Toughness index to characterize a system’s

ability to achieve high performance as measured by “cheap control” [20, 79]. It is well

known that a minimum phase plant has the property that in principle, ignoring nonlinear

effects, a controller can be designed to provide “perfect control” [20,79]. However, there

are examples of systems that are minimum phase but are very difficult in obtaining

“good control” even using high gain controllers. For example, in the study of designing

a controller for a disk drive with a large number of sinusoidal disturbances [10], this

problem arises. Thus in the case of the “perfect control problem”, the binary notion of


minimum/nonminimum phase is replaced by a continuous measure called a Toughness

Index, which for a minimum phase plant, characterizes how difficult it is to achieve high

performance in solving the RSP [18].

1.6 Thesis Outline and Contributions

This thesis contains eight chapters and is organized as follows. The remainder of this

introductory chapter presents some preliminary material, namely: the introduction of

some basic matrix notation related to eigenvalues, singular values, and real perturbation

values; a review of the real representation of complex matrices; and a review of the robust

servomechanism problem (RSP). Instead of reviewing the literature related to the topics

of the thesis here in this first chapter, the reviews are given in their respective chapters.

Chapter 2 reviews the concept of real perturbation values of a single matrix [4], and

generalizes the concept to the generalized real perturbation values of a matrix pair, and

the restricted real perturbation values of a matrix triplet. The main results of Chapter 2

are the development of formulas and lower bounds for computing these values. Chapter 3

introduces the real decentralized fixed mode (DFM) radius, the real transmission zero

at s radius, and the real minimum phase radius. Again, computable formulas for these

radii are presented, along with a list of their properties. Chapter 4 continues the study

of these radii to more general perturbation structures, and extends the results of the

previous chapter to LTI descriptor and time-delay systems. The mobility of closed-

loop eigenvalues and its relationship to the various radii are also studied in Chapter 4.

Chapter 5 presents two efficient algorithms for computing the general real perturbation

value problem, and the general robustness radius problem. Chapter 6 reviews the cheap

LQR control problem and introduces a Toughness Index to characterize a system’s ability

to achieve high performance as measured by the cheap control problem. Chapter 7 studies

a few numerical examples involving the various continuous measures introduced in this


thesis. Finally, the last Chapter 8 summarizes the thesis and discusses possible future

research directions.

This thesis also contains three appendices. Appendix A reviews the singular values

of a single matrix, the generalized singular values of a matrix pair, and the restricted

singular values of a matrix triplet. Appendix B reviews the procedure, as found in [4,44],

for computing real minimum-norm perturbations corresponding to the real perturbation

values. Finally, Appendix C contains the proofs of the various theorems and lemmas

found in the thesis.

The main contributions of this thesis can be summarized as follows:

• A generalization of real perturbation values to account for more general perturba-

tion structures; in particular, we introduce and present formulas and lower bounds

for computing the following:

– the generalized real perturbation values of a matrix pair; and

– the restricted real perturbation values of a matrix triplet;

• An extension of the controllability radius concept to characterize the robustness of

other LTI properties with respect to real parametric perturbations; in particular,

we introduce

– the real decentralized fixed mode (DFM) radius,

– the real transmission zero at s radius, and

– the real minimum phase radius;

• An extension of the various real LTI robustness radii to structured perturbations

and an extension of the controllability radius to LTI descriptor and time-delay

systems;

• The development of an efficient algorithm to compute the various classes of real

perturbation values;


• The development of an efficient algorithm to compute the various LTI robustness

radii; and

• The introduction of the Toughness Index to characterize a system’s ability to

achieve high performance as measured by the cheap control performance index.

1.7 Preliminaries and Notation

Most of the notation used in this thesis is standard and is introduced throughout the

thesis. Nevertheless, some notation are introduced at this point.

1.7.1 Matrix notation

Let the fields of real and complex numbers be denoted by R and C, respectively. Through-

out the thesis, F is used to denote either R or C. The closed right-half and the closed

upper-half of the complex plane are denoted by C+ and CU , respectively, and the non-

negative real line [0,+∞) is denoted by R+. Also, let N = 1, 2, 3, . . .. If n ∈ N, then n

refers to the set 1, 2, . . . , n.

Now, associated with a matrix M ∈ Fm×n, we define the following notation:

M complex conjugate of M

MT transpose of M

MH complex conjugate transpose of M ; i.e., MH = MT

M−H(MH

)−1(asssuming MH is nonsingular)

M+ Moore-Penrose pseudoinverse of M

rank(M) rank of M

nullity(M) nullity of M

ReM real components of M

ImM imaginary components of M .


If M is square (i.e., m = n), then we use the following notation:

λ(M) the set of eigenvalues of M

det(M) determinant of M

trace(M) trace of M .

The i-th singular value of M ∈ Cn×m is denoted by σi(M), where σ1(M) ≥ · · · ≥

σmin(n,m)(M). The spectral norm of M ∈ Cn×m is denoted by ‖M‖2, and is equal to

σ1(M).

1.7.2 Eigenvalues, singular values, real perturbation values no-

tation

In this thesis, we define and refer to a number of different classes of eigenvalues, singular

values, and real perturbation values pertaining to single matrices, matrix pairs, and

matrix triplets. To avoid having to introduce excess notation, we use λ, σ, and τ to

denote eigenvalues, singular values, and real perturbations values, respectively, and it is

clear from the number of matrices involved which class is being referenced.

To clarify, given matrices M ∈ Fn×n and N ∈ Fn×n, we use the following notation:

Eigenvalues:

λi(M) the i-th (ordinary) eigenvalue of M

λi(M,N) the i-th generalized eigenvalue of the matrix pair (M,N).

Recall that the generalized eigenvalues of (M,N), denoted by λ(M,N), are the set of

(complex) values s such thatMx = sNx for some nonzero x ∈ Fn; x is called a generalized

eigenvector of (M,N). Note that when using λi(M), we are assuming thatM is Hermitian

(i.e., M = MH), and hence has only real eigenvalues. It is assumed λ1(M) ≥ · · · ≥

λn(M). Likewise, λi(M,N) also assumes that M and N are both Hermitian and that the

corresponding (real) generalized eigenvalues are ordered nonincreasingly; i.e., λ1(M,N) ≥


· · · ≥ λn(M,N). Finally, it is understood that λ(M) = λ(M, I).

For general matrices M ∈ Fn×m, L ∈ Fn×l, and N ∈ Fp×m, we have the following

notation:

Singular values:

σi(M) the i-th (ordinary) singular value of M

σi(M,N) the i-th generalized singular value of the matrix pair (M,N)

σi(M,L,N) the i-th restricted singular value of the matrix triplet

(M,L,N).

Real perturbation values:

τi(M) the i-th (ordinary) real perturbation value of M

τi(M,N) the i-th generalized real perturbation value of the matrix pair

(M,N)

τi(M,L,N) the i-th restricted real perturbation value of the matrix triplet

(M,L,N).

Similar to the ordering of eigenvalues, the various classes of singular values and real

perturbation values are also assumed to be arranged in an nonincreasingly order.

1.7.3 Real representation of complex matrices

In this thesis, we sometimes make use of the real representation or real form of complex

matrices. Namely, if M ∈ Cn×m, then the representation of M in real form, denoted by

MR, is given by (e.g., see Section A.1.3 in [38])

MR =

ReM − ImM

ImM ReM

. (1.13)

For q ∈ N and defining

Tq :=1√2

Iq jIq

−jIq −Iq

, (1.14)


it can easily be verified that Tq is unitary (i.e., THq Tq = TqT

Hq = Iq), and that

MR =

ReM − ImM

ImM ReM

= Tn

M 0

0 M

Tm. (1.15)

1.7.4 Review of the robust servomechanism problem (RSP)

Consider the following LTI system with disturbances:

x = Ax+Bu+ Ew

y = Cx+Du+ Fw (1.16)

e = yref − y,

where x ∈ Rn is the state of the system, u ∈ Rm is the input, y ∈ Rr is a measurable

output, w ∈ RΩ is an unmeasurable disturbance, e ∈ Rr is the error, and yref ∈ Rr

is a specified tracking signal. Denoting λi as a tracking/disturbance pole [18], where

Reλi ≥ 0, for i = 1, . . . , p, let the coefficients of the tracking/disturbance polynomial be

given byp∏

i=1

(λ− λi) = λp + δpλp−1 + · · ·+ δ2λ+ δ1. (1.17)

The control objective is then to solve the RSP [18] for the system (1.16) with respect to

the class of tracking/disturbance signals described by (1.17); i.e., find a LTI controller

such that

(i) the closed-loop system is asymptotically stable;

(ii) asymptotic tracking/regulation occurs; i.e., limt→∞

e(t) = 0 for all initial conditions;

and

(iii) conditions (i)–(ii) hold for any arbitrary perturbations in the plant model (1.16)

that do not cause the resultant perturbed closed-loop system to be unstable.

The following existence conditions for a solution to the RSP to exist are obtained

from [18].


Lemma 1.7.1. There exists a solution to the RSP for (1.16) with respect to the track-

ing/disturbance poles of (1.17) if and only if the following conditions are all true:

i) (C,A,B) is stabilizable and detectable;

ii) the number of inputs is more than or equal to the number of outputs; i.e., m ≥ r;

and

iii) the transmission zeros of (C,A,B,D) exclude the tracking/disturbance poles (1.17).

Assume now that the existence conditions of Lemma 1.7.1 are satisfied, then this

implies that there exists K0 ∈ Rm×n and K1 ∈ Rm×rp, and a LTI controller of the form

v = K0x+K1η, (1.18)

where x ∈ Rn and η ∈ Rrp are the inputs, and v ∈ Rm are the outputs of the controller,

such that (1.18) stabilizes the following augmented plant-servocompensator system [18]:

˙x

˙η

=

A 0

BC C

x

η

+

B

BD

v (1.19)

z =

[

0 D]

x

η

,

where C, B, and D are given by

C := block diag(C, . . . , C︸︷︷︸

r

)

B := block diag(B, . . . ,B︸︷︷︸

r

)

D := block diag(D, . . . ,D︸︷︷︸

r

),


and where

C :=

0 1 0 · · · 0

0 0 1 · · · 0

......

.... . .

...

0 0 0 · · · 1

−δ1 −δ2 −δ3 · · · −δp

∈ Rp×p, B :=

0

0

...

0

1

∈ Rp×1, and

D :=

[

1 0 0 . . . 0

]

∈ R1×p.

Using the above results, a controller that solves the RSP for system (1.16) is then given

by

η = Cη + Be (1.20a)

u = K0x+K1η, (1.20b)

where e ∈ Rr and u ∈ Rm are the controller inputs and outputs, respectively, K0 and

K1 are the same as given in (1.18), the servocompensator [18] of the controller is given

in (1.20a), and the state x can be estimated using an observer.

Chapter 2

Determination of Structural Real

Perturbation Values

2.1 Introduction

As mentioned in the previous chapter, a key step in deriving formulas for computing

the various LTI robustness radii relies heavily on solving the following problem: Given a

complex matrix M ∈ Cn×m and i ∈ 1, . . . ,min(n,m), compute

inf∆∈Fn×m

‖∆‖2 | rank(M −∆) < i , (2.1)

where F ∈ C,R. When the perturbations ∆ are allowed to be complex (i.e., F = C),

the solution to (2.1) can be computed using the singular values of M ; in particular (see

Appendix A.1)

σi(M) = inf∆∈Cn×m

‖∆‖2 | rank(M −∆) < i .

On the other hand, when only real perturbations are allowed (i.e., F = R), the solution

can be obtained using the so-called real perturbation values of M [4]. Since the system

matrices of LTI systems (e.g., (C,A,B,D) in (1.1)) are real matrices, it is more realistic to

restrict the class of parametric perturbations (i.e., ∆A, ∆B, ∆C , and ∆D in (1.2)) to real

18

Chapter 2. Determination of Structural Real Perturbation Values 19

perturbations. So, a large part of the results in this thesis and particularly in Chapter 3

on real robustness radii is obtained using real perturbation values. A more detailed

review of real perturbation values of a single matrix is given in Section 2.2, including

computable formulas and a procedure for computing complex and real minimum-norm

perturbations when the infimum is achieved.

For the purposes of this thesis, two generalizations are made to the real perturba-

tion values of a single matrix with respect to more general perturbation structures. In

particular, the real perturbation value problem (2.1) is extended to deal with matrix

pairs and matrix triplets. The generalization to matrix pairs is need to study the real

controllability radius of LTI descriptor and time-delay systems, while the generalization

to matrix triplets is need to study the various LTI robustness radii with respect to more

general perturbation structures (e.g., (1.7)). This chapter is organized as follows.

In Section 2.3, we extend the above problem (2.1) to matrix pairs, and consider the

following problem: Given a complex matrix pair M ∈ Cn×m and N ∈ Cp×m, and an

integer i ∈ 1, . . . ,min(n,m), compute

inf∆∈Fn×p

‖∆‖2 | rank(M −∆N) < i . (2.2)

Here we are again interested mainly in real perturbations, as the case of complex per-

turbations is already well studied and is shown to be related to the generalized singular

values of (M,N) (see Appendix A.2). For obvious reasons, we call the set of solutions

to problem (2.2), for real perturbations and i ∈ 1, . . . ,min(n,m), the generalized real

perturbation values of the matrix pair (M,N). Formulas for computing such values are

obtained in Section 2.3.

Then in Section 2.4, we study the generalization of (2.1) to matrix triplets: Given

M ∈ Cn×m, L ∈ Cn×l, N ∈ Cp×m, and i ∈ 1, . . . ,min(n,m), determine

inf∆∈Fl×p

‖∆‖2 | rank(M − L∆N) < i . (2.3)

Again, the case of dealing with complex perturbations has been studied previously and led


to the introduction of so-called restricted singular values (see Appendix A.3). So, we call

the set of solutions to (2.3), for the case of real perturbations and i ∈ 1, . . . ,min(n,m),

the restricted real perturbation values of the matrix triplet (M,L,N). Unlike the previ-

ous cases, however, only (computable) lower bounds are obtained for the restricted real

perturbation values, and are presented in Section 2.4, along with a method for estimating

the restricted real perturbation values.

It should be noted that an area of study closely related to real perturbation values is

the study of structured singular values (see [71] and references therein), which is developed

as a tool for analysis and design of robust feedback systems with structured uncertainties.

In particular, it has been applied to the study of the stability radius (e.g., see [37,74] and

the references therein). However, the study of structure singular values and the stability

radius is beyond the scope of this thesis and the readers are referred to the references

above for further detail.

2.2 Review of Real Perturbation Values

The real perturbation values of a complex matrix is a set of numbers analogous to the

singular values of a matrix, where the former deals with only real perturbations, while

the latter considers both real and complex perturbations. When real perturbation values

were first introduced in [4], two kinds were defined.

Definition 2.2.1 (Real perturbation values [4]). Given M ∈ Cn×m and an integer i ∈

1, . . . ,min(n,m), the i-th real perturbation value of the first kind of M is defined as

τi(M) :=1

inf∆∈Rm×n

‖∆‖2 | nullity(Im −∆M) ≥ i ,

and the i-th real perturbation value of the second kind is defined as

τi(M) := inf∆∈Rn×m

‖∆‖2 | rank (M −∆) < i .


If there does not exist a real ∆ ∈ Rm×n such that nullity(Im −∆M) ≥ i, then τi(M) = 0.

Likewise, if there exists no real ∆ ∈ Rn×m such that rank(M −∆) < i, then τi(M) =∞.

Remark 2.2.1. As mentioned earlier, the real perturbation values of a matrix are closely

related to the singular values of the matrix with the difference that the former takes into

account only real perturbations, whereas the latter considers all complex perturbations. By

comparing Definition 2.2.1 with (A.1) in Appendix A.1, one can easily see the similarities.

Real perturbation values of the first kind were first applied to studying the real

stability radius [74] of LTI systems. On the other hand, real perturbation values of the

second kind are more relevant to studying the real controllability radius [42]. Since this

thesis focuses mainly on LTI robustness radii related to the latter, we deal, in general,

with real perturbation values of the second kind. So for the remainder of this thesis, we

assume the second kind is referred to unless specified otherwise.

2.2.1 Formulas

The real perturbation values of a matrix can be computed using the following formulas [4].

Theorem 2.2.1 ([4]). Let M ∈ Cn×m and i ∈ 1, . . . ,min(n,m), and assume that

τi(M) > 0 and τi(M) <∞. Then,

τi(M) = infγ∈(0,1]

σ2i

ReM −γ ImM

γ−1 ImM ReM

(2.4)

and

τi(M) = supγ∈(0,1]

σ2i−1

ReM −γ ImM

γ−1 ImM ReM

. (2.5)

For ease of presentation, we define the following notation to be used in the remainder


of this thesis:

MR

γ :=

ReM −γ ImM

γ−1 ImM ReM

(2.6)

=

γI 0

0 I

MR

γ−1I 0

0 I

.

2.2.2 Construction of minimum-norm perturbation ∆

Let M ∈ Cn×m and i ∈ 1, . . . ,min(n,m). Also, assume that the i-th singular value

σi(M) and the i-th real perturbation value τi(M) are computed. A complex minimum-

norm perturbation corresponding to σi(M) is then any perturbation ∆ ∈ Cn×m such that

rank(M −∆) < i and ‖∆‖2 = σi(M). Similarly, a real minimum-norm perturbation

corresponding to τi(M) (when the infimum is achieved) is any ∆ ∈ Rn×m such that

rank(M −∆) < i and ‖∆‖2 = τi(M). In this subsection, we discuss how to obtain these

minimum-norm perturbations.

Complex minimum-norm perturbations

Complex minimum-norm perturbations can be constructed using the following general

result on singular value decomposition (e.g., see [44]).

Theorem 2.2.2. Given M ∈ Cn×m and i ∈ 1, . . . ,min(n,m), let a singular value

decomposition (e.g., see Appendix A.1) of M be given as follows:

M =

min(n,m)∑

k=1

σk(M) ukvHk ,

where U = [u1, · · · , un] and V = [v1, · · · , vm] are both unitary matrices. Let X be a vector

space spanned by vi, . . . , vm, and let X = [vi, . . . , vm] be a matrix whose columns form

a basis of X . Then,

xH(σi(M)2 Im −MHM)x ≥ 0, for all x ∈ X .


Furthermore, let

∆ :=

m∑

k=i

σk(M) ukvHk = MXXH . (2.7)

Then,

rank(M −∆) < i and ‖∆‖2 = σi(M) .

Real minimum-norm perturbations

A real minimum-norm perturbation can be constructed using the following theorem found

in [4, 44].

Theorem 2.2.3 ([44]). Let M ∈ Cn×m and i ∈ 1, . . . ,min(n,m). If τi(M) =∞, then

there exists no ∆ ∈ Rn×m such that rank(M −∆) < i. Assume now that τi(M) < ∞,

and let X be any (m − i + 1)-dimensional subspace of Cm such that for all x ∈ X , the

following hermitian-symmetric inequality is satisfied:

xH(τi(M)2 Im −MHM

)x ≥

∣∣xT(τi(M)2 Im −MTM

)x∣∣ . (2.8)

Now let X ∈ Cm×(m−i+1) be any matrix whose columns form a basis of X . Also, let

∆ :=

[

Re (MX) Im (MX)

] [

Re (X) Im (X)

]+

∈ Rn×m. (2.9)

Then,

rank(M −∆) < i and ‖∆‖2 = τi(M) .

So a real minimum-norm perturbation can be constructed by finding a basis of the

subspace that satisfies (2.8). To obtain such a basis, a more detailed procedure can be

found in [4, 44], which is summarized in Appendix B.


2.3 Generalized Real Perturbation Values

2.3.1 Definition

We now make the following definition, which is a generalization of the ordinary real

perturbation values of the second kind in Section 2.2.

Definition 2.3.1 (Generalized real perturbation values). Given M ∈ Cn×m, N ∈ Cp×m,

and i ∈ 1, . . . ,min(n,m), the i-th generalized real perturbation value is defined as

τi(M,N) := inf∆∈Rn×p

‖∆‖2 | rank (M −∆N) < i .

If there does not exist a real matrix ∆ ∈ Rn×p such that rank (M −∆N) < i, then

τi(M,N) =∞.

Note that Definition 2.3.1 should technically be referred to as the definition of gener-

alized real perturbation values of the second kind (cf. Definition 2.2.1). For the purpose

of this thesis, this definition is assumed, unless specified otherwise, when referring to the

generalized real perturbation values of a matrix pair. For completeness, a definition for

the generalized real perturbation value of the first kind is given later in Subsection 2.3.3.

Also, it should be noted that N in Definition 2.3.1 is not necessarily invertible, or even

square. When N is nonsingular, the generalized real perturbation value problem reduces

to the ordinary real perturbation value problem of a single matrix (cf. Definition 2.2.1).

In particular, when N is nonsingular, then

rank(M −∆N) = rank(MN−1 −∆

);

so, it can be shown that τk(M,N) = τk(MN−1) for nonsingular N .

2.3.2 Formula

The generalized real perturbation values of a matrix pair can be computed using the

following result.


Theorem 2.3.1. Let M ∈ Cn×m, N ∈ C

p×m, and i ∈ 1, . . . ,min(n,m), and assume

that τi(M,N) <∞. Then,

τi(M,N) = supγ∈(0,1]

σ2i−1

(MR

γ , NR

γ

)(2.10)

= supγ∈(0,1]

σ2i−1

ReM −γ ImM

γ−1 ImM ReM

,

ReN −γ ImN

γ−1 ImN ReN

.

Proof. See Appendix C.1.1.

2.3.3 Generalized real perturbation values of the first kind

As mentioned earlier, Definition 2.3.1 is actually an extension of ordinary real perturba-

tion values of the second kind. A similar extension can be made to define the generalized

real perturbation values of the first kind.

Definition 2.3.2. Given M ∈ Cn×m, N ∈ C

p×m, and i ∈ 1, . . . ,min(n,m), the i-th

generalized real perturbation value of the first kind of (M,N) is defined as

τi(M,N) :=1

inf∆∈Rp×n

‖∆‖2 | nullity(N −∆M) ≥ i .

If there does not exist ∆ ∈ Rp×n such that nullity(N −∆M) ≥ i, then τi(M,N) = 0.

Using a proof similar to the proof of Theorem 2.3.1, one can show that generalized

real perturbation values of the first kind can be computed as follows.

Theorem 2.3.2. Let M ∈ Cn×m, N ∈ Cp×m, and i ∈ 1, . . . ,min(n,m), and assume

that τi(M,N) > 0. Then,

τi(M,N) = infγ∈(0,1]

σ2i

(MR

γ , NR

γ

)(2.11)

= infγ∈(0,1]

σ2i

ReM −γ ImM

γ−1 ImM ReM

,

ReN −γ ImN

γ−1 ImN ReN

.

Proof. The proof is similar to the proof of Theorem 2.3.1 and so is omitted.


2.4 Restricted Real Perturbation Values

We now make the following definition, which is a further generalization of the ordinary

real perturbation value problem to a perturbation structure involving a matrix triplet.

Definition 2.4.1 (Restricted real perturbation values). Given M ∈ Cn×m, L ∈ C

n×l,

N ∈ Cp×m, and i ∈ 1, . . . ,min(n,m), the i-th restricted real perturbation value of the

matrix triplet (M,L,N) is defined as

τi(M,L,N) := inf∆∈Rl×p

‖∆‖2 | rank(M − L∆N) < i .

If there exists no ∆ ∈ Rl×p such that rank(M − L∆N) < i, then τi(M,L,N) =∞.

Here, L and N in Definition 2.4.1 are both general matrices with no assumptions on

invertibility; i.e., both are not necessarily invertible, or even square. If L and N are both

nonsingular, then

rank(M − L∆N) = rank(L−1MN−1 −∆

);

so, the restricted real perturbation value problem reduces to the ordinary real perturba-

tion value problem (cf. Definition 2.2.1) of the single matrix L−1MN−1. Similarly, if only

L is nonsingular, then the problem can be reduced to the generalized real perturbation

value problem of the matrix pair (L−1M,N) (e.g., see [57]).

2.4.1 Lower bounds for restricted real perturbation values

Since the set of real matrices is a subset of the set of complex matrices, one can observe

from Definition A.3.1 and 2.4.1 that the restricted singular values of (M,L,N) can be

used as lower bounds for the restricted real perturbation values of (M,L,N); i.e., for

i ∈ 1, . . . ,min(n,m),

τi(M,L,N) ≥ σi(M,L,N) . (2.12)


However, as shown in an example in the next Subsection 2.4.3, the lower bound (2.12)

can be rather conservative. Instead, a much tighter bound is provided by the following

result.

Theorem 2.4.1. Let M ∈ Cn×m, L ∈ Cn×l, N ∈ Cp×m, and i ∈ 1, . . . ,min(n,m), and

assume that τi(M,L,N) <∞. Then,

τi(M,L,N) ≥ supγ∈(0,1]

σ2i−1

(MR

γ , LR

γ , NR

γ

). (2.13)


The following theorem guarantees that the lower bounds provided by Theorem 2.4.1

are always tighter than the lower bounds given by the restricted singular values. In fact,

we see in an example later in Subsection 2.4.3 that the difference can be quite large.

Theorem 2.4.2. Let M ∈ Cn×m, L ∈ C

n×l, N ∈ Cp×m, and i ∈ 1, . . . ,min(n,m), and

assume that τi(M,L,N) <∞. Then,

τi(M,L,N) ≥ supγ∈(0,1]

σ2i−1

(MR

γ , LR

γ , NR

γ

)≥ σi(M,L,N) . (2.14)


2.4.2 Approximation of restricted real perturbation values

We now propose a method of approximating the restricted real perturbation values of

(M,L,N). Recall that if L is nonsingular, then

τi(M,L,N) = τi(L−1M,N

); (2.15)

i.e., the restricted real perturbation value problem is reduced to the generalized real

perturbation value problem. Since a formula to compute the latter is available (see


Theorem 2.3.1), one may approximate restricted real perturbation values by first ap-

proximating L (if not nonsingular) with a nonsingular matrix, and then applying (2.15).

To be able to approximate a general (not necessarily square) matrix L with a nonsin-

gular (square) matrix, it should be noted that the matrices (M,L,N) in Definition 2.4.1

can actually all be assumed, without loss of generality, to be square matrices with

the same dimensions. This is because letting M :=

M 0

0 0

, L :=

L 0

0 0

, and

N :=

N 0

0 0

such that all are square matrices with the same dimensions, the follow-

ing is true:

rank

M−L

∆1 ∆2

∆3 ∆4

N

= rank(M − L∆1N) .

It can then be shown that for all i ∈ 1, . . . ,min(n,m),

τi(M,L,N) = τi(M,L,N ) .

The proof is straightforward and involves showing that a minimum-norm perturbation

of (M,L,N ), i.e.,

∆1 ∆2

∆3 ∆4

, occurs at ∆2 = ∆3 = ∆4 = 0 and some ∆1 ∈ Rl×p such

that ‖∆1‖2 = τi(M,L,N); the details are omitted.

Now, to approximate the singular square matrix L with a nonsingular matrix, an

approach similar to the singular value decomposition approach that is often used to

approximate a matrix by one of lower rank (e.g., see [83]) can be used. In particular, let

L ∈ Cn×n be a singular matrix, and

L = U diag(σ1, . . . , σn) V

be a singular value decomposition of L, where U and V are unitary, σ1, . . . , σn are the

singular values of L, and σ1 ≥ · · · ≥ σn (see Appendix A.1). Since L is singular, then

there are one or more zero singular values; i.e., σj = σj+1 = . . . = σn = 0 for some


j ∈ 1, . . . , n. We then approximate L with L by perturbing the zero singular values of

L by a small amount ǫ≪ 1; i.e.,

L = U diag(σ1, . . . , σj−1, ǫ, . . . , ǫ)V.

It can be shown that∥∥∥L − L

∥∥∥2= ǫ, and that the condition number of L is σ1/ǫ.

1 Hence,

an advantage of this approach is its ease in choosing an approximation with a particular

“distance” and condition number.

Hence, the restricted real perturbation values of (M,L,N) can be approximated by

τi(M,L,N) ≈ τi

(

L−1M,N)

. (2.16)

2.4.3 Numerical example

In the following example, the restricted real perturbation value approximations given by

(2.16) are computed and compared to the lowers bounds obtained via Theorem 2.4.1.

In general, the approximations τi

(

L−1M,N)

are different than the lower bounds, but

for some cases, the two are extremely close, suggesting that the lower bounds in Theo-

rem 2.4.1 may in fact be achievable for those particular cases. As an example, consider

the following matrix triplet:

M0 =

−5 − j1 10− j13 −16 + j4 4 + j4

5− j11 6 + j7 −11 + j16 −6 + j19

13− j1 11 + j1 7 + j33 −2 + j8

L0 =

−4 − j5 16− j11 −2− j18 −5 + j6

−5 + j5 10 −3− j11 3 + j10

3 + j2 7 + j1 −4 −1 + j7

1Assuming all the nonzero singular values of L are greater than ǫ.


N0 =

10− j2 −1 − j16 −14 + j4 −6 + j10

−6 + j13 6 + j22 −6 − j9 3− j1

17 + j12 −7 + j1 −5 − j1 2 + j7

−j2 12 + j5 −24 + j8 −1 + j9

8− j11 −5 + j9 −4 + j8 −14− j1

.

Using the approximation technique outlined above with ǫ = 10−6, we obtain

L0 =

−4 − j5 16− j11 −2− j18 −5 + j6

−5 + j5 10 + j6.8310−15 −3− j11 3 + j10

3 + j2 7 + j1 −4 + j1.3310−15 −1 + j7

6.5710−7 −4.6110−7 − j3.4310−8 1.8410−7 − j4.7510−7 7.2610−8 + j3.0010−7

.

Table 2.1 shows the lower bounds of τi(M0, L0, N0) provided by Theorem 2.4.1 (see second

column) and the estimates of τi(M0, L0, N0) obtained by computing τi

(

L−10 M0,N0

)

(see

first column), where L0 denotes a nonsingular approximations of L0 =

L0 0

0 0

, N0

equals

N0 0

0 0

, andM0 equals

M0 0

0 0

such thatM0 conforms with the dimen-

sions of L0 and N0. Table 2.1 also provides the restricted singular values of (M,L,N) in

the third column.

As expected, we see from Table 2.1 (we study Case 0 for now) that the lower bounds

provided by Theorem 2.4.1 are always tighter than using the restricted singular values

σi(M0, L0, N0). In some cases (e.g., for i = 1), the difference can be quite large.

Also, comparing the approximations τi

(

L−10 M0,N0

)

with the other lower bounds,

we see that they are in general different. From the case for i = 2, we see that the

approximations can vary greatly from the lower bounds. This suggests that either the

approximation is very inaccurate (possibly due to inverting L0), or the lower bounds are

too conservative. However, some cases are found where the lower bounds provided by


Theorem 2.4.1 are very close to the approximations by (2.16), which suggests that the

bounds may be achievable in those cases. For instance, consider the following two cases

and examples:

Case 1: L has full column rank; e.g.,

M1 = M0, L1 =

−4− j5 16− j11

−5 + j5 10

3 + j2 7 + j1

, and N1 = N0,

and

Case 2: L is real; e.g.,

M2 = M0, L2 =

−4 16 −2 −5

−5 10 −3 3

3 7 −4 −1

, and N2 = N0.

From Table 2.1 (Case 1 and 2), we see that when L has full column rank or is

real, the estimates of the restricted real perturbation values of (M,L,N) obtained by

τi

(

L−10 M0,N0

)

are very close to the lower bounds provided by Theorem 2.4.1 (shown in

the second column). From experiment, this seems to be true for any L with full column

rank and/or is real, and not just for the particular examples shown here. Therefore

this suggests that for these two cases, the lower bound may actually be achievable; i.e.,

equality in (2.13) holds. The same can be said about N with full row rank and/or is real.

2.5 Summary

In this chapter, two generalizations to the ordinary real perturbation value problem have

been made to deal with general perturbation structures involving matrix pairs and matrix

triplets. The first generalization leads to the introduction of generalized real perturba-

tion values, while the second leads to the introduction of restricted real perturbation

values. Computable formulas for computing generalized real perturbation values have


been obtained, and lower bounds for restricted real perturbation values have been found.

A summary of the various singular and real perturbation value problems and their cor-

responding solutions is illustrated in Table 2.2.

From Table 2.2, it is observed that when (M,L,N) = (M, I, I) or when (M,L,N) =

(M, I, L), the formula (2.13) for computing lower bounds of restricted real perturbation

Table 2.1: Approximation and lower bounds of τi(Mj, Lj , Nj) for j = 0, 1, 2

τi

(

L−1j Mj,Nj

)

τi(Mj , Lj , Nj) σi(Mj , Lj , Nj)

(by (2.16))∗ (by Theorem 2.4.1)∗

Case 0: (M0, L0, N0)

i = 1 15345 15339 0.19336

2 1912.3 0.076982 0.073709

3 0.027633 0.023693 0.018815

∗∗Case 1: (M1, L1, N1)

i = 1 8.535510+10 ∞ ∞

2 1073.3 1073.3 0.095529

3 0.038480 0.038480 0.026681

∗∗Case 2: (M2, L2, N2)

i = 1 20822 20822 0.38061

2 204.70 204.70 0.11524

3 0.033333 0.033333 0.029250

∗ For numerical purposes, the search in (2.10) of Theorem 2.3.1 and (2.13) of Theorem 2.4.1 for

computing generalized and restricted real perturbation values, respectively, is in γ ∈[10−5, 1

].

