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Adaptive Fuzzy Dynamic Surface Control for Uncertain Nonlinear Systems

By Phillip Reynolds,2014-01-26 10:02
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Adaptive Fuzzy Dynamic Surface Control for Uncertain Nonlinear Systems

    Adaptive Fuzzy Dynamic Surface Control

    for Uncertain Nonlinear Systems InternationalJournalofAutomationandComputing6(4),November2009,385390

    DOh10.1007/s116330090385z

    AdaptiveFuzzyDynamicSurfaceControlforUncertain

    NonlinearSystems

    Xiao-YuanLuoZhiHaoZhuXinPingGuan

    InstituteofElectricalEngineering,YanshanUniversity,Qinhuangdao066004,PRC Abstract:Inthispaper,arobustadaptivefuzzydynamicsurfacecontrolforaclassofuncertainnonlinearsystemsisproposed.A

    noveladaptivefuzzydynamicsurfacemodelisbuilttoapproximatetheuncertainnonlinearfunctionsbyonlyonefuzzylogicsystem.

    Theapproximationcapabilityofthismodelisprovedandthemodelisimplementedtosolvetheproblemthattoomanyapproximators

    areusedinthecontrollerdesignofuncertainnonlinearsystems.Theshortageof"explosionofcomplexity''inbacksteppingdesign

    procedureisovercomebyusingtheproposeddynamicsurfacecontrolmethod.ItisprovedbyconstructingappropriateLyapunov

    candidatesthatallsignalsofclosedloopsystemsaresemi

    globallyuniformlyultimatebounded.Also,thisnovelcontrollerstabilizes thestatesofuncertainnonlinearsystemsfasterthantheadaptiveslidingmodecontrollerfSMC).Twosimulationexamplesareprovided

    toillnstratetheeffectivenessofthecontrolapproachproposedinthispaper. Keywords:Uncertainnonlinearsystems,fuzzylogicsystem,dynamicsurfacecontrol,backsteppingdesign

    1Introduction

    Recently.interestintheadaptivecontrolofnonlinear

systemshasbeeneverincreasing.andmanysignificantde

    velopmentshavebeenachieved.Intheearlystageofthe research.undertherestrictionsinthegrowthrateofnon

    linearitiesandmatchingconditionst.adaptivecontrolal

    gorithmswerefirstdevelopedforlinearizablenonlinearsys

    temswithunknownparameters.Theserestrictionswere subsequentlyrelaxedbytheintroductionofintegratorback

    steppingdesignin[2.Inanefforttoenlargetheclassof

    uncertainnonlinearsystemsforwhichadaptivebackstep

    pingcontrolcanbedesigned,aseriesofstudieshavebeen focusedonrobustadaptivecontrolforaclassofnonlinear systemswhoseuncertaintiesincludenonlinearparametric uncertainties.uncertainnonlinearitiesaswellasunmea- rn,

    suredinputtostatestabledynamicst.

    However,adrawbackofthebacksteppingtechniqueis thephenomenonof"explosionofcomplexity"1.Thatis.

    thecomplexityofthecontrollergrowsdrasticallyastheor

    derofthesystemincreases.Theexplosionofcomplexity iscausedbytherepeateddifrerentiationsofcertainnonlin

    earfunctions.Inf581,adynamicsurfacecontrol(DSC) techniquewasproposedtoeliminatethisproblembyintro

    ducingafirstorderfilteringofthesyntheticinputateach stepofthetraditionalbacksteppingapproach.In[912,

    adaptivefuzzycontrolapproachandadaptiveneuralnet

    workcontrolmethodwereproposedforuncertainnonlinear systemswiththestrictfeedbackform.However,somenew

    shortagesappeared,thatis,toomanyapproximatorswere usedintheseresearchesandtheystillcannotbesolvedcom

    pletelyinexistingreferences.

