Adaptive Control- Stability, Convergence, and Robu

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详细说明：Adaptive control: stability, convergence and robustnessLibrary of Congress Cataloging-in-Publication Data Sasiry, Shankar( Sosale Shankara Shankar Sastry anEl Mare bodson m. Prentice Hall information and systcm sciences cries entice hall advanced rcference series Bibliography: p Includes in ISRN0-13-001326-5 1. Adaptive cortrol system I. Bodson marc II. Title T217.S271989 88-28940 629836dci9 Prentice-Hall Advanced Reference Series o our parents Prentice-Hall Information and System Sciences Series 1989 by Prentice-Hall, Inc A Division of simon schuster Englewood Cliffs, New Jersey 07632 Al righted a e red No par of this book may be without permission in writing from the publisher rinted in the United States neila 1098765432 工SBN囗一1彐-口Du3己-5 Prentice-Hall International (UK) Limited, Londo Prentice- nall of Australia Pty. I imited sydnie Prentice-Hall Canada Inc. Toronto Prenticc-Hall Hispanoamericana, S.A., Mexico Prentice-Hall of India Private Limited, New Deihi Prentice-Hall of Japan, Inc, tokyo Simon Schuster Asia Pte. Itd., singapore Exliora Prentice-Hall do Brasil. Ltda. Rio de janeiro CONTENTS Preface Chapter 0 Introduction 0.1 Identification and Adaptive control 0.2 Approaches to Adaptive Control 4 0.2.1 Gain Scheduler 0.2.2 Modcl Reference Adaptive systems 2 3 Self tuning rcgt 0.2. 4 Stochastic Adaptive Control 0. 3 A Simple example Chapter 1 Preliminaries 17 1.1 Notat Norms 1.3 Positive Definite matr 19 1.4 Stability of Dynamic Systems 20 1.4.1 DiFerential Equations 1. 4.2 Stabilitv definitions 1.4.3 Lyapunov Stability Theory 1.5 Exponential Stability Theorems 8 1 Exponential Stability of Nonlinear Systems 1.5.2 Exponential Stability of Linear Time-Varying steins 1.5.3 Exponential Stability of Linear Timc Invariant Systems 38 1.6 Generalized Harmonic Analysis 3.8 Exponential Parameter Convergence 154 Chapter 2 Identification 45 3. 9 Conclusions 156 2.0 Introduction 2.1 Identification Problem 2.2 Identifier Structure 5237 Chapter 4 Parameter Convergence Using Averaging Techniques 158 4.0 Introduction 158 2.3 Linear Error Equation and Identification Algorithms 4.1 Examples of Averaging analysis 2.3.1 Gradient Algorithms 58 4.2 Averaging Theory-One-Time Scale 166 2.3.2 Least-Squares algorithm 4.3 Application to Identification 175 2. 4 Properties of the Identification Algorithms 4.4 Averaging Theory-Two-Time Scales 179 Identifier stabilily 4.4.1 Separated Time scales 183 2.4.1 Gradient Algorithms 63 4.4.2 Mixed Time Scales 2.4.2 Least-Squares algorithms 66 4.5 Applications to Adaptive Control 18 2.4.3 Stability of the Identifier 69 4.5. 1 Output error Scheme-Linearized equations 188 2.5 Persistent Excitation and Exponential parametcr 4.5.2 Output Error Scheme-Nonlinear equations 192 Convergence 4.5.3 Input Error Scheme 202 2.6 Model Reference Identifiers-SPR Error Equation 76 4.6 Conclusions 207 2.6.1 Model Reference Identifiers 76 2.6.2 Strictly Positive Real error equation Chapter 5 robustness 209 and Identification algorithms 5. 1 Structured and Unstructured uncertainty 2.6. 3 Exponential Convergence of the Gradient 5.2 The rohrs Examples 215 Algorithms with SPR Error Fquations 85 5.3 Robustness of Adaptive algorithms with Persistency 2.7 Frequency Domain Conditions for Parameter of excitation 219 Convergence 0 5.3. 1 Exponential Convergence and robustness 221 2.7. 1 Parameter Convergence 5.3. 2 Robustness of an Adaptive control scheme 225 2.7.2 Partial Parameter Convergence 9 5.4 Heuristic Analysis of the rohrs examples 231 2. 8 Conclusions 5.5 Averaging Analysis of Slow Drift Instability 236 Chapter 3 Adaptive Control 99 5.5.1 Instability Theorems Using Averaging 236 3.0 Introduction 99 5.5.2 Application to the output Error Scheme 241 3. 1 Model Reference Adaptive control Problem 103 5.6 Methods for Improving robustness 3.2 Controller Structure 104 Qualitative discussion 248 3.3 Adaptive Control Schemes 110 5.6.1 Robust Identification Schemes 248 3.3. 