Continuity properties of solutions to H 2 and H #

ContinuitypropertiesofsolutionstoH2andH∞RiccatiequationsA.A.Stoorvogel∗A.SaberiDept.ofMathematicsandComp.ScienceSchoolofElectricalEngineeringEindhovenUniversityofTechnologyandComputerScienceP.O.Box513WashingtonStateUniversity5600MBEindhovenPullman,WA99164-2752TheNetherlandsU.S.A.Fax:[phone]Fax:[phone]E-mail:[e-mail]-mail:[e-mail]18,1995AbstractInH2andH∞optimalcontrol(semi-)stabilizingsolutionsofalgebraicRiccatiequationsplayanessentialrole.Itiswell-knownthatthesesolutionsmighthavediscontinuitiesasafunctionofthesystemparameters.Thepapershowsthatthesediscontinuitiesaredirectlylinkedtonon-left-invertibilityand,incontrasttowhatonemightthink,unrelatedtozerosontheimaginaryaxis.Keywords:algebraicRiccatiequations,quadraticmatrixinequalities.1IntroductionInmostH2andH∞controlproblemssolutionsofthealgebraicRiccatiequationplayacrucialrole.NotethatingeneralforcontinuoustimesystemswehavetousequadraticmatrixinequalitiesinsteadofRiccatiequations.Fordetailswereferto[5].Inparticularweareinterestedinthestabilizingso-lutionoftheseRiccatiequationsandquadraticmatrixinequalities.However,ifthesystemhaszerosontheimaginaryaxis(continuoustime)orontheunitcircle(discretetime),wehavetostudysemi-stabilizingsolutions.ThesearesolutionsoftheRiccatiequation/quadraticmatrixinequalityassoci-atedtoeigenvaluesintheclosedleft-halfplane(continuoustime)orintheclosedunitcircle(discretetime).Thestandardwaytoobtainsemi-stabilizingsolutionsisacheapcontrolargumentwhereweperturbthesystemparameterstoobtainasystemwithoutproblemsinducedbyforinstancethezerosontheboundaryofthestabilitydomain.Anaturalquestionisthenwhetherthesemi-stabilizingsolu-tionsdependcontinuouslyonthesystemparameters.Therearesimpleexampleswherethesolutiondoesnotdependcontinuouslyonthesystemparameters(seee.g.[2]).Ontheotherhand,[8]iden-tifiesaclassofperturbationswhichguaranteeacontinuousbehaviour.Wewouldliketostudythisquestioninmoredetail.Wewillclearlyidentifywhatkindofperturbationscanyielddiscontinuousbehaviourandintheprocessshowthatforaverylargeclassofsystemsdiscontinuitiesneveroccur.Wewillconsiderbothcontinuousanddiscretetimesystems.Notationinthispaperismostlystandard.ByM†wedenotetheMoore-PenroseinverseofM.∗Theresearchofdr.A.A.StoorvogelhasbeenmadepossiblebyafellowshipoftheRoyalNetherlandsAcademyofSciencesandArts.12Discretetimesystems2.1ProblemformulationConsiderthefollowingdiscretetimeRiccatiequation:P=ATPA+CTC−BTPA+DT1CETPA+DT2C!TG(P)†BTPA+DT1CETPA+DT2C!,(2.1)whereG(P):=DT1D1DT1D2DT2D1DT2D2−γ2I!+BTET!PBE,(2.2)subjecttoDT2D2+ETPE−(DT2D1+ETPB)(DT1D1+BTPB)†(DT1D2+BTPE)γ2I.(2.3)Weareinterestedinrealsymmetricsemi-stabilizingsolutionsofthisalgebraicRiccatiequation.ThesearesolutionsofthealgebraicRiccatiequationwherethezerosofthematrixpencilzI−A−B−EBTPA+DT1CDT1D1+BTPBDT1D2+BTPEETPA+DT2CDT2D1+ETPBDT2D2+ETPE−γ2I(2.4)areinsideorontheunitcircle.IfthezerosarestrictlyinsidetheunitcirclewewillcallPastabilizingsolutionoftheRiccatiequation.ThisRiccatiequationisassociatedtothefollowingsystem:6:(x(k+1)=Ax(k)+Bu(k)+Ew(k),z(k)=Cx(k)+D1u(k)+D2w(k).(2.5)Basicallythereexistsastabilizingfeedbacku=F1x+F2wsuchthattheclosedloopH∞normislessthanγifandonlyifthereexistsapositivesemi-definitesemi-stabilizingsolutionoftheaboveRiccatiequationandsomeadditionalconditions(see[6]).Forγ=∞thegeneralRiccatiequation(2.1)reducestotheH2Riccatiequation:P=ATPA+CTC−(ATPB+CTD1)(BTPB+DT1D1)†(BTPA+DT1C).(2.6)Moreover,theextracondition(2.3)becomesvoid.Finally,thestabilityrequirementisimposedonthefollowingmatrixpencil:zI−A−BBTPA+DT1CBTPB+DT1D1!.(2.7)TheRiccatiequationisassociatedtothesystem6whichisparameterizedby(A,B,E,C,D1,D2).WedefinethesetDtobetheclassofsystems6forwhich(A,B,C,D1)isleft-invertibleandforwhichthereexistsmatricesF1,F2suchthatA+BF1isstableandk(C+DF1)(zI−A−BF1)−1(E+BF2)+(D2+D1F2)k∞γ(2.8)2FortheH2problem,whereγ=∞thesetDconsistsofsystems6forwhich(A,B,C,D1)isleft-invertibleand(A,B)isstabilizable.Inthatcaseitisknown(see[7])thatthereexistsauniquerealsymmetricsemi-stabilizingsolutionPoftheRiccatiequation.Moreover,thissolutionispositivesemi-definite.FortheH∞controlproblem(i.e.finiteγ)thesemi-stabilizingsolutionneednotbeunique.However,forelementsofthesetDthereexistsasemi-stabilizing,positivesemi-definitesolutionoftheRiccatiequation.Inthispaper,wewillstudythesmallest,positivesemi-definiterank-minimizingsolutionPofthequadraticmatrixinequalitywhichalwaysexistsandisobviouslyunique.Themainobjectiveistostudywhetherthisparticularsolutiondependscontinuouslyonthesystemparameters.InthenextsubsectionwestudythebehaviourofPwhenwevarythesystemparametersoverthesetD.WewillshowthatPdependscontinuouslyonthesystemparametersbothforfiniteγandforγ=∞.Sinceweallowforzerosontheunitcircle,thiscontinuityisfarfromobvious.WeconsidersystemsoutsidethesetDinthesubsection2.3.ForelementsofthesetD,thesystem(A,B,C,D1)isleft-invertible.Thisimpliesthatthegener-alizedinversesin(2.1),(2.3)and(2.6)becomestandardinverses.Moreoverforsemi-stabilizingandstabilizingsolutionsoftheRiccatiequationwecansimplystudytheeigenvaluesofthefollow