A-lifting-technique-for-linear-periodic-systems-wi

Systems&ControlLetters17(1991)79-8879North-HollandAriftingtechniqueforlinearperiodicsystemswithapplicationstosampled-datacontrol*BassamBamiehandJ.BoydPearsonDepartmentofElectricalEngineering,RiceUniversity,Houston,TX77251,USABruceA.FrancisDepartmentofElectricalEngineering,UniversityofToronto,Toronto,CanadaM5S1A4AllenTannenbaumDepartmentofElectricalEngineering,UniversityofMinnesota,Minneapolis,MN55455,USAandTechnion,Haifa,IsraelReceived14January1991Revised19April1991Abstract:Aliftingtechniqueisdevelopedforperiodiclinearsystemsandappliedtothe.,~o¢and.~2sampled-datacontrolproblems.Keywords:Sampled-datasystem;liftedsystem;.~oooptimalcontrol;Riccatiequation;operatornorm.I.IntroductionGiventhesuccessofJff°°-normbasedoptimi-zationmethodsforanalogcontrolsystems,therehasrecentlybeeninterestinapplyingsuchtech-niquestosampled-datasystems[3,4,15,18].Thekeypointinutilizingsuchmethodswouldbeintheirextensiontocertainperiodictime-varyingsystems.Anexampleofsuchasystemisthesampled-datacontrolsystemshowninFigure1below.*ThisresearchwassupportedinpartbygrantsfromtheNationalScienceFoundation(NSFECS-8914467,DMS-8811084),theAirForceOfficeofScientificResearch(AFOSR-90-0024,AFOSR-91-0036),theArmyResearchOffice(DAAL03-91-G-0019),andNSERC.ThegeneralizedplantGisacontinuous-time,time-invariantsystem,Kdisdiscrete-time,time-invariant,Sistheidealperiodicsamplerwithperiodh,andHthesynchronizedzero-orderhold.Continuous-timesignalsarerepresentedbycon-tinuouslines,discrete-timesignalsbydottedlines.Thebehaviorofthesystemfromtheexogenousinputwtothecontrolledoutputzisingeneraltime-varying,infact,periodicwithperiodh.Toanalyzethebehaviorofcontinuous-timeperiodicsystems,weusealiftingtechniquesimilartothatusedfordiscrete-timeperiodicsystemsin[16].Oncewedeveloptheliftingtechnique,weapplyittodescribeacompletesolutiontotheanalysisproblemofverifyingthatagivencon-trollerconstrainsthe.L,e2-inducednormofthesampled-datasystemtobelessthansomepre-specifiedlevel.Wewillalsoshowthattheliftingtechniqueisapplicableinfacttoallnorm-basedoptimizationproblems,andinparticulartosam-pied-dataversionsofthequadraticregulatorandoptimalfilteringproblems.Giventhesuccessofo~°~-normbasedoptimi-zationmethodsforanalogcontrolsystems,therehasrecentlybeeninterestinapplyingsuchtech-niquestosampled-datasystems[3,4,15,18,25].Thepurposeofthisnoteistointroducetheliftingtechniqueitselfandsketchhowitcanbeappliedtotwooptimalcontrolproblems.Toourknowledge,suchaliftingprocedurewasintro-ducedintosampled-datasystemsbyToivonen[25],whoalsotreatstheJ£,~sampled-dataproblem.Thedetailsoftheliftingin[25]aredifferentfromthosegivenhere([25]representscertainfinite-rankoperatorsviaSVD,whichisavoidedinourwork).Themathematicalbasisofsuchliftingtechniquesmaybefoundin[21].Reference[1]givesade-tailedaccountoftheapplicationoftheliftingtechniquetothe~sampled-dataproblem.Yamamoto[28]alsousesliftingforsampled-data0167-6911/91/$03.50©1991-ElsevierSciencePublishersB.V.(North-Holland)80B.Bamiehetal./Liftingtechniqueforlinearperiodicsystemssystems,butheliftsthestateaswellastheinputandoutput.Consequently,hisstatespaceisin-finite-dimensional,whereasoursistheoriginalfinite-dimensionalone.Also,Yamamototreatsasymptotictrackingproblems,whileoptimizationproblemsarestudiedhere.Whilethispaperwasbeingreviewed,severalotherscameintoexistence.Forcompletenesswementionthemhere:asampled-dataJt~2problem(differentfromthatin[4])in[2,17];sample-data~1(i.e.,£,oinducednorm)in[23,8];androbuststabilityofsampled-datasystemsin[24].Intheoperatornormdesignframework,thisliftingtechniquewasdevelopedindependentlybythefirsttwoandthelattertwoauthors.Reference[1]givesadetailedaccountoftheapplicationofthistechniquetotheJd'~sampled-dataproblem.2.Liftingcontinuous-timesignalsInthissectionweintroduceaconstructionwherebyonemay'lift'acontinuous-timesignaltoadiscrete-timeone.Thisconstructionwillalsobeusedtoassociateatime-invariantdiscrete-timesystemtoacontinuous-timeperiodicone.Theutilityofthistechniqueinfeedbackcontrolisthatallnormsarepreserved,aswellasthefeedbackinterconnectionstructure.Wewillfirstworkinarathergeneralframe-workbeforespecializingtothecaseofinterest.LetXdenoteaBanachspaceequippedwithnormIII1,,-.Foreveryintegerp_1weset{j0}:=u'[0,~)~f:Ilu(t)llf'~dtoo.Asiswell-known,.~0p(f)isaBanachspacewithnormIIulip.,.:=Itu(t)llgdt.Forp=2,£,oz(f)maybegiventhestructureofaHilbertspaceintheusualway.Forp=ac,wehaveZe~(X):={u[0,m)--,Xesssupllu(t)11,,oc}(see,e.g.,[22]).Finally,toeachofthespaces~P(X)wemayassociatetheextendedspace5°f(X)inthestandardway.Forallthedefini-tionssee[61.Thesametypesofdefinitionsareofcoursevalidinthediscrete-timecaseforsequences.Asequencewillbewrittenasacolumnvector,forexample,=Ii:AgainforanyBanachspaceX,defineIP(X)={J/:~iEX,~I,~,ll~-~c},i=0lpve,ThenormsaregivenbyII~Ill,,?,-~=II4~iI1~-,1_p~,i=0II+Iliads,-supII+iII~i-.!EquippedwiththisnormlP(X)isaBanachspaceforalll__poc.Onceagainforp=2,12(X)maybegivenaHilbertstructureintheusualway[22],andtheassociatedextendedspacelP(X)maybedefined:itisjustthelinearspaceofallsequencesinY'.Wearenowready