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IEEETRANSACTIONSONSYSTEMS,MAN,ANDCYBERNETICS—PARTB:CYBERNETICS,VOL.29,NO.3,JUNE1999371

AnalysisofDirectActionFuzzyPIDControllerStructures

GeorgeK.I.Mann,Bao-GangHu,Member,IEEE,andRaymondG.Gosine,Member,IEEE

Abstract—ThemajorityoftheresearchworkonfuzzyPIDcontrollersfocusesontheconventionaltwo-inputPIorPDtypecontrollerproposedbyMamdani[1].However,fuzzyPIDcontrollerdesignisstillacomplextaskduetotheinvolvementofalargenumberofparametersindeﬁningthefuzzyrulebase.ThispaperinvestigatesdifferentfuzzyPIDcontrollerstructures,includingtheMamdani-typecontroller.Byexpressingthefuzzyrulesindifferentforms,eachPIDstructureisdistinctlyidentiﬁed.Forpurposeofanalysis,alinear-likefuzzycontrollerisdeﬁned.Asimpleanalyticalprocedureisdevelopedtodeducetheclosedformsolutionforathree-inputfuzzyinference.ThissolutionisusedtoidentifythefuzzyPIDactionofeachstructuretypeinthedissociatedform.Thesolutionforsingle-input–single-outputnonlinearfuzzyinferencesillustratestheeffectofnonlinearitytuning.ThedesignofafuzzyPIDcontrolleristhentreatedasatwo-leveltuningproblem.TheﬁrstleveltunesthenonlinearPIDgainsandthesecondleveltunesthelineargains,includingscalefactorsoffuzzyvariables.Byassigningaminimumnumberofrulestoeachtype,thelinearandnonlineargainsarededucedandexplicitlypresented.ThetuningcharacteristicsofdifferentfuzzyPIDstructuresareevaluatedwithrespecttotheirfunctionalbehaviors.Theruledecoupledandone-inputrulestructuresproposedinthispaperprovidegreaterﬂexibilityandbetterfunctionalpropertiesthantheconventionalfuzzyPIDstructures.IndexTerms—Apparentlineargains,apparentnonlineargains,fuzzycontrol,linear-likefuzzy,PIDstructures,two-leveltuning.

I.INTRODUCTION

VERTHEpasttwodecades,theﬁeldoffuzzycon-trollerapplicationshasbroadenedtoincludemanyin-dustrialcontrolapplications,andsigniﬁcantresearchworkhassupportedthedevelopmentoffuzzycontrollers.In1974,Mamdani[1]pioneeredtheinvestigationofthefeasibilityofusingcompositionalruleofinferencethathasbeenproposedbyZadeh[2],forcontrollingadynamicplant.Ayearlater,MamdaniandAssilian[3]developedtheﬁrstfuzzylogiccontroller(FLC),anditsuccessfullyimplementedtocontrolalaboratorysteamengineplant.Inastrictsense,theﬁrstfuzzycontrollershownin[3]wasequivalenttotwo-inputfuzzyPI(orPI-like)controllerswhereerroranderrorchange,wereusedastheinputsfortheinference.Mamdani’spioneeringworkalsointroducedthemostcommonandrobustfuzzy

ManuscriptreceivedFebruary14,1998;revisedNovember20,1998.ThisworkwassupportedbytheNaturalSciencesandEngineeringResearchCouncilofCanada,theCanadianSpaceAgency,andthePerto-CanadaResources.ThispaperwasrecommendedbyAssociateEditorA.Kandel.G.K.I.MannandR.G.GosinearewiththeC-COREandFacultyofEngineeringandAppliedScience,MemorialUniversityofNewfoundland,St.John’s,NF,Canada(e-mail:mann@engr.mun.caandrgosine@engr.mun.ca).B.-G.HuiswiththeNationalLaboratoryofPatternRecognition,InstituteofAutomation,Beijing100080,China(e-mail:hubg@prlsun3.ia.ac.cn).PublisherItemIdentiﬁerS1083-4419(99)03532-3.

reasoningmethod,calledZadeh–Mamdanimin–maxgravityreasoning.Also,asigniﬁcantnumberofin-depththeoreticalandanalyticalinvestigationsrelatedtothisstructurehavebeenreportedin[4]–[8].TakagiandSugeno[9]introducedadifferentlinguisticdescriptionoftheoutputfuzzysets,andanumericaloptimizationapproachtodesignfuzzycontrollerstructures.

