Stability and Stabilization is the first intermediate-level textbook that covers stability and stabilization of equilibria for both linear and nonlinear time-invariant systems of ordinary differential equations. Designed for advanced undergraduates and beginning graduate students in the sciences, engineering, and mathematics, the book takes a unique modern approach that bridges the gap between linear and nonlinear systems.
Presenting stability and stabilization of equilibria as a core problem of mathematical control theory, the book emphasizes the subject's mathematical coherence and unity, and it introduces and develops many of the core concepts of systems and control theory. There are five chapters on linear systems and nine chapters on nonlinear systems; an introductory chapter; a mathematical background chapter; a short final chapter on further reading; and appendixes on basic analysis, ordinary differential equations, manifolds and the Frobenius theorem, and comparison functions and their use in differential equations. The introduction to linear system theory presents the full framework of basic state-space theory, providing just enough detail to prepare students for the material on nonlinear systems.
Focuses on stability and feedback stabilization
Bridges the gap between linear and nonlinear systems for advanced undergraduates and beginning graduate students
Balances coverage of linear and nonlinear systems
Covers cascade systems
Includes many examples and exercises
Review:
"This book takes a unique modern approach that bridges the gap between linear and nonlinear systems. . . . Clear formulated definitions and theorems, correct proofs and many interesting examples and exercises make this textbook very attractive."--Ferenc Szenkovits, Mathematica
Endorsement:
"This book is a pleasant surprise. William Terrell selects and presents the field's key results in a fresh and unbiased way. He is enthusiastic about the material and his goal of setting forth linear and nonlinear stabilization in a unified format."--Miroslav Krstic, University of California, San Diego
"This textbook has very positive features. The arguments are complete; it does not shy away from making correct proofs one of its main goals; it strikes an unusually good balance between linear and nonlinear systems; and it has many examples and exercises. It is also mathematically sophisticated for an introductory text, and it covers very recent material."--Jan Willems, coauthor of Introduction to Mathematical Systems Theory
Author(s): William J. Terrell
Publisher: Princeton University Press
Year: 2009
Language: English
Pages: C,xvi+, 457, B
Cover
Front Matter
S Title
Stability and Stabilization: An Introduction
Copyright (c) 2009 by Princeton University Press
ISBN-13: 978-0-691-13444-4
Library of Congress Control Number: 2008926510
Contents
List of Figures
Preface
Chapter One Introduction
1.1 OPEN LOOP CONTROL
1.2 THE FEEDBACK STABILIZATION PROBLEM
1.3 CHAPTER AND APPENDIX DESCRIPTIONS
1.4 NOTES AND REFERENCES
Chapter Two Mathematical Background
2.1 ANALYSIS PRELIMINARIES
2.2 LINEAR ALGEBRA AND MATRIX ALGEBRA
2.3 MATRIX ANALYSIS
2.4 ORDINARY DIFFERENTIAL EQUATIONS
2.4.1 Phase Plane Examples: Linear and Nonlinear
2.5 EXERCISES
2.6 NOTES AND REFERENCES
Chapter Three Linear Systems and Stability
3.1 THE MATRIX EXPONENTIAL
3.2 THE PRIMARY DECOMPOSITION AND SOLUTIONS OF LTI SYSTEMS
3.3 JORDAN FORM AND MATRIX EXPONENTIALS
3.3.1 Jordan Form of Two-Dimensional Systems
3.3.2 Jordan Form of n-Dimensional Systems
3.4 THE CAYLEY-HAMILTON THEOREM
3.5 LINEAR TIME VARYING SYSTEMS
3.6 THE STABILITY DEFINITIONS
3.6.1 Motivations and Stability Definitions
3.6.2 Lyapunov Theory for Linear Systems
3.7 EXERCISES
3.8 NOTES AND REFERENCES
Chapter Four Controllability of Linear Time Invariant Systems
4.1 INTRODUCTION
4.2 LINEAR EQUIVALENCE OF LINEAR SYSTEMS
4.3 CONTROLLABILITY WITH SCALAR INPUT
4.4 EIGENVALUE PLACEMENT WITH SINGLE INPUT
4.5 CONTROLLABILITY WITH VECTOR INPUT
4.6 EIGENVALUE PLACEMENT WITH VECTOR INPUT
4.7 THE PBH CONTROLLABILITY TEST
4.8 LINEAR TIME VARYING SYSTEMS: AN EXAMPLE
4.9 EXERCISES
4.10 NOTES AND REFERENCES
Chapter Five Observability and Duality
5.1 OBSERVABILITY, DUALITY, AND A NORMAL FORM
