new stable adaptive control nonlinear systems · 2013. 7. 23. · adaptive and learning systems for...

STABLE ADAPTIVE CONTROLAND ESTIMATION FORNONLINEAR SYSTEMS

Adaptive and Learning Systems for Signal Processing,Communications, and ControlEditor: Simon Haykin

Beckerman / ADAPTIVE COOPERATIVE SYSTEMS

Chen and Gu / CONTROL-ORIENTED SYSTEM IDENTIFICATION: An Hx

Approach

Cherkassky and Mulier / LEARNING FROM DATA: Concepts,Theory, andMethods

Diamantaras and Rung / PRINCIPAL COMPONENT NEURAL NETWORKS:Theory and Applications

Haykin / UNSUPERVISED ADAPTIVE FILTERING: Blind Source Separation

Haykin / UNSUPERVISED ADAPTIVE FILTERING: Blind Deconvolution

Haykin and Puthussarypady / CHAOTIC DYNAMICS OF SEA CLUTTER

Hrycej / NEUROCONTROL: Towards an Industrial Control Methodology

Hyvarinen,Karhunen,and Oja / INDEPENDENT COMPONENT ANALYSIS

Kristic,Kanellakopoulos,and Kokotovic / NONLINEAR AND ADAPTIVECONTROL DESIGN

Mann / INTELLIGENT IMAGE PROCESSING

Nikias and Shao / SIGNAL PROCESSING WITH ALPHA-STABLE DISTRIBUTIONSAND APPLICATIONS

Passino and Burgess / STABILITY ANALYSIS OF DISCRETE EVENT SYSTEMS

Sanchez-Pena and Sznaier / ROBUST SYSTEMS THEORY AND APPLICATIONS

Sandberg, Lo,Fancourt, Principe, Katagiri, and Haykin / NONLINEARDYNAMICAL SYSTEMS: Feedforward Neural Network Perspectives

Spooner, Maggiore, Ordonez, and Passino / STABLE ADAPTIVE CONTROL ANDESTIMATION FOR NONLINEAR SYSTEMS: Neural and Fuzzy ApproximatorTechniques

Tao and Kokotovic / ADAPTIVE CONTROL OF SYSTEMS WITH ACTUATOR ANDSENSOR NONLINEARITIES

Tsoukalas and Uhrig / FUZZY AND NEURAL APPROACHES IN ENGINEERING

Van Hulle / FAITHFUL REPRESENTATIONS AND TOPOGRAPHIC MAPS: FromDistortion- to Information-Based Self-Organization

Vapnik / STATISTICAL LEARNING THEORY

Werbos / THE ROOTS OF BACKPROPAGATION: From Ordered Derivatives toNeural Networks and Political Forecasting

Yee and Haykin / REGULARIZED RADIAL BIAS FUNCTION NETWORKS: Theoryand Applications

STABLE ADAPTIVE CONTROLAND ESTIMATION FORNONLINEAR SYSTEMS

Neural and Fuzzy Approximator Techniques

Jeffrey T. SpoonerSandia National Laboratories

Manfredi MaggioreUniversity of Toronto

Raul OrdonezUniversity of Dayton

Kevin M. PassinoThe Ohio State University

A JOHN WILEY & SONS, INC., PUBLICATION

WILEY-INTERSCIENCE

This text is printed on acid-free paper. ©

Copyright © 2002 by John Wiley & Sons, Inc., New York. All rights reserved.

Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in anyform or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise,except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, withouteither the prior written permission of the Publisher, or authorization through payment of theappropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should beaddressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York,NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ @ WILEY.COM.

For ordering and customer service, call 1-800-CALL-WILEY.

Library of Congress Cataloging-in-Publication Data is available.

ISBN 0-471-41546-4

Printed in the United States of America.

1 0 9 8 7 6 5 4 3 2 1

To our families

This page intentionally left blank

Contents

Preface xv

1 Introduction 1

1.1 Overview 11.2 Stability and Robustness 21.3 Adaptive Control: Techniques and Properties 4

1.3.1 Indirect Adaptive Control Schemes 41.3.2 Direct Adaptive Control Schemes 5

1.4 The Role of Neural Networks and Fuzzy Systems 61.4.1 Approximator Structures and Properties 61.4.2 Benefits for Use in Adaptive Systems 8

