1 ZEROS OF SPECTRAL FACTORS, THE GEOMETRY OF SPLIT

1ZEROSOFSPECTRALFACTORS,THEGEOMETRYOFSPLITTINGSUBSPACES,ANDTHEALGEBRAICRICCATIINEQUALITY*ANDERSLINDQUIST†,GY¨ORGYMICHALETZKY‡,ANDGIORGIOPICCI§Abstract.Inthispaperweshowhowthezerodynamicsof(notnecessarilysquare)spectralfactorsrelatetothesplittingsubspacegeometryofstationarystochasticmodelsandtothecorre-spondingalgebraicRiccatiinequality.Weintroducethenotionofoutput-inducedsubspaceofamin-imalMarkoviansplittingsubspace,whichisthestochasticanalogueofthesupremaloutput-nullingsubspaceingeometriccontroltheory.Throughthisconcepttheanalysiscanbemadecoordinate-free,andstraightforwardgeometricmethodscanbeapplied.WeshowhowthezerostructureofthefamilyofspectralfactorsrelatestothegeometricstructureofthefamilyofminimalMarkoviansplittingsubspacesinthesensethattherelationshipbetweenthezerosofdifferentspectralfactorsisreflectedinthepartialorderingofminimalsplittingsubspaces.Finally,wegeneralizethewell-knowncharacterizationofthethesolutionsofthealgebraicRiccatiequationintermsofLagrangiansubspacesinvariantunderthecorrespondingHamiltoniantothelargersolutionsetofthealgebraicRiccatiinequality.Keywords.Zerodynamics,Markoviansplittingsubspaces,minimalspectralfactors,matrixRiccatiinequality,algebraicRiccatiequation,geometriccontroltheory.AMSsubjectclassifications.93E03,93B27,60G10.1.Introduction.Bynowitshouldbefairlywell-knownthatthereisaone-onecorrespondencebetweeneachtwothefollowingthreefundamentalareasofsystemstheory.(i)Minimalspectralfactorizationofarational(full-rank)m×mspectraldensitymatrixΦ.Theproblemistofindall(squareandrectangular)rationalfunctionsW(s)=C(sI−A)−1B+D,(1.1)(whereprimedenotestransposition)withpolesintheopenlefthalfplane,satisfyingthefactorizationequationW(s)W(−s)=Φ(s),(1.2)andbeingminimalinthesensethattheMcMillandegreeofWisexactlyhalfofthatofΦ.Theclassofallsuchminimalspectralfactors,eachdefinedmodulorightmultiplicationbyaconstantorthogonalmatrix,willbedenotedbyW.ThesubclassofsquarespectralfactorswillbedenotedW0.Throughoutthispaperweshallal-waysconsiderrepresentationsforwhich(A,B,C)isaminimaltripletandBDhasindependentcolumns.Thisresultsinnolossofgenerality[16].(ii)FindingallsymmetricsolutionsofthealgebraicRiccatiinequalityΛ(P)≤0,(1.3)∗ThisworkwaspartiallysupportedbytheSwedishBoardforTechnicalDevelopment,theG¨oranGustafssonFoundation,LADSEB-CNR,andOTKAundergrantNo2042.†DivisionofOptimizationandSystemsTheory,RoyalInstituteofTechnology,10044Stockholm,Sweden‡DepartmentofProbabilityTheoryandStatistics,E¨otv¨osLor´andUniversity,Budapest§DepartmentofElectronicsandInformatics,UniversityofPadova,andLADSEB-CNR,Padova,Italy12A.LINDQUIST,G.MICHALETZKYANDG.PICCIwhereΛ:Rn×n→Rn×nisgivenbyΛ(P)=AP+PA
+(¯C−CP)
R−1(¯C−CP),(1.4)thematricesA∈Rn×n,C,¯C∈Rm×nandR∈Rm×mbeingdefinedthroughaminimalrealizationΦ+(s)=C(sI−A)−1¯C
+12R(1.5)ofthepositiverealpartΦ+ofthespectraldensityΦ,i.e.ofallrationalmatrixfunctionssatisfyingΦ(s)=Φ+(s)+Φ+(−s)
(1.6)Φ+istheonehavingallitspolesintheopenlefthalfplane.HereweassumethatR:=Φ(∞)0.LetusdenotebyPthesolutionsetof(1.3).TheneachP∈Pcorrespondstoaspectralfactor(1.1)whoseB-andD-matricesaredeterminedbyafull-rankmatrixfactorizationofthetypeBD[B
,D
]=−AP−PA
¯C
−PC
¯C−CPR(1.7)Obviouslythecorrespondenceisone-onemodulotrivialcoordinatetransformations([1],[9]).(iii)Describingallminimalstochasticrealizationsofanm-dimensionalstation-ary-incrementsprocess{y(t);t∈R}havingthe(incremental)spectraldensityΦ.Eachstochasticrealizationisobtainedbypassingasuitable”whitenoise”throughafilterdw−→Wdy−→(1.8)havinganm×pminimalspectralfactorasitstransferfunction,thusyieldingalineardynamicalmodel(Σ)dx=Axdt+Bdwdy=Cxdt+Ddw(1.9)fordy,definedonthewholerealline.Moreprecisely,wisavectorWienerprocessonRofadimensionpequaltothenumberofcolumnsofW.ThesystemΣisinstatisticalsteadystatesothatthen-dimensionalstateprocessxandtheincrementsofthem-dimensionaloutputprocessyarejointlystationary.ThemodelΣisaminimalstochasticrealizationinthesensethatthereisnootherrepresentationofdyoftype(1.9)withastateprocesswithfewercomponents.Inregardtotopic(iii),itisactuallymorenaturaltoconsideracoordinate-freerepresentationbyassigningtoeachmodelΣthen-dimensionalspaceX={ax(0)|a∈Rn}(1.10)ofrandomvariables.ThisspaceisthesubspaceofanambientspaceHofthemodel(1.9),definedastheclosureofthelinearhullofthefollowingrandomvariables{wi(t)−wi(τ);i=1,2,...,p;t,τ∈R}inthetopologyoftheinnerproductξ,η=E{ξη},(1.11)ZEROS,SPLITTINGSUBSPACES,ANDTHEARI3whereE{·}standsformathematicalexpectation.TheambientspaceHisnaturallyequippedwiththeshiftinducedbydw,i.e.thestronglycontinuousgroupofunitaryoperators{Ut;t∈R}onHsuchthatUt[wi(τ)−wi(σ)]=wi(τ+t)−wi(σ+t)foralli=1,2,...,pandt,τ,σ∈R.AllrandomvariablesofΣbelongtoH,andmoreovertheprocessesxanddyarestationarywithrespectto{Ut},i.e.Utxi(τ)=xi(τ+t)foralli=1,2,...,nandt,τ∈RandUt[yi(τ)−yi(σ)]=yi(τ+t)−yi(σ+t)foralli=1,2,...,mandt,τ,σ∈R.MinimalityofΣcorrespondstominimalityofthesubspaceXinthesenseofsubspaceinclusion,andhencealsointhesenseofdimension[16].DefiningthepastandfutureoutputspacesasH−=closure{a[y(t)−y(s)]|a∈Rm,t,s≤0}andH+=closure{a[y(t)−y(s)]|a∈Rm,t,s≥0}