∗∗ In these cases, the lower bounds obtained by Theorem 2.4.1 are very close to the approximated

values (see bold values).


values reduces to the formulas for computing ordinary and generalized real perturbation

values, respectively (i.e., (2.5) and (2.10), respectively), where equality actually holds;

i.e.,

τi(M, I, I) = τi(M) = supγ∈(0,1]

σ2i−1

(MR

γ

)

and

τi(M, I,N) = τi(M,N) = supγ∈(0,1]

σ2i−1

(MR

γ , NR

γ

).

Equality holding for both of these two cases leads one to conjecture that equality may

also hold in Theorem 2.4.1; i.e., the lower bounds proposed may actually be exact bounds!

Furthermore, even if equality does not hold in general, it is possible that it may hold

for specific cases, such as when L has full column rank or is real, as suggested by the

numerical example studied in Subsection 2.4.3. This is left for future investigation.

Table 2.2: Summary of various singular value and real perturbation value problems for

(M,L,N) and i ∈ 1, 2, . . .Problem: Solution:

Find inf∆‖∆‖2 such that For complex ∆ For real ∆

(F = C) (F = R)

rank(M −∆) < i σi(M) τi(M) = supγ∈(0,1]

σ2i−1

(MR

γ

)

rank(M −∆N) < i σi(M,N) τi(M,N) = supγ∈(0,1]

σ2i−1

(MR

γ , NR

γ

)

rank(M − L∆N) < i σi(M,L,N) τi(M,L,N) ≥ supγ∈(0,1]

σ2i−1

(MR

γ , LR

γ , NR

γ

)

Chapter 3

LTI Robustness Radii I

3.1 Introduction

One of the main contributions of this thesis is the study of continuous measures for LTI

systems, similar to the controllability radius, with an emphasis on real perturbations to

the system matrices. In particular, we introduce three new LTI robustness radii:

i) the real decentralized fixed modes (DFM) radius;

ii) the real transmission zero at s radius; and

iii) the real minimum phase radius.

The controllability radius, which measures the distance from a controllable pair to the

nearest uncontrollable pair, has extensively been studied in the past by the computer sci-

ence and control community (see [5,24,42,46,67,72] and the references therein). Perhaps

one of the earliest motivations for such study can be found in the work by Paige [72],

who studied properties of numerical algorithms related to determining controllability.

The formulation in [72] was solved by Miminis [67] and Eising [24] for complex pertur-

bations, where the controllability radius is characterized in terms of a 2-D minimization

problem. The minimization problem, however, is nonconvex and subsequently, bounds

34

Chapter 3. LTI Robustness Radii I 35

(e.g. [5, 30, 36]) and algorithms (e.g. [7, 8, 34, 35, 66, 89]) were proposed to estimate the

distance to uncontrollability.

A study of the controllability radius with respect to the more realistic case of real

parametric perturbations can be found in [42,89]. The characterization in [89] is compu-

tationally tedious for high-order systems, while the characterization in [42] is based on

real perturbation values [4], where a more readily computable formula and a procedure

for constructing minimum-norm perturbations can be obtained. The development of the

three aforementioned radii introduced in this thesis is based on a characterization similar

to [42] involving real perturbation values.

Further study of the controllability radius with respect to the class of Hermitian,

symmetric, and skew-symmetric perturbations can be found in [45].

The DFM radius was also previously studied in [84] for complex perturbations. The

formulation used in [84] is similar to the one used in [24, 67] for studying the complex

controllability radius. In this chapter, we extend the DFM radius to deal with real

perturbations using a formulation similar to [42] and the results on real perturbation

values.

In terms of the latter two radii introduced in this thesis (i.e., the transmission zero

at s radius and the minimum phase radius), no similar study can be found in the cur-

rent literature. In fact, to the best of the author’s knowledge, these two radii are the

first robustness measures applied to studying the robustness of a system’s transmission

zero properties; i.e., all previous work of this nature, such as the controllability radius,

observability radius, DFM radius, and stability radius [74], deals with pole properties.

So for completeness, we discuss the transmission zero at s and minimum phase radii in

terms of both complex and real perturbations.

This chapter is organized as follows. First, the controllability radius is reviewed in

Section 3.2, where a more formal definition is given, and the formulas derived for complex

perturbations [24, 67] and for real perturbations [42] are presented. In Section 3.3, de-


centralized LTI systems, DFMs and the DFM radius given in [84] are reviewed, followed

by the development of a computable formula for computing the DFM radius for real per-

turbations. A procedure for computing the DFM radius for decentralized LTI systems

with general information flow constraints is also discussed. Section 3.4 reviews the defi-

nitions of transmission zeros and the minimum phase property, and also introduces the

transmission zero at s radius and the minimum phase radius. Properties of these three

radii and procedures for computing the corresponding minimum-norm perturbations are

discussed in Sections 3.5 and 3.6, respectively. Finally, numerical examples on computing

the three radii are presented in Section 3.7.

3.2 Review of Controllability Radius

Consider the LTI multivariable system given by

x = Ax+Bu (3.1)

y = Cx+Du,

where x ∈ Rn, u ∈ Rm, and y ∈ Rr are respectively the system states, inputs, and

outputs.

The controllability radius introduced in [72] is a measure of the distance between a

controllable pair (A,B) and the set of uncontrollable pairs. The distance between two

matrix pairs (A1, B1) and (A2, B2) was defined in [72] as

dist〈(A1, B1) , (A2, B2)〉 :=∥∥∥∥

[

A1 −A2, B1 −B2

]∥∥∥∥2

, (3.2)

which leads to the following two definitions.

Definition 3.2.1 (Modal controllability radius with respect to s). Given a LTI system

(A,B) in (3.1) and s ∈ C, the modal controllability radius with respect to s, denoted


rc,sF, is defined to be

rc,sF(A,B, s) = inf

∆A ∈ Fn×n

∆B ∈ Fn×m

∥∥∥∥

[

∆A ∆B

]∥∥∥∥2

∣∣∣∣(A +∆A, B +∆B) is uncontrollable at s

,

where F ∈ C,R.

Definition 3.2.2 (Controllability radius [72]). Given a LTI system (A,B) in (3.1), the

controllability radius, denoted rcF, is defined to be

rcF(A,B) = inf

∆A ∈ Fn×n

∆B ∈ Fn×m

∥∥∥∥

[

∆A ∆B

]∥∥∥∥2

∣∣∣∣(A+∆A, B +∆B) is uncontrollable

,

where F ∈ C,R.

To distinguish which field is consider, rcC(i.e., F = C) is called the complex controlla-

bility radius, and rcR(i.e., F = R) is called the real controllability radius. In [24, 67], the

complex controllability radius is given by

rcC(A,B) = inf

s∈Cσn

([

A− sI B

])

. (3.3)

In [42], the real controllability radius is given by

rcR(A,B) = inf

s∈Cτn

([

A− sI B

])

(3.4)

= infs∈C

supγ∈(0,1]

σ2n−1

ReW −γ ImW

γ−1 ImW ReW

,

where W =

[

A− sI B

]

. The proofs of (3.3) and (3.4) are straightforward, and are

given below to provide insight into some of the proofs to follow in the thesis.1

1The proof given here for the real controllability radius (3.4) is not the same as the original prooffound in [42]. The proof given here is in terms of real perturbation values, whereas the proof in [42] isby direct construction of a minimum-norm perturbation.


Proof. Let s ∈ C be fixed. Then for any ∆A ∈ Fn×n and ∆B ∈ F

n×m, the perturbed

system (A +∆A, B +∆B) is uncontrollable at s if and only if

rank

([

A+∆A − sI B +∆B

])

= rank

([

A− sI B

]

+

[

∆A ∆B

])

< n.

From results on singular values (see Appendix A.1), we obtain

σn

([

A− sI B

])

= inf

∆A ∈ Cn×n

∆B ∈ Cn×m

∥∥∥∥

[

∆A ∆B

]∥∥∥∥2

∣∣∣∣rank

([

A +∆A − sI B +∆B

])

< n

.

Similarly, using results on real perturbation values (see Section 2.2), we have

τn

([

A− sI B

])

= inf

∆A ∈ Rn×n

∆B ∈ Rn×m

∥∥∥∥

[

∆A ∆B

]∥∥∥∥2

∣∣∣∣rank

([

A +∆A − sI B +∆B

])

< n

.

So from the above results, we see that the modal complex and real controllability radius

with respect to s are given by σn

([

A− sI B

])

and τn

([

A− sI B

])

, respec-

tively. The complex and real controllability radii are then the minimization of these two

modal radii over the complex variable s in the whole complex plane.

Similar to Definition 3.2.2 of the controllability radius, one can define the stabiliz-

ability radius as follows.

Definition 3.2.3 (Stabilizability radius [24, 42]). Given a LTI system (A,B) in (3.1),

the stabilizability radius, denoted rc+F, is defined to be

rc+F(A,B) = inf

∆A ∈ Fn×n

∆B ∈ Fn×m

∥∥∥∥

[

∆A ∆B

]∥∥∥∥2

∣∣∣∣(A+∆A, B +∆B) is unstabilizable

,

where F ∈ C,R.


Using a similar proof as the one for (3.3) and (3.4), one can easily show that the

complex and real stabilizability radii can be computed using the following formulas:

rc+C(A,B) = inf

s∈C+

σn

([

A− sI B

])

(3.5)

and

rc+R(A,B) = inf

s∈C+

τn

([

A− sI B

])

. (3.6)

The observability and detectability radii can similarly be defined, and their formulas

are similar in form to (3.3)-(3.4) and (3.5)-(3.6), respectively (e.g., see [42]).

We now make similar extensions to characterize a system’s robustness related to

decentralized fixed modes, transmission zeros, and the minimum phase property of a LTI

system. Before we proceed, however, we make one more definition that will be useful

later when discussing results related to the DFM radius.

Definition 3.2.4. Given the LTI system (3.1), the centralized fixed mode radius, denoted

rCFMF

, is defined as

rCFMF (A,B,C,D) =

inf

∆A ∈ Fn×n,∆B ∈ Fn×m

∆C ∈ Fr×n,∆D ∈ Fr×m

∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣∣∣∣

(A+∆A, B +∆B) is uncontrollable

or

(C +∆C , A+∆A) is unobservable

,

where F ∈ C,R.

It can easily be shown that the the centralized fixed mode (CFM) radius can be

obtained as follows.

Theorem 3.2.1. Given the LTI system (3.1),

rCFMF = min(rcF (A,B) , roF (C,A)) ,

where F ∈ C,R, and rcFand ro

Fare the controllability and observability radius, respec-

tively.


Proof. The smallest perturbations (∆A,∆B) such that the perturbed system is uncon-

trollable is given by rcF(A,B), while the smallest perturbations (∆C ,∆A) such that the

perturbed system is unobservable is given by roF(C,A). The rest of the proof follows

immediately.

3.3 Decentralized Fixed Mode Radius

As mentioned earlier, decentralized fixed modes (DFM) are generalizations of centralized

control systems’ uncontrollable and unobservable modes to decentralized systems. In this

section, we review decentralized LTI systems and DFMs, and then define and provide a

formula for computing the real DFM radius.

3.3.1 Review of decentralized LTI systems and DFMs

In this thesis, we consider decentralized LTI multivariable systems described by

x = Ax+v∑

i=1

Biui (3.7)

yi = Cix+

v∑

j=1

Dijuj, for i = 1, . . . , v,

where v is the number of local control stations, x ∈ Rn is the state vector, and for

i ∈ 1, . . . , v, ui ∈ Rmi and yi ∈ Rri are respectively the inputs and outputs of the i-th

control station. In the remainder of this thesis, we use the notation

A B

C D

v

(3.8)

to represent a decentralized LTI system of the form (3.7), where B := [B1 · · ·Bv], C :=

C1

...

Cv

, and D :=

D11 · · · D1v

......

Dv1 · · · Dvv

. Also, let m :=

v∑

i=1

mi and r :=

v∑

i=1

ri.


Now, suppose one wants to design a decentralized LTI controller of the following form:

ui(s) = Ki(s)yi(s), for i = 1, . . . , v. (3.9)

Such a decentralized controller is said to have a diagonal information flow constraint K;

i.e.,

u = Ky,

where u =[u1

T , · · ·, uvT]T, y =

[y1

T , · · ·, yvT]T, and

K ∈K ∈ R

m×r|K = block diag(K1, K2, . . . , Kv) ,

Ki ∈ Rmi×ri, i = 1, . . . , v, det (I −DK) 6= 0

. (3.10)

A decentralized fixed mode of system (3.7) with a diagonal information flow constraint

(3.10) is then defined as follows [87].

Definition 3.3.1 (Decentralized fixed mode (DFM)). The LTI system (3.7) is said to

have a DFM s ∈ C if s ∈ λ(A) and for all conformal Ki ∈ Rmi×ri (i ∈ 1, . . . , v) with

the property that (I −DK)−1 exists, where K := block diag (K1, . . . , Kv),

s ∈ λ(A+BK (I −DK)−1C

). (3.11)

An algebraic characterization of a DFM can be found in [16].

Theorem 3.3.1 ([16]). Given the LTI system (3.7), s ∈ λ(A) is a DFM of (3.7) if and

only if there exists a subset P = i1, . . . , ik of v (i.e., P ⊆ v), and a complementary

subset P = j1, . . . , jv−k (i.e., P = v −P) such that

rank

A− sI BP

CP DPP

< n, (3.12)


where the above notation is defined as follows:

A− sI BP

CP DPP

:=

[

A− sI B1 . . . Bv

]

if k = 0

A− sI Bj1 · · · Bjv−k

Ci1 Di1j1 · · · Di1jv−k

......

...

Cik Dikj1 · · · Dikjv−k

if k ∈ 1, . . . , v − 1

A− sI

C1

...

Cv

if k = v

,

(3.13)

where the dimensions of BP and CP are given by n×mP and rP × n, respectively, with

mP :=∑

i∈P

mi and rP :=∑

i∈Pri.

3.3.2 Definition

A continuous measure to characterize the decentralized eigenvalue assignability of a sys-

tem was previously studied in [84]. The DFM radius proposed in [84] considered complex

parametric perturbations of the following form:

A → A+∆A

Bi → Bi +∆Bi(3.14)

Ci → Ci +∆Ci

Dij → Dij +∆Dij, for i, j = 1, . . . , v,

where ∆A, ∆Bi, ∆Ci

, and ∆Dijare complex perturbations, and the distance between two

decentralized systems are defined to be

dist

⟨

A1 B1

C1 D1

v

,

A2 B2

C2 D2

v

⟩

=

∥∥∥∥∥∥∥

A1 − A2 B1 − B2

C1 − C2 D1 −D2

∥∥∥∥∥∥∥2

.


In this section, we extend the results of [84] to the real DFM radius, where only real

perturbations of the form (3.14) are considered. We follow the same approach as [84] and

the proof of the real controllability radius (see Section 3.2), and define the modal DFM

radius with respect to s and the DFM radius as follows.

Definition 3.3.2 (Modal DFM radius with respect to s). Given a decentralized LTI

system (3.7) that has no DFMs and s ∈ C, the modal DFM radius of (3.7) with respect

to s, denoted by rDFM,sR

, is defined as

rDFM,sR

A B

C D

v

= inf

∆A ∈ Fn×n,∆B ∈ F

n×m


∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥

∣∣∣∣∣∣∣

(3.15)

A +∆A B +∆B

C +∆C D +∆D

v

has at least one DFM at s

,

where F ∈ R,C, m =

v∑

i=1

mi, r =

v∑

i=1

ri, ∆B := [∆B1· · ·∆Bv

], ∆C :=

∆C1

...

∆Cv

, and

∆D :=

∆D11· · · ∆D1v

......

∆Dv1· · · ∆Dvv

.

Definition 3.3.3 (DFM radius). Given a decentralized LTI system (3.7) that has no

DFMs, the DFM radius of the system, denoted rDFMF

, is defined as

rDFMF

A B

C D

v

= inf



∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣

(3.16)

A+∆A B +∆B

C +∆C D +∆D

v

has at least one DFM

,


where F ∈ R,C.

Again, the type of perturbations (i.e., complex or real) is specified by referring to

either the complex DFM radius (rDFMC

) or the real DFM radius (rDFMR

).

We also have the following definition.

Definition 3.3.4 (Unstable DFM radius). Given a decentralized LTI system (3.7) that

has no unstable DFMs, the unstable DFM radius, denoted rDFM+

F, is defined as

rDFM+

F

A B

C D

v

= inf



∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣

(3.17)

A+∆A B +∆B

C +∆C D +∆D

v

has at least one unstable DFM

,

where F ∈ R,C.

3.3.3 Formula

It was shown in [84] that the complex DFM radius is given by

rDFMC

A B

C D

v

= inf

s∈CminP⊆v

σn

A− sI BP

CP DPP

. (3.18)

Applying the real perturbation values results (i.e., Theorem 2.2.1), it can be shown that

the real DFM radius is given by the following.

Theorem 3.3.2. Given a decentralized LTI system (3.7) that has no DFMs, the real

DFM radius of system (3.7) is given by

rDFMR

A B

C D

v

= inf

s∈CminP⊆v

τn

A− sI BP

CP DPP

(3.19)

= infs∈C

minP⊆v

supγ∈(0,1]

σ2n−1

ReW −γ ImW

γ−1 ImW ReW

,


where W :=

A− sI BP

CP DPP

.


Corollary 3.3.1. Given a decentralized LTI system (3.7) that has no unstable DFMs,

the real unstable DFM radius of system (3.7) is given by

rDFM+

R

A B

C D

v

= inf

s∈C+

minP⊆v

τn

A− sI BP

CP DPP

. (3.20)

Remark 3.3.1. As mentioned earlier, the notion of DFMs is a generalization of un-

controllable and unobservable modes of centralized control problems. Here, it can easily

be verified that Theorem 3.3.2 for one control station (i.e., v = 1) simplifies to Theo-

rem 3.2.1 corresponding to computing the real CFM radius (cf. Definition 3.2.4).

3.3.4 Extension to general information flow constraints

In the above, the real DFM radius (3.19) is defined for the decentralized system (3.7) with

the diagonal information flow constraint given in (3.10). In this subsection, we discuss

how to compute the real DFM radius of decentralized LTI systems with a more general

information flow constraint. In particular, suppose one wants to design a decentralized

LTI controller of the form

ui(s) = Ki1(s)y1(s) + · · ·+Kiv(s)yv(s), for i = 1, . . . , v, (3.21)

where any of the elements Kij may be constrained to be zero. Such a decentralized

controller is said to have a general information flow constraint K given by

u = Ky,


where u =[u1

T , · · ·, uvT]T, y =

[y1

T , · · ·, yvT]T, and

K ∈

K ∈ Rm×r

∣∣∣∣∣∣∣∣∣∣

K =

K11 · · · K1v

......

Kv1 · · · Kvv

, Kij ∈ Rmi×rj , for i, j = 1, . . . , v

. (3.22)

To compute the real DFM radius with respect to the general information flow constraint

(3.22), we first present the following result.

Lemma 3.3.1. The decentralized LTI system (3.7) with a general information flow con-

straint K given by (3.22) can be represented as an alternative decentralized system

x = Ax+v∑

i=1

Biui

yi = Cix+

v∑

j=1

Dij uj, for i = 1, . . . , v, (3.23)

subject to a block diagonal information flow constraint K as described by (3.10); i.e.,

u = Ky,

where u =[uT1 , · · ·, uT

v

]T, y =

[yT1 , · · ·, yTv

]T, and

K = block diag(K1,1, . . . , K1,v, K2,1, . . . , K2,v, . . . , Kv,1, . . . , Kv,v) . (3.24)


Hence by Lemma 3.3.1, one can compute the real DFM radius of the decentralized sys-

tem (3.7) with the general information flow constraint (3.22) by first using Lemma 3.3.1

to reformulate the system such that a diagonal information flow constraint is used in-

stead. The previous results (e.g., (3.19) and (3.20)) can then be applied to the new

alternative system after removing repeated columns and rows in

A− sI BP

CP DPP

that

arise from applying Theorem 3.3.2. To illustrate the latter point, it is best to use an

example.


Example 3.3.1. Consider the following system with two control stations and subject to

the general information flow constraint K =

K11 K12

0 K22

:

x = Ax+B1u1 +B2u2

y1 = C1x+D11u1 +D12u2 (3.25)

y2 = C2x+D21u1 +D22u2.

Using Lemma 3.3.1, the system (3.25) can be rewritten as

x = Ax+B1u1 +B1u2 +B2u3

y1 = C1x+D11u1 +D11u2 +D12u3 (3.26)

y2 = C2x+D21u1 +D21u2 +D22u3

y3 = C2x+D21u1 +D21u2 +D22u3,

where the information flow constraint is now given by K = block diag(K11, K12, K22).

Fixing s ∈ C, a direct application of (3.15) to compute the modal real DFM radius

with respect to s (see Definition 3.3.2) for the system (3.26) requires the computation

of the n-th real perturbation values of the following matrices:

[

A− sI B1 B1 B2

]

,

A− sI B1

C1 D11

C2 D21

,

A− sI B1

C2 D21

C2 D21

, etc.

However, note that B1 appears twice in the first matrix. Therefore to avoid perturbing

each B1’s independently, the n-th real perturbation value of

[

A− sI B1 B2

]

should

be computed instead.2 Likewise, all of the other matrices that have repeated columns

and/or rows obtained from applying Lemma 3.3.1 (e.g., the third matrix has repeated

C2 and D21) should first have the repeated columns and rows removed before computing

the corresponding n-th real perturbation value.

2Note that[A− sI B1 B1 B2

]and

[A− sI B1 B2

]both have the same rank.


3.4 Transmission Zero at s Radius and Minimum

Phase Radius

In this section, we review transmission zeros of a system and the minimum phase property,

and then define and derive formulas for computing the transmission zero at s radius, and

the minimum phase radius.

3.4.1 Review of transmission zeros and minimum phase

First recall the LTI multivariable system (3.1) given by

x = Ax+Bu (3.27)

y = Cx+Du.

For the remainder of this thesis, we introduce the notation (C,A,B,D) to represent the

above LTI system (3.1). There are various definitions of transmission zeros of the LTI

system (3.27) (see [62,80] for a survey), and in this thesis, we use the following definition

made in [21].

Definition 3.4.1 (Transmission zeros). Given a LTI system (3.27), the transmission

zeros (TZ) are defined to be the set of complex numbers s ∈ C that satisfy the following

inequality:

rank

A− sI B

C D

< n +min(r,m). (3.28)

The system (3.27) is called degenerate if the set of transmission zeros include the whole

complex plane. For the reminder of the paper, it is, in general, assumed that the system

(3.27) is non-degenerate.

Definition 3.4.2 (Minimum phase). A LTI system (3.27) is said to be minimum phase

(MP) if there are no transmission zeros in the closed right-half of the complex plane;

otherwise the system is said to be nonminimum phase (nonMP).


3.4.2 Transmission zero at s radius - definition

Similar to (3.2), which defines the distance between two matrix pairs, we define the

distance between two LTI systems as follows:

dist〈(C1, A1, B1, D1) , (C2, A2, B2, D2)〉 =

∥∥∥∥∥∥∥

A1 − A2 B1 −B2

C1 − C2 D1 −D2

∥∥∥∥∥∥∥2

. (3.29)

The transmission zero at s radius, which measures how close a LTI system (3.27) is to

having a transmission zero at a particular point s in the complex plane, is defined as

follows.

Definition 3.4.3 (Transmission zero at s radius). Given the LTI system (3.27) and

s ∈ C, the transmission zero at s radius, denoted rTZF

, is defined to be

rTZF (C,A,B,D, s) = inf



∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣

(3.30)

(C +∆C , A+∆A, B +∆B, D +∆D) has a TZ at s

,

where F ∈ C,R. Similar to the other radii, rTZC

(i.e. F = C) is called the complex

transmission zero at s radius, and rTZR

is called the real transmission zero at s radius.

3.4.3 Minimum phase radius - definition

Similarly, the minimum phase radius, which measures the distance (as described by

(3.29)) between a minimum phase system and the closest nonminimum phase system, is

defined as follows.

Definition 3.4.4 (Minimum phase radius). Given a LTI system (3.27) that is minimum


phase, the minimum phase radius, denoted rMPF

, is defined to be

rMPF

(C,A,B,D) = inf


∆C ∈ Fr×n,∆D ∈ F

r×m

∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣

(3.31)

(C +∆C , A+∆A, B +∆B, D +∆D) is nonMP

,

where F ∈ C,R.

3.4.4 Formulas

Applying an approach similar to deriving the formula for the controllability radius, one

can obtain the following formulas for computing the transmission zero at s radius and

the minimum phase radius.

Theorem 3.4.1. Given a LTI system (3.27) and s ∈ C,

rTZC

(C,A,B,D, s) = σn+min(r,m)

A− sI B

C D

(3.32)

and

rTZR

(C,A,B,D, s) = τn+min(r,m)

A− sI B

C D

(3.33)

= supγ∈(0,1]

σ2(n+min(r,m))−1

ReW −γ ImW

γ−1 ImW ReW

,

where W =

A− sI B

C D

.



Theorem 3.4.2. Given a LTI system (3.27),

rMPC

(C,A,B,D) = infs∈C+

rTZC

(C,A,B,D, s) (3.34)

and

rMPR

(C,A,B,D) = infs∈C+

rTZR

(C,A,B,D, s) . (3.35)

Proof. Theorem 3.4.2 follows immediately from the minimization of rTZC

(C,A,B,D, s)

and rTZR

(C,A,B,D, s) over the closed right-half of the complex plane.

3.5 Properties

Some properties of the various LTI robustness radii can be derived directly from the

following properties of singular values (e.g., see [39]) and real perturbation values (see

[3]).

Lemma 3.5.1 (Properties of σi(M) and τi(M)). Given a matrix M ∈ Cn×m and i ∈

1, . . . ,min(n,m), then

(1) σi

(M)= σi(M) and τi

(M)= τi(M);

(2) σi(Q1MQ2) = σi(M) and τi(Q1MQ2) = τi(M); and

(3) σi(M) ≤ τi(M),

where Q1 and Q2 are real orthogonal matrices,3 and where equality in Property (3) is

achieved for a real matrix M ∈ Rn×m.

3The matrices Q1 and Q2 can actually be complex and unitary.


3.5.1 Properties of the DFM radius

Lemma 3.5.2 (Properties of DFM radius). Given the decentralized LTI system (3.7),

the DFM radius has the following properties:

(1) rDFM,sF

A B

C D

v

= rDFM,s

F

A B

C D

v

;

(2) rDFMF

QAQT QB

CQT D

v

= rDFM

F

A B

C D

v

; and

(3) rDFMC

A B

C D

v

≤ rDFM

R

A B

C D

v

,

where F ∈ C,R, Q is an orthogonal coordinate transformation, and rDFM,sF

denotes the

modal DFM radius with respect to s ∈ C (see Definition 3.3.2).

Proof. The proof is straightforward and follows directly from Lemma 3.5.1, and so is

omitted.

3.5.2 Properties of the TZ and MP radius

Lemma 3.5.3 (Properties of rTZC

and rTZR

). Given the LTI system (3.27), the transmis-

sion zero at s radius for s ∈ C has the following properties:

(1) rTZF

(C,A,B,D, s) = rTZF

(C,A,B,D, s);

(2) rTZF

(C,A,B,D, s) = rTZF

(CQT , QAQT , QB,D, s

); and

(3) rTZC

(C,A,B,D, s) ≤ rTZR

(C,A,B,D, s),

where F ∈ C,R, Q is an orthogonal coordinate transformation, and equality in Property

(3) is achieved for s ∈ R.

Proof. Again, the proof follows directly from Lemma 3.5.1, and so is omitted.


Lemma 3.5.4 (Properties of rMPC

and rMPR

). Given the LTI system (3.27), the minimum

phase radius has the following properties:

(1) rMPF

(C,A,B,D) = rMPF

(CQT , QAQT , QB,D

); and

(2) rMPC

(C,A,B,D) ≤ rMPR

(C,A,B,D),

where F ∈ C,R, and Q is an orthogonal coordinate transformation.

Proof. Omitted.

Remark 3.5.1. An immediate result of Property (1) of Lemma 3.5.2 and Property (1)

of Lemma 3.5.3 is that if a given s ∈ C achieves the modal DFM radius or the trans-

mission zero at s radius, then s also achieves it. Therefore rDFM,sF

A B

C D

v

and

rTZF

(C,A,B,D, s) are mirrored images with respect to the real axis, and so the search for

the global minimizer that achieves the DFM radius or the minimum phase radius can be

restricted to either the closed upper- or lower-half of the complex plane. This result is

applied to Algorithm II in Chapter 5 for computing the various LTI robustness radii.

The following is an interesting property of the minimum phase radius.

Lemma 3.5.5. Given a minimum phase LTI system (3.27) with D = 0, the system is

arbitrarily close to a large real nonminimum phase zero; i.e.,

rTZF

(C,A,B, 0, s)→ 0 as s→∞, (3.36)

where s ∈ R and F ∈ C,R. Hence, given a LTI minimum phase system (3.27) with

D = 0, the system is arbitrarily close to a nonminimum phase system; i.e.,

rMPF (C,A,B, 0) = 0, (3.37)

where F ∈ C,R.



Remark 3.5.2. Perhaps the result of Lemma 3.5.5 is not surprising. As an example,

consider the following minimum phase SISO system H(s) with D = 0:

H(s) =1

s− 1. (3.38)

Now perturb the D matrix by some small ǫ ∈ R; i.e., (∆C ,∆A,∆B,∆D) = (0, 0, 0,−ǫ).

The perturbed system H(s) becomes

H(s) =1

s− 1− ǫ = −ǫs−

(1ǫ+ 1)

s− 1, (3.39)

which has a zero at s =(1ǫ+ 1), where s→∞ as ǫ→ 0.

From Lemma 3.5.5, we see that for minimum phase systems with D = 0, one can

immediately obtain the minimum phase radius to be zero without performing any com-

putations. Therefore Lemma 3.5.5 can be applied to Algorithm II in Chapter 5 as a

preliminary check to avoid performing the full optimization in computing the minimum

phase radius.

However, knowing that a system with D = 0 is arbitrarily close to a system with a

large nonminimum phase transmission zero is often not particularly interesting. This is

because a large real nonminimum phase zero that is in the far right of the complex plane

does not pose much of an issue in terms of performance limitations as compared to a

zero that is close to the origin (see Chapter 6). Hence, when considering systems with

D = 0, one may wish to restrict the minimization in (3.34) or (3.35) in computing the

minimum phase radius to a specific region in the complex plane. For instance, one may

wish to restrict the computation of the minimum phase radius to a region bounded by

some given real x > 0, as shown in the following:

rMPC

(C,A,B,D) = inf

s ∈ C+

|s| < x

rTZC

(C,A,B,D, s) (3.40)


or

rMPR

(C,A,B,D) = inf

s ∈ C+

|s| < x

rTZR

(C,A,B,D, s) . (3.41)

Here, x represents the largest magnitude that all the transmission zeros of interest can

have.

3.5.3 Bounds on the global minimizers

The formulas (3.19) and (3.35) for computing the real DFM radius and the real minimum

phase radius, respectively, both require solving a 2-D minimization problem over the

complex plane. In the following, an upper bound on the global minimizer of the real

DFM radius is obtained, which can be used (e.g., in Algorithm II of Chapter 5) to

narrow the search domain of the minimization problem. A similar bound for the real

minimum phase radius, however, still remains an open problem.

Bounds for the real DFM radius

In what follows, the global minimizer of the real DFM radius is denoted as s∗ ∈ C. Also,

the following result on singular values is required.

Lemma 3.5.6 ([39]). Let R ∈ Cn×(n+m), Q ∈ C

n×(n+m), and i ∈ 1, . . . , n, then the

following inequality holds:

σi(R)− σ1(Q) ≤ σi(R +Q) . (3.42)

It can easily be shown that the perturbed system

A+∆A B +∆B

C +∆C D +∆D

v

has a DFM

if the perturbed input and output matrices are zero; i.e., if

∆A ∆B

∆C ∆D

∈

0 −B

0 0

,

0 0

−C 0

.


Hence,

rDFMR

A B

C D

v

= rDFM,s∗

R

A B

C D

v

≤ minσ1(B) , σ1(C) .