1.5 Summary 10

1 Foundations 11

2 Mathematical Foundations 13

2.1 Overview 132.2 Vectors, Matrices, and Signals: Norms and Properties 13

2.2.1 Vectors 142.2.2 Matrices 152.2.3 Signals 19

2.3 Functions: Continuity and Convergence 212.3.1 Continuity and Differentiation 212.3.2 Convergence 23

2.4 Characterizations of Stability and Boundedness 242.4.1 Stability Definitions 26

vii

viii CONTENTS

2.4.2 Boundedness Definitions 302.5 Lyapunov's Direct Method 31

2.5.1 Preliminaries: Function Properties 322.5.2 Conditions for Stability 342.5.3 Conditions for Boundedness 36

2.6 Input-to-State Stability 382.6.1 Input-to-state Stability Definitions 382.6.2 Conditions for Input-to-state Stability 39

2.7 Special Classes of Systems 412.7.1 Autonomous Systems 412.7.2 Linear Time-Invariant Systems 43

2.8 Summary 452.9 Exercises and Design Problems 45

3 Neural Networks and Fuzzy Systems 493.1 Overview 493.2 Neural Networks 50

3.2.1 Neuron Input Mappings 523.2.2 Neuron Activation Functions 543.2.3 The Mulitlayer Perceptron 573.2.4 Radial Basis Neural Network 583.2.5 Tapped Delay Neural Network 59

3.3 Fuzzy Systems 603.3.1 Rule-Base and Fuzzification 613.3.2 Inference and Defuzzification 643.3.3 Takagi-Sugeno Fuzzy Systems 67


4 Optimization for Training Approximators 734.1 Overview 734.2 Problem Formulation 744.3 Linear Least Squares 76

4.3.1 Batch Least Squares 774.3.2 Recursive Least Squares 80

4.4 Nonlinear Least Squares 844.4.1 Gradient Optimization: Single Training Data Pair 854.4.2 Gradient Optimization: Multiple Training Data Pairs 874.4.3 Discrete Time Gradient Updates 92

CONTENTS ix

4.4.4 Constrained Optimization 944.4.5 Line Search and the Conjugate Gradient Method 95


5 Function Approximation 1055.1 Overview 1055.2 Function Approximation 106

5.2.1 Step Approximation 1075.2.2 Piecewise Linear Approximation 1135.2.3 Stone-Weierstrass Approximation 115

5.3 Bounds on Approximator Size 1195.3.1 Step Approximation 1195.3.2 Piecewise Linear Approximation 120

5.4 Ideal Parameter Set and Representation Error 1225.5 Linear and Nonlinear Approximator Structures 123

5.5.1 Linear and Nonlinear Parameterizations 1235.5.2 Capabilities of Linear vs. Nonlinear Approximators 1245.5.3 Linearizing an Approximator 126

5.6 Discussion: Choosing the Best Approximator 1285.7 Summary 1305.8 Exercises and Design Problems 130

II State-Feedback Control 133

6 Control of Nonlinear Systems 1356.1 Overview 1356.2 The Error System and Lyapunov Candidate 137

6.2.1 Error Systems 1376.2.2 Lyapunov Candidates 140

6.3 Canonical System Representations 1416.3.1 State-Feedback Linearizable Systems 1416.3.2 Input-Output Feedback Linearizable Systems 1496.3.3 Strict-Feedback Systems 153

6.4 Coping with Uncertainties: Nonlinear Damping 1596.4.1 Bounded Uncertainties 1606.4.2 Unbounded Uncertainties 1616.4.3 What if the Matching Condition Is Not Satisfied? 162

6.5 Coping with Partial Information: Dynamic Normalization 163

x CONTENTS

6.6 Using Approximators in Controllers 1656.6.1 Using Known Approximations of System Dynamics 1656.6.2 When the Approximator Is Only Valid on a Region 167


7 Direct Adaptive Control 1797.1 Overview 1797.2 Lyapunov Analysis and Adjustable Approximators 1807.3 The Adaptive Controller 184

7.3.1 -modification 1857.3.2 e-modification 198

7.4 Inherent Robustness 2017.4.1 Gain Margins 2017.4.2 Disturbance Rejection 202

7.5 Improving Performance 2037.5.1 Proper Initialization 2047.5.2 Redefining the Approximator 205

7.6 Extension to Nonlinear Parameterization 2067.7 Summary 2087.8 Exercises and Design Problems 210

8 Indirect Adaptive Control 2158.1 Overview 2158.2 Uncertainties Satisfying Matching Conditions 216

8.2.1 Static Uncertainties 2168.2.2 Dynamic Uncertainties 227

8.3 Beyond the Matching Condition 2368.3.1 A Second-Order System 2368.3.2 Strict-Feedback Systems with Static Uncertainties 2398.3.3 Strict-Feedback Systems with Dynamic

