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LinearMatrixInequalitiesinSystemandControlTheory SIAMStudiesinAppliedMathematicsThisseriesofmonographsfocusesonmathematicsanditsapplicationstoproblemsofcurrentconcerntoindustry,government,andsociety.Thesemonographswillbeofinteresttoappliedmathematicians,numericalanalysts,statisticians,engineers,andscientistswhohaveanactiveneedtolearnusefulmethodology.SeriesListVol.1Lie-BÄacklundTransformationsinApplicationsRobertL.AndersonandNailH.IbragimovVol.2MethodsandApplicationsofIntervalAnalysisRamonE.MooreVol.3Ill-PosedProblemsforIntegrodi®erentialEquationsinMechanicsandElectromagneticTheoryFrederickBloomVol.4SolitonsandtheInverseScatteringTransformMarkJ.AblowitzandHarveySegurVol.5FourierAnalysisofNumericalApproximationsofHyperbolicEquationsRobertVichnevetskyandJohnB.BowlesVol.6NumericalSolutionofEllipticProblemsGarrettBirkho®andRobertE.LynchVol.7AnalyticalandNumericalMethodsforVolterraEquationsPeterLinzVol.8ContactProblemsinElasticity:AStudyofVariationalInequalitiesandFiniteElementMethodsN.KikuchiandJ.T.OdenVol.9AugmentedLagrangianandOperator-SplittingMethodsinNonlinearMechanicsRolandGlowinskiandP.LeTallecVol.10BoundaryStabilizationofThinPlateSplinesJohnE.LagneseVol.11Electro-Di®usionofIonsIsaakRubinsteinVol.12MathematicalProblemsinLinearViscoelasticityMauroFabrizioandAngeloMorroVol.13Interior-PointPolynomialAlgorithmsinConvexProgrammingYuriiNesterovandArkadiiNemirovskiiVol.14TheBoundaryFunctionMethodforSingularPerturbationProblemsAdelaidaB.Vasil'eva,ValentinF.Butuzov,andLeonidV.KalachevVol.15LinearMatrixInequalitiesinSystemandControlTheoryStephenBoyd,LaurentElGhaoui,EricFeron,andVenkataramananBalakrishnan StephenBoyd,LaurentElGhaoui,EricFeron,andVenkataramananBalakrishnanLinearMatrixInequalitiesinSystemandControlTheorySocietyforIndustrialandAppliedMathematics¨Philadelphia Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics.Allrightsreserved.Nopartofthisbookmaybereproduced,stored,ortransmittedinanymannerwithoutthewrittenpermissionofthePublisher.Forinformation,writetotheSocietyforIndustrialandAppliedMathematics,3600UniversityCityScienceCenter,Philadelphia,Pennsylvania19104-2688.TheroyaltiesfromthesalesofthisbookarebeingplacedinafundtohelpstudentsattendSIAMmeetingsandotherSIAMrelatedactivities.ThisfundisadministeredbySIAMandquali¯edindividualsareencouragedtowritedirectlytoSIAMforguidelines.LibraryofCongressCataloging-in-PublicationDataLinearmatrixinequalitiesinsystemandcontroltheory/StephenBoyd...[etal.].p.cm.--(SIAMstudiesinappliedmathematics;vol.15)Includesbibliographicalreferencesandindex.ISBN0-89871-334-X1.Controltheory.2.Matrixinequalities.3.Mathematicaloptimization.I.Boyd,StephenP.II.Series:SIAMstudiesinappliedmathematics:15.QA402.3.L4891994515'.64--dc2094-10477 ContentsPrefaceviiAcknowledgmentsix1Introduction11.1Overview::::::::::::::::::::::::::::::::::11.2ABriefHistoryofLMIsinControlTheory:::::::::::::::21.3NotesontheStyleoftheBook::::::::::::::::::::::41.4OriginoftheBook:::::::::::::::::::::::::::::52SomeStandardProblemsInvolvingLMIs72.1LinearMatrixInequalities:::::::::::::::::::::::::72.2SomeStandardProblems:::::::::::::::::::::::::92.3EllipsoidAlgorithm:::::::::::::::::::::::::::::122.4Interior-PointMethods:::::::::::::::::::::::::::142.5StrictandNonstrictLMIs:::::::::::::::::::::::::182.6MiscellaneousResultsonMatrixInequalities::::::::::::::222.7SomeLMIProblemswithAnalyticSolutions::::::::::::::24NotesandReferences:::::::::::::::::::::::::::::::273SomeMatrixProblems373.1MinimizingConditionNumberbyScaling::::::::::::::::373.2MinimizingConditionNumberofaPositive-De¯niteMatrix::::::383.3MinimizingNormbyScaling:::::::::::::::::::::::383.4RescalingaMatrixPositive-De¯nite:::::::::::::::::::393.5MatrixCompletionProblems:::::::::::::::::::::::403.6QuadraticApproximationofaPolytopicNorm:::::::::::::413.7EllipsoidalApproximation:::::::::::::::::::::::::42NotesandReferences:::::::::::::::::::::::::::::::474LinearDi®erentialInclusions514.1Di®erentialInclusions:::::::::::::::::::::::::::514.2SomeSpeci¯cLDIs:::::::::::::::::::::::::::::524.3NonlinearSystemAnalysisviaLDIs:::::::::::::::::::54NotesandReferences:::::::::::::::::::::::::::::::565AnalysisofLDIs:StateProperties615.1QuadraticStability:::::::::::::::::::::::::::::615.2InvariantEllipsoids:::::::::::::::::::::::::::::68v viContentsNotesandReferences:::::::::::::::::::::::::::::::726AnalysisofLDIs:Input/OutputProperties776.1Input-to-StateProperties:::::::::::::::::::::::::776.2State-to-OutputProperties::::::::::::::::::::::::846.3Input-to-OutputProperties::::::::::::::::::::::::89NotesandReferences:::::::::::::::::::::::::::::::967State-FeedbackSynthesisforLDIs997.1StaticState-FeedbackControllers:::::::::::::::::::::997.2StateProperties::::::::::::::::::::::::::::::1007.3Input-to-StateProperties:::::::::::::::::::::::::1047.4State-to-OutputProperties::::::::::::::::::::::::1077.5Input-to-OutputProperties::::::::::::::::::::::::1097.6Observer-BasedControllersforNonlinearSystems:::::::::::111NotesandReferences:::::::::::::::::::::::::::::::1128Lur'eandMultiplierMethods1198.1AnalysisofLur'eSystems:::::::::::::::::::::::::1198.2IntegralQuadraticConstraints::::::::::::::::::::::1228.3MultipliersforSystemswithUnknownParameters:::::::::::124NotesandReferences:::::::::::::::::::::::::::::::1269SystemswithMultiplicativeNoise1319.1AnalysisofSystemswithMultiplicativeNoise::::::::::::::1319.2State-FeedbackSynthesis:::::::::::::::::::::::::134NotesandReferences:::::::::::::::::::::::::::::::13610MiscellaneousProblems14110.1OptimizationoveranA±neFamilyofLinearSystems:::::::::14110.2AnalysisofSystemswithLTIPerturbations:::::::::::::::14310.3PositiveOrthantStabilizability::::::::::::::::::::::14410.4LinearSystemswithDelays::::::::::::::::::::::::14410.5InterpolationProblems:::::::::::::::::::::::::::14510.6TheInverseProblemofOptimalControl:::::::::::::::::14710.7SystemRealizationProblems:::::::::::::::::::::::14810.8Multi-CriterionLQG::::::::::::::::::::::::::::15010.9NonconvexMulti-CriterionQuadraticProblems:::::::::::::151NotesandReferences:::::::::::::::::::::::::::::::152Notation157ListofAcronyms159Bibliography161Index187Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. PrefaceThebasictopicofthisbookissolvingproblemsfromsystemandcontroltheoryusingconvexoptimization.Weshowthatawidevarietyofproblemsarisinginsystemandcontroltheorycanbereducedtoahandfulofstandardconvexandquasiconvexoptimizationproblemsthatinvolvematrixinequalities.Forafewspecialcasesthereareanalyticsolutions"totheseproblems,butourmainpointisthattheycanbesolvednumericallyinallcases.Thesestandardproblemscanbesolvedinpolynomial-time(by,e.g.,theellipsoidalgorithmofShor,Nemirovskii,andYudin),andsoaretractable,atleastinatheoreticalsense.Recentlydevelopedinterior-pointmethodsforthesestandardproblemshavebeenfoundtobeextremelye±cientinpractice.Therefore,weconsidertheoriginalproblemsfromsystemandcontroltheoryassolved.Thisbookisprimarilyintendedfortheresearcherinsystemandcontroltheory,butcanalsoserveasasourceofapplicationproblemsforresearchersinconvexop-timization.Althoughwebelievethatthemethodsdescribedinthisbookhavegreatpracticalvalue,weshouldwarnthereaderwhoseprimaryinterestisappliedcontrolengineering.Thisisaresearchmonograph:Wepresentnospeci¯cexamplesornu-mericalresults,andwemakeonlybriefcommentsabouttheimplicationsoftheresultsforpracticalcontrolengineering.Toputitinamorepositivelight,wehopethatthisbookwilllaterbeconsideredasthe¯rstbookonthetopic,notthemostreadableoraccessible.Thebackgroundrequiredofthereaderisknowledgeofbasicsystemandcontroltheoryandanexposuretooptimization.Sontag'sbookMathematicalControlThe-ory[Son90]isanexcellentsurvey.FurtherbackgroundmaterialiscoveredinthetextsLinearSystems[Kai80]byKailath,NonlinearSystemsAnalysis[Vid92]byVidyasagar,OptimalControl:LinearQuadraticMethods[AM90]byAndersonandMoore,andConvexAnalysisandMinimizationAlgorithmsI[HUL93]byHiriart{UrrutyandLemar¶echal.WealsohighlyrecommendthebookInterior-pointPolynomialAlgorithmsinCon-vexProgramming[NN94]byNesterovandNemirovskiiasacompaniontothisbook.Thereaderwillsoonseethattheirideasandmethodsplayacriticalroleinthebasicideapresentedinthisbook.vii AcknowledgmentsWethankA.Nemirovskiiforencouragingustowritethisbook.WearegratefultoS.Hall,M.Gevers,andA.Packardforintroducingustomultipliermethods,realizationtheory,andstate-feedbacksynthesismethods,respectively.ThisbookwasgreatlyimprovedbythesuggestionsofJ.Abedor,P.Dankoski,J.Doyle,G.Franklin,T.Kailath,R.Kosut,I.Petersen,E.Pyatnitskii,M.Rotea,M.Safonov,S.Savastiouk,A.Tits,L.Vandenberghe,andJ.C.Willems.ItisaspecialpleasuretothankV.Yakubovichforhiscommentsandsuggestions.TheresearchreportedinthisbookwassupportedinpartbyAFOSR(underF49620-92-J-0013),NSF(underECS-9222391),andARPA(underF49620-93-1-0085).L.ElGhaouiandE.FeronweresupportedinpartbytheD¶el¶egationG¶en¶eralepourl'Armement.E.FeronwassupportedinpartbytheCharlesStarkDraperCareerDevelopmentChairatMIT.V.BalakrishnanwassupportedinpartbytheControlandDynamicalSystemsGroupatCaltechandbyNSF(underNSFDCDR-8803012).ThisbookwastypesetbytheauthorsusingLATEX,andmanymacrosoriginallywrittenbyCraigBarratt.StephenBoydStanford,CaliforniaLaurentElGhaouiParis,FranceEricFeronCambridge,MassachusettsVenkataramananBalakrishnanCollegePark,Marylandix LinearMatrixInequalitiesinSystemandControlTheory Chapter1Introduction1.1OverviewTheaimofthisbookistoshowthatwecanreduceaverywidevarietyofprob-lemsarisinginsystemandcontroltheorytoafewstandardconvexorquasiconvexoptimizationproblemsinvolvinglinearmatrixinequalities(LMIs).Sincetheseresult-ingoptimizationproblemscanbesolvednumericallyverye±cientlyusingrecentlydevelopedinterior-pointmethods,ourreductionconstitutesasolutiontotheoriginalproblem,certainlyinapracticalsense,butalsoinseveralothersensesaswell.Incom-parison,themoreconventionalapproachistoseekananalyticorfrequency-domainsolutiontothematrixinequalities.Thetypesofproblemsweconsiderinclude:²matrixscalingproblems,e.g.,minimizingconditionnumberbydiagonalscaling²constructionofquadraticLyapunovfunctionsforstabilityandperformanceanal-ysisoflineardi®erentialinclusions²jointsynthesisofstate-feedbackandquadraticLyapunovfunctionsforlineardi®erentialinclusions²synthesisofstate-feedbackandquadraticLyapunovfunctionsforstochasticanddelaysystems²synthesisofLur'e-typeLyapunovfunctionsfornonlinearsystems²optimizationoverana±nefamilyoftransfermatrices,includingsynthesisofmultipliersforanalysisoflinearsystemswithunknownparameters²positiveorthantstabilityandstate-feedbacksynthesis²optimalsystemrealization²interpolationproblems,includingscaling²multicriterionLQG/LQR²inverseproblemofoptimalcontrolInsomecases,wearedescribingknown,publishedresults;inothers,weareextendingknownresults.Inmanycases,however,itseemsthattheresultsarenew.Byscanningthelistaboveorthetableofcontents,thereaderwillseethatLya-punov'smethodswillbeourmainfocus.Herewehaveasecondarygoal,beyondshowingthatmanyproblemsfromLyapunovtheorycanbecastasconvexorquasi-convexproblems.ThisistoshowthatLyapunov'smethods,whicharetraditionally1 2Chapter1Introductionappliedtotheanalysisofsystemstability,canjustaswellbeusedto¯ndboundsonsystemperformance,providedwedonotinsistonananalyticsolution".1.2ABriefHistoryofLMIsinControlTheoryThehistoryofLMIsintheanalysisofdynamicalsystemsgoesbackmorethan100years.Thestorybeginsinabout1890,whenLyapunovpublishedhisseminalworkintroducingwhatwenowcallLyapunovtheory.Heshowedthatthedi®erentialequa-tiondx(t)=Ax(t)(1.1)dtisstable(i.e.,alltrajectoriesconvergetozero)ifandonlyifthereexistsapositive-de¯nitematrixPsuchthatATP+PA<0:(1.2)TherequirementP>0,ATP+PA<0iswhatwenowcallaLyapunovinequalityonP,whichisaspecialformofanLMI.Lyapunovalsoshowedthatthis¯rstLMIcouldbeexplicitlysolved.Indeed,wecanpickanyQ=QT>0andthensolvethelinearequationATP+PA=¡QforthematrixP,whichisguaranteedtobepositive-de¯niteifthesystem(1.1)isstable.Insummary,the¯rstLMIusedtoanalyzestabilityofadynamicalsystemwastheLyapunovinequality(1.2),whichcanbesolvedanalytically(bysolvingasetoflinearequations).Thenextmajormilestoneoccursinthe1940's.Lur'e,Postnikov,andothersintheSovietUnionappliedLyapunov'smethodstosomespeci¯cpracticalproblemsincontrolengineering,especially,theproblemofstabilityofacontrolsystemwithanonlinearityintheactuator.Althoughtheydidnotexplicitlyformmatrixinequalities,theirstabilitycriteriahavetheformofLMIs.Theseinequalitieswerereducedtopolynomialinequalitieswhichwerethencheckedbyhand"(for,needlesstosay,smallsystems).Neverthelesstheywerejusti¯ablyexcitedbytheideathatLyapunov'stheorycouldbeappliedtoimportant(anddi±cult)practicalproblemsincontrolengineering.FromtheintroductionofLur'e's1951book[Lur57]we¯nd:Thisbookrepresentsthe¯rstattempttodemonstratethattheideasex-pressed60yearsagobyLyapunov,whichevencomparativelyrecentlyap-pearedtoberemotefrompracticalapplication,arenowabouttobecomearealmediumfortheexaminationoftheurgentproblemsofcontemporaryengineering.Insummary,Lur'eandotherswerethe¯rsttoapplyLyapunov'smethodstopracticalcontrolengineeringproblems.TheLMIsthatresultedweresolvedanalytically,byhand.Ofcoursethislimitedtheirapplicationtosmall(second,thirdorder)systems.Thenextmajorbreakthroughcameintheearly1960's,whenYakubovich,Popov,Kalman,andotherresearcherssucceededinreducingthesolutionoftheLMIsthataroseintheproblemofLur'etosimplegraphicalcriteria,usingwhatwenowcallthepositive-real(PR)lemma(seex2.7.2).ThisresultedinthecelebratedPopovcriterion,circlecriterion,Tsypkincriterion,andmanyvariations.Thesecriteriacouldbeappliedtohigherordersystems,butdidnotgracefullyorusefullyextendtosystemscontainingmorethanonenonlinearity.Fromthepointofviewofourstory(LMIsincontroltheory),thecontributionwastoshowhowtosolveacertainfamilyofLMIsbygraphicalmethods.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 1.2ABriefHistoryofLMIsinControlTheory3TheimportantroleofLMIsincontroltheorywasalreadyrecognizedintheearly1960's,especiallybyYakubovich[Yak62,Yak64,Yak67].Thisisclearsimplyfromthetitlesofsomeofhispapersfrom1962{5,e.g.,Thesolutionofcertainmatrixinequalitiesinautomaticcontroltheory(1962),andThemethodofmatrixinequalitiesinthestabilitytheoryofnonlinearcontrolsystems(1965;Englishtranslation1967).ThePRlemmaandextensionswereintensivelystudiedinthelatterhalfofthe1960s,andwerefoundtoberelatedtotheideasofpassivity,thesmall-gaincriteriaintroducedbyZamesandSandberg,andquadraticoptimalcontrol.By1970,itwasknownthattheLMIappearinginthePRlemmacouldbesolvednotonlybygraphicalmeans,butalsobysolvingacertainalgebraicRiccatiequation(ARE).Ina1971paper[Wil71b]onquadraticoptimalcontrol,J.C.WillemsisledtotheLMI"#ATP+PA+QPB+CT¸0;(1.3)BTP+CRandpointsoutthatitcanbesolvedbystudyingthesymmetricsolutionsoftheAREATP+PA¡(PB+CT)R¡1(BTP+C)+Q=0;whichinturncanbefoundbyaneigendecompositionofarelatedHamiltonianmatrix.(Seex2.7.2fordetails.)ThisconnectionhadbeenobservedearlierintheSovietUnion,wheretheAREwascalledtheLur'eresolvingequation(see[Yak88]).Soby1971,researchersknewseveralmethodsforsolvingspecialtypesofLMIs:direct(forsmallsystems),graphicalmethods,andbysolvingLyapunovorRiccatiequations.Fromourpointofview,thesemethodsareallclosed-form"oranalytic"solutionsthatcanbeusedtosolvespecialformsofLMIs.(MostcontrolresearchersandengineersconsidertheRiccatiequationtohaveananalytic"solution,sincethestandardalgorithmsthatsolveitareverypredictableintermsofthee®ortrequired,whichdependsalmostentirelyontheproblemsizeandnottheparticularproblemdata.Ofcourseitcannotbesolvedexactlyina¯nitenumberofarithmeticstepsforsystemsof¯fthandhigherorder.)InWillems'1971paperwe¯ndthefollowingstrikingquote:ThebasicimportanceoftheLMIseemstobelargelyunappreciated.Itwouldbeinterestingtoseewhetherornotitcanbeexploitedincompu-tationalalgorithms,forexample.HereWillemsreferstothespeci¯cLMI(1.3),andnotthemoregeneralformthatweconsiderinthisbook.Still,Willems'suggestionthatLMIsmighthavesomeadvantagesincomputationalalgorithms(ascomparedtothecorrespondingRiccatiequations)foreshadowsthenextchapterinthestory.Thenextmajoradvance(inourview)wasthesimpleobservationthat:TheLMIsthatariseinsystemandcontroltheorycanbeformulatedasconvexoptimizationproblemsthatareamenabletocomputersolution.Althoughthisisasimpleobservation,ithassomeimportantconsequences,themostimportantofwhichisthatwecanreliablysolvemanyLMIsforwhichnoanalyticsolution"hasbeenfound(orislikelytobefound).Thisobservationwasmadeexplicitlybyseveralresearchers.PyatnitskiiandSko-rodinskii[PS82]wereperhapsthe¯rstresearcherstomakethispoint,clearlyandcompletely.TheyreducedtheoriginalproblemofLur'e(extendedtothecaseofmul-tiplenonlinearities)toaconvexoptimizationprobleminvolvingLMIs,whichtheyThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 4Chapter1Introductionthensolvedusingtheellipsoidalgorithm.(Thisproblemhadbeenstudiedbefore,butthesolutions"involvedanarbitraryscalingmatrix.)PyatnitskiiandSkorodinskiiwerethe¯rst,asfarasweknow,toformulatethesearchforaLyapunovfunctionasaconvexoptimizationproblem,andthenapplyanalgorithmguaranteedtosolvetheoptimizationproblem.Weshouldalsomentionseveralprecursors.Ina1976paper,HorisbergerandBe-langer[HB76]hadremarkedthattheexistenceofaquadraticLyapunovfunctionthatsimultaneouslyprovesstabilityofacollectionoflinearsystemsisaconvexprobleminvolvingLMIs.Andofcourse,theideaofhavingacomputersearchforaLya-punovfunctionwasnotnew|itappears,forexample,ina1965paperbySchultzetal.[SSHJ65].The¯nalchapterinourstoryisquiterecentandofgreatpracticalimportance:thedevelopmentofpowerfulande±cientinterior-pointmethodstosolvetheLMIsthatariseinsystemandcontroltheory.In1984,N.Karmarkarintroducedanewlinearprogrammingalgorithmthatsolveslinearprogramsinpolynomial-time,liketheellip-soidmethod,butincontrasttotheellipsoidmethod,isalsoverye±cientinpractice.Karmarkar'sworkspurredanenormousamountofworkintheareaofinterior-pointmethodsforlinearprogramming(includingtherediscoveryofe±cientmethodsthatweredevelopedinthe1960sbutignored).Essentiallyallofthisresearchactivitycon-centratedonalgorithmsforlinearand(convex)quadraticprograms.Thenin1988,NesterovandNemirovskiidevelopedinterior-pointmethodsthatapplydirectlytocon-vexproblemsinvolvingLMIs,andinparticular,totheproblemsweencounterinthisbook.Althoughthereremainsmuchtobedoneinthisarea,severalinterior-pointalgorithmsforLMIproblemshavebeenimplementedandtestedonspeci¯cfamiliesofLMIsthatariseincontroltheory,andfoundtobeextremelye±cient.AsummaryofkeyeventsinthehistoryofLMIsincontroltheoryisthen:²1890:FirstLMIappears;analyticsolutionoftheLyapunovLMIviaLyapunovequation.²1940's:ApplicationofLyapunov'smethodstorealcontrolengineeringprob-lems.SmallLMIssolvedbyhand".²Early1960's:PRlemmagivesgraphicaltechniquesforsolvinganotherfamilyofLMIs.²Late1960's:ObservationthatthesamefamilyofLMIscanbesolvedbysolvinganARE.²Early1980's:RecognitionthatmanyLMIscanbesolvedbycomputerviaconvexprogramming.²Late1980's:Developmentofinterior-pointalgorithmsforLMIs.ItisfairtosaythatYakubovichisthefatherofthe¯eld,andLyapunovthegrandfatherofthe¯eld.Thenewdevelopmentistheabilitytodirectlysolve(general)LMIs.1.3NotesontheStyleoftheBookWeuseaveryinformalmathematicalstyle,e.g.,weoftenfailtomentionregularityorothertechnicalconditions.Everystatementistobeinterpretedasbeingtruemoduloappropriatetechnicalconditions(thatinmostcasesaretrivialto¯gureout).Weareveryinformal,perhapsevencavalier,inourreductionofaproblemtoanoptimizationproblem.Wesometimesskipdetails"thatwouldbeimportantiftheoptimizationproblemweretobesolvednumerically.Asanexample,itmaybeCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 1.4OriginoftheBook5necessarytoaddconstraintstotheoptimizationproblemfornormalizationortoensureboundedness.Wedonotdiscussinitialguessesfortheoptimizationproblems,eventhoughgoodonesmaybeavailable.Therefore,thereaderwhowishestoimplementanalgorithmthatsolvesaproblemconsideredinthisbookshouldbepreparedtomakeafewmodi¯cationsoradditionstoourdescriptionofthesolution".Inasimilarway,wedonotpursueanytheoreticalaspectsofreducingaproblemtoaconvexprobleminvolvingmatrixinequalities.Forexample,foreachreducedproblemwecouldstate,probablysimplify,andtheninterpretinsystemorcontroltheoretictermstheoptimalityconditionsfortheresultingconvexproblem.Anotherfascinatingtopicthatcouldbeexploredistherelationbetweensystemandcontroltheorydualityandconvexprogrammingduality.Oncewereduceaproblemarisingincontroltheorytoaconvexprogram,wecanconsidervariousdualoptimizationproblems,lowerboundsfortheproblem,andsoon.Presumablythesedualproblemsandlowerboundscanbegiveninterestingsystem-theoreticinterpretations.Wemostlyconsidercontinuous-timesystems,andassumethatthereadercantranslatetheresultsfromthecontinuous-timecasetothediscrete-timecase.Weswitchtodiscrete-timesystemswhenweconsidersystemrealizationproblems(whichalmostalwaysariseinthisform)andalsowhenweconsiderstochasticsystems(toavoidthetechnicaldetailsofstochasticdi®erentialequations).Thelistofproblemsthatweconsiderismeantonlytoberepresentative,andcertainlynotexhaustive.Toavoidexcessiverepetition,ourtreatmentofproblemsbecomesmoreterseasthebookprogresses.Inthe¯rstchapteronanalysisoflin-eardi®erentialinclusions,wedescribemanyvariationsonproblems(e.g.,computingboundsonmarginsanddecayrates);inlaterchapters,wedescribefewerandfewervariations,assumingthatthereadercouldworkouttheextensions.EachchapterconcludeswithasectionentitledNotesandReferences,inwhichwehideproofs,precisestatements,elaborations,andbibliographyandhistoricalnotes.Thecompletenessofthebibliographyshouldnotbeoverestimated,despiteitssize(over500entries).Theappendixcontainsalistofnotationandalistofacronymsusedinthebook.Weapologizetothereaderforthesevennewacronymsweintroduce.Tolightenthenotation,weusethestandardconventionofdroppingthetimeargumentfromthevariablesindi®erentialequations.Thus,x_=Axisourshortformfordx=dt=Ax(t).HereAisaconstantmatrix;whenweencountertime-varyingcoe±cients,wewillexplicitlyshowthetimedependence,asinx_=A(t)x.Similarly,wedropthedummyvariablefromde¯niteintegrals,writingforexample,RTTRTTuydtforu(t)y(t)dt.Toreducethenumberofparenthesesrequired,weadopt00theconventionthattheoperatorsTr(traceofamatrix)andE(expectedvalue)have¡¢lowerprecedencethanmultiplication,transpose,etc.Thus,TrATBmeansTrATB.1.4OriginoftheBookThisbookstartedoutasasectionofthepaperMethodofCentersforMinimizingGeneralizedEigenvalues,byBoydandElGhaoui[BE93],butgrewtoolargetobeasection.Forafewmonthsitwasamanuscript(thatpresumablywouldhavebeensubmittedforpublicationasapaper)entitledGeneralizedEigenvalueProblemsAris-inginControlTheory.ThenFeron,andlaterBalakrishnan,startedaddingmaterial,andsoonitwasclearthatwewerewritingabook,notapaper.Theorderoftheauthors'namesre°ectsthishistory.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. Chapter2SomeStandardProblemsInvolvingLMIs2.1LinearMatrixInequalitiesAlinearmatrixinequality(LMI)hastheformXm¢F(x)=F0+xiFi>0;(2.1)i=1wherex2RmisthevariableandthesymmetricmatricesF=FT2Rn£n,i=ii0;:::;m,aregiven.Theinequalitysymbolin(2.1)meansthatF(x)ispositive-de¯nite,i.e.,uTF(x)u>0forallnonzerou2Rn.Ofcourse,theLMI(2.1)isequivalenttoasetofnpolynomialinequalitiesinx,i.e.,theleadingprincipalminorsofF(x)mustbepositive.WewillalsoencounternonstrictLMIs,whichhavetheformF(x)¸0:(2.2)ThestrictLMI(2.1)andthenonstrictLMI(2.2)arecloselyrelated,butaprecisestatementoftherelationisabitinvolved,sowedeferittox2.5.InthenextfewsectionsweconsiderstrictLMIs.TheLMI(2.1)isaconvexconstraintonx,i.e.,thesetfxjF(x)>0gisconvex.AlthoughtheLMI(2.1)mayseemtohaveaspecializedform,itcanrepresentawidevarietyofconvexconstraintsonx.Inparticular,linearinequalities,(convex)quadraticinequalities,matrixnorminequalities,andconstraintsthatariseincontroltheory,suchasLyapunovandconvexquadraticmatrixinequalities,canallbecastintheformofanLMI.MultipleLMIsF(1)(x)>0;:::;F(p)(x)>0canbeexpressedasthesingleLMIdiag(F(1)(x);:::;F(p)(x))>0.ThereforewewillmakenodistinctionbetweenasetofLMIsandasingleLMI,i.e.,theLMIF(1)(x)>0;:::;F(p)(x)>0"willmeantheLMIdiag(F(1)(x);:::;F(p)(x))>0".WhenthematricesFiarediagonal,theLMIF(x)>0isjustasetoflinearinequalities.Nonlinear(convex)inequalitiesareconvertedtoLMIformusingSchurcomplements.Thebasicideaisasfollows:theLMI"#Q(x)S(x)>0;(2.3)S(x)TR(x)7 8Chapter2SomeStandardProblemsInvolvingLMIswhereQ(x)=Q(x)T,R(x)=R(x)T,andS(x)dependa±nelyonx,isequivalenttoR(x)>0;Q(x)¡S(x)R(x)¡1S(x)T>0:(2.4)Inotherwords,thesetofnonlinearinequalities(2.4)canberepresentedastheLMI(2.3).Asanexample,the(maximumsingularvalue)matrixnormconstraintkZ(x)k<1,p£qwhereZ(x)2Randdependsa±nelyonx,isrepresentedastheLMI"#IZ(x)>0Z(x)TI(sincekZk<1isequivalenttoI¡ZZT>0).Notethatthecaseq=1reducestoageneralconvexquadraticinequalityonx.Theconstraintc(x)TP(x)¡1c(x)<1,P(x)>0,wherec(x)2RnandP(x)=P(x)T2Rn£ndependa±nelyonx,isexpressedastheLMI"#P(x)c(x)>0:c(x)T1Moregenerally,theconstraintTrS(x)TP(x)¡1S(x)<1;P(x)>0;whereP(x)=P(x)T2Rn£nandS(x)2Rn£pdependa±nelyonx,ishandledbyintroducinganew(slack)matrixvariableX=XT2Rp£p,andtheLMI(inxandX):"#XS(x)TTrX<1;>0:S(x)P(x)ManyotherconvexconstraintsonxcanbeexpressedintheformofanLMI;seetheNotesandReferences.2.1.1MatricesasvariablesWewilloftenencounterproblemsinwhichthevariablesarematrices,e.g.,theLya-punovinequalityATP+PA<0;(2.5)whereA2Rn£nisgivenandP=PTisthevariable.InthiscasewewillnotwriteouttheLMIexplicitlyintheformF(x)>0,butinsteadmakeclearwhichmatricesarethevariables.ThephrasetheLMIATP+PA<0inP"meansthatthematrixPisavariable.(Ofcourse,theLyapunovinequality(2.5)isreadilyputintheform(2.1),asfollows.LetP1;:::;Pmbeabasisforsymmetricn£nmatrices(m=n(n+1)=2).ThentakeF=0andF=¡ATP¡PA.)LeavingLMIsinacondensedform0iiisuchas(2.5),inadditiontosavingnotation,mayleadtomoree±cientcomputation;seex2.4.4.Asanotherrelatedexample,considerthequadraticmatrixinequalityATP+PA+PBR¡1BTP+Q<0;(2.6)Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 2.2SomeStandardProblems9whereA,B,Q=QT,R=RT>0aregivenmatricesofappropriatesizes,andP=PTisthevariable.NotethatthisisaquadraticmatrixinequalityinthevariableP.Itcanbeexpressedasthelinearmatrixinequality"#¡ATP¡PA¡QPB>0:BTPRThisrepresentationalsoclearlyshowsthatthequadraticmatrixinequality(2.6)isconvexinP,whichisnotobvious.2.1.2LinearequalityconstraintsInsomeproblemswewillencounterlinearequalityconstraintsonthevariables,e.g.P>0;ATP+PA<0;TrP=1;(2.7)k£kwhereP2Risthevariable.Ofcoursewecaneliminatetheequalityconstrainttowrite(2.7)intheformF(x)>0.LetP1;:::;Pmbeabasisforsymmetrick£kmatriceswithtracezero(m=(k(k+1)=2)¡1)andletP0beasymmetrick£kmatrixwithTrP=1.ThentakeF=diag(P;¡ATP¡PA)andF=diag(P;¡ATP¡PA)00000iiiifori=1;:::;m.Wewillrefertoconstraintssuchas(2.7)asLMIs,leavinganyrequiredeliminationofequalityconstraintstothereader.2.2SomeStandardProblemsHerewelistsomecommonconvexandquasiconvexproblemsthatwewillencounterinthesequel.2.2.1LMIproblemsGivenanLMIF(x)>0,thecorrespondingLMIProblem(LMIP)isto¯ndxfeassuchthatF(xfeas)>0ordeterminethattheLMIisinfeasible.(Byduality,thismeans:FindanonzeroG¸0suchthatTrGFi=0fori=1;:::;mandTrGF0·0;seetheNotesandReferences.)Ofcourse,thisisaconvexfeasibilityproblem.WewillsaysolvingtheLMIF(x)>0"tomeansolvingthecorrespondingLMIP.AsanexampleofanLMIP,considerthesimultaneousLyapunovstabilityprob-n£nlem"(whichwewillseeinx5.1):WearegivenAi2R,i=1;:::;L,andneedto¯ndPsatisfyingtheLMIP>0;ATP+PA<0;i=1;:::;L;iiordeterminethatnosuchPexists.DeterminingthatnosuchPexistsisequivalentto¯ndingQ0;:::;QL,notallzero,suchthatXL¡¢Q¸0;:::;Q¸0;Q=QAT+AQ;(2.8)0L0iiiii=1whichisanother(nonstrict)LMIP.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 10Chapter2SomeStandardProblemsInvolvingLMIs2.2.2EigenvalueproblemsTheeigenvalueproblem(EVP)istominimizethemaximumeigenvalueofamatrixthatdependsa±nelyonavariable,subjecttoanLMIconstraint(ordeterminethattheconstraintisinfeasible),i.e.,minimize¸subjectto¸I¡A(x)>0;B(x)>0whereAandBaresymmetricmatricesthatdependa±nelyontheoptimizationvariablex.Thisisaconvexoptimizationproblem.EVPscanappearintheequivalentformofminimizingalinearfunctionsubjecttoanLMI,i.e.,minimizecTx(2.9)subjecttoF(x)>0withFana±nefunctionofx.InthespecialcasewhenthematricesFiarealldiagonal,thisproblemreducestothegenerallinearprogrammingproblem:minimizingthelinearfunctioncTxsubjecttoasetoflinearinequalitiesonx.AnotherequivalentformfortheEVPis:minimize¸subjecttoA(x;¸)>0whereAisa±nein(x;¸).Weleaveittothereadertoverifythattheseformsareequivalent,i.e.,anycanbetransformedintoanyother.AsanexampleofanEVP,considertheproblem(whichappearsinx6.3.2):minimize°"#¡ATP¡PA¡CTCPBsubjectto>0;P>0BTP°In£nn£pm£nwherethematricesA2R,B2R,andC2Raregiven,andPand°aretheoptimizationvariables.Fromourremarksabove,thisEVPcanalsobeexpressedintermsoftheassociatedquadraticmatrixinequality:minimize°subjecttoATP+PA+CTC+°¡1PBBTP<0;P>02.2.3GeneralizedeigenvalueproblemsThegeneralizedeigenvalueproblem(GEVP)istominimizethemaximumgeneralizedeigenvalueofapairofmatricesthatdependa±nelyonavariable,subjecttoanLMIconstraint.ThegeneralformofaGEVPis:minimize¸subjectto¸B(x)¡A(x)>0;B(x)>0;C(x)>0whereA,BandCaresymmetricmatricesthatarea±nefunctionsofx.Wecanexpressthisasminimize¸max(A(x);B(x))subjecttoB(x)>0;C(x)>0Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 2.2SomeStandardProblems11where¸max(X;Y)denotesthelargestgeneralizedeigenvalueofthepencil¸Y¡XwithY>0,i.e.,thelargesteigenvalueofthematrixY¡1=2XY¡1=2.ThisGEVPisaquasiconvexoptimizationproblemsincetheconstraintisconvexandtheobjective,¸max(A(x);B(x)),isquasiconvex.Thismeansthatforfeasiblex,x~and0·µ·1,¸max(A(µx+(1¡µ)x~);B(µx+(1¡µ)x~))·maxf¸max(A(x);B(x));¸max(A(x~);B(x~))g:NotethatwhenthematricesarealldiagonalandA(x)andB(x)arescalar,thisproblemreducestothegenerallinear-fractionalprogrammingproblem,i.e.,minimiz-ingalinear-fractionalfunctionsubjecttoasetoflinearinequalities.Inaddition,manynonlinearquasiconvexfunctionscanberepresentedintheformofaGEVPwithappropriateA,B,andC;seetheNotesandReferences.AnequivalentalternateformforaGEVPisminimize¸subjecttoA(x;¸)>0whereA(x;¸)isa±neinxfor¯xed¸anda±nein¸for¯xedx,andsatis¯esthemonotonicitycondition¸>¹=)A(x;¸)¸A(x;¹).AsanexampleofaGEVP,considertheproblemmaximize®subjectto¡ATP¡PA¡2®P>0;P>0wherethematrixAisgiven,andtheoptimizationvariablesarethesymmetricmatrixPandthescalar®.(Thisproblemarisesinx5.1.)2.2.4AconvexproblemAlthoughwewillbeconcernedmostlywithLMIPs,EVPs,andGEVPs,wewillalsoencounterthefollowingconvexproblem,whichwewillabbreviateCP:minimizelogdetA(x)¡1(2.10)subjecttoA(x)>0;B(x)>0whereAandBaresymmetricmatricesthatdependa±nelyonx.(NotethatwhenA>0,logdetA¡1isaconvexfunctionofA.)Remark:ProblemCPcanbetransformedintoanEVP,sincedetA(x)>¸canberepresentedasanLMIinxand¸;see[NN94,x6.4.3].Aswedowithvariableswhicharematrices,weleavetheseproblemsinthemorenaturalform.AsanexampleofCP,considertheproblem:minimizelogdetP¡1(2.11)subjecttoP>0;vTPv·1;i=1;:::;Liiherev2RnaregivenandP=PT2Rn£nisthevariable.iWewillencounterseveralvariationsofthisproblem,whichhasthefollowinginterpretation.LetEdenotetheellipsoidcenteredattheorigindeterminedbyP,Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 12Chapter2SomeStandardProblemsInvolvingLMIs¢©¯ªE=z¯zTPz·1.Theconstraintsaresimplyvi2E.SincethevolumeofEispro-portionalto(detP)¡1=2,minimizinglogdetP¡1isthesameasminimizingthevolumeofE.Sobysolving(2.11),we¯ndtheminimumvolumeellipsoid,centeredattheori-gin,thatcontainsthepointsv1;:::;vL,orequivalently,thepolytopeCofv1;:::;vLg,whereCodenotestheconvexhull.2.2.5SolvingtheseproblemsThestandardproblems(LMIPs,EVPs,GEVPs,andCPs)aretractable,fromboththeoreticalandpracticalviewpoints:²Theycanbesolvedinpolynomial-time(indeedwithavarietyofinterpretationsforthetermpolynomial-time").²Theycanbesolvedinpracticeverye±ciently.Bysolvetheproblem"wemean:Determinewhetherornottheproblemisfea-sible,andifitis,computeafeasiblepointwithanobjectivevaluethatexceedstheglobalminimumbylessthansomeprespeci¯edaccuracy.2.3EllipsoidAlgorithmWe¯rstdescribeasimpleellipsoidalgorithmthat,roughlyspeaking,isguaranteedtosolvethestandardproblems.Wedescribeitherebecauseitisverysimpleand,fromatheoreticalpointofview,e±cient(polynomial-time).Inpractice,however,theinterior-pointalgorithmsdescribedinthenextsectionaremuchmoree±cient.Althoughmoresophisticatedversionsoftheellipsoidalgorithmcandetectin-feasibleconstraints,wewillassumethattheproblemwearesolvinghasatleastoneoptimalpoint,i.e.,theconstraintsarefeasible.(Inthefeasibilityproblem,weconsideranyfeasiblepointasbeingoptimal.)Thebasicideaofthealgorithmisasfollows.WestartwithanellipsoidE(0)thatisguaranteedtocontainanoptimalpoint.Wethencomputeacuttingplaneforourproblemthatpassesthroughthecenterx(0)ofE(0).Thismeansthatwe¯ndanonzerovectorg(0)suchthatanoptimalpointliesinthe©¯ª©¯ªhalf-spacez¯g(0)T(z¡x(0))·0(orinthehalf-spacez¯g(0)T(z¡x(0))<0,dependingonthesituation).(Wewillexplainhowtodothisforeachofourproblemslater.)Wethenknowthattheslicedhalf-ellipsoidn¯o(0)¯(0)T(0)Ez¯g(z¡x)·0containsanoptimalpoint.NowwecomputetheellipsoidE(1)ofminimumvolumethatcontainsthisslicedhalf-ellipsoid;E(1)isthenguaranteedtocontainanoptimalpoint.Theprocessisthenrepeated.Wenowdescribethealgorithmmoreexplicitly.AnellipsoidEcanbedescribedas©¯ªE=z¯(z¡a)TA¡1(z¡a)·1whereA=AT>0.Theminimumvolumeellipsoidthatcontainsthehalf-ellipsoid©¯ªz¯(z¡a)TA¡1(z¡a)·1;gT(z¡a)·0isgivenbyn¯oE~=z¯¯(z¡a~)TA~¡1(z¡a~)·1;Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 2.3EllipsoidAlgorithm13whereµ¶Ag~m22a~=a¡;A~=A¡Ag~g~TA;m+1m2¡1m+1pandg~=g=gTAg.(Wenotethattheseformulasholdonlyform¸2.Inthecaseofonevariable,theminimumlengthintervalcontainingahalf-intervalisthehalf-intervalitself;theellipsoidalgorithm,inthiscase,reducestothefamiliarbisectionalgorithm.)Theellipsoidalgorithmisinitializedwithx(0)andA(0)suchthatthecorrespondingellipsoidcontainsanoptimalpoint.Thealgorithmthenproceedsasfollows:fork=1;2;:::computeag(k)thatde¯nesacuttingplaneatx(k)¡¢¡1=2g~:=g(k)TA(k)g(k)g(k)x(k+1):=x(k)¡1A(k)g~m+1³´2A(k+1):=mA(k)¡2A(k)g~g~TA(k)m2¡1m+1Thisrecursiongeneratesasequenceofellipsoidsthatareguaranteedtocontainanoptimalpoint.Itturnsoutthatthevolumeoftheseellipsoidsdecreasesgeometrically.Wehave(k)¡k(0)vol(E)·e2mvol(E);andthisfactcanbeusedtoprovepolynomial-timeconvergenceofthealgorithm.