Furthermore, since the complex DFM radius is smaller than or equal to the real DFM

radius (i.e., Property (3) of Lemma 3.5.2), we get

σn

A− s∗I BP∗

CP∗ DP∗P∗

≤ rDFM,s∗

R

A B

C D

v

for some P∗ ⊆ v that achieves rDFM,s∗

R

A B

C D

v

. By applying Lemma 3.5.6, we

obtain that

σn

−s∗I 0

0 0

− σ1

A BP∗

CP∗ DP∗P∗

≤ σn

A− s∗I BP∗

CP∗ DP∗P∗

|s∗| ≤ σ1

A BP∗

CP∗ DP∗P∗

+ σn

A− s∗I BP∗

CP∗ DP∗P∗

.

Finally, combining the above results and noting from (C.11) that

σ1

A BP∗

CP∗ DP∗P∗

≤ σ1

A B

C D

,

we obtain that the optimal value s∗, which minimizes (3.19), has the following bound:

|s∗| ≤ minσ1(B) , σ1(C)+ σ1

A B

C D

. (3.43)

3.6 Construction of Minimum-Norm Parametric Per-

turbations

Minimum-norm parametric perturbations that achieve the various LTI robustness radii

can be constructed using the results discussed in Section 2.2.2. In this section, we briefly


explain how one may obtain such perturbations, which may be useful in trying to gain

physical insight of the system uncertainties; e.g., there may be dominant terms in the

system matrices (C,A,B,D) that need to be perturbed in order for the perturbed system

to be uncontrollable, unobservable, etc.

3.6.1 Minimum-norm parametric perturbations - TZ at s radius

Given the LTI system (3.27) and s ∈ C, the minimum-norm perturbation matrices that

achieve the transmission zero at s radius rTZF

are given by ∆A ∈ Fn×n, ∆B ∈ Fn×m,

∆C ∈ Fr×n, and ∆D ∈ F

r×m, where

rank

A− sI B

C D

+

∆A ∆B

∆C ∆D

< n +min(r,m) (3.44)

and

∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

= rTZF

, for F ∈ C,R. To compute the complex perturbations

that achieve the complex radius rTZC

, the singular value decomposition of

A− sI B

C D

can be used, while for real perturbations that achieve rTZR

, Theorem 2.2.3 can be used.

Perturbations that achieve rTZC

Applying Theorem 2.2.2, the complex system parametric perturbations that achieve rTZC

can be obtained from the singular value decomposition of T :=

A− sI B

C D

; i.e., let

T =

n+min(r,m)∑

k=1

σk(T )ukvHk (3.45)

be a singular value decomposition of T , where U = [u1, · · · , un+r] and V = [v1, · · · , vn+m]

are both unitary matrices, and let X =[vn+min(r,m), · · · , vn+m

]. The system perturba-


tions that achieve rTZC

are given by

∆A ∆B

∆C ∆D

= −TXXH . (3.46)

Perturbations that achieve rTZR

Similarly, on applying Theorem 2.2.3, the real system parametric perturbations that

achieve rTZR

can be constructed by

∆A ∆B

∆C ∆D

=

[

Re (TX) Im (TX)

] [

Re (X) Im (X)

]+

,

where T :=

A− sI B

C D

, and X is a complex matrix whose columns form the basis

for a subspace that satisfies

xH((

rTZR

)2I − THT

)

x ≥∣∣∣xT

((rTZR

)2I − T TT

)

x∣∣∣ .

Such a basis (i.e., X) can be obtained by performing a simultaneous block diagonalization

of(rTZR

)2I − THT and

(rTZR

)2I − T TT . For more information, please see Appendix B,

as well as [44].

3.6.2 Minimum-norm parametric perturbations - MP radius

To compute the minimum-norm parametric perturbations that achieve the minimum

phase radius of (C,A,B,D), one can use the same procedure as outlined previously

in Section 3.6.1. This is because if s∗ ∈ C+ is a point that achieves the minimum

phase radius of (C,A,B,D), then the parametric perturbations that achieve the radius

are exactly the same as the perturbations that achieve the transmission at s radius for

s = s∗.


3.6.3 Minimum-norm parametric perturbations - DFM radius

Recall from Definition 3.3.3 that given a decentralized LTI system

A B

C D

v

, the

minimum-norm parametric perturbations that achieve the real DFM radius rDFMR

are ∆A,

∆B, ∆C , and ∆D that have the property that the perturbed system

A+∆A B +∆B

C +∆C D +∆D

v

has a DFM, and

∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

= rDFMR

. To construct such perturbation matri-

ces, first let the DFM radius be computed and be achieved at s∗, P∗, and P∗. The

perturbation ∆ :=

∆A ∆B

∆C ∆D

can then be directly derived from any matrix ∆s :=

∆A ∆BP∗

∆CP∗∆D

P∗P∗

that satisfies

rank

A− sI BP

CP DPP

+

∆A ∆BP∗

∆CP∗∆D

P∗P∗

< n,

where ∆A, ∆Bi, ∆Ci

, and ∆Dij, for i, j = 1, . . . , v, are defined in (3.14), and similar to

(3.13), we denote

∆A − sI ∆BP

∆CP∆D

PP

:=

[

∆A − sI ∆B1. . .∆Bv

]

if k = 0

∆A − sI ∆Bj1· · · ∆Bjv−k

∆Ci1∆Di1j1

· · · ∆Di1jv−k

......

...

∆Cik∆Dikj1

· · · ∆Dikjv−k

if k ∈ 1, . . . , v − 1

∆A − sI

∆C1

...

∆Cv

if k = v

.

(3.47)


More specifically, ∆ can be obtained from ∆s by setting the rows/columns of ∆B, ∆C , and

∆D corresponding to P and P, as defined by (3.47), to be equal to ∆BP, ∆CP

, and ∆DPP

,

while the remaining elements are set to zero. Hence, the main effort in constructing ∆

is in finding ∆s, which can be obtained using Theorem 2.2.3; i.e.,

∆A ∆BP∗

∆CP∗∆D

P∗P∗

=

[

Re (TX) Im (TX)

] [

Re (X) Im (X)

]+

,

where T :=

A− sI BP

CP DPP

, and where X is a complex matrix whose columns form a

basis of the subspace that satisfies

xH((

rDFMR

)2I − THT

)

x ≥∣∣∣xT

((rDFMR

)2I − T TT

)

x∣∣∣ .

Again, such a basis (i.e., X) can be obtained by performing a simultaneous block diago-

nalization of(rDFMR

)2I − THT and

(rDFMR

)2I − T TT , as shown in Appendix B.

3.7 Numerical Examples

In this section, we study two examples to illustrate the results of this chapter. The

first example computes the real DFM radius of a third-order decentralized system with

two control stations and a diagonal information flow constraint. System perturbations

that achieve the radius are found. The radius of the same system with a more general

information flow is also computed. In the second example, we compute the minimum

phase radius of a minimum phase system, where results are given for both complex and

real perturbations. For the same system, the transmission zero at s radius, for s = 0, is

also computed.


3.7.1 Example 1: real DFM radius

Consider the following decentralized system with two control stations and subject to the

information flow constraint K =

× 0

0 ×

:

x =

0 −1 −1

1 1 1

2 3 1

x+

1

0

0

u1 +

0

0.1

0

u2 (3.48)

y1 =

[

0 0.01 0

]

x

y2 =

[

1 0 0.01

]

x.

It can easily be verified that the system has eigenvalues −0.6956,−1.358± j1.029. By

(3.19) of Theorem 3.3.2, the real DFM radius of (3.48) is found to be 7.90210−2 , which is

obtained when s = 1.336±j1.034, P = 1, and P = 2. Using the procedure presented

in Section 3.6.3, the corresponding minimum-norm system perturbations are found to be

∆ :=

∆A ∆b1 ∆b2

∆c1 ∆d11 ∆d12

∆c2 ∆d21 ∆d22

=

−2.1410−2 4.4510−2 −5.4210−2 0 −1.7810−2

0 0 0 0 −6.2410−2

8.4510−3 −1.7610−2 2.1410−2 0 −4.5010−2

0 −1.0010−2 0 0 0

0 0 0 0 0

,

(3.49)

where it can easily be verified that the norm of the perturbation in (3.49) is equal to the

real DFM radius 7.90210−2, and that the perturbed system has a DFM at s = 1.336 ±

j1.034 as expected.

Now consider the same system (3.48) with the information flow constraints K =

× ×

0 ×

. Using the method presented in Section 3.3.4, the real DFM radius is found


to be 0.1107, which is achieved at s = −0.6981, P = and P = 1, 2, and the

corresponding minimum-norm perturbations that achieve a real DFM radius of 0.1107

are computed to be

∆ =

1.8810−3 −6.1910−4 −9.3110−4 −2.8910−3 −4.7210−3

3.0610−2 −1.0110−2 −1.5210−2 −4.7210−2 −7.7110−2

−1.6610−2 5.4710−3 8.2310−3 2.5610−2 4.1710−2

0 0 0 0 0

0 0 0 0 0

. (3.50)

It may be verified that the norm of (3.50) is equal to 0.1107, and that the perturbed

system has a DFM at s = −0.6981 with respect to the information flow constraint

K =

× ×

0 ×

.

3.7.2 Example 2: real TZ at s and MP radius

Consider the following minimum phase system:

x =

0.74 −0.69 −2.08

−0.12 1.62 0.63

−0.38 −0.21 0.14

x+

−1.23 −0.26

1.02 2.51

−0.66 1.13

u (3.51)

y =

[

1.06 0.71 0.61

]

x+

[

1.33 −2.89]

u.

By (3.34) of Theorem 3.4.2, the complex minimum phase radius is found to be rMPC

=

8.90210−2, which is achieved at s = 0.8158 ± j0.2517. Using the procedure presented

in Section 3.6.1, the corresponding complex minimum-norm system perturbations are


computed to be

∆ :=

∆A ∆B

∆C ∆D

=

−1.0710−2 + j8.5910−3 1.6110−2 − j1.1610−2

−4.5810−3 + j5.4310−3 6.9610−3 − j7.4910−3

4.2810−2 − j2.9010−3 −6.1710−2 + j1.2110−3

1.3910−2 + j2.7410−3 −1.9910−2 − j4.9110−3

(3.52)

−7.9810−4 + j9.8110−4 −7.4310−3 + j1.8010−3 −3.7410−3 + j1.5110−3

−3.0610−4 + j5.7710−4 −3.5810−3 + j1.6610−3 −1.7410−3 + j1.1410−3

3.7810−3 − j1.1010−3 2.2210−2 + j8.5710−3 1.2310−2 + j2.7610−3

1.3010−3 − j2.5710−5 6.3710−3 + j4.6510−3 3.6810−3 + j1.9310−3

,

where it can easily be verified that the norm of the perturbation in (3.52) is equal to

rMPC

= 8.90210−2, and that the perturbed system has a nonminimum phase zero at

s = 0.8158± j0.2517.

Likewise, by (3.35) of Theorem 3.4.2, the real minimum phase radius is found to be

rMPR

= 8.98310−2, which is achieved at s = 0.8209 ± j0.2329. The corresponding real

perturbations that achieve rMPR

= 8.98310−2 are

∆ =

3.7210−3 −5.4910−3 3.2810−4 −6.3810−2 −2.8610−2

5.2010−3 −7.6710−3 4.5810−4 −4.2010−2 −1.8710−2

4.4910−2 −6.6310−2 3.9510−3 1.9810−2 1.1010−2

2.1810−2 −3.2110−2 1.9210−3 −2.2010−2 −8.8910−3

. (3.53)

Again, it can be confirmed that the norm of (3.53) is equal to rMPR

= 8.98310−2 , and that

the perturbed system has a nonminimum phase zero at s = 0.8209± j0.2329.

Now suppose s = 0. By (3.32) and (3.33) of Theorem 3.4.1, the transmission zero at

s radius is obtained to be rMPC

= rMPR

= 0.2882. The system perturbations that achieve


this radius is given by

∆ =

−1.8610−2 −9.4910−3 −1.6910−2 2.3810−2 −3.9610−4

−7.3010−3 −3.7210−3 −6.6110−3 9.3210−3 −1.5510−4

−1.3610−1 −6.9510−2 −1.2410−1 1.7410−1 −2.9010−3

−5.7910−2 −2.9510−2 −5.2410−2 7.3910−2 −1.2310−3

. (3.54)

So for the system (3.51), a solution to the RSP for constant tracking and disturbance sig-

nals is guaranteed to exist for all parametric perturbations such that

∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

<

0.2882.

3.8 Summary

In this chapter, we have introduced three new real LTI robustness radii, namely: the

real decentralized fixed mode (DFM) radius; the real transmission zero at s radius; and

the real minimum phase radius. Some of these radii’s properties are discussed in this

chapter, including bounds on the global minimizers for computing these radii, and how

to construct minimum-norm system perturbations. Like the real controllability radius,

these three radii are defined in terms of unstructured real parametric perturbations,

where computable formulas can be derived using just ordinary real perturbation values.

In the next chapter, we apply the generalizations of real perturbation values made in

Chapter 2 to extend the results of this chapter to more general perturbation structures.

Chapter 4

LTI Robustness Radii II

This chapter is a continuation of the previous chapter and addresses four topics related to

the controllability radius and the three real LTI robustness radii introduced in the previ-

ous chapter. Firstly in Section 4.1, we apply the results of generalized and restricted real

perturbation values to extend the characterization of the various LTI robustness radii to

more general parametric perturbation structures. The more general structures consid-

ered here have the advantages of allowing the original state structure to be maintained

in the perturbed system, and also allowing the system perturbations to be normalized

by the system matrices. Then in Section 4.2 and 4.3, we study the real controllability

radius of LTI descriptor systems, and the real spectral controllability radius of time-delay

systems, respectively. The former was previously investigated in [9, 45] for complex per-

turbations, while the latter still remains an open problem (see [40, Chapter 6]). Finally,

in Section 4.4, we discuss the relationship between the real controllability radius and

the ability to shift the closed-loop eigenvalues using all controller gains of any specified

size. We refer to the latter as the system’s eigenvalue mobility, or eigenvalue assignability

(see [84]). Similar studies are also made in terms of the real DFM radius and the real

transmission zero at s radius, where the latter is used to study the eigenvalue mobility

of the RSP.

65

Chapter 4. LTI Robustness Radii II 66

4.1 Structured LTI Robustness Radii

Up to this point, we have discussed the various LTI robustness radii (e.g., controllability

radius, DFM radius, etc.) in terms of only unstructured parametric perturbations; i.e.

perturbations of the form (1.2). In this section, we extend the results to structured

perturbations described by the following:

A → A+ E∆AF , B → B + E∆BG, (4.1)

C → C +H∆CF , D → D +H∆DG,

where E , F , G, and H are complex matrices of conformal dimensions. The perturbation

structure given by (4.1) is similar to that used in [51], where the structured controllability

radius was defined for nonsingular matrices E , F , G, and H. In this section, we present

results for the structured controllability radius where E , F , G, and H are allowed to be

general matrices that are not necessarily nonsingular or even square. Similarly, we also

study the structured version of the other real robustness radii introduced in Chapter 3,

and present formulas based on the generalized and restricted real perturbation values.

In this section, we consider only real perturbations, but a similar discussion also applies

to the case of complex system perturbations.

4.1.1 Structured real controllability radius

In [51], the structured real controllability radius of the system (A,B) with the perturbation

structure (4.1) was defined as

rc,structR

(A,B, E ,F ,G) = inf

∆A ∈ Rn×n

∆B ∈ Rn×m

∥∥∥∥

[

∆A ∆B

]∥∥∥∥2

∣∣∣∣∣∣∣

(A + E∆AF , B + E∆BG)

is uncontrollable

.

(4.2)

In [51], however, E , F , and G were limited to the class of (square) nonsingular matrices,

which allowed the structured real controllability radius to be reduced to the unstructured


case; i.e.,

rc,structR

(A,B, E ,F ,G) = mins∈C

τn

E−1

[

A− sI B

]

F 0

0 G

−1

. (4.3)

Using the results of Chapter 2, one can now relax the nonsingular restrictions of E ,

F , and G, and compute the structured real controllability radius (4.2) for the following

cases:

i) F and G are arbitrary, but E is nonsingular; and

ii) all E , F and G are arbitrary.

Case i) F and G are arbitrary, but E is nonsingular

Using the results on generalized real perturbation values presented in Section 2.3, we

obtain the following theorem.

Theorem 4.1.1. Given the LTI system (A,B), and matrices E , F , and G, where only

E is required to be nonsingular, the structured real controllability radius (4.2) is given by

rc,structR

(A,B, E ,F ,G) = infs∈C

τn

E−1

[

A− sI, B

]

,

F 0

0 G

. (4.4)

Proof. The proof is similar to the proof in Section 3.2 for obtaining the formula for the

real controllability radius. For more detail, see Appendix C.3.1.

Case ii) all E , F and G are arbitrary

Using the results on restricted real perturbation values presented in Section 2.4, we have

the following theorem.

Theorem 4.1.2. Given the LTI system (A,B), and general matrices E , F , and G of

conformal dimensions, the structured real controllability radius (4.2) is given by

rc,structR

(A,B, E ,F ,G) = infs∈C

τn

[

A− sI B

]

, E ,

F 0

0 G

. (4.5)


Proof. Again the proof is similar to the proof of the real controllability radius formula in

Section 3.2. See Appendix C.3.2 for more detail.

Remark 4.1.1. It is to be noted that (4.5) requires computing the n-th restricted real

perturbation value of the matrix triplet

[

A− sI B

]

, E ,

F 0

0 G

. Since we only

have results for computing a lower bound (e.g., see Theorem 2.4.1), this implies that (4.5)

is not a computable formula, and one may need to resort to some sort of approximation,

such as the one used in Section 2.4.2, to estimate the structured real controllability radius

for this case.

4.1.2 Structured real TZ at s radius and minimum phase radius

Similar to the structured real controllability radius, we have the following definitions.

Definition 4.1.1. Given the LTI system (C,A,B,D), s ∈ C, constant matrices E , F ,

G, and H, and the perturbation structure (4.1), the structured real transmission zero at

s radius is defined as

rTZ,structR

(C,A,B,D, s, E ,F ,G,H) = (4.6)

inf

∆A ∈ Rn×n,∆B ∈ Rn×m

∆C ∈ Rr×n,∆D ∈ Rr×m

∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣∣∣∣

(C +H∆CF , A+ E∆AF ,

B + E∆BG, D +H∆DG)

has a TZ at s

.

Definition 4.1.2. Similarly, the structured real minimum phase radius is defined as

rMP,structR

(C,A,B,D, E ,F ,G,H) =

inf



∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣∣∣∣



is nonminimum phase

.


Again, we consider two cases:

i) F and G are arbitrary, but E and H are nonsingular; and

ii) all E , F , G, and H are arbitrary.

Case i) F and G are arbitrary, but E and H are nonsingular

Applying the generalized real perturbation values results in Section 2.3, we have the

following result.

Theorem 4.1.3. Given the LTI system (C,A,B,D), s ∈ C, and matrices E , F , G,

and H of conformal dimensions, where E and H are nonsingular, the structured real

transmission zero at s radius and minimum phase radius are given by

rTZ,structR

(C,A,B,D, s, E ,F ,G,H) = τn

E 0

0 H

−1

A− sI B

C D

,

F 0

0 G

(4.7)

and

rMP,structR

(C,A,B,D, E ,F ,G,H) = infs∈C+

τn

E 0

0 H

−1

A− sI B

C D

,

F 0

0 G

.

(4.8)

Proof. The proof is similar to the proof of Theorem 4.1.1, and so is omitted.

Case ii) all E , F , G, and H are arbitrary

Again, using the results of restricted real perturbation values presented in Section 2.4,

we obtain the following theorem.

Theorem 4.1.4. Given the LTI system (C,A,B,D), s ∈ C, and general matrices E , F ,

G, and H of conformal dimensions, the structured real transmission zero at s radius and


minimum phase radius are given by

rTZ,structR

(C,A,B,D, s, E ,F ,G,H) = τn

A− sI B

C D

,

E 0

0 H

,

F 0

0 G

(4.9)

and

rMP,structR

(C,A,B,D, E ,F ,G,H) = infs∈C+

τn

A− sI B

C D

,

E 0

0 H

,

F 0

0 G

.

(4.10)

Proof. The proof is similar to the proof of Theorem 4.1.2, and so is omitted.

4.1.3 Structured real DFM radius

For completeness, the structured real DFM radius can also similarly be defined as follows.

Definition 4.1.3. Given the decentralized LTI system

A B

C D

v

, constant matrices E ,

F , G, and H, and subject to the perturbation structure (4.1), the structured real DFM

radius is given by

rDFM,structR

A B

C D

v

, E ,F ,G,H

=

inf



∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣∣∣∣

A+ E∆AF B + E∆BG

C +H∆CF D +H∆DG

v

has a DFM

.

For the general case where all E , F , G, and H are arbitrary matrices, we have the

following result.


Theorem 4.1.5. The structured real DFM radius is given by

rDFM,structR

A B

C D

v

, E ,F ,G,H

= (4.11)

infs∈C

minP⊆v

τn

A− sI BP

CP DPP

,

E 0

0 HP

,

F 0

0 GP

,

where for partitions P := i1, . . . , ik and P := j1, . . . , jv−k, we denote HP to be the

matrix formed from the i1, . . . , ik-th rows of H, and GP to be the matrix formed from the

(j1, . . . , jv−k)-th columns of G; i.e.,

HP =

H(i1 , :)...

H(ik , :)

and GP =

[

G(: , j1) · · · G(: , jv−k)

]

,

where we denote M(i , :) to be the i-th row of the matrix M , and M(: , i) to be the i-th

column of M .1

Proof. Again, the proof is similar to the proof of Theorem 4.1.2, and so is omitted.

4.1.4 Discussion

The extended perturbation structure given in (4.1) has the significant advantage that

it allows for two very interesting applications: i) maintaining the original state/system

structure in a perturbed system; and ii) allowing one to normalize the plant perturbations

in terms of the system matrices. We now discuss each one in more detail.

Maintaining original state structure in the perturbed system

A LTI state-space model of the form (3.1) is often derived directly from differential

equations or transfer functions. As a result, the nominal system matrices (C,A,B,D)

1This is similar to MATLAB’s notation for rows and columns of a matrix.


may contain hard 0’s and 1’s that can arise from the relationship between a state and its

higher-order derivatives, and that should not be perturbed. For example, consider the

following differential equation:

y + 2y + 3y = u. (4.12)

On defining the state vector x =

[

y y

]T

, the following state-space representation of

(4.12) is obtained:

x =

0 1

−3 −2

x+

0

1

u (4.13)

y =

[

1 0

]

x.

In order to preserve the structure of this system, it is undesirable to perturb the hard 0’s

and 1’s of (4.13) (i.e., in the first row of (A,B) and in the C matrix).

The perturbation structure in (4.1) allows us to preserve such system structure. Con-

tinuing with the above example (4.13), we only wish to perturb the system matrices

according to the following:

A →

0 1

−3 −2

+

0 0

× ×

B →

0

1

+

0

×

(4.14)

C →[

1 0

]

+

[

0 0

]

,

where × represents perturbation elements that may be nonzero. Rewriting (4.14) as

A →

0 1

−3 −2

+

0

1

[

× ×]

1 0

0 1

B →

0

1

+

0

1

[×] [1]


C →[

1 0

]

+ [0]

[

× ×]

1 0

0 1

,

we see that the system structure can be preserved using the perturbation structure given

in (4.1) with E =

0

1

, F =

1 0

0 1

, G = 1, and H = 0.

Normalization of the system perturbations

A second application of the perturbation structure (4.1) is in the normalization of the

system perturbations with respect to the system matrices themselves. As a motivation

for doing so, consider the following scalar system:

x = ax+ bu (4.15)

y = cx,

where a ∈ R, b ∈ R, and c ∈ R. Now let the perturbations be of the following form:

a → (1 + δa) a

b → (1 + δb) b (4.16)

c → c (i.e., unperturbed),

where δa and δb are the corresponding normalized perturbations of a and b. If, for

instance, δa = 0.1, then this can be interpreted to be a 10% perturbation to the nominal a;

likewise for δb. Therefore, one may wish to normalize the system parametric perturbations

according to the following structure:

A → (I +∆A)A = A+∆AA

B → (I +∆B)B = B +∆BB (4.17)

C → C (i.e., unperturbed)

D → D (i.e., unperturbed),

which can be written in the form (4.1) with E = I, F = A, G = B, and H = 0.


4.2 Extension to LTI Descriptor Systems

In this section, we extend our study of the real controllability radius to LTI descriptor

systems as described by the following:

Ex = Ax+Bu (4.18)

y = Cx+Du,

where as before, x ∈ Rn, u ∈ R

m, and y ∈ Rr are the system states, inputs, and outputs,

respectively. However, E ∈ Rn×n is a singular matrix with rank(E) < n. Also, to avoid

including impulsive modes [86], which do not occur in real physical systems, it is assumed

that the following condition on E is satisfied [11, 86]:

rank(E) = deg(det(sE − A)) ,

where deg(p(s)) denotes the degree of the polynomial p(s).

Definitions of controllability similar to that of the regular state-space model (i.e.,

E = I) has been defined (e.g., see [50]).

Theorem 4.2.1 (Controllability [50]). Given the LTI descriptor system (4.18), the sys-

tem is controllable if and only if the following conditions are both satisfied:

i-a) rank

([

sE −A B

])

= n, for all s ∈ C; and

ii-a) rank

([

E B

])

= n.

One can then define a real controllability radius for descriptor systems (4.18) as

follows.

Definition 4.2.1. Given a controllable LTI descriptor system (4.18), denoted by (E,A,B),

the real controllability radius rc,dsc.R

is defined as


rc,dsc.R

(E,A,B) = inf

∆E ∈ Rn×n

∆A ∈ Rn×n

∆B ∈ Rn×m

∥∥∥∥

[

∆E ∆A ∆B

]∥∥∥∥2

∣∣∣∣∣∣∣

(E +∆E , A+∆A, B +∆B)

is uncontrollable

.

(4.19)

To obtain a formula for the real controllability radius of LTI descriptor systems (4.18),

first note from Theorem 4.2.1 that for any ∆E ∈ Rn×n, ∆A ∈ Rn×n, and ∆B ∈ Rn×m,

the perturbed system (E +∆E , A+∆A, B +∆B) is uncontrollable if and only if either

i-b) rank

([

s (E +∆E)− (A+∆A) , B +∆B

])

< n, for some s ∈ C; or

ii-b) rank

([

E +∆E B +∆B

])

< n.

So to compute the real controllability radius of (E,A,B), we consider the following two

problems and define two additional radii:

Problem (1): Given (E,A,B) and s ∈ C, find

rc,dsc.R,s (E,A,B, s) = inf

∆E ∈ Rn×n

∆A ∈ Rn×n

∆B ∈ Rn×m

∥∥∥∥

[

∆E ∆A ∆B

]∥∥∥∥2

∣∣∣∣∣∣∣

rank

([

s (E +∆E)−A−∆A, B +∆B

])

< n

;

(4.20)

and


Problem (2): Given (E,A,B), find

rc,dsc.R,f (E,A,B) = inf

∆E ∈ Rn×n

∆A ∈ Rn×n

∆B ∈ Rn×m

∥∥∥∥

[

∆E ∆A ∆B

]∥∥∥∥2

∣∣∣∣∣∣∣

rank

([

E +∆E B +∆B

])

< n

.

(4.21)

Using the results on generalized real perturbation values, these two radii can be

computed using the following formulas.

Theorem 4.2.2. Given controllable (E,A,B) and s ∈ C, rc,dsc.R,s in (4.20) and rc,dsc.

R,f in

(4.21) can be computed as follows:

rc,dsc.R,s (E,A,B, s) = τn

[

sE −A B

]

,

−sI 0

I 0

0 −I

(4.22)

and

rc,dsc.R,f (E,A,B) = τn

[

E B

]

,

−I 0

0 0

0 −I

. (4.23)


Finally, it can easily be shown that the real controllability radius of (E,A,B) can be

obtained by the minimum of the two radii above, as given in the following.

Theorem 4.2.3. Given controllable (E,A,B),

rc,dsc.R

(E,A,B) = min

(

infs∈C

rc,dsc.R,s (E,A,B, s) , rc,dsc.

R,f (E,A,B)

)

. (4.24)

Proof. Given perturbations ∆E ∈ Rn×n, ∆A ∈ Rn×n, and ∆B ∈ Rn×m, the perturbed

system (E +∆E , A+∆A, B +∆B) is uncontrollable if and only if either one of the above


conditions (i-b) or (ii-b) is satisfied. Hence, the proof follows immediately by the fact

that the norm of the smallest perturbations (∆E ,∆A,∆B) that satisfies (i-b) is given by

infs∈C

rc,dsc.R,s (E,A,B, s), while the norm of the smallest perturbation that satisfies (ii-b) is

given by rc,dsc.R,f (E,A,B).

It should be noted that in Definition 4.2.1, the controllability radius is defined such

that E can be perturbed arbitrarily. Since E determines the number of differential

equations and algebraic constraints in the nominal system (4.18), it may be undesirable

to allow arbitrary perturbations to E since these features are often fixed and not subject

to uncertainties. Hence, one may wish to include additional constraints on ∆E (e.g., see

[9]), or not allow E to be perturbed at all (i.e., ∆E = 0). These extensions can easily be

done using the results on generalized and restricted real perturbation values.

4.3 Extension to LTI Time-Delay Systems

In this section, we consider LTI time-delay systems described as follows:

x(t) = A0x(t) + A1x(t− τ) +Bu(t− τ) (4.25)

y = Cx(t− τ),


tively, and τ > 0 represents a time-delay.

There are various definitions of controllability for such systems (e.g. see [76] and

the reference therein), but one that can be characterized by a rank condition is spectral

controllability [63, 64].

Theorem 4.3.1 (Spectral controllability [63, 64]). Given the LTI time-delay system

(4.25), the system is spectrally controllable if and only if

rank

([

sI −A0 − A1e−τs, Be−τs

])

= n, for all s ∈ C.


Now, suppose we perturb the system (4.25) as follows:

x(t) = (A0 +∆A0)x(t) + (A1 +∆A1

) x(t− τ) + (B +∆B) u(t− τ) (4.26)

y = (C +∆C)x(t− τ).

where ∆A0∈ R

n×n, ∆A1∈ R

n×n, ∆B ∈ Rn×m, and ∆C ∈ R

r×n. One can then define a

real spectral controllability radius for the time-delay systems (4.25).

Definition 4.3.1. Given a spectrally controllable time-delay system (4.25), the real spec-

tral controllability radius, denoted rc,delayR

, is given by

rc,delayR

(A0, A1, B) = inf

∆Ai∈ Rn×n

∆B ∈ Rn×m

i = 1, 2

∥∥∥∥

[

∆A0∆A1

∆B

]∥∥∥∥2

∣∣∣∣∣∣∣∣∣∣

the perturbed system

(4.26) is spectrally

uncontrollable

.

(4.27)

Using the results on generalized real perturbation values, the real spectral controlla-

bility radius of (4.25) can be computed as follows.

Theorem 4.3.2. Given the LTI time-delay system (4.25),

rc,delayR

(A0, A1, B) = infs∈C

τn

[

sI − A0 −A1e−τs, Be−τs

]

,

I 0

e−τsI 0

0 −e−τsI

.


Similarly, the above results can be extended to more general LTI time-delay systems

described by

x(t) =

N∑

k=0

(Akx(t− kτ) +Bku(t− kτ)) (4.28)

y =N∑

k=0

Ckx(t− kτ),


where x ∈ Rn, u ∈ R

m, and y ∈ Rr are the system states, inputs, and outputs, respec-

tively, τ > 0 represents a time-delay, and N ∈ 0, 1, 2, . . ..

4.4 Eigenvalue Mobility

In [84], the complex DFM radius was used to bound the mobility of the closed-loop

eigenvalues, where mobility was defined as the relative ability to shift the eigenvalues

using all decentralized controller gains of a specified norm. A DFM therefore has no

mobility, while an eigenvalue s ∈ λ(A) with a relatively small modal DFM radius with

respect to s may have low mobility. In other words, in the former, no decentralized

controller gain can shift a DFM, while in the latter, the eigenvalue s can barely be

shifted if the controller gain is not large enough.

In this section, we discuss the eigenvalue mobility of the LTI system (3.1) in terms

of the real controllability radius, and extend the results to the transmission zero at s

radius, as it applies to the RSP. We also revisit the eigenvalue mobility bound provided

in [84] and obtain an improved bound.

4.4.1 Controllability radius and mobility of λ(A+BK)

The following result provides a measure of difficulty in shifting an eigenvalue of A using

any controller gain K of a specified “size”.

Theorem 4.4.1. Consider the LTI system (A,B) in (3.1) and a given real M > 0. Let

s ∈ C and assume that the modal real controllability radius of (3.1) with respect to s is

computed to be equal to ǫ. Then for all controller gains K ∈ Rm×n with ‖K‖2 ≤M , there

exists a λ ∈ λ(A+BK) such that

∣∣∣λ− s

∣∣∣ ≤ (2 ‖A‖2 + 2 ‖B‖2M + ǫ (M + 1))1−

1

n (ǫ (M + 1))1

n . (4.29)



Therefore suppose we are interested in the closed-loop spectrum of (A +BK) for all

controller gains K such that, say, ‖K‖2 ≤ 106. Also, let s ∈ λ(A) and suppose that

the system has a real controllability radius with respect to s that is sufficiently small

such that the bound in (4.29) is also considered small. Then from Theorem 4.4.1, we

know that the closed-loop spectrum will always contain an eigenvalue close to s for all

controller gains K with ‖K‖2 ≤ 106. So s, in this case, is considered to have low mobility.

