Extending H°° Control to Nonlinear Systems Advances in Design and Control SIAM's Advances in Design and Control series consists of texts and monographs dealing with all areas of design and control and their applications. Topics of interest include shape optimization, multidisciplinary design, trajectory optimization, feedback, and optimal control. The series focuses on the mathematical and computational aspects of engineering design and control that are usable in a wide variety of scientific and engineering disciplines. Editor-in-Chief John A. Burns, Virginia Polytechnic Institute and State University Editorial Board H. Thomas Banks, North Carolina State University Stephen L. Campbell, North Carolina State University Eugene M. Cliff, Virginia Polytechnic Institute and State University Ruth Curtain, University of Groningen Michel C. Delfour, University of Montreal John Doyle, California Institute of Technology Max D. Gunzburger, Iowa State University Rafael Haftka, University of Florida Jaroslav Haslinger, Charles University J. William Helton, University of California at San Diego Art Krener, University of California at Davis Alan Laub, University of California at Davis Steven I. Marcus, University of Maryland Harris McClamroch, University of Michigan Richard Murray, California Institute of Technology Anthony Patera, Massachusetts Institute of Technology H. Mete Soner, Carnegie Mellon University Jason Speyer, University of California at Los Angeles Hector Sussmann, Rutgers University Allen Tannenbaum, University of Minnesota Virginia Torczon, William and Mary University Series Volumes Helton, J. William and James, Matthew R., Extending H°° Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives Extending H°° Control to Nonlinear Systems Control of Nonlinear Systems to Achieve Performance Objectives J. William Helton University of California, San Diego San Diego, California Matthew R. James Australian National University Canberra, Australia 51HJTL Society for Industrial and Applied Mathematics Philadelphia Copyright © 1999 by the Society for Industrial and Applied Mathematics. 10987654321 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Sdence Center, Philadelphia, PA 19104-2688. Library of Congress Catalogingnn-Publkation Data Helton, J. William, 1944- Extending H°° control to nonlinear systems : control of nonlinear systems to achieve performance objectives / J. William Helton, Matthew R. James. p. cm. - (Advances in design and control) Includes bibliographical references and index. ISBN 0-89871-440-0 (pbk.) 1. H°° control. 2. Nonlinear control theory. I. James, Matthew R. (Matthew Ronald) II. Title. III. Series. QA402.35.H45 1999 629.8'312-dc21 99-35569 w ^^ 5JL3JTL is a registered trademark. Contents Preface xiii How to Read this Book xiv Acknowledgments xv Notation xvii 1 Introduction 1 1.1 The Standard Problem of Nonlinear//00 Control 1 1.1.1 The Plant 2 1.1.2 The Class of Controllers 2 1.1.3 Control Objectives 2 1.1.4 A Classic Example 3 1.2 The Solution for Linear Systems 4 1.2.1 Problem Formulation 4 1.2.2 Background on Riccati Equations 5 1.2.3 Standard Assumptions 5 1.2.4 Problem Solution 6 1.2.5 The Linear Solution from a Nonlinear Viewpoint 7 1.3 The Idea of the Nonlinear Solution 8 1.3.1 The State Feedback Control Problem 8 1.3.1.1 Problem Statement 8 1.3.1.2 Problem Solution 9 1.3.1.3 The State Feedback Central Controller 9 1.3.2 The Information State 11 1.3.2.1 Reversing Arrows 11 1.3.2.2 Definition 12 1.3.3 The Central Controller 14 1.3.4 Equilibrium Information States 14 1.3.5 Finding u* and Validating the Controller 15 1.3.5.1 Construction of the Central Controller 16 1.3.5.2 Validating the Controller 16 1.3.5.3 Storage Functions 17 1.3.6 Example: Linear Systems 18 1.3.6.1 W(p) for Linear Systems 18 1.3.6.2 The Information State 18 V vi Contents 1.4 Singular Functions 19 1.4.1 Singular Equilibrium Information States 21 1.4.2 The Central Controller Dynamics 21 1.4.2.1 Computational Requirements 22 1.5 Attractors for the Information State 23 1.6 Solving the PDE and Obtaining u* 23 1.6.1 Certainty Equivalence 24 1.6.2 Bilinear Systems 25 1.7 Factorization 26 1.8 A Classical Perspective on H°° 26 1.8.1 Control 27 1.8.2 Broadband Impedance Matching 29 1.9 Nonlinear "Loop Shaping" 30 1.10 Other Performance Functions 33 1.11 History 35 1.11.1 Linear Frequency Domain Engineering 35 1.11.2 Linear State Space Theory 36 1.11.3 Factorization 37 1.11.4 Game Theory 37 1.11.5 Nonlinear H°° Control and Dissipative Systems 37 1.11.6 Filtering and Measurement Feedback Control 38 1.11.7 H°° Control, Dynamic Games, and Risk-Sensitive Control . .. 39 1.11.8 Nonlinear Measurement Feedback H°° Control 40 1.11.9 Prehistory 41 1.12 Comments Concerning PDEs and Smoothness 41 1 Basic Theory for Nonlinear H°° Control 43 2 The H°° Control Problem 45 2.1 Problem Formulation 45 2.2 Appendix: Some Technical Definitions 49 2.2.1 Spaces, Convergence 49 2.2.1.1 Singular and Nonsingular Functions and Convergence . . 