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Control of switched systems with actuator saturation

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Control of switched systems with actuator saturation

    Control of switched systems with actuator

    saturation

    JournalofContmlTheoryandApplications1(2006)3843

    Controlofswitchedsystemswithactuatorsaturation

    YangSONG,ZhengrongXIANG,QingweiCHEN,WeiliHU

    (DepartmentofAutomation,NamingUniversityofScience&Technology,JiangsuNanjing210094,China)

    Abstract:Thispaperisconcernedwithcontrollersynthesisforlinearswitchedsystemswithactuatorsaturation.

    BasedoncommonLyapunovfunctiontechniqueandmultiple

    Lyapunovfunctiontechnique,twomethodsfordesigning

    statefeedbackcontrollerareproposedrespectivelyintermsoflinearmatrixinequalitiesfortheswitchedsystemswith

    saturation.Anapproachonenlargingtheattractivedomainistheninvestigated.Theapplicationofthepresentedapproach

    isillustratedfinallybyanumericalexample.

    Keywords:Saturatingcontrol;Switchedsystems;CommonLyapunovfunction;MultipleLyapunovfunctions

    1Introduction

    Switchedsystemsarecomposedofasetofsubsystems

    andaswitchingruledesignatingwhichsubsystemtobe

    actuatedateachmoment.Switchedsystemsarecommonly

    foundinvariousengineeringpractice,suchasinauto-

    motiveenginecontrolsystems,chemicalprocesscontrol,

    roboticmanufacture,multiple-modelsystemsandsoon.

    trollerdesignisnecessary.Thestudyonsaturatingcontrol

    hasattractedattentionofmanyresearchersinthepastfew

    years[7,-O].Animportantissueofsaturatingcontrolisto expandthedomainofattractionaslargeaspossible,Huet a1.[8,9havemadeaseriesofinvestigationsonthisaspect, Fortheswitchedsystemswithactuatorsaturation,theanaly- sisandcontrolofsuchsystemsbecamemoredifficultdueto Inthepastdecade,mallyliteratureshaVebeendevotedtotheinteatjonofwithingandthenonli

    nearityofsatura-

    thestudyofswitchedsystems.Stabilityisanimportant issueintheanalysisandsynthesisofswitchedsystems 16.CommonLyapunovfunctiontechnique(CLF)[1

    andmultiple-Lyapunovfunctiontechnique(MLF)[2were

    introducedforcheckingthestabilityofswitchedsystems. Convexcombinationwasaddressedtostabilizeswitched systems3.BasedonweakLyapunovlikefunctions,Ye

    eta1.4proposedastabilityanalysisresultwhichisless conservativethantheresultsof[2.Daafouzeta1.5inves

    tigatedcontrolsynthesisofswitchedsystemsindiscrete- timedomainbyusingLinearMatrixInequalities(LMI). Acomprehensivesurveyontheanalysisandsynthesisof switchedsystemswasprovidedbySun&Ge[6.

    Inpracticalcontrolengineering,thephenomenaofactu- atorsaturationexistsubiquitously.Saturationoftenmakes acontrolsystemdifficultlytoachieveglobalstability. Hence,takingsaturationeffectintoaccountduringcon- tion.Inthispaper,wedeveloptwostabilizationapproaches byCLFandMLFforswitchedsystemswithsaturation,and thendiscusshowtoenlargetheattractivedomain. 2Problemformulation

    Consideringsingleinputswitchedsystemssubjectto actuatorsaturation:

t=Aix+B~sat(),i?M={1,2,,?}(1)

    where?R"issystemstate.constantmatricesAi?

    R,Bi?R"and?Rdenotethestatematrix,input

    matrixandthecontrolinputofsubsystemi,respectively.N isthetotalnumberofsubsystems.sat(?1:R__?Risthe

    followingstandardsaturationfunction:

    sat()=sgn()rnin{wi,I_)_i?M(2)

    where>0istheupperlimitofsat().

    Applyingstatefeedbackcontrollaw=x(t1to Eq.(1),weobtain

    =

    Aix+sat(),i?M.(3)

    Received30September2005;revised14December2005 ThisworkwassupportedbytheNationalNaturalScienceFoundationofChina(No.6047403

    4)

YSONGeta1./JournalofControlTheoryandApplications1(2006)3843

    Introduce()todenotethemagnifyingratiofromthe inputtotheoutputofanactuator,

    ),>

    ()={1,1l?

