nonlinear control systems analysis and design - horacio j. marquez

CONTROi I

naiysis ana uesign

(4)WILEY

NONLINEARCONTROL SYSTEMS

HORACIO J. MARQUEZDepartment of Electrical and Computer Engineering,

University ofAlberta, Canada

WILEY-INTERSCIENCE

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright © 2003 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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109876543

To my wife, Goody (Christina);son, Francisco;

and daughter, Madison

Contents

1 Introduction 1

1.1 Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 First-Order Autonomous Nonlinear Systems . . . . . . . . . . . . . . . . . 5

1.5 Second-Order Systems: Phase-Plane Analysis . . . . . . . . . . . . . . . . . 8

1.6 Phase-Plane Analysis of Linear Time-Invariant Systems . . . . . . . . . . . 10

1.7 Phase-Plane Analysis of Nonlinear Systems . . . . . . . . . . . . . . . . . . 18

1.7.1 Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.8 Higher-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.8.1 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.9 Examples of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.9.1 Magnetic Suspension System . . . . . . . . . . . . . . . . . . . . . . 23

1.9.2 Inverted Pendulum on a Cart . . . . . . . . . . . . . . . . . . . . . . 25

1.9.3 The Ball-and-Beam System . . . . . . . . . . . . . . . . . . . . . . . 26

1.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Mathematical Preliminaries 31

2.1 Sets ........................................ 31

2.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

vii

viii CONTENTS

2.3 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 Linear Independence and Basis . . . . . . . . . . . . . . . . . . . . . 34

2.3.2 Subspaces .................... . . . . ......... 36

2.3.3 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4.1 Eigenvalues, Eigenvectors, and Diagonal Forms . . . . . . . . . . . . 40

2.4.2 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 Basic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5.1 Basic Topology in ]It" . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.6 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.7 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.7.1 Bounded Linear Operators and Matrix Norms . . . . . . . . . . . . . 48

2.8 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.8.1 Some Useful Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.9 Lipschitz Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.10 Contraction Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.11 Solution of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 56

2.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3 Lyapunov Stability I: Autonomous Systems 65

3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2 Positive Definite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.3 Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.5 Asymptotic Stability in the Large . . . . . . . . . . . . . . . . . . . . . . . . 77

3.6 Positive Definite Functions Revisited . . . . . . . . . . . . . . . . . . . . . . 80

3.6.1 Exponential Stability ................ . . ......... 82

3.7 Construction of Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . 82

CONTENTS ix

3.8 The Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.9 Region of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.10 Analysis of Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . 96

3.10.1 Linearization of Nonlinear Systems . . . . . . . . . . . . . . . . . . . 99

3.11 Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4 Lyapunov Stability II: Nonautonomous Systems 107

4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2 Positive Definite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.3 Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.4 Proof of the Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.5 Analysis of Linear Time-Varying Systems . . . . . . . . . . . . . . . . . . . 119

4.5.1 The Linearization Principle . . . . . . . . . . . . . . . . . . . . . . . 120

4.6 Perturbation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.7 Converse Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.8 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.9 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.10 Stability of Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . 130

4.10.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.10.2 Discrete-Time Positive Definite Functions . . . . . . . . . . . . . . . 131

4.10.3 Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5 Feedback Systems 137

5.1 Basic Feedback Stabilization . . . . . . . . . . . ....... . .... .... 138

5.2 Integrator Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.3 Backstepping: More General Cases . . . . . . . . . . . . . . . . . . . . . . . 145

x CONTENTS

5.3.1 Chain of Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.3.2 Strict Feedback Systems . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6 Input-Output Stability 155

6.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.1.1 Extended Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.2 Input-Output Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.3 Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.4 Lp Gains for LTI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.4.1 L. Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.4.2 G2 Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.5 Closed-Loop Input-Output Stability . . . . . . . . . . . . . . . . . . . . . . 168

6.6 The Small Gain Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.7 Loop Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.8 The Circle Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7 Input-to-State Stability 183

7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

7.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.3 Input-to-State Stability (ISS) Theorems . . . . . . . . . . . . . . . . . . . . 186

7.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.4 Input-to-State Stability Revisited . . . . . . . . . . . . . . . . . . . . . . . . 191

7.5 Cascade-Connected Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

8 Passivity 201

CONTENTS xi

8.1 Power and Energy: Passive Systems ...................... 2018.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

8.3 Interconnections of Passivity Systems . . . . . . . . . . . . . . . . . . . . . 208

8.3.1 Passivity and Small Gain . . . . . . . . . . . .. . ...... . ... 210

8.4 Stability of Feedback Interconnections . . . . . . . . . . . . . . . . . . . . . 211

8.5 Passivity of Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . 214

8.6 Strictly Positive Real Rational Functions . . . . . . . . . . . . . . . . . . . . 217

8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

9 Dissipativity 223

9.1 Dissipative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

9.2 Differentiable Storage Functions . . . . . . . . . . . . . . . . . . . . . . . . . 225

9.2.1 Back to Input-to-State Stability . . .. . . . .... . .. .. .... 226

9.3 QSR Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

9.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

9.4.1 Mass-Spring System with Friction . . . . . . . . . . . . . . . . . . . 229

9.4.2 Mass-Spring System without Friction . . . . . . . . . . . . . . . . . 231

9.5 Available Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

9.6 Algebraic Condition for Dissipativity . . . . . . . . . . . . . . . . . . . . . . 233

9.6.1 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

9.7 Stability of Dissipative Systems . . . . . . . . . . . . . . . . . . . . . . . . . 237

9.8 Feedback Interconnections ................. . . . . . . . . . . . 239

9.9 Nonlinear L2 Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

9.9.1 Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . 245

9.9.2 Strictly Output Passive Systems . . . . . . . . . . . . . . . . . . . . 246

9.10 Some Remarks about Control Design . . . . . . . . . . . . . . . . . . . . . . 247

9.11 Nonlinear L2-Gain Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

9.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

xii CONTENTS

10 Feedback Linearization 255

10.1 Mathematical Tools ... . . . .. . . . . . . . .. . . . . ......... . 255

10.1.1 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

10.1.2 Lie Bracket . . .. ........................ . . . . 257

10.1.3 Diffeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

10.1.4 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . 259

10.1.5 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

10.2 Input-State Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

10.2.1 Systems of the Form x = Ax + Bw(x) [u - O(x)] . . . . . . . . . . . . 265

10.2.2 Systems of the Form i = f (x) + g(x)u . . . . . . . . . . . . . . . . . 267

10.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

10.4 Conditions for Input-State Linearization ......... . . . ........ 27310.5 Input-Output Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

10.6 The Zero Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

10.7 Conditions for Input-Output Linearization . . . . . . . . . . . . . . . . . . 287

10.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

11 Nonlinear Observers 291

11.1 Observers for Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . 291

11.1.1 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

11.1.2 Observer Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

11.1.3 Observers for Linear Time-Invariant Systems . . . . . . . . . . . . . 294

11.1.4 Separation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

11.2 Nonlinear Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

11.2.1 Nonlinear Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

11.3 Observers with Linear Error Dynamics . . . . . . . . . . . . . . . . . . 298

11.4 Lipschitz Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

11.5 Nonlinear Separation Principle . . . . . . . . . . . . . . . . . . . . . . . . . 303

CONTENTS xiii

A Proofs 307

A.1 Chapter 3 .. .. . . . . . . . . . .. ...... . .. . . . . . . . . . . .. . 307

A.2 Chapter 4 .. . . ....... .. . . . . . . . . . . . . . .. .. ...... . 313

A.3 Chapter 6 . . .. . . . .. . . . ......... ....... . . . . . . ... 315

A.4 Chapter 7 . . ........ . . . . . . . . . . . . . . . . . . . . . . . .. . . 320

A.5 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

A.6 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

A.7 Chapter 10 . . . . ........ . . . . . . .. . . . . . ... . .. ..... 330

Bibliography

List of Figures

Index

337

345

349

Preface

I began writing this textbook several years ago. At that time my intention was to write aresearch monograph with focus on the input-output theory of systems and its connectionwith robust control, including a thorough discussion of passivity and dissipativity of systems.In the middle of that venture I began teaching a first-year graduate-level course in nonlinearcontrol, and my interests quickly shifted into writing something more useful to my students.The result of this effort is the present book, which doesn't even resemble the original plan.I have tried to write the kind of textbook that I would have enjoyed myself as a student.My goal was to write something that is thorough, yet readable.

The first chapter discusses linear and nonlinear systems and introduces phase planeanalysis. Chapter 2 introduces the notation used throughout the book and briefly summarizesthe basic mathematical notions needed to understand the rest of the book. This materialis intended as a reference source and not a full coverage of these topics. Chapters 3and 4 contain the essentials of the Lyapunov stability theory. Autonomous systems arediscussed in Chapter 3 and nonautonomous systems in Chapter 4. I have chosen thisseparation because I am convinced that the subject is better understood by developing themain ideas and theorems for the simpler case of autonomous systems, leaving the moresubtle technicalities for later. Chapter 5 briefly discusses feedback stabilization basedon backstepping. I find that introducing this technique right after the main stabilityconcepts greatly increases students' interest in the subject. Chapter 6 considers input-output systems. The chapter begins with the basic notions of extended spaces, causality, andsystem gains and introduces the concept of input-output stability. The same chapter alsodiscusses the stability of feedback interconnections via the celebrated small gain theorem.The approach in this chapter is classical; input-output systems are considered withoutassuming the existence of an internal (i.e. state space) description. As such, Chapters 3-5and 6 present two complementary views of the notion of stability: Lyapunov, where thefocus is on the stability of equilibrium points of unforced systems (i.e. without externalexcitations); and the input-output theory, where systems are assumed to be relaxed (i.e.with zero initial conditions) and subject to an external input. Chapter 7 focuses on theimportant concept of input-to-state stability and thus starts to bridge across the twoalternative views of stability. In Chapters 8 and 9 we pursue a rather complete discussionof dissipative systems, an active area of research, including its importance in the so-called nonlinear L2 gain control problem. Passive systems are studied first in Chapter

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8, along with some of the most important results that derive from this concept. Chapter 9generalizes these ideas and introduces the notion of dissipative system. I have chosen thispresentation for historical reasons and also because it makes the presentation easier andenhances the student's understanding of the subject. Finally, Chapters 10 and 11 providea brief introduction to feedback linearization and nonlinear observers, respectively.

Although some aspects of control design are covered in Chapters 5, 9, and 10, theemphasis of the book is on analysis and covers the fundamentals of the theory of nonlinearcontrol. I have restrained myself from falling into the temptation of writing an encyclopediaof everything ever written on nonlinear control, and focused on those parts of the theorythat seem more fundamental. In fact, I would argue that most of the material in this bookis essential enough that it should be taught to every graduate student majoring in controlsystems.

There are many examples scattered throughout the book. Most of them are not meantto be real-life applications, but have been designed to be pedagogical. My philosophy isthat real physical examples tend to be complex, require elaboration, and often distract thereader's attention from the main point of the book, which is the explanation of a particulartechnique or a discussion of its limitations.

I have tried my best to clean up all the typographical errors as well as the moreembarrassing mistakes that I found in my early writing. However, like many before me, andthe many that will come after, I am sure that I have failed! I would very much appreciateto hear of any error found by the readers. Please email your comments to

[email protected]

I will keep an up-to-date errata list on my website:

http://www.ee.ualberta.ca/-marquez

Like most authors, I owe much to many people who directly or indirectly had aninfluence in the writing of this textbook. I will not provide a list because I do not want toforget anyone, but I would like to acknowledge four people to whom I feel specially indebted:Panajotis Agathoklis (University of Victoria), Chris Damaren (University of Toronto), ChrisDiduch, and Rajamani Doraiswami (both of the University of New Brunswick). Each oneof them had a profound impact in my career, and without their example this book wouldhave never been written. I would also like to thank the many researchers in the field, mostof whom I never had the pleasure to meet in person, for the beautiful things that theyhave published. It was through their writings that I became interested in the subject. Ihave not attempted to list every article by every author who has made a contribution tononlinear control, simply because this would be impossible. I have tried to acknowledgethose references that have drawn my attention during the preparation of my lectures andlater during the several stages of the writing of this book. I sincerely apologize to every

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author who may feel that his or her work has not been properly acknowledged here andencourage them to write to me.

I am deeply grateful to the University of Alberta for providing me with an excellentworking environment and to the Natural Sciences and Engineering Research Council ofCanada (NSERC) for supporting my research. I am also thankful to John Wiley and Son'srepresentatives: John Telecki, Kristin Cooke Fasano, Kirsten Rohstedt and Brendan Cody,for their professionalism and assistance.

I would like to thank my wife Goody for her encouragement during the writing of thisbook, as well as my son Francisco and my daughter Madison. To all three of them I owemany hours of quality time. Guess what guys? It's over (until the next project). TonightI'll be home early.

Horacio J. Marquez

Edmonton, Alberta

Chapter 1

Introduction

This first chapter serves as an introduction to the rest of the book. We present severalsimple examples of dynamical systems, showing the evolution from linear time-invariant tononlinear. We also define the several classes of systems to be considered throughout the restof the book. Phase-plane analysis is used to show some of the elements that characterizenonlinear behavior.

1.1 Linear Time-Invariant Systems

In this book we are interested in nonlinear dynamical systems. The reader is assumed tobe familiar with the basic concepts of state space analysis for linear time-invariant (LTI)systems. We recall that a state space realization of a finite-dimensional LTI system has thefollowing form

x = Ax + Bu (1.1)

y=Cx+Du (1.2)

where A, B, C, and D are (real) constant matrices of appropriate dimensions. Equation(1.1) determines the dynamics of the response. Equation (1.2) is often called the read outequation and gives the desired output as a linear combination of the states.

Example 1.1 Consider the mass-spring system shown in the Figure 1.1. Using Newton'ssecond law, we obtain the equilibrium equation

1

2 CHAPTER 1. INTRODUCTION

my = E forces= f(t) - fk - fp

where y is the displacement from the reference position, fp is the viscous friction force, andfk represents the restoring force of the spring. Assuming linear properties, we have thatfp = Qy, and fk = ky. Thus,

my+I3'+ky=mg.Defining states x1 = y, x2 = y, we obtain the following state space realization

Figure 1.1: mass-spring system.

i1 = x2

1 x2=-mkxl-mx2+gor

i1

X2 I1

If our interest is in the displacement y, then

y=x1 = [1 0

X1

X2

thus, a state space realization for the mass-spring systems is given by

i = Ax+Buy = Cx+Du

1.2. NONLINEAR SYSTEMS 3

with

A=[-o -13 ] B= [0] C=[1 0] D=[0]

1.2 Nonlinear Systems

Most of the book focuses on nonlinear systems that can be modeled by a finite number offirst-order ordinary differential equations:

±1 = fl(xl,...,xn,t,u1,...,Up)

xn = fn(xl, ... , xn, t, ul, ... , up)

Defining vectors

x=ul fl (x, t, u)

f (t, x)u) _

UP fn(x, t, u)

we can re write equation (1.3) as follows:

x = f (x, t, u). (1.4)

Equation (1.4) is a generalization of equation (1.1) to nonlinear systems. The vector x iscalled the state vector of the system, and the function u is the input. Similarly, the systemoutput is obtained via the so-called read out equation

y = h(x, t, u). (1.5)

Equations (1.4) and (1.5) are referred to as the state space realization of the nonlinearsystem.

Special Cases:

An important special case of equation (1.4) is when the input u is identically zero. Inthis case, the equation takes the form

x = f (x, t, 0) = f (x, t). (1.6)


This equation is referred to as the unforced state equation. Notice that, in general,there is no difference between the unforced system with u = 0 or any other givenfunction u(x,t) (i.e., u is not an arbitrary variable). Substituting u = 'y(x, t) inequation (1.4) eliminates u and yields the unforced state equation.

The second special case occurs when f (x, t) is not a function of time. In this case wecan write

x = f (x) (1.7)

in which case the system is said to be autonomous. Autonomous systems are invariantto shifts in the time origin in the sense that changing the time variable from t toT = t - a does not change the right-hand side of the state equation.

Throughout the rest of this chapter we will restrict our attention to autonomous systems.

Example 1.2 Consider again the mass spring system of Figure 1.1. According to Newton'slaw

my = E forces= f(t) - fk - fO-

In Example 1.1 we assumed linear properties for the spring. We now consider the morerealistic case of a hardening spring in which the force strengthens as y increases. We canapproximate this model by taking

fk = ky(1 + a2y2).

With this constant, the differential equation results in the following:

my+Qy+ky+ka2y3 = f(t).

Defining state variables xl = y, x2 = y results in the following state space realization

r xl = X2(l

x2 = -kM - La2X3- Ax2 +(e

t

which is of the form i = f (x, u). In particular, if u = 0, then

J it = x2ll x2 = -m x1 - ma2xi

'M m X2

or i = f(x).

1.3. EQUILIBRIUM POINTS 5

1.3 Equilibrium Points

An important concept when dealing with the state equation is that of equilibrium point.

Definition 1.1 A point x = xe in the state space is said to be an equilibrium point of theautonomous system

x = f(x)

if it has the property that whenever the state of the system starts at xe1 it remains at xe forall future time.

According to this definition, the equilibrium points of (1.6) are the real roots of the equationf (xe) = 0. This is clear from equation (1.6). Indeed, if

dxx

_dt

= f(xe) = 0

it follows that xe is constant and, by definition, it is an equilibrium point. Equilibriumpoint for unforced nonautonomous systems can be defined similarly, although the timedependence brings some subtleties into this concept. See Chapter 4 for further details.

Example 1.3 Consider the following first-order system

± = r + x 2

where r is a parameter. To find the equilibrium points of this system, we solve the equationr + x2 = 0 and immediately obtain that

(i) If r < 0, the system has two equilibrium points, namely x = ff.

(ii) If r = 0, both of the equilibrium points in (i) collapse into one and the same, and theunique equilibrium point is x = 0.

(iii) Finally, if r > 0, then the system has no equilibrium points.

1.4 First-Order Autonomous Nonlinear Systems

It is often important and illustrative to compare linear and nonlinear systems. It willbecome apparent that the differences between linear and nonlinear behavior accentuate asthe order of the state space realization increases. In this section we consider the simplest

case, which is that of first order (linear and nonlinear) autonomous systems. In other words,we consider a system of the form

x = f (x) (1.8)

where x(t) is a real-valued function of time. We also assume that f () is a continuousfunction of x. A very special case of (1.8) is that of a first-order linear system. In this case,f (x) = ax, and (1.8) takes the form

x = ax. (1.9)

It is immediately evident from (1.9) that the only equilibrium point of the first-order linearsystem is the origin x = 0. The simplicity associated with the linear case originates in thesimple form of the differential equation (1.9). Indeed, the solution of this equation with anarbitrary initial condition xo # 0 is given by

x(t) = eatxo (1.10)

A solution of the differential equation (1.8) or (1.9) starting at xo is called a trajectory.According to (1.10), the trajectories of a first order linear system behave in one of twopossible ways:

Case (1), a < 0: Starting at x0, x(t) exponentially converges to the origin.

Case (2), a > 0: Starting at x0, x(t) diverges to infinity as t tends to infinity.

Thus, the equilibrium point of a first-order linear system can be either attractive or repelling.Attractive equilibrium points are called stables, while repellers are called unstable.

Consider now the nonlinear system (1.8). Our analysis in the linear case was guidedby the luxury of knowing the solution of the differential equation (1.9). Unfortunately,most nonlinear equations cannot be solved analytically2. Short of a solution, we look fora qualitative understanding of the behavior of the trajectories. One way to do this is toacknowledge the fact that the differential equation

x = f(x)

represents a vector field on the line; that is, at each x, f (x) dictates the "velocity vector"x which determines how "fast" x is changing. Thus, representing x = f (x) in a two-dimensional plane with axis x and x, the sign of ± indicates the direction of the motion ofthe trajectory x(t).

'See Chapter 3 for a more precise definition of the several notions of stability.2This point is discussed in some detail in Chapter 2.

1.4. FIRST-ORDER AUTONOMOUS NONLINEAR SYSTEMS

2 / \ 5ir/2 x

Figure 1.2: The system i = cos x.

Example 1.4 Consider the systemth = cosx

7

To analyze the trajectories of this system, we plot i versus x as shown in Figure 1.2. Fromthe figure we notice the following:

The points where ± = 0, that is where cos x intersects the real axis, are the equilibriumpoints o f (1.11). Thus, all points o f the form x = ( 1 + 2k)7r/2, k = 0, +1, ±2, areequilibrium points of (1.11).

Whenever ± > 0, the trajectories move to the right hand side, and vice versa. Thearrows on the horizontal axis indicate the direction of the motion.

From this analysis we conclude the following:

1. The system (1.11) has an infinite number of equilibrium points.

2. Exactly half of these equilibrium points are attractive or stable, and the other half areunstable, or repellers.

The behavior described in example 1.4 is typical of first-order autonomous nonlinear sys-tems. Indeed, a bit of thinking will reveal that the dynamics of these systems is dominatedby the equilibrium points, in the sense that the only events that can occur to the trajec-tories is that either (1) they approach an equilibrium point or (2) they diverge to infinity.In all cases, trajectories are forced to either converge or diverge from an equilibrium pointmonotonically. To see this, notice that ± can be either positive or negative, but it cannotchange sign without passing through an equilibrium point. Thus, oscillations around anequilibrium point can never exist in first order systems. Recall from (1.10) that a similarbehavior was found in the case of linear first-order systems.

x

Figure 1.3: The system ± = r + x2, r < 0.

Example 1.5 Consider again the system of example 1.3, that is

x=r+x2Plotting ± versus x under the assumption that r < 0, we obtain the diagram shown in Figure1.3. The two equilibrium points, namely xe, and xe1 are shown in the Figure. From theanalysis of the sign of ±, we see that xe is attractive, while xe is a repeller. Any trajectorystarting in the interval x0 = (-oo, -r) U (-r, r) monotonically converge to xe. Trajectoriesinitiating in the interval xo = (r, oo), on the other hand, diverge toward infinity.

1.5 Second-Order Systems: Phase-Plane Analysis

In this section we consider second-order systems. This class of systems is useful in the studyof nonlinear systems because they are easy to understand and unlike first-order systems, theycan be used to explain interesting features encountered in the nonlinear world. Consider asecond-order autonomous system of the form:

i1 = fl(xl,x2) (1.12)

-t2 = f2(xl,x2) (1.13)

or

i = f(x). (1.14)

Throughout this section we assume that the differential equation (1.14) with initial conditionx(0) = xo = [x10,x20 ]T has a unique solution of the form x(t) = [x1(t),x2(t)]T. As in the

1.5. SECOND-ORDER SYSTEMS: PHASE-PLANE ANALYSIS 9

case of first-order systems, this solution is called a trajectory from x0 and can be representedgraphically in the xl - x2 plane. Very often, when dealing with second-order systems, it isuseful to visualize the trajectories corresponding to various initial conditions in the xl - x2plane. The technique is known as phase-plane analysis, and the xl - x2 plane is usuallyreferred to as the phase-plane.

From equation (1.14), we have that

±2

111 (X)

= f().f2 (X)

The function f (x) is called a vector field on the state plane. This means that to each pointx' in the plane we can assign a vector with the amplitude and direction of f (x'). Foreasy visualization we can represent f (x) as a vector based at x; that is, we assign to x thedirected line segment from x to x + f (x). Repeating this operation at every point in theplane, we obtain a vector field diagram. Notice that if

x(t) = xl(t)X2(t)

is a solution of the differential equation i = f (t) starting at a certain initial state x0i theni = f (x) represents the tangent vector to the curve. Thus it is possible to construct thetrajectory starting at an arbitrary point x0 from the vector field diagram.

Example 1.6 Consider the second-order system

1 = x222 = -xi - x2.

Figure 1.4 shows a phase-plane diagram of trajectories of this system, along with the vectorfield diagram. Given any initial condition x0 on the plane, from the phase diagram it is easyto sketch the trajectories from x0. In this book we do not emphasize the manual constructionof these diagrams. Several computer packages can be used for this purpose. This plot, aswell as many similar ones presented throughout this book, was obtained using MAPLE 7.


N \\'l' `.\ \ . 1 ;

Figure 1.4: Vector field diagram form the system of Example 1.6.

1.6 Phase-Plane Analysis of Linear Time-Invariant Systems

Now consider a linear time-invariant system of the form

i=Ax, AEII22x2 (1.15)

where the symbol 1R2x2 indicates the set of 2 x 2 matrices with real entries. These systemsare well understood, and the solution of this differential equation starting at t = 0 with aninitial condition xo has the following well-established form:

x(t) = eAtxo

We are interested in a qualitative understanding of the form of the trajectories. To this end,we consider several cases, depending on the properties of the matrix A. Throughout thissection we denote by Al, A2 the eigenvalues of the matrix A, and vl, v2, the correspondingeigenvectors.

CASE 1: Diagonalizable Systems

Consider the system (1.15). Assume that the eigenvalues of the matrix A are real, anddefine the following coordinate transformation:

x = Ty, T E IIt2x2, T nonsingular. (1.16)

1.6. PHASE-PLANE ANALYSIS OF LINEAR TIME-INVARIANT SYSTEMS 11

Given that T is nonsingular, its inverse, denoted T-1 exists, and we can write

y = T-1ATy = Dy (1.17)

Transformation of the form D = T-'AT are very well known in linear algebra, and thematrices A and D share several interesting properties:

Property 1: The matrices A and D share the same eigenvalues Al and A2. For this reasonthe matrices A and D are said to be similar, and transformations of the form T-1AT arecalled similarity transformations.

Property 2: Assume that the eigenvectors v1i v2 associated with the real eigenvalues A1, A2are linearly independent. In this case the matrix T defined in (1.17) can be formed by placingthe eigenvectors vl and v2 as its columns. In this case we have that

DAl0]

0 A2

that is, in this case the matrix A is similar to the diagonal matrix D. The importance ofthis transformation is that in the new coordinates y = [y1 y2]T the system is uncoupled,i.e.,

[y20 '\2J [Y2or

yl = Alyl (1.18)

y2 = A2y2 (1.19)

Both equations can be solved independently, and the general solution is given by (1.10).This means that the trajectories along each of the coordinate axes yl and Y2 are independentof one another.

Several interesting cases can be distinguished, depending the sign of the eigenvaluesAl and A2. The following examples clarify this point.

Example 1.7 Consider the system

[± 12]-[ 0 -2] [x2 I .

The eigenvalues in this case are Al = -1,A2 = -2. The equilibrium point of a system whereboth eigenvalues have the same sign is called a node. A is diagonalizable and D = T-1ATwith

T=[0 1 -3- 1

0

1], D-[ 0 -2].


Figure 1.5: System trajectories of Example 1.7: (a) original system, (b) uncoupled system.

In the new coordinates y = Tx, the modified system is

yi = -Yiy2 = -2Y2

or y = Dy, which is uncoupled. Figure 1.5 shows the trajectories of both the original system[part (a)] and the uncoupled system after the coordinate transformation [part (b)]. It isclear from part (b) that the origin is attractive in both directions, as expected given that botheigenvalues are negative. The equilibrium point is thus said to be a stable node. Part (a)retains this property, only with a distortion of the coordinate axis. It is in fact worth nothingthat Figure 1.5 (a) can be obtained from Figure 1.5 (b) by applying the linear transformationof coordinates x = Ty.


[ 22 j = [ 0 2 ] [ x2 ] .

The eigenvalues in this case are Al = 1, A2 = 2. Applying the linear coordinate transforma-tion x = Ty, we obtain

y1 = yiy2 = 2Y2

Figure 1.6 shows the trajectories of both the original and the uncoupled systems after thecoordinate transformation. It is clear from the figures that the origin is repelling on both


Figure 1.6: System trajectories of Example 1.8 (a) uncoupled system, (b) original system

directions, as expected given that both eigenvalues are positive. The equilibrium point in thiscase is said to be an unstable node.

Example 1.9 Finally, consider the system.

[ x2 ] - [ 0 2 ] [ x2 J.

The eigenvalues in this case are Al = -1, 1\2 = 2. Applying the linear coordinate transfor-mation x = Ty, we obtain

yl = -yiy2 = 2112

Figure 1.7 shows the trajectories of both the original and the uncoupled systems after thecoordinate transformation. Given the different sign of the eigenvalues, the equilibrium pointis attractive in one direction but repelling in the other. The equilibrium point in this caseis said to be a saddle.

CASE 2: Nondiagonalizable Systems

Assume that the eigenvalues of the matrix A are real and identical (i.e., Al = A2 = A). Inthis case it may or may not be possible to associate two linearly independent eigenvaluesvl and v2 with the sole eigenvalue A. If this is possible the matrix A is diagonalizableand the trajectories can be analyzed by the previous method. If, on the other hand, onlyone linearly independent eigenvector v can be associated with A, then the matrix A is not

Figure 1.7: System trajectories of Example 1.9 (a) uncoupled system, (b) original system

diagonalizable. In this case, there always exist a similarity transformation P such that

P-1AP = J = Al1

0 A2L ] .

The matrix J is in the so-called Jordan canonical form. In this case the transformed systemis

or

y = P-'APy

yl = Ayl + y2y2 = 42

the solution of this system of equations with initial condition yo = [ylo, y20]T is as follows

yl = yloeAt + y2oteat

y2 = y2oeAc

The shape of the solution is a somewhat distorted form of those encountered for diagonal-izable systems. The equilibrium point is called a stable node if A < 0 and unstable node ifA>0.


0 -2][x2The eigenvalues in this case are Al = A2 = A = -2 and the matrix A is not diagonalizable.Figure 1.8 shows the trajectories of the system. In this example, the eigenvalue A < 0 andthus the equilibrium point [0, 0] is a stable node.


Figure 1.8: System trajectories for the system of Example 1.10.

CASE 3: Systems with Complex Conjugate Eigenvalues

The most interesting case occurs when the eigenvalues of the matrix A are complex conju-gate, .\1,2 = a ±13. It can be shown that in this case a similarity transformation M can befound that renders the following similar matrix:

M-1AM=Q= Q

/3 a

Thus the transform system has the form

y1 = aY1 -)3Y2 (1.20)

y2 = i3y1 + aye (1.21)

The solution of this system of differential equations can be greatly simplified by introducingpolar coordinates:

P - yi + y2

0 = tan-1(1).

yl

Converting (1.20) and (1.21) to polar coordinates, we obtain

which has the following solution:

p = Poe" (1.22)

0 = 0o + Qt. (1.23)

Figure 1.9: Trajectories for the system of Example 1.11

From here we conclude that

In the polar coordinate system p either increases exponentially, decreases exponentially,or stays constant depending whether the real part a of the eigenvalues A1,2 is positive,negative, or zero.

The phase angle increases linearly with a "velocity" that depends on the imaginarypart /3 of the eigenvalues \1,2-

In the yl - y2 coordinate system (1.22)-(1.23) represent an exponential spiral. Ifa > 0, the trajectories diverge from the origin as t increases. If a < 0 on the otherhand, the trajectories converge toward the origin. The equilibrium [0, 0] in this caseis said to be a stable focus (if a < 0) or unstable focus (if a > 0).

If a = 0 the trajectories are closed ellipses. In this case the equilibrium [0, 0] is saidto be a center.

Example 1.11 Consider the following system:

1

0 -0 1[ X2

(1.24)

The eigenvalues of the A matrix are A1,2 = 0 f 3f . Figure 1.9 shows that the trajectoriesin this case are closed ellipses. This means that the dynamical system (1.24) is oscillatory.The amplitude of the oscillations is determined by the initial conditions.


Figure 1.10: Trajectories for the system of Example 1.12

±2l_r0.10.5j X2-1 X,

The eigenvalues of the A matrix are

A1,2

= 0.5 ± j; thus the origin is an unstable focus.Figure 1.10 shows the spiral behavior of the trajectories. The system in this case is alsooscillatory, but the amplitude of the oscillations grow exponentially with time, because ofthe presence of the nonzero o: term.

The following table summarizes the different cases:

Eigenvalues Equilibrium pointA1, A2 real and negative stable nodeA1, A2 real and positive unstable nodeA1, A2 real, opposite signs saddleA1, A2 complex with negative real part stable focusA1, A2 complex with positive real part unstable focusA1, A2 imaginary center

As a final remark, we notice that the study of the trajectories of linear systems aboutthe origin is important because, as we will see, in a neighborhood of an equilibrium pointthe behavior of a nonlinear system can often be determined by linearizing the nonlinearequations and studying the trajectories of the resulting linear system.


1.7 Phase-Plane Analysis of Nonlinear Systems

We mentioned earlier that nonlinear systems are more complex than their linear coun-terparts, and that their differences accentuate as the order of the state space realizationincreases. The question then is: What features characterize second-order nonlinear equa-tions not already seen the linear case? The answer is: oscillations! We will say that a systemoscillates when it has a nontrivial periodic solution, that is, a non-stationary trajectory forwhich there exists T > 0 such that

x(t +T) = x(t) dt > 0.

Oscillations are indeed a very important phenomenon in dynamical systems.

In the previous section we saw that if the eigenvalues of a second-order linear time-invariant(LTI) system are imaginary, the equilibrium point is a center and the response is oscilla-tory. In practice however, LTI systems do not constitute oscillators of any practical use.The reason is twofold: (1) as noticed in Example 1.11, the amplitude of the oscillationsis determined by the initial conditions, and (2) the very existence and maintenance of theoscillations depend on the existence of purely imaginary eigenvalues of the A matrix in thestate space realization of the dynamical equations. If the real part of the eigenvalues isnot identically zero, then the trajectories are not periodic. The oscillations will either bedamped out and eventually dissappear, or the solutions will grow unbounded. This meansthat the oscillations in linear systems are not structurally stable. Small friction forces orneglected viscous forces often introduce damping that, however small, adds a negative com-ponent to the eigenvalues and consequently damps the oscillations. Nonlinear systems, onthe other hand, can have self-excited oscillations, known as limit cycles.

1.7.1 Limit Cycles

Consider the following system, commonly known as the Van der Pol oscillator.

y-µ(1-y2)y+y=0defining state variables xl = y, and x2 = y we obtain

IL > 0

xl = X2 (1.25)

x2 = xl + µ(1 - xi)x2. (1.26)

Notice that if µ = 0 in equation (1.26), then the resulting system is

[22] _[-10]

[x2

1.7. PHASE-PLANE ANALYSIS OF NONLINEAR SYSTEMS 19

Figure 1.11: Stable limit cycle: (a) vector field diagram; (b) the closed orbit.

which is linear time-invariant. Moreover, the eigenvalues of the A matrix are A1,2 = ±3,which implies that the equilibrium point [0, 0] is a center. The term µ(1- x1)x2 in equation(1.26) provides additional dynamics that, as we will see, contribute to maintain the oscilla-tions. Figure 1.11(a) shows the vector field diagram for the system (1.25)-(1.26) assumingµ = 1. Notice the difference between the Van der Pol oscillator of this example and thecenter of Example 1.11. In Example 1.11 there is a continuum of closed orbits. A trajectoryinitiating at an initial condition xo at t = 0 is confined to the trajectory passing through x0for all future time. In the Van der Pol oscillator of this example there is only one isolatedorbit. All trajectories converge to this trajectory as t -> oo. An isolated orbit such as thisis called a limit cycle. Figure 1.11(b) shows a clearer picture of the limit cycle.

We point out that the Van der Pol oscillator discussed here is not a theoreticalexample. These equations derive from simple electric circuits encountered in the first ra-dios. Figure 1.12 shows a schematic of such a circuit, where R in represents a nonlinearresistance. See Reference [84] for a detailed analysis of the circuit.

As mentioned, the Van der Pol oscillator of this example has the property that alltrajectories converge toward the limit cycle. An orbit with this property is said to be astable limit cycle. There are three types of limit cycles, depending on the behavior of thetrajectories in the vicinity of the orbit: (1) stable, (2) unstable, and (3) semi stable. Alimit cycle is said to be unstable if all trajectories in the vicinity of the orbit diverge fromit as t -> oo. It is said to be semi stable if the trajectories either inside or outside the orbitconverge to it and diverge on the other side. An example of an unstable limit cycle can beobtained by modifying the previous example as follows:

±1 = -x2±2 = xl - µ(1 - xi)x2.


C T

Figure 1.12: Nonlinear RLC-circuit.

Figure 1.13: Unstable limit cycle.

Figure 1.13 shows the vector field diagram of this system with µ = 1. As can be seen in thefigure, all trajectories diverge from the orbit and the limit cycle is unstable.

1.8 Higher-Order Systems

When the order of the state space realization is greater than or equal to 3, nothingsignificantly different happens with linear time-invariant systems. The solution of the stateequation with initial condition x0 is still x = eATxo, and the eigenvalues of the A matrixstill control the behavior of the trajectories. Nonlinear equations have much more room inwhich to maneuver. When the dimension of the state space realization increases from 2 to3 a new phenomenon is encountered; namely, chaos.

1.8. HIGHER-ORDER SYSTEMS

1.8.1 Chaos

Consider the following system of nonlinear equations

= a(y - x)rx-y-xzxy-bz

21

where a, r, b > 0. This system was introduced by Ed Lorenz in 1963 as a model of convectionrolls in the atmosphere. Since Lorenz' publication, similar equations have been found toappear in lasers and other systems. We now consider the following set of values: or = 10,b = 3 , and r = 28, which are the original parameters considered by Lorenz. It is easy toprove that the system has three equilibrium points. A more detailed analysis reveals that,with these values of the parameters, none of these three equilibrium points are actuallystable, but that nonetheless all trajectories are contained within a certain ellipsoidal regionin ii .

Figure 1.14(a) shows a three dimensional view of a trajectory staring at a randomlyselected initial condition, while Figure 1.14(b) shows a projection of the same trajectoryonto the X-Z plane. It is apparent from both figures that the trajectory follows a recurrent,although not periodic, motion switching between two surfaces. It can be seen in Figure1.14(a) that each of these 2 surfaces constitute a very thin set of points, almost defininga two-dimensional plane. This set is called an "strange attractor" and the two surphases,which together resemble a pair of butterfly wings, are much more complex than they appearin our figure. Each surface is in reality formed by an infinite number of complex surfacesforming what today is called a fractal.

It is difficult to define what constitutes a chaotic system; in fact, until the present timeno universally accepted definition has been proposed. Nevertheless, the essential elementsconstituting chaotic behavior are the following:

A chaotic system is one where trajectories present aperiodic behavior and arecritically sensitive with respect to initial conditions. Here aperiodic behaviorimplies that the trajectories never settle down to fixed points or to periodicorbits. Sensitive dependence with respect to initial conditions means that verysmall differences in initial conditions can lead to trajectories that deviate expo-nentially rapidly from each other.

Both of these features are indeed present in Lorenz' system, as is apparent in the Figures1.14(a) and 1.14(b). It is of great theoretical importance that chaotic behavior cannot existin autonomous systems of dimension less than 3. The justification of this statement comesfrom the well-known Poincare-Bendixson Theorem which we state below without proof.


Figure 1.14: (a) Three-dimensional view of the trajectories of Lorenz' chaotic system; (b)two-dimensional projection of the trajectory of Lorenz' system.

Theorem 1.1 [76] Consider the two dimensional system

x=f(x)where f : IR2 -> j 2 is continuously differentiable in D C 1R2, and assume that

(1) R C D is a closed and bounded set which contains no equilibrium points of x = f (x).

(2) There exists a trajectory x(t) that is confined to R, that is, one that starts in R andremains in R for all future time.

Then either R is a closed orbit, or converges toward a closed orbit as t - oo.

According to this theorem, the Poincare-Bendixson theorem predicts that, in two dimen-sions, a trajectory that is enclosed by a closed bounded region that contains no equilibriumpoints must eventually approach a limit cycle. In higher-order systems the new dimen-sion adds an extra degree of freedom that allows trajectories to never settle down to anequilibrium point or closed orbit, as seen in the Lorenz system.

1.9 Examples of Nonlinear Systems

We conclude this chapter with a few examples of "real" dynamical systems and theirnonlinear models. Our intention at this point is to simply show that nonlinear equationarise frequently in dynamical systems commonly encountered in real life. The examples inthis section are in fact popular laboratory experiments used in many universities aroundthe world.

1.9. EXAMPLES OF NONLINEAR SYSTEMS 23

Figure 1.15: Magnetic suspension system.

1.9.1 Magnetic Suspension System

V (t)

Magnetic suspension systems are a familiar setup that is receiving increasing attention inapplications where it is essential to reduce friction force due to mechanical contact. Magneticsuspension systems are commonly encountered in high-speed trains and magnetic bearings,as well as in gyroscopes and accelerometers. The basic configuration is shown in Figure1.15.

According to Newton's second law of forces, the equation of the motion of the ball is

my = -fk + mg + F (1.27)

where m is the mass of the ball, fk is the friction force, g the acceleration due to gravity,and F is the electromagnetic force due to the current i. To complete the model, we need tofind a proper model for the magnetic force F. To this end we notice that the energy storedin the electromagnet is given by

E = 1Li2

where L is the inductance of the electromagnet. This parameter is not constant since itdepends on the position of the ball. We can approximate L as follows:

L = L(y) = 1+µy. (1.28)


This model considers the fact that as the ball approaches the magnetic core of the coil, theflux in the magnetic circuit is affected, resulting in an increase of the value of the inductance.The energy in the magnetic circuit is thus E = E(i, y) = 2L(y)i2, and the force F = F(i, y)is given by

F(i, y) = 8E _ i2 8L(y) - -1 µz2(1.29)

8y 2 ay 2(1+µy)2Assuming that the friction force has the form

fk = ky (1.30)

where k > 0 is the viscous friction coefficient and substituting (1.30) and (1.29) into (1.27)we obtain the following equation of motion of the ball:

1 .\µi2my=-ky+mg-2(1+µy)2 (1.31)

To complete the model, we recognize that the external circuit obeys the Kirchhoff's voltagelaw, and thus we can write

v=Ri+at-(Li)

where

(1.32)

d _ d Mi

dt (LZ) dt 1 + µy

y(1+µy) dt+o9 (1+µy) dt_ .µi dy .\ di

(1+µy)2 dt + 1+µy dt. (1.33)

Substituting (1.33) into (1.32), we obtain

v = Ri -(1 + Uy)2 dt + l + µy dt

(1.34)

Defining state variables xl = y, x2 = y, x3 = i, and substituting into (1.31) and (1.34), weobtain the following state space model:

it = x2

2

x3 =

k \µx3m9

x2 2m(1+µx1)2l11 + p xl

[_RX3+ (l + xl)2x2x3 + vJ .

1.9. EXAMPLES OF NONLINEAR SYSTEMS 25

Figure 1.16: Pendulum-on-a-cart experiment.

1.9.2 Inverted Pendulum on a Cart

Consider the pendulum on a cart shown in Figure 1.16. We denote by 0 the angle withrespect to the vertical, L = 21, m, and J the length, mass, and moment of inertia about thecenter of gravity of the pendulum; M represents the mass of the cart, and G represents thecenter of gravity of the pendulum, whose horizontal and vertical coordinates are given by

x = X + 2 sing = X +lsinO (1.35)

y 2 cos 0 = l cos 0 (1.36)

The free-body diagrams of the cart and pendulum are shown in Figure 1.7, whereFx and Fy represent the reaction forces at the pivot point. Consider first the pendulum.Summing forces we obtain the following equations:

Fx = mX + ml6 cos O - mlO2 sin O (1.37)

Fy - mg = -mlO sin O - m162 cos 6 (1.38)

Fyl sinO - Fxl cosO = J6. (1.39)

Considering the horizontal forces acting on the cart, we have that

MX = fx - Fx. (1.40)

Substituting (1.40) into (1.37) taking account of (1.38) and (1.40), and defining statevariables xl = 0, x2 = 6 we obtain

±1 = x2


Figure 1.17: Free-body diagrams of the pendulum-on-a-cart system.

22

where we have substituted

g sin x1 - amlx2 sin(2x1) - 2a cos(xl) fx41/3 - 2aml cos2(xl)

1z

=fJ= 2 , and a 2(m+M)

1.9.3 The Ball-and-Beam System

The ball-and-beam system is another interesting and very familiar experiment commonlyencountered in control systems laboratories in many universities. Figure 1.18 shows anschematic of this system. The beam can rotate by applying a torque at the center ofrotation, and the ball can move freely along the beam. Assuming that the ball is always incontact with the beam and that rolling occurs without slipping, the Lagrange equations ofmotion are (see [28] for further details)

0 = (R2 + m)r + mgsin 0 - mrO2

r = (mr2+J+Jb)B+2mr7B+mgrcos0

where J represents the moment of inertia of the beam and R, m and Jb are the radius, massand moment of inertia of the ball, respectively. The acceleration of gravity is represented by

1.10. EXERCISES 27

Figure 1.18: Ball-and-beam experiment.

g, and r and 0 are shown in Figure 1.18. Defining now state variables xl = r, x2 = r, x3 = 0,and x4 = 0, we obtain the following state space realization:

x2

-mg sin x3 + mx1x4

m+ RX4T - rngx1 COS X3 - 2mxlx2x4

mxi+J+Jb

1.10 Exercises

(1.1) For the following dynamical systems, plot ± = f (x) versus x. From each graph, findthe equilibrium points and analyze their stability:

(a) a=x2-2.(b) ±=x3-2x2-x+2.(c) x=tanx -

2<x < 2

(1.2) Given the following linear systems, you are asked to

(i) Find the eigenvalues of the A matrix and classify the stability of the origin.

(ii) Draw the phase portrait and verify your conclusions in part (i).

28

(a)

(b)

(c)

(d)

(e)

(f)

(g)

CHAPTER 1. INTRODUCTION

[i2J-[0 4J[x2]

[i2J-[ 0 -4][x2

[x2J-[ 01 41 [xz

Iiz1 -[0 41[x2J

1

1

[2] =[2 01][X2]

[x2J-[2 -2J[x2

[ xl

±2 -[1

1 1 lIxl2 -1 X21

(1.3) Repeat problem (1.2) for the following three-dimensional systems:

(a)

(b)

zl 2 -6 5 xlX2 = 0 -4 5 X2

±3 0 0 6 x3

xl -2 -2 -1 xli2 I = 0 -4 -1 x223 0 0 -6 x3

(c)

-2 6 1 xlxlx2 0 4 1 X2

0 0 6 X323

1.10. EXERCISES 29

m2

Figure 1.19: Double-pendulum of Exercise (1.5).

(1.4) For each of the following systems you are asked to (i) find the equilibrium points, (ii)find the phase portrait, and (iii) classify each equilibrium point as stable or unstable,based on the analysis of the trajectories:

(a)

{21 -X 1 +X2

-22

(b)

(c)

xi

22

{±1 =i2 =

-x2 + 2x1(21 + x2)

xl + 2x2(21 + x2)

21 = cos 22i2 sin x1

(1.5) Find a state space realization for the double-pendulum system shown in Figure 1.19.Define state variables xl = 01, x2 = 91, x3 = 02, 24 = 92.


Notes and ReferencesWe will often refer to linear time-invariant systems throughout the rest of the book. Thereare many good references on state space theory of LTI systems. See, for example, references[15], [2], [64], or [40]. Good sources on phase-plane analysis of second-order systems areReferences [32] and [59]. See also Arnold [3] for an excellent in-depth treatment of phase-plane analysis of nonlinear systems. First-order systems are harder to find in the literature.Section 1.4 follows Strogatz [76]. The literature on chaotic systems is very extensive. SeeStrogatz [76] for an inspiring, remarkably readable introduction to the subject. Section1.8 is based mainly on this reference. Excellent sources on chaotic dynamical systems areReferences [26], [22] and [58]. The magnetic suspension system of Section 1.9.1 is a slightlymodified version of a laboratory experiment used at the University of Alberta, prepared byDrs. A. Lynch and Q. Zhao. Our modification follows the model used in reference [81].The pendulum-on-a-cart example follows References [9] and [13]. Our simple model of theball-and-beam experiment was taken from reference [43]. This model is not very accuratesince it neglects the moment of inertia of the ball. See Reference [28] for a more completeversion of this model.

Chapter 2

Mathematical Preliminaries

This chapter collects some background material needed throughout the book. As thematerial is standard and is available in many textbooks, few proofs are offered. Theemphasis has been placed in explaining the concepts, and pointing out their importancein later applications. More detailed expositions can be found in the references listed at theend of the chapter. This is, however, not essential for the understanding of the rest of thebook.

2.1 Sets

We assume that the reader has some acquaintance with the notion of set. A set is a collectionof objects, sometimes called elements or points. If A is a set and x is an element of A, wewrite x E A. If A and B are sets and if every element of A is also an element of B, we saythat B includes A, and that A is a subset of B, and we write A C B or B D A. As it hasno elements, the empty set, denoted by 0, is thus contained in every set, and we can write0 C A. The union and intersection of A and B are defined by

AUB = {x:xEAorxEB} (2.1)

AnB = {x:xEAandxEB}. (2.2)

Assume now that A and B are non empty sets. Then the Cartesian product A x Bof A and B is the set of all ordered pairs of the form (a, b) with a E A and b E B:

AxB={(a,b):aeA, andbEB}. (2.3)

31

32 CHAPTER 2. MATHEMATICAL PRELIMINARIES

2.2 Metric Spaces

In real and complex analysis many results depend solely on the idea of distance betweennumbers x and y. Metric spaces form a natural generalization of this concept. Throughoutthe rest of the book, IR and C denote the field of real and complex numbers, respectively.Z represents the set of integers. R+ and Z+ represent the subsets of non negative elementsof IR and Z, respectively. Finally Ilt"" denotes the set of real matrices with m rows and ncolumns.

Definition 2.1 A metric space is a pair (X, d) of a non empty set X and a metric ordistance function d : X x X -4 ]R such that, for all x, y, z c X the following conditions hold:

(i) d(x, y) = 0 if and only if x = y.

(ii) d(x, y) = d(y, x).

(iii) d(x, z) < d(x, y) + d(y, z).

Defining property (iii) is called the triangle inequality. Notice that, letting x = z in(iii) and taking account of (i) and (ii) we have, d(x, x) = 0 < d(x, y) + d(y, x) = 2d(x, y)from which it follows that d(x, y) > 0 Vx, y E X.

2.3 Vector Spaces

So far we have been dealing with metric spaces where the emphasis was placed on the notionof distance. The next step consists of providing the space with a proper algebraic structure.If we define addition of elements of the space and also multiplication of elements of thespace by real or complex numbers, we arrive at the notion of vector space. Alternativenames for vector spaces are linear spaces and linear vector spaces.

In the following definition F denotes a field of scalars that can be either the real orcomplex number system, IR, and C.

Definition 2.2 A vector space over F is a non empty set X with a function "+ ": X x X ->X, and a function ". ": F x X --* X such that, for all A, p E F and x, y, z E X we have

(1) x + y = y + x (addition is commutative).

(2) x + (y + z) = (x + y) + z (addition is associative).

2.3. VECTOR SPACES 33

(3) 3 0 E X : x + 0 = x ("0" is the neutral element in the operation of addition).

(4) 3 - x E X : x + (-x) = 0 (Every x E X has a negative -x E X such that their sumis the neutral element defined in (3)).

(5) 31 E F : 1.x = x ("1 " is the neutral element in the operation of scalar multiplication).

(6) \(x + y) = .\x + Ay (first distributive property).

(7) (A + µ)x = Ax + µx (second distributive property).

(8) A (µx) = (Aµ) x (scalar multiplication is associative).

A vector space is called real or complex according to whether the field F is the real or complexnumber system.

We will restrict our attention to real vector spaces, so from now on we assume that F =R. According to this definition, a linear space is a structure formed by a set X furnishedwith two operations: vector addition and scalar multiplication. The essential feature of thedefinition is that the set X is closed under these two operations. This means that whentwo vectors x, y E X are added, the resulting vector z = x + y is also an element of X.Similarly, when a vector x E X is multiplied by a scalar a E IR, the resulting scaled vectorax is also in X.

A simple and very useful example of a linear space is the n-dimensional "Euclidean"space 1Rn consistent of n-tuples of vectors of the following form:

xl

X2x

xn

More precisely, if X = IRn and addition and scalar multiplication are defined as the usualcoordinatewise operation

AxnL xn Jn xn + yn xn J

xl yl xl + yl xl Axl

X2 + y2 de f x2 + y2 Ax = A X2 de f Ax2

then it is straightforward to show that 1Rn satisfies properties (1)- (8) in Definition 2.2. Inthe sequel, we will denote by xT the transpose of the vector x, that is, if

xl

X2x

then xT is the "row vector" xT = [xi, x2, , x,,]. The inner product of 2 vector in x, y ER is xT y = E° 1 x: y:

Throughout the rest of the book we also encounter function spaces, namely spaceswhere the vectors in X are functions of time. Our next example is perhaps the simplestspace of this kind.

Example 2.1 Let X be the space of continuous real functions x = x(t) over the closedinterval 0 < t < 1.

It is easy to see that this X is a (real) linear space. Notice that it is closed with respectto addition since the sum of two continuous functions is once again continuous.

2.3.1 Linear Independence and Basis

We now look at the concept of vector space in more detail. The following definitionintroduces the fundamental notion of linear independence.

Definition 2.3 A finite set {x2} of vectors is said to be linearly dependent if there exists acorresponding set {a,} of scalars, not all zero, such that

On the other hand, if >2 atx, = 0 implies that a2 = 0 for each i, the set {x,} is said to belinearly independent.

Example 2.2 Every set containing a linearly dependent subset is itself linearly dependent.

Example 2.3 Consider the space R" and let

0

et= 1 1

0


where the 1 element is in the ith row and there are zeros in the other n - 1 rows. Then{el, e2i , en} is called the set of unit vectors in R. This set is linearly independent, since

AlAlel+...+Anen

An

Thus,

is equivalent to

Ale, +...+Anen=0

A1=A2=...=An=0

Definition 2.4 A basis in a vector space X, is a set of linearly independent vectors B suchthat every vector in X is a linear combination of elements in 13.

Example 2.4 Consider the space lR'. The set of unit vectors ei, i = 1, , n forms a basisfor this space since they are linearly independent, and moreover, any vector x E R can beobtain as linear combination of the e, values, since

Al

Ale, + ... + Anen=

A.n

It is an important property of any finite dimensional vector space with basis {bl, b2, , bn}that the linear combination of the basis vectors that produces a vector x is unique. To provethat this is the case, assume that we have two different linear combinations producing thesame x. Thus, we must have that

n n

x = E labz ==1 z=1

But then, by subtraction we have that

n

r7:)bz=0i=1


and since the b,'s, being a basis, are linearly independent, we have that 71, = 0 for1 = 1, , n, which means that the It's are the same as the it's.

In general, a finite-dimensional vector space can have an infinite number of basis.

Definition 2.5 The dimension of a vector space X is the number of elements in any of itsbasis.

For completeness, we now state the following theorem, which is a corollary of previousresults. The reader is encouraged to complete the details of the proof.

Theorem 2.1 Every set of n + 1 vectors in an n-dimensional vector space X is linearlydependent. A set of n vectors in X is a basis if and only if it is linearly independent.

2.3.2 Subspaces

Definition 2.6 A non-empty subset M of a vector space X is a subspace if for any pair ofscalars A and µ, Ax + µy E M whenever x and y E M.

According to this definition, a subspace M in a vector space X is itself a vector space.

Example 2.5 In any vector space X, X is a subspace of itself.

Example 2.6 In the three-dimensional space 1R3, any two-dimensional plane passing throughthe origin is a subspace. Similarly, any line passing through the origin is a one-dimensionalsubspace of P3. The necessity of passing through the origin comes from the fact that alongwith any vector x, a subspace also contains 0 = 1 x + (-1) X.

Definition 2.7 Given a set of vectors S, in a vector space X, the intersection of all sub-spaces containing S is called the subspace generated or spanned by S, or simply the span ofS.

The next theorem gives a useful characterization of the span of a set of vectors.

Theorem 2.2 Let S be a set of vectors in a vector space X. The subspace M spanned byS is the set of all linear combinations of the members of S.

Proof: First we need to show that the set of linear combinations of elements of S is asubspace of X. This is straightforward since linear combinations of linear combinations ofelements of S are again linear combinations of the elements of S. Denote this subspaceby N. It is immediate that N contains every element of S and thus N C M. For theconverse notice that M is also a subspace which contains S, and therefore contains alllinear combinations of the elements of S. Thus M C N, and the theorem is proved.

2.3.3 Normed Vector Spaces

As defined so far, vector spaces introduce a very useful algebraic structure by incorporatingthe operations of vector addition and scalar multiplication. The limitation with the conceptof vector spaces is that the notion of distance associated with metric spaces has been lost.To recover this notion, we now introduce the concept of norrned vector space.

Definition 2.8 A normed vector space (or simply a nonmed space) is a pairconsisting of a vector space X and a norm 11 11: X -* IR such that

(z) 11 x lI= 0 if and only if x = 0.

(ii) 1 1 Ax 111 x11 VA ER,dxEX.

(iii) 11x+yj<-11x11 +11y11 Vx,yEX.

(X,II - I

Notice that, letting y = -x # 0 in (iii), and taking account of (i) and (ii), we have

IIx + (-x) II < Ix 11 + 11 -x II

011=0 < 11 x11 +1-1111x11=211x11

=>11 X 11 >-0

thus, the norm of a vector X is nonnegative. Also, by defining property (iii), the triangleinequality,

I1 x-y11=11 x-z+z-yII511x-z11+11z-y 11 dx,yEX (2.4)

holds. Equation (2.4) shows that every normed linear space may be regarded as a metricspace with distance defined by d(x, y) _II x - y 11.

The following example introduces the most commonly used norms in the Euclideanspace R.

Example 2.7 Consider again the vector space IRn. For each p, 1 < p < oo, the function11

lip, known as the p-norm in Rn, makes this space a normed vector space, where

IlxiiP f (Ix1IP + ... + IxnIP)1/P

In particular

Ilxllldef

Ix11 +... Ixnl. (2.6)

IIxlI2def I21Iz+...+Ixnl2. (2.7)

The 2-norm is the so-called Euclidean norm. Also, the oo-norm is defined as follows:

IIxiI.dcf maxlx,I. (2.8)

4

By far the most commonly used of the p-norms in IRn is the 2-norm. Many of the theoremsencountered throughout the book, as well as some of the properties of functions andsequences (such as continuity and convergence) depend only on the three defining propertiesof a norm, and not on the specific norm adopted. In these cases, to simplify notation, itis customary to drop the subscript p to indicate that the norm can be any p-norm. Thedistinction is somewhat superfluous in that all p-norms in 1R are equivalent in the sensethat given any two norms II - IIa and II -

IIb on ]Rn, there exist constants k1 and k2 such that(see exercise (2.6))

ki llxlla <_ Ilxllb <_ k2IIxIIa, `dx E >Rn

Two frequently used inequalities involving p-norms in Rn are the following:

Holder's inequality: Let p E 1R, p > 1 and let q E R be such that

1

-+1-=1

p q

then

IIxTyll1 <_IIxIIP Ilyllq , dx, y E IRn. (2.9)

Minkowski's inequality Let p E 1R, p > 1. Then

llx + yIIP IIx1IP + IIyIIP , Vx, y E IRn. (2.10)

2.4. MATRICES 39

2.4 Matrices

We assume that the reader has some acquaintance with the elementary theory of matricesand matrix operations. We now introduce some notation and terminology as well as someuseful properties.

Transpose: If A is an m x n matrix, its transpose, denoted AT, is the n x m matrix obtainedby interchanging rows and columns with A. The following properties are straightforwardto prove:

(AT)T = A.

(AB)T = BT AT (transpose of the product of two matrices)-

(A + B)T =A T +B T (transpose of the sum of two matrices).

