Settingt1t0t2tfPtfFitisseenthatJxTt0Pt0xt0sincethe naltermisnonnegativeNotealsothatonsettingt2tfanduDxitfollowsthatifQCandF0thenPt10forallt1tfbecausetheleftsideisnonnegativeAlsomul ID: 143185
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SolutionoftheInverseProblemofLinearOptimalControlwithPositivenessConditionsandRelationtoSensitivityAntonyJamesonandElizerKreindlerJune19711FormulationLet_xAxBu;(1.1)wherethedimensionsofxanduaremandn,andletuDx;(1.2)beagivencontrol.Itisdesiredto ndaperformanceindexJtft0(xTQxuTRu)dtxT(tf)Fx(tf);(1.3)withRRT0,QQT0,whichisminimizedbyu.ThesolutionofthisproblemwithoutpositivenessconditionsonQisgivenin([?]).Ifaperformanceindex(1.3)existswhichisminimizedby(1.2),thenRDBTP;(1.4)wherePisasymmetricmatrixsatisfying_PATPPADTRDQ;P(tf)=F:(1.5)If(1.5)ismultipliedontheleftbyxTandontherightbyx,thensubstitutingfrom(1.1)and(1.4)xTQxuTRuddt(xTPx)+(uDx)TR(uDx)Integratingfromt1tot2t2t1(xTQxuTRu)dtxT(t2)P(t2)x(t2)=xT(t1)P(t1)x(t1)+t2t1((uDx)TR(uDx))dt:(1.6)1 Settingt1t0,t2tf,P(tf)=F,itisseenthatJxT(t0)P(t0)x(t0);sincethe naltermisnon-negative.Notealsothatonsettingt2tfanduDxitfollowsthatifQCandF0,thenP(t1)0forallt1tfbecausetheleftsideisnon-negative.Alsomultiplying(1.4)ontherightbyB,thesymmetryofPisseentoimplythesymmetryofRDB.TheconditionfortheexistenceofRRT0andPPT0satisfying(1.4)aregivenin([?]).Theyare:A1:DBhasnindependentrealeigen-vectors.A2:Theeigen-valuesofDBarenon-positive.A3:rDBrD,whererDdenotestherankofD,etc.ForP&#x-410;&#x.126;0A3isreplacedbyA3*:rDBrDrB.IfRRT0isgiven,thentheconditionsforasolutionP0to(1.4)areB1:RDBissymmetric.B2:RDB0.B3:rRDBrRD.ForP&#x-410;&#x.126;0B3isreplacedbyB3*:rRDBrRDrBHereB3andB3*aresimplyrestatementsofA3andA3*,butB3andB3*wouldstillbeneededforacasewhereRisonlynon-negative.2TheSensitivityInequalityThepropertythatthefeedbackcontrolminimizessomeperformanceindex(1.3)withQ0isofconsiderableinterestbecauseofitsconnectionwiththeabilityofthecontroltoreducethesensitivityofthesystemfoparametervariations.Letxcbethetrajectorydeviationresultingfromplantvariations,AandB,whenthefeedbackcontrol(1.2)isused,andletxobethedeviationwhen(1.2)isreplacedbyanopenloopcontrolwhichwouldgivethesametrajectoryintheabsenceofparameterdeviations.AlsoletAA;BB;andde nex=lim!0.UsingtheequivalenceofcontrolswhenAB0wehave_xc=(ABD)xc+(ABD)x;(2.1)_x0Axo+(ABD)x:(2.2)2 Whencexcxo(2.3)where_ABDxc(2.4)Thentt0xTcDTRDxcdttt0xT0DTRDxodttt0TYdt;(2.5)foralltifthefollowingconditionissatis edC:ThesensitivityinequalitySy(t)=tt0(uDx)TR(uDx)uTRu dttt0xTYxdt0(2.6)holds,wherexisthesolutionof(1.1)withx(t0)=0underanarbitraryinputu.ThisfollowsonsettinguDxcandinterpretingxas.Nowsettingt1t0,x(t0)=0,andt2tin(1.6)itisseenthatCholdswithYQwhenthecontrol(1.2)minimizestheperformanceindex(1.3),providedthatP(t)0.ThisisinturnensuredifQ0andF0.Thenon-negativenessofQandFthusguaranteesareductioninsensitivitytoparametervariationsinthesenseof2.5.3SolutionoftheInverseProblemwithQItwasremarkedinSection1thatQ0impliesP0.IfRisnotspeci