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Genetic DesignofFuzzy Controllers

MarkG. Cooper&JacquesJ.Vidal

UniversityofCalifornia, LosAngeles

3531 BoelterHall

LosAngeles, California 90024

cooper@cs.ucla.edu vidal@cs.ucla.edu

Abstract

This paper considers the application of genetic algorithms as a

general methodology for automatically generating fuzzy process

controllers. In contrast to prior genetic-fuzzy systems which

require that every input-output combination be enumerated, we

propose anovel encodingscheme whichmaintainsonlythose rules

necessary to control the target system. Thekey toour method,

demonstratedonthe classiccart-pole problem, istorepresent each

fuzzysystemasanunorderedlistofanarbitrarynumberofrules.

Boththecompositionandsizeoftherule baseevolvefromaninitialrandom state. By using a compact rule base representation,

successful systems evolve quickly, increasing the likelihood of

successwhenappliedtomore complexproblems.

1. Introduction

The abilitytotranslate informationsupplied in linguisticterms into

a computer-usable form has made fuzzy logic systems popular for

implementingcomplexprocesscontrol. However, mostonlyrepresentthe

"bestguesses"ofhuman experts, and can therefore be improvedusing

feedback to tune the initial fuzzy rule bases. Automatic rule base

generation has been attempted using gradient descent techniques,

PresentedattheSecondInternationalConferenceonFuzzyTheoryandTechnology;Durham, NC, October, 1993.

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feedforward neural networks, radial basis function networks, fuzzy

clustering, andgeneticalgorithms. Thesesystems learntocontrol the

targeted process through trainingexamples are presented and the

system ismodified baseduponperformance evaluation. Unfortunately,

thenumberof rulesmustgenerally bespecifiedapriori, and learningcapacity isoftensmall. We believe, however, thatthese limitationsare

notinherenttotheproblem, butratherlieinthedetailsofthelearning

procedure employed.

To overcome these deficiencies, our approach uses a genetic

algorithmtodeterminethenumberofrulesinthefuzzyrule baseaswell

astheircomposition. Additionally, anovel encodingscheme forthe fuzzy

rulesresults inamore compactrule base thanusedpreviously, allowing

morecomplexitytoevolve.

We begin by introducingthe fuzzy inference model, followed bya

briefdescriptionofgeneticalgorithms(foracomplete treatmentofgenetic

algorithms, see Goldberg [1989]), and of the physical systemused for

demonstrationthe cart-pole problem. Followingareviewoftwo earlier

genetic-fuzzy proposals, we describe ouralternative approach, present

experiment results, and discuss the properties and limitations of our

technique.

1.

1.

TheFuzzyAssociativeMemoryModelFuzzyrulestypicallyappear inthe form"Ifx1 isv1 andx2 isv2

and..., thena1isw1 anda2isw2 and..."wherexnareinputvariables, an

are outputvariables, andvnandwnare descriptors like "small positive",

"large negative", "medium", etc. Associated with each descriptor is a

membership functionspecifying thedegree towhichan inputsatisfies

the descriptor. AFuzzy Associative Memory (orFAM) (Kosko, 1992)

appliesthese rulestoasetof inputs, combinesthe consequentsof each

rule, andproducesavalue for eachoutputvariable. To illustrate this

process, considerthe followingrule base (excerptedfromKosko):

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1. Ifthe pendulumangle, ! , hasasmall positive value (PS)

andtheangularvelocity, "!, isnearzero(ZE), thenadjust

the motorcurrent, #, byasmall negative value (NS).

2. If the pendulum angle, ! , is near zero (ZE) and theangularvelocity, "! , is near zero (ZE), thenadjust the

motorcurrent, #, byavalue nearzero(ZE).

TheserulesaredisplayedgraphicallyinFigure 1. Thetopthreegraphs

correspond to the first rule, and the bottom three correspond to the

second. The firsttwocolumns(fromlefttoright)representthe antecedent

clauses, whilethethirdrepresentstheconsequentclause.

! "! #

0 0 0

000

+ +

++ +

+

ZE ZE

ZE NS

ZE

PS

!= 15 "!!=-10

.8

.2.5

.5

Figure1. FuzzyReasoning (fromKosko)

Instantiationsofthe inputvariablesare presentedtothe rule base

in parallel. The membership function for each antecedent clause is

appliedto itscorresponding inputtoproduce a fitvalue forthe clause.