Symmetric matrix: A is symmetric if A = AT.

Skew symmetric matrix: A is skew symmetric if A = -AT.

Orthogonal matrix: A matrix Q is orthogonal if QTQ = QQT = I. or equivalently, if

QT =Q-1

Inverse matrix: A matrix A-1 E R"' is said to be the inverse of the square matrixA E R""" if

AA-1 = A-1A = I

It can be verified that

(A-1)-' = A.

(AB)-1 = B-'A-1, provided that A and B are square of the same size, and invertible.

Rank of a matrix: The rank of a matrix A, denoted rank(A), is the maximum numberof linearly independent columns in A.

Every matrix A E 1R""" can be considered as a linear function A : R' -4 IR", that isthe mapping

y = Ax

maps the vector x E ]R" into the vector y E R'.


Definition 2.9 The null space of a linear function A : X --+ Y zs the set N(A), defined by

N(A)={xEX: Ax=0}

It is straightforward to show that N(A) is a vector space. The dimension of thisvector space, denoted dimA((A), is important. We now state the following theorem withoutproof.

Theorem 2.3 Let A E .Pn". Then A has the following property:

rank(A) + dimM(A) = n.

2.4.1 Eigenvalues, Eigenvectors, and Diagonal Forms

Definition 2.10 Consider a matrix A E j,,xn. A scalar A E F is said to be an eigenvalueand a nonzero vector x is an eigenvector of A associated with this eigenvalue if

Ax = Ax

or

(A - AI)x = 0

thus x is an eigenvector associated with A if and only if x is in the null space of (A - AI).

Eigenvalues and eigenvectors are fundamental in matrix theory and have numerousapplications. We first analyze their use in the most elementary form of diagonalization.

Theorem 2.4 I f A E R... with eigenvalues Al, A 2 ,--', An has n linearly independenteigenvectors vl, V2i , vn, then it can be expressed in the form

A = SDS-1

where D = diag{Al, A2i . . . , An}, and

S= vl ... vn

L

2.4. MATRICES 41

Proof: By definition, the columns of the matrix S are the eigenvectors of A. Thus, we have

AS=A :l vn Alvl ... Anon

and we can re write the last matrix in the following form:

AlAlvl ... Anvn = V1 ... vn ...

An

Therefore, we have that AS = SD. The columns of the matrix S are, by assumption,linearly independent. It follows that S is invertible and we can write

A = SDS-l

or alsoD = S-IAS.

This completes the proof of the theorem. El

Special Case: Symmetric Matrices

Symmetric matrices have several important properties. Here we mention two of them,without proof:

(i) The eigenvalues of a symmetric matrix A E IIF"' are all real.

(ii) Every symmetric matrix A is diagonalizable. Moreover, if A is symmetric, then thediagonalizing matrix S can be chosen to be an orthogonal matrix P; that is, if A = AT,then there exist a matrix P satisfying PT = P-l such that

P-'AP =PTAP=D.

2.4.2 Quadratic Forms

Given a matrix A E R""n, a function q : lR - IR of the form

q(x) = xT Ax, x E R"

is called a quadratic form. The matrix A in this definition can be any real matrix. There is,however, no loss of generality in restricting this matrix to be symmetric. To see this, notice

any matrix A E 1R"' can be rewritten as the sum of a symmetric and a skew symmetricmatrix, as shown below:

A= 2(A+AT)+2(A-AT)

Clearly,

B = 2(A+AT) = BT, and

C = 2(A-AT)=-CT

thus B is symmetric, whereas C is skew symmetric. For the skew symmetric part, we havethat

xTCx = (xTCx)T = xTCTx = -xTCx.

Hence, the real number xT Cx must be identically zero. This means that the quadratic formassociated with a skew symmetric matrix is identically zero.

Definition 2.11 Let A E ]R""" be a symmetric matrix and let xbe:

EIIF".

(i) Positive definite if xTAx > OVx # 0.

(ii) Positive semidefinite if xT Ax > OVx # 0.

(iii) Negative definite if xT Ax < OVx # 0.

(iv) Negative semidefinite if xT Ax < OVx # 0.

(v) Indefinite if xT Ax can take both positive and negative values.

Then A is said to

It is immediate that the positive/negative character of a symmetric matrix is determinedcompletely by its eigenvalues. Indeed, given a A = AT, there exist P such that P-'AP =PT AP = D. Thus defining x = Py, we have that

yTPTAPy

yT P-1APy

yTDy'\l yl + A2 y2 + ... + .1"y2n

where A,, i = 1, 2, n are the eigenvalues of A. From this construction, it follows that

2.4. MATRICES 43

(i) A is positive definite if and only if all of its (real) eigenvalues are positive.

(ii) A is positive semi definite if and only if \, > 0, Vi = 1, 2, n.

(iii) A is negative definite if and only if all of its eigenvalues are negative.

(iv) A is negative semi definite if and only if \, < 0, Vi = 1, 2,- n.

(v) Indefinite if and only if it has positive and negative eigenvalues.

The following theorem will be useful in later sections.

Theorem 2.5 (Rayleigh Inequality) Consider a nonsingular symmetric matrix Q E Rnxn,and let Aman and A,,,ay be respectively the minimum and maximum eigenvalues of Q. Underthese conditions, for any x E IItn,

amin(Q)IIXI12 < xTQx < am.(Q)11_112. (2.11)

Proof: The matrix Q, which is symmetric, is diagonalizable and it must have a full set oflinearly independent eigenvectors. Moreover, we can always assume that the eigenvectorsu1i U2, , un associated with the eigenvalues ,11, a2, . , ,1n are orthonormal. Given thatthe set of eigenvectors Jul, u2i , un} form a basis in Rn, every vector x can be written asa linear combination of the elements of this set. Consider an arbitrary vector x. We canassume that I xji = 1 (if this is not the case, divide by 11x11). We can write

x = x1u1 +x2U2 +"' +xnun

for some scalars x1i x2, , xn. Thus

xT Qx = XT Q [XI U1 +... + xnun]

= xT [Alxlu1 + ... + Anxnun]

= AixT x1u1 + ....+.nxTxnun

= Ai x112+...+Anlxn12

which implies thatAmin(Q) < xTQx < A.(Q) (2.12)

sincen

1x112=1.

a=1

Finally, it is worth nothing that (2.12) is a special case of (2.11) when the norm of x is 1.

2.5 Basic Topology

A few elements of basic topology will be needed throughout the book. Let X be a metricspace. We say that

(a) A neighborhood of a point p E X is a set Nr (p) C X consisting of all points q E Xsuch that d(p,q) < r.

(b) Let A C X and consider a point p E X. Then p is said to be a limit point of A ifevery neighborhood of p contains a point q # p such that q E A.

It is important to notice that p itself needs not be in the set A.

(c) A point p is an interior point of a set A c X if there exist a neighborhood N of psuch that N C E.

(d) A set A c X is said to be open if every point of A is an interior point.

(e) The complement of A C X is the set A` = {p E X : p V A}.

(f) A set A C X is said to be closed if it contains all of its limit points. Equivalently, Ais closed if and only if and only if A` is open.

(g) A set A C X is bounded if there exist a real number b and a point q E A such thatd(p, q) < b, dp E A.

2.5.1 Basic Topology in 1R"

All the previous concepts can be specialized to the Euclidean space 1R". We emphasize thoseconcepts that we will use more frequently. Now consider a set A C 1R".

Neighborhood: A neighborhood of a point p E A C R" is the set Br (p) defined asfollows:

Br(p) = {x E R' :Iix - p1I < r}.

Neighborhoods of this form will be used very frequently and will sometimes be referredto as an open ball with center p and radius r.

Open set: A set A C 1R" is said to be open if for every p E A one can find aneighborhood Br(p) C A.

Bounded set: A set A C ]R" is said to be bounded if there exists a real numberM > 0 such that

IIxli < M Vx E A.

2.6. SEQUENCES 45

Compact set: A set A c R" is said to be compact if it is closed and bounded.

Convex set: A set A C iR' is said to be convex if, whenever x, y E A, then

9x1+(1-0)x2i 0<0<1

also belongs to A.

2.6 Sequences

Definition 2.12 A sequence of vectors x0i x1, x2, in a metric space (X, d), denoted {xn}is said to converge if there is a point x0 E X with the property that for every real numberl; > 0 there is an integer N such that n > N implies that d(xn, x0) < . We then writex0 = limxn, or xn -* x0 and call x the limit of the sequence {xn}.

It is important to notice that in Definition 2.12 convergence must be taking place inthe metric space (X, d). In other words, if a sequence has a limit, lim xn = x' such thatx` V X, then {xn} is not convergent (see Example 2.10).

Example 2.8 Let X1 = IR, d(x, y) = Ix - yl, and consider the sequence

{xn} = {1,1.4,1.41,1.414, }

(each term of the sequence is found by adding the corresponding digit in v'-2). We have that

f- x(n) l < e for n> N

and since i E IR we conclude that xn is convergent in (X1, d).

Example 2.9 Let X2 = Q, the set of rational numbers (x E Q = x = b, with a, b E 7G, b #0). Let d(x, y) = Ix - yl, and consider the sequence of the previous example. Once againwe have that x, is trying to converge to f. However, in this case f Q, and thus weconclude that xn is not convergent in (X2, d).

Definition 2.13 A sequence {xn} in a metric space (X, d) is said to be a Cauchy sequenceif for every real > 0 there is an integer N such that d(xn, x,n) < 1;, provided that n, m > N.

It is easy to show that every convergent sequence is a Cauchy sequence. The converse is,however, not true; that is a Cauchy sequence in not necessarily convergent, as shown in thefollowing example.

Example 2.10 Let X = (0,1) (i.e., X = {x E R : 0 < x < 1}), and let d(x,y) =I x - y 1.

Consider the sequence {xn} = 1/n. We have

d(xn,xm.) =I1- 1 I< 1+ 1 < 2

n m n m Nwhere N = min(n,m). It follows that {xn} is a Cauchy sequence since d(xn, xm) < e,provided that n, m > 2/e. It is not, however, convergent in X since limn, 1/n = 0, and

An important class of metric spaces are the so-called complete metric spaces.

Definition 2.14 A metric space (X, d) is called complete if and only if every Cauchy se-quence converges (to a point of X). In other words, (X, d) is a complete metric space if forevery sequence {xn} satisfying d(xn,xm) < e for n, m > N there exists x E X such thatd(xn, x) -> 0 as n -> oo.

If a space is incomplete, then it has "holes". In other words, a sequence might be "trying" toconverge to a point that does not belong to the space and thus not converging. In incompletespaces, in general, one needs to "guess" the limit of a sequence to prove convergence. Ifa space is known to be complete, on the other hand, then to check the convergence of asequence to some point of the space, it is sufficient to check whether the sequence is Cauchy.The simplest example of a complete metric space is the real-number system with the metricd =1 x - y 1. We will encounter several other important examples in the sequel.

2.7 Functions

Definition 2.15 Let A and B be abstract sets. A function from A to B is a set f of orderedpairs in the Cartesian product A x B with the property that if (a, b) and (a, c) are elementsof f, then b = c.

In other words, a function is a subset of the Cartesian product between A and B whereeach argument can have one and only one image. Alternative names for functions used inthis book are map, mapping, operator, and transformation. The set of elements of A thatcan occur as first members of elements in f is called the domain of f. The set of elementsof B that can occur as first members of elements of f is called the range of f. A functionf is called injective if f (xl) = f (X2) implies that xl = x2 for every x1, x2 E A. A functionf is called surjective if the range of f is the whole of B. It is called bijective if it is bothinjective and surjective.

2.7. FUNCTIONS 47

If f is a function with domain D and D1 is a subset of D, then it is often useful todefine a new function fl with domain D1 as follows:

fl = {(a,b) E f : a E Dl}.

The function fl is called a restriction of f to the set D1.

Definition 2.16 Let D1, D2 C iR0, and consider functions fl and f2 of the form fl : D1 -4II8n and f2 : D1 -> IR". If fl(Di) C D2, then the composition of f2 and fl is the functionf2 o fl, defined by

(f2 0 f1)(x) = f2[fl(x)] xEDI

Definition 2.17 Let (X, di), (Y, d2) be metric spaces and consider a function f : X -> Y.We say that f is continuous at xo if for every real e > 0 there exists a real 6 = 8(e, xo) suchthat d(x, xo) < 6 implies that d2(f (x), f (xo)) < e.

Equivalently, f : X -> Y is continuous if for every sequence {xn} that converges to x,the corresponding sequence If (xn)} converges to y = f (x). In the special case of functionsof the form f : IR0 -* IR', that is, functions mapping Euclidean spaces, then we say that fis continuous at x E 1R" if given e > 0 there exists b > 0 such that

iix -yMM 1f(x) - f(y) I < E.

This definition is clearly local, that is, it corresponds to pointwise convergence. If f iscontinuous at every point of X, then f is said to be continuous on X.

Definition 2.18 Let (X, d1) and (Y, d2) be metric spaces. Then f : X -i Y is calleduniformly continuous on X if for every e > 0 there exists b = 6(e) > 0, such that d1 (x, y) < bimplies that d2 (f (x), f (y)) < e, where x, y c X.

Remarks: The difference between ordinary continuity and uniform continuity is that inthe former 6(e, x0) depends on both a and the particular x0 E X, while in the latter b(e)is only a function of E. Clearly, uniform continuity is stronger than continuity, and if afunction is uniformly continuous on a set A, then it is continuous on A. The converse isin general not true; that is, not every continuous function is uniformly continuous on thesame set. Consider for example the function f (x) = 2, x > 0, is continuous over (0, oo) butnot uniformly continuous. The exception to this occurs when working with compact sets.Indeed, consider a function f mapping a compact set X into a metric space Y. Then f isuniformly continuous if and only if it is continuous.

2.7.1 Bounded Linear Operators and Matrix Norms

Now consider a function L mapping vector spaces X into Y.

Definition 2.19 A function L : X --4 Y is said to be a linear operator (or a linear map,or a linear transformation) if and only if given any X1, X2 E X and any A, p E R

L(Axi + µx2) = .L(x1) + pL(x2). (2.13)

The function L is said to be a bounded linear operator if there exist a constant M such that

JIL(x)Ij < MIjxII Vx E X. (2.14)

The constant M defined in (2.14) is called the operator norm. A special case ofinterest is that when the vector spaces X and Y are Rn and 1R', respectively. In this caseall linear functions A : Rn -* R'n are of the form

y=Ax, xER', yERm

where A is a m x n of real elements. The operator norm applied to this case originates amatrix norm. Indeed, given A E II8"`n, this matrix defines a linear mapping A : Rn -* R'nof the form y = Ax. For this mapping, we define

def IIAxp =zoo IIx11P 1121 1

(2.15)

Where all the norms on the right-hand side of (2.15) are vector norms. This norm issometimes called the induced norm because it is "induced" by the p vector norm. Importantspecial cases are p = 1, 2, and oo. It is not difficult to show that

m

IIAIIc

max IAxjIi = maxE lazal (2.16)IlxIIi=1 7 x=1

max IIAxII2 = Amax(ATA) (2.17)IIXI12=1

n

max IIAxII. = maxE lain (2.18)Ilxlloo=1 i

J=1

where Amax (AT A) represents the maximum eigenvalue of AT A.

2.8. DIFFERENTIABILITY 49

2.8 Differentiability

Definition 2.20 A function f : R -> R is said to be differentiable at x if f is defined in anopen interval (a, b) C R and

f'(x) = l

af(x + hh - f(x)

exists.

The limit f'(x) is called the derivative of f at x. The function f is said to bedifferentiable if it is differentiable at each x in its domain. If the derivative exists, then

f(x + h) - f(x) = f'(x)h + r(h)

where the "remainder" r(h) is small in the sense that

lim r(h) = 0.h,O h

Now consider the case of a function f : R" -+ R.

Definition 2.21 A function f : R' -4 R' is said to be differentiable at a point x if f isdefined in an open set D C R" containing x and the limit

lim If(x+h) - f(x) - f'(x)hIl = 0h-+0 lhll

exists.

Notice, of course, in Definition 2.21 that h E R'. If IhII is small enough, thenx + h E D, since D is open and f (x + h) E JRt.

The derivative f'(x) defined in Definition 2.21 is called the differential or the totalderivative of f at x, to distinguish it from the partial derivatives that we discuss next. Inthe following discussion we denote by f2, 1 < i < 7n the components of the function f, andby {el, e2, , en} the standard basis in R"

fi (x)f2(x)

el =f(x) = I

fol f°1 fBl,e2 =

0 0Lfm(x)I

en =

1

Definition 2.22 Consider a function f : ]R" -4 IItm and let D be an open set in ]R". ForxEDCIR" andl<i<m, 1 <j <n we define

Of,= D2f: def lim f:(x + Deb) - f:(x) = 0, AC R

Oxj o-+o

provided that the limit exists.

The functions D3 f; (or -) are called the partial derivatives of f .

Differentiability of a function, as defined in Definition 2.21 is not implied by theexistence of the partial derivatives of the function. For example, the function f : ]R2 -> IR

f(xl,x2) =

is not continuous at (0, 0) and so it is not differentiable. However, both A and t x exist atevery point in ]R2. Even for continuous functions the existence of all partial derivatives doesnot imply differentiability in the sense of Definition 2.21. On the other hand, if f is knownto be differentiable at a point x, then the partial derivatives exist at x, and they determinef'(x). Indeed, if f : R' -> R' is differentiable, then f'(x) is given by the Jacobian matrixor Jacobian transformation [f'(x)]:

9 '1...

X1 §Xn

[f'(x)] _xi x

If a function f : 1R - ]R"` is differentiable on an open set D C 1R", then it is continuous onD. The derivative of the function, f', on the other hand, may or may not be continuous.The following definition introduces the concept of continuously differentiable function.

Definition 2.23 A differentiable mapping f of an open set D C IIt" into R' is said to becontinuously differentiable in D if f' is such that

IT(y) - f'(x)II < e

provided that x, y E D and II x - yII < d.

The following theorem, stated without proof, implies that the continuously differentiableproperty can be evaluated directly by studying the partial derivatives of the function.

2.8. DIFFERENTIABILITY 51

Theorem 2.6 Suppose that f maps an open set D C IR' into IRm. Then f is continuouslydifferentiable in D if and only if the partial derivatives , 1 Rm is said to be continuously differentiable at a pointxo if the partial derivatives ', 1 < i < m, 1 < j < n exist and are continuousat xo. A function f : IR" -- 1W is said to be continuously differentiable on a setD C IR'`, if it is continuously differentiable at every point of D.

Summary: Given a function f : R" -i 1R' with continuous partial derivatives, abusing thenotation slightly, we shall use f'(x) or D f (x) to represent both the Jacobian matrix andthe total derivative of f at x. If f : IR' -+ R, then the Jacobian matrix is the row vector

of of off1(X) axl' ax2'

...axn

This vector is called the gradient of f because it identifies the direction of steepest ascentof f. We will frequently denote this vector by either

axor V f (x)

vf(x) = of=

of of.

ofOx axl'ax2' axn

It is easy to show that the set of functions f : IR'2 -+ IR' with continuous partialderivatives, together with the operations of addition and scalar multiplication, form a vectorspace. This vector space is denoted by Cl. In general, if a function f : R -> lR' hascontinuous partial derivatives up to order k, the function is said to be in Ck, and we writef E Ck. If a function f has continuous partial derivatives of any order, then it is said to besmooth, and we write f E C. Abusing this terminology and notation slightly, f is said tobe sufficiently smooth where f has continuous partial derivatives of any required order.

2.8.1 Some Useful Theorems

We collect a number of well known results that are often useful.

Property 2.1 (Chain Rule) Let fl : D1 C R" -+ R", and f2 : D2 C R" -+ IR". If f2 isdifferentiable at a E D2 and fl is differentiable at f2(a), then fl o f2 is differentiable at a,and

D(fl o f2)(a) = DF1(f2(a))Df2(a).

Theorem 2.7 (Mean-Value Theorem) Let f : [a, b] -> R be continuous on the closedinterval [a, b] and differentiable in the open interval (a, b). Then there exists a point c in(a, b) such that

f(b) - f(a) = f'(c) (b - a).

A useful extension of this result to functions f : R" -+ R' is given below. In the followingtheorem S2 represents an open subset of lR'.

Theorem 2.8 Consider the function f : 0 C R" --> R'" and suppose that the open set wcontains the points a, and b and the line segment S joining these points, and assume thatf is differentiable at every point of S. The there exists a point c on S such that

If (b) - f (a)lb = Ilf'(c)(b - a) 11.

Theorem 2.9 (Inverse Function Theorem) Let f : Rn -> R" be continuously differentiablein an open set D containing the point xo E R", and let f'(xo) # 0. Then there exist an openset Uo containing xo and an open set Wo containing f(xo) such that f : Uo -+ Wo has acontinuous inverse f -1 : WO - Uo that is differentiable and for all y = f (x) E WO satisfies

Df-1(y) = [Df(x)] 1 = [Df-1(y)]-1.

2.9 Lipschitz Continuity

We defined continuous functions earlier. We now introduce a stronger form of continuity,known as Lipschitz continuity. As will be seen in Section 2.10, this property will play amajor role in the study of the solution of differential equations.

Definition 2.24 A function f (x) : Rn - lRm is said to be locally Lipschitz on D if everypoint of D has a neighborhood Do C D over which the restriction of f with domain D1satisfies

If(XI) - f (X2)11 < LJJx1 - x2j1. (2.19)

It is said to be Lipschitz on an open set D C R" if it satisfies (2.19) for all x1i x2 E D withthe same Lipschitz constant. Finally, f is said to be globally Lipschitz if it satisfies (2.19)with D = Rn.

2.9. LIPSCHITZ CONTINUITY 53

Notice that if f : 1R' -4 1R' is Lipschitz on D C 1R", then, given e > 0, we can define8 = e/L, and we have that

I41 -x211<8 => 1f(xi)-f(x2)11<Lllxl-x211<LLe, xi,x2ED

which implies that f is uniformly continuous. However, the converse is not true; thatis, not every uniformly continuous function is Lipschitz. It is in fact very easy to findcounterexamples that show that this in not the case. The next theorem gives an importantsufficient condition for Lipschitz continuity.

Theorem 2.10 If a function f : IR" x R' is continuously differentiable on an open setD C IIE", then it is locally Lipschitz on D.

Proof: Consider xo E D and let r > 0 be small enough to ensure that Br(xo) C D, where

B,.(xo)={xED:lx-xol <r}and consider two arbitrary points xl, x2 E Br. Noticing that the closed ball Br is a convexset, we conclude that the line segment y(O) = Oxl + (1 - O)x2i 0 < 0Br(xo). Now define the function 0 : [0, 1] --r R' as follows:

0(0)=fo'r=f(7(O))By the mean-value theorem 2.8, there exist 0l E (0, 1) such that

II0(1) - 0(0)11= 110'(Bl)11-

Thus, calculating 0'(O) using the chain rule, we have that

I0(1) - 0(0)II = ax(xl -x2)

< 1, is contained in

and, substituting 0(1) = f (xi), 0(0) = f (X2)

If(XI) - f (X2)11 ax 11x1 - x211= LIIxi - x211. (2.20)

Theorem 2.10 provides a mechanism to calculate the Lipschitz constant (equation(2.20). Indeed, L can be estimated as follows:

11af11

ax< L, VX E Br(xo).

If the function f is also a function of t, that is, f = f (x, t), then the previous definition canbe extended as follows

Definition 2.25 A function f (x, t) : lR' x R -> IRn is said to be locally Lipschitz in x on anopen set D x [to, T] C In x IR if every point of D has a neighborhood D1 C D over which therestriction off with domain D1 x [to, T] satisfies (2.20). It is said to be locally Lipschitz onD C 1Rn x (to, oo) if it is locally Lipschitz in x on every D1 x [to, T] C D x [to, oo). It is saidto be Lipschitz in x on D x [to, T] if it satisfies (2.20) for all x1, x2 E D and all t E [to, T.

Theorem 2.11 Let f : 1R' x IR -* IR' be continuously differentiable on D x [to, Tj andassume that the derivative of f satisfies

of (x, t) < L (2.21)

on D x [to, T]. The f is Lipschitz continuous on D with constant L:

11f(x,t)_f(y,t)1I <LIIx - yll , Vx, y E D, Vt E [to,T].

2.10 Contraction Mapping

In this section we discuss the contraction mapping principle, which we use later to analyzethe existence and uniqueness of solutions of a class of nonlinear differential equations.

Definition 2.26 Let (X, d) be a metric space, and let S C X. A mapping f : S -* S issaid to be a contraction on S if there exists a number £ < 1 such that

d(f(x),f(y)) : e d(x, y)

for any two points x, y c S.

(2.22)

It is an straightforward exercise that every contraction is continuous (in fact, uniformlycontinuous) on X.

Theorem 2.12 (Contraction Mapping Principle) Let S be a closed subset of the completemetric space (X, d). Every contraction mapping f : S -> S has one and only one x E Ssuch that f (x) = x.

Proof: A point xo E X satisfying f (xo) = xo is called a fixed point. Thus, the contractionmapping principle is sometimes called the fixed-point theorem. Let xo be an arbitrary pointin S, and denote x1 = f (xo), x2 = A X0, , xn+1 = f (xn). This construction defines asequence {xn} = { f (xn_ 1) }. Since f maps S into itself, xk E S Vk. We have

d(xn+1, xn) = d(f (xn), f (xn-1) d(xn, xn-1) (2.23)

2.10. CONTRACTION MAPPING 55

and then, by inductiond(xn+l,xn) < End(xl,xo) (2.24)

Now suppose m > n > N. Then, by successive applications of the triangle inequality weobtain

m

d(xn, xm) E d(xt, xi-1)t=n+1

< (Cn + `n+l +... + m-1) d(xi, xo)

d(xn, xm) < n(1 + 1 + ... + Cm-n-1) d(xl, x0). (2.25)

The series (1 + x + x2 + ) converges to 1/(1 - x), for all x < 1. Therefore, noticingthat all the summands in equation (2.25) are positive, we have:

1+e+...+Cm-n-1 <1

(1 - E)

d(xn, xm) <n

d(xl, xo)

defining 1 £ d(xl, xo) = E, we have that

d(xn, xm) < e n, m > N.

In other words, {xn} is a Cauchy sequence. Since the metric space (X, d) is complete,{xn} has a limit, limn,,,.(xn) = x, for some x E X. Now, since f is a contraction, f iscontinuous. It follows that

f(x) f(xn) = rnn (xn+1) = x.

Moreover, we have seen that xn E S C X, Vn, and since S is closed it follows that x E S.Thus, the existence of a fixed point is proved. To prove uniqueness suppose x and y aretwo different fixed points. We have

f(x)=x, f(y)=y (2.26)

d(f(x),f(y)) = d(x, y) by (2.26) (2.27)

= d(x, y) < d(x, y) because f is a contraction (2.28)

where < 1. Since (2.28) can be satisfied if and only if d(x, y) = 0, we have x = y. Thiscompletes the proof.

2.11 Solution of Differential Equations

When dealing with ordinary linear differential equations, it is usually possible to derive aclosed-form expression for the solution of a differential equation. For example, given thestate space realization of a linear time-invariant system

x = Ax + Bu, x(0) = xo

we can find the following closed form solution for a given u:

Ltx(t) = eAtxo + eA(t-Bu(rr) dr.

In general, this is not the case for nonlinear differential equations. Indeed, when dealingwith nonlinear differential equations two issues are of importance: (1) existence, and (2)uniqueness of the solution.

In this section we derive sufficient conditions for the existence and uniqueness of thesolution of a differential equation of the form

.t= f (t, x) x(to) = xo (2.29)

Theorem 2.13 (Local Existence and Uniqueness) Consider the nonlinear differential equa-tion

x = f (x, t), x(to) = xo (2.30)

and assume that f (x, t) is piecewise continuous in t and satisfies

ll f(x1, t) - f (x2, t)ll < L 11x1 - x211

Vx1i x2 E B = {x E Rn : llx - xo11 < r},Vt E [to, t1]. Thus, there exist some b > 0 such that(2.30) has a unique solution in [to, to + 61.

Proof: Notice in the first place that if x(t) is a solution of (2.30), then x(t) satisfies

x(t) = xo + J f [x (T), r] dr (2.31)tto

which is of the formx(t) = (Fx)(t) (2.32)

where Fx is a continuous function of t. Equation (2.32) implies that a solution of thedifferential equation (2.30) is a fixed point of the mapping F that maps x into Fx. The

2.11. SOLUTION OF DIFFERENTIAL EQUATIONS 57

existence of a fixed point of this map, in turn, can be verified by the contraction mappingtheorem. To start with, we define the sets X and the S C X as follows:

X = C[to, to + 8]

thus X is the set of all continuous functions defined on the interval [to,to+a]. Given x E X,we denote

and finally

Ilxlic =tE m, x

1X(t)11 (2.33)

S={xEX:IIx-xoMIc<r}.Clearly, S C X. It can also be shown that S is closed and that X with the norm (2.33)is a complete metric space. To complete the proof, we will proceed in three steps. In step(1) we show that F : S -> S. In step (2) we show that F is a contraction from S into S.This, in turn implies that there exists one and only one fixed point x = Fx E S, and thus aunique solution in S. Because we are interested in solutions of (2.30) in X (not only in S).The final step, step (3), consists of showing that any possible solution of (2.30) in X mustbe in S.

Step (1): From (2.31), we obtaint

fo(Fx)(t) - xo = f (r, x(r)) d7-

The function f is bounded on [to, tl] (since it is piecewise continuous). It follows that wecan find such that

tmaxjf(xo,t)jI

IIFx-xoll < j[Mf(x(r)r)_f(xor)+ If(xo,r)I] drto

-xoll)+C]drf[L(Ix(r)o

and since for each x E S, IIx - xo II < r we have

t

IFx - xoll < f[Lr+]dr< (t - to) [Lr + ].

58

It follows that

CHAPTER 2. MATHEMATICAL PRELIMINARIES

IFx - xollc= max IFx - xoll < 6(Lr+tE[TT,t0+6]

and then, choosing 6 < means that F maps S into S.

Step (2):To show that F is a contraction on S, we consider x1, x2 E S and proceed asfollows:

II (Fx1)(t) - (Fx2)(t)II = tt

o,[f(x1(T), T) - f(x2(T), T) ] dr

J t 11 f (x1(T ), 1) - f (x2 (T ), T) 11 dTtot

< L Ilxl(T) - x2(T)II d7to

t

< LIlx1-x2IIc ft dTo

IIFxi - Fx211c < L6II xl - x211, < PIIx1 - x211c for 6 <L

Choosing p < 1 and 6 < i , we conclude that F is a construction. This implies that thereis a unique solution of the nonlinear equation (2.30) in S.

Step (3): Given that S C X, to complete the proof, we must show that any solution of(2.30) in X must lie in S. Starting at xo at to, a solution can leave S if and only if at somet = t1, x(t) crosses the border B for the first time. For this to be the case we must have

Ilx(tl)-xoll=r.

Thus, for all t < tl we have that

IIx(t) -x011 < ft[ IIf(x(r),r) -f(xo,T)II + Ilf(xo,T)II J dTo

< j[L(Ilx(r)-xoII)+]drt

< j[Lr+]dr.o

It follows thatr = 11x(ti) - xo II < (Lr + .) (tl - to).

2.12. EXERCISES 59

Denoting t1 = to + p, we have that if u is such that

rµ Lr+

then the solution x(t) is confined to B. This completes the proof.

It is important to notice that Theorem 2.13 provides a sufficient but not necessarycondition for the existence and uniqueness of the solution of the differential equation (2.30).According to the theorem, the solution is guaranteed to exist only locally, i.e. in the interval[to, to+s]. For completeness, below we state (but do not prove) a slightly modified version ofthis theorem that provides a condition for the global existence and uniqueness of the solutionof the same differential equation. The price paid for this generalization is a stronger andmuch more conservative condition imposed on the differential equation.

Theorem 2.14 (Global Existence and Uniqueness) Consider again the nonlinear differen-tial equation (2.30) and assume that f (x, t) is piecewise continuous in t and satisfies

Il f (x1, t) - f (x2, t) II <- Lllxl - x211

1f(xo,t)II <_ 0

dx1, x2 E R", dt E [to, t1]. Then, (2.30) has a unique solution in [to, ti].

Theorem 2.14 is usually too conservative to be of any practical use. Indeed, noticethat the conditions of the theorem imply the existence of a global Lipschitz constant, whichis very restrictive. The local Lipschitz condition of theorem 2.13 on the other hand, is notvery restrictive and is satisfied by any function f (x, t) satisfying somewhat mild smoothnessconditions, as outlined in Theorem 2.11

2.12 Exercises

(2.1) Consider the set of 2 x 2 real matrices. With addition and scalar multiplication definedin the usual way, is this set a vector space over the field of real numbers? Is so, finda basis.

(2.2) Let x, y and z be linearly independent vectors in the vector space X. Is it correct toinfer that x + y, y + z, and z + x are also linearly independent?

(2.3) Under what conditions on the scalars a and 3 in C2 are the vectors [1, a]T, and [1,131Tlinearly dependent?

60 CHAPTER 2. N ATHEMATICAL PRELIMINARIES

(2.4) Prove Theorem 2.1.

(2.5) For each of the norms 11x111,11x112, and IlXlloo in lR , sketch the "open unit ball" centeredat 0 = [0,0]T, namely, the sets IIxiii < 1, lIx112 < 1, and IIxII,,, < 1.

(2.6) Show that the vector norms II 111, II - 112, and 11 - Iloo satisfy the following:

IIxl12 < 1141 :5 v1 11x112

Ilxll- 5 1x111

IIx1Io <_ IIxII2

< n 11x112

< V"n IIxI12

(2.7) Consider a matrix A E R1 ' and let )q, , An be its eigenvalues and X1, , xn its(not necessarily linearly independent) eigenvecti irs.

(i) Assuming that A is nonsingular, what can be said about its eigenvalues?

(ii) Under these assumptions, is it possible to ex press the eigenvalues and eigenvectorsof A-' in terms of those of A? If the am wer is negative, explain why. If theanswer is positive, find the eigenvalues and eigenvectors of A-'.

(2.8) Consider a matrix A E R' '

(i) Show that the functions IIAll1, IIA112, and IIAIIm defined in equations (2.16),(2.17), and (2.18), respectively, are norms, i.e. satisfy properties (i)-(iii) inDefinition 2.8.

(ii) Show that IIIIIP = 1 for any induced normI - lip.

(iii) Show that the function

IIAII = trace(ATA)n

-1: 1a,,12

is a matrix norm' (i.e. it satisfies properties (i)-(iii) in Definition 2.8). Showthat this norm, however, in not an operato norm since it is not induced by anyvector norm.

(2.9) Let (X, d) be a metric space. Show that

(i) X and the empty set 0 are open.

(ii) The intersection of a finite number of open sets is open.

(iii) The union of any collection of open sets is open.

'This norm is called the Frobeneus norm.

2.12. EXERCISES 61

(2.10) Show that a set A in a metric space (X, d) is closed if and only if its complement isopen.

(2.11) Let (X, d) be a metric space. Show that

(i) X and the empty set 0 are closed.

(ii) The intersection of any number of closed sets is closed.

(iii) The union of a finite collection of closed sets is closed.

(2.12) Let (X, dl) and (Y, d2) be metric spaces and consider a function f : X -* Y. Showthat f is continuous if and only if the inverse image of every open set in Y is open inX.

(2.13) Determine the values of the following limits, whenever they exist, and determinewhether each function is continuous at (0, 0):

x2 _y2(1)

x lOm0 1 + x2 + y2

(2.14) Consider the function

(ii) lim2

x2

(iii) lim(1 + y2) sin x

x-*0,y-i0 x+ Y2X

O X

x°+y` for (x, y) # (0, 0)

0 for x = y = 0

Show that f (x, y) is not continuous (and thus not differentiable) at the origin. Proceedas follows:

(i) Show thatlim f (x, y) = 0

x-40,y=x 0

that is, if x - 0 and y --> 0 along the parabola y = x2, then limx 0.

(ii) Show that1

lira f (x, y) = -x-+O,y=x-+0 2

thus the result.

(2.15) Given the function f (x, y) of exercise (2.13), show that the partial derivatives andboth exist at (0, 0). This shows that existence of the partial derivatives does not

imply continuity of the function.

(2.16) Determine whether the function

x2 2for (x, y) # (0, 0)

0 for x=y=0

is continuous at (0, 0). (Suggestion: Notice that y 7 < x2.)

(2.17) Given the following functions, find the partial derivatives and

(i) f (x, y) = ex' cos x sin yx2y2

(ii)f (X,

y) = x2 + y2

(2.18) Use the chain rule to obtain the indicated partial derivatives.

(i) z = x3 + y3, x = 2r + s, y = 3r - 2s, N, and.(ii) z = xr+--y7 x = r cos s, y = r sin s, az, and a

(2.19) Show that the function f = 1/x is not uniformly continuous on E = (0, 1).

(2.20) Show that f = 1/x does not satisfy a Lipschitz condition on E = (0, 1).

(2.21) Given the following functions f : IR -- R, determine in each case whether f is (a)continuous at x = 0, (b) continuously differentiable at x = 0, (c) locally Lipschitz atx=0.

(i) f (x) = ex2(ii) f (x) = Cos X.

(iii) f (x) = satx.(iv) f (x) = sin(' ).

(2.22) For each of the following functions f : 1R2 -+ IIY2, determine whether f is (a) continuousat x = 0, (b) continuously differentiable at x = 0, (c) locally Lipschitz at x = 0, (d)Lipschitz on some D C R2.

(i)

x1 = x2 - x1(xl +x2)22 = -x1 - x2(xl + x2)

(ii):i1 =x2-}'xl(N2-x1 -x2)

{ -x2x2 = -x1 + x2 /32 - x1 2( )l

2.12. EXERCISES 63

J r1 = X2Sl 22 = m21 - 12

J ±1 = X2

ll k,22

Notes and ReferencesThere are many good references for the material in this chapter. For a complete,

remarkably well-written account of vector spaces, see Halmos [33]. For general backgroundin mathematical analysis, we refer to Bartle [7], Maddox [51], and Rudin [62]. Section 2.9,including Theorem 2.10, is based on Hirsch [32]. The material on existence and uniqueness ofthe solution of differential equations can be found on most textbooks on ordinary differentialequations. We have followed References [32], [88] and [41]. See also References [59] and[55].

Chapter 3

Lyapunov Stability I: AutonomousSystems

In this chapter we look at the important notion of stability in the sense of Lyapunov.Indeed, there are many definitions of stability of systems. In all cases the idea is: given aset of dynamical equations that represent a physical system, try to determine whether sucha system is well behaved in some conceivable sense. Exactly what constitutes a meaningfulnotion of good behavior is certainly a very debatable topic. The problem lies in how toconvert the intuitive notion a good behavior into a precise mathematical definition that canbe applied to a given dynamical system. In this chapter, we explore the notion of stabilityin the sense of Lyapunov, which applies to equilibrium points. Other notions of stabilitywill be explored in Chapters 6 and 7. Throughout this chapter we restrict our attention toautonomous systems. The more general case of nonautonomous systems is treated in thenext chapter.

3.1 Definitions

Consider the autonomous systems

x=f(x) f:D -Rnwhere D is an open and connected subset of Rn and f is a locally Lipschitz map from Dinto Rn. In the sequel we will assume that x = xe is an equilibrium point of (3.1). In otherwords, xe is such that

f(xe) = 0.'Notice that (3.1) represents an unforced system

65

66 CHAPTER 3. LYAPUNOV STABILITY I. AUTONOMOUS SYSTEMS

Figure 3.1: Stable equilibrium point.

We now introduce the following definition.

Definition 3.1 The equilibrium point x = xe of the system (3.1) is said to be stable if foreach e > 0, 3b=6(e)>0

IIx(0) - xell < 6 IIx(t) - X, 11 < e Vt > to

otherwise, the equilibrium point is said to be unstable .

This definition captures the following concept-we want the solution of (3.1) to be near theequilibrium point xe for all t > to. To this end, we start by measuring proximity in termsof the norm II II, and we say that we want the solutions of (3.1) to remain inside the openregion delimited by 11x(t) - xell < e. If this objective is accomplished by starting from aninitial state x(0) that is close to the equilibrium xef that is, I1x(0) - xell < 6, then theequilibrium point is said to be stable (see Figure 3.1).

This definition represents the weakest form of stability introduced in this chapter.Recall that what we are trying to capture is the concept of good behavior in a dynamicalsystem. The main limitation of this concept is that solutions are not required to convergeto the equilibrium xe. Very often, staying close to xe is simply not enough.

Definition 3.2 The equilibrium point x = xe of the system (3.1) is said to be convergentif there exists 61 > 0 :

IIx(0) - xeII < 6 = lim x(t) = xe.t +00

3.1. DEFINITIONS 67

Figure 3.2: Asymptotically stable equilibrium point.

Equivalently, x, is convergent if for any given cl > 0, IT such that

IIx(0) - xell < dl 11x(t) - xell < el Vt > to + T.

A convergent equilibrium point xe is one where every solution starting sufficientlyclose to xe will eventually approach xe as t -4 oo. It is important to realize that stabilityand convergence, as defined in 3.1 and 3.2 are two different concepts and neither one of themimplies the other. Indeed, it is not difficult to construct examples where an equilibrium pointis convergent, yet does not satisfy the conditions of Definition 3.1 and is therefore not stablein the sense of Lyapunov.

Definition 3.3 The equilibrium point x = xe of the system (3.1) is said to be asymptoticallystable if it is both stable and convergent.

Asymptotic stability (Figure 3.2) is the desirable property in most applications. Theprincipal weakness of this concept is that it says nothing about how fast the trajectoriesapproximate the equilibrium point. There is a stronger form of asymptotic stability, referredto as exponential stability, which makes precise this idea.

Definition 3.4 The equilibrium point x = xe of the system (3.1) is said to be (locally)exponentially stable if there exist two real constants a, A > 0 such that

11x(t) - xell < a Ilx(0) - xell e- At Vt > 0 (3.2)

whenever Ilx(0) - xell < 6. It is said to be globally exponentially stable if (3.2) holds for anyxER".


Clearly, exponential stability is the strongest form of stability seen so far. It is also imme-diate that exponential stability implies asymptotic stability. The converse is, however, nottrue.

The several notions of stability introduced so far refer to stability of equilibrium points.In general, the same dynamical system can have more than one isolated equilibrium point.Very often, in the definitions and especially in the proofs of the stability theorems, it isassumed that the equilibrium point under study is the origin xe = 0. There is no loss ofgenerality in doing so. Indeed, if this is not the case, we can perform a change of variablesand define a new system with an equilibrium point at x = 0. To see this, consider theequilibrium point xe of the system (3.1) and define,

y = x-xey = i = f(x)f(x) = f(y + xe) f 9(y)

Thus, the equilibrium point ye of the new systems y = g(y) is ye = 0, since

9(0) = f(0 + xe) = f(xe) = 0.

According to this, studying the stability of the equilibrium point xe for the system x = f (x)is equivalent to studying the stability of the origin for the system y = g(y).

Given this property, in the sequel we will state the several stability theoremsassuming that xe = 0.

Example 3.1 Consider the mass-spring system shown in Figure 3.3. We have

my + fly + ky = mg

defining states x1 = y, x2 = y, we obtain the following state space realization,

J ±1 = X2ll x2=-mkxl-mx2+9

which has a unique equilibrium point xe = (k , 0). Now define the transformation z = x-xe.According to this

z1 = x1mg

-k

,= Z1 = 21

Z2 = x2 , : i 2 -

3.2. POSITIVE DEFINITE FUNCTIONS 69

Thus,

or

Figure 3.3: Mass-spring system.

zl = z2

z2 = - m (zl + k ) - mz2 + g

I zl = Z2Sl z2 = -mk zl - mz2

Thus, z = g(x), and g(x) has a single equilibrium point at the origin.

3.2 Positive Definite Functions

Now that the concept of stability has been defined, the next step is to study how to analyzethe stability properties of an equilibrium point. This is the center of the Lyapunov stabilitytheory. The core of this theory is the analysis and construction of a class of functions tobe defined and its derivatives along the trajectories of the system under study. We start byintroducing the notion of positive definite functions. In the following definition, D representsan open and connected subset of R".

Definition 3.5 A function V : D -> IR is said to be positive semi definite in D if it satisfiesthe following conditions:

(1) 0 E D and V (O) = 0.

(ii) V(x) > 0, Vx in D - {0}.

V : D -4 IR is said to be positive definite in D if condition (ii) is replaced by (ii')

(ii') V(x) > 0 in D - {0}.

Finally, V : D -+ R is said to be negative definite (semi definite)in D if -V is positivedefinite (semi definite).

We will often abuse the notation slightly and write V > 0, V > 0, and V < 0 in D toindicate that V is positive definite, semi definite, and negative definite in D, respectively.

Example 3.2 The simplest and perhaps more important class of positive definite functionis defined as follow:s,

V(x):IR"->1R=xTQx, QEIRnxn, Q=QT.

In this case, defines a quadratic form. Since by assumption, Q is symmetric (i.e.,Q = QT), we know that its eigenvalues Ai, i = 1, n, are all real. Thus we have that

positive definite

positive semi definite

V(.) negative definite

negative semi definite

xTQx>O,dx0O b A,>O,di=1,..., nxTQx>0,Vx#0 = A,>0,Vi=1,..., nxTQx < 0, Vx # 0 b At < 0, Vi = 1, ... , nxTQx<O,dx#0 n

Thus, for example:

1 rI >0, b'a, b > 0Vi (x) 1R R: 2 --> = axi + bx2 = [x1, x2] [

0

ab

J L X2

1

V2(x) : IIt2 1R = axi = [xl, x21 L0 0 ] [ X2 . > 0, Va > 0.

is not positive definite since for any x2 # 0, any x of the form x' = [0,x2]T # 0;however, V2(x*) = 0.

Positive definite functions (PDFs) constitute the basic building block of the Lyapunovtheory. PDFs can be seen as an abstraction of the total "energy" stored in a system, as wewill see. All of the Lyapunov stability theorems focus on the study of the time derivative of apositive definite function along the trajectories of 3.1. In other words, given an autonomous

3.3. STABILITY THEOREMS 71

system of the form 3.1, we will first construct a positive definite function V(x) and studyV (x) given by

V(x) = dV aV dxd - ax d = VV f(x)

av av av fl (x)[axl' axe ...' axn]

f, (x)

The following definition introduces a useful and very common way of representing thisderivative.

Definition 3.6 Let V : D -+ R and f : D -> R'. The Lie derivative of V along f , denotedby LfV, is defined by

LfV(x) = a f (x).

Thus, according to this definition, we have that

V(x) = as f(x) = VV f(x) = LfV(x).

Example 3.3 Letax1

bx2 + cos xl

and define V = x1 + x2. Thus, we have

V(x) = LfV(x) = [2x1,2x2] axlbx2 + cos xl 1

taxi + 2bx2 + 2x2 cos x1.

11

It is clear from this example that the V(x) depends on the system's equation f (x) and thusit will be different for different systems.

3.3 Stability Theorems

Theorem 3.1 (Lyapunov Stability Theorem) Let x = 0 be an equilibrium point of ± =f (x), f : D -+ R", and let V : D -> R be a continuously differentiable function such that

72 CHAPTER 3. LYAPUNOV STABILITY I: AUTONOMOUS SYSTEMS

(i) V(O) = 0,

(ii) V(x) > 0 in D - {0},

(iii) V(x) < 0 in D - {0},

thus x = 0 is stable.

In other words, the theorem implies that a sufficient condition for the stability of theequilibrium point x = 0 is that there exists a continuously differentiable-positive definitefunction V (x) such that V (x) is negative semi definite in a neighborhood of x = 0.

As mentioned earlier, positive definite functions can be seen as generalized energyfunctions. The condition V (x) = c for constant c defines what is called a Lyapunov surface.A Lyapunov surface defines a region of the state space that contains all Lyapunov surfacesof lesser value, that is, given a Lyapunov function and defining

S21 = {x E Br : V(x) < c1}522 = {x E Br : V(x) < c2}

where Br = {x E R" : JJxlJ < rl }, and c1 > c2 are chosen such that 522 C Br, i = 1, 2,then we have that 522 C 521. The condition V < 0 implies that when a trajectory crosses aLyapunov surface V (x) = c, it can never come out again. Thus a trajectory satisfying thiscondition is actually confined to the closed region SZ = {x : V(x) < c}. This implies thatthe equilibrium point is stable, and makes Theorem 3.1 intuitively very simple.

Now suppose that V(x) is assumed to be negative definite. In this case, a trajectorycan only move from a Lyapunov surface V (x) = c into an inner Lyapunov surface withsmaller c. This clearly represents a stronger stability condition.

Theorem 3.2 (Asymptotic Stability Theorem) Under the conditions of Theorem 3.1, if

V (O) = 0,

(ii) V(x) > 0 in D - {0},

(iii) V(x) < 0 in D - {0},

thus x = 0 is asymptotically stable.

In other words, the theorem says that asymptotic stability is achieved if the conditionsof Theorem 3.1 are strengthened by requiring V (x) to be negative definite, rather than semidefinite.

The discussion above is important since it elaborates on the ideas and motivationbehind all the Lyapunov stability theorems. We now provide a proof of Theorems 3.1 and3.2. These proofs will clarify certain technicalities used later on to distinguish between localand global stabilities, and also in the discussion of region of attraction.

Proof of theorem 3.1: Choose r > 0 such that the closed ball

Br={xE III": IxMI<r}

is contained in D. So f is well defined in the compact set Br. Let

a = min V (x) (thus a > 0, by the fact that V (x) > 0 E D).IxJJ=r

Now choose Q E (0, a) and denote

0,3={xEBr:V(x)<31-

Thus, by construction, f2p C B,.. Now suppose that x(0) E S20. By assumption (iii) of thetheorem we have that

V(x) < 0 = V(x) < V(x(0)) <'3 Vt > 0.

It then follows that any trajectory starting in Slp at t = 0 stays inside Sup for all t > 0.Moreover, by the continuity of V(x) it follows that 3b > 0 such that

< ' 3< Q (B6 C 00 C Br).

It then follows thatIX(O)II < 6 = x(t) E Q0 C Br Vt > 0

and then1x(O)11 Ix(t)II < T < f Vt > 0

which means that the equilibrium x = 0 is stable. ED

Proof of theorem 3.2: Under the assumptions of the theorem, V (x) actually decreasesalong the trajectories of f (x). Using the same argument used in the proof of Theorem 3.1,for every real number a > 0 we can find b > 0 such that Ib C Ba, and from Theorem 3.1whenever the initial condition is inside f2b, the solution will remain inside 52b. Therefore,to prove asymptotic stability, all we need to show is that 1b reduces to 0 in the limit. In

Figure 3.4: Pendulum without friction.

other words, 52b shrinks to a single point as t -> oo. However, this is straightforward, sinceby assumption

f7(x) < 0 in D.

Thus, V (x) tends steadily to zero along the solutions of f (x). This completes the proof.

Remarks: The first step when studying the stability properties of an equilibrium pointconsists of choosing a positive definite function Finding a positive definite function isfairly easy; what is rather tricky is to select a whose derivative along the trajectoriesnear the equilibrium point is either negative definite, or semi definite. The reason, of course,is that is independent of the dynamics of the differential equation under study, whileV depends on this dynamics in an essential manner. For this reason, when a functionis proposed as possible candidate to prove any form of stability, such a is said to be aLyapunov function candidate. If in addition happens to be negative definite, then Vis said to be a Lyapunov function for that particular equilibrium point.

3.4 Examples

Example 3.4 (Pendulum Without Friction)

Using Newton's second law of motion we have,

ma = -mg sin 9a = la = l9

3.4. EXAMPLES

where l is the length of the pendulum, and a is the angular acceleration. Thus

mlO+mgsin0 = 0or B+ l sing = 0

choosing state variables

we have

c

{xl = 0X2 = 0

.j1 = x222 = -gsinxl

75

which is of the desired form i = f (x). The origin is an equilibrium point (since f (0) = 0).To study the stability of the equilibrium at the origin, we need to propose a Lyapunov functioncandidate V (x) and show that satisfies the properties of one of the stability theorems seenso far. In general, choosing this function is rather difficult, however, in this case we proceedinspired by our understanding of the physical system. Namely, we compute the total energyof the pendulum (which is a positive function), and use this quantity as our Lyapunovfunction candidate. We have

E = K + P (kinetic plus potential energy)

= 2m(wl)2 + mgh

where

Thus

w = 0 = x2h = l(1 - cosO) = l(1 - cosxl).

E = 2ml2x2 + mgl(1 - cosxl).

We now define V(x) = E and investigate whether and its derivative satisfy theconditions of Theorem 3.1 and/or 3.2. Clearly, V(0) = 0; thus, defining property (i) issatisfied in both theorems. With respect to (ii), we see that because of the periodicity ofcos(xj), we have that V(x) = 0 whenever x = (xl, x2)T = (2krr, 0) T, k = 1, 2, . Thus,

is not positive definite. This situation, however, can be easily remedied by restricting thedomain of xl to the interval (-27r, 21r); i. e., we take V : D -4 R, with D = ((-27r, 27r), 1R)T.


With this restriction, V : D -> R is indeed positive definite. There remains to evaluate thederivative of along the trajectories of f(t). We have

V(x) _ VV f(x)IV aV [fl(x),f2(x)1T=axl ax2

_ [mgl sin xl, ml2x2] [x2, -9 sinxl]T

mglx2 sin x1 - mglx2 sin xl = 0.

Thus V (x) = 0 and the origin is stable by Theorem 3.1.

The result of Example 3.4 is consistent with our physical observations. Indeed, a simplependulum without friction is a conservative system. This means that the sum of the kineticand potential energy remains constant. The pendulum will continue to balance withoutchanging the amplitude of the oscillations and thus constitutes a stable system. In ournext example we add friction to the dynamics of the pendulum. The added friction leadsto a loss of energy that results in a decrease in the amplitude of the oscillations. In thelimit, all the initial energy supplied to the pendulum will be dissipated by the friction forceand the pendulum will remain at rest. Thus, this version of the pendulum constitutes anasymptotically stable equilibrium of the origin.

Example 3.5 (Pendulum with Friction) We now modify the previous example by addingthe friction force klO

ma = -mg sin 0 - klO

defining the same state variables as in example 3.4 we have

X2

X2

-S1Ilxl - nk,x2

Again x = 0 is an equilibrium point. The energy is the same as in Example 3.4. Thus

V(x) = 2m12x2 + mgl(1 - cosxl) >0 in D - {0}

V(x) _ VV . f(x)IV IV [fl (x), f2(x)]T=axl

,

ax2

[mgl sin xl, m12x2] [x2, -9 sin xlk x21T

m_ -k12x2.

3.5. ASYMPTOTIC STABILITY IN THE LARGE 77

Thus V(x) is negative semi-definite. It is not negative definite since V(x) = 0 for x2 = 0,regardless of the value of xl (thus V (x) = 0 along the xl axis). According to this analysis, weconclude that the origin is stable by Theorem 3.1, but cannot conclude asymptotic stabilityas suggested by our intuitive analysis, since we were not able to establish the conditions ofTheorem 3.2. Namely, V(x) is not negative definite in a neighborhood of x = 0. The resultis indeed disappointing since we know that a pendulum with friction has an asymptoticallystable equilibrium point at the origin.

This example emphasizes the fact that all of the theorems seen so far provide sufficient butby no means necessary conditions for stability.


±2 -XI + x2(x1 + x - a2).

To study the equilibrium point at the origin, we define V(x) = 1/2(x2 +x2). We have

V (x) = VV f (x)[xl//, x2] [xl (X2 + x - /32) + x2, -xlx2 (2 + x2 - Q2)]T

xllxl+2-'32)+x2(xi+2-'32)

(xi + x2)(xl + 2 - Q2).

Thus, V(x) > 0 and V(x) < 0, provided that (xl +x2) < Q2, and it follows that the originis an asymptotically stable equilibrium point.

3.5 Asymptotic Stability in the Large

A quick look at the definitions of stability seen so far will reveal that all of these conceptsare local in character. Consider, for example, the definition of stability. The equilibrium xeis said to be stable if

JIx(t) - xell < e, provided that 11x(0) - xell < b

or in words, this says that starting "near" xei the solution will remain "near" xe. Moreimportant is the case of asymptotic stability. In this case the solution not only stays withine but also converges to xe in the limit. When the equilibrium is asymptotically stable,it is often important to know under what conditions an initial state will converge to theequilibrium point. In the best possible case, any initial state will converge to the equilibriumpoint. An equilibrium point that has this property is said to be globally asymptotically stable,or asymptotically stable in the large.

Definition 3.7 The equilibrium state xe is said to be asymptotically stable in the large, orglobally asymptotically stable, if it is stable and every motion converges to the equilibriumast-3oo.

At this point it is tempting to infer that if the conditions of Theorem 3.2 hold inthe whole space PJ', then the asymptotic stability of the equilibrium is global. Whilethis condition is clearly necessary, it is however not sufficient. The reason is that theproof of Theorem 3.1 (and so also that of theorem 3.2) relies on the fact that the positivedefiniteness of the function V (x) coupled with the negative definiteness of V (x) ensure thatV(x) < V(xo). This property, however, holds in a compact region of the space defined inTheorem 3.1 by ft = {x E B,. V (x) <,3}. More precisely, in Theorem 3.1 we started bychoosing a ball B,. _ {x E R' Jxii < r} and then showed that Stp C B,.. Both sets S2pand B, are closed set (and so compact, since they are also bounded). If now B, is allowedto be the entire space lR the situation changes since the condition V(x) < 3 does not, ingeneral, define a closed and bounded region. This in turn implies that no is not a closedregion and so it is possible for state trajectories to drift away from the equilibrium point,even if V(x) < Q. The following example shows precisely this.

Example 3.7 Consider the following positive definite function:

V(x)= X2

1+X2+ x2.

1

The region V(x) < Q is closed for values of 3 < 1. However, if Q > 1, the surface is open.Figure 3.5 shows that an initial state can diverge from the equilibrium state at the originwhile moving towards lower energy curves.

The solution to this problem is to include an extra condition that ensures that V (x) =Q is a closed curve. This can be achieved by considering only functions V(.) that growunbounded when x -> oo. This functions are called radially unbounded.

Definition 3.8 Let V : D -4 R be a continuously differentiable function. Then V (x) issaid to be radially unbounded if

V(x) ->oo as iixli-goo.

Theorem 3.3 (Global Asymptotic Stability) Under the conditions of Theorem 3.1, if V(.)is such that

3.5. ASYMPTOTIC STABILITY IN THE LARGE 79

Figure 3.5: The curves V(x) = 0.

(i) V(0) = 0.

(ii) V(x)>0 Vx#0.

(iii) V(x) is radially unbounded.

(iii) V (x) < 0 Vx # 0.

then x = 0 is globally asymptotically stable.

Proof: The proof is similar to that of Theorem 3.2. We only need to show that given anarbitrary /3 > 0, the condition

S2p = {x E Rn : V (X) < /3}

defines a set that is contained in the ball B,. = {x E R' : IIxII < r}, for some r > 0. To seethis notice that the radial unboundedness of implies that for any 0 > 0, 3r > 0 suchthat V(x) > 0 whenever IIxII > r, for some r > 0. Thus, S20 C B,., which implies that S20is bounded.

Example 3.8 Consider the following system2xl = x2 - x1(xl + X2

2 2x2 = -21 - 2201 + 22).

