Fuzzy Controllers
Fuzzy ControllersPRELIMS.PM516/13/97, 10:19 AM
Tomy son Dmitryand other studentsPRELIMS.PM526/13/97, 10:19 AM
Fuzzy ControllersLEONID REZNIKVictoria University of Technology,Melbourne, AustraliaPRELIMS.PM536/13/97, 10:19 AM
NewnesAn imprint of Butterworth-HeinemannLinacre House, Jordan Hill, Oxford OX2 8DPA division of Reed Educational and Professional Publishing LtdA member of the Reed Elsevier plc groupOXFORD BOSTON JOHANNESBURGMELBOURNE NEW DELHI SINGAPOREFirst published 1997 Leonid Reznik 1997All rights reserved. No part of this publication may bereproduced in any material form (including photocopyingor storing in any medium by electronic means and whetheror not transiently or incidentally to some other use ofthis publication) without the written permission of thecopyright holder except in accordance with the provisionsof the Copyright, Designs and Patents Act 1988 or underthe terms of a licence issued by the Copyright LicensingAgency Ltd, 90 Tottenham Court Rd, London, England W1P 9HE.Applications for the copyright holder’s written permissionto reproduce any part of this publication should beaddressed to the publishers.British Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryISBN 0 7506 3429 4Library of Congress Cataloguing in Publication DataA catalogue record for this book is available from the Library of CongressTypeset by The Midlands Book Typesetting Company, Loughborough,Leicestershire, EnglandPrinted in Great Britain by Biddles Ltd, Guildford and King’s LynnPRELIMS.PM546/13/97, 10:19 AM
nPart I123PRELIMS.PM55ixxixiiixvHow Does it Work? or The Theory ofFuzzy Control1Fuzzy sets, logic and control1.1 Why do we need this new theory, whatare the advantages of fuzzy control?1.2 Where does fuzzy logic come from?1.3 What are the main areas of fuzzylogic applications?3Basic mathematical concepts of fuzzy sets2.1 Fuzzy sets versus crisp sets2.2 Operations on fuzzy sets2.3 Extension principle and fuzzy algebra2.3.1 Extension principle2.3.2 Fuzzy numbers2.3.3 Arithmetic operations withintervals of confidence2.3.4 Arithmetic operations withfuzzy numbers2.4 Linguistic variables and hedges2.5 Fuzzy relations191930343437The structure and operation of a fuzzy controller3.1 The reasons to apply fuzzy controllers3.2 Fuzzy rules processing3.2.1 Mamdani-type fuzzy processing3.2.2 Linguistic variables3.2.3 Fuzzy rules firing3.2.4 Calculating the applicability degree3.2.5 Clipping and scaling a fuzzy output59596161636567686/13/97, 10:19 AM35938414451
viCONTENTS3.2.6 Sugeno-type fuzzy processing3.3 Fuzzy controllers operation3.4 Structure of a simple open-loop fuzzy controller3.5 Structure of a feedback PID-like fuzzy controller3.5.1 Fuzzy controllers as a part of a feedbacksystem3.5.2 PD-like fuzzy controller3.5.3 Rules table notation3.5.4 PI-like fuzzy controller3.5.5 PID-like fuzzy controller3.5.6 Combination of fuzzy and conventionalPID controllers3.6 Stability and performance problems for a fuzzycontrol system3.6.1 Stability and performance evaluationby observing the response3.6.2 Stability and performance indicators3.6.3 Stability evaluation by observing thetrajectory3.6.4 Hierarchical fuzzy controllersPart II4PRELIMS.PM5How to Make it Work or The Designand Implementation of Fuzzy ControllersFuzzy controller parameter choice4.1 Practical examples4.1.1 Fuzzy autopilot for a small marine vessel4.1.2 Smart heater control4.1.3 Active noise control4.2 Iterative nature of a fuzzy controller design process4.3 Scaling factor choice4.3.1 What is a scaling factor?4.3.2 Where should the tuning start?4.3.3 Application example4.4 Membership function choice4.4.1 Distributing membership functionson the universe of discourse?4.4.2 An evaluation of the membershipfunction width4.4.3 Application example4.5 Fuzzy rule formulation4.5.1 Where do rules come from?4.5.2 How do we get rules?4.5.3 How do we check if the rules are OK?66/13/97, 10:19 121124124126129131131132135136136138139