As an example, consider the following.

Example: Eigenvalue mobility Consider the following system

x =

0 1

−2 2

x+

0

10−12

u, (4.30)

which has eigenvalues 1 ± j1. The modal real controllability radius with respect to

s = 1+ j1 is found to be 1.00010−12 (radius with respect to s = 1− j1 is also 1.00010−12).

Table 4.1 lists the various mobility bounds obtained using Theorem 4.4.1 for M =

109, 1010, 1011, 1012 (see column 2). To interpret the results, we look at, for instance,

M = 1010, where the mobility bound given by Theorem 4.4.1 is found to be 0.24231.

This implies that for all K such that ‖K‖2 ≤ 1010, the closed-loop system (A+BK) has

at least one eigenvalue λ ∈ λ(A+BK) such that∣∣∣s− λ

∣∣∣ ≤ 0.24231.

As a preliminary check, we used MATLAB’s random generator randn.m to generate

10, 000 random gains K such that ‖K‖2 ≤ 1010, and it is found that for all cases,

the closed-loop system (A+BK) always has an eigenvalue λ ∈ λ(A +BK) such that∣∣∣s− λ

∣∣∣ ≤ 0.031160 (see column 3 of Table 4.1), where recall s = 1 + j1. So we see that

the bound 0.24231 is off by an order of magnitude compared to 0.031160 obtained by

simulation. However, from Table 4.1, we see that the bound given by Theorem 4.4.1 is

not too conservative for the other cases.

As another check, we arbitrarily place (complex conjugate) poles on the two circles

centered at 1 ± j1 with radius 0.24231, and it is found that a gain K with a minimum


Table 4.1: Mobility bounds for closed-loop eigenvalues of example system

M Mobility Bound Mobility Bound min ‖K‖2 ∗

by Theorem 4.4.1 by random generator

1010 0.24231 0.031160 2.679510+11

1011 0.78368 0.37295 6.228610+11

1012 2.9735 1.8766 2.170610+12

1013 18.932 5.9312 3.059910+14

∗ K is chosen so that λ(A+BK) is on two circles centered at s = 1± j1 of radius given in column 2.

norm of 2.67951011 is required (see column 4), which is greater than (or equal to) 1010 as

expected.

Note that the bound 0.24231 obtained by Theorem 4.4.1 implies that if one wishes to

shift the poles of (4.30) to desired locations with distances greater than 0.24231, then a

controller gain with norm greater than 1010 is required, and from the simulation results,

an even greater norm is needed. Since for this example the open-loop poles are at 1± j1,

which are in the right-half plane and away from the imaginary axis by a distance of

1 > 0.24231, this implies that if one wishes to stabilize the system (4.30), a controller

gain with norm greater than 1010 must be used! In fact, if one wishes to place the poles

at ±j1, the controller gain is K =

[

1 −2]

× 1012, which has norm ‖K‖2 = 2.23611012 .

4.4.2 Transmission zero at s radius and eigenvalue mobility of

RSP

Recall that one of the existence conditions for a solution to the RSP to exists for the LTI

system (C,A,B,D) is that the transmission zeros of the system do not coincide with the

tracking/disturbance poles (see Section 1.7.4). If they do coincide, the following is true.

Theorem 4.4.2. Let s ∈ C be a given tracking and/or disturbance pole. If the LTI system

(C,A,B,D) has a transmission zero at s, then the augmented plant-servocompensator


system [18] as given by (1.19) has an uncontrollable eigenvalue at s.


In other words, if one of the tracking/disturbance poles, say s, matches one of the

transmission zeros of (C,A,B,D), then the augmented plant-servocompensator system

has an eigenvalue s that cannot be shifted using any controller gains. On the other hand,

if s is not a transmission zero, then one can find a controller gain to place s at another

location. However, a large gain may be required if the mobility of s is low. The following

result provides a bound on the mobility of the augmented system’s eigenvalues in terms

of the real transmission zero at s radius.

Theorem 4.4.3. Given (C,A,B,D), real M > 0, and a tracking/disturbance pole s ∈ C,

if the real transmission zero at s radius is rTZR

(C,A,B,D, s) = ǫ, then for all controller

gains K0 ∈ Rm×n and K1 ∈ Rm×rp such that

∥∥∥∥

[

K0 K1

]∥∥∥∥2

≤ M , there exists a

λ ∈ λ

A 0

BC C

+

B

BD

[

K0 K1

]

such that

∣∣∣λ− s

∣∣∣ ≤

2

∥∥∥∥∥∥∥

A 0

BC C

∥∥∥∥∥∥∥2

+ 2

∥∥∥∥∥∥∥

B

BD

∥∥∥∥∥∥∥2

M + ǫ (M + 1)

1− 1

n

(ǫ (M + 1))1

n .

(4.31)


4.4.3 DFM radius and eigenvalue mobility

In [84], the following result was presented.

Theorem 4.4.4 ([84]). Given the decentralized system

A B

C D

v

, if s ∈ λ(A) and the


modal real DFM radius2 with respect to s is found to be rDFM,sR

A B

C D

v

= ǫ < 1,

then for all M := maxi∈v‖Ki‖2 <

1

2ǫ, there exists a λ ∈ λ

(

A+v∑

i=1

BiKiCi

)

such that

∣∣∣s− λ

∣∣∣ ≤

(2(α + α2M + βǫ

))1− 1

n n1

n (βǫ)1

n ,

where α := max(‖A‖2 , ‖B‖2 , ‖C‖2), and β := 1 + 2αM + ǫM + 2 (α+ ǫ)2M2.

The result of Theorem 4.4.4 is based on a bound of spectra variation between two

matrices that is found in [26]. In [27], an improved bound can be found, of which

Theorems 4.4.1 and 4.4.3 of this thesis are based on. So for completeness, we apply the

results of [27] to obtain the following improved bound for Theorem 4.4.4.

Theorem 4.4.5. Given the same conditions as Theorem 4.4.4,

∣∣∣s− λ

∣∣∣ ≤

(2(α + α2M

)+ βǫ

)1− 1

n (βǫ)1

n .

Proof. The proof is similar to the proof found in [84] with the exception that Lemma C.3.1

is applied instead of [84, Lemma 3]. The details are omitted.

4.5 Summary

Continuing the study of real LTI robustness radii from the previous chapter, this chapter

makes a number of extensions, namely: i) extensions to the structured real controllability

radius and the structured versions of the other LTI robustness radii in Chapter 3; and ii)

extensions of the real controllability radius to LTI descriptor and time-delay systems. In

fact, given the results on real perturbation values and its generalizations in Chapter 2, one

can easily see that other robustness radii can similarly be obtained for any other system

2In [84], the complex DFM radius was used. With little modification, the real DFM radius is usedhere instead.


definitions that have an algebraic characterization in the form of a rank condition. Such

similar extensions is left for future studies.

The mobility of system eigenvalues is also discussed in this chapter. Since a system

with low eigenvalue mobility means that large controller gains are required to shift the

system’s open-loop eigenvalues, being able to characterize eigenvalue mobility is impor-

tant. In this chapter, mobility bounds are obtained using the various LTI robustness

radii.

Chapter 5

Algorithms

5.1 Problem Statements

In this chapter, we present two algorithms for computing the various types of real per-

turbation values and the various LTI robustness radii, where the former is a 1-D max-

imization problem and the latter is a 2-D minimization problem in the complex plane.

The proposed algorithms have the advantage that they are insensitive to initial search

points, and as a result are more efficient than generic nonlinear search techniques.

Recall from Chapter 2 that computing the various types of real perturbation values (or

computing the lower bounds for the case of restricted real perturbation values) involves

solving the following general 1-D nonlinear optimization problem1:

Problem (1): Given M ∈ Cn×m, L ∈ Cn×l, N ∈ Cp×m and i ∈ 1, . . . ,min(n,m), find

τi(M,L,N) = supγ∈(0,1]

σ2i−1

(MR

γ , LR

γ , NR

γ

)(5.1)

= supγ∈(0,1]

σ2i−1

([ReM −γ ImM

γ−1 ImM ReM

]

,[

ReL −γ ImLγ−1 ImL ReL

]

,[

ReN −γ ImNγ−1 ImN ReN

])

.

In [42], it was shown that for the case when L = I, N = I, and i = min(n,m), which

corresponds to computing the smallest ordinary real perturbation values of a single matrix

1Recall that the ordinary and generalized real perturbation value problem can be obtained by setting(M,L,N) = (M, I, I) and (M,L,N) = (M, I,N), respectively.

85

Chapter 5. Algorithms 86

M , the function to be maximized in (5.1) is quasiconcave. So for this particular case,

efficient unimodal algorithms, such as golden-section search methods, can be used. For

the general case, however, the function to be maximized in (5.1) can have multiple local

maxima. So, in general, one cannot use such unimodal techniques and must resort to

nonlinear optimization techniques.

Also, recall from Chapter 3 that computing the various complex and real LTI robust-

ness radii involves solving a 2-D nonlinear optimization problem in the complex plane. In

particular, the various complex radii can be computed by solving the following standard

general problem:

Problem (2a): Given the system matrices (C,A,B,D) of a LTI system (1.1), i ∈

1, . . . , n+min(r,m), and general conformal complex matrices E and F , find

riC(C,A,B,D) = inf

s∈Cσi

A− sI B

C D

, E, F

. (5.2)

Similarly, the various real robustness radii are computed by solving the following:

Problem (2b): Find

riR(C,A,B,D) = inf

s∈Cτi

A− sI B

C D

, E, F

(5.3)

≥ infs∈C

supγ∈(0,1]

σ2i−1

(

[G(s)]Rγ , ER

γ , FR

γ

)

,

where we denote G(s) :=

A− sI B

C D

.2 Both Problem (2a) and (2b) generally have

multiple local minima with respect to the s ∈ C search, so again one must resort to

nonlinear techniques to compute the various types of LTI robustness radii.

There are many generic nonlinear search algorithms (such as MATLAB’s fmincon.m,

or other gradient search methods) that can be used to solve these optimization problems,

but they tend to be somewhat inefficient. This is because nonlinear search algorithms

2For simplicity, we will assume for the rest of the chapter that the inequality in (5.3) is an equality.


generally only find the local extremum that is closest to the initial search point. The

algorithms are therefore often executed multiple times with different initial points in

order to confirm that a computed extremum is global. Since the number of initial search

points to be tested is arbitrary, a large number is generally selected to ensure accuracy

and confidence in the results, leading to a relatively time-consuming process.

Specialized algorithms (see [7,8,25,34,35,66,89] and the references therein) had been

developed over the past two decades to compute the complex controllability radius (i.e.,

Problem (2a) for A, B, E = I, F = I, i = n, and where C and D are empty matrices).

The works of [25] and [89], for instance, proposed various optimization techniques that

mainly consist of minimizing some functional based on various new characterizations of

the controllability radius. These algorithms, however, provide only local minima. On the

other hand, global minima can be obtained using grid-based techniques such as the work

of [8, 31, 36]. These techniques generally work by first partitioning the search domain

into subdivisions (e.g., rectangles or simplices), and then iteratively reducing the search

domain by solving various subproblems and discarding certain subdivisions. In [34], a

bisection algorithm was proposed that can actually obtain the global minimum to a

factor of two without using a grid. The technique was later improved by [7] who used a

trisection algorithm to obtain the minimum to any arbitrary precision. These techniques

work by maintaining bounds on the radius and then iteratively tightening these bounds

by solving various alternative eigenvalue problems.

In terms of algorithms for solving Problem (1), [82] proposed an algorithm for solving

the real stability radius [74], which can be adapted to solve Problem (1).

In this chapter, we present two efficient algorithms for i) computing the various types

of real perturbation values, and ii) computing the various LTI robustness radii. We refer

to the first algorithm as Algorithm I, and the second as Algorithm II. Both algorithms are

similar in nature and are based on a similar method found in [82] for solving the real sta-

bility radius [74], and on the grid-based techniques mentioned above. In particular, both


algorithms are based on the idea of first finding a set, called a minimizing/maximizing

set, that is guaranteed to contain the global minimizer/maximizer; i.e., the point in the

domain that achieves the global minimum/maximum. The minimizing/maximizing set

is then reduced at each iteration – by solving various related eigenvalue problems – to

narrow down and pinpoint the location of the minimizer/maximizer. An advantage of

this method over generic nonlinear techniques is that the proposed algorithms are not

dependent on the initial search point. Hence, once an extremum is found, it is indeed

global, and there is no need to rerun the algorithms multiple times.

This chapter is organized in three main sections. In Section 5.2, a basic template

of Algorithm I and II is given. The development of the two algorithms for solving the

general form of Problem (1) and Problem (2b) are then given in Section 5.3 and 5.4,

respectively.

5.2 Basic Idea of Algorithm I and II

In the following, we give a brief description of the underlying concepts of Algorithm I

and II, which will later help in the development of the algorithms themselves. For the

moment, we assume that we are searching for a global maximum, denoted by ymax, with

the corresponding maximizer, xmax, which is the point in the search domain that achieves

ymax. The discussion applies with minimal change if we are searching for a minimum

instead. For illustration, we search for the maximum and maximizer of the following

scalar function:

y = f(x) = 1− (1− x)2 , (5.4)

which is plotted in Figure 5.1(a). Here, the maximum is ymax = 1, which is achieved at

xmax = 1.

The basic structure of the two algorithms is given in Algorithm 5.1, from which we

see that the algorithm is iterative and contains the following four key parts:


• keeping track of the variables yk and xk of (5.4), which are the current (i.e., at

iteration k) approximations of the maximum and the corresponding maximizer,

respectively;

• computing the maximizing set, denoted by Sk;

• updating the approximations yk and xk; and

• checking the stopping criteria and terminating the algorithm.

More details of the general overall algorithm structure and how each of the four parts

plays a role are presented below.

5.2.1 Current approximation of maximum

At the start of the k-th iteration, we denote the current approximation of ymax and

xmax by yk−1 and xk−1, respectively, where xk−1 achieves yk−1. In our example, for

instance, suppose we found that yk−1 = 0.64, corresponding to xk−1 = 1.6 as shown in

Figure 5.1(a). A key part of the algorithm is then to iteratively update and improve

these approximations; i.e., to obtain at the next iteration yk and xk, where xk achieves

yk and yk > yk−1. This is done by first computing the maximizing set as described next.

5.2.2 Maximizing set

Given the current approximations yk−1 and xk−1, the maximizing set, denoted by Sk, is

defined to be a set in the domain with the following property:

Sk = x in domain|x achieves y with y > yk−1 .

In other words, the maximizing set contains all points (including xmax) in the domain

that achieves a value larger than our current approximation yk−1. Hence the significance

of computing Sk is that after Sk is obtained, we can improve our current approximation

yk−1 by simply choosing any point xk ∈ Sk, which is guaranteed to achieve yk > yk−1.


Algorithm 5.1: Basic structure of Algorithm I and II.

Input: matrices (M,L,N) (Alg. I), or LTI system matrices (C,A,B,D) (Alg. II)

Input Tolerance:: tolerance for stopping criteria

Output: maximum corresponding to real perturbation values (Alg. I) or

minimum corresponding to LTI robustness radii (Alg. II)

1 Initialization

2 • Choose arbitrary point x0 in domain.

3 • Compute y0, achieved by x0.

4 • Set initial maximizing (minimizing) set S0 as whole domain.

5 for iteration k = 1, 2, . . . do

// Current estimate of maximum (minimum): yk−1

// Current estimate of maximizer (minimizer): xk−1

// Current maximizing (minimizing) set: Sk−1

6 • Using yk−1 and xk−1, reduce the maximizing (minimizing) set to obtain new

maximizing (minimizing) set Sk ⊆ Sk−1.

7 Update approximations

8 Select any point xk ∈ Sk that achieves yk > yk−1 (yk < yk−1).

// Hence:

// - new estimate of maximum (minimum): yk

// - new estimate of maximizer (minimizer): xk

// - new maximizing (minimizing) set: Sk

9 Quit if

10 • Stopping criteria are met; e.g., Sk and/or |yk−1 − yk| are smaller than

given tolerance.


1

y

x 0 2 1

1

y

x 0 2 1

yk-1

xk-1

(a) (b)

1

y

x 0 2 1

yk-1

xk-1 s

1

y

x 0 2 1

yk-1

xk-1 s xk

yk

(c) (d)

Figure 5.1: Plot of (a) y = 1− (1− x)2, (b) the current approximation of the maximum,

yk−1, and the corresponding maximizer xk−1, (c) all points that achieve yk−1 and the

maximizing set Sk (shaded area), and (d) the new approximation yk and xk.


In our approach to compute Sk when given yk−1, we first find all points in the domain

that achieve a value exactly equal to yk−1. Then by continuity, these points divide the

domain into regions, where points in a particular region either all achieve a value greater

than yk−1, or a value less than yk−1.

To demonstrate this, consider again our example. In the example, we are given

xk−1 = 1.6 and yk−1 = 0.64. To find all other values in the domain that achieve yk−1,

one can solve the following equation:

1− (1− x)2 = yk−1

x = 1±√

1− yk−1

= 1±√1− 0.64 = 0.4, 1.6 .

Hence yk−1 = 0.64 is achieved at two points: i) the original point xk−1 = 1.6; and ii)

a new point s = 0.4. Therefore there are three intervals of interest: (−∞, s); (s, 1.6);

and (1.6,+∞) (see Figure 5.1(c)). From Figure 5.1(c), we see that all values in (−∞, s)

and (1.6,+∞) are less than yk−1 = 0.64, while all values in (s, 1.6) are greater than

yk−1 = 0.64; hence the maximizing set is found to be Sk = (0.4, 1.6).

It should be noted that determining how to compute the maximizing set Sk is the

most difficult part of designing the overall algorithm. The actual computation of Sk at

each iteration will take up the largest fraction of the overall computation time. In the

case of solving Problem (1), (2a), and (2b), one can obtain points in the domain that

achieve values exactly equal to yk−1 by converting the associated singular value problems

into corresponding generalized eigenvalue problems. This will be more evident later in

the upcoming sections.

Also, note from Algorithm 5.1 that we are keeping track of the maximizing set from

the previous iteration (i.e., Sk−1). This is actually not necessary, but helps to improve

the overall computational time of the algorithm. This is because since Sk ⊂ Sk−1, where

Sk−1 itself is a subset of the whole domain, computational operations and time can be


reduced by obtaining Sk from Sk−1, as opposed to the whole domain.

5.2.3 Updating the current approximation yk

After computing the maximizing set Sk, the current estimates yk−1 and xk−1 can be

updated by choosing any point xk ∈ Sk. By the construction of Sk, it is then true that

yk > yk−1, where yk is achieved by xk.

In the example, we previously found Sk = (0.4, 1.6). Arbitrarily choosing a point

in Sk, say, at xk = 1.27, we obtain a better approximation of the maximum to be

yk = 0.9271.

Note, however, that to reduce the number of overall iterations, it is important to be

able to obtain a good update at each iteration. For instance, in the above example, if we

had chosen xk to be the middle of Sk (i.e., xk = 1), then we would have obtained a better

approximation of the maximum.3 In the algorithms to follow, we discuss various ways

(such as using a cubic fit) of choosing the next approximations to improve the overall

efficiency of the algorithms.

5.2.4 Stopping criteria

We propose two stopping criteria, which are used to terminate the algorithms when either

one is met. The first condition is related to Sk, while the second depends on yk. Since

Sk ⊂ Sk−1, for k = 1, 2, . . ., the size of the maximizing set shrinks with each iteration.

Therefore the algorithms can terminate when the size of the maximizing set is smaller

than some given tolerance. Also, as the approximation of yk gets closer and closer to

ymax, the subsequent difference between yk−1 and yk becomes smaller. Hence, if the

difference between yk−1 and yk is smaller than some tolerance, then this may indicate

that no significant improvements (e.g., in significant figures) may be achieved in the

3For this particular example, the middle xk = 1 turns out to be the actual maximum ymax = 1.


approximation of ymax; so, the algorithm may also terminate.

5.3 Algorithm I: Computing Real Perturbation Val-

ues

We now go into more detail regarding Algorithm I for solving (5.1) of Problem (1). Note

that if (M,L,N) are all real matrices, then the various real perturbation value problems

simplify to various singular value problems. So to avoid talking about this trivial case,

we assume in the rest of the development that (M,L,N) are not all real matrices.

First of all, let the global maximum of (5.1) be denoted by

r∗ = supγ∈(0,1]

σ2i−1

([ReM −γ ImM

γ−1 ImM ReM

]

,[

ReL −γ ImLγ−1 ImL ReL

]

,[

ReN −γ ImNγ−1 ImN ReN

])

with the global maximizer being γ∗ ∈ (0, 1]; i.e., r∗ = σ2i−1

(MR

γ∗ , LR

γ∗ , NR

γ∗

). At the start

of the k-th iteration, for k = 1, 2, . . ., define the current approximations of r∗ and γ∗ to

be rk−1 and γk−1, respectively, where rk−1 = σ2i−1

(

MR

γk−1, LR

γk−1, NR

γk−1

)

. Also, define the

maximizing set Sk to be

Sk =γ ∈ (0, 1] |σ2i−1

(MR

γ , LR

γ , NR

γ

)> rk−1

.

5.3.1 Obtaining the maximizing set Sk

Recall the current approximations of r∗ and γ∗ are rk−1 and γk−1, respectively. Due to

continuity, Sk is a set of intervals in (0, 1], where the endpoints (i.e., boundary) of these

intervals are all γ ∈ (0, 1] such that σ2i−1

(MR

γ , LR

γ , NR

γ

)= rk−1, with also possibly 0 and

1.

For example, consider the illustration given in Figure 5.2. In Figure 5.2(a), it is shown

that the current approximation rk−1 is achieved at γk−1. Now suppose that all of the other

values of γ ∈ (0, 1] that also achieve a radius equal to rk−1 are found; i.e., s1 and s2 in


rk-1

γ 0 1 γk-1

rk-1

γ 0 1 γk-1 s2 s1

(a) (b)

Figure 5.2: Computing the maximizing set Sk: (a) current approximation rk−1 which is

achieved at γk−1; and (b) other points that achieve rk−1 and Sk = (0, s1) ∪ (γk−1, s2).

Figure 5.2(b), where σ2i−1

(MR

s1 , LRs1, N

Rs1

)= rk−1 and σ2i−1

(MR

s2 , LRs2, N

Rs2

)= rk−1. Hence,

there are four intervals that may belong to the maximizing set Sk: (0, s1); (s1, γk−1);

(γk−1, s2); and (s2, 1]. To determine which interval(s) belong to Sk, a test point, or in

certain cases derivative information at the endpoints (more on this later), can be used.

In the example in Figure 5.2, Sk = (0, s1) ∪ (γk−1, s2).

To actually find these endpoints (e.g., s1 and s2 in Figure 5.2(b)), we have the fol-

lowing key result.

Theorem 5.3.1. Given M ∈ Cn×m, L ∈ Cn×l, N ∈ Cp×m, γ ∈ (0, 1], and real x > 0,

then

x ∈ σ(MR

γ , LR

γ , NR

γ

)⇔ γ2 ∈ λ(A(x,M,L,N) ,B(x,M,L,N)) , (5.5)

where A(x,M,L,N) and B(x,M,L,N) are defined as

A(x,M,L,N) :=

−x[LLH LLT

LLH LLT

]0

2[M 00 M

]H −x[

NHN −NHN−NTN NTN

]

(5.6a)

and

B(x,M,L,N) :=

x[

LLH −LLT

−LLH LLT

]

−2[M 00 M

]

0 x[NHN NHNNTN NTN

]

. (5.6b)



So given the current approximation rk−1, then by Theorem 5.3.1, all γ ∈ (0, 1] that

achieve a restricted singular value of(MR

γ , LR

γ , NR

γ

)equal to rk−1 are among the set of

generalized eigenvalues λ(A(x,M,L,N) ,B(x,M,L,N)); however, at this point one does

not know which restricted singular value it is and can only say that σt

(MR

γ , LR

γ , NR

γ

)=

rk−1 for some t ∈ 1, . . . ,min(n,m). We denote the set of these points by

Γ(rk−1) :=γ ∈ (0, 1]

∣∣∃t ∈ 1, . . . ,min(n,m) , σt

(MR

γ , LR

γ , NR

γ

)= rk−1

(5.7)

=γ ∈ (0, 1]

∣∣γ2 ∈ λ(A(x,M,L,N) ,B(x,M,L,N))

.

Note that to obtain Sk, we are interested only in γ ∈ Γ(rk−1) that achieves a (2i−1)-th re-

stricted singular value of(MR

γ , LR

γ , NR

γ

)equal to rk−1. So to determine which γ ∈ Γ(rk−1)

satisfies this, one can simply compute the restricted singular value of(MR

γ , LR

γ , NR

γ

)for

all γ ∈ Γ(rk−1), and compare to see which γ’s result in a (2i − 1)-th restricted singular

value equal to rk−1. For later reference in upcoming sections, this is referred to as the

direct test method (note that this method can be inefficient, and a more efficient method

can be used for the case when (M,L,N) = (M, I,N)). Now define the set

Γ(rk−1) :=γ ∈ Γ(rk−1)

∣∣σ2i−1

(MR

γ , LR

γ , NR

γ

)= rk−1

. (5.8)

The maximizing set Sk is hence bounded by the elements of Γ(rk−1) (and also possibly

by 0 and 1); i.e., Γ(rk−1) contains the endpoints of the interval(s) in (0, 1] that achieves

σ2i−1

(MR

γ , LR

γ , NR

γ

)greater than rk−1. To determine which of these interval(s) belong to

Sk, arbitrary test points can again be used. As mentioned earlier, this may be inefficient

and a more efficient method, as described next, can be used for the case when (M,L,N) =

(M, I,N).

Obtaining Sk when (M,L,N) = (M, I,N)

When (M,L,N) = (M, I,N) the following simplifications to Theorem 5.3.1 can be made.


Theorem 5.3.2. Given M ∈ Cn×m, N ∈ C

p×m, γ ∈ (0, 1] and real x > 0, then,

x ∈ σ(MR

γ , NR

γ

)⇔ γ2 − 1

γ2 + 1∈ λ(

A(x,M,N) , B(x,M,N))

, (5.9)

where A(x,M,N) and B(x,M,N) are defined as

A(x,M,N) := −

0 MHM − x2NHN

MTM − x2NTN 0

(5.10a)

and

B(x,M,N) :=

MHM − x2NHN 0

0 MTM − x2NTN

. (5.10b)


Hence when (M,L,N) = (M, I,N), one only needs to compute the generalized

eigenvalues of a smaller-sized matrix pair(


instead of the pair

(A(x,M,L,N) ,B(x,M,L,N)) in (5.6). Furthermore, the first matrix pair has the fol-

lowing property.

Lemma 5.3.1. The generalized eigenvalues of(


in (5.10) are

symmetrical about both the imaginary axis and the real axis.


So by Lemma 5.3.1, there can only be at most min(n,m) elements in Γ(rk−1) of (5.7)

for a given M ∈ Cn×m and N ∈ Cp×m (L = I). However, computing the generalized

singular values of(MR

γ , NR

γ

)for all γ ∈ Γ(rk−1) can still be relatively costly. Instead, we

can apply the following results to compute the derivative information at each γ ∈ Γ(rk−1)

to determine which generalized singular value of(MR

γ , NR

γ

)equals rk−1.

Lemma 5.3.2. ([22])

Let A(γ) ∈ Cn×n and B(γ) ∈ Cn×n both be analytic symmetric matrix functions on

an open set Γ ∈ R. Also, let (λ(γ) , y(γ)) be a generalized eigenvalue and eigenvector pair


such that

A(γ) y(γ) = λ(γ)B(γ) y(γ) (5.11a)

y(γ)T B(γ) y(γ) = 1. (5.11b)

Then,

dλ(γ)

dγ= y(γ)T

(A(γ)

dγ− λ(γ)

B(γ)

dγ

)

y(γ) . (5.12)

Lemma 5.3.3. Given M ∈ Cn×m, N ∈ Cp×m and w(γ) ∈ σ(MR

γ , NR

γ

), then,

dw(γ)

dγ=

2γ

w(γ) (γ2 + 1)2×

y(γ)T

(γ2−1γ2+1

)

I I

I(

γ2−1γ2+1

)

I

y(γ)

−1

, (5.13)

where(

γ2−1γ2+1

, y(γ))

is a generalized eigenvalue and eigenvector pair of (A(w(γ) ,M,N),

B(w(γ) ,M,N)), as defined in (5.9), such that

A(w(γ) ,M,N) y(γ) =γ2 − 1

γ2 + 1B(w(γ) ,M,N) y(γ)

y(γ)T B(w(γ) ,M,N) y(γ) = 1.


The procedure on how to apply the derivative information at each γ ∈ Γ(rk−1) to de-

termine which generalized singular value of(MR

γ , NR

γ

)equals rk−1 is described as follows,

which is similar to a technique found in [82].

First, let Γ(rk−1) of (5.7) be sorted in ascending order; i.e., Γ(rk−1) = g1, g2, . . .

such that g1 ≤ g2 ≤ · · · . Also, let gw = γk−1 for some w ∈ N (note that w always exists

since we are given that rk−1 = σ2i−1

(

MR

γk−1, NR

γk−1

)

, which implies γk−1 ∈ Γ(rk−1)). Now

suppose that the derivative (as computed by (5.13) of Lemma 5.3.3) of the generalized

singular value of(MR

gw , NR

gw

), denoted by

dσ(MRgw

,NRgw)

dγ, is positive. Then, this implies


that the plot of the (2i − 1)-th generalized singular value of(MR

γ , NR

γ

)as a function of

γ ∈ (0, 1], denoted y2i−1(γ) = σ2i−1

(MR

γ , NR

γ

), crosses upward at gw.

Next, consider the derivative at gw+1; i.e.,dσ(MR

gw+1,NR

gw+1)

dγ. If the derivative is again

positive, then this means that there is another upward crossing at gw+1. Such an upward

crossing can only occur from the plot y2i−2(γ) = σ2i−2

(MR

γ , NR

γ

)(i.e., the (2i − 2)-th

generalized singular value plot). This is because if one considers all the singular value

plots yt(γ) = σt

(MR

γ , NR

γ

), for γ ∈ (0, 1] and t = 1, . . . , 2 × min(n,m), then by the

ordering of the generalized singular values, we have yt1(γ) ≥ yt2(γ) for all γ ∈ (0, 1] and

for all t1 ≤ t2. In other words, the plot yt1(γ) is always above (or equal to) the plot yt2(γ)

for all t1 ≤ t2. Hence, an upward crossing at gw+1 can only be by the next plot below

y2i−1(γ), which is y2i−2(γ).

On the other hand, if the derivative at gw+1 is negative, then there is a downward

crossing at gw+1, which can only occur from y2i−1(γ) for a similar reason.

Hence, starting from gw and working in the manner as described above, the upward

and downward crossings can be used to determine which singular value plot crosses at

each g ∈ Γ(rk−1). Figure 5.3 illustrates a possible example scenario, where the ‘+’ and

‘–’ signs depict an upward and downward crossing, respectively.

As mentioned earlier, the maximizing set Sk is obtained by determining all γ ∈ Γ(rk−1)

such that σ2i−1

(MR

γ , NR

γ

)= rk−1; i.e., all g ∈ Γ(rk−1) where there is a crossing by the

y2i−1(γ) plot. Define such a set as

Γ(rk−1) :=γ ∈ Γ(rk−1)

∣∣σ2i−1

(MR

γ , NR

γ

)= rk−1

, (5.14)

which is the same as (5.8) for (M,L,N) = (M, I,N). As before, the maximizing set

Sk is bounded by the elements of Γ(rk−1) (and also possibly by 0 and 1); i.e., Γ(rk−1)

contains the endpoints of the interval(s) in (0, 1] that achieves σ2i−1

(MR

γ , NR

γ

)greater

than rk−1. Instead of using test points to determine which interval(s) belong to Sk (i.e.,

using the direct test method), the derivative information obtained previously can be used

to determine which elements in Γ(rk−1) are at the beginning of an interval and which are


-

0 1

2i

–

gw−2

2i− 1

–

gw−1

2i− 1

+

gw

2i− 2

+

gw+1

2i− 2

–

gw+2

2i− 1

–

gw+3

· · · · · ·

Figure 5.3: Example of using the derivative information of g ∈ Γ(rk−1) to determine the

maximizing set, Sk−1.

at the end. In particular, since we are searching for a maximum, all beginning endpoints

thus have an upward crossing and all ending endpoints have a downward crossing.

Referring back to the example scenario shown in Figure 5.3, the maximizing set is

hence Sk = (0, gw−1) ∪ (gw, gw+3) (assuming there are no more upward σ2i−1 crossings in

(0, gw−2) and (gw+3, 1]).

