[ac02++++]A descriptor system approach to H∞ contr

IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.47,NO.2,FEBRUARY2002253ADescriptorSystemApproachtoHControlofLinearTime-DelaySystemsE.FridmanandUriShaked,Fellow,IEEEAbstract—Theoutput-feedbackcontrolproblemissolvedforcontinuous-time,linear,retardedandneutraltypesystems.Adelay-dependentsolutionisobtainedintermsoflinearmatrixin-equalities(LMIs)byusingadescriptormodeltransformationofthesystemandbyapplyingPark’sinequalityforboundingcrossterms.Astate-feedbacksolutionisderivedforsystemswithpoly-topicparameteruncertainties.Anoutput-feedbackcontrolleristhenfoundbysolvingtwoLMIs,oneofwhichisassociatedwithadescriptortime-delay“innovationfilter.”Thecasesofinstanta-neousanddelayedmeasurementsareconsidered.Numericalexam-plesaregivenwhichillustratetheeffectivenessofthenewtheory.IndexTerms—Delay-dependentcriteria,descriptorsystems,-control,linearmatrixinequalities(LMIs),time-delaysys-tems.I.INTRODUCTIONTIME-DELAYoftenappearsinmanycontrolsystems(suchasaircraft,chemicalorprocesscontrolsystems)eitherinthestate,thecontrolinput,orthemeasurements(see[1]–[5]andthereferencestherein).Time-delayis,inmanycases,asourceofinstability.Thestabilityissueandtheperformanceoflinearcon-trolsystemswithdelayare,therefore,oftheoreticalandprac-ticalimportance.Itiswellknown(see,e.g.,[6]–[9])thatthechoiceofanappro-priateLyapunov–Krasovskiifunctionaliscrucialforderivingstabilityandboundedrealcriteriaand,asaresult,forobtainingasolutiontovariouscontrolproblems.Thegeneralformofthisfunctionalleads,inthestate-feedbackcontrollerdesign,toacomplicatedsystemofRiccatitypepartialdiffer-entialequations[10],[11]orinequalities[12].SpecialformsofLyapunov–Krasovskiifunctionalsleadtosimplerdelay-in-dependent[13]–[16]and(lessconservative)delay-dependent[16]–[19],[9]Riccatiequationsorlinearmatrixinequalities(LMIs),for-gainanalysisorformemorylessstate-feedbackcontrollerdesign.Recently,increasingattentionhasbeenpaidtotheproblemsofobservation,output-feedbackstabilizationandthedesignofobserver-basedcontrollersforsystemswithstatedelay[20]–[27].Theonlysolutionsthathavebeenderived,sofar,fortheoutput-feedbackcontrolhavebeendelay-independent[20],[21],[26].Allthedelay-independentandthedelay-de-ManuscriptreceivedNovember6,2000;revisedNovember27,2000andJuly10,2001.RecommendedbyAssociateEditorY.Yamamoto.Thisworkwassup-portedbytheMinistryofAbsorptionofIsrael,andbytheCandMMausChairatTelAvivUniversity.TheauthorsarewiththeDepartmentofElectricalEngineering-Systems,TelAvivUniversity,TelAviv69978,Israel(e-mail:[e-mail]).PublisherItemIdentifierS0018-9286(02)02064-0.pendentresultsmentionedabovetreatthecaseofretardedtypesystems.Inthemoregeneralcaseofneutraltypesystems,wherethedelayappearsinthestatederivativeandinthestate,stabilityconditionsbaseduponLMIsorRiccatiequationshavebeenobtainedforboththedelay-independent[28],[5]andthedelay-dependent[29]–[31],[8]cases.Notethatunlikeretardedtypesystems,neutralsystemsmaybedestabilizedbysmallchangesindelays[32],[33].Concerningthecontrolproblemforneutralsystems,onlyadelay-independentstate-feedbacksolutionhasbeenachieved[34].Theconservatismofthedelay-dependentconditionsof[17]–[24],[31]istwofold:thetransformedandtheoriginalsys-temsarenotequivalent(see[35])andtheboundsplaceduponcertainterms,whendevelopingtherequiredcriteria,arequitewasteful.Recently,anewdescriptormodeltransformationwasintroducedin[8]forstabilityanalysisandhasitbeenappliedtostate-feedbackcontrolandfilteringofretardedtypesystemsin[8],[9],[19],and[27].Thisapproachsignificantlyreducestheoverdesignentailedintheexistingmethodssinceitisbasedonamodelthatisequivalenttotheoriginalsystemandsincefewerboundsareapplied.Theseboundscannowbemadetighterusingtherecent(lessconservative)boundoncrosstermsthatwasintroducedin[36].Inthepresentpaper,we,forthefirsttime,introduceadelay-dependentsolutiontotheoutput-feedbackcontrolproblemofsystemswithstatedelays.Wesolvethestate-feedbackandtheoutput-feedbackcontrolproblemsforneutraltypelinearsystemsbycombiningthedescriptorsystemapproachwiththenewboundingmethod.Newboundedrealcriteriaandstate-feedbacksolutionsaregivenintermsofLMIsforsystemswhichmaycontaindiscreteanddistributeddelaysandpolytopicpa-rameteruncertainties.Thesolutionswederivearedelay-depen-dent,howeverdelay-independentresultscanbeobtained,asaparticularcase,forcertainvaluesofthedesignparameters.Anoutput-feedbackcontrollerisderivedfromtwoLMIsbyap-plyingan“innovationfilter”intheformofadescriptorsystem.Solutionsareofferedforcasesofonlineanddelayedmeasure-ments.Thetheorydevelopedisdemonstratedthroughoutthetextviasixnumericalexamples.Theseexamplesillustratetheeffectivenessofoursolutionsascomparedtoresultsobtainedbyothermethods.Notation:Throughoutthepaperthesuperscript“”standsformatrixtransposition,denotesthedimensionalEu-clideanspacewithvectornorm,isthesetofallrealmatrices,andthenotation,formeansthatissymmetricandpositivedefinite.Thespaceoffunc-tionsinthataresquareintegrableoverisdenotedby0018–9286/02$17.00©2002IEEE254IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.47,NO.2,FEBRUARY2002withthenorm,thespaceofcontinuousfunc-t