Uncertainties 2488.4 Summary 2548.5 Exercises and Design Problems 254

9 Implementations and Comparative Studies 2579.1 Overview 2579.2 Control of Input-Output Feedback Linearizable Systems 258

9.2.1 Direct Adaptive Control 2589.2.2 Indirect Adaptive Control 261

CONTENTS xi

9.3 The Rotational Inverted Pendulum 2639.4 Modeling and Simulation 2649.5 Two Non-Adaptive Controllers 266

9.5.1 Linear Quadratic Regulator 2679.5.2 Feedback Linearizing Controller 268

9.6 Adaptive Feedback Linearization 2719.7 Indirect Adaptive Fuzzy Control 274

9.7.1 Design Without Use of Plant Dynamics Knowledge 2749.7.2 Incorporation of Plant Dynamics Knowledge 282

9.8 Direct Adaptive Fuzzy Control 2859.8.1 Using Feedback Linearization as a Known Controller 2869.8.2 Using the LQR to Obtain Boundedness 2909.8.3 Other Approaches 296


III Output-Feedback Control 305

10 Output-Feedback Control 30710.1 Overview 30710.2 Partial Information Framework 30810.3 Output-Feedback Systems 31010.4 Separation Principle for Stabilization 317

10.4.1 Observability and Nonlinear Observers 31710.4.2 Peaking Phenomenon 32510.4.3 Dynamic Projection of the Observer Estimate 32710.4.4 Output-Feedback Stabilizing Controller 333

10.5 Extension to MIMO Systems 33710.6 How to Avoid Adding Integrators 33910.7 Coping with Uncertainties 34710.8 Output-Feedback Tracking 350

10.8.1 Practical Internal Models 35310.8.2 Separation Principle for Tracking 357

10.9 Summary 359lO.lOExercises and Design Problems 360

11 Adaptive Output Feedback Control 36311.1 Overview 36311.2 Control of Systems in Adaptive Tracking Form 364

xii CONTENTS

11.3 Separation Principle for Adaptive Stabilization 37111.3.1 Full State-Feedback Performance Recovery 37411.3.2 Partial State-Feedback Performance Recovery 381

11.4 Separation Principle for Adaptive Tracking 38711.4.1 Practical Internal Models for Adaptive Tracking 39011.4.2 Partial State-Feedback Performance Recovery 394


12 Applications 40112.1 Overview 40112.2 Nonadaptive Stabilization: Jet Engine 402

12.2.1 State-Feedback Design 40312.2.2 Output-Feedback Design 406

12.3 Adaptive Stabilization: Electromagnet Control 41112.3.1 Ideal Controller Design 41312.3.2 Adaptive Controller Design 41712.3.3 Output-Feedback Extension 422

12.4 Tracking: VTOL Aircraft 42412.4.1 Finding the Practical Internal Model 42612.4.2 Full Information Controller 43012.4.3 Partial Information Controller 431


IV Extensions 435

13 Discrete-Time Systems 43713.1 Overview 43713.2 Discrete-Time Systems 438

13.2.1 Converting from Continuous-Time Representations 43813.2.2 Canonical Forms 442

13.3 Static Controller Design 44413.3.1 The Error System and Lyapunov Candidate 44413.3.2 State Feedback Design 44613.3.3 Zero Dynamics 45113.3.4 State Trajectory Bounds 452

13.4 Robust Control of Discrete-Time Systems 45413.4.1 Inherent Robustness 454

CONTENTS xiii

13.4.2 A Dead-Zone Modification 45613.5 Adaptive Control 458

13.5.1 Adaptive Control Preliminaries 45813.5.2 The Adaptive Controller 460


14 Decentralized Systems 47314.1 Overview 47314.2 Decentralized Systems 47414.3 Static Controller Design 476

14.3.1 Diagonal Dominance 47614.3.2 State-Feedback Control 47814.3.3 Using a Finite Approximator 484

14.4 Adaptive Controller Design 48514.4.1 Unknown Subsystem Dynamics 48514.4.2 Unknown Interconnection Bounds 489


15 Perspectives on Intelligent Adaptive Systems 49915.1 Overview 49915.2 Relations to Conventional Adaptive Control 50015.3 Genetic Adaptive Systems 50115.4 Expert Control for Adaptive Systems 50315.5 Planning Systems for Adaptive Control 50415.6 Intelligent and Autonomous Control 50615.7 Summary 509

For Further Study 511

Bibliography 521

Index 541

Preface

A key issue in the design of control systems has long been the robustnessof the resulting closed-loop system. This has become even more critical ascontrol systems are used in high consequence applications in which certainprocess variations or failures could result in unacceptable losses. Appropri-ately, the focus on this issue has driven the design of many robust nonlinearcontrol techniques that compensate for system uncertainties.