(WereferthereadertothepaperscitedintheNotesandReferencesforprecisestatementsofwhatwemeanbypolynomial-time"andindeed,convergence,"aswellasproofs.)Wenowshowhowtocomputecuttingplanesforeachofourstandardproblems.LMIPs:ConsidertheLMIXmF(x)=F0+xiFi>0:i=1IfxdoesnotsatisfythisLMI,thereexistsanonzerousuchthatÃ!XmuTF(x)u=uTF+xFu·0:0iii=1De¯negbyg=¡uTFu,i=1;:::;m.ThenforanyzsatisfyinggT(z¡x)¸0weiihaveÃ!XmuTF(z)u=uTF+zFu=uTF(x)u¡gT(z¡x)·0:0iii=1Itfollowsthateveryfeasiblepointliesinthehalf-spacefzjgT(z¡x)<0g,i.e.,thisgde¯nesacuttingplanefortheLMIPatthepointx.EVPs:ConsidertheEVPminimizecTxsubjecttoF(x)>0Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 14Chapter2SomeStandardProblemsInvolvingLMIsSuppose¯rstthatthegivenpointxisinfeasible,i.e.,F(x)6>0.Thenwecancon-structacuttingplaneforthisproblemusingthemethoddescribedaboveforLMIPs.Inthiscase,wearediscardingthehalf-spacefzjgT(z¡x)¸0gbecauseallsuchpointsareinfeasible.Nowassumethatthegivenpointxisfeasible,i.e.,F(x)>0.Inthiscase,thevectorg=cde¯nesacuttingplanefortheEVPatthepointx.Here,wearediscardingthehalf-spacefzjgT(z¡x)>0gbecauseallsuchpoints(whetherfeasibleornot)haveanobjectivevaluelargerthanx,andhencecannotbeoptimal.GEVPs:Weconsidertheformulationminimize¸max(A(x);B(x))subjecttoB(x)>0;C(x)>0PmPmPmHere,A(x)=A0+i=1xiAi,B(x)=B0+i=1xiBiandC(x)=C0+i=1xiCi.Nowsupposewearegivenapointx.Iftheconstraintsareviolated,weusethemethoddescribedforLMIPstogenerateacuttingplaneatx.Supposenowthatxisfeasible.Pickanyu6=0suchthat(¸max(A(x);B(x))B(x)¡A(x))u=0:De¯negbyg=¡uT(¸(A(x);B(x))B¡A)u;i=1;:::;m:imaxiiWeclaimgde¯nesacuttingplaneforthisGEVPatthepointx.Toseethis,wenotethatforanyz,uT(¸(A(x);B(x))B(z)¡A(z))u=¡gT(z¡x):maxHence,ifgT(z¡x)¸0we¯ndthat¸max(A(z);B(z))¸¸max(A(x);B(x))whichestablishesourclaim.CPs:WenowconsiderourstandardCP(2.10).Whenthegivenpointxisinfeasiblewealreadyknowhowtogenerateacuttingplane.Soweassumethatxisfeasible.Inthiscase,acuttingplaneisgivenbythegradientoftheobjectiveÃ!Xm¡logdetA(x)=¡logdetA0+xiAii=1atx,i.e.,g=¡TrAA(x)¡1:Sincetheobjectivefunctionisconvexwehaveforalliiz,logdetA(z)¡1¸logdetA(x)¡1+gT(z¡x):Inparticular,gT(z¡x)>0implieslogdetA(z)¡1>logdetA(x)¡1,andhencezcannotbeoptimal.2.4Interior-PointMethodsSince1988,interior-pointmethodshavebeendevelopedforthestandardproblems.Inthissectionwedescribeasimpleinterior-pointmethodforsolvinganEVP,givenafeasiblestartingpoint.Thereferencesdescribemoresophisticatedinterior-pointmethodsfortheotherproblems(includingtheproblemofcomputingafeasiblestartingpointforanEVP).Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 2.4Interior-PointMethods152.4.1AnalyticcenterofanLMIThenotionoftheanalyticcenterofanLMIplaysanimportantroleininterior-pointmethods,andisimportantinitsownright.ConsidertheLMIXmF(x)=F0+xiFi>0;i=1whereF=FT2Rn£n.Thefunctionii(logdetF(x)¡1F(x)>0¢Á(x)=(2.12)1otherwise,is¯niteifandonlyifF(x)>0,andbecomesin¯niteasxapproachestheboundaryofthefeasiblesetfxjF(x)>0g,i.e.,itisabarrierfunctionforthefeasibleset.(WehavealreadyencounteredthisfunctioninourstandardproblemCP.)Wesupposenowthatthefeasiblesetisnonemptyandbounded,whichimpliesthatthematricesF1;:::;Fmarelinearlyindependent(otherwisethefeasiblesetwillcontainaline).ItcanbeshownthatÁisstrictlyconvexonthefeasibleset,soithasauniqueminimizer,whichwedenotex?:x?=¢argminÁ(x):(2.13)xWerefertox?astheanalyticcenteroftheLMIF(x)>0.Equivalently,x?=argmaxdetF(x);(2.14)F(x)>0thatis,F(x?)hasmaximumdeterminantamongallpositive-de¯nitematricesoftheformF(x).Remark:TheanalyticcenterdependsontheLMI(i.e.,thedataF0;:::;Fm)andnotitsfeasibleset:WecanhavetwoLMIswithdi®erentanalyticcentersbutidenticalfeasiblesets.Theanalyticcenteris,however,invariantundercongruenceToftheLMI:F(x)>0andTF(x)T>0havethesameanalyticcenterprovidedTisnonsingular.WenowturntotheproblemofcomputingtheanalyticcenterofanLMI.(ThisisaspecialformofourproblemCP.)Newton'smethod,withappropriatesteplengthselection,canbeusedtoe±cientlycomputex?,startingfromafeasibleinitialpoint.Weconsiderthealgorithm:x(k+1):=x(k)¡®(k)H(x(k))¡1g(x(k));(2.15)where®(k)isthedampingfactorofthekthiteration,andg(x(k))andH(x(k))denotethegradientandHessianofÁ,respectively,atx(k).In[NN94,x2.2],NesterovandNemirovskiigiveasimplesteplengthruleappropriateforageneralclassofbarrierfunctions(calledself-concordant),andinparticular,thefunctionÁ.Theirdampingfactoris:(1if±(x(k))·1=4;®(k):=(2.16)1=(1+±(x(k)))otherwise,Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 16Chapter2SomeStandardProblemsInvolvingLMIswhereq±(x)=¢g(x)TH(x)¡1g(x)iscalledtheNewtondecrementofÁatx.NesterovandNemirovskiishowthatthissteplengthalwaysresultsinx(k+1)feasible,i.e.,F(x(k+1))>0,andconvergenceofx(k)tox?.Indeed,theygivesharpboundsonthenumberofstepsrequiredtocomputex?toagivenaccuracyusingNewton'smethodwiththesteplength(2.16).TheyprovethatÁ(x(k))¡Á(x?)·²whenever³´k¸c+cloglog(1=²)+cÁ(x(0))¡Á(x?);(2.17)123wherec1,c2,andc3arethreeabsoluteconstants,i.e.,speci¯cnumbers.The¯rstandsecondtermsontheright-handsidedonotdependonanyproblemdata,i.e.,thematricesF0;:::;Fm,andthenumbersmandn.Thesecondtermgrowssoslowlywithrequiredaccuracy²thatforallpracticalpurposesitcanbelumpedtogetherwiththe¯rstandconsideredanabsoluteconstant.Thelasttermontheright-handsideof(2.17)dependsonhowcentered"theinitialpointis.2.4.2ThepathofcentersNowconsiderthestandardEVP:minimizecTxsubjecttoF(x)>0Let¸optdenoteitsoptimalvalue,soforeach¸>¸opttheLMIF(x)>0;cTx<¸(2.18)isfeasible.WewillalsoassumethattheLMI(2.18)hasaboundedfeasibleset,andthereforehasananalyticcenterwhichwedenotex?(¸):µ¶?¢¡11x(¸)=argminlogdetF(x)+log:¸¡cTxxThecurvegivenbyx?(¸)for¸>¸optiscalledthepathofcentersfortheEVP.Itcanbeshownthatitisanalyticandhasalimitas¸!¸opt,whichwedenotexopt.Thepointxoptisoptimal(ormoreprecisely,thelimitofaminimizingsequence)sincefor¸>¸opt,x?(¸)isfeasibleandsatis¯escTx?(¸)<¸.Thepointx?(¸)ischaracterizedby¯µ¶@¯1¯logdetF(x)¡1+log@xi¯x?(¸)¸¡cTx(2.19)?¡1ci=¡TrF(x(¸))Fi+=0;i=1;:::;m:¸¡cTx?(¸)2.4.3MethodofcentersThemethodofcentersisasimpleinterior-pointalgorithmthatsolvesanEVP,givenafeasiblestartingpoint.Thealgorithmisinitializedwith¸(0)andx(0),withF(x(0))>0andcTx(0)<¸(0),andproceedsasfollows:¸(k+1):=(1¡µ)cTx(k)+µ¸(k)x(k+1):=x?(¸(k+1))Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 2.4Interior-PointMethods17whereµisaparameterwith0<µ<1.Theclassicmethodofcentersisobtainedwithµ=0.Inthiscase,however,x(k)doesnot(quite)satisfythenewinequalitycTx<¸(k+1).Withµ>0,however,thecurrentiteratex(k)isfeasiblefortheinequalitycTx<¸(k+1),F(x)>0,andthereforecanbeusedastheinitialpointforcomputingthenextiteratex?(¸(k+1))viaNewton'smethod.SincecomputingananalyticcenterisitselfaspecialtypeofCP,wecanviewthemethodofcentersasawayofsolvinganEVPbysolvingasequenceofCPs(whichisdoneusingNewton'smethod).(k)optWenowgiveasimpleproofofconvergence.Multiplying(2.19)by(xi¡xi)andsummingoveriyields:³´(k)¡1(k)opt1T(k)optTrF(x)F(x)¡F(x)=c(x¡x):¸(k)¡cTx(k)SinceTrF(x(k))¡1F(xopt)¸0,weconcludethat1T(k)optn¸(cx¡¸):¸(k)¡cTx(k)ReplacingcTx(k)by(¸(k+1)¡µ¸(k))=(1¡µ)yields(k+1)optn+µ(k)opt(¸¡¸)·(¸¡¸);n+1whichprovesthat¸(k)approaches¸optwithatleastgeometricconvergence.NotethatwecanalsoexpresstheinequalityaboveintheformcTx(k)¡¸opt·n(¸(k)¡cTx(k));whichshowsthatthestoppingcriterion³´until¸(k)¡cTx(k)<²=nguaranteesthatonexit,theoptimalvaluehasbeenfoundwithin².Wemakeafewcommentshere,andreferthereadertotheNotesandReferencesforfurtherelaboration.First,thisvariationonthemethodofcentersisnotpolynomial-time,butmoresophisticatedversionsare.Second,andperhapsmoreimportant,wenotethattwosimplemodi¯cationsofthemethodofcentersasdescribedaboveyieldanalgorithmthatisfairlye±cientinpractice.Themodi¯cationsare:²InsteadoftheNesterov{Nemirovskiisteplength,asteplengthchosentomini-mizeÁalongtheNewtonsearchdirection(i.e.,anexactline-search)willyieldfasterconvergencetotheanalyticcenter.²Insteadofde¯ningx?(¸)tobetheanalyticcenteroftheLMIF(x)>0,cTx<¸,wede¯neittobetheanalyticcenteroftheLMIF(x)>0alongwithqcopiesofcTx<¸,whereq>1isaninteger.Inotherwordsweuseµ¶?¢¡11x(¸)=argminlogdetF(x)+qlog:¸¡cTxxn£nUsingq>1,sayq¼nwhereF(x)2R,yieldsmuchfasterreductionof¸periteration.Amonginterior-pointmethodsforthestandardproblems,themethodofcentersisnotthemoste±cient.Themoste±cientalgorithmsdevelopedsofarappeartobeprimal-dualalgorithms(andvariations)andtheprojectivemethodofNemirovskii;seetheNotesandReferences.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 18Chapter2SomeStandardProblemsInvolvingLMIs2.4.4Interior-pointmethodsandproblemstructureAnimportantfeatureofinterior-pointmethodsisthatproblemstructurecanbeex-ploitedtoincreasee±ciency.Theideaisveryroughlyasfollows.Ininterior-pointmethodsmostofthecomputationale®ortisdevotedtocomputingtheNewtondirec-tionofabarrierorsimilarfunction.ItturnsoutthatthisNewtondirectioncanbeexpressedasthesolutionofaweightedleast-squaresproblemofthesamesizeastheoriginalproblem.Usingconjugate-gradientandotherrelatedmethodstosolvetheseleast-squaressystemsgivestwoadvantages.First,byexploitingproblemstructureintheconjugate-gradientiterations,thecomputationale®ortrequiredtosolvetheleast-squaresproblemsismuchsmallerthanbystandarddirect"methodssuchasQRorCholeskyfactorization.Second,itispossibletoterminatetheconjugate-gradientiter-ationsbeforeconvergence,andstillobtainanapproximationoftheNewtondirectionsuitableforinterior-pointmethods.SeetheNotesandReferencesformorediscussion.Anexamplewilldemonstratethee±cienciesthatcanbeobtainedusingthetech-niquessketchedabove.TheproblemisanEVPthatwewillencounterinx6.2.1.n£nWearegivenmatricesA1;:::;AL2R,symmetricmatricesD1;:::;DL;E2n£nR.WeconsidertheEVPminimizeTrEP(2.20)subjecttoATP+PA+D<0;i=1;:::;LiiiInthisproblemthevariableisthematrixP,sothedimensionoftheoptimizationvariableism=n(n+1)=2.WhentheLyapunovinequalitiesarecombinedintooneN£NlargeLMIF(x)>0,we¯ndthatF(x)2RwithN=Ln.ThisLMIhasmuchstructure:Itisblock-diagonalwitheachblockaLyapunovinequality.VandenbergheandBoydhavedevelopeda(primal-dual)interior-pointmethodthatsolves(2.20),exploitingtheproblemstructure.Theyprovetheworst-caseesti-mateofO(m2:75L1:5)arithmeticoperationstosolvetheproblemtoagivenaccuracy.Incomparison,theellipsoidmethodsolves(2.20)toagivenaccuracyinO(m3:5L)arithmeticoperations(moreover,theconstanthiddenintheO(¢)notationismuchlargerfortheellipsoidalgorithm).Numericalexperimentsonfamiliesofproblemswithrandomlygenerateddataandfamiliesofproblemsarisinginsystemandcontroltheoryshowthattheactualperformanceoftheinterior-pointmethodismuchbetterthantheworst-caseestimate:O(m®L¯)arithmeticoperations,with®¼2:1and¯¼1:2.Thiscomplexityestimateisremarkablysmall.Forexample,thecostofsolvingLLyapunovequationsofthesamesizeisO(m1:5L).Therefore,therelativecostofsolvingLcoupledLyapunovinequalities,comparedtothecostofsolvingLindependentLyapunovinequalitiesisO(m0:6L0:2).Thisexampleillustratesoneofourpoints:Thecomputationalcostofsolvingoneofthestandardproblems(whichhasnoanalyticsolution")canbecomparabletothecomputationalcostofevaluatingthesolutionofasimilarproblemthathasananalyticsolution".2.5StrictandNonstrictLMIsWehavesofarassumedthattheoptimizationproblemsLMIP,EVP,GEVP,andCPinvolvestrictLMIs.WewillalsoencountertheseproblemswithnonstrictLMIs,ormoregenerally,withamixtureofstrictandnonstrictLMIs.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 2.5StrictandNonstrictLMIs19AsanexampleconsiderthenonstrictversionoftheEVP(2.9),i.e.minimizecTxsubjecttoF(x)¸0IntuitionsuggeststhatwecouldsimplysolvethestrictEVP(by,say,aninterior-pointmethod)toobtainthesolutionofthenonstrictEVP.Thisiscorrectinmostbutnotallcases.IfthestrictLMIF(x)>0isfeasible,thenwehavemmfx2RjF(x)¸0g=fx2RjF(x)>0g;(2.21)i.e.,thefeasiblesetofthenonstrictLMIistheclosureofthefeasiblesetofthestrictLMI.Itfollowsthat©ª©ªinfcTxjF(x)¸0=infcTxjF(x)>0:(2.22)Sointhiscase,wecansolvethestrictEVPtoobtainasolutionofthenonstrictEVP.ThisistruefortheproblemsGEVPandCPaswell.WewillsaythattheLMIF(x)¸0isstrictlyfeasibleifitsstrictversionisfeasible,mi.e.,ifthereissomex02RsuchthatF(x0)>0.WehavejustseenthatwhenanLMIisstrictlyfeasible,wecanreplacenonstrictinequalitywithstrictinequalityintheproblemsEVP,GEVP,andCPinordertosolvethem.Inthelanguageofoptimizationtheory,therequirementofstrictfeasibilityisa(verystrong)constraintquali¯cation.WhenanLMIisfeasiblebutnotstrictlyfeasible,(2.21)neednothold,andtheEVPswiththestrictandnonstrictLMIscanbeverydi®erent.Asasimpleexample,considerF(x)=diag(x;¡x)withx2R.Theright-handsideof(2.22)is+1sincethestrictLMIF(x)>0isinfeasible.Theleft-handside,however,isalways0,sincetheLMIF(x)¸0hasthesinglefeasiblepointx=0.Thisexampleshowsoneofthetwopathologiesthatcanoccur:Thenonstrictinequalitycontainsanimplicitequalityconstraint(incontrastwithanexplicitequalityconstraintasinx2.1.2).TheotherpathologyisdemonstratedbytheexampleF(x)=diag(x;0)withx2Randc=¡1.Onceagain,thestrictLMIisinfeasiblesotheright-handsideof(2.22)is+1.ThefeasiblesetforthenonstrictLMIistheinterval[0;1)sotheright-handsideis¡1.TheproblemhereisthatF(x)isalwayssingular.Ofcourse,thenonstrictLMIF(x)¸0isequivalent(inthesenseofde¯ningequalfeasiblesets)tothereduced"LMIF~(x)=x¸0.NotethatthisreducedLMIsatis¯estheconstraintquali¯cation,i.e.,isstrictlyfeasible.2.5.1ReductiontoastrictlyfeasibleLMIItturnsoutthatanyfeasiblenonstrictLMIcanbereducedtoanequivalentLMIthatisstrictlyfeasible,byeliminatingimplicitequalityconstraintsandthenreducingtheresultingLMIbyremovinganyconstantnullspace.n£nTheprecisestatementis:LetF0;:::;Fm2Rbesymmetric.Thentherem£pmisamatrixA2Rwithp·m,avectorb2R,andsymmetricmatricesF~;:::;F~2Rq£qwithq·nsuchthat:0ppx=Az+bforsomez2RwithXpF(x)¸0()¢F~(z)=F~0+ziF~i¸0i=1Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 20Chapter2SomeStandardProblemsInvolvingLMIswheretheLMIF~(z)¸0iseitherinfeasibleorstrictlyfeasible.SeetheNotesandReferencesforaproof.ThematrixAandvectorbdescribetheimplicitequalityconstraintsfortheLMIF(x)¸0.Similarly,theLMIF~(z)¸0canbeinterpretedastheoriginalLMIwithitsconstantnullspaceremoved(seetheNotesandReferences).Inmostoftheprob-lemsencounteredinthisbook,therearenoimplicitequalityconstraintsornontrivialcommonnullspaceforF,sowecanjusttakeA=I,b=0,andF~=F.Usingthisreductionwecan,atleastinprinciple,alwaysdealwithstrictlyfeasibleLMIs.Forexamplewehave©¯ªn¯¯oinfcTx¯F(x)¸0=infcT(Az+b)¯F~(z)¸0n¯oT¯=infc(Az+b)¯F~(z)>0sincetheLMIF~(z)¸0iseitherinfeasibleorstrictlyfeasible.2.5.2Example:LyapunovinequalityToillustratethepreviousideasweconsiderthesimplestLMIarisingincontroltheory,theLyapunovinequality:ATP+PA·0;P>0;(2.23)k£kwhereA2Risgiven,andthesymmetricmatrixPisthevariable.NotethatthisLMIcontainsastrictinequalityaswellasanonstrictinequality.WeknowfromsystemtheorythattheLMI(2.23)isfeasibleifandonlyifalltrajectoriesofx_=Axarebounded,orequivalently,iftheeigenvaluesofAhavenonpositiverealpart,andthosewithzerorealpartarenondefective,i.e.,correspondtoJordanblocksofsizeone.WealsoknowfromsystemtheorythattheLMI(2.23)isstrictlyfeasibleifandonlyifalltrajectoriesofx_=Axconvergetozero,orequivalently,alltheeigenvaluesofAhavenegativerealpart.ConsidertheinterestingcasewheretheLMI(2.23)isfeasiblebutnotstrictlyfeasible.Fromtheremarksabove,weseethatbyachangeofcoordinateswecanputAintheformA~=¢T¡1ATÃ"#"#!0!1Ik10!rIkr=diag;:::;;0kr+1;Astab¡!1Ik10¡!rIkr0where00=8¯Ã"#"#!9>>¯¯P1Q1PrQr>>>>¯P~=diag;:::;;Pr+1;Pstab>>>>¯QTPQTP>>>>¯11rr>>>>¯>>>><¯Pi2Rki£ki;i=1;:::;r+1;QTi=¡Qi;i=1;:::;r>>=¯T¡TP~T¡1¯:>>¯¯"#>>>>>>¯PiQi>>>>>>¯¯T>0;i=1;:::;r;Pr+1>0;Pstab>0>>>>¯QiPi>>>>¯>>:¯ATP+PA·0;stabstabstabstabFromthischaracterizationwecan¯ndareducedLMIthatisstrictlyfeasible.WecantakethesymmetricmatricesP1;:::;Pr+1;Pstabandtheskew-symmetricmatricesQ1;:::;Qrasthefree"variablez;thea±nemappingfromzintoxsimplymapsthesematricesintoÃ"#"#!¡TP1Q1PrQr¡1P=Tdiag;:::;;Pr+1;PstabT:(2.24)QTPQTP11rrPutanotherway,theequalityconstraintsimplicitintheLMI(2.23)arethatPmusthavethisspecialstructure.NowwesubstitutePintheform(2.24)backintotheoriginalLMI(2.23).We¯ndthat¡¢ATP+PA=TTdiag0;ATP+PATstabstabstabstabwherethezeromatrixhassize2k1+¢¢¢+2kr+kr+1.Weremovetheconstantnullspacetoobtainthereducedversion,i.e.,ATP+PA·0.Thus,thereducedstabstabstabstabLMIcorrespondingto(2.23)is"#PiQi>0;i=1;:::;r;Pr+1>0;QTP(2.25)iiP>0;ATP+PA·0:stabstabstabstabstabThisreducedLMIisstrictlyfeasible,sincewecantakeP1;:::;Pr+1asidentitymatri-ces,Q1;:::;Qraszeromatrices,andPstabasthesolutionoftheLyapunovequationATP+PA+I=0.stabstabstabstabInsummary,theoriginalLMI(2.23)hasonesymmetricmatrixofsizekasvariable(i.e.,thedimensionoftheoriginalvariablexism=k(k+1)=2).ThereducedLMI(2.25)hasasvariablethesymmetricmatricesP1;:::;Pr+1;Pstabandtheskew-symmetricmatricesQ1;:::;Qr,sothedimensionofthevariablezinthereducedLMIisXr2kr+1(kr+1+1)s(s+1)p=ki++0;(2.26)wecaneliminateanyvariablesforwhichthecorrespondingcoe±cientinFissemidef-inite.SupposeforexamplethatFm¸0andhasrankr0()U~UF(x)U~U>0:Since"#hiThiU~TF~(x~)U~U~TF~(x~)UU~UF(x)U~U=;UTF~(x~)U~UTF~(x~)U+x(UTU)2mT¢wherex~=[x1¢¢¢xm¡1]andF~(x~)=F0+x1F1+¢¢¢+xm¡1Fm¡1,weseethatF(x)>0ifandonlyifU~TF~U~(x~)>0,andxislargeenoughthatm³´¡1UTF~(x~)U+x(UTU)2>UTF~(x~)U~U~TF~(x~)U~U~TF~(x~)U:mTherefore,theLMIP(2.26)isequivalentto¯ndx;:::;xsuchthatU~TF~(x;:::;x)U~>0:1m¡11m¡12.6.2EliminationofmatrixvariablesWhenamatrixinequalityhassomevariablesthatappearinacertainform,wecanderiveanequivalentinequalitywithoutthosevariables.ConsiderG(z)+U(z)XV(z)T+V(z)XTU(z)T>0;(2.27)n£nwherethevectorzandthematrixXare(independent)variables,andG(z)2R,U(z)andV(z)donotdependonX.Matrixinequalitiesoftheform(2.27)ariseinthecontrollersynthesisproblemsdescribedinChapter7.Supposethatforeveryz,U~(z)andV~(z)areorthogonalcomplementsofU(z)andV(z)respectively.Then(2.27)holdsforsomeXandz=z0ifandonlyiftheinequalitiesU~(z)TG(z)U~(z)>0;V~(z)TG(z)V~(z)>0(2.28)holdwithz=z0.Inotherwords,feasibilityofthematrixinequality(2.27)withvariablesXandzisequivalenttothefeasibilityof(2.28)withvariablez;wehaveeliminatedthematrixvariableXfrom(2.27)toform(2.28).NotethatifU(z)orV(z)hasranknforallz,thenthe¯rstorsecondinequalityin(2.28)disappears.WeprovethislemmaintheNotesandReferences.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 2.6MiscellaneousResultsonMatrixInequalities23Wecanexpress(2.28)inanotherformusingFinsler'slemma(seetheNotesandReferences):G(z)¡¾U(z)U(z)T>0;G(z)¡¾V(z)V(z)T>0forsome¾2R.Asanexample,wewillencounterinx7.2.1theLMIPwithLMIQ>0;AQ+QAT+BY+YTBT<0;(2.29)whereQandYarethevariables.ThisLMIPisequivalenttotheLMIPwithLMIQ>0;AQ+QAT<¾BBT;wherethevariablesareQand¾2R.ItisalsoequivalenttotheLMIP¡¢Q>0;B~TAQ+QATB~<0;withvariableQ,whereB~isanymatrixofmaximumranksuchthatB~TB=0.ThuswehaveeliminatedthevariableYfrom(2.29)andreducedthesizeofthematricesintheLMI.2.6.3TheS-procedureWewilloftenencountertheconstraintthatsomequadraticfunction(orquadraticform)benegativewheneversomeotherquadraticfunctions(orquadraticforms)areallnegative.Insomecases,thisconstraintcanbeexpressedasanLMIinthedatade¯ningthequadraticfunctionsorforms;inothercases,wecanformanLMIthatisaconservativebutoftenusefulapproximationoftheconstraint.TheS-procedureforquadraticfunctionsandnonstrictinequalitiesnLetF0;:::;Fpbequadraticfunctionsofthevariable³2R:¢TTFi(³)=³Ti³+2ui³+vi;i=0;:::;p;whereT=TT.WeconsiderthefollowingconditiononF;:::;F:ii0pF0(³)¸0forall³suchthatFi(³)¸0;i=1;:::;p:(2.30)Obviouslyifthereexist¿1¸0;:::;¿p¸0suchthatXp(2.31)forall³;F0(³)¡¿iFi(³)¸0;i=1then(2.30)holds.Itisanontrivialfactthatwhenp=1,theconverseholds,providedthatthereissome³0suchthatF1(³0)>0.Remark:IfthefunctionsFiarea±ne,then(2.31)and(2.30)areequivalent;thisistheFarkaslemma.Notethat(2.31)canbewrittenas"#"#XpT0u0Tiui¡¿i¸0:uTvuTv00i=1iiThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 24Chapter2SomeStandardProblemsInvolvingLMIsTheS-procedureforquadraticformsandstrictinequalitiesWewilluseanothervariationoftheS-procedure,whichinvolvesquadraticformsandn£nstrictinequalities.LetT0;:::;Tp2Rbesymmetricmatrices.WeconsiderthefollowingconditiononT0;:::;Tp:³TT³>0forall³6=0suchthat³TT³¸0;i=1;:::;p:(2.32)0iItisobviousthatifXpthereexists¿1¸0;:::;¿p¸0suchthatT0¡¿iTi>0;(2.33)i=1then(2.32)holds.Itisanontrivialfactthatwhenp=1,theconverseholds,providedthatthereissome³suchthat³TT³>0.Notethat(2.33)isanLMIinthevariables0010T0and¿1;:::;¿p.Remark:The¯rstversionoftheS-proceduredealswithnonstrictinequalitiesandquadraticfunctionsthatmayincludeconstantandlinearterms.Thesecondversiondealswithstrictinequalitiesandquadraticformsonly,i.e.,quadraticfunctionswithoutconstantorlinearterms.Remark:SupposethatT0,u0andv0dependa±nelyonsomeparameterº.Thenthecondition(2.30)isconvexinº.Thisdoesnot,however,meanthattheproblemofcheckingwhetherthereexistsºsuchthat(2.30)holdshaslowcomplexity.Ontheotherhand,checkingwhether(2.31)holdsforsomeºisanLMIPinthevariablesºand¿1;:::;¿p.Therefore,thisproblemhaslowcomplexity.SeetheNotesandReferencesforfurtherdiscussion.S-procedureexampleInChapter5wewillencounterthefollowingconstraintonthevariableP:forall»6=0and¼satisfying¼T¼·»TCTC»,"#T"#"#»ATP+PAPB»(2.34)<0:¼BTP0¼ApplyingthesecondversionoftheS-procedure,(2.34)isequivalenttotheexistenceof¿¸0suchthat23ATP+PA+¿CTCPB45<0:BTP¡¿IThustheproblemof¯ndingP>0suchthat(2.34)holdscanbeexpressedasanLMIP(inPandthescalarvariable¿).2.7SomeLMIProblemswithAnalyticSolutionsThereareanalyticsolutionstoseveralLMIproblemsofspecialform,oftenwithim-portantsystemandcontroltheoreticinterpretations.Webrie°ydescribesomeoftheseresultsinthissection.Atthesametimeweintroduceandde¯neseveralimportanttermsfromsystemandcontroltheory.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 2.7SomeLMIProblemswithAnalyticSolutions252.7.1Lyapunov'sinequalityWehavealreadymentionedtheLMIPassociatedwithLyapunov'sinequality,i.e.P>0;ATP+PA<0n£nwherePisvariableandA2Risgiven.LyapunovshowedthatthisLMIisfeasibleifandonlythematrixAisstable,i.e.,alltrajectoriesofx_=Axconvergetozeroast!1,orequivalently,alleigenvaluesofAmusthavenegativerealpart.TosolvethisLMIP,wepickanyQ>0andsolvetheLyapunovequationATP+PA=¡Q,whichisnothingbutasetofn(n+1)=2linearequationsforthen(n+1)=2scalarvariablesinP.ThissetoflinearequationswillbesolvableandresultinP>0ifandonlyiftheLMIisfeasible.Infactthisprocedurenotonly¯ndsasolutionwhentheLMIisfeasible;itparametrizesallsolutionsasQvariesoverthepositive-de¯nitecone.2.7.2Thepositive-reallemmaAnotherimportantexampleisgivenbythepositive-real(PR)lemma,whichyieldsafrequency-domain"interpretationforacertainLMIP,andundersomeadditionalassumptions,anumericalsolutionprocedureviaRiccatiequationsaswell.Wegiveasimpli¯eddiscussionhere,andreferthereadertotheReferencesformorecompletestatements.TheLMIconsideredis:"#ATP+PAPB¡CTP>0;·0;(2.35)BTP¡C¡DT¡Dn£nn£pp£np£pwhereA2R,B2R,C2R,andD2Raregiven,andthematrixP=PT2Rn£nisthevariable.(WewillencounteravariationonthisLMIinx6.3.3.)NotethatifD+DT>0,theLMI(2.35)isequivalenttothequadraticmatrixinequalityATP+PA+(PB¡CT)(D+DT)¡1(PB¡CT)T·0:(2.36)WeassumeforsimplicitythatAisstableandthesystem(A;B;C)isminimal.Thelinkwithsystemandcontroltheoryisgivenbythefollowingresult.TheLMI(2.35)isfeasibleifandonlyifthelinearsystemx_=Ax+Bu;y=Cx+Du(2.37)ispassive,i.e.,ZTu(t)Ty(t)dt¸00forallsolutionsof(2.37)withx(0)=0.Passivitycanalsobeexpressedintermsofthetransfermatrixofthelinearsys-tem(2.37),de¯nedas¢¡1H(s)=C(sI¡A)B+Dfors2C.PassivityisequivalenttothetransfermatrixHbeingpositive-real,whichmeansthatH(s)+H(s)¤¸0forallRes>0:Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 26Chapter2SomeStandardProblemsInvolvingLMIsWhenp=1thisconditioncanbecheckedbyvariousgraphicalmeans,e.g.,plottingthecurvegivenbytherealandimaginarypartsofH(i!)for!2R(calledtheNyquistplotofthelinearsystem).Thuswehaveagraphical,frequency-domainconditionforfeasibilityoftheLMI(2.35).Thisapproachwasusedinmuchoftheworkinthe1960sand1970sdescribedinx1.2.Withafewfurthertechnicalassumptions,includingD+DT>0,theLMI(2.35)canbesolvedbyamethodbasedonRiccatiequationsandHamiltonianmatrices.WiththeseassumptionstheLMI(2.35)isfeasibleifandonlyifthereexistsarealmatrixP=PTsatisfyingtheAREATP+PA+(PB¡CT)(D+DT)¡1(PB¡CT)T=0;(2.38)whichisjustthequadraticmatrixinequality(2.36)withequalitysubstitutedforin-equality.NotethatP>0.TosolvetheARE(2.38)we¯rstformtheassociatedHamiltonianmatrix"#A¡B(D+DT)¡1CB(D+DT)¡1BTM=:¡CT(D+DT)¡1C¡AT+CT(D+DT)¡1BTThenthesystem(2.37)ispassive,orequivalently,theLMI(2.35)isfeasible,ifandonlyifMhasnopureimaginaryeigenvalues.ThisfactcanbeusedtoformaRouth{Hurwitz(Sturm)typetestforpassivityasasetofpolynomialinequalitiesinthedataA,B,C,andD.WhenMhasnopureimaginaryeigenvalueswecanconstructasolutionPareas2n£nfollows.PickV2RsothatitsrangeisabasisforthestableeigenspaceofM,e.g.,V=[v1¢¢¢vn]wherev1;:::;vnareasetofindependenteigenvectorsofMassociatedwithitsneigenvalueswithnegativerealpart.PartitionVas"#V1V=;V2¡1whereV1andV2aresquare;thensetPare=V2V1.ThesolutionParethusobtainedistheminimalelementamongthesetofsolutionsof(2.38):ifP=PTsatis¯es(2.38),thenP¸Pare.Muchmorediscussionofthismethod,includingtheprecisetechnicalconditions,canbefoundintheReferences.2.7.3Thebounded-reallemmaThesameresultsappearinanotherimportantform,thebounded-reallemma.HereweconsidertheLMI"#ATP+PA+CTCPB+CTDP>0;·0:(2.39)BTP+DTCDTD¡In£nn£pp£np£pwhereA2R,B2R,C2R,andD2Raregiven,andthematrixP=PT2Rn£nisthevariable.ForsimplicityweassumethatAisstableand(A;B;C)isminimal.ThisLMIisfeasibleifandonlythelinearsystem(2.37)isnonexpansive,i.e.,ZTZTy(t)Ty(t)dt·u(t)Tu(t)dt00Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences27forallsolutionsof(2.37)withx(0)=0,ThisconditioncanalsobeexpressedintermsofthetransfermatrixH.NonexpansivityisequivalenttothetransfermatrixHsatisfyingthebounded-realcondition,i.e.,H(s)¤H(s)·IforallRes>0:ThisissometimesexpressedaskHk1·1where¢kHk1=supfkH(s)kjRes>0giscalledtheH1normofthetransfermatrixH.Thisconditioniseasilycheckedgraphically,e.g.,byplottingkH(i!)kversus!2R(calledasingularvalueplotforp>1andaBodemagnitudeplotforp=1).OnceagainwecanrelatetheLMI(2.39)toanARE.Withsomeappropriatetech-nicalconditions(seetheNotesandReferences)includingDTD0;AP+PAT0anddiagonal.Inequality(2.41)holdsifandonlyif23"#"#"#QS1S2I0QSI06T7TT=4S1§05¸0;0USR0UTS200where[S1S2]=SU,withappropriatepartitioning.WemustthenhaveS2=0,whichholdsyifandonlyifS(I¡RR)=0,and"#QS1¸0;TS1§yTwhichholdsifandonlyifQ¡SRS¸0.FormulatingconvexproblemsintermsofLMIsTheideathatLMIscanbeusedtorepresentawidevarietyofconvexconstraintscanbefoundinNesterovandNemirovskii'sreport[NN90b](whichisreallyapreliminaryversionCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences29oftheirbook[NN94])andsoftwaremanual[NN90a].Theyformalizetheideaofapositive-de¯niterepresentable"function;seex5.3,x5.4,andx6.4oftheirbook[NN94].Theideaisalsodiscussedinthearticle[Ali92b]andthesis[Ali91]ofAlizadeh.In[BE93],BoydandElGhaouigivealistofquasiconvexfunctionsthatcanberepresentedasgeneralizedeigenvaluesofmatricesthatdependa±nelyonthevariable.InfeasibilitycriterionforLMIsmTheLMIF(x)>0isinfeasiblemeansthea±nesetfF(x)jx2Rgdoesnotintersectthepositive-de¯nitecone.Fromconvexanalysis,thisisequivalenttotheexistenceofalinearfunctionalÃthatispositiveonthepositive-de¯niteconeandnonpositiveonthea±nesetofmatrices.Thelinearfunctionalsthatarepositiveonthepositive-de¯niteconeareoftheformÃ(F)=TrGF,whereG¸0andG6=0.FromthefactthatÃisnonpositiveonthema±nesetfF(x)jx2Rg,wecanconcludethatTrGFi=0,i=1;:::;mandTrGF0·0.ThesearepreciselytheconditionsforinfeasibilityofanLMIthatwementionedinx2.2.1.ForthespecialcaseofmultipleLyapunovinequalities,theseconditionsaregivenBellmanandKyFan[BF63]andKamenetskiiandPyatnitskii[KP87a,KP87b].Itisstraightforwardtoderiveoptimalitycriteriafortheotherproblems,usingconvexanal-ysis.Somegeneralreferencesforconvexanalysisarethebooks[Roc70,Roc82]andsurveyarticle[Roc93]byRockafellar.Therecenttext[HUL93]givesagoodoverviewofconvexanalysis;LMIsareusedasexamplesinseveralplaces.ComplexityofconvexoptimizationTheimportantroleofconvexityinoptimizationisfairlywidelyknown,butperhapsnotwellenoughappreciated,atleastoutsidetheformerSovietUnion.Inastandard(Western)treatmentofoptimization,ourstandardproblemsLMIP,EVP,GEVP,andCPwouldbeconsideredverydi±cultsincetheyarenondi®erentiableandnonlinear.Theirconvexityproperties,however,makethemtractable,bothintheoryandinpractice.In[Roc93,p194],Rockafellarmakestheimportantpoint:Onedistinguishingideawhichdominatesmanyissuesinoptimizationtheoryisconvexity...Animportantreasonisthefactthatwhenaconvexfunctionisminimizedoveraconvexseteverylocallyoptimalsolutionisglobal.Also,¯rst-ordernecessaryconditionsturnouttobesu±cient.Avarietyofotherpropertiesconducivetocomputationandinterpretationofsolutionsrideonconvexityaswell.Infactthegreatwatershedinoptimizationisn'tbetweenlinearityandnonlinearity,butconvexityandnonconvexity.Detaileddiscussionofthe(low)complexityofconvexoptimizationproblemscanbefoundinthebooksbyGrÄotschel,Lov¶asz,Schrijver[GLS88],NemirovskiiandYudin[NY83],andVavasis[Vav91].Completeanddetailedworst-casecomplexityanalysesofseveralalgorithmsforourstandardproblemscanbefoundinChapters3,4,and6ofNesterovandNemirovskii[NN94].EllipsoidalgorithmTheellipsoidalgorithmwasdevelopedbyShor,Nemirovskii,andYudininthe1970s[Sho85,NY83].ItwasusedbyKhachiyanin1979toprovethatlinearprogramscanbesolvedinpolynomial-time[Kha79,GL81,GLS88].Discussionoftheellipsoidmethod,aswellasseveralextensionsandvariations,canbefoundin[BGT81]and[BB91,ch14].Adetailedhistoryofitsdevelopment,includingEnglishandRussianreferences,appearsinchapter3ofAkgulÄ[Akg84].Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 30Chapter2SomeStandardProblemsInvolvingLMIsOptimizationproblemsinvolvingLMIsOneoftheearliestpapersonLMIsisalso,inouropinion,oneofthebest:OnsystemsoflinearinequalitiesinHermitianmatrixvariables,byBellmanandKyFan[BF63].Inthispaperwe¯ndmanyresults,e.g.,adualitytheoryforthemultipleLyapunovinequalityEVPandatheoremofthealternativeforthemultipleLyapunovinequalityLMIP.Thepaperconcentratesontheassociatedmathematics:thereisnodiscussionofhowtosolvesuchproblems,oranypotentialapplications,althoughtheydocommentthatsuchinequalitysystemsariseinLyapunovtheory.GivenBellman'slegendaryknowledgeofsystemandcontroltheory,andespeciallyLyapunovtheory,itistemptingtoconjecturethatBellmanwasawareofthepotentialapplications.Butsofarwehavefoundnoevidenceforthis.RelevantworkincludesCullumetal.[CDW75],CravenandMond[CM81],PolakandWardi[PW82],Fletcher[Fle85],Shapiro[Sha85],Friedlandetal.[FNO87],GohandTeo[GT88],Panier[Pan89],Allwright[All89],OvertonandWomersley[Ove88,Ove92,OW93,OW92],Ringertz[Rin91],FanandNekooie[FN92,Fan93]andHiriart{UrrutyandYe[HUY92].LMIPsaresolvedusingvariousalgorithmsforconvexoptimizationinBoydandYang[BY89],PyatnitskiiandSkorodinskii[PS83],KamenetskiiandPyatnitskii[KP87a,KP87b].AsurveyofmethodsforsolvingproblemsinvolvingLMIsusedbyresearchersincontroltheory+canbefoundinthepaperbyBeck[Bec91].Thesoftwarepackages[BDG91]and[CS92a]useconvexprogrammingtosolvemanyrobustcontrolproblems.Boyd[Boy94]outlineshowinterior-pointconvexoptimizationalgorithmscanbeusedtobuildrobustcontrolsoftwaretools.ConvexoptimizationproblemsinvolvingLMIshavebeenusedincontroltheorysinceabout1985:Gilbert'smethod[Gil66]wasusedbyDoyle[Doy82]tosolveadiagonalscalingproblem;anotherexampleisthemusolprogramofFanandTits[FT86].AmongothermethodsusedincontroltosolveproblemsinvolvingmultipleLMIsisthemethodofalternatingconvexprojections"describedinthearticlebyGrigoriadisandSkel-ton[GS92].Thismethodisessentiallyarelaxationmethod,andrequiresananalyticex-pressionfortheprojectionontothefeasiblesetofeachLMI.Itisnotapolynomial-timealgorithm;itscomplexitydependsontheproblemdata,unliketheellipsoidmethodortheinterior-pointmethodsdescribedin[NN94],whosecomplexitiesdependontheproblemsizeonly.However,thealternatingprojectionsmethodisreportedtoworkwellinmanycases.Interior-pointmethodsforLMIproblemsInterior-pointmethodsforvariousLMIproblemshaverecentlybeendevelopedbyseveralre-searchers.The¯rstwereNesterovandNemirovskii[NN88,NN90b,NN90a,NN91a,NN94,NN93];othersincludeAlizadeh[Ali92b,Ali91,Ali92a],Jarre[Jar93c],VandenbergheandBoyd[VB93b],Rendl,Vanderbei,andWolkowocz[RVW93],andYoshise[Yos94].Ofcourse,generalinterior-pointmethods(andthemethodofcentersinparticular)havealonghistory.EarlyworkincludesthebookbyFiaccoandMcCormick[FM68],themethodofcentersdescribedbyHuardetal.[LH66,Hua67],andDikin'sinterior-pointmethodforlinearprogramming[Dik67].Interestininterior-pointmethodssurgedin1984whenKarmarkar[Kar84]gavehisinterior-pointmethodforsolvinglinearprograms,whichappearstohaveverygoodpracticalperformanceaswellasagoodworst-casecomplexitybound.SincethepublicationofKarmarkar'spaper,manyresearchershavestudiedinterior-pointmethodsforlinearandquadraticprogramming.Thesemethodsareoftendescribedinsuchawaythatextensionstomoregeneral(convex)constraintsandobjectivesarenotclear.However,NesterovandNemirovskiihavedevelopedatheoryofinterior-pointmethodsthatappliestomoregeneralconvexprogrammingproblems,andinparticular,everyproblemthatarisesinthisbook;see[NN94].Inparticular,theyderivecomplexityboundsformanydi®erentinterior-pointalgorithms,includingthemethodofcenters,Nemirovskii'sprojectivealgorithmandprimal-dualmethods.Themoste±cientalgorithmsseemtobeNemirovskii'sprojectivealgorithmandprimal-dualmethods(forthecaseoflinearprograms,see[Meh91]).Otherrecentarticlesthatconsiderinterior-pointmethodsformoregeneralconvexprogram-mingincludeSonnevend[Son88],Jarre[Jar91,Jar93b],Kortaneketal.[KPY91],DenHertog,Roos,andTerlaky[DRT92],andthesurveybyWright[Wri92].Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences31Interior-pointmethodsforGEVPsaredescribedinBoydandElGhaoui[BE93](variationonthemethodofcenters),andNesterovandNemirovskii[NN91b]and[NN94,x4.4](avariationonNemirovskii'sprojectivealgorithm).SinceGEVPsarenotconvexproblems,devisingareliablestoppingcriterionismorechallengingthanfortheconvexproblemsLMIP,EVP,andCP.Adetailedcomplexityanalysis(inparticular,astatementandproofofthepolynomial-timecomplexityofGEVPs)isgivenin[NN91b,NN94].Seealso[Jar93a].Severalresearchershaverecentlystudiedthepossibilityofswitchingfromaninterior-pointmethodtoaquadraticallyconvergentlocalmethodinordertoimproveonthe¯nalconver-gence;see[Ove92,OW93,OW92,FN92,NF92].Interior-pointmethodsforCPsandotherrelatedextremalellipsoidvolumeproblemscanbefoundin[NN94,x6.5].Interior-pointmethodsandproblemstructureManyresearchershavedevelopedalgorithmsthattakeadvantageofthespecialstructureoftheleast-squaresproblemsarisingininterior-pointmethodsforlinearprogramming.Asfarasweknow,the¯rst(andsofar,only)interior-pointalgorithmthattakesadvantageofthespecial(Lyapunov)structureofanLMIproblemarisingincontrolisdescribedinVandenbergheandBoyd[VB93b,VB93a].Nemirovskii'sprojectivemethodcanalsotakeadvantageofsuchstructure;thesetwoalgorithmsappeartobethemoste±cientalgorithmsdevelopedsofarforsolvingtheLMIsthatariseincontroltheory.SoftwareforsolvingLMIproblemsGahinetandNemirovskiihaverecentlydevelopedasoftwarepackagecalledLMI-Lab[GN93]basedonanearlierFORTRANcode[NN90a],whichallowstheusertodescribeanLMIprobleminahigh-levelsymbolicform(notunliketheformulasthatappearthroughoutthisbook!).LMI-LabthensolvestheproblemusingNemirovskii'sprojectivealgorithm,takingadvantageofsomeoftheproblemstructure(e.g.,blockstructure,diagonalstructureofsomeofthematrixvariables).Recently,ElGhaouihasdevelopedanothersoftwarepackageforsolvingLMIproblems.Thisnoncommercialpackage,calledLMI-tool,canbeusedwithmatlab.Itisavailableviaanony-mousftp(formoreinformation,sendmailtoelghaoui@ensta.fr).AnotherversionofLMI-tool,developedbyNikoukhahandDelebecque,isavailableforusewiththematlab-likefreewarepackagescilab;inthisversion,LMI-toolhasbeeninterfacedwithNemirovskii'sprojectivecode.scilabcanbeobtainedviaanonymousftp(formoreinformation,sendmailtoScilab@inria.fr).AcommercialsoftwarepackagethatsolvesafewspecializedcontrolsystemanalysisanddesignproblemsviaLMIformulation,calledoptin,wasrecentlydevelopedbyOlasandAssociates;see[OS93,Ola94].ReductiontoastrictlyfeasibleLMIn£nInthissectionweprovethefollowingstatement.LetF0;:::;Fm2Rbesymmetricm£pmmatrices.ThenthereisamatrixA2Rwithp·m,avectorb2R,andsymmetricq£qmatricesF~0;:::;F~p2Rwithq·nsuchthat:Xpp¢F(x)¸0ifandonlyifx=Az+bforsomez2RandF~(z)=F~0+ziF~i¸0:i=1Inaddition,iftheLMIF(x)¸0isfeasible,thentheLMIF~(z)¸0isstrictlyfeasible.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 32Chapter2SomeStandardProblemsInvolvingLMIsConsidertheLMIXmF(x)=F0+xiFi¸0;(2.42)i=1n£nwhereFi2R,i=0;:::;m.LetXdenotethefeasiblesetfxjF(x)¸0g.IfXisempty,thereisnothingtoprove;wecantakep=m,A=I,b=0,F~0=F0;:::;F~p=Fp.HenceforthweassumethatXisnonempty.IfXisasingleton,sayX=fx0g,thenwithp=1,A=0,b=x0,F~0=1,andF~1=0,thestatementfollows.NowconsiderthecasewhenXisneitheremptynorasingleton.Then,thereisana±nesubspaceAofminimaldimensionp¸1thatcontainsX.Leta1;:::apabasisforthelinearpartofA.Theneveryx2Acanbewrittenasx=Az+bwhereA=[a1¢¢¢ap]Pisfull-rankmpandb2R.De¯ningG0=F(b),Gi=F(ai)¡F0,i=1;:::;p,andG(z)=G0+ziGi,i=1pweseethatF(x)¸0ifandonlyifthereexistsz2Rsatisfyingx=Az+bandG(z)¸0.