Fuzzy Control and Identification E-Book

John H Lilly

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 290

Veröffentlichungsjahr: 2011

Beschreibung

This book gives an introduction to basic fuzzy logic and Mamdani and Takagi-Sugeno fuzzy systems. The text shows how these can be used to control complex nonlinear engineering systems, while also also suggesting several approaches to modeling of complex engineering systems with unknown models. Finally, fuzzy modeling and control methods are combined in the book, to create adaptive fuzzy controllers, ending with an example of an obstacle-avoidance controller for an autonomous vehicle using modus ponendo tollens logic.

Leseprobe

Table of Contents

Cover

Table of Contents

Title page

Copyright page

Dedication

PREFACE

CHAPTER 1 INTRODUCTION

1.1 FUZZY SYSTEMS

1.2 EXPERT KNOWLEDGE

1.3 WHEN AND WHEN NOT TO USE FUZZY CONTROL

1.4 CONTROL

1.5 INTERCONNECTION OF SEVERAL SUBSYSTEMS

1.6 IDENTIFICATION AND ADAPTIVE CONTROL

1.7 SUMMARY

CHAPTER 2 BASIC CONCEPTS OF FUZZY SETS

2.1 FUZZY SETS

2.2 USEFUL CONCEPTS FOR FUZZY SETS

2.3 SOME SET-THEORETIC AND LOGICAL OPERATIONS ON FUZZY SETS

2.4 EXAMPLE

2.5 SINGLETON FUZZY SETS

2.6 SUMMARY

CHAPTER 3 MAMDANI FUZZY SYSTEMS

3.1 IF-THEN RULES AND RULE BASE

3.2 FUZZY SYSTEMS

3.3 FUZZIFICATION

3.4 INFERENCE

3.5 DEFUZZIFICATION

3.6 EXAMPLE: FUZZY SYSTEM FOR WIND CHILL

3.7 SUMMARY

CHAPTER 4 FUZZY CONTROL WITH MAMDANI SYSTEMS

4.1 TRACKING CONTROL WITH A MAMDANI FUZZY CASCADE COMPENSATOR

4.2 TUNING FOR IMPROVED PERFORMANCE BY ADJUSTING SCALING GAINS

4.3 EFFECT OF INPUT MEMBERSHIP FUNCTION SHAPES

4.4 CONVERSION OF PID CONTROLLERS INTO FUZZY CONTROLLERS

4.5 INCREMENTAL FUZZY CONTROL

4.6 SUMMARY

CHAPTER 5 MODELING AND CONTROL METHODS USEFUL FOR FUZZY CONTROL

5.1 CONTINUOUS-TIME MODEL FORMS

5.2 MODEL FORMS FOR DISCRETE-TIME SYSTEMS

5.3 SOME CONVENTIONAL CONTROL METHODS USEFUL IN FUZZY CONTROL

5.4 SUMMARY

CHAPTER 6 TAKAGI–SUGENO FUZZY SYSTEMS

6.1 TAKAGI–SUGENO FUZZY SYSTEMS AS INTERPOLATORS BETWEEN MEMORYLESS FUNCTIONS

6.2 TAKAGI–SUGENO FUZZY SYSTEMS AS INTERPOLATORS BETWEEN CONTINUOUS-TIME LINEAR STATE-SPACE DYNAMIC SYSTEMS

6.3 TAKAGI–SUGENO FUZZY SYSTEMS AS INTERPOLATORS BETWEEN DISCRETE-TIME LINEAR STATE-SPACE DYNAMIC SYSTEMS

6.4 TAKAGI–SUGENO FUZZY SYSTEMS AS INTERPOLATORS BETWEEN DISCRETE-TIME DYNAMIC SYSTEMS DESCRIBED BY INPUT–OUTPUT DIFFERENCE EQUATIONS

