DOC

Delay-dependent Criteria for Robust Stability of Uncertain Switched Hopfield Neural Networks

By Brittany Arnold,2014-02-18 01:57
11 views 0
Delay-dependent Criteria for Robust Stability of Uncertain Switched Hopfield Neural Networks

    Delay-dependent Criteria for Robust

    Stability of Uncertain Switched Hopfield

    Neural Networks

    InternationalJournalofAutomationandComputing04(3),July2007,304-314 DOI:10.1007/s11633-0070304-0

    Delay..dependentCriteriaforRobustStabilityof

    UncertainSwitchedHopfieldNeuralNetworks

    XuYangLouBao-TongCui

    CollegeofCommunicationandControlEngineering,JiangnanUniversity,1800LihuRd.,Wuxi214122,PRC

    Abstract:Thispaperdealswiththeproblemofdelay-dependentrobuststabilityforaclassofswitchedHopfieldneuralnetworkswith

    time-varyingstructureduncertaintiesandtime-varyingdelay.SomeLyapunov-Kraso~skiifunctionalsal'econstructedandthelinear

    matrixinequality(LMI)approachandfreeweightingmatrixmethodareemployedtodevisesomedelay-dependentstabilitycriteria

    whichguaranteetheexistence,uniquenessandglobalexponentialstabilityoftheequilibriumpointforalladmissibleparametric

    uncertainties.ByusingLeibniz.Newtonformula,freeweightingmatricesaxeemployedtoexpressthisrelationship,whichimpliesthat

    thenewcriteriaarelessconservativethanexistingones.Someexamplessuggestthattheproposedcriteriaareeffectiveandareall

    improvementoverpreviousones.

    Keywords:Delay-dependentcriteria,robuststability,switchedsystems,Hopfieldneuralnetworks,timevaryingdelay,linearmatrix

    inequality.

1Introduct[on

    Inthepasttwodecades,theanalysisofneuralnetworks hasbeenasubjectofgreatpracticalimportancethathas attractedagreatdealofresearchinterest,andhasbeen foundapplicationsinmanyareassuchassignalprocess? ing,patternrecognition,staticimageprocessing,etc.Such applicationsheavilydependonthedynamicalbehaviors. UPtonow,mostworksondelayedneuralnetworkshave focusedonthestabilityanalysisproblemforneuralnet. workswithconstantortime-varyingdelays.Sufficientcon. ditions,eitherdelay.dependentordelay.independent,have beenproposedtoguaranteetheasymptoticalorexponential stabifityfortheneuralnetworkst1.Manyofthecondi.

    tionsebasedonlinearmatrixinequality(LMI)techniques whichhavebeensuccessfullyusedtotacklevariousstabil. ityproblemsforneuralnetworkswithtimedelays[2,11--161. ThemainadvantageoftheLMI.basedapproachesisthat theLMIstabilityconditionscanbesolvednumericallyus. ingtheeffectiveinteriorpointalgorithmt"1.

    Ontheotherhand.parameterfluctuationinneuralnet. workimplementationonverylargescaleintegration(VLSI) chipsisalsounavoidable.Thisfactimpliesthatagood neuralnetworkshouldhavecertainrobustnesswhichpaves thewayforintroducingthetheoryofintervalmatricesand intervaldynamicstoinvestigatetheglobalstab[1[tyofin? tervalneuralnetworks.Thereex[stseveralrelatedresults onrobuststability[.

