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Ill-conditioned stable inversion arising from singularly perturbed zero dynamics

By Frederick Allen,2014-06-02 02:10
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Ill-conditioned stable inversion arising from singularly perturbed zero dynamicsIll-co

Ill-conditioned stable inversion arising from

    singularly perturbed zero dynamics JControlTheoryAppl20086(4)385391

    D0I10.1007/sl1768.008.6122.2

    IU-conditionedstableinversionarisingfrom

    -一一一一'-

    singularlyperturbedzerodynamics

    XuemeiGUO.GuoliWANG

    (SchoolofInformationScienceandTechnology,SunYat

    SenUniversity,GuangzhouGuangdong510275,China)

    Abstract:Theil1..conditionedstableinversionisstudiedforslightlynonminimumphasesyst

    emswhosezerodynam.

    icsissingularlyperturbed.thatis.therelativedegreeisill

    defined.Forthesesystems,weshowthatthereexistsaninherent limitationinthebandwidthofareferencetrajectorytobetrackedwhenawell

    conditionedfeedforwardinputviastable

    inversionissought.weassertthat.whentheviolationofthislimitationoccm's.theso

    calledreferencetrajectoryredesign

    iscalledfor.Ouranalysisresultscanprovideanexplicitassessmentaswellasusefu1guidance

    forthereferencetmiectory

    redesignifneeded.

    Keywords:Nonminimumphase;Il1.definedrelativedegree;Stableinversion;Singularlyperturbedzerodynamics

    1Introduction

    Annonminimumphasefeatureposesfundamentallimi.

    tationsontransienttrackingperformanceoffeedback.based regulators.Ithasbeenshownthatplantnonminimumphase

zeroshaveanegativeeffectonafeedbacksystem'sabil

    itvtoreducetrackingerrors[1.Noncausalstableinversion

    techniquescanbeusedtoovercomethetransienterrorphe

    nomenon,andtoachieveasymptoticallyexactoutr}uttrack

    ingcontrolintegratedwiththecausalregulatortohandle modellingerrorsanddisturbances.Thisresearchlineprig. inatedfromtheearlierworksontheinversedynamiccon

    trolofflexiblemanipulators,see21and[31.Amajorad

    vanceintheformalizedtreatmentwasmadebyintroduc. ingtheframeworkofthestableinversionbasedapproach in41and51:furtherextensionofthisideawasmade by[6]and[7sothatthisapproachbecameapplicableto aircraftguidancecontrolsystems[8,9].Thekeytostable inversionisthattheinput.outputdecouplinggain/matrixis nonsingular,thatis.thesystem'srelativedegreeiswell

    defined101.Thepresentworkaddressestheinherentlim. itationofstableinversionbasedontputtrackingwhenthe input?'outputdecouplinggain/matrixisclosetobeingslngu.. larandtherelativedegreeisi11.defined.Wleshowthatthe stableinversionisillconditionedathighfrequenciesand

    awel1.conditionedfeedforwardinputviastableinversion canincreasethesmoothnessofareferencetrajectorytobe tracked.

    Earlierstudiesinfll1and12showedthatasingular

    perturbationsofzerodynamicscanbecausedbyregular perturbationsdecreasingthesystem'Srelativedegree.One oftheimportantresultsin12ismatthereisnopeaking

    phenomenoneventhoughthezerodynamicsissingularly perturbed.Motivatedbytheflightguidancecontrolofpla

    narVertical/ConventionalTakeOfrandLanding(v/cTOL)

    aircraft,131revisitedsuchsystemsbutforthepulposeof outputtrackingcontrolviastableinversion,andconfirmed theabsenceofpeakingintheboundedfeedforwardinput again.However,thereseemstoexistagapbetweenthese eleganttheoreticalresultsandsomeobservationsfromreal

    isticapplications.