¢pLetZ=fz2RjG(z)¸0g.Byconstruction,Zhasnonemptyinterior.Letz0apointinTtheinteriorofZandletvlieinthenullspaceofG(z0).Now,vG(z)visanonnegativea±nefunctionofz2ZandiszeroataninteriorpointofZ.Therefore,itisidenticallyzerooverpZ,andhenceoverR.Therefore,vbelongstotheintersectionBofthenullspacesoftheTpGi,i=0;:::;p.Conversely,anyvbelongingtoBwillsatisfyvG(z)v=0foranyz2R.Bmaythereforebeinterpretedastheconstant"nullspaceoftheLMI(2.42).LetqbethedimensionofB.Ifq=n(i.e.,G0=¢¢¢=Gp=0),thenobviouslyF(x)¸0ifandonlyifx=Az+b.Inthisp£pcase,thestatementissatis¯edwithF~0=I2R,F~i=0,i=1;:::;p.Ifq0.Thisconcludestheproof.EliminationprocedureformatrixvariablesWenowprovethematrixeliminationresultstatedinx2.6.2:GivenG,U,V,thereexistsXsuchthatTTTG+UXV+VXU>0(2.43)ifandonlyifU~TGU~>0;V~TGV~>0(2.44)holds,whereU~andV~areorthogonalcomplementsofUandVrespectively.Itisobviousthatif(2.43)holdsforsomeX,sodoinequalities(2.44).Letusnowprovetheconverse.Supposethatinequalities(2.44)arefeasible.Supposenowthatinequality(2.43)isnotfeasibleforanyX.Inotherwords,supposethatTTTG+UXV+VXU6>0;foreveryX.Byduality,thisisequivalenttotheconditionTthereexistsZ6=0withZ¸0,VZU=0,andTrGZ·0.(2.45)Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences33TNowletusshowthatZ¸0andVZU=0implythatZ=V~HHTV~T+U~KKTU~T(2.46)forsomematricesH,K(atleastoneofwhichisnonzero,sinceotherwiseZ=0).Thiswill¯nishourproof,since(2.44)impliesthatTrGZ>0forZoftheform(2.46),whichcontradictsTrGZ·0in(2.45).TTLetRbeaCholeskyfactorofZ,i.e.Z=RR.TheconditionVZU=0isequivalenttoTTTTTVRRU=(VR)(UR)=0:ThismeansthatthereexistsaunitarymatrixT,andmatricesMandNsuchthatTTVRT=[0M]andURT=[N0];wherethenumberofcolumnsofMandNadduptothatofR.Inotherwords,RTcanbewrittenasRT=[AB]TTforsomematricesA;BsuchthatVA=0,UB=0.Fromthede¯nitionofV~andU~,matricesA,BcanbewrittenA=V~H,B=U~KforsomematricesH;K,andwehaveZ=RRT=(RT)(RT)T=AAT+BBT=V~HHTV~T+U~KKTU~T;whichisthedesiredresult.TheequivalencebetweentheconditionsU~(z)TG(z)U~(z)>0andTG(z)¡¾U(z)U(z)>0forsomerealscalar¾TisthroughFinsler'slemma[Fin37](seealsox3.2.6of[Sch73]),whichstatesthatifxQx>0TforallnonzeroxsuchthatxAx=0,whereQandAaresymmetric,realmatrices,thenthereexistsarealscalarsuchthatQ¡¾A>0.Finsler'slemmahasalsobeendirectlyusedtoeliminatevariablesincertainmatrixinequali-ties(seeforexample[PH86,KR88,BPG89b]);itiscloselyrelatedtotheS-procedure.Wealsonoteanotherresultoneliminationofmatrixvariables,duetoParrott[Par78].Theeliminationlemmaisrelatedtoamatrixdilationproblemconsideredin[DKW82],which+wasusedincontrolproblemsin[PZPB91,PZP92,Gah92,Pac94,IS93a,Iwa93](seealsotheNotesandReferencesofChapter7).TheS-procedureTheproblemofdeterminingifaquadraticformisnonnegativewhenotherquadraticformsarenonnegativehasbeenstudiedbymathematiciansforatleastseventyyears.Foracompletediscussionandreferences,wereferthereadertothesurveyarticlebyUhlig[Uhl79].SeealsothebookbyHestenes[Hes81,p354-360]andHornandJohnson[HJ91,p78-86]forproofsofvariousS-procedureresults.TheapplicationoftheS-proceduretocontrolproblemsdatesbackto1944,whenLur'eandPostnikovusedittoprovethestabilityofsomeparticularnonlinearsystems.ThenameS-procedure"wasintroducedmuchlaterbyAizermanandGantmacherintheir1964Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 34Chapter2SomeStandardProblemsInvolvingLMIsbook[AG64];sincethenYakubovichhasgivenmoregeneralformulationsandcorrectederrorsinsomeearlierproofs[Yak77,FY79].TheproofofthetwoS-procedureresultsdescribedinthischaptercanbefoundinthearticlesbyYakubovich[FY79,Yak77,Yak73].RecentandimportantdevelopmentsabouttheS-procedurecanbefoundin[Yak92]andreferencestherein.ComplexityofS-procedureconditionSupposethatT0,u0,andv0dependa±nelyonsomevariableº;thecondition(2.30)isconvexinº.Indeed,for¯xed³,theconstraintF0(³)¸0isalinearconstraintonºandtheconstraint(2.30)issimplyanin¯nitenumberoftheselinearconstraints.Wemightthereforeimaginethattheproblemofcheckingwhetherthereexistsºsuchthat(2.30)holdshaslowcomplexitysinceitcanbecastasaconvexfeasibilityproblem.Thisiswrong.Infact,merelyverifyingthatthecondition(2.30)holdsfor¯xeddataTi,ui,andvi,isashardassolvingageneralinde¯nitequadraticprogram,whichisNP-complete.Wecoulddeterminewhetherthereexistsºsuchthat(2.30)holdswithpolynomiallymanystepsoftheellipsoidalgorithm;theproblemisthat¯ndingacuttingplane(whichisrequiredforeachstepoftheellipsoidalgorithm)isitselfanNP-completeproblem.Positive-realandbounded-reallemmaIn1961,PopovgavethefamousPopovfrequency-domainstabilitycriterionfortheabsolutestabilityproblem[Pop62].Popov'scriterioncouldbecheckedviagraphicalmeans,byverify-ingthattheNyquistplotofthelinearpart"ofthenonlinearsystemwascon¯nedtoaspeci¯cregioninthecomplexplane.Yakubovich[Yak62,Yak64]andKalman[Kal63a,Kal63b]establishedtheconnectionbetweenthePopovcriterionandtheexistenceofapositive-de¯nitematrixsatisfyingcertainmatrixinequalities.ThePRlemmaisalsoknownbyvariousnamessuchastheYakubovich{Kalman{Popov{AndersonLemmaortheKalman{YakubovichLemma.ThePRlemmaisnowstandardmaterial,describedinseveralbooksoncontrolandsystemstheory,e.g.,NarendraandTaylor[NT73],Vidyasagar[Vid92,pp474{478],FaurreandDepeyrot[FD77],AndersonandVongpanitlerd[AV73,ch5{7],Brockett[Bro70],andWillems[Wil70].¡1Initsoriginalform,thePRLemmastatesthatthetransferfunctionc(sI¡A)bofthesingle-inputsingle-outputminimalsystem(A;b;c)ispositive-real,i.e.¡1Rec(sI¡A)b¸0forallRes>0(2.47)TTifandonlyifthereexistsP>0suchthatAP+PA·0andPb=c.Thecondition(2.47)canbecheckedgraphically.Anderson[And67,And66b,And73]extendedthePRlemmatomulti-inputmulti-outputsystems,andderivedsimilarresultsfornonexpansivesystems[And66a].Willems[Wil71b,Wil74a]describedconnectionsbetweenthePRlemma,certainquadraticoptimalcontrolproblemsandtheexistenceofsymmetricsolutionstotheARE;itisin[Wil71b]thatwe¯ndWillems'quoteontheroleofLMIsinquadraticoptimalcontrol(seepage3).Theconnectionsbetweenpassivity,LMIsandAREscanbesummarizedasfollows.WeconsiderA,B,C,andDsuchthatalleigenvaluesofAhavenegativerealpart,(A;B)isTcontrollableandD+D>0.Thefollowingstatementsareequivalent:1.Thesystemx_=Ax+Bu;y=Cx+Du;x(0)=0ispassive,i.e.,satis¯esZTTu(t)y(t)dt¸00foralluandT¸0.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences35¡12.ThetransfermatrixH(s)=C(sI¡A)B+Dispositive-real,i.e.,¤H(s)+H(s)¸0forallswithRes¸0.3.TheLMI"#TTAP+PAPB¡C·0(2.48)TTBP¡C¡(D+D)TinthevariableP=Pisfeasible.T4.ThereexistsP=PsatisfyingtheARETTT¡1TTAP+PA+(PB¡C)(D+D)(PB¡C)=0:(2.49)5.ThesizesoftheJordanblockscorrespondingtothepureimaginaryeigenvaluesoftheHamiltonianmatrix"#T¡1T¡1TA¡B(D+D)CB(D+D)BM=TT¡1TTT¡1T¡C(D+D)C¡A+C(D+D)Barealleven.Theequivalenceof1,2,3,and4canbefoundinTheorem4of[Wil71b]andalso[Wil74a].Theequivalenceof5isfoundin,forexample,Theorem1of[LR80].OriginsofthisresultcanbetracedbacktoReid[Rei46]andLevin[Lev59];itisexplicitlystatedinPotter[Pot66].Anexcellentdiscussionofthisresultanditsconnectionswithspectralfactorizationtheorycanbefoundin[AV73];seealso[Fra87,BBK89,Rob89].Connectionsbetweentheex-tremalpointsofthesetofsolutionsoftheLMI(2.48)andthesolutionsoftheARE(2.49)areexploredin[Bad82].ThemethodforconstructingParefromthestableeigenspaceofM,describedinx2.7.2,isin[BBK89];thismethodisanobviousvariationofLaub'salgo-rithm[Lau79,AL84],orVanDooren's[Doo81].Alessstablenumericalmethodcanbefoundin[Rei46,Lev59,Pot66,AV73].Forotherdiscussionsofstrictpassivity,seethearticlesbyWen[Wen88],Lozano{LealandJoshi[LLJ90].Itisalsopossibletocheckpassivity(ornonexpansivity)ofasystemsymbolicallyusingSturmmethods,whichyieldsRouth{Hurwitzlikealgorithms;seeSiljak[Sil71,Sil73]andBoyd,Balakrishnan,andKabamba[BBK89].ThePRlemmaisusedinmanyareas,e.g.,interconnectedsystems[MH78]andstochasticprocessesandstochasticrealizationtheory[Fau67,Fau73,AM74].Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. Chapter3SomeMatrixProblems3.1MinimizingConditionNumberbyScalingp£qTheconditionnumberofamatrixM2R,withp¸q,istheratioofitslargestandsmallestsingularvalues,i.e.,µ¶1=2¸(MTM)¢max·(M)=¸min(MTM)forMfull-rank,and·(M)=1otherwise.Forsquareinvertiblematricesthisreducesto·(M)=kMkkM¡1k.Weconsidertheproblem:minimize·(LMR)p£p(3.1)subjecttoL2R,diagonalandnonsingularq£qR2R,diagonalandnonsingularp£qwhereLandRaretheoptimizationvariables,andthematrixM2Risgiven.WewillshowthatthisproblemcanbetransformedintoaGEVP.Weassumewithoutlossofgeneralitythatp¸qandMisfull-rank.p£pq£qLetus¯x°>1.Thereexistnonsingular,diagonalL2RandR2Rp£psuchthat·(LMR)·°ifandonlyiftherearenonsingular,diagonalL2Randq£qR2Rand¹>0suchthat¹I·(LMR)T(LMR)·¹°2I:pSincewecanabsorbthefactor1=¹intoL,thisisequivalenttotheexistenceofp£pq£qnonsingular,diagonalL2RandR2RsuchthatI·(LMR)T(LMR)·°2I;whichisthesameas(RRT)¡1·MT(LTL)M·°2(RRT)¡1:(3.2)p£pq£qThisisequivalenttotheexistenceofdiagonalP2R,Q2RwithP>0,Q>0andQ·MTPM·°2Q:(3.3)37 38Chapter3SomeMatrixProblemsp£pq£qToseethis,¯rstsupposethatL2RandR2Rarenonsingularanddiagonal,and(3.2)holds.Then(3.3)holdswithP=LTLandQ=(RRT)¡1.Conversely,p£pq£qsupposethat(3.3)holdsfordiagonalP2RandQ2RwithP>0,Q>0.Then,(3.2)holdsforL=P1=2andR=Q¡1=2.Hencewecansolve(3.1)bysolvingtheGEVP:minimize°2p£psubjecttoP2Randdiagonal;P>0;q£qQ2Randdiagonal;Q>0Q·MTPM·°2QRemark:Thisresultisreadilyextendedtohandlescalingmatricesthathaveagivenblock-diagonalstructure,andmoregenerally,theconstraintthatoneor¤moreblocksareequal.ComplexmatricesarereadilyhandledbysubstitutingMT(complex-conjugatetranspose)forM.3.2MinimizingConditionNumberofaPositive-De¯niteMatrixArelatedproblemisminimizingtheconditionnumberofapositive-de¯nitematrixMthatdependsa±nelyonthevariablex,subjecttotheLMIconstraintF(x)>0.ThisproblemcanbereformulatedastheGEVPminimize°(3.4)subjecttoF(x)>0;¹>0;¹I0;I<ºM0+x~iMi<°Ii=1i=13.3MinimizingNormbyScalingn£nTheoptimaldiagonallyscalednormofamatrixM2Cisde¯nedas¢©°°DMD¡1°°¯¯D2Cn£nªº(M)=infisdiagonalandnonsingular;wherekMkisthenorm(i.e.,largestsingularvalue)ofthematrixM.Wewillshowthatº(M)canbecomputedbysolvingaGEVP.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 3.4RescalingaMatrixPositive-Definite39Notethat(¯¯¡¢¤¡¢)DMD¡1DMD¡1<°2I¯º(M)=inf°¯¯forsomediagonal,nonsingularD©¯ª=inf°¯M¤D¤DM·°2D¤Dforsomediagonal,nonsingularD©¯ª=inf°¯M¤PM·°2PforsomediagonalP=PT>0:Thereforeº(M)istheoptimalvalueoftheGEVPminimize°subjecttoP>0anddiagonal;M¤PM<°2PRemark:Thisresultcanbeextendedinmanyways.Forexample,themoregeneralcaseofblock-diagonalsimilarity-scaling,withequalityconstraintsamongtheblocks,isreadilyhandled.Asanotherexample,wecansolvetheproblemofsimultaneousoptimaldiagonalscalingofseveralmatricesM1;:::;Mp:Tocom-pute(¯)max°°¯¯D2Cn£nisdiagonalinf°DM¡1°¯;iDi=1;:::;p¯andnonsingularwesimplycomputetheoptimalvalueoftheGEVPminimize°¤2subjecttoP>0anddiagonal;MiPMi<°P;i=1;:::;pAnothercloselyrelatedquantity,whichplaysanimportantroleinthestabilityanalysisoflinearsystemswithuncertainparameters,is(¯)¯¤¤2¯P>0;MPM+i(MG¡GM)<°P;inf°¯;(3.5)¯PandGdiagonalandrealwhichcanalsobecomputedbysolvingaGEVP.3.4RescalingaMatrixPositive-De¯niten£nWearegivenamatrixM2C,andaskwhetherthereisadiagonalmatrixD>0suchthattheHermitianpartofDMispositive-de¯nite.ThisistrueifandonlyiftheLMIM¤D+DM>0;D>0(3.6)isfeasible.Sodeterminingwhetheramatrixcanberescaledpositive-de¯niteisanLMIP.WhenthisLMIisfeasible,wecan¯ndafeasiblescalingDwiththesmallestconditionnumberbysolvingtheEVPminimize¹subjecttoDdiagonal;M¤D+DM>0;I0,thequadraticnormde¯nedbykzk=¢zTPz=kP1=2zksatis¯esPpp1=®kzkP·kzkpl·®kzkPforallz;pforsomeconstant®¸1.Thus,thequadraticnormzTPzapproximateskzkplwithinafactorof®.Wewillshowthattheproblemof¯ndingPthatminimizes®,i.e.,theproblemofdeterminingtheoptimalquadraticnormapproximationofk¢kpl,isanEVP.Letv1;:::;vLbetheverticesoftheunitballBplofk¢kpl,sothat¢Bpl=fzjkzkpl·1g=Cofv1;:::;vLgandletBPdenotetheunitballofk¢kP.Thenpp1=®kzkP·kzkpl·®kzkP(3.7)isequivalenttopp1=®BPµBplµ®BP:pThe¯rstinclusion,1=®BPµBpl,isequivalenttoaTP¡1a·®;i=1;:::;p;iipandthesecond,Bplµ®BP,isequivalenttovTPv·®;i=1;:::;L:iiThereforewecanminimize®suchthat(3.7)holdsforsomeP>0bysolvingtheEVPminimize®"#TPaisubjecttoviPvi·®;i=1;:::;L;T¸0;i=1;:::;pai®Remark:ThenumberofverticesLoftheunitballofk¢kplcangrowexponen-tiallyinpandn.Thereforetheresultisuninterestingfromthepointofviewofcomputationalcomplexityandoflimitedpracticaluseexceptforproblemswithlowdimensions.pTheoptimalfactor®inthisproblemneverexceedsn.Indeed,bycomputingthemaximumvolumeellipsoidthatliesinsidetheunitballofk¢kpl,(whichdoesnotrequireusto¯ndtheverticesvi,i=1;:::;L),wecan¯nd(inpolynomial-ptime)anormforwhich®islessthann.SeetheNotesandReferencesformoredetails.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 42Chapter3SomeMatrixProblems3.7EllipsoidalApproximationnTheproblemofapproximatingsomesubsetofRwithanellipsoidarisesinmany¯eldsandhasalonghistory;seetheNotesandReferences.Toposesuchaproblempreciselyweneedtoknowhowthesubsetisdescribed,whetherweseekaninnerorouterapproximation,andhowtheapproximationwillbejudged(volume,majororminorsemi-axis,etc.).InsomecasestheproblemcanbecastasaCPoranEVP,andhencesolvedexactly.Asanexampleconsidertheproblemof¯ndingtheellipsoidcenteredaroundtheoriginofsmallestvolumethatcontainsapolytopedescribedbyitsvertices.Wesawinx2.2.4thatthisproblemcanbecastasaCP.Inothercases,wecancomputeanapproximationoftheoptimalellipsoidbysolvingaCPoranEVP.Insomeofthesecases,theproblemisknowntobeNP-hard,soitisunlikelythattheproblemcanbereduced(polynomially)toanLMIproblem.ItalsosuggeststhattheapproximationsobtainedbysolvingaCPorEVPwillnotbegoodforallinstancesofproblemdata,althoughtheapproximationsmaybegoodontypical"problems,andhenceofsomeuseinpractice.Asanexampleconsidertheproblemof¯ndingtheellipsoidofsmallestvolumethatcontainsapolytopedescribedbyasetoflinearinequalities,i.e.,fxjaTx·ibi;i=1;:::;pg.(Thisisthesameproblemasdescribedabove,butwithadi®erentdescriptionofthepolytope.)ThisproblemisNP-hard.Indeed,considertheproblemofsimplyverifyingthatagiven,¯xedellipsoidcontainsapolytopedescribedbyasetoflinearinequalities.Thisproblemisequivalenttothegeneralconcavequadraticprogrammingproblem,whichisNP-complete.InthissectionweconsidersubsetsformedfromellipsoidsE1;:::;Epinvariousways:union,intersection,andaddition.WeapproximatethesesetsbyanellipsoidE0.WedescribeanellipsoidEintwodi®erentways.The¯rstdescriptionusesconvexquadraticfunctions:E=fxjT(x)·0;g;T(x)=xTAx+2xTb+c;(3.8)whereA=AT>0andbTA¡1b¡c>0(whichensuresthatEisnonemptyanddoesnotreducetoasinglepoint).Notethatthisdescriptionishomogeneous,i.e.,wecanscaleA,bandcbyanypositivefactorwithouta®ectingE.ThevolumeofEisgivenby¡¢vol(E)2=¯det(bTA¡1b¡c)A¡1where¯isaconstantthatdependsonlyonthedimensionofx,i.e.,n.ThediameterofE(i.e.,twicethemaximumsemi-axislength)isq2(bTA¡1b¡c)¸max(A¡1):Wewillalsodescribeanellipsoidastheimageoftheunitballunderana±nemappingwithsymmetricpositive-de¯nitematrix:©¯ªE=x¯(x¡x)TP¡2(x¡x)·1=fPz+xjkzk·1g;(3.9)cccwhereP=PT>0.(Thisrepresentationisunique,i.e.,PandxareuniquelycdeterminedbytheellipsoidE.)ThevolumeofEisthenproportionaltodetP,anditsdiameteris2¸max(P).EachofthesedescriptionsofEisreadilyconvertedintotheother.Givenarepre-sentation(3.8)(i.e.,A,b,andc)weformtherepresentation(3.9)withpP=bTA¡1b¡cA¡1=2;x=¡A¡1b:cCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 3.7EllipsoidalApproximation43Giventherepresentation(3.9),wecanformarepresentation(3.8)ofEwithA=P¡2;b=¡P¡2x;c=xTP¡2x¡1:ccc(ThisformsonerepresentationofE;wecanformeveryrepresentationofEbyscalingtheseA,b,cbypositivefactors.)3.7.1OuterapproximationofunionofellipsoidsWeseekasmallellipsoidE0thatcoverstheSunionofellipsoidsE1;:::;Ep(orequiva-plently,theconvexhulloftheunion,i.e.,Coi=1Ei).WewilldescribetheseellipsoidsviatheassociatedquadraticfunctionsT(x)=xTAx+2xTb+c.Wehaveiiii[pE0¶Ei(3.10)i=1ifandonlyifEiµE0fori=1;:::;p.Thisistrueifandonlyif,foreachi,everyxsuchthatTi(x)·0satis¯esT0(x)·0.BytheS-procedure,thisistrueifandonlyifthereexistnonnegativescalars¿1;:::;¿psuchthatforeveryx;T0(x)¡¿iTi(x)·0;i=1;:::;p;or,equivalently,suchthat"#"#A0b0Aibi¡¿i·0;i=1;:::;p:bTcbTc00iiSinceourrepresentationofE0ishomogeneous,wewillnownormalizeA0,b0andc0inaconvenientway:suchthatbTA¡1b¡c=1.Inotherwordsweset0000c=bTA¡1b¡1(3.11)0000andparametrizeE0byA0andb0alone.Thusourconditionbecomes:"#"#A0b0Aibi¡¿i·0;i=1;:::;p:bTbTA¡1b¡1bTc0000iiUsingSchurcomplements,weobtaintheequivalentLMI2323A0b00Aibi06TT76T74b0¡1b05¡¿i4bici05·0;i=1;:::;p;(3.12)0b0¡A0000withvariablesA0,b0,and¿1,...,¿p.Tosummarize,thecondition(3.10)isequivalenttotheexistenceofnonnegative¿1;:::;¿psuchthat(3.12)holds.(SinceA0>0,itfollowsthatthe¿imust,infact,bepositive.)q¡1Withthenormalization(3.11),thevolumeofE0isproportionaltodetA0.Thuswecan¯ndthesmallestvolumeellipsoidcontainingtheunionofellipsoidsE1,...,EpbysolvingtheCP(withvariablesA0,b0,and¿1,...,¿p)¡1minimizelogdetA0subjecttoA0>0;¿1¸0;:::;¿p¸0;(3:12)Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 44Chapter3SomeMatrixProblemsSpThisellipsoidiscalledtheLÄowner{Johnellipsoidfori=1Ei,andsatis¯esseveralniceproperties(e.g.,a±neinvariance,certainboundsonhowwellitapproximatesSCoEi);seetheNotesandReferences.Wecan¯ndtheellipsoid(orequivalently,sphere)ofsmallestdiameterthatcon-S¡1tainsEibyminimizing¸max(A0)subjecttotheconstraintsofthepreviousCP,whichisequivalenttosolvingtheEVPmaximize¸subjecttoA0>¸I;¿1¸0;:::;¿p¸0;(3:12)pTheoptimaldiameteris2=¸opt,where¸optisanoptimalvalueoftheEVPabove.3.7.2OuterapproximationoftheintersectionofellipsoidsSimplyverifyingthatpE0¶Ei(3.13)i=1holds,givenE0,E1,...,Ep,isNP-complete,sowedonotexpecttorecastitasoneofourLMIproblems.Afortiori,wedonotexpecttocasttheproblemof¯ndingthesmallestvolumeellipsoidcoveringtheintersectionofsomeellipsoidsasaCP.WewilldescribethreewaysthatwecancomputesuboptimalellipsoidsforthisproblembysolvingCPs.The¯rstmethodusestheS-proceduretoderiveanLMIthatisasu±cientconditionfor(3.13)tohold.OnceagainwenormalizeourrepresentationofE0sothat(3.11)holds.FromtheS-procedureweobtainthecondition:thereexistpositivescalars¿1;:::;¿psuchthat"#"#XpA0b0Aibi¡¿i·0;bTbTA¡1b¡1bTc0000i=1iiwhichcanbewrittenastheLMI(invariablesA0,b0,and¿1;:::;¿p):2323A0b00XpAibi06TT7¿6T74b0¡1b05¡i4bici05·0:(3.14)0b¡Ai=100000ThisLMIissu±cientbutnotnecessaryfor(3.13)tohold,i.e.,itcharacterizessomebutnotalloftheellipsoidsthatcovertheintersectionofE1,...,Ep.Wecan¯ndthebestsuchouterapproximation(i.e.,theoneofsmallestvolume)bysolvingtheCP(withvariablesA0,b0,and¿1;:::;¿p)¡1minimizelogdetA0(3.15)subjecttoA0>0;¿1¸0;:::;¿p¸0;(3:14)TpRemark:TheintersectionEiisemptyorasinglepointifandonlyiftherei=1existnonnegative¿1;:::;¿p,notallzero,suchthat"#XpAibi¿i¸0:Tbicii=1Inthiscase(3.15)isunboundedbelow.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 3.7EllipsoidalApproximation45Anothermethodthatcanbeusedto¯ndasuboptimalsolutionfortheproblemofdeterminingtheminimumvolumeellipsoidthatcontainstheintersectionofE1,...,Episto¯rstcomputethemaximumvolumeellipsoidthatiscontainedintheintersection,whichcanbecastasaCP(seethenextsection).Ifthisellipsoidisscaledbyafactorofnaboutitcenterthenitisguaranteedtocontaintheintersection.SeetheNotesandReferencesformorediscussion.Yetanothermethodforproducingasuboptimalsolutionisbasedontheideaoftheanalyticcenter(seex2.4.1).SupposethatE1,...,Eparegivenintheform(3.9),withx1;:::;xpandP1,...,Pp.TheLMIÃ"#¡1IP1(x¡x1)F(x)=diag;:::(x¡x)TP¡1111"#!IP¡1(x¡x)pp:::;>0(x¡x)TP¡11ppTpinthevariableTxhasasfeasiblesettheinteriorofi=1Ei.Weassumenowthatthepintersectioni=1Eihasnonemptyinterior(i.e.,isnonemptyandnotasinglepoint).Letx?denoteitsanalyticcenter,i.e.,x?=argminlogdetF(x)¡1F(x)>0andletHdenotetheHessianoflogdetF(x)¡1atx?.Thenitcanbeshownthat©¯ªpE=¢x¯(x¡x?)TH(x¡x?)·1µE:0ii=1InfactwecangiveaboundonhowpoorlyE0approximatestheintersection.Itcanbeshownthat©¯ªpx¯(x¡x?)TH(x¡x?)·(3p+1)2¶E:ii=1TpThus,whenE0isenlargedbyafactorof3p+1,itcoversi=1Ei.SeetheNotesandReferencesformorediscussionofthis.3.7.3InnerapproximationoftheintersectionofellipsoidsTheellipsoidE0iscontainedintheintersectionoftheellipsoidsE1;:::;Epifandonlyifforeveryxsatisfying(x¡x)TP¡2(x¡x)·1;000wehave(x¡x)TP¡2(x¡x)·1;i=1;:::;p:iiiUsingtheS-procedure,thisisequivalenttotheexistenceofnonnegative¸1;:::;¸psatisfyingforeveryx;(x¡x)TP¡2(x¡x)iii(3.16)¡¸(x¡x)TP¡2(x¡x)·1¡¸;i=1;:::;p:i000iThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 46Chapter3SomeMatrixProblemsWeproveintheNotesandReferencessectionthat(3.16)isequivalentto23¡P2x¡xPii006T74(xi¡x0)¸i¡105·0;i=1;:::;p:(3.17)P00¡¸iITherefore,weobtainthelargestvolumeellipsoidcontainedintheintersectionoftheellipsoidsE1;:::;EpbysolvingtheCP(withvariablesP0,x0,¸1;:::;¸p)¡1minimizelogdetP0subjecttoP0>0;¸1¸0;:::;¸p¸0;(3:17)AmongtheellipsoidscontainedintheintersectionofE1,...,Ep,wecan¯ndonethatmaximizesthesmallestsemi-axisbysolvingminimize¸subjectto¸I>P0>0;¸1¸0;:::;¸p¸0;(3:17)Equivalently,thisyieldsthesphereoflargestdiametercontainedintheintersection.3.7.4OuterapproximationofthesumofellipsoidsThesumoftheellipsoidsE1;:::;Episde¯nedas¢A=fx1+¢¢¢+xpjx12E1;:::;xp2Ep:gWeseekasmallellipsoidE0containingA.TheellipsoidE0containsAifandonlyifforeveryx1;:::;xpsuchthatTi(xi)·0,i=1;:::;p,wehaveÃ!XpT0xi·0:i=1ByapplicationoftheS-procedure,thisconditionistrueifthereexist¿i¸0,i=1;:::;psuchthatÃ!XpXp(3.18)foreveryxi,i=1;:::;p;T0xi¡¿iTi(xi)·0:i=1i=1OnceagainwenormalizeourrepresentationofE0sothat(3.11)holds.Letxdenotethevectormadebystackingx1;:::;xp,i.e.,23x16.7x=6..7:45xpPpPpThenbothT0(i=1xi)andi=1¿iTi(xi)arequadraticfunctionsofx.Indeed,letEidenotethematrixsuchthatxi=Eix,andde¯neXpE=E;A~=ETAE;~b=ETb;i=0;:::;p:0iiiiiiiii=1Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences47Withthisnotation,weseethatcondition(3.18)isequivalenttothereexist¿i¸0,i=1;:::;psuchthat"#"#A~~bXpA~~b(3.19)00ii¡¿i·0:~bTbTA¡1b¡1~bTc0000i=1iiThisconditionisreadilywrittenasanLMIinvariablesA0,b0,¿1;:::;¿pusingSchurcomplements:2323A~0~b00XpA~i~bi064~bT¡1bT75¡¿64~bTc075·0:(3.20)00iii0b¡Ai=100000Therefore,wecomputetheminimumvolumeellipsoid,proventocontainthe¡1sumofE1;:::;EpviatheS-procedure,byminimizinglogdetA0subjecttoA0>0and(3.20).ThisisaCP.NotesandReferencesMatrixscalingproblemsTheproblemofscalingamatrixtoreduceitsconditionnumberisconsideredinForsytheandStraus[FS55](whodescribeconditionsforthescalingTtobeoptimalintermsofa¡¤¡1newvariableR=TT,justaswedo).Otherreferencesontheproblemofimprovingtheconditionnumberofamatrixviadiagonalorblock-diagonalscalingare[Bau63,GV74,Sha85,Wat91].Weshouldpointoutthattheresultsofx3.1arenotpracticallyusefulinnumericalanalysis,sincethecostofcomputingtheoptimalscalings(viatheGEVP)exceedsthecostofsolvingtheproblemforwhichthescalingsarerequired(e.g.,solvingasetoflinearequations).Aspecialcaseoftheproblemofminimizingtheconditionnumberofapositive-de¯nitematrixisconsideredin[EHM92,EHM93].Finally,KhachiyanandKalantari([KK92])considertheproblemofdiagonallyscalingapositivesemide¯nitematrixviainterior-pointmethodsandgiveacompletecomplexityanalysisfortheproblem.Theproblemofsimilarity-scalingamatrixtominimizeitsnormappearsoftenincontrolapplications(seeforexample[Doy82,Saf82]).ArecentsurveyarticleonthistopicisbyPackardandDoyle[PD93].Arelatedquantityistheone-sidedmultivariablestabilitymargin,describedin[TSC92];theproblemofcomputingthisquantitycanbeeasilytransformedintoaGEVP.Thequantity(3.5),usedintheanalysisoflinearsystemswithparameteruncertainties,wasintroducedbyFan,TitsandDoylein[FTD91](seealso[BFBE92]).Somecloselyrelatedquantitieswereconsideredin[SC93,SL93a,CS92b];seealsoChapter8inthisbook,whereadi®erentapproachtothesameproblemanditsextensionsisconsidered.WealsomentionsomeotherrelatedquantitiesfoundinthearticlesbyKouvaritakisandLatchman[KL85a,KL85b,LN92],andRoteaandPrasanth[RP93,RP94],whichcanbecomputedviaGEVPs.DiagonalquadraticLyapunovfunctionsTheproblemofrescalingamatrixMwithadiagonalmatrixD>0suchthatitsHer-mitianpartispositive-de¯niteisconsideredinthearticlesbyBarkeretal.[BBP78]andKhalil[Kha82].TheproblemisthesameasdeterminingtheexistenceofdiagonalquadraticLyapunovfunctionforagivenlinearsystem;see[Ger85,Hu87,BY89,Kra91].In[KB93],KaszkurewiczandBhayagiveanexcellentsurveyonthetopicofdiagonalLyapunovfunc-tions,includingmanyreferencesandexamplesfrommathematicalecology,powersystems,Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 48Chapter3SomeMatrixProblemsanddigital¯lterstability.DiagonalLyapunovfunctionsariseintheanalysisofpositiveorthantstability;seex10.3.MatrixcompletionproblemsMatrixcompletionsproblemsareaddressedinthearticleofDymandGohberg[DG81],inwhichtheauthorsexamineconditionsfortheexistenceofcompletionsofbandmatricessuchthattheinverseisalsoabandmatrix.In[GJSW84],theirresultsareextendedtoamoregeneralclassofpartiallyspeci¯edmatrices.Inbothpapersthesparsitypattern"resultmentionedinx3.5appears.In[BJL89],Barrettetal.provideanalgorithmfor¯ndingthe?completionmaximizingthedeterminantofthecompletedmatrix(i.e.,theanalyticcenterAmentionedinx3.5).In[HW93],HeltonandWoerdemanconsidertheproblemofminimumnormextensionofHankelmatrices.In[GFS93],Grigoriadis,FrazhoandSkeltonconsidertheproblemofapproximatingagivensymmetricmatrixbyaToeplitzmatrixandproposeasolutiontothisproblemviaconvexoptimization.Seealso[OSA93].Dancis([Dan92,Dan93]andJohnsonandLunquist[JL92])studycompletionproblemsinwhichtherankorinertiaofthecompletedmatrixisspeci¯ed.Asfarasweknow,suchproblemscannotbecastasLMIPs.Aclassicpaperonthecontractivecompletionproblem(i.e.,completingamatrixsoitsnormislessthanone)isDavis,Kahan,andWeinbergerin[DKW82],inwhichananalyticsolutionisgivenforcontractivecompletionproblemswithaspecialblockform.Theresultinthispaperiscloselyrelatedto(andalsomoregeneralthan)thematrixeliminationproceduregiveninx2.6.2.See[Woe90,NW92,BW92]formorereferencesoncontractivecompletionproblems.MaximumvolumeellipsoidcontainedinasymmetricpolytopeInthissection,weestablishsomepropertiesofthemaximumvolumeellipsoidcontainedinasymmetricpolytopedescribedas:©¯¯¯ªP=z¯¯aTz¯·1;i=1;:::;p:i¯ConsideranellipsoiddescribedbyE=fx¯xTQ¡1x·1gwhereQ=QT>0.ItsvolumeispTproportionaltodetQ.TheconditionEµPcanbeexpressedaiQai·1fori=1;:::;p.HenceweobtainthemaximumvolumeellipsoidcontainedinPbysolvingtheCP¡1minimizelogdetQTsubjecttoQ>0;aiQai·1;i=1;:::;pSupposenowthatQisoptimal.Fromthestandardoptimalityconditions,therearenonneg-ative¸1;:::;¸psuchthatXp¡1¸iTQ=aiai;TQaaiii=1T1=2and¸iP=0ifaiQai=1.MultiplyingbyQontheleftandrightandtakingthetracepyields¸i=n.i=1nForanyx2R,wehavepX(xTa2T¡1i)xQx=¸i:TQaaiii=1Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences49TForeachi,either¸i=0oraiQai=1,soweconcludeXpT¡1T2xQx=¸i(xai):i=1SupposenowthatxbelongstoP.ThenXpT¡1xQx·¸i=n:i=1pEquivalently,PµnE,sowehave:pEµPµnE;pi.e.,themaximumvolumeellipsoidcontainedinPapproximatesPwithinthescalefactorn.EllipsoidalapproximationnTheproblemofcomputingtheminimumvolumeellipsoidinRthatcontainsagivenconvexsetwasconsideredbyJohnandLÄowner,whoshowedthattheoptimalellipsoid,whenshrunkaboutitscenterbyafactorofn,iscontainedinthegivenset(see[Joh85,GLS88]).SuchellipsoidsarecalledpLÄowner{Johnellipsoids.Inthecaseofsymmetricsets,thefactorncanbeimprovedton(usinganargumentsimilartotheoneintheNotesandReferencessectionabove).Acloselyrelatedproblemis¯ndingthemaximumvolumeellipsoidcontainedinaconvexset.Similarresultsholdfortheseellipsoids.Specialtechniqueshavebeendevelopedtocomputesmallestspheresandellipsoidscontainingagivensetofpointsinspacesoflowdimension;seee.g.,PreparataandShamos[PS85,p.255]orPost[Pos84].Minimumvolumeellipsoidscontainingagivensetariseoftenincontroltheory.SchweppeandSchlaepfer[Sch68,SS72]useellipsoidscontainingthesumandtheintersectionoftwogivenellipsoidsintheproblemofstateestimationwithnorm-boundeddisturbances.SeealsothearticlesbyBertsekasandRhodes[BR71],Chernousko[Che80a,Che80b,Che80c]andKahan[Kah68].In[RB92],ellipsoidalapproximationsofrobotlinkagesandworkspaceareusedtorapidlydetectoravoidcollisions.Minimumvolumeellipsoidshavebeenwidelyusedinsystemidenti¯cation;seethear-ticlesbyFogel[Fog79],FogelandHuang[FH82],Deller[Del89],Belforte,BonaandCerone[BBC90],Lau,KosutandBoyd[LKB90,KLB90,KLB92],Cheung,YurkovichandPassino[CYP93]andthebookbyNorton[Nor86].Intheellipsoidalgorithmweencounteredtheminimumvolumeellipsoidthatcontainsagivenhalf-ellipsoid;thereferencesontheellipsoidalgorithmdescribevariousextensions.E±cientinterior-pointmethodsforcomputingminimumvolumeouterandmaximumvolumeinnerellipsoidsforvarioussetsaredescribedinNesterovandNemirovskii[NN94].Seealso[KT93].EllipsoidalapproximationviaanalyticcentersEllipsoidalapproximationsviabarrierfunctionsariseinthetheoryofinterior-pointalgo-rithms,andcanbetracedbacktoDikin[Dik67]andKarmarkar[Kar84]forpolytopesdescribedbyasetoflinearinequalities.ForthecaseofmoregeneralLMIs,seeNesterovandNemirovskii[NN94],BoydandElGhaoui[BE93],andJarre[Jar93c].Theseapproximationshavethepropertythattheyareeasytocompute,andcomewithprovableboundsonhowsuboptimaltheyare.Theseboundshavethefollowingform:ifThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 50Chapter3SomeMatrixProblemstheinnerellipsoidisstretchedaboutitscenterbysomefactor®(whichdependsonlyonthedimensionofthespace,thetypeandsizeoftheconstraints)thentheresultingellipsoidcoversthefeasibleset.Thepaperscitedabovegivedi®erentvaluesof®;onemaybebetterthananother,dependingonthenumberandtypeofconstraints.Incontrast,themaximumvolumeinnerellipsoid,i.e.,theLÄowner{Johnellipsoid,ismoredif-¯culttocompute(stillpolynomial,however)butthefactor®dependsonlyonthedimensionofthespaceandnotonthetypeornumberofconstraints,andisalwayssmallerthantheonesobtainedviabarrierfunctions.Soanellipsoidalapproximationfromabarrierfunctioncanserveasacheapsubstitute"fortheLÄowner{Johnellipsoid.Proofoflemmainx3.7.3WeprovethatthestatementT¡2T¡2foreveryx,(x¡x1)P1(x¡x1)¡¸(x¡x0)P0(x¡x0)·1¡¸(3.21)whereP0,P1arepositivematrices,¸¸0,isequivalenttothematrixinequality232¡P1x1¡x0P06T74(x1¡x0)¸¡105·0:(3.22)P00¡¸IThemaximumoverxofT¡2T¡2(x¡x1)P1(x¡x1)¡¸(x¡x0)P0(x¡x0)(3.23)22¤is¯niteifandonlyifP0¡¸P1·0(whichimpliesthat¸>0)andthereexistsxsatisfying¡2¤¡2¤P1(x¡x1)=¸P0(x¡x0);(3.24)or,equivalently,satisfying22¡2¤(P0=¸¡P1)P1(x¡x1)=x1¡x0:(3.25)¤Inthiscase,xisamaximizerof(3.23).Using(3.24),condition(3.21)implies¤T¡222¡2¤(x¡x1)P1(P1¡P0=¸)P1(x¡x1)·1¡¸:UsingSchurcomplements,thislastconditionisequivalentto"#¤T¡222¸¡1(x¡x1)P1(P0=¸¡P1)·0:22¡2¤22(P0=¸¡P1)P1(x¡x1)P0=¸¡P1Using(3.25),thisimplies"#T¸¡1(x1¡x0)·0;22x1¡x0P0=¸¡P1whichisequivalentto(3.22).Conversely,itiseasytoshowthat(3.22)implies(3.21).Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. Chapter4LinearDi®erentialInclusions4.1Di®erentialInclusionsAdi®erentialinclusion(DI)isdescribedby:x_2F(x(t);t);x(0)=x0;(4.1)nnwhereFisaset-valuedfunctiononR£R+.Anyx:R+!Rthatsatis¯es(4.1)iscalledasolutionortrajectoryoftheDI(4.1).Ofcourse,therecanbemanysolutionsoftheDI(4.1).Ourgoalistoestablishthatvariouspropertiesaresatis¯edbyallsolutionsofagivenDI.Forexample,wemightshowthateverytrajectoryofagivenDIconvergestozeroast!1.ByastandardresultcalledtheRelaxationTheorem,wemayaswellassumeF(x;t)isaconvexsetforeveryxandt.TheDIgivenbyx_2CoF(x(t);t);x(0)=x0(4.2)iscalledtherelaxedversionoftheDI(4.1).SinceCoF(x(t);t)¶F(x(t);t),everytrajectoryoftheDI(4.1)isalsoatrajectoryofrelaxedDI(4.2).Veryroughlyspeak-ing,theRelaxationTheoremstatesthatformanypurposestheconverseistrue.(SeetheReferencesforprecisestatements.)Asaspeci¯candsimpleexample,itcanbeshownthatforeveryDIweencounterinthisbook,thereachableorattainablesetsoftheDIanditsrelaxedversioncoincide,i.e.,foreveryT¸0fx(T)jxsatis¯es(4:1)g=fx(T)jxsatis¯es(4:2)g:InfactwewillnotneedtheRelaxationTheorem,orrather,wewillalwaysgetitforfree"|everyresultweestablishinthenexttwochaptersextendsimmediatelytotherelaxedversionoftheproblem.ThereasonisthatwhenaquadraticLyapunovfunctionisusedtoestablishsomepropertyfortheDI(4.1),thenthesameLyapunovfunctionestablishesthepropertyfortherelaxedDI(4.2).4.1.1Lineardi®erentialinclusionsAlineardi®erentialinclusion(LDI)isgivenbyx_2•x;x(0)=x0;(4.3)n£nwhere•isasubsetofR.WecaninterprettheLDI(4.3)asdescribingafamilyoflineartime-varyingsystems.EverytrajectoryoftheLDIsatis¯esx_=A(t)x;x(0)=x0;(4.4)51 52Chapter4LinearDifferentialInclusionsforsomeA:R+!•.Conversely,foranyA:R+!•,thesolutionof(4.4)isatrajectoryoftheLDI(4.3).Inthelanguageofcontroltheory,theLDI(4.3)mightbedescribedasanuncertaintime-varyinglinearsystem,"withtheset•describingtheuncertainty"inthematrixA(t).4.1.2AgeneralizationtosystemsWewillencounterageneralizationoftheLDIdescribedabovetolinearsystemswithinputsandoutputs.Wewillconsiderasystemdescribedbyx_=A(t)x+Bu(t)u+Bw(t)w;x(0)=x0;(4.5)z=Cz(t)x+Dzu(t)u+Dzw(t)wwherex:R!Rn,u:R!Rnu,w:R!Rnw,z:R!Rnz.xisreferred++++toasthestate,uisthecontrolinput,wistheexogenousinputandzistheoutput.Thematricesin(4.5)satisfy"#A(t)Bu(t)Bw(t)2•:(4.6)Cz(t)Dzu(t)Dzw(t)forallt¸0,where•µR(n+nz)£(n+nu+nw).Wewillbemorespeci¯cabouttheformof•shortly.Insomeapplicationswecanhaveoneormoreoftheintegersnu,nw,andnzequaltozero,whichmeansthatthecorrespondingvariableisnotused.Forexample,theLDIx_2•xresultswhennu=nw=nz=0.Inordernottointroduceanothertermtodescribethesetofallsolutionsof(4.5)and(4.6),wewillcall(4.5)and(4.6)asystemdescribedbyLDIsorsimply,anLDI.4.2SomeSpeci¯cLDIsWenowdescribesomespeci¯cfamiliesofLDIsthatwewillencounterinthenexttwochapters.4.2.1Lineartime-invariantsystemsWhen•isasingleton,theLDIreducestothelineartime-invariant(LTI)systemx_=Ax+Buu+Bww;x(0)=x0;z=Czx+Dzuu+Dzww;where("#)ABuBw•=:(4.7)CzDzuDzwAlthoughmostoftheresultsofChapters5{7arewell-knownforLTIsystems,somearenew;wewilldiscusstheseindetailwhenweencounterthem.