    Lw/(),<w.

    ThenEq.(3)canbetransformedto

    =

    A+B()(),i?M.

    Obviously,Hi()liesbetweentheinterval(0,1and

    ()=1impliesthattheactuatorofsubsystemidoesnot saturate.Itcanalsobeseenthatthesaturationeffectwould becomemoredramaticalongwiththedecreaseof(). Thequadraticdomainofattraction(orcalledtheLyapunov

    levelset7)ofswitchedsystems(5)isgivenas (P,P)={?R:xTPx<1/p,P>0,P>0).(6) W_edenote

    Then

    

    hi:=min{H~(x),?(P,p))

    :={1,).

    =

    {?R:l()l?i/).

    39

    ProofConstructLyapunovfunctionV(x1=TPf_0r switchedsystems(5).FromEq.(7)andbytheknowledge aboutintervalmatrix11],weknowthatforx(t)?,

    T(t)[(+BHi(x)Fi)TP

    +P(Ai+())()<0,(9)

    whichimpliesthatLyapunovfunctionV(x)=xTPx decreaseinalongthetrajectoryofsubsystemi. Eq.(8)givestheparameterPoftheattractivedomainand ensurescinthat

    x(t)?Y2iIF~x(t)l/w<1.(1o)

    ByusingSchurcomplement,ltfollowsfrom(8)that h_

    i2IF~x(t)l/=

    hilP/.P/x(t)l/,fJ

    ?hiP/IP/x(t)ll/

    =

    PT(t)Px(t)/w

    ?pxT(t)Px(t)<1.(11)

    Therefore.wehave

IF~x(t)l/,fJ<pxT(t)P(t).(12)

    isthep.lyhedroninwhichthemagnifyingrati.()isn.Th

    ?

    means?f2i.N.ticing??,wecan.bain

    lessthanht.?n.Thiscompletestheproof.

    =1

    3MainresultsRemark1Forconciseness,weextendthenotationH: in(7)torepresenttheelementsof.Inotherwords, Inthissection.thestatefeedbackcontrollawisdesigned

    Eq.(7)infactincludestwoinequalitiesfor1and forlinearswitchedsystemswithactuatorsaturationby.. =,respectively.

    usingCLFandMLFmethodsrespectively.

    一一Remark2Thematrixinequalities(7)and(8)isnon

    3.1ControllerdesignbyCLF

    Theorem1Consideringswitchedsystems(5),ifthere Eqs.(7)--(8)aresatisfied,thentheswitchedsystemswith initialconditionx0?arelocallyquadraticallyasymptot- icallystable(LQAS)forarbitraryswitchingrule. (t+Bi)TP+P(A+B)<0,(7)

    [.?wherei?M,theattractivedomainis=f:xTP<

    1/p).

    convexwhichcanbesolvedbythealgorithmin8

    Fig.1illustratestheattractivedomainandthenon

    saturationdomainf2i(thebelttypedomain)foraswitched systemconsistingoffoursubsystems+Wecanseefrom Fig.1thatthedomainofattractiongivenbyTheorem l(thelittleellipse)isrestrictedwithintheintersectionofall .Furthermore,theswitchedsystemscanstillbeLQAS underappropriateswitchingruleiftheattractivedomainis

expandedaslargeasbeingtangentialtotheunionof(the

    bigellipse).

    Remark3Theoremhasconsjderab1econservative

    YSONGeta1./JournalofContwlTheoryandApplications1(2006)38_43 nessbecausetheattractivedomainshouldliewithin3.2ControllerdesignbyMLF ?

    n.Iftheswitchingrulescanbedesignated,thentheInthissection,wewilldiscussthecontrolofs

    witched

    i=1

    attractivedomaincanbeenlargeddramaticallyinsomesystemsbyMLFtechniquewhichisfl

    exibleinchoosing

    CaSeS

    Fig.1Illustrationofenlargingdomainofattraction.