6.5 SUMMARY

CHAPTER 7 PARALLEL DISTRIBUTED CONTROL WITH TAKAGI–SUGENO FUZZY SYSTEMS

7.1 CONTINUOUS-TIME SYSTEMS

7.2 DISCRETE-TIME SYSTEMS

7.3 PARALLEL DISTRIBUTED TRACKING CONTROL

7.4 PARALLEL DISTRIBUTED MODEL REFERENCE CONTROL

7.5 SUMMARY

CHAPTER 8 ESTIMATION OF STATIC NONLINEAR FUNCTIONS FROM DATA

8.1 LEAST-SQUARES ESTIMATION

8.2 BATCH LEAST-SQUARES FUZZY ESTIMATION IN MAMDANI FORM

8.3 RECURSIVE LEAST-SQUARES FUZZY ESTIMATION IN MAMDANI FORM

8.4 LEAST-SQUARES FUZZY ESTIMATION IN TAKAGI–SUGENO FORM

8.5 GRADIENT FUZZY ESTIMATION IN MAMDANI FORM

8.6 GRADIENT FUZZY ESTIMATION IN TAKAGI–SUGENO FORM

8.7 SUMMARY

CHAPTER 9 MODELING OF DYNAMIC PLANTS AS FUZZY SYSTEMS

9.1 MODELING KNOWN PLANTS AS T–S FUZZY SYSTEMS

9.2 IDENTIFICATION IN INPUT–OUTPUT DIFFERENCE EQUATION FORM

9.3 IDENTIFICATION IN COMPANION FORM

9.4 SUMMARY

CHAPTER 10 ADAPTIVE FUZZY CONTROL

10.1 DIRECT ADAPTIVE FUZZY TRACKING CONTROL

10.2 DIRECT ADAPTIVE FUZZY MODEL REFERENCE CONTROL

10.3 INDIRECT ADAPTIVE FUZZY TRACKING CONTROL

10.4 INDIRECT ADAPTIVE FUZZY MODEL REFERENCE CONTROL

10.5 ADAPTIVE FEEDBACK LINEARIZATION CONTROL

10.6 SUMMARY

REFERENCES

APPENDIX COMPUTER PROGRAMS

Index

Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Lilly, John H., 1949–

Fuzzy control and identification / John H. Lilly.

p. cm.

 ISBN 978-0-470-54277-4 (cloth)

 ISBN 978-1-118-09781-6 (ebk)

1. Fuzzy automata. 2. System identification. 3. Automatic control–Mathematics. I. Title.

TJ213.L438 2010

629.8–dc22

2010007956

For Faith, Jack, and Sarah

PREFACE

In 1982, when I obtained my Ph.D. specializing in adaptive control (the nonfuzzy kind), fuzzy control had not been explored to a very great extent as a research area. There had been only a handful of papers (probably <100) published on the subject up to that time, and some of us “serious researchers” did not take fuzzy seriously as a control method. Since then, of course, the number of papers and books written on some application of fuzzy sytstems has grown to tens of thousands, and many of us “serious researchers,” after realizing the potential of the fuzzy approach, have partially or completely redirected our research efforts to some aspect or application of fuzzy identification, classification, or control.

Roughly 10 years after graduating, I started reading anything I could find on the subjects of fuzzy identification and control, culminating in the creation of a graduate-level course on the subject at the University of Louisville. This book is an outgrowth of lectures I presented in this course over the past 10 years, plus some new material that I have not presented yet, but probably will at some point.

I wrote this book to present an introductory-level exposure to two of the principal uses for fuzzy logic: identification and control. This book was written to include topics that I deem important to the subject, but that I could not find all together in any one text. I kept finding myself borrowing material from several sources to teach my course, which is suboptimal for teacher and student alike. In addition, I found that many texts, although excellent, were written on too high a level to be useful as introductory texts. (It is ironic that a subject ridiculed by many as “too easy” quickly becomes so complex as to turn most people away once the basics are covered.) Consequently, I wrote this book, which includes subjects that I think important at hopefully not such a high level as to “blow away” most students.

The book is intended for seniors and first-year graduate students. Some background in control is helpful, but many topics covered in introductory controls courses are of little use here, such as gain and phase margins, root locus, Bode and Nyquist plots, transient and steady-state response, and so on. On the other hand, some of the subjects addressed in this book, such as tracking, model reference, adaptive identification and control, are only covered in advanced-level controls courses. This is in part what makes this subject difficult to teach.

The most helpful preparation would be some understanding of continuous- and discrete-time dynamic systems, and an appreciation of the basic aims and methods of control (i.e., stabilization, tracking, and model reference control). There is little in the way of advanced mathematics beyond differential and difference equations, transfer functions, and linear algebra required to read and understand this book.