When the ruleantecedent is madeup ofa conjunction of clauses, the

minimum fitvalue among the clauses is takenas the fit for the entire

rule, i.e., asthe degree ofapplicabilityofthe rule tothe overall inference.

Inthe example, the inputvalues! = 15 and"!=-10 are appliedtothe

rule base withthe antecedentfitvaluesof 0.8 and 0.5 forthe firstrule,

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and 0.2 and 0.5 for the second. The minimumof eachpairyields the

overallfitforeachrule.

Next, the consequentofeachrule iscalculatedasthe regionunder

itsmembership function belowthe antecedent fitvalue. These regions

constitutethefuzzyoutputoftheFAM. Theoutputofthefirstruleistheregionofthe consequent'smembershipfunction belowthe antecedentfit

value 0.5. Similarly, theoutputofthesecondruleistheregionunder 0.2.

The collectionofregionsgeneratedasthe outputofthe fuzzyrules

isthe qualifiedoutputofthe FAM. Togenerate asingle value for each

consequentvariable, theseregionsmust becombined. Thisprocess, called

defuzzification, is generally accomplished by overlaying the output

regionsandcomputingthe centroid(Figure 2).

0 +

#!=!$#

#

Figure2. Defuzzification byComputingthe Centroid

1.2. GeneticAlgorithms

Geneticalgorithmsare probabilisticsearchtechniquesthat emulate

the mechanics of evolution. They are capable of globally exploring a

solutionspace, pursuingpotentially fruitfulpathswhilealsoexamining

randompoints toreduce the likelihoodofsettling fora local optimum.

The system being evolved is encoded into a long bit string called a

chromosome. Sites on the chromosome correspond to specific

characteristicsofthe encodedsystemandare calledgenes. Initially, a

randomset, orpopulation, ofthesestringsisgenerated. Eachstring isthen evaluatedaccordingtoagivenperformance criterionandassigneda

fitness score. The strings with the best scores are used in the

reproductionphasetoproducethenextgeneration.

The reproductionofapairofstringsproceeds bycopying bitsfrom

one stringuntil arandomlytriggeredcrossoverpoint, afterwhich bitsare

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copied from the other string. As each bit is copied, there is also the

probabilitythatamutationwilloccur. Mutations include inversionof

the copied bit, andthe additionordeletionofan entire rule (Figure 3).

These latter two mutations permit the size of a system's rule base to

evolve. The cycle of evaluation and reproduction continues for apredeterminednumberofgenerations, oruntil anacceptable performance

levelisachieved.

10111 000

1111 000 0

1011 0001

1 11 00000}

101 001 0 1}

10111 000 } 10111 000 1 00 1

10111 000 1 00 1 } 1 0 1 0 1000

Crossover

Mutation

RuleInsertion

Rule Deletion

Figure 3. GeneticOperators

1.3. PoleBalancing

We willapplythe evolvedfuzzysystemstoaclassiccontrolproblem

variously referred to as the "cart-pole", "inverted pendulum", or "pole

balancing"problem. Thisproblem isan example ofan inherentlyunstable

dynamic system, and is an established literature benchmark for

evaluatingcontrol schemes. The objective istocontrol translational forces

that position a cart at the center of a finite width track while

simultaneously balancingapolehingedonthecart'stop(Figure 4). The

physical plant can be simulated from a set of nonlinear differential

equationsdescribingthe cartandpole dynamics(Wieland, 1991).