ed,conditionsA1,A2,andA3arenecessaryforasolutionoftheinverseproblem.AlsoA1,A2,andA3*arenecessaryandsucientforP�0,anditwillnowbeshownthattheexistenceofasolutionP�0to(1.4)issucientforasolutionwithQ0.WethushaveTheorem3.1ConditionsA1,A2andA3arenecessaryforasolutionoftheinverseproblemwithQ0.ConditionsA1,,A2andA3*arealsosucient.ProofNecessityhasalreadybeenestablished.Toprovesuciencyobservethatfrom(1.1)ddt(xe t)=(A I)xe tBue t:(3.1)Thusthecontrol(1.2)minimizesthepreformanceindexJtft0e 2 t(xTQoxuTRou)dte 2 txT(tf)Fox(tf):(3.2)3 If(1.4)holdstogetherwith_P=(A I)TPP(A I)DTRoDQo;P(tf)=Fo:(3.3)UnderconditionsA1,A2andA3*itispossibletoconstructRRT0andPPT0satisfying(1.4).ThenQomaybeconstructedasQoQ1+2 P;(3.4)whereQ1DTRoDATPPA_P:(3.5)SinceP�0andQ1isa xedfunctionoft,itisalwayspossibletochoose �0sucientlylargethatQo0.Alsocomparisonof(3.3)and(3.4)with(3.3)showsthatthecontrol(1.2)minimizes(3.2)andhence(1.3)onsettingQQoe 2 t,RRoe 2 t,FFoe 2 t.Corollary3.2Ifthecontrol(1.2)satis esconditionsA1,A2andA3*thenitsatis esthecriterion(2.5)forsensitivityreductionforsomeY0.ProofObeservethattheprocedureofTheorem(3.1)givesajointsolutionforQandR,butcannotbeusedtoconstructQ0whenRisgiven.Inparticular,ifthesystemsiconstantitleadstoaperformancewithanexponentialtimeweightingfactore 2 t, �0,andestablishesthesensitivitycriterion(2.5)withasimilarfactor.4SolutionoftheInverseProblemwithQ0forgivenWenowconsidermethodsofsolvingforQwhenRisagivenmatrixsatisfyingconditionsB1,B2andB3.To ndqdditionalrequirementsonRforQ0multiply(1.5)ontheleftbyBT.Thenusing(1.4)BT_PBTATPRDABTDTRDBTQ:Butdi erentiating(1.4)BT_P_BTPddt(RD):ThusBTQL;(4.1)whereLBTDTRDRDA(BTAT_BT)Pddt(RD):(4.2)Alsomultiplying(4.2)ontherightbyBandagainusing(1.4)BTQBM;(4.3)4 whereMBTDTRDBRDABBTATDTRddt(RDB)RD_B_BTDTR:(4.4)SinceMdependsonlyonthesystemmatricesandR,andBTQB0ifQ0,thenecessaryconditionforQ0is:B4:M0whereMisde nedin(4.4)InordertoconstructQ0weshallassumethestrongercondition:B4*M�0Aslongas(1.4)holdsBTLTM:(4.5)Thusif(4.1)isregardedasanequationforQithasasymmetricsolutionQoLTM 1L:MoreoverifQisanyothersolutionthenBT(QQo)=0:ThusthegeneralsymmetricsolutionforQisQLTM 1LY;(4.6)whereYisanysymmetricmatrixsuchthatBTY=0:(4.7)ConditionB4*ensuresthatQ0ifY0.Alsoletx=(IBM 1L)z;wherezisanarbitraryvector.ThenxTQxzTYz:ThusY0isalsonecessaryforQ0.Considerthedi erentialequationthatisobtainedwhen(4.6)issubstitutedforQin(1.5):_PATPPADTRDLTM 1LY:(4.8)SinceMisgivenandLislinearinP,thisisaRicattiequationwhichcanbeintegratedtodeterminePandhenceL.Weshallverifythat(4.8)hassolutionsthatsatisfy(1.4).De neKbyKBTPRD(4.9)5 sothatK=0if(1.4)holds.Thenwhen(1.4)isnolongerassumedtohold(4.2)and(4.4)yieldBTLTMK(AB_B)(4.10)insteadof(4.5).Alsousing(4.8)_KBT_P_BTPddt(RD)BTATPBTPABTDTRDBTLTM 1LBTY_BTPddt(RD);whenceinviewof(4.2),(4.7)and(4.10)_KLKAhMK(AB_B)iM 1L(4.11)KhA(AB_B)M 1Li:UnderconditionsB1,B2andB3itispossibletochooseP(tf)0satisfying(1.4).ThenK(tf)=0andthesolutionof(4.11)whenintegratedbackwardsisK=0.Thecorrespondingsolutionof(4.8)thereforesatis es(1.4).Theorem4.1GivenRRT0,conditionsB1-B4arenecessaryforasolu-tionoftheinverseproblemwithQ0.ConditionsB1-B3aresucientforasolutionoversome