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    CONTENTS
    PREFACE
    CHAPTER DEPENDENCE CHART
    1 MATHEMATICAL NOTATION AND REVIEW
    Vectors
    Matrices
    Quadratic Forms
    Matrix Calculus
    Convergence
    Laplace Transform
    z-Transform
    Exercises
    Notes
    2 STATE EQUATION REPRESENTATION
    Examples
    Linearization
    State Equation Implementation
    Exercises
    Notes
    3 STATE EQUATION SOLUTION
    Existence
    Uniqueness
    Complete Solution
    Additional Examples
    Exercises
    Notes
    4 TRANSITION MATRIX PROPERTIES
    Two Special Cases
    General Properties
    State Variable Changes
    Exercises
    Notes
    5 TWO IMPORTANT CASES
    Time-Invariant Case
    Periodic Case
    Additional Examples
    Exercises
    Notes
    6 INTERNAL STABILITY
    Uniform Stability
    Uniform Exponential Stability
    Uniform Asymptotic Stability
    Lyapunov Transformations
    Additional Examples
    Exercises
    Notes
    7 LYAPUNOV STABILITY CRITERIA
    Introduction
    Uniform Stability
    Uniform Exponential Stability
    Instability
    Time-Invariant Case
    Exercises
    Notes
    8 ADDITIONAL STABILITY CRITERIA
    Eigenvalue Conditions
    Perturbation Results
    Slowly-Varying Systems
    Exercises
    Notes
    9 CONTROLLABILITY AND OBSERVABILITY
    Controllability
    Observability
    Additional Examples
    Exercises
    Notes
    10 REALIZABILITY
    Formulation
    Realizability
    Minimal Realization
    Special Cases
    Time-Invariant Case
    Additional Examples
    Exercises
    Notes
    11 MINIMAL REALIZATION
    Assumptions
    Time-Varying Realizations
    Time-Invariant Realizations
    Realization from Markov Parameters
    Exercises
    Notes
    12 INPUT-OUTPUT STABILITY
    Uniform Bounded-Input Bounded-Output Stability
    Time-Invariant Case
    Exercises
    Notes
    13 CONTROLLER AND OBSERVER FORMS
    Controllability
    Controller Form
    Observability
    Observer Form
    Exercises
    Notes
    14 LINEAR FEEDBACK
    Effects of Feedback
    State Feedback Stabilization
    Eigenvalue Assignment
    Noninteracting Control
    Additional Examples
    Exercises
    Notes
    15 STATE OBSERVATION
    Observers
    Output Feedback Stabilization
    Reduced-Dimension Observers
    Time-Invariant Case
    A Servomechanism Problem
    Exercises
    Notes
    16 POLYNOMIAL FRACTION DESCRIPTION
    Right Polynomial Fractions
    Left Polynomial Fractions
    Column and Row Degrees
    Exercises
    Notes
    17 POLYNOMIAL FRACTION APPLICATIONS
    Minimal Realization
    Poles and Zeros
    State Feedback
    Exercises
    Notes
    18 GEOMETRIC THEORY
    Subspaces
    Invariant Subspaces
    Canonical Structure Theorem
    Controlled Invariant Subspaces
    Controllability Subspaces
    Stabilizability and Detectability
    Exercises
    Notes
    19 APPLICATIONS OF GEOMETRIC THEORY
    Disturbance Decoupling
    Disturbance Decoupling with Eigenvalue Assignment
    Noninteracting Control
    Maximal Controlled Invariant Subspace Computation
    Exercises
    Notes
    20 DISCRETE TIME: STATE EQUATIONS
    Examples
    Linearization
    State Equation Implementation
    State Equation Solution
    Transition Matrix Properties
    Additional Examples
    Exercises
    Notes
    21 DISCRETE TIME: TWO IMPORTANT CASES
    Time-Invariant Case
    Periodic Case
    Exercises
    Notes