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x =F $ $c sign( x ) + F

M+ m(1)

! = $3

4 l( x cos ! + g sin ! +

$p!

ml) (2)

where

F = ml!2 sin ! +3

4m cos !(

$p !

ml+ g sin !) (1a)

m = m(1 $3

4cos2 !) (1b)

Table1. Symbols Used inEquations 1 and 2

Symbol Description Value

x Cartposition [-1.0, 1.0]m! Poleanglefromvertical [-0.26, 0.26]rad

F Forceappliedtocart [-10, 10]N

g Forceofgravity 9.8 m/s2

l Half-lengthofpole 0.5 m

M Massofthe cart 1.0 kg

m Massofthe pole 0.1 kg

$c Frictionofcartontrack 0.0005N

$p Frictionofpole'shinge 0.000002 kgm2

Thecartandpoleare initiallyplacedatrestatapredetermined

position. Asimulationsucceedswhensixtysecondsofsimulated time

elapse withoutthe cartreachingthe endofthe trackorthe pole falling

over. Forourpurposes, weconsiderapoleangleof0.26 radians(about 14

degrees) the point at which the pole "falls over." Wieland (1991)

investigatestheeffectofvaryingtheangleatwhichafailureoccursonthe

processof evolvinganeural networkthatcontrolsthe cart-pole system

and concludes that increasing this parameter does not necessarily

generatesuperiorsolutions, andmakestrainingslower.

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!

F

x

Figure 4. The Cart-Pole System

2. GeneratingFuzzySystemsUsing GeneticAlgorithms

Acentral issue infuzzycontrol isthe selectionand encodingofthe

fuzzy rules. Sufficient system information must be encoded and therepresentation must be amenable to evaluation and, in the genetic

framework, reproduction. The representationmustallowflexibility both

in thenumber and content of the discovered rules. Finally, a proper

scoringfunction is essential. We will describe two earlierapplicationsof

geneticalgorithmstothe automaticgenerationofafuzzyrule base, and

comparethesetoourapproach.

One ofthe earliestapplicationsofgeneticalgorithmstothe design

offuzzysystemswasdeveloped byKarr(1991), againtosolve the cart-pole

problem. The productionofthe fuzzycontroller beginswiththe definition

ofthe fuzzysetsusedtodescribe each inputvariable. Eachofthe four

inputvariablesarecharacterized bythree fuzzysetsNEGATIVE, ZERO,

and POSITIVEyielding 81 possible combinations. The fuzzy system

designerthenassignsone ofsevenchoices forthe outputto each input

combination. The resulting fuzzysystem represents the expert's "best

guess". The membership function extremaare then encoded intoa bit

string, and a genetic algorithm is applied to shift the membership

functionssoastofindlocationswhichimproveperformance. Theevolvedsystemconsistentlyoutperformsthe original, beingcapable ofrecovering

frominitialpositionsthatfailunderthe originalrule base.

Lee&Takagi(1992)improveonKarr'smethod bypermitting both

the size andthe compositionofthe rule base to emerge asthe resultofthe

genetic search, instead of being specified by the system designer.

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However, the maximumnumberoffuzzysetspermittedfor eachvariable

isstillfixed. Eachmembershipfunctionspecificationconsistsofcenter,

right base, and left base values, eachrequiringone byte ofstorage. Each

outputvariable hasathree-byte descriptionofthe actionto be takenin

response to every inputvariable combination. There are %midescriptionsassociatedwitheachoutputvariable, wheremiisthemaximumnumber

ofmembership functionspermitted for inputvariable i. Therefore, the

byte lengthofthe entire string is 3 (&mi+v %mi), where v isthe number

ofoutputvariables. Intheir example, each inputvariable isassigneda

maximum of 10 membership functions, and each system description

requires 360 bytes. Hence, 30,120 bytesare requiredto encode the four-

inputversionofthe problem. Asthe numberofvariables increasesthe

lengthofthestringencodingthesystemsizeincreasesexponentially, with

a corresponding exponential increase in the complexity of the search

space. Suchasystem isunlikelytoscale well formore complexproblems.

3. A CompactRuleBaseApproach

3.1. SystemRepresentation

It isa longrecognized limitationofgeneticalgorithmsthat, asa

global searchtechnique, the encoded bitstring lengthsshould be keptas

small as possible as they evolve, since the size of the search space

increasesexponentiallywiththesizeofthestrings. Therefore, itishighlydesirable to maintainonly the rulesnecessary toaccomplish the task.

This is the goal pursued in our approach. The two evolutionary

requirements listed inLee &Takagi are alsomaintained;namely, thatthe

numberofrulesneededtoaccomplishthe taskmust be evolved, andthat

the membership functions comprising those rules must evolve from a

random initial state.