The subjects of fuzzy identification and control are quite heavy in computer programming. In order to implement or simulate fuzzy systems, it is almost unavoidable to write computer programs, so it is assumed that the reader is comfortable with at least basic computer programming and computer simulation of dynamic systems. In this book, Matlab is used exclusively for simulations due to its ease of programming matrix manipulations and plotting. I have not relied on any Matlab “canned” programs (e.g., the Matlab differential equation solvers ode23, ode45, etc.) or toolboxes (e.g., the Fuzzy Logic Toolbox). One exception is the use of the LMI Control Toolbox used in Chapter 7 to solve a linear matrix inequality. The avoidance of these very powerful specialized tools that Matlab provides was done to give a measure of transparency in the example programs provided in the Appendix, and also because whatever computer language is used to implement these controllers may not (in fact, probably will not) have them.

ARRANGEMENT OF THIS BOOK

The arrangement of this book may seem strange to some. Chapter 5, which presents some well-known nonfuzzy modeling and control methods, may look out of place in the middle of the other chapters, which have to do with only fuzzy topics. It was suggested to me that the material in Chapter 5 either be placed in an introductory chapter or relegated to an appendix. However, I felt there is good reason to place it where it is.

Chapters 2–4 cover basic concepts of fuzzy logic, fuzzy sets, fuzzy systems, and control with Mamdani fuzzy systems. All controllers presented in Chapter 4 are designed on the basis of “expert knowledge.” Their design is not based on any mathematical model of the system they control, nor do they use any formal control method (pole placement, tracking, etc.). Therefore, there is no need to study mathematical modeling or control methods to utilize anything through Chapter 4.

On the other hand, Chapters 6 and 7 introduce Takagi–Sugeno (T–S) fuzzy systems, which do necessitate the utilization of a plant model along with choice of some formal control methodology. Thus, the introduction of some standard modeling and control techniques seemed well placed between the Mamdani and T–S developments. I felt that placing this material in either an introductory chapter or an appendix would reduce its chances of being read. At any rate, the chapters are as follows.

Chapter 1 is an introduction to fuzzy logic, fuzzy control, and adaptive fuzzy control. We introduce the concept of expert knowledge, which is the basis for much of fuzzy control. We talk briefly about when fuzzy methods may be justified, when they may not, and why. We discuss the plants used in the examples to illustrate various principles taught in this book. Also included in Chapter 1 are brief descriptions of the identification and control problems. Finally, these are combined to discuss the concept of adaptive fuzzy control.

Chapter 2 covers basic concepts of fuzzy sets, such as membership functions, universe of discourse, linguistic variables, linguistic values, support, α-cut, and convexity. We also discuss some set theoretic and logical operations on fuzzy sets, such as fuzzy subset, fuzzy complement, fuzzy intersection, and fuzzy Cartesian product.

Chapter 3 introduces Mandani fuzzy systems, which were historically the first type of fuzzy system used for control. We discuss the various processes that make up fuzzy systems, including fuzzification, inference, defuzzification, and rule base. We discuss the two most common types of defuzzification: center of gravity and center average. In this chapter, we also introduce the concept of the input–output characteristic of a fuzzy controller. Finally, we introduce the singleton fuzzy set, which is used in all subsequent fuzzy identifiers and controllers in this book.

Chapter 4 discusses closed-loop fuzzy control with Mamdani fuzzy systems. It is shown how an effective controller can be designed for many complex nonlinear systems using only common sense. We discuss how the controller can be tuned for improved performance by scaling universes of discourse. We also discuss how fuzzy controllers can be redesigned (again on the basis of common sense) to increase robustness. Chapter 4 includes a method of converting a nonfuzzy proportional-integral-derivative (PID) controller into a fuzzy controller for the purpose of redesigning it to increase robustness. This chapter also includes an introduction to incremental fuzzy control.

Chapter 5 is nonfuzzy. It contains a summary of some common modeling and control techniques that will be used in the rest of the book. It is shown how continuous-time nonlinear systems, which most real-world systems are, can be modeled as fuzzy systems in several forms (continuous-time feedback linearizable form, continuous- and discrete-time linear state-space form, and discrete-time input–output difference equation form). All of these forms are used later in the book. Also included in Chapter 5 are some conventional control methods used in fuzzy control, such as pole placement control, tracking control, model reference control, and feedback linearization. Again, these are introduced because they are used later in the book.

Chapter 6 introduces T–S fuzzy systems as interpolators between memoryless functions, continuous- and discrete-time dynamic systems described in state-space form, and discrete time linear input–output dynamic systems.