1 Input Error Direct Adaptive Control 111 5.6.2 Specification of the Closed Loop control 3.3.2 Output Error Direct Adaptive Control 118 Objective-Choice of Control Model and of 3.3.3 Indirect Adaptive Control 123 Reference input 250 3.3.4 Alternate model reference schemes 127 5.6.3 The Usage of Prior Information 250 3.3.5 Adaptive Pole Placement Control 129 5.6.4 Time variation of parameters 51 3. 4 The Stability problem in adaptive control 130 5.7 Robustness via Update Law Modifications 251 3.5 Analysis of the Model Reference Adaptive 5.7.1 Dcadzonc and relative deadzone 25l ControI system ? 5.7.2 253 6 Useful lem 138 5.7.3R r vector Filtering 3. 7 Stability proofs 142 5.7.4 Slow Adaptation, Averaging and Hybrid 3.7.1 Stability -Input Error Direct Adaptive Control 142 Updale law 254 3.7, 2 Stability-Output Error Direct Adaptive C 8 Conclusions 254 3.7. 3 Stability Indirect Adaptive Control 151 Chapter 6 Advanced Topics in Identification and Adaptive control 7 Chapter 8 Conclusions 324 24 6.1 Use of Prior Information 8.1 General Conclusions 6. 1. 1 Identification of Partially Known Systems 257 8.2 Future Research 6. 1.2 Effect of Unmodeled dynamics 263 2 Global Stability of Indirect Adaptive Control Schemes 266 appendix 331 6.2. 1 Indirect Adaptive Control Scheme References 359 6.2.2 Indirect Adaptive Pole Placement ) 6.2.3 Indirect Adaptive Stabilization- Index The Factorization Approach 71 6.3 Multivariable Adaptive control 277 277 6.3.2 Preliminaries 278 6.3.2.1 Factorization of Transfer Function matrices 278 6.3.2.2 Interactor Matrix and Hermite Form 282 6.3.3 Model Reference Adaptive Control Controller structure 286 6.3.4M ference Adaptive control- Input Error Scheme 6.3.5 Alternate Schemes 292 6. 4 Conclusions 293 Chapter 7 Adaptive Control of a Class of Nonlinear Systems 294 7.1 Introduction 294 7. 2 Linearizing Control for a Class of Nonlinear Systems A Review 295 7. 2. 1 Basic Theory 7.2.2 Minimum Phase Nonlinear Systems 299 7.2.2.1 The Single-Input Single-Output Case 7.2.2.2 The Multi-Input Multi-Output cas 30 7.2.3 Modcl Reference Control for Nonlinear Systems 307 7. 3 Adaptive Control of Lincarizable Minimum Phase ystems 7.3. 1 Single-Input Single-Output, Relativc Degre One case 309 7.3.2 Extensions to Higher Relative Degree SISO Systems 312 7.3.3 Adaptive Control of MIMO Systems Decouplable by Static Statc Feedback 320 7.4 Conclusions PREFACE The objective of this book and unified fashion. the major results, techniques of analysis and ncw directions of research in adaptive systems. Such a treatment is particularly timely given the rapid advances in microprocessor and multi-processor technol- ogy which make it possible to implement the fairly complicated non linear and time varying control laws associated with adaptive control Indeed, limitations to future growth can hardly be expected to be com putational, but rather from a lack of a fundamental understanding of the methodologies for the design, evaluation and testing of the algorithms Our objective has been to give a clear, conceptual presentation of adap tive mcthods. to enable a critical evaluation of these techniques and sug gest avenues of further dcvclopment daptive control has becn the subject of active research for over three decades now. There have been many theoretical successes, includ ing the development of rigorous proofs of stability and an undcrstanding of the dynamical properties of adaptive schemes, Several successful applications have been reported and the last ten years have seen an impressive growth in the availability of commercial adaptive controllers In this book, we present the deterministic theory of identification and adaptive control. For the most part the focus is on linear continu ous time, single-input single-output systems. The presentation includes the algorithms, their dynamical propertics and tools for analysis including the recently introduced averaging techniques. Current rescarch in the adaptive control of multi-input, multi-output linear systems and a reface XVIl class of nonlinear systems is also covered. Although continuous time and contributed to this work: Erwei Bai, Stephen boyd, Michel de algorithms occupy the bulk of our interest, they are presented in such a Mathelin, Li-Chen Fu, Ping hsu, Jeff