To study the equilibrium point at the origin, we define V (x) = x2 + x2. We have

(2) = 7f(x)2 22[xlix2][x2 -21(21 +22),-21 -22(21 -f-22)]T

-2(x1 + 22)2

Thus, V(x) > 0 and V(x) < 0 for all x E R2. Moreover, since is radially unbounded,it follows that the origin is globally asymptotically stable.

3.6 Positive Definite Functions Revisited

We have seen that positive definite functions play an important role in the Lyapunov theory.We now introduce a new class of functions, known as class 1C, and show that positive definitefunctions can be characterized in terms of this class of functions. This new characterizationis useful in many occasions.

Definition 3.9 A continuous function a : [0, a) -> R+ is said to be in the class K if

(i) a(0) = 0.

(ii) It is strictly increasing.

a is said to be in the class KQO if in addition a : ]EF+ -+ IIt+ and a(r) -+ oo as r -+ oo.

In the sequel, B,. represents the ball

Br= {xER":IxII<r}.

Lemma 3.1 V : D --> R is positive definite if and only if there exists class K functions a1and a2 such that

al(II2II) 5 V(x) < a2(IIxII) Vx E Br C D.

Moreover, if D = 1R' and is radially unbounded then a1 and a2 can be chosen in theclass K.

Proof: See the Appendix.

3.6. POSITIVE DEFINITE FUNCTIONS REVISITED 81

Example 3.9 Let V (x) = xT Px, where P is a symmetric matrix. This function is positivedefinite if and only if the eigenvalues of the symmetric matrix P are strictly positive. Denote)m%n(P) and ),ax(P) the minimum and maximum eigenvalues of P, respectively. It thenfollows that

Am.m(P)Ilxll2 < xTPx < Amax(P)Ilxll2.m,n(P)Ilxll2 < V(x) < Amax(P)Ilxll2.

Thus, al, a2 : [0, oc) --> R+, and are defined by

al(x) = .min(P)Ilxl12a2(x) = \max(P)Ilxll2.

For completeness, we now show that it is possible to re state the stability definitionin terms of class 1C of functions.

Lemma 3.2 The equilibrium xe of the system (3.1) is stable if and only if there exists aclass IC function a(.) and a constant a such that

1x(0) - xell < a = lx(t) - xell <_ a(I1*0) - xell) Vt > 0. (3.4)


A stronger class of functions is needed in the definition of asymptotic stability.

Definition 3.10 A continuous function 3 : (0, a) x pg+ -+ R+ is said to be in the class KLif

(i) For fixed s, 33(r, s) is in the class IC with respect to r.

(ii) For fixed r, 3(r, s) is decreasing with respect to s.

(iii) 0(r,s)---*0 ass -oc.

Lemma 3.3 The equilibrium xe of the system (3.1) is asymptotically stable if and only ifthere exists a class ICL function and a constant e such that

1x(0) - xell < S = 1x(t) - xell <_ 0(I1x(0) - xell, t) Vt > 0. (3.5)

3.6.1 Exponential Stability

As mentioned earlier, exponential stability is the strongest form of stability seen so far. Theadvantage of this notion is that it makes precise the rate at which trajectories convergeto the equilibrium point. Our next theorem gives a sufficient condition for exponentialstability.

Theorem 3.4 Suppose that all the conditions of Theorem 3.2 are satisfied, and in additionassume that there exist positive constants K1, K2, K3 and p such that

Klllxllp < V(x) < K21IxljPV(x) -K3IIxIlP.

Then the origin is exponentially stable. Moreover, if the conditions hold globally, the x = 0is globally exponentially stable.

Proof: According to the assumptions of Theorem 3.4, the function V (x) satisfies Lemma3.1 with al and a2(), satisfying somewhat strong conditions. Indeed, by assumption

Klllxllp < V(x) <_ K2llxjIPV(x) < -K3IIxllP

< -KV(x)

V(x) < -V(x)V(x) < V(xo)e-(K3/K2)t

= Ilxll < V x ]1/p < [V xo e-cK3/K2)t]1/p1 ,

or

lix(t)ll <_ llxoll [K2]1/P e-(K3/2K2)t.

3.7 Construction of Lyapunov Functions

The main shortcoming of the Lyapunov theory is the difficulty associated with the construc-tion of suitable Lyapunov functions. In this section we study one approach to this problem,known as the "variable gradient." This method is applicable to autonomous systems andoften but not always leads to a desired Lyapunov function for a given system.

3.7. CONSTRUCTION OF LYAPUNOV FUNCTIONS 83

The Variable Gradient: The essence of this method is to assume that the gradient ofthe (unknown) Lyapunov function V(.) is known up to some adjustable parameters, andthen finding itself by integrating the assumed gradient. In other words, we start outby assuming that

V V (x) = 9(x), (= V (x) = VV (x) . f (x) = g(x) . f (x))

and propose a possible function g(x) that contains some adjustable parameters. An exampleof such a function, for a dynamical system with 2 states x1 and x2, could be

9(X)=[91,921 = [hixl + hix2, hzxl + h2x2]. (3.6)

The power of this method relies on the following fact. Given that

VV(x) = g(x) it follows that

g(x)dx = VV(x)dx = dV(x)

thus, we have that

V(Xb) - V (xa) =Xb

VV(x) dx =Xb

g(x) dxf J2

that is, the difference V(Xb) - V(xa) depends on the initial and final states xa and xb andnot on the particular path followed when going from xa to xb. This property is often usedto obtain V by integrating VV(x) along the coordinate axis:

X

xl

V (X) = f 9(x) dx =0

J gl (s1, 0, ... 0) dsl0

fzZ x,

+J

92(x1, S2, 0, ... 0) dS2 + ... + 1 9n(x1, x2, ... Sn) dSn. (3.7)0 0

The free parameters in the function g(x) are constrained to satisfy certain symmetry con-ditions, satisfied by all gradients of a scalar function. The following theorem details theseconditions.

Theorem 3.5 A function g(x) is the gradient of a scalar function V(x) if and only if thematrix

ax,

is symmetric.



We now put these ideas to work using the following example.


.i1 = -axl2 = bx2 + xlx2.

Clearly, the origin is an equilibrium point. To study the stability of this equilibrium point,we proceed to find a Lyapunov function as follows.

Step 1: Assume that VV(x) = g(x) has the form

x1 + h2x2]. (3.8)g(x) = [hlxl + hix2, hl2 2

Step 2: Impose the symmetry conditions,

202V= a V

or, equivalentlyas,, - ag,

axtax, ax, ax, ax; - ax,

In our case we have

ahl1991 h2ahlxl + x2+ax2 ax2 ax2

1 2

19x1= h2 + x1

ax,

+ x2 axl .

To simplify the solution, we attempt to solve the problem assuming that the J, 's are constant.If this is the case, then

ahi = ahi = ah2 = ah2= 0

ax2 ax2 19x1 axl

and we have that:1991 = 1992 2 =1 =ax2 19x1

hl h2 k

g(x) = [hlxl + kx2i kxl + h2x2].

In particular, choosing k = 0, we have

9(x) = [91, 92] = [hlxl, h2x2]

3.8. THE INVARIANCE PRINCIPLE

Step 3: Find V:

V(x) = VV f(x)= 9(x) f(x)= [hix1, h2x2]f (x)= -ahixi +h2 (b + xlx2)x2.

Step 4: Find V from VV by integration. Integrating along the axes, we have that

Step 5:

V(x) = I hix2 + 1 h2x2V(x) -ahixi+h2(b+xlx2)x2

85

From (3.9), V (x) > 0 if and only if hi, h2 > 0. Assume then that hi = h2 = 1. In this case

V(x) = -axe + (b+x1x2)x2

assume now that a > 0, and b < 0. In this case

V(x) = -axi - (b - x1x2)x2

and we conclude that, under these conditions, the origin is (locally) asymptotically stable.

3.8 The Invariance Principle

Asymptotic stability is always more desirable that stability. However, it is often the casethat a Lyapunov function candidate fails to identify an asymptotically stable equilibriumpoint by having V (x) negative semi definite. An example of this is that of the pendulumwith friction (Example 3.5). This shortcoming was due to the fact that when studying the

JO

x1

ix1

91(sl, 0) dsl +fx2

92(x1, s2) ds20

hiss dsl +

1 1 22h1x1

10

X2

h2s2 ds2

1 2X2+ 2h2x2.

Verify that V > 0 and V < 0. we have that

properties of the function V we assumed that the variables xl and x2 are independent. Thosevariables, however, are related by the pendulum equations and so they are not independentof one another. An extension of Lyapunov's theorem due to LaSalle studies this problem ingreat detail. The central idea is a generalization of the concept of equilibrium point calledinvariant set.

Definition 3.11 A set M is said to be an invariant set with respect to the dynamical systemi = f(x) if'

x(0)EM = x(t)EM VtER+.

In other words, M is the set of points such that if a solution of i = f (x) belongs to M atsome instant, initialized at t = 0, then it belongs to M for all future time.

Remarks: In the dynamical system literature, one often views a differential equation asbeing defined for all t rather than just all the nonnegative t, and a set satisfying the definitionabove is called positively invariant.

The following are some examples of invariant sets of the dynamical system i = f (x).

Example 3.11 Any equilibrium point is an invariant set, since if at t = 0 we have x(0) =x, then x(t) = xe Vt > 0.

Example 3.12 For autonomous systems, any trajectory is an invariant set.

Example 3.13 A limit cycle is an invariant set (this is a special case of Example 3.12).

11

Example 3.14 If V(x) is continuously differentiable (not necessarily positive definite) andsatisfies V(x) < 0 along the solutions of i = f (x), then the set Sgt defined by

SZi={xE1R' :V(x)<l}

is an invariant set. Notice that the condition V < 0 implies that if a trajectory crosses aLyapunov surface V(x) = c it can never come out again.

Example 3.15 The whole space R" is an invariant set.

3.8. THE INVARIANCE PRINCIPLE 87

Definition 3.12 Let x(t) be a trajectory of the dynamical system ± = f (x). The set N iscalled the limit set (or positive limit set) of x(t) if for any p E N there exist a sequence oftimes {tn} E [0, oc) such that

x(t,,)->p as t -*ocor, equivalently

urn IJx(tn) -p1l = 0.

Roughly speaking, the limit set N of x(t) is whatever x(t) tends to in the limit.

Example 3.16 An asymptotically stable equilibrium point is the limit set of any solutionstarting sufficiently near the equilibrium point.

Example 3.17 A stable limit cycle is the positive limit set of any solution starting suffi-ciently near it.

Lemma 3.4 If the solution x(t, x0i to) of the system (3.1) is bounded for t > to, then its(positive) limit set N is (i) bounded, (ii) closed, and (iii) nonempty. Moreover, the solutionapproaches N as t - oc.


The following lemma can be seen as a corollary of Lemma 3.4.

Lemma 3.5 The positive limit set N of a solution x(t, x0i to) of the autonomous system(3.1) is invariant with respect to (3.1).


Invariant sets play a fundamental role in an extension of Lyapunov's work producedby LaSalle. The problem is the following: recall the example of the pendulum with friction.Following energy considerations we constructed a Lyapunov function that turned out tobe useful to prove that x = 0 is a stable equilibrium point. However, our analysis, basedon this Lyapunov function, failed to recognize that x = 0 is actually asymptotically stable,something that we know thanks to our understanding of this rather simple system. LaSalle'sinvariance principle removes this problem and it actually allows us to prove that x = 0 isindeed asymptotically stable.

We start with the simplest and most useful result in LaSalle's theory. Theorem 3.6can be considered as a corollary of LaSalle's theorem as will be shown later. The difference

between theorems 3.6 and theorem 3.2 is that in Theorem 3.6 is allowed to be onlypositive semi-definite, something that will remove part of the conservativism associatedwith certain Lyapunov functions.

Theorem 3.6 The equilibrium point x = 0 of the autonomous system (3.1) is asymptoti-cally stable if there exists a function V(x) satisfying

(i) V(x) positive definite Vx E D, where we assume that 0 E D.

(ii) 1 (x) is negative semi definite in a bounded region R C D.

(iii) V(x) does not vanish identically along any trajectory in R, other than the null solutionx=0.

Example 3.18 Consider again the pendulum with friction of Example 3.5:

xl

x2

X2

g.

k- l slnx1 - -x2.m

Again

V (X) > 0 Vx E (-7r, -7r) x IR,

(x) _ -kl2x2 (3.13)

which is negative semi definite since V (x) = 0 for all x = [x1i 0]T . Thus, with V short ofbeing negative definite, the Lyapunov theory fails to predict the asymptotic stability of theorigin expected from the physical understanding of the problem. We now look to see whetherapplication of Theorem 3.6 leads to a better result. Conditions (i) and (ii) of Theorem 3.6are satisfied in the region

RX2

l

with -7r < xl < 7r, and -a < X2 < a, for any a E IR+. We now check condition (iii) ofthe same theorem, that is, we check whether V can vanish identically along the trajectoriestrapped in R, other than the null solution. The key of this step is the analysis of the conditionV = 0 using the system equations (3.11)-(3.12). Indeed, assume that V(x) is identicallyzero over a nonzero time interval. By (3.13) we have

l2xV (x) = 0 0 = -k2 b X2 = 0

and by (3.12), we obtain

thus x2 = 0 Vt = x2 = 0

sinxl k- x2 and thus x2 = 0 = sin x1 = 00 = 9-n

restricting xl to the interval xl E (-7r, Tr) the last condition can be satisfied if and only ifxl = 0. It follows that V (x) does not vanish identically along any solution other than x = 0,and the origin is (locally) asymptotically stable by Theorem 3.6.

Proof of Theorem 3.6: By the Lyapunov stability theorem (Theorem 3.1), we know thatfor each e > 0 there exist b > 0

1x011 lx(t)II < E

that is, any solution starting inside the closed ball Bs will remain within the closed ball B,.Hence any solution x(t, x0i t0) of (3.1) that starts in B6 is bounded and tends to its limitset N that is contained in B, (by Lemma 3.4). Also V(x) is continuous on the compact setB, and thus is bounded from below in B,. It is also non increasing by assumption and thustends to a non negative limit L as t -* oo. Notice also that V (x) is continuous and thus,V(x) = L Vx in the limit set N. Also by lemma 3.5, N is an invariant set with respect to(3.1), which means that any solution that starts in N will remain there for all future time.But along that solution, V (x) = 0 since V (x) is constant (= L) in N. Thus, by assumption,N is the origin of the state space and we conclude that any solution starting in R C B6converges to x = 0 as t -4 oo.

Theorem 3.7 The null solution x = 0 of the autonomous system (3.1) is asymptoticallystable in the large if the assumptions of theorem 3.6 hold in the entire state space (i.e.,R = R), and V(.) is radially unbounded.

Proof: The proof follows the same argument used in the proof of Theorem 3.3 and isomitted.


To study the equilibrium

xl = x2

x2 = -x2 - axl - (x1 + x2)2x2.

point at the origin we define V (x) = axe + x2. We have

(x) = 8- f (x)

2[ax1, x2] [x2, -x2 - ax1 - (x1 + x2)2x2]T

-2x2[1 + (x1 +X2)2].

Thus, V(x) > 0 and V(x) < 0 since V(x) = 0 for x = (xii0). Proceeding as in the previousexample, we assume that V = 0 and conclude that

V=0 x2=0, x2=0 Vt 22=0

X2=0 = -x2 - a X1 - (x1 + x2)2x2 = 0

and considering the fact that x2 = 0, the last equation implies that xl = 0. It follows thatV (x) does not vanish identically along any solution other than x = [0, 0]T . Moreover, sinceV(.) is radially unbounded, we conclude that the origin is globally asymptotically stable.

Theorem 3.8 (LaSalle's theorem) Let V : D -* R be a continuously differentiable functionand assume that

(i) M C D is a compact set, invariant with respect to the solutions of (3.1).

(ii) V < 0 in M.

(iii) E : {x : x E M, and V = 0}; that is, E is the set of all points of M such that V = 0.

(iv) N: is the largest invariant set in E.

Then every solution starting in M approaches N as t -+ oo.

Proof: Consider a solution x(t) of (3.1) starting in M. Since V(x) < 0 E M, V(x) is adecreasing function of t. Also, since V(.) is a continuous function, it is bounded from belowin the compact set M. It follows that V(x(t)) has a limit as t -+ oo. Let w be the limitset of this trajectory. It follows that w C M since M is (an invariant) closed set. For anyp E w3 a sequence to with to -+ oo and x(tn) -+ p. By continuity of V(x), we have that

V(p) = lim V(x(tn)) = a (a constant).n- oo

Hence, V (x) = a on w. Also, by Lemma 3.5 w is an invariant set, and moreover V (x) = 0on w (since V(x) is constant on w). It follows that

wCNCEcM.Since x(t) is bounded, Lemma 3.4 implies that x(t) approaches w (its positive limit set) ast -+ oo. Hence x(t) approaches N as t -> oo.

Remarks: LaSalle's theorem goes beyond the Lyapunov stability theorems in two impor-tant aspects. In the first place, V(-) is required to be continuously differentiable (and so

bounded), but it is not required to be positive definite. Perhaps more important, LaSalle'sresult applies not only to equilibrium points as in all the Lyapunov theorems, but also tomore general dynamic behaviors such as limit cycles. Example 3.20, at the end of thissection, emphasizes this point.

Before looking at some examples, we notice that some useful corollaries can be foundif is assumed to be positive definite.

Corollary 3.1 Let V : D -4 R be a continuously differentiable positive definite function ina domain D containing the origin x = 0, and assume that V (x) < 0 E D. Let S = {x ED : V (x) = 0} and suppose that no solution can stay identically in S other than the trivialone. Then the origin is asymptotically stable.

Proof: Straightforward (see Exercise 3.11).

Corollary 3.2 If D = Rn in Corollary 3.1, and is radially unbounded, then the originis globally asymptotically stable.

Proof: See Exercise 3.12.

Example 3.20 [68] Consider the system defined by

it = x2 + xl(Q2 - x1 - x2)i2 = -xi + x2(02 - xi - x2)

It is immediate that the origin x = (0, 0) is an equilibrium point. Also, the set of pointsdefined by the circle x1 + x2 = (32 constitute an invariant set. To see this, we compute thetime derivative of points on the circle, along the solution of i = f (x):

T [x2 + x2 - Q2] = (2x1, 2x2)f (x)

2(xi + x2)(b2 - xi - x2)

for all points on the set. It follows that any trajectory initiating on the circle stays on thecircle for all future time, and thus the set of points of the form {x E R2 : xl + x2 = 32}constitute an invariant set. The trajectories on this invariant set are described by thesolutions of

i = f(x)I. ztx2=R2r x1 = x2

x.x 1 2 l x2 = -xl.

Thus, the circle is actually a limit cycle along which the trajectories move in the clockwisedirection.

We now investigate the stability of this limit cycle using LaSalle's theorem. To thisend, consider the following function:

V(x) = 1(x2 +x2 -o2)2

Clearly, is positive semi definite in R2. Also

1av' avf(x)axl axl

-(x2 +x 2)(xl + x2 - a2)2 < 0.

Moreover, V(x) = 0 if and only if one of the following conditions is satisfied:

(a)xi+x2=0(b) xl+x2-a2=0.

In other words, V = 0 either at the origin or on the circle of radius p. We now applyLaSalle's theorem.

Step #1: Given any real number c > f3, define the set M as follows:

M={xEIR2:V(x)<c}.

By construction M is closed and bounded (i.e., compact). Also V(x) < OVx E M, andthus any trajectory staring from an arbitrary point xo E M will remain inside M and M istherefore an invariant set.

Step #2: Find E = {x E M : V(x) = 0}. Clearly, we have

E=(0,0)U{xER2:xi+x2=j32}

that is, E is the union of the origin and the limit cycle.

Step #3: Find N, the largest invariant set in E.

This is trivial, since E is the union of the origin (an invariant set, since (0, 0) is anequilibrium point), and the invariant set x1 +x2 = Q2. Thus, we conclude that N = E, andby LaSalle's theorem, every motion staring in M converges to either the origin or the limitcycle.

3.9. REGION OF ATTRACTION 93

We can refine our argument by noticing that the function V was designed to measurethe distance from a point to the limit cycle:

V(x) = 2(x1 +x2 - 32)2

r V (x) = 0 whenever x2 + x2 = /j25l V (x) = 2/34 when x = (0, 0).

Thus, choosing c : 0 < c < 1/204 we have that M = {x E JR2 : V(x) < c} includes thelimit cycle but not the origin. Thus application of LaSalle's theorem with any c satisfyingc : e < c < 1/234 with e arbitrarily small shows that any motion starting in M convergesto the limit cycle, and the limit cycle is said to be convergent, or attractive. The sameargument also shows that the origin is unstable since any motion starting arbitrarily near(0, 0) converges to the limit cycle, thus diverging from the origin.

3.9 Region of Attraction

As discussed at the beginning of this chapter, asymptotic stability if often the most desirableform of stability and the focus of the stability analysis. Throughout this section we restrictour attention to this form of stability and discuss the possible application of Theorem 3.2to estimate the region of asymptotic stability. In order to do so, we consider the followingexample,

Example 3.21 Consider the system defined by

r 1 = 3x21. 22 = -5x1 + x1 - 2x2.

This system has three equilibrium points: xe = (0, 0), xe = (-f , 0), and xe = (v', 0). Weare interested in the stability of the equilibrium point at the origin. To study this equilibriumpoint, we propose the following Lyapunov function candidate:

V(x) = axe - bxl + cx1x2 + dx2

with a, b, c, d constants to be determined. Differentiating we have that

V = (3c - 4d)x2 + (2d - 12b)x3 x2 + (6a - 10d - 2c)xlx2 + cxi - 5cxi

we now choose{ 2d - 12b = 0

(3.14)6a-10d-2c = 0

which can be satisfied choosing a = 12, b = 1, c = 6, and d = 6. Using these values, weobtain

V (x) = 12x2 - xi + 6x1x2 + 6x2= 3(x1 + 2x2)2 + 9x2 + 3x2 - x1 (3.15)

V(x) = -6x2 - 30x2 + 6xi. (3.16)

So far, so good. We now apply Theorem 3.2. According to this theorem, if V(x) > 0 andV < 0 in D - {0}, then the equilibrium point at the origin is "locally" asymptotically stable.The question here is: What is the meaning of the word "local"? To investigate this issue,we again check (3.15) and (3.16) and conclude that defining D by

D = {xEIIF2:-1.6<xl <1.6} (3.17)

we have that V(x) > 0 and V < 0, Vx E D - {0}. It is therefore "tempting" to conclude thatthe origin is locally asymptotically stable and that any trajectory starting in D will movefrom a Lyapunov surface V(xo) = cl to an inner Lyapunov surface V(xl) = c2 with cl > C2,thus suggesting that any trajectory initiating within D will converge to the origin.

To check these conclusions, we plot the trajectories of the system as shown in Figure3.6. A quick inspection of this figure shows, however, that our conclusions are incorrect.For example, the trajectory initiating at the point x1 = 0, x2 = 4 is quickly divergent fromthe origin even though the point (0, 4) E D. The problem is that in our example we triedto infer too much from Theorem 3.2. Strictly speaking, this theorem says that the origin islocally asymptotically stable, but the region of the plane for which trajectories converge tothe origin cannot be determined from this theorem alone. In general, this region can be avery small neighborhood of the equilibrium point. The point neglected in our analysis is asfollows: Even though trajectories starting in D satisfy the conditions V(x) > 0 and V < 0,thus moving to Lyapunov surfaces of lesser values, D is not an invariant set and there areno guarantees that these trajectories will stay within D. Thus, once a trajectory crosses theborder xiI = f there are no guarantees that V(x) will be negative.

In summary: Estimating the so-called "region of attraction" of an asymptotically stableequilibrium point is a difficult problem. Theorem 3.2 simply guarantees existence of apossibly small neighborhood of the equilibrium point where such an attraction takes place.We now study how to estimate this region. We begin with the following definition.

Definition 3.13 Let 1li(x, t) be the trajectories of the systems (3.1) with initial condition xat t = 0. The region of attraction to the equilibrium point x, denoted RA, is defined by

RA={xED:1/i(x,t)-->xei as t --goo}.

3.9. REGION OF ATTRACTION 95

Figure 3.6: System trajectories in Example 3.21.

In general, the exact determination of this region can be a very difficult task. In this sectionwe discuss one way to "estimate" this region. The following theorem, which is based entirelyon the LaSalle's invariance principle (Theorem 3.8), outlines the details.

Theorem 3.9 Let xe be an equilibrium point for the system (3.1). Let V : D -4 IR be acontinuous differentiable function and assume that

(i) M C D is a compact set containing xe, invariant with respect to the solutions of (3.1).

(ii) V is such that

V < 0 VX 34 X, E M.V = 0 ifx=xe.

Under these conditions we have that

M C RA.

In other words, Theorem 3.9 states that if M is an invariant set and V is such that V < 0inside M, then M itself provides an "estimate" of RA.

Proof: Under the assumptions, we have that E _ {x : x E M, and V = 0} = xe. It thenfollows that N = largest invariant set in E is also xe, and the result follows from LaSalle'sTheorem 3.8.

Example 3.22 Consider again the system of Example 3.21:

I i1 = 3x2i2 = -5x1 + x1 - 2x2

V (x) = 12x1 - xl + 6x1x2 + 6x2V(x) = -6x2 - 30x1 +6x1

We know that V > 0 and V < 0 for all {x E R2 : -1.6 < xl < 1.6}. To estimate the regionof attraction RA we now find the minimum of V(x) at the very edge of this condition (i.e.,x = ±1.6). We have

VIx1=1.6 24.16 + 9.6x2 + 6x2 = z1dzl

= 9.6 + 12x2 = 0 a x2 = -0.8axe

lS imilar y

Vlx1=-1.6 = 24.16 - 9.6x2 + 6x2 = z2dz2 = -9.6 + 12x2 = 0 a x2 = 0.8.axe

Thus, the function V(±1.6,x2) has a minimum when x2 = ±0.8. It is immediate thatV(1.6, -0.8) = V(-1.6, 0.8) = 20.32. From here we can conclude that given any e > 0, theregion defined by

M = {x E IIY2: V(x) <20.32-e}

is an invariant set and satisfies the conditions of Theorem 3.9. This means that M C RA.

3.10 Analysis of Linear Time-Invariant Systems

Linear time-invariant systems constitute an important class, which has been extensivelyanalyzed and is well understood. It is well known that given an LTI system of the form

2 = Ax, A E 1Rnxn x(0) = x0 (3.18)

the origin is stable if and only if all eigenvalues A of A satisfy Me(at) < 0, where te(A )represents the real part of the eigenvalue )i, and every eigenvalue with Me(at) = 0 has anassociated Jordan block of order 1. The equilibrium point x = 0 is exponentially stable if

3.10. ANALYSIS OF LINEAR TIME-INVARIANT SYSTEMS 97

and only if all the eigenvalues of the matrix A satisfy Re(.i) < 0. Moreover, the solutionof the differential equation (3.18) can be expressed in a rather simple closed form

x(t) = eAtxo

Given these facts, it seems unnecessary to investigate the stability of LTI systems viaLyapunov methods. This is, however, exactly what we will do in this section! There ismore than one good reason for doing so. In the first place, the Lyapunov analysis permitsstudying linear and nonlinear systems under the same formalism, where LTI is a specialcase. Second, we will introduce a very useful class of Lyapunov functions that appears veryfrequently in the literature. Finally, we will study the stability of nonlinear systems via thelinearization of the state equation and try to get some insight into the limitations associatedwith this process.

Consider the autonomous linear time-invariant system given by

x=Ax, AERnxn

and let V(.) be defined as follows

(3.19)

V (X) = xT Px (3.20)

where p E Rnxn is (i) symmetric and (ii) positive definite. With these assumptions, V(.)is positive definite. We also have that

V=xTPx+xTP±

by (3.19), 2T = XT AT . Thus

XTATPi+XTPAxXT(ATP+PA)x

or

V = _XTQX (3.21)

PA+ATP = _Q. (3.22)

Here the matrix Q is symmetric, since

QT = -(PA +ATP)T = -(ATP+AP) = Q

If Q is positive definite, then is negative definite and the origin is (globally) asymp-totically stable. Thus, analyzing the asymptotic stability of the origin for the LTI system(3.19) reduces to analyzing the positive definiteness of the pair of matrices (P, Q). This isdone in two steps:

(i) Choose an arbitrary symmetric, positive definite matrix Q.

(ii) Find P that satisfies equation (3.22) and verify that it is positive definite.

Equation (3.22) appears very frequently in the literature and is called Lyapunov equation.

Remarks: There are two important points to notice here. In the first place, the approachjust described may seem unnecessarily complicated. Indeed, it seems to be easier to firstselect a positive definite P and use this matrix to find Q, thus eliminating the need forsolving the Lyapunov equation. This approach may however lead to inconclusive results.Consider, for example, the system with the following A matrix,

= 0 4

`4 -8 -12

taking P = I, we have that

-Q = PA + AT P = [4

0 -4-24 ]

and the resulting Q is not positive definite. Therefore no conclusion can be drawn from thisregarding the stability of the origin of this system.

The second point to notice is that clearly the procedure described above for thestability analysis based on the pair (P, Q) depends on the existence of a unique solutionof the Lyapunov equation for a given matrix A. The following theorem guarantees theexistence of such a solution.

Theorem 3.10 The eigenvalues at of a matrix A E R T satisfy te(A1) < 0 if and onlyif for any given symmetric positive definite matrix Q there exists a unique positive definitesymmetric matrix P satisfying the Lyapunov equation (3.22)

Proof: Assume first that given Q > 0, 2P > 0 satisfying (3.22). Thus V = xTPx > 0and V = -xT Qx < 0 and asymptotic stability follows from Theorem 3.2. For the converseassume that te(A1) < 0 and given Q, define P as follows:

P =10

eATtQeAt dt00

this P is well defined, given the assumptions on the eigenvalues of A. The matrix P is alsosymmetric, since (eATt)T = eAt. We claim that P so defined is positive definite. To see

3.10. ANALYSIS OF LINEAR TIME-INVARIANT SYSTEMS 99

that this is the case, we reason by contradiction and assume that the opposite is true, thatis, Ix # 0 such that xT Px = 0. But then

xTPx=0 xTeATtQeAtx dt = 010

Ja

ryT Qy dt = 0 with y = eAtx

y=eAtx=O dt>0x=0

since eat is nonsingular Vt. This contradicts the assumption, and thus we have that P isindeed positive definite. We now show that P satisfies the Lyapunov equation

PA + ATP = j eATtQeAtA dt + JATeATtQeAt dt

0

= J' t (eA-tQeAt) dt

eATtQeAt IU = -Q

which shows that P is indeed a solution of the Lyapunov equation. To complete the proof,there remains to show that this P is unique. To see this, suppose that there is anothersolution P # P. Then

(P-P)A+AT(P-P) = 0eAT t [(P - P)A+ AT (P - P)] eAt = 0

=>dt

[eAT t (P - P)eAt] = 0

which implies that eAT t (P - P)e_At is constant Vt. This can be the case if and only ifP - P = 0, or equivalently, P = P. This completes the proof.

3.10.1 Linearization of Nonlinear Systems

Now consider the nonlinear system

x = f (x), f : D -4 R' (3.23)

assume that x = xe E D is an equilibrium point and assume that f is continuously differ-entiable in D. The Taylor series expansion about the equilibrium point xe has the form

f (x) = f (xe) + ax (xe) (x - xe) + higher-order terms

Neglecting the higher-order terms (HOTs) and recalling that, by assumption, f (xe) = 0,we have that

rd (x) =

C72(Xe)(x - xe). (3.24)

Now defining

x = x - xe1 A = f (Xe) (3.25)

we have that X = i, and moreoverAte. (3.26)

We now whether it is possible to investigate the local stability of the nonlinear system(3.23) about the equilibrium point xe by analyzing the properties of the linear time-invariantsystem (3.26). The following theorem, known as Lyapunov's indirect method, shows that ifthe linearized system (3.26) is exponentially stable, then it is indeed the case that for theoriginal system (3.23) the equilibrium xe is locally exponentially stable. To simplify ournotation, we assume that the equilibrium point is the origin.

Theorem 3.11 Let x = 0 be an equilibrium point for the system (3.23). Assume that f iscontinuously differentiable in D, and let A be defined as in (3.25). Then if the eigenvalues) of the matrix A satisfy ate(.\a) < 0, the origin is an exponentially stable equilibrium pointfor the system (3.23).

The proof is omitted since it is a special case of Theorem 4.7 in the next chapter (see Section4.5 for the proof of the time-varying equivalent of this result).

3.11 Instability

So far we have investigated the problem of stability. All the results seen so far are, however,sufficient conditions for stability. Thus the usefulness of these results is limited by ourability to find a function V(.) that satisfies the conditions of one of the stability theoremsseen so far. If our attempt to find this function fails, then no conclusions can be drawn withrespect to the stability properties of the particular equilibrium point under study. In thesecircumstances it is useful to study the opposite problem, namely; whether it is possible toshow that the origin is actually unstable. The literature on instability is almost as extensiveas that on stability. Perhaps the most famous and useful result is a theorem due to Chetaevgiven next.

Theorem 3.12 (Chetaev) Consider the autonomous dynamical systems (3.1) and assumethat x = 0 is an equilibrium point. Let V : D -+ IR have the following properties:

3.11. INSTABILITY 101

(i) V(0) = 0

(ii) lix0 E R", arbitrarily close to x = 0, such that V(xo) > 0

(iii) V > OVx E U, where the set U is defined as follows:

U = {x E D : IIxI1 < e, and V(x) > 0}.

Under these conditions, x = 0 is unstable.

Remarks: Before proving the theorem, we briefly discuss conditions (ii) and (iii). Accord-ing to assumption (ii), V(.) is such that V(xo) > 0 for points inside the ball B6 = {x ED : Ix1I < d}, where 6 can be chosen arbitrarily small. No claim however was made about

being positive definite in a neighborhood of x = 0. Assumptions (iii) says that is

positive definite in the set U. This set consists of all those points inside the ball Bf (i.e.,the set of points satisfying Ix1I < c), which, in addition, satisfy V(x) > 0.

Proof: The proof consists of showing that a trajectory initiating at a point xo arbitrarilyclose to the origin in the set U will eventually cross the sphere defined by IxII = E. Giventhat e is arbitrary, this implies that given c > 0, we cannot find 6 > 0 such that

14011 < 6 = 1X(t)II < e

Notice first that condition (ii) guarantees that the set U is not empty. U is clearly boundedand moreover, its boundary consists of the points on the sphere 1x11 = e, and the surfacedefined by V(x) = 0. Now consider an interior point x0 E U. By assumption (ii) V(xo) > 0,and taking account of assumption (iii) we can conclude that

V(xo) = l; > 0

V(xo) > 0

but then, the trajectory x(t) starting at x0 is such that V(x(t)) > ,Vt > 0. This conditionsmust hold as long as x(t) is inside the set U. Define now the set of points Q as follows:

Q = {xEU:IxII <eandV(x)> }.

This set is compact (it is clearly bounded and is also closed since it contains its boundarypoints IIxII = c and V(x) = l;). It then follows that V(x), which is a continuous functionin a compact set Q, has a minimum value and a maximum value in Q. This argument alsoshows that V (x) is bounded in Q, and so in U. Define

-y = min{V(x) : x E Q} > 0.

This minimum exists since V(x) is a continuous function in the compact set Q. Thus, forthe trajectory starting at x0, we can write

tV(x(t) = V(xo) +J

V (x(r)) dr0

rtV (x0) + J ry d-r = V (xo) + ryt.

0

It then follows that x(t) cannot stay forever inside the set U since V(x) is bounded in U.Thus a trajectory x(t) initiating arbitrarily close to the origin must intersect the boundaryof U. The boundaries of this set are IIxii = E and the surface V(x) = 0. However, thetrajectory x(t) is such that V(x) > and we thus conclude that x(t) leaves the set Uthrough the sphere IIxii = E. Thus, x = 0 is unstable since given E > 0, we cannot find b > 0such that

IIxoli < b = 11x(t)II < E.

This completes the proof.

Example 3.23 Consider again the system of Example 3.20

21 = x2+x1(/32-x1-x2)-t2 = -XI + x2(N2 - x1 - x2)

We showed in Section 3.8 that the origin of this system is an unstable equilibrium point.We now verify this result using Chetaev's result. Let V(x) = 1/2(x2 + x2). Thus we havethat V (O) = 0, and moreover V (x) > OVx E 1R2 # 0, z. e., V(-) is positive definite. Also

V = (x1,x2)f(x)

(xl + x2)(/32 - xi - x2).

Defining the set U byU={xEIR2: IIxiI <E, 0<E<)3}

we have that V (x) > OVx E U, x # 0, and V > OVx E U, x # 0. Thus the origin is unstable,by Chetaev's result.

0

3.12 Exercises

(3.1) Consider the following dynamical system:

X1 = X2x2 = -x1 + xi - x2

3.12. EXERCISES 103

(a) Find all of its equilibrium points.

(b) For each equilibrium point xe different from zero, perform a change of variablesy = x - xef and show that the resulting system y = g(y) has an equilibrium pointat the origin.

(3.2) Given the systems (i) and (ii) below, proceed as follows:

(a) Find all of their equilibrium points.

(b) Find the linear approximation about each equilibrium point, find the eigenvaluesof the resulting A matrix and classify the stability of each equilibrium point.

(c) Using a computer package, construct the phase portrait of each nonlinear systemand discuss the qualitative behavior of the system. Make sure that your analysiscontains information about all the equilibrium points of these systems.

(d) Using the same computer package used in part (c), construct the phase portraitof each linear approximation found in (c) and compare it with the results in part(c). What can you conclude about the "accuracy" of the linear approximationsas the trajectories deviate from the equilibrium points.

J(Z) ll x2 = x, - 2 tan 1(x1 + x2) , (ii) I x2

23 X2

-x, +x2(1 - 3x1 - 2x2)

(3.3) Consider the magnetic suspension system of Section 1.9.1:

X = g -z M I 2m(1 +µx,)21+µx1 Aµ 1X3 =

A[_+

(1 + px,)2x2x3+ vj

(a) Find the input voltage v = vo necessary to keep the ball at an arbitrary posi-tion y = yo (and so x, = yo). Find the equilibrium point xe = [xel , xe2, xe3]corresponding to this input.

(b) Find the linear approximation about this equilibrium point and analyze its sta-bility.

(3.4) For each of the following systems, study the stability of the equilibrium point at theorigin:

-X + x,x21(Z)

f 1 = -X1 - x2x2 , (ii) 2 3-xlx2 - x2l 22 = -X2 - X1X2 22

(3.5) For each of the following systems, study the stability of the equilibrium point at theorigin:

2X1 = 12 - 2xl (x2 + x2) { i1 = -11 + 11x2():i2 = -xl - 212(x2 + x2) '

(ii)1 :i2 = -x2

(3.6) Consider the following system:

xl = -x2 + ax1(x2 + x2)±2 = xl + ax2(x2 + x2)

(a) Verify that the origin is an equilibrium point.

(b) Find the linear approximation about the origin, find the eigenvalues of the re-sulting A matrix, and classify the stability of the equilibrium point at the origin.

(c) Assuming that the parameter a > 0, use a computer package to study the trajec-tories for several initial conditions. What can you conclude about your answersfrom the linear analysis in part (b)?

(d) Repeat part (c) assuming a < 0.

(3.7) It is known that a given dynamical system has an equilibrium point at the origin.For this system, a function V(.) has been proposed, and its derivative has beencomputed. Assuming that V(.) and are given below you are asked to classifythe origin, in each case, as (a) stable, (b) locally asymptotically stable, (c) globallyasymptotically stable, (d) unstable, and/or (e) inconclusive information. Explain youranswer in each case.

(a) V (x) = (x2 + x2) , V(x) = -x2.(b) V(x) _ (x2 +x2 -1) , V(x) _ -(x2 +x2).(c) V(x) = (x2 + x2 - 1)2 , V(x) = -(x2 + x2).

(d) V(x) _ (x2 + x2 -1)2 , V(x) _ (x2 + x2).(e) V (X) = (x2 + x2 - 1) , V (X) _ (x2 + x2).

(f) V (X) = (x2 - x2) , V (x) = -(x2 + x2)

(g) V (X) _ (x2 - x2) , V (x) _ (x2 + x2)(h) V (X) = (xl + 12) , V (x) = (x2 - x2)G) = (xl + 12) ,V (

(X)V (x) = (x2 + x2).

3.12. EXERCISES

(3.8) Prove the following properties of class IC and K, functions:

(i) If a : [0, a) -> IRE K, then a-1 : [0, a(a)) --> R E K.

(ii) If al, a2 E K, then al o a2 E K.(iii) If a E Kim, then a-1 E K.

(iv) If al, a2 E Kim, then a1 o a2 E K.

(3.9) Consider the system defined by the following equations:

xl

(X3x2+Q31 -xl

105

2 = -x1Study the stability of the equilibrium point xe = (0, 0), in the following cases:

(i) /3 > 0.

(ii) /3 = 0.

(iii) Q < 0.

(3.10) Provide a detailed proof of Theorem 3.7.

(3.11) Provide a detailed proof of Corollary 3.1.

(3.12) Provide a detailed proof of Corollary 3.2.


Il = x212 = -x2 - axl - (2x2 + 3x1)2x2

Study the stability of the equilibrium point xe = (0, 0).


(i)

(ii) Study the stability of the origin x = (0, 0).

(iii) Study the stability of the invariant set 1 - (3x1 + 2x2) = 0.

211 = 3x2

±2 = -xl + x2(1 - 3x1 - 2x2)

Show that the points defined by (a) x = (0, 0) and (b) 1 - (3x1 + 2x2) = 0 areinvariant sets.

(3.15) Given the following system, discuss the stability of the equilibrium point at the origin:

X2

xix2 + 2x1x2 + xi

_ -X3 + x2

(3.16) (Lagrange stability) Consider the following notion of stability:

Definition 3.14 The equilibrium point x = 0 of the system (3.1) is said to be boundedor Lagrange stable if there exist a bound A such that

lx(t)II<A Vt>0.

Prove the following theorem

Theorem 3.13 [49] (Lagrange stability theorem) Let fl be a bounded neighborhood ofthe origin and let 1 be its complement. Assume that V(x) : R^ -* R be continuouslydifferentiable in S2' and satisfying:

(i) V(x) > 0 VxE52C.

(i) V(x) < 0 Vx E Q`.

(i) V is radially unbounded.

Then the equilibrium point at the origin is Lagrange stable.

Notes and ReferencesGood sources for the material of this chapter are References [48], [27], [41] [88] [68]

and [95] among others. The proof of theorem 3.1 is based on Reference [32]. Section 3.7as well as lemmas 3.4 and 3.5 follow closely the presentation in Reference [95]. Section3.8 is based on LaSalle, [49], and Khalil, [41]. The beautiful Example 3.20 was taken fromReference [68].

Chapter 4

Lyapunov Stability II:Nonautonomous Systems

In this chapter we extend the results of Chapter 3 to nonautonomous system. We start byreviewing the several notions of stability. In this case, the initial time instant to warrantsspecial attention. This issue will originate several technicalities as well as the notion ofuniform stability, to be defined.

4.1 Definitions

We now extend the several notions of stability from Chapter 3, to nonautonomous systems.For simplicity, in this chapter we state all our definitions and theorems assuming that theequilibrium point of interest is the origin, xe = 0.

Consider the nonautonomous systems

x=f(x,t) f:DxIIF+->1R' (4.1)

where f : D x [0, oo) -a IR is locally Lipschitz in x and piecewise continuous in t onD x [0, oc). We will say that the origin x = 0 E D is an equilibrium point of (4.1) at t = toif

f(0,t) = 0 dt > to.

For autonomous systems equilibrium points are the real roots of the equation f (xe) = 0.Visualizing equilibrium points for nonautonomous systems is not as simple. In general,

'Notice that, as in Chapter 3, (4.1) represents an unforced system.

107

108 CHAPTER 4. LYAPUNOV STABILITY H. NONAUTONOMOUS SYSTEMS

xe = 0 can be a translation of a nonzero trajectory. Indeed, consider the nonautonomoussystem

th = f(x,t) (4.2)

and assume that x(t) is a trajectory, or a solution of the differential equation (4.2) for t > 0.Consider the change of variable

y = X(t) - x(t).

We have

f (x, t) - x(t)

def= g(y, t)

But i = f (x(t), t). Thus

g(y,t) = f(y + x(t),t) - f(x(t),t)

0 if y=0

that is, the origin y = 0 is an equilibrium point of the new system y = g(y, t) at t = 0.

Definition 4.1 The equilibrium point x = 0 of the system (4.1) is said to be

Stable at to if given c > 0, 36 = b(e, to) > 0 :

11x(0)11 < S = 11x(t)II < c Vt > to > 0

Convergent at to if there exists 51 = 51(to) > 0 :

11x(0)11 < Si = slim x(t) = 0.400

Equivalently (and more precisely), xo is convergent at to if for any given cl > 0, 3T =T(elito) such that

1x(0)11 < Si = Ix(t)11 < El dt > to +T (4.5)

Asymptotically stable at to if it is both stable and convergent.

Unstable if it is not stable.

4.1. DEFINITIONS 109

All of these definitions are similar to their counterpart in Chapter 3. The difference is inthe inclusion of the initial time to. This dependence on the initial time is not desirable andmotivates the introduction of the several notions of uniform stability.


Uniformly stable if any given e > 0, 16 = b(e) > 0 :

X(0)11 < 6 1x(t)jj < e Vt > to > 0 (4.6)

Uniformly convergent if there is 61 > 0, independent of to, such that

jxoII<61 = x(t)-+0 ast ->oo.Equivalently, x = 0 is uniformly convergent if for any given E1 > 0, IT = T(El) suchthat

lx(0)II < 61 Ix(t)II < El Vt > to + T

Uniformly asymptotically stable if it is uniformly stable and uniformly convergent.

Globally uniformly asymptotically stable if it is uniformly asymptotically stable andevery motion converges to the origin.

As in the case of autonomous systems, it is often useful to restate the notions of uniformstability and uniform asymptotic stability using class IC and class ICL functions. The fol-lowing lemmas outline the details. The proofs of both of these lemmas are almost identicalto their counterpart for autonomous systems and are omitted.

Lemma 4.1 The equilibrium point x = 0 of the system (4.1) is uniformly stable if and onlyif there exists a class IC function a constant c > 0, independent of to such that

Ix(0)II < c => Ix(t)II <_ 44(0)II) Vt > to. (4.7)

Lemma 4.2 The equilibrium point x = 0 of the system (4.1) is uniformly asymptoticallystable if and only if there exists a class ICL and a constant c > 0, independentof to such that

1x(0)11 < c => lx(t)II <- i30Ix(o)II,t - to) Vt > to. (4.8)

Definition 4.3 The equilibrium point x = 0 of the system (4.1) is (locally) exponentiallystable if there exist positive constants a and A such that

whenever jx(0)

11x(t)II < aIlxollke-at (4.9)

I < 6. It is said to be globally exponentially stable if (4.9) is satisfied foranyxER".

110 CHAPTER 4. LYAPUNOV STABILITY II: NONAUTONOMOUS SYSTEMS

4.2 Positive Definite Functions

As seen in Chapter 3, positive definite functions play a crucial role in the Lyapunov theory.In this section we introduce time-dependent positive definite functions.

In the following definitions we consider a function W : D x pt+ -> IIt, i.e., a scalarfunction W (x, t) of two variables: the vector x E D and the time variable t. Furthermorewe assume that

(i) 0 E D.

(ii) W (x, t) is continuous and has continuous partial derivatives with respect to all of itsarguments.

Definition 4.4 is said to be positive semi definite in D if

(i) W(0,t) = 0 Flt E R+.

(ii) W(x,t)>0 Vx#0,xED.

Definition 4.5 W(., ) is said to be positive definite in D if

(i) W(0, t) = 0.

(ii) 3 a time-invariant positive definite function V1(x) such that

V1(x) < W(x,t) Vx E D.

Definition 4.6 W(., ) is said to be decrescent in D if there exists a positive definite func-tion V2(x) such that

IW(x,t)I < V2(x) Vx E D.

The essence of Definition 4.6 is to render the decay of toward zero a functionof x only, and not of t. Equivalently, W (x, t) is decrescent in D if it tends to zero uniformlywith respect to t as Ixii -+ 0. It is immediate that every time-invariant positive definitefunction is decrescent.

Definition 4.7 is radially unbounded if

W(x,t)--goo as IIxii->oo

4.2. POSITIVE DEFINITE FUNCTIONS 111

uniformly on t. Equivalently, W(., ) is radially unbounded if given M, AN > 0 such that

W (x, t) > M

for all t, provided that Ixjj > N.

Remarks: Consider now function W (x, t). By Definition 4.5, W (, ) is positive definite inD if and only if 3V1 (x) such that

Vi (x) < W (x, t) , Vx E D (4.10)

and by Lemma 3.1 this implies the existence of such that

al(IjxII) < Vl(x) < W(x,t) , Vx E Br C D. (4.11)

If in addition is decrescent, then, according to Definition 4.6 there exists V2:

W (x, t) < V2(x) , Vx E D (4.12)

and by Lemma 3.1 this implies the existence of such that

W(x,t) < V2(x) < 012(IIxII) , Vx E Br C D. (4.13)

It follows that is positive definite and decrescent if and only if there exist a (time-invariant) positive definite functions and V2(.), such that

Vi(x) < W(x,t) < V2 (x) , Vx E D (4.14)

which in turn implies the existence of function and E IC such that

al(IIxII) < W(x,t) < a2(1IxII) , Vx E Br C D. (4.15)

Finally, is positive definite and decrescent and radially unbounded if and only ifal(.) and a2() can be chosen in the class ]Cc,,,.

4.2.1 Examples

In the following examples we assume that x = [x1, x2]T and study several functions W (x, t).

Example 4.1 Let Wl (x, t) _ (x1 + x2)e ,t a > 0. This function satisfies

(i) Wl (0, t) = 0 e"t = 0

(ii) Wl(x,t)>O Vx540, VtE1R.

However, limt,, Wl (x, t) = 0 Vx. Thus, Wl is positive semi definite, but not positivedefinite.

Example 4.2 Let

W(xt) = (x1 + x2)(12 + 1)

V2(x)(t2 + 1),V2(x) = ((x1

+2))

Thus, V2(x) > 0 Vx E 1R2 and moreover W2(x,t) > V2(x) Vx E ]R2, which implies thatis positive definite. Also

lim W2(x,t) = oo Vx E 1R2.taco

Thus it is not possible to find a positive definite function V() such that IW2(x, t)I < V(x) Vx,and thus W2(x,t) is not decrescent. IW2(x,t)l is not radially unbounded since it does nottend to infinity along the xl axis.

Example 4.3 Let

W3(x, t) _ (x1+x2)(12 + 1)V3(x)(12+1) V3(x)deJ(xl+x2)

Following a procedure identical to that in Example 4.2 we have that W3(x) is positive definite,radially unbounded and not decrescent.

Example 4.4 Let2 2

)W4(x t) _ (x 2+X2(xl + 1) .

Thus, W4(., ) > OVx E 1R2 and is positive definite. It is not time-dependent, and so it isdecrescent. It is not radially unbounded since it does not tend to infinity along the xl axis.

Example 4.5 Let

W5(x t) _ (x2 +x2)(12 + 1)(t2 + 2)

= V5(x)(t+ 2),

V5 (x)def

(xl + x2)2

Thus, W5(x, t) > ki V5 (x) for some constant k1, which implies that is positive defi-nite. It is decrescent since

IW5(x,t)I <_ k2V5(x)Vx E R2

It is also radially unbounded since

W5(x,t) -* oo as llxil --p oc.

4.3 Stability Theorems

We now look for a generalization of the stability results of Chapter 3 for nonautonomoussystems. Consider the system (4.1) and assume that the origin is an equilibrium state:

f(O,t)=0 VtER.

Theorem 4.1 (Lyapunov Stability Theorem) If in a neighborhood D of the equilibriumstate x = 0 there exists a differentiable function W(., ) : D x [0, oo) -* IR such that

(i) W (x, t) is positive definite.

(ii) The derivative of W(., ) along any solution of (4.1) is negative semi definite in D,then

the equilibrium state is stable. Moreover, if W(x,t) is also decrescent then the origin is

uniformly stable.

Theorem 4.2 (Lyapunov Uniform Asymptotic Stability) If in a neighborhood D of theequilibrium state x = 0 there exists a differentiable function W(.,) : D x [0, oo) -> R suchthat

(i) W (x, t) is (a) positive definite, and (b) decrescent, and

(ii) The derivative of W (x, t) is negative definite in D, then

the equilibrium state is uniformly asymptotically stable.

Remarks: There is an interesting difference between Theorems 4.1 and 4.2. Namely, inTheorem 4.1 the assumption of W(., ) being decrescent is optional. Indeed, if W(., ) is

decrescent we have uniform stability, whereas if this is not the case, then we settle forstability. Theorem 4.2 is different; if the decrescent assumption is removed, the remainingconditions are not sufficient to prove asymptotic stability. This point was clarified in 1949by Massera who found a counterexample.

Notice also that, according to inequalities (4.14)-(4.15), given a positive definite func-tion W(x, t) there exist positive definite functions V1 (x) and V2 (x), and class K functionsal and a2 such that

V1(x) < W(x, t) < V2(x) Vx E D (4.16)

a1(1Ix1I) < W(x, t) < a2(I1xII) Vx E Br (4.17)

with this in mind, Theorem 4.2 can be restated as follows:

Theorem 4.2 (Uniform Asymptotic Stability Theorem restated) If in a neighborhood D ofthe equilibrium state x = 0 there exists a differentiable function W(., ) : D x [0, oo) -+ IRsuch that

(z) V1 (X) < W(x, t) < V2 (X) Vx E D,Vt

(ii) "W + VW f (x, t) < -V3(x) Vx E D, Vt

where U, i = 1, 2,3 are positive definite functions in D, then the equilibrium state is uni-formly asymptotically stable.

Theorem 4.3 (Global Uniform Asymptotic Stability) If there exists a differentiable func-tion 1R" x [0, oo) -+ 1R such that

(i) W (x, t) is (a) positive definite, and (b) decrescent, and radially unbounded Vx E ]R°b,and such that

(ii) The derivative of W(x, t) is negative definite Vx E 1R", then

the equilibrium state at x = 0 is globally uniformly asymptotically stable.

For completeness, the following theorem extends Theorem 3.4 on exponential stabilityto the case of nonautonomous systems. The proof is almost identical to that of Theorem3.4 and is omitted.

4.4. PROOF OF THE STABILITY THEOREMS 115

Theorem 4.4 Suppose that all the conditions of theorem 4.2 are satisfied, and in additionassume that there exist positive constants K1, K2, and K3 such that

KllIxMMP < W(x,t) < K2IIxllP

W(x) < -KsllxI '.

Then the origin is exponentially stable. Moreover, if the conditions hold globally, the x = 0is globally exponentially stable.


21 = -xl - e-2tx2{ ±2 = x1 - x2.

To study the stability of the origin for this system, we consider the following Lyapunovfunction candidate:

W (X, t) = X2 + (1 + e-2i)x2.

Clearly

Vi (x)

thus, we have that

(x1 + x2) < W(x,t) < (x1 + 2x2) = V2(x)

W (x, t) is positive definite, since Vl (x) < W (x, t), with V1 positive definite in R2.

W (x, t) is decrescent, since W (x, t) > V2 (x), with V2 also positive definite in R2.

Then

W (x' t)ax f (x, t) + at

-2[x1 - X1X2 + x2(1 + 2e-2t)]

< -2[x1 - x1x2 + 3x21.

It follows that W (x, t) is negative definite and the origin is globally asymptotically stable.

11

4.4 Proof of the Stability Theorems

We now elaborate the proofs of theorems 4.1-4.3.

Proof of theorem 4.1: Choose R > 0 such that the closed ball

BR = {xER':IIxII <R}

is contained in D. By the assumptions of the theorem is positive definite andthus there exist a time-invariant positive definite function Vl and a class 1C function alsatisfying (4.11):

ai(11x1I) < Vj(x) < W(x, t) dx E BR. (4.18)

Moreover, by assumption, W (x, t) < 0, which means that W (x, t) cannot increase alongany motion. Thus

W (x(t), t) < W (xo, to) t > to

al(IIxII) < W(x(t),t) < W(xo,to).

Also W is continuous with respect to x and satisfies W(0, to) = 0. Thus, given to, we canfind 6 > 0 such that

11x011 < 6 = W(xo, to) < a(R)

which means that if 1xoHI < 6, then

ai(I1xII) < a(R) => Ix(t)II < R Vt > to.

This proves stability. If in addition W (x, t) is decreasing, then there exists a positivefunction V2(x) such that

I W(x, t)1 < V2(x)

and then E3a2 in the class K such that

ai(IIxH) < V1(x) < W(x, t) < V2(x) < a2(11xI1) Vx E BR, Vt > to.

By the properties of the function in the class K , for any R > 0, 3b = f (R) such that

a2(6) < a, (R)

=> ai(R) > a2(a) > W(xo,to) > W(x,t) > a(1Ix(t)II)which implies that

11x(t)II < R `dt > to.

However, this 6 is a function of R alone and not of to as before. Thus, we conclude that thestability of the equilibrium is uniform. This completes the proof.

Proof of theorem 4.2: Choose R > 0 such that the closed ball

BR={XE]R":11x11 <R}

4.4. PROOF OF THE STABILITY THEOREMS 117

is contained in D. By the assumptions of the theorem there exist class IC functions al, a2,and a3 satisfying

a1(IIxII) < W(x,t) < a2(IIXII) Vt,Vx E BR (4.19)

a3(114) < -W(x,t) Vt,Vx E BR. (4.20)

Given that the at's are strictly increasing and satisfy a,(0) = 0, i = 1, 2, 3, given e > 0, wecan find 5i, b2 > 0 such that

x2(81) < al(R)a2(52) < min[a1(c),a2(b1)]

(4.21)

(4.22)

where we notice that b1 and b2 are functions of e but are independent of to. Notice alsothat inequality (4.22) implies that 61 > 62. Now define

T = C" (R)a3(a2)

Conjecture: We claim that

11x011 < 61 = lx(t')II < b2

(4.23)

for some t = t` in the interval to < t' < to + T.

To see this, we reason by contradiction and assume that 1xo11 < 61 but llx(t`)f1 > b2for all t in the interval to < t' < to + T. We have that

0 < a,(52) al is class IC

al(b2) <_ W(x,t) dto < t < to + T (4.24)

where (4.24) follows from (4.19) and the assumption llx(t`)ll > b2. Also, since W is adecreasing function of t this implies that

0< al(a2) < W(x(to +T), to +T)to+T

< W(x(to), to) + J W(x(t), t) dtto

< W(x(to),to) -Ta3(d2) from (4.20) and (4.23).

ButW(x(to),to) < a2(11x(to)11) a2(5i), since 11xoll < 61, by assumption.

by (4.19)

118 CHAPTER 4. LYAPUNOV STABILITY IL NONAUTONOMOUS SYSTEMS

Thus, we have

0 < W(x(to),to) - Ta3(82)

0 < a2(81) - al(R) by (4.23)

which contradicts (4.21). It then follows that our conjecture is indeed correct. Now assumethat t > t'. We have

ai(IIx(t)II) < W(x(t),t) <W(x(t'),t*)because is a decreasing function of t

< a2(IIx(t')II) < a2(b2) < al(E)-

This implies that 11x(t)II < E. It follows that any motion with initial state IIxoII < 8 is

uniformly convergent to the origin.