    Aclassofhybridsystemsthathasattractedsignificant attention.becauseitcanmodelseveralpracticalcontrol

problemsthatinvolvetheintegrationofsupervisorylogic

    ManuscriptreceiveddateNovember6,2006;reviseddateApril12, 2007

    ThisworkissupportedbytheNationalNaturalScienceFoundation ofChina(No.60674026)andtheKeyResearchFoundationofScience andTechnologyoftheMinistryofEducationofChina(No.107058). C0rresp0ndingauthor.E-mailaddress:btcui~vip.sohu.COrn basedcontrolschemesandfeedbackcontrolalgorithms, istheclassofswitchedormultimodal1systems[.For thisclass,resultshavebeendevelopedforstabilityanal? ysisusingthetoolsofmultipleLyapunovfunctionsfor lineari.andnonlinearsystems[.andtheconceptof

    dwel1.time[;thereadermayreferto[291forasurveyof

    resultsinthisarea.Theseresultshavemotivatedthedeve1. opmentofmethodsforcontrolofvariousclassesofswitched systems[....

    Despitemuchprogress.significantresearch

    remaiastobedoneinthedirectionofnonlinearandcon. strainedcontrolofswitchedsystems.

    ForswitchedHopfieldneuralnetworks.basedonthe Lyapunov?KrasovskiimethodandLMIapproach.Huang.J

    derivedsomeconditionsforglobalexponentialstabilityof aclassofuncertainswitchedHopfieldneuralnetworkswith time-varyingdelay.However,theproposedcriteriain[33] areonlyapplicabletosystemswithadmissiblesometime delay.Asiswellknown.ifthetimedelayisactuallysmall, thedelay.independentconditionstendtobeconservative. Inaddition,forneuralnetworkswithtime.varyingdelay, theglobalrobustexponentialstabilitycriteriaproposedin f331arenotapplicableforsystemswiththederivativeof

    time-varyingdelaybeinglargerthanone.Moreover,Huang didnotusethefree.weightingmatrixmethod[00which

    hastwoadvantages.First.itdealswiththesystemmodel directlyanddoesnotemployanysystemtransformation, thusavoidingtheconservatismthatresultsfromsucha transformation;second.itdoesnotuseanyinequalitiesor theimprovedinequalitiestoestimatethecrosstermsl. Motivatedbytheabovediscussion,theaimsofthispaper aretostudyalargerclassofswitchedHopfieldneuralnet. workswithtime-varyingstructureduncertaintiesandtime- varyingdelaybyintegratingthetheoryofswitchedsystems withneuralnetworks.Afree-weightingmatrixapproach combinedwithLMItechniquesisemployedtoderivesome criteriaforglobalrobustexponent[alstab[1[tyofneuralnet.

x.Y.LouandB.T.Cui/Delay-dependentCriteriaforRobustStabilityofUncertainSwitchedH

    opfieldNeuralNetworks305

    workswithtime-varyingdelay.Theproposedmethodnot onlyislessconservativethanthosein331,butalsoallows

    thederivativeofthetime-varyingdelaytotakeanyvalue. Therestofpaperisorganizedasfollows.Themathemat. icalmodelofalargerclassofuncertainswitchedHopfield neuralnetworks(uSHNN)withtime-varyingdelayispro- posedbyintroducingthetheoryofswitchedsystemsinto neuralnetworksandsomepreliminariesaregiveninsection 2.Insection3.somelessconservativeandlessrestrictive conditionsintermsofLMIsareestablishedfortheclass ofuncertainswitchedHopfieldneuralnetworks.Insection 4,twonumericalexamplesaxegiventovalidatetheadvan. tagesofourresults.Finally,conclusionsfollowinsection

5.

    Inthispaper.ATandAdenotethetransposeandthe

    inverseofanysquarematrixA.W_euseA>0fA<01 todenoteapositive-(negative-)definitematrixA;andI (respectively,0)denotestheidentitymatrix(respectively, zeromatrix)ofappropriatedimension.IIAIIdenotesthe EuclideannormofamatrixA,i.e.,IIAII=V/Amax(ATA), wherema,(?)(respectively,min(?))meansthelargest(re- spectively,smallest)eigenvalueofA.LetRdenotetheset ofrealnumbers;R"denotesthe.dimensionalEuclidean

    sDace;R"denotesthesetofall佗×mrealmatri.

    ces.diag(?)denotesablockdiagonalmatrix.ForT>0, c([_T,0];R")denotesthefamilyofcontinuousfunctions from【一T,0]towiththenormII=sup(t'(0lv(d)l,

    wherelI()lI=,,/?n-l()isanorminR".Thesymbol"-

    k"withinamatrixrepresentsthesymmetrictermofthe matrix.