ThereareseveraltypesofcontrolsystemsthatuseFLCasanessentialsystemcomponent.ThemajorityofapplicationsduringthepasttwodecadesbelongtotheclassoffuzzyPIDcontrollers.Thesefuzzycontrollerscanbefurtherclassiﬁedintothreetypes:thedirectaction(DA)type,thegainsched-uling(GS)typeandacombinationofDAandGStypes.ThemajorityoffuzzyPIDapplicationsbelongtotheDAtype;herethefuzzyPIDcontrollerisplacedwithinthefeedbackcontrolloop,andcomputesthePIDactionsthroughfuzzyinference.InGStypecontrollers,fuzzyinferenceisusedtocomputetheindividualPIDgainsandtheinferenceiseithererrordrivenself-tuning[10]orperformance-basedsupervisorytuning[11].InadditiontothecommonMamdani-typePIstructure,severalotherstructuresusingone-orthree-inputcontrollershavebeenreported.Forcomparison,afewselectederrordrivenfuzzyPIDapplicationsarelistedinTableI.Itisclearfromthisliteraturereviewthatthemajorityoftheseapplicationsbelongtotheclassoftwo-inputfuzzyPIDtypestructures.ThemajorityofotherrelatedfuzzyPIDreferences,whichhavenotbeenincludedinthistable,fallintothecategoryoftwo-inputMamdani-typePIDstructures.Inourrecentwork[38],aone-inputfuzzyPIDstructurewasusedtocontrolseveralﬁrst-andsecond-orderplantmodels.Theone-inputFLCwithfewerruleshasnotbeencommonlyusedforsimultaneouslyderivingthethreefuzzyPIDactions.Basedonthisliteraturereview,wecanarguethatdifferentfuzzyPIDstructuresarepossibleinthecontextofknowledgerepresentation,andthattheyshouldbeevaluatedwithrespecttotheirfunctionalbehaviors.Therefore,inthispaperweintendtodeduceandevaluatedifferentfuzzyPIDstructures,includingthecommonlyavailablefuzzyPIDcontrollers.SincetheDAtypefuzzyPIDisthemostcommonlyused,ourstudyisrestrictedtothosecontrollersonly.

ThelinearPIDcontrollersareeasytoimplement,andsufﬁcienttuningrulesareavailabletocoverwiderrangeofprocessspeciﬁcations.Moreover,theavailablePIDtuningheuristicsareeasytounderstandandimplementforpracticalcontrolproblems.Fuzzycontrollersgenerallyprovidethenonlineartransferelementsfornonlinearcontrol[39].Thesystemofif-thenrulesinthefuzzyknowledgebasesystem

372IEEETRANSACTIONSONSYSTEMS,MAN,ANDCYBERNETICS—PARTB:CYBERNETICS,VOL.29,NO.3,JUNE1999

TABLEI

DIFFERENTFUZZYPIDSTRUCTURESINTHELITERATURE,e-ERROR,1e-CHANGEOFERROR,12e-RATEOFCHANGEOFERROR,y-PLANTRESPONSE,1y-CHANGEOFPLANTRESPONSE,+GSTYPE,3COMBINEDDAANDGSTYPES.OTHERSDATYPE

istransformedintothisnonlineartransferelements.AsaresulttheFLChasbeensuccessfullyimplementedinthepasttoformanylinearandnonlinearprocesses[9],[11],[18],[36].ThenaturalrepresentationofcontrolknowledgethroughfuzzyparadigmsallowsthecontrolactiontobeeitherlinearornonlinearandprovidesimprovedcontrolincomparisonwithaconventionalPIDcontrollerusinglinearcontrolpolicy.Theﬁnaltuningoffuzzycontrollers,however,isstilladifﬁculttask.Manyoff-linetechniqueshavebeendevelopedinthepastfordecidingthenonlineartransferelementsofthefuzzycontrollers.Asanexample,cell-to-cellmapping[30],trainingalgorithmsusinginput/outputdata[40],andgeneticsearchalgorithms[38],[41]arecapableofgeneratingtheoptimumornearoptimumsolutionstothefuzzysystemsinahighdimensionalspace,butatthecostofextensivecomputersimulationsandtime.Althoughthegeneticalgorithmsarequitepowerfulinhandlingalargenumberofvariables,thenumberofiterationcyclesandtheaccuracydeﬁnitions(orresolution)allowsonetoreachonlyanearoptimumratherthanglobaloptimum.Duetothecomplexityofthenonlinearcontrolsurfacethatisgeneratedbyconventionaltwo-inputfuzzycontrollers,identifyingandsolvingalargenumberoftuningparametersbyananalyticalmeansisextremelydifﬁcult.InthispaperweproposesimplefuzzyPIDcontrollerstructuresforreducingthedimensionalityindesigns.Functionalbehaviorsofthesefuzzycontrollersareevaluatedtoshowthemaindrawbacksoftheconventionaltwo-inputfuzzyPIDcontrollers.

Inthispaperwedescribethreecontributions.First,newfuzzyPIDstructuresareidentiﬁedinadditiontocommonly

Fig.1.CascadetypefeedbackPIDcontrolledsystem.

availableconventionalfuzzyPIDstructures.Theseincludeaone-input–three-outputsfuzzycontrollerusingerrormappingforgeneratingindividualfuzzyPIDactions.Second,anewanalyticalprocedureispresentedforthegeneralthree-inputLLFLCinferencebasedonmin–maxgravityreasoning.Twoandone-inputsimpliﬁcationsareincludedtocoveraspectrumoffuzzyPIDstructures.Third,theapparentnonlinearandapparentlinearPIDgainanalysisispresentedforidentifyingthetwo-leveltuningoffuzzyPIDstructures.Thereforetheworkinthispaperisarrangedasfollows.