CONTENTS5PRELIMS.PM57vii4.5.4 Application examples4.6. Choice of the defuzzification procedure4.6.1 Centre-of area/gravity4.6.2 Centre-of-largest-area4.6.3 First-of-maxima/last-of-maxima4.6.4 Middle-of-maxima4.6.5 Mean-of-maxima4.6.6 Height4.6.7 Compare different uzzy controller parameter adjustment5.1 Self-organising, adaptive, and learning fuzzycontrollers: main principles and methods5.1.1 What do we need adjustments for?5.1.2 Self-organising fuzzy controllers5.1.3 Performance/robustness problemand solutions5.1.4 Adaptive fuzzy controllers5.1.5 Features of different controller types5.1.6 Learning fuzzy controllers5.2 Tuning of the fuzzy controller scaling factors5.2.1 On-line and off-line tuning5.2.2 Off-line tuning of the outputscaling factors5.2.3 On-line tuning of the input and outputscaling factors5.2.4 Application example5.3 Artificial neural networks andneuro-fuzzy controllers5.3.1 What is a neural network?5.3.2 ANN structure5.3.3 ANN types5.3.4 ANN application in fuzzy controller design5.3.5 ANFIS architecture5.3.6 Adaptive neuro-fuzzy controller5.3.7 Application examples5.4 Adjustment procedures with genetic/evolutionaryalgorithms5.4.1 How does it work?5.4.2 GA and EA application in fuzzycontroller design5.4.3 Application example1536/13/97, 10:19 171174175176177180180182184
viiiCONTENTS6Fuzzy system design software tools6.1 Fuzzy technology products classification6.2 Main features of the fuzzy software tools6.3 Realisation examples1871871901917Fuzzy controller implementation7.1 How do we implement a fuzzy controller?7.2 Implementation of a digital general purposeprocessor7.3 Implementation of a digital specialisedprocessor7.4 Specialised processor development system7.5 Implementation on analog devices7.6 Integration of fuzzy and conventionalcontrol hardware201201Part III8PRELIMS.PM5What Else Can I Use? or SupplementaryInformation202205209211214219A brief manual to fuzzy controller design8.1 When to apply fuzzy controllers8.2. When not to apply fuzzy controllers8.3 Fuzzy controller operation8.4 Which fuzzy controller type to choose?8.5 Fuzzy controller structure andparameter choice8.6 How to find membership functions8.7 How to find rules?8.8 How to implement a fuzzy controller8.9 How to test a fuzzy controller8.10 How to fix a fuzzy controller8.11 How to choose a design ms and assignment topics23910Design projects25311Glossary26712Bibliography273List of examples283Index28586/13/97, 10:19 AM
ForewordLeonid Reznik’s Fuzzy Controllers is unlike any other book onfuzzy control. In its own highly informal, idiosyncractic and yetvery effective way, it succeeds in providing the reader with awealth of information about fuzzy controllers. It does so with aminimum of mathematics and a surfeit of examples, illustrationsand insightful descriptions of practical applications.To view Fuzzy Controllers in a proper perspective a bit ofhistory is in order. When I wrote my paper on fuzzy sets in 1965,my expectation was that the theory of fuzzy sets would find itsmain applications in fields such as economics, biology, medicine,psychology and linguistics – fields in which the conventional,differential-equation-based approaches to systems analysis arelacking in effectiveness. The reason for ineffectiveness, as I sawit, is that in such fields the standard assumption that classes havesharply defined boundaries is not a good fit to reality. In thiscontext, it is natural to generalise the concept of a set byintroducing the concept of grade of membership or, equivalently,allowing the characteristic function of a set to take valuesintermediate between 0 and 1.Since my background was in systems analysis, it did not takeme long to realise that the theory of fuzzy sets is of substantialrelevance to systems analysis and, especially, to control. Thisperception was articulated in my 1971 paper ‘Toward a theory offuzzy systems’, and 1972 paper, ‘A rationale for fuzzy control’.The pivotal paper was my 1973 paper, ‘Outline of a newapproach to the analysis of complex systems and decisionprocesses’, in which the basic concepts and techniques thatunderlie most of the practical applications of fuzzy set theory (orfuzzy logic, as we call it today), were introduced. The conceptsin question are those of linguistic variable, fuzzy if-then rule andfuzzy rule sets. These concepts serve as the point of departure forwhat I call the theory of fuzzy information granulation. Thistheory postulates that in the context of fuzzy logic there are threebasic modes of generalisation of a theory, method or approach:(a) fuzzification, in which one or more crisp sets are replaced byPRELIMS.PM596/13/97, 10:19 AM