nitetimeinterval.EverysolutionwithQ0isthengivenbythesolutionof(4.2)and(4.8)forsomeY0satisfying(4.7).Ifequation(4.11)isunstablewhenintegratedbackwardsthentheintegra-tionof(4.8)wouldtendtodriftawayfromsatisfying(1.4).ToovercomethisdicultywemayintroduceinsteadofLandMthematricesL1BTDTRDRDA1BTP(A1A)(BTAT_BT)Pddt(RD);(4.12)andM1BTL1(4.13)whereA1isamatrixtobeselected.ThenL1LK(A1A);andfrom(4.10)M1BTLTBT(AT1AT)KTMK(AB_B)(BTATBTAT1)KT:ThusL1LandM1MwhenK=0.Wenowintegrate_PATPPADTRDL1M 11L1Y(4.14)6 Thenfrom(4.7)and(4.13)_KBTATPBTPABTDTRDL1_BTPddt(RD)whence(4.12)yields_KKA1ThusifA1ischosenasanystablematrix,backwardintegrationof(4.14)willpreserveK=0withoutdangerofdrift.ButthenL1LandM1M,sothatalongtheintegrationpathM1MT1andunderconditionB4*,M10.WhileconditionsB1-B3andB4*establishtheexistenceofQ0oversome nitetimeinterval,andeveryQcanthenbeconstructedfromequation(4.6)or(4.14),theseconditionsdonotestablishtheexistenceofQ0overanarbitrarilylargetimeinterval,becauseequation(4.8)or(4.14)whichfollowsthesameintegrationpath,mayhavea niteescapetime.Inparticular,conditionsB1-B3andB4*arenotsucientfortheexistenceofasolutionwithconstantQ0andR�0,whenA,BandDareconstant.Thisiseasilyseenfromanexample.Considerthesystem_x0110x10u;withu212x;whereisisdesiredto ndaperformanceindexJ10(xTQxu2)dt;whichisminimizedbyu.RDBandMarescalars,RDB2;andM=(BTDT)2+2DAB=5:SoconditionsB1-B3andB4*allhold.Ontheotherhand(1.4)maybesolvedforPtogiveP21212P22;whereP22istheonlyundeterminedelementinP.ThensubstitutingforPin(1.5)with_P=0givesQDDATPPA51P221P2234;sothatitisnotpossibletoobtainQ0bychoiceofP22whenPisconstant.7 Thisexampleindicatestheneedtoexaminetheconditionsunderwhich(4.8)or(4.14)canbeintegratedoveranarbitraryinterval.Since(4.14)followsthesamepathas(4.8)whenintegratedfroma nalvalueofPwhichsatis es(1.4),itsucestoconsider(4.8).Substitutingfrom(4.2),itmaybewrittenas_P1ATP1P1ADTRDYDT1MD1;(4.15)whereP1P;(4.16)andD1M 1(BTATP1_BTDTRDRDAddt(RD)_BTP1)(4.17)Let_x1Ax1ABu1_Bu1:(4.18)Thenequations(4.15)and(4.17)areequationsfordeterminingthecontrolu1D1x1;(4.19)whichminimizesJ1tft1xT1(DTRDY)x1(4.20)+2uT1(BTDTRDRDAddt(RD))x1uT1Mu1dtxT1(tf)P1(tf)x1(tf):SubstitutingforMfrom(4.4)andusing(4.7),(4.20)becomesJ1tft1(x1Bu1)T(DTRDY)(x1Bu1)dt(4.21)+2tft1uT1(RDAddt(RD))(x1Bu1)dttft1uT1ddt(RDB)u1dtxT1(tf)P1(tf)x1(tf):Setxx1Bu1:(4.22)Thenxsatis es(1.1)where_u1u,oru1tft1udt;(4.23)therebeingnoconstantif(4.18)and(1.1)arebothtobeinequilibriumwitzerocontrol.8 Thesecondandthirdtermsin(4.21)become2tft1uT1(RDAddt(RD))(x)dttft1uT1ddt(RDB)u1dttft12uT1RD_x+2uT1ddt(RD)x2uT1RDB_u1uT1ddt(RDB)u1dt2uT1RDxuT1RDBu1tft12tft1uRDxdt:Alsousing(1.4)and(4.16)xT1P1x1+2uT1RDx1uT1RDBu1xTP1xxTPx:Thus(4.21)becomesJ1SyxT(tf)P(tf)x(tf)+I(t1);(4.24)whereSyisde nedby(2.6)andI2u1RDx1u1RDBu1:(4.25)Now(4.23)showsthatanon-zerovalueofu1attt1correspondstoanimpulseattt1inuuu1(t1)(tt1):Butundersuchanimpulsexisshiftedfromx 0tox+0x 0Bu1(t1)withacontributiontoSyexactlyequaltoI(t1),sincex1iscontinuoussothatx1(t1)=x 0.Thusthe rstandthirdtermsinJ1equalSyevaluatedfromt 1incaseofaninitialimpulseinu.ItfollowsthatxT(t1)P(t1)x(t1)istheminimumvalueofSywhenx(t1)=x 0.Supposethatthisquantityisnotbounded.Thensincethesystemiscontrollable,itcanbebroughtfromx(t0)=0tox(t1)=x 0witha nitecontributiontoSy,andhenceovertheinterval(to;tf)conditionCwouldbeviolated.WededucethatconditionCissucientfortheexistenceofasolutionto(4.8).9