Proof of theorem 4.3: Let al, a2i and a3 be as in Theorem 4.2. Given that in this caseis radially unbounded, we must have that

ai(IIxII) -> oo as IIxII -4 oo.

Thus for any a > 0 there exists b > 0 such that

al(b) > a2(a).

If IIxoII < a, then

ai(b) > a2(a) ? W(xo,to) >_ W(x(t),t) >- al(IIxII)

Thus, IIxII < a = IIxII < b, and we have that all motions are uniformly bounded. ConsiderEl, 8, and T as follows:

a2 (b) < al (c) T = al (b)a3 (a)

and using an argument similar to the one used in the proof of Theorem 4.2, we can showthat for t > to + T

11X(011 < -E

provided that IIxoII < a. This proves that all motions converge uniformly to the origin.Thus, the origin is asymptotically stable in the large.

4.5. ANALYSIS OF LINEAR TIME-VARYING SYSTEMS 119

4.5 Analysis of Linear Time-Varying Systems

In this section we review the stability of linear time-varying systems using Lyapunov tools.Consider the system:

± = A(t)x (4.25)

where A(.) is an n x n matrix whose entries are real-valued continuous functions of t E R.It is a well-established result that the solution of the state equation with initial conditionxo is completely characterized by the state transition matrix <1(., ). Indeed, the solution of(4.25) with initial condition xo is given by,

x(t) = 4i(t, to)xo. (4.26)

The following theorem, given without proof, details the necessary and sufficient conditionsfor stability of the origin for the nonautonomous linear system (4.25).

Theorem 4.5 The equilibrium x = 0 of the system (4.25) is exponentially stable if andonly if there exist positive numbers kl and k2 such that

IF(t,to)II < kie-kz(t-t0) for any to > 0, Vt > to (4.27)

where

II'(t, to) II = Iml x II4(t, to)xll.

Proof: See Chen [15], page 404.

We now endeavor to prove the existence of a Lyapunov function that guaranteesexponential stability of the origin for the system (4.25). Consider the function

W (x' t) = xT P(t)x (4.28)

where P(t) satisfies the assumptions that it is (i) continuously differentiable, (ii) symmetric,(iii) bounded, and (iv) positive definite. Under these assumptions, there exist constantsk1, k2 E R+ satisfying

kixTx < xTP(t)x < k2xTx Vt > 0,Vx E R"

or

k1IIx1I2 < W(x,t) < k2IIXII2 Vt > 0,Vx c 1R".

This implies that W (x, t) is positive definite, decrescent, and radially unbounded.

W (x, t) = iT P(t)x + xT Pi + xT P(t)x= xTATPX+xTPAx+xTP(t)x= xT[PA+ATP+P(t)]x

or

W(x,t) = -xTQ(t)xT

where we notice that Q(t) is symmetric by construction. According to theorem 4.2, if Q(t)is positive definite, then the origin is uniformly asymptotically stable. This is the case ifthere exist k3, k4 E 1(P+ such that

k3xTx < xTQ(t)x < k4xTx Vt > 0,Vx E IRT.

Moreover, if these conditions are satisfied, then

k3IIxII2 < Q(t) < k4IIxII2 Vt > 0,Vx E Rn

and then the origin is exponentially stable by Theorem 4.4.

Theorem 4.6 Consider the system (4.25). The equilibrium state x = 0 is exponentiallystable if and only if for any given symmetric, positive definite, continuous, and bounded ma-trix Q(t), there exist a symmetric, positive definite, continuously differentiable, and boundedmatrix P(t) such that

-Q(t) = P(t)A(t) + AT(t)p(t) + P(t). (4.29)


4.5.1 The Linearization Principle

Linear time-varying system often arise as a consequence of linearizing the equations of anonlinear system. Indeed, given a nonlinear system of the form

x = f (x, t) f : D x [0, oo) -> R' (4.30)

with f (x, t) having continuous partial derivatives of all orders with respect to x, then it ispossible to expand f (x, t) using Taylor's series about the equilibrium point x = 0 to obtain

X = f (x, t) = f (0, t) + of Ix-o x + HOT

where HOT= higher-order terms. Given that x = 0 is an equilibrium point, f (0, t) = 0,thus we can write

f(x,t) N Of x=O

or

x = A(t)x + HOT (4.31)

4.5. ANALYSIS OF LINEAR TIME-VARYING SYSTEMS 121

where

A(t) = ofl==o

f of (o, t)ax ax

since the higher-order terms tend to be negligible "near" the equilibrium point, (4.31) seemsto imply that "near" x = 0, the behavior of the nonlinear system (4.30) is similar to thatof the so-called linear approximation (4.31). We now investigate this idea in more detail.More explicitly, we study under what conditions stability of the nonlinear system (4.30) canbe inferred from the linear approximation (4.31). We will denote

g(x, t)de f f (x, t) - A(t)x.

Thusx = f (x, t) = A(t)x(t) + g(x, t).

Theorem 4.7 The equilibrium point x = 0 of the nonlinear system

x = A(t)x(t) + g(x, t)

is uniformly asymptotically stable if

(i) The linear system a = A(t)x is exponentially stable.

(ii) The function g(x, t) satisfies the following condition. Given e > 0, there exists 5 > 0,independent of t, such that

Ilxll <6 Ilg(x, t) 11 <E.ilxll

This means that

uniformly with respect to t.

lim g(x, t) = 0i=u-4a Ilxil

(4.32)

Proof: Let 4) (t, to) be the transition matrix of x = A(t)x, and define P as in the proof ofTheorem 4.6 (with Q = I):

P(t) =J

4T (T, t) 4) (r, t) d7c

according to Theorem 4.6, P so defined is positive and bounded. Also, the quadraticfunction

W (x, t) = xT Px

is (positive definite and) decrescent. Moreover

W (x, t) = thT p(t)x= xT +xTP(t)x+xTPx= (xTAT +gT)Px+XTP(Ax+g)+XTPX= XT (ATP+PA+P)x+gTPx+xTPg

(_ -I)_ -XT x+9T (x,t)Px+xTPg(x,t).

By (4.32), there exist o > 0 such that

119(x, t) II provided that II x l l < b< E .,xii

Thus, given that P(t) is bounded and that condition (4.32) holds uniformly in t, there existb > 0 such that

119(x, t)II 1

Ilxll < IIPII

thus, we can write that

2119(x, t)II11PII < IIx11

2119(x, t)11lIPll f Ollxll < Ilxll with 0 < 9 < 1.

It then follows that

W(x,t) < -11x112+21191111PlIlIxll=-11x112+OIIx112 0<9<1

< (1- 9)11x112.

Hence, W (x, t) is negative definite in a neighborhood of the origin and satisfies the conditionsof Theorem 4.4 (with p = 2). This completes the proof.

4.6 Perturbation Analysis

Inspired by the linearization principle of the previous section we now take a look at animportant issue related to mathematical models of physical systems. In practice, it is notrealistic to assume that a mathematical model is a true representation of a physical device.At best, a model can "approximate" a true system, and the difference between mathematicalmodel and system is loosely referred to as uncertainty. How to deal with model uncertainty

4.6. PERTURBATION ANALYSIS 123

and related design issues is a difficult problem that has spurred a lot of research in recentyears. In our first look at this problem we consider a dynamical system of the form

i = f(x,t) + g(x, t) (4.33)

where g(x, t) is a perturbation term used to estimate the uncertainty between the true stateof the system and its estimate given by i = f (x, t). In our first pass we will seek for ananswer to the following question: Suppose that i = f (x, t) has an asymptotically stableequilibrium point; what can it be said about the perturbed system ± = f (x, t) + g(x, t)?

Theorem 4.8 [41] Let x = 0 be an equilibrium point of the system (4.33) and assume thatthere exist a differentiable function W(., ) : D x [0, oo) -p R such that

(i) k1IIxII2 <<W(x,t) < IxII2.

(ii) "W + VW f (x, t) < -k3 11X 112.

(iii) IIVWII < k4IIxI12.

Then if the perturbation g(x, t) satisfies the bound

(iv) I1g(x,t)II < k511x1I, (k3 - k4k5) > 0

the origin is exponentially stable. Moreover, if all the assumptions hold globally, then theexponential stability is global.

Proof: Notice first that assumption (i) implies that W (x, t) is positive definite and de-crescent, with a1(IJxJI) = k1IIxII2 and a2(IIxJI) = k2JIxI12. Moreover, the left-hand side ofthis inequality implies that W(., ) is radially unbounded. Assumption (ii) implies that,ignoring the perturbation term g(x, t), W (X, t) is negative definite along the trajectories ofthe system i = f (x, t). Thus, assumptions (i) and (ii) together imply that the origin is auniformly asymptotically stable equilibrium point for the nominal system i = f (x, t).

We now find along the trajectories of the perturbed system (4.33). We have

W = +VWf(x,t)+VWg(

< k3IIx112 < k4ks1IxI12

W < -(k3 - k4ks)IIXII2 < 0

since (k3 - k4k5) > 0 by assumption. The result then follows by Theorems 4.2-4.3 (alongwith Theorem 4.4).

The importance of Theorem 4.8 is that it shows that the exponential stability is robustwith respect to a class of perturbations. Notice also that g(x, t) need not be known. Onlythe bound (iv) is necessary.

Special Case: Consider the system of the form

(4.34)Ax + g(x, t)

where A E 1Rn"n. Denote by Ai, i = 1, 2, . , n the eigenvalues of A and assume thatt(.\i) < 0, Vi. Assume also that the perturbation term g(x, t) satisfies the bound

119(X,0112 <- 711x112 Vt > 0, Vx E R"-

Theorem 3.10 guarantees that for any Q = QT > 0 there exists a unique matrix P = PT > 0,that is the solution of the Lyapunov equation

PA+ATP=-Q.

The matrix P has the property that defining V(x) = xTPx and denoting ,\,,,in(P) andAmax(P) respectively as the minimum and maximum eigenvalues of the matrix P, we havethat

-amin(P)IIx112 < V(x) < amax(P)Ilx112

- aV Ax = -xTQx < -Amin(Q)IIXI12

Thus V (x) is positive definite and it is a Lyapunov function for the linear system k = Ax.Also

avax

IlOx

It follows that

Px+xTP = 2xTP

112xTP112 = 2Amax(P)Ilx112

x axaAx + 9(x, t)

< -amin(Q)IIx112 + [2Amax(P)IIx112] [711x112]

< -Amin(Q)Ilxl12+27., (P)Ilx1I2

fl (x) < [-Amin(Q) + 27'\, (P)] IIxI12

It follows that the origin is asymptotically stable if

-Amin(Q) + 2'yA.x(P) < 0

4.7. CONVERSE THEOREMS

that is

or, equivalently

Amin(Q) > 2ry Amax(P)

125

min (Q)(4.35)ti

2Amax(P)

From equation (4.35) we see that the origin is exponentially stable provided that the valueof ry satisfies the bound (4.35)

4.7 Converse Theorems

All the stability theorems seen so far provide sufficient conditions for stability. Indeed, allof these theorems read more or less as follows:

if there exists a function (or that satisfies ... , then the equilibriumpoint xe satisfies one of the stability definitions.

None of these theorems, however, provides a systematic way of finding the Lyapunovfunction. Thus, unless one can "guess" a suitable Lyapunov function, one can never concludeanything about the stability properties of the equilibrium point.

An important question clearly arises here, namely, suppose that an equilibrium pointsatisfies one of the forms of stability. Does this imply the existence of a Lyapunov functionthat satisfies the conditions of the corresponding stability theorem? If so, then the searchfor the suitable Lyapunov function is not in vain. In all cases, the question above can beanswered affirmatively, and the theorems related to this questions are known as conversetheorems. The main shortcoming of these theorems is that their proof invariably relies on theconstruction of a Lyapunov function that is based on knowledge of the state trajectory (andthus on the solution of the nonlinear differential equation). This fact makes the conversetheorems not very useful in applications since, as discussed earlier, few nonlinear equationcan be solved analytically. In fact, the whole point of the Lyapunov theory is to provide ananswer to the stability analysis without solving the differential equation. Nevertheless, thetheorems have at least conceptual value, and we now state them for completeness.

Theorem 4.9 Consider the dynamical system i = f (x, t). Assume that f satisfies a Lip-schitz continuity condition in D C Rn, and that 0 E D is an equilibrium state. Then wehave

If the equilibrium is uniformly stable, then there exists a function W(-, ) : D x [0, oo) -aIIt that satisfies the conditions of Theorem 4.1 .

126 CHAPTER 4. LYAPUNOV STABILITY H. NONAUTONOMOUS SYSTEMS

(a)

(b)

u(t)

u(k)...................«

E

Ex(k)

.................. W

Figure 4.1: (a) Continuous-time system E; (b) discrete-time system Ed-

e If the equilibrium is uniformly asymptotically stable, then there exists a functionW(., ) : D x [0, oo) -4 ]R that satisfies the conditions of Theorem 4.2.

If the equilibrium is globally uniformly stable, then there exists a function W(., )D x [0, oo) - JR that satisfies the conditions of Theorem 4.3.

Proof: The proof is omitted. See References [88], [41], or [95] for details.

4.8 Discrete-Time Systems

So far our attention has been limited to the stability of continuous-time systems, that is,systems defined by a vector differential equation of the form

i = f(x,t) (4.36)

where t E 1R (i.e., is a continuous variable). In this section we consider discrete-time systemsof the form

x(k + 1) = f(x(k),k) (4.37)

where k E Z+, x(k) E 1R", and f : ]R" x Z -+ 1R", and study the stability of these systems usingtools analogous to those encountered for continuous-time systems. Before doing so, however,we digress momentarily to briefly discuss discretization of continuous-time nonlinear plants.

4.9. DISCRETIZATION 127

4.9 Discretization

Often times discrete-time systems originate by "sampling" of a continuous-time system.This is an idealized process in which, given a continuous-time system E : u - x (a dynamicalsystem that maps the input u into the state x, as shown in Figure 4.1(a)) we seek a newsystem Ed : u(k) ---> x(k) (a dynamical system that maps the discrete-time signal u(k) intothe discrete-time state x(k), as shown in Figure 4.1(b)). Both systems E and Ed are relatedin the following way. If u(k) is constructed by taking "samples" every T seconds of thecontinuous-time signal u, then the output x(k) predicted by the model Ed corresponds to thesamples of the continuous-time state x(t) at the same sampling instances. To develop sucha model, we use the scheme shown in Figure 4.2, which consists of the cascade combinationof the blocks H, E, and S, where each block in the figure represents the following:

S represents a sampler, i.e. a device that reads the continues variable x every Tseconds and produces the discrete-time output x(k), given by x(k) = x(kT). Clearly,this block is an idealization of the operation of an analog-to-digital converter. Foreasy visualization, we have used continuous and dotted lines in Figure 4.2 to representcontinuous and discrete-time signal, respectively.

E represents the plant, seen as a mapping from the input u to the state x, that is, givenu the mapping E : u -a x determines the trajectory x(t) by solving the differentialequation

x = f (x, u).

H represents a hold device that converts the discrete-time signal, or sequence u(k),into the continuous-time signal u(t). Clearly this block is implemented by using adigital-to-analog converter. We assume the an ideal conversion process takes place inwhich H "holds" the value of the input sequence between samples (Figure 4.3), givenby

u(t) = u(k) for kT < t < (k + 1)T.

The system Ed is of the form (4.37). Finding Ed is fairly straightforward when thesystem (4.36) is linear time-invariant (LTI), but not in the present case.

Case (1): LTI Plants. If the plant is LTI, then we have that

e=f(x,u)=Ax+Bu (4.38)

finding the discrete-time model of the plant reduces to solving the differential equation withinitial condition x(kT) at time t(O) = kT and the input is the constant signal u(kT). The


H U(t) x(t)K S

Figure 4.2: Discrete-time system Ed-

1 2 3 4 5

x

01 2 3T4T5T t

Figure 4.3: Action of the hold device H.

4.9. DISCRETIZATION 129

differential equation (4.38) has a well-known solution of the form

x(t) = eA(t to)x(to) + it eA(t-T)Bu(T) dT.o

In our case we can make direct use of this solution with

t(O) = KTx(to) = x(k)

t = (k+1)Tx(t) = x[(k + 1)T]u(T) = u(k) constant for kT < t < k(T + 1).

Therefore

f(k+1)T

x(k + 1) = x[(k + 1)T] = eAT x(kT) + eA[1)T-r1 dT Bu(kT)T

/T= eATx(k) + J eAT dT Bu(k).0

Case (2): Nonlinear Plants. In the more general case of a nonlinear plant E given by (4.36)the exact model is usually impossible to find. The reason, of course, is that finding the exactsolution requires solving the nonlinear differential equation (4.36), which is very difficult ifnot impossible. Given this fact, one is usually forced to use an approximate model. Thereare several methods to construct approximate models, with different degrees of accuracyand complexity. The simplest and most popular is the so-called Euler approximation, whichconsists of acknowledging the fact that, if T is small, then

dx x(t + AT) - x(t) x(t + T) - x(t)x=-= limdt AT-+O AT T

Thus

can be approximated by

x= f(x,u)

x(k + 1) x(k) + T f[x(k),u(k)].

4.10 Stability of Discrete-Time Systems

In this section we consider discrete-time systems of the form (4.37); that is, we assume that

x(k + 1) = f(x(k),k)

where k E Z+, x(k) E 1R", and f : l x Z -> II?1. It is refreshing to notice that, unlike thecontinuous case, this equation always has exactly one solution corresponding to an initialcondition xo. As in the continuous-time case we consider the stability of an equilibriumpoint xe, which is defined exactly as in the continuous-time case.

4.10.1 Definitions

In Section 4.2 we introduced several stability definitions for continuous-time systems. Forcompleteness, we now restate these definitions for discrete-time systems.


Stable at ko if given e > 0, 38 = 8(e, ko) > 0 :

llxojl < 8 = jx(k)JI < e Vk > ko > 0. (4.39)

Uniformly stable at ko if given any given c > 0, E38 = 8(e) > 0 :

jjxolI < 8 IIx(k)jl < e Vk > ko > 0. (4.40)

Convergent at ko if there exists 81 = 81(ko) > 0 :

jjxojj < 81 => lim x(k) = 0. (4.41)t+00

Uniformly convergent if for any given e1 > 0, 3M = M(e1) such that

IIxo11 <81 = IIx(k)11 <c1 Vk>ko+M.

Asymptotically stable at ko if it is both stable and convergent.

Uniformly asymptotically stable if it is both stable and uniformly convergent.

Unstable if it is not stable.

4.10. STABILITY OF DISCRETE-TIME SYSTEMS 131

4.10.2 Discrete-Time Positive Definite Functions

Time-independent positive functions are uninfluenced by whether the system is continuousor discrete-time. Time-dependent positive function can be defined as follows.

Definition 4.9 A function W : R" X Z --+ P is said to be

Positive semidefinite in D C R' if

(i) W(0,k)=0 Vk>0.(ii) W(x,k)>0 `dx#0, xED.

Positive definite in D C P" if

(i) W (O, k) = 0 Vk > 0, and

(ii) I a time invariant positive definite function Vi(x) such that

Vl (x) < W (x, k) dx E D V.

It then follows by lemma 3.1 that W (x, k) is positive definite in B,. C D if andonly if there exists a class 1C function such that

al(jjxMI) <W(x,t), Vx E B,. C D.

is said to be decrescent in D C P" if there exists a time-invariant positivedefinite function V2(x) such that

W(x, k) < V2(x) Vx E D, Vk.

It then follows by Lemma 3.1 that W(x, k) is decrescent in Br C D if and only if thereexists a class 1C function a2() such that

W(x,t) <a2(IlX ), Vx E B,. C D.

W(., ) is said to be radially unbounded if W(x, k) -4 oo as jxjj -4 oo, uniformly onk. This means that given M > 0, there exists N > 0 such that

W (x, k) > M

provided that jjxjj > N.


4.10.3 Stability Theorems

Definition 4.10 The rate of change, OW (x, k), of the function W (x, k) along the trajec-tories of the system (4.37) is defined by

OW (x, k) = W(x(k + 1), k + 1) - W (x, k).

With these definitions, we can now state and prove several stability theorems for discrete-time system. Roughly speaking, all the theorems studied in Chapters 3 and 4 can berestated for discrete-time systems, with AW (x, k) replacing W (x, t). The proofs are nearlyidentical to their continuous-time counterparts and are omitted.

Theorem 4.10 (Lyapunov Stability Theorem for Discrete-Time Systems). If in a neigh-borhood D of the equilibrium state x = 0 of the system (4.37) there exists a functionW(., ) : D x Z+ -+ R such that

(i) W (x, k) is positive definite.

(ii) The rate of change OW(x, k) along any solution of (4.37) is negative semidefinite inD, then

the equilibrium state is stable. Moreover, if W (x, k) is also decrescent, then the origin isuniformly stable.

Theorem 4.11 (Lyapunov Uniform Asymptotic Stability for Discrete- Time Systems). If ina neighborhood D of the equilibrium state x = 0 there exists a function W(-, ) : D x Z+ ](P

such that

(i) W (x, k) is (a) positive definite, and (b) decrescent.

(ii) The rate of change, AW(x, k) is negative definite in D, then

the equilibrium state is uniformly asymptotically stable.

Example 4.7 Consider the following discrete-time system:

xl(k + 1) = xl(k) + x2(k) (4.42)

X2 (k + 1) = axi(k) + 2x2(k). (4.43)

4.11. EXERCISES 133

To study the stability of the origin, we consider the (time-independent) Lyapunov functioncandidate V(x) = 2xi(k) + 2x1(k)x2(k) + 4x2(k), which can be easily seen to be positivedefinite. We need to find AV(x) = V(x(k + 1)) - V(x(k)), we have

V(x(k+1)) = 2xi(k+1)+2x1(k+1)x2(k+1)+4x2(k+1)

2 [xi(k) + x2(k)]2 + 2(x1(k) +x2(k)][ax3

(k) + 2x2(k)]

+4[axl(k) + 2x2(k)]2

V(x(k)) = 2x2 + 2xix2 + 4x2.

From here, after some trivial manipulations, we conclude that

AV(x) = V(x(k + 1)) - V(x(k)) _ 2x2 + 2ax4 + 6ax3 x2 + 4a2x6.

Therefore we have the following cases of interest:

a < 0. In this case, AV(x) is negative definite in a neighborhood of the origin, andthe origin is locally asymptotically stable (uniformly, since the system is autonomous).

a = 0. In this case AV(x) = V(x(k + 1)) - V(x(k)) = -2x2 < 0, and thus the originis stable.

4.11 Exercises

(4.1) Prove Lemma 4.1.



(4.4) Characterize each of the following functions W : 1R2 x R --4 R as: (a) positive definiteor not, (b) decreasing or not, (c) radially unbounded or not.

(i) W1 (X, t) = (xi + x2)(ii) W2(x, t) = (xl + x2)et.


(iii) W3(x, t) = (xi + x2)e-t.

(iv) W4(x, t) = (xi + x2)(1 + e-t).

(V) W5(x, t) = (xi + x2) cos2 wt.

(vi) W6(x,t) = (xi + x2)(1 + cost wt).

(Vii) W7(x,t) =(x2+x2)(l+e t).

xl{'1)

(4.5) It is known that a given dynamical system has an equilibrium point at the origin.For this system, a function has been proposed, and its derivative has beencomputed. Assuming that and are given below you are asked to classify theorigin, in each case, as (a) stable, (b) locally uniformly asymptotically stable, and/or(c) globally uniformly asymptotically stable. Explain you answer in each case.

(i) W1 (X, t) = (xl + x2), W1(x, t) = -xi.

(ii) W2 (X, t) = (xi + x2), Wz(x, t) = -(xl + x2)e-t.(iii) W3(x, t) = (xi + x2), W3(x, t) = -(xl + x2)et.

(iv) W4 (X, t) = (xi + x2)et, W4 (X, t) _ -(xl + x2)(1 + sine t).

(v) W5(x, t) = (xl + x2)e t W'5 (X, t) _ -(xi + x2)(vi) W6 (X, t) = (xi + x2)(1 + e-t), W6 (x, t) _ -xle-t.

(Vii) W7(x, t) = (xi + x2) cos2 wt, W7(x, t) _ -(xi + x2).

(viii) W8 (X, t) _ (xl + x2)(1 + cos2 wt), 4Vs(x, t) _ -xi.

-(x2 + x2)e-t.(ix) W9(x, t) = (xi +x2)(1 + cos2 wt), Ws(x, t) =

(x) Wio(x,t) = (x2 + x2)(1 + cos2 wt), W10 (x,t) _ -(xi + x2)(1 + e-t).

(xi) W11(x, t) =(x2+ )+1 , W11(x) t) _

(x1 + x2).

(4.6) Given the following system, study the stability of the equilibrium point at the origin:

21 = -x1 - x1x2 cos2 t22 = -x2 - X1X2 sin2 t



(4.9) Given the following system, study the stability of the equilibrium point at the origin:

4.11. EXERCISES 135

xl

±2

= x2 + xi + xz= -2x1 - 3X2

Hint: Notice that the given dynamical equations can be expressed in the formAx + g(x)-

Notes and ReferencesGood sources for the material of this chapter include References [27], [41], [88], [68],

and [95]. Section 4.5 is based on Vidyasagar [88] and Willems [95]. Perturbation analysishas been a subject of much research in nonlinear control. Classical references on the subjectare Hahn [27] and Krasovskii [44]. Section 4.6 is based on Khalil [41].

Chapter 5

Feedback Systems

So far, our attention has been restricted to open-loop systems. In a typical control problem,however, our interest is usually in the analysis and design of feedback control systems.Feedback systems can be analyzed using the same tools elaborated so far after incorporatingthe effect of the input u on the system dynamics. In this chapter we look at several examplesof feedback systems and introduce a simple design technique for stabilization known asbackstepping. To start, consider the system

x = f(x,u) (5.1)

and assume that the origin x = 0 is an equilibrium point of the unforced system ± = f (x, 0).Now suppose that u is obtained using a state feedback law of the form

u = cb(x)

To study the stability of this system, we substitute (5.2) into (5.1) to obtain

± = f (x, O(x))

According to the stability results in Chapter 3, if the origin of the unforced system (5.3) isasymptotically stable, then we can find a positive definite function V whose time derivativealong the trajectories of (5.3) is negative definite in a neighborhood of the origin.

It seems clear from this discussion that the stability of feedback systems can be studiedusing the same tool discussed in Chapters 3 and 4.

137

138 CHAPTER 5. FEEDBACK SYSTEMS

5.1 Basic Feedback Stabilization

In this section we look at several examples of stabilization via state feedback. These exam-ples will provide valuable insight into the backstepping design of the next section.

Example 5.1 Consider the first order system given by

we look for a state feedback of the form

u = O(x)

that makes the equilibrium point at the origin "asymptotically stable." One rather obviousway to approach the problem is to choose a control law u that "cancels" the nonlinear term

ax2. Indeed, settingu = -ax2 - x

and substituting (5.5) into (5.4) we obtain

±= -x

which is linear and globally asymptotically stable, as desired. 0

We mention in passing that this is a simple example of a technique known as feedbacklinearization. While the idea works quite well in our example, it does come at a certainprice. We notice two things:

(i) It is based on exact cancelation of the nonlinear term ax2. This is undesirable sincein practice system parameters such as "a" in our example are never known exactly.In a more realistic scenario what we would obtain at the end of the design processwith our control u is a system of the form

i=(a-a)x2-xwhere a represents the true system parameter and a the actual value used in thefeedback law. In this case the true system is also asymptotically stable, but onlylocally because of the presence of the term (a - a)x2.

(ii) Even assuming perfect modeling it may not be a good idea to follow this approachand cancel "all" nonlinear terms that appear in the dynamical system. The reasonis that nonlinearities in the dynamical equation are not necessarily bad. To see this,consider the following example.

5.1. BASIC FEEDBACK STABILIZATION 139

Example 5.2 Consider the system given by

Lb =ax2-x3+u

following the approach in Example 5.1 we can set

u ui = -ax 2 + x3 - x

which leads to

Once again, ul is the feedback linearization law that renders a globally asymptotically stablelinear system. Let's now examine in more detail how this result was accomplished. Noticefirst that our control law ul was chosen to cancel both nonlinear terms ax2 and -x3. Thesetwo terms are, however, quite different:

The presence of terms of the form x2 with i even on a dynamical equation is neverdesirable. Indeed, even powers of x do not discriminate sign of the variable x andthus have a destabilizing effect that should be avoided whenever possible.

Terms of the form -x2, with j odd, on the other hand, greatly contribute to thefeedback law by providing additional damping for large values of x and are usuallybeneficial.

At the same time, notice that the cancellation of the term x3 was achieved by incor-porating the term x3 in the feedback law. The presence of this variable in the controlinput u can lead to very large values of the input. In practice, the physical charac-teristics of the actuators may place limits on the amplitude of this function, and thusthe presence of the term x3 on the input u is not desirable.

To find an alternate solution, we proceed as follows. Given the system

x= f(x,u) xEr,uE1R, f(0,0)=0we proceed to find a feedback law of the form

u = q5(x)

such that the feedback system± = f (x, O(x)) (5.6)

has an asymptotically stable equilibrium point at the origin. To show that this is the case,we will construct a function Vl = Vl (x) : D -* IR satisfying

(i) V(0) = 0, and V1(x) is positive definite in D - {0}.

(ii) V1(x) is negative definite along the solutions of (5.6). Moreover, there exist a positivedefinite function V2(x) : D --> R+ such that

Vi (x) = ax- l f (x, O(x)) < -V2 (x) Vx E D.

Clearly, if D = ]R" and V1 is radially unbounded, then the origin is globally asymptoticallystable by Theorems 3.2 and 3.3.

Example 5.3 Consider again the system of example 5.2.

=ax2-x3+udefining Vl(x) = 2x2 and computing V1, we obtain

Vl = x f (x, u)ax3 - x4 + xu.

In Example 5.2 we chose u = ul = -axe + x3 - x. With this input function we have that

V1 = ax3 - x4 + x(-ax2 + x3 - x) = -x2 'f - V2(x).

It then follows that this control law satisfies requirement (ii) above with V2(x) = x2. Nothappy with this solution, we modify the function V2(x) as follows:

V1 = ax3 - x4 + xu < -V2(x) dEf - (x4 + x2).

For this to be the case, we must have

ax3 - x4 + xu < -x4 - x2xu < -x2 - ax3 = -x(x + ax2)

which can be accomplished by choosing

u = -x - axe.

With this input function u, we obtain the following feedback system

2 = ax2-x3+u-x-x3

which is asymptotically stable. The result is global since V1 is radially unbounded and D = R.

5.2. INTEGRATOR BACKSTEPPING 141

5.2 Integrator Backstepping

Guided by the examples of the previous section, we now explore a recursive design techniqueknown as backstepping. To start with, we consider a system of the form

= f(x) + g(x)f (5.7)= U. (5.8)

Here x E ]Rn, E ]R, and [x, e]T E ]Rn+1 is the state of the system (5.7)-(5.8). The functionu E R is the control input and the functions f,g : D -> R" are assumed to be smooth. Aswill be seen shortly, the importance of this structure is that can be considered as a cascadeconnection of the subsystems (5.7) and (5.8). We will make the following assumptions (seeFigure 5.1(a)):

(i) The function f Rn -> ]Rn satisfies f (0) = 0. Thus, the origin is an equilibriumpoint of the subsystem i = f (x).

(ii) Consider the subsystem (5.7). Viewing the state variable as an independent "input"for this subsystem, we assume that there exists a state feedback control law of theform

_ Ox), 0(0) = 0

and a Lyapunov function V1 : D -4 IIF+ such that

ov,

x[f (x) + g(x) - O(x)] < -V,,(x) < 0 Vx E DV1(x) 8

where D R+ is a positive semidefinite function in D.

According to these assumptions, the system (5.7)-(5.8) consists of the subsystem (5.7),for which a known stabilizing law already exists, augmented with a pure integrator (thesubsystems (5.8)). More general classes of systems are considered in the next section. Wenow endeavor to find a state feedback law to asymptotically stabilize the system (5.7)-(5.8).To this end we proceed as follows:

We start by adding and subtracting g(x)o(x) to the subsystem (5.7) (Figure 1(b)).We obtain the equivalent system

x = f(x) + g(x)0(x) + g(x)[ - 0(x)1 (5.9)

= U. (5.10)


a)

b)U

U

-O(x)

c)U z

x

x

x

L f (') + 9(')O(')

d)v=z z x

Figure 5.1: (a) The system (5.7)-(5.8); (b) modified system after introducing -O(x); (c)"backstepping" of -O(x); (d) the final system after the change of variables.

5.2. INTEGRATOR BACKSTEPPING 143

Define

z = - O(x) (5.11)

z = - fi(x) = u- fi(x) (5.12)

where

q= a0x=

ao[f(x)+9(x).](5.13)

This change of variables can be seen as "backstepping" -O(x) through the integrator,as shown in Figure 1(c). Defining

v = z (5.14)

the resulting system is

i = f(x) + 9(x)O(x) + 9(x)z (5.15)

i = v (5.16)

which is shown in Figure 1(d). These two steps are important for the following reasons:

(i) By construction, the system (5.15)-(5.16) is equivalent to the system (5.7)-(5.8).

(ii) The system (5.15)-(5.16) is, once again, the cascade connection of two subsys-tems, as shown in Figure 1(d). However, the subsystem (5.15) incorporates thestabilizing state feedback law and is thus asymptotically stable when theinput is zero. This feature will now be exploited in the design of a stabilizingcontrol law for the overall system (5.15)-(5.16).

To stabilize the system (5.15)-(5.16) consider a Lyapunov function candidate of theform

V = V (x, ) = Vi (x) + Z z2. (5.17)

We have that

-av,V _ -

[f(x) + 9(x)O(x) + 9(x)z] + ziE---x

_AX f (x) + - a-x g(x)z + zv.ax

We can choose

v - ((x) + kz) , k > 0 (5.18)

Thus

V - 8V1f (x) +E-1 g(x)cb(x) - kz2

ax 8x

al[f(x) + g(x).O(x)] - kz2

< -Va(x) - kz2. (5.19)

It then follows by (5.19) that the origin x = 0, z = 0 is asymptotically stable. More-over, since z = l; - O(x) and 0(0) = 0 by assumption, the result also implies thatthe origin of the original system x = 0, l; = 0 is also asymptotically stable. If allthe conditions hold globally and V1 is radially unbounded, then the origin is globallyasymptotically stable. Finally, notice that, according to (5.12), the stabilizing statefeedback law is given by

u = i + q5 (5.20)

and using (5.13), (5.18), and (5.11), we obtain

u = LO [f (x) + g(x) - k[ -(5.21)Example 5.4 Consider the following system, which is a modified version of the one inExample 5.3:

it = axi - x1 + X2 (5.22)

X2 = u. (5.23)

Clearly this system is of the form (5.15)-(5.16) with

x` = xlS = x2

f(x) = f(xi) = axi - x3

g(x) = 1

Step 1: Viewing the "state" as an independent input for the subsystem (5.15), find astate feedback control law _ q(x) to stabilize the origin x = 0. In our case we haveit =axi - xi + x2. Proceeding as in Example 5.3, we define

1V1(xl) = 1xi

Vl(xl) = axi - x1 + x1x2 < -Va(x1)deI

- (x1 + x1

5.3. BACKSTEPPING: MORE GENERAL CASES 145

which can be accomplished by choosing

x2 = 0(x1) = -XI - axi

leading to

21 = -xl -X3X.

Step 2: To stabilize the original system (5.22)- (5.23), we make use of the control law(5.21). We have

u8x [f (x) + g(xX] - -5X 1 g(x) - k[C - O(x)]

= -(1+2axi)[ax2 i-x3 l+x2]-xl-k[x2+xl+ax2i].

With this control law the origin is globally asymptotically stable (notice that Vl is radiallyunbounded). The composite Lyapunov function is

V=Vi+2z2 = 2X2+2[x2-O(xl)]2

= 2X2 + 2[x2 - xl + axT.

5.3 Backstepping: More General Cases

In the previous section we discussed integrator backstepping for systems with a state of theform [x, l;], x E 1R", E 1R, under the assumption that a stabilizing law 0. We now look atmore general classes of systems.

5.3.1 Chain of Integrators

A simple but useful extension of this case is that of a "chain" of integrators, specifically asystem of the form

th = f(x) + g(X)1

41 = 6


G-1bk u

Backstepping design for this class of systems can be approached using successive iterationsof the procedure used in the previous section. To simplify our notation, we consider, withoutloss of generality, the third order system

= f(x) + 9(x)1 (5.24)

1 = 6 (5.25)

u (5.26)

and proceed to design a stabilizing control law. We first consider the first two "subsystems"

= f(x) + 9(x)1 (5.27)

S1 = 6 (5.28)

and assume that 1 = O(x1) is a stabilizing control law for the system

th = f(x) + g(x)q5(x)

Moreover, we also assume that V1 is the corresponding Lyapunov function for this subsys-tem. The second-order system (5.27)-(5.28) can be seen as having the form (5.7)-(5.8) with6 considered as an independent input. We can asymptotically stabilize this system usingthe control law (5.21) and associated Lyapunov function V2:

6 = O(x, bl) = aa(x) [f (x) + 9(x) 1] - ax1 g(x) - k > 0

V2 = V1 + 2 [S1 - O(x)]2

We now iterate this process and view the third-order system given by the first three equa-tions as a more general version of (5.7)-(5.8) with

x= 1 z, f= f f(x)+09(x) 1 1, 9=L 0 J

Applying the backstepping algorithm once more, we obtain the stabilizing control law:

at(x) . aV2

u = ax xCC- V 9(x[) - k[f2[- O(x)], k > 0

[ C

[a0OxSl) 90(x, WI [.t ,51]T - [ a 22, a`1 ] [0 , 11T - k[S2 - O(x,i)], k > ol ac,


or

u = a0(x,W [f (x) +9(x)1] + a0(x,S1)b2 - aV2- k[l;2 - 0(x,51)], k> 0.

ax al, ailThe composite Lyapunov function is

1V =CC I CC

= V1 + 2 [S1 - O(x)]2 + 2 [S2 - O(x, bl)]2.

We finally point out that while, for simplicity, we have focused attention on third-ordersystems, the procedure for nth-order systems is entirely analogous.

Example 5.5 Consider the following system, which is a modified version of the one inExample 5.4:

±1 = axi + x2x2 = x3X3 = u.

We proceed to stabilize this system using the backstepping approach. To start, we considerthe first equation treating x2 is an independent "input" and proceed to find a state feedbacklaw O(xi) that stabilizes this subsystem. In other words, we consider the system it =ax2 2+ O(xi) and find a stabilizing law u = O(x1). Using the Lyapunov function Vl = 1/2x1,it is immediate that 0(xl) = -xl - axe is one such law.

We can now proceed to the first step of backstepping and consider the first two sub-systems, assuming at this point that x3 is an independent input. Using the result in theprevious section, we propose the stabilizing law

O(x1,x2)(= x3) =11)[f(xl)

+9(xl)x2] -aav,x19(xl) - k[x2 - O(xl)], k > 0

with associated Lyapunov function

V2 = Vl+_z2

Vl +1

2[x2 - 0(xl )]

= VI +

2

[x2 + XI + axi]2.


In our case

a¢(xl )ax,

avl

-(1 + 2axi)

x1

=> O(xl, x2) = -(1 + 2ax1) [axi + x2] - x1 - [X2 + x1 + axi]

where we have chosen k = 1. We now move on to the final step, in which we consider thethird order system as a special case of (5.7)-(5.8) with

x=I

fL f(x1) Og(x1)x2

9x2 J J L 0 J

From the results in the previous section we have that

A,I,U = 090(x1, x2)[f (XI) + 9(xl)x2] +

aO(xl, x2) X3 _ aV2- k[x3 - W(xl, x2)], k > 0

axi 09x2 09x2

is a stabilizing control law with associated Lyapunov function

V +21

0

5.3.2 Strict Feedback Systems

We now consider systems of the form

f (x) +CC 9(x)S1CC CC

S1 = fl (x, Si) + 91(x, Sl)SCC2

2 = f2(x,e1,6) + 92(x,6,6)e3

Sk-1 = A-1 (X, 6, S2, Sk-1) + 9k-1(x, SL 61- ' GAG6 = A(Xi61 2) ...7 G)+A(xi6)6,.,Ck)U

where x E R', Ct E R, and fi, g= are smooth, for all i = 1, , k. Systems of this formare called strict feedback systems because the nonlinearities f, fzf and gz depend only onthe variables x, C1, that are fed back. Strict feedback systems are also called triangular


systems. We begin our discussion considering the special case where the system is of orderone (equivalently, k = 1 in the system defined above):

= f(x)+g(x)

= fa(x,0+9a(x,0u.(5.29)

(5.30)

Assuming that the x subsystem (5.29) satisfies assumptions (i) and (ii) of the backsteppingprocedure in Section 5.2, we now endeavor to stabilize (5.29)-(5.30). This system reduces tothe integrator backstepping of Section 5.2 in the special case where f" (x, ) - 0, ga(x, ) - 1.To avoid trivialities we assume that this is not the case. If ga(x, ) # 0 over the domain ofinterest, then we can define

U = O(x, S)def

9.(x,1

S)[Ul - Mx' S)].

Substituting (5.31) into (5.30) we obtain the modified system

Cx = f (x) + 9(xX41 = t1

(5.31)

(5.32)

(5.33)

which is of the form (5.7)-(5.8). It then follows that, using (5.21), (5.17), and (5.31), thestabilizing control law and associated Lyapunov function are

9a(x, ){[f(x)+g(x)] - 19(x) - k1 [ - fi(x)] - fa(x, kl > 0

(5.34)

V2 = V2 (x, S) = Vl (x) + 2 [S -O(x)]2. (5.35)

We now generalize these ideas by moving one step further and considering the system

[[ = f (x) +[ 9(x)1[ C

S[[1 = f1(x,S[[1)[[+ 91(x,Sl)S2

S2 = f2(x,S1,S2) + 92(x,1,&)6

which can be seen as a special case of (5.29)-(5.30) with

x= l C1

J, e=e2, l

f `9 1 ] , 9 = [ 0 ],fa=f2, 9a=92f, 91


With these definitions, and using the control law and associated Lyapunov function (5.34)-(5.35), we have that a stabilizing control law and associated Lyapunov function for thissystems are as follows:

02(x, 6, Sz) = 9z 1'x - afi (fi(x)+9t a i91-kz[fz-01]-fz}k2 > 0 (5.36)

V3 (X, S1, 2) = V2(x) + 2 [6 - 01(x, 1)]2.

The general case can be solved iterating this process.

Example 5.6 Consider the following systems:

(5.37)

x1 = axz - xl + x 21 1x2

x2 = XI + x2 + (1 + xz)u.

We begin by stabilizing the x subsystem. Using the Lyapunov function candidate V1 = 1/2x2we have that

V1 = x1 [ax2 - x1 + x2x2]3 2 3ax1 - x1 +X 1x2.

Thus, the control law x2 = O(x1) = -(x1 + a) results in

V-(x1+xi)which shows that the x system is asymptotically stable. It then follows by (5.34)-(5.35)that a stabilizing control law for the second-order system and the corresponding Lyapunovfunction are given by

01(x1, x2)

V2 =

1

(1 + x2)f -(1 +a)[ax2 - X1 + x2 x2] - xi - k1[x2 + x1 + a] - (x1 + X2)},

2x2+2[x1+x2+a]2.

k1>0

5.4. EXAMPLE 151

8Figure 5.2: Current-driven magnetic suspension system.

5.4 Example

Consider the magnetic suspension system of Section 1.9.1, but assume that to simplifymatters, the electromagnet is driven by a current source 72, as shown in Figure 5.2. Theequation of the motion of the ball remain the same as (1.31):

AµI2my = -ky + mg -

2(1 + µy)2 .

Defining states zl = y, X2 = y, we obtain the following state space realization:

X1 = X2 (5.38)

k \I2X2 = 9 - m X2 2m(1 +px1)2. (5.39)

A quick look at this model reveals that it is "almost" in strict feedback form. It is not in theproper form of Section 5.3.2 because of the square term in 72. In this case, however, we canignore this matter and proceed with the design without change. Notice that I representsthe direct current flowing through the electromagnet, and therefore negative values of theinput cannot occur.

We are interested in a control law that maintains the ball at an arbitrary positiony = yo. We can easily obtain the current necessary to achieve this objective. Setting x1 = 0


along with it = x2 = 0 in equations (5.38)-(5.39), we obtain

2'm.Io = µ (1 + µyo)2.

It is straightforward to show that this equilibrium point is unstable, and so we look for astate feedback control law to stabilize the closed loop around the equilibrium point. We startby applying a coordinate transformation to translate the equilibrium point to = (yo, 0)T tothe origin. To this end we define new coordinates:

X1 = :i - Yo, = X1=:1x2 x2 X2 = x2

I2 2mg(1 + µyo)2Ay

In the new coordinates the model takes the form:

21 = X2 (5.40)

k _ 9(1 + µyo)2 _ Ay (5.41)1'2 9_ mx2[1 + µ(x1 + yo]2 2m[1 + µ(x1 + yo)]2

u

which has an equilibrium point at the origin with u = 0. The new model is in the form(5.29)-(5.30) with

x = X1, S = X2, f (X) = 0, 9(x) = 1

k 9(1 + µyo)2 aµMx, 0 = g - mx2 - [1 + µ(x1 + yo]2'

and g. (x,[1 + µ(x1 + yo]2

.

Step 1: We begin our design by stabilizing the xl-subsystem. Using the Lyapunov functioncandidate V1 = 2x? we have

V1 = xjx2

and setting x2 = -x1 we obtain O(x1) _ -xl, which stabilizes the first equation.

Step 2: We now proceed to find a stabilizing control law for the two-state system usingbacksteppping. To this end we use the control law (5.34) with

Ap 009a = -2m[1 + (xl + yo)]2' 8x [f (x) + 9(x).] = x2

1 2 8V1 -V1 =2

x1, ex - x1, 9(x) = 1

5.5. EXERCISES 153

k 2

[S - 0(x)] = x1 + x2, Ja(x, S) = 9 -[1

+ µ(x1+ yo]2 .

Substituting values, we obtain

2 ku = 01(x1, x2) _

_2m[1 + p(xl + yo)] [_(1+ki)(xi+x2)+_x2,

(1 + µyo)2 1I

+g [1+ µ(x1 + yo)]2 J

The corresponding Lyapunov function is

V2 = V2(xl, x2) = V1 + 2 [x2 - O(x)12

X 2)

.

-9

(5.42)

Straightforward manipulations show that with this control law the closed-loop system re-duces to the following:

xl = -X1x2 = -(1 + k)(xl + x2)

5.5 Exercises


r xl=xl+cosxl-1+x25l x2 =U

Using backstepping, design a state feedback control law to stabilize the equilibriumpoint at the origin.


{x1 = x2

x2 = x1 +X3 + x3X2 = U

Using backstepping, design a tate feedback control law to stabilize the equilibriumpoint at the origin.


(5.3) Consider the following system, consisting of a chain of integrators:

xl = xl + ext - 1 + X2x2 = X3X2 =u



f it = xl +X 1 + xlx2

ll a2 = xi + (1 + x2)u



zl = x1 + x2{ :2 = x2 ix2 - xi + u


Notes and ReferencesThis chapter is based heavily on Reference [47], with help from chapter 13 of Khalil [41].The reader interested in backstepping should consult Reference [47], which contains a lotof additional material on backstepping, including interesting applications as well as theextension of the backstepping approach to adaptive control of nonlinear plants.

Chapter 6

Input-Output Stability

So far we have explored the notion of stability in the sense of Lyapunov, which, as discussedin Chapters 3 and 4, corresponds to stability of equilibrium points for the free or unforcedsystem. This notion is then characterized by the lack of external excitations and is certainlynot the only way of defining stability.

In this chapter we explore the notion of input-output stability, as an alternative tothe stability in the sense of Lyapunov. The input-output theory of systems was initiated inthe 1960s by G. Zames and I. Sandberg, and it departs from a conceptually very differentapproach. Namely, it considers systems as mappings from inputs to outputs and definesstability in terms of whether the system output is bounded whenever the input is bounded.Thus, roughly speaking, a system is viewed as a black box and can be represented graphicallyas shown in Figure 6.1.

To define the notion of mathematical model of physical systems, we first need tochoose a suitable space of functions, which we will denote by X. The space X must be

uH y

Figure 6.1: The system H.

155

156 CHAPTER 6. INPUT-OUTPUT STABILITY

sufficiently rich to contain all input functions of interest as well as the corresponding outputs.Mathematically this is a challenging problem since we would like to be able to considersystems that are not well behaved, that is, where the output to an input in the spaceX may not belong to X. The classical solution to this dilemma consists of making use ofthe so-called extended spaces, which we now introduce.

6.1 Function Spaces

In Chapter 2 we introduced the notion of vector space, and so far our interest has beenlimited to the nth-dimensional space ]R". In this chapter we need to consider "functionspaces," specifically, spaces where the "vectors," or "elements," of the space are functionsof time. By far, the most important spaces of this kind in control applications are theso-called Lp spaces which we now introduce.

In the following definition, we consider a function u : pp+ --> R q, i.e., u is of the form:

ui (t)

u(t) = u2(t)

uq(t)

Definition 6.1 (The Space ,C2) The space .C2 consists of all piecewise continuous functionsu : 1R -> ]R9 satisfying

IIuIIc2 =f f -[Iu1I2 + Iu2I2 +... + Iu9I2] dt < oo. (6.1)

The norm IIuII1C2 defined in this equation is the so-called C2 norm of the function u.

Definition 6.2 (The Space Coo) The space Goo consists of all piecewise continuous functionsu : pt+ -1 1R4 satisfying

IIuIIc-dcf

sup IIu(t)II. < oo. (6.2)tEIR+

The reader should not confuse the two different norms used in equation (6.2). Indeed, thenorm IIuIIc_ is the ,Coonorm of the function u, whereas IIu(t)lloo represents the infinity normof the vector u(t) in 1R4, defined in section 2.3.1. In other words

IIuIIcm 4f sup (max Iuil) < oo 1 < i < q.tE]R+

i

6.1. FUNCTION SPACES 157

Both L2 and L,,. are special cases of the so-called LP spaces. Given p : 1 < p < oo, thespace LP consists of all piecewise continuous functions u : IIt+ -4 Rq satisfying

00 1/P

IIUIIcp f (f [Iu1I' + IU2IP + ... + IugIP] dt) < oo. (6.3)(fAnother useful space is the so-called L1. From (6.3), L1 is the space of all piecewise

continuous functions u : pg+ -4 Rq satisfying:

IIUIIc1f (jiuii + IU2I + + jugI] dt) < oo. (6.4)

Property: (Holder's inequality in Lp spaces). If p and q are such that 1 + 1 = 1 with1 < p < oo and if f E LP and g E Lq, then f g E L1, and

I(f9)TII1=Tf f(t)9(t)dt < (fT f(t)Idt) (fT I9(t)I

dt) . (6.5)o

For the most part, we will focus our attention on the space L2, with occasionalreference to the space L. However, most of the stability theorems that we will encounterin the sequel, as well as all the stability definitions, are valid in a much more general setting.To add generality to our presentation, we will state all of our definitions and most of themain theorems referring to a generic space of functions, denoted by X.

6.1.1 Extended Spaces

We are now in a position to introduce the notion of extended spaces.

Definition 6.3 Let u E X. We define the truncation operator PT : X -> X by

u(t), t < Tdef

(PTU)(t) = UT(t) =0, t > T

t, T E R+ (6.6)

Example 6.1 Consider the function u : [0, oo) -+ [0, oo) defined by u(t) = t2. The trunca-tion of u(t) is the following function:

_ t2, 0 < t < TuT(t) 0, t > T

Notice that according to definition 6.3, PT satisfies

(1) [PT(u + v)J(t) = UT(t) + VT(t) Vu, V E Xe .

(ii) [PT(au)](t) = auT(t) Vu E Xe , a E R.

Thus, the truncation operator is a linear operator.

Definition 6.4 The extension of the space X, denoted Xe is defined as follows:

Xe = {u : ]R+ - IR9, such that XT E X VT E R+}. (6.7)

In other words, X. is the space consisting of all functions whose truncation belongs to X,regardless of whether u itself belongs to X. In the sequel, the space X is referred to as the"parent" space of Xe. We will assume that the space X satisfy the following properties:

(i) X is a normed linear space of piecewise continuous functions of the form u : IR+ -> IR9.The norm of functions in the space X will be denoted II - IIx

(ii) X is such that if u E X, then UT E X VT E IR+, and moreover, X is such thatu = limTc,, UT. Equivalently, X is closed under the family of projections {PT}.

(iii) If u E X and T E R+, then IIUTIIX 5 IIxIIx ; that is, IIxTIIxfunction of T E IR+.

(iv) If u E Xe, then u E X if and only if limT-+,, IIxTIIx < oo.

is a nondecreasing

It can be easily seen that all the Lp spaces satisfy these properties. Notice that althoughX is a normed space, Xe is a linear (not normed) space. It is not normed because ingeneral, the norm of a function u E Xe is not defined. Given a function u E Xe , however,using property (iv) above, it is possible to check whether u E X by studying the limitlimT,,,. IIUTII

Example 6.2 Let the space of functions X be defined by

X = {x x : 1R -+ 1R x(t) integrable and Jx x(t) dt < o0

In other words, X is the space of real-valued function in Li. Consider the function x(t) = t.We have

XT(t) =f t, 0 < t < T1f0, t>T

TIIXTII = J I XT(t) I dt = J T t dt =

22

6.2. INPUT-OUTPUT STABILITY 159

Thus XT E Xe VT E R+. However x ¢ X since IXT I = oo.

Remarks: In our study of feedback systems we will encounter unstable systems, i.e., sys-tems whose output grows without bound as time increases. Those systems cannot be de-scribed with any of the LP spaces introduced before, or even in any other space of functionsused in mathematics. Thus, the extended spaces are the right setting for our problem. Asmentioned earlier, our primary interest is in the spaces £2 and G,. The extension of thespace LP , 1 < p < oo, will be denoted Lpe. It consists of all the functions u(t) whosetruncation belongs to LP .

6.2 Input-Output Stability

We start with a precise definition of the notion of system.

Definition 6.5 A system, or more precisely, the mathematical representation of a physicalsystem, is defined to be a mapping H : Xe -> Xe that satisfies the so-called causalitycondition:

[Hu(')]T = Vu E Xe and VT E R. (6.8)

Condition (6.8) is important in that it formalizes the notion, satisfied by all physical systems,that the past and present outputs do not depend on future inputs. To see this, imaginethat we perform the following experiments (Figures 6.2 and 6.3):

(1) First we apply an arbitrary input u(t), we find the output y(t) = Hu(t), and fromhere the truncated output yT(t) = [Hu(t)]T. Clearly yT = [Hu(t)]T(t) represents theleft-hand side of equation (6.8). See Figure 6.3(a)-(c).

(2) In the second experiment we start by computing the truncation u = uT(t) of the inputu(t) used above, and repeat the procedure used in the first experiment. Namely, wecompute the output y(t) = Hu(t) = HuT(t) to the input u(t) = uT(t), and finally wetake the truncation yT = [HuT(t)]T of the function y. Notice that this correspondsto the right-hand side of equation (6.8). See Figure 6.3(d)-(f).

The difference in these two experiments is the truncated input used in part (2). Thus,if the outputs [Hu(t)]T and [HUT(t)]T are identical, the system output in the interval0 < t < T does not depend on values of the input outside this interval (i.e., u(t) for t > T).All physical systems share this property, but care must be exercised with mathematicalmodels since not all functions behave like this.


u(t)H

y(t) = Hu(t)

u(t) =UT(t)H

y(t) = HUT(t)

Figure 6.2: Experiment 1: input u(t) applied to system H. Experiment 2: input fl(t) = UT(t)applied to system H.

We end this discussion by pointing out that the causality condition (6.8) is frequentlyexpressed using the projection operator PT as follows:

PTH = PTHPT (i.e., PT(Hx) = PT[H(PTx)] dx E Xe and VT E W1 .

It is important to notice that the notion of input-output stability, and in fact, thenotion of input-output system itself, does not depend in any way on the notion of state.In fact, the internal description given by the state is unnecessary in this framework. Theessence of the input-output theory is that only the relationship between inputs and outputsis relevant. Strictly speaking, there is no room in the input-output theory for the existenceof nonzero (variable) initial conditions. In this sense, the input-output theory of systemsin general, and the notion of input-output stability in particular, are complementary tothe Lyapunov theory. Notice that the Lyapunov theory deal with equilibrium points ofsystems with zero inputs and nonzero initial conditions, while the input-output theoryconsiders relaxed systems with non-zero inputs. In later chapters, we review these conceptsand consider input-output systems with an internal description given by a state spacerealization.

We may now state the definition of input-output stability.

Definition 6.6 A system H : Xe -> Xe is said to be input-output X-stable if whenever theinput belongs to the parent space X, the output is once again in X. In other words, H isX-stable if Hx is in X whenever u E X.


(a)

y(t) = Hu(t)

(b)

Hu(t)]T

t

t

(c)

(d)

[HuT(t)]

(e)

[HUT(t)1T

(f )

T

t

t

t

t

Figure 6.3: Causal systems: (a) input u(t); (b) the response y(t) = Hu(t); (c) truncationof the response y(t). Notice that this figure corresponds to the left-hand side of equation(6.8); (d) truncation of the function u(t); (e) response of the system when the input is thetruncated input uT(t); (f) truncation of the system response in part (e). Notice that thisfigure corresponds to the right-hand side of equation (6.8).

For simplicity we will usually say that H is input-output stable instead of input-output X-stable whenever confusion is unlikely. It is clear that input-output stability is anotion that depends both on the system and the space of functions.

One of the most useful concepts associated with systems is the notion of gain.

Definition 6.7 A system H : Xe -4 Xe is said to have a finite gain if there exists aconstant y(H) < oo called the gain of H, and a constant,3 E pl+ such that

II(Hu)TIIX <_y(H) IIuTIIX +a. (6.9)

Systems with finite gain are said to be finite-gain-stable. The constant /j in (6.9) is calledthe bias term and is included in this definition to allow the case where Hu # 0 when u = 0.A different, and perhaps more important interpretation of this constant will be discussedin connection with input-output properties of state space realizations. If the system Hsatisfies the condition

Hu = 0 whenever u = 0

then the gain y(H) can be calculated as follows

y(H) = sup II (IHu IIXI X

where the supremum is taken over all u E X. and all T in lR+ for which UT # 0.

(6.10)

Example 6.3 Let X = Goo, and consider the nonlinear operator N(-) defined by the graphin the plane shown in Figure 6.4, and notice that N(0) = 0. The gain y(H) is easilydetermined from the slope of the graph of N.

y(H) = sup II(Hu)TIIG- = 1.IIuTIIL-

Systems such as in example 6.3 have no "dynamics" and are called static ormemoryless systems, given that the response is an instantaneous function of the input.

We conclude this section by making the following observation. It is immediatelyobvious that if a system has finite gain, then it is input-output stable. The converse is,however, not true. For instance; any static nonlinearity without a bounded slope does nothave a finite gain. As an example, the memoryless systems Hi : Gone - Gooe, i = 1, 2 shownin Figure 6.5 are input-output-stable, but do not have a finite gain.


