Fuzzy inference systems

These slides present about Fuzzy Inference Systems.

1. Fuzzy Inference Systems
Course: Computational Intelligence Engineering (Soft Computing)
Prof. (Dr.) Pravat Kumar Rout
Department of EEE, ITER,
Siksha ‘O’ Anusandhan (Deemed to be University),
Bhubaneswar, Odisha, India
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2. Definition 2
✓Fuzzyinference(reasoning)isthe
actualprocessofmappingfromagiven
inputtoanoutputusingfuzzylogic.
✓Theprocessinvolvesallthepiecesthat
wehavediscussedintheprevious
sections:membershipfunctions,fuzzy
logicoperators,andif-thenrules

3. Fuzzy Inference System
Fuzzyinferenceisamethodthatinterpretsthevaluesintheinputvectorand,based
onsomesetsofrules,assignsvaluestotheoutputvector.Infuzzylogic,thetruthof
anystatementbecomesamatterofadegree.
Fuzzyinferenceistheprocessofformulatingthemappingfromagiveninputtoan
outputusingfuzzylogic.Themappingthenprovidesabasisfromwhichdecisions
canbemadeorpatternsdiscerned.
Theprocessoffuzzyinferenceinvolvesallofthepiecesdescribedsofar,i.e.,
membershipfunctions,fuzzylogicoperators,andif-thenrules.
Twomaintypesoffuzzyinferencesystemscanbeimplemented:Mamdani-type
(1977)andSugeno-type(1985).Thesetwotypesofinferencesystemsvary
somewhatinthewayoutputsaredetermined.
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Structure

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Basic Structure

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FUZZIFIER•Convertsthecrispinputtoa
linguisticvariableusingthemembership
functionsstoredinthefuzzyknowledge
base.Thisprocessisknownasfuzzification
.
Step-1 FuzzifyInputs

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Aftertheinputsarefuzzified,youknowthedegreetowhicheachpartoftheantecedent
issatisfiedforeachrule.Iftheantecedentofagivenrulehasmorethanonepart,the
fuzzyoperatorisappliedtoobtainonenumberthatrepresentstheresultofthe
antecedentforthatrule.
Thisnumberisthenappliedtotheoutputfunction.
Theinputtothefuzzyoperatoristwoormoremembershipvaluesfromfuzzifiedinput
variables.Theoutputisasingletruthvalue.
Step-2 Apply Fuzzy
Operators

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Step-3 Apply Implication Method

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Infuzzylogicsystems,thefuzzy
knowledgebaserepresentsthefactsof
therulesandlinguisticvariablesbasedon
thefuzzysettheorysothattheknowledge
base systems will allow
approximatereasoning.

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Step-4 Aggregate all inputs

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Step-5 Defuzzify

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•Defuzzificationistheprocessofproducing
aquantifiableresultinCrisplogic,given
fuzzysetsandcorrespondingmembership
degrees.
•Itistheprocessthatmapsafuzzysettoa
crispset.
•Itistypicallyneeded infuzzy
controlsystems.Thesewillhaveanumber
ofrulesthattransformanumberof
variablesintoafuzzyresult,thatis,the
resultisdescribedintermsofmembership
infuzzysets.
•Forexample,rulesdesignedtodecide
howmuchpressuretoapplymightresultin
"Decrease Pressure(15%),Maintain
Pressure(34%),IncreasePressure(72%)".
Defuzzificationisinterpretingthe
membershipdegreesofthefuzzysetsinto
aspecificdecisionorrealvalue.

14. 14 Overall Fuzzy Inference Diagram

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Afuzzyinferencesystem(FIS)is
asystemthatusesfuzzysettheory
tomapinputs(featuresinthecase
offuzzyclassification)tooutputs
(classes in the case
offuzzyclassification).

17. Steps of Fuzzy Inference System
The steps of fuzzy reasoning (inference operations upon fuzzy IF–THEN rules)
performed by FISs are:
1.Compare the input variables with the membership functions on the
antecedent part to obtain the membership values of each linguistic label. (this
step is often calledfuzzification.)
2. Combine (usually multiplication or min) the membership values on the
premise part to get firing strength (dereeof fullfillment) of each rule.
3. Generate the qualified consequents (either fuzzy or crisp) or each rule
depending on the firing strength.
4. Aggregate the qualified consequents to produce a crisp output. (This step is
called defuzzification.)
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Steps of Fuzzy Inference System...