1)FuzzyPIDelementsareproposedandthensixdifferentfuzzyPIDstructures,includingcommonlyavailablestructures,areconstructed.

2)ClosedformsolutionsfortheoutputsoffuzzyPIDelementsarededucedbasedonalinear-likefuzzylogiccontroller(LLFLC).AlsotheoutputofaSISOnonlinearlikefuzzycontrollerwiththreerulesisdeduced.

3)Usingtheclosed-formexpressions,apparentnonlinearandapparentlinearfuzzyPIDgainsarededucedwhileconsideringtwo-levelsoftuning.

4)Thestructuresareevaluatedintermsoftwo-levelsoftuning.Nonlineartuningisevaluatedwithrespecttothefunctionalbehaviorsofstructures.

II.FUZZYPIDSTRUCTURALELEMENTS

ThelinearPIDcontrollerscanbeclassiﬁedintodifferentcategorieswithrespecttothepositioningofthethreetermsintheclosed-loopcontrolsystem.Incomputercontrolledsingle-inputsingle-output(SISO)plantsystems,thecascade-formPIDcontrolleriscommonlyused.ThereforeinthisstudywerestrictourclassiﬁcationtocascadetypePIDcontrollersasshowninFig.1.Otherforms[35]–[37]canbeobtainedbyextendingthefundamentalprincipleweproposeinthisstudy.ConsideringalinearPIDcontrollerinFig.1,thecontrollersignalatanygiventimeinstance

(1)

and

MANNetal.:FUZZYPIDCONTROLLERSTRUCTURESFig.2.FuzzyPIDstructuralelements.

aredeﬁnedas:error,errorchange

with

beingthefeedbackresponsesignal,andthedesiredresponseorreferenceinputatthe

fromallthecontrolvariables.Thescalefactors(

,,and,arethenormalizederrorvari-ablescorrespondingtotheerrorterms

,and)then(output).”Inthecaseoftwo-inputconﬁgurations,onlyPDandPIcontrollerelementsareconsidered.AsubscriptwiththenormalizedoutputvariableisusedforidentifyingthecorrespondingactioninafuzzyPIDcontroller.

InderivingapracticalfuzzyPIDstructurethefollowingremarksaremade.

373

Remark1:Itisdifﬁculttoformulatecontrolruleswiththeinputvariablesum-of-error

)is

consideredthenecessaryinputforderivinganyPIDstructure.Theerrorinputprovidesthenonlinearproportionalactionsthroughthefuzzyinference.Foranysystemtodrivefromadeadstate,proportionalcontrolisthebasicactionrequiredfromthethree-termPIDcontroller.Forexample,incaseofasteadyoffsetinthesystemresponse,orincaseofatime-delayprocess,themagnitudeofallerrorderivativesbecomesnegligible.Inthosecircumstancesthesteadyerroristheonlyavailableinformationthatcanprovideaﬁnitecontrolactiontodiverttheoutputfromadeadsituation.

III.FUZZYPIDSTRUCTURES

BytakingdifferentcombinationsofthefuzzyPIDstructuralelementsdeﬁnedintheprevioussection,wecannowconstructfuzzycontrollerstorepresentPIDactionsinanonlinearform.BasedonRemarks1and2,someofthestructuralelementscanbeconsideredtobe“bad”andcanbeeliminatedinbuildingafuzzyPIDstructure.Thereforeinthissystematicinvestigationweevaluatesixtypesofcontrollersandcomparetheirperformance.In1975,Zadehpublishedathree-partpaper[42]describingthefundamentalsoffuzzylogicprinciplesforusingindecision-makingsystems.Zadehhasincludedmanydeﬁnitionsandconceptstogeneralizethebroaderperspectivesofhumanisticsystems.TheFLCsystemsusessomeofthoseconceptsfordescribingtheknowledgebase.

Deﬁnethelinguisticvariablesthatcorrespondtothein-putscaledvariables,

,,and,respectively.Theindices,and

,where

istheﬁnal

controlleroutput.Assignlinguisticvariablesforthecontroller

outputas

,orforincrementalsignal

.Thevalue

isusedto

denotethenonlinearmappingbetweeninputsandoutput.TypeI—Three-InputFLCStructurewithCoupledRules:Itispracticallydifﬁculttoassignlinguisticvaluesortermsfortheinput

,and,correspondingtoanincrementaltypefuzzyPID

374IEEETRANSACTIONSONSYSTEMS,MAN,ANDCYBERNETICS—PARTB:CYBERNETICS,VOL.29,NO.3,JUNE1999

Fig.3.Three-inputfuzzyPID(TypeI).

