INFORMATIONSCIENCES45,129-151(1988)129 Fuzzyf hnt r ok UnderVariousFuzzyReasonhqMethods MASAHARUMLZUMOTO DepartmentofManagementEngineering, OsakaElectra-CommunicationUniversity,Neyagawa,Osaka572,J apan ABSTRACT Afuzzylogiccontrollerconsistsofiinguisticcontrolrulestiedtogetherbymeansoftwo concepts:fuzzyimplicationsandacompositionalruleofinference.Mostoftheexistingfuzzy logiccontrollersarebasedontheapproximatereasoningmethodduetoMamdani.Thispaper introducesotherfuzzyimplications,suchasthearithmeticruleandmaximinrule,forlinguistic controlrulesandcomparescontrolresultsforaplantmodelwithfirstorderdelayunder variousapproximatereasoningmethods.Moreover,controlresultsarecomparedwhenthe widthsoffuzzysetsoflinguisticcontrolrulesarechanged. 1.~~ODUC~ON Anumberofstudiesonfuzzylogiccontrollershavebeenreportedsince Mamdani[l} implementedafuzzylogiccontrolleronaboilersteamengine.A fuzzylogiccontrollerconsistsoflinguisticcontrolrulestiedtogetherbymeans oftwoconcepts:fuzzyimplicationsanda compositionalruleofinference.Most oftheexistingfuzzylogiccontrollersarebasedontheapproximatereasoning methodduetoMamdani,inwhichtheimplicationforacontrolruleIfxisA thenyisBisexpressedasthedirectproductAX BofthefuzzysetsA andB. Inthispaperweintroduceotherfuzzyimplications,suchasthearithmetic ruleandthemaximinrule,forlinguisticcontrolrulesduetoYamazakiand Sugeno[4], andcomparecontrolresultsforaplantmodelwithfirstorderdelay underthevariousfuzzyreasoningmethods.Furthermore,weinvestigatehow controlresultsareinfluencedwhenthewidthsoffuzzysetsoflinguisticcontrol rulesarechanged. @F,lsevier SciencePublishingCo.,Inc.1988 52 VanderbiltAve.,NewYork,NY10017OOZO-0255,88/$03.50 130~S~RUMIZUMOTO 2.FUZZYREASONINGMETHODS Weshallconsiderthefollowingformofinferenceinwhichafuzzyimplica- tionis contained,whereA,Aarefuzzysets inU, andB,BarefuzzysetsinV, Anti:IfxisAthenyisB Ant2:xisA cons:y isB TheconsequenceBisdeducedfromAnt1andAnt2bytakingthemax-min composition0ofthefuzzysetAandthefuzzyrelationA-+ Bobtainedfrom thefuzzyimplicationifAthenB.Namely,we have B-Ao(A+B), WhenthefuzzysetAisasingletonuo,thatis,pLAP(ua)= 1 andpA,(u)=0 foru #u,,,theconsequenceBissimplifiedas WhenthefuzzyimplicationA+Bisrepresentedbythedirectproduct AxBoffuzzysetsAandBasinthecaseofMamdanismethod[l],Bis givenas WelistseveralfuzzyimplicationsA+BinTable1[2]whichwill beusedin thediscussionoffuzzylogiccontrols. EXAMPLE1,LetAand3befuzzysets inU andV,respectively,as inFigure 1.Thentheconsequence3of(1)atA =u.underthefuzzyimplications A+BinTable1aredepictedinFigure2,wherepA( us)=owithu =0.3 (dottedline)and(I = 0.7 (solidline). VARIOUSFUZZYREASONINGMETHODS131 TABLE1 FuzzyImplicationspA _& uo, 0) = BA(UO)-+ Ps(U)PI Rc: Rp: Rbp: Rdp: Ra: Rm: Rb: R*: Rft: Rs: Rg: RA: PAUO)A Pa(U) a(uo)~Pa(u) o[P,(~o)+cs(u)-ll i PA(UO). Pa(U)=l P&?