Stability and Stabilization: An Introduction

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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

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