xFOREWORDfuzzy sets; (b) granulation, in which an object is partitioned intoa collection of granules, with a granule being a clump of points(objects) drawn together by indistinguishability, similarity orfunctionality; and (c) fuzzy granulation, in which a crisp or fuzzyobject is partioned into fuzzy granules. In effect, fuzzyinformation granulation (f-granulation) is a combination offuzzification and granulation.What has not been recognised to the extent that it should is thatthe successes of fuzzy logic involve not just fuzzification but,more importantly, fuzzy granulation. Furthermore, fuzzy logic isthe only methodology which provides a machinery for fuzzyinformation granulation. As we alluded to already, the keyconcepts underlying this machinery are those of linguisticvariable, fuzzy if-then rule and fuzzy rule sets. Basically, fuzzyrule sets or, equivalently, fuzzy graphs, serve to provide a way ofapproximating to a function or a relation by a disjunction ofCartesian products of values of linguistic variables.Viewed against this backdrop, it is – in effect, though not byname – the machinery of fuzzy information granulation that isemployed in fuzzy controllers to explain – with high expositoryskills – what fuzzy controllers are, how they are designed, andhow they are used in real-world applications. One cannot but begreatly impressed by the profusion of examples, the up-todatedness of information, lucidity of style and reader-friendlinessof Leonid Reznik’s exposition. His work should have strongappeal to anyone who is looking for a very informative and easyto understand introduction to fuzzy controllers and their role inthe conception, design and deployment of intelligent systems.An issue of key importance in the design of fuzzy controllersis that of induction of rules from input-output data and tuning oftheir parameters. In the past, this was done by trial and error. Morerecently, techniques drawn from neurocomputing and geneticcomputing have been employed for this purpose. In FuzzyControllers, these techniques are discussed briefly but with insightin the last chapters. In these chapters, the reader will also find avery useful discussion of fuzzy system design software tools, theircapabilities and their applications.In sum, this book is an unconventional and yet very informative,self-contained and reader-friendly introduction to the basics offuzzy logic and its application to the design of fuzzy controllers.Leonid Reznik deserves high marks for his achievement.Lotfi A. ZadehBerkeley, CAPRELIMS.PM5106/13/97, 10:19 AM
PREFACESince fuzzy logic was introduced by Lotfi Zadeh in 1965, ithas had many successful applications mostly in control. This‘fuzzy’ boom has generated strong interest in this area togetherwith a boom in studying and teaching of fuzzy theory andtechnology. Although a few books which can be used forteaching and learning are available on the market, what is stillmissed is an introductory textbook suitable for both under- andpostgraduate students, as well as for a beginner.The aim of the book is to teach a reader how to design a fuzzycontroller and to share some experience in design andapplications. It can be used as a textbook by both teachers andstudents. Being an introduction this book tends to explain thingsstarting from basics roots and does not require any preliminaryknowledge in fuzzy theory and technology. I wanted to make thisbook different from other books available on the market. My goalswere: to write a textbook that is intelligible even to a nonspecialist;to pay attention, first of all, to practical aspects of fuzzycontroller design;to facilitate the learning and teaching process for both astudent and a teacher.The structure of the book includes a description of the theoreticalfundamentals of fuzzy logic as well as study of practical aspectsof fuzzy technology. Consideration of all topics is practicallyoriented. This means that all the chapters work on achieving thefinal goal: to give a reader the knowledge necessary to design afuzzy control system. To become a real textbook which can beused for self-assessment and teaching, this book contains the listof problems, assignment topics and design projects.The style of the book changes from a textbook at the beginning(when it discusses theoretical aspects of fuzzy control) to aPRELIMS.PM5116/13/97, 10:19 AM
xiiPREFACEhandbook (when it describes software and hardware tools whichcan be used in a fuzzy controller design). The book is written(especially at the beginning) as a discussion between a teacher andstudents who come from various educational and practicalbackgrounds and are supposed to be interested in different aspectsof fuzzy control theory and technology.I wanted to avoid making this book dull and boring, so I havetried to apply ordinary (not scientific) language without losing acorrectness of mathematical determinations. It was very hardsometimes. That’s why a few chapters (especially Chapter 2)contain a number of mathematical definitions and otherconstructions. However, I tried to provide the reader with someexplanation about what all this mathematical stuff meant.PRELIMS.PM5126/13/97, 10:19 AM
