Rulebased Expert Systems Ajith Abraham Oklahoma State University Stillwater OK USA Problem - PDF document

Rule-basedExpertSystemsAjithAbrahamOklahomaStateUniversity,Stillwater,OK,USA 1ProblemSolvingUsingHeuristics9092WhatareRule-basedSystems?9103InferenceEngineinRule-basedSystems9114ExpertSystemDevelopment9115FuzzyExpertSystems9126ModelingFuzzyExpertSystems9147IllustrationofFuzzyExpertSystemDesign9148AdaptationofFuzzyInferenceSystems9189Summary918References919 HandbookofMeasuringSystemDesign,editedbyPeterH.SydenhamandRichardThorn.2005JohnWiley&Sons,Ltd.ISBN:0-470-02143-8. Elements:B–SignalConditioning schedulingtheLongestJobontheFastestResource(LJFR),whichwouldminimizeasksthattheaveragejobÞnishesquickly,attheexpenseofthelargestjobtakingalongtime,whereasminimizing,asksthatnojobtakestoolong,attheexpenseofmostjobstakingalongtime.Insummary,minimizationofwillresultinmaximizationof,whichmakestheproblemmoreBycontrast,algorithmsarestraightforwardproceduresthatareguaranteedtoworkeverytimefortheyarefullydeterminateandtimeinvariant.Forexample,certaindailyroutinetaskscouldbeformulatedinastrictalgorithmformat(example,startingupanautomobile).However,foraÔproblemsolverÕtobemoreadaptive,novelelementsornewcircumstancesmustbeintroduced.Manyreal-worldproblemscannotbereducedtoalgorithms,whichleadsustothequesttoÞndmorepowerfultechniques.2WHATARERULE-BASEDSYSTEMS?Conventionalproblem-solvingcomputerprogramsmakeuseofwell-structuredalgorithms,datastructures,andcrispreasoningstrategiestoÞndsolutions.ForthedifÞcultproblemswithwhichexpertsystemsareconcerned,itmaybemoreusefultoemployheuristics:strategiesthatoftenleadtothecorrectsolution,butthatalsosometimesfail.Conventionalrule-basedexpertsystems,usehumanexpertknowledgetosolvereal-worldproblemsthatnormallywouldrequirehumanintelligence.Expertknowledgeisoftenrepresentedintheformoforaswithinthecomputer.Dependingupontheproblemrequirement,theserulesanddatacanberecalledtosolveproblems.Rule-basedexpertsystemshaveplayedanimportantroleinmodernintelligentsystemsandtheirapplicationsinstrategicgoalsetting,planning,design,scheduling,faultmonitoring,diagnosisandsoon.Withthetechnologicaladvancesmadeinthelastdecade,todayÕsuserscanchoosefromdozensofcommercialsoftwarepackageshavingfriendlygraphicuserinterfaces(Ignizio,1991).Conventionalcomputerprogramsperformtasksusingadecision-makinglogiccontainingverylittleknowledgeotherthanthebasicalgorithmforsolvingthatspeciÞcproblem.Thebasicknowledgeisoftenembeddedaspartoftheprogrammingcode,sothatastheknowledgechanges,theprogramhastoberebuilt.Knowledge-basedexpertsystemscollectthesmallfragmentsofhumanknow-howintoaknowledgebase,whichisusedtoreasonthroughaproblem,usingtheknowledgethatisappropriate.Animportantadvantagehereisthatwithinthedomainoftheknowledgebase,adifferentproblemcanbesolvedusingthesameprogramwithoutreprogrammingefforts.Moreover,expertsystemscouldexplainthereasoningprocessandhandlelevelsofconÞdenceanduncertainty,whichconventionalalgorithmsdonothandle(GiarratanoandRiley,1989).Someoftheimportantadvantagesofexpertsystemsareasfollows:abilitytocaptureandpreserveirreplaceablehumanabilitytodevelopasystemmoreconsistentthanhumanminimizehumanexpertiseneededatanumberoflocationsatthesametime(especiallyinahostileenvironmentthatisdangeroustohumanhealth);solutionscanbedevelopedfasterthanhumanexperts.ThebasiccomponentsofanexpertsystemareillustratedinFigure1.Theknowledgebasestoresallrelevantinfor-mation,data,rules,cases,andrelationshipsusedbytheexpertsystem.Aknowledgebasecancombinetheknowl-edgeofmultiplehumanexperts.Aruleisaconditionalstatementthatlinksgivenconditionstoactionsorout-comes.Aframeisanotherapproachusedtocaptureandstoreknowledgeinaknowledgebase.Itrelatesanobjectoritemtovariousfactsorvalues.Aframe-basedrepre-sentationisideallysuitedforobject-orientedprogrammingtechniques.Expertsystemsmakinguseofframestostoreknowledgearealsocalledframe-basedexpertsystemsThepurposeoftheinferenceengineistoseekinfor-mationandrelationshipsfromtheknowledgebaseandtoprovideanswers,predictions,andsuggestionsinthewayahumanexpertwould.TheinferenceenginemustÞndtherightfacts,interpretations,andrulesandassemblethemcorrectly.TwotypesofinferencemethodsarecommonlyusedÐBackwardchainingistheprocessofstartingwithconclusionsandworkingbackwardtothesupportingfacts.Forwardchainingstartswiththefactsandworksforwardtotheconclusions. Knowledge baseUser interfaceFigure1.Architectureofasimpleexpertsystem. Rule-basedExpertSystems Theexplanationfacilityallowsausertounderstandhowtheexpertsystemarrivedatcertainresults.TheoverallpurposeoftheknowledgeacquisitionfacilityistoprovideaconvenientandefÞcientmeansforcapturingandstoringallcomponentsoftheknowledgebase.Veryoftenspecializeduserinterfacesoftwareisusedfordesigning,updating,andusingexpertsystems.Thepurposeoftheuserinterfaceistoeaseuseoftheexpertsystemfordevelopers,users,andadministrators.3INFERENCEENGINEINRULE-BASEDArule-basedsystemconsistsofif-thenrules,abunchof,andaninterpretercontrollingtheapplicationoftherules,giventhefacts.rulestatementsareusedtoformulatetheconditionalstatementsthatcomprisethecompleteknowledgebase.AsingleruleassumestheformÕandtheif-partoftheruleÔÕiscalledthepremise,whilethethen-partoftheruleÔÕiscalledthe.Therearetwobroadkindsofinferenceenginesusedinrule-basedforwardchainingbackwardchainingInaforwardchainingsystem,theinitialfactsarepro-cessedÞrst,andkeepusingtherulestodrawnewconclu-sionsgiventhosefacts.Inabackwardchainingsystem,thehypothesis(orsolution/goal)wearetryingtoreachispro-cessedÞrst,andkeeplookingforrulesthatwouldallowtoconcludethathypothesis.Astheprocessingprogresses,newsubgoalsarealsosetforvalidation.Forwardchainingsystemsareprimarilydata-driven,whilebackwardchain-ingsystemsaregoal-driven.ConsideranexamplewiththefollowingsetofRule1IfAandCthenYRule2IfAandXthenZRule3IfBthenXRule4IfZthenDIfthetaskistoprovethatistrue,givenaretrue.Accordingtoforwardchaining,startwithRule1andgoondownwardtillarulethatÞresisfound.Rule3istheonlyonethatÞresintheÞrstiteration.AftertheÞrstiteration,itcanbeconcludedthatA,B,andaretrue.Theseconditerationusesthisvaluableinformation.Aftertheseconditeration,Rule2Þresaddingistrue,whichinturnhelpsRule4toÞre,provingthatistrue.Forwardchainingstrategyisespeciallyappropriateinsituationswheredataareexpensivetocollect,butfewinquantity.However,specialcareistobetakenwhentheserulesareconstructed,withthepreconditionsspecifyingaspreciselyaspossiblewhendifferentrulesshouldÞre.Inthebackwardchainingmethod,processingstartswiththedesiredgoal,andthenattemptstoÞndevidenceforprovingthegoal.Returningtothesameexample,thetasktoprovethatistruewouldbeinitiatedbyÞrstÞndingarulethatproves.Rule4doesso,whichalsoprovidesasubgoaltoprovethatistrue.NowRule2comesintoplay,andasitisalreadyknownthatistrue,thenewsubgoalistoshowthatistrue.Rule3providesthenextsubgoalofprovingthatistrue.Butthattrueisoneofthegivenassertions.Therefore,itcouldbeconcludedthatistrue,whichimpliesthatistrue,whichinturnalsoimpliesthatistrue.BackwardchainingisusefulinsituationswherethequantityofdataispotentiallyverylargeandwheresomespeciÞccharacteristicofthesystemunderconsiderationisofinterest.Ifthereisnotmuchknowledgewhattheconclusionmightbe,orthereissomespeciÞchypothesistotest,forwardchainingsystemsmaybeinefÞcient.Inprinciple,wecanusethesamesetofrulesforbothforwardandbackwardchaining.Inthecaseofbackwardchaining,sincethemainconcerniswithmatchingtheconclusionofaruleagainstsomegoalthatistobeproved,theÔthenÕ(consequent)partoftheruleisusuallynotexpressedasanactiontotakebutmerelyasastate,whichwillbetrueiftheantecedentpart(s)aretrue(Donald,1986).4EXPERTSYSTEMDEVELOPMENTStepsintheexpertsystemsdevelopmentprocessincludedeterminingtheactualrequirements,knowledgeacquisi-tion,constructingexpertsystemcomponents,implement-ingresults,andformulatingaprocedureformaintenanceandreview.Knowledgeacquisitionisthemostimportantelementinthedevelopmentofexpertsystem(Niwa,SasakiandIhara,1988).Knowledgecouldbeobtainedbyinterviewingdomainexpertsand/orlearningbyexperience.Veryoftenpeopleexpressknowledgeasnaturallanguage(spokenlanguage),orusinglettersorsymbolicterms.Thereexistseveralmethodstoextracthumanknowledge.CognitiveWorkAnalysis(CWA)andtheCognitiveTaskAnalysis(CTA)provideframeworkstoextractknowledge.TheCWAisatechniquetoanalyze,design,andevaluatethehumancomputerinteractivesystems(Vicente,1999).TheCTAisamethodtoidentifycognitiveskill,mentaldemands,andneedstoperformtaskproÞciency(MilitalloandHutton,1998).Thisfocusesondescribingtherepresen-tationofthecognitiveelementsthatdeÞnesgoalgenerationanddecision-making.Itisareliablemethodforextracting Elements:B–SignalConditioning humanknowledgebecauseitisbasedontheobservationsoraninterview.Mostexpertsystemsaredevelopedusingspecializedsoftwaretoolscalled.Theseshellscomeequippedwithaninferencemechanism(backwardchaining,forwardchaining,orboth),andrequireknowledgetobeenteredaccordingtoaspeciÞedformat.Oneofthemostpopularshellswidelyusedthroughoutthegovernment,industry,andacademiaistheCLIPS(CLIPS,2004).CLIPSisanexpertsystemtoolthatprovidesacompleteenvironmentfortheconstructionofrule-and/orobject-basedexpertsystems.CLIPSprovidesacohesivetoolforhandlingawidevarietyofknowledgewithsupportforthreedifferentprogrammingparadigms:rule-based,object-oriented,andprocedural.CLIPSiswritteninCforportabilityandspeedandhasbeeninstalledonmanydifferentoperatingsystemswithoutcodechanges.5FUZZYEXPERTSYSTEMSTheworldofinformationissurroundedbyuncertaintyandimprecision.Thehumanreasoningprocesscanhandleinexact,uncertain,andvagueconceptsinanappropriatemanner.Usually,thehumanthinking,reasoning,andper-ceptionprocesscannotbeexpressedprecisely.Thesetypesofexperiencescanrarelybeexpressedormeasuredusingstatisticalorprobabilitytheory.Fuzzylogicprovidesaframeworktomodeluncertainty,thehumanwayofthink-ing,reasoning,andtheperceptionprocess.FuzzysystemswereÞrstintroducedbyZadeh(1965).Afuzzyexpertsystemissimplyanexpertsystemthatusesacollectionoffuzzymembershipfunctionsandrules,insteadofBooleanlogic,toreasonaboutdata(Schneideretal.,1996).Therulesinafuzzyexpertsystemareusuallyofaformsimilartothefollowing:mediumareinputvariables,isanoutputvariable.Herelow,high,andmediumarefuzzysetsdeÞnedonA,Brespectively.Theantecedent(theruleÕspremise)describestowhatdegreetheruleapplies,whiletheruleÕsconsequentassignsamembershipfunctiontoeachofoneormoreoutputvariables.beaspaceofobjectsandbeagenericelement.AclassicalsetA,A,isdeÞnedasacollectionofelementsorobjects,suchthatcaneitherbelongornotbelongtotheset.AfuzzysetisdeÞnedasasetoforderedpairs(x,µ(x))(x)iscalledthemembershipfunction(MF)forthefuzzyset.TheMFmapseachelementoftoamembershipgrade(ormembershipvalue)betweenzeroandone.Obviously(1)isasimpleextensionofthedeÞnitionofaclassicalsetinwhichthecharacteristicfunctionispermittedtohaveanyvaluesbetweenzeroandone.TheintersectionoftwofuzzysetsisspeciÞedingeneralbyafunction:[0,1]0,1][0,1],whichaggregatestwomembershipgradesasfollows:(x)T(µ(x),µ(x))(x)(x)(isabinaryoperatorforthefunction.ThisclassoffuzzyintersectionoperatorsareusuallyreferredtoasT-normoperators(Jang,SunandMizutani,1997).FourofthemostfrequentlyusedT-normoperatorsare(a,b)(a,b)Algebraicproduct:(a,b)ab(Boundedproduct:(a,b)Drasticproduct:(a,b)a,bLikeintersection,thefuzzyunionoperatorisspeciÞedingeneralbyafunction:[0,1]0,1][0,1],whichaggregatestwomembershipgradesasfollows:(x)S(µ(x),µ(x))(x)(x)(isthebinaryoperatorforthefunction.Thisclassoffuzzyunionoperatorsareoftenreferredtoas)operators(Jang,SunandMizutani,1997).FourofthemostfrequentlyusedT-conormoperatorsare(a,b)(a,b)Algebraicsum:(a,b)ab(Boundedsum:(a,b)b)(Drasticsum:(a,b),ifa,b�Boththeintersectionandunionoperatorsretainsomepropertiesoftheclassicalsetoperation.Inparticular,theyareassociativeandcommutative.Figure2illustratesthebasicarchitectureofafuzzyexpertsystem.ThemaincomponentsareafuzziÞcationinterface,afuzzyrulebase(knowledgebase),aninferenceengine(decision-makinglogic),andadefuzziÞcationinter-face.TheinputvariablesarefuzziÞedwherebythemember-shipfunctionsdeÞnedontheinputvariablesareappliedtotheiractualvalues,todeterminethedegreeoftruthforeachruleantecedent.Fuzzyrulesandfuzzyreasoningarethebackboneoffuzzyexpertsystems,whicharethemost Rule-basedExpertSystems Figure2.Basicarchitectureofafuzzyexpertsystem. (COA) Figure3.MamdanifuzzyinferencesystemusingminandmaxforT-normandT-conormoperators.importantmodelingtoolsbasedonfuzzysettheory.Thefuzzyrulebaseischaracterizedintheformofinwhichtheantecedentsandconsequentsinvolvelinguis-ticvariables.Thecollectionofthesefuzzyrulesformstherulebaseforthefuzzylogicsystem.Usingsuitableinfer-enceprocedure,thetruthvaluefortheantecedentofeachruleiscomputed,andappliedtotheconsequentpartofeachrule.Thisresultsinonefuzzysubsettobeassignedtoeachoutputvariableforeachrule.Again,byusingsuitablecom-positionprocedure,allthefuzzysubsetsassignedtoeachoutputvariablearecombinedtogethertoformasinglefuzzysubsetforeachoutputvariable.Finally,defuzziÞcationisappliedtoconvertthefuzzyoutputsettoacrispoutput.Thebasicfuzzyinferencesystemcantakeeitherfuzzyinputsorcrispinputs,buttheoutputsitproducesarealwaysfuzzysets.ThedefuzziÞcationtaskextractsthecrispoutputthatbestrepresentsthefuzzyset.Withcrispinputsandoutputs,afuzzyinferencesystemimplementsanonlinearmappingfromitsinputspacetooutputspacethroughanumberoffuzzyInwhatfollows,thetwomostpopularfuzzyinferencesystemsareintroducedthathavebeenwidelydeployedinvariousapplications.Thedifferencesbetweenthesetwofuzzyinferencesystemslieintheconsequentsoftheirfuzzyrules,andthustheiraggregationanddefuzziÞcationproceduresdifferaccordingly.AccordingtoMamdani,fuzzyinferencesystem(Mam-daniandAssilian,1975)ÐseeFigure3Ðtheruleante-cedentsandconsequentsaredeÞnedbyfuzzysetsandhasthefollowingstructure:ThereareseveraldefuzziÞcationtechniques.ThemostwidelyuseddefuzziÞcationtechniqueusesthecentroidofareamethodasfollowsCentroidofarea(z)z (z)(z)istheaggregatedoutputMF.TakagiandSugeno(1985)proposedaninferenceschemeinwhichtheconclusionofafuzzyruleisconstitutedbyaweightedlinearcombinationofthecrispinputsratherthanafuzzyset.AbasicTakagiÐSugenofuzzyinferencesystemisillustratedinFigure4andtherulehasthefollowing,andarelinearparameters.TSKTak-agiÐSugenoKangfuzzycontrollerusuallyneedsasmaller Elements:B–SignalConditioning Input (x,y)A1B1XYmA2XXmmB2YYmw1w2zp1*x + q1*y + r1zZ p2*x + q2*y + r2w1*z1 + w2*z2w w2= Figure4.TakagiÐSugenofuzzyinferencesystemusingaminorproductasT-normoperator.numberofrules,becausetheiroutputisalreadyalinearfunctionoftheinputsratherthanaconstantfuzzyset.6MODELINGFUZZYEXPERTSYSTEMSFuzzyexpertsystemmodelingcanbepursuedusingthefollowingsteps.Selectrelevantinputandoutputvariables.Determinethenumberoflinguistictermsassociatedwitheachinput/outputvariable.Also,choosetheappropriatefam-ilyofmembershipfunctions,fuzzyoperators,reasoningmechanism,andsoon.ChooseaspeciÞctypeoffuzzyinferencesystem(forexample,Mamdani,TakagiÐSugenoetc.).Inmostcases,theinferenceofthefuzzyrulesiscarriedoutusingtheÔminÕandÔmaxÕoperatorsforfuzzyintersectionandunion.Designacollectionoffuzzyrules(knowledgebase).Toformulatetheinitialrulebase,theinputspaceisdividedintomultidimensionalpartitionsandthenactionsareassignedtoeachofthepartitions.Inmostapplications,thepartitioningisachievedusingone-dimensionalmembershipfunctionsusingfuzzyasillustratedinFigure5.Theconsequentpartsoftherulerepresenttheactionsassociatedwitheachpartition.ItisevidentthattheMFsandthenumberofrulesaretightlyrelatedtothepartitioning.7ILLUSTRATIONOFFUZZYEXPERTSYSTEMDESIGNThissectionillustratesthedevelopmentofareactivepowerpredictionmodelusingMamdaniandTakagiÐSugenofuzzyinferenceexpertsystems.TheMatLabfuzzylogictool-boxwasusedtosimulatethevariousexperiments(FuzzyLogicToolBox,2004). 7R8R9R4R3R5R2R6R1LargeMediumInput-2Input-1SmallLargeMediumSmall Figure5.Exampleshowinghowthetwo-dimensionalspacesarepartitionedusingthreetrapezoidalmembershipfunctionsperinputdimension.Asimpleif-thenrulewillappearasIfinput-1ismediumandinput2islarge,thenruleisredThetaskistodevelopafuzzyexpertsystemtoforecastthereactivepower(attime1byknowingtheloadcurrent()andvoltage()attime.Theexperimentsystemconsistsoftwostages:developingthefuzzyexpertsystem,andperformanceevaluationusingthetestdata.ThemodelhastwoinÐoutvariables()andoneoutputvariable().Trainingandtestingdatasetswereextractedrandomlyfromthemasterdataset.Sixtypercentofdatawasusedfortrainingandtheremaining40%fortesting(AbrahamandKhan,2003).7.1Designandexperiments:fuzzyexpertsystemsFirst,theeffectsof(a)shapeandquantityofmem-bershipfunctions(b)T-normandT-conormoperators(c)defuzziÞcationmethodsand(d)inferencemethodfor