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 4.2SomeSpecificLDIs534.2.2PolytopicLDIsWhen•isapolytope,wewillcalltheLDIapolytopicLDIorPLDI.Mostofourresultsrequirethat•bedescribedbyalistofitsvertices,i.e.,intheform("#"#)A1Bu;1Bw;1ALBu;LBw;LCo;:::;:(4.8)Cz;1Dzu;1Dzw;1Cz;LDzu;LDzw;Lwherethematrices(4.8)aregiven.Ifinstead•isdescribedbyasetofllinearinequalities,thenthenumberofvertices,i.e.,L,willgenerallyincreaseveryrapidly(exponentially)withl.ThereforeresultsforPLDIsthatrequirethedescription(4.8)areoflimitedinterestforproblemsinwhich•isdescribedintermsoflinearinequalities.4.2.3Norm-boundLDIsAnotherspecialclassofLDIsistheclassofnorm-boundLDIs(NLDIs),describedbyx_=Ax+Bpp+Buu+Bww;x(0)=x0;q=Cqx+Dqpp+Dquu+Dqww;(4.9)z=Czx+Dzpp+Dzuu+Dzww;p=¢(t)q;k¢(t)k·1:where¢:R!Rnp£nq,withk¢(t)k·1forallt.Wewilloftenrewritethe+conditionp=¢(t)q,k¢k·1intheequivalentformpTp·qTq.FortheNLDI(4.9)theset•hastheformn¯o•=A~+B~¢(I¡D¢)¡1C~¯¯k¢k·1;qpwhere"#"#hiA~=ABuBw;B~=Bp;C~=CDD:(4.10)qquqwCzDzuDzwDzpThus,•istheimageofthe(matrixnorm)unitballundera(matrix)linear-fractionalmapping.Notethat•isconvex.Theset•iswellde¯nedifandonlyifDTD0when¢takesonits2nqextremevalues,i.e.,j¢j=1.Althoughthisconditionisexplicit,theiinumberofinequalitiesgrowsexponentiallywithnq.Infact,determiningwhetheraDNLDIiswell-posedisNP-hard;seetheNotesandReferences.ItturnsoutthatforaDNLDI,Co•isapolytope.Infact,n¯oCo•=CoA~+B~¢(I¡D¢)¡1C~¯¯¢diagonal;j¢;qpiij=1i.e.,theverticesofCo•areamong2nqimagesoftheextremepointsof¢underthematrixlinear-fractionalmapping.ThereforeaDNLDIcanbedescribedasaPLDI,atleastinprinciple.Butthereisaveryimportantdi®erencebetweenthetwodescriptionsofthesameLDI:thenumberofverticesrequiredtogivethePLDIrepresentationofanLDIincreasesexponentiallywithnq(seetheNotesandReferencesofChapter5formorediscussion).Remark:MoreelaboratevariationsontheNLDIaresometimesuseful.Doyleintroducedtheideaofstructuredfeedback,"inwhichthematrix¢isrestrictedtohaveagivenblock-diagonalform,andinaddition,therecanbeequalitycon-straintsamongsomeoftheblocks.Whenwehaveasingleblock,wehaveanNLDI;whenitisdiagonal(withnoequalityconstraints),wehaveaDNLDI.MostoftheresultswepresentforNLDIsandDNLDIscanbeextendedtothesemoregeneraltypesofLDIs.4.3NonlinearSystemAnalysisviaLDIsMuchofourmotivationforstudyingLDIscomesfromthefactthatwecanusethemtoestablishvariouspropertiesofnonlinear,time-varyingsystemsusingatechniqueknownasgloballinearization".TheideaisimplicitintheworkonabsolutestabilityoriginatingintheSovietUnioninthe1940s.Considerthesystemx_=f(x;u;w;t);z=g(x;u;w;t):(4.13)Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 4.3NonlinearSystemAnalysisviaLDIs55Supposethatforeachx,u,w,andtthereisamatrixG(x;u;w;t)2•suchthat23"#xf(x;u;w;t)67=G(x;u;w;t)4u5(4.14)g(x;u;w;t)wwhere•µR(n+nz)£(n+nu+nw).Thenofcourseeverytrajectoryofthenonlinearsystem(4.13)isalsoatrajectoryoftheLDIde¯nedby•.IfwecanprovethateverytrajectoryoftheLDIde¯nedby•hassomeproperty(e.g.,convergestozero),thenafortioriwehaveprovedthateverytrajectoryofthenonlinearsystem(4.13)hasthisproperty.4.3.1AderivativeconditionConditionsthatguaranteetheexistenceofsuchaGaref(0;0;0;0)=0,g(0;0;0;0)=0,and23@f@f@f66@x@u@w772•forallx;u;w;t:(4.15)4@g@g@g5@x@u@wInfactwecanmakeastrongerstatementthatlinksthedi®erencebetweenapairoftrajectoriesofthenonlinearsystemandthetrajectoriesoftheLDIgivenby•.Supposewehave(4.15)butnotnecessarilyf(0;0;0;0)=0,g(0;0;0;0)=0.Thenforanypairoftrajectories(x;w;z)and(x;~w~;z~)wehave23"#x¡x~x_¡x~_672Co•4u¡u~5;z¡z~w¡w~i.e.,(x¡x~;w¡w~;z¡z~)isatrajectoryoftheLDIgivenbyCo•.Theseresultsfollowfromasimpleextensionofthemean-valuetheorem:ifÁ:nnR!Rsatis¯es@Á2•@xnthroughoutR,thenforanyxandx~wehaveÁ(x)¡Á(x~)2Co•(x¡x~):(4.16)nToseethis,letc2R.Bythemean-valuetheoremwehaveTT@Ác(Á(x)¡Á(x~))=c(³)(x¡x~)@xforsome³thatliesonthelinesegmentbetweenxandx~.Sincebyassumption@Á(³)2•;@xweconcludethatcT(Á(x)¡Á(x~))·supcTA(x¡x~):A2Co•Sincecwasarbitrary,weseethatÁ(x)¡Á(x~)isineveryhalf-spacethatcontainstheconvexsetCo•(x¡x~),whichproves(4.16).Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 56Chapter4LinearDifferentialInclusions4.3.2SectorconditionsSometimesthenonlinearsystem(4.13)canbeexpressedintheformx_=Ax+Bpp+Buu+Bww;q=Cqx+Dqpp+Dquu+Dqww;(4.17)z=Czx+Dzpp+Dzuu+Dzww;pi=Ái(qi;t);i=1;:::;nq;whereÁi:R£R+!Rsatisfythesectorconditions®q2·qÁ(q;t)·¯q2iiiforallqandt¸0,where®iand¯iaregiven.Inwords,thesystemconsistsofalinearparttogetherwithnqtime-varyingsector-boundedscalarnonlinearities.Thevariablespiandqicanbeeliminatedfrom(4.17)providedthematrixI¡Dqp¢isnonsingularforalldiagonal¢with®i·¢ii·¯i.Inthiscase,wesaythesystemiswell-posed.Assumenowthatthesystem(4.17)iswell-posed,andlet(4.13)betheequationsobtainedbyeliminatingthevariablespandq.De¯nen¯o•=A~+B~¢(I¡D¢)¡1C~¯¯¢diagonal,¢;qpii2[®i;¯i]whereA~,B~andC~aregivenby(4.10).Then,thecondition(4.14)holds.NotesandReferencesDi®erentialinclusionsThebooksbyAubinandCellina[AC84],Kisielewicz[Kis91],andtheclassictextbyFilip-pov[Fil88]coverthetheoryofdi®erentialinclusions.SeealsothearticlebyRoxin[Rox65]andreferencestherein.Backgroundmaterial,e.g.,set-valuedanalysis,canbefoundinthebookbyAubinandFrankowska[AF90].Thetermlineardi®erentialinclusionisabitmisleading,sinceLDIsdonotenjoyanyparticu-larlinearityorsuperpositionproperties.Weusethetermonlytopointouttheinterpretationasanuncertaintime-varyinglinearsystem.ManyauthorsrefertoLDIsusingsomesubsetofthephraseuncertainlineartime-varyingsystem".Amoreaccurateterm,suggestedtousandusedbyA.FilippovandE.Pyatnitskii,isselector-lineardi®erentialinclusion.PLDIscomeupinseveralarticlesoncontroltheory,e.g.,[MP89,KP87a,KP87b,BY89].IntegralquadraticconstraintsTTThepointwisequadraticconstraintp(t)p(t)·q(t)q(t)thatoccurinNLDIscanbegener-RRalizedtotheintegralquadraticconstrainttpTpd¿·tqTqd¿;seex8.2andx8.3.00GloballinearizationTheideaofreplacinganonlinearsystembyatime-varyinglinearsystemcanbefoundinLiuetal.[Liu68,LSL69].Theycallthisapproachgloballinearization.Ofcourse,approximatingthesetoftrajectoriesofanonlinearsystemviaLDIscanbeveryconservative,i.e.,therearemanytrajectoriesoftheLDIthatarenottrajectoriesofthenonlinearsystem.WewillseeinChapter8howthisconservatismcanbereducedinsomespecialcases.TheideaofgloballinearizationisimplicitintheearlySovietliteratureontheabsolutestabilityproblem,e.g.,Lur'eandPostnikov[LP44,Lur57]andPopov[Pop73].Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences57ModelingsystemsasPLDIsOneofthekeyissuesinrobustcontroltheoryishowtomodelormeasureplantuncertainty"orvariation".Weproposeasimplemethodthatshouldworkwellinsomecases.Attheleast,themethodisextremelysimpleandnatural.Supposewehavearealsystemthatisfairlywellmodeledasalinearsystem.Wecollectmanysetsofinput/outputmeasurements,obtainedatdi®erenttimes,underdi®erentop-eratingconditions,orperhapsfromdi®erentinstancesofthesystem(e.g.,di®erentunitsfromamanufacturingrun).Itisimportantthatwehavedatasetsfromenoughplantsorplantconditionstocharacterizeoratleastgivesomeideaoftheplantvariationthatcanbeexpected.Foreachdatasetwedevelopalinearsystemmodeloftheplant.Tosimplifytheproblemwewillassumethatthestateinthismodelisaccessible,sothedi®erentmodelsrefertothesamestatevector.Thesemodelsshouldbefairlyclose,butofcoursenotexactlythesame.Wemight¯ndforexamplethatthetransfermatricesofthesemodelsats=0di®erbyabout10%,butathighfrequenciestheydi®erconsiderablymore.Moreimportantly,thiscollectionofmodelscontainsinformationaboutthestructure"oftheplantvariation.WeproposetomodelthesystemasaPLDIwiththeverticesgivenbythemeasuredorestimatedlinearsystemmodels.Inotherwords,wemodeltheplantasatime-varyinglinearsystem,withsystemmatricesthatcanjumparoundamonganyofthemodelsweestimated.ItseemsreasonabletoconjecturethatacontrollerthatworkswellwiththisPLDIislikelytoworkwellwhenconnectedtotherealsystem.SuchcontrollerscanbedesignedbythemethodsdescribedinChapter7.Well-posednessofDNLDIsSee[BY89]foraproofthataDNLDIiswell-posedifandonlyifdet(I¡Dqp¢)>0forevery¢withj¢iij=1;theideabehindthisproofcanbetracedbacktoZadehandDesoer[ZD63,x9.17];seealso[Sae86].AnequivalentconditionisintermsofP0matrices(amatrixisP0ifeveryprincipalminorisnonnegative;see[FP62,FP66,FP67]):ADNLDIiswell-posedif¡1andonlyifI+Dqpisinvertible,and(I+Dqp)(I¡Dqp)isaP0matrix[Gha90].Anotherequivalentconditionisthat(I+Dqp;I¡Dqp)isaW0pair(see[Wil70]).TheproblemofdeterminingwhetherageneralmatrixisP0isthoughttobeverydi±cult,i.e.,ofhighcomplexity[CPS92,p149].Standardbranch-and-boundtechniquescanbeappliedtotheproblemofdeterminingwell-posednessofDNLDIs;seeforexample[Sd86,GS88,SP89,BBB91,BB92].Ofcoursethesenqmethodshavenotheoreticaladvantageoversimplycheckingthatthe2determinantsarepositive.Butinpracticetheymaybemoree±cient,especiallywhenusedwithcomputation-allycheapsu±cientconditions(seebelow).Thereareseveralsu±cientconditionsforwell-posednessofaDNLDIthatcanbecheckedinpolynomial-time.Fan,Tits,andDoyle[FTD91]showthatiftheLMITDqpPDqp0;(4.18)isfeasible,thentheDNLDIiswell-posed.Condition(4.18)isequivalenttotheexistenceofa¡1diagonal(scaling)matrixRsuchthatkRDqpRk<1.In[FTD91],theauthorsalsogiveasu±cientconditionforwell-posednessofmoregeneralstructured"LDIs,e.g.,whenequalityconstraintsareimposedonthematrix¢in(4.11).TheseconditionscanalsobeobtainedusingtheS-procedure;see[Fer93].RepresentingDNLDIsasPLDIsForawell-posedDNLDI,Co•isapolytope,whichmeansthattheDNLDI(4.12)canalsoberepresentedasaPLDI.Morespeci¯cally,wehave©¯ªCo•=CoA~+B~¢(I¡D¡1C~¯¢diagonal,j¢;qp¢)iij=1Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 58Chapter4LinearDifferentialInclusionswhereA~,B~andC~aregivenby(4.10).Notethatthesetontheright-handsideisapolytopenqwithatmost2vertices.Wenowprovethisresult.De¯ne©¯ªK=A~+B~¢(I¡D¡1C~¯¢diagonal;j¢;qp¢)iij=1nqwhichisasetcontainingatmost2points.Byde¯nition©¯ªCo•=CoA~+B~¢(I¡D¡1C~¯¢diagonal,j¢:qp¢)iij·1ClearlywehaveCo•¶CoK.WewillnowshowthateveryextremepointofCo•iscontainedinCoK,whichwillimplythatCoK¶Co•,completingtheproof.LetWextbeanextremepointofCo•.ThenthereexistsalinearfunctionÃsuchthatWextistheuniquemaximizerofÃoverCo•.Infact,Wext2•,sinceotherwise,thehalf-spacefWjÃ(W)<Ã(Wext)¡²g,where²>0issu±cientlysmall,contains•butnotWext,whichcontradictsWext2Co•.Now,forW2•,Ã(W)isaboundedlinear-fractionalfunctionofthe¯rstdiagonalentry¢11of¢,for¯xed¢22;:::;¢n,i.e.,itequals(a+b¢11)=(c+d¢11)forsomea,b,candqnqd.Moreover,thedenominatorc+d¢11isnonzerofor¢112[¡1;1],sincetheDNLDIiswell-posed.Therefore,forevery¯xed¢22;:::;¢n,thefunctionÃachievesamaximumatqnqanextremevalueof¢11,i.e.,¢11=§1.Extendingthisargumentto¢22;:::;¢nleadsqnqustothefollowingconclusion:Wext=A~+B~S(I¡DqpS)¡1C~,whereSsatis¯esSii=§1.Inotherwords,Wext2CoK,whichconcludesourproof.WehavealreadynotedthatthedescriptionofaDNLDIasaPLDIwillbemuchlargerthanitsdescriptionasaDNLDI;seetheNotesandReferencesofChapter5,andalso[BY89]and[PS82,KP87a,Kam83].ApproximatingPLDIsbyNLDIsItisoftenusefultoconservativelyapproximateaPLDIasanNLDI.Onereason,amongmany,isthepotentiallymuchsmallersizeofthedescriptionasanNLDIcomparedtothedescriptionofthePLDI.WeconsiderthePLDI¢x_=A(t)x;A(t)2•PLDI=CofA1;:::;ALg;(4.19)nwithx(t)2R,andtheNLDIx_=(A+Bp¢(t)Cq)x;k¢(t)k·1;(4.20)¢withassociatedset•NLDI=fA+Bp¢Cqjk¢k·1g.Ourgoalisto¯ndA,Bp,andCqsuchthat•PLDIµ•NLDIwiththeset•NLDIassmallaspossibleinsomeappropriatesense.Thiswillgiveane±cientouterapproximationofthePLDI(4.19)bytheNLDI(4.20).Since•PLDIµ•NLDIimpliesthateverytrajectoryofthePLDIisatrajectoryoftheNLDI,everyresultweestablishforalltrajectoriesoftheNLDIholdsforalltrajectoriesofthePLDI.ForLverylarge,itismucheasiertoworkwiththeNLDIthanthePLDI.n£nForsimplicity,wewillonlyconsiderthecasewithBp2Randinvertible.EvidentlythereissubstantialredundancyinourrepresentationoftheNLDI(4.20):WecanreplaceBpby®BpandCqbyCq=®where®6=0.(WecanalsoreplaceBpandCqwithBpUandVCqwhereUandVareanyorthogonalmatrices,withouta®ectingtheset•NLDI,butwewillnotusethisfact.)Wehave•PLDIµ•NLDIifforevery»andk=1;:::;L,thereexists¼suchthatTTTBp¼=(Ak¡A)»;¼¼·»CqCq»:Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences59ThisyieldstheequivalentconditionT¡T¡1T(Ak¡A)BpBp(Ak¡A)·CqCq;k=1;:::;L;which,inturn,isequivalentto"#TTTCqCq(Ak¡A)BpBp>0;¸0;k=1;:::;L:TAk¡ABpBpTTIntroducingnewvariablesV=CqCqandW=BpBp,wegettheequivalentLMIinA,VandW"#TV(Ak¡A)W>0;¸0;k=1;:::;L:(4.21)Ak¡AWThuswehave•PLDIµ•NLDIifcondition(4.21)holds.Thereareseveralwaystominimizethesizeof•NLDI¶•PLDI.ThemostobviousistominimizeTrV+TrWsubjectto(4.21),whichisanEVPinA,VandW.Thisobjectiveisclearlyrelatedtoseveralmeasuresofthesizeof•NLDI,butwedonotknowofanysimpleinterpretation.Wecanformulatetheproblemofminimizingthediameterof•NLDIasanEVP.Wede¯nethen£ndiameterofaset•µRasmaxfkF¡GkjF;G2•g.Thediameterof•NLDIisequalpto2¸max(V)¸max(W).WecanexploitthescalingredundancyinCqtoassumewithoutlossofgeneralitythat¸max(V)·1,andthenminimize¸max(W)subjectto(4.21)andV·I,optoptwhichisanEVP.ItcanbeshownthattheresultingoptimalVsatis¯es¸max(V)=1,sothatwehaveinfactminimizedthediameterof•NLDIsubjectto•PLDIµ•NLDI.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. Chapter5AnalysisofLDIs:StatePropertiesInthischapterweconsiderpropertiesofthestatexoftheLDIx_=A(t)x;A(t)2•;(5.1)n£nwhere•µRhasoneoffourforms:²LTIsystems:LTIsystemsaredescribedbyx_=Ax.²PolytopicLDIs:PLDIsaredescribedbyx_=A(t)x,A(t)2CofA1;:::;ALg.²Norm-boundLDIs:NLDIsaredescribedbyx_=Ax+Bpp;q=Cqx+Dqpp;p=¢(t)q;k¢(t)k·1;whichwewillrewriteasx_=Ax+Bp;pTp·(Cx+Dp)T(Cx+Dp):(5.2)pqqpqqpWeassumewell-posedness,i.e.,kDqpk<1.²DiagonalNorm-boundLDIs:DNLDIsaredescribedbyx_=Ax+Bpp;q=Cqx+Dqpp;(5.3)pi=±i(t)qi;j±i(t)j·1;i=1;:::;nq:whichcanberewrittenasx_=Ax+Bpp;q=Cqx+Dqpp;jpij·jqij;i=1;:::;nq:Again,weassumewell-posedness.5.1QuadraticStabilityWe¯rststudystabilityoftheLDI(5.1),thatis,weaskwhetheralltrajectoriesofsystem(5.1)convergetozeroast!1.Asu±cientconditionforthisistheexistenceofaquadraticfunctionV(»)=»TP»,P>0thatdecreasesalongeverynonzerotrajectoryof(5.1).IfthereexistssuchaP,wesaytheLDI(5.1)isquadraticallystableandwecallVaquadraticLyapunovfunction.61 62Chapter5AnalysisofLDIs:StatePropertiesSinced¡¢V(x)=xTA(t)TP+PA(t)x;dtanecessaryandsu±cientconditionforquadraticstabilityofsystem(5.1)isP>0;ATP+PA<0forallA2•:(5.4)Multiplyingthesecondinequalityin(5.4)ontheleftandrightbyP¡1,andde¯ninganewvariableQ=P¡1,wemayrewrite(5.4)asQ>0;QAT+AQ<0forallA2•:(5.5)Thisdualinequalityisanequivalentconditionforquadraticstability.WenowshowthatconditionsforquadraticstabilityforLTIsystems,PLDIs,andNLDIscanbeexpressedintermsofLMIs.²LTIsystems:Condition(5.4)becomesP>0;ATP+PA<0:(5.6)Therefore,checkingquadraticstabilityforanLTIsystemisanLMIPinthevariableP.Thisispreciselythe(necessaryandsu±cient)LyapunovstabilitycriterionforLTIsystems.(Inotherwords,alinearsystemisstableifandonlyifitisquadraticallysta-ble.)Alternatively,using(5.5),stabilityofLTIsystemsisequivalenttotheexistenceofQsatisfyingtheLMIQ>0;AQ+QAT<0:(5.7)Ofcourse,eachofthese(veryspecial)LMIPscanbesolvedanalyticallybysolvingaLyapunovequation(seex1.2andx2.7).²PolytopicLDIs:Condition(5.4)isequivalenttoP>0;ATP+PA<0;i=1;:::;L:(5.8)iiThus,determiningquadraticstabilityforPLDIsisanLMIPinthevariableP.Thedualcondition(5.5)isequivalenttotheLMIinthevariableQQ>0;QAT+AQ<0;i=1;:::;L:(5.9)ii²Norm-boundLDIs:Condition(5.4)isequivalenttoP>0and"#T"#"#»ATP+PAPB»p<0(5.10)¼BTP0¼pforallnonzero»satisfying"#T23"#¡CTC¡CTD»qqqqp»45·0:(5.11)¼¡DTCI¡DTD¼qpqqpqpInordertoapplytheS-procedureweshowthatthesetA=f(»;¼)j»6=0;(5:11)gCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 5.1QuadraticStability63equalsthesetB=f(»;¼)j(»;¼)6=0;(5:11)g:Itsu±cestoshowthatf(»;¼)j»=0;¼6=0;(5:11)g=;.Butthisisimmediate:If¼6=0,thencondition(5.11)cannotholdwithouthaving»6=0,sinceI¡DTD>0.qpqpTherefore,theconditionthatdV(x)=dt<0forallnonzerotrajectoriesisequivalentto(5.10)beingsatis¯edforanynonzero(»;¼)satisfying(5.11).(Thisargumentrecursthroughoutthischapterandwillnotberepeated.)UsingtheS-procedure,we¯ndthatquadraticstabilityof(5.2)isequivalenttotheexistenceofPand¸satisfyingP>0;¸¸0;23ATP+PA+¸CTCPB+¸CTD(5.12)qqpqqp45<0:(PB+¸CTD)T¡¸(I¡DTD)pqqpqpqpThus,determiningquadraticstabilityofanNLDIisanLMIP.SinceLMI(5.12)impliesthat¸>0,wemay,byde¯ningP~=P=¸,obtainanequivalentcondition23ATP~+P~A+CTCP~B+CTDqqpqqpP~>0;45<0;(5.13)(P~B+CTD)T¡(I¡DTD)pqqpqpqpanLMIinthevariableP~.Thus,quadraticstabilityoftheNLDIhasthefrequency-domaininterpretationthattheH1normoftheLTIsystemx_=Ax+Bpp;q=Cqx+Dqppislessthanone.WiththenewvariablesQ=P¡1,¹=1=¸,quadraticstabilityofNLDIsisalsoequivalenttotheexistenceof¹andQsatisfyingtheLMI¹¸0;Q>0;"#TTTT(5.14)AQ+QA+¹BpBp¹BpDqp+QCq<0:(¹BDT+QCT)T¡¹(I¡DDT)pqpqqpqpRemark:Notethatourassumptionofwell-posednessisinfactincorporatedinTtheLMI(5.12)sinceitimpliesI¡DqpDqp>0.ThereforetheNLDI(5.2)iswell-posedandquadraticallystableifandonlyiftheLMIP(5.12)hasasolution.IntheremainderofthischapterwewillassumethatDqpin(5.2)iszero;thereadershouldbearinmindthatallthefollowingresultsholdfornonzeroDqpaswell.²DiagonalNorm-boundLDIs:FortheDNLDI(5.3),wecanobtainonlyasu±cientconditionforquadraticstability.TheconditiondV(x)=dt<0forallnonzerotrajectoriesisequivalentto»T(ATP+PA)»+2»TPB¼<0pforallnonzero(»;¼)satisfying2T¼i·(Cq;i»+Dqp;i¼)(Cq;i»+Dqp;i¼);i=1;:::;L;Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 64Chapter5AnalysisofLDIs:StatePropertieswherewehaveusedCq;iandDqp;itodenotetheithrowsofCqandDqprespec-tively.UsingtheS-procedure,weseethatthisconditionisimpliedbytheexistenceofnonnegative¸1;:::;¸nqsuchthat»T(ATP+PA)»+2»TPB¼pXL³´T2+¸i(Cq;i»+Dqp;i¼)(Cq;i»+Dqp;i¼)¡¼i<0i=1forallnonzero(»;¼).With¤=diag(¸;:::;¸nq),thisisequivalentto"#ATP+PA+CT¤CPB+CT¤Dqqpqqp<0:(5.15)BTP+DT¤CDT¤D¡¤pqpqqpqpTherefore,ifthereexistP>0anddiagonal¤¸0satisfying(5.15),thentheDNLDI(5.3)isquadraticallystable.Checkingthissu±cientconditionforquadraticstabilityisanLMIP.Notethatfrom(5.15),wemusthave¤>0.Equivalently,quadraticstabilityisimpliedbytheexistenceofQ>0,M=diag(¹1;:::;¹nq)>0satisfyingtheLMI"#AQ+QAT+BMBTQCT+BMDTppqpqp<0;(5.16)CQ+DMBT¡M+DMDTqqppqpqpanotherLMIP.Remark:QuadraticstabilityofaDNLDIviatheS-procedurehasasimplefrequency-domaininterpretation.DenotingbyHthetransfermatrixH(s)=¡1Dqp+Cq(sI¡A)Bp,andassuming(A;Bp;Cq)isminimal,quadraticstabil-ityisequivalenttothefactthat,forsomediagonal,positive-de¯nitematrix¤,1=2¡1=2¡1=2k¤H¤k1<1.¤canthenbeinterpretedasascaling.(SeetheNotesandReferencesformoredetailsontheconnectionbetweentheS-procedureandscaling.)Remark:Heretooourassumptionofwell-posednessisinfactincorporatedintheLMI(5.15)or(5.16).Thebottomrightblockispreciselythesu±cientconditionforwell-posednessmentionedonpage57.ThereforetheDNLDI(5.2)iswell-posedandquadraticallystableiftheLMI(5.15)or(5.16)isfeasible.NotethesimilaritybetweenthecorrespondingLMIsfortheNLDIandtheDNLDI;theonlydi®erenceisthatthediagonalmatrixappearingintheLMIassociatedwiththeDNLDIis¯xedasthescaledidentitymatrixintheLMIassociatedwiththeNLDI.Ingeneralitisstraightforwardtoderivethecorresponding(su±cient)conditionforDNLDIsfromtheresultforNLDIs.Intheremainderofthebook,wewilloftenrestrictourattentiontoNLDIs,andpayonlyoccasionalattentiontoDNLDIs;weleavetothereaderthetaskofgeneralizingalltheresultsforNLDIstoDNLDIs.Remark:AnLTIsystemisstableifandonlyifitisquadraticallystable.FormoregeneralLDIs,however,stabilitydoesnotimplyquadraticstability(seetheNotesandReferencesattheendofthischapter).ItistruethatanLDIisstableifandonlyifthereexistsaconvexLyapunovfunctionthatprovesit;seetheNotesandReferences.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 5.1QuadraticStability655.1.1CoordinatetransformationsWecangiveanotherinterpretationofquadraticstabilityintermsofstate-spacecoor-dinatetransformations.Considerthechangeofcoordinatesx=Tx¹,wheredetT6=0.Inthenewcoordinatesystem,(5.1)isdescribedbythematrixA¹(t)=T¡1A(t)T.Weaskthequestion:DoesthereexistTsuchthatinthenewcoordinatesystem,all(nonzero)trajectoriesoftheLDIarealwaysdecreasinginnorm,i.e.,dkx¹k=dt<0?ItiseasytoshowthatthisistrueifandonlyifthereexistsaquadraticLyapunovfunctionV(»)=»TP»fortheLDI,inwhichcasewecantakeanyTwithTTT=P¡1,forexample,T=P¡1=2(theseT'sareallrelatedbyrightorthogonalmatrixmulti-plication).Withthisinterpretation,itisnaturaltoseekacoordinatetransformationmatrixTthatmakesallnonzerotrajectoriesdecreasinginnorm,andhasthesmall-estpossibleconditionnumber·(T).ThisturnsouttobeanEVPforLTIsystems,polytopicandnorm-boundLDIs.Toseethis,letP=T¡TT¡1,sothat·(T)2=·(P).MinimizingtheconditionnumberofTsubjecttotherequirementthatdkx¹k=dt<0forallnonzerotrajectoriesisthenequivalenttominimizingtheconditionnumberofPsubjecttod(xTPx)=dt<0.(AnyTwithTTT=P¡1wherePisanoptimalsolution,isoptimalfortheoriginaloptoptproblem.)Foreachofoursystems(LTIsystems,PLDIsandNLDIs),thechangeofcoordinateswithsmallestconditionnumberisobtainedbysolvingthefollowingEVPs:²LTIsystems:TheEVPinthevariablesPand´isminimize´subjectto(5:6);I·P·´I(Inthisformulation,wehavetakenadvantageofthehomogeneityoftheLMI(5.6)inthevariableP.)²PolytopicLDIs:TheEVPinthevariablesPand´isminimize´subjectto(5:8);I·P·´I²Norm-boundLDIs:TheEVPinthevariablesP,´and¸isminimize´subjectto(5:12);I·P·´I5.1.2QuadraticstabilitymarginsQuadraticstabilitymarginsgiveameasureofhowmuchtheset•canbeexpandedaboutsomecenterwiththeLDIremainingquadraticallystable.²PolytopicLDIs:ForthePLDIx_=(A0+A(t))x;A(t)2®CofA1;:::;ALg;wede¯nethequadraticstabilitymarginasthelargestnonnegative®forwhichitisquadraticallystable.ThisquantityiscomputedbysolvingthefollowingGEVPinPThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 66Chapter5AnalysisofLDIs:StatePropertiesand®:maximize®subjecttoP>0;®¸0;¡¢ATP+PA+®ATP+PA<0;i=1;:::;L00ii²Norm-boundLDIs:Thequadraticstabilitymarginofthesystemx_=Ax+Bp;pTp·®2xTCTCx;pqqisde¯nedasthelargest®¸0forwhichthesystemisquadraticallystable,andiscomputedbysolvingtheGEVPinP,¸and¯=®2:maximize¯"#ATP+PA+¯¸CTCPBqqpsubjecttoP>0;¯¸0;<0BTP¡¸IpDe¯ningP~=P=¸,wegetanequivalentEVPinP~and¯:maximize¯"#ATP~+P~A+¯CTCP~BqqpsubjecttoP~>0;¯¸0;<0BTP~¡IpRemark:ThequadraticstabilitymarginobtainedbysolvingthisEVPisjust¡11=kCq(sI¡A)Bpk1.5.1.3DecayrateThedecayrate(orlargestLyapunovexponent)oftheLDI(5.1)isde¯nedtobethelargest®suchthatlime®tkx(t)k=0t!1holdsforalltrajectoriesx.Equivalently,thedecayrateisthesupremumof¡logkx(t)kliminft!1toverallnonzerotrajectories.(Stabilitycorrespondstopositivedecayrate.)WecanusethequadraticLyapunovfunctionV(»)=»TP»toestablishalowerboundonthedecayrateoftheLDI(5.1).IfdV(x)=dt·¡2®V(x)foralltrajectories,thenV(x(t))·V(x(0))e¡2®t,sothatkx(t)k·e¡®t·(P)1=2kx(0)kforalltrajectories,andthereforethedecayrateoftheLDI(5.1)isatleast®.²LTIsystems:TheconditionthatdV(x)=dt·¡2®V(x)foralltrajectoriesisequivalenttoATP+PA+2®P·0:(5.17)Therefore,thelargestlowerboundonthedecayratethatwecan¯ndusingaquadraticLyapunovfunctioncanbefoundbysolvingthefollowingGEVPinPand®:maximize®(5.18)subjecttoP>0;(5:17)Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 5.1QuadraticStability67Thislowerboundissharp,i.e.,theoptimalvalueoftheGEVP(5.18)isthedecayrateoftheLTIsystem(whichisthestabilitydegreeofA,i.e.,negativeofthemaximumrealpartoftheeigenvaluesofA).Asanalternatecondition,thereexistsaquadraticLyapunovfunctionprovingthatthedecayrateisatleast®ifandonlyifthereexistsQsatisfyingtheLMIQ>0;AQ+QAT+2®Q·0:(5.19)²PolytopicLDIs:TheconditionthatdV(x)=dt·¡2®V(x)foralltrajectoriesisequivalenttotheLMIATP+PA+2®P·0;i=1;:::;L:(5.20)iiTherefore,thelargestlowerboundonthedecayrateprovableviaquadraticLyapunovfunctionsisobtainedbymaximizing®subjectto(5.20)andP>0.ThisisaGEVPinPand®.Asanalternatecondition,thereexistsaquadraticLyapunovfunctionprovingthatthedecayrateisatleast®ifandonlyifthereexistsQsatisfyingtheLMIQ>0;AQ+QAT+2®Q·0;i=1;:::;L:(5.21)ii²Norm-boundLDIs:ApplyingtheS-procedure,theconditionthatdV(x)=dt·¡2®V(x)foralltrajectoriesisequivalenttotheexistenceof¸¸0suchthat"#ATP+PA+¸CTC+2®PPBqqp·0:(5.22)BTP¡¸IpTherefore,weobtainthelargestlowerboundonthedecayrateof(5.2)bymaximizing®overthevariables®,Pand¸,subjecttoP>0,¸¸0and(5.22),aGEVP.Thelowerboundhasasimplefrequency-domaininterpretation:De¯nethe®-shiftedH1normofthesystemby¢kHk1;®=supfkH(s)kjRes>¡®g:ThentheoptimalvalueoftheGEVPisequaltothelargest®suchthatkHk1;®<1.Thiscanbeseenbyrewriting(5.22)as"#(A+®I)TP+P(A+®I)+¸CTCPBqqp·0;BTP¡¸IpandnotingthattheH1normofthesystem(A+®I;Bp;Cq)equalskHk1;®.Analternatenecessaryandsu±cientconditionfortheexistenceofaquadraticLyapunovfunctionprovingthatthedecayrateof(5.2)isatleast®isthatthereexistsQand¹satisfyingtheLMIQ>0;¹¸0;"#TTT(5.23)AQ+QA+¹BpBp+2®QQCq·0:CqQ¡¹IThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 68Chapter5AnalysisofLDIs:StateProperties5.2InvariantEllipsoidsQuadraticstabilitycanalsobeinterpretedintermsofinvariantellipsoids.ForQ>0,letEdenotetheellipsoidcenteredattheorigin©¯ªE=»2Rn¯»TQ¡1»·1:TheellipsoidEissaidtobeinvariantfortheLDI(5.1)ifforeverytrajectoryxoftheLDI,x(t0)2Eimpliesx(t)2Eforallt¸t0.ItiseasilyshownthatthisisthecaseifandonlyifQsatis¯esQAT+AQ·0;forallA2•;orequivalently,ATP+PA·0;forallA2•;(5.24)whereP=Q¡1.Thus,forLTIsystems,PLDIsandNLDIs,invarianceofEischarac-terizedbyLMIsinQorP,itsinverse.Remark:Condition(5.24)isjustthenonstrictversionofcondition(5.4).²LTIsystems:ThecorrespondingLMIinPisP>0;ATP+PA·0;(5.25)andtheLMIinQisQ>0;AQ+PAT·0;(5.26)²PolytopicLDIs:ThecorrespondingLMIinPisP>0;ATP+PA·0;i=1;:::;L;(5.27)iiandtheLMIinQisQ>0;QAT+AQ·0;i=1;:::;L:(5.28)ii²Norm-boundLDIs:ApplyingtheS-procedure,invarianceoftheellipsoidEisequivalenttotheexistenceof¸suchthatP>0;¸¸0;23ATP+PA+¸CTCPB+¸CTD(5.29)qqpqqp45·0:(PB+¸CTD)T¡¸(I¡DTD)pqqpqpqpInvarianceoftheellipsoidEisequivalenttotheexistenceof¹suchthat¹¸0;Q>0;and"#TTTT(5.30)AQ+QA+¹BpBp¹BpDqp+QCq·0:(¹BDT+QCT)T¡¹(I¡DDT)pqpqqpqpInsummary,theconditionthatEbeaninvariantellipsoidcanbeexpressedineachcaseasanLMIinQoritsinverseP.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 5.2InvariantEllipsoids695.2.1SmallestinvariantellipsoidcontainingapolytopeConsiderapolytopedescribedbyitsvertices,P=Cofv1;:::;vpg.TheellipsoidEcontainsthepolytopePifandonlyifvTQ¡1v·1;j=1;:::;p:jjThisconditionmaybeexpressedasanLMIinQ"#1vTj¸0;j=1;:::;p;(5.31)vjQorasanLMIinP=Q¡1asvTPv·1;j=1;:::;p:(5.32)jjMinimumvolumepThevolumeofEis,uptoaconstantthatdependsonn,detQ.WecanminimizethisvolumebysolvingappropriateCPs.²LTIsystems:TheCPinPisminimizelogdetP¡1subjectto(5:25);(5:32)²PolytopicLDIs:TheCPinPisminimizelogdetP¡1(5.33)subjectto(5:27);(5:32)²Norm-boundLDIs:TheCPinPand¸isminimizelogdetP¡1subjectto(5:29);(5:32)MinimumdiameterpThediameteroftheellipsoidEis2¸max(Q).WecanminimizethisquantitybysolvingappropriateEVPs.²LTIsystems:TheEVPinthevariablesQand¸isminimize¸subjectto(5:26);(5:31);Q·¸I²PolytopicLDIs:TheEVPinQand¸isminimize¸(5.34)subjectto(5:28);(5:31);Q·¸I²Norm-boundLDIs:TheEVPinQ,¸and¹isminimize¸subjectto(5:30);(5:31);Q·¸IThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 70Chapter5AnalysisofLDIs:StatePropertiesRemark:Theseresultshavethefollowinguse.ThepolytopePrepresentsourknowledgeofthestateattimet0,i.e.,x(t0)2P(thisknowledgemayre°ectmeasurementsorpriorassumptions).AninvariantellipsoidEcontainingPthengivesaboundonthestatefort¸t0,i.e.,wecanguaranteethatx(t)2Eforallt¸t0.5.2.2LargestinvariantellipsoidcontainedinapolytopeWenowconsiderapolytopedescribedbylinearinequalities:©¯ªP=»2Rn¯aT»·1;k=1;:::;q:kTheellipsoidEiscontainedinthepolytopePifandonlyif©ªmaxaT»j»2E·1;k=1;:::;q:kThisisequivalenttoaTQa·1;k=1;:::;q;(5.35)kkwhichisasetoflinearinequalitiesinQ.ThemaximumvolumeofinvariantellipsoidscontainedinPcanbefoundbysolvingCPs:²LTIsystems:ForLTIsystems,theCPinthevariableQisminimizelogdetQ¡1subjectto(5:26);(5:35)p(SincethevolumeofEis,uptoaconstant,detQ,minimizinglogdetQ¡1willmaximizethevolumeofE).²PolytopicLDIs:ForPLDIs,theCPinthevariableQisminimizelogdetQ¡1subjectto(5:28);(5:35)²Norm-boundLDIs:ForNLDIs,theCPinthevariablesQand¹isminimizelogdetQ¡1subjectto(5:30);(5:35)Wealsocan¯ndthemaximumminordiameter(thatis,thelengthoftheminoraxis)ofinvariantellipsoidscontainedinPbysolvingEVPs:²LTIsystems:ForLTIsystems,theEVPinthevariablesQand¸ismaximize¸subjectto(5:26);(5:35);¸I·Q²PolytopicLDIs:ForPLDIs,theEVPinthevariablesQand¸ismaximize¸subjectto(5:28);5:35);¸I·QCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 5.2InvariantEllipsoids71²Norm-boundLDIs:ForNLDIs,theEVPinthevariablesQ,¹and¸ismaximize¸subjectto(5:30);(5:35);¸I·QRemark:Theseresultscanbeusedasfollows.ThepolytopePrepresentstheallowable(orsafe)operatingregionforthesystem.Theellipsoidsfoundabovecanbeinterpretedasregionsofsafeinitialconditions,i.e.,initialconditionsforwhichwecanguaranteethatthestatealwaysremainsinthesafeoperatingregion.5.2.3BoundonreturntimeThereturntimeofastableLDIforthepolytopePisde¯nedasthesmallestTsuchthatifx(0)2P,thenx(t)2Pforallt¸T.UpperboundsonthereturntimecanbefoundbysolvingEVPs.²LTIsystems:Letusconsiderapositivedecayrate®.IfQ>0satis¯esQAT+AQ+2®Q·0;thenEisaninvariantellipsoid,andmoreoverx(0)2Eimpliesthatx(t)2e¡®tEforallt¸0.ThereforeifTissuchthate¡®TEµPµE;wecanconcludethatifx(0)2P,thenx(t)2Pforallt¸T,sothatTisanupperboundonthereturntime.Ifweusebothrepresentationsofthepolytope,©¯ªP=x2Rn¯aTx·1;k=1;:::;q=Cofv;:::;vg;k1ptheconstrainte¡®TEµPisequivalenttotheLMIinQand°aTQa·°;k=1;:::;q;(5.36)iiwhere°=e2®T,andtheproblemof¯ndingthesmallestsuch°andthereforethesmallestT(fora¯xed®)isanEVPinthevariables°andQ:minimize°subjectto(5:19);(5:31);(5:36)²PolytopicLDIs:ForPLDIs,thesmallestboundonthereturntimeprovableviainvariantellipsoids,withagivendecayrate®,isobtainedbysolvingtheEVPinthevariablesQand°minimize°subjectto(5:21);(5:31);(5:36)²Norm-boundLDIs:ForNLDIs,thesmallestboundonthereturntimeprovableviainvariantellipsoids,withagivendecayrate®,isobtainedbysolvingtheEVPinthevariablesQ,°and¹minimize°subjectto(5:23);(5:31);(5:36)Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 72Chapter5AnalysisofLDIs:StatePropertiesNotesandReferencesQuadraticstabilityforNLDIsLur'eandPostnikov[LP44]gaveoneoftheearlieststabilityanalysesforNLDIs:Theyconsideredthestabilityofthesystemx_=Ax+bpp;q=cqx;p=Á(q;t)q;(5.37)wherepandqarescalar.Thisproblemcametobepopularlyknownastheproblemofabsolutestabilityinautomaticcontrol".IntheoriginalsettingofLur'eandPostnikov,Áwasassumedtobeatime-invariantnonlinearity.Subsequently,variousadditionalassumptionsweremadeaboutÁ,generatingdi®erentspecialcasesandsolutions.(Thefamilyofsystemsoftheform(5.37),whereÁ(q;t)canbeanysector[¡1;1]nonlinearity,isanNLDI.)Pyatnitskii,in[Pya68],pointsoutthatby1968,over200papershadbeenwrittenaboutthesystem(5.37).Amongthese,theonesmostrelevanttothisbookareundoubtedlyfromYakubovich.Asfarasweknow,Yakubovichwasthe¯rsttomakesystematicuseofLMIsalongwiththeS-proceduretoprovestabilityofnonlinearcontrolsystems(seereferences[Yak62,Yak64,Yak67,Yak66,Yak77,Yak82]).ThemainideaofYakubovichwastoexpressthere-¡1sultingLMIsasfrequency-domaincriteriaforthetransferfunctionG(s)=cq(sI¡A)bp.(Yakubovichcallsthismethodthemethodofmatrixinequalities".)Thesecriteriaaremostusefulwhendealingwithexperimentaldataarisingfromfrequencyresponsemeasurements;theyaredescribedindetailinthebookbyNarendraandTaylor[NT73].Seealso[Bar70b].Popov[Pop73]andWillems[Wil71b,Wil74a]outlinetherelationshipbetweentheproblemofabsolutestabilityofautomaticcontrolandquadraticoptimalcontrol.Thecasewhenpandqarevector-valuedsignalshasbeenconsideredmuchmorerecently;seeforinstance,[ZK88,ZK87,KR88,BH89].In[KPZ90,PD90],theauthorsremarkthatquadraticstabilityofNLDIsisequivalenttotheH1condition(5.13).However,asstatedin[PD90],thereisnofundamentaldi®erencebetweentheseandtheolderresultsofYakubovichandPopov;thenewerresultscanberegardedasextensionsoftheworksofYakubovichandPopovtofeedbacksynthesisandtothecaseofstructuredperturba-+tions"[RCDP93,PZP92].WhilealltheresultspresentedinthischapterarebasedonLyapunovstabilitytechniques,theyhaveverystrongconnectionswiththeinput-outputapproachtostudythestabilityofnonlinear,uncertainsystems.ThisapproachwassparkedbyPopov[Pop62],whousedittostudytheLur'esystem.OthermajorcontributorsincludeZames[Zam66a,Zam66b,ZF67],Sandberg[San64,San65a,San65b]andBrockett[Bro65,Bro66,BW65].SeealsothepapersandthebookbyWillems[Wil69a,Wil69b,Wil71a].Theadvantageoftheinput-outputapproachisthatthesystemsarenotrequiredtohave¯nite-dimensionalstate[Des65,DV75,Log90];thecorrespondingstabilitycriteriaareusuallymosteasilyexpressedinthefrequencydomain,andresultinin¯nite-dimensionalconvexoptimizationproblems.ModernapproachesfromthisviewpointincludeDoyle's¹-analysis[Doy82]andSafonov'sKm-analysis[Saf82,SD83,SD84],implementedintherobustnessanalysissoft-+warepackages¹-toolsandKm-tools[BDG91,CS92a],whichapproximatelysolvethesein¯nite-dimensionalproblems.Seealso[AS79,SS81].QuadraticstabilityforPLDIsThistopicismuchmorerecentthanNLDIs,thelikelyreasonbeingthattheLMIexpressingquadraticstabilityofageneralPLDI(whichisjustanumberofsimultaneousLyapunovinequalities(5.8))cannotbeconvertedtoaRiccatiinequalityorafrequency-domaincriterion.Oneofthe¯rstoccurrencesofPLDIsisin[HB76],whereHorisbergerandBelangernotethattheproblemofquadraticstabilityforaPLDIisconvex(theauthorswritedownthecorrespondingLMIs).ForotherresultsandcomputationalproceduresforPLDIssee[PS82,PS86,BY89,KP87a,Kam83,EZKN90,KB93,AGG94];thearticle[KP87a],wheretheauthorsdevelopasubgradientmethodforprovingquadraticstabilityofaPLDI,whichisfurtherre¯nedin[Mei91];thepaper[PS86],wherethediscrete-timecounterpartofthisCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences73problemisconsidered.SeealsothearticlesbyGuetal.[GZL90,GL93b,Gu94],Garofaloetal.[GCG93],andthesurveyarticlebyBarmishetal.[BK93].Necessaryandsu±cientconditionsforLDIstabilityMuchattentionhasalsofocusedonnecessaryandsu±cientconditionsforstabilityofLDIs(asopposedtoquadraticstability);seeforexample[PR91b]or[MP86].Earlierreferencesonthistopicinclude[Pya70b]whichconnectstheproblemofstabilityofanLDItoanoptimalcontrolproblem,and[Pya70a,BT79,BT80],wherethediscrete-timecounterpartofthisproblemisconsidered.VectorLyapunovfunctionsThetechniqueofvectorLyapunovfunctionsalsoyieldssearchproblemsthatcanbeexpressedasLMIPs.Oneexampleisgivenin[CFA90].HerewealreadyhavequadraticLyapunovfunctionsforanumberofsubsystems;theproblemisto¯ndanappropriatepositivelinearcombinationthatproves,e.g.,stabilityofaninterconnectedsystem.Finally,letuspointoutthatquadraticLyapunovfunctionshavebeenusedtodetermineestimatesofregionsofstabilityforgeneralnonlinearsystems.Seeforexample[Gha94,BD85].StableLDIsthatarenotquadraticallystableAnLTIsystemisstableifandonlyifitisquadraticallystable;thisisjustLyapunov'sstabilitytheoremforlinearsystems(seee.g.,[Lya47,p277]).Itispossible,however,foranLDItobestablewithoutbeingquadraticallystable.HereisaPLDIexample:x_=A(t)x;A(t)2CofA1;A2g;"#"#¡10008¡9A1=;A2=:0¡1120¡18From(2.8),thisPLDIisnotquadraticallystableifthereexistQ0¸0,Q1¸0andQ2¸0,notallzero,suchthatTTQ0=A1Q1+Q1A1+A2Q2+Q2A2:Itcanbeveri¯edthatthematrices"#"#"#5:220:132:71Q0=;Q1=;Q2=22439011satisfythesedualityconditions.However,thepiecewisequadraticLyapunovfunction©ªTTV(x)=maxxP1x;xP2x;"#"#14¡100(5.38)P1=;P2=;¡1101provesthatthePLDIisstable.Toshowthis,weusetheS-procedure.Anecessaryandsu±cientconditionfortheLyapunovfunctionVde¯nedin(5.38)toprovethestabilityofThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 74Chapter5AnalysisofLDIs:StatePropertiesthePLDIistheexistenceoffournonnegativenumbers¸1,¸2,¸3,¸4suchthatTA1P1+P1A1¡¸1(P2¡P1)<0;T(5.39)A2P1+P1A2¡¸2(P2¡P1)<0;TTA1P2+P2A1+¸3(P2¡P1)<0;A2P2+P2A2+¸4(P2¡P1)<0:Itcanbeveri¯edthat¸1=50,¸2=0,¸3=1,¸4=100aresuchnumbers.ForotherexamplesofstableLDIsthatarenotquadraticallystable,seeBrockett[Bro65,Bro77,Bro70]andPyatnitskii[Pya71].In[Bro77],BrockettusestheS-proceduretoprovethatacertainpiecewisequadraticformisaLyapunovfunctionand¯ndsLMIssimilarto(5.39);seealso[Vid78].Finally,anLDIisstableifandonlyifthereisaconvexLyapunovfunctionthatprovesit.OnesuchLyapunovfunctionisV(»)=supfkx(t)kjxsatis¯es(5.1);x(0)=»;t¸0g:SeeBraytonandTong[BT79,BT80],andalso[Zub55,MV63,DK71,VV85,PR91b,PR91a,Rap90,Rap93,MP89,Mol87].Ingeneral,computingsuchLyapunovfunctionsiscomputationallyintensive,ifnotintractable.NonlinearsystemsandfadingmemoryConsiderthenonlinearsystemx_=f(x;w;t);(5.40)with@f2•forallx;w;t@xwhere•isconvex.SupposetheLDIx_2•xisstable,i.e.,alltrajectoriesconvergetozeroast!1.Thisimpliesthesystemhasfadingmemory,thatis,for¯xedinputw,thedi®erencebetweenanytwotrajectoriesof(5.40)convergestozero.Inotherwords,thesystemforgets"itsinitialcondition.(See[BC85].)Fixaninputwandletxandx~denoteanytwosolutionsof(5.40).Usingthemean-valuetheoremfromx4.3.1,wehaved(x¡x~)=f(x;w;t)¡f(x~;w;t)=A(t)(x¡x~)dtforsomeA(t)2•.SincetheLDIisstable,x(t)¡x~(t)convergestozeroast!1.RelationbetweenS-procedureforDNLDIsandscalingWediscusstheinterpretationofthediagonalmatrix¤in(5.15)asascalingmatrix.EverytrajectoryoftheDNLDI(5.3)isalsoatrajectory(andviceversa)oftheDNLDI¡1x_=Ax+BpTp;¡1(5.41)q=TCqx+TDqpTp;pi=±i(t)qi;j±i(t)j·1;i=1;:::;nqforanydiagonalnonsingularT.Therefore,ifDNLDI(5.41)isstableforsomediagonalnonsingularT,thensoisDNLDI(5.3).Tisreferredtoasascalingmatrix.