Nonlinear dynamical systems and control: A Lyapunov-based approach

976 Pages·2008·6.34 MB·English

by  Haddad W.

Nonlinear Dynamical Systems
and Control

Nonlinear Dynamical Systems
and Control
A Lyapunov-Based Approach
Wassim M. Haddad
VijaySekhar Chellaboina
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright c 2008byPrincetonUniversityPress
(cid:13)
PublishedbyPrincetonUniversityPress
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LibraryofCongressCataloging-in-PublicationData
Haddad,WassimM.,1961–
Nonlinear dynamical systems and control : a Lyapunov-based approach / Wassim M.
Haddad,VijaySekharChellaboina.
p. cm.
Includes bibliographicalreferencesandindex.
ISBN-9780691133294 (alk. paper)
1. Nonlinearcontroltheory. 2. Automaticcontrol. 3. Lyapunovfunctions. I.Chellaboina,
VijaySekhar,1970-II.Title.
QA402.35.H332008
629.8′36–dc22 2007061026
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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.

Arq n te ka tèlo
èqon oudèn, oÔte aòdion oÔte ˆpeiron.
Anything that has a beginning and an end cannot at the same time be
infinite and everlasting.
—Anaximander of Miletus, Ionia, Greece
Kìsmon (tìnde), tän aÎtän (cid:129)pˆntwn, oÖte ti
jeÀn, oÖte ‚njr¸pwn
âpohsen, ‚ll' ®n ‚eÈ kaÈ êstin kaÈ êstai pÜr ‚ezwon, (cid:129)ptìmenon mètra
kaÈ ‚posbennÔmenon mètra.
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.
Ta pˆnta re ka oÔdèn mènei.
Everything is in a state of flux and nothing is stationary.
Potame
to
auto
embanomen te ka ouk embanomen, emen te ka ouk
emen.
Man cannot step into the same river twice, because neither the man nor
the river is the same.
—Herakleitos of Ephesus, Ionia, Greece
Dì
moi poÔ st¸ kai tan gˆn kinˆsw.
Give me a place to stand and I will move the Earth.
—Archimedes of Syracuse, Sicily, Greater Greece

Contents
Conventions and Notation xv
Preface xxi
Chapter 1. Introduction 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 Rn 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 Dynamical Systems 94
2.12 Limit Points, Limit Sets, and Attractors 96
2.13 Periodic Orbits, Limit Cycles, and Poincar´e-Bendixson
Theorems 103
2.14 Problems 111
2.15 Notes and References 133
Chapter 3. Stability Theory for Nonlinear Dynamical Systems 135
3.1 Introduction 135
3.2 Lyapunov Stability 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