5.3.2 Updating the current maximum rk

By the construction of Sk, any γ ∈ Sk achieves σ2i−1

(MR

γ , LR

γ , NR

γ

)greater than rk−1.

Hence, the next approximation γk of the global maximizer can arbitrarily be chosen

in Sk, and then set rk = σ2i−1

(MR

γk, LR

γk, NR

γk

)> rk−1. From experience, however, the

performance of the algorithm seems to improve by choosing γk as the middle of the

largest interval of Sk. If the derivative information is available (i.e., in the case where

(M,L,N) = (M, I,N)), γk is chosen as the maximizer of a cubic fit through the endpoints

(and their derivatives) of the intervals of Sk (see [82] and Algorithm 5.2).


For simplicity, the algorithm terminates when the largest interval of Sk is smaller than a

specified tolerance.


Algorithm 5.2: Algorithm I for computing real perturbation values.

Input: Matrices (M,L,N) and i ∈ 1, 2 . . .

Input Tolerance:: TOLSk

Output: r∗ and γ∗, where r∗ = σ2i−1

(MR

γ∗ , LR

γ∗ , NR

γ∗

)

1 Initialization

2 • Choose arbitrary γ0 ∈ (0, 1] and compute r0 = σ2i−1

(MR

γ0 , LR

γ0 , NR

γ0

).

3 • Set initial maximizing set S0 as whole domain (0, 1].


// Current estimate of maximum and maximizer: rk−1 and γk−1

// Current maximizing set: Sk−1

5 • Compute Λr = λ(A(rk−1,M, L,N) ,B(rk−1,M, L,N)) via Theorem 5.3.1.

6 • Compute the set (sorted in ascending order) Γ(rk−1) = γ ∈ (0, 1] |γ2 ∈ Λr.

7 • Determine (either using derivative information or the direct test method) the

set Γ(rk−1) =γ ∈ Γ(rk−1)

∣∣σ2i−1

(MR

γ , LR

γ , NR

γ

)= rk−1

.

8 • Using Γ(rk−1) (and possibly derivative information), determine the

maximizing set Sk =γ ∈ (0, 1] |σ2i−1

(MR

γ , LR

γ , NR

γ

)> rk−1

.


10 • 1) Randomly select γk ∈ Sk and compute rk = σ2i−1

(MR

γk, LR

γk, NR

γk

).

11 • OR 2) if possible, use cubic fit as described below:

12 for each interval sj ∈ Sk do

13 • Find cubic fit through the endpoints (and their derivatives) of sj.

14 • Define mj := maximizer of cubic fit within sj .

15 • Let γk be whichever mj that achieves largest σ2i−1

(

MR

mj, LR

mj, NR

mj

)

.

16 • Set rk = σ2i−1

(MR

γk, LR

γk, NR

γk

).

17 Quit if

18 • the largest interval of Sk is less than TOLSk.


Table 5.1: Summary of computational requirements of Algorithm I

Line # of (σ,λ) Arguments

2 1× σ(MR

γ0 , LRγ0 , N

Rγ0

)

5 1× λ (A(rk−1,M, L,N) ,B(rk−1,M, L,N))

(Theorem 5.3.1)

7 at most 2 (n+m)× σ(MR

γ , LR

γ , NR

γ

)

8 at most (2 (n +m) + 1)× σ(MR

γ , LR

γ , NR

γ

)

10 or 11 at least 1× σ(MR

γ , LR

γ , NR

γ

)

Here, σ and λ denote the number of singular value and eigenvalue problem(s).

5.3.4 Computational requirements

To provide an idea of the computational requirements of using Algorithm I for computing

various real perturbation values, the number of operations required are counted in terms

of the number of singular value (σ) and eigenvalue (λ) problems that need to be solved,

and are listed in Table 5.1.

As it can be seen from Table 5.1, only a small number of eigenvalue and singular

value problems are solved at each iteration. In particular, only one generalized eigen-

value problem of size 2 (m+ n) is solved at step (5). Subsequently, Γ(rk−1) in step (6)

contains at most 2 (m+ n) elements. Therefore at most 2 (m+ n) restricted singular

value problems need to be solved in step (7), resulting in at most 2 (m+ n) + 1 intervals

to be tested in step (8) to determine the maximizing set. If derivative information is

used, then no additional singular value problems need to be solved, but if no derivative

information is available, then step (8) adds up to an additional 2 (m+ n) + 1 restricted

singular value problems to be solved. Finally in step (10) or (11), at least one restricted

singular value problem is solved in order to obtain the next approximation update.

It should be noted that the derivative information at step (7)-(8) is computed at

a relatively low cost. As shown by Lemma 5.3.3, the derivatives are simple functions


of the generalized eigenvectors of(

A(rk−1,M,N) , B(rk−1,M,N))

as defined in Theo-

rem 5.3.2. Since these eigenvectors are already known from the eigendecomposition of(

A(rk−1,M,N) , B(rk−1,M,N))

in step (5), the derivatives can be obtained very easily.


Consider the following matrix

M =

0.95+j0.88 0.21 0.62 0.73

0.47 1.51+j0.88 0 0.32

0.81 0.15 0.52+j0.88 0.42

0.37 0.63 0.77 0.68

,

where we are interested in computing τ3(M).

Using a direct grid search, the third real perturbation value of M is found to be

τ3(M) = 1.2222, achieved at γ = 0.73461.

Using the proposed algorithm with the (arbitrary) starting point γ0 = 0.5, τ3(M)

is found to within 3 significant figures in 1 iteration and to 9 significant figures in 2

iterations. Table 5.2 lists rk, γk, and the maximizing set Sk for each iteration k =

0, . . . , 4. Figure 5.4 plots the maximizing set Sk obtained at Iterations 1 and 2. As

it can be seen that in this example, the maximizing set is significantly reduced by the

second iteration. Furthermore, the proposed algorithm required solving only a total of

4 generalized eigenvalue and 10 singular value problems for this example, which is much

less demanding than performing a fine grid search.

5.4 Algorithm II: Computing LTI Robustness Radii

We now go into detail and present an efficient algorithm for solving the two 2-D op-

timization Problem (2a) and (2b), for given constant real matrices (A,B,C,D) and

i ∈ 1, . . . , n+min(m, r). Note that to avoid the trivial case where the radius is zero


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

Iteration 1

γ

σ 2i−

1(P(γ

,M))

σ2i−1

(P(γ,M))

cubic fitmaximizing set, S

k

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

Iteration 2

γ

σ 2i−

1(P(γ

,M))

σ2i−1

(P(γ,M))

cubic fitmaximizing set, S

k

Figure 5.4: Intermediate results at Iteration 1 (top) and Iteration 2 (bottom).


Table 5.2: Estimates of the global maximum (rk), the maximizer (γk), and the maximizing

set (Sk) at each iteration k

Iteration (k) γk rk Sk

0 0.50000000 1.01008415 (0.00000000, 1.00000000]

1 0.72067356 1.22015737 (0.00000000, 0.08621154)

(0.50000000, 1.00000000]

2 0.73460319 1.22219350 (0.72067356, 0.74818690)

3 0.73460892 1.22219350 (0.73460319, 0.73461465)

4 0.73460892 1.22219350 (0.73460892, 0.73460893)

for all s ∈ C, we assume that the following assumption is satisfied:

rank

A B

C D

≥ i. (5.15)

In the following development, we focus mainly on the algorithm for solving Problem

(2b). It is shown later that with a slight modification, the development also applies

to the algorithm for solving Problem (2a). Also, for the sake of simplicity, we denote

G(s) :=

A− sI B

C D

.

To start, let the global minimum of (5.3) be denoted by

r∗ = infs∈C

supγ∈(0,1]

σ2i−1

(

[G(s)]Rγ , ER

γ , FR

γ

)

and is achieved at s∗ ∈ C and γ∗ ∈ (0, 1], i.e., r∗ = σ2i−1

(

[G(s∗)]Rγ∗ , ER

γ∗ , FR

γ∗

)

. At the start

of the k-th iteration, for k = 1, 2, . . ., we define the current approximations of r∗, γ∗, and

s∗ to be rk−1, γk−1, and sk−1, respectively, where rk−1 = σ2i−1

(

[G(sk−1)]R

γk−1, ER

γk−1, FR

γk−1

)

.

The minimizing set Sk−1 is defined as follows.


5.4.1 The minimizing set Sk

For Algorithm II, we define the minimizing set to be the union of particular sectors in the

closed upper-half of the complex plane, where the angles of the sectors are described by

Sk ⊆ [0, π]. The reason why the minimizing set is constrained to the closed upper-half of

the complex plane is because of the following result given by Property (1) of Lemma 3.5.1:

σi

A− sI B

C D

= σi

A− sI B

C D

and

σ2i−1

(

[G(s)]Rγ , ER

γ , FR

γ

)

= σ2i−1

(

[G(s)]Rγ , ER

γ , FR

γ

)

for any s ∈ C and γ ∈ (0, 1]. In other words, the radii in (5.2) and (5.3) are both

symmetrical with respect to the real axis; hence one only needs to search within either

the closed upper- or lower-half of the complex plane for the global minimum.

It should be pointed out that Sk is actually a subset of [0, π], and not the actual

minimizing set itself. Recall from above that the actual minimizing set is the set of

sectors of the complex plane described by Sk; i.e., the minimizing set is s ∈ C|∠s ∈ Sk,

where ∠s denotes the angle of s ∈ C. For convenience, however, we sometimes refer to Sk

as the minimizing set, although it should be understood that the set itself is not actually

Sk.

To reduce the size of Sk based on the given rk−1, γk−1, and sk−1, we first construct

the set Rk given by

Rk =

s ∈ CU

∣∣∣σ2i−1

(

[G(s)]Rγk−1, ER

γk−1, FR

γk−1

)

< rk−1

.

The significance of Rk is that for all points not in Rk, the radius can never be smaller

than rk−1. This is because for all s /∈ Rk,

supγ∈(0,1]

σ2i−1

(

[G(s)]Rγ , ER

γ , FR

γ

)

≥ σ2i−1

(

[G(s)]Rγk−1, ER

γk−1, FR

γk−1

)

≥ rk−1.


Hence, one only needs to search within Rk to find points in the complex plane that

achieve a radius smaller than rk−1. Note that Rk can also be used as a minimizing set,

which is actually tighter than the minimizing set described by Sk. However, for ease of

implementation, we choose the minimizing set to be the sectors containing Rk (i.e., as

described by Sk) instead.

To obtain Rk, we first fix a particular θ ∈ [0, π], and consider only the ray Rθ :=weiθ ∈ C |w ∈ R+

. We then find the set Rθ

k, which is the set of points within the ray

Rθ that achieves a radius less than the current approximation rk−1; i.e.,

Rθk =

weiθ ∈ C

∣∣∣w ∈ R+, σ2i−1

([G(weiθ

)]R

γk−1

, ER

γk−1, FR

γk−1

)

< rk−1

⊆ Rθ. (5.16)

Therefore,

Rk =⋃

θ∈[0,π]Rθ

k. (5.17)

Now, to obtain the set Rθk, we have the following result.

Theorem 5.4.1. Given the system matrices (C,A,B,D) of a LTI system (1.1), general

conformal complex matrices E and F , θ ∈ [0, 2π], γ ∈ (0, 1], a real x > 0, and a real

w > 0, then denoting G(weiθ

):=

A− weiθI B

C D

, we have

x ∈ σ([

G(weiθ

)]R

γ, ER

γ , FR

γ

)

⇔ w ∈ λ(A(x, γ) ,B(x, γ)) , (5.18)

where A(x, γ) and B(x, γ) are defined by4

A(x, γ) =

[ A BC D ] 0

0 [ A BC D ]

−1

2x

[(γ−2+1)EEH (γ−2−1)EET

(γ−2−1)EEH (γ−2+1)EET

]

−12x

[(γ2+1)FHF (γ2−1)FHF

(γ2−1)FTF (γ2+1)FTF

]

[ A BC D ]

T0

0 [ A BC D ]

T

(5.19a)

4Here, A(x, γ) and B(x, γ) are also dependent on (C,A,B,D), E, F , and θ. For notational simplicity,however, the arguments are dropped and is assumed throughout the remainder of the chapter.


and

B(x, γ) =

[eiθI 00 0

]0

0[e−iθI 0

0 0

]

0

0

[e−iθI 0

0 0

]0

0[eiθI 00 0

]

. (5.19b)


In other words, Theorem 5.4.1 states that the points along the ray Rθ that achieve a

radius equal to rk−1 are among the real nonnegative generalized eigenvalues of the matrix

pair (A(x, γ) ,B(x, γ)). Hence, by solving for Λ = λ(A(x, γ) ,B(x, γ)) and then finding

all real nonnegative w ∈ Λ such that

σ2i−1

([G(weiθ

)]R

γk−1

, ER

γk−1, FR

γk−1

)

= rk−1, (5.20)

we obtain the endpoints of the intervals along the ray Rθ that achieve a radius greater

than rk−1 (i.e., σ2i−1

([G(weiθ

)]R

γk−1

, ER

γk−1, FR

γk−1

)

> rk−1) and a radius less than rk−1

(i.e., σ2i−1

([G(weiθ

)]R

γk−1

, ER

γk−1, FR

γk−1

)

< rk−1). To determine which interval is the

latter (i.e., the one of interest), one can simply pick a trial point, wt, within a particular

interval and evaluate σ2i−1

([G(wte

iθ)]R

γk−1

, ERγk−1

, FRγk−1

)

(cf. the direct test method in

Section 5.3.1).

As an illustration, see Figure 5.5, where θ is fixed and we assume the current ap-

proximation of the radius is rk−1. In Figure 5.5(a), we find (i.e., via Theorem 5.4.1)

that the points w1eiθ and w2e

iθ achieve a radius equal to rk−1. Hence we have three

intervals of interest: [0, w1); (w1, w2); and (w2,∞). To determine if, say, the first interval

belongs to the minimizing set, one can simply pick a point inside the first interval (e.g.,

w1

2) and test if σ2i−1

([G(w1

2eiθ)]R

γk−1

, ER

γk−1, FR

γk−1

)

is greater than or less than rk−1. If

it is the latter, then the interval (0, w1) belongs to the minimizing set, otherwise it does

not. Figure 5.5(b) shows, for example, that the second interval (w1, w2) belongs to the

minimizing set. Note that if there are no real nonnegative eigenvalues w ∈ Λ such that


Re

Im

Achieves rk-1

θ

w1

w2

Re

Im

θ Achieves < rk-1

Achieves > rk-1

w1

w2

(a) (b)

Figure 5.5: Illustration of computing Rθk: (a) endpoints that achieve a radius equal to

rk−1; and (b) interval(s) that achieve a radius less than rk−1; i.e., Rθk.

σ2i−1

([G(weiθ

)]R

γk−1

, ER

γk−1, FR

γk−1

)

= rk−1, then either Rθk = R+, or R

θk = ∅ (i.e., empty

set). Again, a straightforward test using a trial point can confirm which is true.

Finally by varying θ from 0 to π and doing a “radar-like” sweep, one can obtain the

set Rk; e.g., see Figure 5.6.

Remark 5.4.1. In terms of implementation, note that if Rθk = ∅ for a particular

θ ∈ [0, π], then this implies that all the values along the ray Rθ achieve a radius larger

than rk−1; hence θ can be eliminated from the minimizing set Sk. Therefore at the next

iteration, there is no need to sweep this particular θ again. So for efficiency, the mini-

mizing set for the next iteration is updated as follows. Let Θk ⊆ [0, π] be the set of angles

that describes the smallest union of sectors that contains Rk; i.e.,

Θk =θ ∈ [0, π] |Rθ

k 6= ∅. (5.21)

Then update Sk = Sk−1 ∩Θk.


Re

Im

θ Re

Im

θ

(a) (b)

Figure 5.6: Illustration of computing Rk: (a) the set Rkθ is obtained for a particular θ;

and (b) the set Rθ is obtained by varying θ from 0→ π.

5.4.2 Updating the current minimum rk

Note that by construction of the minimizing set Sk (or even Rk), not all points in the

set achieve a radius smaller than rk−1.5 So to obtain a better approximation of the

global minimum r∗ (i.e., to find a radius smaller than rk−1), one may need to perform

a search (e.g., a grid or random search) within the setRθ

k|θ ∈ Sk

. From experience

though, it is found that one can often obtain rk by letting sk = wmeiθm (i.e., rk =

supγ∈(0,1]

σ2i−1

(

[G(sk)]R

γ , ER

γ , FR

γ

)

), where θm is the midpoint of the largest interval of Sk,

and wm is the midpoint of the largest interval of Rθmk .


Algorithm II has two main stopping criteria. Firstly, the algorithm stops when the size

of the minimizing set Sk, and the size of Rk in (5.17) are both smaller than the user-

specified tolerances, TOLSkand TOLRk

, respectively, where the size of Sk is chosen to

5Note that this is different than the case for Algorithm I, where all points in the maximizing setdefined for Algorithm I actually do achieve a value greater than the current approximation.


be the length of the largest interval in Sk, and the size of Rk is defined to be

size(Rk) = maxθ∈[0,π]

length of largest interval in Rθ

k

.

Secondly, Algorithm II terminates when rk is smaller than the user-specified tolerance,

TOLrk . This second stopping criteria is used to handle the special case when the radius

is zero in spite of the fact that assumption (5.15) is satisfied. For instance, it is shown in

Lemma 3.5.5 that the complex and real minimum phase radius are both 0 when D = 0;

i.e.,

rn+min(r,m)F

(A,B,C, 0) = 0,

where F ∈ C,R, and this is achieved as (real) s→∞. Therefore, the second stopping

criteria is added to prevent the algorithm from searching off into infinity.

5.4.4 Algorithm for solving Problem (2a)

To solve Problem (2a), only two slight modifications to Algorithm 5.3 are needed, which

actually results in a simpler algorithm. Firstly, instead of computing the eigenvalues of

(A(x, γk−1),B(x, γk−1)) in step (7) to obtain Rθk, one is only required to compute the

generalized eigenvalues of

λ

A B

C D

−xEEH

−xFHF

A B

C D

T

,

eiθI 0

0 0

0

0

e−iθI 0

0 0

(the proof is trivial and is similar to the proof of Theorem 5.4.1). Furthermore, when

solving (5.2) in Problem (2a), all the values within Rk actually do achieve a radius smaller

than rk−1 (and all the values outside Rk do not). This is not true when solving (5.3) in

Problem (2b), where Rk only achieves a bound. Hence, step (14) for solving Problem (2a)

can very easily be accomplished by selecting any point in Rk, so no search is required.


Algorithm 5.3: Algorithm II for computing LTI robustness radii.

Input: (A,B,C,D) and i ∈ 1, 2, . . ., where assumption (5.15) is satisfied

Input Tolerance:: TOLSk, TOLSk

, and TOLrk

Output: r∗, s∗, and γ∗, where r∗ = σ2i−1

(

[G(s∗)]Rγ∗ , ER

γ∗ , FR

γ∗

)

1 Initialization

2 • Set S0 = [0, π].

3 • Pick an arbitrary point s0 ∈ CU , and compute r0 and γ0, which achieves

r0 = supγ∈(0,1]

σ2i−1

(

[G(s0)]R

γ , ER

γ , FR

γ

)

= σ2i−1

(

[G(s0)]R

γ0, ER

γ0, FR

γ0

)

.


// Current estimate of maximum and maximizer: rk−1, γk−1, sk−1

// Current maximizing set: Sk−1

5 • Reset Θk = ∅.

6 for θ ∈ Sk−1 (θ is discretized steps of Sk−1) do

7 • Compute Rθk by finding all real nonnegative w ∈ λ(A(x, γk−1) ,B(x, γk−1))

in (5.19) such that σ2i−1

([G(weiθ

)]R

γk−1

, ER

γk−1, FR

γk−1

)

= rk−1, and the

intervals that achieve σ2i−1

([G(weiθ

)]R

γk−1

, ER

γk−1, FR

γk−1

)

< rk−1.

8 • If Rθk 6= ∅, then update Θk ← Θk ∪ θ.


10 • Sk = Sk−1 ∩Θk.

11 Quit if

12 • the length of the largest interval of Sk < TOLSkand size(Rk) < TOLRk

.


14 • Find (e.g., search for) one point sk ∈ Rθk, where θ ∈ Sk and such that

rk = supγ∈(0,1]

σ2i−1

(

[G(sk)]R

γ , ER

γ , FR

γ

)

< rk−1 (note: rk is achieved at γk).

15 Quit if

16 • rk < TOLrk .


5.4.5 Algorithm for computing the real stabilizability radius

To compute the real stabilizability radius, the real unstable DFM radius, and the real

minimum phase radius, where the search is for a minimum in the closed right-half of the

complex plane – i.e., by solving the following problem:

riR(A,B,C,D) = inf

s∈C+

τi

A− sI B

C D

(5.22)

for given (A,B,C,D) and i ∈ 1, . . . , n+min(m, r) – only one very simple modification

to Algorithm 5.3 is needed. In particular, one only needs to change the initialization step

(2) of Algorithm 5.3 from S0 = [0, π] to S0 =[0, π

2

](the proof is trivial).

5.4.6 Computational requirements

To provide an idea of the computational requirements of using Algorithm II, the number

of operations required is counted in terms of the number of singular value (σ), real

perturbation value (τ), and eigenvalue (λ) problems that need to be solved and are listed

in Table 5.3.

It is to be noted that every time the radius is evaluated at a particular point s ∈ C, a

restricted real perturbation value problem6 is solved. This occurs in step (3) and (14). In

step (3), the restricted real perturbation value problem is evaluated only once, but step

(14) may require multiple evaluations, depending on the number of trials needed before

rk < rk−1 is obtained. However, it is found from experiment that the number of trials is

typically small (i.e., close to 1) when using the method outlined in Section 5.4.2.

It is also to be noted that the main portion of the total operations count arises from

computing Rθk for all θ ∈ Sk in step (7). In terms of implementation, Sk is discretized into

a number of points. From experiment, it is found that it is generally sufficient to discretize

Sk into 15 to 20 points, or a minimum of 0.5 (i.e., a maximum of 360 (= π0.5

) points),

6Here, recall that we assume that the inequality in (5.3) is an equality.


Table 5.3: Summary of computational requirements of Algorithm II

Line # of (σ,τ ,λ) Matrices

3 1× τ (G(s0) , E, F )

7 (for each 1× λ (A(x, γk−1) ,B(x, γk−1))

θ ∈ Sk−1) in (5.19)

at most (4 (2n+m+ r) + 1)× σ([

G(weiθ

)]R

γk−1

, ERγk−1

, FRγk−1

)

14 at least 1× τ (G(sk) , E, F )

Here, we denote σ, τ , and λ as singular value, real perturbation value, and eigenvalue

problems.

whichever results in a smaller angular division. Then for a given θ ∈ Sk, computing Rθk

requires: i) solving the 2 (2n+m+ r) × 2 (2n+m+ r) generalized eigenvalue problem

Λ = λ(A(x, γk−1) ,B(x, γk−1)) (see (5.19)); and ii) evaluating a number of restricted

singular value problems to verify which real nonnegative eigenvalues s ∈ Λ (there are

a maximum of 2 × (2n+m+ r)) satisfy (5.20). Furthermore, since there are at most

2× (2n+m+ r) real nonnegative eigenvalues s ∈ Λ that may satisfy (5.20), Rθk contains

at most 2× (2n+m+ r)+1 intervals. Hence, to determine which intervals in Rθk achieve

a radius less than rk−1, at most 2 × (2n+m+ r) + 1 points are tested, resulting in

2× (2n +m+ r) + 1 additional restricted singular value problems.

From Table 5.3, we see that Algorithm II solves only a limited number of restricted

real perturbation value problems, which by itself is a 1-D optimization problem involving

multiple restricted singular value problems. Hence, this proves to be an advantage of

Algorithm II, as compared to say, a gradient search method, which requires solving

a restricted real perturbation value problem at each iteration point, and at each step

involved with computing the steepest gradient.



The following example can be found in [42], where the real controllability radius of

A =

1 1 1

0.1 3 5

0 −1 −1

B =

1

0.1

0

is found to be 0.0492, and is achieved at s = 0.972 + j0.982. Using Algorithm II with

the (arbitrary) starting point s0 = j1, the same radius and global minimizer are found

to within 3 significant figures in 5 iterations and to 6 significant figures in 7 iterations.

Table 5.4 lists rk, sk, and the minimizing set Sk for each iteration k = 0, . . . , 9. Figure 5.7

provides a grid plot of the real controllability radius with respect to a given s ∈ C. Fig-

ure 5.8 plots Rk (outlined by the dots) obtained at iterations 1, 3, 4, and 8, superimposed

on a contour plot of Figure 5.7. The straight lines originating from the origin depict Sk.

It is interesting to note that on a computer with a Pentium IV 2.0GHz processor,

512MB of RAM, and MATLAB 7.0, Algorithm II took a total of about 6 sec. to complete,

which was approximately the same amount of time a gradient search method took to

obtain a local minimum from a single starting point. Hence, if the gradient search

method had to test 20 initial points in order to achieve a certain level of confidence

that the obtained minimum is global, then the gradient search method would have taken

approximately 20 times longer to run than Algorithm II.

5.5 Summary

In this chapter, two efficient algorithms are presented for i) solving the various real

perturbation value problem, and ii) computing the various LTI robustness radii. Both

algorithms work by iteratively reducing the so-called minimizing/maximizing set, which

contains the global minimizer/maximizer. The advantage of this approach compared

to general nonlinear techniques, such as steepest-descent methods, is that the proposed


Table 5.4: Estimates of the global minimum radius (rk), the minimizer (sk), and the

minimizing set (Sk) at each iteration k

Iteration (k) rk sk Sk

0 0.745637 j1 [0.00, 180.00]

1 0.740724 3.061610−17 + j0.5 [0.00, 180.00]

2 0.218632 0.46766 [0.00, 180.00]

3 0.117352 0.98098 + j0.58561 [0.00, 61.67]

4 5.3300410−2 0.97584 + j0.91703 [30.33, 56.11]

5 4.9230410−2 0.97060 + j0.98023 [42.70, 47.86]

6 4.9219110−2 0.97214 + j0.98179 [45.02, 45.55]

7 4.9218610−2 0.97176 + j0.98203 [45.27, 45.33]

8 4.9218610−2 0.97186 + j0.98194 [45.29, 45.30]

9 4.9218610−2 0.97184 + j0.98197 [45.30, 45.30]

algorithms are insensitive to the initial search points and return the global extrema after

one run, whereas a steepest-descent method may require multiple runs with different

initial points. This results in a saving in computation time.

However, further improvements to the proposed algorithms can actually be made

by combining them with such gradient search techniques. In particular, one may first

use the minimizing/maximizing set approach to narrow down the search domain until it

contains only one – and hence global – minimum/maximum. One can then apply the

gradient search techniques to promptly pinpoint the minimum/maximum. Of course,

this approach assumes that one is able to somehow know (or at least make further

approximations of) when a minimizing/maximizing set contains only one extremum.

This is left as a topic for future studies.


−5

0

5

0

1

2

3

4

50

1

2

3

4

5

6

7

real axis

Grid plot of real controllability radius with respect to s

imaginary axis

radi

us

Figure 5.7: Grid plot of the real controllability radius with respect to s ∈ C.


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

real axis

imag

inar

y ax

is

Iteration 1

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

real axis

imag

inar

y ax

is

Iteration 3

(a) (b)

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

real axis

imag

inar

y ax

is

Iteration 4

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

real axis

imag

inar

y ax

is

Iteration 8

(a) (b)

Figure 5.8: Plot of Rk at (a) iteration 1; (b) iteration 3; (c) iteration 4; and (d) iteration

8.

Chapter 6

Toughness Index for High

Performance Control

In this chapter, we study another type of continuous measure that is different than

the LTI robustness radii discussed so far. Using the new measure, we are interested

in characterizing a system’s limitation in achieving high performance, as measured by

the cheap control performance index. We call the new continuous measure a Toughness

Index.

6.1 Motivation

It is well known in both classical and modern linear control theory that a closed-loop

system can often achieve high performance if the feedback controller gains are allowed

to be sufficiently large. In fact, under certain criteria, “perfect control” [20, 79] can be

achieved if the control effort is allowed to be arbitrarily large. Of course in practice,

one cannot implement extremely large controller gains, and practical restrictions such as

controller actuator size limits, impose a limitation on the fastest response achievable by

the closed-loop system. Some systems, however, may be more difficult to control than

others, and we are interested in studying and being able to characterize what are some

119

Chapter 6. Toughness Index for High Performance Control 120

of the inherent difficulties a system may have in achieving high performance.

In our approach, we consider high performance controllers obtained via the cheap

control problem. In particular, consider the following stabilizable and detectable LTI

system modelled by

x = Ax+Bu, x(0) = x0, (6.1)

y = Cx,


tively.1 The cheap control problem consists of finding a stabilizing feedback controller

that minimizes the following quadratic performance index:

Jǫ = minu

∫ ∞

0

(yTQy + ǫuTRu

)dt, (6.2)

where ǫ > 0 is a small positive scalar, Q is positive definite, and R is positive definite. In

the case as ǫ → 0, the cost of the control effort decreases, subsequently allowing larger

controller gains, and resulting in faster output response times.

The cheap LQR problem has been extensively studied in the past (e.g., see [28,43,48,

49,77,79]). One of the most well-known results is that perfect regulation (i.e., Jǫ→0 = 0)

can be achieved if and only if the system is minimum phase and right invertible2 (e.g., see

[28,48,79]). If the system is nonminimum phase, then there exists a fundamental perfor-

mance limitation in controller design, which in some cases, such as the servomechanism

problem, can be characterized in terms of the number and location of the nonminimum

phase zeros [75]. Another system limitation is the rate at which a closed-loop system’s

faraway poles approach infinity as ǫ→ 0, which is the focus of our study. For the cheap

control problem, it is well known that as ǫ → 0, the closed-loop poles behave in such a

way that i) some poles asymptotically approach the system’s stable finite transmission

zeros, ii) some approach the mirror-image of the unstable transmission zeros, and iii) all

1In this chapter, we mainly consider systems with equal number of inputs and outputs (i.e., m = r).2The system (C,A,B,D) is said to be right invertible if rank(C (sI −A)B +D) = r for some s ∈ R.


the other poles approach infinity in various Butterworth patterns at a rate relative to 1ǫk

for some k > 0 (e.g., see [48]). Since the closed-loop poles have a direct effect on the

system’s response time, the rate at which the faraway poles approach infinity as ǫ → 0

poses a limitation on system performance. Other inherent difficulties a system may have

in transient control include unbounded peaking as ǫ → 0 [29, 47], and singular initial

behaviour [28].

As mentioned earlier, the focus of this chapter is to investigate system limitations due

to the faraway closed-loop poles. In particular, it is noted that if the rate at which the

faraway poles approach infinity is slow, then in order to achieve high performance, one

is required to choose ǫ to be extremely small, resulting in impractically large controller

gains. So subsequently if one has limited controller gains or control effort, then a slow

rate poses a fundamental performance limitation, as we will see in some examples later

on. The Toughness Index introduced in this chapter is based on this observation.

This chapter is organized as follows. First, some topics related to cheap control are

reviewed in Section 6.2. Section 6.3 proposes a method for determining the asymptotic

rates at which the faraway poles approach infinity for the cheap control problem, and

defines a Toughness Index for the system based on the these rates. A few numerical

examples are then looked at in Section 6.4. Finally, Section 6.5 applies the Toughness

Index to study the robust servomechanism problem [18], and shows that as the number

of tracking/disturbance poles increases, there are fundamental limits in obtaining high

performance in the resultant closed-loop system, even if the system is minimum phase.

6.2 Review: Asymptotic Locations of the Optimal

Closed-Loop Poles

In this section, we review the results from [48] regarding the asymptotic pole locations

of the cheap control problem as ǫ→ 0.


In the following, a transfer matrix realization, H(s), of the system (6.1) is given by

H(s) = C (sI − A)−1B. (6.3)

The finite transmission zeros [21] of (6.1) are the roots of the numerator polynomials of

the nonzero elements of the Smith-McMillan form of (6.3), and correspond to the points

in the complex plane that satisfy

rank

A− sI B

C 0

< n+min(m, r)

(cf. Definition 3.4.1). The transfer matrix H(s) is said to have an infinite transmission

zero of order k (i.e., 1/sk) if H(1/s) has a finite transmission zero of precisely that

order at s = 0. System (6.1) and (6.3) are said to be minimum phase if they have no

transmission zeros in the closed right-half of the complex plane, otherwise they are said

to be nonminimum phase (see Definition 3.4.2).