Mason, Niklas Nordstrom, Andy way as to enable their transcription to the discrete time case Packard, Brad Paden and Tim Salcudean. Many of them have now a bricf outline of the book is as follows: Chapter o is a brief hi adapted to new environments, and we wish them good luck. torical overview of adaptive control and identification, and an introduc We are indebted to many colleagues for stimulating discussions at tion to various approaches. Chapter I is a chapter of mathcmatical pre onferences, workshops and mcctings. Thcy have helped us broaden our luminaries containing most of the key stability results used later in the view and understanding of the field We would particularly like to men book. In Chapter 2, we develop several adaptive identification algo Lion Anu Annaswamy, Michael Athans, Bob Bitmead, Soura Dasgupta rithms along with their stability and convergence properties. Chapter 3 Graham Goodwin. Petros Ioannou. Alberto Isidori. rick Johnson. ed is a corresponding development for model reference adaptive control Kamen. Bob Kosut, Jim Krause, rogelio lozano- Leal. Iven mareels Chapter 4, we give a self contained presentation of averaging techniques Sanjoy Mitter, Bob Narendra, Dorothee Normand-Cyrot, Romeo Ortega and we analyze the rates of convergence of the schemes of chapters Laurent Praly, Brad Riedle, Charles rohrs, Fathi salam and Lena vala and 3. Chapter 5 deals with robustness properties of the adaptive schemes, how to analyze their potential instability using averaging tech- We acknowledge the support of several organizations, including niques and how to make the schemes more robust. Chapter 6 covers NASA (Grant NAG-243), the Army Research Office (Grant DAAG 29- some advanced topics: the use of prior information in adaptive 85-K0072) the IBM Corporation(Faculty Development Award), and th identification schemes, indirect adaptive control as an extension of National Science Foundation(Grant ECS-8810145). Special thanks are robust non-adaptive control and multivariable adaptive control. due to George Meyer and Jagdish Chandra: their continuous support of Chapter 7 gives a brief introduction to the control of a class of nonlinear our research made this book possible systems, explicitly linearizable by state feedback and their adaptive con trol using thc techniques of Chapter 3. Chapter 8 concludes with some We are also grateful for the logistical support received from the of our suggestions about the areas of future exploration administration of the Department of Electrical Engineering and Com puter Scicnces at the University of California at Berkeley and of the This book is intended to introduce rescarchcrs and practitioners to Electrical and Computer Enginecring Departmcnt of Carnegie Mellon the current theory of adaptive control, we have used the book as a text University. Part of our work was also done at the Laboratory for Infor- several times for a one-semester graduate course at the University of mation and Decision systems. in the massachussetts institute of tech- California at Berkeley and at Carnegie-Mellon University. Some back- nology thanks to the hospitality of Sanjoy mitler ground in basic control systems and in linear systems theory at the gra duatc lcvcl is assumed. Background in stability theory for nonlinear sys We wish to express our gratitude to Carol Block and Tim Burns tems is desirable, but the presentation is mostly sclf-containcd for their diligent typing and layout of the manuscript in the presence of uncertainty. The figures were drafted by Osvaldo Garcia, Cynthia Bil- Acknowledgents rey and craig t the electr Research Laboratory at Bcrkc ley. Simulations were executed using the package SIMNON, and we It is a pleasure to acknowledge the contributions of the people who thank Karl Astrom for providing us with a copy of this software pack- helped us in the writing of this book. We are especially appreciative of age We also acknowledge Bernard Goodwin of Prentice Hall for his the detailed and thoughtful reviews given by Charles desoer of the origi friendly management of this enterprise and Elaine Lynch for coordinat- nal ph.d. dissertation of the second