(U)>PA(UO)=l 0, PA(UOhUB(U)Be(U) i 1, P(UO)4Pa(U) PB(U>YPA(UO)>BB(U) 1, P(UO)satJ(u> CB(UVP(UO)lBA(UO)BB(U) [byMamdani] [byLarsen] [boundedproduct] [drasticproduct] [arithmeticrulebyZadeh] [maximinrulebyZadeh] [Booleanimplication] [byBandler] [byBandler] [standardsequence] [Giidelianlogic] [by Gougen] Fig.1.FuzzysetsAandB. Weshallnextconsiderthefollowingformofinferenceinwhichthehypothe- sisofafuzzyconditionalpropositionIf... then... containstwofuzzyprop- ositionsxisAandyisB combinedusingtheconnectiveand. Ant1:IfxisAandyisBthenzisC Ant2:xisAandyisB (2) Cons:zisC whereA,AarefuzzysetsinU,andB,BinV,andC,CinW. 132MASAHARUMIZUMOTO G @i .r _-.-._. -5 VARIOUSFUZZYREASONINGMETHODS133 TheconsequenceCcanbededucedfromAnt1andAnt2bytakingthe max-mincomposition0ofafuzzyset(AandB)inUxYandafuzzy relation(AandB)+CinUXVXW.Namely,wehave C=(AandB)o[(AandB)-C] InthecaseofMamdanismethodRcinTable1,thefuzzyimplication [(AandB)+C]istranslatedintopA(pB(u)Ap=(w)invirtueofa+b= aAb.Thus,theconsequenceCisgivenas PC(W)= v {[PA)uN3441 A[C1A(U)ACLg(U)A~UC(W)l}.(4) U. LetRc(A,B;C)=(AandB)+C,Rc(A;C)=A +C,andRc(B;C)=B+C befuzzyimplicationsbyMamdanismethodRc.ThentheconsequenceCis reducedfrom(4)asfollows: r&)=V( PA]> = M~o)~PcLg(~cl)I-+Pc(W).(7) Forexample,inthecaseofRcandRawehavetheconsequencesCatA =ZQ, andB=u,asfollows(thesamecanbeobtainedfromotherfuzzyimplications inTable1): RC:~A(UO)PB(UO)Ikb)~(8) Intheabovediscussion,theoperationA( = min)isusedasthemeaningof andintheapproximatereasoningof(2).Itispossibletointroduceother operations,say,algebraicproduce.and,moregenerally,t-normsasand.For example,theconsequenceCatA=u,,andB=u,willbe whenthealgebraicproduct.isusedasand.IfthefuzzyimplicationRpof Table1isusedin4,theconsequenceCbecomes PlfO=cL(1(o)~cI-~(~~)~c1~(w)~(11) EXAMPLE2.Figure3(a)and(b)showtheconsequencesCbyRc(8)andRa (9)atA=u.andB=u,.Figure3(c)indicatestheconsequenceCby(11).In asimilarway,wecanobtainconsequencesCatA =u.andB =u,, byother fuzzyimplicationsinTable1from(7)and(10)bylettingpA(pB(u,,)= a orpA( z+,).~,(uO) =ainFigure2. VARIOUSFUZZYREASONINGMETHODS (a> 135 C ,* * (b) Fig.3.InferenceresultsCatpA(u,,)=0.8andpe(uo)=0.6:(a)pc,(w)=(pA(pB(uo)] AC&W)of(8);(b)~c,(w)=lA[l-(~,(u,)A~,(u,))+~~c(w)lof(9);(4PC,(W)= ~P~~O~~PS~~O~l~PC~~~of (11). 136MASAHARUMIZUMOTO Asageneralizedformofapproximatereasoningof(2),weshallconsider approximatereasoningwithseveralfuzzyconditionalpropositionscombined withelse: Arul:IfxisA,andyisB,thenzisC,else Ant2:ifxisA,andyisB,thenzisC,else ..................................... Antn:ifxisA,,andyisB,,thenzisC,. Antn+l:xisAandyisB. (12) Cons:zisC. Forexample,theconsequenceCbyMamdanismethodRcisgivenas