GUARANTEEDPERFORMANCEROBUSTGAIN-SCHEDULINGCONTROLWITHUNCERTAINSCHEDULINGPARAMETERSByAliKhudhairAl-JibooryADISSERTATIONSubmittedtoMichiganStateUniversityinpartialoftherequirementsforthedegreeofMechanicalEngineeringŒDoctorofPhilosophy2016ABSTRACTGUARANTEEDPERFORMANCEROBUSTGAIN-SCHEDULINGCONTROLWITHUNCERTAINSCHEDULINGPARAMETERSByAliKhudhairAl-JibooryOneofthemainobjectivesincontroltheoryistodevelopcontrolstrategiesandsynthesiscon-ditionsthatnotonlyguaranteeclosed-loopstabilitybutalsoachieveguaranteedperformance.Inthisresearch,novelRobustGain-Scheduling(RGS)controlsynthesisconditionsaredevelopedforLinearParameter-Varying(LPV)systems.Incontrasttotheconventionalgain-schedulingsynthe-sismethods,theschedulingparametersareassumedtobeinexactlymeasured.Thisisapracticalassumptionsincemeasurementnoiseisunavoidableinpracticalengineeringapplications.ThecontributionsofthisdissertationarethecharacterizationofnovelsynthesisconditionsintermsofParametrizedLinearMatrixInequalities(PLMIs)andParametrizedBilinearMatrixInequalities(PBMIs)fordesigningRGScontrollerswithguaranteedstabilityandperformance.Multi-simplexmodelingapproachisutilizedtomodeltheschedulingparametersandtheirun-certaintiesinaconvexdomain.SynthesisconditionsforRGSState-Feedback(SF),full-orderDynamicOutput-Feedback(DOF),andStaticOutput-Feedback(SOF)controllersaredevelopedinaframework.MatrixcoefcheckapproachisusedtorelaxthePLMIsconditionsintodimensionalsetofLinearMatrixInequalities(LMIs)toobtaintheoptimalorsubopti-malcontroller.Theresultingcontrollernotonlyensuresrobustnessagainstschedulingparametersuncertaintiesbutalsoguaranteesclosed-loopperformanceundertheseuncertaintiesintermsofH2andH¥performance.Bythevirtueofintroducingextraslackvariables,controllersynthe-sisisindependentofLyapunovvariables,thatassuresimprovedperformanceandviabilityformulti-objectivecontrollersynthesiswithoutintroducingadditionalconservativeness.SincePB-MIsproblemsarenon-tractableingeneral,numericalalgorithmisdevelopedtosolvethePBMIsconditions.Numericalillustrativeexamplesandcomparisonswiththeexistingapproachesthatthedevelopedcontrolapproachoutperformstheexistingones.Furthermore,experimentalvalidationofthedevelopedRGScontrollershasbeenconductedonthetestbenchoftheElectricVariableValveTiming(EVVT)actuatorofautomotiveengines.En-ginespeedandvehiclebatteryvoltageareusedasnoisyschedulingparameters.TheexperimentsareperformedatMSUAutomotiveControlsLabataroomtemperatureof25.Experimentalresultsdemonstratetheeffectivenessofthedevelopedapproach.ToMyDadwithloveivACKNOWLEDGEMENTSGettingyourPh.D.degreedoneisexactlylikeyoureachedthepeakofahighmountainafteryouspentahardjourneytoclimbit.Theanalogyisreallyvalidbecausewhenyouclimbamountainyouwillneveraclearpavedwaytogetthere.Youhavetobreakrockstomakeyourownway.Additionally,sometimesasyougothroughyourway,unexpectedwindblowupandpullyoubackmanystepsbehind.Thenyouhavetotakeadeepbreath,lookforabetterway,andproceedclimbingagain.Insum,Ph.D.andmountainclimbingcannotbedonewithoutpassionandpersistence.Overthepastfouryears,Ihavereceivedsupport,encouragement,andassistancefrommanypeoplearoundmetogetthisdissertationdone.First,Iwouldliketoexpressmydeepgratitudeandappreciationtomyadvisor,ProfessorGuomingZhu,forhisguidance,advise,andsupportduringmyPh.D.studyatMichiganStateUniversity.Hesharedwithmehiswidediverseknowledgeandmadeconsiderableefforttomakemylifeeasier.Inmyopinion,Iwillneverevermeetapersonhassweetheartlikehim.IwouldliketothankProfessorJonguenChoifortheusefuldiscussionsduringyearofmystudy.GreatthankstoProfessorHassanKhalilandProfessorRanjanMukherjeeforservingonmyguidancecommittee.IhavegainedawealthofknowledgefromthelecturestaughtbyProfessorHassanKhalilthroughmystudyyears.Last,butnotleast,Iwouldliketoexpresssinceregratitudetomyfamilyfortheirpersistentencouragementandsupportovertheroughyears.Theybelievedinmeandmademebelieveinmyselfduringsomepainfultimes.vTABLEOFCONTENTSLISTOFTABLES.......................................viiiLISTOFFIGURES.......................................ixKEYTOABBREVIATIONS..................................xiCHAPTER1INTRODUCTION...............................11.1Background......................................11.2LPV,LTV,andLTISystems..............................51.3MotivationoftheWork................................71.4LiteratureSurvey...................................91.5Contributions.................................101.6Organization......................................11CHAPTER2PRELIMINARIESOFMULTI-SIMPLEXMODELING...........132.1Notations.......................................132.2andTerminologies............................132.3PolynomialCompletionandHomogenization....................152.4HomogeneousPolynomialLyapunovMatrix.....................172.5GeneralityoftheModelingApproach........................182.5.1Schedulingvariablesparameterization....................182.5.2Schedulingvariablesdependency.......................232.6Summary.......................................25CHAPTER3PROBLEMFORMULATIONANDSOLUTIONAPPROACH.......263.1ProblemFormulation.................................263.2SolutionApproach..................................313.2.1AftoMulti-SimplexTransformation...................313.2.1.1Singleschedulingparameter....................343.2.1.2Twoschedulingparameters....................343.2.1.3Multiplenumbersofschedulingparameters............363.2.2RateofVariationModeling..........................373.2.3PLMIsConditions..............................393.2.4PLMIsRelaxation...............................393.2.5PBMIAlgorithm...............................443.2.6InverseTransformation............................443.3Summary.......................................46CHAPTER4RGSSTATE-FEEDBACKCONTROL....................474.1SFSynthesisProblem.................................474.2RGSH2Control...................................484.3RGSH¥Control...................................53vi4.4ExtensiontoUnmeasurableParameters........................564.5NumericalExamples.................................594.6Summary.......................................65CHAPTER5RGSDYNAMICOUTPUT-FEEDBACKCONTROL............675.1DOFSynthesisProblem................................675.2DOFH2Control...................................695.3DOFH¥Control...................................805.4PBMIAlgorithm...................................885.5NumericalExamples.................................895.6Summary.......................................98CHAPTER6RGSSTATICOUTPUT-FEEDBACKCONTROL..............996.1SOFSynthesisProblem................................996.2ModelingApproach..................................1016.3PLMIsSynthesisConditionswithH2performance.................1026.3.1State-Feedbackcontrol............................1036.3.2StaticOutput-FeedbackControl.......................1046.4SynthesisConditionswithH¥Performance.....................1086.4.1State-FeedbackH¥control..........................1086.4.2StaticOutput-FeedbackH¥control.....................1096.5IllustrativeExamples.................................1146.5.1AcademicExample..............................1156.5.2EVVTActuator................................1166.6Summary.......................................122CHAPTER7EXPERIMENTALVALIDATIONONEVVTACTUATOR.........1237.1EVVTEngineCam-PhasingActuator........................1247.1.1ActuatorComponents.............................1257.1.2Testbenchset-up...............................1267.2SystemandLPVModeling.......................1267.2.1SystemTestsofEVVTActuator................1277.2.2LPVmodelconstruction...........................1287.2.3State-SpaceRepresentation..........................1307.3RGSControllerDesign................................1327.4ExperimentalResults.................................1347.5Summary.......................................136CHAPTER8CONCLUSIONSANDFUTURERESEARCH................1408.1Conclusions......................................1408.2RecommendationsforFutureResearch........................142APPENDIX...........................................144BIBLIOGRAPHY........................................153viiLISTOFTABLESTable4.1:GuaranteedH2performance:Theorem4.1.....................60Table4.2:GuaranteedH2performance:methodof[1]....................60Table4.3:H¥Guaranteedcostg¥usingTheorem4.2.....................63Table4.4:H¥Guaranteedcostg¥usingmethodof[1]....................63Table5.1:PossibleformulationsforCorollary5.2.......................80Table5.2:PossibleformulationsforCorollary5.4.......................88Table5.3:ComparisonoftheguaranteedH2boundnforCorollary5.2.eisthenumbergiveninparentheses().Actualclosed-loopH2-normisgivenbythenumberbetweenthesquarebrackets[]...........................91Table5.4:ComparisonofguaranteedH2performancewithothermethodsfromliterature..93Table5.5:ComparisonoftheguaranteedH¥boundg¥forCorollary5.2.eisthenum-bergiveninparentheses().Actualclosed-loopH¥-normisgivenbythenumberbetweenthesquarebrackets[].......................95Table5.6:ComparisonofguaranteedH¥performancewithmethodof[2]..........96Table6.1:H2performancewithdifferentboundsofmeasurementnoise.e=0:01,h=0:1.117Table6.2:Rangeofthetime-varyingparameters........................119Table6.3:H2performanceforstage1andstage2.e=105,h=0:01...........121Table7.1:FixedvaluesofenginespeedandbatteryvoltageusedinthesystemIDtests...127Table7.2:Coefq1(t)&q2(t)........................129Table7.3:Rangeofthetime-varyingparameters........................132viiiLISTOFFIGURESFigure1.1:ofGain-schedulingcontrol.....................2Figure1.2:LPVsysteminpolytopicstructure.........................4Figure1.3:LPVsysteminLFTstructure............................5Figure1.4:Topicofthedissertation..............................8Figure1.5:Dissertation'schaptersroadmap..........................12Figure2.1:Comparisonbetweendifferentmodelingapproach................24Figure3.1:Uncertaintydomainformeasuredschedulingparameter..............28Figure3.2:Closed-loopsystemwithoutput-feedbackgain-schedulingcontrol........28Figure3.3:Sixstagessolutionapproach............................32Figure4.1:H2guaranteedcost.................................61Figure4.2:Linesearchforetoobtaintheoptimalcontrollerforz=0:5andk=0:01....62Figure4.3:H¥guaranteedperformance............................63Figure4.4:H¥performancevs.ewithz=0:2........................64Figure4.5:Simulation:A)Measuredandexactschedulingparameters,B)Disturbanceattenuationresponsesassociatedwithexactandnoisyschedulingparameter...65Figure5.1:Algorithmconvergencefordifferentboundsofmeasurementnoise(withe=0:02).........................................90Figure5.2:ComparisonofH2guaranteedperformancevs.uncertaintysizebetweenthedevelopedconditionsandthemethodof[2]....................92Figure5.3:ComparisonoftheguaranteedH2performancevs.uncertaintyboundbe-tweenthedevelopedconditionsandmethodof[3].................93Figure5.4:Simulation:A)Measuredandactualschedulingparameters,B)Disturbanceattenuation.....................................94Figure5.5:ComparisonofguaranteedH¥performancebetweenTheorem5.2and[3]....96ixFigure5.6:Simulation:A)Measuredandactualschedulingparameters,B)Disturbanceattenuation.....................................97Figure6.1:Closed-loopsystemwithRGScontrol.......................100Figure6.2:Thedevelopedsynthesisapproach.........................101Figure6.3:AlgorithmConvergence..............................117Figure6.4:EVVTcam-phaseactuatorschematicdiagram...................118Figure6.5:Time-domainsimulationsfortheEVVTactuator(piecewise-constantschedul-ingsignals).....................................120Figure6.6:Time-domainsimulationsfortheEVVTactuator(sinusoidalschedulingsignals).121Figure7.1:FlowchartforthedesignandimplementationofaRGScontrolleronEVVTsystem........................................124Figure7.2:ElectricplanetarygearEVVTsystem.......................125Figure7.3:EngineExperimentsetup..............................127Figure7.4:Bodeplotofthe9localLTImodels........................128Figure7.5:Thevaryingparametersasfunctionofenginespeedandbatteryvoltage.