author on which this book is based ing production matters His advice and support from thc beginning are gratefully acknowledged Brian anderson and Petar Kokotovic offered excellent critical comments Last, but not least, wc would like to express our sincere apprecia that were extremely helpful both in our research, and in the revisions of tion to Nirmala and Cecilia for their patience and encouragement the manuscript. we also thank Karl Astrom and steve Morse for their Despite distance, our families have been a source of continuous support thusiasm about adaptive control, and for fruitful interactions and deserve our deepest gratitude The persistently exciting inputs of students at Berkeley and Carne- gie Mellon have helped us refine much of the material of this book. We Shankar sas are especially thankful to those who collaborated with us in research Berkeley, Calilornia Marc bodson CHAPTER 0 INTRODUCTION 0.1 IDENTIFICATION AND ADAPTIVE CONTROL Most current techniques for designing control systems are based on a good understanding of the plant under study and its environment. How- ever, in a number of instances, the plant to be controlled is too complex and the basic physical processes in it are not fully understood. Control design techniques then need to be augmented with an identification tech nique aimed at obtaining a progressively better understanding of the plant to be d,It is thus int intuitive te to aggregate system identification and control, Often, the two steps will be taken separately If the system idcntification is recursive-that is the plant model is periodically updated on the basis of previous estimates and new data- identification and control may be performed concurrently. We will see adaptive control, pragmatically, as a direct aggregation of a (nor adaptive) control methodology with some form of recursive system identifcation Abstractly, system identification could be aimed at determining if the plant to be controlled is linear or nonlinear, finite or infinite dimen sional, and has continuous or discrete event dynamics. Here we will res- trict our attention to finite dimensional, single-input single-output linear plants, and some classes of multivariable and nonlinear plants. Then, the primary step of system identification (structural identifi Ication h lread been taken, and only parameters of a fixed type of madel need to be determined. Implicitly, we will thus be limiting ourselves to parametric system identification, and parametric adaptive control introduction Section o1 ldentification and ade Applications of such systems arise in several contexts: advanced flight control systems for aircraft or spacecraft, robot manipulators, process thoroughly researched and understood. Further, Parks [1966] found a control, power systems, and others ay of redesigning the update laws proposed in the 1950s for mode Adaptive control, then, is a technique of applying some system reference schemes so as to be able to prove convergence of his controller. dentification technique to obtain a model of the process and its environ In the 1970s owing to the culmination of determined efforts b ment from input-output experiments and using this model to design a several teams of researchers, complete proofs of stability for several controller. The parameters of the controller are adjusted during the ptive schemes appeared. State space(Lyapunoy based )proofs of sta- operation of the plant as the amount of data available for plant bility for model reference adaptive schemes appeared in the work of Identification increases. For a number of simple PID(proportional Narendra, Lin,& Valavani [1980] and Morse [1980]. In the late 1970s, Integral+ derivative)controllers in process control, this is often done input output (Popov hyperstability based)proofs appeared in Egardt manually. However, when the number of parameters is larger than three [ 1979] and Landau [1979]. Stability proofs in the discrete time deter- or four, and they vary with time, automatic adjustment is needed. The ministic and stochastic case (due to Goodwin, Ramadge, Caines design techniques for adaptive systems are studied and analyzed