Fig.4.Three-inputfuzzyPID(TypeII).

controller.Usingtherulebasenotationof[11],Type-IfuzzyPIDstructurecanbeexpressedby

ELSE

AND

THENIS

ELSE

)astheuseful

PIDelementsforfuzzycontrol.TheyarecorrespondingtotheincrementalPIorabsolutePDsignals.Theothertwo-inputcontrolelementsshownintheFig.2areeliminatedaccordingtotheRemarks1and2.BycombiningbothPIandPDactionsasshowninFig.5,atwo-inputfuzzyPIDcontrollercanbeformed.TherulebasestructureisidenticaltoMamdani-typefuzzyPIcontroller.Thebasicrulebaseofthisconventionaltypeisgivenby

ELSE

ISIS

THEN

andand

and

THEN

MANNetal.:FUZZYPIDCONTROLLERSTRUCTURESFig.6.Two-inputfuzzyPID(TypeIV).

Fig.7.One-inputfuzzyPID(TypeV).

fuzzyproportionalaction.Therulebaseoftheone-inputfuzzyproportionalcontrolelementisgivenby

ELSEIF

(12)

Similartothepreviouscase,wecaninferfromone-input

elements

andbyassumingtheanalogybetweentheproportionalandderivativeactionsas,

ELSEIFIS

THEN

ELSE

375

376IEEETRANSACTIONSONSYSTEMS,MAN,ANDCYBERNETICS—PARTB:CYBERNETICS,VOL.29,NO.3,JUNE1999

thegeneralsolutionisexpressedwithonlytwodifferentnonlinearterms.Inadditiontotheabovewehaveusedthestandardcenterofarea(COA)defuzziﬁcationmethodratherthancenterofheights(COH)[39]orcenteraveragedefuzziﬁcation[40]thatwasusedin[5],[6],and[35].TheCOHmethodisaconvenientwaytoobtainoutputsolutionwithleastnumberofexpressions.However,theCOHmethodignorestheeffectoffuzziness[11]associatedwiththeoutputlinguisticvariablesandisequivalenttotakingfuzzysingletonfunctions.Asanexample,theCOHmethodignoresthewidthofthesupportsetorthepartitioningoftheoutputmembershipfunctionsduringthedefuzziﬁcations.AsaresulttheCOHproduceslessnonlinearitythantheCOAmethod,particularlyforone-inputfuzzyinferences.OntheotherhandCOHisbetterforobtainingpiece-wiselinearity.Forhighdegreeofnonlinearity,itrequiresalargenumberofrules.Thisparticularcharacteristicshasbeenexploitedtoobtainthenonlinearfunctionapproximations[40],butattheexpenseoflargernumberofrules.HowevertheCOGmethodisdifﬁculttoanalyzeforahighlynonlinearrulebases.Thenonlinearlikeanalysisweperforminthelatterpartofthispaper(SectionIV-D)clearlydemonstratesthebeneﬁtsofCOGmethod.A.Deﬁnition—Linear-LikeFuzzyLogicController

Letthethreeerrorinputsinanyorderbedeﬁnedas

1,1]asshownin

Fig.9(a).Thetotalnumbersoflinguisticvariablesused

for

,and,where

rules,therulebaseisdeﬁnedas

ELSE

AND

THEN

and

round

MANNetal.:FUZZYPIDCONTROLLERSTRUCTURESTABLEII

NONLINEARTERMFORTHETHREE-INPUTLLFLCOUTPUT

Step3:Deﬁnetheincrementalinputvectors.NormalizedincrementalinputvectorandNormalizedabsoluteincrementalinputvectorarerespectivelygivenby

(26)

377

modiﬁed

Step7:ComputetheLLFLCoutput

)andanonlinearcontrolleroutput().The

linearcontrollerisdeﬁnedastheequivalentlinearcontroller(ELC)oftheLLFLCsystem

and

.From(15),.Thetriangular

membershipfunctiondeﬁnedforthesinglelinguisticvariablewillnowhaveaninﬁnitelongsupportsetasshowninFig.10.Thefuzzymembershipfunctionwillbeahorizontallinewithaunitgradeofmembershipheight.Themodalpositionofthe

singlefuzzysetbecomeswith.Thusforanyinput

conditionsthe

whichimplies

linearrulesandisobtainedby

simplifyingthethree-inputrulebasein(17)asELSEIF

THEN

.Sincewenowhaveonlytwo

inputvariables,theeightcasesinTableIIreducetofourcases

and

iseliminated.Foratwo-inputfuzzycontroller,Steps1–7areusedwhileequatingoneoftheinputvariablestozero.