Recall that given a LTI system (6.1), the optimal control law that minimizes the

performance index (6.2) is

u = −1ǫR−1BTPǫx, (6.4)

where Pǫ is the unique positive semidefinite solution to the algebraic Riccati equation

(ARE)

ATPǫ + PǫA + CTQC − 1

ǫPǫBR−1BTPǫ = 0. (6.5)

The optimal closed-loop poles (i.e., λ(A− 1

ǫBR−1BTPǫ

)) are the stable eigenvalues of

the Hamiltonian matrix (e.g., see [48])

Z =

A −1ǫBR−1BT

−CTQC −AT

, (6.6)

or equivalently, the stable roots of

det(sI − Z) = (−1)n φ (s)φ (−s) det(

I +1

ǫR−1HT (−s)QH(s)

)

= 0, (6.7)


or also equivalently, the stable roots of

det

(

I +1


)

= 0, (6.8)

where H(s) is the transfer matrix of (6.1) as given by (6.3), and

φ (s) = det(sI −A)

is the system characteristic equation. In the case when the system (6.1) is square (i.e.,

m = r), det(H(s)) is given by

det(H(s)) =

α

p∏

i=1

(s− zi)

n∏

i=1

(s− pi)

,

where α is a constant, pi, for i = 1, . . . , n, are the eigenvalues of system (6.1), and zi, for

i = 1, . . . , p, are the finite transmission zeros of (6.1). The following result is obtained

[48].

Lemma 6.2.1. ([48]) Given (6.1), assume that m = r; then it follows from (6.7) that

as ǫ→ 0, the following is true:

• p of the optimal closed-loop poles approach zi, where

zi =

zi if Re zi ≤ 0

−zi if Re zi > 0;

• the remaining (faraway) closed-loop poles approach infinity in various Butterworth

configurations of different orders and radii; and

• a “rough estimate” of the faraway poles’ distance to the origin as ǫ→ 0 is given by

(α2

ǫm

)1/(2(n−p))

. (6.9)


6.3 Toughness Index

In this section, we study how to compute the exact asymptotic rates at which the indi-

vidual faraway closed-loop poles approach infinity as ǫ → 0. As shown in the previous

section, a “rough estimate” of the rates is given by (6.9); however, the estimate is rather

crude (e.g., see [48, Example 3.21]), and also does not provide an estimate for each in-

dividual faraway pole. To compute the individual faraway poles for a given ǫ, one can

directly solve the Hamiltonian matrix (6.6) for the stable eigenvalues. However, for sys-

tems with large dimensions (e.g., n > 50), this method may have very severe numerical

problems when ǫ is chosen to be small (e.g., ǫ = 10−12), and so cannot be used. Instead,

we use an alternative method described next, which determines the faraway poles by

computing the closed-loop eigenvalues of a reduced model based on the system’s infi-

nite transmission zeros. In the remainder of this chapter, we assume, unless specified

otherwise, that the LTI system has equal inputs and outputs.

6.3.1 Reduced model based on H(s) approximation

Let H(s) be a strictly proper transfer matrix realization of (6.1), and expand H(s) in a

Laurent series about the origin s = 0 as follows:

H(s) =

∞∑

k=1

CAk−1B

sk. (6.10)

Assuming that the infinite transmission zeros of H(s) are [1/sp1, . . . , 1/spk ], denote p∗ =

max(p1, . . . , pk). Now approximate H(s) by the truncated series H(s) given by

H(s) =

p∗∑

k=1

CAk−1B

sk, (6.11)

where the approximation is valid for large s. It can easily be shown that H(s) and H(s)

have identical infinite transmission zeros. Also, as ǫ → 0, the faraway closed-loop poles

approach infinity and the approximation H(s) for large s improves. So on replacing H(s)


with H(s) in (6.8) for a sufficiently small ǫ, we obtain

0 = det

(

I +1


)

= det

(

I +1

ǫR−1

(p∗∑

k=1

BT(AT)k−1

CT

sk

)

Q

(p∗∑

k=1

CAk−1B

sk

))

= det

(

s2p∗

I +1

ǫ

2p∗−2∑

k=0

(−1)kMksk

)

, (6.12)

where Mk, for k = 0, . . . , 2p∗ − 2, is given by3

Mk =k∑

j=0

R−1(CAp∗−1−jB

)TQCAp∗−1−k+jB.

In this case, on applying controller (6.4), the faraway poles for a given ǫ can be obtained

by computing the stable roots of the matrix polynomial (6.12), or equivalently, the stable

eigenvalues of the matrix Aǫ, where

Aǫ =

0 I 0 · · · 0 0

0 0 I · · · 0 0

......

.... . .

......

0 0 0 · · · 0 I

1ǫM0 −1

ǫM1

1ǫM2 · · · 1

ǫM2(p∗−1) 0

. (6.13)

6.3.2 Toughness Index - definition

Using the same derivation for obtaining the estimate (6.9), one can show that the dis-

tances of the faraway poles of (6.12) from the origin, or equivalently, the distances of the

faraway eigenvalues of (6.13), asymptotically approach the following as ǫ→ 0:

(αi

ǫ

)βi

(6.14)

for some constant αi > 0 and βi > 0, for i = 1, 2, . . . , p∗. To compute αi > 0 and βi > 0,

the following procedure is used.

3Note that for notational convenience, we assume that Ai = 0 for i < 0.


First compute the two sets of (faraway) eigenvalues of Aǫ, for ǫ = ǫ1 and ǫ = ǫ2; i.e.,

λ(Aǫ1) =

s(ǫ1)1 , s

(ǫ1)2 , . . . , s

(ǫ1)p∗

, for ǫ = ǫ1,

λ(Aǫ2) =

s(ǫ2)1 , s

(ǫ2)2 , . . . , s

(ǫ2)p∗

, for ǫ = ǫ2,

where ǫ1 > ǫ2, and ǫ1 and ǫ2 are chosen sufficiently small such that the distances of the

faraway eigenvalues for ǫ1 and ǫ2 are closely described by (6.14). So for sufficiently small

ǫ1 and ǫ2, one can approximate that

∣∣∣s

(ǫ1)1

∣∣∣ =

(αi

ǫ1

)βi

(6.15)

∣∣∣s

(ǫ2)2

∣∣∣ =

(αi

ǫ2

)βi

, for i = 1, 2, . . . , p∗.

So from (6.15), one can obtain αi and βi to be

βi =log10

(∣∣∣s

(ǫ2)i

∣∣∣ /∣∣∣s

(ǫ1)i

∣∣∣

)

log10(ǫ1/ǫ2), for i = 1, 2, . . . , p∗ (6.16a)

and

αi = ǫ1

(∣∣∣s

(ǫ1)i

∣∣∣

)1/βi

, for i = 1, 2, . . . , p∗. (6.16b)

We can now make the following definition.

Definition 6.3.1 (Toughness Indices). Given (C,A,B) of system (6.1), positive definite

Q, positive definite R, and 0 < ǫ ≪ 1, let p∗ denote the number of optimal faraway

closed-loop poles. Then for the given ǫ, define the set of Toughness Indices for system

(6.1) with the performance index (6.2) as follows:

Indexi =(αi

ǫ

)βi

, for i = 1, 2, . . . , p∗, (6.17)

where αi and βi, for i = 1, . . . , p∗, are given by (6.16). Also, for the given ǫ, define the

Toughness Index for the overall system (6.1) as

ToughnessIndex = mini

(αi

ǫ

)βi

. (6.18)


The Toughness Index as defined in (6.18) gives an indication of a system’s difficulty

in achieving high performance via solving the cheap control problem. In particular, the

Toughness Index (6.18) corresponds to the dominant faraway closed-loop pole’s distance

from the origin for a given 0 < ǫ ≪ 1. So if the Toughness Index is small, even for

small ǫ, then it implies that the dominant faraway pole is very close to the origin, even

when the control effort is allowed to be relatively cheap. This in turn implies that the

closed-loop system will have a slow response time even when large controller gains are

used.

Remark 6.3.1. The rate of increase in the Toughness Index as ǫ→ 0 may also be used

as a measure to characterize a system’s difficulty in achieving high performance via the

cheap control problem. From (6.18), we see that the Toughness Index is monotonically

increasing as ǫ → 0. Intuitively, this is because as ǫ → 0, larger controller gains are

allowed, and so the resulting closed-loop system will have faster response times. However,

if the Toughness Index’s rate of increase as ǫ→ 0 is “small”, then this means that little

improvement in the closed-loop response time can be achieved when one tries to increase

the size of the controller gain. Hence, a low rate of increase implies that the system

may have difficulty in achieving high performance, and vice versa. Further study of using

the Toughness Index’s rate of increase as a measure is of interest and is left for future

investigation.

6.4 Numerical Examples

We now study three examples. The first example demonstrates how the proposed algo-

rithm of Section 6.3 is used to compute a system’s faraway closed-loop poles, while the

second and third example demonstrate how the Toughness Index is applied to study a

system’s difficulty in achieving high performance via cheap control.


6.4.1 Example 1: longitudinal control of an airplane

Consider the following LTI system given in Example 3.21 of [48]:

A =

−0.01580 0.02633 −9.810 0

−0.1571 −1.030 0 120.5

0 0 0 1

5.27410−4 −0.01652 0 −1.466

, B =

6.05610−4 0

0 −9.496

0 0

0 −5.565

C =

1 0 0 0

0 0 1 0

. (6.19)

The system (6.19) has a finite transmission zero at s = −1.002 and infinite transmission

zeros [1/s, 1/s2].

Using the approach in Section 6.3, the magnitudes of the stable eigenvalues of (6.13)

for ǫ equal to 10−9, 10−10, 10−11, and 10−12 are given below:

ǫ1 = 10−9 ǫ2 = 10−10 ǫ3 = 10−11 ǫ4 = 10−12

∣∣∣s

(ǫ)i

∣∣∣ 4.19710+2 7.46410+2 1.32710+3 2.36010+3

4.19710+2 7.46410+2 1.32710+3 2.36010+3

1.91310+1 6.05010+1 1.91310+2 6.05010+2

On computing the Toughness Indices of Section 6.3.2, we obtain the following two tables

for αi and βi:

ǫ2 = 10−10 ǫ3 = 10−11 ǫ4 = 10−12

βi 0.25 0.25 0.25

0.25 0.25 0.25

0.50 0.50 0.50

and


ǫ2 = 10−10 ǫ3 = 10−11 ǫ4 = 10−12

αi 3.10310+1 3.10410+1 3.10610+1

3.10310+1 3.10410+1 3.10610+1

3.66010−7 3.66010−7 3.65910−7

.

Hence for given 0 < ǫ≪ 1, the Toughness Indices (6.17), which correspond to the optimal

faraway closed-loop poles’ distances from the origin, are given by

(31.0

ǫ

)0.25

,

(31.0

ǫ

)0.25

, and

(3.6610−7

ǫ

)0.5

, (6.20)

and the Toughness Index for the system is given by min

((31.0ǫ

)0.25,(

3.6610−7

ǫ

)0.5)

. To

verify these results, we compare the faraway poles as given by (6.20) with the optimal

closed-loop poles obtained by solving for the actual optimal controller (6.4), and we

observe that there is a strong agreement, as shown below.

ǫ s ∈ λ(A− 1

ǫBR−1BTPǫ

)|s|

(αi

ǫ

)βi

10−10 −5.27810+2 + j5.27810+2 7.46410+2 7.46610+2

−5.27810+2 − j5.27810+2 7.46410+2 7.46610+2

−6.08710+1 6.08710+1 6.05010+1

−1.009 1.009 –∗

10−11 −9.38510+2 + j9.38510+2 1.32710+3 1.32710+3

−9.38510+2 − j9.38510+2 1.32710+3 1.32710+3

−1.91410+2 1.91410+2 1.91310+2

−1.003 1.003 –∗

10−12 −1.66910+3 + j1.66910+3 2.36010+3 2.36110+3

−1.66910+3 − j1.66910+3 2.36010+3 2.36110+3

−6.05010+2 6.05010+2 6.04910+2

−1.002 1.002 –∗

∗ Not estimated by procedure since it is not a faraway pole.


6.4.2 Example 2: a simple example

Given the following two unstable plants described by

H1(s) =1

s50and H2(s) =

(s− 1)40

s50,

it is desired to determine the asymptotic distances of the two systems’ closed-loop poles

controlled using optimal state feedback and minimizing the cheap control performance

index (6.2) for Q = I, R = I, and some 0 < ǫ≪ 1.

In this case, the standard optimization algorithm lqr.m of MATLAB fails for all

ǫ < 1 due to numerical problems. Therefore it is not possible to study the asymptotic

behaviour for ǫ → 0 by simply using lqr.m to compute the closed-loop poles. However,

using the proposed numerical procedure of Section 6.3, it is immediately obtained that

the Toughness Index for H1(s) is given by (1/ǫ)0.01, and for H2(s) is given by (1/ǫ)0.05.

So for H1(s) and a sufficiently small ǫ, at least one of the faraway closed-loop poles of

the resultant closed-loop system (if implemented) would be at a distance D from the

origin, where D = (1/ǫ)0.01; i.e., if ǫ = 10−10, then D = 1.259, and if ǫ = 10−100, then

D = 10. A closed-loop system having a pole with a distance of D = 10 from the origin

means the system with have a mode with a time constant of 0.1 (1/D) or more. So it

is therefore virtually impossible to obtain a relatively fast settling time for H1(s) using

cheap control. In comparison, the distance of the faraway closed-loop poles of H2(s) for

ǫ = 10−10 and ǫ = 10−100 would be D = 3.1623 and D = 105, respectively.

One could also have studied the asymptotic distance of the closed-loop poles by com-

puting the eigenvalues of the Hamiltonian matrix (6.6), instead of solving the full LQR

problem. However, it should be noted that computing the eigenvalues of Aǫ (6.13) in the

proposed procedure of Section 6.3 is often more numerically sound than computing the

eigenvalues of the Hamiltonian matrix (6.6), especially for systems with infinite trans-

mission zeros of lower order. In this example, for instance, a state-space realization of the

second plant H2(s) has 50 states (n = 50), and infinite transmission zeros [1/s10]. The


dimension of the corresponding Hamiltonian matrix (6.6) is of order 100. On the other

hand, the dimension of the Aǫ matrix in (6.13) is only 20, so computing the eigenvalues

of Aǫ is numerically more reliable than computing the eigenvalues of the Hamiltonian

matrix.

6.4.3 Example 3: distillation column

In the previous examples, the Toughness Index was found to be “bad” due to a small βi

term. Now consider the following example with a small Toughness Index due to a small

αi term. In particular, consider the distillation column [14] described by

A=

−0.014 4.3010−3 0 0 0 0 0 0 0 0 0

9.5010−3 −0.0138 4.60

10−3 0 0 0 0 0 0 0 5.0010−4

0 9.5010−3 −0.0141 6.30

10−3 0 0 0 0 0 0 2.0010−4

0 0 9.5010−3 −0.0158 0.0110 0 0 0 0 0 0

0 0 0 9.5010−3 −0.0312 0.0150 0 0 0 0 0

0 0 0 0 0.0202 −0.0352 0.0220 0 0 0 0

0 0 0 0 0 0.0202 −0.0422 0.0280 0 0 0

0 0 0 0 0 0 0.0202 −0.0482 0.0370 0 2.0010−4

0 0 0 0 0 0 0 0.0202 −0.0572 0.0420 5.0010−4

0 0 0 0 0 0 0 0 0.0202 −0.0483 5.0010−4

0.0255 0 0 0 0 0 0 0 0 0.0255 −0.0185

B=

0 0 0

5.0010−6 −4.0010−5 2.5010−3

2.0010−6 −2.0010−5 5.0010−3

1.0010−6 −1.0010−5 5.0010−3

0 0 5.0010−3

0 0 5.0010−3

−5.0010−6 1.0010−5 5.0010−3

−1.0010−5 3.0010−5 5.0010−3

−4.0010−5 5.0010−6 2.5010−3

−2.0010−5 2.0010−6 2.5010−3

4.6010−4 4.6010−4 0

, C =

0 0 0 0 0 0 0 0 0 1 0

1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1

,


which has infinite transmission zeros [1/s, 1/s, 1/s2], and where the Toughness Index is

found to be (4.15010−14/ǫ)0.25 for Q = I, R = I, and 0 < ǫ ≪ 1. So for ǫ = 10−10, the

dominant faraway pole is at a distance of D = 0.1427 from the origin, and for ǫ = 10−20,

D = 45.14. Hence again, the Toughness Index for the system indicates (this time with

a small αi term) that it may be difficult to achieve fast performance on the distillation

column system.

Here, the dimension of the Aǫ matrix in (6.13) is 12, whereas the dimension of the

corresponding Hamiltonian matrix is 22. So in this example, it is again more favor-

able to use the proposed approach in Section 6.3 than computing the eigenvalues of the

Hamiltonian matrix.

6.5 Application to the Robust Servomechanism Prob-

lem

In this section, we use the Toughness Index to study the robust servomechanism problem

(RSP) [18] for (1.16) for the case when there is a large number of tracking/disturbance

poles to be tracked/regulated. Recall from the review in Section 1.7.4 that if the exis-

tence conditions of Lemma 1.7.1 are satisfied, then there exists a controller of the form

(1.18) to stabilize the augmented plant-servocompensator system (1.19). Assume that

the controller (1.18) is found to minimize the cheap performance index (see Section 1.7.4)

J =

∫ ∞

0

(zT z + ǫvT v

)dτ. (6.21)

The augmented system (1.19) has the following property.

Lemma 6.5.1. Let [1/sp1, . . . , 1/spk ] be the infinite transmission zeros of (C,A,B,D) in

(1.16), then[1/s(p1+N), . . . , 1/s(pk+N)

]are the infinite transmission zeros of

[

0 D]

,

A 0

BC C

,

B

BD

,


where N is the number of disturbance/tracking poles.


From Lemma 6.5.1, we see that if the Toughness Index of the original plant (C,A,B)

is found to be(αǫ

)(1/(2k)), for some α > 0 and k > 0, then the Toughness Index of the

augmented plant-servocompensator system (1.19) is(γǫ

)(1/(2k+2N))for some γ > 0. So if

one is to design a RSP controller with a large number of tracking/disturbance poles (as

is often the case), then one may encounter difficulty trying to achieve high performance,

as measured by the Toughness Index. We will see some examples of this latter point in

Section 7.6 of the following chapter.

6.6 Summary

In this chapter, a Toughness Index is proposed to characterize a LTI system’s difficulty

in using cheap control to achieve high performance. The Toughness Index measures

the dominant faraway optimal closed-loop pole’s distance from the origin, and indicates

that there exists a performance limitation if the distance is small, even when the cost

of the control effort is made relatively cheap. An important observation is then made,

which is that in solving the RSP, the Toughness Index becomes worst as the number of

tracking/disturbance poles increases. Thus even for minimum phase systems, there is a

performance limitation that occurs when trying to solve the RSP for a large number of

tracking/disturbance poles.

Chapter 7

Applications

In this chapter, we apply the various results of this thesis to a number of examples.

In Section 7.1, we compute the real DFM radius for a number of industrial system

models and compare it with the corresponding real CFM radius. It is interesting to

observe in these examples that for many of the models, the robustness, with respect to

parametric uncertainties, of the systems using decentralized control is no worse than for

the case of using centralized control.

In Section 7.2, we use the real DFM radius to study a particular class of DFMs called

unstructured DFMs, which have the property that (under certain conditions) they can be

removed by sampling using almost any sampling period. In other words, if a continuous

decentralized LTI system has an unstable unstructured DFM, then a discrete controller

can be used to stabilize the system. However, we observe that for small sampling periods,

the resulting real DFM radius is small, indicating that there may still remain problems

in shifting this unstructured DFM.

In Section 7.3, we propose to use the real DFM radius to choose the optimal input-

output pairing for a decentralized controller. Comparing our results with the pairings

obtained using the relative gain array (RGA) approach, which is an approach widely used

in industry, we observe that the RGA approach can sometimes lead to a decentralized

134

Chapter 7. Applications 135

control structure that has an unstable DFM!

In Section 7.4, we revisit the study in [51] of a multi-link inverted pendulum system’s

structured real controllability radius. In this section, we apply the various new general-

izations of real perturbation values obtained in this thesis to improve the computation

of the structured radius for the inverted pendulum system.

In Section 7.5, we study the RSP for a number of industrial system models using the

transmission zero at s radius and the corresponding mobility bounds of the closed-loop

eigenvalues.

Finally, in Section 7.6, we apply the Toughness Index of Chapter 6 to study the

difficulty in achieving high performance for the RSP for two systems, a mass-spring

system and a commercial hard drive.

7.1 DFM Radius of Industrial Plants

In this study, the real DFM radii for a number of representative industrial system models

are computed and compared with the corresponding real CFM radii (see Definition 3.2.4).

In all of the examples considered, the system has 2 inputs and 2 outputs, and a diagonal

information flow constraint (3.9) is assumed; i.e., input ui is paired with output yi, for

i = 1, 2.

7.1.1 DFM radius vs. CFM radius

A listing of the calculated real DFM radii for a number of industrial system models is

presented in Table 7.1, where the following notation is used:

• n is the order of the plant model;

• S and US denote a stable and unstable plant, respectively; and

• M and NM denote a minimum phase and nonminimum phase plant, respectively.


Table 7.1: Comparison of the real DFM radius and the real CFM radius of various

industrial system models

Plant Real DFM Real CFM Remark

radius radius

A. distillation [14] n = 11, S, M 1.74310−6 5.30210−5

B. gas turbine [65] n = 4, S, M 0.5693 0.5693 identical

C. turbo [60] n = 6, S, NM 0.1986 0.3884

D. helicopter [69] n = 4, US, M 9.35710−2 9.35710−2 identical

E. thermal [19] n = 9, S, NM 9.56210−3 3.01210−2

F. pilot [78] n = 6, US, M 2.14310−3 2.14310−3 identical

It is observed in Table 7.1 that for many of the examples, the real DFM radius is iden-

tical (i.e., plants B, D, and F) or close to (i.e., plants C and E) the real CFM radius.

This implies for these industrial systems that the robustness, with respect to parametric

perturbations, of a system using decentralized control is no worse than the case of using

centralized control. This perhaps is somewhat surprising.

7.1.2 Unstable DFM radius vs. unstable CFM radius

The real unstable DFM radii of some of the above unstable industrial systems are cal-

culated (using Corollary 3.3.1) and listed in Table 7.2. A listing of the real unstable

CFM radius is also included. Again we see that in all four cases, the real unstable DFM

radius is either identical or very close to the centralized radius, which implies that the

constraint of using decentralized control on these unstable systems does not introduce

any additional robustness issues as compared to centralized control. Again, the result is

perhaps surprising.


Table 7.2: Comparison of the real unstable DFM radius and the real unstable CFM

radius of various industrial system models

Plant Real unstable Real unstable Remark

DFM radius CFM radius

G. boiler [12] n = 9, US, NM 1.01610−6 1.01610−6 identical

H. mass [17] n = 6, US, M 9.74710−2 9.74710−2 identical

I. 2-link n = 36, US, NM 7.13110−3 8.54810−3

J. 2-cart n = 8, US, NM 2.35710−2 2.35710−2 identical

7.1.3 Real DFM radius wrt s vs. real CFM radius wrt s

In the previous subsection, the real unstable DFM radius is used to measure how close

a given system is to having any unstable DFM. In general, the resulting unstable DFM

that achieves the DFM radius is not equal to any of the system’s original modes. So in

contrast, this example studies how close a system’s unstable mode is to being a DFM.

This can be directly measured by computing the modal real DFM radius with respect to

s, where s is set equal to the unstable mode of interest.

The results are listed in Table 7.3 and it can be seen that except for one case (i.e., the

3.280 pole of plant J), the modal real DFM radius with respect to a system’s unstable

pole is identical to the modal real CFM radius. Again, this implies that the use of

decentralized control on a system does not introduce any additional robustness issues as

compared to centralized control.

7.2 Unstructured Decentralized Fixed Modes

Several classifications of DFMs have been introduced including the notion of quotient

fixed modes (QFM) [33], structurally fixed modes [81], and structured DFM/unstructured

DFM [70] (also, see [2] for a brief summary of all three classifications). In this example,


Table 7.3: Comparison of the modal real DFM radius and the modal real CFM radius

with respect to the given unstable pole s

Plant Unstable pole Modal real DFM Modal real CFM

(s) radius wrt s radius wrt s

D. helicopter [69] 0.2758± j0.2576 0.8854 0.8854

F. pilot [78] 0.6888± j0.2466 0.2248 0.2248

G. boiler [12] 0 1.01610−6 1.01610−6

H. mass [17] ±j1.76810−8 0.1218 0.1218

I. 2-link 0 8.55510−3 8.55510−3

J. 2-cart 3.269 2.35810−2 2.35810−2

3.280 2.40110−2 2.42410−2

6.97810−10 9.05310−2 9.05310−2

we apply the real DFM radius to study structured and unstructured DFMs.

Given a plant (C,A,B) with a diagonal A, the structured DFMs are those modes (if

any) that continue to be DFMs after perturbing the nonzero values of the system matrices

(B,C), while unstructured DFMs are those modes (if any) that disappear for almost all

perturbations to the nonzero values of (B,C). It is shown in [70] that for almost all

sampling periods, a sampled discrete-time LTI system has the same number of structured

DFMs as the original continuous-time system and no unstructured DFMs, provided that

all of the unstructured DFMs are nonzero and distinct. Therefore an important result

is that if a continuous-time LTI system has unstable DFMs that are all unstructured,

distinct, and nonzero, then even though no continuous-time decentralized LTI controller

can stabilize the system, one can use a discrete-time decentralized controller to do so.

This is because all (distinct and nonzero) unstructured DFMs of the continuous-time

system disappear in the discrete-time model, so one can then proceed to stabilize the

system using standard discrete-time design approaches.


However, it is noted in [2] that even though the distinct nonzero unstructured DFMs

disappear in the discrete-time equivalent model, there may still exist approximate DFMs

(ADFMs) [85], which also pose a control difficulty as large controller gains are required

to shift ADFMs. However, the conditioning measure used to characterize a ADFM may

be rather ad-hoc, so instead, we perform a similar study in terms of the real DFM radius.

In the following, we assume that the continuous-time decentralized LTI system (3.7),

denoted

A B

C D

v

, is given and that the corresponding sampled system

x[k + 1] = Adx[k] +v∑

i=1

Bdiui[k] (7.1)

yi[k] = Cdix[k] +

v∑

j=1

Ddijuj[k], for i = 1, . . . , v,

with sampling period T > 0 (i.e., x[k] = x(kT )) is obtained via the zero-order hold

approximation; i.e.,

Ad = eAT , Bdi =

∫ T

0

eAτdτBi,

Cdi = Ci , Ddij = Dij , for i, j = 1, . . . , v.

Denoting the sampled system (7.1) by

Ad Bd

C D

v

(cf. see (3.8)), the real DFM radius

of the sampled system (7.1) is then given by the following (cf. Theorem 3.3.2), whose

proof directly follows from the continuous-time case result.

Theorem 7.2.1. Given a discrete-time decentralized LTI system (7.1) that has no DFMs,

the real DFM radius of system (7.1) is given by

rDFMR

Ad Bd

C D

v

= inf

s∈CminP⊆v

τn

Ad−sIT

BdP

T

CP DPP

(7.2)

= infs∈C

minP⊆v

supγ∈(0,1]

σ2n−1

ReW −γ ImW

γ−1 ImW ReW

,


where W :=

Ad−sIT

BdP

T

CP DPP

.

Example 7.2.1. Consider the following continuous-time decentralized LTI system:

x =

1 0 1

0 2 0

0 0 3

x+

−1

0

−1

u1 +

0

−1

0

u2 (7.3)

y1 =

[

0 1 0

]

x

y2 =

[

1 0 0

]

x,

which has eigenvalues at 1, 2, 3. It can easily be verified that (7.3) has an unstructured

DFM at λ = 2. In other words, λ = 2 is a DFM of system (7.3), but e2T is not a DFM

of the sampled system for almost all T > 0 (in fact, in this case it is true for all T > 0).

Therefore one is expected to be able to use a discrete-time decentralized controller with

any T > 0 to stabilize (7.3) and satisfy any arbitrary specification.

However, from Figure 7.1 that plots the real DFM radius of the sampled system of

(7.3) for 0 < T ≤ 5, one can observe that this may not necessarily be the case in a

practical setting. Figure 7.1 shows that as the sampling period approaches zero, the real

DFM radius becomes arbitrarily small. Therefore, to avoid large controller gains, one is

required to choose a sufficiently large sampling period, which in turn results in a more

sluggish response. So the unstructured DFM in this case still remains to be an issue, and

in particular, prevents the closed-loop system from achieving a fast response.

7.3 Pairing of Inputs/Outputs: Real DFM Radius

vs. Relative Gain Array

Given a multi-input multi-output system that is to be controlled by a decentralized LTI

controller, the real DFM radius can be used to determine how the inputs and outputs


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16Real DFM Radius of Sampled System with respect to mode e2T

Rea

l DF

M R

adiu

s

Sampling Period, T (sec.)

Figure 7.1: Real DFM radius of the sampled system of (7.3) with respect to the mode

e2T .

should be paired in order to obtain a robust decentralized control system structure with

respect to parametric perturbations. In particular, one can compute the real DFM radius

of the system with various input/output pairing combinations, and choose the pairing

that maximizes the real DFM radius. As a preliminary study, this section compares the

pairing decisions obtained using the real DFM radius versus using the so-called Relative

Gain Array (RGA) approach (see [6]), which is an approach widely used in industry.

7.3.1 Review: relative gain array (RGA)

Given a stable system (C,A,B,D), the Relative Gain Array is defined as

ΓRGA = G(0)⊗(G(0)T

)−1, (7.4)

where G(s) := C (sI − A)−1B + D is the transfer matrix, and ⊗ denotes element-by-

element product (Hadamard product).


All elements in any row or column of the RGA sum up to 1, and the “optimal” pairing

of the inputs and outputs is selected based on the RGA elements that are positive and

closest to 1.0. For instance, if the RGA is found to be

−0.5 1.5

1.5 −0.5

,

then the RGA approach suggests that the following pairing should be used:

u1

u2

=

0 ×

× 0

y1

y2

. (7.5)

An advantage of the RGA approach is that it requires only the steady-state properties

of a given system’s input/output behaviour and is scaling independent. However, a

limitation is that the approach is valid only for stable systems. Also, as the following

example demonstrates, the RGA approach can sometimes lead to erroneous pairings.

Example 7.3.1 (Counterexample to the RGA approach). Consider the following stable

system:

x =

−1 0 0

0 −0.01 0

0 0 −3

x+

1

0

0

u1 +

0

1

1

u2

y1 =

[

1 1 0

]

x (7.6)

y2 =

[

0 0 1

]

x,

where the RGA is obtained to be ΓRGA =

1 0

0 1

, which suggests that the following

pairing should be used to control the system (7.6) ([6]):

u1

u2

=

× 0

0 ×

y1

y2

. (7.7)


Table 7.4: Input/Output pairing using real DFM radius vs. RGA approach

Plant rDFMR

rDFMR

RGA Pairing

× 0

0 ×

0 ×

× 0

Comparison

A. distillation [14] 1.74310−6 5.30210−5

1.572 −0.572

−0.572 1.572

different

B. gas turbine [65] 0.5693 0.5693

−0.406 1.406

1.406 −0.406

same

C. turbo [60] 0.1986 0.2394

1.505 −0.505

−0.505 1.505

different

Note: input/output pairing by the real DFM radius approach is suggested based on the larger

real DFM radius (bold values).

However, the real DFM radius of (7.6) for the information flow constraint (7.7) is zero

corresponding to the mode -0.01; i.e., the decentralized system with the pairing (7.7)

has a decentralized fixed mode! In contrast, the real DFM radius of (7.6) using the

information flow constraint

u1

u2

=

0 ×

× 0

y1

y2

(7.8)

is 0.2333 corresponding to the mode -0.7668. This implies that if the pairing (7.7) is used,

as recommended by the RGA method, then the resulting controller would be unable to

control the dominant mode (-0.01) of the plant! On the other hand, if the pairing (7.8),

as suggested by the real DFM radius is used, then all modes of the system can easily be

controlled.

Table 7.4 compares the input/output pairings obtained by using the real DFM radius

versus the RGA approach. In the former, the input/output pairing is selected based on


whichever configuration results in the largest radius, which is shown by bold values. As

it is observed from Table 7.4, the RGA approach does not always suggest a configuration

that is “optimal” in terms of robustness with respect to parametric uncertainties.

7.4 Structured Real Controllability Radius of the

Pendulum

In [51], the real controllability radius was used to study the difficulty of balancing a

multi-link inverted pendulum system. In particular, it was claimed that the difficulty is

related to the pendulum’s controllability robustness, and was numerically shown that as

the number of pendulum links increases, the real controllability radius becomes smaller,

indicating that the system gets closer to becoming uncontrollable.

In [51], perturbations with a particular structure were considered, which led to a

controllability radius problem that is difficult to solve. Subsequently, a somewhat ad-hoc

normalization method involving a random matrix was used in [51] to compute an estimate

u

θθθθ1

θθθθ2

θθθθv

m1

m2

mv

l1

lv

l2

Figure 7.2: Model of a multi-link inverted pendulum with v links [51].


of the corresponding radius. In this section, the results on generalized and restricted real

perturbation values are applied to obtain more accurate estimates of the structured real

controllability radius of the multi-link inverted pendulum system.