At the same time neural networks and fuzzy systems have found theirway into control applications and in sub-fields of almost every engineeringdiscipline. Even though their implementations have been rather ad hocat times, the resulting performance has continued to excite and capturethe attention of engineers working on today's "real-world" systems. Theseresults have largely been due to the ease of implementation often possiblewhen developing control systems that depend upon fuzzy systems or neuralnetworks.

In this book we attempt to merge the benefits from these two approachesto control design (traditional robust design and so called "intelligent con-trol" approaches). The result is a control methodology that may be verifiedwith the mathematical rigor typically found in the nonlinear robust controlarea while possessing the flexibility and ease of implementation tradition-ally associated with neural network and fuzzy system approaches. Withinthis book we show how these methodologies may be applied to state feed-back, multi-input multi-output (MIMO) nonlinear systems, output feed-back problems, both continuous and discrete-time applications, and evendecentralized control. We attempt to demonstrate how one would applythese techniques to real-world systems through both simulations and ex-perimental settings.

This book has been written at a first-year graduate level and assumessome familiarity with basic systems concepts such as state variables andstability. The book is appropriate for use as a text book and homeworkproblems have been included.

XV

xvi Preface

Organization of the Book

This book has been broken into four main parts. The first part of the bookis dedicated to background material on the stability of systems, optimiza-tion, and properties of fuzzy systems and neural networks. In Chapter 1a brief introduction to the control philosophy used throughout the book ispresented. Chapter 2 provides the necessary mathematical background forthe book (especially needed to understand the proofs), including stabilityand convergence concepts and methods, and definitions of the notation wewill use. Chapter 3 provides an introduction to the key concepts from neuralnetworks and fuzzy systems that we need. Chapter 4 provides an introduc-tion to the basics of optimization theory and the optimization techniquesthat we will use to tune neural networks and fuzzy systems to achieve theestimation or control tasks. In Chapter 5 we outline the key propertiesof neural networks and fuzzy systems that we need when they are used asapproximators for unknown nonlinear functions.

The second part of the book deals with the state-feedback control prob-lem. We start by looking at the non-adaptive case in Chapter 6 in whichan introduction to feedback linearization and backstepping methods arepresented. It is then shown how both a direct (Chapter 7) and indirect(Chapter 8) adaptive approach may be used to improve both system ro-bustness and performance. The application of these techniques is furtherexplained in Chapter 9, which is dedicated to implementation issues.

In the third part of the book we look at the output-feedback problem inwhich all the plant state information is not available for use in the designof the feedback control signals. In Chapter 10, output-feedback controllersare designed for systems using the concept of uniform complete observ-ability. In particular, it is shown how the separation principle may beused to extend the approaches developed for state-feedback control to theoutput-feedback case. In Chapter 11 the output-feedback methodology isdeveloped for adaptive controllers applicable to systems with a great degreeof uncertainty. These methods are further explained in Chapter 12 whereoutput-feedback controllers are designed for a variety of case studies.

The final part of the book addresses miscellaneous topics such as discrete-time control in Chapter 13 and decentralized control in Chapter 14. Finally,in Chapter 15 the methods studied in this book will be compared to conven-tional adaptive control and to other "intelligent" adaptive control methods(e.g., methods based on genetic algorithms, expert systems, and planningsystems).

Acknowledgments

The authors would like to thank the various sponsors of the research thatformed the basis for the writing of this textbook. In particular, we wouldlike to thank the Center for Intelligent Transportation Systems at The Ohio

Preface xvii

State University, Litton Corp., the National Science Foundation, NASA,Sandia National Laboratories, and General Electric Aircraft Engines fortheir support throughout various phases of this project.

This manuscript was prepared using IATEX. The simulations and manyof the figures throughout the book were developed using MATLAB.

As mentioned above, the material in this book depends critically onconventional robust adaptive control methods, and in this regard it wasespecially influenced by the excellent books of P. loannou and J. Sun, and S.Sastry and M. Bodson (see Bibliography). As outlined in detail in the "ForFurther Study" section of the book, the methods of this book are also basedon those developed by several colleagues, and we gratefully acknowledgetheir contributions here. In particular, we would like to mention: J. Farrell,H. Khalil, F. Lewis, M. Polycarpou, and L-X. Wang. Our writing processwas enhanced by critical reviews, comments, and support by several personsincluding: A. Bentley, Y. Diao, V. Gazi, T. Kim, S. Kohler, M. Lau, Y. Liu,and T. Smith. We would like to thank B. Codey, S. Paracka, G. Telecki,and M. Yanuzzi for their help in producing and editing this book. Finally,we would like to thank our families for their support throughout this entireproject.