N(u)

1.25

0.25 1.25 U

Figure 6.4: Static nonlinearity NO.

y = elUl

u

Figure 6.5: The systems H1u = u2, and H2u = ek"I.

6.3 Linear Time-Invariant Systems

So far, whenever dealing with LTI systems we focused our attention on state space real-izations. In this section we include a brief discussion of LTI system in the context of theinput-output theory of systems. For simplicity, we limit our discussion to single-input-single-output systems.

Definition 6.8 We denote by A the set of distributions (or generalized functions) of theform

At) = 1 foo(t) + fa(t),t < 0

where fo E R, denotes the unit impulse, and fa(.) is such that

JrIfa(r)I

dr<oo

namely, fa E L. The norm off c A is defined by

11f 11A =1 fo I + jW I fa(t) I dt.0

(6.11)

We will also denote by A the set consisting of all functions that are Laplace transforms ofelements of A. Note that, with the norm of f defined in (6.11), if f E L1 (i.e., if fo = 0),then IIf IIA = IIf 111. The extension of the algebra A, denoted Ae, is defined to be the set ofall functions whose truncation belongs to A.

We now introduce the following notation:

R[s]: set of polynomials in the variable s.

R(s): field of fractions associated with R[s], i.e. R(s) consists of all rational functionsin s with real polynomials. A rational function M E R(s) will be said to be proper ifit satisfies

It is said to be strictly proper if

lim M < oo.e-+0a

lim M < 0.8-+00

Notice that, according to this, if H(s) is strictly proper, then it is also proper. The converseis, however, not true.

6.3. LINEAR TIME-INVARIANT SYSTEMS 165

Theorem 6.1 Consider a function F(s) E R(s). Then F(s) E A if and only if (i) F(s) isproper, and (ii) all poles of F(s) lie in the left half of the complex plane.


Definition 6.9 The convolution of f and g in A, denoted by f * g, is defined by

(f * g)(t) = f 00 f (T)g(t - T) dT =J

00 f (t - T)g(T) dT. (6.12)0 0

It is not difficult to show that if f, g c A, then f * g E A and g * f E A.

We can now define what will be understood by a linear time-invariant system.

Definition 6.10 A linear time-invariant system H is defined to be a convolution operatorof the form

(Hu)(t) = h(t) * u(t) =J

h(T)u(t - T) dT = FOO h(t - T)u(T) dr (6.13)

where h(.) E A. The function h(.) is called the "kernel" of the operator H.

Given the causality assumptionption h(T) = 0 for r < 0, we have that, denoting y(t)I

Hu(t)

y(t) = hou(0) + J r h(T)u(t - T) dT = hou(0) + / t h(t - T)u(T) dT (6.14)00o

If in addition, u = 0 for t f< 0, then

h(t - T)u(T) dT (6.15)y(t) = hou(0) +J

t h(r)u(t - T) dT = hou(0) + fo t0

Definition 6.10 includes the possible case of infinite-dimensional systems, such as systemswith a time delay. In the special case of finite-dimensional LTI systems, Theorem 6.1 impliesthat a (finite-dimensional) LTI system is stable if and only if the roots of the polynomialdenominator lie in the left half of the complex plane. It is interesting, however, to considera more general class of LTI systems. We conclude this section with a theorem that givesnecessary and sufficient conditions for the Lp stability of a (possibly infinite-dimensional)linear time-invariant system.

Theorem 6.2 Consider a linear time-invariant system H, and let represent its impulseresponse. Then H is GP stable if and only if h(.) = h06(t) + ha(t) E A and moreover, if His LP stable, then IHuIjc, < Jjh11Al1u1lcp .

6.4 Cp Gains for LTI Systems

Having settled the question of input-output stability in ,Cp spaces, we focus our attentionon the study of gain. Once again, for simplicity we restrict our attention to single-input-single-output (SISO) systems. It is clear that the notion of gain depends on the space ofinput functions in an essential manner.

6.4.1 L Gain

By definition

7(H),,. = sup IIII IIII =Ilullc

p_1 IIHuIIc_ (6.16)U

This is a very important case. The space Cc,, consists of all the functions of t whose absolutevalue is bounded, and constitutes perhaps the most natural choice for the space of functionsX. We will show that in this case

ti(H). = Il h(t) II A (6.17)

Consider an input u(t) applied to the system H with impulse response hoo(t)+ha(t) EA. We have

(h * u)(t) = hou(t) + J ha(T)u(t - T)drt0

tIhollu(t)l + fc

sip Iu(t)I {IhoI + f Iha(T)Idr}

= IIuIIc, IIhIIA

Thus

Ilyll. < IIuII.IIhIIA or

IIhIIA > IIHuIIc-IIuIIc-

This shows that IIhIIA ? 7(H).. To show that IIhIIA = ry(H). we must show that equalitycan actually occur. We do this by constructing a suitable input. For each fixed t, let

u(t - T) = sgn[h(T)] VT

6.4. L GAINS FOR LTI SYSTEMS 167

where

It follows that lullc_ = 1, and

sgn[h(t)]1 if h(T) > 0

l 0 if h(T) < 0

y(t) = (h * u)(t) = bou(t) +J

ha,(T)U(t - r)drt0

t

Ihol + f Iha(T)ldr = IhIIA0

and the result follows.

6.4.2 G2 Gain

This space consists of all the functions of t that are square integrable or, to state this indifferent words, functions that have finite energy. Although from the input-output point ofview this class of functions is not as important as the previous case (e.g. sinusoids and stepfunctions are not in this class), the space G2 is the most widely used in control theory becauseof its connection with the frequency domain that we study next. We consider a linear time-invariant system H, and let E A be the kernel of H., i.e., (Hu)(t) = h(t) *u(t), du E G2.We have that

-Y2(H) = sup IIIxIIIC2

(6.18)x

1/2

IIxIIc2 = { f Ix(t)I2 dt}

(6.19)J00

We will show that in this case, the gain of the system H is given by

'Y2(H) = sup I H(.)If

IIHIIoo (6.20)

where H(tw) = .F[h(t)], the Fourier transform of h(t). The norm (6.20) is the so-calledH-infinity norm of the system H. To see this, consider the output y of the system to aninput u

ii:Ilylicz = IIHuIIcz = f [h(t) * u(t)]dt =21r

where the last identity follows from Parseval's equality. From here we conclude that

IIyIIcz < {supIH(Jw)I}2 I2r1 00

r i- IU(JW)12dw}

168

but

CHAPTER 6. INPUT-OUTPUT STABILITY

sup lH(.7w)IfIIHII.

W

IlylI 2 <_ (6.21)

yz(H) < IIUI12. (6.22)

W

Equation (6.21) or (6.22) proves that the H-infinity norm of H is an upper bound for thegain 7(H). As with the G,,. case, to show that it is the least upper bound, we proceed toconstruct a suitable input. Let u(t) be such that its Fourier transform, .P[u(t)] = U(yw),has the following properties:

U()I_{ A if Iw - woI< .w, orIw+woI<Ow

0 otherwise

In this case-W+OWO _ W+AWO

IIYIIc, = 1 I

fA21 H(7w)I2dw +

JAZI H(M)I2dw2ir }

w-&0 W pW0

Therefore, as Ow -4 0, ft(iw) -+ R(p) and

IIyIIc, -> 1 A2IH(.n))I2 4 Aw

Thus, defining A = {7r/2Aw}1/2, we have that IIyII2 --+ IIHI12 as Aw -+ oo, which completesthe proof.

It is useful to visualize the L2 gain (or H-infinity norm) using Bode plots, as shownin Figure 6.6.

6.5 Closed-Loop Input-Output Stability

Until now we have concentrated on open-loop systems. One of the main features of theinput-output theory of systems, however, is the wealth of theorems and results concerningstability of feedback systems. To study feedback systems, we first define what is understoodby closed loop input-output stability, and then prove the so-called small gain theorem.As we will see, the beauty of the input-output approach is that it permits us to drawconclusions about stability of feedback interconnections based on the properties of theseveral subsystems encountered around the feedback loop. The following model is generalenough to encompass most cases of interest.

6.5. CLOSED-LOOP INPUT-OUTPUT STABILITY 169

H(34

IIHII.

w

Figure 6.6: Bode plot of H(jw)I, indicating the IIHII. norm of H.

Definition 6.11 We will denote by feedback system to the interconnection of the subsystemsH1 and H2 : Xe -* Xe that satisfies the following assumptions:

(i) e1, e2i yl, and y2 E Xe for all pairs of inputs ul, u2 E Xe .

(ii) The following equations are satisfied for all u1, u2 E Xe :

el = ul - H2e2 (6.23)

e2 = u2 + H1e1. (6.24)

Here ul and u2 are input functions and may represent different signals of interest such ascommands, disturbances, and sensor noise. el and e2 are outputs, usually referred to aserror signals and yl = H1e1, Y2 = H2e2 are respectively the outputs of the subsystems H1and H2. Assumptions (i) and (ii) ensure that equations (6.23) and (6.24) can be solvedfor all inputs U1, U2 E Xe . If this assumptions are not satisfied, the operators H1 and H2do not adequately describe the physical systems they model and should be modified. Itis immediate that equations (6.23) and (6.24) can be represented graphically as shown inFigure 6.7.

In general, the subsystems H1 and H2 can have several inputs and several outputs.For compatibility, it is implicitly assumed that the number of inputs of H1 equals thenumber of outputs of 112, and the number of outputs of H1 equals the number of inputsof H2. We also notice that we do not make explicit the system dimension in our notation.For example, if X = ,C,,,, the space of functions with bounded absolute value, we writeH : Xe --> Xe (or H : £ooe -* L.,) regardless of the number of inputs and outputs of thesystem.


Figure 6.7: The Feedback System S.

In the following definition, we introduce vectors u, e, and y, defined as follows:

u(t)= [

uul (t)

2 (t) I I e(t) - [

eel (t)

2 (t) I , y(t)Y 2

yi (t)]

(6.25)

Definition 6.12 Consider the feedback interconnection of subsystems H1 and H2. For thissystem we introduce the following input-output relations. Given Ui, U2 E Xe , i, j = 1, 2,and with u, e, and y given by (6.25) we define

E = {(u,e) E Xe x Xe and e satisfies (6.23) and (6.24)} (6.26)

F = {(u, y) E Xe x Xe and y satisfies (6.23) and (6.24)}. (6.27)

In words, E and F are the relations that relate the inputs u2 with ei and y2, respectively.Notice that questions related to the existence and uniqueness of the solution of equations(6.23) and (6.24) are taken for granted. That is the main reason why we have chosen towork with relations rather than functions.

Definition 6.13 A relation P on Xe is said to be bounded if the image under P of everybounded subset of XX E dom(P) is a bounded subset of Xe .

In other words, P is bounded if Pu E X for every u c X , u E dom(P).

Definition 6.14 The feedback system of equations (6.23) and (6.24) is said to be boundedor input-output-stable if the closed-loop relations E and F are bounded for all possible u1, u2in the domain of E and F.

6.6. THE SMALL GAIN THEOREM 171

In other words, a feedback system is input-output-stable if whenever the inputs u1 and u2are in the parent space X, the outputs y1 and y2 and errors e1 and e2 are also in X. In thesequel, input-output--stable systems will be referred to simply as "stable" systems.

Remarks: Notice that, as with open-loop systems, the definition of boundedness dependsstrongly on the selection of the space X. To emphasize this dependence, we will sometimesdenote X-stable to a system that is bounded in the space X.

6.6 The Small Gain Theorem

In this section we study the so-called small gain theorem, one of the most important resultsin the theory of input-output systems. The main goal of the theorem is to provide open-loopconditions for closed-loop stability. Theorem 6.3 given next, is the most popular and, insome sense, the most important version of the small gain theorem. It says that if the productof the gains of two systems, H1 and H2, is less than 1, then its feedback interconnection isstable, in a sense to be made precise below. See also remark (b) following Theorem 6.3.

Theorem 6.3 Consider the feedback interconnection of the systems H1 and H2 : Xe -4 Xe .

Then, if y(Hj)7(H2) < 1, the feedback system is input-output-stable.

Proof: To simplify our proof, we assume that the bias term 3 in Definition 6.7 is identicallyzero (see Exercise 6.5). According to Definition 6.14 we must show that u1i u2 E X implythat e1 i e2, y1 and Y2 are also in X. Consider a pair of elements (u1, el), (u2i e2) that belongto the relation E. Then, u1, u2, e1, e2 must satisfy equations (6.23) and (6.24) and, aftertruncating these equations, we have

e1T = u1T - (H2e2)T (6.28)

e2T = u2T + (Hlel)T. (6.29)

Thus,

IIelTII < IIu1TII + II(H2e2)TII < IIu1TII + 7(H2)IIe2TII

Ie2TII Iu2TII + II(Hlel)TII < IIu2TII + 'Y(H1)II e1TIl.

Substituting (6.31) in (6.30) we obtain

IIe1TII <_ IIu1TII + 7(H2){IIu2TII +'Y(Hl)IIe1TII}

5 IIu1T + ry(H2)IIu2TII +' Y(H1)'Y(H2)II e1TII

(6.30)

(6.31)

= [1 - 1(Hl)1'(H2)]II e1TII 5 IIu1TII +7(H2)IIu2TII (6.32)

and since, by assumption, y(Hl)y(H2) < 1, we have that 1 - y(H1)y(H2) # 0 and then

IIe1TII <_ [1 - y(H1)y(H2)]-1{IIu1TII +y(H2)IIu2TII}.

Similarly

(6.33)

IIe2TII < [1-y(H1)y(H2)]-1{IIu2TII +y(Hl)IIu1TII}. (6.34)

Thus, the norms of e1T and e2T are bounded by the right-hand side of (6.34) and (6.33).If, in addition, u1 and u2 are in X (i.e., IIui1I < oc for i = 1, 2), then (6.34) and (6.33) mustalso be satisfied if we let T -4 oo. We have

Ie1II < [1 - y(H1)y(H2)]-1{IIu1II + 1(H2)IIu2II}

IIe2II < [1 - y(H1)y(H2)]-1{IIu2II +y(H1)IIu1II}.

(6.35 )

(6.36)

It follows that e1 and e2 are also in X (see the assumptions about the space X) and theclosed-loop relation E is bounded. That F is also bounded follows from (6.35) and (6.36)and the equation

evaluated as T -+ oo.

I(Hiei)TII < y(Hi)IIeiTII, i = 1,2 (6.37)

0Remarks:

(a) Theorem 6.3 says exactly that if the product of the gains of the open-loop systemsH1 and H2 is less than 1, then each bounded input in the domain of the relations Eand F produces a bounded output. It does not, however, imply that the solution ofequations (6.23) and (6.24) is unique. Moreover, it does not follow from Theorem 6.3that for every pair of functions u1 and u2 E X the outputs e1i e2, y1 y2 are bounded,because the existence of a solution of equations (6.23) and (6.24) was not proved forevery pair of inputs u1 and u2. In practice, the question of existence of a solution canbe studied separately from the question of stability. If the only thing that is knownabout the system is that it satisfies the condition y(H1)y(H2) < 1, then Theorem 6.3guarantees that, if a solution of equations (6.23) and (6.24) exists, then it is bounded.Notice that we were able to ignore the question of existence of a solution by makinguse of relations. An alternative approach was used by Desoer and Vidyasagar [21],who assume that e1 and e2 belong to Xe and define u1 and u2 to satisfy equations(6.23) and (6.24).

(b) Theorem 6.3 provides sufficient but not necessary conditions for input-output stabil-ity. In other words, it is possible, and indeed usual, to find a system that does notsatisfy the small gain condition y(H1)y(H2) < 1 and is nevertheless input-output-stable.

6.6. THE SMALL GAIN THEOREM 173

N(x)

x

Figure 6.8: The nonlinearity N(.).

Examples 6.4 and 6.5 contain very simple applications of the small gain theorem. We assumethat X = G2.

Example 6.4 Let Hl be a linear time-invariant system with transfer function

G(s) = -2,s+2s+4and let H2 be the nonlinearity N() defined by a graph in the plane, as shown in Figure 6.8.

We apply the small gain theorem to find the maximum slope of the nonlinearity N() thatguarantees input-output stability of the feedback loop. First we find the gain of Hl and H2.For a linear time-invariant system, we have

-y(H) = sup 16(3w) IW

In this case, we have

1

I(4-w2)+23w11

[(4 - w2)2 + 4w2]

Since JG(3w_) I 0 as w -* oo, the supremum must be located at some finite frequency,and since IG(3w)I is a continuous function of w, the maximum values of lG(jw)I exists andsatisfies

7(Hi) = maxlG(jw)IW

IG(7w')I.

dIG )Iw w =0, and

d2I2w) I Iw- < 0.

Differentiating IC(.jw)I twice with respect to w we obtain w = v, and IG(j-w`)j = 1/ 12.

It follows that y(H1) = 1/ 12. The calculation of y(H2) is straightforward. We havey(H2) = IKL. Applying the small gain condition y(Hl)y(H2) < 1, we obtain

y(Hl)y(H2) < 1 = IKI< 12.

Thus, if the absolute value of the slope of the nonlinearity N(.) is less than 12 the systemis closed loop stable.

Example 6.5 Let Hl be as in Example 6.4 and let H2 be a constant gain (i.e., H2 is lineartime-invariant and H2 = k). Since the gain of H2 is y(H2) = IkI, application of the smallgain theorem produces the same result obtained in Example 6.4, namely, if IkI < 12, thesystem is closed loop stable. However, in this simple example we can find the closed looptransfer function, denoted H(s), and check stability by obtaining the poles of H(s). Wehave

G(s)H(s) =

1 + kG(s) s2 + 2s + (4 + k)

The system is closed-loop-stable if and only if the roots of the polynomial denominator ofH(s) lie in the open left half of the complex plane. For a second-order polynomial this issatisfied if and only if all its coefficients have the same sign. It follows that the system isclosed-loop-stable if and only if (4 + k) > 0, or equivalently, k > -4. Comparing the resultsobtained using this two methods, we have

Small gain theorem: - 12 < k < 12.

Pole analysis: -4 < k < oo.

We conclude that, in this case, the small gain theorem provides a poor estimate of thestability region.

6.7 Loop Transformations

As we have seen, the small gain theorem provides sufficient conditions for the stability ofa feedback loop, and very often results in conservative estimates of the system stability.

6.7. LOOP TRANSFORMATIONS 175


The same occurs with any other sufficient but not necessary stability condition, such as thepassivity theorem (to be discussed in Chapter 8). One way to obtain improved stabilityconditions (i.e. less conservative) is to apply the theorems to a modified feedback loop thatsatisfies the following two properties: (1) it guarantees stability of the original feedbackloop, and (2) it lessens the overall requirements over H1 and H2. In other words, it ispossible that a modified system satisfies the stability conditions imposed by the theorem inuse whereas the original system does not. The are two basic transformations of feedbackloops that will be used throughout the book, and will be referred to as transformations ofTypes I and Type II.

Definition 6.15 (Type I Loop Transformation) Consider the feedback system S of Figure6.9. Let H1, H2, K and (I + KH1)-1 be causal maps from Xe into X. and assume that Kis linear. A loop transformation of Type I is defined to be the modified system, denotedSK, formed by the feedback interconnection of the subsystems Hi = Hl(I + KHl)-1 andH2 = H2 - K, with inputs u'1 = ul - Ku2 and u'2 = u2i as shown in Figure 6.10. Theclosed-loop relations of SK will be denoted EK and FK.

The following theorem shows that the system S is stable if and only if the system SKis stable. In other words, for stability analysis, the system S can always be replaced by thesystem SK.

Theorem 6.4 Consider the system S of Figure 6.9 and let SK be the modified systemobtained after a type I loop transformation and assume the K and (I+KH1)-1 : X -+ X.Then (i) The system S is stable if and only if the system SK is stable.


...........................................................

ul - Ku2 +>Hl

K

Hi=Hl(I+KHl)-l

yl

:..................... ..................................... :...........................................................

Z2 I e2H2

K

..........................................................

H2 =

+ u2

H2 - K

Figure 6.10: The Feedback System SK.

6.7. LOOP TRANSFORMATIONS 177

U l M-1

Z2

M

M-1 Y2

H1

H2

yl

e2

Figure 6.11: The Feedback System SM.

+ U2

Proof: The proof is straightforward, although laborious, and is omitted . Notice, however,that the transformation consists essentially of adding and subtracting the term Kyl at thesame point in the loop (in the bubble in front of H1), thus leading to the result. O

Definition 6.16 (Type II Loop Transformation) Consider the feedback system S of Figure6.9. Let H1, H2 be causal maps of Xe into XQ and let M be a causal linear operator satisfying

(i)M:X -4X.(ii) IM-1 : X --1 X : MM-1 = I, M-1 causal.

(iii) Both M and M-1 have finite gain.

A type II loop transformation is defined to be the modified system SM, formed bythe feedback interconnection of the subsystem Hi = H1M and H2 = M-1H2, with inputsui = M-lu1 and u2 = a2i as shown in Figure 6.11. The closed-loop relation of this modifiedsystem will be denoted EM and FM.

Theorem 6.5 Consider the system S of Figure 6.9 and let SM be the modified systemobtained after a type II loop transformation. Then the system S is stable if and only if thesystem SM is stable.

Proof: The proof is straightforward and is omitted (see Exercise 6.9).

Figure 6.12: The nonlinearity 0(t*, x) in the sector [a, 3].

6.8 The Circle Criterion

Historically, one of the first applications of the small gain theorem was in the derivation ofthe celebrated circle criterion for the G2 stability of a class of nonlinear systems. We firstdefine the nonlinearities to be considered.

Definition 6.17 A function 0 : R+ x 1l -a IR is said to belong to the sector [a, Q] wherea<0if

axe < xO(t, x) < f3x2 Vt > 0, Vx E R. (6.38)

According to this definition, if 0 satisfies a sector condition then, in general, it is time-varying and for each fixed t = t', t(t', x) is confined to a graph on the plane, as shown inFigure 6.12.

We assume that the reader is familiar with the Nyquist stability criterion. The Nyquistcriterion provides necessary and sufficient conditions for closed-loop stability of lumpedlinear time-invariant systems. Given a proper transfer function

G(s) = p(s)q(s)

where p(s) and q(s) are polynomials in s with no common zeros, expanding d(s) in partialfractions it is possible to express this transfer function in the following form:

G(s) = 9(s) + d(s)

where

6.8. THE CIRCLE CRITERION 179

(i) g(s) has no poles in the open left-half plane (i.e., g(s) is the transfer function of anexponentially stable system).

(ii) n(s) and d(s) are polynomials, andn(s)d(s)

is a proper transfer function.

(iii) All zeros of d(s) are in the closed right half plane. Thus,

n(s)d(s)

contains the unstable part of G. The number of open right half plane zeros of d(s)will be denoted by v.

With this notation, the Nyquist criteria can be stated as in the following lemma.

Lemma 6.1 (Nyquist) Consider the feedback interconnection of the systems Hl and H2.Let Hl be linear time-invariant with a proper transfer function d(s) satisfying (i)-(iii)above, and let H2 be a constant gain K. Under these conditions the feedback system isclosed-loop-stable in LP , 1 < p < oo, if and only if the Nyquist plot of G(s) [i.e., the polarplot of G(3w) with the standard indentations at each 3w-axis pole of G(3w) if required] isbounded away from the critical point (-1/K + 30), Vw E R and encircles it exactly v timesin the counterclockwise direction as w increases from -oo to oo.

The circle criterion of Theorem 6.6 analyzes the L2 stability of a feedback systemformed by the interconnection of a linear time-invariant system in the forward path and anonlinearity in the sector [a, ,Q] in the feedback path. This is sometimes referred to as theabsolute stability problem, because it encompasses not a particular system but an entireclass of systems.

In the following theorem, whenever we refer to the gain -y(H) of a system H, it willbe understood in the L2 sense.

Theorem 6.6 Consider the feedback interconnection of the subsystems Hl and H2 : Gee -+Gee. Assume H2 is a nonlinearity 0 in the sector [a, Q], and let Hl be a linear time-invariant system with a proper transfer function G(s) that satisfies assumptions (i)-(iii)above. Under these assumptions, if one of the following conditions is satisfied, then thesystem is £2-stable:

(a) If 0 < a < f3: The Nyquist plot of d(s) is bounded away from the critical circle C',centered on the real line and passing through the points (-a-1 + 30) and (-(3-1 +,y0)and encircles it v times in the counterclockwise direction, where v is the number ofpoles of d(s) in the open right half plane.

(b) I_f 0 = a <,3: G(s) has no poles in the open right half plane and the Nyquist plot ofG(s) remains to the right of the vertical line with abscissa -f3-1 for all w E R.

(c) If a < 0 < 3: G(s) has no poles in the closed right half of the complex plane andthe Nyquist plot of G(s) is entirely contained within the interior of the circle C'.


6.9 Exercises

(6.1) Very often physical systems are combined to form a new system (e.g. by addingtheir outputs or by cascading two systems). Because physical systems are representedusing causal operators, it is important to determine whether the addition of two causaloperators [with addition defined by (A + B)x = Ax + Bx and the composition product(defined by (AB)x = A(Bx)], are again causal. With this introduction, you are askedthe following questions:

(a) Let A : Xe -> Xe and B : Xe -4 Xe be causal operators. Show that the sumoperator C : Xe --4 Xe defined by C(x) _ (A + B)(x) is also causal.

(b) Show that the cascade operator D : Xe -> Xe defined by D(x) = (AB)(x) is alsocausal.

(6.2) Consider the following alternative definition of causality

Definition 6.18 An operator H : Xe 4 Xe is said to be causal if

PTU1 = PTU2 = PTHui = PTHU2 VUI, U2 E Xe and VT E IIF+ (6.39)

According to this definition, if the truncations of ul and U2 are identical (i.e., ifU1 = U2, Vt < T), then the truncated outputs are also identical.

You are asked to prove the following theorem, which states that the two definitionsare equivalent.

Theorem 6.7 Consider an operator H : Xe 4 Xe. Then H is causal according toDefinition 6.5 if and only if it is causal according to Definition 6.18.

6.9. EXERCISES 181

(6.3) Prove the following theorem, which states that, for bounded causal operators, all ofthe truncations in the definition of gain may be dropped if the input space is restrictedto the space X.

Theorem 6.8 Consider a causal operator H : Xe -* Xe satisfying HO = 0, andassume that 3y'(H) = sup jjHxIj/jIxjI < oodx E X , x 0. Then H has a finite gain,according to Definition 6.7, and y*(H) = y(H).

(6.4) Prove the following theorem.

Theorem 6.9 Let H1, H2 : X --> X be causal bounded operator satisfying H1O =0, i = 1, 2. Then

y(H1H2) < y(Hl)^t(H2) (6.40)

(6.5) Prove the small gain theorem (Theorem 6.3) in the more general case when Q # 0.



(6.8) In Section 6.2 we introduced system gains and the notion of finite gain stable system.We now introduce a stronger form of input-output stability:

Definition 6.19 A system H : Xe -> Xe is said to be Lipschitz continuous, or simplycontinuous, if there exists a constant I'(H) < oo, called its incremental gain, satisfying

I(Hul)T - (Hu2)TIIr(H) =SupI(u1)T - u2)TII

(6.41)

where the supremurn, is taken over all u1, u2 in Xe and all T in pt+ for which u1T # u2T.

Show that if H : Xe 4 Xe is Lipschitz continuous, then it is finite-gain-stable.

(6.9) Find the incremental gain of the system in Example 6.3.

(6.10) For each of the following transfer functions, find the sector [a, /3] for which the closed-loop system is absolutely stable:

(2) H(s)(s+ 1)(s+3) '

(ii) H(s) _(s + 2)(s - 3)


Notes and References

The classical input/output theory was initiated by Sandberg, [65], [66], (see also the morecomplete list of Sandberg papers in [21]), and Zames [97], [98]. Excellent general referencesfor the material of this chapter are [21], [88] and [92]. Our presentation follows [21]) as wellas [98].

Chapter 7

Input-to-State Stability

So far we have seen two different notions of stability: (1) stability in the sense of Lyapunovand (2) input-output stability. These two concepts are at opposite ends of the spectrum.On one hand, Lyapunov stability applies to the equilibrium points of unforced state spacerealizations. On the other hand, input-output stability deals with systems as mappingsbetween inputs and outputs, and ignores the internal system description, which may ormay not be given by a state space realization.

In this chapter we begin to close the gap between these two notions and introduce theconcept of input-to-state-stability. We assume that systems are described by a state spacerealization that includes a variable input function, and discuss stability of these systems ina way to be defined.

7.1 Motivation

Throughout this chapter we consider the nonlinear system

x = f(x,u) (7.1)

where f : D x Du -+ 1R' is locally Lipschitz in x and u. The sets D and Du are defined byD = {x E 1R : jxjj < r}, Du = {u E Ift"` : supt>0 IIu(t)JI = 1jullc_ < ru}.

These assumptions guarantee the local existence and uniqueness of the solutions ofthe differential equation (7.1). We also assume that the unforced system

x = f(x,0)

has a uniformly asymptotically stable equilibrium point at the origin x = 0.

183

184 CHAPTER 7. INPUT-TO-STATE STABILITY

Under these conditions, the problem to be studied in this chapter is as follows. Giventhat in the absence of external inputs (i.e., u = 0) the equilibrium point x = 0 is asymptot-ically stable, does it follow that in the presence of nonzero external input either

(a) limt_,,,,, u(t) = 0 =: limt__,,,,) x(t) = 0 ?

(b) Or perhaps that bounded inputs result in bounded states ?, specifically, IIuT(t)Ile_ <J, 0<T <t = suptlIx(t)II <e ?

Drawing inspiration from the linear time invariant (LTI) case, where all notions of stabilitycoincide, the answer to both questions above seems to be affirmative. Indeed, for LTIsystems the solution of the state equation is well known. Given an LTI system of the form

= Ax + Bu

the trajectories with initial condition x0 and nontrivial input u(t) are given by

x(t) = eAtxp +J

eA(t-T)Bu(T) dr (7.2)t0

If the origin is asymptotically stable, then all of the eigenvalues of A have negative realparts and we have that IleAtll is bounded for all t and satisfies a bound of the form

IleAtll < keAt, A < 0


tIIx(t)II <- keAtlIxoll + 1

kea(t_T)IIBII IIu(T)II dr

0

< keAtllxoll + kIIBA

II supt

IIu(t)II

= keAtllxoII + kIIBII IIUT(t)IIf_ , 0 < T < t

Thus, it follows trivially that bounded inputs give rise to bounded states, and that iflimt__, , u(t) = 0, then limt, 0 x(t) = 0.

The nonlinear case is, however a lot more subtle. Indeed, it is easy to find counterex-amples showing that, in general, these implications fail. To see this, consider the followingsimple example.

Example 7.1 Consider the following first-order nonlinear system:

x=-x+(x+x3)u.

Setting u = 0, we obtain the autonomous LTI system ± = -x, which clearly has an as-ymptotically stable equilibrium point. However, when the bounded input u(t) = 1 is applied,the forced system becomes ± = x3, which results in an unbounded trajectory for any initialcondition, however small, as can be easily verified using the graphic technique introduced inChapter 1.

7.2 Definitions

In an attempt to rescue the notion of "bounded input-bounded state", we now introducethe concept of input-to-state stability (ISS).

Definition 7.1 The system (7.1) is said to be locally input-to-state-stable (ISS) if thereexist a ICE function 0, a class K function -y and constants k1, k2 E ][i;+ such that

IIx(t)II <_ Q(IIxoll,t) +y(IIuT(.)Ilc,), Vt > 0, 0 <T < t (7.3)

for all xo E D and u E Du satisfying : Ilxoll < ki and sups>o IUT(t)II = IluTllc_ < k2i 0 <T < t. It is said to be input-to-state stable, or globally ISS if D = R", Du = Rm and (7.3)is satisfied for any initial state and any bounded input u.

Definition 7.1 has several implications, which we now discuss.

Unforced systems: Assume that i = f (x, u) is ISS and consider the unforced systemx = f (x, 0). Given that y(0) = 0 (by virtue of the assumption that -y is a class ICfunction), we see that the response of (7.1) with initial state xo satisfies.

Ilx(t)II < Q(Ilxoll, t) Vt > 0, Ilxoll < ki,

which implies that the origin is uniformly asymptotically stable.

Interpretation: For bounded inputs u(t) satisfying Ilullk < b, trajectories remainbounded by the ball of radius 3(Ilxoll,t) +y(b), i.e.,

lx(t) I1 <_ 0040 11, t) + y(b).

As t increases, the term 3(I1 xo11,t) -* 0 as t -> oo, and the trajectories approach theball of radius y(5), i.e.,

lim Ilx(t)II <_ y(b)t

For this reason, is called the ultimate bound of the system 7.1.

Alternative Definition: A variation of Definition 7.1 is to replace equation (7.3) withthe following equation:

Iix(t)II <_ maX{)3(IIxoII, t),'Y(IIuT(')IIG,)}, Vt > 0, 0 < T < t. (7.4)

The equivalence between (7.4) and (7.3) follows from the fact that given 0 > 0 andy > 0, max{Q, y} < Q + -y < {2$, 2y}. On occasions, (7.4) might be preferable to(7.3), especially in the proof of some results.

It seems clear that the concept of input-to-state stability is quite different from thatof stability in the sense of Lyapunov. Nevertheless, we will show in the next section thatISS can be investigated using Lyapunov-like methods. To this end, we now introduce theconcept of input-to-state Lyapunov function (ISS Lyapunov function).

Definition 7.2 A continuously differentiable function V : D -> Ift is said to be an ISSLyapunov function on D for the system (7.1) if there exist class AC functions al, a2, a3,and X such that the following two conditions are satisfied:

al(IIXII) <_ V(x(t)) < a2(IIxII) Vx E D, t > 0 (7.5)

aa(x) f (x, u) < -a3(IIxII) Vx E D, u E Du : IIxII >_ X(IIuII). (7.6)

V is said to be an ISS Lyapunov function if D = R", Du = R, and al, a21 a3 E 1Coo .

Remarks: According to Definition 7.2, V is an ISS Lyapunov function for the system (7.1)if it has the following properties:

(a) It is positive definite in D. Notice that according to the property of Lemma 3.1,given a positive definite function V, there exist class IC functions al and a2 satisfyingequation (7.5).

(b) It is negative definite in along the trajectories of (7.1) whenever the trajectories areoutside of the ball defined by IIx*II = X(IIuPI).

7.3 Input-to-State Stability (ISS) Theorems

Theorem 7.1 (Local ISS Theorem) Consider the system (7.1) and let V : D -a JR be anISS Lyapunov function for this system. Then (7.1) is input-to-state-stable according to

7.3. INPUT-TO-STATE STABILITY (ISS) THEOREMS 187

Definition 7.1 with

aj1oa2oX (7.7)

k1 = a21(al(r)) (7.8)

k2 = X-1(min{k1iX(r,,)}). (7.9)

Theorem 7.2 (Global ISS Theorem) If the preceeding conditions are satisfied with D = 1R'and Du = 1Wt, and if al, 02, a3 E )Coo, then the system (7.1) is globally input-to-state stable.

Proof of Theorem 7.1: Notice first that if u = 0, then defining conditions (7.5) and(7.6) of the ISS Lyapunov function guarantee that the origin is asymptotically stable. Nowconsider a nonzero input u, and let

ru f SupIuTII = IuIIG_,t>O

0<T<t.

Also define

fc a2(X(ru))

S2c {x E D : V (x) < c}

and notice that 1C, so defined, is bounded and closed (i.e., Qc is a compact set), and thus itincludes its boundary, denoted O(Qc). It then follows by the right-hand side of (7.5) thatthe open set of points

{xCR'i:IIxII <X(ru)<c}CQ,CD.

Moreover, this also implies that IIxHI > X(IIu(t)II) at each point x in the boundary of III.Notice also that condition (7.6) implies that V(x(t)) < 0 Vt > 0 whenever x(t) ¢ 1 C. Wenow consider two cases of interest: (1) xp is inside S2, and (2) xo is outside Q,.

Case (1) (xo e 1, ): The previous argument shows that the closed set 1 is surrounded by3(1L) along which V(x(t)) is negative definite. Therefore, whenever x0 E 1C, x(t) is lockedinside 0, for all t > 0. Trajectories x(t) are such that

al(IIxII) < V(x(t))

IIx(t)II : aj 1(V(x(t))) < al 1(c) = ai 1(a2(X(ru)))

defining

yfa11oa2oX(IuTIG_) 0<T<t

we conclude that whenever xo E Sao,

1Ix(t)II. <'Y(IIuTIIG-) Vt >_ 0, 0 <T < t. (7.10)

Case (ii) (xo ¢ S1,): Assume that xo E Q,, or equivalently, V(xo) > c. By condition (7.6),IIxIl > X(IIull) and thus V(x(t)) < 0. It then follows that for some tl > 0 we must have

V(x(t)) > 0 for 0 < t < tlV(x(tl)) = c.

But this implies that when t = t1, then x(t) E 8(1k), and we are back in case (1). Thisargument also shows that x(t) is bounded and that there exist a class /CL function 0 suchthat

11x(t)JI. < /3(Ilx,,t)

Combining (7.10) and (7.11), we obtain

for 0 < t < tl (7.11)

JIx(t)1Ioo < -Y(Ilulloo) for t > tjIlx(t)lIc < Q(IIxoII,t) for 0<t<tj

and thus we conclude that

lx(t)II < max{p(Ilxoll,t),7(IIuII.)} Vt > 0

or thatIIx(t)II <_ 3GIxoII, t) + 7(IIuII.) Vt > 0.

To complete the proof, it remains to show that kl and k2 are given by (7.8) and (7.9),respectively. To see (7.8), notice that, by definition

c = a2(X(ru)), andS2c = {x E D : V(x) < c}.

Assuming that xo V Q. we must have that V(x) > c. Thus

and

Thus

al(IIxII) < V(x) S a2(IIXII)al(r) < V(xo) <_ a2(IIxOI)

lixol1 al-l(ai(r)).

kld=f

Ixoll al l(ai(r))

7.3. INPUT-TO-STATE STABILITY (ISS) THEOREMS

which is (7.8). To see (7.9) notice that

IMx(t)JI < max{Q(IIxol(,t),y(Ilutlloo)} Vt> 0, 0 <T <tX(k2) < IjxII < min{kl, X(ru))}.

Thus

7.3.1 Examples

k2 = X-1(min{k1,X(ru)}).

189

We now present several examples in which we investigate the ISS property of a system. Itshould be mentioned that application of Theorems 7.1 and 7.2 is not an easy task, somethingthat should not come as a surprise since even proving asymptotic stability has proved tobe quite a headache. Thus, our examples are deliberately simple and consist of variousalterations of the same system.


±=-ax3+u a>0.

To check for input-to-state stability, we propose the ISS Lyapunov function candidate V (x) =2x2. This function is positive definite and satisfies (7.5) with al(IIxII) = 020411) = 2x2.

We haveV = -x(ax3 - u). (7.12)

We need to find and X(.) E K such that V(x) < -a3(IIxII), whenever IIxHj > X(IjuII).To this end we proceed by adding and subtracting aOx4 to the right-hand side of (7.12),where the parameter 0 is such that 0 < 0 < 1

V = -ax4 + aOx4 - aOx4 + xu= -a(1 - 9)x4 - x(aOx3 - u)

< -a(1- 9)x4 = a3(IIx1I)

provided that

This will be the case, provided that

x(aOx3 - u) > 0.

aOIxI3 > Iul

or, equivalently

IxI > ( )1/3a

( B

1/3It follows that the system is globally input-to-state stable with ^j(u) _ "

a

0

Example 7.3 Now consider the following system, which is a slightly modified version ofthe one in Example 7.2:

x=-ax3+x2u a>0Using the same ISS Lyapunov function candidate used in Example 7.2, we have that

= -ax4 +x3u= -ax4 + aOx4 - aBx4 + x3u 0 < 0 < 1

= -a(1 - 0)x4 - x3(a0x - u)< -a(1 - 0)x4, provided

x3(a0x - u) > 0 or,

lxI > I .

Thus, the system is globally input-to-state stable with 7(u) = .

0

Example 7.4 Now consider the following system, which is yet another modified version ofthe one in Examples 7.2 and 7.3:

i=-ax3+x(1+x2)u a>0

Using the same ISS Lyapunov function candidate in Example 7.2 we have that

-ax 4 + x(1 + x2)u-ax4 + aOx 4 - aOx 4 + x(1 + x2)u-a(1 - 0)x4 - x[aOx 3 - (1 + x2)u]

< 4-a(1 - 9)x, provided

7.4. INPUT-TO-STATE STABILITY REVISITED 191

x[aOx3 - (1 + x2)u] > 0. (7.13)

Now assume now that the sets D and Du are the following: D = {x E 1R" : jxIj < r},D = R. Equation (7.13) is satisfied if

Ixl > ( 1+r2 u 1/3

Therefore, the system is input-to-state stable with k1 = r:

(1+(5)1-12

/ 1/37(u) =

k2 = X-1[min{k1iX(ru)}]= X 1(k1) = X-1(r)

= k2aOr

(1 + r2).

7.4 Input-to-State Stability Revisited

In this section we provide several remarks and further results related to input-to-statestability. We begin by stating that, as with all the Lyapunov theorems in chapters 3 and4, Theorems 7.1 and 7.2 state that the existence of an ISS Lyapunov function is a sufficientcondition for input-to-state stability. As with all the Lyapunov theorems in Chapters 3and 4, there is a converse result that guarantees the existence of an ISS Lyapunov functionwhenever a system is input-to-state-stable. For completeness, we now state this theoremwithout proof.

Theorem 7.3 The system (7.1) is input-to-state-stable if and only if there exist an ISSLyapunov function V : D -> R satisfying the conditions of Theorems 7.1 or 7.2.

Proof: See Reference [73].

Theorems 7.4 and 7.5 are important in that they provide conditions for local and globalinput-to-state stability, respectively, using only "classical" Lyapunov stability theory.

Theorem 7.4 Consider the system (7.1). Assume that the origin is an asymptoticallystable equilibrium point for the autonomous system x = f (x, 0), and that the function f (x, u)is continuously differentiable. Under these conditions (7.1) is locally input-to-state-stable.

Theorem 7.5 Consider the system (7.1). Assume that the origin is an exponentially stableequilibrium point for the autonomous system i = f (x, 0), and that the function f (x, u) iscontinuously differentiable and globally Lipschitz in (x, u). Under these conditions (7.1) isinput-to-state-stable.

Proof of theorems 7.4 and 7.5: See the Appendix.

We now state and prove the main result of this section. Theorem 7.6 gives an alter-native characterization of ISS Lyapunov functions that will be useful in later sections.

Theorem 7.6 A continuous function V : D -> R is an ISS Lyapunov function on D forthe system (7.1) if and only if there exist class 1C functions al, a2i a3, and or such that thefollowing two conditions are satisfied:

al(IIxII) 5 V(x(t)) < a2(IIxII) `dx E D, t > 0 (7.14)

OV(x) f(x,u) < -a3(IIxJI) +a(IIuII) Vx E D,u E Du (7.15)

V is an ISS Lyapunov function if D = R', Du = W' , and al, a2i a3, and a E 1C,,..

Proof: Assume first that (7.15) is satisfied. Then we have that

ax f(x,u) < -a3(IIxII)+a0IuID Vx E D,du E Du

< -(1 - 9)a3(IIxII) - Oa3(IIxII) + a(IIuII) 9 < 1

< -(1- O)a3(IIxII)

VIxII>x31ro u 1

which shows that (7.6) is satisfied. For the converse, assume that (7.6) holds:

aw f (x, u) <- -a3(IIxII) Vx E D, u E Du : IIxii >- X(IIuii)

To see that (7.15) is satisfied, we consider two different scenarios:

(a) IIxII > X(IIuii): This case is trivial. Indeed, under these conditions, (7.6) holds forany a(.) -

7.4. INPUT-TO-STATE STABILITY REVISITED 193

(b) IIxii < X(IIuiD: Define

0(r) = max [ a7x f (x, u) + a3(X(IIu1I))] (lull = r, IxII <_ X(r)

then

Now, definingeX

f (X, u) < -a3(Ilxll) +O(r)

o = max{0, O(r)}

we have that or, so defined, satisfies the following: (i) 0(0) = 0, (ii) it is nonnegative,and (iii) it is continuous. It may not be in the class /C since it may not be strictlyincreasing. We can always, however, find a E /C such that a(r) > Q.


The only difference between Theorem 7.6 and definition 7.2 is that condition (7.6) inDefinition 7.2 has been replaced by condition (7.15). For reasons that will become clear inChapter 9, (7.15) is called the dissipation inequality.

Remarks: According to Theorems 7.6 and 7.3, we can claim that the system (7.1) isinput-to-state-stable if and only if there exists a continuous function V : D -> R satisfyingconditions (7.14) and (7.15). Inequality (7.15) permits a more transparent view of theconcept and implications of input-to-state stability in terms of ISS Lyapunov functions. Tosee this, notice that, given ru > 0, there exist points x E R" such that

013(IIXIi) = a(ru).

This implies that 3d E 1R such that

a3(d) = a(ru), or

d = a 1(a(ru))

Denoting Bd = {x E 1 : IIxii < d}, we have that for any Ilxll > d and any u : Ilullc_ < ru:

jx f (x,u) < -a(Ilxll) +a(IIuIU

<_ -a(Ildll)+a(Ilulle_)

This means that the trajectory x(t) resulting from an input u(t) : lullL- < ru will eventually(i.e., at some t = t') enter the region

SZd = max V(x).IxII<d

Once inside this region, the trajectory is trapped inside Std, because of the condition on V.

According to the discussion above, the region Std seems to depend on the compositionof and It would then appear that it is the composition of these two functionsthat determines the correspondence between a bound on the input function u and a boundon the state x. The functions a(.) and a(.) constitute a pair [a(.), referred to as anISS pair for the system (7.1). The fact that Std depends on the composition of

a given system, the pair [a(.), o(.)] is perhaps nonunique. Our nexttheorem shows that this is in fact the case.

In the following theorem we consider the system (7.1) and we assume that it is globallyinput-to-state-stable with an ISS pair [a, o].

a(IIxII) < V(x) < a(IIxII) Vx E 1R", (7.16)

VV(x) f(x,u) < -a(IIx1I) +o(IIu1I) Vx E 1R",u E 1W7 (7.17)

for some a, a, d E and or E 1C.

We will need the following notation. Given functions x(.), R -> IR, we say that

x(s) = O(y(s)) as s --+ oo+

if

Similarly

if

lim 1x(s)1 < oo.8-100 ly(s)I

x(s) = O(y(s)) as s -+ 0+

limIx(s)I

< 00.s-+0 I y(s) I

Theorem 7.7 Let (a, a) be a supply pair for the system (7.1). Assume that & is a 1C,,.function satisfying o(r) = O(&(r)) as r -a oo+. Then, there exist & E 1C,,. such that (a,&)is a supply pair.

Theorem 7.8 Let (a, a) be a supply pair for the system (7.1). Assume that & is a 1C,,.function satisfying a(r) = O(a(r)) as r -a 0+. Then, there exist & E 1C. such that (&, ii)is a supply pair.

Proof of theorems 7.7 and 7.8: See the Appendix.

Remarks: Theorems 7.7 and 7.8 have theoretical importance and will be used in the nextsection is connection with the stability of cascade connections if ISS systems. The proof oftheorems 7.7 and 7.8 shows how to construct the new ISS pairs using only the bounds aand a, but not V itself.

7.5. CASCADE-CONNECTED SYSTEMS 195

UE2

zE1

x

Figure 7.1: Cascade connection of ISS systems.

7.5 Cascade-Connected Systems

Throughout this section we consider the composite system shown in Figure 7.1, where Eland E2 are given by

E1 : ± = f (x, z) (7.18)

E2 : z = 9(z, u) (7.19)

where E2 is the system with input u and state z. The state of E2 serves as input to thesystem E1.

In the following lemma we assume that both systems E1 and E2 are input-to-state-stable with ISS pairs [al, 01] and [a2, o2], respectively. This means that there exist positivedefinite functions V1 and V2 such that

VV1 f(x,z) < -al(1Ix1I)+a1(I1z11) (7.20)

VV2 9(Z, U) G -a2(IIZIj) +J2(jIUID) (7.21)

The lemma follows our discussion at the end of the previous section and guarantees theexistence of alternative ISS pairs [&1, Si] and [&2i Q2] for the two systems. As it turns out,the new ISS-pairs will be useful in the proof of further results.

Lemma 7.1 Given the systems E1 and E2, we have that

(i) Defining

a2a2(s) fors "small"a2(S) for s "large"{

then there exist &2 such that (&2, a2) is an ISS pair for the system E2.

CHAPTER 7. INPUT-TO-STATE STABILITY

al(s) = 2a2

there exist al : [al, all = [al, 1&2] is an ISS pair for the system El.

Proof: A direct application of Theorems 7.7 and 7.8.

We now state and prove the main result of this section.

Theorem 7.9 Consider the cascade interconnection of the systems El and E2. If bothsystems are input-to-state-stable, then the composite system E

is input-to-state-stable.

Proof: By (7.20)-(7.21) and Lemma 7.1, the functions V1 and V2 satisfy

VV1 f(x,z) < -al(IIxII)+2612(IIZII)

VV2 9(z,u) < -a2(IIzII) +&2(IIUII)

Define the ISS Lyapunov function candidate

V = V1 + V2

for the composite system. We have

V ((x, z), u) = VV1 f (X, z) + VV2 f (Z, u)1

-a1(IIxII) - 2a2(IIZII)+a2(IIuII)

It the follows that V is an ISS Lyapunov function for the composite system, and the theoremis proved.

This theorem is somewhat obvious and can be proved in several ways (see exercise7.3). As the reader might have guessed, a local version of this result can also be proved.For completeness, we now state this result without proof.

Theorem 7.10 Consider the cascade interconnection of the systems El and E2. If bothsystems are locally input-to-state-stable, then the composite system E

xE:u -+z

is locally input-to-state-stable.

7.5. CASCADE-CONNECTED SYSTEMS 197

u=0E2

zE1

x

Figure 7.2: Cascade connection of ISS systems with input u = 0.

The following two corollaries, which are direct consequences of Theorems 7.9 and 7.10are also important and perhaps less obvious. Here we consider the following special case ofthe interconnection of Figure 7.1, shown in Figure 2:

El : i = f (x, z) (7.22)

E2 : z = g(z) = g(z,0) (7.23)

and study the Lyapunov stability of the origin of the interconnected system with state

z

Corollary 7.1 If the system El with input z is locally input-to-state-stable and the originx = 0 of the system E2 is asymptotically stable, then the origin of the interconnected system(7.22)-(7.23)

is locally asymptotically stable.

Corollary 7.2 Under the conditions of Corollary 7.1, if El is input-to-state-stable andthe origin x = 0 of the system E2 is globally asymptotically stable, then the origin of theinterconnected system (7.22)-(7.23).

is globally asymptotically stable.

Proof of Corollaries 7.1 and 7.2: The proofs of both corollaries follow from the fact that,under the assumptions, the system E2 is trivially (locally or globally) input-to-state-stable,and then so is the interconnection by application of Theorem 7.9 or 7.10.


7.6 Exercises

(7.1) Consider the following 2-input system ([70]):

i= -x3 + x2U1 - XU2 + U1162

Is it input-to-state stable?

(7.2) Sketch the proof of theorem 7.5.

(7.3) Provide an alternative proof of Theorem 7.9, using only the definition of input-to-state-stability, Definition 7.1.

(7.4) Sketch a proof of Theorem 7.10.


.t1 = -x1 + x2Ix2{ x2 = -x2 + 21x2 + u

(i) Is it locally input-to-state-stable?

(ii) Is it input-to-state-stable?


I i1 = -21 - 22ll i2 = x1-x2+u



(7.7) Consider the following cascade connection of systems

2 = -x3 + x2u{ z = -z3 + z(1 + Z2)x



(7.8) Consider the following cascade connection of systems:

f

2 = -x3 + x2u1 - XU2 + u1u2

1 z -z3 +z 2 X

7.6. EXERCISES 199



(7.9) Consider the following cascade connection of systems:

-21 + 222 + 71,1-22 + U2-Z3 + 2221 - Z22 + 2122



Notes and ReferencesThe concept of input-to-state stability, as presented here, was introduced by Sontag [69].Theorems 7.3 and 7.6 were taken from reference [73]. The literature on input-to-statestability is now very extensive. See also References [70], [74], [71], and [72] for a thoroughintroduction to the subject containing the fundamental results. See also chapter 10 ofReference [37] for a good survey of results in this area. ISS pairs were introduced inReference [75]. Section 7.5 on cascade connection of ISS systems, as well as Theorems 7.7and 7.8 are based on this reference.

Chapter 8

Passivity

The objective of this chapter is to introduce the concept of passivity and to present someof the stability results that can be obtained using this framework. Throughout this chapterwe focus on the classical input-output definition. State space realizations are considered inChapter 9 in the context of the theory of dissipative systems. As with the small gain theo-rem, we look for open-loop conditions for closed loop stability of feedback interconnections.

8.1 Power and Energy: Passive Systems

Before we introduce the notion of passivity for abstract systems, it is convenient to motivatethis concept with some examples from circuit theory. We begin by recalling from basicphysics that power is the time rate at which energy is absorbed or spent,

P(t) =dwd(t)

(8.1)

where

Then

power

w(.) . energy

t . time

w(t) =J

t p(t) dt. (8.2)to

Now consider a basic circuit element, represented in Figure 8.1 using a black box.

201

202 CHAPTER 8. PASSIVITY

i

v

Figure 8.1: Passive network.

In Figure 8.1 the voltage across the terminals of the box is denoted by v, and thecurrent in the circuit element is denoted by i. The assignment of the reference polarity forvoltage, and reference direction for current is completely arbitrary. We have

p(t) = v(t)i(t) (8.3)

thus, the energy absorbed by the circuit at time "t" is

tw(t) = J00

v(t)i(t) dt = J00

v(t)i(t) dt + J v(t)i(t) dt. (8.4)

The first term on the right hand side of equation (8.4) represents the effect of initialconditions different from zero in the circuit elements. With the indicated sign convention,we have

(i) If w(t) > 0, the box absorbs energy (this is the case, for example, for a resistor).

(ii) If w(t) < 0, the box delivers energy (this is the case, for example, for a battery, withnegative voltage with respect to the polarity indicated in Figure 8.1).

In circuit theory, elements that do not generate their own energy are called passive,i.e., a circuit element is passive if

ft v(t)i(t) dt > 0. (8.5)-00

Resistors, capacitors and inductors indeed satisfy this condition, and are therefore calledpassive elements. Passive networks, in general, are well behaved, in an admittedly ambiguous

8.1. POWER AND ENERGY: PASSIVE SYSTEMS 203

e(t)

Ri

v

Figure 8.2: Passive network.

sense. It is not straightforward to capture the notion of good behavior within the contextof a theory of networks, or systems in a more general sense. Stability, in its many forms,is a concept that has been used to describe a desirable property of a physical system, andit is intended to capture precisely the notion of a system that is well behaved, in a certainprecise sense. If the notion of passivity in networks is to be of any productive use, then weshould be able to infer some general statements about the behavior of a passive network.

To study this proposition, we consider the circuit shown in Figure 8.2, where weassume that the black box contains a passive (linear or not) circuit element. Assuming thatthe network is initially relaxed, and using Kirchhoff voltage law, we have

e(t) = i(t)R + v(t)Assume now that the electromotive force (emf) source is such that

T

we have,

e2(t)dt<oo

IT Te2(t) dt =

Jo

(i(t)R + v(t))2 dt

T T= R2 J i2(t) dt + 2RJ i(t)v(t) dt + J T v2(t) dt0 0 0

and since the black box is passive, fo i(t)v(t) dt > 0. It follows that

JT T

e2(t) dt > R2J

i2(t) dt + f T v2(t) dt.0 0 0

Moreover, since the applied voltage is such that fo e2 (t) dt < oo, we can take limits asT -+ o0 on both sides of

jthinequality,, and we have

R2 i(t)2 dt+ J v2(t) dt <0 e2(t) dt < 0000

0)0 0

which implies that both i and v have finite energy. This, in turn, implies that the energyin these two quantities can be controlled from the input source e(.), and in this sense wecan say that the network is well behaved. In the next section, we formalize these ideas inthe context of the theory of input-output systems and generalize these concepts to moregeneral classes of systems. In particular, we want to draw conclusions about the feedbackinterconnection of systems based on the properties of the individual components.

8.2 Definitions

Before we can define the concept of passivity and study some of its properties we needto introduce our notation and lay down the mathematical machinery. The essential toolneeded in the passivity definition is that of an inner product space.

Definition 8.1 A real vector space X is said to be a real inner product space, if for every2 vectors x, y E X, there exists a real number (x, y) that satisfies the following properties:

(Z) (x, y) = (y, x)

(ii) (x + y, z) = (x, z) + (y + z) dx, y, z E X.

(iii) (ax, y) = 01 (y, x) Vx, y E X, We E R.

(iv) (x, x) > 0.

(v) (x, x) = 0 if and only if x = 0.

The function X x X -+ IR is called the inner product of the space X. If the space X iscomplete, then the inner product space is said to be a Hilbert space. Using these properties,we can define a norm for each element of the space X as follows:

llxjfx = (x,x).An important property of inner product space is the so-called Schwarz inequality:

I (x)y)l <_ IIxIIx IlylIx Vx, y E X. (8.6)

Throughout the rest of this chapter we will assume that X is a real inner product space.

Example 8.1 Let X be Rn. Then the usual "dot product" in IRn, defined by

x.y=xTy=xlyl+x2y2+...±xnyn

defines an inner product in IRn. It is straightforward to verify that defining properties (i)-(v)are satisfied.

For the most part, our attention will be centered on continuous-time systems. Virtually allof the literature dealing with this type of system makes use of the following inner product

(x, y) = f000 x(t) y(t) dt (8.7)

where x y indicates the usual dot product in Htn. This inner product is usually referred toas the natural inner product in G2. Indeed, with this inner product, we have that X = G2,and moreover

00rIIxIIc, = (x, x) = f IIx(t)II2 dt. (8.8)

We have chosen, however, to state our definitions in more general terms, given that ourdiscussion will not be restricted to continuous-time systems. Notice also that, even forcontinuous time systems, there is no a priori reason to assume that this is the only interestinginner product that one can find.

In the sequel, we will need the extension of the space X (defined as usual as the spaceof all functions whose truncation belongs to X), and assume that the inner product satisfiesthe following:

(XT, y) = (x, yT) = (xT, yT) (x, y)T

Example 8.2 : Let X = £2, the space of finite energy functions:

X = {x : IR -+ ]R, and satisfy}

IIxIIc2 = (x, x) =f00

x2(t) dt < oo0

Thus, X,, = Lee is the space of all functions whose truncation XT belongs to £2, regardlessof whether x(t) itself belongs to £2. For instance, the function x(t) = et belongs to Gee eventhough it is not in G2. El

Definition 8.2 : (Passivity) A system H : Xe 4 Xe is said to be passive if

(u, Hu)T > 0 Vu E Xe, VT E R+. (8.10)


Definition 8.3 : (Strict Passivity) A system H : Xe -4 Xe is said to be strictly passive ifthere exists 6 > 0 such that

(u,Hu)T > 611UT111 +Q Vu E Xe1 VT E IR+. (8.11)

The constant 3 in Definitions 8.2 and 8.3 is a bias term included to account for thepossible effect of energy initially stored in the system at t = 0. Definition 8.2 states thatonly a finite amount of energy, initially stored at time t = 0, can be extracted from a passivesystem. To emphasize these ideas, we go back to our network example:

Example 8.3 Consider again the network of Figure 8.1. To analyze this network as anabstract system with input u and output y = Hu, we define

u = v(t)y = Hu = i(t).

