Type-2 Fuzzy Logic in Control of Nonsmooth Systems: Theoretical Concepts and Applications

126 Pages·2019·5.6 MB·English

by  Oscar Castillo

Studies in Fuzziness and Soft Computing
Oscar Castillo
Luis T. Aguilar
Type-2 Fuzzy
Logic in Control
of Nonsmooth
Systems
Theoretical Concepts and Applications
Studies in Fuzziness and Soft Computing
Volume 373
Series editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
e-mail: [email protected]
The series “Studies in Fuzziness and Soft Computing” contains publications on
various topics in the area of soft computing, which include fuzzy sets, rough sets,
neural networks, evolutionary computation, probabilistic and evidential reasoning,
multi-valuedlogic,andrelatedfields.Thepublicationswithin“StudiesinFuzziness
and Soft Computing” are primarily monographs and edited volumes. They cover
significant recent developments in the field, both of a foundational and applicable
character. An important feature of the series is its short publication time and
world-wide distribution. This permits a rapid and broad dissemination of research
results.
More information about this series at http://www.springer.com/series/2941
Oscar Castillo Luis T. Aguilar
(cid:129)
Type-2 Fuzzy Logic
in Control of Nonsmooth
Systems
Theoretical Concepts and Applications
123
Oscar Castillo LuisT. Aguilar
Division of Graduate Studies CentrodeInvestigacióny Desarrollo
andResearch deTecnologíaDigital
TijuanaInstitute of Technology Instituto Politécnico Nacional
Tijuana, BajaCalifornia, Mexico Tijuana, BajaCalifornia, Mexico
ISSN 1434-9922 ISSN 1860-0808 (electronic)
Studies in FuzzinessandSoft Computing
ISBN978-3-030-03133-6 ISBN978-3-030-03134-3 (eBook)
https://doi.org/10.1007/978-3-030-03134-3
LibraryofCongressControlNumber:2018960196
©SpringerNatureSwitzerlandAG2019
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Fuzzylogiccontrolapproachhasbeenanactiveresearchandapplicationareasince
the 60s of the past century. This approach has been well understood when dealing
with dynamic models described by linear or continuously differentiable nonlinear
time-invariant or time-variant differential equations. This book presents the syn-
thesis and analysis offuzzy controllers and its application to a class of mechanical
systems. The purpose of this monograph goes beyond on addressing type-1 fuzzy
logic controllers, since thecore material about synthesis and Lyapunovanalysisof
type-2fuzzylogiccontrollersanditsapplicationtosystemsgovernedbydifferential
equations with discontinuous right-hand side or with non-differentiable terms, also
known as nonsmooth systems, constitute the main topic of this book.
FuzzyLogicSystemsarecomprisedofrules.Quiteoften,theknowledgethatis
used to build these rules is uncertain. Such uncertainty leads to rules whose ante-
cedents or consequents are uncertain, which translates into uncertain antecedent or
consequent membership functions [1]. Type-1 fuzzy systems, whose membership
functionsaretype-1fuzzysets,areunabletodirectlyhandlesuchuncertainties.We
describe in this chapter, type-2 fuzzy systems in which the antecedent or conse-
quent membership functions are type-2 fuzzy sets. Such sets are fuzzy sets whose
membership grades themselves are type-1 fuzzy sets; they are very useful in cir-
cumstances where it is difficult to determine an exact membership function for a
fuzzy set.
Theoriginalfuzzylogic,foundedbyLotfiZadeh,hasbeenaroundformorethan
30 years, and yet it is unable to handle uncertainties [2]. That the original fuzzy
logic (type-1 fuzzy logic) cannot do this sounds paradoxical because the word
“fuzzy”hastheconnotationofuncertainty.Theexpandedfuzzylogic(type-2fuzzy
logic) is able to handle uncertainties because it can model and minimize their
effects.
Theconceptofatype-2fuzzyset,wasintroducedbyZadeh[3]asanextension
of the concept of an ordinary fuzzy set (henceforth called a “type-1 fuzzy set”).
A type-2 fuzzy set is characterized by a fuzzy membership function, i.e., the
membership grade for each element of this set is a fuzzy set in [0, 1], unlike a
type-1 set where a membership grade is a crisp number in [0, 1]. Such sets can be
v
vi Preface
used in situations where there is uncertainty about the membership grades them-
selves, e.g., uncertainty in the shape of the membership function or some of its
parameters. Consider the transition from ordinary sets to fuzzy sets. When we
cannotdeterminethemembershipofanelementinasetas0or1,weusefuzzysets
oftype-1.Similarly,whenthesituationissofuzzythatwehavetroubledetermining
themembershipgradeevenasacrispnumberin[0,1],weusefuzzysetsoftype-2.
Togetherwithfuzzylogic,inthisbook,wepresentgeneticalgorithms,whichis
a basic search methodology that can be used for modeling and simulation of
complexnon-ineardynamicalsystems.Sincethesetechniquescanbeconsideredas
general purpose optimization methodologies, we can use them to find the mathe-
matical model which minimizes the fitting errors for a specific problem. On the
otherhand,wecanalsousethistechniqueforsimulationifweexploittheirefficient
search capabilities to find the appropriate parameter values for a specific mathe-
matical model. We can use a genetic algorithm to optimize the number of rules or
the membership functions of a fuzzy system for a specific problem. These are two
importantapplicationsofgeneticalgorithms,whichwillbeusedinlaterchaptersto
design intelligent systems for controlling real-world dynamical systems.
Whynonsmooth systems? Mechanical systems are typically affectedby external
forces such as friction, dead-zone, and backlash which adversely affect the per-
formanceoftheplant.Thosekindsofdisturbancesareknownashardornonsmooth
nonlinearitiesbecausetheirmodelsareaccuratelydescribedbynon-differentiableor
discontinuous functions. Coulomb friction modeled by a signum function is an
exampleofdiscontinuity.Mechanicalsystemscomposedofinteractingrigidbodies
belong to this class of systems [4] (e.g., biped robots). Instant impacts [5] are also