    Inthispaper,wepresentanovelrobustadaptivefuzzy dynamicsurfacecontroldesignprocedureforaclassofan

    certainnonlinearsystems.Byapproximatingtheunknown ManuscriptreceivedJanuary9,2009;revisedMarch11.2009 ThisworkwassupportedbyNationalNaturalScienceR0undati0n ofChina(No.60525303and60704009)andKeyResearchProgram ofHebeiEducationDepartment(No.ZD200908). Correspondingauthor.Emailaddress:xylu@ysu.

    edu.cn

    nonlinearfunctionswithonlyonefuzzylogicsystem,we incorporatethedynamicsurfacetechniqueintotheexist

    ingfuzzylogicsystembasedonadaptivecontroldesign framework.Furthermore,itisprovedbyconstructingap

    propriateLyapunovfunctionsthatallsignalsofclosedloop

    systemsaresemigloballyuniformlyultimateboundedand thisproposedmethodstabilizesthestatesofuncertainnon- linearsystemsfasterthantheadaptiveslidingmodecon

    trolmethodwhichcannotdealwithunknownnonlinear functions["1.Finally.simulationexamplesareusedtoil- lustratetheemciencyofourapproach.

    Inthefollowing,lrepresents2-normofx,isestima-

    tionofx.andisdefinedas=x一岔.

    2Problemformulation

    2.1Systemformulation

    Consideraclassofsingleinputsingleoutput(SIS0)an

    certainnonlinearsystems

    』圣=)+6(1)y

    wherex=x1,.,n]T?Risthestatevector,?R

    andY?Rarethecontrolinputandoutput,respectively,

,(x)=[,1,,^,,l,,,nareunknownsmoothnon-

    linearfunctionsandsatisfy^(0)=0,b=0,,0,1.

    Assumption1.Thevectorf(x)canbeexpressedas f(x)=f0(x)+A/(x)

    wheref0(x)=[x2,,n,A0],f0()isthenominal

    partoff(),A0isanominalfunctionof,Af=

    【?,l(1),,?^n)isanuncertainsmoothfunc.

    tion,x=Ixi,?,nl?Risthestatevector,t=

    z1,?,xi]?R.,andspecially,n=x.

    Remark1.Assumption1specifiestheclassofuncer. taintiesconsideredinthispaperanditisdifferentfromthe

    assumptionin[13]inwhichAf=[0,,0,?A()In

    386InternationalJournalofAutomationandComputing6(4),November2009

    thispaperweconsideraclassof

    

    unknownuncertaintiesand

    ?.=fAll(1),,?ln()sotheresultsofthispa- peraremorepopularthanthosein[13].

    AccordingtoAssumption1,system(1)canbedescribed by

    '

    ,…一1

    Assumption2.ThesmoothfunctionsAfl?l

    {nsatisfythefollowingconditions:

    (3)

    (4)

    whereaandbaretunablepositiveparameters,1(x1)and 2(X1)areunknownnonlinearfunctions. 2.2Fuzzylogicsystemformulation

Inthissection,thedesignoffuzzylogicsystemwillbede

    scribed.Thefuzzyrulebasesarecomposedofthefollowing fuzzyrules

    R(':ifx1isand

    andxis

    thenYisG(5)

    whereisafuzzysetof?RandGisafuzzysetof?

    R.x?(Xl,z2,???,xn)?U1×U2×...×nandY?Vare

    theinputandoutputofthefuzzylogicsystem,respectively,

    f=1,2,?,M,Misthenumberoffuzzyrules.Using thesingletonfuzzifier,theproductinferenceengine, center

    defuzzifierandGaussianmembershipfunctions, weobtain

    theoutputofthefuzzysystemintheform: h(x)

    ?兀F()

    Z=ii=1

    Mn

    ?兀UFZ(Xi)

    (6)

    where01=maxE:RGf(z).Thefuzzybasisfunctionis definedas

    uF()

    E音—一

    ?兀?(xi)

    Sothefuzzylogicsystemcanbedescribedasfollows (7)

    (1)=oTEX1)+A(z)(8)

where0=(01,02,,0M),Ex1)(E1,E2,EM)

    ?(X1)istheapproacherrorandsatisfieslA(X1)l<

    whereisapositiveconstant.