49 2.2.1.2 Growth at Infinity 50 2.2.2 Some Basic Properties of Functions 51 2.2.2.1 Domain 51 2.2.2.2 Structure 51 2.2.3 Differentiation 51 2.2.4 Transition Operators and Generators 52 2.2.5 Stability 53 2.2.6 Stabilizability 55 2.2.7 Hyperbolicity 55 2.2.8 Observability/Detectability 56 2.3 Notes 57 Contents vii 3 Information States 59 3.1 Differential Games and Information States 59 3.1.1 Cost Function 59 3.1.2 The Information State 61 3.1.3 Information States and Closed-Loop Dissipation 66 3.2 Equilibrium Information States 69 3.2.1 Quadratic Upper Limiting 72 3.3 Information State Dynamics and Attractors 72 3.4 Adjoint Information State 75 3.5 Notes 76 4 Information State Control 77 4.1 Introduction 77 4.2 Information State Controllers 80 4.3 Dynamic Programming 82 4.4 The Dynamic Programming PDE 85 4.4.1 Smooth Nonsingular Information States and Frechet Derivatives 86 4.4.2 Directional Derivatives 88 4.4.3 General Information States 92 4.5 Solving the Dynamic Programming PDE and Dissipation PDI 93 4.5.1 Smoothness 93 4.5.2 Admissibility 95 4.5.3 Solutions of the Dynamic Programming PDE and Dissipation PDI 95 4.5.4 The Value Function Solves the Dynamic Programming PDE . .. 96 4.5.5 Dissipation 100 4.6 Optimal Information State Controllers 101 4.6.1 Direct Minimization and Dynamic Programming 101 4.7 Necessity of an Optimai Information State Solution 105 4.8 Definition of Central Controller 106 4.9 Initialization of Information State Controllers 106 4.9.1 Coupling 107 4.9.2 Null Initialization 114 4.10 Solution of the H°° Control Problem 115 4.11 Further Necessity Results 121 4.12 Optimal Control and Observation 122 4.12.1 Stabilizing Property 122 4.12.2 Zero Dynamics 123 4.13 List of Properties of the Value Function 123 4.14 Notes 124 5 State Feedback H°° Control 127 5.1 Dissipative Systems 127 5.2 Bounded Real Lemma 130 5.3 Strict Bounded Real Lemma 133 viii Contents 5.3.1 Main Results 134 5.3.2 Proofs of Main Results 135 5.4 State Feedback H°° Control 139 5.4.1 The State Feedback Problem 139 5.4.2 A State Feedback H2 Assumption 139 5.4.3 Necessity 140 5.4.4 Sufficiency 143 5.4.5 State Feedback and Its Relation to Output Feedback 144 5.5 Notes 146 6 Storage Functions 147 6.1 Storage Functions for the Information State Closed Loop 147 6.2 Explicit Storage Functions 150 7 Special Cases 155 7.1 Bilinear Systems 155 7.2 Linear Systems 158 7.2.1 Coupling 160 7.2.2 Storage Function 161 7.3 Certainty Equivalence Principle 161 7.3.1 Breakdown of Certainty Equivalence 164 7.4 Notes 165 8 Factorization 167 8.1 Introduction 167 8.2 The Problem 167 8.2.1 Factoring 167 8.2.2 The Setup 169 8.2.3 Dissipation, Losslessness, and Being Outer 169 8.3 The Information State and Critical Feedback 171 8.4 RECIPE for the Factors 171 8.5 Properties of the Factors 172 8.5.1 The Factoring PDE 172 8.5.2 Factoring Assumptions 173 8.5.3 The Outer Factor E° 174 8.5.4 The Inner Factor E7 174 8.5.5 The Inverse Outer Factor (E0)"1 176 8.5.6 Necessity of the RECIPE Formulas 176 8.5.7 Singular Cases 176 8.6 Examples 176 8.6.1 Certainty Equivalence 176 8.6.2 A Stable 178 8.6.3 A Strictly Antistable 178 8.6.4 Bilinear Systems 179 8.6.5 Linear Systems 180 8.7 Factoring and Control 181 Contents ix 8.7.1 RECIPE for Solving the Control Problem 186 8.7.2 Parameterizing All Solutions 187 8.8 Necessity of the RECIPE 187 8.9 State Reading Factors 188 8.9.1 RECIPE for State Reading Factors 188 8.9.2 Properties of State Reading Factors 190 8.9.3 Separation Principle 190 8.10 Nonsquare Factors and the Factoring PDE 191 8.10.1 Nonsquare Factoring PDE 191 8.10.2 Reversing Arrows on One Port 192 8.10.3 Proof of Factoring PDE 195 8.11 Notes 199 9 The Mixed Sensitivity Problem 201 9.1 Introduction 201 9.2 Notation and Other Details 202 9.3 Choosing the Weights 202 9.4 Standard Form 203 9.5 Formula for the Controller 205 9.6 Notes 205 II Singular Information States and Stability 207 10 Singular Information States 209 10.1 Introduction 209 10.2 Singular Information State Dynamics 210 10.2.1 Geometrical Description of Information State Dynamics . . .. 210 10.2.2 Computational Complexity 211 10.3 Interpreting the Dynamic Programming PDE 212 10.3.1 Transition Operators and Generators 212 10.3.2 Certainty Equivalence Case 223 10.3.3 Pure Singular Case 224 10.4 Formulas for the Central Controller 226 10.4.1 General Singular Case 227 10.4.2 Hyperbolic 1 and 2A Block Systems 227 10.4.3 Purely Singular 1 and 2A Block Systems 227 10.4.4 Nonsingular 1 and 2A Block Systems 227 10.4.5 Certainty Equivalence Controller for Hyperbolic 1 and 2A Block Systems 228 10.5 Notes 228 11 Stability of the Information State Equation 231 11.1 Introduction 231 11.1.1 Nonsingular Cases 231 11.1.2 Singular Cases 232