nonlinear phenomena which belong to the class of nonsmooth systems.
This book is a research monograph that summarizes the investigation we made
since the middle of the past decade on type-2 fuzzy logic and the control of
nonsmoothmechanicalsystems.Thetheoreticalresultsareillustratedbysimulation
andexperimentsmadeforthecontrolofservomotorswithdead-zoneandCoulomb
friction, the control of wheeled mobile robots, and a biped robot. The closed-loop
systems are analyzed in the framework of smooth and nonsmooth Lyapunov
functions.
Stability is one of the more important aspects of the traditional knowledge of
automatic control. Type-2 fuzzy logic is an emerging and promising area for
achievingintelligentcontrol(in thiscase,fuzzy control).Inthisbook,wereferthe
fuzzyLyapunovsynthesisasproposedbyMargaliotandLangholz[6].Stabilityhas
been one of the central issues concerning fuzzy control since Mamdani’s pioneer
works[7,8].Mostofthecriticalcommentstofuzzycontrolareduetothelackofa
general method for its stability analysis which this book covers.
This book first introduces a background of type-1 and type-1 fuzzy logic sys-
tems. Part II is devoted to genetic algorithms used for modeling and simulation of
complex nonlinear systems. Part III includes the synthesis and analysis of fuzzy
logic control to a servomotor with friction up to biped robots. This monograph is
addressed to engineers, researchers, and practitioners interested in both the fuzzy
logic control approach and the stabilization of electromechanical systems.
Preface vii
This work took advantage of the collaborative research developed with Nohé
Cazárez and Selene Cárdenas.
The work was partially supported by CONACYT under grant 285279.
Tijuana, Mexico Oscar Castillo
Luis T. Aguilar
References
1. Karnik,N.,Mendel,J.:Anintroductiontotype-2fuzzylogicsystems.Tech.Rep.,University
ofSouthernCalifornia(1998)
2. Mendel, J.: Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions.
PrenticeHall,UpperSaddleRiver,(2001)
3. Zadeh,L.:Theconceptofalinguisticvariableanditsapplicationtoapproximatereasoning—I.
Info.Sci.8(3),199–249(1975)
4. Brogliato,B.:NonsmoothMechanics.Springer,London(1999)
5. Yang,T.:ImpulsiveControlTheory.Springer,London(2000)
6. Margaliot, M., Langholz, G.: New Approaches to Fuzzy Modeling and Control: Design and
Analysis.WorldScientific,Singapore(2000)
7. Mamdani,E.:Advancesinthelinguisticsynthesisoffuzzycontrollers.J.Man.Mach.Stud.8,
669–678(1976)
8. Mamdani, E., Assilian, S.: An experiment in linguistic synthesis with fuzzy logic controller.
J.Man.Mach.Stud.7(1),1–13(1975)
Contents
Part I Introduction
1 Background on Type-1 and Type-2 Fuzzy Logic . . . . . . . . . . . . . . . 5
1.1 Type-2 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Operations of Type-2 Fuzzy Sets. . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Type-2 Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Singleton Type-2 Fuzzy Logic Systems . . . . . . . . . . . . . . 14
1.3.2 Non-singleton Fuzzy Logic Systems . . . . . . . . . . . . . . . . . 18
1.3.3 Sugeno Type-2 Fuzzy Systems. . . . . . . . . . . . . . . . . . . . . 18
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Part II Genetic Algorithms
2 Genetic Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Genetic Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Modifications to Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Chromosome Representation . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2 Objective Functions and Fitness . . . . . . . . . . . . . . . . . . . . 31
2.2.3 Selection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.4 Genetic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.5 Parallel Genetic Algorithm. . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Applications of Genetic Algorithms. . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Part III Applications
3 Fuzzy Lyapunov Synthesis for Nonsmooth Mechanical Systems. . . . 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Fuzzy Logic Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
ix
x Contents
3.2.1 Type-1 Fuzzy Logic Control . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Type-2 Fuzzy Logic Control . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Systematic and Practical Design of the Stable Fuzzy
Controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Fuzzy Control for Systems with Dead-Zone and Backlash . . . . . . . . 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.1 The Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Fuzzy Sets and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.1 Type-1 Fuzzy Sets and Systems . . . . . . . . . . . . . . . . . . . . 59
4.3.2 Type-2 Fuzzy Sets and Systems . . . . . . . . . . . . . . . . . . . . 60
4.4 Design of the Fuzzy Logic Controller . . . . . . . . . . . . . . . . . . . . . 61
4.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Fuzzy Control Synthesis for Systems with Discontinuous
Friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3 Tools for Stability Analysis of Nonsmooth Systems . . . . . . . . . . . 76
5.4 Fuzzy Logic Control Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4.1 Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4.2 Fuzzy Logic Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5 Stability Analysis for the DC Motor with Backlash
and Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5.1 Comparison Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 Fuzzy Control for Wheeled Mobile Robots. . . . . . . . . . . . . . . . . . . . 85
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 Dynamic Model and Problem Statement . . . . . . . . . . . . . . . . . . . 86
6.3 Fuzzy Control Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3.1 Control of the Kinematic Model. . . . . . . . . . . . . . . . . . . . 88
6.3.2 Type-1 Velocity Fuzzy Control Synthesis . . . . . . . . . . . . . 90
6.3.3 Type-2 Velocity Fuzzy Control Synthesis . . . . . . . . . . . . . 92
6.4 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95