Chapter 7 introduces parallel distributed control with T–S fuzzy systems. We introduce the concept of linear matrix inequalities, by which stability can be proved for closed-loop systems involving fuzzy controllers. We discuss how fuzzy tracking and model reference control can be realized for nonlinear systems using parallel distributed controllers.

Chapter 8 discusses the estimation of static nonlinear functions from data using the batch least squares, recursive least squares, and gradient methods. The gradient parameter update equations, similar to backpropagation in neural networks, are derived. We address the importance of choice of input data, as well as model validation for these methods.

Chapter 9 uses the principles discussed in Chapter 8 to obtain T–S fuzzy models of dynamic plants for the purpose of using these for closed-loop control. The chapter begins by giving a method of modeling time-invariant nonlinear systems with known mathematical models, with T–S fuzzy systems. The remainder of the chapter is concerned with identification from data of nonlinear continuous time systems as T–S fuzzy systems in either feedback linearizeable or input–output difference equation form.

Chapter 10 uses the principles given in Chapter 9 to develop direct and indirect adaptive fuzzy controllers. These methods are applied to several different systems including a motor-driven robot arm, a ball-and-beam system, and a gantry.

WHAT IS NOT COVERED IN THIS BOOK

A thorough presentation of the topics covered in this book would be quite involved technically and would include a lot of complicated notation. While this is certainly valuable and even indispensable to one wanting to be fully versed in fuzzy control and identification, I have tried to streamline things somewhat in this introductory exposition of the subject by omitting some topics. Specifically, the following topics, which are well known to fuzzy practitioners, are not covered here.

1. Fuzzification Methods Other Than Singleton Fuzzification

Fuzzification is the method by which measured quantities from real-world systems are converted into fuzzy sets. There are many methods of fuzzification, depending on how the measured quantities are to be interpreted. The most straightforward fuzzification philosophy is singleton fuzzification, in which the measured quantities are simply taken to be exact as measured. Since most fuzzification in practice is of the singleton type, this is the only fuzzification strategy considered here.

2. Nonsingleton Membership Function Shapes Other Than Triangles and Gaussians

In general, the membership function characterizing a fuzzy set should accurately reflect membership in the set. Thus the choice of shapes is virtually unlimited. In practice, however, only a few membership shapes tend to be used much. Of these, we have chosen triangular and Gaussian shapes (triangular because of the possibility of partitions of unity, and Gaussian because of their usefulness in fuzzy identifiers and adaptive controllers).

3. Defuzzification Methods Other Than Center of Gravity and Center Average

Defuzzification is the method by which fuzzy sets are converted into crisp numbers to be delivered to the outside world. There are numerous defuzzification strategies, but we have chosen to present only the two most common ones: center of gravity and center average defuzzification.

4. Overall Implied Fuzzy Sets for Defuzzification

Overall implied fuzzy sets are not covered for defuzzification here in favor of the more straightforward individual implied fuzzy sets.

5. Universal Approximation Property of Fuzzy Systems

The universal approximation property of fuzzy systems states that any function can be approximated by a fuzzy system to arbitrary accuracy. It is important in that it justifies the use of fuzzy identifiers for nonlinear systems. However, it can be found in many references, and seemed slightly off-topic for the purposes of this book. Therefore, it is omitted.

6. Identification Methods Other Than Least Squares and Gradient

We concentrate on batch and recursive least squares (RLS) and gradient methods and omit other methods, such as orthogonal projection, least-mean squares, clustering, and learning from examples. Also omitted are methods that add momentum to identification techniques.

ACKNOWLEDGMENTS

I would like to acknowledge the help of several people who directly or indirectly influenced me in the writing of this book. First, I have learned much from my Ph.D. students Jerry Branson, Liang Yang, and Jie Liu, all of whom expanded my thinking in fuzzy control in various ways. Second, my colleague Jacek Zurada, who himself is the author of several excellent books on neural networks and computational intelligence, has helped me immeasurably in my career and has given me excellent advice and inspiration. I am grateful to my own Ph.D. advisor, Prof. Mark Balas, for giving me an appreciation of mathematics and rigor in engineering. Finally, I would like to thank Prof. Chi-Tsong Chen, who is the author of numerous excellent textbooks in control, linear systems, and signals, for indirectly inspiring me to write this book. His inspiration came several decades ago when I taught at SUNY Stony Brook, but I have never forgotten it.