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Stable adaptive fuzzy control of nonlinear systems using small-gain theorem and LMI approach

By Sandra Holmes,2014-02-18 23:09
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Stable adaptive fuzzy control of nonlinear systems using small-gain theorem and LMI approachof,using,fuzzy,Using

    Stable adaptive fuzzy control of nonlinear systems using small-gain theorem and LMI

    approach

    JControlTheoryAppl20108(4)527532

    DoI10.1oo7,s1176801080350

    Stableadaptivefuzzycontrolofnonlinearsystems

    .

    ;mall'theorcandLMIuslnRsmall-gaintheoremanaLIVIIapproach'

    HaiboJIANG,JianjiangYU,CaigenZHOU.

    (1.SchoolofMathematics,YanchengTeachersUniversity,YanchengJiangsu224051,China;

    2.SchoolofInformationScienceandTechnology,YanchengTeachersUniversitBYanchengJiangsu224002,China)

    Abstract:Anewdesignschemeofstableadaptivefuzzycontrolforaclassofnonlinearsystemsisproposedinthis

    paper.TheT-Sfuzzymodelisemployedtorepresentthesystems.First,theconceptoftheso

    calledparalleldistributed

    compensation(PDC)andlinearmatrixinequality(LMI)approachareemployedtodesignthestatefeedbackcontroller

    withoutconsideringtheerrorcausedbyfuzzymodeling.Su

    cientconditionswithrespecttodecayrateotarederivedin

    thesenseofLvapunovasymptoticstability.Finally,theerrorcausedbyfuzzymodelingisconsideredandtheinputto

    statestable(iss)methodisusedtodesigntheadaptivecompensationtermtoreducetheeffectofthemodelingerror.By

    thesmallgaintheorem.theresultingclosedloopsystemisprovedtobeinputto

statestable.Theoreticalanalysisverifies

    thatthestateconvergestozeroandallsignalsoftheclosed

    loopsystemsarebounded.Theeffecfivenessoftheproposed controllerdesignmethodologyisdemonstratedthroughnumericalsimulationonthechaotic

    Henonsystem.

    Keywords:Nonlinearcontrol;Fuzzycontrol;Adaptivecontrol;Smal1.

    gaintheorem;Input--to?statestability

    1lntroduction

    Inrecentyears,fuzzylogiccontrolofnonlinearsystems hasreceivedmuchattention[1,21.Usingtheapproximation capabilitvOffuzzysystems,variousstableadaptivecontrol schemesforaclassofnonlinearcanonicalsystemswere proposedinf1,3,4.Amongvariouskindsoffuzzymeth

    ods.fuzzy.mode1.basedcontrO1iswidelyusedbecausethe designandanalysisoftheoverallfuzzysystemcanbesys

    tematicallyperformedusingthewellestablishedclassical

    1inearsystemstheory5?7.Thestabilityanalysisandde

    signoffuzzycontrolsystemsbasedonT-Sfuzzymodel werediscussedin[571.In[71,arelaxedquadraticsta

    bilityconditionoffuzzycontrolsystemswasderived.The robustfuzzycontrolproblemforT-Sfuzzymodelsystems withparametricuncertaintieswasstudiedin8,9.How.

    ever,theuncertaintiesarerequiredtosatisfycertainmatch

    ingconditions.Becausetheeflfectofapproximationerror duetofuzzymodelinghasnotbeenconsidered,theresults in591areonlyusefulforthesystemsthatarerepresented bysfuzzymode1.In[10],amixedH2/H..fuzzyOHtput feedbackcontroldesignmethodfornonlinearsystemswas introducedandthemodelingerrorwasconsidered.How

    ever,theupperboundoftheapproximationerrorisneeded

    inthemethod.whichisdifficulttobedeterminedinprac. tice.Therefore.itisimportanttoconsiderthefuzzymod. elingerrorinfuzzymode1basedcontroltechniqueandthe

    fuzzymodelingerrorneednotberequiredtosarisfycer- tainconstraintormatchingconditions.Inrecentyearsmany effortshavebeendevotedtoanalyzeanddesignastable adaptivecontroilerwiththefuzzymodelingerrorconsid. ered[11N171.In11and[121,arobustH..fuzzycontrol

    formultipletime.delayuncertainnonlinearsystemsbased onfuzzymodelwasproposed.wheretheRBFneuralnet. workwasusedtoeliminatetheinfluenceofthefuzzymod

    elingerroranduncertainties.