Jeff SpoonerManfredi Maggiore

Raul OrdonezKevin Passino

March, 2002

Chapter X

Introduction

1.1 Overview

The goal of a control system is to enhance automation within a systemwhile providing improved performance and robustness. For instance, wemay develop a cruise control system for an automobile to release driversfrom the tedious task of speed regulation while they are on long trips. Inthis case, the output of the plant is the sensed vehicle speed, y, and theinput to the plant is the throttle angle, w, as shown in Figure 1.1. Typically,control systems are designed so that the plant output follows some referenceinput (the driver-specified speed in the case of our cruise control example)while achieving some level of "disturbance rejection." For the cruise controlproblem, a disturbance would be a road grade variation or wind. Clearly wewould want our cruise controller to reduce the effects of such disturbanceson the quality of the speed regulation that is achieved.

Figure 1.1. Closed loop control.

In the area of "robust control" the focus is on the development of con-trollers that can maintain good performance even if we only have a poormodel of the plant or if there are some plant parameter variations. In thearea of adaptive control, to reduce the effects of plant parameter variations,robustness is achieved by adjusting (i.e., adapting) the controller on-line.

2 Introduction

For instance, an adaptive controller for the cruise control problem wouldseek to achieve good speed tracking performance even if we do not have agood model of the vehicle and engine dynamics, or if the vehicle dynamicschange over time (e.g., via a weight change that results from the addition ofcargo, or due to engine degradation over time). At the same time it wouldtry to achieve good disturbance rejection. Clearly, the performance of agood cruise controller should not degrade significantly as your automobileages or if there are reasonable changes in the load the vehicle is carrying.

We will use adaptive mechanisms within the control laws when certainparameters within the plant dynamics are unknown. An adaptive controllerwill thus be used to improve the closed-loop system robustness while meet-ing a set of performance objectives. If the plant uncertainty cannot beexpressed in terms of unknown parameters, one may be able to reformu-late the problem by expressing the uncertainty in terms of a fuzzy system,neural network, or some other parameterized nonlinearity. The uncertaintythen becomes recast in terms of a new set of unknown parameters that maybe adjusted using adaptive techniques.

1.2 Stability and Robustness

Often, when given the challenge of designing a control system for a par-ticular application, one is provided a model of the plant that contains thedominant dynamic characteristics. The engineer responsible for the designof a control system may then proceed to formulate a control algorithm as-suming that when the model is controlled to within specifications, then thetrue plant will also be controlled within specifications. This approach hasbeen successfully applied to numerous systems. More often, however, thecontroller may need to be adjusted slightly when moving from the designmodel to the actual implementation due to a mismatch between the modeland true system. There are also cases when a control system performswell for a particular operating region, but when tested outside that region,performance degrades to unacceptable levels.

Figure 1.2. Robust control of a plant with unmodeled dynamics.

Sec. 1.2 Stabil i ty and Robustness 3

These issues, among others, are addressed by robust control design.When developing a robust control design, the focus is often on maintainingstability even in the presense of unmodeled dynamics or external distur-bances applied to the plant. Figure 1.2 shows the situation in which thecontroller must be designed to operate given any possible plant variation A.Unmodeled dynamics are typically associated with every control problemin which a controller is designed based upon a model. This may be due toany one of a number of reasons:

• It may be the case that only a nominal set of parameters are availablefor the control design. If a controller is to be incorporated into a mass-produced product, for example, it may not be practical to measurethe exact parameter values for each plant so that a controller can becustomized to each particular system.

• It may not be cost effective to produce a model that exactly (or evenclosely) represents the plant's dynamics. It may be possible to spendfewer resources on a robust control design using an incomplete modelthan developing a high fidelity model so that traditional non-robusttechniques may be used.

Hence, the approach in robust control is to accept a priori that there willbe model uncertainty, and try to cope with it.

The issue of robustness has been studied extensively in the control lit-erature. When working with linear systems, one may define phase and gainmargins which quantify the range of uncertainty a closed-loop system maywithstand before becoming unstable. In the world of nonlinear control de-sign, we often investigate the stability of a closed-loop system by studyingthe behavior of a Lyapunov function candidate. The Lyapunov functioncandidate is a mathematical function designed to provide a simplified mea-sure of the control objectives allowing complex nonlinear systems to beanalyzed using a scalar differential equation. When a controller is designedthat drives the Lyapunov function to zero, the control objectives are met. Ifsome system uncertainty tends to drive the Lyapunov candidate away fromzero, we will often simply add an additional stabilizing term to the controlalgorithm that dominates the effect of the uncertainty, thereby making theclosed-loop system more robust.