(a)q1(N;V);(b)q2(N;V)..............................131Figure7.6:Engineexperimentaloperatingtrajectoryinparameterspace...........135Figure7.7:Measuredenginespeedandbatteryvoltage,andcam-phasingangletrackingwithstepreferenceof40degree..........................136Figure7.8:Measuredenginespeedandbatteryvoltage,andcam-phasingangletrackingwithstepchangereference.............................137Figure7.9:Speedandvoltagevariationswithperfectmeasurement(left)andnoisymea-surement(right)andcorrespondingcam-phaseresponses.............138Figure7.10:Speedandvoltagevariationswithperfectmeasurement(left)andnoisymea-surement(right)andcorrespondingcam-phaseresponses.............138Figure7.11:Speedandvoltagevariationswithperfectmeasurement(left)andnoisymea-surement(right)andcorrespondingcam-phaseresponses.............139xKEYTOABBREVIATIONSASPActualSchedulingParameterBMIBilinearMatrixInequalityDCDirectCurrentDOFDynamicOutput-FeedbackEVVTElectricVariableValveTimingGSGain-SchedulingICInternalCombustionISOFDIterativeStaticOutput-FeedbackDesignLFTLinearFractionalTransformationLMILinearMatrixInequalityLPVLinearParameter-VaryingLTVLinearTime-VaryingLTILinearTime-InvariantMSPMeasuredSchedulingParameterPDLFParameter-DependentLyapunovFunctionPLMIParametrizedLinearMatrixInequalityPBMIParametrizedBilinearMatrixInequalityqLPVquasi-LinearParameterVaryingRGSRobustGain-SchedulingROLMIPRobustLMIParserSFState-FeedbackSOSSum-Of-SquaresSVSlackVariableSOFStaticOutput-FeedbackTDCTopDeadCenterxiCHAPTER1INTRODUCTION"Intheblack-and-whiterobustversusgain-scheduledcontrolworld,agrayareashouldbecreatedthatallowstotrade-offclosed-loopperformancewithrobustnessagainstuncertaintyintheschedulingparameter."JanDeCaigny1.1BackgroundLinearParameter-Varying(LPV)systemsarealargeclassofdynamicalsystemsforwhichthefutureevolutionofthestatesdependsonthecurrentstatesofthesystemplussomeadditional(time-varying)parameterscalledschedulingparameters.Thesesystemsemergedfromnon-linearsystemtheoryandbecameoneofthemostsuccessfuldirectionsinthepost-moderncontrolera.Inthepastfewdecades,classical(orconventional)gain-schedulingcontrolapproachhadbeensuccessfullyappliedtowidevarietycontrolapplicationsfornonlinearandtime-varyingsystems.Theclassicalapproachescanbegenerallydescribedasdivideandconquertechniques,wherethecontrolproblemofnonlinearsystemsisdecomposedintoanumberoflinearsubproblems[4].Themajordif,atthattime,wasthelackofgeneraltheoryforanalyzingstabilityofLPVsystemsandforefdesigninggain-scheduledcontrollaws.Duetotheabsenceofaconcretetheoryforanalysisandsynthesis,theclassicalgain-scheduledcontrolmethodscomewithnoguar-anteesonstability,performanceorrobustness,aspointedoutinthepioneeringworkofShammaandAthans[5,6,7].Asaresultoftheseshortcomings,analysisandsynthesistheorieshavebeenpersistentlyconsideredandrevisitedbythecontrolcommunityoverthepastveyears,andacontinuingeffortisevidenttodevelopsolidtheorythatguaranteestabilityandperformanceof1Gain-SchedulingControlModernGain-SchedulingClassicalGain-SchedulingLinearisation-basedApproachPolytopicApproachLFTApproachFigure1.1:ofGain-schedulingcontrol.LPVsystems.Consequently,cleardifferencesaremadeintheliteraturesbetweenclassicalandtheso-calledmoderngain-schedulingapproach(seeFigure1.1).Intheclassicalapproach,thedesignproceduretoobtaingain-schedulingcontrollerconsistsofthefollowingadhocsteps.Initially,afamilyoflocalLinearTime-Invariant(LTI)modelsisdeterminedbyselectingdifferentoperatingpointsofthedynamicalsystemthatcovertheentirerangeofparametersvariations.Then,localLTIcontrollersaredesignedforeachLTImodelindividually.Next,basedonthevaluesoftheparam-eters(measuredorestimatedon-line),schedulethelocalcontrollersusingsomeinterpolation(orswitching)methods.Finally,extensivesimulationsareconductedtocheckandverifyclosed-loopstabilityandperformance.Thus,theclassicalgain-schedulingapproachhasthefollowingcriticaldrawbacksExhaustiveandcostlysimulationsandvalidationsaremandatorybecauseadhocstepsareusedinthedesignprocedure.Itisachallengingtasktoguaranteestabilityandperformancegloballywheninterpolating(orswitching)overafamilyofseparatelydesigned(local)controllers.Sinceclassicalapproachesrelyonlocalgridingoftheoperatingdomain,suchapproachesimplyaseverrisktomisscriticalsystemMoreimportantly,thesetechniquesimplicitlyassumethattheschedulingparametersare2frozenintimeandignorethenon-stationarynatureofparametervariations.Inotherwords,thedesignedcontrollerdoesnotprovideanyguaranteesinthefaceofrapidchangesintheschedulingparameters.Thesephenomenarepresentamajorsourceoffailureandmayde-stroytheoverallcontrolscheme.Inresponsetotheseshortcomings,moderngain-schedulingapproachesemergedasapromis-ingalternativeandreceivedaconsiderableattentionincontrolcommunity.Generally,theyoffercapabilitiestohandlethewholeoperatingdomainwithoutrecoursetogridtheparameterspace.Furthermore,robuststabilityandperformanceareguaranteedagainstparametervariations.Andasakeyingredient,theyofferanindisputabledegreeofcomputationalandoperationalsimplicitysincethecontrollercanbesynthesizeddirectlywithoutusinganysortofgridingscheme.Moreconcretely,modernLPVproblemsareconvexandamenabletoLMIcomputations,thelatterbeingsupportedbyefandreliablesoftwaretools.Altogether,thismakesthese(modern)tech-niquesanexcellentcandidatesforpracticalengineeringapplications.ModernGSapproachescanbefurtherintotwodistinctcategories.LinearFractionalTransformation(LFT)basedstructurethatusesmall-gaintheoryapproach,andpolytopicstructurethatbasedonLyapunovtheoryapproach.ThefollowingoverviewgivesashortsurveyonthemaindevelopmentsinliteratureforboththeLFTandpolytopicstructure,withoutanin-depthdiscussion.PolytopicLPVstructure(seeFigure1.2)startsfromstate-spacerepresentationofthesystemandappliesLyapunov'sdirectmethod(seeKhalil[8])toderiveanalysisandsynthesisconditions.OneofthemostcriticalissuesinthepolytopicapproachistheparametrizationoftheLyapunovfunction(asafunctionofschedulingparameters)usedtoestablishstabilityandperformance.Ini-tially,manyoftheresearchersadoptedtheconceptofquadraticstabilitywhereconstantLyapunovmatrixisconsideredbecausethischoiceresultsinnumericallytemptingandtractableoptimizationproblems[9,10].In[9],sufconditionswerederivedfortheexistenceofoutput-feedbackcontrollerthatstabilizesclosed-loopsystemexponentiallyforarbitrarilyfastparametervariations.Theexistenceconditionswereintheformofafeasibilityproblemwithconstraints.Al-3Figure1.2:LPVsysteminpolytopicstructure.thoughthereis,ingeneral,nosystematicmethodtosolvethisproblem,simplcanbemadeforsomeclassesofLPVmodels.ForafLPVmodelswithparametervaluesbelongingtoaconvexpolytope,thesolvabilityconditionsreducetoafeasibilityproblemwithanumberofLMIconstraints.,itissuftoevaluatetheconstraintsassociatedwiththeverticesofthepolytopeofparametervaluessincethisensuresthattheconstraintsholdforeveryparametervaluewithinthepolytope[11].However,aspointedoutin[12],quadraticstabilityapproachleadstoconservativeresultssinceitassumesthattherateofchangesoftheschedulingparametersareConsequently,manyresearchersstudiedParameter-DependentLyapunovFunctions(PDLF)toalleviatetheconservatismassociatedwiththequadraticstability-basedap-proach[13,14,15].Ontheotherhand,Packard[16]developedtheLPVcontroldesignviaLFTstructure(seeFigure1.3)usingsmall-gaintheory[17]fordiscrete-timesystems.Then,ApkarianandGahinet4Figure1.3:LPVsysteminLFTstructure.extendtheworkbydevelopingaunifyingLMIapproachforsynthesizingdynamicoutput-feedbackGScontrollersforbothcontinuous-anddiscrete-timeLPVsystemswithH¥performance[18].Forplantmodelswithparameter-dependentLFTstructure,thescaledsmall-gainsolvabilitycondi-tionscanbereformulatedequivalentlyasanumericallytractableconvexfeasibilityproblemwithanumberofLMIs.Althoughtheapproachprescribedinthesepapers([16,18])isveryattrac-tiveandfullycharacterizedintermsofanumberofLMIs,itsuffersfromconservativenessduetostructuredscalingmatrices.However,suchconservativenesscanbereducedbythenewmethodbasedonnon-structured(full-block)scalingmatricesdevelopedbyScherer[19].Toconclude,thisdissertationisconcernedwithpolytopicstructureGSsynthesismethods.Indirectcontrasttotheliteraturesmentionedabove,theschedulingparametersareassumedtobepollutedbynoise,whichisrelativelynewtopicintheofGScontrolaswillbeillustratedshortlyinSection1.3andSection1.4.1.2LPV,LTV,andLTISystemsTheterminologylinearparameter-varyingwasintroducedin[20]todistinguishLPVsystemsfrombothLinearTime-Invariant(LTI)andLinearTime-Varying(LTV)systems.Generally,an5LPVsystemisasystemthatcanbegovernedbythefollowingstate-spacerepresentationx(t)=A(q(t))x(t)+B(q(t))u(t)y(t)=C(q(t))x(t)+D(q(t))u(t);(1.1)whereq(t)isatime-varyingvectorofplantparametersbelongtoaknownset,andthematricesA(q(t)),B(q(t)),C(q(t)),D(q(t))arefunctionsofq(t).AcommonassumptionintheLPVsys-temtheoryisthatschedulingparametersareunknownduringcontrollersynthesisstage,however,theyareavailableinreal-time(bymeasurementorestimation)forgain-scheduling.Clearly,forafrozenparameters(q(t)=constant),theLPVsystemin(1.1)turnsintoanLTIsystem,i.e.x(t)=Ax(t)+Bu(t)y(t)=Cx(t)+Du(t):(1.2)Thus,thedistinctionbetweenLTIandLPVisclearsinceLPVsystemsarenon-stationarysystems.Ontheotherhand,thedistinctionbetweenLPVandLTVislessapparent.RecallthatLTVplantisanylinearsystemgovernedbystateequationsoftheformx(t)=A(t)x(t)+B(t)u(t)y(t)=C(t)x(t)+D(t)u(t);(1.3)wherethestate-spacematricesA(t);B(t);C(t);D(t)aretime-varyingmatrices.Itisworthnotingthatforanygivenschedulingparametertrajectory,q(t),thedynamic(1.1)representsLTVsys-tembutthereverseisnottrue,sincetheLTVsystemin(1.3)iscompletelyknowninadvance.Thus,theoreticaltreatmentofLPVandLTVsystemsisnotthesamefromanalysisandsynthesisperspective.Inthesamecontext,thisdissertationprovidesynthesisconditionsforgain-schedulingcon-trollerswithguaranteedperformanceintermsofH2andH¥norms.SinceH2andH¥normsarewellforLTIsystems,specialcareneedtobetakenwhendealingwiththeseperformanceindicesintheLPVframework.However,weuseH2andH¥normsherewithslightlyabusedterminologysothatthereadercaneasilygraspourproblemsettingbysimpleanalogytoLTIsys-6tems.ThestrictofthecontrolproblemwillbegivenlaterinChapter3sincenecessaryandnotationsneedtobeintroducedinthenextchapter.ItisworthmentioningthatthetermsLPVandparameter-dependentsystems(andgain-schedulinginthecaseofcontroller)areusedinterchangeablyinthisdissertationtorefertoasysteminthestructureof(1.1).1.3MotivationoftheWorkThemainmotivationbehindGain-Scheduling(GS)controlisthedirectextensionofthewelles-tablishedlinearcontroldesigntoolstononlinearandtime-varyingsystems.Theoflinearparameter-varyingsystemshasevolvedrapidlyinthelasttwodecadesandbecameoneofthemostpromisingframeworkformodernindustrialcontrolwithagrowingnumberofapplications(see[21]forarecentsurvey).Althoughschedulingparametersareunknownduringcontrollerdesignstage,itisimplicitlyassumedthattheyareavailableforon-linemeasurementtobeusedforcontroladaptation.Thecanceofthiscontrolstrategyisattributedtothefactthatthedynamicsofmanyphysicalsystemscanbeefmodeledasafunctionofatime-varyingparameters.Moreover,awideclassofnonlinearsystemscanberepresentedasquasi-LPV(qLPV)systems[22]thatexploitthesimplicityoflinearcontroltheoryinsteadofsophisticatednonlineardesignmethodologies.Practicalexamplesthatprovedtheeffectivenessofgainschedulingcontrolincludespacecrafts[23],Hypersonicvehicles[24],windturbines[25],automotiveengines[26],roboticmanipulators[27],activemagneticbearings[28],andmiscellaneousmechatronicsystems[29,30,31].Acommonassumptionconsideredinthevastmajorityoftheexistingworksisthatanexactmeasurementofschedulingparametersisavailableinreal-timeforcontrollerscheduling.Gener-ally,thisassumptionisnottrueforpracticalapplications.Sinceuncertaintiesinschedulingpa-rametersareunavoidable,perfectmeasurementisimpossibletoobtain.Duetothismeasurementnoise,discrepancyalwaysexistbetweentheActualSchedulingParameters(ASPs)andtheMea-7RobustcontrolLPVsystemsLMIsFigure1.4:Topicofthedissertation.suredSchedulingParameters(MSPs).Thisdiscrepancynotonlyleadstoperformancedegradationbutcouldalsoleadtoinstabilityproblems.Inotherwords,whenapplyingthecontrollerdesignedusingtraditionaltechniquestoapracticalapplication,theclosed-loopperformancewillbeworsethantheexpectedtheoreticalperformancesincemeasurementnoiseintheschedulingparametershadnotbeenconsideredduringcontrollersynthesisstage.Furthermore,theoverallstabilityofthesystemcouldbelostbecausethemismatchbetweentheASPsandtheMSPs.Therefore,thiscontrolproblemisoneofthemostimportantcontroldesignproblemsinthecommunityofgain-schedulingcontrolandLPVsystems.Motivatedbytheimportanceofthisproblem,thisdissertationdealswithgain-schedulingcon-trolwithguaranteedperformancesubjecttouncertainschedulingparameters.Thus,thetopicofthisdissertationiswellillustratedbytheVenndiagramshowninFigure1.4thatrepresentstheintersectionofthefollowingareas,LPVsystems,robustcontrol,andLMIs.Asaresultofthisintersection,RobustGain-Scheduling(RGS)techniquesariseinordertonotonlycopewiththistypeofuncertaintybutalsotoguaranteeclosed-loopperformance.81.4LiteratureSurveyThevastmajorityoftheavailableworkingain-schedulingcontrolliteratureassumeperfectknowl-edgeofschedulingparameters[9,10,32,11,13,14,33,15,34].Althoughtherearemanyattemptsinliteraturetoaddressuncertaintiesinschedulingparameterstheoretically,thisproblemisstillundisclosedandbarelyinvestigated.In[35],output-feedbacksynthesisconditionsarederivedwiththeassumptionthatonlysomeoftheschedulingparametersareavailableforfeedbackcon-trolwithoutconsideringuncertaintiesintheschedulingparameters.TheworkthataddressuncertaintiesintheschedulingparametersexplicitlyisproposedbyDaafouzetal.[2].Inthispaper,gain-schedulingsynthesisconditionsthatguaranteeaprescribedperformancelevelinthepresenceofuncertaintiesintheschedulingparametersarederived.However,thewholeapproachpresentedin[2]isimpracticalsinceuncertaintiesaremodeledtobeproportionaltothevaluesoftheschedulingparameters,whichisnotcommontoanymeasurementsystem.Furthermore,thesynthesisconditionsareverysensitivetotheuncertaintybound.After[2],severalpapersthatad-dressedthesamecontrolproblemhavebeenpublishedbySatoetal.