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences75TreatingDNLDI(5.41)asanNLDI,andapplyingthecondition(5.13)forquadraticstability,werequire,forsomediagonalnonsingularT,"#TT2¡1T2¡1AP+PA+CqTCqPBpT+CqTDqpTP>0;<0;¡1T2¡1T¡1T2¡1(PBpT+CqTDqpT)¡(I¡TDqpTDqpT)2Anobviouscongruence,followedbythesubstitutionT=¤,yieldstheLMIconditioninP>0anddiagonal¤>0:"#TTAP+PA+Cq¤CqPBp+Cq¤Dqp<0:TTTBPP+Dqp¤CqDqp¤Dq¡¤ThisispreciselyLMI(5.15),theconditionforstabilityoftheDNLDI(5.3)obtainedusingtheS-procedure.ThisrelationbetweenscalingandthediagonalmatricesarisingfromtheS-procedureisdescribedinBoydandYang[BY89].PyatnitskiiandSkorodinskii[PS82,PS83],andKamenetskii[Kam83]reducetheproblemofnumericalsearchforappropriatescalingsandtheassociatedLyapunovfunctionstoaconvexoptimizationproblem.SaekiandAraki[SA88]alsoconcludeconvexityofthescalingproblemandusethepropertiesofM-matricestoobtainasolution.RepresentingaDNLDIasaPLDI:ImplicationsforquadraticstabilityWesawinChapter4thataDNLDIcanberepresentedasaPLDI,wherethenumberLofnqverticescanbeaslargeas2.Fromtheresultsofx5.1,checkingquadraticstabilityforthisPLDIrequiresthesolutionofLsimultaneousLyapunovinequalitiesinP>0(LMI(5.8)).IfthesystemisrepresentedasaDNLDI,asu±cientconditionforquadraticstabilityisgivenbytheLMI(5.15),inthevariablesP>0and¤>0.Comparingthetwoconditionsforquadraticstability,weobservethatwhilethequadraticstabilitycondition(5.8)forLDI(5.3),obtainedbyrepresentingitasaPLDI,isbothnecessaryandsu±cient(i.e.,notconservative),thesizeoftheLMIgrowsexponentiallywithnq,thesizeof¢.Ontheotherhand,thesizeoftheLMI(5.15)growspolynomiallywithnq,butthisLMIyieldsonlyasu±cientconditionforquadraticstabilityofLDI(5.3).DiscussionofthisissuecanbefoundinKamenetskii[Kam83]andPyatnitskiiandSkorodinskii[PS82].Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. Chapter6AnalysisofLDIs:Input/OutputProperties6.1Input-to-StatePropertiesWe¯rstconsiderinput-to-statepropertiesoftheLDIx_=A(t)x+Bw(t)w;[A(t)Bw(t)]2•;(6.1)wherewisanexogenousinputsignal.²LTIsystems:Thesystemhastheformx_=Ax+Bww.²PolytopicLDIs:Herethedescriptionisx_=A(t)x+Bw(t)w;[A(t)Bw(t)]2Cof[A1Bw;1];:::;[ALBw;L]g:²Norm-boundLDIs:NLDIshavetheformx_=Ax+Bpp+Bww;q=Cqx;p=¢(t)q;k¢(t)k·1;orequivalently,x_=Ax+Bp+Bw;q=Cx;pTp·qTq:pwq²DiagonalNorm-boundLDIs:DNLDIshavetheformx_=Ax+Bpp+Bww;q=Cqx;pi=±i(t)qi;j±i(t)j·1;i=1;:::;nq;orequivalently,x_=Ax+Bpp+Bww;q=Cqx;jpij·jqij;i=1;:::;nq:6.1.1Reachablesetswithunit-energyinputsLetRuedenotethesetofreachablestateswithunit-energyinputsfortheLDI(6.1),i.e.,8¯9>>¯¯x,wsatisfy(6.1);x(0)=0>><¯=¢¯ZRue=x(T)¯T:>>¯T>>:¯wwdt·1;T¸0;077 78Chapter6AnalysisofLDIs:Input/OutputPropertiesWewillboundRuebyellipsoidsoftheform©ªE=»j»TP»·1;(6.2)whereP>0.SupposethatthefunctionV(»)=»TP»,withP>0,satis¯esdV(x)=dt·wTwforallx;wsatisfying(6.1):(6.3)Integratingbothsidesfrom0toT,wegetZTV(x(T))¡V(x(0))·wTwdt:0NotingthatV(x(0))=0sincex(0)=0,wegetZTV(x(T))·wTwdt·1;0RTforeveryT¸0,andeveryinputwsuchthatwTwdt·1.Inotherwords,the0ellipsoidEcontainsthereachablesetRue.²LTIsystems:Condition(6.3)isequivalenttotheLMIinP"#ATP+PAPBwP>0;·0;(6.4)BTP¡IwwhichcanalsobeexpressedasanLMIinQ=P¡1Q>0;AQ+QAT+BBT·0:(6.5)wwRemark:ForacontrollableLTIsystem,thereachablesetistheellipsoidT¡1f»j»Wc»·1g,whereWcisthecontrollabilityGramian,de¯nedbyZ1¢AtTATtWc=eBwBwedt:0SinceWcsatis¯estheLyapunovequationTTAWc+WcA+BwBw=0;(6.6)weseethatQ=Wcsatis¯es(6.5).ThustheellipsoidalboundissharpforLTIsystems.²PolytopicLDIs:ForPLDIs,condition(6.3)holdsifandonlyif"#A(t)TP+PA(t)PB(t)wP>0and·0forallt¸0.(6.7)B(t)TP¡IwInequality(6.7)holdsifandonlyif"#ATP+PAPBiiw;iP>0;·0;i=1;:::;L:(6.8)BTP¡Iw;iThustheellipsoidEcontainsRueiftheLMI(6.8)holds.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 6.1Input-to-StateProperties79Alternatively,wecanwrite(6.8)asQ>0;QAT+AQ+BBT·0;i=1;:::;L;(6.9)iiw;iw;iwhereQ=P¡1.NotethatthisconditionimpliesthatEisinvariant.²Norm-boundLDIs:Condition(6.3)holdsifandonlyifP>0and»T(ATP+PA)»+2»TP(B¼+B!)¡!T!·0pwholdforevery!andforevery»and¼satisfying¼T¼¡»TCTC»·0:qqUsingtheS-procedure,thisisequivalenttotheexistenceofPand¸satisfying23ATP+PA+¸CTCPBPBqqpw6T7P>0;¸¸0;4BpP¡¸I05·0:(6.10)BTP0¡IwEquivalently,de¯ningQ=P¡1,condition(6.3)holdsifandonlyifthereexistQand¹satisfying"#AQ+QAT+BBT+¹BBTQCTwwppqQ>0;¹¸0;·0:(6.11)CqQ¡¹I²DiagonalNorm-boundLDIs:Condition(6.3)holdsifandonlyifP>0andforall»and¼thatsatisfy2T¼i·(Cq;i»)(Cq;i»);i=1;:::;nq;wehaveforall!,»T(ATP+PA)»+2»TP(B¼+B!)¡!T!·0:pwItfollowsfromtheS-procedurethatthisconditionholdsifthereexistPand¤=diag(¸;:::;¸nq)satisfying23ATP+PA+CT¤CPBPBqqpw6T7P>0;¤¸0;4BpP¡¤05·0:BTP0¡IwEquivalently,de¯ningQ=P¡1,condition(6.3)holdsifthereexistQandadiagonalmatrixMsuchthat"#AQ+QAT+BBT+BMBTQCTwwppqQ>0;M¸0;·0:(6.12)CqQ¡MFortheremainderofthissectionweleavetheextensiontoDNLDIstothereader.TheresultsdescribedabovegiveusasetofellipsoidalouterapproximationsofRue.Wecanoptimizeoverthissetinseveralways.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 80Chapter6AnalysisofLDIs:Input/OutputPropertiesSmallestouterapproximations²LTIsystems:ForLTIsystems,wecanminimizethevolumeoverellipsoidsEwithPsatisfying(6.3)byminimizinglogdetP¡1overthevariablePsubjectto(6.4).ThisisaCP.If(A;Bw)isuncontrollable,thevolumeofthereachablesetiszero,andthisCPwillbeunboundedbelow.If(A;Bw)iscontrollable,thentheCP¯ndstheexactreachableset,whichcorrespondstoP=W¡1.c²PolytopicLDIs:ForPLDIs,theminimumvolumeellipsoidoftheform(6.2)withPsatisfying(6.3),isfoundbyminimizinglogdetP¡1subjectto(6.8).ThisisaCP.²Norm-boundLDIs:ForNLDIs,theminimumvolumeellipsoidoftheform(6.2),satisfying(6.3),isobtainedbyminimizinglogdetP¡1overthevariablesPand¸subjectto(6.10).ThisisagainaCP.Wecanalsominimizethediameterofellipsoidsoftheform(6.2),satisfying(6.3),byreplacingtheobjectivefunctionlogdetP¡1intheCPsbytheobjectivefunction¸(P¡1).ThisyieldsEVPs.maxTestingifapointisoutsidethereachablesetThepointx0liesoutsidethereachablesetifthereexistsanellipsoidalouterapprox-imationofRuethatdoesnotcontainit.ForLTIsystems,x0doesnotbelongtothereachablesetifthereexistsPsatisfyingxTPx>1and(6.4).ThisisanLMIP.Of00course,if(A;Bw)iscontrollable,thenx0doesnotbelongtothereachablesetifandonlyifxTW¡1x>1,whereWisde¯nedbytheequation(6.6).0c0cForPLDIs,asu±cientconditionforapointx0tolieoutsidethereachablesetRcanbecheckedviaanLMIPinthevariableP:xTPx>1and(6.8).ForNLDIs,ue00xliesoutsidethereachablesetRifthereexistPand¸satisfyingxTPx>10ue00and(6.10),anotherLMIP.Testingifthereachablesetliesoutsideahalf-space©¯ªThereachablesetRliesoutsidethehalf-spaceH=»2Rn¯aT»>1ifandonlyueifanellipsoidEcontainingRliesoutsideH,thatis,theminimumvalueof»TQ¡1»ueover»satisfyingaT»>1exceedsone.Weeasilyseethat©¯ªmin»TQ¡1»¯aT»>1=1=(aTQa):Therefore,forLTIsystems,HdoesnotintersectthereachablesetifthereexistsQsatisfyingaTQa<1and(6.5).ThisisanLMIP.If(A;B)iscontrollable,thenHwdoesnotintersectthereachablesetifandonlyifaTWa<1,whereWisde¯nedbycctheequation(6.6).ForPLDIs,asu±cientconditionforHnottointersectthereachablesetRuecanbecheckedviaanLMIPinthevariableQ:aTQa<1and(6.9).ForNLDIs,HdoesnotintersectthereachablesetRifthereexistsQand¹satisfyingaTQa<1ueand(6.11).ThisisanLMIP.TestingifthereachablesetiscontainedinapolytopeSinceapolytopecanbeexpressedasanintersectionofhalf-spaces(thesedetermineitsfaces"),wecanimmediatelyusetheresultsoftheprevioussubsectioninordertoCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 6.1Input-to-StateProperties81checkifthereachablesetiscontainedinapolytope.(Noteweonlyobtainsu±cientconditions.)Weremarkthatwecanusedi®erentellipsoidalouterapproximationstocheckdi®erentfaces.6.1.2Reachablesetswithcomponentwiseunit-energyinputsWenowturntoavariationofthepreviousproblem;weconsiderthesetofreachablestateswithinputswhosecomponentshaveunit-energy,thatis,weconsidertheset8¯9><¯¯x,wsatisfy(6.1);x(0)=0;>=¢¯ZTRuce=x(T)¯:>:¯w2dt·1;i=1;:::;n;T¸0>;¯iw0Supposethereisapositive-de¯nitequadraticfunctionV(»)=»TP»,andR=diag(r1;:::;rnw)suchthatP>0;R¸0;TrR=1;d(6.13)V(x)·wTRwforallxandwsatisfying(6.1):dtThentheellipsoidEgivenby(6.2)containsthereachableset.Toprovethis,weintegratebothsidesofthelastinequalityfrom0toT,togetZTV(x(T))¡V(x(0))·wTRwdt:0NotingthatV(x(0))=0sincex(0)=0,wegetZTV(x(T))·wTRwdt·1:0Letusnowconsidercondition(6.13)forthevariousLDIs:²LTIsystems:ForLTIsystems,condition(6.13)isequivalenttotheLMIinPandR:P>0,R¸0anddiagonal,TrR=1,"#T(6.14)AP+PAPBw·0:BTP¡RwTherefore,ifthereexistPandRsuchthat(6.14)issatis¯ed,thentheellipsoidEcontainsthereachablesetRuce.Alternatively,usingthevariableQ=P¡1,wecanwriteanequivalentLMIinQandR:Q>0,R¸0anddiagonal,TrR=1,"#T(6.15)QA+AQBw·0:BT¡RwRemark:ItturnsoutthatwithLTIsystems,thereachablesetisequaltotheintersectionofallsuchellipsoids.(Formoredetailsonthis,seetheNotesandReferences.)Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 82Chapter6AnalysisofLDIs:Input/OutputProperties²PolytopicLDIs:ForPLDIs,condition(6.13)holdsifandonlyifP>0,R¸0anddiagonal,TrR=1,"#T(6.16)AiP+PAiPBw;i·0;i=1;:::;L:BTP¡Rw;iTherefore,ifthereexistPandRsuchthat(6.16)holds,thentheellipsoidE=©ª»j»TP»·1containsR.uceWithQ=P¡1,condition(6.13)isequivalenttoQ>0,R¸0anddiagonal,TrR=1,"#T(6.17)QAi+AiQBw;i·0;i=1;:::;L:BT¡Rw;i²Norm-boundLDIs:ForNLDIs,applyingtheS-procedure,condition(6.13)holdsifandonlyifthereexistP,¸andRsuchthatP>0,R¸0anddiagonal,TrR=1,¸¸0,23ATP+PA+¸CTCPBPBqqpw(6.18)6T74BpP¡¸I05·0:BTP0¡RwTherefore,ifthereexistsP,¸andRsuchthat(6.18)issatis¯ed,thentheellipsoidEcontainsthereachablesetRuce.WithQ=P¡1,condition(6.13)isalsoequivalenttotheexistenceofQ,Rand¹suchthatQ>0,R¸0anddiagonal,TrR=1,¹¸0,23AQ+QAT+¹BBTQCTBppqw(6.19)674CqQ¡¹I05·0:BT0¡RwAsintheprevioussection,wecan¯ndtheminimumvolumeandminimumdi-ameterellipsoidsamongallellipsoidsoftheform(6.2),satisfyingcondition(6.13),bysolvingCPsandEVPs,respectively.Wecanalsocheckthatapointdoesnotbelongtothereachablesetorthatahalf-spacedoesnotintersectthereachableset,bysolvingappropriateLMIPs.6.1.3Reachablesetswithunit-peakinputsWeconsiderreachablesetswithinputswthatsatisfywTw·1.Thus,weareinter-estedintheset(¯)¯¢¯x,wsatisfy(6.1);x(0)=0;Rup=x(T)¯:¯wTw·1;T¸0SupposethatthereexistsaquadraticfunctionV(»)=»TP»withP>0;anddV(x)=dt·0forallx,w(6.20)satisfying(6.1),wTw·1andV(x)¸1:Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 6.1Input-to-StateProperties83Then,theellipsoidEgivenby(6.2)containsthereachablesetRup.Letusconsidernowcondition(6.20)forthevariousLDIs:²LTIsystems:ForLTIsystems,condition(6.20)isequivalenttoP>0and»T(ATP+PA)»+»TPBw+wTBTP»·0wwforany»andwsatisfyingwTw·1and»TP»¸1:UsingtheS-procedure,weconcludethatcondition(6.20)holdsifthereexist®¸0and¯¸0suchthatforallxandw,"#T"#"#xATP+PA+®PPBxw+¯¡®·0;(6.21)wBTP¡¯Iwworequivalently23ATP+PA+®PPB0w6T74BwP¡¯I05·0:(6.22)00¯¡®Clearlywemusthave®¸¯.Next,if(6.22)holdsforsome(®0;¯0),thenitholdsforall®0¸¯¸¯0.Therefore,wecanassumewithoutlossofgeneralitythat¯=®,andrewrite(6.21)as"#ATP+PA+®PPBw·0:(6.23)BTP¡®IwTherefore,ifthereexistsPand®satisfyingP>0;®¸0;and(6:23);(6.24)thentheellipsoidEcontainsthereachablesetRup.Notethatinequality(6.23)isnotanLMIin®andP;however,for¯xed®itisanLMIinP.Alternatively,condition(6.23)isequivalenttothefollowinginequalityinQ=P¡1and®:"#QAT+AQ+®QBw·0:(6.25)BT¡®Iw²PolytopicLDIs:Condition(6.20)holdsifP>0andthereexists®¸0satisfying"#ATP+PA+®PPBiiw;i·0;i=1;:::;L:(6.26)BTP¡®Iw;iTherefore,ifthereexistsPand®satisfyingP>0;®¸0;and(6:26);(6.27)thentheellipsoidEcontainsthereachablesetRup.Onceagain,wenotethatcondi-tion(6.26)isnotanLMIin®andP,butisanLMIinPfor¯xed®.Wecanalsorewritecondition(6.26)intermsofQ=P¡1astheequivalentinequality"#QAT+AQ+®QBiiw;i·0;i=1;:::;L:(6.28)BT¡®Iw;iThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 84Chapter6AnalysisofLDIs:Input/OutputProperties²Norm-boundLDIs:ForNLDIs,condition(6.20)holdsifandonlyifP>0and»T(ATP+PA)»+2»TP(B!+B¼)·0wpforevery!,»and¼satisfying!T!·1;»TP»¸1;¼T¼¡»TCTC»·0:qqTherefore,usingtheS-procedure,asu±cientconditionfor(6.20)toholdisthatthereexist®¸0and¸¸0suchthat23ATP+PA+®P+¸CTCPBPBqqpw6T74BpP¡¸I05·0:(6.29)BTP0¡®IwIfthereexistsP,®and¸satisfyingP>0;®¸0;¸¸0;and(6:29);(6.30)theellipsoidEP¡1containsthereachablesetRup.Wenoteagainthatinequality(6.29)isnotanLMIinPand®,butisanLMIinPfor¯xed®.De¯ningQ=P¡1,condition(6.20)isalsoimpliedbytheexistenceof¹¸0and®¸0suchthat23QAT+AQ+®Q+¹BTBQCTBppqw674CqQ¡¹I05·0:(6.31)BT0¡®Iw(Infact,theexistenceof¸¸0and®¸0satisfying(6.29)isequivalenttotheexistenceof¹¸0and®¸0satisfying(6.31).)For¯xed®,wecanminimizethevolumeordiameteramongallellipsoidsEsatis-fyingcondition(6.24)forLTIsystems,condition(6.27)forPLDIsandcondition(6.30)forNLDIsbyformingtheappropriateCPs,orEVPs,respectively.Remark:Inequality(6.23)canalsobederivedbycombiningtheresultsofx6.1.1withanexponentialtime-weightingoftheinputw.SeetheNotesandReferencesfordetails.Asinx6.1.2,itispossibletodetermineouterellipsoidalapproximationsofthereachablesetofLTIsystems,PLDIsandNLDIs,subjecttocomponentwiseunit-peakinputs.6.2State-to-OutputPropertiesWenowconsiderstate-to-outputpropertiesfortheLDIx_=A(t)x;z=Cz(t)x(6.32)where"#A(t)2•:(6.33)Cz(t)²LTIsystems:LTIsystemshavetheformx_=Ax,z=Czx.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 6.2State-to-OutputProperties85²PolytopicLDIs:PLDIshavetheformx_=A(t)x,z=Cz(t)x,where"#("#"#)A(t)A1AL2Co;:::;:Cz(t)C1CL²Norm-boundLDIs:NLDIshavetheformx_=Ax+Bpp;q=Cqx;z=Czxp=¢(t)q;k¢(t)k·1;orequivalentlyx_=Ax+Bp;q=Cx;z=Cx;pTp·qTq:pqz²DiagonalNorm-boundLDIs:DNLDIshavetheformx_=Ax+Bpp;q=Cqx;z=Czx;pi=±i(t)qi;j±i(t)j·1;i=1;:::;nq;orequivalentlyx_=Ax+Bpp;q=Cqx;z=Czx;jpij·jqij;i=1;:::;nq:6.2.1BoundsonoutputenergyWeseekthemaximumoutputenergygivenacertaininitialstate,½Z¯¾1¯maxzTzdt¯x_=A(t)x;z=C(t)x;(6.34)¯z0wherex(0)isgiven,andthemaximumistakenoverA(t),Cz(t)suchthat(6.33)holds.SupposethereexistsaquadraticfunctionV(»)=»TP»suchthatdTP>0andV(x)·¡zz;foreveryxandzsatisfying(6.32):(6.35)dtThen,integratingbothsidesofthesecondinequalityin(6.35)from0toT,wegetZTV(x(T))¡V(x(0))·¡zTzdt:0foreveryT¸0.SinceV(x(T))¸0,weconcludethatV(x(0))=x(0)TPx(0)isanupperboundonthemaximumenergyoftheoutputzgiventheinitialconditionx(0).WenowderiveLMIsthatprovideupperboundsontheoutputenergyforthevariousLDIs.²LTIsystems:InthecaseofLTIsystems,condition(6.35)isequivalenttoP>0;ATP+PA+CTC·0:(6.36)zzTherefore,weobtainthebestupperboundontheoutputenergyprovableviaquadraticfunctionsbysolvingtheEVPminimizex(0)TPx(0)(6.37)subjecttoP>0;(6:36)Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 86Chapter6AnalysisofLDIs:Input/OutputPropertiesInthiscasethesolutioncanbefoundanalytically,anditisexactlyequaltotheoutputenergy,whichisx(0)TWx(0),whereWistheobservabilityGramianofthesystem,oode¯nedbyZ1¢ATtTAtWo=eCzCzedt:0SinceWosatis¯estheLyapunovequationATW+WA+CTC=0;(6.38)oozzitsatis¯es(6.35).WithQ=P¡1,condition(6.35)isequivalentto"#AQ+QATQCTzQ>0;·0:(6.39)CzQ¡IIfQsatis¯es(6.39),thenanupperboundontheoutputenergy(6.34)isx(0)TQ¡1x(0).²PolytopicLDIs:ForPLDIs,condition(6.35)isequivalenttoP>0;A(t)TP+PA(t)+C(t)TC(t)·0forallt¸0.(6.40)zzInequalities(6.40)holdifandonlyifthefollowingLMIinthevariablePholds:P>0;ATP+PA+CTC·0;i=1;:::;L:(6.41)iiz;iz;iThereforetheEVPcorrespondingto¯ndingthebestupperboundontheoutputenergyprovableviaquadraticfunctionsisminimizex(0)TPx(0)(6.42)subjecttoP>0;(6:41)De¯ningQ=P¡1,condition(6.35)isalsoequivalenttotheLMIinQ"#AQ+QATQCTiiz;iQ>0;·0:Cz;iQ¡I²Norm-boundLDIs:ForNLDIs,condition(6.35)isequivalenttoP>0and»T(ATP+PA+CTC)»+2»TPB¼·0zzpforevery»and¼thatsatisfy¼T¼·»TCTC».UsingtheS-procedure,anequivalentqqconditionistheexistenceofPand¸suchthatP>0;¸¸0;"#TTT(6.43)AP+PA+CzCz+¸CqCqPBp·0:BTP¡¸IpThereforeweobtainthesmallestupperboundontheoutputenergy(6.34)provableviaquadraticfunctionssatisfyingcondition(6.35)bysolvingthefollowingEVPinthevariablesPand¸minimizex(0)TPx(0)(6.44)subjecttoP>0;¸¸0;(6:43)Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 6.2State-to-OutputProperties87WithQ=P¡1,condition(6.35)isalsoequivalenttotheexistenceof¹¸0satisfying23AQ+QAT+¹BBTQCTQCTppzq67Q>0;4CzQ¡I05·0:CqQ0¡¹I²DiagonalNorm-boundLDIs:ForDNLDIs,condition(6.35)isequivalentto»T(ATP+PA+CTC)»+2»TPB¼·0;zzpforevery»and¼thatsatisfy2T¼i·(Cq;i»)(Cq;i»);i=1;:::;nq:UsingtheS-procedure,asu±cientconditionisthatthereexistsadiagonal¤¸0suchthat"#ATP+PA+CTC+CT¤CPBzzqqp·0:(6.45)BTP¡¤pThereforeweobtainthesmallestupperboundontheoutputenergy(6.34)provableviaquadraticfunctionssatisfyingcondition(6.35)andtheS-procedurebysolvingthefollowingEVPinthevariablesPand¤:minimizex(0)TPx(0)subjecttoP>0;¤¸0anddiagonal;(6:45)WithQ=P¡1,condition(6.35)isalsoequivalenttotheexistenceofadiagonalmatrixM¸0satisfying23AQ+QAT+BMBTQCTQCTppzq674CzQ¡I05·0:CqQ0¡MOnceagain,weleaveittothereadertoextendtheresultsintheremainderofthesectiontoDNLDIs.MaximumoutputenergyextractablefromasetAsanextension,weseekboundsonthemaximumextractableenergyfromx(0),whichisknownonlytolieinapolytopeP=Cofv1;:::;vpg.We¯rstnotethatsinceP>0,x(0)TPx(0)takesitsmaximumvalueononeoftheverticesv1;:::;vp.Therefore,thesmallestupperboundontheextractableenergyfromPprovableviaquadraticLyapunovfunctionsforanyofourLDIsiscomputedbysolving(6.37),(6.42)or(6.44)ptimes,successivelysettingx(0)=vi,i=1;:::;p.Themaximumofthepresultingoptimalvaluesisthesmallestupperboundsought.Asavariation,supposethatx(0)isonlyknowntobelongtotheellipsoidE=f»j»TX»·1g.Then,thesmallestupperboundonthemaximumoutputenergyextractablefromEprovableviaquadraticfunctionssatisfying(6.35)isobtainedbyreplacingtheobjectivefunctionx(0)TPx(0)intheEVPs(6.37),(6.42),and(6.44)bytheobjectivefunction¸(X¡1=2PX¡1=2),andsolvingthecorrespondingEVPs.FormaxLTIsystems,theexactvalueofthemaximumextractableoutputenergyisobtained.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 88Chapter6AnalysisofLDIs:Input/OutputPropertiesFinally,ifxisarandomvariablewithExxT=X,thenweobtainthesmall-0000estupperboundontheexpectedoutputenergyprovableviaquadraticfunctionsbyreplacingtheobjectivefunctionx(0)TPx(0)intheEVPs(6.37),(6.42),and(6.44)bytheobjectivefunctionTrX0PandsolvingthecorrespondingEVPs.6.2.2BoundsonoutputpeakItispossibletoderiveboundsonkz(t)kusinginvariantellipsoids.Assume¯rstthattheinitialconditionx(0)isknown.SupposeE=f»j»TP»·1gisaninvariantellipsoidcontainingx(0)fortheLDI(6.32).Thenz(t)Tz(t)·max»TC(t)TC(t)»zz»2Eforallt¸0.Wecanexpressmax»TC(t)TC(t)»asthesquarerootofthe»2Ezzminimumof±subjectto"#PCTz¸0:(6.46)Cz±I²LTIsystems:ThesmallestboundontheoutputpeakthatcanbeobtainedviainvariantellipsoidsisthesquarerootoftheoptimalvalueoftheEVPinthevariablesPand±minimize±subjecttoP>0;x(0)TPx(0)·1;(6.47)(6:46);ATP+PA·0²PolytopicLDIs:ForPLDIs,weobtainthesmallestupperboundonkzkprovableviainvariantellipsoidsbytakingthesquarerootoftheoptimalvalueofthefollowingEVPinthevariablesPand±minimize±subjecttoP>0;x(0)TPx(0)·1;(6:46);ATP+PA·0;i=1;:::;Lii²Norm-boundLDIs:ForNLDIs,weobtainthesmallestupperboundonkzkprovableviainvariantellipsoidsbytakingthesquarerootoftheoptimalvalueofthefollowingEVPinthevariablesP,¸and±minimize±subjecttoP>0;x(0)TPx(0)·1;¸¸0;(6:46);23ATP+PA+¸CTCPBqqp45·0BTP¡¸IpCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 6.3Input-to-OutputProperties89Remark:Asvariationofthisproblem,wecanimposeadecayrateconstraintontheoutput,thatis,given®>0,wecancomputeanupperboundonthesmallest°suchthattheoutputzsatis¯es¡®tkz(t)k·°e;forallt¸0.ThisalsoreducestoanEVP.6.3Input-to-OutputPropertiesWe¯nallyconsidertheinput-outputbehavioroftheLDIx_=A(t)x+Bw(t)w;x(0)=x0;(6.48)z=Cz(t)x+Dzw(t)w;where"#A(t)Bw(t)2•:(6.49)Cz(t)Dzw(t)6.3.1HankelnormboundsInthissection,weassumethatDzw(t)=0forallt,andconsiderthequantity8¯R9T¸0andthemaximumistakenoverA(t),B(t)andC(t)satisfying(6.49).ForanLTIpsystem,ÁequalstheHankelnorm,anamethatweextendforconveniencetoLDIs.AnupperboundforÁcanbecomputedbycombiningtheellipsoidalboundsonreachablesetsfromx6.1.1andtheboundsontheoutputenergyfromx6.2.1.SupposethatP>0andQ>0satisfyd¡¢d¡¢xTPx·wTw;xTQx·¡zTz(6.50)dtdtforallx,wandzsatisfying(6.48).Then,fromtheargumentsinx6.1.1andx6.2.1,weconcludethat©R1¯ªÁ·supzTzdt¯x(T)TPx(T)·1©T¯ª·supx(T)TQx(T)¯x(T)TPx(T)·1=¸(P¡1=2QP¡1=2):max²LTIsystems:WecomputethesmallestupperboundontheHankelnormprovableviaquadraticfunctionssatisfying(6.50)bycomputingthesquarerootoftheminimum°suchthatthereexistP>0andQ>0satisfyingATP+PA+PBBTP·0;wwATQ+QA+CTC·0;zz°P¡Q¸0:Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 90Chapter6AnalysisofLDIs:Input/OutputPropertiesThisisaGEVPinthevariables°,P,andQ.ItispossibletotransformitintoanEVPbyintroducingthenewvariableQ~=Q=°.ThecorrespondingEVPinthevariables°,P,andQ~isthentominimize°subjecttoATP+PA+PBBTP·0;wwATQ~+QA~+CTC=°·0;zzP¡Q~¸0Inthiscase,theoptimalvalueisexactlytheHankelnorm.Itcanbefoundanalyticallyas¸max(WcWo)whereWcandWoarethecontrollabilityandobservabilityGramians,i.e.,thesolutionsoftheLyapunovequations(6.6)and(6.38),respectively.²PolytopicLDIs:WecomputethesmallestupperboundontheHankelnormprovableviaquadraticfunctionssatisfying(6.50)bycomputingthesquarerootoftheminimum°suchthatthereexistP>0andQ>0satisfyingATP+PA+PBBTP·0;iiw;iw;iATQ+QA+CTC·0;i=1;:::;Liiz;iz;i°P¡Q¸0:ThisisaGEVPinthevariables°,P,andQ.De¯ningQ~=Q=°,anequivalentEVPinthevariables°,P,andQ~istominimize°subjecttoATP+PA+PBBTP·0;iiw;iw;iATQ~+QA~+CTC=°·0;i=1;:::;Liiz;iz;iP¡Q~¸0²Norm-boundLDIs:Similarly,thesmallestupperboundontheHankelnormprovableviaquadraticfunctionssatisfying(6.50)iscomputedbytakingthesquarerootoftheminimum°suchthatthereexistP>0,Q>0,¸¸0,and¹¸0satisfying23ATP+PA+¸CTCPBPBqqpw6T74BpP¡¸I05·0;BTP0¡I"w#ATQ+QA+CTC+¹CTCQBzzqqp·0;BTQ¡¹Ip°P¡Q¸0:ThisisaGEVPinthevariables°,P,Q,¸,and¹.IfweintroducethenewvariablesQ~=Q=°,º=¹=°andequivalentEVPinthevariables°,P,Q~,¸,andºistominimize°suchthat23ATP+PA+¸CTCPBPBqqpw6T74BpP¡¸I05·0;BTP0¡I"w#ATQ~+QA~+CTC=°+ºCTCQ~Bzzqqp·0;BTQ~¡ºIpP¡Q~¸0:Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 6.3Input-to-OutputProperties916.3.2L2andRMSgainsWeassumeDzw(t)=0forsimplicityofexposition.Wede¯netheL2gainoftheLDI(6.48)asthequantitykzk2supkwk26=0kwk2R1wheretheLnormofuiskuk2=uTudt,andthesupremumistakenover220allnonzerotrajectoriesoftheLDI,startingfromx(0)=0.TheLDIissaidtobenonexpansiveifitsL2gainislessthanone.Now,supposethereexistsaquadraticfunctionV(»)=»TP»,P>0,and°¸0suchthatforallt,dT2TV(x)+zz¡°ww·0forallxandwsatisfying(6.48):(6.51)dtThentheL2gainoftheLDIislessthan°.Toshowthis,weintegrate(6.51)from0toT,withtheinitialconditionx(0)=0,togetZT¡¢V(x(T))+zTz¡°2wTwdt·0:0sinceV(x(T))¸0,thisimplieskzk2·°:kwk2TRemark:Itcanbeeasilycheckedthatif(6.51)holdsforV(»)=»P»,P>0,then°isalsoanupperboundontheRMSgainoftheLDI,wheretheroot-mean-square(RMS)valueof»isde¯nedasµZT¶1=2¢1TRMS(»)=limsup»»dt;T!1T0andtheRMSgainisde¯nedasRMS(z)sup:RMS(w)6=0RMS(w)Nowreconsidercondition(6.51)forLDIs.²LTIsystems:Condition(6.51)isequivalentto"#ATP+PA+CTCPBzzw·0:(6.52)BTP¡°2IwTherefore,wecomputethesmallestupperboundontheL2gainoftheLTIsystemprovableviaquadraticfunctionsbyminimizing°overthevariablesPand°satisfyingconditionsP>0and(6.52).ThisisanEVP.Remark:Assumingthat(A;Bw;Cz)isminimal,thisEVPgivestheexactvalueoftheL2gainoftheLTIsystem,whichalsoequalstheH1normofitstransfer¡1matrix,kCz(sI¡A)Bwk1.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 92Chapter6AnalysisofLDIs:Input/OutputPropertiesAssumeBw6=0andCz6=0;thentheexistenceofP>0satisfying(6.52)isequivalenttotheexistenceofQ>0satisfying"#AQ+QAT+BBT=°2QCTwwz·0:CzQ¡I²PolytopicLDIs:Condition(6.51)isequivalentto"#ATP+PA+CTCPBiiz;iz;iw;i·0;i=1;:::;L:(6.53)BTP¡°2Iw;iAssumethereexistsi0forwhichBw;i06=0,andj0forwhichCz;j06=0.ThenthereexistsP>0satisfying(6.53)ifandonlyifthereexistsQ>0satisfying"#QAT+AQ+BBTQCTiiw;iw;iz;i·0;i=1;:::;L:CQ¡°2Iz;iWegetthesmallestupperboundontheL2gainprovableviaquadraticfunctionsbyminimizing°(over°andP)subjectto(6.53)andP>0,whichisanEVP.²Norm-boundLDIs:ForNLDIs,condition(6.51)isequivalentto»T(ATP+PA+CTC)»+2»TP(B¼+Bw)¡°2wTw·0zzpwforall»and¼satisfying¼T¼·»TCTC»:qqThisistrueifandonlyifthereexists¸¸0suchthat23ATP+PA+CTC+¸CTCPBPBzzqqpw6T74BpP¡¸I05·0:(6.54)BTP0¡°2IwTherefore,weobtainthebestupperboundontheL2gainprovableviaquadraticfunctionsbyminimizing°overthevariables°,Pand¸,subjecttoP>0,¸¸0and(6.54).IfBw6=0andCz6=0,thentheexistenceofP>0and¸¸0satisfying(6.54)isequivalenttotheexistenceofQ>0and¹¸0satisfying23QAT+AQ+BBT+¹BBTQCTQCTwwppqz674CqQ¡¹I05·0:CQ0¡°2Iz²DiagonalNorm-boundLDIs:ForDNLDIs,condition(6.51)isequivalentto»T(ATP+PA+CTC)»+2»TP(B¼+Bw)¡°2wTw·0zzpwforall»and¼satisfying¼T¼·(C»)T(C»);i=1;:::;n:iiq;iiq;iiqCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 6.3Input-to-OutputProperties93UsingtheS-procedure,asu±cientconditionisthatthereexistadiagonal¤¸0suchthat23ATP+PA+CTC+CT¤CPBPBzzqqpw6T74BpP¡¤05·0:(6.55)BTP0¡°2IwWeobtainthebestsuchupperboundbyminimizing°overthevariables°,Pand¸,subjecttoP>0,¸¸0and(6.55).IfBw6=0andCz6=0,thentheexistenceofP>0and¸¸0satisfying(6.54)isequivalenttotheexistenceofQ>0andM=diag(¹1;:::;¹nq)¸0satisfying23QAT+AQ+BBT+BMBTQCTQCTwwppqz674CqQ¡M05·0:CQ0¡°2Iz6.3.3DissipativityTheLDI(6.48)issaidtobepassiveifeverysolutionxwithx(0)=0satis¯esZTwTzdt¸00forallT¸0.Itissaidtohavedissipation´ifZT¡¢wTz¡´wTwdt¸0(6.56)0holdsforalltrajectorieswithx(0)=0andallT¸0.Thuspassivitycorrespondstononnegativedissipation.Thelargestdissipationofthesystem,i.e.,thelargestnumber´suchthat(6.56)holds,willbecalleditsdissipativity.SupposethatthereisaquadraticfunctionV(»)=»TP»,P>0,suchthatdTTforallxandwsatisfying(6.48);V(x)¡2wz+2´ww·0:(6.57)dtThen,integrating(6.57)from0toTwithinitialconditionx(0)=0yieldsZT¡¢V(x(T))¡2wTz¡2´wTwdt·0:0SinceV(x(T))¸0,weconcludeZT¡¢wTz¡´wTwdt¸0;0whichimpliesthatthedissipativityoftheLDIisatleast´.Nowreconsidercondi-tion(6.57)forthevariousLDIs.²LTIsystems:ForLTIsystems,condition(6.57)isequivalentto"#ATP+PAPB¡CTwz·0:(6.58)BTP¡C2´I¡(DT+D)wzzwzwWe¯ndthelargestdissipationthatcanbeguaranteedusingquadraticfunctionsbymaximizing´overthevariablesPand´,subjecttoP>0and(6.58),anEVP.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 94Chapter6AnalysisofLDIs:Input/OutputPropertiesAssumingthatthesystem(A;Bw;Cz)isminimal,theoptimalvalueofthisEVPisexactlyequaltothedissipativityofthesystem,whichcanbeexpressedintermsofthetransfermatrixH(s)=C(sI¡A)¡1B+Daszwzw¸(H(s)+H(s)¤)mininf:(6.59)Res>02ThisfollowsfromthePRlemma.De¯ningQ=P¡1,condition(6.58)isequivalentto"#QAT+AQB¡QCTwz·0:(6.60)BT¡CQ2´I¡(DT+D)wzzwzw²PolytopicLDIs:ForPLDIs,condition(6.57)isequivalentto"#ATP+PAPB¡CTiiw;iz;i·0;i=1;:::;L:(6.61)BTP¡C2´I¡(DT+D)w;iz;izw;izw;iWe¯ndthelargestdissipationthatcanbeguaranteedviaquadraticfunctionsbymaximizing´overthevariablesPand´satisfyingP>0and(6.61).ThisisanEVP;itsoptimalvalueisalowerboundonthedissipativityofthePLDI.Ifwede¯neQ=P¡1,condition(6.57)isequivalentto"#QAT+AQB¡QCTiiw;iz;i·0;i=1;:::;L:(6.62)BT¡CQ2´I¡(DT+D)w;iz;izw;izw;i²Norm-boundLDIs:ForNLDIs,usingtheS-procedure,thecondition(6.57)istrueifandonlyifthereexistsanonnegativescalar¸suchthat23ATP+PA+¸CTCPB¡CTPBqqwzp6TT74BwP¡Cz2´I¡(Dzw+Dzw)05·0:(6.63)BTP0¡¸IpThereforewecan¯ndthelargestdissipationprovablewithaquadraticLyapunovfunctionbymaximizing´(over´andP),subjecttoP>0and(6.63).ThisisanEVP.De¯ningQ=P¡1,condition(6.57)isequivalenttotheexistenceof¹¸0satisfying23QAT+AQ+¹BBTB¡QCTQCTppwzq6TT74Bw¡CzQ2´I¡(Dzw+Dzw)05·0:(6.64)CqQ0¡¹I6.3.4DiagonalscalingforcomponentwiseresultsWeassumeforsimplicitythatDzw(t)=0.Assumingthatthesystem(6.48)hasasmanyinputsasoutputs,weconsiderthesystemx_=A(t)x+B(t)T¡1w~;x(0)=0;w(6.65)z~=TCz(t)x:whereTisapositive-de¯nitediagonalmatrix,whichhastheinterpretationofascal-ing.Wewillseethatscalingenablesustoconcludecomponentwise"resultsforthesystem(6.65).Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 6.3Input-to-OutputProperties95Considerforexample,thescaledL2gain"kz~k2®=infsup:Tdiagonal;kw~k26=0kw~k2T>0ThescaledL2gainhasthefollowinginterpretation:kzikmaxsup·®:i=1;:::;nzkwik26=0kwikWeshowthisasfollows:Forevery¯xedscalingT=diag(t1;:::;tnz),Pnkz~k2zt2kzk2sup2=supPiii2kw~k26=0kw~k2kwk26=0nzt2kwk22iii2kzk2i2¸supkwik26=0kwk2:i2SincetheinequalityistrueforalldiagonalnonsingularT,thedesiredinequalityfollows.WenowshowhowwecanobtainboundsonthescaledL2gainusingLMIsforLTIsystemsandLDIs.²LTIsystems:TheL2gainfortheLTIsystemscaledbyTisguaranteedtobelessthan°ifthereexistsP>0suchthattheLMI"#ATP+PA+CTSCPBzzw<0BTP¡°2SwholdswithS=TTT.ThesmallestscaledLgainofthesystemcanthereforebe2computedasaGEVP.²PolytopicLDIs:ForPLDIs,thescaledL2gainisguaranteedtobelowerthan°ifthereexistsS>0,withSdiagonal,andP>0whichsatisfy"#ATP+PA+CTSCPBiiz;iz;iw;i<0;i=1;:::;L:(6.66)BTP¡°2Sw;iTheoptimalscaledL2gain°isthereforeobtainedbyminimizing°over(°,PandS)subjectto(6.66),P>0,S>0andSdiagonal.ThisisaGEVP.²Norm-boundLDIs:ForNLDIs,thescaledL2gain(withscalingT)islessthan°ifthereexist¸¸0,P>0,andS>0,Sdiagonal,suchthatthefollowingLMIholds:23ATP+PA+CTSC+¸CTCPBPBzzqqpw6T74BpP¡¸I05·0:(6.67)BTP0¡°2SwTherefore,minimizing°2subjecttotheconstraint(6.67)isanEVP.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 96Chapter6AnalysisofLDIs:Input/OutputPropertiesNotesandReferencesIntegralquadraticconstraintsInx8.2(andx10.9)weconsiderageneralizationoftheNLDIinwhichthepointwiseconditionTTpp·qqisreplacedbytheintegralconstraintZZ11TTppdt·qqdt:00ManyoftheresultsderivedforNLDIsandDNLDIsthenbecomenecessaryandsu±cient.IntegralquadraticconstraintswereintroducedbyYakubovich.Forageneralstudyofthese,wereferthereadertothearticlesofYakubovich(see[Yak92]andreferencestherein),andalsoMegretsky[Meg93,Meg92a,Meg92b].Reachablesetsforcomponentwiseunit-energyinputsWestudyingreaterdetailthesetofstatesreachablewithcomponentwiseunit-energyinputsforLTIsystems(seex6.1.2).Webeginbyobservingthatapointx0belongstothereachablesetifandonlyiftheoptimalvalueoftheproblemZ12minimizemaxwidti0subjecttox_=¡Ax+Bww;x(0)=x0;limx(t)=0t!1islessthanone.Thisisamulti-criterionconvexquadraticproblem,consideredinx10.8.Inthiscase,theproblemreducestocheckingwhethertheLMI"#TAP+PAPBTx0Px0<1;·0;TBP¡RP>0;R>0;diagonal;TrR=1©ªTisfeasible.Thisshowsthatthereachablesetistheintersectionofellipsoids»j»P»·1satisfyingequation(6.14).AvariationRofthisproblemisconsideredin[SZ92],wheretheauthorsconsiderinputsw1Tsatisfyingwwdt·Q,orequivalently0Z1TwWwdt·TrWQforeveryW¸0.0Theygetnecessaryconditionscharacterizingthecorrespondingreachableset.WenowderiveLMIconditionsthatarebothnecessaryandsu±cient.Asbefore,x0belongstothereachablesetifandonlytheoptimalvalueoftheproblemZ1TminimizewWwdt0subjecttox_=¡Ax+Bww;x(0)=x0;limx(t)=0t!1islessthanTrWQforeveryW¸0.ThisisequivalenttoinfeasibilityoftheLMI(inthevariablesPandW):"#TTAP+PAPBx0Px0¡TrWQ¸0;P>0;·0:TBP¡WCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences97Thesearepreciselytheconditionsobtainedin[SZ92].Reachablesetswithunit-peakinputsSchweppe[Sch73,x4.3.3]considersthemoregeneralproblemoftime-varyingellipsoidalap-proximationsofreachablesetswithunit-peakinputsfortime-varyingsystems;thepositive-de¯nitematrixdescribingtheellipsoidalapproximationsatis¯esamatrixdi®erentialequa-tion,termsofwhichcloselyresemblethoseinthematrixinequalitiesdescribedinthisbook.SabinandSummers[SS90]studytheapproximationofthereachablesetviaquadraticfunc-tions.AsurveyoftheavailabletechniquescanbefoundinthearticlebyGayek[Gay91].Seealso[FG88].ThetechniqueusedtoderiveLMI(6.23)canbeinterpretedasanexponentialtime-weightingprocedure.Fix®>0.ThenforeveryT>0,rewriteLDI(6.1),withnewexponentially®(t¡T)=2®(t¡T)=2time-weightedvariablesxT(t)=ex(t)andv(t)=ew(t)as³´®x_T=A(t)+IxT+Bw(t)v;xT(0)=0:2RSincew(t)Tw(t)·1,wehaveTvTvd¿·1=®.0NowsupposethatP>0satis¯escondition(6.23),whichwerewriteforconvenienceas"#T(A+®I=2)P+P(A+®I=2)PBw·0:(6.68)TBwP¡®ITTTheresultsofx6.1.1implythatxT(T)satis¯esxT(T)©PxT(T)·1;therefore,ªx(T)Px(T)·T1.SinceLMI(6.68)isindependentofT,theellipsoidxjxPx·1containsthereachablesetwithunit-peakinputs.BoundsonovershootTheresultsofx6.1.3canbeusedto¯ndaboundonthestepresponsepeakforLDIs.Considerasingle-inputsingle-outputLDI,subjecttoaunit-stepinput:"#x_=A(t)x+bw(t);x(0)=0;A(t)bw(t)2•:s=cz(t)x;cz(t)0Sinceaunit-stepisalsoaunit-peakinput,anupperboundonmaxt¸0js(t)j(i.e.,theover-shoot)canbefoundbycombiningtheresultsonreachablesetswithunit-peakinputs(x6.1.3)andboundsonoutputpeak(x6.2.2).SupposethereexistQ>0and®>0satisfying(6.25),T1TA(t)Q+QA(t)+®Q+bw(t)bw(t)<0:®2Then,fromx6.2.2,weconcludethatmaxt¸0js(t)jdoesnotexceedM,whereM=maxt¸0Tcz(t)Qcz(t).Thiscanbeusedto¯ndLMI-basedboundsonovershootforLTIsystems,PLDIsandNLDIs.Wenotethattheresultingboundscanbequiteconservative.TheseLMIscanbeusedtodeterminestate-feedbackmatricesaswell(seetheNotesandReferencesofChapter7).Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 98Chapter6AnalysisofLDIs:Input/OutputPropertiesBoundsonimpulseresponsepeakforLTIsystemsAtTheimpulseresponseh(t)=cebofthesingle-inputsingle-outputLTIsystemx_=Ax+bu;x(0)=0;y=cxisjusttheoutputywithinitialconditionx(0)=bandzeroinput.Therefore,wecancom-puteaboundontheimpulseresponsepeakbycombiningtheresultsoninvariantellipsoidsfromx5.2withthoseonthepeakvalueoftheoutputfromx6.2.2.Theboundobtainedthiswaycanbequiteconservative.Forinstance,considertheLTIsystemwithtransferfunctionYn1s¡siH(s)=;s+s1s+sii=2wheresi+1Àsi>0,i.e.,thedynamicsofthissystemarewidelyspaced.TheboundonthepeakvalueoftheimpulseresponseofthesystemcomputedviatheEVP(6.47)turnsouttobe2n¡1timestheactualmaximumvalueoftheimpulseresponse,wherenisthedimensionofaminimalrealizationofthesystem[Fer93].Weconjecturethattheboundcanbenomoreconservative,thatis,foreveryLTIsystem,theboundonthepeakoftheimpulseresponsecomputedviatheEVP(6.47)canbenomorethan2n¡1timestheactualmaximumvalueoftheimpulseresponse.ExamplesforwhichtheboundobtainedviatheEVP(6.47)issharpincludethecaseofpassiveTLTIsystems.Inthiscase,weknowthereexistsapositive-de¯nitePsuchthatAP+PA·0,TPb=c,sothatthemaximumvalueoftheimpulseresponseiscbandisattainedfort=0.Thisisanillustrationofthefactthatpassivesystemshavenicepeaking"properties,whichisusedinnonlinearcontrol[Kok92].Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. Chapter7State-FeedbackSynthesisforLDIs7.1StaticState-FeedbackControllersWeconsidertheLDIx_=A(t)x+Bw(t)w+Bu(t)u;z=Cz(t)x+Dzw(t)w+Dzu(t)u;"#(7.1)A(t)Bw(t)Bu(t)2•;Cz(t)Dzw(t)Dzu(t)where•hasoneofourspecialforms(i.e.,singleton,polytope,imageofaunitballunderamatrixlinear-fractionalmapping).Hereu:R!Rnuisthecontrolinput+andw:R!Rnwistheexogenousinputsignal.+LetK2Rnu£n,andsupposethatu=Kx.Sincethecontrolinputisalinearfunctionofthestate,thisiscalled(linear,constant)state-feedback,andthematrixKiscalledthestate-feedbackgain.Thisyieldstheclosed-loopLDIx_=(A(t)+Bu(t)K)x+Bw(t)w;(7.2)z=(Cz(t)+Dzu(t)K)x+Dzw(t)w:Inthischapterweconsiderthestate-feedbacksynthesisproblem,i.e.,theproblemof¯ndingamatrixKsothattheclosed-loopLDI(7.2)satis¯escertainpropertiesorspeci¯cations,e.g.,stability.Remark:Usingtheideaofgloballinearizationdescribedinx4.3,themethodsofthischaptercanbeusedtosynthesizealinearstate-feedbackforsomenonlinear,time-varyingsystems.Asanexample,wecansynthesizeastate-feedbackforthesystemx_=f(x;w;u;t);z=g(x;w;u;t);providedwehave23@f@f@f6@x@w@u745(x;w;u;t)2•;@g@g@g@x@w@uandf(0;0;0;t)=0,g(0;0;0;t)=0,forallx,t,wandu.99 100Chapter7State-FeedbackSynthesisforLDIs7.2StatePropertiesWeconsidertheLDIx_=A(t)x+Bu(t)u;[A(t)Bu(t)]2•:(7.3)²LTIsystems:ForLTIsystems,(7.3)becomesx_=Ax+Buu:(7.4)²PolytopicLDIs:PLDIsaregivenbyx_=A(t)x+Bu(t)u;[A(t)Bu(t)]2Cof[A1Bu;1];::::[ALBu;L]g:(7.5)²Norm-boundLDIs:ForNLDIs,(7.3)becomesx_=Ax+Buu+Bpp;q=Cqx+Dquu+Dqpp(7.6)p=¢(t)q;k¢(t)k·1:Equivalently,wehavex_=Ax+Bu+Bp;q=Cx+Dp+Du;pTp·qTq:upqqpqu²DiagonalNorm-boundLDIs:ForDNLDIs,equation(7.3)becomesx_=Ax+Buu+Bpp;q=Cqx+Dqpp+Dquu;(7.7)pi=±i(t)qi;j±i(t)j·1;i=1;:::;L:Equivalently,wehavex_=Ax+Buu+Bpp;q=Cqx+Dqpp+Dquu;jpij·jqij;i=1;:::;nq:7.2.1QuadraticstabilizabilityThesystem(7.1)issaidtobequadraticallystabilizable(vialinearstate-feedback)ifthereexistsastate-feedbackgainKsuchthattheclosed-loopsystem(7.2)isquadrat-icallystable(hence,stable).QuadraticstabilizabilitycanbeexpressedasanLMIP.