    }CroLaircraftareslightlynonminimumphasesystems withsingularlyperturbedzerodynamics;theflightguid

    ancecontrOlofsuchsystemshasattractedmuchattention becausetrivialcontroltechniquesarenotdirectlyapplica. blel41.In15],anumericalsimulationshowedthatthe controlmagnitudeistoo1argetobeacceptablewhenthe exactinput.outputlinearizationisappliedtoaVT0Lair- craft.161madecomparisonsbetweenthestableinversion approachandsomeapproximatetrackingapproaches[15

    inthecontextofCTOLaircraftandfoundthatthefeedfor- wardinputsviastableinversionhavemuchlargermagni_ tudesthanthoseusingtheotherapproaches.Inparallel,we havebeeninterestedinthetiptrackingcontrolofaflexible manipulatorwheretherelativedegreeisi11.defined.arising fromnoncolocatedsensor/actuatorstructuresf17.Theend.

    pointtrackingcontrolofaflexiblemanipulatorisnonmin. imumphaseanditszerodynamicsisalsosingularlyper

    turbed18,191.20foundthatthefeedforwardtorqueinput generatedbytheinversedynamicsofaflexiblemanipulator containshighfrequencypeaks.Thishigh.frequencypeak. ingphenomenonhasbeenconfirmedbymanyresearchers 1ater,forexample,3,21,221.A11oftheseobservationsseem tocontradictthewel1.establishedtheoreticalresultsby[12

    and[13].

    Tofilltheabove.mentionedgap,muchsystematically analysisisneededtoofferabridgethatprovidesadeepin. sightintothisissue.Thisworkaimstoachievethisgoal byclarifyingtheinherentlimitationinthebandwidthofa referencetrajectorytobetrackedwhenawellconditioned

    solutionofthestableinversionissought.Inparticular,it isshownthat,whentheviolationofthislimitationoccurs, thesocalledreferencetrajectoryredesigniscalledfor.Our analysisresultscanprovideanexplicitassessmentaswell asusefu1guidanceforthereferencetrajectoryredesignif Received13July2006;revised3December2008.

    ThisworkwassupposedbytheNationalNaturalScienceFoundationofChina(No.60473120

    )andtheNaturalScienceFoundationofGuangdong (No.6023190).

    386XGUOeta1./JControlTheoryAppI20086f41385391

    needed.Thisisanewconsiderationonthisaspect 2Problemstatement

    Considerasingleinputsingleoutput(SISO)nonmini- mumphaselinearsystemwithsingularlyperturbedzerody

    namicsintwotimescales,anditscontrollablestatespace modeltakesthefornl

    =Fz+gu

    Y=hT.

    where?Risthestatevector,thescalarvariablesand Yarethecontrolsignalandtheoutput,respectively,Fisa qxmatrix,gandhareqvectors.Letrbetherelativede

    greeoftheperturbedsystemandr+dtherelativedegreeof thenominal(unperturbed)system.Inotherwords,therela

    tivedegreeisreducedfromr+dtorduetoperturbations.

Precisely,hFJg=0,J=0,1,?,r1,andthetwo-

    timescaleassumptiononthechangeoftherelativedegreeis characterizedas

    TFr+kg=ed--ka

    ,k=0,1,,d.(2)

    HereE>0isasmallperturbationparameter,a0?0and

    0d?0.Definethenewcoordinatesby

    fhTF,i=1,2,,r+d,

    lxr+d+=z,i=1,2,--?,,

    where=mrdand)arearbitrarilychosensothat P=hFTh(FT)+hz112z]T

    isfull-rank.Inthenewcoordinates,(1)canbeputinthe following'normalform':

    r=Xi+l,i=1,2,,r1,

    lr+:r+t+1+~d--iaiu,i=0,1,,d1,

    {r+d=b+adu,(3)

    Ir+d+t=c+ad+iU,i=1,2,,n,

    =1,

    wherebT=FP_.

    ,

    c=IWFp?

    ,0=ZTg,

    i=1,2,.,n.Weassumethatthissystemishyperbolic, thatis,therearenozerosontheimaginaryaxis,irrespective oftheperturbation.Thisistherequirementthatguarantees thesysteminvertibility[23].Notethatthenonlinearcoun- terpartofthisnormalformisacommonpointofdeparture forstableinversionbasedapproaches141.