[36,37,1,38].Synthesisconditionsforstate-feedback[36]anddynamicoutput-feedbackcontrollers[37]arederivedwithnoisyschedulingparameters.However,in[36,37]quadraticstability(constantLyapunovmatrix)approachisusedforcontrollersynthesis.Aspointedoutin[12],suchapproachareextremelyconservativeandcertainsystemsarenotevenquadraticallystabilizable.Toalleviatethisproblem,parameter-dependentLyapunovfunctionapproachwasusedtosynthesizeschedulingcontrollersin[1]forstate-feedback,andin[38]fordynamicoutput-feedbackasaremedyforquadraticsta-bilityapproach.WhilePDLFapproachreduceconservativenessassociatedwithquadraticstabilityapproach,butitintroducesaseriousimplementationdrawback.Thus,thedevelopedcontrollerrequiresnotonlythereal-timemeasurementoftheschedulingparameters,butalsorequirestheirderivativestobeavailableon-lineaswell.Hence,thesynthesizedcontrollerisnotpracticallyvalid[13].Frompracticalviewpoint,thederivativesoftheschedulingparameterscannotbeobtainedinreal-timeduetothefactthatderivativeisverysensitivetomeasurementnoise.Furthermore,some9ofthesystemmatricesarerestrictedtobeindependentonthevaryingparametersin[38]inordertosynthesizeacontroller.Consideringtheexistliterature,theobjectivesofthisresearchistoovercomethedrawbacksassociatedwiththeexistingresultsbydevelopingnovelsynthesisconditionstosynthesizeRGScontrollerswithguaranteedperformanceundernoisyschedulingparameters.1.5ContributionsThecontributionsofthisdissertationcanbesummarizedasfollow:1.CharacterizationofPLMIssynthesisconditionsforsynthesizingRGSstate-feedbackcon-trollerwithguaranteedH2performanceinChapter4.2.Inthesamechapter,RGSstate-feedbacksynthesisconditionswithguaranteedH¥perfor-mancearedeveloped.3.InChapter5,novelconditionsintermsofParametrizedBilinearMatrixInequalities(PB-MIs)havebeenderivedtosynthesizeRGSDynamicOutput-Feedback(DOF)controllerwithguaranteedH2.4.Similarly,synthesisconditionsintermsofPBMIshasbeencharacterizedtosynthesizeRGSDOFcontrollerwithguaranteedH¥inChapter5.ItisworthmentioningthatthesynthesisconditionsoftheRGSDOFcanhandlethecasewherethetime-varyingparametersaffect-ingboththestatematrixandthecontrolinputmatrix.ThisisoneofthecontributionsofChapter5sinceinliteratureonlystatematrixwasallowedtobeaffectedbythetime-varyingparameters.5.DevelopmentofanefnumericalalgorithmtosolvethePBMIsconditionsiteratively.6.NovelconditionsintermsofPLMIshasbeenderivedinChapter6tosynthesizeRGSStaticOutput-Feedback(SOF)controllerwithguaranteedH2.Theseconditionsareutilizethetwo-10stagedesignapproachtosynthesizestate-feedbackschedulingcontrollerinthestage,then,usingthiscontrollerinthesecondstagetosynthesizetheRGSSOFcontroller.TheRGSSOFcontrollerissynthesizedindependentlyofanyoftheopen-loopmatricesorLya-punovmatrix,therefore,withthisnoveldesignthetime-varyingparameterscouldaffectalltheopen-loopmatriceswithoutanyrestrictions.7.CharacterizationsofsynthesisconditionstosynthesizeRGSStaticOutput-Feedback(SOF)controllerwithguaranteedH¥performance.Similarly,theseconditionsutilizethetwo-stagedesignapproachmentionedabovetosynthesizeRGSSOFcontroller.8.ExperimentalvalidationoftheRGScontrollersonthetestbenchofElectricVariableValveTiming(EVVT)actuatorisgiveninChapter7.Enginespeedandvehiclebatteryvoltageareusedasnoisyschedulingparameters.1.6OrganizationFigure1.5showsaroadmapofthedissertation'schapters.Thisdissertationisorganizedasfol-lows:notations,andmulti-simplexmodelingapproacharegiveninChapter2.ReadersarerecommendedtoreadChapter2beforeproceedingtootherchapterssinceitrepresentsthebasicbuildingblockformodelingthetime-varyingparameters.MathematicalformulationsoftheRGScontrolproblemandtheproposedsolutionapproachareoutlinedinChapter3.Inthischap-ter,ageneralframeworkispresentedtohandleuncertaintiesintheschedulingparameters.RGSState-Feedback(SF)PLMIssynthesisconditionsarepresentedinChapter4withH2andH¥performances.Numericalexamples,simulations,andcomparisonswithotherapproachesfromliteraturearepresentedattheendofChapter4.InChapter5,PBMIssynthesisconditionsforRGSDOFcontrollerswithH2andH¥performancesaredevelopedalongwiththenumericalalgorithmnecessarytosolvethePBMIsconditions.Then,theRGSSOFsynthesisconditionsaredevelopedinChapter6.ThesynthesisapproachoftheSOFutilizesthetwo-stagedesignapproach,whereinthestageSFschedulingcontrollerisdesignedtobeusedinthesecondstageforsynthesizing111.Introduction2.PrelamenariesofMulti-simplexModeling3.ProblemFormulationandSolutionApproach4.RGSSFControl6.RGSSOFControl5.RGSDOFControl7.ExperimentalValidationonEVVT8.ConclusionsandFutureResearchFigure1.5:Dissertation'schaptersroadmap.SOFcontroller.Chapter7presentstheexperimentalstudyofapplyingRGScontrolleronElectricVariableValveTiming(EVVT)systemtestbenchofautomotiveenginewithvalidation.Finally,Chapter8presentsconclusionsandrecommendationsforfutureresearch.FundamentalsofLMIsforLTIsystemsaregivenintheappendix.12CHAPTER2PRELIMINARIESOFMULTI-SIMPLEXMODELINGTheaimofthischapteristointroducenecessarynotationsassociatedwiththemulti-simplexmodelingapproachthatrepresentsthefoundationofthisdissertation.Thischapterdoesnotpresentanytheoreticalcontributionsbutitisincludedheretopresentnotations,terminologies,andtionsthatareusedthroughoutthisdissertation.Mostoftheandterminologiesusedinthischaptercanbefoundin[39,40,41].2.1NotationsNotationsusedinthisdissertationarefairlystandard.ThepositiveofamatrixAisdenotedbyA>0.RandNdenotethesetofrealandnaturalnumbers,respectively.Thesymbol?isusedtorepresentsthetransposeoftheoff-diagonalmatrixblock.trace(A)denotesthetraceofthematrixA,whichrepresentsthesumofdiagonalelementsofthematrixA.Inisusedtorefertoidentitymatrixofsizenn.Zeromatrixofsizenpisreferredtoas000np.Thesesubscriptswillbeomittedwhenthesizeofthecorrespondingmatrixcanbeinferredfromthecontext.ThetransposeofmatrixAisrefereedtoasA0;andA+()0=A+A0.Othernotationswillbeexplainedinduecourse.2.2andTerminologies2.1.Unit-simplex[39]:aunit-simplexisasfollowsL`:=(a(t)2R`:`åi=1ai(t)=1;ai(t)0;i=1;2;;`);wherethevariableai(t)variesintheunit-simplexL`thathas`vertices.2.2.Multi-simplex[40]:amulti-simplexListheCartesianproductofanumber13ofqsimplexes,whereLN1LN2LNq=qÕi=1LNi:=L:Thedimensionofthemulti-simplexLisastheindexN=(N1;N2;;Nq)andforsimplicityofnotation,RNdenotesforthespaceRN1+N2++Nq.Thus,anyvariablea(t)inthemulti-simplexdomainLcanbedecomposedas(a1(t);a2(t);;aq(t)),andeachai(t),belongingtoaunit-simplexLNi,canbefurtherdecomposedas(ai1(t);ai2(t);;aiNi(t))fori=1;2;;q.2.3.HomogeneousPolynomial:Givenaunit-simplexLNofdimensionN2N,apoly-nomialp(a)onRNofdegreeg2Niscalledhomogeneousifallofitsmonomialshavethesametotaldegreeg.Example2.1.Leta2L3,thenthefollowingpolynomialp(a)=5a41+a21a232a32a3+6a1a2a23isahomogeneouspolynomialofdegreeg=4.2.4.L-HomogeneousPolynomial:Givenamulti-simplexLofdimensionN2Nq,apolynomialp(a)onRNofdegreeg2NqiscalledL-homogeneousif,foranygivenintegeri0,with1i0q,andforanygivenai2RNi,for1i6=i0q,thepartialapplicationai02RNi0!P(a)isahomogeneouspolynomialinai0.Example2.2.Leta2L,withL=L2L3,thenthefollowingpolynomialp(a)=a311a223a11a212a23a312a21+6a211a12a22isaL-homogeneouspolynomialofpartialdegreeg=(3;1).2.5.Partialdegreeisthedegreeofaparameter-dependentmatrixthatdependsonmulti-simplexparameterswhichisusedtotheindividualdegreeofeachunit-simplexinsidethemulti-simplexdomain.Foraunit-simplex,gisscalar;whileinthemulti-simplexdomain,gisavectorrepresentingthedegreesofeachunit-simplexinsidethemulti-simplex.Thus,thenumberofelementsofvectorgisthesameasthenumberofindividualsimplexesinsidethemulti-simplex.14Lemma2.1.(BinomialExpansion)Foragivennonnegativeintegerg2Nandtwogivennumbersaandb(a+b)g=gåj=0g!j!(gj)!agjbjLemma2.2.(ExpansionofpowersofsumsofNnumbers)ForagivennonnegativeintegergandagivenvectorxofNnumbers Nåi=1xi!g=åk2Q(N;g)g!p(k)xk(2.1)whereQ(N;g)isthesetofN-tuplesobtainedfromallpossiblecombinationsofNnonnegativeintegerski;i=1;2;;N;withsumk1+k2++kN=gandp(k)=(k1!)(k2!)(kN!),suchthatQ(N;g)=(k2NN:Nåi=1ki=g):ThenumberofelementsinQ(N;g)isgivenbyR(N;g):=cardQ(N;g)=(N+g1)!g!(N1)!;withcardQ(N;g)referstothecardinalityofQ(N;g):2.3PolynomialCompletionandHomogenization2.6.(LN-completionofapolynomial)Givenaunit-simplexLNofdimensionN2Nandapolynomialp(a)onRN,theLN-completionofp(a),denotedcompLN(p(a)),isthe(unique)homogeneouspolynomialofminimaldegreeequaltop(a)onLN.TheLN-completionofp(a)canbeeasilyconstructedusingLN-homogenizationprocedure.2.7.(LN-homogenization)Fora2LNandagivenmonomialm(a)ofdegreed2N,theLN-homogenizationofdegreeg2Nofm(a)isobtainedbymultiplyingm(a)with Nåi=1ai!g=1;15then,usingLemma2.2,canbewrittenashomogeneouspolynomial1= Nåi=1ai!g=åk2Q(N;g)g!p(k)ak:Now,theLN-completionofp(a)canbeeasilyconstructedasfollows.Letp(a)consistofMmonomialsofrespectivedegreed`,for`=1;2;;M,andletg=max`d`.Then,theminimaldegreeLN-completionofp(a)isobtainedbyapplyingaLN-homogenizationofdegreegd`toeachmonomialofp(a).Example2.3.Fora2L3theLNcompletionofp(a)=a31+2a1a25isobtainedascompLN(p(a))=a31+2a1a2(a1+a2+a3)5(a1+a2+a3)3:Naturally,theofcompletionandhomogenizationcanbeeasilyextendedtothemulti-simplexcase,asshowninthefollowing2.8.(L-completionofapolynomial)Givenamulti-simplexLofdimensionN2NNandapolynomialp(a)onRN,theL-completionofp(a),denotedcompL(p(a)),isthe(unique)L-homogeneouspolynomialofminimaldegreeequaltop(a)onL.Asintheunit-simplexcase,theL-completionofp(a)canbeconstructedusingL-homogenization.2.9.(L-homogenization)Fora2Landagivenmonomialm(a)ofdegreed2Nq,theL-homogenizationofdegreeg2Nqofm(a)isobtainedbymultiplyingm(a)withqÕi=10@Niåj=1ai;j1Agi=1usingLemma2.2,thisequationisequaltoL-homogeneouspolynomial1=qÕi=10@Niåj=1ai;j1Agi=åk2Q(N;g)p(g)p(k)ak:Now,theL-completionofp(a)canbeeasilyconstructedasfollows.Letp(a)consistofMmonomialsofrespectivedegreed`2Nq,for`=1;2;;M,andletgbetheminimalvectorofqnaturalnumberssuchthatgd`,for`=1;2;;M.Then,theminimaldegreeL-completionofp(a)isobtainedbyapplyingaL-homogenizationofdegreegd`toeachmonomialofp(a).16Example2.4.ConsidertheL-completionofthepolynomialp(a)=9a21;15a1;2a2;1a2;2+2a32;3wherea2L=L2L3.Sincethepolynomialdegreeofthethreemonomialsisd1=(2;0);d2=(1;2)andd3=(0;3),thedegreeoftheL-homogenizationisobtainedasg=(2;3)andconse-quentlycompL(p(a))=9a21;1(a2;1+a2;2+a2;3)35a1;2a2;1a2;2(a1;1+a1;2)(a2;1+a2;2+a2;3)+2a32;3(a1;1+a1;2)2(2.2)2.4HomogeneousPolynomialLyapunovMatrixInordertoprovideasystematicproceduretogeneratesufLMIconditionsofincreasedpre-cision,aquadraticLyapunovfunctionv(x(t))=x(t)0P(a(t))x(t)iswith1P(a)=åk2Q(N;g)ak11ak22akNNPk1k2kN=åk2Q(N;g)akPk;k=k1k2kN;(2.3)wherePk2Rnnisamatrix-valuedcoefandakisthecorrespondingmonomialwithhomo-geneousofdegreeg2N.Example2.5.Considerahomogeneouspolynomialmatrixofdegreeg=3withtwovertices(N=2),thenthepossiblecombinationsofthepartialdegreesareQ(N;g)=Q(2;3)=f03;12;21;30g,soR(2;3)=4correspondingtothegenericpolynomialformP(a)=a32P03+a1a22P12+a21a2P21+a31P30:forg=0=)P(a)=P0,forg=1;N=2=)Q(N;g)=8><>:01109>=>;=)P(a)=a1P01+a2P101Sometimesthedependencyontwillbeomittedfornotationalsimplicity.17g=2;N=2,=)Q(N;g)=8>>>><>>>>:0211209>>>>=>>>>;=)P(a)=a21P02+a1a2P11+a22P20g=3;N=2,=)Q(N;g)=8>>>>>>><>>>>>>>:031221309>>>>>>>=>>>>>>>;=)P(a)=a32P03+a1a22P12+a21a12P21+a31P302.5GeneralityoftheModelingApproachThissectionillustratesthegeneralityofthemulti-simplexLandthecorrespondingL-homogeneouspolynomialparameterization.Itisshownthatpolytopic,afandpolynomialparameterizationscanberecoveredasspecialcasesofthehomogeneouspolynomialparameterization.2.5.1Schedulingvariablesparameterization1.PolytopicparametrizationItiseasytonotethatamatrixwiththefollowingrepresentationA(a(t))=Nåi=1aiAi;witha(t)intheunit-simplexLNofdimensionN2Nisaspecialcaseofthegeneralparame-terization(2.3)bychoosingthemulti-simplexL=LNandthedegreeoftheL-homogeneouspolynomialg=1.2.parametrizationAmatrixwiththefollowingafstructureonqboundedvariables¯qiqi(t)¯qifori=1;2;q,A(q)=A0+qåi=1qiAi;18canbewrittenasaL-homogeneousmatrix-valuedpolynomialbyai1(t)=qi(t)+¯qi2¯qi;ai2(t)=1ai1(t)ai(t)=(ai1(t);ai2(t))= qi(t)+¯qi2¯qi;¯qiqi(t)2¯qi!fori=1;2;;q;suchthatai10;ai20,andai1+ai2=1.Consequently,a=a1;a2;;aqtakesvaluesinsidethemulti-simplexLofdimension(2;2;;2)2Nq.AL-homogeneouspoly-nomial‹A(a),overthismulti-simplex,equaltoA(q)canbeconstructedasfollowsA(q)=A0qåi=1¯qiAi+qåi=12¯qiAiai1=ŸA(a)Obviously,ŸA(a)hasadegreeoneforeachvariableai1.Therefore,theL-completion‹A(a)=compL(ŸA(a))isahomogeneouspolynomialofdegreeg=(1;1;;1)2Nq,overthemulti-simplexLofdimensionN=(2;2;;2)2NqequaltoA(q).3.PolynomialparametrizationAmatrixwiththefollowingpolynomialstructureofdegreegonaboundedvariable¯qq(t)¯qA(q)=gåk=0qkAkcanberewrittenasahomogeneousmatrix-valuedpolynomialasfollows.First,a2Lasa(t)=(a1(t);a2(t))=q(t)+¯q2¯q;¯qq(t)2¯qsuchthata1(t)0;a2(t)0,anda1(t)+a2(t)=1.Sinceq(t)=2¯qa1(t)¯q,itisclearthatA(q(t))=gåk=02¯qa1(t)¯qkAkUsingLemma2.1,itcanbewrittenasA(q(t))=gåk=0 kåj=0k!j!(kj)!(2¯q)jaj1(t)¯qkj!Ak;19whichyields,afterreorderingtheterms,A(q(t))=gåj=0(2¯q)ja1(t)j gåk=jk!j!(kj)!¯qkjAk!