Table Of ContentKaiMichels,FrankKlawonn,RudolfKruse,AndreasNürnberger
FuzzyControl
StudiesinFuzzinessandSoftComputing,Volume 200
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Rule-BasedEvolutionaryOnlineLearning
FuzzyControl,2006
Systems,2006
ISBN3-540-31765-1
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Kai Michels
Frank Klawonn
Rudolf Kruse
Andreas Nürnberger
Fuzzy Control
Fundamentals, Stability
and Design of Fuzzy Controllers
ABC
Dr.KaiMichels ProfessorDr.RudolfKruse
FichtnerGmbH&Co.KG Fak.Informatik,Abt.Wiss.-und
Sarweystr.3,70191Stuttgart Sprachverarbeitung
Germany Otto-von-Guericke-Universität
E-mail:[email protected] Universitätsplatz2
39106Magdeburg
Germany
E-mail:[email protected]
ProfessorDr.FrankKlawonn Dr.AndreasNürnberger
FHBraunschweig/Wolfenbüttel ComputerScienceDivision
FBInformatik UniversityCaliforniaatBerkeley
SalzdahlumerStr.46/48 CoryHall
38302Wolfenbüttel 94720Berkeley
Germany USA
E-mail:[email protected] E-mail:[email protected]
LibraryofCongressControlNumber:2006920240
ISSNprintedition:1434-9922
ISSNelectronicedition:1860-0808
ISBN-10 3-540-31765-1SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-31765-4SpringerBerlinHeidelbergNewYork
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Preface
“Fuzzy Control - the revolutionary computer technology that is changing our
world” - these and other headlines could be read when in the early 90’s news
from Japan came over telling us about the success of fuzzy controllers. The
idea which was put into practice had been suggested by Lotfi A. Zadeh in
Berkeley in 1965. It had been developed and tested in some practical ap-
plications, especially in Europe. In Japan fuzzy control was celebrated as a
technologyreflectingtheJapanesewayofthinkingbyitsunsharpnessandim-
plicitoverlappingofseveralstatements.Anewtechnologyboomwaspredicted
for Japan which would make Europe lose ground.
Consequently, this news created unrest. Research projects were initiated
and development departments were engaged to translate fuzzy control into
products. Adversaries and supporters hurried up to inform themselves and
intensely discussed whether the “conventional” or the fuzzy control were the
better alternative.
Finally,theexcitementcooleddownsinceinrecentyearsfuzzycontrolwas
analyzedfromtheclassicalpointofview.Thus,amoreobjectiveevaluationof
itsstrongandweakpointswaspossible.Furthermore,itwasshownhowfuzzy
systems could be put to use in the steering level which is the level above the
control loop, especially in interaction with other methods of soft computing
and artificial intelligence. Based on these fundamentals, the aim of this book
is to support the convenient use of fuzzy controllers and fuzzy systems in the
branch of control engineering and automation systems. This book addresses
both the control engineers, who should regard fuzzy controllers as additional
option for solving control problems, and computer scientists, for introducing
themintotheworldofcontrolengineeringandtoshowthemsomepossibilities
of applying methods of soft computing and artificial intelligence.
On the one hand, this book is a text book explaining the fundamentals
whicharerequiredforpeopleworkingwithfuzzycontrollers-itaddressesstu-
dents of engineering and computer science after their intermediate exam. On
theotherhand,thisbookservesasacomprehensivereferencebooktoseveral
VI Preface
aspects of fuzzy controllers and the current state of the art. The structure of
this book was chosen according to these aims.
In the first chapter we give an introduction to the theory of fuzzy systems
and describe i.e. the way of connecting fuzzy sets and their semantic back-
ground.Onlytheknowledgeaboutthismakestheuserabletoestimatereally
the possibilities of applying fuzzy systems.
In the second chapter we explain the fundamentals of control engineering
which are important for people working with fuzzy controllers. Thus, this
chapterisactuallynotforcontrolengineersbutalsothesecouldbeinterested
inthesubchaptersabouti.e.hyperstabilitytheoryorslidingmodecontrollers
since,normally,theseissuesarenotpartofcontrolengineeringfundamentals.
In the third chapter we introduce the fuzzy controller; it can be quiet
short because of the well-founded introduction of fuzzy systems in chapter
one. The main issue of this chapter is to present the current types of fuzzy
controllers.Furthermore,wegiveaninterpretationoftheMamdanicontroller
with similarity relations which help us to explain the ideas fuzzy controllers
arebasedon.Attheendofthischapterwediscussinhowfarfuzzycontrollers
are advantageous or disadvantageous with regard to classic controllers.
In the fourth chapter we discuss the stability analysis of fuzzy controllers.