5.2 LYAPUNOV EQUATIONS AND HURWITZ MATRICES
5.3 THE PBH OBSERVABILITY TEST
5.4 EXERCISES
5.5 NOTES AND REFERENCES
Chapter Six Stabilizability of LTI Systems
6.1 STABILIZING FEEDBACKS FOR CONTROLLABLE SYSTEMS
6.2 LIMITATIONS ON EIGENVALUE PLACEMENT
6.3 THE PBH STABILIZABILITY TEST
6.4 EXERCISES
6.5 NOTES AND REFERENCES
Chapter Seven Detectability and Duality
7.1 AN EXAMPLE OF AN OBSERVER SYSTEM
7.2 DETECTABILITY, THE PBH TEST, AND DUALITY
7.3 OBSERVER-BASED DYNAMIC STABILIZATION
7.4 LINEAR DYNAMIC CONTROLLERS AND STABILIZERS
7.5 LQR AND THE ALGEBRAIC RICCATI EQUATION
7.6 EXERCISES
7.7 NOTES AND REFERENCES
Chapter Eight Stability Theory
8.1 LYAPUNOV THEOREMS AND LINEARIZATION
8.1.1 Lyapunov Theorems
8.1.2 Stabilization from the Jacobian Linearization
8.1.3 Brockett's Necessary Condition
8.1.4 Examples of Critical Problems
8.2 THE INVARIANCE THEOREM
8.3 BASIN OF ATTRACTION
8.4 CONVERSE LYAPUNOV THEOREMS
8.5 EXERCISES
8.6 NOTES AND REFERENCES
Chapter Nine Cascade Systems
9.1 THE THEOREM ON TOTAL STABILITY
9.1.1 Lyapunov Stability in Cascade Systems
9.2 ASYMPTOTIC STABILITY IN CASCADES
9.2.1 Examples of Planar Systems
9.2.2 Boundedness of Driven Trajectories
9.2.3 Local Asymptotic Stability
9.2.4 Boundedness and Global Asymptotic Stability
9.3 CASCADES BY AGGREGATION
9.4 APPENDIX: THE POINCARE-BENDIXSON THEOREM207
9.5 EXERCISES
9.6 NOTES AND REFERENCES
Chapter Ten Center Manifold Theory
10.1 INTRODUCTION
10.1.1 An Example
10.1.2 Invariant Manifolds
10.1.3 Special Coordinates for Critical Problems
10.2 THE MAIN THEOREMS
10.2.1 Definition and Existence of Center Manifolds
10.2.2 The Reduced Dynamics
10.2.3 Approximation of a Center Manifold
10.3 TWO APPLICATIONS
10.3.1 Adding an Integrator for Stabilization
10.3.2 LAS in Special Cascades: Center Manifold Argument
10.4 EXERCISES
10.5 NOTES AND REFERENCES
Chapter Eleven Zero Dynamics
11.1 THE RELATIVE DEGREE AND NORMAL FORM
11.2 THE ZERO DYNAMICS SUBSYSTEM
11.3 ZERO DYNAMICS AND STABILIZATION
11.4 VECTOR RELATIVE DEGREE OF MIMO SYSTEMS
11.5 TWO APPLICATIONS
11.5.1 Designing a Center Manifold
11.5.2 Zero Dynamics for Linear SISO Systems
11.6 EXERCISES
11.7 NOTES AND REFERENCES
Chapter Twelve Feedback Linearization of Single-Input Nonlinear Systems
12.1 INTRODUCTION
12.2 INPUT-STATE LINEARIZATION
12.2.1 Relative Degree n
12.2.2 Feedback Linearization and Relative Degree n
12.3 THE GEOMETRIC CRITERION
12.4 LINEARIZING TRANSFORMATIONS
12.5 EXERCISES
12.6 NOTES AND REFERENCES
Chapter Thirteen An Introduction to Damping Control
13.1 STABILIZATION BY DAMPING CONTROL
13.2 CONTRASTS WITH LINEAR SYSTEMS: BRACKETS, CONTROLLABILITY, STABILIZABILITY
13.3 EXERCISES
13.4 NOTES AND REFERENCES
Chapter Fourteen Passivity
14.1 INTRODUCTION TO PASSIVITY
14.1.1 Motivation and Examples
14.1.2 Definition of Passivity
14.2 THE KYP CHARACTERIZATION OF PASSIVITY
14.3 POSITIVE DEFINITE STORAGE
14.4 PASSIVITY AND FEEDBACK STABILIZATION
14.5 FEEDBACK PASSIVITY
14.5.1 Linear Systems
14.5.2 Nonlinear Systems
14.6 EXERCISES
14.7 NOTES AND REFERENCES
Chapter Fifteen Partially Linear Cascade Systems
15.1 LAS FROM PARTIAL-STATE FEEDBACK
15.2 THE INTERCONNECTION TERM
15.3 STABILIZATION BY FEEDBACK PASSIVATION
15.4 INTEGRATOR BACKSTEPPING
15.5 EXERCISES
15.6 NOTES AND REFERENCES
Chapter Sixteen Input-to-State Stability
16.1 PRELIMINARIES AND PERSPECTIVE
16.2 STABILITY THEOREMS VIA COMPARISON FUNCTIONS
16.3 INPUT-TO-STATE STABILITY
16.4 ISS IN CASCADE SYSTEMS
16.5 EXERCISES
16.6 NOTES AND REFERENCES
Chapter Seventeen Some Further Reading
Back Matter
Appendix A Notation: A Brief Key
Appendix B Analysis in R and R^n
B.1 COMPLETENESS AND COMPACTNESS
B.2 DIFFERENTIABILITY AND LIPSCHITZ CONTINUITY
Appendix C Ordinary Differential Equations
C.1 EXISTENCE AND UNIQUENESS OF SOLUTIONS
C.2 EXTENSION OF SOLUTIONS
C.3 CONTINUOUS DEPENDENCE
Appendix D Manifolds and the Preimage Theorem; Distributions and the Frobenius Theorem
D.1 MANIFOLDS AND THE PREIMAGE THEOREM
D.2 DISTRIBUTIONS AND THE FROBENIUS THEOREM
Appendix E Comparison Functions and a Comparison Lemma
E.1 DEFINITIONS AND BASIC PROPERTIES
E.2 DIFFERENTIAL INEQUALITY AND COMPARISON LEMMA
Appendix F Hints and Solutions for Selected Exercises
Bibliography
Index
Back Cover