Takingthespecialcasefor

when378IEEETRANSACTIONSONSYSTEMS,MAN,ANDCYBERNETICS—PARTB:CYBERNETICS,VOL.29,NO.3,JUNE1999

TABLEIII

NONLINEARTERMFORTHETWO-INPUTLLFLCOUTPUT

correspondingnonlinearterm(

,where

(32)

Similartothethree-inputcase,thegeneraloutputexpressionforthetwo-inputLLFLCoutputcanbeobtainedasthesum

oflinear(

)andnonlinear()controlleroutputsISTHENIS

forany(

isgivenby

and

)controlleroutputs,andis

givenby

nowhaveonlytwovalues,0and1,weﬁrstconsider

thepositiveincrementalinputsmeasuredfromthe0indexpositions.Foranygiveninputerrorvector

MANNetal.:FUZZYPIDCONTROLLERSTRUCTURESThe(38)isrewritteninthedissociatedformas

where

379

”isusedtoidentify

thenonlinearlikecontroller.Thesimplestthreelinearrules,R1–R3,foraone-inputPIDcontrollerelementcanbethenrepresentedby

R1:IfisNBthenisNBR2:IfisZEthenisZER3:If

isPBthen

isPB

1,1].In

ordertoreducethecomplexityofthesolution,thefollowingconstraintisimposedforthemembershipvariables.Rangefor

380IEEETRANSACTIONSONSYSTEMS,MAN,ANDCYBERNETICS—PARTB:CYBERNETICS,VOL.29,NO.3,JUNE1999

Fig.13.Fuzzyoutputs(shadedareas)correspondingtodifferentinputcon-

ditions.

CaseII—Overlapping:AND

MANNetal.:FUZZYPIDCONTROLLERSTRUCTURES381

ANGTERMS

TABLEV

DIFFERENTFUZZYPIDSTRUCTURES

questionforinnerloopcontrollerswheretheavailabilityofsuchcontrollerexperienceisminimal[11].Inmanycases,thenonlineartuningiscarriedoutarbitrarilybychangingrulesandmembershipfunctionparameters,andobservingtheeffectincomputersimulations.Agenericanalysisisextremelydifﬁcult,particularlyforcoupledthree-inputortwo-inputrulebases.AsweareprimarilyinterestedincomparingfuzzyPIDstructures,asimplestLLFLCrulebasestructureisassumedforderivingANGtermsofcontrollerstructuretypesI–V.TheANGtermsoftypeVIcontrollerareshownwithrespecttothenonlinearlikefuzzycontroller.

ThenonlinearPIDgains(ANGterms)relatedtonormalizedPIDactionsaredeﬁnedas

and

(46)

where

382IEEETRANSACTIONSONSYSTEMS,MAN,ANDCYBERNETICS—PARTB:CYBERNETICS,VOL.29,NO.3,JUNE1999

TheequivalentformwithANGtermsisthenarrangedas

in(42)–(),theANGtermsthatcorrespondtothearrangementin(53)arethusobtained.Forsmallsamplingtimeintervalstheequivalentnonlinearderivativegainhasbeenfurthersimpliﬁedwhileusingtherelation

6)ANGforTypeVI:SincetypeVstructureisaspecialcaseoftypeVI,withthesimplestLLFLCrulebasesbothtypesareidentical.Apracticalhighperformancefuzzycontrollerrequirestheknowledgebasetohaveanonlinear-likestructure.However,forthenormalizedproportionalcontrolleroutputtobemonotonicwithrespecttoerror,therulesmustbearrangedinthelinearform,asin(34).Insuchcircumstances,themembershipfunctionsareplacednonuniformlytoobtainthenonlineartuning.Inordertoillustratethis,thesolutionofthesimplestnonlinearlikefuzzycontrollershownby(45)is

used.Let

,bethevectorcontainingnonlineartuningparametersoftheone-inputfuzzyknowledgebase.Thenwecandeﬁnethreeseparateproportionalactionswiththreedifferent

(56)

TheexpressionoftheANGtermsarrangementfor(56)isidenticalto(53).Substituting(55)into(56),theANGtermsthatcorrespondtothearrangementin(53)arethusobtained.SimilartothetypeV,thesmallsamplingtimeisassumedforobtainingthederivativeANGterm.B.ApparentLinearGains

Theoveralltuningoffuzzycontrollersisgenerallyachievedbythesecond-leveltuning,wherescalefactorsandothergainsareadjustedtoobtainthedesiredoroptimumresponse.Inpracticethisisatrialanderrorprocedure.Sometuningrulesfortheselineargainsarereportedin[44]forthetwo-inputPIstructure.Theuseofgeneticalgorithmstoselectthesegainsisdescribedin[38]and[41].Inthisanalysis,apparentlinearPIDgainsaredeﬁnedforthefuzzyPIDstructures.ThebehaviorofthosegainsisexpectedtobelinearlyequivalenttoconventionalPIDgains.Inorderfortheapparentgainstobefunctional,withoutlossofgenerality,weimposethefollowingconstraints.

Constraint1:Assumetheuniverseofdiscourseofallin-putvariablesareuniformlypartitionedandthemembershipfunctionsareplacedwith50%overlapsupportsets.Therulesaredeﬁnedinthelinearform.Nonlinearityisallowedbychangingpositionsofoutputmembershipfunctions.Letthe

uniforminputmembershipspacingbegivenby

,and,respectivelyfortheinputs.