    2Problemformulation

    ThemodelofuncertainHopfieldneuralnetworkswith time-varyingdelaycanbeformulatedasfollows: u(t)=(c+AC(t))u(t)+(A+?A(?))(u(?))+

    (B+AB(t))g(u(tr(?)))+J(1)

    whereu(t)=(tl(?),,t(?))T?R"isthestatevec.

    torassociatedwiththeneurons,C=diag(cl,c2,-,c)

    isadiagonalmatrixwithpositiveentries.A=(ao)n×n

    andB:(bo)n×naretheconnectionweightmatrixand

    delayedconnectionweightmatrix,respectively.T(t)isthe timedelaycorrespondingtothefinitespeedofaxonalsignal transmission.(u(?))=(gl(Ul()),,u(?)))isthe

neuronactivationfunctionvector,andJ=(Jl,,^)'is

    aconstantexternalinputvector.?c(?),AA(t)andAB(t)

    areparametricuncertaintiesandcontinuousmatrix.valued functionsoft.

    Asmentionedin[33],itisreasonabletosupposethat Hopfieldneuralnetwork(1)hasonlyoneequilibriumpoint, denotedbyu=(u:,,u)T.Clearly,itsatisfies

    

    (c+?c(?))u+(A+AA(t))g(u)+

    (B+AB(t))g(u)+J=0

    Now,makingatransformationx(t)=u(t)u,thensystem

    (1)canberewrittenas

    x(t)

    where

    =

    (c+AC(t))x(t)+(A+?A(?)),(?))+

    (B+AB(t))f(x(tr(?)))(2)

    ,(z(?))=(/1(Xl/(?)),,,n(?)))T,

    with/~(x/t))=gi(x3(t)+us)9J(u;),J=1,2,,.

    Asin[33,40],alargerclassofUSHNNwithtimevarying

    delayisdescribedasfollows:

    ?(?)=(c+?c(t))z(t)+(A.+AA.(t)),(z(?))+

    (+AB.(?)),(?一r(?)))(3)

    whereQisaswitchingsignalwhichtakesitsvaluesinthe finiteset?={1,2,,}.Thismeansthatthema.

    trices,ace(t),A,AA(?),,AB.(?)areallowedto

    takevalues,atanarbitrarytime,inthefiniteset [(,ACl(t),Al,?Al(t),Bl,ABl(t)1,,

    (,Ace(t),AAM(t),ABM(t))]

    Throughoutthispaper,weassumethattheswitching

ruleQisnotknownapriorianditsinstantaneousvalueis

    availableinrealtime.

    Definetheindicatorfunction((?)=(el(?),,'M(?))T,

    where

    (t)

    1

    0

    whentheswitchedsystemis

    describedbythei-thmodeCz

    ?(?),Ai,AAi(t),Bi,ABi(t)

    otherwise

    (4)

    withi??.Therefore,themodeloftheUSHNN(6)can

    alsobewrittenas

    M

    (?)=??[(+?)z(?)+(A;+AAt)(?))=1

    +(+ABt)(?一r(?)))].(5)

    Asitmustbesatisfiedunderanyswitchingrules.itfollows

    that?1(?)=1.

    Obviously,whenAG=AAi:ABi0,thenominal counterpartofUSHNN(5)canbedescribedby M

    ()=?卜Gz(?)+A(?))+t=1

    ,(z(t-r(?)))].(6)

    Throughoutthepaper,wehavethefollowingassump

    tions:

    A1.0T(t)r,/-.