Ourversionofthe cart-pole problemhas four inputvariables (x-

position, !-position, dx/dt, d!/dt), and one output variable (the force

applied to the cart). The membership function for each variable is a

triangle characterized bythe locationof itscenterandthe half-lengthof

its base. Asinglerule, therefore, consistsoftheconcatenationoftenone-

byte unsignedcharacters(assumingvaluesfrom 0 to 255)specifyingthe

centersandhalf-lengthsofthe membershipfunctions(Figure 5). The rule

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descriptionsforasingle fuzzysystemare thenconcatenatedintoasingle

bitstringwherethenumberofrulesisnotrestricted.

x ! dx/dt d!/dt

Centers

x ! dx/dt d!/dt

Half-Lengths

System 1 System 2 System 3 System 4 System 5

Fuzzy System Population

...

Rule 1 Rule 2 Rule 3 Rule 4 Rule 5 ...

Fuzzy System Description

Fuzzy Rule Description

F F

Figure5. OrganizationoftheFuzzySystemPopulation

One limitationofearlierattemptsatfuzzysystemlearningwasthe

assumptionthat eachrule isdependentuponcombinationsofall ofthe

inputvariables. Inmanycasesthis isundesirable since the appropriate

actionisoftenstronglyconditioned byonlyasmallsubsetofthevariables.

Without the ability to ignore irrelevant variables, rules evolve which

account for everycombinationofspuriousvariableswith eachrequired

value for the relevant ones. We address this problem by ignoring a

variable when itscorrespondinghalf-length fallsoutside ofaparticular

range ([40, 215]outofthe overall range of[0, 255] inour experiments).

3.2. SystemEvaluation

Each evolvedfuzzysystemis evaluatedaccordingtothe lengthof

time that itkeeps the cart-polesystem from failing. Twentydifferent

initial startingpositionsare considered, arrangedsymmetricallyaround

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{x= 0, ! = 0}(i.e., ifthe position{x= 0.5, != 0.07} is inthe testsuite, so

tooare {0.5, -0.07}, {-0.5, -0.07}, and{-0.5, 0.07}). The selectionofatest

suiterepresentativeofallpossiblecombinationsofinputvaluesiscritical

for evolvingarobustsystem.

Each fuzzycontroller'sobjective istokeepthe pole balancedand

thecartpositionednearthecenterofthetrack. Thescoreforaparticular

trial accumulatesas longas the system doesnot fail. Once a failure

occursthe nexttest immediately begins. Performance scores, however,

needonly be baseduponthedeviationofcart'spositionfromthecenter.

The pole'sangle isnotrequired explicitly inthe scoring equationsince it is

alreadyanimplicitconstraintonthe scoreonce the pole angle exceedsa

critical value the simulation is terminated. In fact, when both the x-

positionangle are included inthe scoring equation, the locationofthe cart

isalmost entirely ignoredwhereas includingonly thex-position in the

scoring equation, bothvariablesassume comparable importance.

score = 1. 0 $x

xmax

'

()

*

+,

100updates

second

&60seconds

&20testcases

& (3)

3.3. RuleAlignmentandReproduction

To be meaningful, the geneticparadigmrequiresthatthe rules in

the twostrings be alignedsothatsimilarrulesare combinedwith each

other. Consider, forexample, twofuzzysystemshavingthesameruleset,

butarrangeddifferently in each. Simply combining the strings in the

order they appear does not preserve much information about either

systemandproducesnearlyrandomresults, ratherthanachildsystem

that performs in a manner similar to its parents. Therefore, before

reproduction, the twostringsmust be alignedsothatthe centersofthe

inputvariablesmatchascloselyaspossible. Inouralgorithm, the most

closelymatchingrules(definedasthe sumofthe differences betweenthe

input variable centers) are combined first, followed by thenext most

closelymatchingrulesfromthose thatremain, andsoon(Figure 6). Any

rulesfroma longerstringthatare notmatchedare addedatthe end. This

matchingschemeisnotoptimal, butrequiresconsiderablylesstimeand

performsnearlyaswell asthe optimal methodwhich involvesgenerating

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the difference matrix between each pair of rules, then solving the

"distributionproblem"tofindtheminimalmatching.