in 1980])also appeared at this time, and are contained in the textbook by theory for unk fixed(tha at 1s, time invariant )plants. In practice Goodwin Sin [1984]. Thus, this period was marked by the culmina they are applied to slowly time-varying and unknown plants tion of the analytical efforts of the past twenty years. Given the firnl, analytical footing of the work to this point, the Overview of the Literature 1980s have proven to be a time of critical examination and evaluation of Research in adaptive control has a long and vigorous history. In the the accomplishments to date. It was first pointed out by Rohrs and co- 1950s, it was motivated by the problem of designing autopilots for air- workers [1982] that the assumptions under which stability of adaptive craft operating at a wide range of speeds and altitudes. While the object schemes had been proven were very sensitive to the presence of unmo- of a good fixed-gain controller was to build an autopilot which was deled dynamics, typically high-frequency parasitic modes that were eglected to limit the complexity of the controller. This sparked a flood negl insensitive to these (large) parameter variations, it was frequentl observed that a single constant gain controller would not suffice. Conse of research into the robustness of adaptive algorithms: a re-examination quently, gain scheduling based on some auxiliary measurements of of whether or not adaptive controllers were at least as good as fixed gain airspeed was adopted. With this scheme in place several rudimentary controllers, the development of tools for the analysis of the transient model reference schemes were also attempted-the goal in this scheme behavior of the adaptive algorithms and attempts at implementing the was to build a self-adjusting controller which yielded a closed loop algorithms on practical systems (reactors, robot manipulators, and ship transfer function matching a prescribed reference model. Several steering systems to mention only a few). The implementation of the schemes of self-adjustment of the controller parameters were proposed complicated nonlinear laws inherent in adaptive control has been greatly facilitated by the such as the sensitivity rules and the so-called M I t. rule. and were boom in microelectronics and today, one can talk in verified to perform well under certain conditions. Finally, Kalman terms of custom adaptive controller chips. All this flood of research and [1958] put on a firm analytical footing the concept of a general self- development is bearing fruit and the industrial use of daptive control is growing tuning controller with explicit identification of the parameters of a linear, single-input, single-output plant and the usage of these parameter Adaptive control has a rich and varied literature and it is impossi- estimates to update an optimal lincar quadratic controller ble to do justice to all the manifold publications on the subject. It is a The 1960s marked an important time in the development of con- tribute to the vitality of the field that there are a large number of fairly trol theory and adaptive control in particular, Lyapunov's stability recent books and monographs. Some recent books on recursive estima- theory was firmly established as a lool for proving convergence in adap- tion, which is an important part of adaptive control are by Eykhoff tive control schemes. Stochastic control made giant strides with the [19741, Goodwin Payne [1977], Ljung Soderstrom [ 1983] and ljung understanding of dynamic programming, due to Bellman and others [1987]. Recent books dealing with the theory of adaptive control are by Learning schemes proposed by Tsypkin, Feldbaum and others(see Tsyp Landau [1979], egardt 1979 loannou Kokotovic [1984], Goodwin kin [1971] and [ 1973])were shown to have roots in a single unified Sin [1984, Anderson, Bitmead, Johnson, Kokotovi .osut, Mareels ramework of recursive equations. System identification (off-line)was Praly, Riedle [1986] Kumar and Varaiya [1986], Polderman [1988] and Caines [1988]. An attempt to link the signal processing viewpoint

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