followsbyinterpretingelseasunion(u)andfrom(5): C=(AandB)o[((A,andB,)-C,)u...u((A,andB,,)-C,)] =[(A~oA~-C,)n(BoB,+C,)] u. . . u[(A~A,+C,)n(B~B,+C,)]. (13) NotethatelseisalsointerpretedasunionforthefuzzyimplicationsRp, Rbp,andRdpinTable1,andtheaboveequalityholdsfortheseimplications. WhenA=u,-, andB =u,,theconsequenceCbythemethodRcisgivenas c=c;uc;u-0.UC,, (14 wherefori=l,...,n, (15) Namely,from(8)and(13) Inthesameway,CisobtainedfromRp,Rbp,andRdpasin(14). VARIOUSFUZZYREASONINGMETHODS137 ForthefuzzyimplicationsRa,Rm,Rb,R*, R#,Rs,Rg,andRAinTable1, elsein(12)canbeinterpretedasintersection(n).Thus,theconsequencesC forthesefuzzyimplicationsaredefinedas C=(AandB)e[((A,andB,)+C,)n.-.n&4,andB,)-C,)] c[(AoA,~c,)u(BoB,-,c,)]n .*a n[(AoA,~c,)u(BoB,-,c,)]. (16) ItisnotedthattheconsequenceCisnotequaltobutcontainedinthe intersectionoffuzzyinferenceresults[(A 0 Aj+C;.) U(B0 Bi+C,)](i= 1 9.1.) n).Inthefollowingdiscussion,however,weshallassumethatCisgiven astheintersectionoftheindividualfuzzyinferenceresults,forsimplicityinthe calculationofC. WhenA =u,,andB=u,,theconsequenceC,say,bythemethodRais givenas C=c ;nc ;n...nc ;, (17) whereeachC).l, ill,...,n,isrepresentedfrom(9)as cc;(w) =l~[l-(Pri(~o)~~,,(OO))+Pc,(W)].(18) Inthesameway,wecanhaveCbyRm,Rb,R* ,R#,Rs,Rg,andRAasin (17). Toobtainasingleton%whichisarepresentativepointfortheresulting fuzzysetC,severalmethodshavebeenproposed.Forexample,thepoint whichhasthelargestmembershipgradeofCistakenasadesiredsingleton.In thefollowingdiscussion,themethodisemployedwhichtakesthecenterof gravityofthefuzzysetC,asadesiredsingleton,thatis, Jwc44 dw w=/pc#(w)dw . 3.FUZZYCONTROLSUNDERVARIOUS FUZZYREASONINGMETHODS (19) Weshallconsiderasystemwithfirstorderdelayasasimpleplantmodel whichisrepresentedbyadifferentialequationTdh/dt+h=q,withTbeinga timeconstant. 138MASAHARUMIZUMOTO TABLE 2 Fuzzy ControlRules e,Ae +Aq [4] e1Ae-NBNMNSZOPSPMPB NB NM NS zo PS PM PB PBPMPS PB PM PS zo NS NM NB NSNMNB LeteandAebeinputvariablesofafuzzycontrollerwhichrepresent errorandchangeinerror,andletAqbeanoutputvariablerepresenting changeinaction,whereeandAearedefinedas e=Ah=(presentvalueofh)-(setpoint), Ae=e(k)-e(k-l), andtheactualactionq(k)tobetakenattimekisgivenas q(k)=q(k-l)+Aq. YamazakiandSugeno[4]givefuzzycontrolrulesforasystemwithfirst orderdelayasinTable2.Thistableshows13fuzzycontrolrulesinterpretedas Rl:ejsNBandAeisZO+AqisPB, R2:eisNMandAeisZO+AqisPM, (20) R13:eisZ0andAeisPB+AqisNB. -6-5-4-3-2-10123456 Fig. 4.Fuzzy sets of fuzzy control rules inTable 2. VARIOUSFUZZYREASONINGMETHODS139 whereNB(negativebig),NM(negativemedium),NS(negativesmall),ZO (zero),PS(positivesmall),PM(positivemedium),andPB(positivebig)are fuzzysetsin[ -6,6]asshowninFigure4. Whene=e,,andAe=Ae,aregiventoafuzzycontrollerasapremiseof (20),thechangeofactionAq=Aq,,isobtainedasthecenterofgravity(19)of thefuzzysetwhichisaggregatedfromthefuzzysetsinferredfromeachoffuzzy