Rule 2 Rule 5 Rule 1Rule 4 Rule 3

Rule 1 Rule 2Rule 3 Rule 4Rule 5

System 1

System 2

Rule 2

Rule 5

Rule 1

Rule 4 Rule 3

Rule 1 Rule 2

Rule 3Rule 4

Rule 5

System 1

System 2

Rule 2 Rule 5Rule 1 Rule 4Rule 3

Rule 1 Rule 2 Rule 3 Rule 4 Rule 5

System 1

System 2

Step 1

Step 2

Step 3

Figure 6. Rule Alignment

4. SimulationResults

To demonstrate our training procedure, several fuzzy cart-pole

controllerswere evolved. The populationsize in eachtrialwas 200, with

eachsystemrandomlyassigned betweenoneandthirtyrulesinitialized

with random characters. The fitness of each controller was then

establishedaccordingtoitsperformance onthe cart-pole systemforsixty

s

imu

lat

eds

econ

ds

, with

th

estat

eof

th

esyst

emu

pdat

edus

in

gEu

ler's

method 100 times everysecond. Pairsofstringsfromthe fiftytopscoring

controllers were randomly selected and combined to produce the next

generation(the geneticparametersappear inTable 2).

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Table2. GeneticParameters UsedinFuzzyRule Base Generation

CrossoverRate: 0.0113

MutationRate: 0.0115

AddRuleMutationRate: 0.0040

DeleteRuleMutationRate: 0.0040

Learning typically occurred quickly. By about the fiftieth

generationthe top-scoringcontrollerwasusuallycapable ofmaintaining

thecart-polesystemfortheentiresixtyseconds(asopposedtothe 5000

generations required in Lee & Takagi). The progress of training for

severalrunsappearsinFigure 7.

Toillustrate

the

results

of

a

ty

pical

run

in

detail

,an

evolve

drule

base appears inFigure 8. Eachrowcorrespondstoasingle rule, and each

column corresponds to a system variablex, ! , dx/dt, d!/dt, and F

respectively. Thisrule base wasthenappliedtoarandom initial position

notusedduringtraining(x= 0.54, !=-0.077). Tracesofthex-position, !-

position, and forceappliedappear inFigure 9. Thecontrollerdoesnot

actually bring the system to rest, but rather establishes a stable

oscillation. In fact, the fuzzycontrollermaintainsthe cart-pole system

uprightthrough 24 hoursofsimulatedtime. Itisimpossibleto bringthe

cart-pole system completely to rest (the oscillationsjust keep getting

smallerandsmaller). Inour example, the closestthatthe systemcomes

to beingcenteredatrestafter 24 hourswasx= 0.000067, != 0.00085,

dx/dt= 0.000011, andd!/dt= 0.000027.

5. Discussion

The mostsigificantcontributionofourapproach isthe ability to

evolve afuzzyrule base consistingsolelyofreleventrules, incontrastto

pr

evious

a

pproach

es

wh

ich

r

equ

ir

ethat

ever

yposs

ibleco

mbinat

ion

of

a

fixedsetofinputs be ennumerated, orrequire thatthe relevancyofa large

suite of rules be determined. Since large systemsare generally more

difficultto evolve, aselectionpressure inherenttothe geneticalgorithm

method, no explicitselectionpressure isrequiredtoreduce the numberof

rulesintheevolvedcontrollers(asLee&Takagi[1993]).

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Generation

Fitness

Score

(Percentage

of

Maximum)

0

0. 1

0. 2

0. 3

0. 4

0. 5

0. 6

0. 7

0. 8

0. 9

1

1

1

0

0

Figure7. ProgressofTrainingfor Several EvolutionaryTrials

Figure8. FuzzyRuleBaseProduced bytheGeneticAlgorithm

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Time Steps (0.01 of a second)

X

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0

0. 2

0. 4

0. 6

0. 8

1

4

0

0

0

Time Steps (0.01 of a second)

Theta

- 0 . 1

- 0 . 0 8

- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0

0 . 0 2

0 . 0 4

0 . 0 6

0 . 0 8

0 .1

1

4

0

0

0

Time Steps (0.01 of a second)