=ŸA(a(t))Itisclearthattheg+1monomialtermsofŸA(a)respectivelyhavedegreejina1,forj=0;1;;g.Consequently,theL2-completion‹A(a(t))=compL2Ÿ(A(a(t)))canbeob-tainedbyapplyingaL2-homogenizationofdegreegjtoeachmonomialtermj,forj=0;1;;g.ThisyieldsaL2-homogeneouspolynomial‹A(a(t))ofdegreeg2N,de-overtheunit-simplexL2,thatisequaltoA(q(t)).Example2.6.ConsiderthefollowingLPVsystemwithpolynomialdynamicmatrixx(t)=26401q2(t)q2(t)q3(t)1q21(t)2375x(t)(2.4)withthefollowingbounds,1q1(t)1;1q2(t)1;0:5q3(t)0:5Thissystemcanbemodeledusingmulti-simplexapproachwiththreesimplexesSimplex#1tomodelq1(t)withtwoverticesN1=2andg1=2,Q(N1;g1)=8>>>><>>>>:0211209>>>>=>>>>;,thus,Ÿa(t)=(Ÿa1(t);Ÿa2(t))2L2.Simplex#2tomodelq2(t)with2-verticesN2=2andg2=1,Q(N2;g2)=8><>:01109>=>;;thus,‹a(t)=(‹a1(t);‹a2(t))2L2.Simplex#3tomodelq3(t)with2-verticesN3=2withg3=1,Q(N3;g3)=8><>:01109>=>;,thus,a(t)=(a1(t);a2(t))2L2.20ThematrixA(q)canbewrittenasA(q(t))=2640112375+2640001375q21(t)+2640100375q2(t)+2640010375q2(t)q3(t)(2.5)Nowexpressschedulingvariablesintermsofthemulti-simplexvariablestoobtainŸA(a(t))bysubstitutingthefollowingrelationshipsin(2.5)q1(t)=2¯q1Ÿa1(t)¯q1=2Ÿa1(t)1;withŸa1(t)+Ÿa2(t)=1q2(t)=2¯q2‹a1(t)¯q2=2‹a1(t)1;with‹a1(t)+‹a2(t)=1q3(t)=2¯q3a1(t)¯q3=a1(t)0:5;witha1(t)+a2(t)=1andsimplifyingtogetA(q)=A(Ÿa;‹a;a)=264020:51375+2640004375Ÿa1(t)+2640004375Ÿa21(t)+2640210375‹a1(t)+2640010375a1(t)+2640020375‹a1(t)a1(t):(2.6)NowL-completionof(2.6)canbeeasilyconstructedusingL-homogenizationproceduresuchthatL=L2L2L2;Term#1:The1sttermshouldbemultipliedby(Ÿa21+Ÿa1Ÿa2+Ÿa22)(‹a1+‹a2)(a1+a2)=Ÿa21‹a1a1+Ÿa21‹a1a2+Ÿa21‹a2a1+Ÿa21‹a2a2+Ÿa1Ÿa2‹a1a1+Ÿa1Ÿa2‹a1a2+Ÿa1Ÿa2‹a2a1+Ÿa1Ÿa2‹a2a2+Ÿa22‹a1a1+Ÿa22‹a1a2+Ÿa22‹a2a1+Ÿa22‹a2a2Term#2:ThesecondtermshouldbehomogenizedasŸa1(Ÿa1+Ÿa2)(‹a1+‹a2)(a1+a2)=Ÿa21‹a1a1+Ÿa21‹a1a2+Ÿa21‹a2a1+a21‹a2a2+Ÿa1Ÿa2‹a1a1+Ÿa1Ÿa2‹a1a2+Ÿa1Ÿa2‹a2a1+Ÿa1Ÿa2‹a2a2Term#3:The3rdtermshouldbehomogenizedasŸa21(‹a1+‹a2)(a1+a2)=Ÿa21‹a1a1+Ÿa21‹a1a2+Ÿa21‹a2a1+Ÿa21‹a2a221Term#4:The4thtermwillbehomogenized‹a1(Ÿa21+Ÿa1Ÿa2+Ÿa22)(a1+a2)=Ÿa21‹a1a1+Ÿa21‹a1a2+Ÿa1Ÿa2‹a1a1+Ÿa1Ÿa2‹a1a2+Ÿa22‹a1a1+a22‹a1a2Term#5:The5thtermwillbehomogenizeda1(Ÿa21+Ÿa1Ÿa2+Ÿa22)(‹a1+‹a2)=Ÿa21‹a1a1+Ÿa21‹a2a1+Ÿa1Ÿa2‹a1a1+Ÿa1Ÿa2‹a2a1+Ÿa22‹a1a1+Ÿa22‹a2a1Term#6:The6thtermwillbehomogenized‹a1a1(Ÿa21+Ÿa1Ÿa2+Ÿa22)=Ÿa21‹a1a1+Ÿa1Ÿa2‹a1a1+Ÿa22‹a1a1andsoon.Now,thematrixA(q(t))insystem(2.4),canbewritteninthehomogenizedmulti-simplexvariablesa(t)=(Ÿa(t);‹a(t);a(t))asfollowA(q)=A(Ÿa;‹a;a)=Ÿa21‹a1a1A1+Ÿa21‹a1a2A2+Ÿa21‹a2a1A3+Ÿa21‹a2a2A4+Ÿa1Ÿa2‹a1a1A5+Ÿa1Ÿa2‹a1a2A6+Ÿa1Ÿa2‹a2a1A7+Ÿa1Ÿa2‹a2a2A8+Ÿa22‹a1a1A9+Ÿa22‹a1a2A10+Ÿa22‹a2a1A11+Ÿa22‹a2a2A12=A(a)withthefollowingvertixmatrices(ofthemulti-simplex):A1=264000:51375;A2=264001:51375;A3=264021:51375;A4=264020:51375;A5=2640016375;A6=2640036375;A7=2640436375;A8=2640416375;A9=264000:51375;A10=264001:51375;A11=264021:51375;A12=264020:51375Notethatthenumberofverticesofthemulti-simplexisgivenbyR(N;g)=R((2;2;2);(2;1;1))=3Õi=1R(Ni;gi)=322=12222.5.2SchedulingvariablesdependencyModelinguncertaintydomainwhenoneschedulingparameterdependsonanotherschedulingpa-rametercanbeeasilydoneviamulti-simplexmodelingapproach.Thefollowingexampleisbor-rowedfrom[42]toillustratethisidea.Considerasystemdependingontwoschedulingparametersq1(t)andq2(t)thattakevaluesintheregionindicatedbyFigure2.1awiththegray-shadedarea.Dependingontheavailableinformation,thisregioncanbemodeledinadifferentway.Considerthetwofollowingsituations:1.Theschedulingparametersareboundedby1q1(t)4;0q2(t)5:2.Thesecondschedulingparameterdependsontheonesuchthat,1q1(t)4;q2(q1(t))q2(t)¯q2(q1(t))withq2(q1(t))=8>>>>>><>>>>>>:0:5q1(t)+2:5;if1q1(t)21:5q1(t)+4:5;if2q1(t)33q1(t)9;if3q1(t)4¯q2(q1(t))=8>><>>:0:5q21(t)q1(t)+3:5;if1q1(t)3q1(t)+8;if3q1(t)4(2.7)Inthecase,themulti-simplexcanbemodeledbytreatingbothschedulingparametersindependentlyasintheapproachpresentedinExample2.6,yieldinga(t)=((a1;1;a1;2);(a2;1;a2;2))=q1(t)13;4q1(t)3;q2(t)5;5q2(t)5bytakingvaluesinthemulti-simplexofdimensionN=(2;2),wherea1;2=1a1;1,a2;2=1a2;1,anda(t)2L.Theboundaryoftheresultingregioncanberepresentedinthe(a1;1;a2;1)-space,asshowninthereddashedlinesinFigure2.1b.Sinceonlytheboundsofthescheduling23Figure2.1:Comparisonbetweendifferentmodelingapproachparameterareused,allpointsinsidethis(red-dashed)boxarebeingconsideredinthismodel.Thisleadstoconservativenesssincethegray-shadedareaistheactualregionwhere(a1;1;a2;1)canassumevaluesbasedonthebounds(2.7).Inthesecondcase,anexactrepresentationoftheregioninthemulti-simplexcanbeobtainedbyobservingthatthelowerandupperboundonq2(t)arefunctionsofq1(t),asgivenin(2.7).Usingthesebounds,theparameterb(t)(whereb(t)2L)canbeasb(t)=((b1;1;b1;2);(b2;1;b2;2))= q1(t)13;4q1(t)3; q2(t)q2(q1(t))¯q2(q1(t))q2(q1(t));¯q2(q1(t))q2(t)¯q2(q1(t))q2(q1(t))!!takingvaluesinthemulti-simplexofdimensionN=(2;2).Thered-dashedsquareinFigure2.1cshowstheboundaryoftheregioninthe(b1;1;b2;1)-space.Inthiscase,thesquarecoincideswiththeactualregion(thegrayshadedarea).Thisexampleshowstheeffectivenessofthemulti-simplexmodelingapproachtoutilizealltheavailableinformationaboutschedulingparameterstoreduceconservativenessasmuchaspossible[42].242.6SummaryThischapterintroducedthenotationsandofthemulti-simplexmodelingandhomo-geneouspolynomialsparametrizationusedthroughoutthisdissertation.Itiswell-knownthatthemulti-simplexdomainisageneralizedrepresentationoftheunit-simplex.Thesenotationsandnitionswillbeutilizedinthenextchapterstomodelthetime-varyingparametersandtheassociateduncertainties.25CHAPTER3PROBLEMFORMULATIONANDSOLUTIONAPPROACHInthischapter,problemformulationofgain-schedulingcontrollersynthesiswithuncertainschedul-ingparametersispresentedinaframework.Then,basedontheconceptsgiveninChapter2,stepsofthesolutionapproacharegiveninthischapteraswell.Generally,thesolutionapproachconsistsofsixstages.First,aconvexchangeofvariablesistobeperformedtoconverttheschedul-ingparametersandtheassociateduncertaintiesfromtheiroriginalparameterdomaininto(convex)multi-simplexdomain.Then,theratesofvariationoftheschedulingparametersanduncertaintiesaremodeledinaconvexsetinthesecondstageaswell.ThethirdstageincludesderivationofthePLMIs/PBMIssynthesisconditions.TheproofsofthePLMIs/PBMIssynthesisconditionswillbegiveninthenextthreechapters.However,shortdescriptionofPLMIswillbegiveninthischap-ter.Then,arelaxationschemeisusedtorelaxthedimensionalconstraintsintosetofLMIsconstraints.MatrixcoefcheckrelaxationmethodwillbeillustratedinthischapterasitisusedtorelaxthePLMIsconditionsinthisdissertation.Onceafeasiblesolutionisobtained,thecontrollercoefrecoveredviainversetransformation.Finally,thecontrollerisimplementedusingthesecoefbyutilizingtheMeasuredSchedulingParameters(MSPs).3.1ProblemFormulationConsiderthefollowingLPVsystemSOL:=8>>>>>><>>>>>>:x(t)=A(q(t))x(t)+Bu(q(t))u(t)+Bw(q(t))w(t)z(t)=Cz(q(t))x(t)+Dzu(q(t))u(t)+Dzw(q(t))w(t)y(t)=Cy(q(t))x(t)+Dyw(q(t))w(t);(3.1)wherex(t)2Rnisthestate,u(t)2Rnuisthecontrolinput,w(t)2Rnwisthedisturbanceinput,z(t)2Rnzisthecontrolledoutput,andy(t)2Rnyisthemeasuredoutput.Thesystemmatri-ceshavethefollowingcompatibledimensionsA(q(t))2Rnn,Bu(q(t))2Rnnu,Bw(q(t))226Rnnw,Cz(q(t))2Rnzn,Dzu(q(t))2Rnznu,Dzw(q(t))2Rnznw,Cy(q(t))2Rnyn,andDyw(q(t))2Rnynw.q(t)isarealvectorcontainingthetime-varyingschedulingparameters,whereq(t)=q1(t);q2(t);;qq(t)0;(3.2)andqrepresentsthenumberofschedulingparameters.Thesystemmatricesin(3.1)areassumedtobeafparameter-dependent,i.e.,eachofthesystemmatricescanberepresentedbythefollowingparametrizationA(q(t))=A0+qåi=1qi(t)Ai:Theschedulingparametersin(3.2)areassumedtobeinexactlymeasured(corruptedwithnoise)denotedbyŸq(t),suchthatŸq(t)=Ÿq1(t);Ÿq2(t);;Ÿqq(t)0;d(t)=d1(t);d2(t);;dq(t)0;Ÿq(t)=q(t)+d(t);orinthescalarform,Ÿqi(t)=(qi(t)+di(t));i=1;2;;q;(3.3)wheredi(t)representsuncertaintyofthei-thschedulingparameterandqi(t)isthetruevalue.Theseschedulingparametersanditsuncertaintiesareassumedtobeindependentoneachotherandtheyvarieswithinthefollowingknownbounds(seeFigure3.1)¯qiqi(t)¯qi;¯didi(t)¯di;i=1;2;;q:(3.4)Furthermore,theseparametersareassumedtohaveboundedratesofvariationbqiqi(t)bqi;bdidi(t)bdi;i=1;2;;q:(3.5)Withoutlossofgenerality,theseboundsareassumedtobesymmetric.Notethat(3.4)and(3.5)arenotrestrictive,since(3.4)canalwaysbeachievedbychangeofvariables;while(3.5)representsarealistichypothesisbecausetheratesofvariationoftheparametersarenaturallylimitedinpracticalengineeringapplications.27Figure3.1:Uncertaintydomainformeasuredschedulingparameter.Figure3.2:Closed-loopsystemwithoutput-feedbackgain-schedulingcontrol.Thegoalistosynthesize1.RGSstate-feedbackcontrolleroftheformu(t)=K(Ÿq(t))x(t);(3.6)torobustlystabilizetheclosed-loopsystemSCL:=8>><>>:x(t)=A(q(t);Ÿq(t))x(t)+B(q(t);Ÿq(t))w(t);z(t)=C(q(t);Ÿq(t))x(t)+D(q(t);Ÿq(t))w(t):(3.7)28withA(q(t);Ÿq(t))=A(q(t))+Bu(q(t))K(Ÿq(t));B(q(t);Ÿq(t))=Bw(q(t));C(q(t);Ÿq(t))=Cz(q(t))+Dzu(q(t))K(Ÿq(t));D(q(t);Ÿq(t))=Dzw(q(t));Cy(q(t))=I;Dyw(q(t))=000;(3.8)or2.DynamicOutput-FeedbackcontrolleroftheformKDOF:=8>><>>:xc(t)=Ac(Ÿq(t))xc(t)+Bc(Ÿq(t))y(t);u(t)=Cc(Ÿq(t))x(t);(3.9)torobustlystabilizetheclosed-loopsystem(seeFigure3.2)SCL:=8>><>>:x(t)=A(q(t);Ÿq(t))x(t)+B(q(t);Ÿq(t))w(t);z(t)=C(q(t);Ÿq(t))x(t)+D(q(t);Ÿq(t))w(t):(3.10)withx(t)=x(t)0xc(t)00,and264A(q(t);Ÿq(t))B(q(t);Ÿq(t))C(q(t);Ÿq(t))D(q(t);Ÿq(t))375=266664A(q(t))000Bw(q(t))000000000Cz(q(t))000000377775+2666640Bu(q(t))In000000Dzu(q)377775264Ac(Ÿq(t))Bc(Ÿq(t))Cc(Ÿq(t))000375264000In000Cy(q(t))000Dyw(q(t))375:29withA(q(t);Ÿq(t))=264A(q(t))Bu(q(t))Cc(Ÿq(t))Bc(Ÿq(t))Cy(q(t))Ac(Ÿq(t))375;B(q(t);Ÿq(t))=264Bw(q(t))Bc(Ÿq(t))Dyw(q(t))375;C(q(t);Ÿq(t))=Cz(q(t))Dzu(q(t))Cc(Ÿq(t));D(q(t);Ÿq(t))=Dzw(q(t)):(3.11)or3.StaticOutput-Feedbackcontrolleroftheformu(t)=K(Ÿq(t))y(t)(3.12)torobustlystabilizestheclosed-loopsystemx(t)=A(q;Ÿq)x(t)+B(q;Ÿq)w(t)z(t)=C(q;Ÿq)x(t)+D(q;Ÿq)w(t)A(q;Ÿq):=A(q)+Bu(q)K(Ÿq)Cy(q)B(q;Ÿq):=Bw(q)+Bu(q)K(Ÿq)Dyw(q)C(q;Ÿq):=Cz(q)+Dzu(q)K(Ÿq)Cy(q)D(q;Ÿq):=Dzw(q)+Dzu(q)K(Ÿq)Dyw(q)(3.13)Notethatthecontrollermatricesin(3.6),(3.9),and(3.12)areassumedtohaveafparametriza-tionwithrespecttotheMeasuredSchedulingParameters(MSPs).Inotherwords,thosematrices30K(Ÿq(t)),Ac(Ÿq(t)),Bc(Ÿq(t)),Cc(Ÿq(t)),andK(Ÿq(t))areparameterizedasfollowsK(Ÿq(t))=K0+qåi=1Ÿqi(t)Ki;Ac(Ÿq(t))=Ac0+qåi=1Ÿqi(t)Aci;Bc(Ÿq(t))=Bc0+qåi=1Ÿqi(t)Bci;Cc(Ÿq(t))=Cc0+qåi=1Ÿqi(t)Cci;K(Ÿq(t))=K0+qåi=1Ÿqi(t)Ki:(3.14)Therefore,thegoalistoobtainthecontrollercoefmatricesKi,(Aci;Bci;Cci),andKifori=0;1;;q,suchthattheRGScontrollercanbeimplementedusingonlytheMSPsŸqi.3.2SolutionApproachThesixstagesofthesolutionapproachofRGSsynthesisproblemispresentedinFigure3.3.Ap-propriatetransformationisusedinthestagetoconverttheschedulingparametersandtheuncertaintiesfromtheiroriginalparameterdomainintoa(convex)multi-simplexdomain.Then,theratesofvariationsoftheschedulingparametersanduncertaintiesaremodeledinaconvexsetinthesecondstage.ThethirdstageconsistsderivationofthePLMIs/PBMIssynthesiscon-ditions.Matrixcoefcheckrelaxationscheme[39]isusedtorelaxthedimensionalconstraintsintoconstraintstosolvetheoptimizationproblem.Onceafeasiblesolutionisobtained,inversetransformation(multi-simplex-to-afisusedtoobtaincontrollerimplementationcoefthatutilizethenoisyschedulingparameters.3.2.1toMulti-SimplexTransformationThegoalofthissubsectionistodevelopasuitablechangeofvariablestotransformallthetime-varyingparameters(schedulinganduncertainties)fromtheiroriginalspaceintoaconvexmulti-31Aftomulti-simplextransformationRateofvariationmodelingPLMI/PBMIsynthesisconditionsPLMI/PBMIrelaxation&algorithmInversetransformationControllerimplementationFigure3.3:Sixstagessolutionapproach.simplexdomain.SupposethattheActualSchedulingParameters(ASPs)q(t)areaffectedbytime-varyingmeasurementnoised(t)asgivenby(3.3);andsupposefurtherthatq(t),d(t),q(t),andd(t)areboundedasin(3.4)and(3.5).Sincedi(t)associatedwitheachqi(t)needstobemodeledinaconvexdomain,twounit-simplexesforeachMSPareused.Eachunit-simplexhastwoverticesduetothefactthateachparameterhasupperandlowerboundsasin(3.4).Thus,eachofthose(time-varying)parameters(qi(t)anddi(t))willbemodeledindependentlyintheirownunit-simplexes.Followingtheapproachdepictedin[43],theASPandtheiruncertaintiescanbemodeledasfollow:1.Actualschedulingparameters(qi(t))ai(t));ai1(t)=qi(t)+¯qi2¯qi)qi(t)=2¯qiai1(t)¯qi;(3.15)then,ai2(t)=1ai1(t)=1qi(t)+¯qi2¯qi=¯qiqi(t)2¯qi;32where,ai(t)=(ai1(t);ai2(t))2L2;8i=1;2;;q;a(t)=(a1(t);a2(t);;aq(t)):2.Uncertainties(di(t))‹ai(t));‹ai1(t)=di(t)+¯di2¯di)di(t)=2¯di‹ai1(t)¯di;(3.16)then,‹ai2(t)=1‹ai1(t)=1di(t)+¯di2¯di=¯didi(t)2¯di;where,‹ai(t)=(‹ai1(t);‹ai2(t))2L2;8i=1;2;;q;‹a(t)=(‹a1(t);‹a2(t);;‹aq(t)):Thus,usingthischangeofvariables,theoriginalafparameter-dependentsystem(3.1)aswellasthegain-schedulingcontrollers(3.6),(3.9),and(3.12)canbeconvertedfromq(t)andŸq(t)intonewmulti-simplexvariablesa(t)and‹a(t),respectively.Therefore,themulti-simplexvariablesa(t)canbeas,Ÿa(t)=(ai(t);‹ai(t));i=1;2;;q;a(t)2L;whereL=L2L2L2|{z}2qsimplexes:(3.17)Consideringthecasethatq=1(oneschedulingparameter),a1(t)=(a11(t);a12(t))and‹a1(t)=(‹a11(t);‹a12(t));thehomogeneoustermsinthemulti-simplexvariablescanbewrittenintermsofthenewvariablesŸa(t)=(a11(t);a12(t);‹a11(t);‹a12(t)).Forillustrationpurposes,thedetailsofsuchtransformationwillbegivenfortwocases,single(measured)schedulingparameterandtwo(measured)schedulingparameters.Then,ageneraliza-tionforanynumberofschedulingparameterswillbegivenaswell.333.2.1.1SingleschedulingparameterForinstance,letZ(Ÿq(t))beanymatrixofthecontrollervariablesgivenin(3.14).ThismatrixcanbeexpressedafintermsoftheMSPsasZ(Ÿq(t))=Z0+Ÿq1(t)Z1=Z0+(q1(t)+d1(t))Z1:(3.18)Substitutingforq1(t)andd1(t)from(3.15)and(3.16)yields1,Z(Ÿq(t))=Z0+(2¯q1a11¯q1+2¯d1‹a11¯d1)Z1=Z(Ÿa(t));andapplyinghomogenizationprocedure[39]leadstoZ(Ÿa(t))=Z01z}|{(a11+a12)1z}|{(‹a11+‹a12)+[2¯q1a111z}|{(‹a11+‹a12)(¯q1+¯d1)(a11+a12)|{z}1(‹a11+‹a12)|{z}1+2¯d1‹a11(a11+a12)|{z}1]Z1:Asaresult,Z(Ÿa(t))isaparameter-dependentmatrixthatdependsontimevaryingparametersinsidethemulti-simplexdomainL[43].Inotherwords,theparametersbounds¯qand¯dareusedtoconvertZ(Ÿq(t))intoZ(Ÿa(t)).Thus,thematrixcanbewritteninthehomogenizedtermsasZ(Ÿa(t))=a11‹a11Z1;1+a11‹a12Z1;2+a21‹a11Z2;1+a21‹a21Z2;2;(3.19)wherethecoefZ1;1;Z1;2;Z2;1andZ2;2canbegeneratedusingtheboundsasZ1;1=Z0+(¯q1+¯d1)Z1;Z1;2=Z0+(¯q1¯d1)Z1;Z2;1=Z0+(¯q1+¯d1)Z1;Z2;2=Z0+(¯q1¯d1)Z1:(3.20)3.2.1.2TwoschedulingparametersZ(Ÿq(t))=Z0+Ÿq1(t)Z1+Ÿq2(t)Z2=Z0+(q1(t)+d1(t))Z1+(q2(t)+d2(t))Z2:1Sometimesthedependencyontwillbeomittedfornotationalsimplicity.34then,Z(Ÿq)=Z0+(2¯q1a11¯q1)Z1+(2¯d1‹a11¯d1)Z1+(2¯q2a21¯q2)Z2+(2¯d2‹a21¯d2)Z2=Z(Ÿa):Homogenizingthisequation,Z(Ÿa)=Z0(a11+a12)|{z}1(‹a11+‹a12)|{z}1(a21+a22)|{z}1(‹a21+‹a22)|{z}1+[2¯q1a11(‹a11+‹a12)|{z}1(a21+a22)|{z}1(‹a21+‹a22)|{z}1]Z1(¯q1Z1+¯d1Z1+¯q2Z2+¯d2Z2)(a11+a12)|{z}1(‹a11+‹a12)|{z}1(a21+a22)|{z}1(‹a21+‹a22)|{z}1+[2¯d1a11(‹a11+‹a12)|{z}1(a21+a22)|{z}1(‹a21+‹a22)|{z}1]Z1+[2¯q2a11(‹a11+‹a12)|{z}1(a21+a22)|{z}1(‹a21+‹a22)|{z}1]Z2+[2¯d2a11(‹a11+‹a12)|{z}1(a21+a22)|{z}1(‹a21+‹a22)|{z}1]Z2:(3.21)Asaresult,Z(Ÿa)isaparameter-dependentmatrixwithparametersinthemulti-simplexL,Z(Ÿa)=a11‹a11a21‹a21Z1;1;1;1+a11‹a11a21‹a22Z1;1;1;2+a11‹a11a22‹a21Z1;1;2;1+a11‹a11a22‹a22Z1;1;2;2+a11‹a12a21‹a21Z1;2;1;1+a11‹a12a21‹a22Z1;2;1;2+a11‹a12a22‹a21Z1;2;2;1+a11‹a12a22‹a22Z1;2;2;2+a12‹a11a21‹a21Z2;1;1;1+a12‹a11a21‹a22Z2;1;1;2+a12‹a11a22‹a21Z2;1;2;1+a12‹a11a22‹a22Z2;1;2;2+a12‹a12a21‹a21Z2;2;1;1+a12‹a12a21‹a12Z2;2;1;2+a12‹a12a22‹a21Z2;2;2;1+a12‹a12a22‹a22Z2;2;2;2(3.22)wherethematricesZ1;1;1;1,Z1;1;1;2,Z1;1;2;1,Z1;1;2;2,Z1;2;1;1,Z1;2;1;2,Z1;2;2;1,Z1;2;2;1,Z1;2;2;2,Z2;1;1;1,Z2;1;1;2,Z2;1;2;1,Z2;1;2;2,Z2;2;1;1,Z2;2;1;2,Z2;2;2;1,Z2;2;2;1,andZ2;2;2;235canbegeneratedas,Z1;1;1;1=Z0+¯q1Z1+¯d1Z1+¯q2Z2+¯d2Z2;Z1;1;1;2=Z0+¯q1Z1+¯d1Z1+¯q2Z2¯d2Z2;Z1;1;2;1=Z0+¯q1Z1+¯d1Z1¯q2Z2+¯d2Z2;Z1;1;2;2=Z0+¯q1Z1+¯d1Z1¯q2Z2¯d2Z2;Z1;2;1;1=Z0+¯q1Z1¯d1Z1+¯q2Z2+¯d2Z2;Z1;2;1;2=Z0+¯q1Z1¯d1Z1+¯q2Z2¯d2Z2;Z1;2;2;1=Z0+¯q1Z1¯d1Z1¯q2Z2+¯d2Z2;Z1;2;2;2=Z0+¯q1Z1¯d1Z1¯q2Z2¯d2Z2;Z2;1;1;1=Z0¯q1Z1+¯d1Z1+¯q2Z2+¯d2Z2;Z2;1;1;2=Z0¯q1Z1+¯d1Z1+¯q2Z2¯d2Z2;Z2;1;2;1=Z0¯q1Z1+¯d1Z1¯q2Z2+¯d2Z2;Z2;1;2;2=Z0¯q1Z1+¯d1Z1¯q2Z2¯d2Z2;Z2;2;1;1=Z0¯q1Z1¯d1Z1+¯q2Z2+¯d2Z2;Z2;2;1;2=Z0¯q1Z1¯d1Z1+¯q2Z2¯d2Z2;Z2;2;2;1=Z0¯q1Z1¯d1Z1¯q2Z2+¯d2Z2;Z2;2;2;2=Z0¯q1Z1¯d1Z1¯q2Z2¯d2Z2:(3.23)3.2.1.3MultiplenumbersofschedulingparametersThisprocedurecanbesystematicallyextendedtohandleallthesystemmatricesin(3.1)andcon-trollermatricesin(3.6),(3.9),and(3.12)toconvertthemintothemulti-simplexvariablesŸa(t)=(a(t);‹a(t))foranynumberofschedulingparametersq1.ThematricesZj1;j2;;jq;k1;k2;;kqin(3.23)forj1;j2;;jq;k1;k2;;kq=1;2,canbewritteninageneralizedformasZj1;j2;;jq;k1;k2;;kq=Z0+qåi=1n(1)ji+1¯qi+(1)ki+1¯dioZi:(3.24)36Thus,itisworthmentioningthatthesynthesisvariablesthatusedtoconstructcontrollermatricesin(3.6),(3.9),and(3.12)shouldbeconvertedintothemulti-simplexdomainusingtheproceduredescribedabove.Therefore,thecontrollermatricescanbewrittenintermsofthemulti-simplexparametersasK(Ÿa(t));Ac(Ÿa(t));Bc(Ÿa(t));Cc(Ÿa(t)),andK(Ÿa(t)).Remark3.1.Notethattheopen-loopsystemmatricesin(3.1)areindependentoftheuncertaintiesdi(t).TheyareonlydependontheASPsq(t).However,thesameproceduredescribedabovecanbeusedtotransformthemfromtheoriginalparameterspaceq(t)intomulti-simplexspacea(t)byimposing¯di=0in(3.24).Inthiscase,fornotationalsimplicity,newthemulti-simplexvariablesa(t)isusedinsteadofŸa(t)todistinguishvariablesthatdependonASPsfromvariablesthatdependonMSPs.Thus,theopen-loopsystemmatriceswillbewrittenintermsofthemulti-simplexvariablesasA(a(t));Bw(a(t));Bu(a(t));Cz(a(t));Dzu(a(t));Dzw(a(t));Cy(a(t));andDyw(a(t)):3.2.2RateofVariationModelingTheobjectiveofthissubsectionistoconstructanewconvexparameterspaceh(t)tomodelthederivativesofthevaryingparametersintheconvexdomain.Theratesofchangeofeachparameteranduncertaintyareassumedtobeboundedasin(3.5)forallt0.Sinceeachvaryingparameterbelongstoaunit-simplex,itisclearthatthefollowingrelationisai1(t)+ai2(t)=0i=1;2;;q:(3.25)Sinceai(t)2L2,thetimederivativesoftheparametersaicanassumevaluesthatmodeledbyaconvexpolytopeWi[44,45]Wi=(f2R2:f=2åk=1hikH(k)i;2åk=1Hi(k;j)=0;hi2L2);j=1;2;i=1;2;;2q:(3.26)Giventheboundsbqiandbdiin(3.5),H(k)irepresentsthek-thcolumnofmatrixHi.Sincesim-plexeswithtwoverticeshavebeenconsideredforeachvaryingparameter,asadirectconsequence,37thematricesHiwillhavesizeof22.Noticethat,dueto(3.25),thesumoftheelementsofeachcolumnofH(k)iiszero.Consequently,a(t)2W=W1W2W2q=2qÕi=1Wi:(3.27)Notethattherelationshipbetweentheboundsoftheratesofvariationsofthevaryingparametersqandd,andtheratesofchangesofmulti-simplexvariablesacanbeobtainedusing(3.5)and(3.15)asfollowsbqi2¯qiai1(t)bqi2¯qi;withai2(t)=ai1(t)astheconsequenceof(3.25).Therefore,thetransformationoftheratesofvariationsfromq(t)andd(t)intoai(t)isexactaswell.Asanexampleconsideroneschedulingparameter(q=1)withthefollowingbounds1q(t)1;1q(t)1;themulti-simplexvariablesrateboundsare0:5a11(t)0:5;andconsidering(3.25)attheboundsofa11(t)anda12(t),onecaneasilyconstructthecolumnsofthematrixH1.Takingtheconvexcombinationofthesecolumns[40],yields2640:50:5375h11+2640:50:5375h12=2640:50:50:50:5375|{z}H1264h11h12375:Thus,thederivativeoftheparametricLyapunovmatrixthatdependonatime-varyingparametersinmulti-simplexcanbecomputedthroughthisprocedureasP(a)=¶P(a)¶aa=2qåi=12åj=1¶P(a)¶aijaij=2qåi=12åj=1¶P(a)¶aij2åk=1hikHi(j;k)=¶P(a)¶aij(hi1Hi(j;1)+hi2Hi(j;2)):=P(a;h);hi2L2:(3.28)383.2.3PLMIsConditionsItisknownthatsynthesisproblemsofRGSandrobustcontrollerswithparametricuncertaintiesarefrequentlyariseasoptimizationproblemwithPLMIsconstraints.APLMIsisanextensionofanordinaryLinearMatrixInequalities(LMIs)withthedifferenceisthatitisdependentonatime-varyingparametervector.Generallyspeaking,thesolutiontothistypeofLPVcontrolprob-lemsisformulatedasPLMIs,whichisaspecialtypeofconvexoptimizationproblem.PLMIsareequivalenttonumberofLMIconstraintsand,consequently,requirenumeri-calcomputationstobesolveddirectly.Consider,forexample,thestabilityproblemofacontinuouslydifferentiableparameter-dependentsymmetricmatrixP(a(t))forthefollowingnon-autonomoussystemx(t)=A(a(t))x(t)with(a;a)2LWA(a(t))0P(a(t))+P(a(t))A(a(t))+2qåi=12åj¶P(a(t))¶aij(t)aij(t)<0P(a(t))>0:(3.29)Clearly,inequality(3.29)isaPLMIwithdimensionalspace.Foreveryvalueofthevaryingparameter(a;a)2LW,theinequality(3.29)representsordinaryLMI.Therefore,synthesisconditionsoftheRGScontrollerspresentedinthisdissertationarefor-mulatedasaconvexoptimizationproblemwithPLMIsconstraints.Afterallschedulingparame-tersandtheiruncertainties(withtheirratesofvariations)aremodeledtovarywithinconvexsets(L;W),asillustratedbytheproceduregivenintheprevioustwosub-sections,thedetailedproofsandderivationofthesynthesisPLMIsconditionswillbethecoreofthenexttwochaptersforH2andH¥performance,respectively.3.2.4PLMIsRelaxationAsmentioned,thesynthesisconditionsofRGSstate-feedbackandthestaticoutput-feedbackwithH2andH¥performanceareformulatedintermsofPLMIswhileRGSdynamicoutput-feedbackareformulatedasPBMIs.Althoughsuchconditionsrepresentsconvexoptimizationproblem(for39PLMIs)withdimensionalconstraints,modernrobustoptimizationtechniquesconsiderablystrengthenedthisframeworkbyprovidingrigorouswaysfordealingwithparameter-dependentLMIs[46,47,48].In[12],brutalforcegridingmethoddevelopedtodivideparameterspacetorelaxtheoptimizationproblemintoaproblem.However,usingthismethod,thenumberoftheresultingLMIconstraintsgrowsrapidlyasthenumberofschedulingparametersincreases.Moreover,thismethodonlyprovidesanapproximatedsolutionwhichtheLMIconstraintsatgriddingpointsintheparameterspace.Thus,theresultfromgriddingpointsisunreliable.Ontheotherhand,alternativeapproacheshavebeenactivelysoughttoturnPLMIsintoastandardLMIproblembyconstructingtheirrelaxationforms.InthecaseofLPVsystemsdependingafontheschedulingparameter,vertexmethodwascon-sideredin[49]todetermineconstantLyapunovfunctionssatisfyingafparameter-dependentLMI.Thesolutionisexactbutitpreventsthepossibilityofusingparameter-dependentLyapunovfunctionswhichleadtoaconservativesolution.In[50],convexcoveringtechniqueswereappliedtoPLMIstoobtainparameter-dependentsolutions.However,thesemethodsoftenrequirelargedivisionnumberstoachieveaccurateresults.Multi-convexitypropertieswasimposedin[51]toprovideasetofLMIstosolvePLMIsproblems.Morerecently,manypowerfultheoreti-calandcomputationaltoolshavebeendevelopedandappliedsuccessfully(byseveralresearchersindependently)torelaxPLMIsproblemsintosetofLMIs.Thus,thenotionofSum-Of-Square(SOS)-convexityhasbeenproposedasatractableapproachforconvePLMIsbasedonSOS-decomposition[52].SlackVariable(SV)approach[53],dilatedLMIapproach[54],andcoefcheckapproachusingPólya'stheorem[41]areotherpowerfultechniquesthatappliedsuccessfullyinliteraturetorelaxPLMIs.Thepromisingresultsoftheserelaxationmethodsover-comesthedifofsolvingPLMIstosynthesizegain-schedulingcontrollersasdemonstratedin[55].Therefore,therelaxationapproachthatwasdevelopedin[39]isadoptedinthisdissertationtorelaxthePLMIsconditionsinChapter4,Chapter5,andChapter6sinceitsupportsPLMIsthatdependsonmulti-simplexparameters.In[41],Oliveiraetal.developedasystematicprocedure40toconstructafamilyofLMIrelaxationsforuncertainLTIsystemsinpolytopicdomainmodeledbyunit-simplex.Then,theyextendtheworkin[39]formulti-simplexdomainwithtime-varyingparameters.Forcompleteness,PLMIsrelaxationprocedureforthestabilitytestofthetime-varyingsystemin(3.29)willbeillustratedhere.Forconvenience,thePLMIsarerewrittenagainhereA(a(t))0P(a(t))+P(a(t))A(a(t))+2qåi=12åj¶P(a(t))¶aij(t)aij(t)<0;(3.30)P(a(t))>0;(3.31)with(a;a)2LW.ForL-homogeneousmatricesP(a(t))andA(a(t))ofpartialdegreesg=(g1;g2;:::;gq)andr=(r1;r2;:::;rm)respectively,thetotaldegreeofthetwotermsofin-equality(3.30)is¯g=(g1+r1;g2+r2;:::;gq+rq).Thus,themaintaskistohomogenizeaccord-inglythethirdterm,i.e.2qåi=12åj¶P(a(t))¶aij(t)aij(t):ThegeneralexpressionforthederivativeoftheLyapunovmatrixP(a(t))withrespecttothei-thcomponentofthemulti-simplex,i=1;2;:::;qandthenwithrespecttoitsj-thcomponent,j=1;2isgivenby¶P(a(t))¶aij(t)=åk2Q(N;g)kijak11aki1i1akij1ijaki2i2akqqPk=åk2Q(N;geijq)ak(k+eijqejj2)ijPk+eijqejj2wherebytioneijqisthevectorofdimensionqwithzerocomponents,except1inthei-thposition.To(ona)withthepartialdegrees¯g,thefollowinghomogenizationisnecessary41qåi=1(ai1++ai2)ri+12åj=1¶P(a(t))¶aij(t)=qåi=12åj=1åk2Q(N;geijq)ak0BBBB@å‹k2Q(r+eijq)‹kk(ri+1)!p(‹ki)(k‹k+eijqejj2)ijPk‹k+eijqejj21CCCCA(3.32)wherep(ki)=(ki1!)(ki2!).Nowthethirdtermof(3.30)shouldbehomogenizedtobecomemulti-afonh.ThisisdoneasfollowsqÕp=1p6=i(hp1+hp2)2å`=1hi`Hi(j;`)=2åp1=12åpi=12åpq=1h1p1hipihqpqHi(j;pi):(3.33)Considering(3.32)and(3.33),thethirdtermintheleft-handsideof(3.30)canbeequivalentlywrittenasqåi=12åj=1¶P(a)¶aij2å`=1hi`Hi(j;`)=2åp1=12åpi=12åpq=1h1p1hipihqpq0BBBB@åk2Q(N;g+r)akqåi=12åj=1å‹k2Q(r+eijq)‹kk(ri+1)!p(‹ki)(k‹k+eijqejj2)ijPk‹k+eijqejj2Hi(j;pi)1CCCCA:Now,notethatqÕp=1(hp1+hp2)A(a)0P(a)+P(a)A(a)=2åp1=12åpi=12åpq=1h1p1hipihqpqA(a)0P(a)+P(a)A(a);(3.34)and,(3.30)canbetestedsincealltermshavethesamepartialdegreesforbothaandh.ThenextlemmapresentsLMIrelaxationsfortherobuststability(analysis)problemoftheparameter-varyingmatrixA(a(t))foranypair(a;a)2LW.42Lemma3.1.