Constraint2:Thedefuzziﬁedoutputvalueisscaledtotherange[

where

whereisthe

maximumerrorsignalduringthetransient.Asthesetpointvariesthisvaluealsovaries.

TheConstraint1isdeﬁnedforobtainingrulecompleteness[39].Also,thisallowsonetodeﬁneaparticularcontrollerthatwouldbelinearlyclosesttothenonlinearfuzzycontrolleroutput.Alternatively,alinearsurfaceequivalenttoanexistingnonlinearfuzzyoutputcanbedeterminedbylinearregressionanalysis.Sincethisworkisofamoregeneralnature,thisconstraintisimposedsothattheequivalentrepresentationcanbejustiﬁed.AstheELCisderivedfromLLFLCanditsmaximumoutputisnormalizedwithin[

”denotesthe

equivalentlinearactions.Aftersubstitutingthescalefactors

MANNetal.:FUZZYPIDCONTROLLERSTRUCTURES383

TABLEVI

ALGTERMSOFDIFFERENTFUZZYPIDSTRUCTURES

andassigning,,theELCoutputsshownin(29),(33),and(37)arerewrittenasfollows:Forthree-inputelements

(60)

where

384IEEETRANSACTIONSONSYSTEMS,MAN,ANDCYBERNETICS—PARTB:CYBERNETICS,VOL.29,NO.3,JUNE1999

controlaction.ThebasicdissociationthathasbeendoneforthesimplestLLFLCstructure[see(37)and(39)]isanattempttoidentifytheindividualPIDactionsindissociatedform.Asimilarapproachhasbeenemployedin[6]toidentifyANGtermsofasimplestfuzzyPIcontrollerusingdifferentinferencemethods.Thisisquiteartiﬁcialsincethealgebraicdecomposi-tionofnonlineartermsmaynotshowthetruerepresentationoftheindividualfuzzyPIDoutputs.Furthermore,whentherulesarehighlynonlinearandmembershipsarenonuniform,actionidentiﬁcationinadissociatedformwillbecomeanextremelydifﬁcultmathematicaltask.ThenonlinearPIDgainsbecomenontransparentforindependentnonlineartuning.Theactionassociationisoneofmajorreasonswhynosatisfactoryin-depthanalysishasbeendoneinidentifyingnonlineartuningparametersinanexplicitformforthemostcommonMamdani-typetwo-inputfuzzyPIDcontrollers.

2)InputCoupling:InthecoupledrulebasesweagainseeinputcouplingintheANGterms.InthetypeIcontroller,allthegainsarehighlycoupledbyallthreeerrorterms.Theadvantageofinputcouplingistheinclusionofgeneralizeddamping[47],whichgiveseachnonlineargaintermtheeffectoferrorderivatives.Thedisadvantageisthattheproportionalandintegralactionsareunnecessarilycomplicatedbytheeffectofdampingandthisresultsinamoresluggishresponse.Forexample,whenaprocessisrespondingslowly,thecoupledactionoferrorratestendstoproducelowequivalentgainfortheapparentnonlinearproportionalaction.ThiscanbenumericallyveriﬁedbycomparingthemaximumproportionalANGvalueswhenalltheerrorderivativesareforcedtozero.Thisisoneofthereasonswhyin[7]theconventional(typeIII)fuzzyPIstructurewasunabletoperformbetterthananoptimallydesignedlinearPIcontroller.

3)GainDependency:ThisfunctionalbehaviorcanbeseenwhenonefuzzyactionisgeneratedbyanotherfuzzyactionasintypeIII–Vstructuresandcanbedescribedmathematicallybythefollowinganalysis.

a)DependencybetweencoupledPIandPDcontrollers:ThedependencythatexistinthetypeIIIcontrolleroutputsis

givenby

.ReplacingthenormalizedtermswithANGterms,thegaindependencycanbeexpressedby

.Substitutingthenormalized

termswithANGtermsthegaindependencycanbedescribedby

()

Consideringsmallsamplingintervals,theabovecanbede-scribedinacontinuousformbythefollowingnonlineardifferentialequation:

MANNetal.:FUZZYPIDCONTROLLERSTRUCTURESVII.SUMMARY

AND

CONCLUSIONS

ThispaperdescribesresearchtoprovidecontrolengineerswithfundamentalinformationaboutthedesignaspectsoffuzzyPIDcontrollersandaselectionprocedurebyevaluatingthefunctionalbehaviorsofstructures.Thissystematicanal-ysishasfacilitatedtheidentiﬁcationofdifferentfuzzyPIDcontrollerstructures,particularlydecoupledandone-inputtypecontrollers,whichhavenotbeencommonlyusedinpreviousapplications.Itisknownthatthecurseofdimensionalityisamajorprobleminfuzzycontrollerdesigntoday[45].Incontrollerdesigns,theidentiﬁcationoffuzzycontrollerparametersrelatingtheplantdynamicsorperformanceisparticularlychallenging.Inmostcasesextensivecomputersimulationsorexhaustivenumericalsearchtechniquesareusedforsolvingthemultidimensionalproblem.Inourwork,thishighdimensionaldesignwasidentiﬁedasatwo-leveltuningproblem.ThechoiceofanyfuzzyPIDstructureshouldbedonebasedontheefﬁciencyofthesetuninglevelswhileseekingsuperiorperformance.Ourstudyalsohasshowntheexplicitrepresentationofhigh-leveltuningbyANGterms.ForoptimaldesignonehastochoosethenonlineartuningparametersforvaryingtheANGterms.

ThetypeVcontrolleristhesimplest,withthenonlineartuningaccomplishedthroughthefuzzyproportionalaction.However,thegaindependencyinthiscontrolleravoidsin-dependenttuningofintegralandderivativenonlineargains.Theruledecoupledstructuresandone-inputfuzzystructureshavetheadvantageofidentifyingindividualPIDactionsintermsoftheirnonlineartuningparameters.TypesIIandVIstructuresofferindependentgaincontrolforbothofthetuninglevels.ThetypeVIcontrollerismoreanalogoustoalinearPIDcontroller,whereeachcontrolactionisnonlinearlyrelatedtotheerror.ThesystemcanbemadeexactlylikealinearPIDcontrollerbyselectingnonlineartuningparameterstoproducealinearfunctionfortheproportionalsignal.Asan

example,theproperselectionof

andintheone-inputnonlinearlikefuzzycontrollerelementallowsthefuzzyoutputtobealmostlinear(curveCinFig.14).ThereforeproperselectionofnonlineartuningparameterscanproducethelinearcontrollerasaspecialcaseofthefuzzyPIDcontroller.ThisparticularfeaturemakesthefuzzycontrolleralwaysperformeitherbetterthanorequaltoalinearPIDcontrollerandavoidsthepoorerperformanceofthefuzzycontrollersasexperiencedin[7].Thescalingfactorsfortheerrorcanbereadilycomputedbyknowingitsmaximumdeviation,whichisusuallyavailablewiththeresponsedata.Withproperchoiceofnonlineartuning,thetypeIIcontrolleralsocanbemadewithaperfectincremental(velocity)typePIDcontroller.Duetothederivativeerrorinputs,thisstructureissensitivetonoise[48].Howevertheerrorderivativesprovideadditionalinformationandenhancethegeneralizeddampingofthecontrolsystem[47].ThusthetypeIIstructuremaymakethecontrollermorerobustthanthetypeVIcontroller.

Inthisstudywehaveproposedanequivalentlinearcon-trolleranalysistoidentifysecondleveloroveralltuningterms.TheALGtermsderivedfromtheELCanalysishavethesameeffectasthethreePIDgainsofalinearPIDcontroller.Alsowe

385

haveshownthattheﬁnaloveralltuningtaskcanbesimpliﬁedtoathreetermtuningproblem.ThereforeonecanﬁndsuitabletuningheuristicsfortheALGtuningtermsbycorrelatingexistinglinearPIDtuningmethods.

Allcoupledstructureshavethedisadvantageofusingalargenumberofrulescomparedtodecoupledstructures.Sincethenonlinearitytuningparametersareassociatedwiththerules,theparametergrowthalsoincreaseswiththerulegrowth.Therefore,ruledecoupledstructuresarequiteadvantageousintermsofusingtheleastnumberofnonlinearitytuningparameters,thusenablingonetoperformefﬁcientandeasyhighleveltuningforattainingoptimumperformance.

Thedesignofafuzzycontrollerrequiresthebuildingaknowledgebasedsystemwiththespeciﬁcnonlinearitytogenerateaspeciﬁcperformanceoftheprocessresponse.Thevariationoftuningparametersisalwaysrelatedtotheperformance.Therefore,developmentofasuitabletuningschemeforfuzzyPIDcontrollersrequiresconsiderationofthetwotuninglevels,whereonelevelmatchestheplantdynamicsandthenonlinearbehaviorandthesecondlevelprovidesthenecessarymagniﬁcationstoPIDcontrolactions.

FromthisstudyitcanbeconcludedthattheMamdani-typeconventionaltwo-inputfuzzyPIDstructureproducesaninferiorperformanceintermsoffunctionalbehaviors.Thesedrawbackscanbesummarizedasfollows.

1)ThecoupledrulesproduceanassociatedPIDactionandthereforeidentifyingnonlineartuningparametersforthenonlinearity(orhigh-level)tuningisdifﬁcult.

2)Thecomplexandcouplednatureofbothlinearandnon-lineargainsmakesthetuningoffuzzyPIDcontrollersanextremelyadifﬁculttask,andthereforeitsapplicationsarelimitedtoeitherthePIorPDversions.

