Control Synthesis of T–S Fuzzy Systems Based

ControlSynthesisofT–SFuzzySystemsBased

onaNewControlScheme

JiuxiangDongandGuang-HongYang,SeniorMember,IEEE

Abstract—ThispaperstudiesthecontrolsynthesisproblemofTakagi–Sugeno(T–S)fuzzysystems.Bysplittingthepremisevari-ablespacesandusingthepropertiesoffuzzysets,anewcontrolschemeisproposedbasedonanewclassoffuzzyLyapunovfunc-tions,andaconvexconditionfordesigningfuzzycontrollersisgiven,wherethenewfuzzyLyapunovfunctionsandfuzzycon-trollersareconstructedbasedonthesplitsubspaces.Inparticular,someexistingfuzzyLyapunovfunctionsandcontrolschemesarespecialcasesofthenewLyapunovfunctionandcontrolscheme,respectively.Numericalexamplesaregiventoillustratetheeffec-tivenessoftheproposedmethod.

IndexTerms—Fuzzycontrol,fuzzyLyapunovfunction,linearmatrixinequalities(LMIs),nonlinearsystems,Takagi–Sugeno(T–S)fuzzymodels.

I.INTRODUCTION

I

NTHENONLINEARcontrolarea,thereisnosystem-aticmathematicaltechniquetoobtainnecessaryandsuf- cientconditionstoguaranteethestabilityandperformanceofnonlinearsystems.Ingeneral,controlofnonlinearsys-temsisoftenverydif cult,andvariouscontrolmethodshavebeenexploitedforthenonlinearcontrolsystems[1].Inpartic-ular,animportantapproachtononlinearcontrolsystemde-signistomodeltheconsiderednonlinearsystemsasTak-agiandSugeno(T–S)fuzzysystems,whicharelocallylin-eartime-invariantsystemsconnectedbyIF–THENrules.Asaresult,theconventionallinearsystemtheorycanbeap-pliedforanalysisandsynthesisofthenonlinearcontrolsys-tems.Inrecentyears,controlsynthesisproblemsofT–Sfuzzysystemshavebeenwellstudied,wherequadraticLyapunovfunctionapproaches[2]–[8]arewidelyemployed.SinceacommonLyapunovmatrixisusedforalllocalmodelsoffuzzysystems,thequadraticLyapunovfunctionapproachof-

ManuscriptreceivedApril25,2010;revisedSeptember3,2010;acceptedNovember10,2010.DateofpublicationDecember3,2010;dateofcurrentversionApril4,2011.ThisworkwassupportedinpartbytheFundsforCreativeResearchGroupsofChinaunderGrant60821063,National973Pro-gramofChinaunderGrant2009CB320604,theFundsofNationalScienceofChinaunderGrant60904010andGrant60974043,the111Projectun-derGrantB08015,theFundamentalResearchFundsfortheCentralUniversi-tiesunderGrantN090404016,ChinaPostdoctoralScienceFoundationunderGrant20100470074,andthePostdoctoralScienceFoundationofNortheasternUniversity,China.

TheauthorsarewiththeCollegeofInformationScienceandEngineer-ingandKeyLaboratoryofIntegratedAutomationofProcessIndustry(Min-istryofEducation),NortheasternUniversity,Shenyang110819,China(e-mail:dongjiuxiang@http://www.77cn.com.cn;dong_jiuxiang@http://www.77cn.com.cn;yangguanghong@http://www.77cn.com.cn;yang_guanghong@http://www.77cn.com.cn).

Colorversionsofoneormoreofthe guresinthispaperareavailableonlineathttp://www.77cn.com.cn.

DigitalObjectIdenti er10.1109/TFUZZ.2010.2096470

tenleadstoconservativeresults.Then,parameter-dependentLyapunovfunctions(orcalledfuzzyLyapunovfunctions)[9]–[12],piecewiseLyapunovfunctions[13],[14],andk-samplevariationLyapunovfunctions[15]are,respectively,proposedtoreducetheconservatismintroducedbyusingquadraticLyapunovfunctions.InmostofthefuzzycontroldesignsbasedonT–Sfuzzymodels,theparalleldistributedcompensation(PDC)controlschemein[16],i.e.,thecontrollersharesthesamefuzzyruleswiththeconsideredfuzzymodel,playsanimportantrole.Inaddition,anumberofalternativecontrolschemes,suchasthenon-PDCcontrolschemein[11],theswitchingconstantcontrollergainschemein[17],andtheswitchingPDC(SPDC)controlschemein[18]and[19],arealsodevelopedfordesigningfuzzycontrollers.

Althoughmanyimportantprogresseshavebeenachievedinthefuzzycontrolarea,thepropertiesaboutthestructureorshapeofmembershipfunctionsareoftenneglectedinsomeoftheliterature.Therefore,agreatdealofefforthasrecentlybeendevotedtoexploitingthepropertiesaboutthestructureorshapeofmembershipfunctionsforless-conservativeresults[7],[20]–[30].In[24],byusingtheknowledgeofthemembershipfunctions’shapetointroduceslackvariables,relaxedstabil-ityconditionsarepresented.AsystematicdesignapproachofT–Sfuzzycontrolsystemsispresentedbysearchingacommonpositive-de nitesymmetricmatrixineachmaximaloverlapped-rulesgroupoffuzzyrulesin[28].Byseparatingtheoriginalplantrulesintoseveralfuzzyregions,theT–Sfuzzyregioncontrolapproachandtheregional-membership-function-shape-dependentapproachare,respectively,proposedin[25]and[30].Byexploitingthedependenceofthestabilityuponmembershipfunctions,afuzzycontroldesignapproachisgivenbasedonKharitonov’stheoremin[23].In[26],aline-integralfunctionisintroducedasafuzzyLyapunovfunctionwithouttheassociationwiththetimederivativesofmembershipfunctions,andthen,re-laxedstabilityconditionsareachieved.Moreover,byexploitingthepropertiesofmultidimensionalfuzzysummation,relaxedstabilityanalysisandsynthesisconditionsareproposedin[6],[7],and[31],andanasymptoticallynecessaryandsuf cientconditionisachievedin[6].Inparticular,Caoetal.[20]–[22] rstpartitionthepremisevariablespaceintosomesubspacesbyfuzzymembershipfunctions,andthen,somerelaxedsta-bilityanalysisandsynthesisconditionsarepresentedbasedonpiecewiseLyapunovfunctions.

Motivatedbytheseworks,wherethepropertiesaboutthestructureorshapeofmembershipfunctionsareexploitedforless-conservativeresults,thispaperwillfurtherstudythefuzzycontroldesigntechniquebyexploitingsomenewprop-ertiesaboutthestructureandshapeofmembershipfunctions.

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Accordingtotheroleoffuzzysets,thepremisevariablespaceissplitintoasetofsubspaces,wherethereisoneandonlyonefuzzysetplayingadominantroleoneachpremisevariablevj-axis.Byswitchingtheparametersofaclassofmatrixfunctionswithsomespecialconstraintsbetweenthesplitsubspaces,thecontinuityoftheclassofmatrixfunctionscanbeguaranteed.Further,byusingtheclassofmatrixfunctions,anewfuzzyLya-punovfunction(whichareknownasdominantfuzzyLyapunovfunctions)andanewcontrolschemecanbeobtained.Then,theproposedtechniquecancontinuouslyswitchLyapunovfunc-tionsandcontrolgainsbasedontheroleofthedifferentfuzzysets,whichhasthepotentialtogivele …… 此处隐藏：45511字，全部文档内容请下载后查看。喜欢就下载吧 ……