Gershgorin-Brualdi Perturbations and Riccati Equat

Gershgorin-BrualdiPerturbationsandRiccatiEquationsDiederichHinrichsenZentrumf¨[e-mail][e-mail]PaulFuhrmannontheoccasionofhis70thbirthday.AbstractForuncertainlinearsystemswithcomplexparameterperturbationsofstaticoutputfeedbacktypeaquadraticLiapunovfunctionofmaximalrobustnesswasconstructedin[5].SuchLiapunovfunctionscanbeusedtoensurethestabilityofuncertainsystemsunderarbitrarynonlinearandtime-varyingperturbationswhicharesmallerthanthestabilityradius.InthispaperweestablishanalogousresultsforstructuredGershgorin-Brualditypeperturbationsofdiagonalmatriceswhereallthematrixen-triesatanarbitrarilyprescribedsetofpositionsareindependentlyperturbed.Wealsoderiveexplicitandcomputableformulaefortheassociatedμ-values,stabilityradiiandspectralvaluesets.1IntroductionGershgorin’stheorem[4]hasfoundmanyapplicationsincontroltheory.Forinstance,thedirectandinverseNyquistarraymethodsinmultivariablefeedbackdesign[12]arebaseduponthisresultandtherelatedconceptofdiagonaldominanceplaysanimportantroleinstabilityanalysisandcontroloflargescalesystems[13].However,refinementsofGer-shgorin’sinclusiontheorem,likeBauer’sandBrualdi’stheorems,havehardlyeverbeenappliedinsystemstheory,eventhoughtheymayyieldlessconservativerobustnessresults,forinstance,inthestabilityanalysisofcompositesystemswithagiveninterconnectionstructure,see[11].Foradetailedandup-to-datesurveyoftheresultsavailableinthefieldofGershgorintypeinclusiontheoremsfromalinearalgebraicpointofview,thereaderisreferredtotherecentbookbyR.S.Varga[14].Thisbookalsocontainsanextensivelistofreferences.Inthispaperwewillstudyperturbationsofdiagonalsystemmatriceswherealltheentriesatanarbitrarilyprescribedsetofpositionsareindependentlyperturbed.ThestructureoftheperturbationsisdefinedviaamatrixE=(eij)withentrieseither1or0.Ifeij=1thereisaperturbationatthe(i,j)positionandifeij=0thereisnoperturbation.SuchperturbationswillbecalledGershgorin-Brualdiperturbations.Brualdiwasthefirsttoderiverefinedeigenvalueinclusionregionsbytakingthezeropatternofthematrixintoaccount.Inhispaper[3]heusedthisideainordertosharpenGershgorin’sandBrauer’spreviousinclusiontheorems[4],[2],seealso[9,Ch.6]and[14,Ch.2].Incontrastwiththeaboveauthorswedonotdealwiththeproblemoffindinganinclusionregionforthespectrumofagivenmatrix,butwetrytodeterminethesetofeigenvaluesofallthematriceswhichareperturbationsofafixeddiagonalmatrix,withzeroperturbation1entriesatprescribedpositions,andwheretheperturbationsareboundedinnormbyanarbitrarypositiverealδ0.Theproblemofdeterminingtheseso-calledspectralvaluesetswillbestudiedinthecontextofgeneralizedμ-analysis,see[7].Thefirstaimistoobtainexplicitcomputableformulasfortheassociatedμ-values,stabilityradiiandspectralvaluesets.Ourapproachisdifferentfromthatoftherecentarticle[11]where,moregenerally,Gershgorintypeperturbationsofblock-diagonalmatricesandvariousperturbationnormsareconsidered.Herewestudyaspecificperturbationnormforwhichasurprisingvarietyofconcreteresultscanbeobtained.TheessentialnewtoolisaparametrizedMetzlermatrixwhichpermitsacomputablecharacterizationoftheμ-value.Thisinturnyieldsexplicitformulasforthecorrespondingstabilityradiiandspectralvaluesets.MoreoverinthecasewhereEisreduciblewewillseethattheproblemofcomputingthemcanbedecomposedintocomputingstabilityradiiandspectralvaluesetsofsubsystemscorrespondingtoitsirreduciblecomponents.Inthecaseoffull-blockperturbationscarryingthespectralnormitisknown(see[5],[6]andSection2)thatthestabilityradiuscanbecharacterizedbymeansofaparametrizedRiccatiequation.ThismakesitpossibletoconstructaquadraticLiapunovfunctionofmaximalrobustnessandisakeyforobtainingtightrobustnessresultswithrespecttotime-varyingandnonlinearfull-blockperturbations.Wewillshowthat,althoughtherearesubtledifferences,similarresultscanbeobtainedforarbitraryGershgorin-Brualdiperturbationstructures.TothebestofourknowledgeLiapunovfunctionsofmaximalrobustnesshavenotbeenconstructedforhighlystructuredperturbationsbefore.TheproblemofconstructingsuchLiapunovfunctionsinthecontextofμ-analysishasbeenstatedasanopenproblemin[11].ThisproblemissolvedhereforthespecialcaseofGershgorin-Brualdiperturbations.Weproceedasfollows.InSection2werecallthedefinitionsofμ-value,stabilityradiusandspectralvaluesetsaswellassomeknownresults,especiallyforthefull-blockcase.WealsointroduceaparametrizedalgebraicRiccatiequationandstateatheoremconcerningtheexistenceofHermitiansolutions.InSection3weintroducethecrucialtoolsofouranalysis,theparametrizedMetzlermatrixandthesingularityparameterassociatedwithadiagonalmatrixandagivenGershgorin-Brualdiperturbationstructure.Bymeansofthissingularityparameterwecharacterizethestabilityradiusandspectralvaluesets.InparticularweprovethattherealandthecomplexstabilityradiicoincideforGershgorin-Brualdiperturbationsifthesystemdataarereal.InSection4weintroduceparametrizedalgebraicRiccatiequationsandconstructquadraticLiapunovfunctionsofmaximalro-bustnessfordiagonalsystems