controlrulesof(20)givene,andAe,byuseof(14)or(17). EXAMPLE3.Weshallconsiderthefollowingthreefuzzycontrolsforsimplic- ity: eisNSandAeisZO+A9isPS, eisZ0andAeisZO4AqisZ0, eisZ0andAeisPS-,AqisNS. (21) WhenMamdanismethodof(15)isused,thechangeinactionAqO isobtained asinFigure5.Inthesameway,AqO isgivenasinFigure6bythemethodof Raof(18). Figure7(a)showsAqOate=e0andAe=Ae,whenusingall13fuzzy controlrulesinTable2byMamdanismethodRc.Figure7(b)and(c)showAq, accordingtoRaandRg,respectively. Usingtheabovemethods,weshallfirstindicatecontrolresultsforaplant modelG(s)=e-2/(1+20s)withfirstorderdelayanddeadtimeundervarious approximatereasoningmethodsinTable1.Inthisexperiment,weusethe followingexpression: clc:(Aq)= [pa,(edApB,(Ae,)]+puc,(Aq)(22) [see(7)-(g)],whereandin(20)isinterpretedasA(=min),andA,,B,,C, (i=1,.. . ,13)arefuzzysetsshowninFigure2andTable2.Itisfoundfromthe computersimulationinFigure8(a)-(c)thatalloftheapproximatereasoning methodsexceptRm,Rg,Rs,andRAobtaingoodcontrolresults.Inparticular, Rc,Rp,Rbp,andRdpobtainthebestresults.Notethatthesemethodsare basedonfuzzyproductsknownast-norms.Similarcontrolresultsareobserved inothercomputersimulationsnotshowninthispaper. InthecaseofMamdanismethodRc,whichgetsagoodcontrolresult,itis foundfromFigure7(a)forAq,,ate,andAe,thatwehaveAqO =0ate,=0 andbe,=0(indicatedbyadotinthecenterofthefigure)andthatAqO decreasestominuswhene,and/orbe,increasetoplusintheareaof e,=AqO +0.Ontheotherhand,forthemethodRa[seeFigure7(b)],therate 140MASAHARUMIZUMOTO 3 VARIOUSFUZZYREASONINGMETHODS 141 .- i. .- c- :- .- = .. -. -_ z--.. --_ Q11 ..-- 1 _ . . .- -._p-- -. -_ NI .. WI i w I W Ei _- _. . _- : : 0 _.-- *. I --. *. --. a? 142 MASAHARUMIZUMOTO -6 0 +e0 6 (a> -6d+eo6 (b) Fig.7.Aq,ate,andAe,byfuzzycontrolrulesinTable2:(a)MamdanismethodRc(14): (b)ZadehsmethodRa(18);(c)RgbasedonGadelianlogicinTable1. VARIOUSFUZZYREASONINGMETHODS143 6 6 -6 -6 c)+e 0 6 Fig.I.Continued. ofdecreaseofAq,isobservedtobesmallerthanthatofAq,bythemethodRc. Therefore,theconvergenceonthesetpointofthecontrolresultbythemethod RabecomesslowerthanthatofthemethodRc[seeFigure8(b)]. ItisnotedthatthemethodsRg,Rs,andRAshowtheworstcontrolresults, asinFigure8(c).WeshallanalyzewhythemethodRg,whichisbasedonthe implicationruleofGiidelianlogicandwhichcangetreasonableinference resultsinfuzzyreasoning[2],cannotgetagoodcontrolresult.Asisseenfrom Figures7(c)and9,therateofdecreaseofAqoiszero(flat)ate,,be,,80,that is,Aq,,=0inthearea.Thus,nochangeismadeinthecontrolactionq,andso thesameactioncontinuestobetaken.Moreprecisely,itisseenfromFigure 8(c)thatthecontrolresultofRgconvergesonthepointh=58.3(notat60).In ourcomputersimulationweusetheexpression h-40x6e,=- 40 40 =setpoint,6 =scalefactor, toobtaintheerrore,fromtheoutputhoftheplantmodel.Forexample,we havee,=3ath= 60.WeshallshowwhatvalueofAq,,canbeobtainedat e,