According to Definition 8.2, the network is passive if and only if

(x, Hx)T = (v(t), i(t))T > Q

Choosing the inner product to be the inner product in £2, the last inequality is equivalent tothe following:

J

T Tx(t)y(t)dt =

Jv(t)i(t) dt > 0 Vv(t) E Xe, VT E R.

0

From equation (8.4) we know that the total energy absorbed by the network at time t is

t oftoo v(t)i(t) dt = J v(t)i(t) dt + J v(t)i(t) dt00

(v(t), i(t))T + J v(t)i(t) dt.c

Therefore, according to definition 8.2 the network is passive if and only if

(v(t), i(t))TJ e v(t)i(t) dt.

0

Closely related to the notions of passivity and strict passivity are the concepts ofpositivity and strict positivity, introduced next.


Definition 8.4 : A system H : X -a X is said to be strictly positive if there exists d > 0such that

(u, Hu) > bI uI X + /.1 Vu E X. (8.12)

H is said to be positive if it satisfies (8.12) with b = 0.

The only difference between the notions of passivity and positivity (strict passivityand strict positivity) is the lack of truncations in (8.12). As a consequence, the notions ofpositivity and strict positivity apply to input-output stable systems exclusively. Notice that,if the system H is not input-output stable, then the left-hand side of (8.12) is unbounded.

The following theorem shows that if a system is (i) causal and (ii) stable, then thenotions of positivity and passivity are entirely equivalent.

Theorem 8.1 Consider a system H : X --> X , and let H be causal. We have that

(i) H positive b H passive.

(ii) H strictly positive H strictly passive.

Proof: First assume that H satisfies (8.12) and consider an arbitrary input u E Xe . Itfollows that UT E X, and by (8.12) we have that

(UT, HUT) ? 5U7' + 0but

(UT, HUT) (UT, (HUT)T) by (8.9)(UT, (Hu)T) since H is causal

(u, Hu)T by (8.9).

It follows that (u, HU)T > 6IIuTII2, and since u E Xe is arbitrary, we conclude that (8.12)implies (8.11). For the converse, assume that H satisfies (8.11) and consider an arbitraryinput u E X. By (8.11), we have that

(u,Hu)T ? SIIuTIIX + a

but

(u, Hu)T = (UT, (Hu)T)= (UT, (HUT)T)

= (UT, HUT)


Thus (UT, HUT) > IIUTII2 + /3, which is valid for all T E R+. Moreover, since u E X andH : X - X, we can take limits as T -* oo to obtain

(u, Hu) > 8IIuIIX + Q

Thus we conclude that (8.11) implies (8.12) and the second assertion of the theorem isproved. Part (i) is immediately obvious assuming that b = 0. This completes the proof.

8.3 Interconnections of Passivity Systems

In many occasions it is important to study the properties of combinations of passive systems.The following Theorem considers two important cases

Theorem 8.2 Consider a finite number of systems Hi : Xe - Xe , i = 1, , n. We have

(i) If all of the systems Hi, i = 1, , n are passive, then the system H : Xe + Xe ,defined by (see Figure 8.3)

(8.13)

is passive.

(ii) If all the systems Hi, i = 1, , n are passive, and at least one of them is strictlypassive, then the system H defined by equation (8.13) is strictly passive.

(iii) If the systems Hi, i = 1, 2 are passive and the feedback interconnection defined by theequations (Figure 8.4)

e = u-H2y (8.14)

y = Hie (8.15)

is well defined (i.e., e(t) E Xe and is uniquely determined for each u(t) E Xe), then,the mapping from u into y defined by equations (8.14)-(8.15) is passive.

Proof of Theorem 8.2

Proof of (i): We have

(x,(Hl+...+Hn)x)T = (x,Hlx+...+Hnx)T

(x+ Hlx)T + ... + (x+ Hnx)TdeJ

N.

3.3. INTERCONNECTIONS OF PASSIVITY SYSTEMS 209

u(t)

Hn

H2

H1

Figure8.3:

H1

H2

y(t)

y(t)

Figure 8.4: The Feedback System S1.

Thus, H f (H1 + + Hn) is passive.

Proof of (ii) Assume that k out of the n systems H= are strictly passive, 1 < k < n. By rela-beling the systems, if necessary, we can assume that these are the systems H1, H2,- , Hk.It follows that

(x, Hx)T = (x, H1x + ... + HnX)T= (x Hlx)T+...+(x,Hkx)T+...+(x,Hnx)T

61(x,x)T +... +6k(x,x)T+Q1 +... +/3n


Proof of (iii): Consider the following inner product:

(u, Y)T = (e + H2Y, Y)T

= (e, y)T + (H2y, Y)T

= (e, Hle)T + (y, H2y)T ? (31+02).


Remarks: In general, the number of systems in parts (i) and (ii) of Theorem 8.2 cannot beassumed to be infinite. The validity of these results in case of an infinite sequence of systemsdepends on the properties of the inner product. It can be shown, however, that if the innerproduct is the standard inner product in £2, then this extension is indeed valid. The proofis omitted since it requires some relatively advanced results on Lebesgue integration.

8.3.1 Passivity and Small Gain

The purpose of this section is to show that, in an inner product space, the concept ofpassivity is closely related to the norm of a certain operator to be defined.

In the following theorem Xe is an inner product space, and the gain of a systemH : Xe -+ Xe is the gain induced by the norm IIxII2 = (x, x).

Theorem 8.3 Let H : Xe -4 Xe , and assume that (I + H) is invertible in X, , that is,assume that (I + H)-1 : Xe -+ Xe . Define the function S : Xe -- Xe :

S = (H - I)(I + H)-1. (8.16)

We have:

8.4. STABILITY OF FEEDBACK INTERCONNECTIONS 211

ul(t) + el(t)

H1yi(t)

H2e2(t)

Figure 8.5: The Feedback System S1.

(a) H is passive if and only if the gain of S is at most 1, that is, S is such that

I(Sx)TI x < II-TIIX Vx E Xe ,VT E Xe. (8.17)

(b) H is strictly passive and has finite gain if and only if the gain of S is less than 1.


8.4 Stability of Feedback Interconnections

In this section we exploit the concept of passivity in the stability analysis of feedbackinterconnections. To simplify our proofs, we assume without loss of generality that thesystems are initially relaxed, and so the constant )3 in Definitions 8.2 and 8.3 is identicallyzero.

Our first result consists of the simplest form of the passivity theorem. The simplicityof the theorem stems from considering a feedback system with one single input u1, as shownin Figure 8.5 (u2 = 0 in the feedback system used in Chapter 6).

Theorem 8.4 : Let H1, H2 : Xe 4 Xe and consider the feedback interconnection definedby the following equations:

ei = ul - H2e2 (8.18)

yi = H1e1. (8.19)

Under these conditions, if H1 is passive and H2 is strictly passive, then y1 E X for everyxEX.

Proof: We have

(ul, yl)T = (u1, Hle1)T= (el + H2e2, Hlel)T

= (el, Hlel)T + (H2e2, Hlel)T= (el, Hlel)T + (H2y1, y1)T

but

(e1,Hlel)T > 0

(H2yl, yl)T > 6IIy1TI x

since H1 and H2 are passive and strictly passive, respectively. Thus

2(u1,y1)T > 61IYlTIIX

By the Schwarz inequality, I (ul, yl)T 1< IIu1TIIx IIy1TIIX. Hence

2>_ 6IIy1TIIX

Iy1TIIX <_ 6-11Iu1TIIX (8.20)

Therefore, if ul E X, we can take limits as T tends to infinity on both sides of inequality(8.20) to obtain

I I y 1 I I>_ b 1IIu1II

which shows that if u1 is in X, then y1 is also in X.

Remarks: Theorem 8.4 says exactly the following. Assuming that the feedback system ofequations (8.18)-(8.19) admits a solution, then if H1 is passive and H2 is strictly passive,the output y1 is bounded whenever the input ul is bounded. According to our definition ofinput-output stability, this implies that the closed-loop system seen as a mapping from u1to y1 is input-output-stable. The theorem, however, does not guarantee that the error eland the output Y2 are bounded. For these two signals to be bounded, we need a strongerassumption; namely, the strictly passive system must also have finite gain. We consider thiscase in our next theorem.

Theorem 8.5 : Let H1, H2 : Xe - Xe and consider again the feedback system of equations(8.18)-(8.19). Under these conditions, if both systems are passive and one of them is (i)strictly passive and (ii) has finite gain, then eli e2, y1i and Y2 are in X whenever x E X.

8.4. STABILITY OF FEEDBACK INTERCONNECTIONS 213


Proof: We prove the theorem assuming that H1 is passive and H2 is strictly passive withfinite gain. The opposite case (i.e., H2 is passive and H1 is strictly passive with finite gain)is entirely similar. Proceeding as in Theorem 8.4 (equation (8.20), we obtain

Iy1TIIX < b-1IIU1TIIX (8.21)

so that y1 E X whenever u1 E X. Also Y2 = H2y1. Then IIy2TIIX = II(H2y1)TIIx, and sinceH2 has finite gain, we obtain

Ily2zTlix < ry(H2)IIy1TIIx <_ 'Y(H2)6-1IIu1TIIx

Thus, Y2 E X whenever u1 E X. Finally, from equation (8.18) we have

IIe1TIIX: Iu1TIIX+II(H2y1)TIIX 5 IIN1TIIX+-Y(H2)IIy1TIIX

which, taking account of (8.21), implies that e1 E X whenever u1 E X.

Remarks: Theorem 8.5 is general enough for most purposes. For completeness, we includethe following theorem, which shows that the result of Theorem 8.5 is still valid if the feedbacksystem is excited by two external inputs (Figure 8.6).

Theorem 8.6 : Let H1, H2 : Xe -> Xe and consider the feedback system of equations

e1 = u1 - H2e2 (8.22)

e2 = u2 + H1e1 (8.23)


Under these conditions, if both systems are passive and one of them is (i) strictly passiveand (ii) has finite gain, then el, e2, yl, and y2 are in X whenever xl, x2 E X.

Proof: Omitted.

8.5 Passivity of Linear Time-Invariant Systems

In this section, we examine in some detail the implications of passivity and strict passivityin the context of linear time-invariant systems. Throughout this section, we will restrictattention to the space G2.

Theorem 8.7 Consider a linear time-invariant system H : C2e -> G2e defined by Hx =h* x, where h E A, x CGee. We have

(i) H is passive if and only if te[H(yw)] > 0 Vw E R.

(ii) H is strictly passive if and only if 36 > 0 such that Ste[H(yw)] > b Vw E R.

Proof: The elements of A have a Laplace transform that is free of poles in the closed righthalf plane. Thus, points of the form s = ,yw belong to the region of convergence of H(s),and H(3w) is the Fourier transform of h(t). We also recall two properties of the Fouriertransform:

(a) If f is real-valued, then the real and imaginary parts of P(p), denoted We[F(3w)]and `Zm[F(yw)], respectively, are even and odd functions on w, respectively. In otherwords

te[F(-yw)]+ ` m[F'(7w)] _ -`3m[F(-yw)]We[F'(7w)] =

(b) (Parseval's relation)

P(31)15(31))' dwf f (t)g(t) dt - 2a _C0

where G(yw)` represents the complex conjugate of G(yw).

Also, by assumption, h c A, which implies that H is causal and stable. Thus, accordingto Theorem 8.1, H is passive (strictly passive) if and only if it is positive (strictly positive),and we can drop all truncations in the passivity definition. With this in mind, we have

8.5. PASSIVITY OF LINEAR TIME-INVARIANT SYSTEMS

(x, Hx)

1 te[H(7w)] lX (7w)12 dw +27r

f00

(x,h*x)

Jx(t)[h(t) * x(t)] dt

0000

27rf X (7w)[H(?w)X (7w)] du)

1 IX(.7w)I2 H(7w)' dw27r f00

300

2rrm[H(7w)]00

215

du)

and sincem[H(yw)] is an odd function of w, the second integral is zero. It follows that

(x, Hx) _1

27r

and noticing that

we have that

00

J Re[H(34 dw

(x, x) I I -`f (7w) I2 dw27r

(x,Hx) > inf te[H(yw)]W

from where the sufficiency of conditions (i) and (ii) follows immediately. To prove necessity,assume that 2e[H(3w)] < 0 at some frequency w = w'. By the continuity of the Fouriertransform as a function of w, it must be true that te[H(tw)] < 0 Vu ;E 1w - w' I <c, forsome e > 0. We can now construct 9(y,) as follows:

X(pw) > MOW) < m

VwE 1w-w*I <eelsewhere

It follows that X(yw) has energy concentrated in the frequency interval where lRe[H(3w)]is negative and thus (x, Hx) < 0 for appropriate choice of M and M. This completes theproof.

Theorem 8.7 was stated and proved for single-input-single-output systems. For com-pleteness, we state the extension of this result to multi-input-multi-output systems. Theproof follows the same lines and is omitted.

Theorem 8.8 Consider a multi-input-multi-output linear time-invariant system H : £2e -4£2e defined by Hx = h * x, where h c A , x E L2e. We have,

(i) H is passive if and only if [H(,w)] + H(3w)*] > 0 Vw E R.

(ii) H is strictly passive if and only if 3b > 0 such that

Amin[1 (3w)] + H(3w)'] > b Vw E ]R

It is important to notice that Theorem 8.7 was proved for systems whose impulseresponse is in the algebra A. In particular, for finite-dimensional systems, this algebraconsists of systems with all of their poles in the open left half of the complex plane. Thus,Theorem 8.7 says nothing about whether a system with transfer function

H(s) =s2

asw2 a> 0, w o > 0

0

(8.24)

is passive. This transfer function, in turn, is the building block of a very important classof system to be described later. Our next theorem shows that the class of systems with atransfer function of the form (8.24) is indeed passive, regardless of the particular value ofa and w.

Theorem 8.9 Consider the system H : Gee -> G2e defined by its transfer function

H(s) =s2

asw2

a > 0, w0 > 0.0

Under these conditions, H is passive.

Proof: By Theorem 8.3, H is passive if and only if

1-H

but for the given H(s)

S(s) _

SOW) _

<11+H 00

s2 - as + wos2+as+wo(U)02 - w2) - haw(wo - w2) + haw

which has the form (a- 3b)/(a+.7b). Thus IS(3w)I = 1 Vw, which implies that IIsJi = 1,and the theorem is proved.

Remarks: Systems with a transfer function of the form (8.24) are oscillatory; i.e., if excited,the output oscillates without damping with a frequency w = wo. A transfer function of this

8.6. STRICTLY POSITIVE REAL RATIONAL FUNCTIONS 217

form is the building block of an interesting class of systems known as flexible structures. Alinear time-invariant model of a flexible structure has the following form:

00H(s) _ cxts(8.25)

Examples of flexible structures include flexible manipulators and space structures. They arechallenging to control given the infinite-dimensional nature of their model. It follows fromTheorem 8.2 that, working in the space £2i a system with transfer function as in (8.25) ispassive.

8.6 Strictly Positive Real Rational Functions

According to the results in the previous section, a (causal and stable) LTI system H isstrictly passive if and only if H(3w) > b > 0 Vw E R. This is a very severe restriction,rarely satisfied by physical systems. Indeed, no strictly proper system can satisfy thiscondition. This limitation, in turn, severely limits the applicability of the passivity theorem.Consider, for example, the case in which a passive plant (linear or not) is to be controlledvia a linear time-invariant controller. If stability is to be enforced by the passivity theorem,then only controllers with relative degree zero can qualify as possible candidates. This isindeed a discouraging result. Fortunately, help in on the way: in this section we introducethe concept of strict positive realness (SPRness), which, roughly speaking, lies somewherebetween passivity and strict passivity. It will be shown later that the feedback combinationof a passive system with an SPR one, is also stable, thus relaxing the conditions of thepassivity theorem.

In the sequel, P" denotes the set of all polynomials on nth degree in the undeterminedvariable s.

Definition 8.5 Consider a rational function ft(s) = p(s)/q(s), where p(.) E P", andE Pm. Then H(s) is said to be positive real (PR), if

?Re[H(s)] > 0 for rte[s] > 0.

H(s) is said to be strictly positive real (SPR) if there exists e > 0 such that H(s - e) is PR.

Remarks: Definition 8.5 is rather difficult to use since it requires checking the real part ofH(s) for all possible values of s in the closed right half plane. For this reason, frequency-domain conditions for SPRness are usually preferred. We now state these conditions as analternative definition for SPR rational functions.


Definition 8.6 Consider a rational function H(s) = p(s)/q(s), where E P', andE P'. Then H(s) is said to be in the class Q if

(i) is a Hurwitz polynomial (i.e. the roots of are in the open left half plane), and

(ii) Re[H(jw)] > 0, Vw E [0, oo).

Definition 8.7 H(s), is said to be weak SPR if it is in the class Q and the degree of thenumerator and denominator polynomials differ by at most 1. H(s) is said to be SPR if itis weak SPR and in addition, one of the following conditions is satisfied:

(i) n = m, that is, p and q have the same degree.

(ii) n = m + 1, that is, H(s) is strictly proper, and

lim w2JIe[H(yw)] > 0W-i00

It is important to notice the difference amon,, the several concepts introduced above.Clearly, if H(s) is SPR (or even weak SPR), then H(s) E Q. The converse is, however, nottrue. For example, the function Hl (s) = (s + 1)-1 + s3 E Q, but it is not SPR accordingto Definition 8.5. In fact, it is not even positive real. The necessity of including condition(ii) in Definition 8.7 was pointed out by Taylor in Reference [79]. The importance of thiscondition will became more clear soon.

We now state an important result, known as the Kalman-Yakubovich lemma.

Lemma 8.1 Consider a system of the form

Lb = Ax+Bu, xEl[t", uE1imy = Cx+Du, UERm

and assume that (i) the eigenvalues of A lie in the left half of the complex plane, (ii)(A, B) is controllable, and (iii) (C, A) is observable. Then H = C(sI - A)-1B + D isSPR if and only if there exist a symmetric positive definite matrix P E Rn,n, and matricesQ E Rmxm W E lRm', and e > 0 sufficiently small such that

PA + ATP = _QQT _.EP (8.26)

PB + WTQ = C (8.27)

WTW = D+DT (8.28)

8.6. STRICTLY POSITIVE REAL RATIONAL FUNCTIONS 219

Remarks: In the special case of single-input-single-output systems of the form

i = Ax+Buy = Cx

the conditions of lemma 8.1 can be simplified as follows: H = C(sI - A) -1B is SPR if andonly if there exist symmetric positive definite matrices P and L a real matrix Q, and psufficiently small such that

PA + ATP = _QQT - pL (8.29)

PB = CT (8.30)

Proof: The proof is available in many references and is omitted. In Chapter 9, however,we will state and prove a result that can be considered as the nonlinear counterpart of thistheorem. See Theorem 9.2.

Theorem 8.10 Consider the feedback interconnection of Figure 8.5, and assume that

(i) H1 is linear time-invariant, strictly proper, and SPR.

(ii) H2 is passive (and possibly nonlinear).

Under these assumptions, the feedback interconnection is input-output-stable.

Proof: The proof consists of employing a type I loop transformation with K = -E andshowing that if e > 0 is small enough, then the two resulting subsystem satisfy the followingconditions:

Hi(s) = Hi(s)/[1 - eH1(s)] is passive

Hz = H2 + eI is strictly passive.

Thus, stability follows by theorem 8.4. The details are in the Appendix.

Theorem 8.10 is very useful. According to this result, the conditions of the passivitytheorem can be relaxed somewhat. Indeed, when controlling a passive plant, a linear time-invariant SPR controller can be used instead of a strictly passive one. The significance of theresult stems from the fact that SPR functions can be strictly proper, while strictly passivefunctions cannot. The following example shows that the loop transformation approach usedin the proof of Theorem 8.10 (see the Appendix) will fail if the linear system is weak SPR,thus emphasizing the importance of condition (ii) in Definition 8.7.

Example 8.4 Consider the linear time-invariant system H(s) = (s + c)/[(s + b)(s + b)],and let H'(s) = H(s)/[1 - eH(s)]. We first investigate the SPR condition on the systemH(s). We have

H( )c+,yw

3w

II(t(ab-w2)+,yw(a+b)c(ab - w2) + w2(a + b)

=

e[ 7w)] =

lim w2 te[H(,w)] =

(ab-w2)2+w2(a+b)2

a + b - cW-ioo

from here we conclude that

(i) H(s) is SPR if and only if a + b > c.

(ii) H(s) is weak SPR if a + b = c.

(iii) H(s) is not SPR if a + b < c.

We now consider the system H'(s), after the loop transformation. We need to see whetherH'(s) is passive (i.e., Re[H'(3w)] > 0). To analyze this condition, we proceed as follows

1

[H (7w)+ H(-7w)] =

(abc - ec2) + w2(a + b - c - e) > 0Z (ab-eC-w2)2+(a+b-e)2

if and only if

abc - ec2 > 0 (8.31)

a+b-c-e > 0. (8.32)

If a + b > c, we can always find an e > 0 that satisfies (8.31)-(8.32). However, if fl(s) isweak SPR, then a + b = c, and no such e > 0 exists.

8.7 Exercises

(8.1) Prove Theorem 8.4 in the more general case when Q # 0 in Definitions 8.2 and 8.3.

(8.2) Prove Theorem 8.5 in the more general case when 0 $ 0 in Definitions 8.2 and 8.3.

8.7. EXERCISES 221

Notes and ReferencesThis chapter introduced the concept of passivity in its purest (input-output) form. Ourpresentation closely follows Reference [21], which is an excellent source on the input-outputtheory of systems in general, and passivity results in particular. Early results on stability ofpassive systems can be found in Zames [98]. Strictly positive real transfer functions and theKalman-Yakubovich lemma 8.1, play a very important role in several areas of system theory,including stability of feedback systems, adaptive control, and even the synthesis of passivenetworks. The proof of the Kalman-Yakubovich lemma can be found in several works.See for example Narendra and Anaswami [56], Vidyasagar [88] or Anderson [1]. Reference[1], in particular, contains a very thorough coverage of the Kalman-Yakubovich lemma andrelated topics. See References [35] and [78] for a detailed treatment of frequency domainproperties of SPR functions. Example 8.4 is based on unpublished work by the author incollaboration with Dr. C. Damaren (University of Toronto).

Chapter 9

Dissipativity

In Chapter 8 we introduced the concept of a passive system (FIgure 9.1). Specifically, givenan inner product space X, a system H : Xe -4 Xe is said to be passive if (u, HU)T > Q.This concept was motivated by circuit theory, where u and y are, respectively, the voltagev(t) and current i(t) across a network or vice versa. Thus, in that case (assuming X = G2)

T(u, Hu)T = MO, i(t))T =

100

v(t)i(t) dt

which represents the energy supplied to the network at time T or, equivalently, the energyabsorbed by the network during the same time interval. For more general classes of dy-namical systems, "passivity" is a somewhat restrictive property. Many systems fail to bepassive simply because (u, y) may not constitute a suitable candidate for an energy function.In this chapter we pursue these ideas a bit further and postulate the existence of an inputenergy function and introduce the concept of dissipative dynamical system in terms of a non-negativity condition on this function. We will also depart from the classical input-output

uH y

Figure 9.1: A Passive system.

223

224 CHAPTER 9. DISSIPATIVITY

theory of systems and consider state space realizations. The use of an internal descriptionwill bring more freedom in dealing with initial conditions on differential equations and willalso allow us to study connections between input-output stability, and stability in the senseof Lyapunov.

9.1 Dissipative Systems

Throughout most of this chapter we will assume that the dynamical systems to be studiedare given by a state space realization of the form

(i=f(x,u), uEU, XEXSl y = h(x, u), y c Y

where

U is the input space of functions, u E U : SZ C R --> 1R'. In other words, the functionsin U map a subset of the real numbers into 1R1.

Y the output space, which consists of functions, y E U : SZ C R -> IItp.

X The set X C 1R" represents the state space.

Associated with this system we have defined a function w(t) = w(u(t), y(t)) : U x y -+ ]R,called the supply rate, that satisfies

J

tlIw(t)I dt < oo,

to

that is, is a locally integrable function of the input and output u and y of the system

Definition 9.1 A dynamical system t' 1is said to be dissipative with respect to the supplyrate w(t) if there exists a function ,, : X -+ 1R+, called the storage function, such that forall xo E X and for all inputs u E U we have

O(x1) < q5(xo) + ft,

w(t) dt. (9.2)to

Inequality (9.2) is called the dissipation inequality, and the several terms in (9.2) representthe following:

9.2. DIFFERENTIABLE STORAGE FUNCTIONS 225

the storage function; O(x(t`)) represents the "energy" stored by the system V) attime t'.

fto' w(t) dt: represents the energy externally supplied to the system zG during theinterval [to, t1].

Thus, according to (9.2), the stored energy l4(xi) at time t1 > to is, at most, equal tothe sum of the energy O(xo) initially stored at time to, plus the total energy externallysupplied during the interval [to, t1]. In this way there is no internal "creation" of energy. Itis important to notice that if a motion is such that it takes the system V) from a particularstate to the same terminal state along a certain trajectory in the state space, then we have(since x1 = xo)

0(Xo) < 0(Xo) + i w(t) dt

= J w(t)dt > 0 (9.3)

where f indicates a closed trajectory with identical initial and final states. Inequality (9.3)states that in order to complete a closed trajectory, a dissipative system requires externalenergy.

9.2 Differentiable Storage Functions

In general, the storage function 0 of a dissipative system, defined in Definition 9.1 neednot be differentiable. Throughout the rest of this chapter, however, we will see that manyimportant results can be obtained by strengthening the conditions imposed on 0. First wenotice that if 0 is continuously differentiable, then dividing (9.2) by (t1 - to), and denotingO(xi) the value of O(x) when t = ti, we can write

_(xl) - 0(Xo) < 1

tl - to tl - to

but

w(t) dt

lim O(x1) - 0(Xo) = dO(x) = 00(x)f(x, u)

tl-*to t1 - to dt 8x

and thus (9.4) is satisfied if and only if

0a(x) f (x, u) < w(t) = w(u, y) = w(u, h(x, u)) Vx, it. (9.5)

226 CHAPTER 9. DISSIPATWITY

Inequality (9.5) is called the differential dissipation inequality, and constitutes perhaps themost widely used form of the dissipation inequality. Assuming that 0 is differentiable, wecan restate definition 9.1 as follows.

Definition 9.2 (Dissipativity re-stated) A dynamical system Eli is said to be dissipative withrespect to the supply rate w(t) = w(u, y) if there exist a continuously differentiable function0 : X --> IR+, called the storage function, that satisfies the following properties:

(i) There exist class 1C,, functions al and a2 such that

al(IIxII) <- O(x) <- a2(I14II) Vx E IRn

.90

8x f (x, u) < w(u,y) Vx E R' , u E lRm, and y = h(x, u).

In other words; defining property (i) simply states that 0(.) is positive definite, whileproperty (ii) is the differential dissipation inequality.

9.2.1 Back to Input-to-State Stability

In Chapter 7 we studied the important notion of input-to-state stability. We can now reviewthis concept as a special case of a dissipative system. In the following lemma we assumethat the storage function corresponding to the supply rate w is differentiable.

Lemma 9.1 A system Eli is input-to-state stable if and only if it is dissipative with respectto the supply rate

w(t) = -a3(IIxII) +a(IIuII)where a3 and a are class )C,,. functions.

Proof: The proof is an immediate consequence of Definition 9.2 and Lemma 7.6.

9.3 QSR Dissipativity

So far we have paid little attention to the supply rate w(t). There are, however, severalinteresting candidates for this function. In this section we study a particularly importantfunction and some of its implications. We will see that concepts such as passivity andsmall gain are special cases of this supply rate.

9.3. QSR DISSIPATIVITY 227

Definition 9.3 Given constant matrices Q E 1Rp> , S E R' m, and R E Rmxm with Q andR symmetric, we define the supply rate w(t) = w(u, y) as follows:

w(t) = yTQy+2yTSu+uTRu

[YT, UT) LQ RSi 1

L UI

(9.6)

It is immediately obvious that00

(t) dt = (y, Qy) + 2(y, Su) + (u, Ru) (9.7)f wand

J

Tw(t) dt = (y, QY)T + 2(y, SU)T + (u, Ru)T (9.8)

0

moreover, using time invariance, w(t) defined in (9.6) is such that

to

to+Tw(t) dt =

JO

Thus, we can now state the following definition

Definition 9.4 The system V) is said to be QSR-dissipative if there exist a storage functionX -- ]R+ such that 11x(0) = xo E X and for all u E U we have that

J

Tw(t) dt = (y, Qy)T + 2(y, Su)T + (u, Ru)T ? O(xi) - 1(xo). (9.10)

0

Definition 9.4 is clearly a special case of Definition (9.1) with one interesting twist. Noticethat with this supply rate, the state space realization of the system V) is no longer essentialor necessary. Indeed, QSR dissipativity can be interpreted as an input-output property.Instead of pursuing this idea, we will continue to assume that the input output relationshipis obtained from the state space realization as defined in (9.1). As we will see, doing sowill allow us to study connections between certain input-output properties and stability inthe sense of Lyapunov.

We now single out several special cases of interest, which appear for specific choicesof the parameters QS and R.

1- Passive systems: The system V) is passive if and only if it is dissipative with respectto Q = 0, R = 0, and S = 21. Equivalently, V) is passive if and only if it is (0, 2, 0)-dissipative. The equivalence is immediate since in this case (9.10) implies that

(y, u)T ? O(xi) - O(xo) ? -O(xo) (9.11)(since O(x) > 0 Vx, by assumption)

Tw(t) dt. (9.9)

Thus, defining ,0f

- O(xo), (9.11) is identical to Definition (8.2). This formulationis also important in that it gives 3 a precise interpretation: 3 is the stored energy attime t = 0, given by the initial conditions xo.

2- Strictly passive systems: The system Vi is strictly passive if and only if it isdissipative with respect to Q = 0, R = -b, and S = 2I. To see this, we substitutethese values in (9.10) and obtain

(y, u)T + (U, -bU)T >- 0(x1) - O(x0) >- -O(xO)f

Q

or

(u, y)T b(u, U) T + Q = 5IIuIIT + 3.

3- Finite-gain-stable: The system 7/i is finite-gain-stable if and only if it is dissipativewith respect to Q = -

iI, R = 2 I, and S = 0. To see this, we substitute these values

in (9.10) and obtain

1

2 (y, y)T + 2 ('a, u)T > O(xl) - O(x0) > -qS(xo)

(y,y)T >_ -72(u,u)T - 20(x0)

or (y, y)T < 72 (n,u)T + 20(xo)IIyTIIGZ <- 72IIuTIIGz + 20(xo)

IIyTIIGZ 72IIUTIIG2 + 20(x0)

and since for a, b > 0, a -+b2 < (a + b), then defining 3 = 2 (xo), we concludethat

IIyTIIGZ <- 7IIUTIIGZ + 3

We have already encountered passive, strictly passive, and finite-gain-stable systems inprevious chapters. Other cases of interest, which appear frequently in the literature aredescribed in paragraphs 4 and 5.

4- Strictly output-passive systems: The system V) is said to be strictly output pas-sive if it is dissipative with respect to Q = -EI, R = 0, and S = aI. In this case,substituting these values in (9.10), we obtain

-e(y, y)T + (y, u)T >- 0(x1) - O(xo) ? -O(xo)

orfT

J UT y dt = (u, y)T ? E(y, y)T +0

9.4. EXAMPLES 229

5- Very strictly-passive Systems: The system V) is said to be very strictly passiveif it is dissipative with respect to Q = -eI, R = -bI, and S = I. In this case,substituting these values in (9.10), we obtain

-E(y,Y)T - b(u, U)T + (y, U)T > -O(x1) - O(xo) > -O(x0)I

0

orrT

J U T y dt = (U, Y)T ? CU, U)T + E(y, Y)T + 13-0

The following lemma states a useful results, which is a direct consequence of thesedefinitions.

Lemma 9.2 If -0 is strictly output passive, then it has a finite L2 gain.

Proof: The proof is left as an exercise (Exercise 9.2).

9.4 Examples

9.4.1 Mass-Spring System with Friction

Consider the mass-spring system moving on a horizontal surface, shown in Figure 9.2.Assuming for simplicity that the friction between the mass and the surphase is negligible,we obtain the following equation of the motion:

ml+0±+kx= fwhere m represents the mass, k is the spring constant, p, the viscous friction force associatedwith the spring, and f is an external force. Defining state variables x1 = x, and ±1 = x2,and assuming that the desired output variable is the velocity vector, x2, we obtain thefollowing state space realization:

1 = X22 = -mx1 - P- x2 + 1

Y = X2

To study the dissipative properties of this system, we proceed to find the total energy storedin the system at any given time. We have

E = 1kx2 +21

mx2


Figure 9.2: Mass-spring system.

where zmx2 represents the kinetic energy of the mass and Zkxi is the energy stored bythe spring. Since the energy is a positive quantity, we propose E as a "possible" storagefunction; thus, we define:

¢r=E=2kxi+2mx2.

Since 0 is continuously differentiable with respect to xl and x2i we can compute the timederivative of 0 along the trajectories of i/i:

.a0x8x

[kxl 7fl2]L

-/3x2+xf_Qy2 + Y f.

Thus,

X2

-m x1 - 1x2

Jt 0 dt = E(t) > 0

0

and it follows that the mass-spring system with output i = x2 is dissipative with respectto the supply rate

w(t) = yf - /3y2.

It is immediately evident that this supply rate corresponds to Q = -0, S = 2, and R = 0,from where we conclude that the mass-spring system 0 is strictly output-passive.

9.5. AVAILABLE STORAGE 231

9.4.2 Mass-Spring System without Friction

Consider again the mass-spring system of the previous example, but assume that f3 = 0.In this case, the state space realization reduces to

I it = x2i2 = -mxl+fy = X2

Proceeding as in the previous example, we definef = E = 2kxi + 2rnx2.

Differentiating m along the trajectories of 0, we obtain:

=x2f =yfsince once again,

frt 0 dt = E(t) > 0.0

We conclude that the mass-spring system with output i = x2 is dissipative with respect tothe supply rate

w(t) = yfwhich corresponds to Q = 0, S = 2, and R = 0. This implies that the mass-spring system

is passive.

9.5 Available Storage

Having defined dissipative systems, along with supply rates and storage functions, we nowturn our attention to a perhaps more abstract question. Given a dissipative dynamicalsystem, we ask: What is the maximum amount of energy that can be extracted from it, atany given time? Willems [91] termed this quantity the "available storage." This quantityplays an important conceptual role in the theory of dissipative systems and appears in theproofs of certain theorems. For completeness, we now introduce this concept. This sectionis not essential and can be skipped in a first reading of this chapter.

Definition 9.5 The available storage, cba of a dynamical system,0 with supply rate w isdefined by

fT0a(x) = sup - J w(u(t), y(t)) dt, x(0) = X. (9.12)

u() oT>0

As defined, 0a(x) denotes the energy that can be extracted from ,b, starting from the initialstate x at t = 0.

The following theorem is important in that it provides a (theoretical) way of checkingwhether or not a system is dissipative, in terms of the available storage.

Theorem 9.1 A dynamical system 0 is dissipative if and only if for all x E X the availablestorage (¢a(x) is finite. Moreover, for a dissipative system we have that 0 < Oa < S, andthus cba itself is a possible storage function.

Proof:

Sufficiency: Assume first that Oa is finite. Since the right-hand side of (9.12) contains thezero element (obtained setting T = 0) we have that Oa > 0. To show that Vi is dissipative,we now consider an arbitrary input u' : [0, T] - 1R" that takes the dynamical system V)from the initial state xo at t = 0 to a final state xl at t = T, and show that the cba satisfies

fw(t)T

Oa(xl) < Oa(x0) + dt.

Thus cba is itself a storage function. To see this, we compare Wa(xo) and !aa(xl), the"energies" that can be extracted from V, starting at the states xo and xl, respectively.We have

O(xo)

/Tsup - J w(u(t), y(t)) dt,uca oT>o

Tw(u(t), y(t)) dt,> - J

cl

w(u', y) dt - sup fo0 T"O>0

-J

tlw(u*, y) dt + Oa(xl).

0

This means that when extracting energy from the system Vi at time xo, we can follow an"optimal" trajectory that maximizes the energy extracted following an arbitrary trajectory,or we can force the system to go from xo to xl and then extract whatever energy is left in Viwith initial state x1. This second process is clearly nonoptimal, thus leading to the result.

Necessity: Assume now that 0 is dissipative. This means that 10 > 0 such that for allu(.) we have that

O(xo) + J w(u,y) dt > O(x(T)) > 0.T0

9.6. ALGEBRAIC CONDITION FOR DISSIPATIVITY 233

Thus,T

w(u(t), y(t)) dt, _ 0a(xo)O(xo) ? sup - Jo"OT > 0

9.6 Algebraic Condition for Dissipativity

We now turn our attention to the issue of checking the dissipativity condition for a givensystem. The notion of available storage gave us a theoretical answer to this riddle in Theo-rem 9.1. That result, however, is not practical in applications. Our next theorem providesa result that is in the same spirit as the Kalman-Yakuvovich lemma studied in Chapter 8;specifically, it shows that under certain assumptions, dissipativeness can be characterizedin terms of the coefficients of the state space realization of the system Vi. Moreover, itwill be shown that in the special case of linear passive systems, this characterization ofdissipativity, leads in fact, to the Kalman-Yakuvovich lemma.

Throughout the rest of this section we will make the following assumptions:

1- We assume that both f and in the state space realizationfunctions of the input u, that is, we assume that 0 is of the form

J ±=f(x)+9(x)uy = h(x) + 7(x)u

where x E R", u E R, and y E RP, respectively.

V) are affine

(9.13)

2- The state space of the system (9.13) is reachable from the origin. This means thatgiven any xl E R" and t = tl E R+ there exists a to < tl and an input u E U suchthat the state can be driven from the origin at t = 0 to x = xl at t = t1.

3- Whenever the system V) is dissipative with respect to a supply rate of the form (9.6),the available storage 0a,(x) is a differentiable function of x.

These assumptions, particularly 1 and 3, bring, of course, some restrictions on the class ofsystems considered. The benefit is a much more explicit characterization of the dissipativecondition.

Theorem 9.2 The nonlinear system zb given by (9.13) is QSR-dissipative (i.e. dissipativewith supply rate given by (9.6)) if there exists a differentiable function 0 : R" -+ IR andfunction L : 1R' -4 Rq and W : R" -4 R'> "` satisfying


q(x) > 0 vx $ 0, 0(0) = 0 (9.14)

a-f (x) = hT (x)Qh(x) - LT (x)L(x) (9.15)

1

19T (ax)TSTh(x) - WTL(x) (9.16)

R = WTW (9.17)

for all x, where

and

R = R + jT (x)S + STj (x) +7T(x)Q7(x) (9.18)

S(x) = Qj(x) + S. (9.19)

Proof: To simplify our proof we assume that j (x) = 0 in the state space realization (9.13)(see exercise 9.3). In the case of linear systems this assumption is equivalent to assumingthat D = 0 in the state space realization, something that is true in most practical cases.With this assumption we have that

k = R, and S = S,

We prove sufficiency. The necessity part of the proof can be found in the Appendix. As-suming that S, L and W satisfy the assumptions of the theorem, we have that

w(u,y) = yTQY+UTRu+2yTSu= hT Qh + uT Ru + 2hTSu substituting (9.13)

f (x) + LTL] + UTRu + 2hTSu substituting (9.15)

aof (x) +LTL+uTRu+2uTSTh

a46f(x)+LTL+uTWTWu+2uT[2gT(O )T +WTL]

substituting (9.17) and (9.16)

a[f(x)+gu]+LTL+uTWTWu+2uTWTLox

+ (L + Wu)T (L + Wu)

_ 0+ (L + Wu)T (L + Wu) (9.20)

9.6. ALGEBRAIC CONDITION FOR DISSIPATIVITY 235

J w(t) dt = O(x(t)) - O(xo) + f(L + Wu)T (L + Wu) dtt

2 O(x(t)) - O(xo)

and setting xo = 0 implies that

t w(t) dt > O(x(t)) > 0.

Corollary 9.1 If the system is dissipative with respect to the supply rate (9.6), then thereexist a real function 0 satisfying ¢(x) > 0 Vx # 0, 0(0) = 0, such that

d = -(L+Wu)T(L+Wu)+w(u,y) (9.21)

Proof: A direct consequence of Theorem 9.2. Notice that (9.21) is identical to (9.20) inthe sufficiency part of the proof.

9.6.1 Special Cases

We now consider several cases of special interest.

Passive systems: Now consider the passivity supply rate w(u, y) = uTy (i.e. Q = R = 0,and S = 1 in (9.6)), and assume as in the proof of Theorem (9.2) that j(x) = 0. In thiscase, Theorem 9.2 states that the nonlinear system 0 is passive if and only if

8xf (x) = -LT (x)L(x)

9T(aO)T = h(x)or, equivalently

.90

8xf (x)< 0

ax- 9 = hT(x).

(9.22)

(9.23)

Now assume that, in addition, the system V is linear then f (x) = Ax, g(x) = B, andh(x) = Cx. Guided by our knowledge of Lyapunov stability of linear systems, we definethe storage function O(x) = xT Px, p = PT > 0. Setting u = 0, we have that

T TP5xf(x)=x[A+PA]x

which implies that (9.22)-(9.23) are satisfied if and only if

ATP+PA < 0BTP = CT.

Therefore, for passive systems, Theorem 9.2 can be considered as a nonlinear version of theKalman-Yakubovich lemma discussed in Chapter 8.

Strictly output passive systems: Now consider the strictly output passivity supply ratew(u,y) = uTy - eyTy (i.e., Q = -JI, R = 0, and S =

2Iin (9.6)), and assume once

again that j (x) = 0. In this case Theorem 9.2 states that the nonlinear system z/) is strictlyoutput-passive if and only if

a7 f(x) = -ehT(x)h(x) - LT(x)L(x)

or, equivalently,

a f(x) < -ehT(x)h(x)

ao g hT (x).

(9.24)

(9.25)

Strictly Passive systems: Finally consider the strict passivity supply rate w(u, y) =uT y - 5UT u (i.e., Q = 0, R = -8I, and S = 2I in (9.6)), and assume once again thatj(x) = 0. In this case Theorem 9.2 states that the nonlinear system t,b is strictly passive ifand only if

ax f (x) = -LT (x)L(x)

with

agT (ax) = h(x) - 2WTL

R=R=WTW=-8I

which can never be satisfied since WTW > 0 and 8 > 0. It then follows that no system ofthe form (9.13), with j = 0, can be strictly passive.

9.7. STABILITY OF DISSIPATIVE SYSTEMS 237

9.7 Stability of Dissipative Systems

Throughout this section we analyze the possible implications of stability (in the sense ofLyapunov) for dissipative dynamical systems. Throughout this section, we assume thatthe storage function 4) : X -> R+ is differentiable and satisfies the differential dissipationinequality (9.5).

In the following theorem we consider a dissipative system z/i with storage function 0and assume that xe is an equilibrium point for the unforced systems ?P, that is, f (xe) _f(xe,0)=0.

Theorem 9.3 Let be a dissipative dynamical system with respect to the (continuouslydifferentiable) storage function 0 : X -* IR+, which satisfies (9.5), and assume that thefollowing conditions are satisfied:

(i) xe is a strictly local minimum for 0:

4)(xe) < 4)(x) Vx in a neigborhood of xe

(ii) The supply rate w = w(u, y) is such that

w(0, y) < 0 Vy.

Under these conditions xe is a stable equilibrium point for the unforced systems x = f (x, 0).

Proof: Define the function V (x) tf 4)(x) - 4)(xe). This function is continuously differen-tiable, and by condition (i) is positive definite Vx in a neighborhood of xe. Also, the timederivative of V along the trajectories of 0 is given by

V (x) = aa(x) f (x, u)

thus, by (9.5) and condition (ii) we have that V(x) < 0 and stability follows by the Lyapunovstability theorem.

Theorem 9.3 is important not only in that implies the stability of dissipative systems(with an equilibrium point satisfying the conditions of the theorem) but also in that itsuggests the use of the storage function 0 as a means of constructing Lyapunov functions.It is also important in that it gives a clear connection between the concept of dissipativityand stability in the sense of Lyapunov. Notice also that the general class of dissipativesystems discussed in theorem 9.3 includes QSR dissipative systems as a special case. The

property of QSR dissipativity, however, was introduced as an input-output property. Thus,dissipativity provides a very important link between the input-output theory of system andstability in the sense of Lyapunov.

Corollary 9.2 Under the conditions of Theorem 9.3, if in addition no solution of i = f (x)other than x(t) = xe satisfies w(0, y) = w(0, h(x, 0) = 0, then xe is asymptotically stable.

Proof: Under the present conditions, V(x) is strictly negative and all trajectories of ?p =f (xe, 0), and satisfies V (x) = 0 if and only if x(t) = xe, and asymptotic stability followsfrom LaSalle's theorem.

Much more explicit stability theorems can be derived for QSR dissipative systems.We now present one such a theorem (theorem ), due to Hill and Moylan [30].

Definition 9.6 ([30]) A state space realization of the form V) is said to be zero-state de-tectable if for any trajectory such that u - 0, y - 0 we have that x(t) - 0.

Theorem 9.4 Let the system

i = f(x) +g(x)u1 y = h(x)

be dissipative with respect to the supply rate w(t) = yTQy + 2yT Su + uT Ru and zero statedetectable. Then, the free system i = f (x) is

(i) Lyapunov-stable if Q < 0.

(ii) Asymptotically stable if Q < 0.

Proof: From Theorem 9.2 and Corollary 9.1 we have that if 1i is QSR dissipative, thenthere exists 0 > 0 such that

dt _ -(L+Wu)T(L+Wu)+w(u,y)

and setting u = 0,

dO _ -LT L + hT (x)Qh(x)

along the trajectories of i = f (x). Thus, stability follows from the Lyapunov stabilitytheorem. Assume now that Q < 0. In this case we have that

dO = 0 = hT (x)Qh(x)

h(x) = 0x=0

9.8. FEEDBACK INTERCONNECTIONS 239

by the zero-state detectability condition.

The following corollary is then an immediate consequence of Theorem 9.6 and thespecial cases discussed in Section 9.2.

Corollary 9.3 Given a zero state detectable state space realization

J ± = f(x) + g(x)u

l y = h(x) +j(x)u

we have

(i) If 0

(ii) If 0

(iii) If 0

(iv) If 0

(v) If 0

is passive, then the unforced system i = f (x) is Lyapunov-stable.

is strictly passive, then i = f (x) is Lyapunov-stable.

is finite-gain-stable, then = f (x) is asymptotically stable.

is strictly output-passive, then i = f (x) is asymptotically stable.

is very strictly passive, then i = f (x) is asymptotically stable.

9.8 Feedback Interconnections

In this section we consider the feedback interconnections and study the implication ofdifferent forms of QSR dissipativity on closed-loop stability in the sense of Lyapunov. Wewill see that several important results can be derived rather easily, thanks to the machinerydeveloped in previous sections.

Throughout this section we assume that state space realizations VP1 and 02 each havethe form

-ii = fi(xi) +g(xi)ui, u E Rm, x c R7Wi

yi = hi(xi) +.7(xi)u'ie y E 1R'n

for i = 1, 2. Notice that, for compatibility reasons, we have assumed that both systems are"squared"; that is, they have the same number of inputs and outputs. We will also makethe following assumptions about each state space realization:

Y)1 and V2 are zero-state-detectable, that is, u - 0 and y - 0 implies that x(t) 0.

Y'1 and 02 are completely reachable, that is, for any given xl and tl there exists ato < tl and an input function E U -* R' such that the state can be driven fromx(to) = 0 to x(tl) = x1.


Figure 9.3: Feedback interconnection.

Having laid out the ground rules, we can now state the main theorem of this section.

Theorem 9.5 Consider the feedback interconnection (Figure 9.3) of the systems Y'1 and7'2, and assume that both systems are dissipative with respect to the supply rate

wt(ut, yi) = yT Q;yti + 2yT Slut + u'Rut. (9.26)

Then the feedback interconnection of 01 and 02 is Lyapunov stable (asymptotically stable)if the matrix Q defined by

Ql + aR2 -S1 + aS2-ST + aS2 R1 + aQ2 ]

is negative semidefinite (negative definite) for some a > 0.

(9.27)

Proof of Theorem 9.5: Consider the Lyapunov function candidate

O(x1, X2) = O(x1) + aO(x2) a > 0

where O(x1) and O(x2) are the storage function of the systems a/il and z/b2i respectively. Thus,O(x1i X2) is positive definite by construction. The derivative of along the trajectoriesof the composite state [x1, x2]T is given by

_w1(u1,yl)+aw2(u2,y2)(y1 Qlyl + 2yTSlu1 + u1 R1ul) + a(yz Q2y2 + 2y2 S2u2 + u2 R2u2) (9.28)

9.8. FEEDBACK INTERCONNECTIONS 241

Substituting ul = -Y2 and u2 = yl into (9.28), we obtain

r T T 1 f Qi + aR2 -S1 + aS2 1 r yl 1= L yi y2 J L -S + aS2 Rl + aQ2 J IL

YT

J


This theorem is very powerful and includes several cases of special interest, which wenow spell out in Corollaries 9.4 and 9.5.

Corollary 9.4 Under the conditions of Theorem 9.5, we have that

(a) If both 01 and 02 are passive, then the feedback system is Lyapunov stable.

(b) Asymptotic stability follows if, in addition, one of the following conditions are satisfied:

(1) One of 01 and 02 is very strictly passive.

(2) Both 01 and are strictly passive.

(3) Both 01 and 02 are strictly output passive.

Proof: Setting a = 1 in (9.27), we obtain the following:

(a) If both 01 and 02 are passive, then Qi = 0, Ri = 0, and Si = 2I, for i = 1, 2. Withthese values, Q = 0, and the result follows.

(bl) Assuming that ¢1 is passive and that 02 is very strictly passive, we have

01 passive: Q1 = 0, R1 = 0, and S1 = I.

02 very strictly passive: Q2 = -e21, R2 = -621, and S2 = I.

Thus621 0

Q0 -e2I

which is negative definite, and the result follows by Theorem 9.5.and 01 very strictly passive is entirely analogous.

The case 02 passive

(b2) If both 01 and 02 are strictly passive, we have

Qi=0,R2=-6iI,andSi=2I.

Thusb2I 0

0 -b11

which is once again negative definite, and the result follows by Theorem 9.5.

(b3) If both 01 and 02 are strictly output-passive, we have:

Qt=eiI,R2=0, and Si =21j.

ThusEll 0

0 -e21

which is negative definite, and the result follows.

Corollary 9.4 is significant in that it identifies closed-loop stability-asymptotic stabilityin the sense of Lyapunov for combinations of passivity conditions.

The following corollary is a Lyapunov version of the small gain theorem.

Corollary 9.5 Under the conditions of Theorem 9.5, if both 01 and 02 are finite-gain-stable with gains 'y and 72i the feedback system is Lyapunov-stable (asymptotically stable)if 7172<-1 (7172<1).

Proof: Under the assumptions of the corollary we have that

Qi=I,R1= 2, andSi=0.

Thus Q becomes0 1a72

-1)1QL 0 (7i - a)I J

Thus, Q is negative semidefinite, provided that

(9.29)

(9.30)

and substituting (9.29) into (9.30) leads to the stability result. The case of asymptoticstability is, of course, identical.

0

9.9. NONLINEAR G2 GAIN

9.9 Nonlinear G2 Gain

Consider a system V) of the form

i = f(x) + g(x)u

y = h(x)

243

(9.31)

As discussed in Section 9.2, this system is finite-gain-stable with gain y, if it is dissipativewith supply rate w = 2(y2IIu112 - IIy112) Assuming that the storage function correspondingto this supply rate is differentiable, the differential dissipation inequality (9.5) implies that

fi(t) =aa( )x

< w(t) s 272IIu112 - 2IIyl12 < 0.1

Thus, substituting (9.31) into (9.32) we have that

(9.32)

a0(x) f(x) + ao x)g(x)u < w(t) <- 1y2IIuII2 - 1 IIy112. (9.33)Ox ax 2 2

xloAdding and subtracting 2y2uTu and 21 ggT( )T to the the left hand side of (9.33) weobtain

a_(x)f (x) +

ao(x) g(x)u =ax ax

a6x f(x) + a_(x)g(x)v, + 12

y2'U,T76 - 12

y2uTuax ax

12 aoT (ao)T

12 aOggT

(aO)T+ 2'Y axgg ax 2y ax ax J

1y2 u- 1 gT(a")T +aof(x)2 2y2 ax ax

+2- axggT (ax) + 2y21lu112

Therefore, substituting (9.34) into (9.33) results in

1 2

-27 u- 2729 T(,g,, ) T

+.90

Px) +1 .90

99T

(,,,, )T<- -2 11y112

or, equivalently

(9.34)

T T 2T + 1 IIyll2 < 1'Y2 v - 1 T (ao < 0.f(x) + aax

(LO)99 ax 2 2 ax -ax

(9.35)2 - 2

This result is important. According to (9.35), if the system V) given by (9.31) is finite-gain-stable with gain ry, then it must satisfy the so-called Hamilton-Jacobi inequality:

H def1

a-f(x) + 2,y2 a--gg

(ax )T+

2 IyI 2< 0 (9.36)

Finding a function that maximizes f and satisfies inequality (9.36) is, at best, verydifficult. Often, we will be content with "estimating" an upper bound for y. This can bedone by "guessing" a function 0 and then finding an approximate value for -y; a process thatresembles that of finding a suitable Lyapunov function. The true L2 gain of 0, denoted 'y',is bounded above by -y, i.e.,

0<y'<7


xix2 - 2xlx2 - xi +Qxlu , 3 > 0- xi - x2 + Qx2u

X2

f (x) = I 32- x2 1x2 - xl 9(x) =L

13x2J

, h(x) = (xi + x2)

To estimate the .C2 norm, we consider the storage function "candidate" fi(x) = 2 (xi + x2).With this function, the three terms in the Hamilton-Jacobi inequality (9.36) can be obtainedas follows:

(i) f(x)

T(ii)

2 ggT

1X2 2x 3

a f (X) _ [x1, x21 x l - x2 1x2 -xl

= -(xl + x2)2

= -11x114

ao . . , )3xl 2 2 2T1, x21 [

Qx2I = P 1 2 = PIIXU

1I T q2

? L99T(LO)

ate= 2ry2 IIxI14

9.9. NONLINEAR L2 GAIN 245

(iii)2

IHy1I2

2IIy1I2 = 2(x2 +x2)2 =2

IIxII4

Combining these results, we have that

= -IIxII4 + IIXI14 + 1 IIXI14 < 0

or

9.9.1 Linear Time-Invariant Systems

It is enlightening to consider the case of linear time-invariant systems as a special case.Assuming that this is the case, we have that

( i=Ax+BuSl y=Cx

that is, f (x) = Ax, g(x) = B, and h(x) = Cx. Taking 4(x) = a xT Px (P > p, p = PT )

and substituting in (9.36), we obtain

7{ = xT + 212 PBBT P + 2CTCl x < 0. (9.37)7 J

Taking the transpose of (9.37), we obtain

7IT = xT 1ATp + PBBT P + 2 CT CJ7

Thus, adding (9.38) and (9.38), we obtain

71 + NT = xT

x < 0. (9.38)

PA+ATP+ 212PBBTP+ 2CTC x < 0. (9.39)

R

Thus, the system z/i has a finite G2 gain less than or equal to -y provided that for some y > 0

< 0.PA+ATP+ _ PBBTP+ ZCTC _-2-y2

The left-hand side of inequality (9.40) is well known and has played and will continue toplay a very significant role in linear control theory. Indeed, the equality

R'fPA+ATP+ 212PBBTP+ ZCTC = 0 (9.41)y

is known as the Riccati equation. In the linear case, further analysis leads to a strongerresult. It can be shown that the linear time-invariant system V) has a finite gain less thanor equal to -y if and only if the Riccati equation (9.41) has a positive definite solution. SeeReference [23] or [100].

9.9.2 Strictly Output Passive Systems

It was pointed out in Section 9.3 (lemma 9.2) that strictly output passive systems have afinite G2 gain. We now discuss how to compute the L2 gain y. Consider a system of theform

f x = f (x) + 9(x)uy = h(x)

and assume that there exists a differentiable function 01 > 0 satisfying (9.24)-(9.25), i.e.,

00T

8x f (x) < -eh (x)h(x) (9.42)

hT (x) (9.43)

which implies that tli is strictly output-passive. To estimate the L2 gain of Eli, we considerthe Hamilton-Jabobi inequality (9.36)

H'f axf (x) + 2y2 ax99T (ax)T + 2

111y1J2 < 0 (9.44)

Letting 0 = ko1 for some k > 0 and substituting (9.42)-(9.43) into (9.44), we have that Vhas L2 gain less than or equal to y if and only if

-kehT (x)h(x) + 2y2 k 2 h T (x)h(x) +2

hT (x)h(x) < 0

9.10. SOME REMARKS ABOUT CONTROL DESIGN

or, extracting the common factor hT(x)h(x)

-ke+212k2+2 <0

7>k

2_k - 1

and choosing k = 1/c we conclude that

9.10 Some Remarks about Control Design

247

Control systems design is a difficult subject in which designers are exposed to a multiplicityof specifications and design constraints. One of the main reasons for the use of feedback,however, is its ability to reduce the effect of undesirable exogenous signals over the system'soutput. Therefore, over the years a lot of research has been focused on developing designtechniques that employ this principle as the main design criteria. More explicitly, controldesign can be viewed as solving the following problem: find a control law that

1- Stabilizes the closed-loop system, in some prescribed sense.

2- Reduces the effect of the exogenous input signals over the desired output, also in somespecific sense.

Many problems can be cast in this form. To discuss control design in some generality, itis customary to replace the actual plant in the feedback configuration with a more generalsystem known as the generalized plant. The standard setup is shown in Figure 9.4, where

C= controller to be designed,

G= generalized plant,

y= measured output,

z= output to be regulated (such as tracking errors and actuator signals),

u= control signal, and

d= "exogenous" signals (such as disturbances and sensor noise).


d

G

u I I y

C

Figure 9.4: Standard setup.

z

To fix ideas and see that this standard setup includes the more "classical " feedbackconfiguration, consider the following example:

Example 9.2 Consider the feedback system shown in Figure 9.5.

Suppose that our interest is to design a controller C to reduce the effect of the ex-ogenous disturbance d over the signals Y2 and e2 (representing the input and output of theplant). This problem can indeed be studied using the standard setup of Figure 9.4. To seethis, we must identify the inputs and outputs in the generalized plant G. In our case, theseveral variables in the standard setup of Figure 9.4 correspond to the following variablesin Figure 9.5:

y = el (the controller input)

u = yl (the controller output)

d: disturbance (same signal in both Figure 9.5 and the generalized setup of Figure9.4).

Z=L

e2 J, i.e., z is a vector whose components are input and output of the plant.Y2

9.10. SOME REMARKS ABOUT CONTROL DESIGN 249

Figure 9.5: Feedback Interconnection used in example 9.2.

