Haddad W.M. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach
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NonlinearBook10pt November 20, 2007
Nonlinear Dynamical Systems
and Control
NonlinearBook10pt November 20, 2007
NonlinearBook10pt November 20, 2007
Nonlinear Dynamical Systems
and Control
A Lyapunov-Based Approach
Wassim M. Haddad
VijaySekhar Chellaboina
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
NonlinearBook10pt November 20, 2007
Copyright
c
2008 by Princeton University Press
Published by Princeton University Press
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press
6 Oxford St, Woodstock, Oxfordshire OX20 1TW
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Haddad, Wassim M., 1961–
Nonlinear dynamical systems and control : a Lyapunov-based approach / Wassim M.
Haddad, Vi jaySekhar Chellaboina.
p. cm.
Includes bibliographical references and index.
ISBN-9780691133294 (alk. paper)
1. Nonlinear control theory. 2. Automatic control. 3. Lyapunov functions. I. Chellaboina,
VijaySekhar, 1970- II. Title.
QA402.35 .H33 2008
629.8
′
36–dc22 2007061026
British Library Cataloging-in-Publication Data is available
This book has been composed in L
A
T
E
X
The publisher would like to acknowledge the authors of this volume for providing the camera-ready
copy from which this book was printed.
Printed on acid-free paper. ∞
press.princeton.edu
Printed in the U nited States of America
10 9 8 7 6 5 4 3 2 1
NonlinearBook10pt November 20, 2007
To my parents Mikhael and Sofia Haddad, with gratitude, appreciation,
and love. Through their actions they inspired me to passionately pursue
knowledge throughout my life because it is a more lasting possession
than anything else
W. M. H.
To my wife Padma, who preserves the stability of my family life
despite life’s nonlinearities, turbulence, and uncertainties while I am
deeply entrenched in my own world
V. C.
NonlinearBook10pt November 20, 2007
NonlinearBook10pt November 20, 2007
Anything that has a beginning and an end cannot at the same time be
infinite and everlasting.
—Anaximander of Miletus, Ionia, Greece
This universe which is the same everywhere, and which no one god or
man has made, existed, exists, and will continue to exist as an eternal
source of energy set on fire by its own natural laws, and will dissipate
under its own laws.
Everything is in a state of flux and nothing is stationary.
Man cannot step into the same river twice, because neither the man nor
the river is the same.
—Herakleitos of Ephesus, Ionia, Greece
Give me a place to stand and I will move the Earth.
—Archimedes of Syracuse, Sicily, Greater Greece
NonlinearBook10pt November 20, 2007
NonlinearBook10pt November 20, 2007
Contents
Conventions and Notation xv
Preface xxi
Chapter 1. Intr oduction 1
Chapter 2. Dynamical Systems and Differential Equations 9
2.1 Introduction 9
2.2 Vector and Matrix Norms 14
2.3 Set Theory and Topology 21
2.4 Analysis in R
n
32
2.5 Vector Spaces and Banach Spaces 56
2.6 Dynamical Systems, Flows, and Vector Fields 71
2.7 Nonlinear Differential Equations 74
2.8 Extendability of Solutions 85
2.9 Global Existence and Uniqueness of Solutions 88
2.10 Flows and Dynamical Systems 92
2.11 Time-Varying Nonlinear Dyna mi cal Systems 94
2.12 Limit Points, Limit Sets, and Attractors 96
2.13 Periodic Orbits, Limit Cycles, a n d Poincar´e-Bendixson
Theorems 103
2.14 Problems 111
2.15 Notes and R eferences 133
Chapter 3. S tability Theory for Nonlinear Dynamical Systems 135
3.1 Introduction 135
3.2 Lyapunov Stabil ity Theory 136
3.3 Invariant Set Stability Theorems 147
3.4 Construction of Lyapunov Functions 152
3.5 Converse Lyapunov Theorems 161
3.6 Lyapunov Instability Theorems 167
3.7 Stability of Linear Systems and Lyapunov’s Linearization
Method 171
NonlinearBook10pt November 20, 2007
x CONTENTS
3.8 Problems 182
3.9 Notes and References 204
Chapter 4. Advanced Stability Theory 207
4.1 Introduction 207
4.2 Partial Stability of Nonlinear Dynamical S y stems 207
4.3 Stability Theory for Nonlinear Time-Varying Systems 225
4.4 Lagrange Sta b ility, Boundedness, and Ultimate Boundedness 236
4.5 Input-to-State Stability 245
4.6 Finite-Time Stability of Nonlinear Dynamical Systems 252
4.7 Semistability of Nonlinear Dynamical Systems 259
4.8 Generalized Lyapunov Theorems 268
4.9 Lyapunov and Asymptotic Stability of Sets 283
4.10 Poincar´e Maps and Stability of Periodic Orbits 290
4.11 Stability Theory via Vector Lyapunov Functions 300
4.12 Problems 313
4.13 Notes and R eferences 322
Chapter 5. Dissipativity Theory for Nonlinear Dynamical Systems 325
5.1 Introduction 325
5.2 Dissipativ e and Exponentially Dissipative Dynami cal Systems 328
5.3 Lagrangian and Hamiltonian Dyna mi cal Syst ems 340
5.4 Extended Kalman-Yakubovich-Popov Conditions for
Nonlinear Dynamical Sys tems 346
5.5 Linearizati on of Di s sipative Dynamical Systems 358
5.6 Positive R eal and Bounded Real Dynamical Systems 361
5.7 Absolute Stability Theory 372
5.8 The Positivity T h eorem and the Circle Criterion 375
5.9 The Popov Criterion 381
5.10 Problems 388
5.11 Notes and R eferences 408
Chapter 6. S tability and Optimality of Feedback Dynamical
Systems 411
6.1 Introduction 411
6.2 Feedback Interconnections of Dissipa tive Dynamical Syst ems 411
6.3 Energy-Based Feedback Control 420
6.4 Stability Margins for Nonlinear Feedback Regulators 434
6.5 Control Lyapunov Functions 438
6.6 Optimal Control and the Hamilton-Jacobi-Bellman Equation 442
6.7 Feedback Linearizati on, Zero Dynamics, and Minimum-
Phase Systems 447
6.8 Problems 457
6.9 Notes and References 469