The single-input single-output multi-link inverted pendulum system with v links con-

sidered in [51] is illustrated in Fig. 7.2, where the i-th link is modeled as a point-mass

mi attached via a massless rigid rod of length li, for i = 1, . . . , v. The control input u

is a single torque applied at the pivot of the bottom link. All angles are measured with

respect to the vertical. Defining x =

[

θ1 · · · θv θ1 · · · θv

]T

as the state vector,

and θ1 as the output (i.e., the bottom link’s angle is measured), the linear state-space

model (linearized about the vertical zero equilibrium point (x, u) = (0, 0)) is given by

[51]

A =

0 I

(MvLv)−1Ma 0

, B =

0

(MvLv)−1Mb

, (7.9)

C =

[

1 0 . . . 0

]

, D = 0,

where Ma = diag

(

gv∑

i=1

mi, gv∑

i=2

mi, · · ·, mvg

)

, Mb =

1

l10...0

, Mv =

[m1 m2 ··· mv

0 m2 ··· mv

.........

...0 0 ··· mv

]

, and

Lv =

l1 0 ··· 0l1 l2 ··· 0......... 0

l1 l2 ··· lv

.

From the state-space model of the pendulum system given by (7.9), it can be seen

that the upper portion of the (A,B) matrices are hard 0’s and 1’s. This arises from

the direct relationship between the state variables (θ1, . . . , θv) and their first derivatives(

θ1, . . . , θv

)

. Therefore in [51], only perturbations that affect the lower half of the (A,B)

matrices were considered. Furthermore, the perturbations were normalized by the (A,B)

matrices; i.e., (A,B)→ ((I +∆A)A, (I +∆B)B). As a result, perturbations considered


in [51] are of the following form:

A →

I +

0 0

0 I

∆A

A (7.10)

B →

I +

0 0

0 I

∆B

B.

Referring back to Section 4.1.1, the structured real controllability radius of the pen-

dulum system with respect to the perturbation structure (7.10) is then given by

rc,structR

(A,B, E ,F ,G) = inf

∆A ∈ Rn×n

∆B ∈ Rn×m

∥∥∥∥

[

∆A ∆B

]∥∥∥∥2

∣∣∣∣∣∣∣

(A+ E∆AF , B + E∆BG)

is uncontrollable

(7.11)

where

E =

0 0

0 I

, F = A, and G = B. (7.12)

However, since E in (7.12) is singular, we do not have a formula to compute the exact

value of the structured real controllability radius for the pendulum system with respect

to the perturbation structure (7.10). This is because when E , F , and G are arbitrary

matrices that are not nonsingular, as is the case here, the n-th restricted real perturbation

values of

[

A− sI B

]

, E ,

F 0

0 G

is required (see Remark 4.1.1).

So to estimate the structured controllability radius, various approximations had been

made in [51]. In particular, E =

0 0

0 I

was approximated with a nonsingular matrix

E =

ǫI 0

0 I

, where ǫ > 0 was chosen as small as possible (e.g., ǫ = 10−6) such that

E was not ill-conditioned, and

F 0

0 G

was approximated with another nonsingular


matrix,

F 0

0 G

, which was obtained via an ad-hoc normalization technique involving

a random matrix. These approximations allows the structured real controllability radius

(7.11) to be estimated using the following formula (cf. (4.3)):

rc,structR

(

A,B, E , F , G)

= infs∈C

τn

E−1

[

A− sI B

]

F 0

0 G

−1

. (7.13)

However, with the results on generalized and restricted real perturbation values, an

improved estimate can be obtained. In particular, the structured real controllability

radius (7.11) can be approximated using (4.4)

rc,structR

(

A,B, E ,F ,G)

= infs∈C

τn

E−1

[

A− sI B

]

,

F 0

0 G

, (7.14)

where F and G are given in (7.12), and E =

ǫI 0

0 I

is again a nonsingular approxima-

tion of E =

0 0

0 I

(also see Section 2.4.2 on approximating restricted real perturbation

values).

7.4.1 Results

The results of using the various approximations mentioned above for computing the struc-

tured real controllability radius of the multi-link inverted pendulum system are listed in

Table 7.5. For completeness, we also include the lower bounds of the radius as ob-

tained by computing the lower bounds of the n-th restricted real perturbation values of

[

A− sI B

]

, E ,

F 0

0 G

. Here we see that the values obtained using (7.14) is

a slight improvement over the values obtained in [51] using (7.13). Also, we observe that

the values obtained using (7.14) are almost equal (up to the 11th significant figure) to


Table 7.5: The structured real controllability radius of a multi-link inverted pendulum

system

Number of rc,structR

rc,structR

rc,structR

Links (v) via (7.13) in [51] via (7.14) (lower bounds)

1 1.00010+0 1.00010+0 1.00010+0

2 1.11110−1 1.08510−1 1.08510−1

3 4.68810−2 4.65510−2 4.65510−2

4 2.45610−2 2.45010−2 2.45010−2

5 1.46710−2 1.46610−2 1.46610−2

6 9.56510−3 9.55910−3 9.55910−3

7 6.63510−3 6.63310−3 6.63310−3

the computed lower bounds. Since one may expect E =

ǫI 0

0 I

to be a good approx-

imation of E =

0 0

0 I

for small ǫ, this suggests that for this particular perturbation

structure given in (7.10), the lower bounds of the structured real controllability radius

may be achievable, and that the values obtained via (7.14) may be very close to the true

radius.

7.5 Example - Transmission Zero at s Radius

In this example, we study the RSP for constant signal tracking and disturbance rejection

in terms of the transmission zero at s radius, where s = 0, for a number of industrial

plant models. Recall from [15] that there exists a solution to the RSP for constant

signal tracking and disturbance rejection if and only if the system has no transmission

zeros at s = 0. Hence, the real transmission zero at s radius, for s = 0, can be used


to evaluate the robustness of the system’s servomechanism controller; in particular, if a

plant model’s transmission zero at s = 0 radius is small, then the system is very close to

having a transmission zero at the origin, which implies that the system’s servomechanism

controller may be fragile and not robust.

In this study, we consider normalized perturbations as described by the following

form:

A B

C D

→

(I +∆A)A (I +∆B)B

C D

, (7.15)

where the C and D matrices are unperturbed, and the perturbations affecting A and

B are normalized according to (A,B) → ((I +∆A)A, (I +∆B)B). The reason for not

perturbing C and D is that the C and D matrices of the plant models studied in this

example all contain hard 0’s and 1’s arising from the system structure.

So considering the perturbation structure (7.15), the structured real transmission zero

at s radius as given by (4.6) is

rTZ,structR

(C,A,B,D, s, E ,F ,G,H) = (7.16)

inf

∆A ∈ Rn×n,∆B ∈ R

n×m


∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣∣∣∣



has TZ at s

,

where E = I, F = A, G = B, H = 0, which can be approximated by

rTZ,structR

(C,A,B,D, s) = τn+min(r,m)

I 0

0 ǫI

−1

A− sI B

C D

,

A 0

0 B

for ǫ→ 0.1

1Again, ǫ is chosen as small as possible (e.g., 10−6) such that [ I 00 ǫI

] is not ill-conditioned.


Table 7.6: Transmission zero at s = 0 radius for various industrial examples

Plant rTZR

rTZ,structR

A. distillation [14] n = 11, S, M 1.44710−5 2.21510−2 (= 2.22%)

B. gas turbine [65] n = 4, S, M 0.8062 2.67210−1 (= 26.7%)

C. turbo [60] n = 6, S, NM 0.1521 3.36410−1 (= 33.6%)

D. helicopter [69] n = 4, US, M 1.27410−2 9.24410−2 (= 9.24%)

E. thermal [19] n = 9, S, NM 0 0 (= 0%)

F. pilot [78] n = 6, US, M 3.49110−4 1.71010−3 (= 0.171%)

G. boiler [12] n = 9, US, NM 1.01610−6 2.04310−6 (= 2.0410−4%)

H. mass [17] n = 6, US, M 0.1148 9.92610−2 (= 9.93%)

I. 2-link n = 36, US, NM 8.46110−3 4.62110−5 (= 4.62110−3%)

J. 2-cart n = 8, US, NM 8.76810−2 4.72610−1 (= 47.3%)

7.5.1 Results

Table 7.6 displays both the unstructured and structured real transmission zero at s radius

for s = 0. From the table, we see that there is a large difference in the robustness of the

various models, and it can be seen that some plants such as Plant A, F, I, and especially

G and E, appear to be very fragile; although, Plant E has a zero % radius because the

plant’s nominal system is degenerate and therefore has no solution to the servomechanism

problem.

On further study of Plant G, the boiler system, it is found that the nominal system

has a transmission zero that is very close to the origin, namely, at s = −9.5510−3 , which

implies that a slight perturbation can cause this transmission zero to approach the origin,

resulting in no solution existing to the servomechanism problem for this system. However,

since the boiler system has widely been studied and is often not thought to be “fragile”,

one may be surprised to see that it has a small transmission zero at s radius. So to

further investigate, we study the mobility of the RSP’s closed-loop eigenvalues using the


Table 7.7: Mobility bounds of the RSP’s closed-loop eigenvalues for various industrial

examples

Plant rTZR

∥∥∥∥∥∥∥

A B

C D

∥∥∥∥∥∥∥2

Mobility Bound

by Theorem 4.4.3

A. distillation n = 11, S, M 1.44710−5 1.002 7.491

B. gas turbine n = 4, S, M 0.8062 1.56610+3 3.03210+4

C. turbo n = 6, S, NM 0.1521 23.77 9.51810+3

D. helicopter n = 4, US, M 1.27410−2 11.52 1.99210+3

E. thermal n = 9, S, NM 0 3.87810+3 0

F. pilot n = 6, US, M 3.49110−4 56.96 4.52410+3

G. boiler n = 9, US, NM 1.01610−6 2.28810+4 4.66510+3

H. mass n = 6, US, M 0.1148 2.094 6.98410+2

I. 2-link n = 36, US, NM 8.46110−3 3.06010+6 2.23510+6

J. 2-cart n = 8, US, NM 8.76810−2 10.81 1.42110+2

mobility bounds provide by Theorem 4.4.3.

Bounds on closed-loop eigenvalue mobility

Table 7.7 summarizes the mobility bounds, as obtained by Theorem 4.4.3, of the RSP’s

closed-loop eigenvalues, where the bounds correspond to all controller gains

[

K0 K1

]

with

∥∥∥∥

[

K0 K1

]∥∥∥∥2

≤ 1000. We see from Table 7.7 that although the boiler system has a

small transmission zero at s = 0 radius, the corresponding mobility bound is comparable

to the other systems. Therefore one may be relieved to see that the boiler system is

not as fragile as suggested by the small transmission zero at s radius. Of course, since

the mobility bounds provided by Theorem 4.4.3 are only upper bounds, one still cannot

necessarily conclude that the closed-loop response of the boiler system is fast either.


On the other hand, we observe that Plant A has a relatively low mobility bound (i.e.,

7.491). This implies that for all controller gains of magnitude 1000 or less, the closed-loop

spectrum always has an eigenvalue of magnitude 7.491 or less. Therefore even for “large”

controller gains of 1000, the distillation column system will still have a relatively slow

response.

7.6 Toughness Index: Robust Servomechanism Prob-

lem

In this section, we apply the Toughness Index of Chapter 6 to study the difficulty in

achieving high performance for the RSP for two different systems.

7.6.1 Example 1: RSP for a mass-spring system

Consider a mass-spring system, where two smaller masses are attached to a larger mass

via springs and a damper. The system outputs are the outputs of the two smaller masses

and the inputs are the forces applied to the two smaller masses. The system can be

modeled by the following LTI system:

A =

0 1 0 0 0 0

−0.2 −0.02 0.1 0.01 0.1 0.01

0 0 0 1 0 0

1 0.1 −1 −0.1 0 0

0 0 0 0 0 1

1 0.1 0 0 −1 −0.1

, B =

0 0

0 0

0 0

1 0

0 0

0 1

(7.17)

C =

0 0 1 0 0 0

0 0 0 0 1 0

.


Table 7.8: Summary of the mass-spring example for tracking/disturbance poles = [0,±j1]

ǫ

∥∥∥∥

[

K0 K1

]∥∥∥∥2

trace(Pǫ) TZ∞

10−4 1.58810+2 3.934 [1/s5, 1/s5]

10−8 1.12110+4 1.114 [1/s5, 1/s5]

10−12 1.02110+6 0.4140 [1/s5, 1/s5]

The system (7.17) is open-loop unstable, has two minimum phase finite transmission

zeros at s = −0.0100± j0.447, and infinite transmission zeros [1/s2, 1/s2].

Now suppose we want to solve the RSP for (7.17) with tracking/disturbance poles

[0,±j1]. In this case, the resultant augmented system (which includes a servocompen-

sator for the tracking/disturbance poles) has infinite transmission zeros [1/s5, 1/s5]. On

applying the results of Section 6.3, it is determined that the Toughness Index is given by

(1

ǫ

)1/10

, (7.18)

which is an indication that high performance cannot be obtained in the system; e.g., with

ǫ = 10−10, the Toughness Index = 10, which indicates that a relatively slow transient re-

sponse with a time constant of 0.1 sec. or more will occur. On solving the RSP using state

feedback and minimizing the cheap performance index (6.21) with ǫ = (10−4, 10−8, 10−12),

the results obtained for a tracking reference signal of yref = 1 is presented in Figures 7.3-

7.4. From these figures2, we see that for ǫ = 10−12, the system achieves a response with

a settling time of ∼ 1 sec. For these chosen ǫ, Table 7.8 summarizes the size of the corre-

sponding controller gains (1.18), and the performance cost given by the trace of Pǫ, where

Pǫ is the solution to the associated ARE (6.5), and TZ∞ denotes infinite transmission

zeros.

Now suppose we want to solve the RSP for (7.17) with tracking/disturbance poles

[0,±j1,±j2,±j4,±j8,±j10]. The resultant augmented system now has infinite trans-

2Note that in the figures, Output 1 and Output 2 actually coincide with each other very closely.


0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2Output Response

(a)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

(b)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

Time (s)

(c)

Output 1Output 2

Figure 7.3: Output response of the mass-spring system using cheap servomechanism

control with tracking/disturbance poles [0,±j1], and (a) ǫ = 10−4, (b) ǫ = 10−8, and (c)

ǫ = 10−12.


0 1 2 3 4 5 6 7 8 9 10−5

0

5

10Input Response

(a)

0 1 2 3 4 5 6 7 8 9 10−50

0

50

(b)

0 1 2 3 4 5 6 7 8 9 10−400

−200

0

200

400

Time (s)

(c)

Input 1Input 2

Figure 7.4: Input response of the mass-spring system using cheap servomechanism control

with tracking/disturbance poles [0,±j1], and (a) ǫ = 10−4, (b) ǫ = 10−8, and (c) ǫ =

10−12.


Table 7.9: Summary of the mass-spring example for tracking/disturbance poles =

[0,±j1,±j2,±j4,±j8,±j10]

ǫ

∥∥∥∥

[

K0 K1

]∥∥∥∥

trace(Pǫ) TZ∞

10−12 3.20710+6 34.79 [1/s13, 1/s13]

10−18 2.15910+9 6.249 [1/s13, 1/s13]

10−24 1.42810+12 2.430 [1/s13, 1/s13]

mission zeros [1/s13, 1/s13]. Figures 7.5-7.6 display the input and output responses for

ǫ = (10−12, 10−18, 10−24), and Table 7.9 summarizes the corresponding controller gains

and performance costs. From these figures, it can be seen that with the additional track-

ing/disturbance poles, the system has difficulty achieving a fast response. In particular,

even with a large controller gain corresponding to ǫ = 10−24 (and input magnitudes sim-

ilar to the previous response shown in Figure 7.4), the settling time is about ∼ 7 sec.

On computing the Toughness Indices (6.17), it is determined that all the faraway poles

approach infinity at a rate(1

ǫ

)1/26

, (7.19)

which is much slower than (7.18).

7.6.2 Example 2: control of a commercial hard disc drive

The study of obtaining high performance control of a commercial hard disk drive system

was carried out in [10]. The plant model describing the disk drive model in this case

was a SISO LTI system of high order (≫ 10), and it was desired to reject unknown

and unmeasurable disturbances using a RSP controller design. In this case the class

of disturbances to be rejected were modeled by harmonic sinusoidal signals with five

frequencies given by 180π × (1, 2, 3, 5, 24) radians/sec.

In this case the resultant RSP controller design was highly effective compared to


0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5Output Response

(a)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

(b)

0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

Time (s)

(c)

Output 1Output 2

Figure 7.5: Output response of the mass-spring system using cheap servomechanism

control with tracking/disturbance poles [0,±j1,±j2,±j4,±j8,±j10], and (a) ǫ = 10−12,

(b) ǫ = 10−18, and (c) ǫ = 10−24.


0 1 2 3 4 5 6 7 8 9 10−1

0

1

2Input Response

(a)

0 1 2 3 4 5 6 7 8 9 10−50

0

50

(b)

0 1 2 3 4 5 6 7 8 9 10−400

−200

0

200

400

Time (s)

(c)

Input 1Input 2

Figure 7.6: Input response of the mass-spring system using cheap servomechanism control

with tracking/disturbance poles [0,±j1,±j2,±j4,±j8,±j10], and (a) ǫ = 10−12, (b)

ǫ = 10−18, and (c) ǫ = 10−24.


conventional approaches, but the following observation was made in [10]:

“For this initial servocompensator design, one can see in Figure 3 that an un-

desirably long settling time occurs due to the relatively slow error attenuation

of the sinusoidal harmonic having frequency component ω5.”

This undesirable long settling time is precisely the type of result which would be predicted

by the Toughness Index. Due to the large number of tracking/disturbance poles required

to be rejected in this study (a total of 10), extremely high controller gains would be

required to speed up the disturbance rejection properties of the servo-controller, but such

high controller gains are completely unrealistic to use, and so some of the closed-loop

poles of the system were sluggish.

Chapter 8

Conclusions

8.1 Summary

This thesis studies the problem of defining continuous measures to extend the ‘yes/no’

metrics of traditional LTI system definitions such as controllability, observability, sta-

bilizability, not having DFMs, minimum phase, etc. The need for such measures arises

on recognizing that an ideal system model that proves to be, say, controllable and ob-

servable, may in fact be very “close” to being uncontrollable and unobservable due to

uncertainties in the system parameters. So in this thesis, traditional binary metrics are

extended to the following continuous metrics: the real decentralized fixed mode (DFM)

radius; the real transmission zero at s radius; and the real minimum phase radius, where

these radii are defined in terms of real parametric perturbations in the system matri-

ces. Various problems surrounding the study of these radii are investigated, namely,

the generalization of the real perturbation value problem, the derivation of computable

formulas for the various radii, and the development of efficient algorithms for evaluating

these formulas. In this thesis, a continuous measure called the Toughness Index is also

introduced to characterize a system’s difficulty in achieving high performance that may

occur even when the system is minimum phase. The contributions of each chapter can

160

Chapter 8. Conclusions 161

be summarized as follows.

In Chapter 2, two generalizations of the ordinary real perturbation value problem are

made, which resulted in the introduction of the so-called generalized real perturbation

values and the restricted real perturbation values. Computable formulas are derived for

the former, while lower bounds are obtained for the latter.

In Chapter 3, the three new real LTI robustness radii mentioned above are introduced

and defined. Computable formulas for these radii are derived using ordinary singular

values and ordinary real perturbation values, along with a discussion of the radii’s various

properties.

In Chapter 4, various extensions of the LTI robustness radii are made using the

new results on generalized and restricted real perturbation values. In particular, we

generalized the various radii to account for structured parametric perturbations, and

extended the real controllability radius to LTI descriptor and time-delay systems. The

topic of eigenvalue mobility in relationship to the various robustness radii is also studied.

In Chapter 5, two efficient algorithms are presented for computing the various classes

of real perturbation values, and for evaluating the various LTI robustness radii.

In Chapter 6, we propose a different type of continuous measure, called a Toughness

Index, to characterize a system’s performance limitation, as measured by the cheap LQR

control problem, and apply it to the RSP.

All the results of this thesis are then applied to a number of examples in Chapter 7.

8.2 Future Directions

There are a number of open problems related to the results of this thesis and some are

discussed here.


8.2.1 Restricted real perturbation values formula

A computable formula for restricted real perturbation values still remains to be found.

As discussed in Section 2.5, the similarities of the formulas for ordinary and generalized

real perturbation values to the lower bounds (2.13) of restricted real perturbation values

leads one to suspect that these lower bounds are actually exact, or at least achievable for

certain classes of L and N (as suggested by the approximations made in Section 2.4.2).

Whether the lower bounds (2.13) are exact, or under what conditions they are exact still

remains to be investigated in future studies.

8.2.2 Extensions to other radii

In this thesis, the various radii are defined in terms of parametric perturbations (e.g.,

of the form (1.2) or (4.1)), and formulas for computing these radii are obtained using

the framework involving the various classes of real perturbation values. Using such a

framework, other radii can similarly be defined for all other LTI definitions that can

be characterized by appropriate rank conditions. Two such extensions are made in this

regard to LTI descriptor and time-delay systems, and future extensions to other linear

system definitions are also possible.

A limitation, however, of using the above framework is that the LTI definition must

be characterized by only a single rank condition. Now suppose we want to consider two

or more rank conditions. As an example, let say we want to extend the real transmission

zero at s radius to the real transmission zeros at s1 and s2 radius defined as follows.

Given the LTI system (C,A,B,D), s1 ∈ C and s2 ∈ C, define

rTZR

(C,A,B,D, s1, s2) =

inf



∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣

(C +∆C , A+∆A, B +∆B, D +∆D)

has a TZ at s1 and s2

,


which by definition is equivalent to

rTZR

(C,A,B,D, s1, s2) =

inf

∆A ∈ Rn×n

∆B ∈ Rn×m

∆C ∈ Rr×n

∆D ∈ Rr×m

∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

rank

A− s1I B

C D

< n +min(r,m)

and

rank

A− s2I B

C D

< n +min(r,m)

.

This definition or property consists of two rank conditions to be satisfied simultaneously,

in which case the real perturbation values results cannot be applied directly. So further

study is required to handle this case or other characterizations involving multiple rank

conditions.

8.2.3 Coordinate dependence of LTI robustness radii

From Section 3.5, it is shown that the various robustness radii are independent to orthog-

onal coordinate transformations, but not to nonorthogonal transformations. Depending

on the application, this may or may not be a problem. For example, consider a plant

with the following transfer function:

Y (s) =1

s2 − 2s+ 2U(s),

which has open-loop poles at s = 1 ± j1. Two possible state-space representations are

then given by:


Model #1:

x =

0 1

−2 2

x+

0

1

u

y =

[

1 0

]

x

Model #2:

x =

0 1

−2 2

x+

0

10−12

u

y =

[

1012 0

]

x

where Model #1 and #2 differ by only a scaling factor of 10−12 and 1012 in the input

and output matrices, respectively. The real controllability radius for Model #1 and

#2 is 0.5821 (corresponding to s = 0.6176) and 1.000 × 10−12 (corresponding to s =

1.000+j1.000), respectively, which indicates that Model #2 is more fragile to parametric

perturbations than Model #1. In fact, Model #2 can be further scaled (e.g., 10−16, 10−20,

etc.) to make the controllability radius arbitrarily smaller. From a numerical point of

view, the controllability radius represents a good measure to use as the small 10−12 term

in Model #2 does present a numerical issue (e.g., low eigenvalue mobility). However,

from a system perspective, it may be misleading for the same identical system to have

two (or more) different radii. To remedy this issue, further assumptions are required. For

example, it may be reasonable to assume that some “standard” state-space representation

(e.g., a balanced realization) is obtained before computing the various robustness radii.

However, which representation “makes sense” is up for further discussion and study.

Appendix A

Review of Singular Values

Various types of singular values are reviewed in this section; namely,

i) the ordinary singular values of a single matrix;

ii) the generalized singular values of a matrix pair; and

iii) the restricted singular values of a matrix triplet.

These different types of singular values can all be obtained using their corresponding

singular value decompositions. Such decompositions have many applications such as in

pseudoinverse operations, low-rank matrix approximations, and in obtaining the condi-

tion number of linear equations (e.g., see [32, 39]). For our purpose, we use the various

singular values to compute the norm of the smallest perturbation matrix that can cause

a given matrix to lose a certain degree of rank, where the norm is taken to be the spectral

norm.

A.1 Singular Values of a Matrix

The (ordinary) singular value decomposition of a matrix is given as follows (see [32,39]).

165

Appendix A. Review of Singular Values 166

Theorem A.1.1 (Ordinary Singular Value Decomposition). Given M ∈ Cn×m with

rank(M) = r, there exist unitary matrices U ∈ Cn×n and V ∈ Cm×m, and there exist

σi > 0, for i = 1, . . . , r, such that

M = U

SM 0

0 0

V,

where SM = diag(σ1, · · · , σr), and σ1 ≥ · · · ≥ σr ≥ 0.

The set

σ1, · · · , σr, 0, · · · , 0

︸︷︷︸

min(n,m)−r

1 defines the singular values of M and is denoted

σi(M) := σi, for i = 1, . . . , r, and σi(M) = 0, for i = r + 1, . . . ,min(n,m). It can be

shown that given a matrix M ∈ Cn×m, the singular values of M satisfy the following

property (e.g. see [4]):

σi(M) = inf∆∈Cn×m

‖∆‖2 | rank(M −∆) < i (A.1)

=1

inf∆∈Cm×n

‖∆‖2 | nullity(Im −∆M) ≥ i ,

for i ∈ 1, . . . ,min(n,m). Furthermore, given i ∈ 1, . . . ,min(n,m), a minimum-norm

perturbation ∆ such that rank(M −∆) < i is given by

∆ = U

SM 0

0 0

V,

where SM = diag

0, · · · , 0︸︷︷︸

i−1

, σi, . . . , σr, 0, · · · , 0︸︷︷︸

min(n,m)−r

, and ‖∆‖2 = σi(M).

There are many available algorithms for computing the singular value decomposition.

In this thesis, a (ordinary) singular value decomposition of a matrix is obtained by using

MATLAB’s routine, SVD.m, which uses the LAPACK routines [1].

1In the literature, the set σ1, · · · , σr is sometimes referred to as the set of singular values of Minstead. For notation purposes, we include the min(n,m)− r zero diagonals.


A.2 Generalized Singular Values of a Matrix Pair

An extension of the ordinary singular value decomposition to deal with a matrix pair is

given by the following generalized singular value decomposition (see [61, 73]).

Theorem A.2.1 (Generalized singular value decomposition). Let M ∈ Cn×m, N ∈

Cp×m, and r = rank

M

N

. Then there exist unitary matrices U ∈ Cn×n and

V ∈ Cp×p, a nonsingular matrix Q ∈ Cm×m, and scalars σi, for i = r1 + 1, . . . , r1 + r2,

such that

U∗MQ−1 =

r1 r2 r3

r1

r2

I 0 0 0

0 SM 0 0

0 0 0 0

and V ∗NQ−1 =

r1 r2 r3

r2

r3

0 0 0 0

0 I 0 0

0 0 I 0

,

where SM = diag (σr1+1, · · · , σr1+r2), σr1+1 ≥ · · · ≥ σr1+r2 > 0, and where r1, r2,

and r3 are defined by r1 = rank

M

N

− rank(N), r2 = rank(M) + rank(N) −

rank

M

N

, and r3 = r − r1 − r2.

The diagonal elements of the first r columns of U∗MQ−1 and V ∗NQ−1 consists of

three possible kinds of pairs:

• For i = 1, . . . , r1: (1, 0);

• For i = r1 + 1, . . . , r1 + r2: (σi, 1); and

• For i = r1 + r2 + 1, . . . , r: (0, 1),

and the (nontrivial) generalized singular values of (M,N), denoted by σi(M,N), are

defined to be the ratios of these pairs. In particular, there are r nontrivial generalized


singular values, of which r1 are infinite, r2 are nonzero finite (σi), and r3 are zero. Cor-

responding to the last m − r zero columns of U∗MQ−1 and V ∗NQ−1, there are m − r

trivial generalized singular values.

Recall that for a given matrix M ∈ Cn×m, where n ≥ m, the eigenvalues of MHM are

the squares of the singular values of M . This result extends to the matrix pair (M,N)

by the fact that

MHM = QH

I 0 0 0

0 SHMSM 0 0

0 0 0 0

0 0 0 0

Q and NHN = QH

0 0 0 0

0 I 0 0

0 0 I 0

0 0 0 0

Q.

Therefore, the nontrivial generalized eigenvalues2 of the matrix pair(MHM,NHN

)are

the squares of the nontrivial generalized singular values of (M,N).

Also, it can be shown that the i-th generalized singular value of (M,N), for i ∈

1, . . . ,min(n,m), satisfies

σi(M,N) = inf∆∈Cn×p

‖∆‖2 | rank(M −∆N) < i

=1

inf∆∈Cp×n

‖∆‖2 | nullity(N −∆M) ≥ i .

In this thesis, a generalized singular value decomposition of a matrix pair is obtained

by using MATLAB’s routine, GSVD.m [1].

A.3 Restricted Singular Values of a Matrix Triplet

The restricted singular values of a matrix triplet are introduced in [90].

Definition A.3.1 (Restricted singular values [90]). Given M ∈ Cn×m, L ∈ Cn×l, and

2The trivial generalized eigenvalues are those corresponding to the last m− r columns of Q in MHM

and NHN , which span the common null space of MHM and NHN .


N ∈ Cp×m, the restricted singular values of the triplet (M,L,N) are defined as

σi(M,L,N) := inf∆∈Cl×p

‖∆‖2 | rank(M − L∆N) < i ,

for i ∈ 1, . . . ,min(n,m).

The restricted singular values of a given matrix triplet can be obtained by the re-

stricted singular value decomposition as given in the following theorem (see [68, 90]).

Theorem A.3.1 (Restricted singular value decomposition). Let M ∈ Cn×m, L ∈ Cn×l,

and N ∈ Cp×m, and define the following:

rM = rank(M) , rL = rank(L) , rN = rank(N)

rML = rank

([

M L

])

, rMN = rank

M

N

, rMLN = rank

M L

N 0

.

Then there exist nonsingular matrices X ∈ Cn×n and Y ∈ Cm×m, unitary matrices

U ∈ Cl×l and V ∈ Cp×p, and scalars σi, for i = 1, . . . , η, such that

X−1MY −1 =

r1 r2 r3 µ

r1

r2

r3

ν

I 0 0 0 0

0 I 0 0 0

0 0 I 0 0

0 0 0 SM 0

0 0 0 0 0

X−1LU∗ =

r2 ν

r1

r2

r3

ν

0 0 0

0 I 0

0 0 0

0 0 I

0 0 0

V ∗NY −1 =

r1 r2 r3 µ

r3

µ

0 0 0 0 0

0 0 I 0 0

0 0 0 I 0

,

where SM =

Σ 0

0 0

with Σ = diag (σ1, · · · , ση) and σ1 ≥ · · · ≥ ση > 0, and where r1,


r2, r3, µ, ν, and η are defined by the following:

r1 = rMLN − rL − rN , r2 = rMN + rL − rMLN , r3 = rML + rN − rMLN

µ = rMLN − rML , ν = rMLN − rMN , η = rMLN + rM − rML − rMN .

Let αi, βj, and γk be the “diagonal” elements of X−1MY −1, X−1LU∗, and V ∗NY −1,

respectively, then there are 6 possible kinds of triplets (αi, βjγk):

• r1 triplets of the form (1, 0, 0);



• η triplets of the form (σi, 1, 1), for i = 1, . . . , η;

• min(µ, ν)− η triplets of the form (0, 1, 1); and

• min(n− r1 − r2 − r3 − ν,m− r1 − r2 − r3 − µ) triplets of the form (0, 0, 0).

The restricted singular values of (M,L,N) are defined to be the ratios of αi

βjγk.

Therefore from Theorem A.3.1, the matrix triplet (M,L,N) has r1 + r2 + r3 infinite,

η nonzero finite, and min(µ, ν)− η zero (nontrivial) restricted singular values, and also

has min(n− r1 − r2 − r3 − ν,m− r1 − r2 − r3 − µ) trivial ones.

Consider now the following generalized eigenvalue problem: given M , L, and N , it is

desired to find an eigenvalue λ and eigenvector x that satisfy

0 M

M∗ 0

x = λ

LL∗ 0

0 N∗N

x. (A.2)

It is shown in [90] that the generalized eigenvalues of the matrix pair in (A.2), i.e.,

λ

0 M

M∗ 0

,

LL∗ 0

0 N∗N

,


are related to the positive and negative of the restricted singular values of (M,L,N). In

particular, the solution to (A.2) consists of

• 2 (r1 + r2 + r3) infinite generalized eigenvalues,

• 2η nonzero finite generalized eigenvalues, namely, ± αi

βjγk,

• (µ− η) + (µ− η) zero generalized eigenvalues, and

• (m+ n)− 2 (r1 + r2 + r3 + η)− (µ− η)− (ν − η) trivial generalized eigenvalues.