    Webeginwithaclosewatchonthecharacteristicsofthe

regularperturbationintheforwarddynamics.LetGe(s)de

    notethetransferfunctionsfromutoY,whichcanbeex

    pressedas

    GE(s)=:(sI-F)-lg=hTFkg

    .

    (4)

    Withtheaboveassumptionsontherelativedegree,we seethatthefirstr1termsarezeros.thefollowing

    dtermsarecollectedass-()za(c,s)with?(E,8)=

    ?aies,andtheremainderisdenotedbyG0(E,s)= ,

    .,

    

    (hTFkg

    ).

    Itisinterestingt.n.tethatthere1ative degreeofG0(e,81isr+dandthefeedforwardgainof 8r+dGo(e,8)isad?0,irrespectiveofE=0.Inaddition, a(o,8)=oandGo(0,s)becomesthetransferfunctionof thenomina1system.Withthesenotations.wehave 8r+dGS)=?(E,s)+8r+dGo(E,s).(5)

    Clearly,theregularperturbationcanbecharacterizedin thefrequencydomainbytheterm?(E,81.Forcut.off

    frequencycoc,0<coc<?,itiseasytoseethat

    supIl?Ia(c,_7)I=o(0.HereJ=,/l,isthe

    realfrequencyvariableinradiansperunittime,and0(h) denotesatermoforderh.i.e.1O(h)l?forsomefixed

    >0.Itisnotsurprisingthatsucharegularperturbation makeslittlecontributionatlowfrequenciesandcanbecon

    siderablysignificantonlyathighfrequencies.

    Whathappenswiththisregularperturbationwhenwe seekthestableinversion?Recalling(51,wecanwritethe transferfunctionofthestableinversionoftheperturbedsys

    temaS

    (s):=(8rG(s))8dIto(~,8)__.?.-?__?___?--_.___________.__?

    ____________.__._.

    1+(E,s)H0(E,

    whereH0(E,s)=(8r+dGo(e,s))-.isthetransferfunc- tionofthenominalstableinversionwhenE=0.It iseasytoseethatforcut.offfrequency>0. supIl?lH~(jco)(jco)dHo(c,)I=0(E).Thisim?

    pliesthat!(s)differsslitlyfrom8d](E,8)thusthe

    nominalcaseof8dHo(0,81atlowfrequencies.0nthe otherhand,recalling(5),weseethat8rGE(s)isdomi

    hatedby8--dZa(E,s)athighfrequencies,thus,c(s)bythe term8dAfE,s).Therefore,theeffectoftheperturbation onstableinversioncanbecharacterizedbythebehavior ofs?(E,s)athighfrequencies.Inaddition,thehi曲一

    frequencygainofthestableinversionis.iml(s)l=lsI}..

    .imIs"?(E,s)I=0I,whichisclosetobeing1Sl..

    singular.ThismeansthatI()Itendstobeunbounded athighfrequenciesasEgoestozero.Thestableinversion inwhichthissituationoccursissaidtobeillconditioned.

    Asasummary.weassertthattheperturbationhasanin. significanteffectonthestableinversionatlowfrequencies butcausesthestableinversiontobeillconditionedathigh

    frequencies.

    AsEgoesto0,thesolutiontothestableinversional

    waysremainsboundedatlowfrequenciesbutitsbehav

    iorathighfrequenciesisdependentonthesmoothnessof

    areferencetrajectorytobetracked.Toseethispoint,for agivenreferencetrajectoryYD,weexpressthefeedfor- wardinputviastableinversionas(s)=(s)};(s). Notethatlim(8)=8dHo(0,s).Withthisinmind,we e---*0,

    havethat,asEgoesto0,u(s)approachesu0(s),given byu0(s)=H0(0,s)sd(s)=Ho(0,s)(s).0b

    viously,0isboundedifandonlyifisbounded.