We will find that by adding a static term in the control law that simplydominates the plant uncertainty, it is often easy to simply stabilize anuncertain plant, however, driving the system error to zero may be difficultif not impossible. Consider the case when the plant is defined by

where x £ R is the plant state that we wish to drive to the point x = 1,u 6 R is the plant input, and 9 is an unknown constant. Since 9 is unknown,

4 Introduction

one may not define a static controller that causes x = 1 to be a stableequilibrium point. In order for x = 1 to be a stable equilibrium point, itis necessary that x = 0 when x = 1, so u(x] — —9 when x — 1. Since 9 isunknown, however, we may not define such a controller.

In this case, the best that a static nonlinear controller may do is to keepx bounded in some region around x — I . If dynamics are included in thenonlinear controller, then it turns out that one may define a control systemthat does drive x —> I even if 9 is unknown. In this book we will use theapproach of adaptive control to help us define such a nonlinear dynamiccontroller that will stabilize a certain class of nonlinear uncertain systems.

1.3 Adaptive Control: Techniques and Properties

An adaptive controller can be designed so that it estimates some uncertaintywithin the system, then automatically designs a controller for the estimatedplant uncertainty. In this way the control system uses information gatheredon-line to reduce the model uncertainty, that is, to figure out exactly whatthe plant is at the current time so that good control can be achieved.Considering the system defined by (1.1), an adaptive controller may bedefined so that an estimate of 9 is generated, which we will denote 9. If9 were known, then including a term —9x in the control law would cancelthe effects of the uncertainty. If 9 —> 9 over time, then including the term—9x in the control law would also cancel the effects of the uncertainty overtime. This approach is referred to as indirect adaptive control.

1.3.1 Indirect Adaptive Control Schemes

An indirect approach to adaptive control is made up of an approximator(often referred to as an "identifier" in the adaptive control literature) thatis used to estimate unknown plant parameters and a "certainty equiva-lence" control scheme in which the plant controller is defined ("designed")assuming that the parameter estimates are their true values. The indirectadaptive approach is shown in Figure 1.3. Here the adjustable approximatoris used to model some component of the system. Since the approximationis used in the control law, it is possible to determine if we have a goodestimate of the plant dynamics. If the approximation is good (i.e., we knowhow the plant should behave), then it is easy to meet our control objec-tives. If, on the other hand, the plant output moves in the wrong direction,then we may assume that our estimate is incorrect and should be adjustedaccordingly.

As an example of an indirect adaptive controller, consider the cruisecontrol problem where we have an approximator that is used to estimatethe vehicle mass and aerodynamic drag. Assume that the vehicle dynamics

Sec. 1.3 Adaptive Control: Techniques and Properties 5

Figure 1.3. Indirect adaptive control,

may be approximated by

where m is the vehicle mass, p is the coefficient of aerodynamic drag, x isthe vehicle velocity, and u is the plant input. Assume that an approximatorhas been defined so that estimates of the mass and drag are found such thatm -> m and p —> p. Then the control law

may be used so that x — v(t] when m = m and p = p. Here v(t] maybe considered a new control input that is defined to drive x to any desiredvalue.

Later in this book, we will learn how to define an approximator forrh and p in the above example that allows us to drive x to some desiredvelocity. We will also find that the indirect approach remains stable whenm(0) ^ m and /3(0) ^ p though the initial parameter values may affect thetransient performance of the closed-loop system.

1.3.2 Direct Adaptive Control Schemes

Yet another approach to adaptive control is shown in Figure 1.4. Herethe adjustable approximator acts as a controller. The adaptation mecha-nism is then designed to adjust the approximator causing it to match someunknown nonlinear controller that will stabilize the plant and make theclosed-loop system achieve its performance objectives.

Note that we call this scheme "direct" since there is a direct adjustmentof the parameters of the controller without identifying a model of the plant.

6 Introduction

Figure 1.4. Direct adaptive control.

Direct adaptive control, while a somewhat less popular approach (at least inthe neural/fuzzy adaptive control literature), will be considered each timewe consider an indirect scheme in this book. Part of the reason we givea relatively equal treatment to direct adaptive schemes is that in severalimplementations we have found them to work more effectively than theirindirect adaptive counterparts.

1.4 The Role of Neural Networks and Fuzzy Systems

In this section we outline how neural networks and fuzzy systems can beused as the "approximator" in the adaptive schemes outlined in the previoussection. Then we discuss the advantages of using neural networks or fuzzysystems as approximators in adaptive systems.