Rule-basedExpertSystems designingthefuzzyexpertsystemisanalyzed.Experimentswerecarriedoutusingfourdifferentsettingsusingthesamerulebase.Experiment1(Toevaluatetheeffectonthenumberofmembershipfunctions)Thefollowingsettingswereusedfordesigningtheexpertsystem1.Twotriangularmembershipfunctions(MFs)foreachinputvariableandfourtriangularMFsfortheoutputvariable(power).Usingthegridpartitioningmethod(Figure5),fourrulesweredeveloped.2.ThreetriangularMFsforeachinputvariableandninetriangularMFsfortheoutputvariable(power).TherulebaseconsistedofnineÔminÕandÔmaxÕwereusedasT-normandT-conormoper-atorsandthecentroidmethodofdefuzziÞcationforMam-daniinferencemethodandweightedaveragedefuzziÞcationmethodforTakagiÐSugenoFuzzyInferenceSystem(FIS).ThedevelopedfuzzyinferencesystemsusingMamdaniandTakagiÐSugenomodelsaredepictedinFigures6to9.Table1summarizesthetrainingandtestingRootMeanSquaredError(RMSE)values.Experiment2(ToevaluatetheeffectofshapeofmembershipFortheMamdaniFIS,threeGaussianMFsforTable1.EmpiricalcomparisonoffuzzyinferencesystemsandquantityofMembershipFunctions(MFs). No.ofMamdaniFISTakagiÐSugenoFIS MFsRootmeansquarederror TrainingTestTrainingTest 20.4010.3970.0240.02330.3480.3340.0170.016 eachinputvariableandnineGaussianMFsfortheoutputvariablewereused.Therulebaseconsistedofninerules.ÔminÕandÔmaxÕasT-normandT-conormoperators,andthecentroidmethodofdefuzziÞcationforMamdaniFISandtheweightedaveragedefuzziÞcationmethodforTakagiÐSugenoFISwerealsoused.ThedevelopedfuzzyinferencesystemsusingMamdaniandTakagiÐSugenomodelsaredepictedinFigures10and11.Table2summarizesthetrainingandtestingRMSEvalues.Experiment3(Toevaluatetheeffectoffuzzyoperators)MamdaniFIS,threeGaussianMFsforeachinputvariableandnineGaussianMFsfortheoutputvariablewereused.Therulebaseconsistedofninerules.T-normandT-conormoperatorswereÔproductÕandÔsumÕandthecentroidmethodofdefuzziÞcationforMamdaniFIS,andweightedaveragedefuzziÞcationmethodforTakagiÐSugenoFISwereused.Table3summarizesthetrainingandtestingRMSEvalues.Experiment4(ToevaluatetheeffectofdefuzzicationFortheMamdaniFIS,threeGaussianMFsforeachinputvariableandnineGaussianMFsfortheoutputvariablewereused.Therulebaseconsistedofninerules.T-normandT-conormoperatorswereÔproductÕandÔsumÕandthefollowingdefuzziÞcationoperatorsweretestedforMamdaniFIS.Table2.EmpiricalcomparisonoffuzzyinferencesystemsusingGaussianMFs. MamdaniFISTakagiÐSugenoFIS Rootmeansquarederror TrainingTestTrainingTest 0.2430.2400.0210.019 Voltage = 0.5Current = 0.5Power = 0.51123401010 Figure6.MamdanifuzzyinferencesystemusingtwotriangularMFsforinputvariables. Elements:B–SignalConditioning 0.5Current Figure7.TakagiÐSugenofuzzyinferencesystemusingtwotriangularMFsforinputvariables. Current = 0.5Power = 0.625112301010 Figure8.MamdanifuzzyinferencesystemusingthreetriangularMFsforinputvariables. = 0.5 0.527Power = 0.4291123010Š Figure9.TakagiÐSugenofuzzyinferencesystemusingthreetriangularMFsforinputvariables. Rule-basedExpertSystems = 0.5Current = 0.5Power = 0.58411236710 Figure10.MamdanifuzzyinferencesystemusingthreeGaussianMFsforinputvariables. 