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,