A bit of work shows that the feedback system of Figure 9.5 can be redrawn as shown in Figure9.6, which has the form of the standard setup of Figure 9.4. The control design problem cannow be restated as follows. Find C such that

(i) stabilizes the feedback system of Figure 9.6, in some specific sense, and

(ii) reduces the effect of the exogenous signal d on the desired output z. D

A lot of research has been conducted since the early 1980s on the synthesis of controllers thatprovide an "optimal" solution to problems such as the one just described. The propertiesof the resulting controllers depend in an essential manner on the spaces of functions usedas inputs and outputs. Two very important cases are the following:

L2 signals:

In this case the problem is to find a stabilizing controller that minimizes the L2 gainof the (closed-loop) system mapping d -+ z. For linear time-invariant systems, we sawthat the L2 gain is the H,,. norm of the transfer function mapping d -4 z, and thetheory behind the synthesis of these controllers is known as H, optimization. TheH. optimization theory was initiated in 1981 by G. Zames [99], and the synthesis ofH... controllers was solved during the 1980s, with several important contributions byseveral authors. See References [25] and [100] for a comprehensive treatment of theH. theory for linear time-invariant systems.


e2

Y2

UI I I-y2=el I y

z

Figure 9.6: Standard setup.

9.11. NONLINEAR L2-GAIN CONTROL 251

L, signals:

Similarly, in this case the problem is to find a stabilizing controller that minimizes theL, gain of the (closed-loop) system mapping d -> z. For linear time-invariant systems,we saw that the L. gain is the Ll norm of the transfer function mapping d -- z, andthe theory behind the synthesis of these controllers is known as Ll optimization. TheL optimization theory was proposed in 1986 by M. Vidyasagar [89]. The first solutionof the Li-synthesis problem was obtained in a series of papers by M. A. Dahleh andB. Pearson [16]-[19]. See also the survey [20] for a comprehensive treatment of the Lltheory for linear time-invariant systems.

All of these references deal exclusively with linear time-invariant systems. In theremainder of this chapter we present an introductory treatment of the .C2 control problemfor nonlinear systems.

We will consider the standard configuration of Figure 6.4 and consider a nonlinearsystem of the form

i = f (x, u, d)V) y = 9(x, u, d) (9.45)

z = h(x,u,d)

where u and d represent the control and exogenous input, respectively, and d and y representthe measured and regulated output, respectively. We look for a controller C that stabilizesthe closed-loop system and minimizes the ,C2 gain of the mapping from d to z. This problemcan be seen as an extension of the H. optimization theory to the case of nonlinear plants,and therefore it is often referred to as the nonlinear Ham-optimization problem.

9.11 Nonlinear G2-Gain Control

Solving the nonlinear G2-gain design problem as described above is very difficult. Instead,we shall content ourselves with solving the following suboptimal control problem. Given a"desirable" exogenous signal attenuation level, denoted by ryl, find a control Cl such thatthe mapping from d -* w has an .C2 gain less than or equal to ryl. If such a controller Clexists, then we can choose rye < yl and find a new controller C2 such that the mappingfrom d -> w has an G2 gain less than or equal to 'y2. Iterating this procedure can lead to acontroller C that approaches the "optimal" (i.e., minimum) -y.

We will consider the state feedback suboptimal L2-gain optimization problem. Thisproblem is sometimes referred to as the full information case, because the full state isassumed to be available for feedback. We will consider a state space realization of the form

x = a(x) + b(x)u + g(x)d, u E II8"`, d E R''

z = [h(x)]

where a, b, g and h are assumed to be Ck, with k > 2.

(9.46)

Theorem 9.6 The closed-loop system of equation (9.46) has a finite C2 gain < -y if andonly if the Hamilton-Jacobi inequality 9{ given by

ll

H8xa(x)

+ 2 8 [.g(x)gT(x) - b(x)bT (x)(Ocb)T

J+ hT (x)h(x) < 0 (9.47)x_ i

has a solution 0 > 0. The control law is given byT

u= -bT (x) (a,) (x). (9.48)

Proof: We only prove sufficiency. See Reference [85] for the necessity part of the proof.

Assume that > 0 is a solution of L. Substituting u into the system equations (9.46),we obtain

a(x) - b(x)bT(x)()T

+g(x)d (9.49)

z =L -bT(x)

(9.50)

Substituting (9.49) and (9.50) into the Hamilton-Jacobi inequality (9.36) with

f (x) = a(x) - b(x)bT (x)()T

(x)

h(x) [_bT(X) ( )T ]

results inT T

H = ax(a(x)_b(x)bT(x)_) ) + 2ry2 axg(x)gT (x) (8

hx+2

[h(X)T8xb(x), [bT(X) (g)T ]

which implies (9.47), and so the closed-loop system has a finite G2 gain ry. 0

9.12. EXERCISES 253

9.12 Exercises



(9.3) Consider the pendulum-on-a-cart system discussed in Chapter 1. Assuming for sim-plicity that the moment of inertia of the pendulum is negligible, and assuming thatthe output equation is

y=±=X2

(a) Find the kinetic energy K1 of the cart.(b) Find the kinetic energy K2 of the pendulum.

(c) Find the potential energy P in the cart-pendulum system.

(d) Using the previous results, find the total energy E = K1 + K2 + P = K + Pstored in the cart-pendulum system.

(e) Defining variables

_ x M+m mlcoso fq= 9 M= L ml cos ml2B '

u _0

show that the energy stored in the pendulum can be expressed as

1E = 24TM(q)4 +mgl(cos0 - 1)

(f) Computing the derivative of E along the trajectories of the system, show that thependulum-on-a-cart system with input u = f and output ± is a passive system.

(9.4) Consider the system

-a21-22+u, a>0X1-X X2

= xl

(a) Determine whether i is(i) passive.

(ii) strictly passive.(iii) strictly output-passive.(iv) has a finite G2 gain.

(b) If the answer to part (a)(iv) is affirmative, find an upper bound on the L2 gain.


(9.5) Consider the system

-x1+x2+u, a>0-x1x2 - X2)3x1 Q>0.

(a) Determine whether i is

(i) passive.(ii) strictly passive.(iii) strictly output-passive.(iv) has a finite £2 gain.

(b) If the answer to part (a)(iv) is affirmative, find an upper bound on the G2 gain.

Notes and ReferencesThe theory of dissipativity of systems was initiated by Willems is his seminal paper [91],on which Section 9.1 is based. The beauty of the Willems formulation is its generality.Reference [91] considers a very general class of nonlinear systems and defines dissipativityas an extension of the concept of passivity as well as supply rates and storage functions asgeneralizations of input power and stored energy, respectively. Unlike the classical notionof passivity, which was introduced as an input-output concept, state space realizations arecentral to the notion of dissipativity in Reference [91].

The definition of QSR dissipativity is due to Hill and Moylan [30], [29]. Employingthe QSR supply rate, dissipativity can be interpreted as an input-output property. Section9.3 is based on Hill and Moylan's original work [30]. Theorem 9.1 is from Willems [91],Theorem 1. The connections between dissipativity and stability in the sense of Lyapunovwere extensively explored by Willems, [93], [94], as well as [91]. Theorem 9.3 follows Theo-rem 6 in [91]. See also Section 3.2 in van der Schaft [85]. Section 9.6 follows Reference [30].Stability of feedback interconnections was studies in detail by Hill and Moylan. Section 9.8follows Reference [31] very closely. Nonlinear £2 gain control (or nonlinear H,,. control) iscurrently a very active area of research. Sections 9.9 and 9.11 follow van der Schaft [86],and [87]. See also References [6] and [38] as well as the excellent monograph [85] for acomprehensive treatment of this beautiful subject.

Chapter 10

Feedback Linearization

In this chapter we look at a class of control design problems broadly described as feedbacklinearization. The main problem to be studied is: Given a nonlinear dynamical system,find a transformation that renders a new dynamical system that is linear time-invariant.Here, by transformation we mean a control law plus possibly a change of variables. Oncea linear system is obtained, a secondary control law can be designed to ensure that theoverall closed-loop system performs according to the specifications. This time, however,the design is carried out using the new linear model and any of the well-established linearcontrol design techniques.

Feedback linearization was a topic of much research during the 1970s and 1980s.Although many successful applications have been reported over the years, feedback lin-earization has a number of limitations that hinder its use, as we shall see. Even with theseshortcomings, feedback linearization is a concept of paramount importance in nonlinearcontrol theory. The intention of this chapter is to provide a brief introduction to the sub-ject. For simplicity, we will limit our attention to single-input-single-output systems andconsider only local results. See the references listed at the end of this chapter for a morecomplete coverage.

10.1 Mathematical Tools

Before proceeding, we need to review a few basic concepts from differential geometry.Throughout this chapter, whenever we write D C R", we assume that D is an open andconnected subset of R'.

We have already encountered the notion of vector field in Chapter 1. A vector field

255

256 CHAPTER 10. FEEDBACK LINEARIZATION

is a function f : D C R" -> 1R1, namely, a mapping that assigns an n-dimensional vec-tor to every point in the n-dimensional space. Defined in this way, a vector field is ann-dimensional "column." It is customary to label covector field to the transpose of a vectorfield. Throughout this chapter we will assume that all functions encountered are sufficientlysmooth, that is, they have continuous partial derivatives of any required order.

10.1.1 Lie Derivative

When dealing with stability in the sense of Lyapunov we made frequent use of the notionof "time derivative of a scalar function V along the trajectories of a system d = f (x)." Aswe well know, given V : D -+ 1R and f (x), we have

xf (x) = VV f (x) = LfV(x).

A slightly more abstract definition leads to the concept of Lie derivative.

Definition 10.1 Consider a scalar function h : D C 1R" -* JR and a vector field f : D C]R" -4 1R" The Lie derivative of h with respect to f, denoted Lfh, is given by

Lfh(x) = ax f (x). (10.1)

Thus, going back to Lyapunov functions, V is merely the Lie derivative of V with respectto f (x). The Lie derivative notation is usually preferred whenever higher order derivativesneed to be calculated. Notice that, given two vector fields f, g : D C ]R" -* 1R" we have that

Lfh(x) = axf(x), Lfh(x) = aX-9(x)

and

L9Lfh(x) = L9[Lfh(x)] = 2(8xh)g(x)

and in the special case f = g,

_ 8(Lfh)

Lf Lf h(x)= L2fh(x)

8x f (x)

Example 10.1 Let1

h(x) = 2 (xi + x2)_ _ - 2

XIX22A X ) 9(x) _ -x2 + x Jx2 , .

Then, we have

10.1. MATHEMATICAL TOOLS 257

Lfh(x):

Lfh(x) = axf(x)

= [X1 x2]

Lfh(x):

LfL9h(x):

a-g(x)

x2

[ ]xl - x2 2xl x2x2 + xlx2

-(xl + x2).I

LfL9h(x) = a(Oxh)f(x)

x2-2 [xl xz] I -x1 - µ(1 - xi)xz= 2µ(1 -

xl)x22.

Lfh(x):

Lfh(x)

10.1.2 Lie Bracket

-x2-xl - µ(1 - x2Dx2

µ(1 - xl)x2.

a(8xh) g(x)-xl - xlx2

2-2 [xl x21x2 + x2 Ix2

2(x2 + x2).

Definition 10.2 Consider the vector fields f, g : D C 1R" -> R. The Lie bracket of f andg, denoted by [f,g], is the vector field defined by

,9g afxf (x) axg(x). (10.2)[f,g](x) = 8


Example 10.2 Letting

f(x) =L

we have

-x2 x-XI - µ(1 - x2 j)x2

, g( ) _

[f,g](x) = 8xf(x) 8xg(x)

[1 °1[-x2

0 1 -x1 - all - xl)x2

L O2pxix2

x1

X2 1

0 -1-µ(1 - xi)

xl2µxlx2 X2 1

The following notation, frequently used in the literature, is useful when computing repeatedbracketing:

[f, g](x)1 f

ad fg(x)

and

Thus,

adfg =

adfg =g

[f, ad:f 1g]

adfg = [f, adfg] _ [f, [f, g]]ad3 2= [f, adf g] 9]]]

The following lemma outlines several useful properties of Lie brackets.

Lemma 10.1 : Given f1,2 : D C R -+ 1R , we have

(i) Bilinearity:

(a) [a1 f1 + a2f2, g] = al [fl, g] + a2 [f2, g]

(b) [f, algl + a2g2] = a1 [f, gl] + a2 [f, g2]

(ii) Skew commutativity:

[f,g] = -[g,f](iii) Jacobi identity: Given vector fields f and g and a real-valued function h, we obtain

L[f,gl h = L fLgh(x) - L9Lfh.

where L[f,gj h represents the Lie derivative of h with respect to the vector [f, g].


10.1.3 Diffeomorphism

Definition 10.3 : (Diffeomorphism) A function f : D C Rn -4 Rn is said to be a diffeo-morphism on D, or a local diffeomorphism, if

(i) it is continuously differentiable on D, and

(ii) its inverse f -1, defined by

1-1(f(x)) = x dxED

exists and is continuously differentiable.

The function f is said to be a global diffeomorphism if in addition

(i) D=R , and

(ii) limx', 11f (x) 11 = 00-

The following lemma is very useful when checking whether a function f (x) is a localdiffeomorphism.

Lemma 10.2 Let f (x) : D E lW -+ 1R" be continuously differentiable on D. If the Jacobianmatrix D f = V f is nonsingular at a point xo E D, then f (x) is a diffeomorphism in asubset w C D.

Proof: An immediate consequence of the inverse function theorem.

10.1.4 Coordinate Transformations

For a variety of reasons it is often important to perform a coordinate change to transform astate space realization into another. For example, for a linear time-invariant system of theform

±=Ax+Buwe can define a coordinate change z = Tx, where T is a nonsingular constant matrix. Thus,in the new variable z the state space realization takes the form

z=TAT-1z+TBu = Az+Bu.


This transformation was made possible by the nonsingularity of T. Also because of the ex-istence of T-1, we can always recover the original state space realization. Given a nonlinearstate space realization of an affine system of the form

x = f (x) + g(x)u

a diffeomorphism T(x) can be used to perform a coordinate transformation. Indeed, as-suming that T(x) is a diffeomorphism and defining z = T(x), we have that

z = a x = [f (x) + g(x)u]

Given that T is a diffeomorphism, we have that 3T-1 from which we can recover the originalstate space realization, that is, knowing z

x = T-1(z).

Example 10.3 Consider the following state space realization

0

x1

X2 - 2x1x2 + X1

and consider the coordinate transformation:

x1

x = T(x) = xl +X2

x22+X3

1 0 0OIT

(x) = 2x1 1 0

0 2x2 1

Therefore we have,

U

1 0 0 0 1 0 0 1

z = 2x1 1 0 x1 + 2x1 1 0 - 2x1

0 2x2 1 x2 - 2x1x2 + xi 0 2x2 1 4x1x2

U

and0 1

z = x1 + 0 uX2 +X2 0


and substituting xi = z1i x2 = z2 - zi, we obtain

0 1

i = z1 + 0 u.z2 0

10.1.5 Distributions

Throughout the book we have made constant use of the concept of vector space. Thebackbone of linear algebra is the notion of linear independence in a vector space. We recall,from Chapter 2, that a finite set of vectors S = {xi, x2i , xp} in R' is said to be linearlydependent if there exist a corresponding set of real numbers {) }, not all zero, such that

E Aixi = Aixi + A2x2 + - + APxP = 0.

On the other hand, if >i Aixi = 0 implies that Ai = 0 for each i, then the set {xi} is saidto be linearly independent.

Given a set of vectors S = {x1, X2, , xp} in R', a linear combination of those vectordefines a new vector x E R", that is, given real numbers A1, A2, Ap,

x=Aixl+A2x2+...+APxP Ellen

the set of all linear combinations of vectors in S generates a subspace M of Rn known asthe span of S and denoted by span{S} = span{xi, x2i )XP} I i.e.,

span{x1 i x2, , xp} _ {x E IRn' : x = A1x1 + A2x2 + . + APxp, Ai E R1.

The concept of distribution is somewhat related to this concept.

Now consider a differentiable function f : D C Rn -> Rn. As we well know, thisfunction can be interpreted as a vector field that assigns the n-dimensional vector f (x) toeach point x E D. Now consider "p" vector fields fi, f2,. , fp on D C llPn. At any fixedpoint x E D the functions fi generate vectors fi(x), f2 (x), , fp(x) E and thus

O(x) = span{ fi(x), ... f,(x)}

is a subspace of W'. We can now state the following definition.


Definition 10.4 (Distribution) Given an open set D C 1R' and smooth functions fl, f2i , fpD -> R", we will refer to as a smooth distribution A to the process of assigning the subspace

A = span{f1, f2,. - -fp}

spanned by the values of x c D.

We will denote by A(x) the "values" of A at the point x. The dimension of the distributionA(x) at a point x E D is the dimension of the subspace A(x). It then follows that

dim (A(x)) = rank ([fi (x), f2(x), ... , fp(x)])

i.e., the dimension of A(x) is the rank of the matrix [fi(x), f2(x), , fp(x)].

Definition 10.5 A distribution A defined on D C Rn is said to be nonsingular if thereexists an integer d such that

dim(O(x)) = d Vx E D.

If this condition is not satisfied, then A is said to be of variable dimension.

Definition 10.6 A point xo of D is said to be a regular point of the distribution A if thereexist a neighborhood Do of xo with the property that A is nonsingular on Do. Each pointof D that is not a regular point is said to be a singularity point.

Example 10.4 Let D = {x E ii 2 : xl + X2 # 0} and consider the distribution Aspan{ fl, f2}, where

fl=[O], f2=Lx1 +x2]We have

dim(O(x)) = rank 1 1

0 xl+x2 ])Then A has dimension 2 everywhere in R2, except along the line xl + x2 = 0. It followsthat A is nonsingular on D and that every point of D is a regular point.

Example 10.5 Consider the same distribution used in the previous example, but this timewith D = R2. From our analysis in the previous example, we have that A is not regularsince dim(A(x)) is not constant over D. Every point on the line xl + X2 = 0 is a singularpoint.

Definition 10.7 (Involutive Distribution) A distribution A is said to be involutive if gi E iand 92 E i = [91,92] E A.

It then follows that A = span{ fl, f2,. , fp} is involutive if and only if

rank ([fi(x), ... , fn(x)]) --rank ([fi(x),..., fn(x), [f i, f,]]) , Vx and all i,j

Example 10.6 Let D = R3 and A = span{ fi, f2} where

x21 1

Then it can be verified that dim(O(x)) = 2 Vx E D. We also have that

[f1, f21 = a2 f1 - afl f2 =0

1

0

Therefore i is involutive if and only if

1 0 1 0 0

rank 0 xi = rank 0 xi 1

x2 1 x2 1 0

This, however is not the case, since

rank0 1 0 0

xi = 2 and rank 0 xi 1 = 31 1) ( 1 1 0

and hence A is not involutive.

Definition 10.8 (Complete Integrability) A linearly independent set of vector fields fi, , fpon D C II2n is said to be completely integrable if for each xo E D there exists a neighbor-hood N of xo and n-p real-valued smooth functions hi(x), h2(x), , hn_p(x) satisfying thepartial differentiable equation

8hj8xf=(x)=0

1<i<p, 1<j<n-p

and the gradients Ohi are linearly independent.


The following result, known as the Frobenius theorem, will be very important in later sec-tions.

Theorem 10.1 (Frobenius Theorem) Let fl, f2i , fp be a set of linearly independent vec-tor fields. The set is completely integrable if and only if it is involutive.

Proof: the proof is omitted. See Reference [36].

Example 10.7 [36] Consider the set of partial differential equations

Oh Oh2x3 = 0

-C7x1 8x2Oh 8h i9h-x1-- -2x2a2 +x3ax

3

which can be written as

or

with

= 0

2x3 -X8h 8h oh -1 -2X2 I = 0

0 x3

Vh[f1 f2]

2X3 -xlfl = -1 , f2 = -2x2

0 x3

To determine whether the set of partial differential equations is solvable or, equivalently,whether [fl, f2] is completely integrable, we consider the distribution A defined as follows:

2x3 -xl= span -1 , -2X2

0 x3

It can be checked that A has dimension 2 everywhere on the set D defined by D = {x E IIF3xi + x2 # 0}. Computing the Lie bracket [fl, f2], we obtain

-4x3(fl,fz] = 2

0

10.2. INPUT-STATE LINEARIZATION 265

and thus223 -21 -4X3

fl f2 [fl, f2] 1 = -1 -2X2 2

0 X3 0

which has rank 2 for all x E R3. It follows that the distribution is involutive, and thus it iscompletely integrable on D, by the Frobenius theorem.

10.2 Input-State Linearization

Throughout this section we consider a dynamical systems of the form

± = f(2) + g(x)u

and investigate the possibility of using a state feedback control law plus a coordinatetransformation to transform this system into one that is linear time-invariant. We willsee that not every system can be transformed by this technique. To grasp the idea, we startour presentation with a very special class of systems for which an input-state linearizationlaw is straightforward to find. For simplicity, we will restrict our presentation to single-inputsystems.

10.2.1 Systems of the Form t = Ax + Bw(2) [u - cb(2)]

First consider a nonlinear system of the form

2 = Ax + Bw(2) [u - q5(2)] (10.3)

where A E Ilt"', B E II2ii1, 0 : D C II2" -4 )LP1, w : D C 1(P" -4 R. We also assume thatw # 0 d2 E D, and that the pair (A, B) is controllable. Under these conditions, it isstraightforward to see that the control law

u = O(2) + w-1 (2)v (10.4)

renders the systemi=A2+Bv

which is linear time-invariant and controllable.

The beauty of this approach is that it splits the feedback effort into two componentsthat have very different purpose.

1- The feedback law (10.4) was obtained with the sole purpose of linearizing the originalstate equation. The resulting linear time-invariant system may or may not have"desirable" properties. Indeed, the resulting system may or may not be stable, andmay or may not behave as required by the particular design.

2- Once a linear system is obtained, a secondary control law can be applied to stabilizethe resulting system, or to impose any desirable performance. This secondary law,however, is designed using the resulting linear time-invariant system, thus takingadvantage of the very powerful and much simpler techniques available for controldesign of linear systems.

Example 10.8 First consider the nonlinear mass-spring system of example 1.2

I xl = x2

(l x2 = mxl - ma2xi mx2 +

which can be written in the form

Clearly, this system is of the form (10.3) with w = 1 and O(x) = ka2xi. It then follows thatthe linearizing control law is

u = ka2xi + v

Example 10.9 Now consider the system

1 it = x2

ll x2 = -axl + bx2 + cOS xl (u - x2)

Once again, this system is of the form (10.3) with w = cos xl and O(x) = x2. Substitutinginto (10.4), we obtain the linearizing control law:

u=x2+cos-l X1 V

which is well defined for - z < xl < z . El


10.2.2 Systems of the Form 2 = f (x) + g(x)u

Now consider the more general case of affine systems of the form

x = f(x) + g(x)u. (10.5)

Because the system (10.5) does not have the simple form (10.4), there is no obvious way toconstruct the input-state linearization law. Moreover, it is not clear in this case whethersuch a linearizing control law actually exists. We will pursue the input-state linearization ofthese systems as an extension of the previous case. Before proceeding to do so, we formallyintroduce the notion of input-state linearization.

Definition 10.9 A nonlinear system of the form (10.5) is said to be input-state linearizableif there exist a diffeomorphism T : D C r -> IR defining the coordinate transformation

z = T(x) (10.6)

and a control law of the formu = t(x) + w-1(x)v

that transform (10.5) into a state space realization of the form

i = Az + By.

(10.7)

We now look at this idea in more detail. Assuming that, after the coordinate transformation(10.6), the system (10.5) takes the form

i = Az + Buv(z) [u - (z)]= Az + Bw(x) [u - O(x)] (10.8)

where w(z) = w(T-1(z)) and (z) = O(T-1(z)). We have:

z- axx- ax[f(x)+g(x)u]. (10.9)

Substituting (10.6) and (10.9) into (10.8), we have that

[f (x) + g(x)u] = AT(x) + Bw(x)[u - O(x)] (10.10)

must hold Vx and u of interest. Equation (10.10) is satisfied if and only if

f (x) = AT(x) - Bw(x)O(x) (10.11)

ag(x) = Bw(x). (10.12)


From here we conclude that any coordinate transformation T(x) that satisfies thesystem of differential equations (10.11)(10.12) for some 0, w, A and B transforms, via thecoordinate transformation z = T(x), the system

z = f(x) + g(x)u (10.13)

into one of the formi = Az + Bw(z) [u - O(z)] (10.14)

Moreover, any coordinate transformation z = T(x) that transforms (10.13) into (10.14)must satisfy the system of equations (10.11)-(10.12).

Remarks: The procedure just described allows for a considerable amount of freedom whenselecting the coordinate transformation. Consider the case of single-input systems, andrecall that our objective is to obtain a system of the form

i = Az + Bw(x) [u - ¢(x)]

The A and B matrices in this state space realization are, however, non-unique and thereforeso is the diffeomorphism T. Assuming that the matrices (A, B) form a controllable pair, wecan assume, without loss of generality, that (A, B) are in the following so-called controllableform:

0 1 0

0 0 1

Letting

Ar _

0 1

0 0 0 0 nxn

, B, _

1

Ti(x)

T(x)T2(x)

Tn(x)nx1

with A = Ac, B = B,, and z = T(x), the right-hand side of equations (10.11)-(10.12)becomes

A,T(x) - Bcw(x)4(x) _

0

0

nxl

(10.15)


and

(10.16)

0

L w(x)

Substituting (10.15) and (10.16) into (10.11) and (10.12), respectively, we have that

a lf(x) = T2(x)

a f(x) = T3(x)

8x 1 f (x) = T. (x)

' n f(x) = -O(x)W(x)

and

a 119(x) = 02(x) = 0

(10.17)

(10.18)

19(x) = 0ax

a n9(x) = W(x) # 0.

Thus the components T1, T2, , T, of the coordinate transformation T must be such that

(i)

29(x) = 0 Vi = 1,2,. n - 1.x

n9(x) 71 0.

0

0

(ii)

2f (x) =T,+1 i = 1,2,...,n- 1.


(iii) The functions 0 and w are given by

W(x) = n9(x)(an/ax)f(x)

ax (0'Tn1ax)g(x) '

10.3 Examples


= L exax2 1J

+[01

Ju = f (x) + g(x)u.

To find a feedback linearization law, we seek a transformation T = [T1iT2]T such that

1g = 0 (10.19)

29 # 0 (10.20)

with

alf(x) =T2. (10.21)

In our case, (10.19) implies that

OT1 _ aTi 0Ti 0 _ 0T1 _ 0a7X

g - Fa7X1 axe I 1 axe

so that T1 = T1(x1) (independent of 7X2). Taking account of (10.21), we have that

axf(x)=T2

5T1 aTi aT1 ex' - 1 _[57X1 57X2

f(7X) -57X1

0axi T2

=> T2 = (e22 - 1).1

To check that (10.20) is satisfied we notice that

8772=

oT2 0T2 OT2 = a aT1 xa7X g(7X) 57X1 axe 9(7X)

=57X2 57X2 57X1 (e

- 1) # 0

10.3. EXAMPLES 271

provided that

which results in

0. Thus we can choose

T1(x) = xi

T=[e-x21 1J

Notice that this coordinate transformation maps the equilibrium point at the origin in the xplane into the origin in the z plane.

The functions 0 and w can be obtained as follows:

w = 29(x) = ex2

(aT2/ax)f(x) _(0T2/8x)9(x)

It is easy to verify that, in the z-coordinates

f z1 = x1ll z2=eX2-1

which is of the form

with

{zl = z2z2 = az1 z2 + azl + (z2 + 1)u

z = Az + Bw(z) [u - O(z)]

A A,- [ 0 O I, B=B, = 1 0J

w(z) = z2 + 1, O(z) _ -azi.

11

Example 10.11 (Magnetic Suspension System)

Consider the magnetic suspension system of Chapter 1. The dynamic equations are

x2 0xl

kaµa2

x2 = 9 mX2 2m 1+µa1 +(

0

xg 1+µa1 (-Rx + l µ x )a 3 1+µa1) 2x3 A


which is of the form i = j (x) + g(x)u. To find a feedback linearizing law, we seek atransformation T = [T1 T2 T3]T such that

l5g(x) = 0 (10.22)

2 ( 0 10 23x) =g ( . )

2 ( ) 0 24(10g x ).

with

1f(x) = T2 (10.25)a'2a 1(X) = Ts (10.26)

a 2 AX) = -q(x)w(x). (10.27)

Equation (10.22) implies that

0[] 0r1+ i)

c711

1+µxi

8x3 A) = 0

so T1 is not a function of x3. To proceed we arbitrarily choose T1 = x1. We will need toverify that this choice satisfies the remaining linearizing conditions (10.22)-(10.27).

Equation (10.25) implies that

ax 1(X) = T2 = [1 0 011(X) = T2

and thus T2 = x2. We now turn to equation (10.26). We have that

2af(x)=T3and substituting values, we have that

k aµx3

({tx1)2.T3 =g-mx2 2m1+

10.4. CONDITIONS FOR INPUT-STATE LINEARIZATION 273

To verify that (10.23)-(10.24) we proceed as follows:

0aT2

9(x) = [0 1 0] 0 0ax1( )

u =

so that (10.23) is satisfied. Similarly

39(x) = m,(1µ+i 0x1)

provided that x3 # 0. Therefore all conditions are satisfied in D = {x E R3 : X3 # 0}. Thecoordinate transformation is

xl

T = X2k

g mx2

The function q and w are given by

w(x) _ 0 g(x) O(x)(aT3/ax)f(x)

ax (aT3/ax)g(x)

10.4 Conditions for Input-State Linearization

In this section we consider a system of the form

± = f(x) + g(x)u (10.28)

where f , g : D -+ R' and discuss under what conditions on f and g this system is input-statelinearizable.

Theorem 10.2 The system (10.28) is input-state linearizable on Do C D if and only ifthe following conditions are satisfied:

(i) The vector fields {g(x), adfg(x), , ad f-lg(x)} are linearly independent in Do.Equivalently, the matrix

C = [g(x), adfg(x),... adf-lg(x)]nxn

has rank n for all x E Do.

aµx2

2m(l+µxi)


(ii) The distribution A = span{g, ad fg, , adf-2g} is involutive in Do.



r ex2 - 1axe + 1 u = f(x) + 9(x)u.

L i

We have,

Thus, we have that

adfg = [f,9] = axf(x) - 'fg(x)

x2adfg =

e0

{g,adfg}={LOJ' [er2

J}and

rank(C) = rank0 ex2( 1 0 2, Vx E 1R2.

Also the distribution A is given by

01span{9} = span C

L 1

which is clearly involutive in R2. Thus conditions (i) and (ii) of Theorem 10.2 are satisfiedVxEIR2. E)

Example 10.13 Consider the linear time-invariant system

i=Ax+Bu

i.e., f (x) = Ax, and g(x) = B. Straightforward computations show that

adf9 = [f, 9] = ex f (x) - of g(x) = -AB

af9 = [f, [f, 9]] = A2B

g = (-1)tRB.

10.5. INPUT-OUTPUT LINEARIZATION 275

Thus,

{g(x), ad fg(x), ... , adf-19(x)} _ {B, -AB, A2B, ... (-1)n-'An-'B}

and therefore, the vectors {g(x), adfg(x), , adf-lg(x)} are linearly independent if andonly if the matrix

C = [B, AB, A2B, - -, An-1B]nxn

has rank n. Therefore, for linear systems condition (i) of theorem 10.2 is equivalent to thecontrollability of the pair (A, B). Notice also that conditions (ii) is trivially satisfied forany linear time-invariant system, since the vector fields are constant and so 6 is alwaysinvolutive.

10.5 Input-Output Linearization

So far we have considered the input-state linearization problem, where the interest is inlinearizing the mapping from input to state. Often, such as in a tracking control problem,our interest is in a certain output variable rather than the state. Consider the system

= f (x) + g(x)u f, g : D C Rn -* lR'y=h(x) h:DCRn-4 R (10.29)

Linearizing the state equation, as in the input-state linearization problem, does not neces-sarily imply that the resulting map from input u to output y is linear. The reason, of course,is that when deriving the coordinate transformation used to linearize the state equation wedid not take into account the nonlinearity in the output equation.

In this section we consider the problem of finding a control law that renders a lineardifferential equation relating the input u to the output y. To get a better grasp of thisprinciple, we consider the following simple example.

Example 10.14 Consider the system of the form

- axix2 + (x2 + 1)u

We are interested in the input-output relationship, so we start by considering the output

equation y = x1. Differentiating, we obtain

=x2


which does not contain u. Differentiating once again, we obtain

x2 = -xl -axix2+(x2+1)u.

Thus, letting

we obtain

or

uf

1 [v + axiX2 + 1 x2] (X2 54 1)

y+y=vwhich is a linear differential equation relating y and the new input v. Once this linearsystem is obtained, linear control techniques can be employed to complete the design. 11

This idea can be easily generalized. Given a system of the form (10.29) where f,g : D Cl -4 1R' and h : D C R" -+ R are sufficiently smooth, the approach to obtain a linearinput-output relationship can be summarized as follows.

Differentiate the output equation to obtain

y

There are two cases of interest:

= Lfh(x) + L9h(x)u

Oh.

8x xah Oh

axf (x) + 8xg(x)u

- CASE (1): Qh # 0 E D. In this case we can define the control law

U = L91(x) [-Lfh + v]

that renders the linear differential equation

y=v.

- CASE (2): = 0 E D. In this case we continue to differentiate y until uD-X

appears explicitly:

ydef y(2) dt[axf(x)l = Lfh(x)+L9Lfh(x)u.

If L9Lfh(x) = 0, we continue to differentiate until, for some integer r < n

y(r) = Lfh(x) + L9Lf-llh(x)u

with L9Lf -llh(x) $ 0. Letting

u= 1 [-Lrh+v]L9Lf llh(x)

we obtain the linear differential equation

y(r) = v. (10.30)

The number of differentiations of y required to obtain (10.30) is important and iscalled the relative degree of the system. We now define this concept more precisely.

Definition 10.10 A system of the form

x = f (x) + g(x)u f, g : D C Rn R"y=h(x) h:DCR"R

is said to have a relative degree r in a region Do C D if

L9L fh(x) = 0 Vi, 0 < i < r - 1, Vx E Do

L9Lf lh(x) # 0 Vx E Do

Remarks:

(a) Notice that if r = n, that is, if the relative degree is equal to the number of states,then denoting h(x) = Tl (x), we have that

y = 15[f (x) + g(x)u].

The assumption r = n implies that Og(x) = 0. Thus, we can define

C9T1 de fy = ax f(x) = T2

y(2) = a 22[f(x) + g(x)ul, 22g(x) = 0


) f T32 f ((2) = 8 xy xand iterating this process

(n) = n f ( ) + )n (

Therefore

y X x u.gal5g(x) = 0

) =2 ( 0xg

) =(10xgax

n ( ) 0g x

which are the input-state linearization conditions (10.17). Thus, if the relative degreer equals the number of states n, then input-output linearization leads to input-statelinearization.

(b) When applied to single-input-single-output systems, the definition of relative degreecoincides with the usual definition of relative degree for linear time-invariant systems,as we shall see.


±1 =x222 = -xl - ax21x2 + (x2 + 1)uy=x1

we saw that

X2

-xl - axix2 + (x2 + 1)u

hence the system has relative degree 2 in Do = {x E R2 : x2 # 1}.

Example 10.16 Consider the linear time-invariant system defined by

{ x=Ax+Buy=Cx

where0 1 0 ... 0

0 0 1 0

A=0 0 0 1

-qo -Q1 .. -9n-1

0

0

, B=

nxn

0

1nxl

C= [ Po P1... Pm 0 ... 11xn' Pin 0.

The transfer function associated with this state space realization can be easily seen to be

H(S) = C(SI - A) 1B =Pmsm +Pm-lsm-1 + ... +Po

sn+qn-lss-1+...+qo m<nc

The relative degree of this system is the excess of poles over zeros, that is, r = n - in > 0.We now calculate the relative degree using definition 10.10. We have

y=Ci=CAx+CBu

0

0

CB = [ Po P1 ... Pm 0 .

IO

1

_ pm, if m=n-10, otherwise

Thus, if CB = p.,,,,, we conclude that r = n - m = 1. Assuming that this is not the case, wehave that y = CAx, and we continue to differentiate y:

y(2) = CAi = CA2X + CABu

0

0

CAB = [ Po P1 ... Pm 0 ... ]J Pm, if m=n-2l 0, otherwise

If CAB = pm, then we conclude that r = n - m = 2. Assuming that this is not the case,we continue to differentiate. With every differentiation, the "1" entry in the column matrixA'B moves up one row. Thus, given the form of the C matrix, we have that

CA'-1B= 1 0 for i=l pm for i=n - m



y(?) = CA' x + CA''-'Bu, CA''-1B # 0, r = n - m

and we conclude that r = n - m, that is, the relative degree of the system is the excessnumber of poles over zeros.

Example 10.17 Consider the linear time-invariant system defined by

Ix=Ax+Buy=Cx

where0 1 0 0 0 010 0 1 0 0 0

A= 0 0 0 1 0 , B= 00 0 0 0 1

L

0

-2 -3 -5 -1 -4 5x5 1

C=[ 7 2 6 0 0 11x5

5x1

To compute the relative degree, we compute successive derivatives of the output equation:

y=CAx+CBu, CB=Oy(2) = CA2X + CABu, CAB = 0

y(3) = CA3X + CA2Bu, CA2B = 6

Thus, we have that r = 3. It is easy to verify that the transfer function associated with thisstate space realization is

H(s) = C(sI - A)-1B = 6s2+2s+7s5+4s4+s3+5s2+3s+2

which shows that the relative degree equals the excess number of poles over zeros.

10.6 The Zero Dynamics

On the basis of our exposition so far, it would appear that input-output linearization israther simple to obtain, at least for single-input-single-output (SISO) systems, and that allSISO systems can be linearized in this form. In this section we discuss in more detail theinternal dynamics of systems controlled via input-output linearization. To understand themain idea, consider first the SISO linear time-invariant system

10.6. THE ZERO DYNAMICS 281

i=Ax+Bu(10.31)

y = Cx.

To simplify matters, we can assume without loss of generality that the system is of thirdorder and has a relative degree r = 2. In companion form, equation (10.31) can be expressedas follows:

±1

±2

x3

0 1 0

00 0 1 x2 +- 'I,

0

4o -ql 42 X3 1

y = [ Po P1 0]

The transfer function associated with this system is

U

H(s) =+ Pls

U.qo + q1S + g2S2 + Q3S3

Suppose that our problem is to design u so that y tracks a desired output yd. Ignoringthe fact that the system is linear time-invariant, we proceed with our design using theinput-output linearization technique. We have:

Y = Pox1 +P1x2y = POxl +P1±2

= POx2 + Plx3= P0X2 +P1X3= POx3 + pl(-40x1 - 41x2 - 42x3) + Plu

Thus, the control law

1u = 40x1 + q, X2 + Q2x3 - P0 X3 + - v

P1 P1

produces the simple double integrator

y=v.

Since we are interested in a tracking control problem, we can define the tracking errore = y - yd and choose v = -kle - k2e + yd. With this input v we have that

u = Igoxl + 41x2 + 42x3 - x3J + 1[- kle - k2e + yd]


which renders the exponentially stable tracking error closed-loop system

e + k2e + kle = 0. (10.32)

A glance at the result shows that the order of the closed-loop tacking error (10.32) isthe same as the relative order of the system. In our case, r = 2, whereas the originalstate space realization has order n = 3. Therefore, part of the dynamics of the originalsystem is now unobservable after the input-output linearization. To see why, we reason asfollows. During the design stage, based on the input-output linearization principle, the stateequation (10.31) was manipulated using the input u. A look at this control law shows that uconsists of a state feedback law, and thus the design stage can be seen as a reallocation of theeigenvalues of the A-matrix via state feedback. At the end of this process observability of thethree-dimensional state space realization (10.31) was lost, resulting in the (external) two-dimensional closed loop differential equation (10.32). Using elementary concepts of linearsystems, we know that this is possible if during the design process one of the eigenvalues ofthe closed-loop state space realization coincides with one of the transmission zeros of thesystem, thus producing the (external) second-order differential equation (10.32). This isindeed the case in our example. To complete the three-dimensional state, we can considerthe output equation

y = Pox1 +P1x2= Poxl +pi ;i.

Thus,

±1 = - xl +11 y = Aid xl + Bid y.

P1 P1(10.33)

Equation (10.33) can be used to "complete" the three-dimensional state. Indeed, usingxl along with e and a as the "new" state variables, we obtain full information about theoriginal state variables x1 - x3 through a one-to-one transformation.

The unobservable part of the dynamics is called the internal dynamics, and plays avery important role in the context of the input-output linearization technique. Notice thatthe output y in equation (10.33) is given by y = e + yd, and so y is bounded since e wasstabilized by the input-output linearization law, and yd is the desired trajectory. Thus, xlwill be bounded, provided that the first-order system (10.33) has an exponentially stableequilibrium point at the origin. As we well know, the internal dynamics is thus exponentiallystable if the eigenvalues of the matrix Aid in the state space realization (10.33) are in theleft half plane. Stability of the internal dynamics is, of course, as important as the stabilityof the external tracking error (10.32), and therefore the effectiveness of the input-outputlinearization technique depends upon the stability of the internal dynamics. Notice also


that the associated transfer function of this internal dynamics is

Hid =Po + pis

Comparing Hid and the transfer function H of the original third-order system, we concludethat the internal dynamics contains a pole whose location in the s plane coincides withthat of the zero of H, thus leading to the loss of observability. This also implies that theinternal dynamics of the original system is exponentially stable, provided that the zeros ofthe transfer function Hid are in the left-half plane. System that have all of their transferfunction zeros are called minimum phase.

Our discussion above was based on a three dimensional example; however, our conclu-sions can be shown to be general, specifically, for linear time-invariant systems the stabilityof the internal dynamics is determined by the location of the system zeros.

Rather than pursuing the proof of this result, we study a nonlinear version of thisproblem. The extension to the nonlinear case is nontrivial since poles and zeros go hand inhand with the concept of transfer function, which pertains to linear time-invariant systemonly. We proceed as follows. Consider an nth-order system of the form

x = f(x) + g(x)u(10.34)

y = h(x)and assume that (10.34) has relative degree r. Now define the following nonlinear transfor-mation

µi (x)

{fin-r(x)

z = T(x) =01

Lrwhere

def

V), = h(x)

02 = a 11 f (x)

V)i+i = V)i f (x)

71EII2" r, ERr (10.35)

i = 1

and pi - {Ln-r are chosen so that T(x) is a diffeomorphism on a domain Do C D and

Xg(x)=0, for 1<i<n-r, dxEDo. (10.36)

It will be shown later that this change of coordinates transfers the original system (10.34)into the following so-called normal form:

where

and

fl = fo(rl, ) (10.37)= A, + Bcw[u - O(x)] (10.38)

y = Cc (10.39)

f0(77,0 = allOx 1x=T-1(z)

W = a rr g(x),(rlax) f (x)(00r/ax) g(x)

The normal form is conceptually important in that it splits the original state x into twoparts, namely, 77 and , with the following properties:

f represents the external dynamics, which can be linearized by the input

u' _ O(x) + w-IV (10.40)

77 represents the internal dynamics. When the input u` is applied, equations (10.37)-(10.39) take the form

fl = fo(rl,0= Ac + Bcv

y = Ccy

and thus, 77 is unobservable from the output. From here we conclude that the stabilityof the internal dynamics is determined by the autonomous equation

7 = fo(rl, 0) -

This dynamical equation is referred to as the zero dynamics, which we now formallyintroduce.

Definition 10.11 Given a dynamical system of the form (10.37)-(10.39) (i.e., representedin normal form), the autonomous equation

1 = fo(y,0)

is called the zero dynamics.

Because of the analogy with the linear system case, systems whose zero dynamics is stableare said to be minimum phase.

Summary: The stability properties of the zero dynamics plays a very important rolewhenever input-output linearization is applied. Input-output linearization is achieved viapartial cancelation of nonlinear terms. Two cases should be distinguished:

r = n: If the relative degree of the nonlinear system is the same as the order of thesystem, then the nonlinear system can be fully linearized and, input-output lineariza-tion can be successfully applied. This analysis, of course, ignores robustness issuesthat always arise as a result of imperfect modeling.

r < n: If the relative degree of the nonlinear system is lower than the order of thesystem, then only the external dynamics of order r is linearized. The remaining n - rstates are unobservable from the output. The stability properties of the internaldynamics is determined by the zero dynamics. Thus, whether input-output lineariza-tion can be applied successfully depends on the stability of the zero dynamics. If thezero dynamics is not asymptotically stable, then input-output linearization does notproduce a control law of any practical use.

Before discussing some examples, we notice that setting y - 0 in equations (10.37)-(10.39) we have that

y-0 ty t;-0u(t) _ O(x)

f7 = fo(mI, 0)

Thus the zero dynamics can be defined as the internal dynamics of the system when theoutput is kept identically zero by a suitable input function. This means that the zerodynamics can be determined without transforming the system into normal form.


xl = -kxl - 2x2u4 22 = -x2 + xiuy = X2

First we find the relative order. Differentiating y, we obtain

y=i2=-x2+x1u.

Therefore, r = 1. To determine the zero dynamics, we proceed as follows:

y=0 x2=0 t-- u=0.

Then, the zero dynamics is given by

which is exponentially stable (globally) if k > 0, and unstable if k < 0.


I xl = x2 + xi

i2=x2+ux3 = x1 +X 2 + ax3y = X1

Differentiating y, we obtain

y = LEI =x2+xi2x111 + i2 = 2x1 (x2 + xl) + x2 + u

Therefore

r=2To find the zero dynamics, we proceed as follows:

#-.i1=0=x2+2i =>' X2=0

i2=x2+u=0 u=-x2.

Therefore the zero dynamics is given by

x3 = ax3.

Moreover, the zero dynamics is asymptotically stable if a = 0, and unstable if a > 0.

10.7. CONDITIONS FOR INPUT-OUTPUT LINEARIZATION 287

10.7 Conditions for Input-Output Linearization

According to our discussion in Section 10.6, the input-output linearization procedure de-pends on the existence of a transformation T(.) that converts the original system of theform

fx = f (x) + g(x)u, f, g : D C R" -> R" (10.41)y=h(x), h:DCRn ->Rinto the normal form (10.37)-(10.39). The following theorem states that such a transfor-mation exists for any SISO system of relative degree r < n.

Theorem 10.3 Consider the system (10.41) and assume that it has relative degree r <n Vx E Do C D. Then, for every xo E Do, there exist a neighborhood S2 of xo and smoothfunction It,, , /t,,, _r such that

(i) Lgpi(x) = 0, for 1 < i < n - r, dx E S2

(ii)

f i i ( ) 1

T(x) =An-r(x)

01

Lr J

01 = h(x)02 = Lfh(x)

Or = Lf lh(x)

is a diffeomorphism on Q.


10.8 Exercises

10.1) Prove Lemma 10.2.


(10.2) Consider the magnetic suspension system of Example 10.11. Verify that this systemis input-state linearizable.

(10.3) Consider again the magnetic suspension system of Example 10.11. To complete thedesign, proceed as follows:

(a) Compute the function w and 0 and express the system in the form i = Az + By.

(b) Design a state feedback control law to stabilize the ball at a desired position.

(10.4) Determine whether the systemJ ±1 = X 2 X 3ll X2 = U

is input-state linearizable.


{21 = x1 + x2±2=x3+46X3=x1+x2+x3

(a) Determine whether the system is input-state linearizable.

(b) If the answer in part (a) is affirmative, find the linearizing law.

(10.6) Consider the following system

{21=x1+x3x2 = xix2X3 = x1 sin x2 + U



(10.7) ([36]) Consider the following system:

{xl = X3 + 3.2x3x2 = x1 + (1 + x2)4623=x2(1+x1)-x3+46



10.8. EXERCISES 289


.t1 =21+22±2=2g+UX3 = X2 - (X23

y = 21

(a) Find the relative order.

(b) Determine whether it is minimum phase.

(c) Using feedback linearization, design a control law to track a desired signal y =

yref-

Notes and ReferencesThe exact input-state linearization problem was solved by Brockett [12] for single-inputsystems. The multi-input case was developed by Jakubczyk and Respondek, [39], Su, [77]and Hunt et al. [34]. The notion of zero dynamics was introduced by Byrnes and Isidori[14].

For a complete coverage of the material in this chapter, see the outstanding books ofIsidori [36], Nijmeijer and van der Schaft [57], and Marino and Tomei, [52]. Our presentationfollows Isidory [36] with help from References [68], [88] and [41]. Section 10.1 follows closelyReferences [36] and [11]. The linear time-invariant example of Section 6, used to introducethe notion of zero dynamics follows Slotine and Li [68].

Chapter 11

Nonlinear Observers

So far, whenever discussing state space realizations it was implicitly assumed that thestate vector x was available. For example, state feedback was used in Chapter 5, alongwith the backstepping procedure, assuming that the vector x was measured. Similarly,the results of Chapter 10 on feedback linearization assume that the state x is available formanipulation. Aside from state feedback, knowledge of the state is sometimes importantin problems of fault detection as well as system monitoring. Unfortunately, the state x isseldom available since rarely can one have a sensor on every state variable, and some formof state reconstruction from the available measured output is required. This reconstructionis possible provided certain "observability conditions" are satisfied. In this case, an observercan be used to obtain an estimate 2 of the true state x. While for linear time-invariantsystems observer design is a well-understood problem and enjoys a well-established solution,the nonlinear case is much more challenging, and no universal solution exists. The purposeof this chapter is to provide an introduction to the subject and to collect some very basicresults.

11.1 Observers for Linear Time-Invariant Systems

We start with a brief summary of essential results for linear time-invariant systems. Through-out this section we consider a state space realization of the form

x=Ax+Bv AEII2nXn BEIfPnxly = Cx C E R1xn

where, for simplicity, we assume that the system is single-input-single-output, and thatD = 0 in the output equation.

291

292 CHAPTER 11. NONLINEAR OBSERVERS

11.1.1 Observability

Definition 11.1 The state space realization (11.1) is said to be observable if for any initialstate xo and fixed time tl > 0, the knowledge of the input u and output y over [0, t1] sufficesto uniquely determine the initial state xo.

Once xo is determined, the state x(ti) can be reconstructed using the well-known solutionof the state equation:

tl) = elxo + eA(t-T)Bu(r) dr.Lt1x(

Also

(11.2)

t,y(ti) = Cx(tl) = CeAtlxo + C r eA(t-T)Bu(rr) dr. (11.3)

0

Notice that, for fixed u = u`, equation (11.3) defines a linear transformation L : 1R" -+ ]Rthat maps xo to y; that is, we can write

y(t) = (Lxo)(t).

Thus, we argue that the state space realization (11.1) is observable if and only if the mappingLx0 is one-to-one. Indeed, if this is the case, then the inversion map xo = L-ly uniquelydetermines x0. Accordingly,

,b1 "observable" if and only if Lxl = Lx2 x1 = x2.

Now consider two initial conditions xl and x2. We have,

and

tly(ti) = (Lxl)(t) = CeAt'xl +C J eA(t-T)Bu*(r) d7 (11.4)

0

rt,y(t2) = (Lx2)(t) = CeAtlx2 +CJ eA(t-T)Bu*(r) drr (11.5)

0

yl - y2 = Lx1 - Lx2 = CeAtl (xl - x2).

By definition, the mapping L is one-to-one if

y1 = Y2

For this to be the case, we must have:

CeAtx = 0 x=0

11.1. OBSERVERS FOR LINEAR TIME-INVARIANT SYSTEMS 293

or, equivalentlyN(CeAt) = 0.

Thus, L is one-to-one [and so (11.1) is observable] if and only if the null space of CeAt isempty (see Definition 2.9).

Notice that, according to this discussion, the observability properties of the statespace realization (11.1) are independent of the input u and/or the matrix B. Therefore wecan assume, without loss of generality, that u - 0 in (11.1). Assuming for simplicity thatthis is the case, (11.1) reduces to

x= Ax(11.6)l y=Cx.

Observability conditions can now be easily derived as follows. From the discussion above,setting u = 0, we have that the state space realization (11.6) [or (11.1)] is observable if andonly if

y(t) = CeAtxo - 0 = x - 0.

We now show that this is the case if and only if

rank(O)'Irank

To see this, note that

y(t) = CeAtxo

CCA

CAn-1

= n (11.7)

2

C(I+tA+

By the Sylvester theorem, only the first n - 1 powers of AZ are linearly independent. Thus

y = 0 b Cxo = CAxo = CA2xo = ... = CAn-1xo

or,

y=0

CCA

CAn-1

xp = 0, b Ox0 = 0

and the condition Oxo = 0 xo = 0 is satisfied if and only if rank(O) = n.

The matrix 0 is called the observability matrix. Given these conditions, we canredefine observability of linear time-invariant state space realizations as follows.


Definition 11.2 The state space realization (11.1) for (11.6)] is said to be observable if

CCA

rank = nCAn-1

11.1.2 Observer Form

The following result is well known. Consider the linear time-invariant state space realization(11.1), and let the transfer function associated with this state space realization be

H(s) = C(sI - A)-1B =pn-lsn-1 +pn-2sn-2 + ... +po

yn + qn-lsn-1 + ... + qo

Assuming that (A, C) form an observable pair, there exists a nonsingular matrix T E IItfXfsuch that defining new coordinates t = Tx, the state space realization (11.1) takes theso-called observer form:

r 0 0 0 -qo1 0 0 -q10 1 0 -Q2

0 0 1 -Qn-1J L 2n J

y=[0 ... 11t

+

11.1.3 Observers for Linear Time-Invariant Systems

U

Now consider the following observer structure:

x=Ax+Bu+L(y-Cx)where L E Rn-1 is the so-called observer gain. Defining the observer error

de fx=x - iwe have that

PO

Pi

(11.8)

(11.9)

x = x-i=(Ax+Bu)-(Ai+Bu+Ly-LCi)= (A - LC)(x - i)

11.1. OBSERVERS FOR LINEAR TIME-INVARIANT SYSTEMS 295

or

x = (A - LC).i. (11.10)

Thus, x -+ 0 as t --> oo, provided the so-called observer error dynamics (11.10) is asymp-totically (exponentially) stable. This is, of course, the case, provided that the eigenvaluesof the matrix A - LC are in the left half of the complex plane. It is a well-known resultthat, if the observability condition (11.7) is satisfied, then the eigenvalues of (A - LC) canbe placed anywhere in the complex plane by suitable selection of the observer gain L.

11.1.4 Separation Principle

Assume that the system (11.1) is controlled via the following control law:

u = Kx (11.11)

i = Ax+Bu+L(y-Cx) (11.12)

i.e. a state feedback law is used in (11.11). However, short of having the true state xavailable for feedback, an estimate x of the true state x was used in (11.11). Equation(11.12) is the observer equation. We have

i = Ax+BKxAx+BKx - BK±(A + BK)x - BK±.

Also

Thus

where we have defined

(A - LC)i

A 0BK ABLC] [x]x=Ax

The eigenvalues of the matrix A are the union of those of (A + BK) and (A - LC).Thus, we conclude that

(i) The eigenvalues of the observer are not affected by the state feedback and vice versa.

(ii) The design of the state feedback and the observer can be carried out independently.This is called the separation principle.


11.2 Nonlinear Observability

Now consider the system

x=f(x)+g(x)u f :R"->lR',g:]R'->1R(11.13)

y = h(x) h : PJ - 1R

For simplicity we restrict attention to single-output systems. We also assume that f (),are sufficiently smooth and that h(O) = 0. Throughout this section we will need the followingnotation:

xu(t,xo): represents the solution of (11.13) at time t originated by the input u andthe initial state xo.

y(xu(t, xo): represents the output y when the state x is xu(t, xo).

Clearlyy(xu(t, xo)) = h(xu(t, xo))

Definition 11.3 A pair of states (xo, xo) is said to be distinguishable if there exists aninput function u such that

y(xu(t, xo)) = y(xu(t, xo))

Definition 11.4 The state space realization ni is said to be (locally) observable at xo E R"if there exists a neighborhood U0 of xo such that every state x # xo E fZ is distinguishablefrom xo. It is said to be locally observable if it is locally observable at each xo E R".

This means that 0,,, is locally observable in a neighborhood Uo C Rn if there exists an inputu E li such that

y(xu(t, xo)) y(xu(t, xo)) Vt E [0, t] xo = x2

There is no requirement in Definition 11.4 that distinguishability must hold for allfunctions. There are several subtleties in the observability of nonlinear systems, and, in gen-eral, checking observability is much more involved than in the linear case. In the followingtheorem, we consider an unforced nonlinear system of the form

I x=f(x) f :1 -*'Yn1 y = h(x) h:ii. [F

(11.14)

and look for observability conditions in a neighborhood of the origin x = 0.

11.2. NONLINEAR OBSERVABILITY 297

Theorem 11.1 The state space realization (11.14) is locally observable in a neighborhoodUo C D containing the origin, if

Vh

rank = n dx E Uo

VLf 'h

Proof: The proof is omitted. See Reference [52] or [36].

The following example shows that, for linear time-invariant systems, condition (11.15)is equivalent to the observability condition (11.7).

Example 11.1 Let= Ax

y = Cx.

Then h(x) = Cx and f (x) = Ax, and we have

Vh(x) = CVLfh = V(- ) = V(CAx) = CA

VLf-1h = CAn-1

and therefore '51 is observable if and only if S = {C, CA, CA2, . . , CAn-1 } is linearly inde-pendent or, equivalently, if rank(O) = n.

Roughly speaking, definition 11.4 and Theorem 11.1 state that if the linearizationof the state equation (11.13) is observable, then (11.13) is locally observable around theorigin. Of course, for nonlinear systems local observability does not, in general, implyglobal observability.

We saw earlier that for linear time-invariant systems, observability is independentof the input function and the B matrix in the state space realization. This property is aconsequence of the fact that the mapping xo --1 y is linear, as discussed in Section 11.2.Nonlinear systems often exhibit singular inputs that can render the state space realizationunobservable. The following example clarifies this point.

Example 11.2 Consider the following state space realization:

-k1 = x2(1 - u)0.1 x2 = x1

y = xl


which is of the form.x = f (x) + g(x)uy = h(x)

with

X1 0

If u = 0, we haverank({Oh,VLfh}) = rank({[1 0], [0 1]}) = 2

and thusJ i =f(x)

y = h(x)is observable according to Definition 11.4. Now consider the same system but assume thatu = 1. Substituting this input function, we obtain the following dynamical equations:

{

A glimpse at the new linear time-invariant state space realization shows that observabilityhas been lost.

11.2.1 Nonlinear Observers

There are several ways to approach the nonlinear state reconstruction problem, dependingon the characteristics of the plant. A complete coverage of the subject is outside the scopeof this textbook. In the next two sections we discuss two rather different approaches tononlinear observer design, each applicable to a particular class of systems.

11.3 Observers with Linear Error Dynamics

Motivated by the work on feedback linearization, it is tempting to approach nonlinear statereconstruction using the following three-step procedure:

(i) Find an invertible coordinate transformation that linearizes the state space realization.

(ii) Design an observer for the resulting linear system.

(iii) Recover the original state using the inverse coordinate transformation defined in (i).