    22 DISCRETE TIME: INTERNAL STABILITY
    Uniform Stability
    Uniform Exponential Stability
    Uniform Asymptotic Stability
    Additional Examples
    Exercises
    Notes
    23 DISCRETE TIME: LYAPUNOV STABILITY CRITERIA
    Uniform Stability
    Uniform Exponential Stability
    Instability
    Time-Invariant Case
    Exercises
    Notes
    24 DISCRETE TIME: ADDITIONAL STABILITY CRITERIA
    Eigenvalue Conditions
    Perturbation Results
    Slowly-Varying Systems
    Exercises
    Notes
    25 DISCRETE TIME: REACHABILITY AND OBSERVABILITY
    Reachability
    Observability
    Additional Examples
    Exercises
    Notes
    26 DISCRETE TIME: REALIZATION
    Realizability
    Transfer Function Realizability
    Minimal Realization
    Time-Invariant Case
    Realization from Markov Parameters
    Additional Examples
    Exercises
    Notes
    27 DISCRETE TIME: INPUT-OUTPUT STABILITY
    Uniform Bounded-Input Bounded-Output Stability
    Relation to Uniform Exponential Stability
    Time-Invariant Case
    Exercises
    Notes
    28 DISCRETE TIME: LINEAR FEEDBACK
    Effects of Feedback
    State Feedback Stabilization
    Eigenvalue Assignment
    Noninteracting Control
    Additional Examples
    Exercises
    Notes
    29 DISCRETE TIME: STATE OBSERVATION
    Observers
    Output Feedback Stabilization
    Reduced-Dimension Observers
    Time-Invariant Case
    A Servomechanism Problem
    Exercises
    Notes
    AUTHOR INDEX
    SUBJECT INDEX
    LINEAR SYSTEM THEORY Second Edition WILSON J. RUGH Department of Electrical and Computer Engineering The Johns Hopkins University PRENTICE HALL, Upper Saddle River, New Jersey 07458
    Library of Congress Cataloglng-in•Pubilcatlon Data Linear system theory I Wilson J. Rugh. --2nd ed. p. cot — (Prentice-Hall information and system sciences Includes bibliological references and index. ISBN: 0-13-441205-2 1, Control theory. 2. Linear systems. I. Title. II. Series. Rugh, Wilson I. series) QA402.3R84 1996 003'.74--dc2O 95-21164 CIP Acquisitions editor: Tom Robbins Production editor: Rose Kernan Copy editor: Adrienne Rasmussen Cover designer: Karen Salzbach Donna Suflivan Editorial assistant: PbyIIIs Morgan = — © 1996 by Prentice-Hail, Inc. Simon & Schuster/A Viacom Company Upper Saddle River, NJ 07458 All Tights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs. Printed in the United States of America 10 9 8 7 6 5 4 3 2 ISBN 0—13—441205—2 90000> Prentice-Hall International (UK) Limited, London Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall Canada Inc., Toronto Prentice-Hall Hispanoamencana, S.A., Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo Simon & Schuster Asia Pie. Ltd., Singapore Editora Prentice-Hail do Brasil, Ltda., Rio de Janeiro
    To Terry, David, and Karen
    PRENTICE HALL INFORMATION AND SYSTEM SCIENCES SERIES Thomas Kailath, Editor ANDERSON & MOORE ANDERSON & MOORE ASTROM & BASSEVILLE & NIKIROV BOYD & BARRA'IT DICKINSON FRIEDLAND GARDNER GRAY & DAVISSON GREEN & LIIvIEBEER HAYKIN HAYKIN JAIN JOHANSSON JOHNSON KAILATH KUNG KUNG, W}{ITEHOUSE. & KAILATH, EDS. KWAKERNAAK & SWAN LANDAU LJUNG LIUNG & GLAD MACOVSKI MOSCA NARENDRA & ANNASWAMY RUGH RUGH SASTRY & BODSON SOLIMAN & SRINATH SOLO & KONG SRINATH, RAJASEKARAN, & VISWANATHAN VISWANADHAM & NARAHARI WILLIAMS Optimal Control: Linear Quadratic Methods Optimal Filtering Computer-Controlled Systems: Theory and Design, 2/E Detection of Abrupt Changes: Theory & Application Linear Controller Design: Limits of Perfor,nance Systems: Analysis, Design and Computation Advanced Control System Design Statistical Spectral Analysis: A Nonprobabilistic Theory Random Processes: A Mathematical App roach for Engineers Linear Robust Control Adaptive Filter Theory Blind Deconvolution Fundamentals of Digital Image Processing Modeling and System Identification Lectures on Adaptive Parameter Estimation Linear Systems VLSI Array Processors VLSI and Modern Signal Processing Signals and Systems System Identification and Control Design Using P.I.M. + Software System Identification: Theory for the User Modeling of Dynamic Systems Medical Imaging Systems Stochastic and Predictive Adaptive Control Stable Adaptive Systems Linear System Theory Linear System Theory, Second Edition Adaptive Control: Stability, Convergence, and Robustness Signals and Systems Continuous and Adaptive Signal Processing Algorithms: Stability & Performance Introduction to Statistical Signal Processing with Applications Performance Modeling of Automated Manufacturing Systems Designing Digital Filters
    CONTENTS PREFACE CHAPTER DEPENDENCE CHART 1 MATHEMATICAL NOTATION AND REVIEW 2 3 Vectors Matrices Quadratic Forms Matrix Calculus Convergence Laplace Transform z-Transform 16 Exercises Notes 21 18 8 10 14 11 2 STATE EQUATION REPRESENTATION 24 Examples Linearization State Equation Implementation Exercises Notes 34 38 28 34 3 STATE EQUATION SOLUTION 41 Existence Uniqueness 45 Complete Solution Additional Examples Exercises Notes 53 55 47 50 xiii xv 23 40
    4 TRANSITION MATRIX PROPERTIES Two Special Cases General Properties State Variable Changes Exercises Notes 69 73 58 61 66 5 TWO IMPORTANT CASES 74 Time-Invariant Case Periodic Case 81 Additional Examples Exercises Notes 92 87 96 6 INTERNAL STABILITY 99 Uniform Stability Uniform Exponential Stability Uniform Asymptotic Stability Lyapunov Transformations Additional Examples Exercises Notes 109 110 113 101 106 107 7 LYAPUNOV STABILITY CRITERIA 114 116 Introduction Uniform Stability Uniform Exponential Stability Instability Time-Invariant Case Exercises Notes 123 122 125 129 117 8 ADDITIONAL STABILITY CRITERIA Eigenvalue Conditions Perturbation Results Slowly-Varying Systems Exercises Notes 140 138 131 133 135 9 CONTROLLABILITY AND OBSERVABILITY 142 148 Controllability Observability Additional Examples Exercises Notes 152 155 150 Contents 58 74 99 114 131 142
    Contents 10 REALIZABILITY ix 158 159 160 Formulation Realizability Minimal Realization Special Cases Time-Invariant Case Additional Examples Exercises Notes 180 177 164 162 169 175 11 MINIMAL REALIZATION 182 Assumptions Time-Varying Realizations Time-Invariant Realizations 189 Realization from Markov Parameters Exercises Notes 184 199 201 194 12 INPUT-OUTPUT STABILITY Uniform Bounded-Input Bounded-Output Stability Relation to Uniform Exponential Stability 206 Time-Invariant Case Exercises Notes 216 214 211 13 CONTROLLER AND OBSERVER FORMS Controllability Controller Form Observability Observer Form Exercises Notes 238 234 219 222 231 232 14 LINEAR FEEDBACK Effects of Feedback 241 State Feedback Stabilization Eigenvalue Assignment Noninteracting Control Additional Examples Exercises Notes 258 261 256 244 247 249 15 STATE OBSERVATION 266 Observers Output Feedback Stabilization Reduced-Dimension Observers 269 272 203 182 203 218 240 265
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