IEEE论文-An approach to adaptive control of fuzzy dy

268IEEETRANSACTIONSONFUZZYSYSTEMS,VOL.10,NO.2,APRIL2002AnApproachtoAdaptiveControlofFuzzyDynamicSystemsGangFengAbstract—Thispaperdiscussesadaptivecontrolforaclassoffuzzydynamicmodels.Theadaptivecontrollawisfirstdesignedineachlocalregionandthenconstructedinglobaldomain.Itisshownthattheresultingfuzzyadaptivecontrolsystemisgloballystable.Robustnessissuesoftheadaptivecontrolsystemarealsoaddressed.Asimulationexampleisgivenfordemonstrationoftheapplicationoftheapproach.IndexTerms—Adaptivecontrol,fuzzymodeling,nonlinearsys-tems,stability.I.INTRODUCTIONSINCEthefirstpaperonfuzzysets[1]waspublished,fuzzylogiccontrolhasattractedagreatattentionfromboththeacademicandindustrialcommunities.Manypeoplehavede-votedagreatdealoftimeandefforttoboththeoreticalresearchandimplementationtechniquesforfuzzylogiccontrollers.MuchprogresshasbeenmadeinsuccessfullyapplyingFLCinindustrialcontrolsystems[2]–[5].Duringthepastacoupleofyears,manysystematicfuzzycon-trollerdesignmethodshavebeendeveloped[6]–[12]basedontheTakagi–Sugeno(T–S)model,orthefuzzydynamicmodels.Thebasicideaofthesemethodsis:i)torepresentthecomplexnonlinearsysteminafamilyoflocallinearmodels,eachlinearmodelrepresentsthedynamicsofthecomplexsysteminonelocalregion;ii)toconstructaglobalnonlinearmodelbyaggre-gatingallthelocalmodelsthroughthefuzzymembershipfunc-tions.Theprimaryadvantageofthismodelisthatthecontrollerdesigncanbemainlybasedoneachlocalmodel,whichismucheasierthanthatfornonlinearsystemsintheglobalregion,andthentheglobalcontrollercanbeconstructedfromthelocalcon-trollers.Ithasalsobeenshownthatfuzzysystemscanapproximateanynonlinearfunctionsoveraconvexcompactregion[13].Basedonthisobservation,anumberofattemptshavebeenmadetousefuzzylogicforadaptivecontrolofnonlinearsystems[14]–[17].Thebasicideaofmostoftheseworksistousefuzzybasisfunctionstoapproximatetheunknownnonlinearfunctionsandupdatetheconstantparametersofthefunctiononlineandthenimplementadaptivecontrolusingtheconventionalcontroltechnology.Theauthorin[18]recentlyproposedamodel-basedfuzzycontrolaswellasadaptivecontrolmethod.Inthispaper,wewilldevelopanadaptivecontroldesignmethodforaclassoffuzzydynamicmodels.ThebasicideaisManuscriptreceivedApril18,2001;revisedAugust27,2001andOctober3,2001.ThisworkwassupportedbytheCityUniversityofHongKong(SRG7001245).TheauthoriswiththeDepartmentofMEEM,CityUniversityofHongKong,Kowloon,HongKong(e-mail:[e-mail]).PublisherItemIdentifierS1063-6706(02)02971-5.todesignanadaptivecontrollerineachlocalregionandthenconstructtheglobaladaptivecontrollerbysuitablyintegratingthelocaladaptivecontrollerstogetherinsuchawaythattheglobalclosed-loopadaptivecontrolsystemisstable.Therestofthepaperisorganizedasfollows.SectionIIformu-latesthefuzzysystemmodeling,SectionIIIpresentsthefuzzyadaptivecontroldesignandstabilityproof.Robustnessissuesarealsodiscussedinthesection.OneexampleofsimulationispresentedinSectionIV,whichisfollowedbyconcludingre-marksinSectionV.II.FUZZYSYSTEMMODELINGManyphysicalsystemsareverycomplexinpracticesothattheirrigorousmathematicalmodelscanbeverydifficulttoobtainifnotimpossible.However,manyphysicalsystemscanindeedbeexpressedinsomeformofmathematicalmodelslocally,orthosesystemscanbeexpressedasanaggregationofasetofmathematicalmodels.Variousfuzzymodelshavebeenproposedinthelastfewyears,see,forexample,[3],[6].Here,weconsiderusingthefollowingfuzzymodeltorepresentacomplexsingle-input–single-output(SISO)systemthatincludesbothfuzzyinferencerulesandlocalanalyticlinearmodels:IFisANDisTHEN(2.1)wheredenotesthe-thfuzzyinferencerule,thenumberofinferencerules,arefuzzysets,thesysteminputvariable,theoutputofthesystem,arecoefficientsofthethsubsystem,andsomemeasurablesystemvariables.Themodel(2.1)canalsobedescribedinthestatespaceform.Let...(2.2)then,themodel(2.1)becomesIFisANDisTHEN(2.3)1063-6706/02$17.00©2002IEEEIEEETRANSACTIONSONFUZZYSYSTEMS,VOL.10,NO.2,APRIL2002269where...............Letbethenormalizedmembershipfunctionofthein-ferredfuzzysetwhere(2.4)and(2.5)Itisnotedthatthemodelin(2.1)onlyrepresentsthepropertiesofthesysteminalocalregionandthusisreferredtoasthefuzzydynamiclocalmodel.Inordertoclarifytheconceptofthelocaldynamicmodels,weintroducethefollowingdefinition.Definition2:1:Thethfuzzylocaldynamicmodelisabi-narysetdefinedasFLDMorFLDMwhereorrepresentsthecrispinput–outputrelationshipordynamicpropertiesofthesystematthecrisppoint.Byusingastandardfuzzyinferencemethod,thatis,usingasingletonfuzzifier,productfuzzyinferenceandcenter-averagedefuzzifier,thefollowingdynamicglobalfuzzymodelcanbeobtained:(2.6)whichcanalsoberewrittenas(2.7)Itshouldbenotedthatthemembershipfunctions’sareingeneralthefunctionofthesystemoutputanditsderivatives,andthusthesystemdescribedin(2.6)or(2.7)isanonlinearsystemingeneral.Thestate-spaceformof(2.7)canbeexpressedas(2.8)Itisnotedthatarethesame.Inthispaper,wearegoingtodiscussthedesignofcontrolsystemsfor(2.7)whenthecoefficientsoftheplantareunknown.Itisassumedthatthemembershipfunctionshavebeenchosenaprioribasedontheexpert’sknowledgeortheplantdata.Theobjectiveofthefuzzyadaptivecontrolistofin