AcknowledgementsMany people from different countries and continents havecontributed generously to the creation of this book. Unfortunately,I am not able to name everybody who has made this undertakinghappen. It does not mean I do not remember all of you. Yourvaluable advice, encouragement and support is very muchappreciated. Let me use this small space to give those nameswhich I cannot miss.A number of individuals, organisations and commercialcompanies have granted permission to reprint or adapt somematerial. I gratefully acknowledge the permission given by theInstitute of Electrical and Electronics Engineering, Inc.; AdaptiveLogic Inc.; David K. Kahaner, Director of Asian TechnologyInformation Program, Tokyo, Japan; INFORM GmbH, Aachen,Germany; Inform Software Corporation, Chicago, USA andConstantin von Altrock; CICS Automation, Newcastle, Australia andSam Crisafulli.I wish to thank Professor Lotfi Zadeh for the invention of fuzzylogic and inspiration of this work as well as for finding time toread the manuscript and write this great foreword.I would like to express my gratitude to my family, my parentsCarl and Ninel, my wife Olga and son Dmitry for their patience.Finally I want to acknowledge great work of the editors andpublishers in shaping the manuscript and its preparation to thepublication.FeedbackI shall appreciate receiving any comments regarding this book,especially from teachers and students. Please do not hesitate tocontact me directly or through the publishers.Leonid ReznikVictoria University of TechnologyP.O. Box 14428 MCMC Melbourne 8001 AustraliaEmail: Leon Reznik@vut.edu.auPRELIMS.PM5136/13/97, 10:19 AM
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INTRODUCTIONCould you teach us very quickly how to design a fuzzycontroller?Well, it is not that easy Just explain a general method of design, optimisation, andimplementation.I do not think there is a general method of designing andfinding optimal parameters for a fuzzy controller, because anysuch values always depend on the specific process or objectunder control and the control objectives. This is particularly thecase for a fuzzy controller where the design process is verysubjective.However, I understand you want to start a designimmediately. As you know nothing about fuzzy control andfuzzy controllers, we will base our design just on our humanexperience. Let us consider a classical control problem. Wecontrol the boat movement which we should drive along thestraight line from point A to point B. How will you design acontroller to do it?Well, firstly I must derive a mathematical model of the plant andthen develop a mathematical model of a controller.And how will you develop this model?I will apply one of the design methods, e.g., pole placementdesign, and obtain a transfer function for the controller.OK. Now suppose we do not know exactly the mathematicalmodel of our plant. Moreover, we do not know any classicaldesign method. What can we do then?Nothing. You have to know some theory. Otherwise you cannot doanything.Try to use your own experience. How did you drive a boatin your childhood?INTRO.PM5156/12/97, 1:19 PM
xviINTRODUCTIONI turned the rudder left or right depending on the position of a boat.Great! So we can say that if the boat is situated exactly on theline we should not do anything, if the boat is situated to the leftof the line we should turn the rudder to the right (let us call thisdirection positive), and if the boat is positioned to the right of theline we should turn the rudder to the left.Now let us try to formulate this control law as a set of rules.If deviation is zeroIf deviation is positiveIf deviation is negativethen turn is zero.then turn is negative.then turn is positive.We have formulated the set of the control rules which can bewritten as a table, which is a mathematical model of our controller.No! A mathematical model is either an equation or amathematical function, or something like that.Not necessarily. The mathematical model of a controllershould describe mapping of an input, in our case it is thedifference between the boat’s current position and the desiredone, to an output – control signal, in our case – the rudder angle.By using this table we can find the output corresponding to eachvalue of the input.Are you sure