    BesidestheLyapunovmethod,theISSapproachisalsoa goodtooltoanalyzeanddesignthestabilityofthecontrol systems.TheISStheorywasfirstproposedbySontagl8]

    andthesmall.gaintheoremwasgivenin19].In[20and

    21,anoveldirectrobustadaptivefuzzycontro1scheme wasproposedbyusingISSapproachandsmall?-gaintheo-- rein,wheretheT-Sfuzzymodelwasusedtoapproximate thenonlinearunknownfunction.

    BycombiningthesmallgaintheoremandLMIapproach.

    anewdesignschemeofstableadaptivefuzzycontrolfora classofnonlinearsystemsisproposedinthisPaper.TheT- Sfuzzymodelisemployedtorepresentthesystems.First, theconceptoftheso..calledPDCandLMIapproachareem.. ployedtodesignthestatefeedbackcontrollerwithoutcon. sideringtheerrorcausedbyfuzzymodeling.Sufficientcon. ditionswithrespecttodecayrateolarederivedinthesense OfLvapunovasymptoticstability.Finally,theerrorcaused byfuzzymodelingisconsideredandtheISSmethodisused

    todesigntheadaptivecompensationtermtoreducetheef- fectofthemodelingerror.Bysmallgaintheorem,there

    sultingclosed--loopsystemisprovedtobeinput--to--statesta-- ble.Theoreticalanalysisveiltiesthatthestateconvergesto zeroandallsignalsoftheclosedloopsystemsarebounded.

    Themainadvantagesofourschemeinclude:l1themodel

    ingerrorneedsnotsarisfyanymatchingconditionandthe upperboundofthemodelingerrorneedsnotbeknown;21

    line. onlyonelearningparameterneedstobeadaptedon

    TherestofthePaperisorganizedasfollows.InSection3. theproblemformulationandthebasicassumptionsarepre

    sented.InSection4,anovelstableadaptivefuzzycontroller basedonsmal1.gaintheoremandLMIapproachisgiven.In Received11March2008;revised13January2009.

    ThisworkwaspartlysupposedbytheNaturalScienceFoundationoftheJiangsuHigherEduc

    ationInstitutionsofChina(No.07KJB510125

    08KJD510008),andtheNaturalScienceFoundationofYanchengTeachersUniversity(No.0

    7YCKL062,08YCKL053).

    ?

    SouthChinaUniversityofTechnologyandAcademyofMathematicsandSystemsscience,c

    AsaIldSpringer-VerlBerlinHeidelbe2010

    528HJIANGeta1.

    /JControlTheoryAppl20108(4)527532

    Section5,simulationresultsaredemonstratedtoshowthe effectivenessoftheproposedmethod.Conclusionisdrawn inSection6.

    NotationThroughoutthispaper,thevectornormof x?isEuclidean,ie=.ForamatrixP,

    P>0(P<0)denotesPisasymmetricpositive(neg

    ative)matrix,Amin(P)denotesthesmallesteigenvalueof

P.

    2Mathematicalpreliminaries

    Afunction:++isofclassKifitiscontinu.

    OUS,strictlyincreasingandiszeroatzero.Afunctionis ofclassKooifadditionally,itisunbounded.Afunction: R+×+}+isofclassKiffs,t)isofclassKfor

    everyt?0and(8,t)0ast__+o(3.

    Consideranonlinearsystem

    =

    f(x,);??;?m.(1)

    Definition118~20System(1)issaidtobeISSif

    thereexistafunctionofclassKL,andafunctionof classKsuchthat

    x(t)lI?~(1lx(O)ll,t)+7(1lull~),vt?o.(2)

    theunknowncontinuousfunctioncontrolgainandf1?

    gmin>0,1Sthecontrolinput.

    Thecontrolobjectiveistodesignastableadaptivefuzzy controllersuchthat:1,theresultingclosedloopsystemis

    ISSandthestateofthesystemconvergestozero;21allsig. nalsoftheclosed.1oopsystemsarebounded.

    AfuzzydynamicmodelhasbeenproposedbyTakagi andSugenotorepresentthenonlinearsystemfll.ThisT-S fuzzymodelcombinesthefuzzyinferenceruleandthelocal linearstatemode1.