    Lyapunovfunctions

    Theorem2TheswitchedsystemsEq.(5)consistingof

    twosubsystemsareasymptoticallystablewithinitialcondi- tionx0?,ifthereexist,>0,scalarpi>0,1?0,

    2?0suchthat,

    (1+BlU~F1)TPl

    +P1(A1+B1F1)<1(P2P1),(14)

    (2+B2)TP2

    +P2(A2+B2)<2(P1P2),(15)

    [I'0brieflydescribetheresulsinthefoilowing,severa1wheretheattractiVedomain=1U2?Theswitching notationsareintroducedhere.weden.tethetw.b.undariesrule=~rgmin{V,(x)},i=1,2?

    ofas.:Fix+{/=0,lib:Fix_"Jt/=0.Let

    ?

    :=Us2i.Itcanbeseenthatthesomeoftheintersection

    =l

    pointsbytheboundariesofdifferentS2iarelocatedeither ProofFromEq.(14)(15),wecanobtainthat

    IfxT(P2)z?0and?l,()<0;

    IfxT(P2P1)z?0andx?$22,(z)<0.

    Theswitchedsystemisasymptoticallystableevidentlyfor withinthed.main(suchp.in6Fig?1).r.nt.theboundthegivenswitchingmle. Thiscompletestheproof.

    aryoff2(suchpointainFig?1)whicharecalledtheexternalitcanbeseenthecone

    typeregionsoutsidefsuchas

    intersectionpointsanddenotedbyxiajborthelike,where thesubscriptionia,jbmeanthatthispointistheintersection ofthelinesf.,ljb.

    theconemnrinFig.2)isexcludedfromthedomainof} attractionthatinTheorem1or2.However,insomecases, wecantaketheseconetyperegionsintoaccounttoenlarge

    Corollary1Consideringswitchedsystems(5),ifthereheamaiVedomain exist,P>0andconstantscalarP>0,suchthatEqs.(7) and(13)aresatisfied,theswitchedsystemswithinitialstate Xo?mareLQASundertheswitchingruleo.

    [.,theswitchingruleo-canbechosenas:

    Ifz?(m\)nS2i,subsystemiisdesignatedtowork;

    ifz?,anysubsystemcanbedesignatedtowork.where, =

    {:TPz<1/p},:{x:xTPx<1/p),

    1/,=min{min{x~Px),max{1/p}),

    1/p=min{1/pi},i,J?M,i?J,

    eareexternalintersectionpoints.

    \

    \

    \

    \ \ \ \ \ \ \ \ \ r|

    , ,,2

    , x2

    \ , . , , . , ( , \ \ \ ( \ ,

,

    \

    \,1

    Fig.2IllustrationofstabilizationbyMLE \

    LetaJadenotetheconeoutsidewhichiscreatedby

ONGeta1./JournalofControlTheoryandApplications1(2006)383

    twolinesli?,ljn.?Jncanberepresentedas

    {+n>0)n{Fix+.>0),

    whereFk.equalseitherFk

    .or(.;.equalseitherA.

    .,:,J.Equivalently,theregion8carlbe alsowrittenas

    where

    Rj+}

    =

    >0

    jj:

    ChoosethefollowingLyapunovfunctionforx?

    ,

    Viws()=TQJ"+2qiTw+riwjv.

    Itcanbeseenthat"J>0intheregionJif T

    (17)

    whereAtj

    isasymmetricmatrixofwhichalltheentries arenonnegative.

    Ontheotherhand,ifthefollowinginequalityaresatisfied

    forsubsystemk,=,J,thenwehaved?f/d)<0in theregion

    T(ATQ_uJ+Qi~j.Ak)z+xTQiwjvBk

    +kT上】kTQ+2T"(k+Bkwk)

    (+J)T(EJ+,{j)<0

    whichisholdprovidedthat J//iw~"J+QBkwk%咖咖]

    2J一盘扣咖I

    >0(8)

    whereZjisasymmetricmatrixofwhicha1ltheentries

    arenonnegative;

    J"=ATQi+QJk一砚jzJRJ.

    Toguaranteetheswitchedsystemstobestable

    ,following

    constraintsshouldbeadded 1)theequilibriumpointofsubsystem,=,J,

    iB?,

    2)theLyapunovfunctionsofdifferentsubsystemsare

    equalontheboundary,thatis ()=(),

    ()=J(),

    Noticef=+/=

    metricdescription[1o]as 4l

    for?li_f』】;

    forx?f_j.

    0canbeexpressedbypara

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