    A2.Thereexistsapositivediagonalmatrix?:

    diag(al,o-2,,n)>0suchthattheactivationfunctions

()I?IzI,Vz?R

forJ=1,2,,n

    InternationalJournalofAutomationandComputing04(3),July2007

    A2.Thereexistconstantst>0,i=1,2,,n,such

    that

    0

    ,_112,In'(7)

    forz?0.Wedenote?=diag(al,0"2,,n).

    A3.Theparametricuncertainties?(t),AA~(t)

    ABi(t)aretimevariantandunknown,butnormbounded Theuncertaintiesareofthefollowingform:

    [?(t)?t(t)?2(t)]=DF(t)[EicE]

    inwhichDEFEE}areknownrealconstantmatrices withappropriatedimensions.TheuncertainmatrixF(t) satisfies

    F(t)FT(t),,Vt?R.

    Obviously,conditionA2islessconservativethanA2 andsomegeneralactivationfunctionsinc0nventi0nalneu

    ralnetworks,sucha,sthestandardsigmoidalfunctionand piecewise-linearfunction,satisfyConditionsA2orA2 Inordertoobtainourresults.weneedthefollowing definitionsandlemmas:

    thetermsintheformulaistakenintoaccount.Specifically thetermsontheleftsideoftheequation

    2?(f))_)ds(9)

    where

    (t)=[zT(t)zT(t.r(t))fT(z(t))fT(z(t.r(t)))]T

    areaddedtothederivativeoftheLyapunovKrasovskii

    functional,(t).Inthisequation,thefreeweightingmatrix

N=【孵】Tindicatestherelationship

    betweenthetermsintheLeibniz-Newtonformula.As isshowninthefollowingtheorem,theycaneasilybe determinedbysolvingthecorrespondingLMI. Theorem1.Forsystemf61,supposeA1andA2 hold.GivenscalarsT>0and,theequilibriumu=

    (u,,u)ofthesystem(6)isgloballyexponentially stable,ifthereexistappropriatelydimensionedmatrices P=P>0,Ql=Q}>0,Q2=Q>0,and

    Z=ZT>0

    ,N=【胛孵WTandscalars

    ?>0,=1,2,suchthatthefollowingLMI Definition1.Theparametricuncertainties?,?t,1

    ?BtaresaidtobeadmissibleifA3holds.

    Definition2.FortheUSHNN(5)andevery?

    c(【一.r,0;R"),thetrivialsolutionissaidtobegloballyro- bustlyexponentiallystableifforalladmissibleuncertainties

    ?(,AAi,?B,thereexistpositivescalarsoLandsuch tha

    lIx(t;qo)llorel1.(8)

    Lemma14l.GivenmatricesQ=QT,H,EandR=

    RTofappropriatedimensions,

    Q+HFE+ETFTHT<0

    f0raUFsatisfyingFTFR,

    ifandonlyifthereexists

    some>0suchthat

    Q+AHHT+A-1ETRE<0

    :Givenconstantma- Lemma2.rSchurcomplement)[

    tricesQl,Q2,Q3,whereQl=Qand0<Q2=Q, Ql+QQQ3<0ifandonlyif

()r(嚣曼:)<.

    3Mainresults

    First,thenominalcounterpart(6)ofUSHNN(5)isdis-

    cussed.TheLeibnizNewtonformulaisemployedtoobtain

    adelay-dependentcondition,andtherelationshipbetween

    12

    22

    ?

    ?

    ?

    ?

    TN1TCtZ

    1_N20

    rN3rAWZ

    rN4rBwZ

    

    1_Z0

    ?一1_Z

    holds,where

    ll=PC,P+Ql+N1+?+gl?T? 12=NNl

    l3=PAt+'

    14=PBt+N

    422=(1n)Ql一?2一?+g2?T? 33=Q2一?l,

    44=--g2I(177)Q2. <o,(10)

    Proof.UsingtheLeibnizNewtonformula,wecanwrite

    x(t.r(t))=x(t)厂圣(.)d.JtT(t) (11)

Report this document

For any questions or suggestions please email
cust-service@docsford.com