The stability is the most significant question for every controller and it is a
field in which very interesting developments were made in the recent years.
Therefore,itistheissueofaspecialchapterwherewegiveanoverviewabout
the different approaches of stability analysis and discuss the pros and cons of
the methods in order to provide some help for decision making in practical
applications.
Inthefinalchapterwedescribeapproachesofevaluationandoptimization
of fuzzy controllers, that is, methods of supporting or even automating the
design of fuzzy controllers, especially through the use of neural nets and evo-
lutionary algorithms. Additionally to the fundamental descriptions of these
issues we take recent results of research into account.
Finally, we would like to thank all the people who helped us writing this
book, either by the work of their research projects, by student’s works or by
the discussions we benefited from. We are especially grateful to Prof. Werner
Leonhard for initiating the research project about stability analysis and self
adjustment of fuzzy conrollers, to Prof. Kai Mu¨ller for supporting us so ex-
cellently in questions of control engineering, to Prof. Lotfi A. Zadeh for many
ideasanddiscussions,toAnneKutzeraforhersupportintranslatingpartsof
thisbookandtotheserieseditorJanuszKacprzykforthegoodco-operation.
We are also grateful to many others of our colleagues and friends who – di-
rectly or indirectly – supported us during the work on this book.
Germany Kai Michels
November 2005 Frank Klawonn
Rudolf Kruse
Andreas Nu¨rnberger
Contents
1 Fundamentals of Fuzzy Systems ........................... 1
1.1 Fuzzy Sets.............................................. 2
1.2 Representation of Fuzzy Sets.............................. 5
1.2.1 Definition Using Functions ......................... 5
1.2.2 Level Sets ........................................ 7
1.3 Fuzzy Logic............................................. 9
1.3.1 Propositions and Truth Values ...................... 11
1.3.2 t-Norms and t-Conorms ............................ 13
1.3.3 Basic Assumptions and Problems.................... 18
1.4 Operations on Fuzzy Sets................................. 19
1.4.1 Intersection....................................... 19
1.4.2 Union............................................ 21
1.4.3 Complement...................................... 22
1.4.4 Linguistic Modifiers................................ 23
1.5 The Extension Principle.................................. 24
1.5.1 Mappings for Fuzzy Sets ........................... 25
1.5.2 Mapping of Level Sets ............................. 27
1.5.3 Cartesian Product and Cylindric Extension........... 29
1.5.4 Extension Principle for Multivariate Mappings ........ 30
1.6 Fuzzy Relations ......................................... 31
1.6.1 Crisp Relations ................................... 31
1.6.2 Application of Relations and Deduction .............. 32
1.6.3 Chains of Deductions .............................. 36
1.6.4 Simple Fuzzy Relations ............................ 37
1.6.5 Composition of Fuzzy Relations ..................... 41
1.7 Similarity Relations...................................... 43
1.7.1 Fuzzy Sets and Extensional Hulls.................... 45
1.7.2 Scaling Concepts .................................. 46
1.7.3 Interpretation of Fuzzy Sets ........................ 50
VIII Contents
2 Fundamentals of Control Theory .......................... 57
2.1 Basic Concepts.......................................... 57
2.2 The Model of the Plant .................................. 61
2.2.1 General Task ..................................... 61
2.2.2 Scaling........................................... 62
2.2.3 Fundamental Linear Transfer Elements............... 63
2.2.4 Fundamental Nonlinear Transfer Elements............ 65
2.2.5 First and Second Order Lags........................ 67
2.2.6 Fields of Application............................... 71
2.2.7 Linearization ..................................... 72
2.2.8 Closing Comments ................................ 73
2.3 Transfer Function ....................................... 74
2.3.1 Laplace Transformation ............................ 74
2.3.2 Computation of Transfer Functions .................. 76
2.3.3 Interpretation of the Transfer Function............... 78
2.3.4 Computation of the Step Response .................. 79
2.3.5 Simplification of a Transfer Function................. 83
2.4 Frequency Response ..................................... 84
2.4.1 Introduction to the Frequency Response.............. 84
2.4.2 Nyquist Plot...................................... 86
2.4.3 Bode Diagram .................................... 90
2.5 Stability of Linear Systems ............................... 91
2.5.1 A Definition of Stability............................ 91
2.5.2 Stability of a Transfer Function ..................... 93
2.5.3 Stability of a Control Loop ......................... 94
2.5.4 Criterion of Cremer, Leonhard and Michailow......... 96
2.5.5 Nyquist Criterion ................................. 98
2.6 PID Controller ..........................................102
2.6.1 Demands on a Controller...........................102
2.6.2 Types of Controllers ...............................106
2.6.3 Controller Design..................................113
2.6.4 Structure Expansion ...............................116
2.7 State Space Description and State Space Control ............125
2.7.1 Fundamentals.....................................125