    3Observerdesignanddynamicsurface controllerdesign

    3.1Systemobserverdesign

    Designanobserverwiththeform {2i=:39ui++1+ki1.=1.',_n1(9)

    whereistheestimateofx,isdefinedas=.and

    1,???,narechosenasconstantstomake :O

    

    向一10

    

    0

    1

    O

    aHurwitzmatrix.

    Considering(3)and(9),wecanobtain =A+f(x)

    where/(z)=[All(1),?,?A?1(

    Accordingto(4),wehave

    (z)I

    ),(.)]T

    (10)

    (11)

    3.2Adaptivedynamicsurfacecontroller design

    Accordingto(3)and(9),wecanobtain .

(12)

    Then,theprocedureofsystemcontrollerdesignisdescribed

    a8follows

    Step1.Designthedynamicsurfacefunctionandfirst.. orderfilterasfollows

    S1=Xl(13)

    a+=?KIssoTE()+](14)

    whereK1isthecontrolgain,7-1isthetimeconstant, and

    theadaptivelawsof0and5are

    rr(15)

    whereF=F,>0.andr>0areadaptivecontrol parameters.

    Stepi.Wheni=2,-nl,wedesignthedynamic

    surfacefunctionandfirstorderfilterasfollows

    Si=.oLt1

    ri&i+Q=K&一圣1+a1

    (16)

    (17)

    whereKsisthecontrolgain,andistimeconstant. Stepn.Designthedynamicsurfacefunctionandcon

    trollerasfollows

    Sn=岔一一l

    =

    KnSnkn1+a几一1

    (18)

    (19)

    whereKnisthecontrolgain,andTnistimeconstant. Remark2.Inthecontrollerdesignprocedures,wein-

    troducedonlyonefuzzyapproximator,sothecontrollerde signprocedureissimplerthanthosein[6,7,12inwhichn

    fuzzyapproximatorswereintroduced.Moreover,thepro-

    posedmethodinthispapercanalsoavoidthephenomenon

    withrespecttotheexplosionofcomplexityinthebackstep-

    pingdesignproceOure- ^

    

    +n}

    )

    .

    )

    

    ,

    n11=

    ?

    ?

    (

    ?

    ,???,(,?【

    

    +

    

    ?

    2

    ?

    2

    )r

    

,E

    r

    =li

    ^^n:r.0

    ,??,,

    x.Luoeta1./AdaptiveFuzzyDynamicSurfaceControlforUncertainNonlinearSystems

    4Stabilization Introduceanewvector=f1,2,???,Zn--1]asfollows

    Zl:--7"1(}

    ai+K~Si],t.

    ,

    n.

    +1i1.

    Accordingto(13),(16),and(18),onecaD_obtain

    s:s.+zK1szfE(z)+]+ 2+Afl@1)

    where

    =Si+l+?KiSi,i=2,3,?,几一1 =KnS

    (21)

    (22)

    (23)

    1:+K1+圣『TE(z1)+1+ z

    [OWE川筹针]=(24)

    

    +(S11slzl,,/7_1\ (s,,,)=

    Iqg+[TE(z)+]+

z)OE?

    (25)

    andrh(S1,?)Sn,Z1,',Zi).+1+1/1

    arecontinuousfunctions. Theorem1.FortheSIS0uncertainnonlinearsystem

    (11satisfyingAssumptions1and2,withobserver(9),dy namicsurfacefunctions(13),(16),and(18),firstorderill- ters(14)and(17),adaptivelaws(15),andthecontrollaw

    (19),thereexistK,ki,,r,,andr,i=1,2,',such

    thatallsignalsoftheclosedloopsystemsaresemiglobally uniformlyultimateboundedonthecompactset

    :

    {Li=1

    Thetimederivativeofis :

    面童+2TP()?

    

    +T+9IIPII

    P

    lI

    1I

    (z)1I

    1?I

    

    1-

    )+2pllPll耋础)+b2X2

    

    52

    2(z1)].

    387

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