1123890 Current = 0.510 Š Š Figure11.TakagiÐSugenofuzzyinferencesystemusingthreeGaussianMFsforinputvariables.Table3.Empiricalcomparisonoffuzzyinferencesystemsfordifferentfuzzyoperators. MamdaniFISTakagiÐSugenoFIS Rootmeansquarederror TrainingTestTrainingTest 0.2210.2190.0190.018 BisectorofArea(BOA)MeanofMaximum(MOM)SmallestofMaximum(SOM).FortheTakagiÐSugenoFIS,theweightedsumandweightedaveragedefuzziÞcationmethodswereused.Table4summarizesthetrainingandtestingofRMSEDiscussionsofResultsandProblemSolutionAsdepictedinTable1,whenthenumberofinputMFswereincreasedfromtwotothree,theRMSEvaluesreducedregardlessoftheinferencesystemused.However,whentheshapeoftheMFwaschangedtoGaussian,RMSEvaluesforMamdaniFISdecreasedbuttheRMSEvaluesforTak-agiÐSugenoFISincreased(Table2).UsingGaussianMFs,whentheT-normandT-conormoperatorswerechangedtoÔproductÕandÔsumÕ(insteadofÔminÕandÔmaxÕ)boththeinferencemethodsperformedbetter(Table3).Finally,theselectionofanidealdefuzziÞcationoperatoralsohasadirectinßuenceintheperformanceofFISasshowninTable4. Elements:B–SignalConditioning Table4.EmpiricalcomparisonoffuzzyinferencesystemsfordifferentdefuzziÞcationoperators. MamdaniFISTakagiÐSugenoFIS DefuzziÞcationRMSEDefuzziÞcationRMSE TrainingTestTrainingTest Centroid0.2210.0219Weightedsum0.0190.018MOM0.2300.232Weightedaverage0.0850.084BOA0.2180.216SOM0.2290.232 Thedesignoftherulebase(numberofrulesandhowtheinputsandoutputsarerelated)isalsoveryimportantforthegoodperformanceofFIS.Theroleofweightingfactorsemphasizingtheimportanceofcertainrulesalsobearsaprominentrolefortheoverallperformance.Whentheinput/outputdimensionsbecomelarger,manualdesignbecomestediousandsometimescouldevenleadtopoordesignandimplementation.8ADAPTATIONOFFUZZYINFERENCEExpertknowledgeisoftenthemainsourcetodesignthefuzzyexpertsystems.Figure12illustratesthevariousparametersandcomponentsthatneedtobeadaptedforcon-trollingaprocess.Accordingtotheperformancemeasureoftheproblemenvironment,themembershipfunctions,rulebases,andtheinferencemechanismaretobeadapted(Abraham,2002).Neuralnetworklearning,self-organizingmapsandclus-teringmethodscouldbeusedtogeneraterules.Gradi-entdescentanditsvariantscouldbeappliedtoÞne-tunetheparametersofparameterizedinput/outputmem-bershipfunctionsandfuzzyoperators(Abraham,2001).Adaptationoffuzzyinferencesystemsusingevolutionary inference system if-then rulesFuzzy operatorsKnowledge base Fuzzy inference system measure Process +Š Figure12.Adaptationoffuzzyinferencesystems.computationtechniqueshasbeenwidelyexplored.Evolu-tionaryComputation(EC)isapopulationbasedadaptivemethod,whichmaybeusedtosolveoptimizationproblems,basedonthegeneticprocessesofbiologicalorganisms(MichalewiczandFogel,1999).Overmanygenerations,naturalpopulationsevolveaccordingtotheprinciplesofnaturalselectionandÔsur-vivaloftheÞttestÕ,ÞrstclearlystatedbyCharlesDarwininÔOntheOriginofSpeciesÕ.Bymimickingthispro-cess,ECcouldÔevolveÕsolutionstoreal-worldproblems,iftheyhavebeensuitablyencoded(problemrepresentationischromosome).Automaticadaptationofmembershipfunctionsispopularlyknownasselftuningandthechromo-someencodesparametersoftrapezoidal,triangle,logistic,hyperbolic-tangent,Gaussianmembershipfunctions,andsoon.Evolutionarysearchoffuzzyrulescanbecarriedoutusingthreeapproaches.IntheÞrstmethod(Michi-ganapproach),thefuzzyknowledgebaseisadaptedasaresultofantagonisticrolesofcompetitionandcooperationoffuzzyrules.Thesecondmethod(Pittsburghapproach),evolvesapopulationofknowledgebasesratherthanindividualfuzzyrules.Reproductionoperatorsservetoprovideanewcombinationofrulesandnewrules.Thethirdmethod(iterativerulelearningapproach),isverymuchsimilartotheÞrstmethodwitheachchromosomerepresentingasinglerule,butcontrarytotheMichiganapproach,onlythebestindividualisconsideredtoformpartofthesolution,discardingtheremainingchromo-somesofthepopulation.Theevolutionarylearningprocessbuildsupthecompleterulebasethroughaniterativelearn-ingprocess(Cordetal.,2001).9SUMMARYRule-basedexpertsystemshavebeenappliedinavastnumberofapplicationareas.Animportantadvantageofthefuzzyexpertsystemisthattheknowledgeisexpressedaseasy-to-understandlinguisticrules.Ifwehavedata,thefuzzyexpertsystemcanbetaughtusingneuralnetwork Rule-basedExpertSystems learning,EC,orotheradaptationtechniques.Itistobeexpectedthatthenumberofapplicationswillgrowconsiderablyinthefuturenowthatsuccessisclearlyprovenforthesemethods.Abraham,A.(2001)Neuro-FuzzySystems:State-of-the-ArtMod-elingTechniques,ConnectionistModelsofNeurons,LearningProcesses,andArticialIntelligence,inLectureNotesinCom-puterScience,Vol.2084,(eds.Mira.,JoseandPrieto.,Alberto)SpringerVerlag,Germany(pp.269Ð276).Abraham,A.(2002)IntelligentSystems:ArchitecturesandPer-spectives,RecentAdvancesinIntelligentParadigmsandAppli-cations,inStudiesinFuzzinessandSoftComputing,Chapter1,(edsA.,Abraham,L.,JainandJ.,Kacprzyk),SpringerVerlag,Germany(pp.1Ð35).Abraham,A.andKhan,M.R.(2003)Neuro-FuzzyParadigmsforIntelligentEnergyManagement,InnovationsinIntelligentSystems:Design,ManagementandApplications,inStudiesinFuzzinessandSoftComputing,Chapter12,(edsA.,Abraham,L.,JainandB.,JanvanderZwaag),SpringerVerlag,Germany(pp.285Ð314).CLIPS(2004)ExpertSystemShellon,O.,Herrera,F.,Hoffmann,F.andMagdalena,L.(2001)GeneticFuzzySystems:EvolutionaryTuningandLearningofFuzzyKnowledgeBases,WorldScientiÞcPublishingCompany,Donald,W.A.(1986)AGuidetoExpertSystems,Addison-Wesley,Boston,MA.FuzzyLogicToolbox,TheMathWorks(2004)http://www.Giarratano,J.andRiley,G.(1989)ExpertSystems:PrinciplesandProgramming,PWS-KentPublishingCo,Boston,MA.Ignizio,J.P.(1991)IntroductiontoExpertSystems:TheDevel-opmentandImplementationofRule-BasedExpertSystemsMcGraw-Hill,Inc,USA.Jang,J.S.R.,Sun,C.T.andMizutani,E.(1997)Neuro-FuzzyandSoftComputing:AComputationalApproachtoLearningandMachineIntelligence,PrenticeHallInc,USA.Mamdani,E.H.andAssilian,S.(1975)AnExperimentinLin-guisticSynthesiswithaFuzzyLogicController.JournalofMan-MachineStudies(1),1Ð13.Michalewicz,Z.andFogel,D.B.(1999)HowtoSolveIt:Modern,SpringerVerlag,Germany.Militallo,L.G.,Hutton,R.J.B.(1998)AppliedCognitiveTaskAnalysis(ACTA):APractitionerÕsToolkitforUnderstandingErgonomics(11),1618Ð1642.Niwa,K.,Sasaki,K.andIhara,H.(1988)AnExperimentalCom-parisonofKnowledgeRepresentationSchemes,inPrinciplesofExpertSystems,(EdsA.,GuptaandE.B.,Prasad),IEEEPress,NewYork(pp.133Ð140).Schneider,M.,Langholz,G.,Kandel,A.andChew,G.(1996)FuzzyExpertSystemTools,JohnWiley&Sons,USA.Takagi,T.andSugeno,M.(1985)FuzzyidentiÞcationofsystemsanditsapplicationsofmodelingandcontrol,IEEETransactionsofSystems.ManandCybernetics,USA(pp.116Ð132).Vicente,K.J.(1999)CognitiveWorkAnalysis:TowardsSafe,Productive,andHealthyComputer-BasedWork,LawrenceErl-baumAssociates,Inc.Press,USA.Zadeh,L.A.(1965)FuzzySets.InformationandControl338Ð353.