this table can control the boat movement?Of course, very roughly. To improve the control quality andmake our controller more reactive, we need to increase thenumber of values describing each variable. Until now we havenot distinguished the values of the deviation and turn, but justconsidered their signs. Now let us use small, medium and bigvalues for both the deviation and the turn. Then we obtain therules table:DeviationNBNMNSZPSPMPBTurnPBPMPSZNSNMNBHow can we use this table? It just contains some abbreviationswhich I do not understand.Actually these abbreviations represent the labels of fuzzyvalues. These labels give the ‘fuzzy’ value of the distance: big(PB and NB), medium (PM and NM), small (PS and NS), andINTRO.PM5166/12/97, 1:19 PM
INTRODUCTIONxviivery small (Z). The prefixes P and N represent the side of the line,that our boat deviates to, right or left, positive or negative.So we describe our controller with the help of the rules table.This is just a rules base controller. Where is the fuzzy logichidden?In the processing of these rules. The problem of controllerdesign includes not just the compilation of the rules table, butalso the use of the table to calculate a control output. So weshould also say how to process these rules in order to get theoutput result, and this processing is based on the fuzzy theorymethods examined in Section 3.2. This describes the process ofproducing a fuzzy output from fuzzy inputs, which in the fuzzyset theory is called an inference engine.Fuzzy inputs and outputs? Usually a controller uses measurementresults, doesn’t it? Are t
5.3 Artificial neural networks and neuro-fuzzy controllers 166 5.3.1 What is a neural network? 166 5.3.2 ANN structure 167 5.3.3 ANN types 171 5.3.4 ANN application in fuzzy controller design 174 5.3.5 ANFIS architecture 175 5.3.6 Adaptive neuro-fuzzy controller 176 5.3.7 Application examples 177 5.4 Adjustment procedures with genetic/evolutionary
ing fuzzy sets, fuzzy logic, and fuzzy inference. Fuzzy rules play a key role in representing expert control/modeling knowledge and experience and in linking the input variables of fuzzy controllers/models to output variable (or variables). Two major types of fuzzy rules exist, namely, Mamdani fuzzy rules and Takagi-Sugeno (TS, for short) fuzzy .
fuzzy controller that uses an adaptive neuro-fuzzy inference system. Fuzzy Inference system (FIS) is a popular computing framework and is based on the concept of fuzzy set theories, fuzzy if and then rules, and fuzzy reasoning. 1.2 LITERATURE REVIEW: Implementation of fuzzy logic technology for the development of sophisticated
Different types of fuzzy sets [17] are defined in order to clear the vagueness of the existing problems. D.Dubois and H.Prade has defined fuzzy number as a fuzzy subset of real line [8]. In literature, many type of fuzzy numbers like triangular fuzzy number, trapezoidal fuzzy number, pentagonal fuzzy number,
Fuzzy Logic IJCAI2018 Tutorial 1. Crisp set vs. Fuzzy set A traditional crisp set A fuzzy set 2. . A possible fuzzy set short 10. Example II : Fuzzy set 0 1 5ft 11ins 7 ft height . Fuzzy logic begins by borrowing notions from crisp logic, just as
of fuzzy numbers are triangular and trapezoidal. Fuzzy numbers have a better capability of handling vagueness than the classical fuzzy set. Making use of the concept of fuzzy numbers, Chen and Hwang [9] developed fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) based on trapezoidal fuzzy numbers.
ii. Fuzzy rule base: in the rule base, the if-then rules are fuzzy rules. iii. Fuzzy inference engine: produces a map of the fuzzy set in the space entering the fuzzy set and in the space leaving the fuzzy set, according to the rules if-then. iv. Defuzzification: making something nonfuzzy [Xia et al., 2007] (Figure 5). PROPOSED METHOD
2D Membership functions : Binary fuzzy relations (Binary) fuzzy relations are fuzzy sets A B which map each element in A B to a membership grade between 0 and 1 (both inclusive). Note that a membership function of a binary fuzzy relation can be depicted with a 3D plot. (, )xy P Important: Binary fuzzy relations are fuzzy sets with two dimensional
the traditional fuzzy c-means to a generalized model in convenience of application and research. 2.1 Fuzzy C-Means The basic idea of fuzzy c-means is to find a fuzzy pseudo-partition to minimize the cost function. A brief description is as follows: (1) In above formula, x i is the feature data to be clustered; m k is the center of each clus-ter; u
