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FuzzyInferenceProcess

Fuzzyinferenceistheprocessofformulatingthemappingfromagiveninputtoanoutputusingfuzzylogic.Themappingthenprovidesabasisfromwhichdecisionscanbemade,orpatternsdiscerned.TheprocessoffuzzyinferenceinvolvesallofthepiecesthataredescribedinMembershipFunctions,LogicalOperations,andIfThenRules.

Thissectiondescribesthefuzzyinferenceprocessandusestheexampleofthetwoinput,oneoutput,threeruletippingproblemTheBasicTippingProblemthatyousawintheintroductioninmoredetail.Thebasicstructureofthisexampleisshowninthefollowingdiagram:

Informationflowsfromlefttoright,fromtwoinputstoasingleoutput.Theparallelnatureoftherulesisoneofthemoreimportantaspectsoffuzzylogicsystems.Insteadofsharpswitchingbetweenmodesbasedonbreakpoints,logicflowssmoothlyfromregionswherethesystem'sbehaviorisdominatedbyeitheroneruleoranother.

Fuzzyinferenceprocesscomprisesoffiveparts:

Fuzzificationoftheinputvariables

Applicationofthefuzzyoperator(ANDorOR)intheantecedent

Implicationfromtheantecedenttotheconsequent

Aggregationoftheconsequentsacrosstherules

Defuzzification

Afuzzyinferencediagramdisplaysallpartsofthefuzzyinferenceprocessfromfuzzificationthroughdefuzzification.

Step1.FuzzifyInputsThefirststepistotaketheinputsanddeterminethedegreetowhichtheybelongtoeachoftheappropriatefuzzysetsviamembershipfunctions.InFuzzyLogicToolboxsoftware,theinputisalwaysacrispnumericalvaluelimitedtotheuniverseofdiscourseoftheinputvariable(inthiscasetheintervalbetween0and10)andtheoutputisafuzzydegreeofmembershipinthequalifyinglinguisticset(alwaystheintervalbetween0and1).Fuzzificationoftheinputamountstoeitheratablelookuporafunctionevaluation.

Thisexampleisbuiltonthreerules,andeachoftherulesdependsonresolvingtheinputsintoanumberofdifferentfuzzylinguisticsets:serviceispoor,serviceisgood,foodisrancid,foodisdelicious,andsoon.Beforetherulescanbeevaluated,theinputsmustbefuzzifiedaccordingtoeachoftheselinguisticsets.Forexample,towhatextentisthefood

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reallydelicious?Thefollowingfigureshowshowwellthefoodatthehypotheticalrestaurant(ratedonascaleof0to10)qualifies,(viaitsmembershipfunction),asthelinguisticvariabledelicious.Inthiscase,weratedthefoodasan8,which,givenyourgraphicaldefinitionofdelicious,correspondsto=0.7forthedeliciousmembershipfunction.

Inthismanner,eachinputisfuzzifiedoverallthequalifyingmembershipfunctionsrequiredbytherules.

Step2.ApplyFuzzyOperatorAftertheinputsarefuzzified,youknowthedegreetowhicheachpartoftheantecedentissatisfiedforeachrule.Iftheantecedentofagivenrulehasmorethanonepart,thefuzzyoperatorisappliedtoobtainonenumberthatrepresentstheresultoftheantecedentforthatrule.Thisnumberisthenappliedtotheoutputfunction.Theinputtothefuzzyoperatoristwoormoremembershipvaluesfromfuzzifiedinputvariables.Theoutputisasingletruthvalue.

AsisdescribedinLogicalOperationssection,anynumberofwelldefinedmethodscanfillinfortheANDoperationortheORoperation.Inthetoolbox,twobuiltinANDmethodsaresupported:min(minimum)andprod(product).TwobuiltinORmethodsarealsosupported:max(maximum),andtheprobabilisticORmethodprobor.TheprobabilisticORmethod(alsoknownasthealgebraicsum)iscalculatedaccordingtotheequation

probor(a,b)=a+bab

Inadditiontothesebuiltinmethods,youcancreateyourownmethodsforANDandORbywritinganyfunctionandsettingthattobeyourmethodofchoice.

ThefollowingfigureshowstheORoperatormaxatwork,evaluatingtheantecedentoftherule3forthetippingcalculation.Thetwodifferentpiecesoftheantecedent(serviceisexcellentandfoodisdelicious)yieldedthefuzzymembershipvalues0.0and0.7respectively.ThefuzzyORoperatorsimplyselectsthemaximumofthetwovalues,0.7,andthefuzzyoperationforrule3iscomplete.TheprobabilisticORmethodwouldstillresultin0.7.