[39]LetLbeamulti-simplexofdimensionN=(2;2;:::;2).TheL-homogeneouspolynomialmatrixA(a(t))ofpartialdegreesr=(r1;r2;:::;rq)isrobustlystableforanypair(a;a)2LW,ifthereexistsg=(g1;g2;:::;gq),k2Q(N;g)andmatricesPk=P0k2Rnnsuchthatforall(i1;i2;:::;iq)2f1;2gf1;2gf1;2gthefollowingLMIsarePk>000n;Fk=åŸk2Q(N;r)ŸkkA0ŸkPkŸk+PkŸkAŸk+Yk<000n;8k2Q(N;g+r)whereYk=qåi=12åj=1å‹k2Q(r+eijq)‹kk(ri+1)!p(‹ki)(k‹k+eijqejj2)ijPk‹k+eijqejj2Hi(j;pi):Proof.See[39].Itisclearfromthepreviousexamplethatthealgebraicmanipulationoftherelaxedcondi-tionsrequiresaprioriknowledgeontheformationlawofthemonomials,whichdependsonthenumberofschedulingparametersandonthedegreeoftheparametrizationoftheLyapunovma-trixP(a(t)).Ascanbeeasilyobserved,thepreviousproceduredealswithproductsbetweentwoparameter-dependentmatrices.However,WhentheLMIstobesolvedaremorecomplexandhaveproductsinvolvingthreeormoreparameter-dependentmatrices,therulestocomposethemonomialsbecomemuchmorecomplicated.Moreover,eachPLMIdemandsmanipulationofdifferentpolynomials.Suchtask,aswellasprogrammingtheresultingLMIs,issophisticated,time-demandingandcanbeasourceofprogrammingerrors.Therefore,aspecializedRObustLMIParser(ROLMIP)2[56]hasbeenrecentlydevelopedasatooltoperformsuchmanipulationandLMIrelaxationofthePLMIs.ThispackageworksjointlywiththeLMIparserYALMIP[57]andtheLMIsolverSeDuMi[58]thatisusedinthisdissertationtoobtaintheoptimalsolutionof2Availablefordownloadathttp://www.dt.fee.unicamp.br/~agulhari/rolmip/rolmip.htm43theconvexoptimizationproblemsofthesynthesisconditionsthatwillbegiveninthenextthreechapters.3.2.5PBMIAlgorithmSinceonlythesynthesisconditionsofRGSDOFcontrollerareformulatedasPBMIs,itwillbecoveredinChapter5.3.2.6InverseTransformationOnceafeasiblesolutionisobtainedusingthePLMIsconditionsorthePBMIalgorithm,inversetransformationisrequiredtomapthesolutionfrommulti-simplexdomainŸaintooriginalparame-terspaceŸq.Sincereal-timeimplementationofthegain-schedulingcontrollersfollowsthestructureof(3.18)thatcanbegeneralizesforaq-schedulingparametersasZ(Ÿq(t))=Z0+qåi=1Ÿqi(t)Zi:(3.35)Inotherwords,thekeyistocalculateZj,forj=0;1;2;;q,thatrequiredinreal-timeforcontrollerimplementations.Oneschedulingparameter:Z1;1=Z0+(¯q1+¯d1)Z1;(3.36)Z1;2=Z0+(¯q1¯d1)Z1;(3.37)Z2;1=Z0+(¯q1+¯d1)Z1;(3.38)Z2;2=Z0+(¯q1¯d1)Z1:(3.39)ToobtainZ0andZ1fromthemulti-simplexvariables(afterafeasiblesolutionisfound),adding(3.36),(3.37),(3.38)and(3.39)toobtainZ0=14[Z1;1+Z1;2+Z2;1+Z2;2]:44Then,(3.36)+(3.37)(3.38)(3.39)toobtainZ1=14¯q[Z1;1+Z1;2Z2;1Z2;2]:Twoschedulingparameters:Equation(3.35)canbewrittenasZ(Ÿq(t))=Z0+Ÿq1(t)Z1+Ÿq2(t)Z2:ToobtainZ0,Z1,andZ2fromthemulti-simplexvariables(afterafeasiblesolutionisfound),addingallequationsin(3.23)toobtainZ0=116[Z1;1;1;1+Z1;1;1;2+Z1;1;2;1+Z1;1;2;2+Z1;2;1;1+Z1;2;1;2+Z1;2;2;1+Z1;2;2;2+Z2;1;1;1+Z2;1;1;2+Z2;1;2;1+Z2;1;2;2+Z2;2;1;1+Z2;2;1;2+Z2;2;2;1+Z2;2;2;2]:Then,(the1st8equations)-minus-(thelast8equations)toobtainZ1=116¯q1[Z1;1;1;1+Z1;1;1;2+Z1;1;2;1+Z1;1;2;2+Z1;2;1;1+Z1;2;1;2+Z1;2;2;1+Z1;2;2;2Z2;1;1;1Z2;1;1;2Z2;1;2;1Z2;1;2;2Z2;2;1;1Z2;2;1;2Z2;2;2;1Z2;2;2;2]:Then,(the1st2eqs)-minus-(the2nd2eqs)-plus-(the3rd2eqs)-minus-(the4th2eqs)-plus-...toobtainZ2=116¯q2[Z1;1;1;1+Z1;1;1;2Z1;1;2;1Z1;1;2;2+Z1;2;1;1+Z1;2;1;2Z1;2;2;1Z1;2;2;2+Z2;1;1;1+Z2;1;1;2Z2;1;2;1Z2;1;2;2+Z2;2;1;1+Z2;2;1;2Z2;2;2;1Z2;2;2;2]:Anynumberofschedulingparametersi=1;2;;q:Z0=122q2åj1=12åj2=12åjq=12åki=12åk2=12åkq=1Zj1;j2;;jq;k1;k2;;kq:(3.40)45Zi=122q¯qi2åj1=12åj2=12åjq=12åki=12åk2=12åkq=1(1)ji+iZj1;j2;;jq;k1;k2;;kq:(3.41)3.3SummaryThischapterpresentedmathematicalformulationoftheRGScontrollersynthesisproblemwithuncertainschedulingparameters.Then,asystematicsolutionprocedurewasgivenaswell.ThenextthreechaptersdealwiththesynthesisconditionsofRGSSF,DOF,andSOFcontrollerswithguaranteedH2andH¥performance,respectively.Basedontheprocedurepresentedinthischap-ter,numericalexamples,simulations,andcomparisonswithotherworkfromliteraturearegiveninthenextthreechapterstodemonstratetheeffectivenessofthedevelopedconditions.46CHAPTER4RGSSTATE-FEEDBACKCONTROLInthischapter,characterizationsofsynthesisconditionsforRGSState-Feedback(SF)controlwithguaranteedH2andH¥performancesubjecttonoisyschedulingparametersaredeveloped.Theorganizationofthischapterisasfollows.First,RGSsynthesisproblemformulationwithguaranteedH2andH¥performancearepresentedandthen,PLMIsconditionsforthesynthesisofH2andH¥controllersaredeveloped.Numericalexamples,simulations,andcomparisonswithotherapproachesfromliteratureisgiventodemonstratetheeffectivenessofthedevelopedconditions.Finally,ashortsummaryisgiveninthelastsection.4.1SFSynthesisProblemConsiderthefollowingLPVsystemSOL:=8>><>>:x(t)=A(q(t))x(t)+Bu(q(t))u(t)+Bw(q(t))w(t);z(t)=Cz(q(t))x(t)+Dzu(q(t))u(t)+Dzw(q(t))w(t);(4.1)wherex(t)2Rnisthestate,u(t)2Rnuisthecontrolinput,w(t)2Rnwisthedisturbanceinput,andz(t)2Rnzisthecontrolledoutput.ThesystemmatriceshavethefollowingcompatibledimensionsA(q(t))2Rnn,Bu(q(t))2Rnnu,Bw(q(t))2Rnnw,Cz(q(t))2Rnzn,Dzu(q(t))2Rnznu,andDzw(q(t))2Rnznw.ThegoalistosynthesizeaRGSstate-feedbackcontrolleroftheformu(t)=K(Ÿq(t))x(t);(4.2)tostabilizetheclosed-loopsystem(seeFigure3.2)SCL:=8>><>>:x(t)=A(q(t);Ÿq(t))x(t)+B(q(t);Ÿq(t))w(t);z(t)=C(q(t);Ÿq(t))x(t)+D(q(t);Ÿq(t))w(t):(4.3)47whereA(q(t);Ÿq(t))=A(q(t))+Bu(q(t))K(Ÿq(t));B(q(t);Ÿq(t))=Bw(q(t));C(q(t);Ÿq(t))=Cz(q(t))+Dzu(q(t))K(Ÿq(t));D(q(t);Ÿq(t))=Dzw(q(t)):(4.4)Furthermore,performanceboundsintermsofH2andH¥shouldbeguaranteedaswell.Thecontrollermatrixin(4.2)isassumedtohaveafparametrizationwithrespecttotheMeasuredSchedulingParameters(MSPs).Inotherwords,thecontrollermatrix,K(Ÿq(t)),ispa-rameterizedasK(Ÿq(t))=K0+qåi=1Ÿqi(t)Ki:(4.5)Therefore,thegoalistoobtainthecontrollercoefKifori=0;1;2;;q,toimplementtheRGScontrollerbyusingonlytheMSPsŸqi.4.2RGSH2ControlSinceLPVsystemsisthetopicofthisdissertation,theterm"H2norm"forLPVsystemsshouldbetreatedwithcaresincethestandardH2controltheoryisoriginallydevelopedforLTIsystems.However,severalofH2normforLPVsystemshavebeenproposedinliterature[59].Thesecanessentiallybedividedintothefollowingtwogroups,DeterministicinterpretationwheretheexogenousinputismodeledasunknowndisturbancebelongstoaboundedL2energyset[60],andStochasticinterpretationbasedonthecovarianceoftheoutputduetoGaussianwhitenoise[61].Itiswell-knownthatthesecoincideforLTIsystems,butnotfortheLPVcase.Inpursu-ingtheextensionofthesecondtotheLPVcase,caremustbeexercisedsincetheoutputtostationarynoisemaynolongerbestationary.Thisleadstotwodifferentinterpretationsbased48onwhethertheaverageorworst-caseoutputvarianceareconsidered[59].Inthisdissertation,thestationarywhitenoiseinterpretationisusedandtheworst-caseoutputvarianceisasthesquaredH2normforLPVsystems.SincethisnormcanbecalculatedintermsoftheLyapunovmatrix,suchachoiceofH2norminterpretationappearstobemostappropriateforthesynthesisapproachdevelopedinthisdissertation.Problem4.1.Supposethattheschedulingparametersq(t)areprovidedasŸq(t)withuncertaintyd(t)asin(3.3).SupposefurtherthatD(q(t);Ÿq(t))=0in(4.3).Convertingalltheopen-loopsystemmatricesandsynthesisvariablestothemulti-simplexvariablesaorŸainsteadofqandŸq,respectively,using(3.15)and(3.16).Foragivenpositivescalarn,aRGSstate-feedbackcontrollerintheformof(4.2)tostabilizetheclosed-loopsystem(4.3)foranypair(Ÿa(t);Ÿa(t))2LWandsatisfysup(Ÿa(t);Ÿa(t))2LWEEEˆZT0z(t)0z(t)dt˙<n2;(4.6)forthedisturbanceinputw(t)givenbyw(t)=w0d(t)whered(t)istheDirac'sdeltafunctionandw0isarandomvariablesatisfyingEw0w00=IkandEgdenotesthemathematicalexpectation.Lemma4.1.[62]LetD(q(t);Ÿq(t))=0in(4.3).Foragivenpositivescalarn,ifthereexistacon-tinuouslydifferentiablematrixP(a)=P(a)02Rnnandparameter-dependentmatrixW(a)=W(a)02RnznzsuchthatthefollowingPLMIsare49264A(a;Ÿa)P(a)+P(a)A(a;Ÿa)0P(a)?B(a;Ÿa)0I375<000;(4.7)264P(a)?C(a;Ÿa)P(a)W(a)375>000;(4.8)trace(W(a))<n2;(4.9)thentheclosed-loopsystemin(4.3)isasymptoticallystableforanypairs(Ÿa;Ÿa)2LWand(4.6)iswhereA(a;Ÿa),B(a;Ÿa),andC(a;Ÿa)aretheclosedsystemmatricesin(4.3)withqandŸqreplacedbyaandŸausing(3.15)and(3.16).InviewofLemma4.1andequation(4.6),nrepresentsanupperboundoftheH2normoftheclosed-loopsystem(4.3).BeforepresentingPLMIssynthesisconditionsFinsler'slemmawillbegivensinceitisusedinthederivationofthesynthesisconditions[63,64].Lemma4.2.Letz2Rn,Y2RnnandV2Rmnwithrank(V)<n,V?suchthatVV?=0.Then,thefollowingconditionsareequivalent:i)z0Yz<0;8z6=0:Vz=0;ii)V?0YV?<0;iii)9m2R:YmV0V<0;iv)9X2Rnm:Y+XV+V0X0<0:Theorem4.1.LetD(q(t);Ÿq(t))=0in(4.3).Givenascalarn>0andasufsmallscalare>0.Ifthereexistacontinuouslydifferentiableparameter-dependentmatrix0<P(a)=P(a)02Rnn,parameter-dependentmatricesW(a)=W(a)02Rnznz,Z(Ÿa)2Rnun,G(Ÿa)2Rnnfor50anypair(Ÿa(t);Ÿa(t))2LWsuchthatthefollowingPLMIsare266664A(a)G(Ÿa)+Bu(a)Z(Ÿa)+()0P(a)??P(a)G(Ÿa)+e(A(a)G(Ÿa)+Bu(a)Z(Ÿa))0e(G(Ÿa)+G(Ÿa)0)?Bw(a)0000nwnInw377775<0002n+nw;(4.10)264G(Ÿa)+G(Ÿa)0P(a)?Cz(a)G(Ÿa)+Dzu(a)Z(Ÿa)W(a)375>000n+nz;(4.11)trace(W(a))<n2:(4.12)Then,thegain-schedulingcontrollerK(Ÿa)=Z(Ÿa)G(Ÿa)1;(4.13)stabilizestheclosed-loopsystemwithguaranteedH2performanceboundnsatisfying(4.6).Proof.Usingslackvariableapproach,additionaloptimizationvariableU(Ÿa)canbeintroducedviaFinsler'slemmatodecouplethedynamicmatrixA(a)fromLyapunovmatrixP(a),henceusingiv)inLemma4.2,inequality(4.7)canbewrittenX(a)+U(Ÿa)V(a)+V(a)0U(Ÿa)0<000;(4.14)where,X(a):=266664P(a)P(a)000P(a)000000000000I377775;U(Ÿa):=266664G(Ÿa)0000Y(Ÿa)0000000I377775;V(a):=264A(a)0I000Bw(a)0000I375;suchthatV(a)?0X(a)V(a)?<000;withV(a)?0=[IA(a)Bw(a)].Therefore,substituting51theserelationsinto(4.14),266664P(a)P(a)000P(a)000000000000I377775+266664G(Ÿa)0000Y(Ÿa)0000000I377775264A(a)0I0Bw(a)0000I375+266664A(a)Bw(a)I000000I377775264G(Ÿa)Y(Ÿa)000000000I375<000:(4.15)Atthisend,itisimportanttoimposeparticularstructuretotheslackvariableU(Ÿa)tomaintainconvexparametrization.Therefore,settingY(Ÿa)=eG(Ÿa)issuftokeepconvexityof(4.15),whereeisascalarusedasanextradegreeoffreedomtoperformlinesearchforperformanceimprovement[65].SubstitutingforA(a)asclosed-loopmatrixA(a;Ÿa)givenby(4.4)withaandŸareplacingqandŸq,respectively,andthechangeofvariableZ(Ÿa)=K(Ÿa)G(Ÿa)yields266664P(a)P(a)000P(a)000000000000I377775+266664G(Ÿa)0A(a;Ÿa)0G(Ÿa)0000eG(Ÿa)0A(a;Ÿa)0eG(Ÿa)0000Bw(a)0000I377775+266664A(a;Ÿa)G(Ÿa)eA(a;Ÿa)G(Ÿa)Bw(a)G(Ÿa)eG(Ÿa)000000000I377775<000;thatdirectlyleadto(4.10).Multiplying(4.11)fromleftby[C(a;Ÿa)I]andbyitstransposefromrightwithC(a;Ÿa)in(4.4)toobtainW(a)>C(a;Ÿa)P(a)C(a;Ÿa)0withSchurcomplement,(4.8)canberecovered.ThePLMI(4.12)ensuresthatnistheguaranteedcost(upperbound)oftheH2normoftheclosed-loopsystem.524.3RGSH¥ControlProblem4.2.Supposethattheschedulingparametersq(t)areprovidedasŸq(t)withuncertaintyd(t).Convertingalltheopen-loopsystemmatricesandsynthesisvariablestothemulti-simplexvariablesaorŸainsteadofqandŸq,respectively,using(3.15)and(3.16).Foragivenpositivescalarg¥,aRGSstate-feedbackcontrollerintheformof(4.2)tostabilizetheclosed-loopsystem(4.3)foranypair(Ÿa(t);Ÿa(t))2LWandsatisfysup(Ÿa(t);Ÿa(t))2LWsupw2L2;w6=0kz(t)k2kw(t)k2<g¥:(4.16)ThenextlemmawillbeusedinthederivationsofthePLMIssynthesisconditions.Lemma4.3.[66]Ifthereexistsacontinuouslydifferentiableparameter-dependentsymmetricmatrixP(a)2Rnnforanypair(Ÿa(t);Ÿa(t))2LWsuchthatthefollowingPLMI266664A(a;Ÿa)P(a)+()0P(a)??C(a;Ÿa)P(a)Inz?B(a;Ÿa)0D(a;Ÿa)0g2¥Inw377775<000;(4.17)theclosed-loopsystem(4.3)isasymptoticallystablewith(4.16)whereA(a;Ÿa),B(a;Ÿa),C(a;Ÿa)andD(a;Ÿa)aretheclosedsystemmatricesin(4.3)withqandŸqreplacedbyaandŸausing(3.15)and(3.16).Theorem4.2.Givenascalarg¥>0andasufsmallscalare>0.Ifthereexistacontinu-ouslydifferentiableparameter-dependentmatrix0<P(a)=P(a)02Rnn,parameter-dependentmatricesZ(Ÿa)2Rnun,G(Ÿa)2Rnnforanypair(Ÿa(t);Ÿa(t))2LWsuchthatthefollowingPLMI266666664A(a)G(Ÿa)+Bu(a)Z(Ÿa)+()0P(a)???P(a)G(Ÿa)+e(A(a)G(Ÿa)+Bu(a)Z(Ÿa))0e(G(Ÿa)+G(Ÿa)0)??C(a)G(Ÿa)+Du(a)Z(Ÿa)eC(a)G(Ÿa)+eDu(a)Z(Ÿa)Inz?Bw(a)0000nwnDw(a)0g2¥Inw377777775<000:(4.18)53Then,thegain-schedulingcontrollerK(Ÿa)=Z(Ÿa)G(Ÿa)1;stabilizestheclosed-loopsystemwithguaranteedH¥performanceboundg¥satisfying(4.16).Proof.AdditionalslackvariableU(Ÿa)canbeintroducedtoinequality(4.17)usingLemma4.2todecouplethedynamicmatrixfromLyapunovmatrixP(a).WithA(a;Ÿa)andC(a;Ÿa)aretheclosed-loopmatrices,inequality(4.17)canbewrittenX(a)+F(Ÿa)<0withX(a)=diag0B@264P(a)P(a)P(a)0375;264InzDw(a)Dw(a)0g2¥Inw3751CA=266666664P(a)P(a)00P(a)00000InzDw(a)00Dw(a)0g2¥Inw377777775(4.19)andF(Ÿa)=U(Ÿa)V(a)+V(a)0U(Ÿa)0(4.20)whereU(Ÿa)=266666664G(Ÿa)0IeG(Ÿa)000000377777775;V(a)=264A(a;Ÿa)0IC(a;Ÿa)00000Bw(a)37554suchthatV(a)?0X(a)V(a)?<0,withV(a)?=266666664I0A(a;Ÿa)0C(a;Ÿa)00I00377777775:SubstitutingU(Ÿa)andV(a)in(4.20),F(Ÿa)=U(Ÿa)V(a)+V(a)0U(Ÿa)0=266666664G(Ÿa)0IeG(Ÿa)000000377777775264A(a;Ÿa)0IC(a;Ÿa)00000Bw(a)375+266666664A(a;Ÿa)0I0C(a;Ÿa)00Bw(a)0377777775264G(Ÿa)eG(Ÿa)00I000375;F(Ÿa)=266666664G(Ÿa)0A(a;Ÿa)0G(Ÿa)0G(Ÿa)0C(a;Ÿa)0Bw(a)eG(Ÿa)0A(a;Ÿa)0eG(Ÿa)0eG(Ÿa)0C(a;Ÿa)0000000000377777775+266666664A(a;Ÿa)G(Ÿa)eA(a;Ÿa)G(Ÿa)00G(Ÿa)eG(Ÿa)00C(a;Ÿa)G(Ÿa)eC(a;Ÿa)G(Ÿa)00Bw(a)0000377777775;55F(Ÿa)=266666664G(Ÿa)0A(a;Ÿa)0+A(a;Ÿa)G(Ÿa)G(Ÿa)0+eA(a;Ÿa)G(Ÿa)G(Ÿa)0C(a;Ÿa)0Bw(a)eG(Ÿa)0A(a;Ÿa)0G(Ÿa)eG(Ÿa)0eG(Ÿa)eG(Ÿa)0C(a;Ÿa)00C(a;Ÿa)G(Ÿa)eC(a;Ÿa)G(Ÿa)00Bw(a)0000377777775;(4.21)then,adding(4.19)and(4.21)andsubstitutingforclosed-loopmatriceswiththechangeofvari-ablesK(Ÿa)=Z(Ÿa)G(Ÿa)1,inequality(4.18)canbeobtained.4.4ExtensiontoUnmeasurableParametersTheorem4.1andTheorem4.2addressthedesignproblemofRGSstate-feedbackcontrollersinwhichallthetime-varyingparametersoftheplantaremeasurableon-line.However,therearemanycaseswheresomeoftheparametersareavailableforreal-timemeasurementandothersarenot.Insuchacase,theunmeasurableparameterscanbetreatedasplantuncertainties,thenthesynthesizedgain-schedulingcontrollerwillbeindependentoftheseparametersandtheresultingclosed-loopsystemshouldberobustagainsttheseparametersaswell.Toillustrate,thefollowingnotationshouldbeforpartialdegreesinthemulti-simplexdomain:(i)g=(g1;g2;;g2q)isavectorrepresentingthepartialdegreesassociatedwithLya-punovmatrixP(a)[40].Thus,girepresentsthedegreeofLyapunovmatrixassociatedwiththei-thunit-simplex.