1.4.1 Approximator Structures and Properties

Neural networks are parameterized nonlinear functions. Their parametersare, for instance, the weights and biases of the network. Adjustment ofthese parameters results in different shaped nonlinearities. Typically, theadjustment of the neural network parameters is achieved by a gradientdescent approach on an error function that measures the difference betweenthe output of the neural network and the output of the actual system(function). That is, we try to adjust the neural network to serve as anapproximator for an unknown function that we only know by how it specifiesoutput values for the given input values (i.e., the training data). Or, viewedanother way, we seek to adjust the neural network so that it serves as an"interpolator" for the input-output data so that if it is presented with inputdata, it will produce an output that is close to the actual output that thefunction (system) would create.

Due to the wide range of roles that the neural network can play inadaptive schemes we will simply call them "approximators," and below

Sec. 1.4 The Role of Neural Networks and Fuzzy Systems 7

we will focus on their properties and advantages. It is important to note,however, that neural networks are not unique in their ability to serve asapproximators. There are conventional approximator structures such aspolynomials. Moreover, there is the possibility of using a fuzzy system asan approximator structure as we discuss next.

Historically, fuzzy controllers have stirred a great deal of excitement insome circles since they allow for the simple inclusion of heuristic knowl-edge about how to control a plant rather than requiring exact mathemat-ical models. This can sometimes lead to good controller designs in a veryshort period of time. In situations where heuristics do not provide enoughinformation to specify all the parameters of the fuzzy controller a priori, re-searchers have introduced adaptive schemes that use data gathered duringthe on-line operation of the controller, and special adaptation heuristics, toautomatically learn these parameters.

Hence, fuzzy systems have served not only their originally intendedfunction of providing an approach to nonadaptive control, but also in adap-tive controllers where, for example, the membership functions are adjusted.Fuzzy systems are indeed simply nonlinear functions that are parameter-ized by, for example, membership function parameters. In fact, in somesituations they are mathematically identical to a certain class of radial ba-sis function neural networks. It is then not surprising that we can use fuzzysystems as approximators in the same way that we can use neural networks.It is possible, however, that the fuzzy system can offer an additional ad-vantage in that it may be easier to incorporate heuristic knowledge abouthow the input-output map for which you are gathering data from should beshaped. In some situations this can lead to better convergence properties(simply because it may be easier to initialize the shape of the nonlinearityimplemented by the approximator).

In this book we will provide some insights into how to pick an approx-imator (e.g., based on physical considerations); however, the question ofwhich approximator is best to use is still an open research issue. In ourdiscussions on approximator properties, when we refer to an "approximatorstructure," we mean the nonlinear function that is tuned by the parametersof the approximator. The "size" of the approximator is some measure ofthe complexity of the mapping it implements (e.g., for a neural network itmight be the total number of parameters used to adjust the network). An-other feature that we will use to distinguish among different approximatorsis whether they are "linear in their parameters." For instance, when onlycertain parameters in a neural network are adjusted, these may be ones thatenter in a linear fashion. Clearly, linear in the parameter approximatorsare a special case of nonlinear in the parameter approximators and hencethey can be more limited in what functions that they can approximate.

We will study approximators (neural or fuzzy) that satisfy the "univer-sal approximation property." If an approximator possesses the universal

8 Introduction

approximation property, then it can approximate any continuous functionon a closed and bounded domain with as much accuracy as desired (how-ever, most often, to get an arbitrarily accurate approximation you have tobe willing to increase the size the the approximator structure arbitrarily).It turns out that some approximator structures provide much more efficientparameterized nonlinearities in the sense that to get definite improvementin approximation accuracy they only have to grow in size in a linear fashion.Other approximator structures may have to grow exponentially to achievesmall increases in approximation accuracy. However, it is important tonote that the inclusion of physical domain knowledge may help us to avoidprohibitive increases in the size of the approximator.

The "approximation error" is some suitably defined measure (e.g., themaximum distance between the two functions over their domains) of theerror between the function you are trying to approximate (e.g., the plant)and the function implemented by the approximator. The "ideal approxi-mation error" (also known as the "representation error") is the minimumerror that would result from the best choice of the approximator param-eters (i.e., the "ideal parameters"). For a class of neural networks it canbe shown that the ideal approximation error has definite decreases withan increase in the size of the approximator (i.e., it decreases at a certainrate with a linear increase in the size of the neural network); however, inthis case you must adjust the parameters that enter in a nonlinear fashionand there are no general guarantees for current algorithms that you willfind the ideal parameters. Linear in the parameter approximators provideno such guarantees of reduction of the ideal approximation error; however,when one incorporates physical domain knowledge, experience with appli-cations shows that increases in approximator accuracy can often be foundwith reasonable increases in the size of the approximator.