²LTIsystems:Letus¯xthematrixK.TheLTIsystem(7.4)is(quadratically)stableifandonlyifthereexistsP>0suchthat(A+BK)TP+P(A+BK)<0;uuorequivalently,thereexistsQ>0suchthatQ(A+BK)T+(A+BK)Q<0:(7.8)uuNeitheroftheseconditionsisjointlyconvexinKandPorQ,butbyasimplechangeofvariableswecanobtainanequivalentconditionthatisanLMI.De¯neY=KQ,sothatforQ>0wehaveK=YQ¡1.Substitutinginto(7.8)yieldsAQ+QAT+BY+YTBT<0;(7.9)uuCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 7.2StateProperties101whichisanLMIinQandY.Thus,thesystem(7.4)isquadraticallystabilizableifandonlyifthereexistQ>0andYsuchthattheLMI(7.9)holds.IfthisLMIisfeasible,thenthequadraticfunctionV(»)=»TQ¡1»proves(quadratic)stabilityofsystem(7.4)withstate-feedbacku=YQ¡1x.Remark:WewillusethesimplechangeofvariablesY=KQmanytimesinthesequel.Itallowsustorecasttheproblemof¯ndingastate-feedbackgainasanLMIPwithvariablesthatincludeQandY;thestate-feedbackgainKis¡1recoveredasK=YQ.Analternateequivalentconditionfor(quadratic)stabilizability,involvingfewervariables,canbederivedusingtheeliminationofmatrixvariablesdescribedinx2.6.2:ThereexistQ>0andascalar¾suchthatAQ+QAT¡¾BBT<0:(7.10)uuSincewecanalwaysassume¾>0inLMI(7.10),andsincetheLMIishomogeneousinQand¾,wecanwithoutlossofgeneralitytake¾=1,thusreducingthenumberofvariablesbyone.IfQ>0satis¯estheLMI(7.10),astabilizingstate-feedbackgainisgivenbyK=¡(¾=2)BTQ¡1.uFromtheeliminationprocedureofx2.6.2,anotherequivalentconditionis¡¢B~TAQ+QATB~<0;(7.11)uuwhereB~uisanorthogonalcomplementofBu.ForanyQ>0satisfying(7.11),astabilizingstate-feedbackgainisK=¡(¾=2)BTQ¡1,where¾isanyscalarsuchuthat(7.10)holds(condition(7.11)impliesthatsuchascalarexists).ForLTIsystems,theseLMIconditionsarenecessaryandsu±cientforstabiliz-ability.Intermsoflinearsystemtheory,stabilizabilityisequivalenttotheconditionthateveryunstablemodebecontrollable;itisnothardtoshowthatthisisequivalenttofeasibilityoftheLMIs(7.9),(7.11),or(7.10).²PolytopicLDIs:ForthePLDI(7.5),thesameargumentapplies.QuadraticstabilizabilityisequivalenttotheexistenceofQ>0andYwithQAT+AQ+BY+YTBT<0;i=1;:::;L:(7.12)iiu;iu;i²Norm-boundLDIs:Withu=Kx,theNLDI(7.6)isquadraticallystable(seeLMI(5.14))ifthereexistQ>0and¹>0suchthat2Ã!3AQ+QAT+BKQu¹BDT+Q(C+DK)T6TTTpqpqqu74+QKBu+¹BpBp5<0:¹DBT+(C+DK)Q¡¹(I¡DDT)qppqquqpqpThisconditionhasasimplefrequency-domaininterpretation:TheH1normofthetransfermatrixfromptoqfortheLTIsystemx_=(A+BuK)x+Bpp,q=(Cq+DquK)x+Dqppislessthanone.WithY=KQ,weconcludethattheNLDIisquadraticallystabilizableifthereexistQ>0,¹>0andYsuchthat2Ã!3AQ+QAT+¹BBTpp¹BDT+QCT+YTDT6TTpqpqqu74+BuY+YBu5<0:(7.13)¹DBT+CQ+DY¡¹(I¡DDT)qppqquqpqpThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 102Chapter7State-FeedbackSynthesisforLDIsUsingtheeliminationprocedureofx2.6.2,weobtaintheequivalentconditions:2Ã!3AQ+QATQCT+¹BDT¡¾BDT66TTqpqpuqu774¡¾BuBu+¹BpBp5<0;(7.14)CQ+¹DBT¡¾DBT¡¹(I¡DDT)¡¾DDTqqppquuqpqpququ¡¹(I¡DDT)<0:qpqpThelatterconditionisalwayssatis¯edsincetheNLDIiswell-posed.Byhomogeneity,wecanset¹=1.IfQ>0and¾>0satisfy(7.14)with¹=1,astabilizingstate-feedbackgainisK=(¡¾=2)BTQ¡1.uAnalternate,equivalentconditionforquadraticstabilizabilityisexpressedintermsoftheorthogonalcomplementof[BTDT]T,whichwedenotebyG~:uqu"#AQ+QAT+¹BBTQCT+¹BDTG~TppqpqpG~<0:(7.15)CQ+¹DBT¡¹(I¡DDT)qqppqpqp(Again,wecanfreelyset¹=1inthisLMI.)IntheremainderofthischapterwewillassumethatDqpin(7.6)iszeroinordertosimplifythediscussion;allthefollowingresultscanbeextendedtothecaseinwhichDqpisnonzero.²DiagonalNorm-boundLDIs:Withu=Kx,theDNLDI(7.7)isquadraticallystable(seeLMI(5.16))ifthereexistQ>0,M=diag(¹1;:::;¹nq)>0satisfying2Ã!3AQ+QAT+BKQuBMDT+Q(C+DK)T6TTTpqpqqu74+QKBu+BpMBp5<0:DMBT+(C+DK)Q¡(M¡DMDT)qppqquqpqpThisconditionleadstoasimplefrequency-domaininterpretationforquadraticstabi-lizabilityofDNLDIs:LetHdenotethetransfermatrixfromptoqfortheLTIsystemx_=(A+BuK)x+Bpp,q=(Cq+DquK)x+Dqpp.ThentheDNLDIisquadraticallystabilizableifthereexistsKsuchthatforsomediagonalpositive-de¯nitematrixM,kM¡1=2HM1=2k·1.Thereisnoanalyticalmethodforcheckingthiscondition.1WithY=KQ,weconcludethattheNLDIisquadraticallystabilizableifthereexistQ>0,M>0anddiagonal,andYsuchthat2Ã!3AQ+QAT+BMBTppBMDT+QCT+YTDT6TTpqpqqu74+BuY+YBu5<0:DMBT+CQ+DY¡(M¡DMDT)qppqquqpqpAsbefore,wecaneliminatethevariableYusingtheeliminationprocedureofx2.6.2,andobtainequivalentLMIsinfewervariables.7.2.2HoldableellipsoidsJustasquadraticstabilitycanalsobeinterpretedintermsofinvariantellipsoids,wecaninterpretquadraticstabilizabilityintermsofholdableellipsoids.Wesaythattheellipsoid©¯ªE=»2Rn¯»TQ¡1»·1Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 7.2StateProperties103isholdableforthesystem(7.3)ifthereexistsastate-feedbackgainKsuchthatEisinvariantforthesystem(7.3)withu=Kx.ThereforetheLMIsdescribedintheprevioussectionalsocharacterizeholdableellipsoidsforthesystem(7.3).WiththisparametrizationofquadraticstabilizabilityandholdableellipsoidsasLMIsinQ>0andY,wecansolvevariousoptimizationproblems,suchas¯ndingthecoordinatetransformationthatminimizestheconditionnumberofQ,imposingnormconstraintsontheinputu=Kx,etc.Remark:InChapter5,wesawthatquadraticstabilityandinvariantellipsoids¡1werecharacterizedbyLMIsinavariableP>0andalsoitsinverseQ=P.Incontrast,quadraticstabilizabilityandellipsoidholdabilitycanbeexpressedasLMIsonlyinvariablesQandY;itisnotpossibletorewritetheseLMIsas¡1LMIswithQasavariable.ThisrestrictstheextensionofsomeoftheresultsfromChapter5(andChapter6)tostate-feedbacksynthesis.Asaruleofthumb,resultsintheanalysisofLDIsthatareexpressedasLMIsinthevariableQcanbeextendedtostate-feedbacksynthesis,withafewexceptions;resultsexpressedasLMIsinParenot.Forexample,wewillnotbeabletocomputetheminimumvolumeholdableellipsoidthatcontainsagivenpolytopeP(seeproblem(5.33))asanoptimizationproblemoverLMIs;however,wewillbeabletocomputetheminimumdiameterholdableellipsoid(seeproblem(5.34))containingP.7.2.3ConstraintsonthecontrolinputWhentheinitialconditionisknown,wecan¯ndanupperboundonthenormofthecontrolinputu(t)=Kx(t)asfollows.PickQ>0andYwhichsatisfythequadraticstabilizabilitycondition(either(7.9),(7.12)or(7.13)),andinadditionx(0)TQ¡1x(0)·1.Thisimpliesthatx(t)belongstoEforallt¸0,andconse-quently,maxku(t)k=maxkYQ¡1x(t)kt¸0t¸0·maxkYQ¡1xkx2E=¸(Q¡1=2YTYQ¡1=2):maxTherefore,theconstraintku(t)k·¹isenforcedatalltimest¸0iftheLMIs"#"#1x(0)TQYT¸0;¸0(7.16)x(0)QY¹2Ihold,whereQ>0andYsatisfythestabilizabilityconditions(either(7.9),(7.12)or(7.13)).Wecanextendthistothecasewherex(0)liesinanellipsoidorpolytope.Forexamplesupposewerequire(7.16)toholdforallkx(0k·1.Thisiseasilyshowntobeequivalentto"#QYTQ¸I;¸0:Y¹2IAsanotherextensionwecanhandleconstraintson¢ku(t)kmax=maxjui(t)jiThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 104Chapter7State-FeedbackSynthesisforLDIsinasimilarmanner:maxku(t)k=maxkYQ¡1x(t)kmaxt¸0maxt¸0·maxkYQ¡1xkx2Emax¡¢=maxYQ¡1YT:iiiTherefore,theconstraintku(t)kmax·¹fort¸0isimpliedbytheLMI"#"#1x(0)TXY¸0;¸0;X·¹2;iix(0)QYTQwhereonceagain,Q>0andYsatisfythestabilizabilityconditions(either(7.9),(7.12)or(7.13)).7.3Input-to-StatePropertiesWenextconsidertheLDIx_=A(t)x+Bw(t)w+Bu(t)u:(7.17)²LTIsystems:ForLTIsystems,equation(7.17)becomesx_=Ax+Bww+Buu.²PolytopicLDIs:PLDIsaregivenbyx_=A(t)x+Bw(t)w+Bu(t)u,where[A(t)Bw(t)Bu(t)]2Cof[A1Bw;1Bu;1];:::;[ALBw;LBu;L]g.²Norm-boundLDIs:ForNLDIs,equation(7.17)becomesx_=Ax+Buu+Bpp+Bww;q=Cqx+Dquup=¢(t)q;k¢(t)k·1whichwecanalsoexpressasx_=Ax+Bu+Bp+Bw;q=Cx+Du;pTp·qTq:upwqqu²DiagonalNorm-boundLDIs:ForDNLDIs,equation(7.17)becomesx_=Ax+Buu+Bpp+Bww;q=Cqx+Dquupi=±i(t)qi;j±i(t)j·1;i=1;:::;L:Equivalently,wehavex_=Ax+Buu+Bpp+Bww;q=Cqx+Dquu;jpij·jqij;i=1;:::;nq:7.3.1Reachablesetswithunit-energyinputsForthesystem(7.17)withu=Kx,thesetofstatesreachablewithunitenergyisde¯nedas8¯9><¯¯x,w,usatisfy(7.17);u=Kx;x(0)=0;>=¢¯ZTRue=x(T)¯:>:¯wTwdt·1;T¸0>;¯0Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 7.3Input-to-StateProperties105Wewillnowderiveconditionsunderwhichthere¯existsastate-feedbackgainKguar-anteeingthatagivenellipsoidE=f»2Rn¯»TQ¡1»·1gcontainsR.ue²LTIsystems:Fromx6.1.1,theellipsoidEcontainsRueforthesystem(7.4)forsomestate-feedbackgainKifKsatis¯esAQ+QAT+BKQ+QKTBT+BBT·0:uuwwSettingKQ=Y,weconcludethatE¶RueforsomeKifthereexistYsuchthatQAT+AQ+BY+YTBT+BBT·0:uuwwForanyQ>0andYsatisfyingthisLMI,thestate-feedbackgainK=YQ¡1issuchthattheellipsoidEcontainsthereachablesetfortheclosed-loopsystem.Usingtheeliminationprocedureofx2.6.2,weeliminatethevariableYtoobtaintheLMIinQ>0andthevariable¾:QAT+AQ¡¾BBT+BBT·0:(7.18)uuwwForanyQ>0and¾satisfyingthisLMI,acorrespondingstate-feedbackgainisgivenbyK=¡(¾=2)BTQ¡1.AnotherequivalentconditionforEtocontainRisuue¡¢B~TAQ+QAT+BBTB~·0;(7.19)uwwuwhereB~uisanorthogonalcomplementofBu.²PolytopicLDIs:ForPLDIs,EcontainsRueforsomestate-feedbackgainKifthereexistQ>0andYsuchthatthefollowingLMIholds(seeLMI(6.9)):QAT+AQ+BY+YTBT+BBT<0;i=1;:::;L:iiu;iu;iw;iw;i²Norm-boundLDIs:ForNLDIs,EcontainsRueforsomestate-feedbackgainKifthereexist¹>0andYsuchthatthefollowingLMIholds(seeLMI(6.11)):"#QAT+AQ+BY+YTBT+BBT+¹BBT(CQ+DY)Tuuwwppqqu·0:CqQ+DquY¡¹IWecaneliminateYtoobtaintheLMIinQ>0and¾thatguaranteesthatEcontainsthereachablesetRueforsomestate-feedbackgainK:2Ã!3AQ+QAT¡¾BBTuuQCT+¹BDT¡¾BDT6TTqpqpuqu74+¹BpBp+BwBw5<0:(7.20)CQ+¹DBT¡¾DBT¡¹(I¡DDT)¡¾DDTqqppquuqpqpququThecorrespondingstate-feedbackgainisK=(¡¾=2)BTQ¡1.uAnequivalentconditionisexpressedintermsofG~,theorthogonalcomplementof[BTDT]T:uqu2Ã!3AQ+QATQCT+¹BDTG~T64+¹BBT+BBTqpqp75G~<0:ppwwCQ+¹DBT¡¹(I¡DDT)qqppqpqpThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 106Chapter7State-FeedbackSynthesisforLDIs²DiagonalNorm-boundLDIs:ForDNLDIs,EcontainsRueforsomestate-feedbackgainKifthereexistQ>0,M>0anddiagonal,andYsuchthatthefollowingLMIholds(seeLMI(6.12)):"#QAT+AQ+BY+YTBT+BBT+BMBT(CQ+DY)Tuuwwppqqu·0:CqQ+DquY¡MUsingtheeliminationprocedureofx2.6.2,wecaneliminatethevariableYtoobtainanequivalentLMIinfewervariables.UsingtheseLMIs(seex6.1.1fordetails):²Wecan¯ndastate-feedbackgainKsuchthatagivenpointx0liesoutsidethesetofreachablestatesforthesystem(7.3).²Wecan¯ndastate-feedbackgainKsuchthatthesetofreachablestatesforthesystem(7.3)liesinagivenhalf-space.Thisresultcanbeextendedtocheckifthereachablesetiscontainedinapolytope.Inthiscase,incontrastwiththeresultsinChapter6,wemustusethesameouterellipsoidalapproximationtocheckdi®erentfaces.(ThisisduetothecouplinginducedbythenewvariableY=KQ.)7.3.2Reachablesetswithcomponentwiseunit-energyinputsWithu=Kx,thesetofreachablestatesforinputswithcomponentwiseunit-energyforthesystem(7.17)isde¯nedas8¯9><¯¯x,w,usatisfy(7.17);u=Kx;x(0)=0;>=¢¯ZTRuce=x(T)¯:>:¯wTwdt·1;T¸0>;¯ii0Wenowconsider¯theexistenceofthestate-feedbackgainKsuchthattheellipsoidE=f»2Rn¯»TQ¡1»·1gcontainsRforLTIsystemsandLDIs.uce²LTIsystems:FromLMI(6.15),E¶RucefortheLTIsystem(7.4)ifthereexistsdiagonalR>0withunittracesuchthattheLMI"#QAT+AQ+BY+YTBTBuuw·0:BT¡Rw²PolytopicLDIs:ForPLDIs,E¶Ruceforsomestate-feedbackgainKifthereexistQ>0,Y(=KQ),andadiagonalR>0withunittrace(seeLMI(6.17)),suchthat"#QAT+AQ+BY+YTBTBiiu;iu;iw;i<0;i=1;:::;L:BT¡Rw;i²Norm-boundLDIs:ForNLDIs,E¶Ruceforsomestate-feedbackgainKifthereexistQ>0,¹>0,YandadiagonalR>0withunittrace(seeLMI(6.19)),suchthat23QAT+YTBT+AQ+BY+¹BBTQCT+YTDTBuuppqquw674CqQ+DquY¡¹I05·0:BT0¡RwCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 7.4State-to-OutputProperties107Remark:ForLTIsystemsandNLDIs,itispossibletoeliminatethevariableY.7.3.3Reachablesetswithunit-peakinputsFora¯xedstate-feedbackgainK,thesetofstatesreachablewithinputswithunit-peakisde¯nedas(¯)¯¢¯x,w,usatisfy(7.17);u=Kx;x(0)=0;Rup=x(T)¯:¯w(t)Tw(t)·1;T¸0²LTIsystems:Fromcondition(6.25),EcontainsRucefortheLTIsystem(7.4)ifthereexists®>0suchthatAQ+QAT+BY+YTBT+BBT=®+®Q·0;uuwwwhereY=KQ.²PolytopicLDIs:ForPLDIs,E¶RuceifthereexistQ>0andY(seecondi-tion(6.28))suchthatAQ+QAT+BY+YTBT+®Q+BBT=®·0iiu;iu;iw;iw;ifori=1;:::;L.²Norm-boundLDIs:FromChapter6,EcontainsRuceifthereexistQ>0,Y,®>0and¹>0suchthat2Ã!3AQ+QAT+®Q(CQ+DY)T6TTTqqu74+BuY+YBu+¹BpBp+BwBw=®5·0:CqQ+DquY¡¹IThisconditionisanLMIfor¯xed®.Remark:Again,forLTIsystemsandNLDIs,itispossibletoeliminatethevariableY.7.4State-to-OutputPropertiesWeconsiderstate-feedbackdesigntoachievecertaindesirablestate-to-outputprop-ertiesfortheLDI(4.5).Wesettheexogenousinputwtozeroin(4.5),andconsiderx_=A(t)x+Bu(t)u;z=Cz(t)x:²LTIsystems:ForLTIsystems,thestateequationsarex_=Ax+Buu;z=Czx+Dzuu:(7.21)²PolytopicLDIs:ForPLDIswehave"#("#"#)A(t)Bu(t)A1Bu;1ALBu;L2Co;:::;:Cz(t)Dzu(t)Cz;1Dzu;1Cz;LDzu;LThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 108Chapter7State-FeedbackSynthesisforLDIs²Norm-boundLDIs:NLDIsaregivenbyx_=Ax+Buu+Bpp;z=Czx+Dzuuq=Cqx+Dquup=¢(t)q;k¢(t)k·1;whichcanberewrittenasx_=Ax+Buu+Bpp;z=Czx+Dzuuq=Cx+Du;pTp·qTq:qqu²DiagonalNorm-boundLDIs:Finally,DNLDIsaregivenbyx_=Ax+Buu+Bpp;z=Czx+Dzuu;q=Cqx+Dquu;pi=±i(t)qi;j±i(t)j·1;i=1;:::;L:whichcanberewrittenasx_=Ax+Buu+Bpp;z=Czx+Dzuu;q=Cqx+Dquu;jpij·jqij;i=1;:::;nq:7.4.1BoundsonoutputenergyWe¯rstshowhowto¯ndastate-feedbackgainKsuchthattheoutputenergy(asde¯nedinx6.2.1)oftheclosed-loopsystemislessthansomespeci¯edvalue.Weassume¯rstthattheinitialconditionx0isgiven.²LTIsystems:WeconcludefromLMI(6.39)thatforagivenstate-feedbackgainK,theoutputenergyofsystem(7.21)doesnotexceedxTQ¡1x,whereQ>0and00Y=KQsatisfy"#AQ+QAT+BY+YTBT(CQ+DY)Tuuzzu·0;(7.22)CzQ+DzuY¡IRegardingYasavariable,wecanthen¯ndastate-feedbackgainthatguaranteesanoutputenergylessthan°bysolvingtheLMIPxTQ¡1x·°and(7.22).00Ofcourse,inequality(7.22)iscloselyrelatedtotheclassicalLinear-QuadraticRegulator(LQR)problem;seetheNotesandReferences.²PolytopicLDIs:Inthiscase,theoutputenergyisboundedabovebyxTQ¡1x,00whereQsatis¯estheLMI2Ã!3AQ+QATiiT6(Cz;iQ+Dzu;iY)76+BY+YTBT7·0;i=1;:::;L4u;iu;i5Cz;iQ+Dzu;iY¡IforsomeY.²Norm-boundLDIs:InthecaseofNLDIs,theoutputenergyisboundedabovebyxTQ¡1x,foranyQ>0,Yand¹¸0suchthat002Ã!3AQ+QAT+BYu(CQ+DY)T(CQ+DY)T6zzuqqu76+YTBT+¹BBT76upp7·0:(7.23)674CzQ+DzuY¡I05CqQ+DquY0¡¹ICopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 7.5Input-to-OutputProperties109Remark:Asbefore,wecaneliminatethevariableYfromthisLMI.²DiagonalNorm-boundLDIs:Finally,inthecaseofDNLDIs,theoutputenergyisboundedabovebyxTQ¡1x,foranyQ>0,YandM¸0anddiagonalsuchthat002Ã!3AQ+QAT+BYu(CQ+DY)T(CQ+DY)T6zzuqqu76+YTBT+BMBT76upp7·0:674CzQ+DzuY¡I05CqQ+DquY0¡MGivenaninitialcondition,¯ndingastate-feedbackgainKsoastominimizetheupperboundontheextractableenergyforthevariousLDIsisthereforeanEVP.Wecanextendtheseresultstothecasewhenx0isspeci¯edtolieinapolytopeoranellipsoid(seex6.2.1).IfxisarandomvariablewithExxT=X,EVPsthatyield0000state-feedbackgainsthatminimizetheexpectedvalueoftheoutputenergycanbederived.7.5Input-to-OutputPropertiesWe¯nallyconsidertheproblemof¯ndingastate-feedbackgainKsoastoachievede-siredpropertiesbetweentheexogenousinputwandtheoutputzforthesystem(7.1).AsmentionedinChapter6,thelistofproblemsconsideredhereisfarfromexhaustive.7.5.1L2andRMSgainsWeseekastate-feedbackgainKsuchthattheL2gainkzk2supkzk2=supkwk2=1kwk26=0kwk2oftheclosed-loopsystemislessthanaspeci¯ednumber°.²LTIsystems:Asseenin6.3.2,theL2gainforLTIsystemsisequaltotheH1normofthecorrespondingtransfermatrix.Fromx6.3.2,thereexistsastate-feedbackgainKsuchthattheL2gainofanLTIsystemislessthan°,ifthereexistKandQ>0suchthat,2Ã!3(A+BK)Q+Q(A+BK)TuuT6TQ(Cz+DzuK)74+BwBw5·0:(7.24)(C+DK)Q¡°2IzzuIntroducingY=KQ,thiscanberewrittenas"#AQ+QAT+BY+YTBT+BBT(CQ+DY)Tuuwwzzu·0:(7.25)CQ+DY¡°2IzzuInthecaseofLTIsystems,thisconditionisnecessaryandsu±cient.AssumingCTD=0andDTDinvertible,itispossibletosimplifytheinequality(7.25)zzuzuzuandgettheequivalentRiccatiinequalityAQ+QAT¡B(DTD)¡1BT+BBT+QCTCQ=°2·0:(7.26)uzuzuuwwzzThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 110Chapter7State-FeedbackSynthesisforLDIsThecorrespondingRiccatiequationisreadilysolvedviaHamiltonianmatrices.Theresultingcontroller,withstate-feedbackgainK=¡(DTD)¡1BT,yieldsaclosed-zuzuuloopsystemwithH1-normlessthan°.²PolytopicLDIs:Fromx6.3.2,thereexistsastate-feedbackgainsuchthattheL2gainofaPLDIislessthan°ifthereexistYandQ>0suchthat2Ã!3AQ+QAT+BYiiu;iT6TTT(Cz;iQ+Dzu;iY)74+YBu;i+Bw;iBw;i5·0:(7.27)CQ+DY¡°2Iz;izu;i²Norm-boundLDIs:ForNLDIs,theLMIthatguaranteesthattheL2gainislessthan°forsomestate-feedbackgainKis2013AQ+QAT66BTTCT(CQ+DY)T776@+BuY+YBuA(CzQ+DzuY)qqu76TT766+BwBw+¹BpBp77·0:(7.28)6CQ+DY¡°2I076zzu745CqQ+DquY0¡¹IWecantherefore¯ndstate-feedbackgainsthatminimizetheupperboundontheL2gain,provablewithquadraticLyapunovfunctions,forthevariousLDIsbysolvingEVPs.7.5.2DissipativityWenextseekastate-feedbackgainKsuchthattheclosed-loopsystem(7.2)ispassive;moregenerally,assumingDzw(t)isnonzeroandsquare,wewishtomaximizethedissipativity,i.e.,´satisfyingZT¡¢wTz¡´wTwdt¸00forallT¸0.²LTIsystems:SubstitutingY=KQ,andusingLMI(6.60),weconcludethatthedissipationoftheLTIsystemisatleast´iftheLMIin´,Q>0andYholds:"#AQ+QAT+BY+YTBTB¡QCT¡YTDTuuwzzu·0:BT¡CQ¡DY2´I¡(D+DT)wzzuzwzw²PolytopicLDIs:From(6.62),thereexistsastate-feedbackgainsuchthatthedissipationexceeds´ifthereexistQ>0andYsuchthat2Ã!3AQ+QAT6iiB¡QCT¡YTD76TTw;iz;izu;i74+Bu;iY+YBu;i5·0:BT¡CQ¡DY2´I¡(D+DT)w;iz;izu;izw;izw;iCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 7.6Observer-BasedControllersforNonlinearSystems111²Norm-boundLDIs:From(6.64),thereexistsastate-feedbackgainsuchthatthedissipationexceeds´ifthereexistQ>0andYsuchthat2Ã!Ã!3AQ+QAT+BYB¡QCTuwzT6(CqQ+DquY)76+YTBT+¹BBT¡YTDT76Ãuq!qzu76T766Bw¡CzQT77·06¡(Dzw+Dzw¡2´I)074¡DzuY5CqQ+DquY0¡¹IWecantherefore¯ndstate-feedbackgainsthatmaximizethelowerboundonthedissipativity,provablewithquadraticLyapunovfunctions,forthevariousLDIsbysolvingEVPs.Remark:Aswithx6.3.4,wecanincorporatescalingtechniquesintomanyoftheresultsabovetoderivecomponentwiseresults.SincethenewLMIsthusobtainedarestraightforwardtoderive,wewillomitthemhere.7.5.3Dynamicversusstaticstate-feedbackcontrollersAdynamicstate-feedbackcontrollerhastheformx¹_=A¹x¹+B¹yx;u=C¹ux¹+D¹uyx;(7.29)whereA¹2Rr£r,B¹2Rr£n,C¹2Rnu£r,D¹2Rnu£n.Thenumberriscalledyuuytheorderofthecontroller.Notethatbytakingr=0,thisdynamicstate-feedbackcontrollerreducestothestate-feedbackwehaveconsideredsofar(whichiscalledstaticstate-feedbackinthiscontext).Itmightseemthatthedynamicstate-feedbackcontroller(7.29)allowsustomeetmorespeci¯cationsthancanbemetusingastaticstate-feedback.Forspeci¯cationsbasedonquadraticLyapunovfunctions,however,thisisnotthecase.Forexam-ple,supposetheredoesnotexistastaticstate-feedbackgainthatyieldsclosed-loopquadraticstability.Inthiscasewemightturntothemoregeneraldynamicstate-feedback.WecouldworkoutthemorecomplicatedLMIsthatcharacterizequadraticstabilizabilitywithdynamicstate-feedback.Wewould¯nd,however,thatthesemoregeneralLMIsarealsoinfeasible.SeetheNotesandReferences.7.6Observer-BasedControllersforNonlinearSystemsWeconsiderthenonlinearsystemx_=f(x)+Buu;y=Cyx;(7.30)npwherex:R+!Risthestatevariable,u:R+!Risthecontrolvariable,qandy:R+!Risthemeasuredorsensedvariable.Weassumethefunctionnnf:R!Rsatis¯esf(0)=0and@f2CofA1;:::;AMg@xwhereA1;:::;ALaregiven,whichisthesameas(4.15).Welookforastabilizingobserver-basedcontrolleroftheformx¹_=f(x¹)+Buu+L(Cyx¹¡y);u=Kx;¹(7.31)Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 112Chapter7State-FeedbackSynthesisforLDIsi.e.,wesearchforthematricesK(theestimated-statefeedbackgain)andL(theobservergain)suchthattheclosed-loopsystem"#"#x_f(x)+BuKx¹=(7.32)x¹_¡LCyx+f(x¹)+(BuK+LCy)x¹isstable.Theclosed-loopsystem(7.32)isstableifitisquadraticallystable,whichis2n£2ntrueifthereexistsapositive-de¯nitematrixP~2Rsuchthatforanynonzerotrajectoryx,x¹,wehave:"#T"#dxxP~<0:dtx¹x¹IntheNotesandReferences,weprovethatthisistrueifthereexistP,Q,Y,andWsuchthattheLMIsQ>0;AQ+QAT+BY+YTBT<0;i=1;:::;M(7.33)iiuuandP>0;ATP+PA+WC+CTWT<0;i=1;:::;M(7.34)iiyyhold.ToeveryP,Q,Y,andWsatisfyingtheseLMIs,therecorrespondsastabilizingobserver-basedcontrolleroftheform(7.31),obtainedbysettingK=YQ¡1andL=P¡1W.Usingtheeliminationprocedureofx2.6.2,wecanobtainequivalentconditionsinwhichthevariablesYandWdonotappear.TheseconditionsarethatsomeP>0andQ>0satisfyAQ+QAT<¾BBT;i=1;:::;M(7.35)iiuuandATP+PA<¹CTC;i=1;:::;M(7.36)iiyyforsome¾and¹.Byhomogeneitywecanfreelyset¾=¹=1.ForanyP>0,Q>0satisfyingtheseLMIswith¾=¹=1,wecanobtainastabilizingobserver-basedcontrolleroftheform(7.31),bysettingK=¡(1=2)BTQ¡1andL=¡(1=2)P¡1C.uyAnotherequivalentconditionisthat¡¢B~TAQ+QATB~<0;i=1;:::;Muiiuand¡¢C~ATP+PAC~T<0;i=1;:::;MyiiyholdforsomeP>0,Q>0,whereB~andC~TareorthogonalcomplementsofBanduyuCT,respectively.IfPandQsatisfytheseLMIs,thentheysatisfytheLMIs(7.36)yand(7.35)forsome¾and¹.Theobserver-basedcontrollerwithK=¡(¾=2)BTQ¡1uandL=¡(¹=2)P¡1Cstabilizesthenonlinearsystem(7.30).yNotesandReferencesLyapunovfunctionsandstate-feedbackTheextensionofLyapunov'smethodstothestate-feedbacksynthesisproblemhasalonghistory;seee.g.,[Bar70a,Lef65].Itisclearlyrelatedtothetheoryofoptimalcontrol,inCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences113whichthenaturalLyapunovfunctionistheBellman{Pontryaginmin-cost-to-go"orvaluefunction(see[Pon61,Pon62]).Wesuspectthatthemethodsdescribedinthischaptercanbeinterpretedasasearchforsuboptimalcontrolsinwhichthecandidatevaluefunctionsarerestrictedtoaspeci¯cclass,i.e.,quadratic.Inthegeneraloptimalcontrolproblem,the(exact)valuefunctionsatis¯esapartialdi®erentialequationandishardtocompute;byrestrictingoursearchtoquadraticapproximatevaluefunctionsweendupwithalowcomplexitytask,i.e.,aconvexprobleminvolvingLMIs.SeveralapproachesuseLyapunov-likefunctionstosynthesizenonlinearstate-feedbackcontrollaws.OneexampleistheLyapunovMin-MaxController"describedin[Gut79,GP82,Cor85,CL90,Wei94].Seealso[Zin90,SC91,Rya88,GR88].QuadraticstabilizabilityThetermquadraticstabilizability"seemstohavebeencoinedbyHollotandBarmishin[HB80],wheretheauthorsgivenecessaryandsu±cientconditionsforit;seealso[Lei79,Gut79,BCL83,PB84,Cor85].Petersen[Pet85],showsthatthereexistLDIsthatarequadraticallystabilizable,thoughnotvialinearstate-feedback;seealso[SP94e].Hol-lot[Hol87]describesconditionsunderwhichaquadraticallystabilizableLDIhasin¯nitestabilizabilitymargin;seealso[SRC93].Wei[Wei90]derivesnecessaryandsu±cientsign-pattern"conditionsontheperturbationsfortheexistenceofalinearfeedbackforquadraticstabilityforaclassofsingle-inputuncertainlineardynamicalsystems.Petersen[Pet87b]derivesaRiccati-basedapproachforstabilizingNLDIs,whichisinfactthestate-feedbackH1equation(7.28)(thisisshownin[Pet87a]).¡1¡1ThechangeofvariablesQ=PandY=KP,whichenablestheextensionofmanyoftheresultsofChapters5and6tostate-feedbacksynthesis,isduetoBernussou,PeresandGeromel[BPG89a].Seealso[GPB91,BPG89b,BPG89a,PBG89,PGB93].Thischangeofvariables,thoughnotexplicitlydescribed,isusedinthepaperbyThorpandBarmish[TB81];seealso[HB80,Bar83,Bar85].Forarecentandbroadreviewaboutquadraticstabilizationofuncertainsystems,see[Cor94].In[Son83],Sontagformulatesageneraltheoremgivingnecessaryandsu±cientLyapunovtypeconditionsforstabilizability.ForLTIsystemshiscriteriareducetotheLMIsinthischapter.QuadraticLyapunovfunctionsforprovingperformanceTheadditionofperformanceconsiderationsintheanalysisofLDIscanbetracedbackto1955andevenearlier,whenLetov[Let61]studiedtheperformanceofunknown,nonlinearandpossiblytime-varyingcontrolsystemsandcallsittheproblemofcontrolquality".Atthattime,theperformancecriteriaweredecayrateandoutputpeakdeviationsforsystemssubjecttoboundedpeakinputs(see[Let61,p.234]fordetails).InthecaseofLTIsystems,addingperformanceindicesallowsonetorecovermanyclassicalresultsofautomaticcontrol.Theproblemofmaximizingdecayrateisdiscussedingreatdetailin[Yan92];otherreferencesondecayratesandquadraticstabilitymarginsare[Gu92b,+Gu92a,GPB91,Gu92b,Sko91b,PZP92,EBFB92,Cor90].Arzelieretal.[ABG93]considertheproblemofrobuststabilizationwithpoleassignmentforPLDIsviacutting-planetechniques.TheproblemofapproximatingreachablesetsforLTIsystemsundervariousassumptionsontheenergyoftheexogenousinputhasbeenconsideredforexamplebySkelton[SZ92]andreferencestherein;seealsotheNotesandReferencesofthepreviouschapter.Seealso[OC87].Theideathattheoutputvarianceforwhitenoiseinputs(andrelatedLQR-likeperformancemeasures)canbeminimizedusingLMIscanbefoundin[BH88a,BH89,RK91,KR91,KKR93,BB91,FBBE92,RK93,SI93].Peres,SouzaandGeromel[PSG92,PG93]con-sidertheextensiontoPLDIs.ThecounterpartforNLDIsisinPetersen,McFarlaneandRotea[PM92,PMR93],aswellasinStoorvogel[Sto91].Theproblemof¯ndingastate-feedbackgaintominimizethescaledL2gainforLTIsys-temsisdiscussedin[EBFB92].TheL2gainofPLDIshasbeenstudiedbyObradovicandThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 114Chapter7State-FeedbackSynthesisforLDIsValavani[OV92],byPeres,GeromelandSouza[PGS91]andalsobyOhara,MasubuchiandSuda[OMS93].NLDIshavebeenstudiedinthiscontextbyPetersen[Pet89],andDeSouzaetal.[XFdS92,XdS92,WXdS92,dSFX93],whopointoutexplicitlythatquadraticsta-bilityofNLDIswithanL2gainboundisequivalenttothescaledH1condition(6.54);seealso[ZKSN92,Gu93].ForLTIsystems,itisinterestingtocomparetheinequality(7.25)whichprovidesanecessaryandsu±cientconditionforasystemtohaveL2gainlessthan°withthecorrespondingmatrixinequalityfoundinthearticlebyStoorvogelandTrentel-man[ST90].Inparticular,thequadraticmatrixinequalityfoundthere(expressedinournotation)is"#T2TTTAP+PA+°PBwBwP+CzCzPBu+CzDzu¸0;TTTBuP+DzuCzDzuDzuwhichdoesnotpossessanyobviousconvexityproperty.WenotethattheRiccatiinequal-ity(7.26)isalsoencounteredinfull-statefeedbackH1control[Pet87a].Itsoccurrenceincontroltheoryismucholder;itappearsverbatiminsomearticlesonthetheoryofnonzero-sumdi®erentialgames;seeforexampleStarrandHo[SH68],Yakubovich[Yak70,Yak71]andMageirou[Mag76].Kokotovic[Kok92]considersstabilizationofnonlinearsystems,andprovidesamotivationforrenderinganLTIsystempassiveviastate-feedback(seex6.3.3).In[PP92],theauthorsgiveanalyticconditionsformakinganLTIsystempassiveviastate-feedback;seealso[SKS93].Finally,wementionaninstanceofaLyapunovfunctionwithabuilt-inperformancecriterion.InthebookbyAubinandCellina[AC84,x6],we¯nd:Weshallinvestigatewhetherdi®erentialinclusionsx_2F(x(t));x(0)=x0havetrajectoriessatisfyingthepropertyZt8t>s;V(x(t))¡V(x(s))+W(x(¿);x_(¿))d¿·0:(7.37)sWeshallthensaythatafunctionV[¢¢¢]satisfyingthisconditionisaLyapunovfunctionforFwithrespecttoW.Similarideascanbefoundin[CP72].WeshallencountersuchLyapunovfunctionsinx8.2.Inequality(7.37)isoftencalledadissipationinequality"[Wil71b].Theproblemofminimizingthecontrole®ortgivenaperformanceboundontheclosed-loopsystemcanbefoundinthearticlesofSchÄomig,SznaierandLy[SSL93],andGrigoriadis,Carpenter,ZhuandSkelton[GCZS93].LMIformulationofLQRproblemFortheLTIsystemx_=Ax+Buu,z=Czx+Dzuu,theLinear-QuadraticRegulator(LQR)problemRis:Givenaninitialconditionx(0),¯ndthecontrolinpututhatminimizestheoutput1zTzdt.Weassumeforsimplicitythat(A;B;C)isminimal,DTDenergyzuzuisinvertible0TandDzuCz=0.Itturnsoutthattheoptimalinputucanbeexpressedasaconstantstate-feedbacku=Kx,T¡1TwhereK=¡(DzuDzu)BuPareandPareistheuniquepositive-de¯nitematrixthatsatis¯esthealgebraicRiccatiequationTT¡1TTAPare+PareA¡PareBu(DzuDzu)BuPare+CzCz=0:(7.38)TTheoptimaloutputenergyforaninitialconditionx(0)isthengivenbyx(0)Parex(0).With¡1Qare=Pare,wecanrewrite(7.38)asTT¡1TTAQare+QareA¡Bu(DzuDzu)Bu+QareCzCzQare=0;(7.39)Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences115T¡1andtheoptimaloutputenergyasx(0)Qarex(0).Insectionx7.4.1,weconsideredthecloselyrelatedproblemof¯ndingastate-feedbackgainKthatminimizesanupperboundontheenergyoftheoutput,givenaninitialcondition.ForLTIsystems,theupperboundequalstheoutputenergy,andweshowedthatinthiscase,T¡1theminimumoutputenergywasgivenbyminimizingx(0)Qx(0)subjecttoQ>0and"#TTTTAQ+QA+BuY+YBu(CzQ+DzuY)·0:(7.40)CzQ+DzuY¡IOfcourse,theoptimalvalueofthisEVPmustequaltheoptimaloutputenergygivenviathesolutiontotheRiccatisolution(7.39);theRiccatiequationcanbethusbeinterpretedasyieldingananalyticsolutiontotheEVP.WecanderivethisanalyticsolutionfortheEVPviathefollowingsteps.First,itcanbeshown,usingasimplecompletion-of-squaresargumentT¡1TthatLMI(7.40)holdsforsomeQ>0andYifandonlyifitholdsforY=¡(DzuDzu)Bu;inthiscase,Q>0mustsatisfyTT¡T¢¡1TQA+AQ+QCzCzQ¡BuDzuDzuBu·0:(7.41)Next,itcanbeshownbystandardmanipulationsthatifQ>0satis¯es(7.41),thenQ·Qare.T¡1T¡1Thereforeforeveryinitialconditionx(0),wehavex(0)Qarex(0)·x(0)Qx(0),andT¡1thereforetheoptimalvalueoftheEVPisjustx(0)Qarex(0),andtheoptimalstate-feedback¡1T¡1TgainisKopt=YoptQare=¡(DzuDzu)BuPare.¡1TItisinterestingtonotethatwithP=Q,theEVPisequivalenttominimizingx(0)Px(0)subjecttoTT¡T¢¡1TAP+PA+CzCz¡PBuDzuDzuBuP·0;(7.42)TwhichisnotaconvexconstraintinP.However,theproblemofmaximizingx(0)Px(0)subjectP>0andtheconstraintTT¡1TTAP+PA¡PBu(DzuDzu)BuP+CzCz¸0;whichisnothingotherthan(7.42),butwiththeinequalityreversed,isanEVP;itiswell-known(seeforexample[Wil71b])thatthisEVPisanotherformulationoftheLQRproblem.Staticversusdynamicstate-feedbackAnLTIsystemcanbestabilizedusingdynamicstate-feedbackifandonlyifitcanbestabi-lizedusingstaticstate-feedback(seeforexample,[Kai80]).Infact,similarstatementscanbemadeforallthepropertiesandforalltheLDIsconsideredinthischapter,providedthepropertiesarespeci¯edusingquadraticLyapunovfunctions.Wewilldemonstratethisfactononesimpleproblemconsideredinthischapter.ConsidertheLTIsystemx_=Ax+Buu+Bww;x(0)=0;z=Czx:(7.43)Fromx7.5.1,theexistsastaticstate-feedbacku=KxsuchthattheL2gainfromwtozdoesnotexceed°iftheLMIinvariablesQ>0andYisfeasible:"#TTTTTAQ+QA+BuY+YBu+BwBwQCz·0:2CzQ¡°IThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 116Chapter7State-FeedbackSynthesisforLDIs(Infactthisconditionisalsonecessary,butthisisirrelevanttothecurrentdiscussion.)EliminatingY,wegettheequivalentLMIinQ>0andscalar¾:TTTQCzCzQTAQ+QA+BwBw+2·¾BuBu:(7.44)°Wenextconsideradynamicstate-feedbackforthesystem:x_c=Acxc+Bcx;xc(0)=0;u=Ccxc+Dcx:(7.45)WewillshowthatthereexistmatricesAc,Bc,CcandDcsuchthattheL2gainofthesystem(7.43,7.45)doesnotexceed°ifandonlyiftheLMI(7.44)holds.ThiswillestablishthatthesmallestupperboundontheL2gain,provableviaquadraticLyapunovfunctions,isthesameirrespectiveofwhetherastaticstate-feedbackoradynamicstate-feedbackisemployed.TTTWewillshowonlythepartthatisnotobvious.Withstatevectorxcl=[xxc],theclosed-loopsystemisx_cl=Aclxcl+Bclu;xcl(0)=0;z=Cclxcl;whereAcl=Abig+BbigKbigwith"#"#"#A00BuBcAcAbig=;Bbig=;Kbig=;00I0DcCc"#BwBcl=;Ccl=[Cz0]:0Fromx6.3.2,theL2gainoftheclosed-loopsystemdoesnotexceed°iftheLMIinPbig>0isfeasible:"#TTTTAbigP+PbigA+PBbigKbig+KbigBbigP+CclCclPBcl·0:T2BclP¡°IEliminatingKbigfromthisLMIyieldstwoequivalentLMIsinPbig>0andascalar¿."#TTAbigP+PbigA¡¿I+CclCclPBcl·0;T2BclP¡°I"#TTTAbigP+PbigA¡¿PBbigBbigP+CclCclPBcl·0:T2BclP¡°IItiseasytoverifythatthe¯rstLMIholdsforlargeenough¿,whilethesecondreducesto¿>0andTTBwBwTTAQ11+Q11A+2+Q11CzCzQ11·¿BuBu;°¡1whereQ11istheleadingn£nblockofP.ThelastLMIisprecisely(7.44)withthechangebig22ofvariablesQ=°Q11and¾=°¿.+Asimilarresultisestablished[PZP92];otherreferencesthatdiscussthequestionofwhendynamicornonlinearfeedbackdoesbetterthanstaticfeedbackincludePetersen,Khar-gonekarandRotea[Pet85,KPR88,RK88,KR88,Pet88,RK89].Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences117Dynamicoutput-feedbackforLDIsAnaturalextensiontotheresultsofthischapteristhesynthesisofoutput-feedbacktomeetvariousperformancespeci¯cations.Wesuspect,however,thatoutput-feedbacksynthesisproblemshavehighcomplexityandarethereforeunlikelytoberecastasLMIproblems.Severalresearchershaveconsideredsuchproblems:SteinbergandCorless[SC85],GalimidiandBarmish[GB86],andGeromel,Peres,deSouzaandSkelton[PGS93];seealso[HCM93].Onevariationofthisproblemisthereduced-ordercontrollerproblem",thatis,¯ndingdy-namicoutput-feedbackcontrollerswiththesmallestpossiblenumberofstates.Itisshownin[PZPB91,EG93,Pac94]thatthisproblemcanbereducedtotheproblemofminimizingtherankofamatrixP¸0subjecttocertainLMIs.ThegeneralproblemofminimizingtherankofamatrixsubjecttoanLMIconstrainthashighcomplexity;infact,aspecialrankminimizationproblemcanbeshowntobeequivalenttotheNP-hardzero-onelinearpro-grammingproblem(seee.g.[Dav94]).Neverthelessseveralresearchershavetriedheuristic,localalgorithmsforsuchproblemsandreportpracticalsuccess.OthersstudyingthistopicincludeIwasaki,Skelton,GeromelanddeSouza[IS93b,ISG93,GdSS93,SIG93].Observer-basedcontrollersfornonlinearsystemsWeprovethattheLMIs(7.33)and(7.34)(inthevariablesP>0,Q>0,WandY)ensuretheexistenceofanobserver-basedcontrolleroftheform(7.31)whichmakesthesystem(7.30)quadraticallystable.Westartbyrewritingthesystem(7.32)inthecoordinatesx¹,x¹¡x:"#"#x¹_LCy(x¹¡x)+f(x¹)+BuKx¹=:(7.46)x¹_¡x_f(x¹)¡f(x)+LCy(x¹¡x)Usingtheresultsofx4.3,wecanwrite"#"#x¹_©¯ªx¹2CoA~ij¯1·i·M;1·j·Mx¹_¡x_x¹¡xwith"#Ai+BuKLCyA~ij=;i;j=1;:::;M:0Aj+LCySimplestate-spacemanipulationsshowthesystem(7.32)isquadraticallystableifandonlyifthesystem(7.46)isquadraticallystable.Therefore,thesystem(7.32)isquadraticallystableifthereexistsP~>0suchthatA~TP~+P~A~ijij<0;i;j=1;:::;M:(7.47)WewillnowshowthattheLMI(7.47)hasasolutionifandonlyiftheLMIs(7.33)and(7.34)do.Firstsupposethat(7.47)hasasolutionP~.WithP~partitionedasn£nblocks,"#P11P12P~=;TP12P22weeasilycheckthattheinequality(7.47)impliesT(Ai+BuK)P11+P11(Ai+BuK)<0;i=1;:::;M:(7.48)Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 118Chapter7State-FeedbackSynthesisforLDIs¡1WiththechangeofvariablesQ=P11andY=BuQ,theinequality(7.48)isequivalentto(7.33).¡1De¯neQ~=P~.Theinequality(7.47)becomesA~Q~+Q~A~T<0;i;j=1;:::;M:ijijDenotingbyQ22thelower-rightn£nblockofQ~,weseethatthisinequalityimpliesT(Ai+LCy)Q22+Q22(Ai+LCy)<0;i=1;:::;M:(7.49)¡1WithP=Q22andW=PL,theinequality(7.49)isequivalentto(7.34).Conversely,supposethatP,Q,YandWsatisfytheLMIs(7.34)and(7.33)andde¯ne¡1¡1L=PW,K=YQ.Wenowprovethereexistsapositive¸suchthat"#¡1¸Q0P~=0Psatis¯es(7.47)andthereforeprovestheclosed-loopsystem(7.32)isquadraticallystable.Wecompute2Ã!3T¡1(Ai+BuK)Q¡16¸¡1¸QLCy7A~TP~+P~A~66+Q(Ai+BuK)77;ijij=4(Aj+LCy)TP5T¡1¸(LCy)Q+P(Aj+LCy)fori;j=1;:::;M.Therefore,usingSchurcomplements,theclosed-loopsystem(7.32)isquadraticallystableif¸>0satis¯es¡1T¡1T¡1¸(QLCy((Aj+LCy)P+P(Aj+LCy))(LCy)Q)T¡1¡1¡(A+BuK)Q¡Q(A+BuK)>0fori;j=1;:::;M.Thisconditionissatis¯edforany¸>0suchthat¸min¹j>maxºi;1·j·M1·i·Mwhere¡1T¡1T¡1¹j=¸min(QLCy((Aj+LCy)P+P(Aj+LCy))(LCy)Q);j=1;:::;MandT¡1¡1ºi=¸max((Ai+BuK)Q+Q(Ai+BuK));i=1;:::;M:Since(7.34)and(7.33)aresatis¯ed,sucha¸alwaysexists.Problemsofobserver-basedcontrollerdesignalongwithquadraticLyapunovfunctionsthatprovestabilityhavebeenconsideredbyBernsteinandHaddad[BH93]andYaz[Yaz93].Gain-scheduledorparameter-dependentcontrollersSeveralresearchershaveextendedtheresultsonstate-feedbacksynthesistohandlethecaseinwhichthecontrollerparameterscandependonsystemparameters.Suchcontrollersarecalledgain-scheduledorparameter-dependent.Wereferthereadertothearticlescitedforpreciseexplanationsofwhatthesecontrollawsare,andwhichdesignproblemscanberecastasLMIproblems.Lu,ZhouandDoylein[LZD91]andBeckerandPackard[Bec93,BP91]considertheprobleminthecontextofNLDIs.Otherrelevantreferencesinclude[PB92,Pac94,PBPB93,BPPB93,GA94,IS93a,IS93b,AGB94].Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. Chapter8Lur'eandMultiplierMethods8.1AnalysisofLur'eSystemsWeconsidertheLur'esystemx_=Ax+Bpp+Bww;q=Cqx;(8.1)z=Czx;pi(t)=Ái(qi(t));i=1;:::;np;wherep(t)2Rnp,andthefunctionsÁsatisfythe[0;1]sectorconditionsi0·¾Á(¾)·¾2forall¾2R;(8.2)ior,equivalently,Ái(¾)(Ái(¾)¡¾)·0forall¾2R:ThedatainthisproblemarethematricesA,Bp,Bw,CqandCz.TheresultsthatfollowwillholdforanynonlinearitiesÁisatisfyingthesectorconditions.IncaseswheretheÁiareknown,however,theresultscanbesharpened.Itispossibletohandlethemoregeneralsectorconditions®¾2·¾Á(¾)·¯¾2forall¾2R;iiiwhere®iand¯iaregiven.Suchsystemsarereadilytransformedtotheformgivenin(8.1)and(8.2)byalooptransformation(seetheNotesandReferences).Animportantspecialcaseof(8.1),(8.2)occurswhenthefunctionsÁiarelinear,i.e.,Ái(¾)=±i¾,where±i2[0;1].Intheterminologyofcontroltheory,thisisreferredtoasasystemwithunknown-but-constantparameters.ItisimportanttodistinguishthiscasefromthePLDIobtainedwith(8.1),Ái(¾)=±i(t)¾and±i(t)2[0;1],whichisreferredtoasasystemwithunknown,time-varyingparameters.OuranalysiswillbebasedonLyapunovfunctionsoftheformXnpZCi;q»V(»)=»TP»+2¸Á(¾)d¾;(8.3)iii=10whereCi;qdenotestheithrowofCq.ThusthedatadescribingtheLyapunovfunctionarethematrixPandthescalars¸i,i=1;:::;np.WerequireP>0and¸i¸0,whichimpliesthatV(»)¸»TP»>0fornonzero».NotethattheLyapunovfunction(8.3)dependsonthe(notfullyspeci¯ed)non-linearitiesÁi.ThusthedataPand¸ishouldbethoughtofasprovidingarecipefor119 120Chapter8Lur'eandMultiplierMethodsconstructingaspeci¯cLyapunovfunctiongiventhenonlinearitiesÁi.Asanexam-ple,considerthespecialcaseofasystemwithunknown-but-constantparameters,i.e.,Ái(¾)=±i¾.TheLyapunovfunctionwillthenhavetheform¡¢V(»)=»TP+CT¢¤C»qqwhere¢=diag(±1;:::;±np)and¤=diag(¸1;:::;¸np).Inotherwords,wearereallysynthesizingaparameter-dependentquadraticLyapunovfunctionforourparameter-dependentsystem.8.1.1StabilityWesettheexogenousinputwtozero,andseekPand¸isuchthatdV(x)<0forallnonzeroxsatisfying(8.1)and(8.2).(8.4)dtSinceÃ!XnpdV(x)T=2xP+¸ipiCi;q(Ax+Bpp);dti=1condition(8.4)holdsifandonlyifÃ!Xnp»TP+¸¼C(A»+B¼)<0iii;qpi=1forallnonzero»satisfying¼i(¼i¡ci;q»)·0;i=1;:::;np:(8.5)Itiseasilyshownthatf(»;¼)j»6=0;(8:5)g=f(»;¼)j»6=0or¼6=0;(8:5)g:TheS-procedurethenyieldsthefollowingsu±cientconditionfor(8.4):TheLMIinP>0,¤=diag(¸1;:::;¸np)¸0andT=diag(¿1;:::;¿np)¸0"#ATP+PAPB+ATCT¤+CTTpqq<0(8.6)BTP+¤CA+TC¤CB+BTCT¤¡2Tpqqqppqholds.Remark:Whenweset¤=0weobtaintheLMI"#TTAP+PAPBp+CqT<0;TBpP+TCq¡2TwhichcanbeinterpretedasaconditionfortheexistenceofaquadraticLyapunovfunctionfortheLur'esystem,asfollows.PerformalooptransformationontheLur'esystemtobringthenonlinearitiestosector[¡1;1].ThentheLMIaboveisthesame(towithinascalingofthevariableT)asonethatyieldsquadraticstabilityoftheassociatedDNLDI;seex5.1.Remark:Condition(8.6)isonlyasu±cientconditionfortheexistenceofaLur'eLyapunovfunctionthatprovesstabilityofsystem(8.1).Itisalsonecessarywhenthereisonlyonenonlinearity,i.e.,whennp=1.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 8.1AnalysisofLur'eSystems1218.1.2Reachablesetswithunit-energyinputsWeconsiderthesetreachablewithunit-energyinputs,8¯9>>¯¯x,wsatisfy(8.3);x(0)=0>><¯=¢¯ZRue=x(T)¯T:>>¯T>>:¯wwdt·1;T¸0;0Theset(¯¯npZ)XCi;q»¯TF=»¯»P»+2¸iÁi(¾)d¾·1¯0i=1containsRueifdTV(x)·wwforallxandwsatisfying(8.1)and(8.2).(8.7)dtFromtheS-procedure,thecondition(8.7)holdsifthereexistsT=diag(¿1;:::;¿np)¸0suchthat23ATP+PAPB+ATCT¤+CTTPBpqqw6TTT74BpP+¤CqA+TCq¤CqBp+BpCq¤¡2T¤CqBw5·0(8.8)BTPBTCT¤¡Iwwqholds,where¤=diag(¸1;:::;¸np)¸0.¯Ofcourse,theellipsoidE=f»2Rn¯»TP»·1gcontainsF,sothatE¶F¶Rue.Therefore,EgivesanouterapproximationofRueifthenonlinearitiesarenotknown.IfthenonlinearitiesÁiareknown,Fgivesapossiblybetterouterapproximation.Wecanusetheseresultstoprovethatapointx0doesnotbelongtothereachablesetbysolvingappropriateLMIPs.8.1.3OutputenergyboundsWeconsiderthesystem(8.1)withinitialconditionx(0),andcomputeupperboundsontheoutputenergyZ1J=zTzdt0usingLyapunovfunctionsVoftheform(8.3).IfdTV(x)+zz·0forallxsatisfying(8.1);(8.9)dtthenJ·V(x(0)).Thecondition(8.9)isequivalenttoÃ!Xnp2»TP+¸¼C(A»+B¼)+»TCTC»·0;iii;qpzzi=1forevery»satisfying¼i(¼i¡ci;q»)·0;i=1;:::;np:Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 122Chapter8Lur'eandMultiplierMethodsUsingtheS-procedure,weconcludethatthecondition(8.9)holdsifthereexistsT=diag(¿1;:::;¿np)¸0suchthat"#ATP+PA+CTCPB+ATCT¤+CTTzzpqq·0:(8.10)BTP+¤CA+TC¤CB+BTCT¤¡2TpqqqppqSinceV(x(0))·x(0)T(P+CT¤C)x(0),anupperboundonJisobtainedqqbysolvingthefollowingEVPinthevariablesP,¤=diag(¸1;:::;¸np)andT=diag(¿1;:::;¿np):¡¢minimizex(0)TP+CT¤Cx(0)qqsubjectto(8:10);T¸0;¤¸0;P>0IfthenonlinearitiesÁiareknown,wecanobtainpossiblybetterboundsonJbymodifyingtheobjectiveintheEVPappropriately.8.1.4L2gainWeassumethatDzw=0forsimplicity.IfthereexistsaLyapunovfunctionoftheform(8.3)and°¸0suchthatd2TTV(x)·°ww¡zzforallxandwsatisfying(8.1);(8.11)dtthentheL2gainofthesystem(8.1)doesnotexceed°.Thecondition(8.11)isequivalenttoÃ!Xnp2»TP+¸¼C(A»+B¼)·°2wTw¡»TCTC»iii;qpzzi=1forany»satisfying¼i(¼i¡ci;q»)·0;i=1;:::;np.UsingtheS-procedure,thisissatis¯edifthereexistsT=diag(¿1;:::;¿np)¸0suchthat23ATP+PA+CTCPB+ATCT¤+CTTPBzzpqqw6TTT74BpP+¤CqA+TCq¤CqBp+BpCq¤¡2T¤CqBw5·0:(8.12)BTPBTCT¤¡°2IwwqThesmallestupperboundontheL2gain,provableusingLur'eLyapunovfunctions,isthereforeobtainedbyminimizing°over°,P,¤andTsubjecttoP>0,¤=diag(¸1;:::;¸np)¸0,T=diag(¿1;:::;¿np)¸0and(8.12).ThisisanEVP.8.2IntegralQuadraticConstraintsInthissectionweconsideranimportantvariationontheNLDI,inwhichthepoint-wiseconstraintp(t)Tp(t)·q(t)Tq(t)isreplacedwithaconstraintontheintegralsofp(t)Tp(t)andq(t)Tq(t):x_=Ax+Bpp+Buu+Bww;q=Cqx+Dqpp+Dquu+Dqww;z=Czx+Dzpp+Dzuu+Dzww(8.13)ZtZtp(¿)Tp(¿)d¿·q(¿)Tq(¿)d¿:00Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 8.2IntegralQuadraticConstraints123SuchasystemiscloselyrelatedtotheNLDIwiththesamedatamatrices;indeed,everytrajectoryoftheassociatedNLDIisatrajectoryof(8.13).Thissystemisnot,however,adi®erentialinclusion.Incontroltheoryterms,thesystem(8.13)isdescribedasalinearsystemwith(dynamic)nonexpansivefeedback.Animportantexampleiswhenpandqarerelatedbyalinearsystem,i.e.,x_f=Afxf+Bfq;p=Cfxf+Dfq;xf(0)=0;wherekD+C(sI¡A)¡1Bk·1.ffff1WecanalsoconsiderageneralizationoftheDNLDI,inwhichwehavecompo-nentwiseintegralquadraticconstraints.Moreover,wecanconsiderintegralquadraticconstraintswithdi®erentsectorbounds;asanexamplewewillencounterconstraintsRtoftheformp(¿)Tq(¿)d¿¸0inx8.3.0WewillnowshowthatmanyoftheresultsonNLDIs(andDNLDIs)fromChap-ters5{7generalizetosystemswithintegralquadraticconstraints.Wewilldemon-stratethisgeneralizationforstabilityanalysisandL2gainbounds,leavingotherstothereader.AsinChapters5{7,ouranalysiswillbebasedonquadraticfunctionsofthestateV(»)=»TP»,androughlyspeaking,theverysameLMIswillarise.However,theinterpretationofVisdi®erenthere.Forexample,instabilityanalysis,theVofChapters5{7decreasesmonotonicallytozero.Here,theverysameVdecreasestozero,butnotnecessarilymonotonically.Thus,VisnotaLyapunovfunctionintheconventionalsense.8.2.1StabilityConsidersystem(8.13)withoutwandz.SupposeP>0and¸¸0aresuchthatd¡¢xTPx<¸pTp¡qTq;(8.14)dtorµZt¶d¡¢xTPx+¸q(¿)Tq(¿)¡p(¿)Tp(¿)d¿<0:dt0Notethatthesecondtermisalwaysnonnegative.UsingstandardargumentsfromLyapunovtheory,itcanbeshownthatlimt!1x(t)=0,orthesystemisstable.Now,letusexaminethecondition(8.14).ItisexactlythesameastheconditionobtainedbyapplyingtheS-proceduretotheconditiondTTTxPx<0;wheneverpp·qq;dtwhichinturn,leadstotheLMIconditionforquadraticstabilityofNLDI(5.3)P>0;¸¸0;23ATP+PA+¸CTCPB+¸CTDqqpqqp45<0:(PB+¸CTD)T¡¸(I¡DTD)pqqpqpqp8.2.2L2gainWeassumethatx(0)=0and,forsimplicity,thatDzw=0andDqp=0.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 124Chapter8Lur'eandMultiplierMethodsSupposeP>0and¸¸0aresuchthatd¡¢xTPx<¸pTp¡qTq+°2wTw¡zTz:(8.15)dtIntegratingbothsidesfrom0toT,ZT¡¢ZT¡¢x(T)TPx(T)+¸q(¿)Tq(¿)¡p(¿)Tp(¿)d¿<°2wTw¡zTzdt:00ThisimpliesthattheL2gainfromwtozforsystem(8.13)doesnotexceed°.Condition(8.15)leadstothesameLMI(6.55)whichguaranteesthattheL2gainoftheNLDI(4.9)fromwtozdoesnotexceed°.Remark:AlmostalltheresultsdescribedinChapters5,6and7extendimme-diatelytosystemswithintegralquadraticconstraints.Theonlyexceptionsaretheresultsoncoordinatetransformations(x5.1.1,alsox7.2.2),andonreachablesetswithunit-peakinputs(x6.1.3andx7.3.3).8.3MultipliersforSystemswithUnknownParametersWeconsiderthesystemx_=Ax+Bpp+Bww;q=Cqx+Dqpp;(8.16)pi=±iqi;i=1;:::;np;z=Czx+Dzww;wherep(t)2Rnp,and±,i=1;:::;nareanynonnegativenumbers.Wecanconsideripthemoregeneralcasewhen®i·±i·¯iusingalooptransformation(seetheNotesandReferences).Forreasonsthatwillbecomeclearshortly,webeginbyde¯ningnpLTIsystemswithinputqiandoutputqm;i,whereqm;i(t)2R:x_m;i=Am;ixm;i+Bm;iqi;xm;i(0)=0;(8.17)qm;i=Cm;ixm;i+Dm;iqi;whereweassumethatAm;iisstableand(Am;i;Bm;i)iscontrollable.WefurtherassumethatZtqi(¿)qm;i(¿)d¿¸0forallt¸0.0ThislastconditionisequivalenttopassivityoftheLTIsystems(8.17),whichinturnisequivalenttotheexistenceofPi¸0,i=1;:::;np,satisfying"#ATP+PAPB¡CTm;iiim;i¡im;im;i¢·0;i=1;:::;np:(8.18)BTP¡C¡D+DTm;iim;im;im;i(Seex6.3.3.)¢TThus,qandqm=[qm;1;:::;qm;np]aretheinputandoutputrespectivelyofthediagonalsystemwithrealizationAm=diag(Am;i);Bm=diag(Bm;i);Cm=diag(Cm;i);Dm=diag(Dm;i);Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 8.3MultipliersforSystemswithUnknownParameters125De¯ning"#"#"#A~=A0;B~=Bp;B~=Bw;pwBmCqAmBmDqp0C~q=[DmCqCm];D~qp=DmDqp;C~z=[Cz0]:considerthefollowingsystemwithintegralquadraticconstraints:dx~=A~x~+B~pp+B~ww;dtqm=C~qx~+D~qpp;(8.19)z=C~zx~+D~qpp+Dzww;Ztpi(¿)qm;i(¿)d¿¸0;i=1;:::;np:0Itiseasytoshowthatifxisatrajectoryof(8.16),then[xTxT]Tisatrajectorymof(8.19)forsomeappropriatexm.Therefore,conclusionsabout(8.16)haveimplica-tionsfor(8.19).Forinstance,ifsystem(8.19)isstable,soissystem(8.16).8.3.1StabilityConsidersystem(8.19)withoutwandz.SupposeP>0issuchthatXnpdTx~Px~+2piqm;i<0;(8.20)dti=1orÃnpZt!dXx~TPx~+2p(¿)q(¿)d¿<0:im;idt0i=1Then,itcanbeshownusingstandardargumentsthatlimt!1x~(t)=0,i.e.,thesystem(8.19)isstable.Condition(8.20)isjusttheLMIinP>0,CmandDm:"#A~TP+PA~PB~+C~Tpq<0:(8.21)B~TP+C~D~+D~TpqqpqpRemark:LMI(8.21)canalsobeinterpretedasapositivedissipationcon-ditionfortheLTIsystemwithtransfermatrixG(s)=¡W(s)H(s),where¡1¡1W(s)=Cm(sI¡Am)Bm+DmandH(s)=Cq(sI¡A)Bp+Dqp.Therefore,weconcludethatsystem(8.16)isstableifthereexistsapassiveWsuchthat¡W(s)H(s)haspositivedissipation.HenceWisoftencalledamultiplier".SeetheNotesandReferencesfordetails.8.3.2Reachablesetswithunit-energyinputsThesetofreachablestateswithunit-energyinputsforthesystemx_=Ax+Bpp+Bww;q=Cqx+Dqpp;(8.22)p=±q;±¸0;Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 126Chapter8Lur'eandMultiplierMethodsis8¯9>>¯¯x,wsatisfy(8.22);x(0)=0>><¯=¢¯ZRue=x(T)¯T:>>¯T>>:¯wwdt·1;T¸0;0Wecanobtainasu±cientconditionforapointx0tolieoutsidethereachablesetRue,usingquadraticfunctionsfortheaugmentedsystem.SupposeP>0issuchthatXnpdTTx~Px~+2piqm;i0,CmandDm:23A~TP+PA~PB~PB~+C~Twpq64B~TP¡I075<0:(8.24)wB~TP+C~0D~+D~TpqqpqpNow,thepointx0liesoutsidethesetofreachablestatesforthesystem(8.22)ifthereexistsPsatisfying(8.23)andthereexistsnozsuchthat"#T"#x0x0P·1:zzPartitioningPconformallywiththesizesofx0andzas"#P11P12P=;PTP1222wenotethat"#T"#"#x0P11P12x0T¡1TminzT=x0(P11¡P12P22P12)x0:zP12P22zTherefore,x0doesnotbelongtothereachablesetforthesystem(8.22)ifthereexistP>0,Pi¸0,¸¸0,Cm,andDMsatisfying(8.18),(8.24)and"#xTPx¡1xTP0110012>0:PTxP12022ThisisanLMIP.NotesandReferencesLur'esystemsTheNotesandReferencesofChapters5and7arerelevanttothischapteraswell.Lur'eandPostnikovwerethe¯rsttoproposeLyapunovfunctionsconsistingofaquadraticformCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences127plustheintegralofthenonlinearity[LP44].Thebook[Lur57]followedthisarticle,inwhichtheauthorshowedhowtosolvetheproblemof¯ndingsuchLyapunovfunctionsanalyti-callyforsystemsoforder2or3.AnotherearlybookthatusessuchLyapunovfunctionsisLetov[Let61].WorkonthistopicwascontinuedbyPopov[Pop62],Yakubovich[Yak64].Tsypkin[Tsy64d,Tsy64c,Tsy64a,Tsy64b],Szego[Sze63]andMeyer[Mey66].JuryandLee[JL65]considerthecaseofLur'esystemswithmultiplenonlinearitiesandderiveacorrespondingstabilitycriterion.Seealso[IR64].TheellipsoidmethodisusedtoconstructLyapunovfunctionsfortheLur'estabilityprobleminthepapersbyPyatnitskiiandSkorodinskii[PS82,PS83].Seealso[Sko91a,Sko91b]forgeneralnumericalmethodstoprovestabilityofnonlinearsystemsviathePopovorcirclecriterion.AprecursoristhepaperbyKarmarkarandSiljak[KS75],whousedalocallycon-vergentalgorithmtodeterminemarginsforaLur'e-Postnikovsystemwithonenonlinearity.Indeed,intheearlybooksbyLur'e,Postnikov,andLetovtheproblemofconstructingLya-punovfunctionsreducestothesatisfactionofsomeinequalities.Insomespecialcasestheyevenpointoutgeometricalpropertiesoftheregionsde¯nedbytheinequalities,i.e.,thattheyformaparallelogram.Butasfarasweknow,convexityoftheregionsisnotnotedintheearlywork.Inanycase,itwasnotknowninthe1940'sand1950'sthatsolutionofconvexinequalitiesispracticallyandtheoreticallytractable.ConstructionofLyapunovfunctionsforsystemswithintegralquadraticconstraintsTWesawinx8.2.1thatthequadraticpositivefunctionV(»)=»P»isnotnecessarilyaLyapunovfunctionintheconventionalsenseforthesystem(8.13)withintegralquadraticconstraints;althoughV(x(t))!0ast!1,itdoesnotdosomonotonically.WenowshowhowwecanexplicitlyconstructaLyapunovfunctiongivenP,incaseswhenpandqarerelatedbyx_f=f(xf;q;t);p=g(xf;q;t);(8.25)nfwherexf(t)2R.RRSincetpTpd¿·tqTqd¿,itcanbeshown(seeforexample[Wil72])thatVnff:R!R+00givenby½Z¯¾T¡¢¯¢TT¯Vf(»f)=sup¡qq¡ppdt¯xf(0)=»f;xfsatis¯es(8.25);T¸00isaLyapunovfunctionthatprovesstabilityofsystem(8.25).Moreover,theLyapunovnnffunctionV:R£R!R+givenby¢TV(»;»f)=»P»+Vf(»f)provesstabilityoftheinterconnectionx_=Ax+Bpp;q=Cqx+Dqpp;x_f=f(xf;q;t);p=g(xf;q;t):TNotethatthisconstructionalsoshowsthat»P»provesthestabilityoftheNLDI(5.2).Integralquadraticconstraintshaverecentlyreceivedmuchattention,especiallyintheformerSovietUnion.Seeforexample[Yak88,Meg92a,Meg92b,Yak92,Sav91,SP94b,SP94h,SP94d].Systemswithconstant,unknownparametersProblemsinvolvingsystem(8.16)arewidelystudiedinthe¯eldofrobustcontrol,sometimesunderthenamereal-¹problems";seeSiljak[Sil89]forasurvey.Theresultsofx8.3wereThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 128Chapter8Lur'eandMultiplierMethodsderivedbyFan,TitsandDoylein[FTD91]usinganalternateapproach,involvingtheap-plicationofano®-axiscirclecriterion(seealsox3.3).Wealsomention[TV91],whereTesiandVicinostudyahybridsystem,consistingofaLur'esystemwithunknownparameters.Thereareelegantanalyticsolutionsforsomeveryspecialproblemsinvolvingparameter-dependentlinearsystems,e.g.,Kharitonov'stheorem[Kha78]anditsextensions[BHL89].AnicesurveyoftheseresultscanbefoundinBarmish[Bar93].ComplexityofstabilizationproblemsThestability(andperformance)analysisproblemsconsideredinx8.3havehighcomplexity.Forexample,checkingif(8.16)isstableisNP-hard(seeCoxsonandDemarco[CD91],BraatzandYoung[BYDM93],PoljakandRohn[PR94],andNemirovskii[Nem94]).WeconjecturethatcheckingwhetherageneralDNLDIisstableisalsoNP-hard.Likewise,manystabilizationproblemsforuncertainsystemsareNP-hard.Forinstance,theproblemofcheckingwhethersystem(8.16)isstabilizablebyaconstantstate-feedbackcanbeshowntobeNP-hardusingthemethodof[Nem94].Arelatedresult,duetoBlondelandGevers[BG94],statesthatcheckingwhetherthereexistsacommonstabilizingLTIcontrollerforthreeLTIsystemsisundecidable.Incontrast,checkingwhetherthereexistsastatic,output-feedbackcontrollawforasingleLTIsystemisrationallydecidable[ABJ75].MultipliermethodsSee[Wil69a,NT73,DV75,Sil69,Wil70,Wil76]fordiscussionofandbibliographyonmultipliertheoryanditsconnectionstoLyapunovstabilitytheory.Ingeneralmultipliertheoryweconsiderthesystemx_=Ax+Bpp+Bww;q=Cqx+Dqpp;p(t)=¢(q;t):¡¢¡1ThemethodinvolvescheckingthatHm=WCq(sI¡A)Bp+Dqpsatis¯es¤Hm(j!)+Hm(j!)¸0forevery!2R;forsomechoiceofWfromasetthatisdeterminedbythepropertiesof¢.When¢isanonlineartime-invariantmemorylessoperatorsatisfyingasectorcondition(i.e.,whenwehaveaLur'esystem),themultipliersWareoftheform(1+qs)forsomeq¸0.ThisisthefamousPopovcriterion[Pop62,Pop64];wemustalsociteYakubovich,whoshowed,usingtheS-procedure[Yak77],thatfeasibilityofLMI(8.6)isequivalenttotheexistenceof¡1qnonnegativesuchthat(1+qs)(cq(sI¡A)bp)+1=khaspositivedissipation.Inthecasewhen¢(q;t)=±q(t)forsomeunknownrealnumber±,themultiplierWcanbeanypassivetransfermatrix.ThisobservationwasmadebyBrockettandWillemsin[BW65];givenatransferfunctionHqptheyshowthatthetransferfunctionHqp=(1+kHqp)isstableforallvaluesofkin(0;1)ifandonlyifthereexistsapassivemultiplierWsuchthatWHqpisalsopassive.Thiscasehasalsobeenconsideredin[CS92b,SC93,SL93a,SLC94]wheretheauthorsdeviseappropriatemultipliersforconstantrealuncertainties.Thepaper[BHPD94]byBalakrishnanetal.,showshowvariousstabilitytestsforuncertainsystemscanbered-erivedinthecontextofmultipliertheory,andhowthesetestscanbereducedtoLMIPs.Seealso[LSC94].SafonovandWyetzneruseaconvexparametrizationofmultipliersfoundbyZamesandFalb[ZF68,ZF67]toprovestabilityofsystemssubjecttomonotonic"orodd-monotonic"nonlinearities(see[SW87]fordetails;seealso[GG94]).HallandHow,alongwithBern-steinandHaddad,generalizetheuseofquadraticLyapunovfunctionsalongwithmultipliersforvariousclassesofnonlinearities,andapplythemtoAerospaceproblems;seeforex-ample[HB91a,How93,HH93b,HHHB92,HH93a,HHH93a,HCB93,HB93b,HB93a,HHH93b].Seealso[CHD93].Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences129MultiplenonlinearitiesforLur'esystemsOuranalysismethodforLur'esystemswithmultiplenonlinearitiesinx8.1canbeconser-vative,sincetheS-procedurecanbeconservativeinthiscase.Rapoport[Rap86,Rap87,Rap88]devisesawayofdeterminingifthereexistsLyapunovfunctionsoftheform(8.3)nonconservativelybygeneratingappropriateLMIs.Wenote,however,thatthenumberoftheseLMIsgrowsexponentiallywiththenumberofnonlinearities.Kamenetskii[Kam89]andRapoport[Rap89]derivecorrespondingfrequency-domainstabilitycriteria.Kamenet-skiicallshisresultsconvolutionmethodforsolvingmatrixinequalities".LooptransformationsConsidertheLur'esystemwithgeneralsectorconditions,i.e.,x_=Ax+Bp;q=Cx+Dp;(8.26)22pi(t)=Ái(qi(t));®i¾·¾Ái(¾)·¯i¾:HerewehavedroppedthevariableswandzandthesubscriptsonB,CandDtosimplifythepresentation.Weassumethissystemiswell-posed,i.e.,det(I¡D¢)6=0foralldiagonal¢with®i·¢ii·¯i.De¯ne¢1p¹i=(Ái(qi)¡®iqi)=Á¹i(qi):¯i¡®i2Itisreadilyshownthat0·¾Á¹i(¾)·¾forall¾.Let¤and¡denotethediagonalmatrices¤=diag(®1;:::;®n);¡=diag(¯1¡®1;:::;¯n¡®n);qqq¡1sothatp¹=¡(p¡¤q).Wenowsubstitutep=¡p¹+¤qinto(8.26)and,usingourwell-posednessassumption,solveforx_andqintermsofxandp¹.Thisresultsin¡¢¡1¡1x_=A+B¤(I¡D¤)Cx+B(I¡¤D)¡p;¹¡1¡1q=(I¡D¤)Cx+(I¡D¤)D¡p¹:Wecanthereforeexpress(8.26)asx_=Ax¹+B¹p;¹q=C¹x+D¹p;¹(8.27)2p¹i(t)=Á¹i(qi(t));0·¾Á¹i(¾)·¾:whereA¹=A+B¤(I¡D¤)¡1C;B¹=B(I¡¤D)¡1¡;C¹=(I¡D¤)¡1C;D¹=(I¡D¤)¡1D¡:Notethat(8.27)isinthestandardLur'esystemform.SotoanalyzethemoregeneralLur'esystem(8.26),wesimplyapplythemethodsofx8.1totheloop-transformedsystem(8.27).Inpractice,i.e.,inanumericalimplementation,itisprobablybettertoderivetheLMIsassociatedwiththemoregeneralLur'esystemthantolooptransformtothestandardLur'esystem.Theconstructionaboveisalooptransformationthatmapsnonlinearitiesinsector[®i;¯i]intothestandardsector,i.e.,[0;1].Similartransformationscanbeusedtomapanysetofsectorsintoanyother,includingso-calledin¯nitesectors,inwhich,say,¯i=+1.Thetermlooptransformationcomesfromasimpleblock-diagraminterpretationoftheequa-tionsgivenabove.Fordetaileddescriptionsoflooptransformations,seethebookbyDesoerandVidyasagar[DV75,p50].Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. Chapter9SystemswithMultiplicativeNoise9.1AnalysisofSystemswithMultiplicativeNoise9.1.1Mean-squarestabilityWe¯rstconsiderthediscrete-timestochasticsystemÃ!XLx(k+1)=A0+Aipi(k)x(k);(9.1)i=1wherep(0),p(1),:::,areindependent,identicallydistributedrandomvariableswithEp(k)=0;Ep(k)p(k)T=§=diag(¾;:::;¾):(9.2)1LWeassumethatx(0)isindependentoftheprocessp.De¯neM(k),thestatecorrelationmatrix,as¢TM(k)=Ex(k)x(k):Ofcourse,Msatis¯esthelinearrecursionXLM(k+1)=AM(k)AT+¾2AM(k)AT;M(0)=Ex(0)x(0)T:(9.3)iiii=1Ifthislinearrecursionisstable,i.e.,regardlessofx(0),limk!1M(k)=0,wesaythesystemismean-squarestable.Mean-squarestabilityimplies,forexample,thatx(k)!0almostsurely.Itcanbeshown(seetheNotesandReferences)thatmean-squarestabilityisequivalenttotheexistenceofamatrixP>0satisfyingtheLMIXLATPA¡P+¾2ATPA<0:(9.4)iiii=1Analternative,equivalentconditionisthatthedualinequalityXLAQAT¡Q+¾2AQAT<0(9.5)iiii=1holdsforsomeQ>0.131 132Chapter9SystemswithMultiplicativeNoiseTRemark:ForP>0satisfying(9.4),wecaninterpretthefunctionV(»)=»P»asastochasticLyapunovfunction:EV(x)decreasesalongtrajectoriesof(9.1).Alternatively,thefunctionV(M)=TrMPisa(linear)Lyapunovfunctionforthedeterministicsystem(9.3)(seetheNotesandReferences).Remark:Mean-squarestabilitycanbeveri¯eddirectlybysolvingtheLyapunovequationinPXLT2TAPA¡P+¾iAiPAi+I=0i=1andcheckingwhetherP>0.ThustheLMIPs(9.4)and(9.5)o®ernocomputa-tional(ortheoretical)advantagesforcheckingmean-squarestability.Wecanconsidervariationsonthemean-squarestabilityproblem,forexample,determiningthemean-squarestabilitymargin.Here,wearegivenA0;:::;AL,andaskedto¯ndthelargest°suchthatwith§<°2I,thesystem(9.1)ismean-squarestable.FromtheLMIPcharacterizationsofmean-squarestability,wecanderiveGEVPcharacterizationsthatyieldtheexactmean-squarestabilitymargin.Again,theGEVPo®ersnocomputationalortheoreticaladvantages,sincethemean-squarestabilitymargincanbeobtainedbycomputingtheeigenvalueofa(large)matrix.9.1.2Statemeanandcovarianceboundswithunit-energyinputsWenowaddanexogenousinputwtoourstochasticsystem:XLx(k+1)=Ax(k)+Bww(k)+(Aix(k)+Bw;iw(k))pi(k);x(0)=0:(9.6)i=1Weassumethattheexogenousinputwisdeterministic,withenergynotexceedingone,i.e.,X1w(k)Tw(k)·1:k=0Letx¹(k)denotethemeanofx(k),whichsatis¯esx¹(k+1)=Ax¹(k)+Bww(k),andletTX(k)denotethecovarianceofx(k),i.e.,X(k)=E(x(k)¡x¹(k))(x(k)¡x¹(k)).Wewilldevelopjointboundsonx¹(k)andX(k).SupposeV(»)=»TP»whereP>0satis¯esEV(x(k+1))·EV(x(k))+w(k)Tw(k)(9.7)foralltrajectories.Then,XkEV(x(k))·w(j)Tw(j);k¸0;j=0whichimpliesthat,forallk¸0,Ex(k)TPx(k)=x¹(k)TPx¹(k)+E(x(k)¡x¹(k))TP(x(k)¡x¹(k))=x¹(k)TPx¹(k)+TrX(k)P·1:Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 9.1AnalysisofSystemswithMultiplicativeNoise133Letusnowexaminecondition(9.7).Wenotethat"#T"#x¹(k)x¹(k)EV(x(k+1))=Mw(k)w(k)Ã!XL+TrATPA+¾2ATPAX(k);iiii=1where"#"#AThiXLAThiM=P+¾2iP:TABwiTAiBw;iBwi=1Bw;iNext,theLMIconditions"#"#"#AThiXLAThiP0P+¾2iP·(9.8)TABwiTAiBw;iBwi=1Bw;i0IandXLATPA+¾2ATPA0satisfyingtheLMI(9.8).Forexample,wehavetheboundTrM(k)P·1onthestatecovariancematrix.WecanderivefurtherboundsonM(k).Forexample,sincewehaveTrM(k)P·¸min(P)TrM(k)andthereforeTrM(k)·1=¸min(P),wecanderiveboundsonTrMbymaximizing¸min(P)subjectto(9.8)andP>0.ThisisanEVP.Asanotherexample,wecanderiveanupperboundon¸max(M)bymaximizingTrPsubjectto(9.8)andP>0,whichisanotherEVP.9.1.3BoundonL2gainWenowconsiderthesystemXLx(k+1)=Ax(k)+Bww(k)+(Aix(k)+Bw;iw(k))pi(k);x(0)=0;i=1(9.10)XLz(k)=Czx(k)+Dzww(k)+(Cz;ix(k)+Dzw;iw(k))pi(k):i=1Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 134Chapter9SystemswithMultiplicativeNoisewiththesameassumptionsonp.Weassumethatwisdeterministic.Wede¯netheL2gain´ofthissystemas(¯)X1¯X12¢T¯T´=supEz(k)z(k)¯w(k)w(k)·1:(9.11)¯k=0k=0SupposethatV(»)=»TP»,withP>0,satis¯esEV(x(k+1))¡EV(x(k))·°2w(k)Tw(k)¡Ez(k)Tz(k):(9.12)Then°¸´.Thecondition(9.12)canbeshowntobeequivalenttotheLMI"#T"#"#"#ABwP0ABwP0¡CD0ICD0°2Izzwzzw"#T"#"#(9.13)XLABP0AB2iw;iiw;i+¾i·0:i=1Cz;iDzw;i0ICz;iDzw;iMinimizing°subjecttoLMI(9.13),whichisanEVP,yieldsanupperboundon´.Remark:Itisstraightforwardtoapplythescalingmethoddevelopedinx6.3.4toobtaincomponentwiseresults.9.2State-FeedbackSynthesisWenowaddacontrolinpututooursystem:x(k+1)=Ax(k)+Buu(k)+Bww(k)+XL(Aix(k)+Bu;iu(k)+Bw;iw(k))pi(k);i=1(9.14)z(k)=Czx(k)+Dzuu(k)+Dzww(k)+XL(Cz;ix(k)+Dzu;iu(k)+Dzw;iw(k))pi(k):i=1Weseekastate-feedbacku(k)=Kx(k)suchthattheclosed-loopsystemx(k+1)=(A+BuK)x(k)+Bww(k)+XL((Ai+Bu;iK)x(k)+Bw;iw(k))pi(k);i=1(9.15)z(k)=(Cz+DzuK)x(k)+Dzww(k)+XL((Cz;i+Dzu;iK)x(k)+Dzw;iw(k))pi(k);i=1satis¯esvariousproperties.9.2.1StabilizabilityWeseekKandQ>0suchthat(A+BK)Q(A+BK)T¡QuuXL+¾2(A+BK)Q(A+BK)T<0:iiu;iiu;ii=1Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 9.2State-FeedbackSynthesis135WiththechangeofvariableY=KQ,weobtaintheequivalentcondition(AQ+BY)Q¡1(AQ+BY)T¡QuuXL(9.16)+¾2(AQ+BY)Q¡1(AQ+BY)T<0;iiu;iiu;ii=1whichiseasilywrittenasanLMIinQandY.Thus,system(9.14)ismean-squarestabilizable(withconstantstate-feedback)ifandonlyiftheLMIQ>0,(9.16)isfeasible.Inthiscase,astabilizingstate-feedbackgainisK=YQ¡1.Asanextension,wecanmaximizetheclosed-loopmean-squarestabilitymargin.System(9.15)ismean-squarestablefor§·°2Iifandonlyif(AQ+BY)Q¡1(AQ+BY)T¡QuuXL(9.17)+°2(AQ+BY)Q¡1(AQ+BY)T<0iu;iiu;ii=1holdsforsomeYandQ>0.Therefore,¯ndingKthatmaximizes°reducestothefollowingGEVP:maximize°subjecttoQ>0;(9:17)9.2.2MinimizingtheboundonL2gainWenowseekastate-feedbackgainKwhichminimizestheboundonL2gain(asde¯nedby(9.11))forsystem(9.15).´·°forsomeKifthereexistP>0andKsuchthat"#T"#XLA+BKBA+BKB¾2iu;iw;iP~iu;iw;iii=1Cz;i+Dzu;iKDzw;iCz;i+Dzu;iKDzw;i"#T"#"#+A+BuKBwP~A+BuKBw·P0C+DKDC+DKD0°2Izzuzwzzuzwwhere"#P0P~=:0IWemakethechangeofvariablesQ=P¡1,Y=KQ.ApplyingacongruencewithQ~=diag(Q;I),weget"#T"#XLAQ+BYBAQ+BYB¾2iu;iw;iQ~¡1iu;iw;iii=1Cz;iQ+Dzu;iYDzw;iCz;iQ+Dzu;iYDzw;i"#T"#"#+AQ+BuYBwQ~¡1AQ+BuYBw·Q0:CQ+DYDCQ+DYD0°2IzzuzwzzuzwThisinequalityisreadilytransformedtoanLMI.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 136Chapter9SystemswithMultiplicativeNoiseNotesandReferencesSystemswithmultiplicativenoiseSystemswithmultiplicativenoiseareaspecialcaseofstochasticsystems.ForanintroductiontothistopicseeArnold[Arn74]orºAstrÄom[ºAst70].Arigoroustreatmentofcontinuous-timestochasticsystemsrequiresmuchtechnicalmachin-ery(see[EG94]).Infacttherearemanywaystoformulatethecontinuous-timeversionofasystemsuchas(9.10);seeIt^o[Ito51]andStratonovitch[Str66].StabilityofsystemwithmultiplicativenoiseForasurveyofdi®erentconceptsofstabilityforstochasticsystemssee[Koz69]andthereferencestherein.Thede¯nitionofmean-squarestabilitycanbefoundin[Sam59].Mean-squarestabilityisastrongformofstability.Inparticular,itimpliesstabilityofthemeanEx(k)andthatalltrajectoriesconvergetozerowithprobabilityone[Kus67,Wil73].Forthesystemsconsideredhere,itisalsoequivalenttoL2-stability,whichisthepropertythat(seePT2forexample,[EP92])Ekx(t)khasa(¯nite)limitasT!1forallinitialconditions.t=0Theequationforthestatecorrelationmatrix(9.3)canbefoundin[McL71,Wil73].Analge-braiccriterionformean-squarestabilityofasystemwithmultiplicativewhitenoiseisgivenin[NS72].Itisreminiscentoftheclassical(deterministic)Hurwitzcriterion.Frequency-domaincriteriaaregivenin[WB71].TherelatedLyapunovequationwasgivenbyKlein-mann[Kle69]forthecaseL=1(i.e.,asingleuncertainty).ThecriterionthenreducestoanH2normconditiononacertaintransfermatrix.(Thisisinparallelwiththeresultsof[EP92],seebelow.)KatsandKrasovskiiintroducedin[KK60]theideaofastochasticLyapunovfunctionandprovedarelatedLyapunovtheorem.Athoroughstabilityanalysisof(nonlinear)stochasticsystemswasmadebyKushnerin[Kus67]usingthisideaofastochasticLyapunovfunc-tion.Kushnerusedthisideatoderiveboundsforreturntime,expectedoutputenergy,etc.However,thisapproachdoesnotprovideawaytoconstructtheLyapunovfunctions.ProofofstabilitycriterionWenowprovethatcondition(9.4)isanecessaryandsu±cientconditionformean-squarestability.We¯rstprovethatcondition(9.4)issu±cient.LetP>0satisfy(9.4).Introducethe(linearLyapunov)functionV(M)=TrMP,whichispositiveontheconeofnonnegativematrices.FornonzeroM(k)satisfying(9.3),Ã!XLT2TV(M(k+1))=TrAM(k)A+¾iAiM(k)AiPÃi=1!XLT2T=TrM(k)APA+¾iAiPAii=10,andletN(k)bethecorrespondingsolutionof(9.18).SinceN(k)satis¯esastable¯rst-orderlineardi®erenceequationwithconstantcoe±cients,thefunctionXkP(k)=N(j)j=0hasa(¯nite)limitfork!1,whichwedenotebyP.ThefactthatN(0)>0impliesP>0.Now(9.18)impliesXLT2TP(k+1)=N(0)+AP(k)A+¾iAiP(k)Aii=1Takingthelimitk!1showsthatPsatis¯es(9.4).Mean-squarestabilitymarginSystem(9.10)canbeviewedasalinearsystemsubjecttorandomparametervariations.Forsystemswithdeterministicparametersonlyknowntolieinranges,thecomputationofanexactstabilitymarginisanNP-hardproblem[CD92].Thefactthatastochasticanalogofthedeterministicstabilitymarginiseasiertocomputeshouldnotbesurprising;wearedealingwithaveryspecialformofstability,whichonlyconsiderstheaveragebehaviorofthesystem.Anexactstabilitymargincanalsobecomputedforcontinuous-timesystemswithmultiplica-tivenoiseprovidedtheIt^oframeworkisconsidered;see[EG94].Incontrast,thecomputationofanexactstabilitymarginseemsdi±cultforStratonovitchsystems.In[EP92],ElBouhtouriandPritchardprovideacompleterobustnessanalysisinaslightlydi®erentframeworkthanours,namelyforso-calledblock-diagonal"perturbations.Roughlyspeaking,theyconsideracontinuous-timeIt^osystemoftheformXLdx(t)=Ax(t)dt+Bi¢i(Cx(t))dpi(t);i=1wheretheoperators¢iareLipschitzcontinuous,with¢i(0)=0;theprocessespi,i=1;:::;L,arestandardBrownianmotions(which,roughlyspeaking,areintegralsofwhitenoise).Wenotethatourframeworkcanberecoveredbyassumingthattheoperators¢iarerestrictedtoadiagonalstructureoftheform¢i(z)=¾iz.¡PL2¢1=2Measuringthesize"oftheperturbationtermbyk¢ik,wherek¢kdenotesthei=1Lipschitznorm,leadstheauthorsof[EP92]totherobustnessmarginminimize°TTTsubjecttoP>0;AP+PA+CC<0;BiPBi·°;i=1;:::;LThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 138Chapter9SystemswithMultiplicativeNoiseForL=1,andB1andCarevectors,thatis,whenasinglescalaruncertainparameterperturbsanLTIsystem,boththisresultandourscoincidewiththeH2normof(A;B1;C).ThisisconsistentwiththeearlierresultsofKleinmann[Kle69]andWillems[WB71](seealso[BB91,p114]forarelatedresult).ForL>1however,applyingtheirresultstoourframeworkwouldbeconservative,sinceweonlyconsiderdiagonal"perturbationswithre-peatedelements",¢i=¾iI.NotethattheLMIapproachcanalsosolvethestate-feedbacksynthesisproblemintheframeworkof[EP92].ElBouhtouriandPritchardprovideasolutionofthisproblemin[EP94].Stabilityconditionsforarbitrarynoiseintensitiesaregiveningeometrictermsin[WW83].Inourframework,thishappensifandonlyiftheoptimalvalueofthecorrespondingGEVPiszero.BoundsonstatecovariancewithnoiseinputWeconsidersystem(9.6)inwhichwisaunitwhitenoiseprocess,i.e.,w(k)areindependent,TidenticallydistributedwithEw(k)=0,Ew(k)w(k)=W,andwisindependentofp.Inthiscase,ofcourse,thestatemeanx¹iszero.Wederiveanupperboundonthestatecorrelation(orcovariance)ofthesystem.ThestatecorrelationM(k)satis¯esthedi®erenceequationXL¡¢TT2TTM(k+1)=AM(k)A+BwWBw+¾iAiM(k)Ai+Bw;iWBw;i;i=1withM(0)=0.Sincethesystemismean-squarestable,thestatecorrelationM(k)hasa(¯nite)limitM1whichsatis¯esXL¡¢TT2TTM1=AM1A+BwWBw+¾iAiM1Ai+Bw;iWBw;i:i=1Infact,thematrixM1canbecomputeddirectlyasthesolutionofalinearmatrixequation.However,theaboveLMIformulationextendsimmediatelytomorecomplicatedsituations(forinstance,whenthewhitenoiseinputwhasacovariancematrixwithunknowno®-diagonalelements),whilethedirect"methoddoesnot.SeetheNotesandReferencesforChapter6fordetailsonhowthesemorecomplicatedsituationscanbehandled.ExtensiontootherstochasticsystemsTheLur'estabilityproblemconsideredinx8.1canbeextendedtothestochasticframework;seeWonham[Won66].ThisextensioncanbecastintermsofLMIs.Itisalsopossibletoextendtheresultsinthischaptertouncertainstochasticsystems",oftheformÃ!XLx(k+1)=A(k)+¾iAipi(k)x(k);i=1wherethetime-varyingmatrixA(k)isonlyknowntobelongtoagivenpolytope.State-feedbacksynthesisConditionsforstabilizabilityforarbitrarynoiseintensitiesaregiveningeometrictermsin[WW83].Apartfromthiswork,thebulkofearlierresearchonthistopicconcentratedCopyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences139onLinear-QuadraticRegulatortheoryforcontinuous-timeIt^osystems,whichaddressestheproblemof¯ndinganinpututhatminimizeaperformanceindexoftheformZ1¡¢TTJ(K)=Ex(t)Qx(t)+u(t)Ru(t)dt0whereQ¸0andR>0aregiven.Thesolutionofthisproblem,asfoundin[McL71]or[Ben92],canbeexpressedasa(linear,constant)state-feedbacku(t)=Kx(t)wherethegainKisgivenintermsofthesolutionofanon-standardRiccatiequation.(See[Kle69,Won68,FR75,BH88b]foradditionaldetails.)Theexistingmethods(seee.g.[Phi89,RSH90])usehomotopyalgorithmstosolvetheseequations,withnoguaranteeofglobalcon-vergence.Wecansolvethesenon-standardRiccatiequationsreliably(i.e.,withguaranteedconvergence),byrecognizingthatthesolutionisanextremalpointofacertainLMIfeasibilityset.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. Chapter10MiscellaneousProblems10.1OptimizationoveranA±neFamilyofLinearSystemsWeconsiderafamilyoflinearsystems,x_=Ax+Bww;z=Cz(µ)x+Dzw(µ)w;(10.1)pwhereCzandDzwdependa±nelyonaparameterµ2R.WeassumethatAisstableand(A;Bw)iscontrollable.Thetransferfunction,¢¡1Hµ(s)=Cz(µ)(sI¡A)Bw+Dzw(µ);dependsa±nelyonµ.Severalproblemsarisinginsystemandcontroltheoryhavetheformminimize'0(Hµ)(10.2)subjectto'i(Hµ)<®i;i=1;:::;pwhere'iarevariousconvexfunctionals.TheseproblemscanoftenberecastasLMIproblems.Todothis,wewillrepresent'i(Hµ)<®iasanLMIinµ,®i,andpossiblysomeauxiliaryvariables³i(foreachi):Fi(µ;®i;³i)>0:Thegeneraloptimizationproblem(10.2)canthenbeexpressedastheEVPminimize®0subjecttoFi(µ;®i;³i)>0;i=0;1;:::;p10.1.1H2normTheH2normofthesystem(10.1),i.e.,Z121¤kHµk2=TrHµ(j!)Hµ(j!)d!;2¼0is¯niteifandonlyifD(µ)=0.Inthiscase,itequalsTrCWCT,whereW>0iszwzczcthecontrollabilityGramianofthesystem(10.1)whichsatis¯es(6.6).Therefore,theH2normofthesystem(10.1)islessthanorequalto°ifandonlyifthefollowingconditionsonµand°2aresatis¯ed:D(µ)=0;TrC(µ)WC(µ)T·°2:zwzczThequadraticconstraintonµisreadilycastasanLMI.141 142Chapter10MiscellaneousProblems10.1.2H1normFromthebounded-reallemma,wehavekHµk1<°ifandonlyifthereexistsP¸0suchthat"#"#TThiAP+PAPBwCz(µ)T2+TCz(µ)Dzw(µ)·0:BwP¡°IDzw(µ)Remark:Herewehaveconvertedtheso-calledsemi-in¯nite"convexconstraintkHµ(i!)k<°forall!2Rintoa¯nite-dimensionalconvex(linearmatrix)inequality.10.1.3EntropyThe°-entropyofthesystem(10.1)isde¯nedas82Z1<¡°2¤¢logdet(I¡°Hµ(i!)Hµ(i!))d!;ifkHµk1<°,I°(Hµ)=2¼¡1:1;otherwise.Whenitis¯nite,I(H)isgivenbyTrBTPB,wherePisasymmetricmatrix°µwwwiththesmallestpossiblemaximumsingularvalueamongallsolutionsoftheRiccatiequationTT1TAP+PA+Cz(µ)Cz(µ)+2PBwBwP=0:°Forthesystem(10.1),the°-entropyconstraintI°·¸isthereforeequivalenttoanLMIinµ,P=PT,°2,and¸:23ATP+PAPBC(µ)TwzD(µ)=0;6T27TPB·¸:zw4BwP¡°I05·0;TrBwwCz(µ)0¡I10.1.4DissipativitySupposethatwandzarethesamesize.ThedissipativityofHµ(see(6.59))exceeds°ifandonlyiftheLMIinthevariablesP=PTandµholds:"#ATP+PAPB¡C(µ)Twz·0:BTP¡C(µ)2°I¡D(µ)¡D(µ)TwzzwzwWeremindthereaderthatpassivitycorrespondstozerodissipativity.10.1.5HankelnormTheHankelnormofHislessthan°ifandonlyifthefollowingLMIinQ,µand°2µholds:D(µ)=0;ATQ+QA+C(µ)TC(µ)·0;zwzz°2I¡W1=21=2cQWc¸0;Q¸0:(WcisthecontrollabilityGramiande¯nedby(6.6).)Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 10.2AnalysisofSystemswithLTIPerturbations14310.2AnalysisofSystemswithLTIPerturbationsInthissection,weaddresstheproblemofdeterminingthestabilityofthesystemx_=Ax+Bpp;q=Cqx+Dqpp;pi=±i¤qi;i=1;:::;np;(10.3)wherep=[p¢¢¢p]T,q=[q¢¢¢q]T,and±;i=1;:::;n,areimpulse1np1npipresponsesofsingle-outputsingle-outputLTIsystemswhoseH1normisstrictlylessthanone,and¤denotesconvolution.Thisproblemhasmanyvariations:Someofthe±imaybeimpulseresponsematrices,ortheymaysatisfyequalityconstraintssuchas±1=±2.Ouranalysiscanbereadilyextendedtothesecases.LetH(s)=C(sI¡A)¡1B+D.SupposethatthereexistsadiagonaltransferqpqpqpmatrixW(s)=diag(W1(s);:::;Wnp(s)),whereW1(s);:::;Wnp(s)aresingle-inputsingle-outputtransferfunctionswithnopolesorzerosontheimaginaryaxissuchthat°°sup°W(i!)H(i!)W(i!)¡1°<1(10.4)qp!2RThenfromasmall-gainargument,thesystem(10.3)isstable.SuchaWisreferredtoasafrequency-dependentscaling.Condition(10.4)isequivalenttotheexistenceof°>0suchthatH(i!)¤W(i!)¤W(i!)H(i!)¡W(i!)¤W(i!)+°I·0;qpqpforall!2R.Fromspectralfactorizationtheory,itcanbeshownthatsuchaWexistsifandonlyifthereexistsastable,diagonaltransfermatrixVandpositive¹suchthatH(i!)¤(V(i!)+V(i!)¤)H(i!)¡(V(i!)+V(i!)¤)+°I·0;qpqp(10.5)V(i!)+V(i!)