    Itisnothardtoimagineintuitivelyhowmuchcorruptible thesolutionofthestableinversionbecomeswhenarefler

    encetrajectorycontalnsconsiderablehigh.f1_equencyc0n.

GUOeta1./JControlTheoryAppl20086(4)385391

    tents.Inshort,thereexistsaninherentlimitationonthe bandwidthofareferencetrajectorytobetrackedwhenthe wellconditionedsolutionviastableinversionissoughtfor suchaperturbedsystem.Wewillinvestigatethisinherent bandwidthlimitationofareferencetrajectorywhichistobe trackedmuchpreciselyinnextsection.

    3Mainresults

    3.1Inherentlimitation

    WithYDinsteadofYandginsteadof,i=

    1,2,,r,astheinputsofthestableinversion,thestable inversionprocedureisoutlinedasfollows.Webeginbyin

    vertingtherthequationin(3)for,giving

    =Ed0(吕一Xr+1).(7)

    Substitutingthisintothenextd+nequationsin(3)yields thesocalleddrivendynamics,inwhichthefirstdthequa- tionsare

e"e.+e(

    er+der+d+e+),8

    Xr+d:6T+a

    .

    d

    ,-d(~A

    andthelastnthequationsare

    r+dc+e(D+),(9)

    i=1,2,,n.ForfixedE?0,wecansolvethebounded

    solutiontothedrivendynamicsforthedesiredintemaldy

    namicsstates{Xr+),andsubstitutingtheminto(7) yieldstheboundedsolutionforthedesiredfeedforwardin

    put.Notethatthesuccessfulimplementationoftheabove standardprocedureforseekingtheboundedsolutionre.

    posesonthehypothesisthatYDanditsfirstrthderivatives {)1arebounded,thatis,YD?(It,.Here

    ([cs{,])={,I)?..((】),i=0,1,m)

    isaSobolevspaceofordermequippedwiththenornl llYJI(t,])=maXtY?llL..(t),t?0is

    thestartingtimeandt,>ttheendingtime.

    Nowwetumourattentiontotheinterestingsituation:the regularperturbationintroducesEdintothedecouplinggain intherthequationof(3),andthestableinversionprocedure putsitinthedenominatoroftherightsideoff71.Thenthe questionarisesisthefeedforwardinputgivenbyf71still boundedasEapproacheszero?Accordingtothefrequency

    domainobservationabove,theanswertothisquestionis dependentonthesmoothnessofareferencetrajectorytobe tracked.Inwhatfollows,weverifythefactthat.asEgoesto 0,thesolutiontotheperturbedstableinversionapproaches

    thatofthenominalstableinversionifandonlyifYDand itsfirst(r+d)thderivatives{)arebounded,thatis, YD?w(It,).

    Wlebeginbyrepresentingthehighorderderivative(r+.J

    aS

    (+)=Xr+i+1+~d-iat2t+...+eda0f"()

    387

    i=0,1,,d1.Theimportantthingto(10)isthat,as Egoesto0,~--dq-i(L.)Xr++1)approachesaiu,which

    isboundediftheinputintheforwarddynamicsoff31is bounded.Thus,intheprocedureofthestableinversion.itis expectedthatthedifferencebetweenandxr++1(the internalstate)is0(Ed_.).Ifthisistrue,thenisbounded asEgoesto0.ThismotivatesUStointroducethenewcoor- dinates5i=Ed+(+.)Xr+i),i=1,2,,d.In

    thenewcoordinates,(8)canbeequivalentlywrittenasthe singularperturbationformof

    fE1:21,l

    1fD

    IE:(r+d)6T1,

    inwhichtheithequationisderivedbysubtracting e-d+iy~+)frombothsidesoftheithequationin (8).Let?D=[Dg...g]T,eD=

    gg+?…一r+d--1)lT,=[12…如]T,=

    [12]T,theithcomponentofisdefinedas

    =xrq-d+i.Hereweemphasizethatinthedrivendy

    namicscanbeexplicitlyexpressedin(6,)spaceas

    =

    [?eT(e)r/w]T

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