3)Withlinearrules,[see(17)]thenonlinearityobtainedbychangingmembershipfunctionsoftheconsequentfuzzyvariablesislimited[33].Thereforeanynonlin-earitytuningforbettercontrolperformancerequiresanexhaustivesearchoflargenumbersofrulesforobtaininganoptimumcontrolsurface.

Inthispaper,wehavealsodescribedanewanalyticalsolutionprocedurefortheoutputofageneralthree-inputLLFLCsystem.Theinputtransformationprocedurereducesthenumberofnonlinearexpressionsrequiredtorepresentmulti-phasesolutionsforanyLLFLCstructure.TheLLFLCstructurecanbeusedasthebasiccontrollerstructuretocom-parethedynamiccharacteristicsofdifferentfuzzycontrollerstructures.

APPENDIX

DERIVATION

THENONLINEARTERM

386IEEETRANSACTIONSONSYSTEMS,MAN,ANDCYBERNETICS—PARTB:CYBERNETICS,VOL.29,NO.3,JUNE1999

Fig.15.Relativepositionsofinputs.

Fig.16.Fuzzyoutputshapescorrespondingtodifferentinputconditions.Theincrementalinputsaremeasuredfromthemodalpositions.Thesubscriptijk󰀑i+j+k.

TheshadedareasinFig.15showtheserelativeinputcondi-tions.Thisparticularregionisselectedtogiveasimpleand

conciseexpressionforthenonlinearterm

.Thereferenceoutputiswhen

allcrispinputsareatmembershipmodalpositions.TheinputconditionsandtheresultantfuzzyoutputscorrespondingtoeachcaseareshowninFig.16.Forconveniencethesubscript

isrepresentedby

)ofthetrapezoids

producedforeachruleareshowninTableVII–X.Asanexample,theruleR1showninTableVIIIreadsas“If(

isand)thenis

MANNetal.:FUZZYPIDCONTROLLERSTRUCTURES387

TABLEIX

RULEIMPLICATIONANDFUZZYOUTPUTSFORCASEIII

TABLEX

RULEIMPLICATIONANDFUZZYOUTPUTSFORCASEIV

THE

TABLEXI

NONLINEAROUTPUTTERM

Defuzziﬁcation:TheCOAbaseddefuzziﬁedvaluecanbeexpressedas[11]

.Thisreferstothecenterof

theshadedareasshowninFig.16.Fromthesediagramsthemembershipheightsshowninthe

and

388IEEETRANSACTIONSONSYSTEMS,MAN,ANDCYBERNETICS—PARTB:CYBERNETICS,VOL.29,NO.3,JUNE1999

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GeorgeK.I.MannreceivedtheB.Sc.(honors)degreeinengineeringfromtheUniversityofMoratuwa,SriLanka,in1984andtheM.Sc.degreeincomputerintegratedmanufacturefromLoughboroughUniversityofTechnology,U.K.,in19.HeiscurrentlypursuingthePh.D.degreeatMemorialUniversityofNewfoundland,St.John’s,NF,Canada.

HewaspreviouslyaLecturerintheDepartmentofMechanicalEngineering,UniversityofMoratuwa.Hisresearchinterestsareclassical/intelligentcontrolandmanufacturingengineering.

Bao-GangHu(M’94)receivedtheM.Eng.degreein1983fromtheUniversityofScienceandTechnology,Beijing,China,andthePh.D.degreein1993fromMcMasterUniversity,Hamilton,Ont.,Canada,bothinmechanicalengineering.

From1983to1987,hewaswiththeDepartmentofMechanicalEngineering,UniversityofScienceandTechnology.In1993,hewaswithNEFAB,Inc.,Canada.From1994to1997,hewasaResearchEngineerandSeniorResearchEngineeratC-CORE(CenterforColdOceanResourcesEngineering),Memo-rialUniversityofNewfoundland,St.John’s,NF,Canada.HeiscurrentlyanAssociateProfessor,NationalLaboratoryofPatternRecognition,InstituteofAutomation,ChineseAcademicofScience,Beijing,China.Hisresearchinterestsincludefuzzycontrol,patternrecognition,andintelligentsystems.

RaymondG.Gosine(S’84–M’93)receivedtheB.Eng.(Elect.)degreein1986fromtheMemorialUniversityofNewfoundland,Canada,andthePh.D.degreein1990fromCambridgeUniversity,Cambridge,U.K.

In1990and1991,hewasaResearchAssociateinEngineering,UniversityofCambridge,andaBye-FellowofSelwynCollege,Cambridge.From1991to1993,hewastheNSERCJuniorChairofIndustrialAutomationandanAssistantProfessor,DepartmentofMechanicalEngineering,UniversityofBritishColumbia.,Vancouver,B.C.,Canada.HeisnowanAssociateProfessorofEngineeringatMemorialUniversityofNewfoundlandandistheDirectorofIntelligentSystemsatC-CORE.Hisresearchinterestsareintheareaofindustrialautomation.

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