(ii)s=(s1;s2;;s2q)isavectorrepresentingthepartialdegreesassociatedwiththesynthesismatricesG(Ÿa)andZ(Ÿa)inTheorem4.1.Therefore,choosings=(1;1;;1),rationalcontrollercanbesynthesizedsincethesynthesismatriceshaveafdependenceonthevaryingparameters.Ontheotherhand,arobustcontroller(i.e.,parameter-independent)canbedesignedbysettings=(0;0;;0).Thus,properselectionofpartialdegreesofLyapunovmatrixandsynthesisvariableshasnotonlyontheachievedperformancebutalsoontheparametrizationofthesynthesizedcontroller(robust,aforrational)[40].56Asaruleofthumb,allthepartialdegreesofthesynthesisvariables(s)shouldbechosenthesameforgain-schedulingcontrol,however,settingsomeofthesedegreestozeroresultsinapartiallyscheduledcontroller.Thisremarkablefeaturecouldbeexploitedeftohandlethecaseofunmeasurableschedulingparameters.Inotherwords,thesynthesisvariablesG(Ÿa)andZ(Ÿa)willbeindependentoftheseparameters(correspondingtothezero-degreesimplexes),leadingtothenotionofselectivegain-schedulingcontrollersthatwasaddressedforthetimein[40].SupposeforexampleanLPVsystemthatdependsontwotime-varyingparameters,oneofthemismeasurable(theone)andthesecondoneisnot(uncertain).Thenbysettingthepartialdegreesofthesynthesisvariableass=(1;0)resultsinacontrollerthatisdependentontheparameterandindependentonthesecondone.Thisselectivityfeaturecanalsobeusedtoinvestigatetheimpactofeachoftheschedulingparameterindependentlyontheachievableperformance[40].DuetopresenceoftheadditionalslackvariableG(Ÿa)inTheorem4.1,thecontrollerwillbesynthesizedusingthisvariableinsteadofLyapunovmatrixP(a).Therefore,thepartialdegreesofthesynthesisvariablesandLyapunovmatrixgcouldbedifferent.Thus,robustcontrollerscanbedesignedwithparameter-dependentLyapunovmatrixthatimprovesthecontrollerperformanceconsiderably.Corollary4.1willutilizethisfeaturetosynthesizerobustcontrollersthatwillbeusedinSection4.5forperformancecomparisonwithgain-schedulingcontrollers.Therefore,supposethatallschedulingparametersareunavailableforreal-timemeasurement.BysettingthepartialdegreesofthesynthesisvariablesG(Ÿa)andZ(Ÿa)tobeazerovector,i.e.s=(0;0;;0),robustcontroller(parameter-independent)issynthesizedsinceinthiscaseG(Ÿa)=GandZ(Ÿa)=Z.Inotherwords,Theorem4.1treatsrobustcontrollersynthesisasaspecialcase.Corollary4.1.Givenascalarn>0andasufsmallpositivescalare>0.Ifthereexistacontinuouslydifferentiableparameter-dependentmatrix0<P(a)=P(a)02Rnn,parameter-dependentmatrixW(a)=W(a)02Rnznz,constantmatricesZ2RnunandG2Rnnforany57pairs(a(t);a(t))2LWsuchthatthefollowingPLMIsare266664A(a)G+Bu(a)Z+()0??P(a)G+e(A(a)G+Bu(a)Z)0e(G+G0)?Bw(a)0000nwnInw377775<02n+nw;264G+G0P(a)?Cz(a)G+Dzu(a)ZW(a)375>000n+nz;trace(W(a))<n2;Then,therobustcontrollerK=ZG1;stabilizestheclosed-loopsystemwithguaranteedH2performanceboundednsatisfying(4.6).Proof.TheproofcanbedoneinasimilarwaytothatofTheorem4.1.Corollary4.2.Givenascalarg¥>0andasufsmallscalare>0.Ifthereexistacontin-uouslydifferentiableparameter-dependentmatrix0<P(a)=P(a)02Rnn,constantmatricesZ2Rnun,G2Rnnforanypair(Ÿa(t);Ÿa(t))2LWsuchthatthefollowingPLMI266666664A(a)G+Bu(a)Z+()0???P(a)G+e(A(a)G+Bu(a)Z)0e(G+G0)??C(a)G+Du(a)ZeC(a)G+eDu(a)ZInz?Bw(a)0000nwnDw(a)0g2¥Inw377777775<0002n+nz+nw:Then,therobustcontrollerK=ZG1;stabilizestheclosed-loopsystemwithguaranteedH¥performanceboundg¥satisfying(4.16).Proof.TheproofcanbedoneinasimilarwaytothatofTheorem4.2Remark4.1.AshavebeenmentionedinSubsection3.2.4,matrixcoefcheckrelaxationmethod[41,39]withROLMIP[56]isusedsolvetheconditionsofTheorem4.1andTheorem4.2toobtaintheoptimalcontroller.584.5NumericalExamplesTheobjectiveofthenumericalexamplespresentedinthissectionistoillustratetheeffectivenessofthedevelopedconditions.Tofacilitatecomparisonswithothermethods,twoillustrativeexampleshavebeenborrowedfromliterature,oneforstate-feedbackandtheotherisfordynamicoutput-feedbackcontroller.TheroutinesforthesesexamplesareimplementedinMATLABR(R2013a)usingacomputerequippedwithIntelQuadCorei5(2:4GHz)processor,4GBRAMandWindows7(64-bit)operatingsystem.Example4.1:ConsiderthefollowingLPVsystem[67],[1].Itrepresentsthedynamicsofthemechanicalsystemwithtwo-massesandtwo-springs,264A(q)Bu(q)Bw(q)Cz(q)Dzu(q)375=266666666666400100000010021q1(t)0012202q1(t)100100q2(t)3777777777775;withthefollowingbounds,0:5q1(t)3:5;0:5q2(t)1:5;jqq(t)jk;jdq(t)jz;jdq(t)j10z;q=1;2:AfterapplyingtheproceduredevelopedinSection3.2toconvertthesystemmatricesintomulti-simplexdomain,Theorem4.1isusedtosynthesizegain-schedulingcontrollerforthissys-temwiththegivenbounds.Table4.1illustratestheguaranteedH2performanceboundnwithdifferentuncertaintybounds(z).Asexpected,nisbytheuncertaintyboundzassoci-atedwiththeschedulingparametersqq(t)forq=1;2.Corollary4.1isusedtosynthesizerobuststate-feedbackcontrollerswhichachievesclosed-loopperformancepresentedinthelastrowoftheTable4.1.59Table4.1:GuaranteedH2performance:Theorem4.1.zk0.0010.010.110.10.0920.0930.0970.1290.20.1730.1730.1780.1570.50.2760.2760.2780.28510.2780.2790.2800.28720.2820.2830.2860.295Robust0.2820.2830.2860.295Table4.2:GuaranteedH2performance:methodof[1].zk0.0010.010.110.10.0700.0770.1390.1470.20.1320.1410.1920.2010.50.2580.2630.2990.30010.2840.2890.3000.30020.3000.3000.3000.300Robust0.3000.3000.3000.300Notethat,astheuncertaintybound(measurementnoise)increases,theachievedH2perfor-manceofthegain-schedulingcontrollerdeterioratesandapproachestheperformanceprovidedbytherobustcontroller.Forexample,whenz=2theachievedperformanceisthesameastheper-formanceprovidedbytherobustcontroller.Thisisalogicalobservationsinceastheuncertaintyoftheschedulingparameterincreases,themeasurementwillbeunreliableforschedulingthecon-trolleranymore.Inthiscase,thereisnoforthegain-schedulingcontrollerovertherobustonesincetheachievableperformanceofthetwodesignsisthesame.Table4.2from[1]isgivenheretofacilitatecomparisonwithourcontrollers.Althoughthetwomethodsachievecompetitiveresults,itcanbenotedthatthecontrollerssynthesizedviaTheorem4.1achievebetterperformancethanthecontrollersofmethod[1]forlargeuncertaintyboundzandratesofvariationsofthepa-rametersk.Figure4.1illustratestheachievedperformanceasfunctionofzandk.Thisdemonstratesthataszandkincrease,theperformanceofgain-schedulingcontrollersapproachtheperformanceoftherobustcontroller.Alinesearchforewithalinear-gridof200pointsbetween60Figure4.1:H2guaranteedcost.103and101hasbeenconductedandisshowninFigure4.2.Example4.2:ConsiderthefollowingLPVsystem[1],A(q(t))=26425:960q(t)12040q(t)3464q(t)375;Bu=26432375;Bw=2640:030:47375;Cz=2641100375;Dzw=26400375;Dzu=26401375:Thevaryingparameterq(t)hasthefollowingbounds0q(t)1,jq(t)jk,withmeasurementuncertaintyboundjd(t)jz,andjd(t)j10z.Theorem4.2isusedtosynthesizestate-feedbackRGSH¥controllerforthisexample.TheguaranteedH¥performanceisshowninTable4.3,asexpected,theperformancedeterioratesastheuncertaintybounddandtheratesofchangeoftheparameterskincrease.Robustcontrollerhasbeensynthesizedaswelltostudytheachievedperformancecomparedtotheperformanceofgain-schedulingcontrollersfordifferentnoisebounds.Theperformanceoftherobustcontrolleris61Figure4.2:Linesearchforetoobtaintheoptimalcontrollerforz=0:5andk=0:01.showninthelastrowofTable4.3.Whilethegain-schedulingcontrollershowsbetterperformanceforlowrangeofuncertaintysize,itprovidesnoimprovementovertherobustcontrollerforz0:5.Thus,forz0:5,thereisnopointtoimplementgain-schedulingcontrollerasdemonstratedinFigure4.3thatshowstheguaranteedH¥performanceasfunctionofzandk.Thisisanaturalexpectation,sinceforalargernoiseboundsthemeasurementwouldbeunreliableforcontrollerscheduling.Table4.4showstheresultsof[1]forthesameexample.Comparingthesetwotables,itcanbeobservedthatourapproachachievesverycompetitiveresultswiththoseassociatedwith[1].Alinesearchforewithalinear-gridof350pointsbetween104and101hasbeenconductedandisshowninFigure4.4forz=0:2withdifferentratesofchange(k).SimulationstudyhasbeenconductedforthisexampletoillustraterobustnessofthesynthesizedcontrolleragainstthemismatchbetweentheASPsandMSPs.Aschedulingparameterthatisasq(t)=0:5+0:5sin(0:2t)andanoisyversionofthissignal(Ÿq(t))arebothshowninFigure4.5A.Arandomnoisewithboundsjd(t)j0:075andjd(t)j1hasbeenintentionallyadded62Table4.3:H¥Guaranteedcostg¥usingTheorem4.2.zk0.0010.010.110.010.5830.5840.5890.6520.10.7180.7180.7270.7930.20.7910.7910.7920.7950.50.7950.7950.7950.795Robust0.7950.7950.7950.795Table4.4:H¥Guaranteedcostg¥usingmethodof[1].zk0.0010.010.110.010.4800.4890.5940.7950.10.6120.6220.7320.7950.20.7520.7630.7950.7950.50.7950.7950.7950.795Robust0.7950.7950.7950.795Figure4.3:H¥guaranteedperformance.63Figure4.4:H¥performancevs.ewithz=0:2.totheASPtoimitatemeasurementnoise.Then,Theorem4.2isusedtosynthesizecontroller(withe=0:001)attheverticesofthemulti-simplexdomain.Thenequation(3.24)isutilizedtocomputethecoefmatricesofthecontrollerin(4.5)asK0=[1:5940258:245];K1=[1:076011:5690];whichisusedtoimplementthecontrollerinreal-time.Tosimulatetheclosed-loopsystem,anL2disturbancesignalbyw(t)=exp(0:4t)isgeneratedasdisturbanceinput.TheresponsestothisdisturbanceforbothcasesareshowninFigure4.5B.Clearly,thenoiseamplitudeintheresponse(associatedwiththenoisyschedulingparameter)ismuchlessthanthenoiseamplitudeintheMSP.Thissimulationresultnotonlyshowsgoodrobustnessagainstmeasurementnoisebutalsogooddisturbanceattenuation.64Figure4.5:Simulation:A)Measuredandexactschedulingparameters,B)Disturbanceattenuationresponsesassociatedwithexactandnoisyschedulingparameter.4.6SummaryRGSState-feedbackcontrollersynthesisconditionsforLPVsystemsweredevelopedinthischap-ter.TheseconditionsguaranteeH2andH¥performancesubjecttouncertainschedulingparam-eters.SynthesisconditionsareformulatedintermsofPLMIswithasinglelinesearchparameter65e.Bythevirtueofslackvariableapproach,theformulationofbothcontrollers,H2andH¥,areindependentofLyapunovmatrix.Therefore,considerableperformanceachievementcanbeattainedwiththedevelopedconditions.Numericalexamplesandsimulationresultsaregivenaswell.Therefore,thesynthesisconditionsencompassrobust(parameter-independent)controllersynthesisasaspecialcase.Comparisonstudyisconductedwiththeexistingapproachesfromlit-eratureaswell.Simulationsandcomparisonresultsshowthatthesynthesizedcontrollersnotonlyshowrobustnessagainstuncertaintiesintheschedulingparameters,butalsoachieveimprovedperformance.66CHAPTER5RGSDYNAMICOUTPUT-FEEDBACKCONTROLInthischapter,characterizationofsynthesisconditionsintermsofPBMIsarederivedforcontinuous-timepolytopicLPVsystemswithnoisyschedulingparameters.Both,H2andH¥full-orderdy-namicoutput-feedbackcontrollersareinvestigated.SincethesynthesisconditionsareformulatedasPBMIs,numericalalgorithmhasbeendevelopedtosolvetheseconditionsiteratively.Illustrativeexamples,simulations,andcomparisonswithotherapproachesfromliteraturearealsoincluded.Ashortsummaryisgiveninthelastsection.5.1DOFSynthesisProblemConsiderthefollowingopen-loopsystemSOL:=8>>>>>><>>>>>>:x(t)=A(q(t))x(t)+Bu(q(t))u(t)+Bw(q(t))w(t)z(t)=Cz(q(t))x(t)+Dzu(q(t))u(t)+Dzw(q(t))w(t)y(t)=Cy(q(t))x(t)+Dyw(q(t))w(t);(5.1)wherex(t)2Rnisthestate,u(t)2Rnuisthecontrolinput,w(t)2Rnwisthedisturbanceinput,z(t)2Rnzisthecontrolledoutput,andy(t)2Rnyisthemeasuredoutput.Thesystemmatri-ceshavethefollowingcompatibledimensionsA(q(t))2Rnn,Bu(q(t))2Rnnu,Bw(q(t))2Rnnw,Cz(q(t))2Rnzn,Dzu(q(t))2Rnznu,Dzw(q(t))2Rnznw,Cy(q(t))2Rnyn,andDyw(q(t))2Rnynw.Thegoalistosynthesizefull-orderdynamicoutput-feedbackcontrolleroftheformKDOF:=8>><>>:xc(t)=Ac(Ÿq(t))xc(t)+Bc(Ÿq(t))y(t)u(t)=Cc(Ÿq(t))x(t)(5.2)67tostabilizetheclosed-loopsystemSCL:=8>><>>:x(t)=A(q(t);Ÿq(t))x(t)+B(q(t);Ÿq(t))w(t)z(t)=C(q(t);Ÿq(t))x(t)+D(q(t);Ÿq(t))w(t)(5.3)withx(t)=x(t)0xc(t)00and264A(q(t);Ÿq(t))B(q(t);Ÿq(t))C(q(t);Ÿq(t))D(q(t);Ÿq(t))375=266664A(q(t))000Bw(q(t))000000000Cz(q(t))000000377775+2666640Bu(q(t))In000000Dzu(q)377775264Ac(Ÿq(t))Bc(Ÿq(t))Cc(Ÿq(t))000375264000In000Cy(q(t))000Dyw(q(t))375:whereA(q(t);Ÿq(t))=264A(q(t))Bu(q(t))Cc(Ÿq(t))Bc(Ÿq(t))Cy(q(t))Ac(Ÿq(t))375;B(q(t);Ÿq(t))=264Bw(q(t))Bc(Ÿq(t))Dyw(q(t))375;C(q(t);Ÿq(t))=Cz(q(t))Dzu(q(t))Cc(Ÿq(t));D(q(t);Ÿq(t))=Dzw(q(t)):(5.4)Furthermore,performance1boundsintermsofH2andH¥normsareguaranteed.Lemma5.1.Foragivenparameter-dependentsymmetricmatrixP(a)andmatricesY1(a)andY2(a)withcompatibledimensions.Ifoneofthetwofollowingconditionsholds264P(a)?[Y1(a)H(a)Y2(a)]H(a)375<000;(5.5)1NotethatintheH2controlproblem,thefeed-throughmatrixoftheclosed-loopsystemshouldbezero,i.e.D(q;Ÿq)=000.68264P(a)?[H(a)Y1(a)Y2(a)]H(a)375<000(5.6)forsomeparameter-dependentsymmetricmatrixH(a),theconditionP(a)+264000?Y2(a)0Y1(a)000375<000;(5.7)holds.Proof.ApplyingSchurcomplimentto(5.5)yieldsP(a)+264Y1(a)0Y2(a)0H(a)0375H(a)1Y1(a)H(a)Y2(a)<000thatcanbewrittenasP(a)+264000?Y2(a)0Y1(a)000375<264Y1(a)0H(a)1Y1(a)000000Y2(a)0H(a)Y2(a)375:(5.8)SincetheRHSisnegativ(5.7)holds.Theprooffor(5.6)canbedoneinasimilarmanner.5.2DOFH2ControlProblem5.1.Supposethattheschedulingparametersq(t)areprovidedasŸq(t)withuncertaintyd(t)asin(3.3).SupposefurtherthatD(q(t);Ÿq(t))=0in(5.3).Convertingalltheopen-loopsystemmatricesandsynthesisvariablestothemulti-simplexvariablesaorŸainsteadofqandŸq,respectively,using(3.15)and(3.16).Foragivenpositivescalarn,aRGSdynamicoutput-feedbackcontrollerintheformof(5.2)tostabilizetheclosed-loopsystem(5.3)foranypair(Ÿa(t);Ÿa(t))2LWsuchthatsup(Ÿa(t);Ÿa(t))2LWEEEˆZT0z(t)0z(t)dt˙<n2;(5.9)69forthedisturbanceinputw(t)givenbyw(t)=w0d(t)whered(t)istheDirac'sdeltafunctionandw0isarandomvariablesatisfyingEw0w00=IkandEgdenotesthemathematicalexpectation.ThenexttheoremprovidesPLMIsconditionsforsynthesizingRGSdynamicoutput-feedbackcontrolleroftheform(5.2)thatisrobusttothemeasurementuncertaintieswithaguaranteedH2performance.Assumption5.1.Themeasurementmatrixin(5.1)isconstantmatrix,i.e.Cy(q)=Cy.Theorem5.1.Considerthesystemin(5.1).Givenascalarn>0andasufsmallscalare>0,thereexistsagain-schedulingdynamicoutput-feedbackcontrollerKDOFintheformof(5.2)suchthattheclosed-loopsystemSCLin(5.3)isasymptoticallystablewithaguaranteedH2performanceboundnsatisfying(5.9),ifthereexistcontinuouslydifferentiableparameter-dependentmatrices0<P11(a)=P11(a)02Rnn,0<P22(a)=P22(a)02Rnn,P21(a)2Rnn,andparameter-dependentmatricesW(a)=W(a)02Rnznz,R(Ÿa)2Rnn,S(Ÿa)2Rnn,T(Ÿa)2Rnn,K1(Ÿa)2Rnn,K2(Ÿa)2Rnny,K3(Ÿa)2Rnun,andparameter-dependentmatrixH(Ÿa)2RnnsatisfyingthefollowingPLMIs26664P2(a;Ÿa)?