1.4.2 Benefits for Use in Adaptive Systems

First, for comparison purposes it is useful to point out that we can broadlythink of many conventional adaptive estimation and control approachesfor linear systems as techniques that use linear approximation structuresfor systems with known model order (of course, this is for the state feed-back case and ignores the results for plants where the order is not assumedknown). Most often, in these cases, the problems are set up so that thelinear approximator (e.g., a linear model with tunable parameters) canperfectly represent the underlying unknown function that it is trying toapproximate (e.g., the plant model). However, it may take a certain "per-sistency of excitation" to achieve perfect approximation and conditions forthis were derived for adaptive estimation and control.

Regardless, thinking along these lines, linear robust adaptive controlstudies how to tune linear approximators when it is not possible to per-

Sec. 1.4 The Role of Neural Networks and Fuzzy Systems 9

fectly approximate the unknown function with a linear map. In this sense,it becomes clear why there is such a strong reliance of the methods of on-lineapproximation based control via neural or fuzzy systems on conventional ro-bust control of linear systems. While the universal approximation propertyguarantees that our approximators can represent the unknown function, forpractical reasons we have to limit their size so a finite approximation errorarises and must be dealt with; on-line approximation approaches deal withit in similar (or the same) ways to how it is dealt with in linear robustcontrol.

Now, while there is a strong connection to the conventional robust adap-tive control approaches, the on-line approximation based approach allowsyou to go further since it does not restrict the unknown function to belinear. In this way, it provides a logical extension to create nonlinear ro-bust control schemes where there is no need to assume that the plant is alinear parameterization of known nonlinear functions (as in the early workon adaptive feedback linearization [192] and the more recently developedsystematic approach of adaptive backstepping [115]).

It is interesting to note, however, that while there are strong connec-tions to conventional adaptive schemes, there is an additional interestingcharacteristic of the resulting adaptive systems in that if designed prop-erly they can implement something that is more similar to the way wethink of "learning" than conventional adaptive schemes. Some on-line ap-proximation based schemes (particularly some that are implemented withapproximators that have basis functions with "local support" like radial ba-sis function neural networks and fuzzy systems) achieve local adjustmentsto parameters so that only local adjustments to the tuned nonlinearity takeplace. In this case, if designed properly, the controller can be taught oneoperating condition, then learn a different operating condition, and laterreturn to the first operating condition with a controller that is alreadyproperly tuned for that region. Another way to think of this is that sincewe are tuning nonlinear functions that have an ability to be tuned locally(something a simple linear map cannot do since if you change a parameterit affects the shape of the map over the whole space) they can rememberpast tuning to a certain extent.

To summarize, in many ways, the advantages of using neural networks orfuzzy systems arise as practical rather than theoretical benefits in the sensethat we could avoid their use all together and simply use some conventionalapproximator structure (e.g., a polynomial approximator structure). Thepractical benefits of neural networks or fuzzy systems are the following:

• They offer forms of nonlinearities (e.g., the neural network) that areuniversal approximators (hence more broadly applicable to many ap-plications) and that offer reduced ideal approximation error for onlya linear increase in the number of parameters.

10 Introduction

• They offer convenient ways to incorporate heuristics on how to ini-tialize the nonlinearity (e.g., the fuzzy system).

In addition, to help demonstrate the practical nature of the approaches weintroduce in this book, there will be an experimental component where wediscuss several laboratory implementations of the methods.

1.5 Summary

The general control philosophy used within this book may be summarizedas follows:

1. We use concepts and techniques from robust control theory,

2. Adaptive approaches are used to compensate for unknown systemcharacteristics, and

3. When a system uncertainty may be characterized by a function, theproblem is reformulated in terms of fuzzy systems or neural networksto extend the applicability of the adaptive approaches.

We will use the traditional controller development and analysis approachesused in robust, adaptive, and nonlinear control, with the mathematicalflexibility provided by fuzzy systems and neural networks, to develop apowerful approach to solving many of today's challenging real-world controlproblems.

Overall, while we understand that many people do not read introduc-tions to books, we tried to make this one useful by giving you a broad viewof the lines of reasoning that we use, and by explaining what benefits themethods may provide to you.

Part I

Foundations

new stable adaptive control nonlinear systems · 2013. 7. 23. · adaptive and learning systems for...

Documents