2.7.2 Controllability and Observability ....................134
2.7.3 Lyapunov’s Definition of Stability for Linear Systems ..138
2.7.4 Design of a State Space Controller...................141
2.7.5 Linear Observers ..................................146
2.7.6 Steady-State Accuracy of State Space Controllers .....149
2.7.7 Norm-Optimal Controllers..........................151
2.8 Nonlinear Systems.......................................157
2.8.1 Features of Nonlinear Systems ......................157
2.8.2 Application of Linear Methods to Nonlinear Systems...158
2.8.3 Switching Elements................................165
2.8.4 Definition of Stability for Nonlinear Systems ..........177
Contents IX
2.8.5 Lyapunov’s Direct Method .........................185
2.8.6 Describing Function ...............................189
2.8.7 Popov Criterion...................................202
2.8.8 Circle Criterion ...................................210
2.8.9 Hyperstability ....................................218
2.8.10 Sliding Mode Controllers ...........................228
2.8.11 Nonlinear Observers ...............................232
3 Design of Fuzzy Controllers................................235
3.1 Mamdani Controllers ....................................235
3.1.1 Remarks on Fuzzy Controller Design.................240
3.1.2 Defuzzification Methods............................243
3.2 Takagi-Sugeno-Kang Controllers...........................245
3.3 Logic-based Controllers ..................................246
3.4 Mamdani Controllers and Similarity Relations ..............249
3.4.1 Interpretation of a Controller .......................249
3.4.2 Construction of a Controller ........................251
3.5 Fuzzy Control versus Classical Control .....................253
4 Stability Analysis of Fuzzy Controllers.....................257
4.1 Preliminaries............................................257
4.1.1 Structure of the Closed-Loop System ................257
4.1.2 Analytic Approximation of a Fuzzy Controller ........259
4.1.3 Takagi-Sugeno-Kang Systems .......................261
4.2 Lyapunov’s Direct Method................................266
4.2.1 Application to Conventional Fuzzy Controllers ........266
4.2.2 Application to Takagi-Sugeno-Kang Controllers .......268
4.2.3 Application to Facet Functions......................279
4.3 Describing Function .....................................284
4.4 Popov Criterion .........................................286
4.5 Circle Criterion .........................................287
4.5.1 Controllers with One Input Quantity ................287
4.5.2 Controllers of Multiple Input Quantities..............287
4.5.3 MIMO Controllers.................................290
4.6 Norm-based Stability Analysis ............................291
4.7 Hyperstability Criterion ..................................294
4.8 A Comparison with the Sliding Mode Controller.............295
4.9 Direct Analysis in State Space ............................298
4.9.1 Convex Factorization ..............................298
4.9.2 Cell-to-Cell Mapping...............................300
4.10 Conclusion .............................................304
X Contents
5 Parametrization and Optimization of Fuzzy Controllers....307
5.1 Design of TSK Controllers................................310
5.2 Adaptation of Characteristic Surfaces ......................311
5.2.1 Adaptive Compensation Control ....................311
5.2.2 Adaptive Sliding Mode Control .....................319
5.3 Adaptation of Fuzzy Rules................................321
5.4 Model-Based Control ....................................323
5.4.1 Structure of the Model.............................323
5.4.2 Single Step Control................................327
5.4.3 Optimal Control ..................................333
5.5 Fuzzy Controllers and Fuzzy Clustering ....................338
5.5.1 Fuzzy Cluster Analysis.............................338
5.5.2 Generating Mamdani Controllers ....................343
5.5.3 Generating TSK Models............................343
5.6 Neuro Fuzzy Control.....................................344
5.6.1 Neural Networks ..................................344
5.6.2 Combination of Neural Nets and Fuzzy Controllers ....349
5.6.3 Models with Supervised Learning Methods ...........352
5.6.4 Models with Reinforcement Learning ................356
5.6.5 Recurrent Approaches .............................361
5.6.6 Discussion........................................365
5.7 Fuzzy Controllers and Evolutionary Algorithms .............365
5.7.1 Evolution Strategies ...............................366
5.7.2 Genetic Algorithms................................367
5.7.3 Evolutionary Algorithms for Optimising Fuzzy
Controllers .......................................369
5.7.4 A Genetic Algorithm for Learning a TSK Controller ...372
5.7.5 Discussion........................................374
Correspondence Table for Laplace Transformations ............375
Non-Minimum Phase Systems .................................377
Norms.........................................................381
Lyapunov Equation............................................387
Lie Derivative .................................................389
Positive Real Systems .........................................391
Linear Matrix Inequalities .....................................393
References.....................................................395
Index..........................................................407