¤¸¹I;forall!2R.Inordertoreduce(10.5)toanLMIcondition,werestrictVtobelongtoana±ne,¯nite-dimensionalset£ofdiagonaltransfermatrices,i.e.,oftheformV(s)=D(µ)+C(µ)(sI¡A)¡1B;µVVwhereDVandCVarea±nefunctionsareµ.Then,H(¡s)TV(s)H(s)¡V(s)hasastate-spacerealization(A,B,qpµqpµaugaugCaug(µ),Daug(µ))whereCaugandDaugarea±neinµ,and(Aaug;Baug)iscontrollable.Fromx10.1.4,theconditionthatV(i!)+V(i!)¤¸¹Iforall!2RisequivalentµµtotheexistenceofP1¸0,µand°>0suchthattheLMI"#ATP+PAPB¡C(µ)TV11V1VV·0(PB¡C(µ)T)T¡(D(µ)+D(µ)T)+¹I1VVVVholds.TheconditionH(i!)¤(V(i!)+V(i!)¤)H(i!)¡(V(i!)+V(i!)¤)+°I·0qpµµqpµµisequivalenttotheexistenceofP=PTandµsatisfyingtheLMI22"#ATP+PAPB+C(µ)Taug22aug2augaug·0:(10.6)(PB+C(µ)T)TD(µ)+D(µ)T+°I2augaugaugaug(SeetheNotesandReferencesfordetails.)Thusprovingstabilityofthesystem(10.3)usinga¯nite-dimensionalsetoffrequency-dependentscalingsisanLMIP.Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 144Chapter10MiscellaneousProblems10.3PositiveOrthantStabilizabilitynTheLTIsystemx_=Axissaidtobepositiveorthantstableifx(0)2R+impliesthatnx(t)2R+forallt¸0,andx(t)!0ast!1.Itcanbeshownthatthesystemx_=AxispositiveorthantstableifandonlyifAij¸0fori6=j,andthereexistsadiagonalP>0suchthatPAT+AP<0.ThereforecheckingpositiveorthantstabilityisanLMIP.Wenextconsidertheproblemof¯ndingKsuchthatthesystemx_=(A+BK)xispositiveorthantstable.Equivalently,weseekadiagonalQ>0andKsuchthatQ(A+BK)T+(A+BK)Q<0,withtheo®-diagonalentriesofA+BKbeingnonnegative.SinceQ>0isdiagonal,thelastconditionholdsifandonlyifalltheo®-diagonalentriesof(A+BK)Qarenonnegative.Therefore,withthechangeofvariablesY=KQ,provingpositiveorthantstabilizabilityforthesystemx_=Ax+Buisequivalentto¯ndingasolutiontotheLMIPwithvariablesQandY:Q>0;(AQ+BY)¸0;i6=j;QAT+YTBT+AQ+BY<0:(10.7)ijRemark:Themethodcanbeextendedtoinvarianceofmoregeneral(polyhedral)sets.PositiveorthantstabilityandstabilizabilityofLDIscanbehandledinasimilarway.SeetheNotesandReferences.10.4LinearSystemswithDelaysConsiderthesystemdescribedbythedelay-di®erentialequationXLdx(t)=Ax(t)+Aix(t¡¿i);(10.8)dti=1nwherex(t)2R,and¿i>0.IfthefunctionalXLZ¿iV(x;t)=x(t)TPx(t)+x(t¡s)TPx(t¡s)ds;(10.9)ii=10whereP>0,P1>0;:::;PL>0,satis¯esdV(x;t)=dt<0foreveryxsatisfying(10.8),thenthesystem(10.8)isstable,i.e.,x(t)!0ast!1.Itcanbeveri¯edthatdV(x;t)=dt=y(t)TWy(t),where23XL6ATP+PA+PPA¢¢¢PA723i1L67x(t)6i=176T7676A1P¡P1¢¢¢076x(t¡¿1)7W=6677;y(t)=66.77:6.....74..566.......7745x(t¡¿L)ATP0¢¢¢¡PLLTherefore,wecanprovestabilityofsystem(10.8)usingLyapunovfunctionalsoftheform(10.9)bysolvingtheLMIPW<0,P>0,P1>0;:::;PL>0.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 10.5InterpolationProblems145Remark:Thetechniquesforprovingstabilityofnorm-boundLDIs,discussedinChapter5,canalsobeusedonthesystem(10.8);itcanbeveri¯edthattheLMI(5.15)thatwegetforquadraticstabilityofaDNLDI,inthecasewhenthestatexisascalar,isthesameastheLMIW<0above.Wealsonotethatwecanregardthedelayelementsofsystem(10.8)asconvolutionoperatorswithunitL2gain,andusethetechniquesofx10.2tocheckitsstability.Next,weaddaninpututothesystem(10.8)andconsiderXLdx(t)=Ax(t)+Bu(t)+Aix(t¡¿i):(10.10)dti=1Weseekastate-feedbacku(t)=Kx(t)suchthatthesystem(10.10)isstable.Fromthediscussionabove,thereexistsastate-feedbackgainKsuchthataLya-punovfunctionaloftheform(10.9)provesthestabilityofthesystem(10.10),if23XL6(A+BK)TP+P(A+BK)+PPA¢¢¢PA7i1L676i=176T76A1P¡P1¢¢¢07W=6677<06........766....7745ATP0¢¢¢¡PLLforsomeP,P1;:::;PL>0.MultiplyingeveryblockentryofWontheleftandontherightbyP¡1andsettingQ=P¡1,Q=P¡1PP¡1andY=KP¡1,weobtaintheconditionii23XL6AQ+QAT+BY+YTBT+QAQ¢¢¢AQ7i1L676i=17676QAT¡Q¢¢¢076117X=67<0:676.........76...76745QAT0¢¢¢¡QLLThus,checkingstabilizabilityofthesystem(10.8)usingLyapunovfunctionalsoftheform(10.9)isanLMIPinthevariablesQ,Y,Q1;:::;QL.Remark:Itispossibletousetheeliminationprocedureofx2.6.2toeliminatethematrixvariableYandobtainanequivalentLMIPwithfewervariables.10.5InterpolationProblems10.5.1TangentialNevanlinna-Pickproblem¢Given¸1;:::;¸mwith¸i2C+=fsjRes>0ganddistinct,u1;:::;um,withqpui2Candv1;:::;vm,withvi2C,i=1;:::;m,thetangentialNevanlinna-PickThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 146Chapter10MiscellaneousProblemsp£qproblemisto¯nd,ifpossible,afunctionH:C¡!CwhichisanalyticinC+,andsatis¯esH(¸i)ui=vi;i=1;:::;m,withkHk1·1.(10.11)Problem(10.11)arisesinmulti-inputmulti-outputH1controltheory.FromNevanlinna-Picktheory,problem(10.11)hasasolutionandonlyifthePickmatrixNde¯nedbyu¤u¡v¤vN=ijijij¸¤+¸)ijispositivesemide¯nite.NcanalsobeobtainedasthesolutionoftheLyapunovequationA¤N+NA¡(U¤U¡V¤V)=0:whereA=diag(¸1;:::;¸m),U=[u1¢¢¢um],V=[v1¢¢¢vm].Forfuturereference,wenotethatN=Gin¡Gout,whereA¤G+GA¡U¤U=0;A¤G+GA¡V¤V=0:ininoutoutSolvingthetangentialNevanlinna-PickproblemsimplyrequirescheckingwhetherN¸0.10.5.2Nevanlinna-PickinterpolationwithscalingWenowconsiderasimplevariationofproblem(10.11):Given¸1;:::;¸mwith¸i2ppC+,u1;:::;um,withui2Candv1;:::;vm,withvi2C,i=1;:::;m,theproblemisto¯nd(¯)¯¤¡1¯HisanalyticinC+;D=D>0°opt=infkDHDk1¯;(10.12)¯D2D;H(¸i)ui=vi;i=1;:::;mwhereDisthesetofm£mblock-diagonalmatriceswithsomespeci¯edblockstruc-ture.Problem(10.12)correspondsto¯ndingthesmallestscaledH1normofallinterpolants.Thisproblemarisesinmulti-inputmulti-outputH1controlsynthesisforsystemswithstructuredperturbations.WithachangeofvariablesP=D¤Dandthediscussionintheprevioussection,itfollowsthat°optisthesmallestpositive°suchthatthereexistsP>0,P2Dsuchthatthefollowingequationsandinequalityhold:A¤G+GA¡U¤PU=0;ininA¤G+GA¡V¤PV=0;outout°2G¡G¸0:inoutThisisaGEVP.10.5.3Frequencyresponseidenti¯cationWeconsidertheproblemofidentifyingthetransferfunctionHofalinearsystemfromnoisymeasurementsofitsfrequencyresponseatasetoffrequencies.WeseekHsatisfyingtwoconstraints:Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 10.6TheInverseProblemofOptimalControl147²Consistencywithmeasurements.Forsomeniwithjnij·²,wehavefi=H(j!i)+ni.Here,fiisthemeasurementofthefrequencyresponseatfrequency!i,niisthe(unknown)measurementerror,and²isthemeasurementprecision.²Priorassumption.®-shiftedH1normofHdoesnotexceedM.FromNevanlinna-Picktheory,thereexistHandnwithH(!i)=fi+nisatisfyingtheseconditionsifandonlyifthereexistniwithjnij·²,Gin>0andGout>0suchthatM2G¡G¸0;inout(A+®I)¤G+G(A+®I)¡e¤e=0;inin(A+®I)¤G+G(A+®I)¡(f+n)¤(f+n)=0;outoutwhereA=diag(j!1;:::;j!m),e=[1¢¢¢1],f=[f1¢¢¢fm],andn=[n1¢¢¢nm].ItcanbeshownthattheseconditionsareequivalenttoM2G¡G¸0;inout(A+®I)¤G+G(A+®I)¡e¤e=0;inin(A+®I)¤G+G(A+®I)¡(f+n)¤(f+n)¸0;outoutwithjnij·².Withthisobservation,wecanansweranumberofinterestingquestionsinfre-quencyresponseidenti¯cationbysolvingEVPsandGEVPs.²For¯xed®and²,minimizeM.SolvingthisGEVPanswersthequestionGiven®andaboundonthenoisevalues,whatisthesmallestpossible®-shiftedH1normofthesystemconsistentwiththemeasurementsofthefrequencyresponse?"²For¯xed®andM,minimize².SolvingthisEVPanswersthequestionGiven®andaboundon®-shiftedH1normofthesystem,whatisthesmallest"possiblenoisesuchthatthemeasurementsareconsistentwiththegivenvaluesof®andM.10.6TheInverseProblemofOptimalControlGivenasystem"#"#Q1=20xx_=Ax+Bu;x(0)=x0;z=;0R1=2uwith(A;B)isstabilizable,(Q;A)isdetectableandR>0,theLQRproblemistodetermineuthatminimizesZ1zTzdt:0Thesolutionofthisproblemcanbeexpressedasastate-feedbacku=KxwithK=¡R¡1BTP,wherePistheuniquenonnegativesolutionofATP+PA¡PBR¡1BTP+Q=0:Theinverseproblemofoptimalcontrolisthefollowing.GivenamatrixK,determineifthereexistQ¸0andR>0,suchthat(Q;A)isdetectableandu=KxistheThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 148Chapter10MiscellaneousProblemsoptimalcontrolforthecorrespondingLQRproblem.Equivalently,weseekR>0andQ¸0suchthatthereexistsPnonnegativeandP1positive-de¯nitesatisfying(A+BK)TP+P(A+BK)+KTRK+Q=0;BTP+RK=0andATP+PA1g:Next,supposethatthestatequantizationnoiseismodeledasawhitenoisesequencew(k)withEw(k)Tw(k)=´,injecteddirectlyintothestate,anditse®ectontheoutputismeasuredbyitsRMSvalue,whichisjust´timestheH2normofthestate-to-outputtransfermatrix:°°p=´°C(zI¡A)¡1T°:(10.16)noise2OurproblemisthentocomputeTtominimizethenoisepower(10.16)subjecttotheover°owavoidanceconstraint(10.15).Theconstraint(10.15)isequivalenttotheexistenceofP>0suchthat"#ATPA¡P+T¡TT¡1ATPB·0:BTPABTPB¡I=®2Theoutputnoisepowercanbeexpressedasp2=´2TrTTWT;noiseowhereWoistheobservabilityGramianoftheoriginalsystem(A;B;C),givenbyX1W=(AT)kCTCAk:ok=0Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 10.7SystemRealizationProblems149WoisthesolutionoftheLyapunovequationATWA¡W+CTC=0:ooWithX=T¡TT¡1,therealizationproblemis:Minimize´2TrWX¡1subjectotoX>0andtheLMI(inP>0andX)"#ATPA¡P+XATPB·0:(10.17)BTPABTPB¡I=®2ThisisaconvexprobleminXandP,andcanbetransformedintotheEVPminimize´2TrYWc"#YIsubjectto(10:17);P>0;¸0IXSimilarmethodscanbeusedtohandleseveralvariationsandextensions.InputuhasRMSvalueboundedcomponentwiseSupposetheRMSvalueofeachcomponentuiislessthan®(insteadoftheRMSvalueofubeinglessthan®)andthattheRMSvalueofthestateisstillrequiredtobelessthanone.Then,usingthemethodsofx10.9,anequivalentconditionistheLMI(withvariablesX,PandR):"#ATPA¡P+XATPB·0;P>0;(10.18)BTPABTPB¡R=®2whereR>0isdiagonal,withunittrace,sothattheEVP(overX,Y,PandR)isminimize´2TrYWcsubjectto(10:18);P>0;X>0R>0;Risdiagonal;TrR=1"#YI¸0IXMinimizingstate-to-outputH1normAlternatively,wecanmeasurethee®ectofthequantizationnoiseontheoutputwiththeHnorm,thatis,wecanchooseTtominimizekC(zI¡A)¡1Tksubjecttothe11constraint(10.15).TheconstraintkC(zI¡A)¡1Tk·°isequivalenttotheLMI1(overY>0andX=T¡TT¡1>0):"#ATYA¡Y+CTCATY·0:(10.19)YAY¡X=®2ThereforechoosingTtominimizekC(zI¡A)¡1Tksubjectto(10.15)istheGEVP1minimize°subjecttoP>0;Y>0;X>0;(10:17);(10:19)AnoptimalstatecoordinatetransformationTisanymatrixthatsatis¯esXopt=(TTT)¡1,whereXisanoptimalvalueofXintheGEVP.optThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 150Chapter10MiscellaneousProblems10.8Multi-CriterionLQGConsidertheLTIsystemgivenby"#"#Q1=20xx_=Ax+Bu+w;y=Cx+v;z=;0R1=2uwhereuisthecontrolinput;yisthemeasuredoutput;zistheexogenousoutput;wandvareindependentwhitenoisesignalswithconstant,positive-de¯nitespectraldensitymatricesWandVrespectively;wefurtherassumethatQ¸0andR>0,that(A;B)iscontrollable,andthat(C;A)and(Q;A)areobservable.ThestandardLinear-QuadraticGaussian(LQG)problemistominimizeJ=limEz(t)Tz(t)lqgt!1overu,subjecttotheconditionthatu(t)ismeasurableony(¿)for¿·t.TheoptimalcostisgivenbyJ¤=Tr(XU+QY);lqglqglqgwhereXlqgandYlqgaretheuniquepositive-de¯nitesolutionsoftheRiccatiequationsATX+XA¡XBR¡1BTX+Q=0;lqglqglqglqg(10.20)AY+YAT¡YCTV¡1CY+W=0;lqglqglqglqgandU=YCTV¡1CY.lqglqgInthemulti-criterionLQGproblem,wehaveseveralexogenousoutputsofinterestgivenby"#"#1=2Qi0xzi=1=2;Qi¸0;Ri>0;i=0;:::;p:0RiuWeassumethat(Q0;A)isobservable.Foreachzi,weassociateacostfunctionJi=limEz(t)Tz(t);i=0;:::;p:(10.21)lqgiit!1Themulti-criterionLQGproblemistominimizeJ0overusubjecttothemeasurabil-lqgityconditionandtheconstraintsJi<°;i=1;:::;p.Thisisaconvexoptimizationlqgiproblem,whosesolutionisgivenbymaximizingXpTr(XlqgU+QYlqg)¡°i¿i;i=1overnonnegative¿1;:::;¿p,whereXlqgandYlqgarethesolutionsof(10.20)withXpXpQ=Q0+¿iQiandR=R0+¿iRi:i=1i=1NotingthatXlqg¸XforeveryX>0thatsatis¯esATX+XA¡XBR¡1BTX+Q¸0;weconcludethattheoptimalcostisthemaximumofÃ!XpXpTrXU+(Q0+¿iQi)Ylqg¡°i¿i;i=1i=1Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. 10.9NonconvexMulti-CriterionQuadraticProblems151overX,¿1;:::;¿p,subjecttoX>0,¿1¸0;:::;¿p¸0andÃp!¡1pXXATX+XA¡XBR+¿RBTX+Q+¿Q¸0:0ii0iii=1i=1ComputingthisisanEVP.10.9NonconvexMulti-CriterionQuadraticProblemsInthissection,weconsidertheLTIsystemx_=Ax+Bu;x(0)=x0;where(A;B)iscontrollable.Foranyu,wede¯neasetofp+1costindicesJ0;:::;JpbyZ"#"#1QCxTTiiJi(u)=[xu]dt;i=0;:::;p:CTRu0iiHerethesymmetricmatrices"#QiCi;i=0;:::;p;CTRiiarenotnecessarilypositive-de¯nite.Theconstrainedoptimalcontrolproblemis:minimizeJ0;(10.22)subjecttoJi·°i;i=1;:::;p;x(t)!0ast!1Thesolutiontothisproblemproceedsasfollows:We¯rstde¯neXpXpXpQ=Q0+¿iQi;R=R0+¿iRi;C=C0+¿iCi;i=1i=1i=1where¿1¸0;:::;¿p¸0.Next,with¿=[¿1¢¢¢¿p],wede¯neXpXp¢S(¿;u)=J0+¿iJi¡¿i°i:i=1i=1Then,thesolutionofproblem(10.22),whenitisnot¡1,isgivenby:supinffS(¿;u)jx(t)!0ast!1g¿u(SeetheNotesandReferences.)Forany¯xed¿,thein¯mumoveruofS(¿;u),whenitisnot¡1,iscomputedbysolvingtheEVPinP=PTXpmaximizex(0)TPx(0)¡¿°ii"i=1#(10.23)ATP+PA+QPB+CTsubjectto¸0BTP+CRTherefore,theoptimalcontrolproblemissolvedbytheEVPinP=PTand¿:Xpmaximizex(0)TPx(0)¡¿°ii"i=1#ATP+PA+QPB+CTsubjectto¸0;¿1¸0;:::;¿p¸0:BTP+CRThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 152Chapter10MiscellaneousProblemsNotesandReferencesOptimizationoverana±nefamilyoflinearsystemsOptimizationoverana±nefamilyoftransfermatricesarisesinlinearcontrollerdesignviaYoula'sparametrizationofclosed-looptransfermatrices;seeBoydandBarratt[BB91]andthereferencestherein.Otherexamplesinclude¯nite-impulse-response(FIR)¯lterdesign(seeforexample[OKUI88])andantennaarrayweightdesign[KRB89].In[Kav94]Kavrano¸gluconsiderstheproblemofapproximatingintheH1normagiventransfermatrixbyaconstantmatrix,whichisaproblemoftheformconsideredinx10.1.HecaststheproblemasanEVP.StabilityofsystemswithLTIperturbationsForreferencesaboutthisproblem,wereferthereadertotheNotesandReferencesinChap-ter3.LMI(10.6)isderivedfromTheorems3and4inWillems[Wil71b].PositiveorthantstabilizabilityTheproblemofpositiveorthantstabilityandholdabilitywasextensivelystudiedbyBerman,NeumannandStern[BNS89,x7.4].Theysolvethepositiveorthantholdabilityproblemwhenthecontrolinputisscalar;ourresultsextendtheirstothemulti-inputcase.Thestudyofpositiveorthantholdabilitydrawsheavilyfromthetheoryofpositivematrices,referencesforwhichare[BNS89]andthebookbyBermanandPlemmons[BP79,ch6].DiagonalsolutionstotheLyapunovequationplayacentralroleinthepositiveorthantstabilizabilityandholdabilityproblems.ArelatedpaperisbyGeromel[Ger85],whogivesacomputationalprocedureto¯nddiagonalsolutionsoftheLyapunovequation.DiagonalquadraticLyapunovfunctionsalsoariseinthestudyoflarge-scalesystems[MH78];seealsox3.4.AsurveyofresultsandapplicationsofdiagonalLyapunovstabilityisgiveninKaszkurewiczandBhaya[KB93].Ageneralizationoftheconceptofpositiveorthantstabilityisthatofinvarianceofpolyhedralsets.Theproblemofcheckingwhetheragivenpolyhedronisinvariantaswellastheassociatedstate-feedbacksynthesisproblemhavebeenconsideredbymanyauthors,seee.g.,[Bit88,BBT90,HB91b]andreferencestherein.TheLMImethodofx10.3canbeextendedtothepolyhedralcase,usingtheapproachof[CH93].Finally,wenotethattheresultsofx10.3areeasilygeneralizedtotheLDIsconsideredinChapter4.StabilizationofsystemswithtimedelaysTheintroductionofLyapunovfunctionalsoftheform(10.9)isduetoKrasovskii[Kra56].Skorodinskii[Sko90]observedthattheproblemofprovingstabilityofasystemwithdelaysviaLyapunov{Krasovskiifunctionalsisaconvexproblem.Seealso[PF92,WBL92,SL93b,FBB92].Itisinterestingtonotethatthestructuredstabilityproblem"or¹-analysisproblem"canbecastintermsoflinearsystemswithdelays.ConsiderastableLTIsystemwithtransfermatrixHandLinputsandoutputs,connectedinfeedbackwithadiagonaltransfermatrix¢withH1normlessthanone.Thissystemisstableforallsuch¢ifandonlyifsup¹(H(j!))<1,where¹denotesthestructuredsingularvalue(see[Doy82]).Itcan!2Rbeshown(see[Boy86])thatifthefeedbacksystemisstableforall¢whicharediagonal,¡sTiwith¢ii(s)=e,Ti>0(i.e.,delays)thenthesystemisstableforall¢withk¢k1·1.Thus,verifyingsup¹(H(j!))<1canbecastascheckingstabilityofalinearsystem!2RwithLarbitrary,positivedelays.InparticularweseethatKrasovskii'smethodfrom1956canbeinterpretedasaLyapunov-based¹-analysismethod.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences153InterpolationproblemsTheclassicalNevanlinna-Pickproblemdatesbackatleastto1916[Pic16,Nev19].Therearetwomatrixversionsofthisproblem:thematrixNevanlinna-Pickproblem[DGK79]andthetangentialNevanlinna-Pickproblem[Fed75].AcomprehensivedescriptionoftheNevanlinna-Pickproblem,itsextensionsandvariationscanbefoundinthebookbyBall,GohbergandRodman[BGR90].AnumberofresearchershavestudiedtheapplicationofNevanlinna-Picktheorytoproblemsinsystemandcontrol:Delsarte,GeninandKampdiscusstheroleofthematrixNevanlinna-Pickproblemincircuitandsystemstheory[DGK81].ZamesandFrancis,in[ZF83],studytheimplicationsofinterpolationconditionsincontrollerdesign;alsosee[OF85].ChangandPearson[CP84]solveaclassofH1-optimalcontrollerdesignproblemsusingmatrixNevanlinna-Picktheory.Safonov[Saf86]reducesthedesignofcontrollersforsystemswithstructuredperturbationstothescaledmatrixNevanlinna-Pickproblem.In[Kim87],KimurareducesaclassofH1-optimalcontrollerdesignproblemstothetangentialNevanlinna-Pickproblem;seealso[Bal94].Systemidenti¯cationThesystemidenti¯cationproblemconsideredinx10.5.3canbefoundin[CNF92].Other+systemidenti¯cationproblemscanbecastintermsofLMIproblems;seee.g.,[PKT94].TheinverseproblemofoptimalcontrolThisproblemwas¯rstconsideredbyKalman[Kal64],whosolveditwhenthecontroluisscalar;seealso[And66c].ItisdiscussedingreatdetailbyAndersonandMoore[AM90,x5.6],FujiiandNarazaki[FN84]andthereferencestherein;theysolvetheproblemwhenthecontrolweightingmatrixRisknown,bycheckingthatthereturndi®erenceinequality¡¢¡¢TT¡1T¡1I¡B(¡i!I¡A)KRI¡K(i!I¡A)B¸Rholdsforall!2R.OurresulthandlesthecaseofunknownR,i.e.,themostgeneralformoftheinverseoptimalcontrolproblem.SystemrealizationproblemsForadiscussionoftheproblemofsystemrealizationssee[MR76,Thi84],whichgivean-alyticalsolutionsviabalancedrealizationsforspecialrealizationproblems.ThebookbyGeversandLi[GL93a]describesanumberofdigital¯lterrealizationproblems.Liu,SkeltonandGrigoriadisdiscusstheproblemofoptimal¯niteword-lengthcontrollerimplementa-tion[LSG92].SeealsoRoteaandWilliamson[RW94b,RW94a].WeconjecturethattheLMIapproachtosystemrealizationcanalsoincorporatetheconstraintofstability,i.e.,thatthe(nonlinear)systemnotexhibitlimitcyclesduetoquantizationorover°ow;see,e.g.,[Wil74b].WealsonotethatanH1normconstraintontheinput-to-statetransfermatrixyields,indirectly,aboundonthepeakgainfrominputtostate;BoydandDoyle[BD87].Thiscanbeusedtoguaranteenoover°owgivenamaximumpeakinputlevel.Multi-criterionLQGThemulti-criterionLQGproblemisdiscussedin[BB91];seealsothereferencescitedthere.Rotea[Rot93]callsthisproblemgeneralizedH2control".Heshowsthatonecanrestrictattentiontoobserver-basedcontrollerssuchasdT¡1x^=(A+BK)x^+YCV(y¡Cx^);u=Kx^;dtThiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 154Chapter10MiscellaneousProblemswhereYisgivenby(10.20).Thecovarianceoftheresidualstatex^¡xisYandthecovariancePofthestatex^isgivenbythesolutionoftheLyapunovequation:TT¡1(A+BK)P+P(A+BK)+YCVCY=0:(10.24)iThevalueofeachcostindexJlqgde¯nedby(10.21)istheniTJlqg=TrRiKPK+TrQi(Y+P):Now,takingZ=KPasanewvariable,wereplace(10.24)bythelinearequalityinWandPTTTT¡1AP+PA+BZ+ZB+YCVCY=0andeachcostindexbecomesi¡1TJlqg=TrRiZPZ+TrQi(Y+P):Themulti-criterionproblemisthensolvedbytheEVPinP,ZandXminimizeTrR0X+TrQ0(P+Y)"#XZsubjectto¸0;TZPTTTT¡1AP+PA+BZ+ZB+YCVCY=0;TrRiX+TrQi(P+Y)·°i;i=1;:::;LThisformulationofthemulti-criterionLQGproblemisthedualofourformulation;thetwoformulationscanbeusedtogetherinane±cientprimal-dualmethod(see,e.g.,[VB93b]).WealsomentionanalgorithmdevisedbyZhu,RoteaandSkelton[ZRS93]tosolveasimilarclassofproblems.Foravariationonthemulti-criterionLQGproblemdiscussedhere,see[Toi84];seealso[TM89,Mak91].Nonconvexmulti-criterionquadraticproblemsThissectionisbasedonMegretsky[Meg92a,Meg92b,Meg93]andYakubovich[Yak92].Inourdiscussionweomittedimportanttechnicaldetails,suchasconstraintregularity,whichiscoveredinthesearticles.Also,Yakubovichgivesconditionsunderwhichthein¯mumisactuallyaminimum(thatis,thereexistsanoptimalcontrollawthatachievesthebestperformance).TheEVP(10.23)isderivedfromTheorem3inWillems[Wil71b].Manyofthesu±cientconditionsforNLDIsinChapters5{7canbeshowntobenecessaryandsu±cientconditionswhenweconsiderintegralconstraintsonpandq,byanalyzingthemasnonconvexmulti-criterionquadraticproblems.Thisisillustratedinthefollowingsection.MixedH2{H1problemWeconsidertheanalogofanNLDIwithintegralquadraticconstraint:"#"#1=2Q0xx_=Ax+Bpp+Buu;q=Cqx;z=1=20RuZZ11TTppdt·qqdt00Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. NotesandReferences155R1TDe¯nethecostfunctionJ=zzdt.OurgoalistodetermineminKmaxpJ,overall0possiblelinearstate-feedbackstrategiesu=Kx.(Notethatthisisthesameastheprob-lemconsideredinx7.4.1withintegralconstraintsonpandqinsteadofpointwiseintimeconstraints.)For¯xedK,¯ndingthemaximumofJoveradmissiblep'sisdoneusingtheresultsofx10.9.ThismaximumisobtainedbysolvingtheEVPTminimizex(0)Px(0)23T(A+BuK)P+P(A+BuK)+6TTPBp7subjectto4Q+KRK+¿CqCq5·0;¿¸0TBpP¡¿I¡1TheminimizationoverKisnowsimplydonebyintroducingthenewvariablesW=P,Y=KW,¹=1=¿.Indeed,inthisparticularcase,weknowwecanaddtheconstraintP>0.ThecorrespondingEVPisthenT¡1minimizex(0)Wx(0)2Ã!3TTWA+YBu+AW+BuY+6TT¹Bp7subjectto4WQW+YRY+WCqCqW=¹5·0;¹¸0T¹Bp¡¹IwhichisthesameastheEVP(7.23).Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. Notationkm£nR;R;RTherealnumbers,realk-vectors,realm£nmatrices.R+Thenonnegativerealnumbers.CThecomplexnumbers.Re(a)Therealpartofa2C,i.e.,(a+a¤)=2.nSTheclosureofasetSµR.IkThek£kidentitymatrix.Thesubscriptisomittedwhenkisnotrelevantorcanbedeterminedfromcontext.MTTransposeofamatrixM:(MT)=M.ijjiM¤Complex-conjugatetransposeofamatrixM:(M¤)=M¤,whereijji®¤denotesthecomplex-conjugateof®2C.n£nPnTrMTraceofM2R,i.e.,i=1Mii.M¸0Missymmetricandpositivesemide¯nite,i.e.,M=MTandzTMz¸0forallz2Rn.M>0Missymmetricandpositive-de¯nite,i.e.,zTMz>0forallnonzeronz2R.M>NMandNaresymmetricandM¡N>0.M1=2ForM>0,M1=2istheuniqueZ=ZTsuchthatZ>0,Z2=M.¸(M)ThemaximumeigenvalueofthematrixM=MT.max¸(P;Q)ForP=PT,Q=QT>0,¸(P;Q)denotesthemaximumeigen-maxmaxvalueofthesymmetricpencil(P;Q),i.e.,¸(Q¡1=2PQ¡1=2).max¸(M)TheminimumeigenvalueofM=MT.minpkMkThespectralnormofamatrixorvectorM,i.e.,¸max(MTM).pReducestotheEuclideannorm,i.e.,kxk=xTx,foravectorx.diag(¢¢¢)Block-diagonalmatrixformedfromthearguments,i.e.,23M167diag(M;:::;M)=¢6..7:1m4.5Mm157 158NotationnCoSConvexhullofthesetSµR,givenby(¯)Xp¯¢¯CoS=¸ixi¯xi2S;p¸0:¯i=1(Withoutlossofgenerality,wecantakep=n+1here.)ExExpectedvalueof(therandomvariable)x.Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. AcronymsAcronymMeaningPageAREAlgebraicRiccatiEquation3CPConvexMinimizationProblem11DIDi®erentialInclusion51DNLDIDiagonalNorm-boundLinearDi®erentialInclusion54EVPEigenvalueMinimizationProblem10GEVPGeneralizedEigenvalueMinimizationProblem10LDILinearDi®erentialInclusion51LMILinearMatrixInequality7LMIPLinearMatrixInequalityProblem9LQGLinear-QuadraticGaussian150LQRLinear-QuadraticRegulator114LTILinearTime-invariant52NLDINorm-boundLinearDi®erentialInclusion53PLDIPolytopicLinearDi®erentialInclusion53PRPositive-Real25RMSRoot-Mean-Square91159 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IndexAComplementAbsolutestability,54,56,72orthogonal,22AcronymsSchur,7apologyfortoomany,5Completionproblem,40,48list,159ComplexityA±neconvexoptimization,18,29familyoftransfermatrices,141S-procedure,24matrixinequality,27stabilizationproblem,128AlgebraicRiccatiequation,3,26,110,115ComponentwiseresultsAlgorithmusingscaling,111,134conjugate-gradients,18viascaling,94ellipsoid,49Componentwiseunit-energyinputforLyapunovinequalities,18LDI,81,96interior-point,30state-feedback,106Karmarkar's,4Concavequadraticprogramming,42methodofcenters,16,30Conditionnumber,37NesterovandNemirovskii's,4coordinatetransformation,65,103Newton,15Conjugate-gradients,18projective,30Constrainedoptimalcontrolproblem,151Almostsureconvergence,131Constraintsystemwithmultiplicativenoise,136equality,9,19®-shiftedH1norm,67integralquadraticvs.pointwise,96,Analyticcenter,15122ellipsoidalapproximation,45oncontrolinput,103ellipsoidalbound,49quali¯cation,19Analyticsolution,2,62,115Contractivecompletionproblem,48LMI,24Controlinput,52Approximationnormconstraint,103ellipsoidal,49ControllabilityGramian,78ofPLDI,58ControllerARE,3,26,110,115dynamicfeedback,111Augmentedsystem,124gain-scheduled,118order,111Breduced-order,117state-feedback,99Barrierfunction,15ConvexBellmanfunction,112Lyapunovfunction,74Bounded-reallemma,26ConvexfunctionBrownianmotion,137LMIrepresentation,29Lyapunov,64CConvexhull,12,51,158Centralpath,16ConvexoptimizationCheapellipsoidapproximation,50complexity,18,29Circlecriterion,2duality,5Closed-loopsystemellipsoidalgorithm,12dynamicstate-feedback,116interior-pointmethod,14staticstate-feedback,99problemstructure,18Youlaparametrization,152software,31Co,12,51Convexproblem,11187 188IndexConvolution,143state-feedback,109CoordinatetransformationExponentialtime-weighting,84,97conditionnumber,65Extractableenergydigital¯lterrealization,148LDI,87state-feedback,103state-feedback,109CP,11Cuttingplane,12FFadingmemory,74DFamilyoftransfermatricesDecayrate,66examples,152Delay,144parametrization,143DI,51FeedbackDiagonaldiagonalnorm-bound,54Lyapunovfunction,40,47,144nonexpansive,123norm-boundLDI,54structured,54Diameter,69time-varying,53invariantellipsoidofLDI,80Finsler'slemma,22NLDI,59Frequencymeasurements,146Di®erentialinclusion,51Functionselector-linear,56quasiconvex,29Dissipativityvalue,112LDI,93state-feedback,110GDNLDI,54Gainwell-posedness,57L2orRMS,91Duality,5,9,29,49state-feedback,99Dynamicstate-feedback,111Gain-scheduledcontroller,118°-entropy,142EGeneralizedeigenvalueproblem,10E,11GEVP,10Eigenvalueproblem,10Globallinearization,54Eliminationstate-feedback,99inLMI,22Gramianofvariable,9controllability,78procedure,48observability,85Ellipsoidalgorithm,12,29,49Happroximatingintersection,44Half-spaceapproximatingpolytope,42cuttingplane,12approximatingreachableset,78,104,Hamiltonianmatrix,26106,121,125,132Hankelnormapproximatingsum,46approximatingunion,43LDI,89approximation,49transfermatrix,142diameter,69,80H1norm,27,91,142extractableenergy,87,109Holdability,102holdable,102Holdableellipsoidinvariant,68approximatingreachableset,106LÄowner{John,44,49LDI,102minimumvolume,11,44outputenergy,108poorman's,50H2norm,141viaanalyticcenter,45,49H2{H1problem,154Entropy,142Equalityconstraint,19IEuclideannorm,8Identi¯cation,146EVP,10I°,142Exogenousinput,52Implicitequalityconstraint,19E,5Impulseresponse,98ExpectedoutputenergyInequalityLDI,87return-di®erence,153Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. Index189InfeasibleLMI,29reachableset,77,104Initialconditionreturntime,71extractableenergy,85,108scaling,94,111safe,71stability,61Inputstabilitymargin,65controlorexogenous,52stable,notquadraticallystable,64,unit-energy,77,104,121,125,13273unit-peak,82,107state-feedback,99Input-to-outputpropertiesunit-energyinput,77LDI,89unit-peakinput,82Lur'esystem,122Lemmastate-feedback,109bounded-real,26systemwithmultiplicativenoise,133Finsler,22Input-to-statepropertiesPR,25,93¯lterrealization,148LinearLDI,77inequality,7Lur'esystem,121matrixinequality,7state-feedback,104program,10systemwithmultiplicativenoise,132Lineardi®erentialinclusion,51systemwithunknownparameters,125Linear-fractionalIntegralquadraticconstraint,96,122mapping,53Interior-pointmethod,4,14,30programming,11primal-dual,18LinearizationIntervalmatrix,40global,54InvarianceLinear-QuadraticRegulator,108ofpositiveorthant,152Line-search,28InvariantellipsoidLMIapproximatingreachableset,78,104,analyticcenter,15106analyticsolution,24extractableenergy,87convexfunctionrepresentation,29forLDI,68de¯nition,7outputenergy,108equalityconstraint,9Inverseproblemofoptimalcontrol,147father,4It^ostochasticsystem,136feasibilityproblem,9grandfather,4Kgraphicalcondition,2·,37history,2Kalman-Yakubovich-Popovlemma,2infeasible,29Karmarkar'salgorithm,4interior-pointmethod,4,14multiple,7Lnonstrict,7,18LÄowner{Johnellipsoid,49optimization,30LDIreduction,19,31bettername,56Riccatiequationsolution,3componentwiseunit-energyinput,81semide¯niteterms,22decayrate,66slackvariable,8de¯nition,51software,31diagonalnorm-bound,54standardproblems,9dissipativity,93,110strict,18expectedoutputenergy,87LMI-Lab,31Hankelnorm,89LMIP,9invariantellipsoid,68LMI-tool,31L2gain,91,109Looptransformation,119norm-bound,53sectorcondition,129outputenergy,85,108LÄowner{Johnellipsoid,44outputpeak,88LP,10polytope,53LQGpositiveorthantstabilizability,144multi-criterion,150quadraticallystabilizable,100systemwithmultiplicativenoise,138quadraticstability,61LQR,108,114Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 190IndexLTIsystemtransfer,25de¯nition,52Maximumsingularvalue,8input-to-stateproperties,78Mean-squarestabilityovershoot,97de¯nition,132realization,148state-feedback,135stability,62Methodofcenters,16,30transfermatrix,91Minimumsizeellipsoid,69withdelay,144Minimumvolumeellipsoid,11withnonparametricuncertainty,143MixedH2{H1problem,154L2gainM-matrix,75LDI,91Moore{Penroseinverse,28LTIsystem,142¹-analysis,152Lur'esystem,122Multi-criterionLQGproblem,96,150,153scaled,94MultipleLMI,7state-feedback,109Multiplicativenoise,131systemwithmultiplicativenoise,133MultiplierLur'eresolvingequation,3nonparametricuncertainty,143Lur'esystemsystemwithunknownparameters,124absolutestability,56theory,128de¯nition,119history,126NLur'etermNesterovandNemirovskii'salgorithm,4inLyapunovfunction,119Nevanlinna-Pickinterpolation,145LyapunovNevanlinna-Picktheory,146equation,2,25,62,78,85Newton'smethod,15exponent,66NLDIfunctional,144approximatingPLDI,58inequality,2,8de¯nition,53Lyapunovfunctioninput-to-stateproperties,79convex,64,74quadraticstability,62diagonal,40,47,144state-to-outputproperties,84integralquadraticconstraints,123,127well-posedness,53,63Lur'e,119Noiseparameter-dependent,119ininterpolationproblems,146quadratic,61multiplicative,131stochastic,132,136NonconvexquadraticproblemLyapunovinequalitymixedLQR{L2problem,154reductiontoastrictone,20multi-criterionLQG,151Nonexpansive,26Mfeedback,123MarginLDI,91quadraticstability,65Nonlinearsystemquadraticstabilizability,113fadingmemory,74systemwithmultiplicativenoise,132state-feedback,99matlab,31NonparametricuncertaintyMatrixinLTIsystem,143completionproblem,40NonstrictLMI,7,18eliminationofvariable,22,48NormHamiltonian,26®-shiftedH1,67inequality,7Euclidean,8interval,40Hankel,89linear-fractionalmapping,53H1,27,91M,75H2,141Moore{Penroseinverse,28matrix,8norm,8maximumsingularvalue,8P0,57ofLTIsystem,148pencil,28quadraticapproximationofpiecewisePick,146linear,41problems,37scaledsingularvalue,38quadraticinequality,8Norm-boundLDI,53Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. Index191NotationPerturbationdi®erentialequations,5LTI,143table,157norm-bound,53NP-complete,42stochastic,137NP-hard,128structured,39,54well-posednessofDNLDI,54,57Pickmatrix,146Piecewiselinearnorm,41OPLDI,53ObservabilityGramian,85approximationbyNLDI,58•,51input-to-stateproperties,78Optimalcontrol,112,151quadraticstability,62Optimizationstate-to-outputproperties,84ellipsoidalgorithm,12Pointwiseconstraint,96,122interior-pointmethod,14Polyhedronlinear-fractional,11invariant,152linearprogram,10Polynomial-time,12LMI,30Polytope,12quasiconvex,11approximatedbyinvariantellipsoid,optin,3169Orderofcontroller,111containingreachableset,80Orthantellipsoidapproximation,42invariant,152extractableenergy,87,109stability,144LDI,53Orthogonalcomplement,22returntime,71Outerapproximationofreachablesetsymmetric,48state-feedback,106PolytopicLDI,53Output,52Polytopicnorm,41OutputenergyPopovcriterion,2,119LDI,85Positive-de¯niteLur'esystem,121completion,40state-feedback,108matrix,7Outputpeakrepresentable,29LDI,88Positiveorthantstabilizability,144,152Outputvarianceforwhitenoiseinput,113,Positive-real138lemma,25Over°ow,148transfermatrix,25OvershootPrimal-dualinterior-pointmethod,18LTIsystem,97Procedureeliminationofvariable,22PS,23P0matrix,57Projectivealgorithm,30Parameter-dependentPRlemma,2,25,93controller,118Lyapunovfunction,119Parametricuncertainty,124Qscalingmatrixproblem,39QuadraticParametrizationLyapunovfunction,61familyoftransfermatrices,143matrixinequality,8,114Youla,152QuadraticapproximationPassive,25ofpolytopicnorm,41Passivity,3ofreachableset,105dissipativity,93Quadraticprogrammingstate-feedback,110inde¯nite,42Pathofcenters,16QuadraticstabilityPencil,28invariantellipsoid,68Penroseinverse,28LDI,61Performancemargin,65LQGcost,150nonlinearsystem,74ofLDI,113vs.stability,64systemwithmultiplicativenoise,133Quadraticstabilizability,100Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed. 192IndexQuali¯cationSelf-concordantbarrier,15constraint,19Simultaneousmatrixcompletion,40Quantization,148Singularvalueplot,27QuasiconvexSlackvariableinLMIs,8function,29Small-gaintheorem,3,143problem,11SoftwareforLMIproblems,31Sparsitypattern,40RSphereReachablesetsmallestorlargest,44,46LDI,77S-procedureLur'esystem,121andscaling,64relaxedDI,51description,23state-feedback,104Stabilitysystemwithmultiplicativenoise,132absolute,56,72systemwithunknownparameters,125degree,67Ruce,81LDI,61Rue,77LTIsystem,62Rup,82Lur'esystem,120Realizationofsystem,148positiveorthant,144Reduced-ordercontroller,117systemwithmultiplicativenoise,132Reductionsystemwithunknownparameters,125Lyapunovinequality,20StabilitymarginstrictlyfeasibleLMI,19LDI,65Regularityconditions,4mean-square,132,137RelaxedversionofaDI,51StabilizabilityResponseLDI,100steporimpulse,97positiveorthant,144Return-di®erenceinequality,153systemwithmultiplicativenoise,134ReturntimeStabilizationproblemLDI,71complexity,128Riccatiequation,3,26,115StandardLMIproblems,9Riccatiinequality,110,115¤,143RMSgainState,52LDI,91State-feedbackstate-feedback,109controller,99RMSnorm,148globallinearization,99Root-Mean-Square,91inlinearsystemwithdelay,145Routh{Hurwitzalgorithm,35LDI,99nonlinearsystem,99Spositiveorthantstability,144Safeinitialcondition,71scaling,111ScaledL2gainsynthesis,99LDI,94systemwithmultiplicativenoise,134state-feedback,111Statepropertiessystemwithmultiplicativenoise,134LDI,61Scaledsingularvalue,38Lur'esystem,120Scalingstate-feedback,100andS-procedure,64systemwithmultiplicativenoise,131componentwiseresults,94State-to-outputpropertiesforcomponentwiseresults,111,134¯lterrealization,148inmatrixproblem,37LDI,84interpolation,145Lur'esystem,121Schurcomplementstate-feedback,107nonstrict,28Staticstate-feedback,99strict,7Stepscilab,31length,28Sectorcondition,56response,97looptransformation,129Stochasticstabilitymargin,132Selector-linearStochasticsystemdi®erentialinclusion,56continuous-time,136Copyright°c1994bytheSocietyforIndustrialandAppliedMathematics. Index193discrete-time,131UnknownparametersStoppingcriterionsystemwith,124methodofcenters,17Stratonovitchstochasticsystem,136VStrictfeasibilityValuefunction,112reductionto,19VariableStrictLMI,18elimination,9Structuredmatrix,8feedback,54,152slack,8singularvalue,152VertexStructureduncertaintyofpolytope,41interpolation,146vol,13matrixproblem,39VolumeSturmmethod,26,35ellipsoidalgorithm,13Symmetricpolytope,48invariantellipsoid,69Synthesisofellipsoid,41output-feedback,117,150state-feedback,99,134WSystemWell-posednessrealization,148DNLDI,54,57stochastic,136NLDI,53,63uncertain,56Whitenoisewithmultiplicativenoise,131Brownianmotion,137Systemtheoryduality,5uncertainty,131Systemwithunknownparametersmatrixproblem,39multipliersfor,124YYoulaparametrization,152TTangentialNevanlinna-Pickinterpolation,145Time-varyingfeedbackmatrix,53system,51Tr,5Trace,5Transfermatrix,25LTIsystem,64,91optimization,141positive-real,25Transformationloop,119Tsypkincriterion,2UUncertaintyLTI,143parametric,124random,131time-varying,51Unit-energyinputLDI,77Lur'esystem,121state-feedback,104systemwithmultiplicativenoise,132systemwithunknownparameters,125Unit-intensitywhitenoise,138Unit-peakinput,97LDI,82state-feedback,107Thiselectronicversionisforpersonaluseandmaynotbeduplicatedordistributed.