11.3. OBSERVERS WITH LINEAR ERROR DYNAMICS 299

More explicitly, suppose that given a system of the form

x= f(x)+g(x,u) xE]R", uER(11.16)y=h(x)

there exist a diffeomorphism satisfying

yER

z = T(x), T(0) = 0, z E Rn (11.17)

and such that, after the coordinate transformation, the new state space realization has theform

x =A z + ( u)o 7 y,(11.18)

y = Coz yEllhw ere

0 0 ... 0 71(y,u)1 0 ... 0 I 'Y1 (y, u)

Ao = 0 1 0 , Co=[0 0 ... 0 1], ry= (11.19)

L 0 0 1 0 JL 7n(y,u) J

then, under these conditions, an observer can be constructed according to the following

theorem.

Theorem 11.2 ([52]) If there exist a coordinate transformation mapping (11.16) into(11.18), then, defining

z = Aoz + ry(y, u) - K(y - zn), F E R (11.20)

x = T-1(z) (11.21)

such that the eigenvalues of (Ao+KCo) are in the left half of the complex plane, then x -+ xast-4 oc.

Proof: Let z = z - z, and x = x - i. We have

z = ,i-z= [Aoz +'Y(y, u)] - [Aoz +'Y(y, u) - K(y - zn)]= (Ao + KC0)z.

If the eigenvalues of (Ao+KCo) have negative real part, then we have that z -+ 0 as t -4 oo.Using (11.21), we obtain

i = x-i= T-1(z) -T-1(z - z) - + 0 as t - oo

0


Example 11.3 Consider the following dynamical system

{it = x2 + 2x122 = X1X2 + X U

y=x1

and define the coordinate transformation

j X2z1=x2-2 1z2=x1.

In the new coordinates, the system (11.22) takes the form

I .tl = -2y3 + y3uZ2=Z1+Zy2y=Z2

which is of the form (11.18) with

r 2y3§y3uAo=[ p], Co=(01], andy=L-z

Jzy

The observer is

z=Aoz+y(y,u)-K(y-22)

Lz2J-L1 00Jz2]+[-25y123u]+[K2

l ]The error dynamics is

(11.22)

Lx2] L1 -K2J Lz2Thus, z ---> 0 for any K1i K2 > 0. El

It should come as no surprise that, as in the case of feedback linearization, this ap-proach to observer design is based on cancelation of nonlinearities and therefore assumes"perfect modeling." In general, perfect modeling is never achieved because system pa-rameters cannot be identified with arbitrary precision. Thus, in general, the "expected"cancellations will not take place and the error dynamics will not be linear. The result isthat this observer scheme is not robust with respect to parameter uncertainties and thatconvergence of the observer is not guaranteed in the presence of model uncertanties.

11.4. LIPSCHITZ SYSTEMS 301

11.4 Lipschitz Systems

The nonlinear observer discussed in the previous section is inspired by the work on feedbacklinearization, and belongs to the category of what can be called the differential geometricapproach. In this section we show that observer design can also be studied using a Lyapunovapproach. For simplicity, we restrict attention to the case of Lipschitz systems, definedbelow.

Consider a system of the form:

J i = Ax + f (x, u) (11.23)y=Cx

where A E lRnxn C E Rlxn and f : Rn x R -a Rn is Lipschitz in x on an open set D C R',i.e., f satisfies the following condition:

If(x1,u*) - f(x2,u )lI < 7IIx1 -x211 Vx E D. (11.24)

Now consider the following observer structure

x=Ax+f(x,u)+L(y-Cx) (11.25)

where L E Rnxl. The following theorem shows that, under these assumptions, the estima-tion error converges to zero as t -+ oo.

Theorem 11.3 Given the system (11.23) and the corresponding observer (11.25), if theLyapunov equation

P(A - LC) + (A - LC)T P = -Q (11.26)

where p = PT > 0, and Q = QT > 0, is satisfied with

y< .min (Q)(11.27)

2A, (P)

then the observer error 2 = x - x is asymptotically stable.

Proof:

= [Ax + f (x, u)] - [Ax + f (x, u) + L(y - C2)]= (A - LC)2 + f (x, u) - f (x, u).

To see that I has an asymptotically stable equilibrium point at the origin, consider theLyapunov function candidate:

V iPi + iPx_ -2TQ2+22TP[f(i+2,u) - f(2,u)]

but

and

pTQiII Amin(Q)II.t1I2

II2xT P[f [(.i+ x, u) - f(, u)jjj J12±TPII II f p+ x, u) - fp, u)jj

< 2y\,,

Therefore, V is negative definite, provided that

Amin(Q)II:i1I2 > 2yAmax(P)jjill2

or, equivalently

y)'min (Q)

2A""' (P)

0


LX2J=[12J[x2J+[

y=[10][ 2

Setting

L=[°we have that

A-LC= [ 01

1

2

Solving the Lyapunov equation

0

2 I

P(A - LC) + (A - LC)T P = -Q

11.5. NONLINEAR SEPARATION PRINCIPLE 303

with Q = I, we obtain1.5 -0.5

-0.5 0.5

which is positive definite. The eigenvalues of P are Amjn(P) = 0.2929, and Amax(P) _1.7071. We now consider the function f. Denoting

x1 =2

,

we have that

x2 =[µ2

Il f(x1) - f(x2)112 = (Q2 92)2

= -µ2)I= 1(6 +µ2)(6 -{12)1< 2121 16 - 921 = 2k112- 921

< 2k11x1 - x2112

for all x satisfying 161 < k. Thus, ry = 2k and f is Lipschitz Vx = [l;1 C2]T: I£21 < k, and

we haveS

1

or

ry=2k<

k<

2Amax(P)

1

6.8284

The parameter k determines the region of the state space where the observer is guaranteedto work. Of course, this region is a function of the matrix P, and so a function of theobserver gain L. How to maximize this region is not trivial (see "Notes and References" atthe end of this chapter).

11.5 Nonlinear Separation Principle

In Section 11.1.4 we discussed the well-known separation principle for linear time-invariant(LTI) systems. This principle guarantees that output feedback can be approached in twosteps:

(i) Design a state feedback law assuming that the state x is available.

(ii) Design an observer, and replace x with the estimate i in the control law.


In general, nonlinear systems do not enjoy the same properties. Indeed, if the true stateis replaced by the estimate given by an observer, then exponential stability of the observerdoes not, in general, guarantee closed-loop stability. To see this, consider the followingexample.

Example 11.5 ([47]) Consider the following system:

-x + x4 + x2(11.28)

k>0 (11.29)

We proceed to design a control law using backstepping. Using as the input in (11.28), wechoose the control law cb1(x) = -x2. With this control law, we obtain

2 = -x+x4+ x2(61 (x) = -x

Now define the error state variable

CCz=S-01(x)=1;+x2.With the new variable, the system (11.28)-(11.29) becomes

Letting

and taking

we have that

i = -x +x2z

i = t: - q(x)_ -kt +u+2xi_ -kl; + u + 2x(-x + x4 + x2£)_ -k + u + 2x(-x + x2z)

2(x2+z2)

-x2 + z[x3 - k + u + 2x(-x + x2z)]

(11.30)

(11.31)

(11.32)

(11.33)

(11.34)

u = -cz - x3 + kl; - 2x(-x + x2z), c > 0 (11.35)

V = -x2 - cz2

which implies that x = 0, z = 0 is a globally asymptotically stable equilibrium point of thesystem

Lb = -x +x2zi = -cz - x3

11.5. NONLINEAR SEPARATION PRINCIPLE 305

This control law, of course, assumes that both x and 1; are measured. Now assume that onlyx is measured, and suppose that an observer is used to estimate l;. This is a reduced-orderobserver, that is, one that estimates only the nonavailable state 1;. Let the observer be givenby

1;=-kl;+u.The estimation error is t; = l; - l;. We have

_ -kl;+u+kl-u-kl;

It follows that&)e-kt

which implies that l; exponentially converges toward . Using the estimated in the controllaw (11.35), we obtain

-x+x2z+x21;-cz-x3+2x31;

Even though l; exponentially converges to zero, the presence of the terms x21; and 2x31; leadsto finite escape time for certain initial conditions. To see this, assume that the error variablez - 0. We have

_ -x +x21;&)e-kt

Solving for x, we obtain

x(t) =xo(1 + k)

(1 + k - oxo)et + Eoxoe-kt

which implies that the state x grows to infinity in finite time for any initial condition lxo >1+k.


Notes and ReferencesDespite recent progress, nonlinear observer design is a topic that has no universal solutionand one that remains a very active area of research. Observers with linear error dynamicswere first studied by Krener and Isidori [45] and Bestle and Zeitz [10] in the single-input-single-output case. Extensions to multivariable systems were obtained by Krener and Re-spondek [46] and Keller [42]. Section 11.3 is based on Reference [45] as well as References[52] and [36]. Besides robustness issues, one of the main problems with this approach toobserver design is that the conditions required to be satisfied for the existence of theseobservers are very stringent and are satisfied by a rather narrow class of systems.

Observers for Lipschitz systems were first discussed by Thau [80]. Section 11.4 isbased on this reference. The main problem with this approach is that choosing the observergain L so as to satisfy condition (11.27) is not straightforward. See References [61], [60],and [96] for further insight into how to design the observer gain L.

There are several other approaches to nonlinear observer design, not covered in thischapter. The list of notable omissions includes adaptive observers [52]-[54], high gain-observers, [4], [5], and observer backstepping, [47].

Output feedback is a topic that has received a great deal of attention in recent years.As mentioned, the separation principle does not hold true for general nonlinear systems.Conditions under which the principle is valid were derived by several authors. See References[90], [82], [83], [8], [4] and [5].

Appendix A

Proofs

The purpose of this appendix is to provide a proof of several lemmas and theorems en-countered throughout the book. For easy of reference we re-state the lemmas and theoremsbefore each proof.

A.1 Chapter 3

Lemma 3.1: V : D -> R is positive definite if and only if there exists class IC functions aland a2 such that

ai(Ilxll) < V(x) < a2(IIXII) Vx E Br C D.

Moreover, if D = R" and V() is radially unbounded, then al and a2 can be chosen in theclass

Proof: Given V(x) : D C R" -> R, we show that V(x) is positive definite if and only ifthere exist ai E IC such that ai(IlxI) < V(x) Vx E D. It is clear that the existence of aiis a sufficient condition for the positive definiteness of V. To prove necessity, define

X(y) = min V(x); for 0 < y < rv<_IIxII<r

The function X() is well defined since V(.) is continuous and y < lxll < r defines a compactset in ]R". Moreover, so defined has the following properties:

(i) X < V(x), 0 < l x l l < r.

307

308 APPENDIX A. PROOFS

Figure A.1: Asymptotically stable equilibrium point.

(ii) It is continuous.

(iii) It is positive definite [since V(x) > 0].

(iv) It satisfies X(0) = 0.

Therefore, X is "almost" in the class 1C. It is not, in general, in K because it is not strictlyincreasing. We also have that

X(IIxIj) <_ V(x) for 0 < IIxII < r

is not, however, in the class IC since, in general, it is not strictly increasing. Let al(y)be a class K function such that al (y) < kX(y) with 0 < k < 1. Thus

ai(IIxil) < X(IIxlI) <_ V(x) for IIxII < r.

The function al can be constructed as follows:

al(s) = min [x)] s < y < r

this function is strictly increasing. To see this notice that

al= min [X(y)1 s < y < r

s y J(A.1)

is positive. It is also nondecreasing since as s increases, the set over which the minimum iscomputed keeps "shrinking" (Figure A.2). Thus, from (A.1), al is strictly increasing.

A.1. CHAPTER 3 309

Figure A.2: Asymptotically stable equilibrium point.

This proves that there exists E X such that

al(jlxlj) < V(x) for IxMI < r

The existence of E 1C such that

V(x) < a2(lIXIl) for IxII < r

can be proved similarly.

To simplify the proof of the following lemma, we assume that the equilibrium pointis the origin, xe = 0.

Lemma 3.2: The equilibrium xe of the system (3.1) is stable if and only if there exists aclass 1C function a constant a such that

IIx(0)II < 8 = IIx(t)II < a(IIx(0)II) Vt > 0. (A.2)

Proof: Suppose first that (A.2) is satisfied. We must show that this condition implies thatfor each e1i361 = al(es) > 0:

Ix(0)II < 81 = IIx(t)II < el Vt > to. (A.3)

Given e1, choose 81 = min{8, a-'(fl) }. Thus, for 11x(0)I1 < 61i we have

Ilx(t)II C a(Ilx(O)II) < a(a-1(el)) = el.

For the converse, assume that (A.3) is satisfied. Given el, let s C 1R+ be the set defined asfollows:

A = 161 E 1R+ : IIx(0)II < bl = 11x(t)II < ei}.

This set is bounded above and therefore has a supremum. Let 8' = sup(s). Thus,

IIx(0)II < a' = 11x(t)11 < el Vt > to.

The mapping 4): el -* 5*(el) satisfies the following: (i) 4i(O) = 0, (ii) P(e1) > 0 de > 0, and(iii) it is nondecreasing, but it is not necessarily continuous. We can find a class 1C function

that satisfies fi(r) _< 4i(r), for each r E 1R+. Since the inverse of a function in the classK is also in the same class, we now define:

and a(.) E 1C .

Given e, choose any x(0) such that a(IIx(0)II) = e. Thus IIx(0)II < b', and we have

11x(t)II < e = a(IIx(0)II) provided that IIx(0)II < P.

Lemma 3.3 The equilibrium xe = 0 of the system (3.1) is asymptotically stable if and onlyif there exists a class 1CG function and a constant a such that

IIxoII < 8 = Ix(t)II <_ Q(IIxoII,t) Vt > 0. (A.4)

Proof: Suppose that A.4 is satisfied. Then

IIx(t)II < 3(Ilxo11, 0) whenever IIxoII oo. Thus, IIx(t)II -* 0 as t -4 oo and x = 0 is also convergent. Itthen follows that x = 0 is asymptotically stable. The rest of the proof (i.e., the converseargument) requires a tedious constructions of a 1CL function Q and is omitted. The readercan consult Reference [41] or [88] for a complete proof.

Theorem 3.5 A function g(x) is the gradient of a scalar function V(x) if and only if thematrix j19

xlxlxlL9 0 09

8xz 8x, 8x2

19 L9

9X Rn e

A.1. CHAPTER 3

is symmetric.

Proof: To prove necessity, we assume that g(x) = VV(x). Thus

av

Thus

T=

and the result follows since

a2v a2v a2vx xl

717a'2v acv o vv2 x2 x2 TX. X2

I a2v a2v a2vL x z2 x xn x, J

a2v = a2v To rove sufficiency, assume thatp Y,TX-1 xI x; xl

'99i 0gaxe au

and consider the integration as defined in (3.7):

xl

V (x) =fox

g(x) dx = f g1(si, 0, 0) ds10

X2

+ f g2(xi, s2, 0, ... , 0) ds2 + . +0

We now compute the partial derivatives of this function:

avax-1

10

xn

311

9n(xl,x2,...,sn) dsn(A.5)

gi (xi) o, ... , 0) +fx2

(x1, s2, 0, ... , 0) ds2 +x,9g

n

+ 0axl(xl,x2,...,sn) dsn

X2 a9191(x1,0)...,0) + J 5x2(x1,s2,0,...,0) ds2 + ...

X+ / 5ag,dsn

where we have used the fact that', by assumption. It then follows that

9V

ax-1

xz91(x1, 0, .. , 0) + 91(x1, s2, ... , 0) I0 +

+91 x1, x2,"',sn 091(x)

Proceeding in the same fashion, it can be shown that

OV =a

g= (x) Vi.xe

Lemma 3.4: If the solution x(t, x0i to) of the system (3.1) is bounded for t > to, then its(positive) limit set N is (i) bounded, (ii) closed, and (iii) nonempty. Moreover, the solutionapproaches N as t -+ oo.

Proof: Clearly N is bounded, since the solution is bounded. Now take an arbitrary infinitesequence {tn} that tends to infinity with n. Then x(t, xo, to) is a bounded sequence ofpoints, and by properties of bounded sequences, it must contain a convergent subsequencethat converges to some point p E R'. By Definition 3.12 p E N, and thus N is nonempty.To prove that it is closed, we must show that it contains all its limit points. To show thatthis is the case, consider the sequence of points {qi} in N, which approaches a limit pointgofNasi -+oo. Given e > 0, 8q E N such that

Ilgk - qjI < 2

Since qk is in N there is a sequence {tn} such that to -4 oo as n -3 oo, and

Ix(tn, xo, to) - gkli <2

Thus,IIx(tn, x0, t0) - qII < e

which implies that q E N and thus N contains its limit points and it is closed. It remainsto show that x(t, xo, t0) approaches N as t -> oo. Suppose that this is not the case. Thenthere exists a sequence {tn} such that

I Ip - x(tn, x0, to) II > e as n -+ oo

for any p in the closed set N. Since the infinite sequence x(tn, x0i to) is bounded, it containssome subsequence that converges to a point q N. But this is not possible since itcontradicts Definition 3.12.

A.2. CHAPTER 4 313

Lemma 3.5: The positive limit set N of a solution x(t, xo, to) of the autonomous system

x = f(x) (A.6)

is invariant with respect to (A.6).

Proof: We need to show that if xo E N, then x(t, xo, to) E N for all t > to. Let p c N.Then there exists a sequence {tn} such that to -+ oc and x(tn, xo, to) -> p as n -> oo. Sincethe solution of (A.6) is continuous with respect to initial conditions, we have

lim x[t, x(tn, xo, to), to] = x(t,p, to).n-+oo

Also,

It follows that

x[t, x(tn, x0, to), to] = x[t + tn, xo, to).

lim x[t + tn, xo, to] = x(t, P, to) (A.7)n ,oo

and since x(t, xo, to) approaches N as t - oo, (A.7) shows that x(t, p, to) is in NVt.

A.2 Chapter 4

Theorem 4.6: Consider the system i = A(t)x. The equilibrium state x = 0 is expo-nentially stable if and only if for any given symmetric, positive definite continuous andbounded matrix Q(t), there exist a symmetric, positive definite continuously differentiableand bounded matrix P(t) such that

-Q(t) = P(t)A(t) + AT(t)p(t) + P(t) (A.8)

Proof: Sufficiency is straightforward. To prove necessity, let

P(t) = j D'(T, t)Q(T),D(T, t) dr.t

Thus,

00

xT 4T (T, t)Q(T)4)(T, t)x dT

itoo r4)T(T,

t)x(t)]T`w (T) [4(T, t)x(t)]dr

TO

J

O

x(T)T Q(T)x(T)dT.

Since the equilibrium is globally asymptotically stable, there exist k1, k2 such that (see, forexample, Chen [15] pp. 404)

Ix(t)II < kle-k2(t-to)

Thus

xTPx = f kle-k2(T-t) Q(T)kle-k2(T-t) dT

t

ftk2Q(T)e-2k2(T-t) dr.

001

The boundedness of Q(t) implies that there exist M: 11Q(t)II < M. Thus

or

t

2

xTPx < kM2

Also, this implies that xTPx < A2IIx1I2 On the other hand, since Q is positive definite,3a > 0 such that xT Qx > axT x. Also, A bounded implies that I I A(t) I I < N, Vt E R. Thus

00

xT4T (T, t)Q(T)d)(T, t)x dT

tx(T)TQ(T)x(T) d7

f' aIIAN)IIIIx(T)II2 dT

Thus,

or

xTPx = Jk2Me-2k2(T-t) dr.

"01

f' N II x(T)T A(T)x(T) II dT

> II f Nx(T)T drx(T) dTII NxTx.

xTPx > aNxTx

AlIIxII2 << xTPx

A1IIxII2 < xTPx < A2IIxII2

which shows that P is positive definite and bounded. That P is symmetric follows fromthe definition. Thus, there remains to show that P satisfies (A.8). We know that (see Chen[15], Chapter 4,

at) = -45(T, t)A(t).

A.3. CHAPTER 6 315

Thus,

f IfP =J

(DT(T,t)Q(T)a(D(T,t) dr+J 4T(T,t)Q(T)41(T,t)dT - Q(t)

itrJ V-l A(t)-AT(t) 004pT (T, t)Q(T)4D (T, t) dTl - Q(t)

J

which implies that

A.3 Chapter 6

P = -P(t)A(t) - AT(t)p(t) - Q(t).

Theorem 6.1 Consider a function P(s) E R(s). Then F(s) E A if and only if (i) P(s) isproper, and (ii) all poles of F(s) lie in the left half of the complex plane.

Proof: Assume first that P(s) satisfies (i) and (ii), and let n(s)andd(s) E R[s] respectively,be the numerator and denominator polynomials of F(s). Dividing n(s) by d(s), we canexpress F(s) as:

F(s) = k + d s) = k + G(s) (A.9)

where k E 1R and d(s) is strictly proper. Expanding G(s) in partial fractions and antitrans-forming (A.9), we have

n

f (t) = G-1{F(s)} = kb(t) + E g2(t)2=1

where n is the degree of d(s) and each g2 has one of the following forms [u(t) representsthe unit step function : (i) ktii-1e)tu(t), if A < 0 is a real pole of multiplicity m; and(ii) ke(°+.3.+)tu(t), a < 0, if A = (v +,yw) is a complex pole. It follows that f (t) E A, andso F(s) E A. To prove the converse, notice that if (i) does not hold, then f (-) containsderivatives of impulses and so does not belong to A._Also, if (ii) does not hold, then, clearly,for some i, g2 ¢ G1 and then f (t) V A and F(s) il A, which completes the proof.

Theorem 6.2 Consider a linear time-invariant system H, and let represent its impulseresponse. Then H is L stable if and only if hoo(t) + ha,(t) E A and moreover, if His Gp stable, then JjHxjjr < jh11A11x11p.

Proof: The necessity is obvious. To prove sufficiency, assume that E A and considerthe output of H to an input u E Cp . We have

h(t) * u(t) = hou(t) + J halt - T)u(T) dTt0

= 91+92We analyze both terms separately. Clearly

IIg1TIIc, = Ilhou(t)IIc,= IhoIIu(t)IIc,.

For the second term we have that

t192(t)I=I I ha(t-T)u(T)dTI <ftlha(t-T)I

Iu(T)I dr0

and choosing p and q E 1R+ such that 1/p + 1/q = 1, it follows that

(A.10)

t t

I92(t)I <- f Iha(t - T)I Iu(T)I dT = f Iha(t -T)I1/p Iha(t - T)I1/gIu(T)I dT. (A.11)

Thus, by Holder's inequality (6.5) with f I ha(t - T)I1/PIu(T)I and I halt - 7-)I11qwe have that

1I92(t)I C (ft Iha(t-T)IdTJ(ft -T)u(T)dT)

1/p(II(ha)t!IGI )1/q (ft lha(t-T)IU(T)IdT)<-

(II(g2)TIIG,)pfT

= I92(3)Ip ds

s JOT (II(ha)TIIG, )p/q (f t Iha(s -T)IIu(T)I' dT) ds

(II(ha)TIIGI )p/q j (ft ha(s -T)IIu(r)IP dT) ds

and reversing the order of integration, we obtain

fIUMIPt T

1(II(92)TIIG,)' -< (II(ha)TIIG1)p/q (f Iha(s - T)I ds) dr

0

A.3. CHAPTER 6 317

< (II(ha)TIIGI )p/q II(ha)TIIC1 (IIuTIIc,)p

= (II(ha)TIIC, )P (IIuTIIG,)p.

It follows thatII(92)TIIc, <- II(ha)TIIG, IIuTIIc,. (A.12)

Thus, from (A.12) and (A.10) we conclude that

II(h * u)TIIc, < II(91)TII C, + II(92)TIIc,

<- (Ihol + II(ha)TIIc,) IkTIic,

and since, by assumption, u E LP and E A, we can take limits as T -a oo. Thus

IIHullc, = IIh * ullc, = (Ihol + IIhaIIc1) IIUIIC,.

11

Theorem 6.6 Consider the feedback interconnection of the subsystems H1 and H2 : £2e --->Gee. Assume H2 is a nonlinearity 0 in the sector [a, /3], and let H1 be a linear time-invariantsystem with transfer function G(s) of the form:

G(s) = 9(s) + d(s)

Where G(s) satisfies assumptions (i)-(iii) stated in Theorem 6.6. Under these assumptions,if one of the following conditions is satisfied then the system is G2 stable:

(a) If 0 < a < /3: The Nyquist plot of G(s) is bounded away from the critical circle C',centered on the real line and passing through the points (-a-1 +30) and (-/3-1 +30),and encircles it v times in the counterclockwise direction, where v is the number ofpoles of d(s) in the open right half plane.

(b) If 0 = a < /3: G(s) has no poles in the open right half plane and the Nyquist plot ofG(s) remains to the right of the vertical line with abscissa -0-1 for all w E IR.

(c) If a < 0 < 0: G1s) has no poles in the closed right half of the complex plane and theNyquist plot of G(s) is contained entirely within the interior of the circle C'.

Proof: Consider the system S of equations (6.23) and (6.24) (Fig. 6.6) and define thesystem SK by applying a loop transformation of the Type I with K = q defined as follows:

f (l3 + a)o .(A. 13)2

x

Figure A.3: Characteristics of H2 and H2.

We have

Hi ]-s =H= H [1+G(s)

(A 14)is q[1 + qd (s)]

.

H2' = H2-q. (A.15)

By Theorem 6.4, the stability of the original system S can be analyzed using themodified system SK. The significance is that the type I loop transformation has a con-siderable effect on the nonlinearity 0. Namely, if H2 = ¢ E [a, I3], it is easy to see thatH2 = 0' E [-r, r], where

f (Q a)r 2 (see Figure A.3.)

Thus, the gain of H2 is-y(H2) = r (A.16)

According to the small gain theorem, if the following two conditions are satisfied, then thesystem is stables.

(i) Hl and H2 are G2 stable.

(ii) 7(Hi)7(Hz) < 1.

Condition (i): H2 is bounded by assumption, and moreover, 7(H2) = r. By lemma 6.9.1,Hi is G2 stable if and only if the Nyquist plot of d(s) encircles the point (-q-s + 30) vtimes in counterclockwise direction.

'Here condition (ii) actually implies condition (i). Separation into two conditions will help clarify therest of the proof.

A.3. CHAPTER 6 319

Condition (ii): H1 and Hi are linear time-invariant systems; thus, the ,C2 gain of H1 is

'Y(H1) = j1G1j. = sup IG(7w)I-

Thus

w

ry(Hl)ry(H2) < 1 (A.17)

if and only if

or

G(7w)r [sup l[1+gG(.7w)J < 1 (A.18)

rlG(yw)l < l1+gG(.7w)l dwER. (A.19)

Let G(3w) = x + jy. Then equation (A.19) can be written in the form

r2(x2 + y2)

x2(82 - r2) + y2(g2 - r2) + 2qx + 1

a,3(x2 + y2) + (a + 3)x + 1

We now analyze conditions (a)-(c) separately:

< (1 + qx)2 + g2y2

> 0> 0.

(a) 0 < a <,3: In this case we can divide equation (A.20) by a,3 to obtain

x2 + y2 + (0, a)3)x + > 0

which can be expressed in the following form

(x + a)2 + y2 > R2

where_ a+Q _ (/3-a)

a 2af R2a,3

(A.20)

(A.21)

(A.22)

(A.23)

Inequality (A.22) divides the complex plane into two regions separated by the circle Cof center (-a + 30) and radius R. Notice that if y = 0, then C satisfies (x + a)2 = R2,which has the following solutions: x1 = (-/3-1 + 30) and x2 = (-a-1 + 30). ThusC = C', i.e. C is equal to the critical circle C' defined in Theorem 6.6. It is easy to seethat inequality (A.22) is satisfied by points outside the critical circle. Therefore, thegain condition y(Hi)y(H2) < 1 is satisfied if the Nyquist plot of G(s) lies outside thecircle C*. To satisfy the stability condition (i), the Nyquist plot of d(s) must encirclethe point (-q-1 +30), v times in counterclockwise direction. Since the critical point(-q-1 +, j0) is located inside the circle C*, it follows that the Nyquist plot of d(s)must encircle the entire critical circle C' v times in the counterclockwise direction,and part (a) is proved.

(b) 0 = a <)3: In this case inequality (A.22) reduces to x >,3-1, or Q[((3w)] > -Q-1

Thus, condition (ii) is satisfied if the Nyquist plot of d(s) lies entirely to the right ofthe vertical line passing through the point (-/3-1 + 30). But -q-1 = -2(a +,0)-',and if a = 0, -q-1 = -2,0-1. Thus, the critical point is inside the forbidden regionand therefore d(s) must have no poles in the open right half plane.

(c) a < 0 <,3: In this case dividing (A.22) by a/3, we obtain

x2 + y2 + (a+ 0) x + a < 0 (A.24)

where we notice the change of the inequality sign with respect to case (a). Equation(A.24) can be written in the form

(x + a)2 + y2 < R2 (A.25)

where a and R2 are given by equation (A.23). As in case (a) this circle coincides withthe critical circle C`, however, because of the change in sign in the inequality, (A.24)is satisfied by points inside C. It follows that condition (ii) is satisfied if and only ifthe Nyquist plot of d(s) is contained entirely within the interior of the circle C'. Tosatisfy condition (i) the Nyquist plot of C(s) must encircle the point (-q-1 + 30) vtimes in the counterclockwise direction. However, in this case (-q-1 +30) lies outsidethe circle C'. It follows that d(s) cannot have poles in the open right half plane.Poles on the imaginary axis are not permitted since in this case the Nyquist diagramtends to infinity as w tends to zero, violating condition (A.24). This completes theproof of Theorem 6.6.

A.4 Chapter 7

Theorem 7.4: Consider the system (7.1). Assume that the origin is an asymptoticallystable equilibrium point for the autonomous system ,k = f (x, 0), and that the function f (x, u)is continuously differentiable. Under these conditions (7.1) is locally input-to-state-stable.

Proof of theorem 7.4: We will only sketch the proof of the theorem. The reader shouldconsult Reference [74] for a detailed discussion of the ISS property and its relationship toother forms of stability. Given that x = 0 is asymptotically stable, the converse Lyapunovtheorem guarantees existence of a Lyapunov function V satisfying

al(IIxII) < V(x(t)) < a2(jjxjj) dx E D (A.26)

OV (x)8x -

f (x 0) < -c' (11X11) Vx E D (A 27)

A.4. CHAPTER 7 321

andavax

< ki lxll, kl >0 (A.28)

We need to show that, under the assumptions of the theorem, there exist a E K such that

a f(x,u) < -a(IIxID, dx : lxll > a(IIull)

To this end we reason as follows. By the assumptions of the theorem, f (,) satisfies

ll f (x, u) - f (x, 0)II < L Ilull, L > 0 (A.29)

the set of points defined by u : u(t) E Rm and lull 0.

Now consider a function a E IC with the property that

a(llxll) >- a3(I1xll) +E, dx : IIxII >- X(Ilull)

With a so defined, we have that,

eXf(x,u) < -a(IIxII), Vx : II4II ? X(Ilull)


Theorem 7.5: Consider the system (7.1). Assume that the origin is an exponentially stableequilibrium point for the autonomous system t = f (x, 0), and that the function f (x, u) iscontinuously differentiable and globally Lipschitz in (x, u). Under these conditions (7.1) isinput-to-state stable.

Proof of Theorem 7.5: The proof follows the same lines as that of Theorem 7.4 and isomitted.

Proof of theorem 7.7: Assume that (a, a) is a supply pair with corresponding ISS Lya-punov function V, i.e.,

VV(x) f(x,u) < -a(Ilxll)+a(Ilull) !1x E Rn,u E Rm. (A.30)

We now show that given o E 1C,, I&

VV(x) f(x,u) < -a(Ilxll) +a(Ilull) Vx E W ,u E R'n. (A.31)

To this end we consider a new ISS Lyapunov function candidate W, defined as follows

W = p o V = p(V(x)), where (A.32)

p(s) = f q(t) dt (A.33)0

where pt+ -+ R is a positive definite, smooth, nondecreasing function.

We now look for an inequality of the form (A.30) in the new ISS Lyapunov functionW. Using (A.32) we have that

W = aW f(x,u) = P (V(x))V(x,n)= q[V(x)) [a(IIuII)-a(IIxII)l. (A.34)

Now define

that is,

9=aoa-1o(2o)

0(s) = d(a-1(2a)).

We now show that the right hand-side of (A.34) is bounded by

q[9(IIuIUa(IIuII) - 1q[V(x)1a(IIxII)-

To see this we consider two separate cases:

(i) a(IIuII) < za(IIxII): In this case, the right hand-side of (A.34) is bounded by

-2q[V(x)]a(IIxII)

(A.35)

(A.36)

(ii) 2a(IIxII) < a(IIuII): In this case, we have that V(x) 5 a(IIxII) < 9(IIuII).

From (i) and (ii), the the right-hand side of (A.34) is bounded by (A.36).

The rest of the proof can be easily completed if we can show that there exist & E 1C,,.such that

q[9(r)]o(r) - 2q[a(s))a(s) < a (r) - a(s) Vr,s > 0. (A.37)

To this end, notice that for any /3, /3 E 1C,,. the following properties hold:

Property 1: if /3 = O(13(r)) as r -+ oo, then there exist a positive definite, smooth, andnondecreasing function q:

q(r)/3(r) < 4(r) Vr E [0, oo).

A.4. CHAPTER 7 323

With this property in mind, consider a E K. [so that 8, defined in (A.35) is 1C.], anddefine

/3(r) = a(B-1(r)), / (r) = &(O-'(r))-

With these definitions, /3(.) and E K. By property 1, there exist a positive definite,smooth, and nondecreasing function q(-):

q(r)/(r) < /3(r) Vr E [0, oo).

Thus

q[9(r)]a(r) < &. (A.38)

Finally, defining

a(s) = 1 q(c (s))a(s) (A.39)

we have thatq[a(s)]a(s) > 2&(s) (A.40)

j q[B(s)]a(s) < &(r)

and the theorem is proved substituting (A.40) into (A.37).

Proof of Theorem 7.8: As in the case of Theorem 7.8 we need the following property

Property 2: if /3 = 0(13(r)) as r -f 0+, then there exist a positive definite, smooth, andnondecreasing function q:

a(s) < q(s),3(s) Vs E [0, oc).

Defining /3(r) = 2a[9-1(s)] we obtain /3(r) = d[9-1(s)]. By property 2, there exist q:

/3 < q(s)Q(s) Vs E [0, oo).

Thus

-2q[9(s)]a(s) (A.41)

Finally, defining

a(r) = q[9(r)]a(s) (A.42)

we have that (A.40) is satisfied, which implies (A.37).

A.5 Chapter 8

Theorem 8.3: Let H : Xe -> Xe , and assume that (I + H) is invertible in Xe , i.e., assumethat (I + H)-1 : Xe --> Xe . Define the function S : Xe --> Xe :

S = (H - I)(I + H)-1 (A.43)

We have

(a) H is passive if and only if the gain of S is at most 1, that is, S is such that

II(Sx)TIIx s IIxTIIx Vx E XefVT E Xe (A.44)

(b) H is strictly passive and has finite gain if and only if the gain of S is less than 1.

(a) : By assumption, (I + H) is invertible, so we can define

xr (I + H)-1y (A.45)

(I + H)x = y (A.46)

Hx = y - x (A.47)

= Sy = (H - I)(I + H)-1y = (H - I)x (A.48)

IISyIIT == ((H - I)x, (H - I)x >T(Hx-x,Hx-x>T

= IIHxliT + IIxIIT - 2(x, Hx >T (A.49)

IIyIIT = (y, y >T=((I + H)x, (I + H)x >T

IIHxIIT+IIxIIT+2(x,Hx>T (A.50)

Thus, subtracting (A.50) from (A.49), we obtain

IISyIIT = IIyIIT - 4(x, Hx >T (A.51)

Now assume that H is passive. In this case, (x, Hx >T> 0, and (A.51) implies that

IISyIIT - IIyIIT = IISyIIT <- IIyIIT (A.52)

which implies that (A.44) is satisfied. On the other hand, if the gain of S is less thanor equal to 1, then (A.51) implies that

4(x, Hx >T= IIyIIT - IISyIIT - 0

and thus H is passive. This completes the proof of part (a).

A.5. CHAPTER 8 325

(b) : Assume first that H is strictly passive and has finite gain. The finite gain of Himplies that

IIHxIIT << y(H)IIxIIT

Substituting (A.47) in the last equation, we have

II Y - XIIT y(H)IIXIIT

IIYIIT - IIyIIT I<- y(H)IIxIIT

IIyIIT (1+y(H))IIxIIT

or

(1 + y(H))-l IIyIIT <_ IIyIIT.

Also, if H is strictly passive, then

(A.53)

(x,Hx>T >_ SIxIT4(x, Hx >T > 461I x I T (A.54)

Substituting (A.53) in (A.54), we obtain

4(x, Hx >T > 45(1 + y(H))-211YIITor 4(x,Hx >T > b'IIyIIT

where0 < 5' < min[l, 45(1 + -y(H) )-2].

Thus, substituting (A.55) in (A.51), we see that

ISyIIT +5'IIyIIT <- IIyIIT

= IISyIIT <- (1- 5')112IIyIIT

(A.55)

(A.56)

and since 0 < 5' < 1, we have that 0 < (1 - 51)1/2 < 1, which implies that S has gainless than 1.

For the converse, assume that S has gain less than 1. We can write

ISyIIT (1 - 5')IIyIIT 0 < 5' < 1.

Substituting IISyIIT by its equivalent expression in (A.51), we have

IIyIIT - 4(x, Hx >T IIyIIT - 51IIyII2T

= 4(x,Hx >T > 5'11yI

(A.57)

and substituting IIyIIT using (A.50), we obtain

4(x, Hx >T 6'(II HxIIT + IIxIIT + 2(x, Hx >T)

(4 - 26')(x,Hx>T 6'(IIHxJIT+IIxIIT)26'IIxIIT

and since 0 < 6' < 1, we have

(x, Hx >T>it

IIxIIT(4 - 26')

so that H is strictly passive. To see that H has also finite gain, substitute IISyJIT andIIyIIT in (A.56) by their equivalent expressions in (A.49) and (A.50) to obtain

6'II HxIIT < (4- 26')(x, Hx >T -6'IIxIIT< (4 - 26')IIHx1ITIIxjIT - 6'II x112 by Schwartz' inequality

and since 6'11x117. > 0,

so that

6'II HxIIT < (4 - 26')IIHxIITIIxIIT

II HxIIT (4 626/) IIxIIT

from where it follows that H has finite gain. This completes the proof.

Theorem 8.10 Consider the feedback interconnection of Figure 7.5, and assume that

(i) Hl is linear time-invariant, strictly proper, and SPR.

(ii) H2 is passive (and possibly nonlinear).

Under these assumptions, the feedback interconnection is input-output-stable.

Proof: We need to show that y E X whenever u E X. To see this, we start by applying aloop transformation of the type I, with K = -e, e > 0, as shown in Figure A.4.

We know that the feedback system S of Figure 7.5 is input-output stable if andonly if the system SE of Figure A.4 is input-output stable. First consider the subsystemH2' = H2 + EI. We have,

(XT, (H2 + EI)xT >= (XT, H2xT > +e(XT, XT » e(xT, XT > .

A.5. CHAPTER 8 327

Hi = H1(I - eHl)-1

H1

:.......................................................... :

........ - ..............................................

H2 14

Yi

H2 =

:.......................................................... :

H2+e

Figure A.4: The feedback system S.

It follows that HZ is strictly passive for any e > 0. We now argue that Hi is SPR forsufficiently small E. To see this, notice that, denoting 7(H1), then by the small gain theorem,Hi is stable for any e < '. Also, since H1 is SPR, there exist positive definite matrices Pand L, a real matrix Q and µ sufficiently small such that (8.29)-(8.30) are satisfied. Butthen it follows that, given P, L, Q, µ,

P(A + eBC) + (A + eBC)T P = _QQT - µL + 2CCT C

and provided that 0 < e < e' = p)tmjn[L]/(2A,,,ny[CTC]), the matrixµL-2eCTC is positivedefinite. We conclude that for all e E (0, min(c*, -y- 1)), Hi is strong SPR, hence passive,and HZ is strictly passive. Thus, the result follows from Theorem 8.4.

A.6 Chapter 9

Theorem 9.2: The nonlinear system Eli given by (9.13) is QSR-dissipative [i.e., dissipativewith supply rate given by (9.6)] if there exists a differentiable function 0 : ]Rn -, R andfunction L : Rn -> Rq and W : 1Rn Rgxm satisfying

fi(x) > 0, 0(0) = 0 (A.58)

81-31-f (x) =ax

gT

(O)ThT (x)Qh(x) - LT(x)L(x)

ST h(x) - WT L(x)

(A.59)

(A.60)

R = WTW (A.61)

Proof: Sufficiency was proved in Chapter 9. To prove necessity we show that the availablestorage defined in Definition 9.5 is a solution of equations (A.61)(A.61) for appropriatefunctions L and W.

Notice first that for any state x0 there exist a control input u E U that takes the statefrom any initial state x(t_1) at time t = t-1 to xo at t = 0. Since the system is dissipative,we have that

t

t'In particular

,

w(t) dt = O(x(tl)) - 4(x(to)) > 0.

w(t) dt + w(t) dt = cb(x(T)) - O(x(t_1)) > 0/

Tw(t) dt =

/o

, foT

t, t

A.6. CHAPTER 9 329

that isff f

JT w(t) dt > -

fow(t) dt

o Jt,The right-hand side of the last inequality depends only on xo, whereas u can be chosenarbitrarily on [0, T]. Hence there exists a bounded function C : R' --4 IR such that

JO

t

which implies that 0a is bounded. By Theorem 9.1 the available storage is itself a storagefunction, i.e.,

i w(s) ds > 4a(x(t)) - 0a(xo) Vt > 0

which, since 0a is differentiable by the assumptions of the theorem, implies that

d(x,u) fw - d dtx) > 0.

Substituting (9.13), we have that

d(x,u) +w(u,y)

a0axx) [f(x) +g(x)u] +yTQy+2yTSu+uTRu

00"2x) f(x) -aa(x)g(x)u+hTQh+2hTSu+UTRu

we notice that d(x, u), so defined, has the following properties: (i) d(x, u) > 0 for all x, u;and (ii) it is quadratic in u. It then follows that d(x, u) can be factored as

d(x,u) = [L(x) + Wu]T [L(x) + Wu]= LTL+2LTWu+uTWTWu

which implies that

R = WTWaSa

f = hT (x)Qh(x) - LT (x)L(x)a

2 ax

Tw(t) dt > C(xo) > -oc

T TL(x).h(x) - W-g - = Sx

1 T aSa



A.7 Chapter 10

Theorem 10.2: The system (10.28) is input-state linearizable on Do C D if and only ifthe following conditions are satisfied:

(i) The vector fields {g(x), adfg(x), , adf-lg(x)} are linearly independent in Do.Equivalently, the matrix

C = [g(x), adfg(x), ... , adf-lg(x)}nxn

has rank n for all x E Do.

(ii) The distribution A = span{g, ad fg, , adf-2g} is involutive in Do.

Proof: Assume first that the system (10.28) is input-state linearizable. Then there existsa coordinate transformation z = T(x) that transforms (10.28) into a system of the form.z = Az + By, with A = Ac and B = B,, as defined in Section 10.2. From (10.17) and(10.18) we know that T is such that

ax f (x) = T2(x)

axf(x) = T3(x)

ax 1 f W = Tn(x)

T. f(x) i4 0

and

ag(x) = 0l

ag(x) = 02

(A.62)

(A.63)

8x 1 g(x) = 0

a ng(x) 54 0.

A. 7. CHAPTER 10 331

Equations (A.62) and (A.63) can be rewritten as follows:

LfTi=LfTi+1, (A.64)

L9T1 = L9T2 = = L9Tn_1 = 0 L9Tn # 0 (A.65)

By the Jabobi identity we have that

VT1[f,g] _ V(L9T1)f -V(LfT1)g0-L9T2=0

or

Similarly

VTladfg = 0.

OT1adfg = 0 (A.66)

VT1ad f-1g # 0. (A.67)

We now claim that (A.66)-(A.67) imply that the vector fields g, adf g, , adf-1g are lin-early independent. To see this, we use a contradiction argument. Assume that (A.66)(A.67) are satisfied but the g, adfg, , adf-1g are not all linearly independent. Then, forsome i < n - 1, there exist scalar functions Al (X), A2(x), , .i_1(x) such that

i-1adfg = Ak adfg

k=0

and thenn-2

adf-19= adfgk=n-i-1

and taking account of (A.66). we conclude that

n-2

OT1adf 1g = > Ak VT1 adf g = 0k=n-i-l

which contradicts (A.67). This proves that (i) is satisfied. To prove that the second propertyis satisfied notice that (A.66) can be written as follows

VT1 [g(x), adfg(x), ... , adf 2g(x)] = 0 (A.68)

that is, there exist T1 whose partial derivatives satisfy (A.68). Hence A is completelyintegrable and must be involutive by the Frobenius theorem.

Assume now that conditions (i) and (ii) of Theorem 10.2 are satisfied. By the Frobe-nius theorem, there exist Tl (x) satisfying

L9T1(x) = LadfgTl = ... = Lad,._2gT1 = 0

and taking into account the Jacobi identity, this implies that

L9T1(x) = L9L fT1(x) _ ... = L9L' -2Ti (x) = 0

but then we have that

VT1(x)C=OT1(x)[9, adf9(x), ..., adf-lg(x)] = [0, ... 0, Ladf-19Ti(x)]

The columns of the matrix [g, adfg(x), , adf-lg(x)] are linearly independent on Doand so rank(C) = n, by condition (i) of Theorem 10.2, and since VT1(x) # 0, it must betrue that

adf-19Ti(x) 0

which implies, by the Jacobi identity, that

LgL fn-1T1(x) 4 0.

Proof of theorem 10.3:

The proof of Theorem 10.3 requires some preliminary results. In the following lemmaswe consider a system of the form

f x = f (x) + g(x)u, f, g : D C R" - RnS

(A.69)l y=h(x), h:DClg"->R

and assume that it has a relative degree r < n.

Lemma A.1 If the system (A.69) has relative degree r < n in S2, then

k - f 0Lad)fgLfh(x) t (-1) L9Lf lh(x) 54 0

0<j+k<r-1j+ k = r - 1 (A.70)

VxEc, Vj<r-1, k>0.

Proof: We use the induction algorithm on j.

A.7. CHAPTER 10 333

(i) j = 0: For j = 0 condition (A.70) becomes

LAkh(x)= (0 0<k<r-1tl L9Lf 1h(x) k = r - 1

which is satisfied by the definition of relative degree.

(ii) j = is Continuing with the induction algorithm, we assume that

k _ 0 0<i+k<r-1(-1)'L9Lf lh(x) i + k = r - 1

is satisfied, and show that this implies that it must be satisfied for j = i + 1.

From the Jacobi identity we have that

(A.71)

Lad1gR. = LfLpa - LOLfA

for any smooth function A(x) and any smooth vector fields f (x) and 3(x). Defining

A = Lf h(x)= adfg

we have thatLaf i9Lf h(x) = LfLa fgLfh(x) - Lad;I9Lf+lh(x) (A.72)

Now consider any integer k that satisfies i + 1 + k < r - 1. The first summand inthe right-hand side of (A.72) vanishes, by (A.71). The second term on the right-handside is

k+ 1h() 10 0<i+k+1<r-1-LadisLf x = (-1)'L9Lf lh(x) i + k + 1 = r - 1

and thus the lemma is proved.

Lemma A.2 If the relative degree of the system (A.69) is r in 1, then Vh(x), VLfh(x),, VL f lh(x) are linearly independent in Q.

Proof: Assume the contrary, specifically, assume that Vh(x), VLfh(x), , VLf'h(x)are not linearly independent in 0. Then there exist smooth functions a1(x), , ar (x)such that

aiVh(x) + a2VLfh(x) + + arVLf lh(x) = 0. (A.73)

Multiplying (A.73) by adfg = g we obtain

a1Lgh(x) + a2L9Lfh(x) + + arLgLf 1h(x) = 0. (A.74)

By assumption, the system has relative degree r. Thus

L9L fh(x) = 0 for 0 < i < r - 1LgLf lh(x) 0.

Thus (A.74) becomesarLgLr 1h(x) = 0

and since L9Lr 1 76 0 we conclude that ar must be identically zero in Q.

Next, we multiply (A.73) by ad fg and obtain

aladfgh(x) + a2adfgL fh(x) + + aradfgLr 1h(x) = -ar-lad fgLf lh(x) = 0

where lemma A.1 was used. Thus, ar-1 = 0. Continuing with this process (multiplyingeach time by ad fg, ... , adr-1g we conclude that a1 i , ar must be identically zero on0, and the lemma is proven.

Theorem 10.3: Consider the system (10.41) and assume that it has relative degree r <n Vx E Do C D. Then, for every x0 E Do there exist a neighborhood S of x0 and smoothfunction µ1i , such that

(i) L9uj(x)=0, for 1 -<i <n-r, VxC-Q, and

(i2)

F Ai (x) 1

T(x) =01

L Or J

01 = h(x)02 = Lfh(x)

Or = Lf 1h(x)

is a diffeomorphism on I.

A.7. CHAPTER 10 335

Proof of Theorem 10.3: To prove the theorem, we proceed as follows.

The single vector g is clearly involutive and thus, the Frobenius theorem guaranteesthat for each xp E D there exist a neighborhood SZ of xo and n - 1 linearly independentsmooth functions µl, . ,µ._1(x) such that

Lyµi(x) = 0 for 1 < i < n - 1 dx E SZ

also, by (A.73) in Lemma A.2, Vh(x), , VLf-2h(x) are linearly independent. Thus,defining

r µ1(x) 1

T(x) =

i.e., with h(x), , Lf lh(x) in the last r rows of T, we have that

VLFlh(xo) Vspan{Vµ1, ..., phn-1}

= rank (15 (xo)J I = n.

which implies that (xo) # 0. Thus, T is a diffeomorphism in a neighborhood of xo, andthe theorem is proved.

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List of Figures

1.1 mass-spring system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The system i = cos x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 The system ± = r + x2, r < 0 .. . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Vector field diagram form the system of Example 1.6 . . . . . . . . . . . . . 10

1.5 System trajectories of Example 1.7: (a) original system, (b) uncoupled sys-tem ......................................... 12

1.6 System trajectories of Example 1.8 (a) uncoupled system, (b) original system 13

1.7 System trajectories of Example 1.9 (a) uncoupled system, (b) original system 14

1.8 System trajectories for the system of Example 1.10 . . . . . . . . . . . . . . 15

1.9 Trajectories for the system of Example 1.11 . . . . . . . . . . . . . . . . . . 16

1.10 Trajectories for the system of Example 1.12 . . .......... ...... 17

1.11 Stable limit cycle: (a) vector field diagram; (b) the closed orbit.. . . . . . . 19

1.12 Nonlinear RLC-circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.13 Unstable limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.14 (a) Three-dimensional view of the trajectories of Lorenz' chaotic system; (b)two-dimensional projection of the trajectory of Lorenz' system. . . . . . . . 22

1.15 Magnetic suspension system . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.16 Pendulum-on-a-cart experiment . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.17 Free-body diagrams of the pendulum-on-a-cart system .. . . . . . . . . . . . 26

1.18 Ball-and-beam experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.19 Double-pendulum of Exercise (1.5) .. . . . . . . ... . . . .......... 29

345

346 LIST OF FIGURES

3.1 Stable equilibrium point .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2 Asymptotically stable equilibrium point . . . . . . . . . . . . . . . . . . . . . 67

3.3 Mass-spring system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4 Pendulum without friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.5 The curves V(x) =,3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.6 System trajectories in Example 3.21 . . . . . . . . . . . . . . . . . . . . . . . 95

4.1 (a) Continuous-time system E; (b) discrete-time system Ed. . . . . . . . . . 126

4.2 Discrete-time system Ed .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.3 Action of the hold device H . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.1 (a) The system (5.7)-(5.8); (b) modified system after introducing -O(x); (c)"backstepping" of -O(x); (d) the final system after the change of variables. 142

5.2 Current-driven magnetic suspension system .. . . . . . . . . . . . . . . . . . 151

6.1 The system H . .. . . . . .. ..... . .. . ... . .. ........... 155

6.2 Experiment 1: input u(t) applied to system H. Experiment 2: input u(t) =uT(t) applied to system H . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.3 Causal systems: (a) input u(t); (b) the response y(t) = Hu(t); (c) truncationof the response y(t). Notice that this figure corresponds to the left-hand sideof equation (6.8); (d) truncation of the function u(t); (e) response of thesystem when the input is the truncated input uT(t); (f) truncation of thesystem response in part (e). Notice that this figure corresponds to the right-hand side of equation (6.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.4 Static nonlinearity N(.) . . ..... .......... ............. 163

6.5 The systems Hlu = u2, and H2u = e1" 1 . . . . . . . . . . . . . . . . . . . . . 163

6.6 Bode plot of JH(yw)j, indicating the IIHIIoo norm of H...... ...... . 169

6.7 The Feedback System S .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.8 The nonlinearity N(.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.9 The Feedback System S .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.10 The Feedback System SK ......................... .... 176

6.11 The Feedback System 5M .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

LIST OF FIGURES 347

6.12 The nonlinearity 0(t*, x) in the sector [a,,3] .. . . . . . . . . . . . . . . . . . 178

7.1 Cascade connection of ISS systems...... ...... . . . .. . . . . . . . 195

7.2 Cascade connection of ISS systems with input u = 0 . . . . . . . . . . . . . . 197

8.1 Passive network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

8.2 Passive network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

8.3 H=H1+H2+...+H.. . . ........... .. . . . . . . . . . . . . . 209

8.4 The Feedback System S1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

8.5 The Feedback System S1 ............................. 2118.6 The Feedback System S .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

9.1 A Passive system .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

9.2 Mass-spring system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

9.3 Feedback interconnection . . . . .... ... . . . . . . . . .......... 2409.4 Standard setup .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

9.5 Feedback Interconnection used in example 9.2 . . . . . . . . . . . . . . . . . 249

9.6 Standard setup .... . . . . . . . . . . . . . . . . . . . . ........... 250A.1 Asymptotically stable equilibrium point .. . . . . ......... .. . . . . 308

A.2 Asymptotically stable equilibrium point . . . . . . . . . . . . . . . . . . . . . 309

A.3 Characteristics of H2 and H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

A.4 The feedback system Sf . . ... ......... . . . . . . . .. ... . . . . 327

Index

Absolute stability, 179Asymptotic stability in the large, 77Autonomous system, 4Available storage, 231

Backsteppingchain of integrators, 145integrator backstepping, 141strict feedback systems, 148

Ball-and-beam system, 26Basis, 34Bounded linear operators, 48Bounded set, 44

Cartesian product, 46Cauchy sequence, 45Causality, 159Center, 16Chaos, 21Chetaev's theorem, 100Circle criterion, 178Class )CL function, 81Class TC,,. function, 80Class JC function, 80Closed-loop input-output stability, 168Compact set, 45Complement (of a set), 44Complete integrability, 263Contraction mapping, 54

theorem, 54Converse theorems, 125Convex set, 45

Coordinate transformations, 259Covector field, 256

Diagonal form of a matrix, 40Diffeomorphism, 259Differentiability, 49Differentiable function, 49Differential, 49Discrete-time systems, 126

stability, 130Discretization, 127Dissipation inequality, 224

for input-to-state stability, 193Dissipative system, 224Dissipativity, 223

QSR, 226algebraic condition for, 233

Distinguishable states, 296Distribution

involutive, 263nonsingular, 262regular point, 262singular point, 262variable dimension, 262

Distributions, 261

Eigenvalues, 40Eigenvectors, 40Equilibrium point, 5

asymptotically stable, 67convergent, 66exponentially stable, 67

349

350

globally uniformly asymptotically sta-ble, 109

stable, 66uniformly asymptotically stable, 109uniformly convergent, 109uniformly stable, 109unstable, 66

Euclidean norm, 38Exponential stability, 82Extended spaces, 157

Feedback interconnections, 239Feedback linearization, 255Feedback systems, 137

basic feedback stabilization,Finite-gain-stable system, 228Fixed-point theorem, 54Focus

stable, 16unstable, 16

Fractal, 21Frobenius theorem, 264Function spaces, 156

,C2, 156L,, 156

Functions, 46bijective, 46continuous, 47domain, 46injective, 46range, 46restriction, 47surjective, 46uniformly continuous,

Gradient, 51

Hilbert space, 204Holder's inequality, 38

Inner product, 204

47

138

INDEX

Inner product space, 204Input-output linearization, 275Input-output stability, 160Input-output systems, 155Input-State linearization, 265Input-output stability, 159Input-to-state stability, 183, 226Instability, 100Interior point, 44Internal dynamics, 282Invariance principle, 85Invariant set, 86Inverse function theorem, 52Inverted pendulum on a cart, 25ISS Lyapunov function, 186

Jacobian matrix, 50

Kalman-Yakubovich lemma, 218

Lagrange stable equilibrium point, 106LaSalle's Theorem, 90Lie bracket, 257Lie derivative, 256Limit cycles, 18Limit point, 44Limit set, 86Linear independence, 34Linear map, operator, transformation, 48Linear time-invariant systems, 1

£ stability, 164analysis, 96

Linearizationinput-state, 265

Linearization principle, 120Lipschitz continuity, 52Lipschitz systems, 301Loop transformations, 174Lyapunov function, 74Lyapunov stability

autonomous systems, 65, 71

INDEX

discrete-time, 132nonautonomous, 113

Magnetic suspension system, 23, 271Mass-spring system, 1Matrices, 39

inverse, 39orthogonal, 39skew symmetric, 39symmetric, 39transpose, 39

Matrix norms, 48Mean-value theorem, 52Memoryless systems, 162Metric spaces, 32Minimum phase systems, 285Minkowski's inequality, 38

Neighborhood, 44Node, 11

stable, 12unstable, 13

Nonautonomous systems, 107Nonlinear observability, 296Nonlinear observers, 291Nonlinear systems

first order., 5linearization, 99

Normed vector spaces, 37

Observabilitylinear systems, 292nonlinear, 296

ObserverLipschitz systems, 301with linear error dynamics, 298

Observers, 291Open set, 44

Partial derivatives, 50Passive system, 205, 227

351

interconnections, 208Passivity, 201

of linear time-invariant systems, 214theorem, 211

Passivity and small gain, 210Perturbation analysis, 122Phase-plane analysis, 8Poincare-Bendixson theorem., 21Positive definite functions, 69

decrescent, 110discrete-time, 131time dependent, 110

Positive real rational functions, 217Positive system, 207

Quadratic forms, 41

Radially unbounded function, 78Rayleigh inequality, 43Region of attraction, 93Relative degree, 277Riccati equation, 246

Saddle, 13Schwarz's inequality, 204Second order nonlinear systems, 8Sequences, 45Sets, 31Small gain theorem, 171Stability

input-to-state, 183of dissipative systems, 237

State vector, 3Static systems, 162Storage function, 225Strange attractor, 21Strict feedback systems, 148Strictly output-passive system, 228Strictly passive system, 206, 228Strictly positive real rational functions, 217Strictly positive system, 207

352

Subspaces, 36Supply rate, 224System gain, 162

G2 gain of a nonlinear system, 243G2 gain of an LTI system, 167G,,. gain of an LTI system, 166

Topology, 44in R", 44

Total derivative, 49

Ultimate bound, 185Unforced system, 4

Van der Pol oscillator, 19Vector field, 9, 256

diagram, 9Vector spaces, 32Very-strictly-passive system, 229

INDEX

Zero dynamics, 280, 284

nonlinear control systems analysis and design - horacio j. marquez

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