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Step3.ApplyImplicationMethodBeforeapplyingtheimplicationmethod,youmustdeterminetherule'sweight.Everyrulehasaweight(anumberbetween0and1),whichisappliedtothenumbergivenbytheantecedent.Generally,thisweightis1(asitisforthisexample)andthushasnoeffectatallontheimplicationprocess.Fromtimetotimeyoumaywanttoweightonerulerelativetotheothersbychangingitsweightvaluetosomethingotherthan1.

Afterproperweightinghasbeenassignedtoeachrule,theimplicationmethodisimplemented.Aconsequentisafuzzysetrepresentedbyamembershipfunction,whichweightsappropriatelythelinguisticcharacteristicsthatareattributedtoit.Theconsequentisreshapedusingafunctionassociatedwiththeantecedent(asinglenumber).Theinputfortheimplicationprocessisasinglenumbergivenbytheantecedent,andtheoutputisafuzzyset.Implicationisimplementedforeachrule.Twobuiltinmethodsaresupported,andtheyarethesamefunctionsthatareusedbytheANDmethod:min(minimum),whichtruncatestheoutputfuzzyset,andprod(product),whichscalestheoutputfuzzyset.

Step4.AggregateAllOutputs

BecausedecisionsarebasedonthetestingofalloftherulesinaFIS,therulesmustbecombinedinsomemannerinordertomakeadecision.Aggregationistheprocessbywhichthefuzzysetsthatrepresenttheoutputsofeachrulearecombinedintoasinglefuzzyset.Aggregationonlyoccursonceforeachoutputvariable,justpriortothefifthandfinalstep,defuzzification.Theinputoftheaggregationprocessisthelistoftruncatedoutputfunctionsreturnedbytheimplicationprocessforeachrule.Theoutputoftheaggregationprocessisonefuzzysetforeachoutputvariable.

Aslongastheaggregationmethodiscommutative(whichitalwaysshouldbe),thentheorderinwhichtherulesareexecutedisunimportant.Threebuiltinmethodsaresupported:

max(maximum)

probor(probabilisticOR)

sum(simplythesumofeachrule'soutputset)

Inthefollowingdiagram,allthreeruleshavebeenplacedtogethertoshowhowtheoutputofeachruleiscombined,oraggregated,intoasinglefuzzysetwhosemembershipfunctionassignsaweightingforeveryoutput(tip)value.

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Step5.DefuzzifyTheinputforthedefuzzificationprocessisafuzzyset(theaggregateoutputfuzzyset)andtheoutputisasinglenumber.Asmuchasfuzzinesshelpstheruleevaluationduringtheintermediatesteps,thefinaldesiredoutputforeachvariableisgenerallyasinglenumber.However,theaggregateofafuzzysetencompassesarangeofoutputvalues,andsomustbedefuzzifiedinordertoresolveasingleoutputvaluefromtheset.

Perhapsthemostpopulardefuzzificationmethodisthecentroidcalculation,whichreturnsthecenterofareaunderthecurve.Therearefivebuiltinmethodssupported:centroid,bisector,middleofmaximum(theaverageofthemaximumvalueoftheoutputset),largestofmaximum,andsmallestofmaximum.

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FuzzyInferenceDiagramThefuzzyinferencediagramisthecompositeofallthesmallerdiagramspresentedsofarinthissection.Itsimultaneouslydisplaysallpartsofthefuzzyinferenceprocessyouhaveexamined.Informationflowsthroughthefuzzyinferencediagramasshowninthefollowingfigure.

Inthisfigure,theflowproceedsupfromtheinputsinthelowerleft,thenacrosseachrow,orrule,andthendowntheruleoutputstofinishinthelowerright.Thiscompactflowshowseverythingatonce,fromlinguisticvariablefuzzificationallthewaythroughdefuzzificationoftheaggregateoutput.

Thefollowingfigureshowstheactualfullsizefuzzyinferencediagram.Thereisalottoseeinafuzzyinferencediagram,butafteryoubecomeaccustomedtoit,youcanlearnalotaboutasystemveryquickly.Forinstance,fromthisdiagramwiththeseparticularinputs,youcaneasilyseethattheimplicationmethodistruncationwiththeminfunction.ThemaxfunctionisbeingusedforthefuzzyORoperation.Rule3(thebottommostrowinthediagramshownpreviously)ishavingthestrongestinfluenceontheoutput.andsoon.TheRuleViewerdescribedinTheRuleViewerisaMATLABimplementationofthefuzzyinferencediagram.

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MoreAboutMembershipFunctions

LogicalOperations

IfThenRules

TypesofFuzzyInferenceSystems