    PIantrulez

    IFzl(t)isandz2(t)is,,

    andZg()is

    THEN(t)=Aix(t)+Biu(t)+Q(Aa~x(t)

    +Abiu(t)),

i=1,2,,r,(12)

    wherez(t)=zx(t),2(t),,zg(t)】?R9isthepremise

    variable,isfuzzysets,risthenumberoftherules, Ai??凡×他andB?Rn×areconstantmatriceswhich areofthefollowingform:

    whereIlull~isthe..nOITUofu,restrictedto[0,tf]. Definition2[18~20ACfunctionVissaidtobean ISSLyapunovfunctionforsystem(1)if 1)thereexistfunctions1,2ofclassKo.suchthatA ~l(1lxl1)?Y(x)?~(1lxl1),Vx?,(3)

    2)thereexistfunctionsog3,4ofclassKsuchthat Or(),(

    ,)?一3(1lxl1)+4(IIII),(4)

    andthen,itholdsthatonemaypickanonlinear?gain

    oftheform

    -y(s)=QooL2.i.4(s),Vs>0.(5)

    Theorem1(small-gaintheorem[19,20])Considerfl systemincompositefeedbackform

    .

    1圣乞'I

    2

    f(x,w)

    H();(6)

    :

    {

    oftwoISSsystems.Inparticular,thereexistfunctions

    .8?18:ofclassKLandfunctions1lIofclassKsuch

    that

    l1((;w,))1I?(Itxll,t)+(1l(t)II..),(8)

IIK(Y(y;,t))ll?(1lyll,t)+()_I..),(9)

    jf

    ((s))<s,Vs>o

    andthen,thesolutiontothecompositesystems(6)

    isISS.

    (10)

    and(7)

    3Problemstatementandbasicassumptions

    010.

    001.

    000.

    000.

    ai1ai2ai3. .

    00

    ?

    00

    .

    10

    .

    01

    .

    ai(n1)ain 0

    0

    :

    ?

    0

    0

    6t

AlsoQ=[0,,0,1Aai=[AaAa,Aai】?

    andAbi?denotetheapproximationerrorwitll f(x)=?hi(z(t))(ai+Aai)x(t),

    g(x)=?(())(6+Abi)

    t==1

    Byusingthefuzzyinferencemethodwithasingleton

    fuzzification,productinference,andcenteraveragedefuzzi

    fication,theoverallfuzzymodelisofthefollowingform:

    r

    ()=?hi(z(t))[Aix(t)+Biu(t)+Q(Aaix(t) i=1

    where

    Considerthenonlinearsysteminthefollowingform:

    Xi+l

    ,

    ~lj,)

    I:f(x)+g(x)u,4

    where=(Xl,2,,Xn)T??isthesystemstate

    vector,f(x)istheunknowncontinuousfunction,9(x)is

    +?6iu())],

    hi(z(t))=t((t))

    r

    ?

    i=1

    i(z())

    t(z(t))gMj(())

    J=1

    (13)

    r

Weassume(z())?oand?i(())>o,i==

    1

    2,?,r,forallt.Therefore,weget

    (z(t))?0,

    Fuzzy?-model??basedcontroldesign Thesystem(11)canberepresentedbyfollowingtheT-S

    fuzzymodelwithoutconsideringthemodelingerror. r

    2

    1

    lI

    t

    =

    @

    ?

    JIANGela1./JControlTheoryAppl20108(4)527-532 Plantrulei

    IFzl(t)isMiandz2(t)is,,

    andZg()is

    THEN()=A~x(t)+(),

    i=1,2,.,r.(14)

    Byusingthefuzzyinferencemethodwithasingleton fuzzification,productinference,andcenteraveragedefuzzi

    fication,theoverallfuzzymodelisofthefollowingform:

    r

    e(t):Ehi(z(t))[Aix(t)+BMt)].(15) i=1

    BasedontheideaofPDC,thestatefeedbackcontroller isdesignedasfollows:

Plantrulei

    IFzl(t)isandz2(t)is,,

    andzg(t)is

    THENUeq()=Kix(t),

    i=1,2,?,r.(16)

    Byusingthefuzzyinferencemethodwithasingleton fuzzification,productinference,andcenteraveragedefuzzi

    fication,theoverallfuzzymodelisofthefollowingform: r

    Ueq(t)=一?hi(z(t))Kix(t).(17)

    i=1

    Substituting(17)into(15),wehave

    rr

    (t)=EEi(z())j(z(t))【—Bt()

    i=1J=1

    =

    Eh~(z(t))Giix(t)+2?hi(z(t))h5(())

    i=11?<j?

    .

    [(t),(18)

    whereGij=AiBi,i,J=1,,7'.

    Inordertoderivethestabilityconditionsofthefuzzysys? tem(18),nowwepresentthefollowingdefinition. Definition3Thefuzzysystem(18)isgloballyasymp

    toticallystablewithd

    .

    ecayrateoL,ifthereexistsascalar

    >0,suchthat())?-2aV(x(t)),wherethe

    Lyapunovfunctioncandidateis((t))':xT()P(t), Theorem2Thereexistsfuzzycontroller(17)sothat

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