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20. Mamdani-type inference
Mamdani-typeinferenceexpectstheoutputmembershipfunctionstobefuzzysets.
Aftertheaggregationprocess,thereisafuzzysetforeachoutputvariable,which
needsdefuzzification.
Itispossible,andsometimesmoreefficient,touseasinglespikeastheoutput
membershipfunctionratherthanadistributedfuzzyset.
This,sometimescalledasingletonoutputmembershipfunction,canbeconsidered
apre-defuzzifiedfuzzyset.
Itenhancestheefficiencyofthedefuzzificationprocessbecauseitgreatlysimplifies
thecomputationrequiredbythemoregeneralMamdanimethod,whichfindsthe
centroidofatwo-dimensionalfunction.Insteadofintegratingacrossthetwo-
dimensionalfunctiontofindthecentroid,theweightedaverageofafewdata
pointscanbeused.
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21. MamdaniFuzzy Inference Systems
Mamdanifuzzyinferencewasfirstintroducedasamethodtocreateacontrol
systembysynthesizingasetoflinguisticcontrolrulesobtainedfromexperienced
humanoperators.InaMamdanisystem,theoutputofeachruleisafuzzyset.
SinceMamdanisystemshavemoreintuitiveandeasiertounderstandrulebases,
theyarewell-suitedtoexpertsystemapplicationswheretherulesarecreatedfrom
humanexpertknowledge,suchasmedicaldiagnostics.
Theoutputofeachruleisafuzzysetderivedfromtheoutputmembershipfunction
andtheimplicationmethodoftheFIS.Theseoutputfuzzysetsarecombinedintoa
singlefuzzysetusingtheaggregationmethodoftheFIS.Then,tocomputeafinal
crispoutputvalue,thecombinedoutputfuzzysetisdefuzzifiedusingoneofthe
methodsdescribedinDefuzzificationMethods.
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1.DeterminethesetofFuzzyRules
2.Maketheinputsfuzzyusinginput
fuzzymembershipfunctions
3.Combined thefuzzifiedinputs
accordingtothefuzzyrulesfor
establishingarulestrength
4.Determinetheconsequentofthe
rulebycombiningtherulestrength
andtheoutputmembershipfunction
5.Combinealltheconsequentstoget
anoutputdistribution
6.Finally,a defuzzifiedoutput
distributionisobtained

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26. MamdaniFuzzy Inference System
Intuitive
Well-suited to human input
More interpretable rule base
Have widespread acceptance
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27. Sugenomethod of fuzzy inference
TheSugenomethodoffuzzyinferenceissimilartotheMamdanimethodin
manyrespects.
Thefirsttwopartsofthefuzzyinferenceprocess,fuzzifyingtheinputsand
applyingthefuzzyoperator,areexactlythesame.
ThemaindifferencebetweenMamdani-typeandSugeno-typefuzzy
inferenceisthattheoutputmembershipfunctionsareonlylinearorconstant
fortheSugeno-typefuzzyinference.
Atypicalfuzzyruleinafirst-orderSugenofuzzymodelhastheform.whereA
andBarefuzzysetsintheantecedent,whilep,q,andrareallconstants.
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28. Continue...
Higher-orderSugenofuzzymodelsarepossible,buttheyintroducesignificant
complexitywithlittleobviousmerit.
Becauseofthelineardependence ofeachruleonthesystem’sinput
variables,theSugenomethodisidealforactingasaninterpolatingsupervisor
ofmultiplelinearcontrollersthataretobeapplied,respectively,todifferent
operatingconditionsofadynamicnonlinearsystems.
ASugenofuzzyinferencesystemisextremelywellsuitedtothetaskof
smoothlyinterpolatingthelineargainsthatwouldbeappliedacrosstheinput
space,i.e.,itisanaturalandefficientgainscheduler.
Similarly,aSugenosystemissuitableformodelingnonlinearsystemsby
interpolatingmultiplelinearmodels.
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29. SugenoFuzzy Inference System
Sugenofuzzyinference,alsoreferredtoasTakagi-Sugeno-Kangfuzzyinference,
usessingletonoutputmembershipfunctionsthatareeitherconstantoralinear
functionoftheinputvalues.
ThedefuzzificationprocessforaSugenosystemismorecomputationallyefficient
comparedtothatofaMamdanisystem,sinceitusesaweightedaverageor
weightedsumofafewdatapointsratherthancomputeacentroidofatwo-
dimensionalarea.
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34. SugenoFuzzy Inference System
Computationally efficient
Work well with linear techniques, such as PID control
Work well with optimization and adaptive techniques
Guarantee output surface continuity
Well-suited to mathematical analysis
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Comparison between MamdaniFIS and SugenoFIS

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