Force

- 1 0

- 8

- 6

- 4

- 2

0

2

4

6

8

1 0

1

4

0

0

0

Figure 9. TracesofSelectedVariables inthe Trial Run

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The fuzzy controllers evolved by our method, like most systems

generatedusinggeneticalgorithms, are verysensitive to the selection

pressuresimposed bythe systemdesigner. These include the trainingset

and the performance criteria. With regard to the composition of thetraining set, the range and distribution of the test cases must be

representative of the expected system input. If the test cases are

unevenlydistributed, e.g., mostofthe initial positionshave positive x-

positions while very few have negative x-positions, then the evolved

systemswillfailtolearnthecorrectcontrolactionfortherarercases. In

fact, continued trainingafter learning to correctlyhandle the common

cases still fails to improve performance. This is because in early

generations thesystemsperformingcorrectlyonthecommoncasesare

more likelytoreproduce because theyperformwell onahigherpercentage

of the training set. In later generations, the evolved systems cannot

change sufficiently to successfully handle the rarer cases without

degrading performance to the point where they are not selected to

reproduce. Hence, the rarercasesare neveradequatelylearned.

Additionally, careful consideration must be made in choosing a

performance criterion. Intrialswhere the divergence ofbothxand!!were

considered in the scoring equation, rule bases were generated which

successfully balanced thepole, butwhich ignored thex-positionof thecart. Since success in balancingthe pole isthe primaryprerequisite for

thecontinuanceofeachtestcase, thex-position becameanunimportant

evolutionaryfactorandoften ignored. Therefore, we usedthe x-position

asanexplicittrainingcriterionwhilethe!-positionremainedanimplicit

trainingcriterion.

Despite the fact that the rule bases generated by this method

include onlythose rulesrequiredtosolve the problem, there are instances

whererulesareduplicated, orwhereonerule issubsumed byanother.

One directionforcontinuedresearch isto be able toautomatically identify

instanceswhere rulescan be combinedor eliminated. Thiswill allowthe

rule bases to be further consolidated. Due to the compactness of the

representations, we expectthatthismethodwill be capable ofaddressing

more complexproblemsthanpreviousapproaches. Ourfuture research

will more fully characterize this method by observing the effects of

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changingsystemparameters (e.g. pole length)during evolutionand by

examiningindetailtheproblemsolvingmethodologiesdiscovered bythe

geneticsearch. We will alsoattemptmore difficultproblemssuchasthe

multiple pole andthe jointedpole systemsto expandthe limitsofgenetic-

fuzzysystemgeneration.In short, the fuzzy systems produced by genetic search are

extremely faithfulto boththetrainingsetandtheperformancecriteria

usedduring evolution. Inmanycases, the resultingsystemsperform ina

mannerconsistentwiththese parameters, yetstill differfromthe visionof

thesystemdesigner. Therefore, althoughtheobjectiveofautomaticfuzzy

systemgeneration isto eliminate the needforahuman expert, learning

solelythroughthe observationandfeedback, an expert isstill requiredto

shapetheprogressoflearning, andtointerpretandevaluatetheresulting

systems.

References

Karr, C.L. (1991) "DesignofaCart-PoleBalancingFuzzyLogicControllerUsingaGeneticAlgorithm", ProceedingsoftheSPIEConferenceontheApplicationsofArtificialIntelligence .Bellingham,WA: SPIEThe International SocietyforOpticalEngineering, pp 26-36.

Goldberg, D.E. (1989) GeneticAlgorithmsinSearch, Optimization,andMachineLearning . Reading, MA: Addison-Wesley.

Kosko, B. (1992). NeuralNetworksandFuzzySystems: ADynamicalSystemsApproachtoMachineIntelligence.EnglewoodCliffs, NJ: Prentice-Hall, Inc.

Lee, M.A. &H. Takagi (1993) "IntegratingDesign StagesofFuzzySystemsusingGeneticAlgorithms", ProceedingsoftheSecondIEEE

Internati

onal

Conferenceo

nFuzz

ySys

tems. New

York

,NY: InstituteofElectricalandElectronicsEngineers, pp 612-617.

Wieland, A.P. (1991) "EvolvingControlsfor Unstable Systems", In S.Touretzky et. al. (ed.), ConnectionistModels: Proceedingsofthe1990SummerSchool. SanMateo, CA: MorganKaufmannPublishers, Inc., pp 91-102.