In this thesis, a MATLAB implementation of the algorithm presented in [13] is used

to compute a restricted singular value decomposition of a matrix triplet.

Appendix B

Construction of a Minimum-Norm

Perturbation

This section provides a detailed procedure to compute a real minimum-norm perturbation

that achieves a corresponding real perturbation value. A minimum-norm perturbation

can be constructed using the following theorem found in [4] (also see [44]).

Theorem B.0.2 ([4]). Let M ∈ Cn×m and i ∈ 1, . . . ,min(n,m). If τi(M) =∞, then

there exists no ∆ ∈ Rn×m such that rank(M +∆) < i. Suppose τi(M) < ∞. Let X be

any (m − k + 1)-dimensional subspace of Cm such that all x ∈ X satisfy the hermitian-

symmetric inequality

xH(τk(M)2 Im −MHM

)x ≥

∣∣xT(τk(M)2 Im −MTM

)x∣∣ . (B.1)

Now let X ∈ Cm×(m−k+1) be any matrix whose columns form a basis of X . Set

∆ :=

[

Re (MX) Im (MX)

] [

Re (X) Im (X)

]+

∈ Rn×m. (B.2)

Then,

rank(M −∆) < i and ‖∆‖2 = τi(M) . (B.3)

172

Appendix B. Construction of a Minimum-Norm Perturbation 173

To obtain the basis of the subspace that satisfies (B.1) (or the matrix X), one can

perform the following simultaneous block diagonalization of H and S, where H :=

τi(M)2 Im − MHM and S := τi(M)2 Im − MTM . In particular, first denote the set

of n× n hermitian-symmetric matrix pairs by

HSn :=(H,S) ∈ C

n×n × Cn×n|HH = H,ST = S

,

and also denote

HS+n := (H,S) ∈ HSn| det(H) 6= 0, C is condiagonalizable1, where C := S−1H

.

It is shown in [44] that the set HS+n is an open and dense subset of HSn. Also, the

following is true.

Theorem B.0.3 ([44]). Let (H,S) ∈ HS+n and C = S−1H. Then CC has nonnegative

real eigenvalues, the nonreal eigenvalues of CC occur in complex conjugate pairs, and

there exists an invertible matrix P ∈ Cn×n such that

PHHP = block diagλ1, . . . , λρ, λρ+1, . . . , λρ+l,

λρ+l+1, . . . , λr,

0 ν1

ν1 0

, . . . ,

0 νs

νs 0

P TSP = In, (B.4)

where

• λi ∈ R and λ2i is a real eigenvalue of CC, for i = 1, . . . , r;

• Im νi ∈ C\R with Im νi > 0, and ν2i and νi

2 are the nonreal eigenvalues of CC, for

i = 1, . . . , s;

• 1 > λ1 ≥ · · · ≥ λρ ≥ −1;

• λi ≥ 1 for ρ+ 1 ≤ i ≤ ρ+ l; and

1i.e., there exists a nonsingular R ∈ Cn×n such that R−1CR is diagonal.


• λi < −1 for ρ+ l + 1 ≤ i ≤ r.

Furthermore, the columns of P form a basis of eigenvectors for CC.

Note that for H and S given above,(

H, S)

∈ HS+m. So by Theorem B.0.3, there

exists a P that can simultaneously block diagonalize H and S into the form given by

(B.4). Such a P in Theorem B.0.3 can be constructed by a procedure given in [44], which

is summarized below.

First, let µ1, . . . , µr be the real eigenvalues of CC, where C = S−1H , and µr+1, . . . , µr+s

be the eigenvalues of CC with positive imaginary part. Now, let u1, . . . , ur be the eigen-

vectors corresponding to µ1, . . . , µr, and v1, . . . , vs be the eigenvectors corresponding to

µr+1, . . . , µr+s. (Note: Cv1, . . . , Cvs are eigenvectors corresponding to µr+1, . . . , µr+s.)

Define

Q = [u1, . . . , ur, v1, Cv1, . . . , vs, Cvs]

and let P be as follows:

P = Q diag(

α−11 , . . . , α−1

r , β−11 , γ−1

1 β−11 , . . . , β−1

s , γ−1s β−1

s

)

,

where αj, βj , γj ∈ C\0 satisfy

α2j = uT

j Suj, for 1 ≤ j ≤ r,

β2j = vTj Svj, for 1 ≤ j ≤ s,

γ2j = µr+j, for 1 ≤ j ≤ s.

Such a P will simultaneously block diagonalize H and S in the form given by (B.4).2

Denote the columns of P as P := [p1, . . . , pr+2s]. Then,

• for each λj ≥ 1, define aj := pρ+j, for j = 1, . . . , l;

• for each complex pair (νj , νj), define bj := pr+2j−1 + ipr+2j, for j = 1, . . . , s; and

2To achieve the particular ordering of λ1, . . . , λr as specified in (B.4), the first r columns of P mayneed to be rearranged in a different order.


• for the set of eigenvalues λ1, . . . , λρ, there exists w := m − l − s + 1 pairs such

that λj + λ2w+1−j ≥ 0; define cj := pj + ip2w+1−j , for j = 1, . . . , w.

Finally, let X = [a1, . . . , al, b1, . . . , bs, c1, . . . , cw]. By construction, X satisfies (B.1),

which can then be substituted into (B.2) to obtain a real minimum-norm perturbation

∆.

Appendix C

Proofs of Theorems and Lemmas

C.1 Proofs of Theorems of Chapter 2

C.1.1 Proof of Theorem 2.3.1

The proof of Theorem 2.3.1 is similar to the proof of Theorem 2.2.1 found in [4], which

is based on the following results, namely Lemma C.1.1 and Theorem C.1.1.

Lemma C.1.1 ([4]). Given matrices T1 ∈ Cp×m and T2 ∈ Cn×m, there exists a real

contraction ∆ ∈ Rn×p (i.e., ‖∆‖2 ≤ 1) such that ∆T1 = T2 if and only if

[

T2 T2

]H [

T2 T2

]

≤[

T1 T1

]H [

T1 T1

]

.

Theorem C.1.1 ([4]). Given A = AH ∈ Cn×n and B = BT ∈ C

n×n, the following

conditions are equivalent for k = 1, . . . , n:

i) There exists a complex matrix Sk of rank ≥ k such that

Sk 0

0 Sk

H

A B

B A

Sk 0

0 Sk

≥ 0

176

Appendix C. Proofs of Theorems and Lemmas 177

ii) The matrix

A αB

αB A

has at least 2k nonnegative eigenvalues for every real

|α| ≤ 1; i.e.,

inf|α|≤1

λ2k

A αB

αB A

≥ 0, for α ∈ R.

Now consider a real τ ≥ 0. From Definition 2.3.1, τ ≥ τi(M,N) if and only if

there exists a real ∆ ∈ Rn×p such that rank(M −∆N) < i and ‖∆‖2 ≤ τ . The latter

implies that there exists some complex matrix S with rank(S) ≥ m − (i − 1) such

that (M −∆N)S = 0, or equivalently, (∆/τ)NS = MS/τ . By Lemma C.1.1, this is

equivalent to

τ 2[

NS NS

]H [

NS NS

]

≥[

MS MS

]H [

MS MS

]

. (C.1)

Let Ar := τ 2NHN −MHM and Br := τ 2NTN −MTM . Then (C.1) is equivalent to

S 0

0 S

H

Ar Br

Br Ar

S 0

0 S

≥ 0. (C.2)

Therefore by Theorem C.1.1, τ ≥ τi(M,N) if and only if

Ar αBr

αBr Ar

has at least

2 (m− (i− 1)) nonnegative eigenvalues for α ∈ (−1, 0];1 i.e., for α ∈ (−1, 0],

λ2(m−(i−1))

Ar αBr

αBr Ar

≥ 0,

or equivalently,

λ2i−1

(MHM − τ 2NHN

)α(

MTM − τ 2NTN)

α(MTM − τ 2NTN

)MHM − τ 2NHN

≤ 0, (C.3)

1Due to symmetry, it is sufficient to take only α ∈ (−1, 0] instead of all values |α| ≤ 1 (e.g., see [4]).


since for Hermitian matrix H = HH ∈ Cn×n, λk(H) = −λn−k+1(−H). Now define

Tα,n :=1√2

1√1+α

Inj√1−α

In

1√1+α

In − j√1−α

In

, (C.4)

which satisfies T−Hα,n T−1

α,n =

In αIn

αIn In

. Furthermore,

T−1α,n

M 0

0 M

Tα,n = MR

√

1+α1−α

, (C.5)

where the notation used in (C.5), in particular MR

γ for γ ∈ (0, 1], is defined in (2.6).

Finally, the following equivalences are true for all τ ≥ 0:

τ ≥ τi(M,N)

⇔ λ2i−1

M 0

0 M

H

In αIn

αIn In

M 0

0 M

−τ 2

N 0

0 N

H

Ip αIp

αIp Ip

N 0

0 N

≤ 0,

for α ∈ (−1, 0] (from (C.3)),

⇔ λ2i−1

TH

α,m

M 0

0 M

H

T−Hα,n T−1

α,n

M 0

0 M

−τ 2

N 0

0 N

H

T−Hα,p T−1

α,p

N 0

0 N

Tα,m

≤ 0,

for α ∈ (−1, 0],

⇔ λ2i−1

((

MR√

1+α1−α

)H (

MR√

1+α1−α

)

− τ 2(

NR√

1+α1−α

)H (

NR√

1+α1−α

))

≤ 0, for α ∈ (−1, 0],

⇔ λ2i−1

((MR

γ

)HMR

γ ,(NR

γ

)HNR

γ

)

≤ τ 2, for γ ∈ (0, 1]

⇔ σ2i−1

(MR

γ , NR

γ

)≤ τ, for γ ∈ (0, 1]

⇔ supγ∈(0,1]

σ2i−1

(MR

γ , NR

γ

)≤ τ.


So it is concluded that (2.10) must be true.


We prove Theorem 2.4.1 by showing that for any real τ ≥ 0 and i ∈ 1, . . . ,min(n,m),

the following is true:

τi(M,L,N) ≤ τ ⇒ supγ∈(0,1]

σ2i−1

(MR

γ , LR

γ , NR

γ

)≤ τ. (C.6)

In particular, if τi(M,L,N) ≤ τ , then there exists ∆ ∈ Rl×p such that ‖∆‖2 ≤ τ and

rank(M − L∆N) < i, or equivalently,

rank

M 0

0 M

−

L 0

0 L

∆ 0

0 ∆

N 0

0 N

< 2i− 1 (C.7)

⇔ rank

MR − LR

∆ 0

0 ∆

NR

< 2i− 1.

Pre- and post-multiplying the left-hand side of (C.7) by

γI 0

0 I

and

γ−1I 0

0 I

,

respectively, for γ ∈ (0, 1], we obtain

rank

MR

γ − LR

γ

∆ 0

0 ∆

NR

γ

< 2i− 1.

Hence

∆ 0

0 ∆

is a perturbation to

(MR

γ , LRγ , N

Rγ

). So recalling the definition of re-

stricted singular values (i.e., Definition A.3.1), we see that

σ2i−1

(MR

γ , LR

γ , NR

γ

)≤

∥∥∥∥∥∥∥

∆ 0

0 ∆

∥∥∥∥∥∥∥2

= ‖∆‖2 ≤ τ

for γ ∈ (0, 1]. Hence (C.6) follows immediately.



Let the following be true for any real τ ≥ 0 and i ∈ 1, . . . ,min(n,m):

supγ∈(0,1]

σ2i−1

(MR

γ , LR

γ , NR

γ

)≤ τ.

Then, σ2i−1

(MR, LR, NR

)≤ τ , or equivalently, there exists ∆ ∈ C2l×2p such that ‖∆‖2 ≤

τ and rank(MR − LR∆NR

)< 2i− 1. By (1.13), this is equivalent to

rank

M 0

0 M

−

L 0

0 L

Tl∆Tp

N 0

0 N

< 2i− 1

where the notation Tl is defined in (1.14). Since Tl and Tp are unitary, then ‖Tl∆Tp‖2 =

‖∆‖2 ≤ τ . Therefore,

σ2i−1

M 0

0 M

,

L 0

0 L

,

N 0

0 N

≤ τ,

which implies that σi(M,L,N) ≤ τ .

C.2 Proofs of Theorems and Lemmas of Chapter 3


The proof is similar to the proof of the real controllability radius, except a little care

is required to handle the permutations P and P in (3.19). The proof consists of two

parts. In the first part, the modal real DFM radius with respect to a given s ∈ C (see

Definition 3.3.2) is computed. In particular, we have the following result.

Lemma C.2.1. Given

A B

C D

v

that has no DFMs and s ∈ C, the modal real DFM

radius with respect to s is given by

rDFM,sR

A B

C D

v

= min

P⊆vτn

A− sI BP

CP DPP

. (C.8)


Proof. Let s ∈ C. Then by Theorem 3.3.1, the perturbed system

A+∆A B +∆B

C +∆C D +∆D

v

,

for any real

∆A ∆B

∆C ∆D

, has a DFM at s if and only if there exists a partition P and

P such that

rank

A+∆A − sI BP +∆BP

CP +∆CPDPP +∆D

PP

< n, (C.9)

where ∆BP, ∆CP

, and ∆DPP

are understood to be the rows/columns of ∆B, ∆C , and ∆D

corresponding to P and P defined in a similar manner as in (3.13). Hence, rDFM,sR

can

equivalently be written as follows:

rDFM,sR

A B

C D

v

= min

P⊆vinf

∆A ∈ Rn×n,∆B ∈ R

n×m


∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣

(C.10)

rank


CP +∆CPDPP +∆D

PP

< n

.

Furthermore, since∥∥∥∥∥∥∥

∆A ∆BP

∆CP∆D

PP

∥∥∥∥∥∥∥

≤

∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥

, (C.11)

then, rDFM,sR

can be written as

rDFM,sR

A B

C D

v

= min

P⊆vinf

∆A ∈ Rn×n,∆BP∈ Rn×m

P

∆CP∈ RrP×n,∆D

PP∈ RrP×m

P

∥∥∥∥∥∥∥

∆A ∆BP

∆CP∆D

PP

∥∥∥∥∥∥∥2

∣∣∣∣∣∣∣

rank


CP +∆CPDPP +∆D

PP

< n

. (C.12)

Finally, by applying Theorem 2.2.1 to (C.12), Lemma C.2.1 immediately follows.


Then in the second part of the overall proof, the computation of the real DFM radius

is simply a minimization of the modal real DFM radius over s ∈ C.

C.2.2 Proof of Lemma 3.3.1

Given the the decentralized LTI system (3.7) with the general information flow constraint

K in (3.22), then applying the feedback gain u = Ky, the closed-loop system is

x = Ax+BKy,

which is equivalent to

x = Ax+ BKy,

where K is the diagonal information flow constraint (3.24) and

B =

B1, . . . , B1︸︷︷︸

v

, B2, . . . , B2︸︷︷︸

v

, · · · , Bv, . . . , Bv︸︷︷︸

v

y =

yT1 , · · · , yTv︸︷︷︸

v

, yT1 , · · · , yTv︸︷︷︸

v

· · ·, yT1 , · · · , yTv︸︷︷︸

v

T

.

Therefore the decentralized LTI system (3.7) with a general information flow constraint

can alternatively be represented as (3.23), where for i, j ∈ 1, 2, . . . , v2 (note: K in

(3.24) has v2 diagonals),

Bi = B⌊ i−1

v⌋+1

Ci = C((i−1)mod v)+1

Dij = D(⌊ i−1

v⌋+1)(((i−1) mod v)+1)

where ⌊·⌋ and mod denote the floor and modulo function, respectively.


Given ∆A ∈ Fn×n, ∆B ∈ Fn×m, ∆C ∈ Fr×m, and ∆D ∈ Fr×m, for F ∈ C,R, the

perturbed system (C +∆C , A+∆A, B +∆B, D +∆D) has a transmission zero at s if


and only if

rank

A+∆A − sI B +∆B

C +∆C D +∆D

= rank

A− sI B

C D

+

∆A ∆B

∆C ∆D

< n +min(m, r) .

Therefore the proof follows directly from the property of the (n + min(m, r))-th singu-

lar value (see Appendix A.1), and the (n + min(m, r))-th real perturbation value (see

Chapter 2.2) of the matrix

A− sI B

C D

.


The following proof is for rTZC

(C,A,B, 0, s), but it also applies to rTZR

(C,A,B, 0, s) by

Property (3) of Lemma 3.5.3. Given a minimum phase system (3.27) with D = 0, let

p > 0 be a given real number, and consider the following perturbed system:

x = Ax+Bu (C.13)

y = Cx+ Du,

where D = ǫC(ǫA − pI)−1B for ǫ > 0. It can be verified that the perturbed system

(C.13) has m transmission zeros at pǫ> 0 since

rank

A− pǫI B

C ǫC(ǫA− pI)−1B

= rank

In (A− pǫI)−1B

C ǫC(ǫA− pI)−1B

= rank

In ǫ(ǫA− pI)−1B

0 0

< n +min(m, r)

So it can be concluded that as ǫ → 0, an arbitrary small perturbation D gives rise to a

perturbed system that is nonminimum phase with unstable transmission zeros given by

pǫ> 0.


C.3 Proofs of Theorems of Chapter 4


Given s ∈ C and perturbation matrices ∆A and ∆B of conformal dimensions, the per-

turbed system (A+ E∆AF , B + E∆BG) is uncontrollable at s if and only if

rank

([

A + E∆AF − sI B + E∆BG])

< n

⇔ rank

E−1

[

A− sI B

]

+

[

∆A ∆B

]

F 0

0 G

< n.

Using the generalized real perturbation values results, the norm of the smallest real per-

turbations

[

∆A ∆B

]

such that the perturbed system is uncontrollable at s is given by

τn

E−1

[

A− sI B

]

,

F 0

0 G

. The structured real controllability radius rc,struct

R

is then the minimization of this function over the complex plane.


Again, given s ∈ C and matrices ∆A and ∆B of conformal dimensions, the perturbed

system (A + E∆AF , B + E∆BG) is uncontrollable at s if and only if

rank

([

A+ E∆AF − sI B + E∆BG])

< n

⇔ rank

[

A− sI B

]

+ E[

∆A ∆B

]

F 0

0 G

< n.

Therefore the norm of the smallest perturbations

[

∆A ∆B

]

such that the perturbed

system is uncontrollable at s is given by τn

[

A− sI B

]

, E ,

F 0

0 G

. The struc-

tured real controllability radius rc,structR

is then the minimization of this function over s

in the complex plane.



Using the results on generalized real perturbation values in Section 2.3, the proof of

(4.22) follows directly from the fact that for a given s ∈ C and the perturbed system

(E +∆E , A+∆A, B +∆B), then

rank

([

s (E +∆E)− (A+∆A), B +∆B

])

< n

⇔ rank

[

sE − A, B

]

−[

∆E ∆A ∆B

]

−sI 0

I 0

0 −I

< n.

In other words, a minimum-norm perturbation (∆E ,∆A,∆B) that solves Problem (1)

have a norm

∥∥∥∥

[

∆E ∆A ∆B

]∥∥∥∥2

, which is given by the n-th generalized real pertur-

bation value of

[

sE −A, B

]

and

−sI 0

I 0

0 −I

.

Similarly, the proof of (4.23) follows from the fact that

rank

([

E +∆E B +∆B

])

< n

⇔ rank

[

E B

]

−[

∆E ∆A ∆B

]

−I 0

0 0

0 −I

< n.


Given a fixed s ∈ C, then according to Theorem 4.3.1, the perturbed system (4.26) is

spectrally uncontrollable at s if and only if

rank

([

sI −A0 −∆A0−A1e

−τs −∆A1e−τs, Be−τs +∆Be

−τs

])

= (C.14)


rank

[

sI −A0 − A1e−τs, Be−τs

]

−[

∆A0∆A1

∆B

]

I 0

e−τsI 0

0 −e−τsI

< n.

Therefore the proof follows immediate by computing the n-th generalized real perturba-

tion value of

[

sI − A0 −A1e−τs, Be−τs

]

and

I 0

e−τsI 0

0 −e−τsI

, and then minimiz-

ing s ∈ C over the complex plane.


The proof is similar to that found in [84, Theorem 4]. First we recall the following result.2

Lemma C.3.1 ([27]). Let A ∈ Cn×n and B ∈ Cn×n, and define

λ(A)1 , . . . , λ

(A)n

and

λ(B)1 , . . . , λ

(A)n

as the spectra of A and B, respectively. Then

maxj

mini

∣∣∣λ

(A)i − λ

(B)j

∣∣∣ ≤ (‖A‖2 + ‖B‖2)

1− 1

n ‖A−B‖1

n

2 .

From the assumption that the modal real controllability radius with respect to s

is computed to be equal to ǫ, then there exists

∥∥∥∥

[

∆A ∆B

]∥∥∥∥2

= ǫ such that the

perturbed system (A+∆A, B +∆B) is uncontrollable at s. So if we apply any controller

gain K ∈ Rm×n with ‖K‖2 ≤M to the perturbed system, then we have

‖A+∆A + (B +∆B)K‖2 =

∥∥∥∥∥∥∥

A+BK +

[

∆A ∆B

]

I

K

∥∥∥∥∥∥∥2

≤ ‖A‖2 +M ‖B‖2 + ǫ (M + 1) .

Since (A+∆A, B +∆B) is uncontrollable at s, then s ∈ λ(A +∆A + (B +∆B)K). So

2Note that the bound used in the proof of [84, Theorem 4] is from [26] and is not the same as thebound given here in Lemma C.3.1. The bound in Lemma C.3.1 is an improved bound found in [27].


from Lemma C.3.1, there exists λ ∈ λ(A +BK) such that

∣∣∣λ− s

∣∣∣ ≤ (‖A+BK‖2 + ‖A +∆A + (B +∆B)K‖2)

1− 1

n

[

∆A ∆B

]

I

K

1

n

≤ (2 ‖A‖2 + 2 ‖B‖2M + ǫ (M + 1))1−1

n (ǫ (M + 1))1

n .


It is obvious that the tracking/disturbance pole s is also an eigenvalue of the augmented

system

[

0 D]

,

A 0

BC C

,

B

BD

, so it suffice to show that s is uncontrollable

by showing that

rank

A− sI 0 B

BC C − sI BD

< n+ rp. (C.15)

To do so, we recall the following result.

Lemma C.3.2 ([39]). Given A ∈ Cm×k and B ∈ Ck×n, then

rank(AB) ≤ min(rank(A) , rank(B)) .

So we then have

A− sI B 0

BC BD C − sI

=

In 0 0

0 B Irp

A− sI B 0

C D 0

0 0 C − sI

, (C.16)

where it can easily be shown that

rank

In 0 0

0 B Irp

= n+ rp.

Furthermore, since s is a tracking and/or disturbance pole, then rank(C − sI) ≤ p − 1,

which implies that rank(

C − sI)

≤ r(p − 1). Since s is also a transmission zero of

(C,A,B,D), then


rank

A− sI B

C D

≤ n + r − 1 and

rank

A− sI B 0

C D 0

0 0 C − sI

≤ (n+ r)− 1 + r(p− 1) = n + rp− 1.

Hence, applying Lemma C.3.2 to (C.16), (C.15) follows immediately.


The proof is similar to the proof of Theorem 4.4.1. In particular, given that the trans-

mission zero at s radius is rTZR

(C,A,B,D, s) = ǫ, then there exists

∥∥∥∥∥∥∥

∆A ∆B

∆C ∆D

∥∥∥∥∥∥∥2

= ǫ

such that the perturbed system (C +∆C , A+∆A, B +∆B, D +∆D) has a transmission

zero at s. So if we apply any controller gain

[

K0 K1

]

, with

∥∥∥∥

[

K0 K1

]∥∥∥∥2

≤M , to

the perturbed augmented system

A+∆A 0

B (C +∆C) C

,

B +∆B

B (D +∆D)

, (C.17)

we then have∥∥∥∥∥∥∥

A +∆A 0

B (C +∆C) C

+

B +∆B

B (D +∆D)

[

K0 K1

]

∥∥∥∥∥∥∥2

=

∥∥∥∥∥∥∥

A 0

BC C

+

B

BD

[

K0 K1

]

+

I 0

0 B

∆A ∆B

∆C ∆D

I 0

K0 K1

∥∥∥∥∥∥∥2

≤

∥∥∥∥∥∥∥

A 0

BC C

∥∥∥∥∥∥∥2

+M

∥∥∥∥∥∥∥

B

BD

∥∥∥∥∥∥∥2

+ ǫ (M + 1) ,

where∥∥∥B∥∥∥2= 1 (see (1.19)).


It can easily be shown that the tracking/disturbance pole s is also an eigenvalue

of the perturbed augmented system (C.17). So by Theorem 4.4.2, s is uncontrollable,

and hence s ∈ λ

A +∆A 0

B (C +∆C) C

+

B +∆B

B (D +∆D)

[

K0 K1

]

. Then from

Lemma C.3.1, there exists λ ∈ λ

A 0

BC C

+

B

BD

[

K0 K1

]

such that

∣∣∣λ− s

∣∣∣ ≤

∥∥∥∥∥∥∥

A 0

BC C

+

B

BD

[

K0 K1

]

∥∥∥∥∥∥∥2

+

∥∥∥∥∥∥∥

A+∆A 0

B (C +∆C) C

+

B +∆B

B (D +∆D)

[

K0 K1

]

∥∥∥∥∥∥∥2

1− 1

n

×

∥∥∥∥∥∥∥

I 0

0 B

∆A ∆B

∆C ∆D

I 0

K0 K1

∥∥∥∥∥∥∥2

1

n

≤

2

∥∥∥∥∥∥∥

A 0

BC C

∥∥∥∥∥∥∥2

+ 2

∥∥∥∥∥∥∥

B

BD

∥∥∥∥∥∥∥2

M + ǫ (M + 1)

1− 1

n

(ǫ (M + 1))1

n .

C.4 Proofs of Theorems and Lemmas of Chapter 5


From (A.2) in Appendix A.3, we see that for x > 0,

x ∈ σ(MR

γ , LR

γ , NR

γ

)

⇔ det

0 MR

γ

(MR

γ

)H0

− x

LR

γ

(LR

γ

)H0

0(NR

γ

)HNR

γ

= 0


⇔ det

0[M 00 M

]

[M 00 M

]H0

−x

12

[L 00 L

][(γ−2+1)I (γ−2−1)I(γ−2−1)I (γ−2+1)I

][L 00 L

]H0

0 12

[N 00 N

]H[(γ2+1)I (γ2−1)I(γ2−1)I (γ2+1)I

][N 00 N

]

= 0

⇔ det

−x[LLH LLT

LLH LLT

]0

2[M 00 M

]H −x[

NHN −NHN−NTN NTN

]

− γ2

x[

LLH −LLT

−LLH LLT

]

−2[M 00 M

]

0 x[NHN NHNNTN NTN

]

= 0

⇔ det(A(x,M,L,N)− γ2B(x,M,L,N)

),

where A(x,M,L,N) and B(x,M,L,N) are given by (5.6).


Let α = γ2−1γ2+1

. It can easily be shown (e.g. see [44]) that

MR

γ = T−1α,q

M 0

0 M

Tα,l, (C.18)

where for n ∈ N, we denote

Tα,n :=1√2

1√1+α

Inj√1−α

In

1√1+α

In − j√1−α

In

, (C.19)

which satisfies Tα,nTHα,n =

I αI

αI I

−1

. The rest of the proof then follows from

the following:

x ∈ σ(MR

γ , NR

γ

)

⇔ det

0 MR

γ

(MR

γ

)H0

− x

I 0

0(NR

γ

)HNR

γ

= 0


⇔ det

−xI2q MR

γ

(MR

γ

)H −x(NR

γ

)HNR

γ

= 0

⇔ det

−xTα,qTHα,q

M 0

0 M

MH 0

0 MT

NH 0

0 NT

− xT−H

α,l T−1α,l

N 0

0 N

= 0

⇔ det(−xTα,qT

Hα,q

)× det

NH 0

0 NT

− xT−H

α,l T−1α,l

N 0

0 N

−

MH 0

0 MT

(−xTα,qT

Hα,q

)−1

M 0

0 M

= 0

⇔ det

−x

NHN αNHN

αNTN NTN

+

1

x

MHM αMHM

αMTM MTM

= 0

⇔ det(A(x,M)− αB(x,M)) = 0,

where A(x,M) and B(x,M) are given by (5.10).


Let α ∈ C be a generalized eigenvalue of(


in (5.10), and define

J1 :=

0 −I

I 0

. Then,

det(

A(x,M,N)− αB(x,M,N))

= 0

⇔ det(

J1

(


J−11

)

= 0

⇔ det

0 MTM − x2NTN

MHM − x2NHN 0


−α

MTM − x2NTN 0

0 MHM − x2NHN

= 0

⇔ det(

A(x,M,N) + αB(x,M,N))

= 0.

Hence, −α is also a generalized eigenvalue of (A(x,M) ,B(x,M)). Similarly, define J2 :=

0 I

I 0

and note that

det(


= 0 ⇔

det(

J2

(


J2

)

= 0.

Hence, α is also a generalized eigenvalue of(


.


Let α(γ) = γ2−1γ2+1

. Since w(γ) is a generalized singular value of(MR

γ , NR

γ

), then by The-

orem 5.3.2, α(γ) is a generalized eigenvalue of(

A(w(γ) ,M,N) , B(w(γ) ,M,N))

. Let

y(γ) be a corresponding generalized eigenvector such that y(γ)T B(w(γ) ,M,N) y(γ) = 1

(such a y(γ) can always be obtained by the appropriate scaling of any eigenvector corre-

sponding to α(γ)). Then by Lemma 5.3.2,

dα(γ)

dw(γ)= y(γ)T

(

dA(w(γ) ,M,N)

dw(γ)− α(γ)

dB(w(γ) ,M,N)

dw(γ)

)

y(γ) .

Noting that

dA(w(γ) ,M,N)

dw(γ)= 2w(γ)×

0 I

I 0

and

dB(w(γ) ,M,N)

dw(γ)= −2w(γ)×

I 0

0 I

,

then the proof follows from

dw(γ)

dγ=

dα(γ)

dγ

dw(γ)

dα(γ).



The proof proceeds as follows:

x ∈ σ([

G(weiθ

)]R

γ, ER

γ , FR

γ

)

⇔ x ∈ λ

0[G(weiθ

)]R

γ([

G(weiθ

)]R

γ

)H

0

,

ER

γ

(ER

γ

)H0

0(FRγ

)HFRγ

⇔ det

−xER

γ

(ER

γ

)H [G(weiθ

)]R

γ([

G(weiθ

)]R

γ

)H

−x(FR

γ

)HFR

γ

= 0

⇔ det

−12x

[(γ−2+1)EEH (γ−2−1)EET

(γ−2−1)EEH (γ−2+1)EET

]

[A−weiθI B

C D

]0

0[A−weiθI B

C D

]

[A−weiθI B

C D

]0

0[A−weiθI B

C D

]

H

−12x

[

(γ2+1)FHF (γ2−1)FHF

(γ2−1)FTF (γ2+1)FTF

]

= 0

⇔ det(A(x, γ)− wB(x, γ)) = 0,

where A(x, γ) and B(x, γ) are defined in (5.19).

C.5 Proofs of Lemmas of Chapter 6


Denote

H(s) =

[

0 D]

sI −

A 0

BC C

−1

B

BD

.

From the diagonal structure of B, C, and D in (1.19), it can easily be shown that

H(s) = diag(H2(s) , . . . , H2(s)︸︷︷︸

r

)H1(s) , (C.20)

where H1(s) := C (sI − A)−1 + D and H2(s) := D (sI − C)−1 B. Now given that the

infinite transmission zeros of H1(s) are [1/sp1, . . . , 1/spk ], then there exists the following


Smith-McMillan factorization at infinity for (C,A,B,D) (see [23]):

H1(s) = B1(s)

Λ(s) 0

0 0

B2(s), (C.21)

where Λ(s) = diag(s−p1, . . . , s−pk), and B1(s) and B2(s) are respectively (r × r) and

(m×m) bicausal isomorphisms. Furthermore, it can easily be shown that the infinite

transmission zeros of H2(s) are[1/sN

], so there exists a factorization at infinity given by

H2(s) = B3(s)

[1

sN

]

B4(s) = B5(s)1

sN, (C.22)

where B3(s), B4(s), and B5(s) := B3(s)B4(s) are all (1 × 1) bicausal isomorphisms.

From (C.20)-(C.22), it can then be seen that H(s) has the following Smith-McMillan

factorization at infinity:

H(s) = diag(B5(s), . . . , B5(s)︸︷︷︸

r

)B1(s)

Λ(s) 0

0 0

B2(s) (C.23)

where Λ(s) = diag(s−(p1+N), . . . , s−(pk+N)

); so the infinite transmission zeros of H(s) are

given by[1/s(p1+N), . . . , 1/s(pk+N)

].

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