[Y1(Ÿa)H(Ÿa)Y2(Ÿa)]H(Ÿa)37775<0005n+nw(5.10)or26664P2(a;Ÿa)?[H(Ÿa)Y1(Ÿa)Y2(Ÿa)]H(Ÿa)37775<0005n+nw(5.11)70withY1(Ÿa)=[DA(a;Ÿa)R(Ÿa)+DBu(a;Ÿa)K3(Ÿa)000nn];Y2(Ÿa)=S(Ÿa)0000nn000nnw;DA(a;Ÿa):=A(Ÿa)A(a);DBu(a;Ÿa):=Bu(Ÿa)Bu(a)(5.12)andtrace(W(a))<n2(5.13)264P11(a)?P21(a)P22(a)375>000nn;(5.14)266664R(Ÿa)+R(Ÿa)0P11(a)??In+T(Ÿa)P21(a)S(Ÿa)+S(Ÿa)0P22(a)?Cz(a)R(Ÿa)+Dzu(a)K3(Ÿa)Cz(a)W(a)377775>0002n+nz(5.15)P2(a;Ÿa)=:266666666664A(a)R(Ÿa)+Bu(a)K3(Ÿa)+()0P11(a)?A(a)0+K1(Ÿa)P21(a)S(Ÿa)A(a)+K2(Ÿa)Cy+()0P22(a)P11(a)R(Ÿa)+e(R(Ÿa)0A(a)0+K3(Ÿa)0Bu(a)0)P21(a)0In+eK1(Ÿa)0P21(a)T(Ÿa)+eA(a)0P22(a)S(Ÿa)+e(A(a)0S(Ÿa)0+C0yK2(Ÿa)0)Bw(a)0Bw(a)0S(Ÿa)0+Dyw(a)0K2(Ÿa)0??????e(R(Ÿa)+R(Ÿa)0)??e(T(Ÿa)+In)e(S(Ÿa)+S(Ÿa)0)?000nwn000nwnInw377777777775;(5.16)foranypairs(Ÿa(t);Ÿa(t))2LW.Then,thematricesofthegain-schedulingdynamicoutput-71feedbackcontrollerKDOFin(5.2)canbeobtainedasfollowsCc(Ÿa)=K3(Ÿa)F(Ÿa)1;Bc(Ÿa)=X(Ÿa)1K2(Ÿa);Ac(Ÿa)=X(Ÿa)1[K1(Ÿa)S(Ÿa)A(Ÿa)R(Ÿa)S(Ÿa)Bu(Ÿa)K3(Ÿa)F(Ÿa)X(Ÿa)K2(Ÿa)CyR(Ÿa)]F(Ÿa)1;(5.17)whereF(Ÿa)2RnnandX(Ÿa)2Rnncanbeobtainedbytakinganyfull-rankmatrixfactorizationofX(Ÿa)F(Ÿa)=T(Ÿa)S(Ÿa)R(Ÿa).Proof.Tosimplifynotationsintheproof,closed-loopsystemmatricesA(a;Ÿa),B(a;Ÿa),andC(a;Ÿa)inLemma4.1willbedenotedasA,B,andC,respectively.Usingslackvariableap-proach,additionaloptimizationvariablesU(Ÿa)canbeintroducedtoinequality(4.7)viaFinsler'sLemma[64]todecouplethedynamicmatrixAfromLyapunovmatrixP(a).Thisleadstothefollowingsufconditionof(4.7)X(a)+U(Ÿa)V(Ÿa)+V(Ÿa)0U(Ÿa)0<000;(5.18)where,X(a):=266664P(a)P(a)000P(a)000000000000I377775;U(Ÿa):=266664G(Ÿa)0000Y(Ÿa)0000000I377775;V(a):=264A0I000B0000I375;suchthatV(Ÿa)?0X(a)V(Ÿa)?<000andV(Ÿa)?0=[IAB].Therefore,substitutingtheserelationsinto(5.18)leadsto,266664P(a)P(a)000P(a)000000000000I377775+266664G(Ÿa)0000Y(Ÿa)0000000I377775264A0I000B0000I375+266664ABI000000I377775264G(Ÿa)Y(Ÿa)000000000I375<000:(5.19)NotethatitisimportanttoimposeparticularstructuretotheslackvariableU(Ÿa)tomaintainconvexparametrization.Therefore,settingY(Ÿa)=eG(Ÿa)issuftokeep(5.19)convex,72wheree>0isascalarusedasanextradegreeoffreedomtoperformlinesearchforreducingconservativeness[65].Thisyields266664P(a)P(a)000P(a)000000000000I377775+266664G(Ÿa)0A0G(Ÿa)0000eG(Ÿa)0A0eG(Ÿa)0000B0000I377775+266664AG(Ÿa)eAG(Ÿa)BG(Ÿa)eG(Ÿa)000000000I377775<000;thatcanbewrittenas266664AG(Ÿa)+()0P(a)??P(a)G(Ÿa)+e(G(Ÿa)0A0)e(G(Ÿa)+G(Ÿa)T)?B0000nwnInw377775<0002n+nw:(5.20)Sinceblock(2;2)of(5.20)impliesG(Ÿa)+G(Ÿa)0>0,thematrixG(Ÿa)isinvertibleandcanbepartitionedas,G(Ÿa)=264R(Ÿa)G1(Ÿa)F(Ÿa)G2(Ÿa)375;J(Ÿa):=G(Ÿa)1=264S(Ÿa)0J1(Ÿa)X(Ÿa)0J2(Ÿa)375:thefollowingnon-singularcongruencetransformationmatrices,Qg(Ÿa):=264R(Ÿa)IF(Ÿa)000375;Qv(Ÿa):=264IS(Ÿa)0000X(Ÿa)0375;(5.21)suchthat,G(Ÿa)Qv(Ÿa)=Qg(Ÿa);J(Ÿa)Qg(Ÿa)=Qv(Ÿa):(5.22)Inordertoguaranteethatcongruencetransformationsin(5.21)havefull-rank,blockmatricesF(Ÿa)andX(Ÿa)shouldbenon-singular.Ifthisisnotthecase,smallperturbationofF(Ÿa)andX(Ÿa)intermsofnormscanalwaysbedonesuchthatF(Ÿa)+DF(Ÿa)andX(Ÿa)+DX(Ÿa)areinvertible.73TheLyapunovmatrixcanbepartitionedasP(a):=264P11(a)?P21(a)P22(a)375>000;andQv(Ÿa)0P(a)Qv(Ÿa):=P(a)=264P11(a)?P21(a)P22(a)375>000:(5.23)LetT1(Ÿa)=266664Qv(Ÿa)000000000Qv(Ÿa)000000000Inw377775;Multiplying(5.20)byT1(Ÿa)fromrightandbyT1(Ÿa)0fromleftandusing(5.22)leadto266664Q0v(Ÿa)AQg(Ÿa)+()0P(a)??P(Ÿa)Q0v(Ÿa)Qg(Ÿa)+e(Q0v(Ÿa)AQg(Ÿa))0e(Q0v(Ÿa)Qg(Ÿa)+()0)000B0Qv(Ÿa)000Inw377775<000;(5.24)Then,substitutingclosed-loopmatrices(5.4)into(5.24)andconsidering(5.21)and(5.23)withthefollowingrelationships,Qv(Ÿa)0Qg(Ÿa)=264R(Ÿa)IT(Ÿa)S(Ÿa)375;T(Ÿa):=S(Ÿa)R(Ÿa)+X(Ÿa)F(Ÿa);Qv(Ÿa)0AQg(Ÿa)=264A(a)R(Ÿa)+Bu(a)K3(Ÿa)A(a)K1(Ÿa)S(Ÿa)A(a)+K2(Ÿa)Cy375:=F(Ÿa);Qv(Ÿa)0B=264Bw(a)S(Ÿa)Bw(a)+K2(Ÿa)Dyw(a)375;74inequality(5.24)canbewrittenas2666666666664F(Ÿa)+F(Ÿa)0264P11(a)P21(a)0P21(a)P22(a)375??P(a)264R(Ÿa)InT(Ÿa)S(Ÿa)375+eF(Ÿa)0e0B@264R(Ÿa)InT(Ÿa)S(Ÿa)375+()01CA?Bw(a)0Bw(a)0S(Ÿa)0+Dyw(a)0K2(Ÿa)00nw2nInw3777777777775<04n+nw;(5.25)whereK1(Ÿa),K2(Ÿa),andK3(Ÿa)areintermediatecontrollervariablesas264K1(Ÿa)K2(Ÿa)K3(Ÿa)000375:=264X(Ÿa)S(Ÿa)Bu(Ÿa)000I375264Ac(Ÿa)Bc(Ÿa)Cc(Ÿa)000375264F(Ÿa)000CyR(Ÿa)I375+264S(Ÿa)000375A(Ÿa)R(Ÿa)000:(5.26)SincecontrollermatricesshouldonlydependontheMSPs,theopen-loopmatricesin(5.26)areallowedtodependonthemulti-simplexvariablesŸanota.However,A(Ÿa)andBu(Ÿa)canbewrittenasA(Ÿa)=A(Ÿa)+A(a)A(a)=A(a)+DA(a;Ÿa)Bu(Ÿa)=Bu(Ÿa)+Bu(a)Bu(a)=Bu(a)+DBu(a;Ÿa);(5.27)whereDA(a;Ÿa):=A(Ÿa)A(a),andDBu(Ÿa;a):=Bu(Ÿa)Bu(a).Hence,thissubstitutionallowsustoconstructcontrollermatricesbasedonlyontheMSPs.Therefore,Substituting(5.27)into(5.26)toobtain264K1(Ÿa)K2(Ÿa)K3(Ÿa)000375=264¯K1(Ÿa)¯K2(Ÿa)¯K3(Ÿa)000375+S(5.28)whereS:=264S(Ÿa)DA(a;Ÿa)R(Ÿa)+S(Ÿa)DBu(a;Ÿa)Cc(Ÿa)F(Ÿa)000000000375;75and264¯K1(Ÿa)¯K2(Ÿa)¯K3(Ÿa)000375:=264X(Ÿa)S(Ÿa)Bu(a)000I375264Ac(Ÿa)Bc(Ÿa)Cc(Ÿa)000375264F(Ÿa)000CyR(Ÿa)I375+264S(Ÿa)A(a)R(Ÿa)000000000375:SubstitutingCc(Ÿa)=¯K3(Ÿa)F(Ÿa)1=K3(Ÿa)F(Ÿa)1intoSyieldsS=264S(Ÿa)DA(a;Ÿa)R(Ÿa)+S(Ÿa)DBu(a;Ÿa)K3(Ÿa)000000000375:(5.29)Substituting(5.29)into(5.28),andtheninto(5.25)andnotingthat¯K2=K2and¯K3=K3leadsto2666666666664F(Ÿa)+F(Ÿa)0264P11(a)P21(a)0P21(a)P22(a)375??P(a)264R(Ÿa)InT(Ÿa)S(Ÿa)375+eF(Ÿa)0e0B@264R(Ÿa)InT(Ÿa)S(Ÿa)375+()01CA?Bw(a)0Bw(a)0S(Ÿa)0+Dyw(a)0K2(Ÿa)0000nw2nInw3777777777775+2640002n2n?Y2(Ÿa)0Y1(Ÿa)0002n+nw2n375<0004n+nw:Notethattheaboveinequalityisintheformof(5.7)ofLemma5.1withY2(Ÿa)0Y1(Ÿa)=266664S(Ÿa)DA(a;Ÿa)R(Ÿa)+S(Ÿa)DBu(a;Ÿa)K3(Ÿa)000nn000nn000nn000nwn000nwn377775thatcanbefactorizedintoY1(Ÿa)=[DA(a;Ÿa)R(Ÿa)+DBu(a;Ÿa)K3(Ÿa)000nn];Y2(Ÿa)=S(Ÿa)0000nn000nnw:76thatdirectlyleadsto(5.10)or(5.11).Since(4.8)canbewrittenas264G(Ÿa)+()0P(a)?C(Ÿa)G(Ÿa)W(a)375>0002n+nz;(5.30)multiplying(5.30)byT2(Ÿa)fromtherightandbyT2(Ÿa)0fromtheleftwithT2(Ÿa)=264Qv(Ÿa)000000Inz375;leadsto264Qv(Ÿa)0Qg(Ÿa)+()0Qv(Ÿa)0P(a)Qv(Ÿa)?C(Ÿa)Qg(Ÿa)W(a)375>0002n+nz:Considering(5.21)and(5.22)andsubstitutingC(Ÿa)Qg(Ÿa)=Cz(a)R(Ÿa)+Dzu(a)K3(Ÿa)Cz(a);result2666664264R(Ÿa)IT(Ÿa)S(Ÿa)375+()0264P11(a)P21(a)0P21(a)P22(a)375?Cz(a)R(Ÿa)+Dzu(a)K3(Ÿa)Cz(a)W(a)3777775>0002n+nz;whichleadsto(5.15).Ontheotherhand,solving(5.26)forthevariablesK1(Ÿa),K2(Ÿa),andK3(Ÿa)yieldsthefollowingrelationsK1(Ÿa)=X(Ÿa)Ac(Ÿa)F(Ÿa)+S(Ÿa)A(Ÿa)R(Ÿa)+S(Ÿa)Bu(Ÿa)Cc(Ÿa)F(Ÿa)+K2(Ÿa)CyR(Ÿa);K2(Ÿa)=X(Ÿa)Bc(Ÿa);K3(Ÿa)=Cc(Ÿa)F(Ÿa):ControllermatricescanbesolvedinthefollowingorderCc(Ÿa),Bc(Ÿa),andAc(Ÿa),leadingto(5.17).77Remark5.1.Theorem5.1addressesthegeneralcasewhentheuncertaintermsDA(a;Ÿa)andDBu(a;Ÿa)arebothincluded(foraCy)2.Tomybestknowledge,thisisamoregeneraltheorythathandlesthisproblemsinceonlyDA(a;Ÿa)wasconsideredinliterature[2,38,3,37].ThenexttwocorollariesarespecialcasesofTheorem5.1.Corollary5.1.Supposethattheinputmatrixin(5.1)isconstant,i.e.Bu(q)=Bu.Givenascalarn>0andasufsmallscalare>0,thereexistsagain-schedulingdynamicoutput-feedbackcontrollerKDOFintheformof(5.2)suchthattheclosed-loopsystemSCLin(5.3)isasymptoticallystablewithaguaranteedH2performanceboundnsatisfying(4.6),ifthereex-istcontinuouslydifferentiableparameter-dependentmatrices0<P11(a)=P11(a)02Rnn,0<P22(a)=P22(a)02Rnn,P21(a)2Rnn,andparameter-dependentmatricesW(a)=W(a)02Rnznz,R(Ÿa)2Rnn,S(Ÿa)2Rnn,T(Ÿa)2Rnn,K1(Ÿa)2Rnn,K2(Ÿa)2Rnny,K3(Ÿa)2Rnun,andparameter-dependentmatrixH(Ÿa)2Rnnsatisfying(5.10)or(5.11)withY1(Ÿa)=[R(Ÿa)000nn];Y2(Ÿa)=DA(a;Ÿa)0S(Ÿa)0+DCy(a;Ÿa)0K2(Ÿa)0000nn000nnw;DA(a;Ÿa):=A(Ÿa)A(a);DCy(a;Ÿa):=Cy(Ÿa)Cy(a)(5.31)andtheconditions(5.13),(5.14),and(5.15)arewithP2(a;Ÿa)givenby(5.16)foranypairs(Ÿa(t);Ÿa(t))2LW.Then,thematricesofthegain-schedulingdynamicoutput-feedbackcontrollerKDOFin(5.2)canbeobtainedusing(5.17)byreplacingBu(Ÿa)andCywithBuandCy(Ÿa),respectively.Remark5.2.Theparameter-dependentmatrixH(Ÿa)inTheorem5.2canbeviewedasascalingmatrixtoover-boundtheuncertaintermsDA(a;Ÿa)andDBu(a;Ÿa)(orDA(a;Ÿa)andDCy(a;Ÿa)2Ideally,thegeneralproblemwithalluncertaintermsincludingDA(a;Ÿa),DBu(a;Ÿa),andDCy(a;Ÿa)shouldbesolved,butunfortunately,itendsupwithahighlynonlinearterminthesyn-thesisconditionssuchthatnoexistingalgorithmcanbeusedtosolvethisproblemwithguaranteedconvergence.Therefore,thecasewitheitherconstantCyorBuaresolvedinthischapter.78inCorollary5.1).Theintroductionofthisweightingmatriximprovestheachievedperformance.Remark5.3.Again,basedon(5.10),(5.11),and(5.12),twodifferentformulationsofTheorem5.2canbeobtained(similarly,basedon(5.10),(5.11),and(5.31),twodifferentformulationsofCorol-lary5.1canbeobtained).Atthisstage,itisnotclearwhichformulationachievesbetterperfor-mance.Therefore,foragivendesignproblem,thesetwoformulationsshouldbetriedtoobtainthebestpossibleperformance.Corollary5.2.SupposethatBu(q)andCy(q)areconstantmatricesin(5.1).Givenascalarn>0andasufsmallscalare>0,thereexistsagain-schedulingdynamicoutput-feedbackcon-trollerKDOFintheformof(5.2)suchthattheclosed-loopsystemSCLin(5.3)isasymptoticallystablewithaguaranteedH2performanceboundnsatisfying(5.9),ifthereexistcontinuouslydifferentiableparameter-dependentmatrices0<P11(a)=P11(a)02Rnn,0<P22(a)=P22(a)02Rnn,P21(a)2Rnn,andparameter-dependentmatricesW(a)=W(a)02Rnznz,R(Ÿa)2Rnn,S(Ÿa)2Rnn,T(Ÿa)2Rnn,K1(Ÿa)2Rnn,K2(Ÿa)2Rnny,K3(Ÿa)2Rnunandparameter-dependentmatrixH(Ÿa)2Rnnsatisfying(5.10)or(5.11)withY1(Ÿa)andY2(Ÿa)givenbyY1(Ÿa)=[R(Ÿa)000nn];Y2(Ÿa)=DA(a;Ÿa)0S(Ÿa)0000nn000nnw:(5.32)orY1(Ÿa)=[DA(a;Ÿa)R(Ÿa)000nn];Y2(Ÿa)=S(Ÿa)0000nn000nnw;(5.33)andtheconditions(5.13),(5.14),and(5.15)arewithP2(a;Ÿa)givenby(5.16)foranypairs(Ÿa(t);Ÿa(t))2LW.Then,thematricesoftheRGSdynamicoutput-feedbackcontrollerKDOFin(5.2)canbeobtainedusing(5.17)byreplacingBu(Ÿa)withBu.Proof.TheproofwillbeomittedheresinceitcanbeshowninasimilarwaytotheproofofTheorem5.1.79Table5.1:PossibleformulationsforCorollary5.2.FormulationAFormulationBFormulationCFormulationD(5.10),(5.32)(5.10),(5.33)(5.11),(5.32)(5.11),(5.33)Remark5.4.Basedon(5.10),(5.11),(5.32),and(5.33),fourdifferentformulationsofCorol-lary5.2canbeobtainedasillustratedinTable5.1.Similarly,itisnotclearwhichformulationachievesthebestperformance.Therefore,foragivendesignproblem,allformulationsshouldbetriedtoobtainthebestpossibleperformance.5.3DOFH¥ControlProblem5.2.Supposethattheschedulingparametersq(t)areprovidedasŸq(t)withuncertaintyd(t).Convertingalltheopen-loopsystemmatricesandsynthesisvariablestodependonthemulti-simplexvariablesaorŸainsteadofqandŸq,respectively,using(3.15)and(3.16).Foragivenpositivescalarg¥,aRGSdynamicoutput-feedbackcontrollerintheformof(5.2)tostabilizetheclosed-loopsystem(5.3)foranypair(Ÿa(t);Ÿa(t))2LWsuchthatsup(Ÿa(t);Ÿa(t))2LWsupw2L2;w6=0kz(t)k2kw(t)k2<g¥:(5.34)Assumption5.2.Themeasurementmatrixin(5.1)isconstantmatrix,i.e.Cy(q)=Cy.Theorem5.2.Considerthesystemin(5.1).Givenascalarg¥>0andasufsmallscalare>0,thereexistsagain-schedulingdynamicoutput-feedbackcontrollerKDOFintheformof(5.2)suchthattheclosed-loopsystemSCLin(5.3)isasymptoticallystablewithaguaranteedH¥performanceboundg¥satisfying(5.34),ifthereexistcontinuouslydifferentiableparameter-dependentmatrices0<P11(a)=P11(a)02Rnn,0<P22(a)=P22(a)02Rnn,P21(a)2Rnn,andparameter-dependentmatricesR(Ÿa)2Rnn,S(Ÿa)2Rnn,T(Ÿa)2Rnn,K1(Ÿa)2Rnn,K2(Ÿa)2Rnny,K3(Ÿa)2Rnun,andparameter-dependentmatrixH(Ÿa)280RnnsatisfyingthefollowingPLMIs26664P¥(a;Ÿa)?[Y1(Ÿa)H(Ÿa)Y2(Ÿa)]H(Ÿa)37775<0005n+nz+nw(5.35)or26664P¥(a;Ÿa)?[H(Ÿa)Y1(Ÿa)Y2(Ÿa)]H(Ÿa)37775<0005n+nz+nw(5.36)withY1(Ÿa)=[DA(a;Ÿa)R(Ÿa)+DBu(a;Ÿa)K3(Ÿa)000nn];Y2(Ÿa)=S(Ÿa)0000nn000nnz000nnw;DA(a;Ÿa):=A(Ÿa)A(a);DBu(a;Ÿa):=Bu(Ÿa)Bu(a)(5.37)264P11(a)?P21(a)P22(a)375>000nn;P¥(a;Ÿa):=2666666666666664A(a)R(a)+Bu(a)K3(a)+()0P11(a)?K1(a)+A(a)0P21(a)S(a)A(a)+K2(a)Cy+()0P22(a)P11(a)R(a)+e(R(a)0A(a)0+K3(a)0Bu(a)0)P21(a)0In+eK1(a)0P21(a)T(a)+eA(a)0P22(a)S(a)+e(A(a)0S(a)0+C0yK2(a)0)Cz(a)R(a)+Dzu(a)K3(a)Cz(a)Bw(a)0Bw(a)0S(a)0+Dyw(a)0K2(a)0????????e(R(a)+R(a)0)???e(T(a)+In)e(S(a)+S(a)0)??e(Cz(a)R(a)+Dzu(a)K3(a))eCz(a)Inz?0nwn0nwnDzw(a)0g2¥Inw3777777777777775;(5.38)81foranypairs(Ÿa(t);Ÿa(t))2LW.Then,thematricesofthegain-schedulingdynamicoutput-feedbackcontrollerKDOFin(5.2)canbeobtainedasfollowsCc(Ÿa)=K3(Ÿa)F(Ÿa)1;Bc(Ÿa)=X(Ÿa)1K2(Ÿa);Ac(Ÿa)=X(Ÿa)1[K1(Ÿa)S(Ÿa)A(Ÿa)R(Ÿa)S(Ÿa)Bu(Ÿa)K3(Ÿa)F(Ÿa)X(Ÿa)K2(Ÿa)CyR(Ÿa)]F(Ÿa)1;(5.39)whereF(Ÿa)2RnnandX(Ÿa)2Rnncanbeobtainedbytakinganyfull-rankmatrixfactorizationofX(Ÿa)F(Ÿa)=T(Ÿa)S(Ÿa)R(Ÿa).Proof.FollowingtheproofofTheorem4.2,inequality(4.18)canbewrittenintermsoftheclosed-loopmatricesas266666664AG(Ÿa)+()0P(a)???P(a)G(Ÿa)+e(G(Ÿa)0A0)e(G(Ÿa)+G(Ÿa)0)??CG(Ÿa)eCG(Ÿa)Inz?B0000nwnD0g2¥Inw377777775<0002n+nw;(5.40)wherefornotationssimplicityintheproof,closed-loopsystemmatricesA(a;Ÿa),B(a;Ÿa),C(a;Ÿa)andD(a;Ÿa)inLemma4.3willbedenotedasA,B,CandD,respectively.P(a):=264P11(a)?P21(a)P22(a)375>000:Block(2;2)of(5.40)impliesG(Ÿa)+G(Ÿa)0>0.Therefore,thematrixG(Ÿa)isinvertibleandcanbepartitionedas,G(a)=264R(Ÿa)G1(a)F(Ÿa)G2(a)375;V(Ÿa):=G(Ÿa)1=264S(Ÿa)0V1(a)X(Ÿa)0V2(a)375:thefollowingnon-singularcongruencetransformationmatrices,82Qg(Ÿa)=264R(Ÿa)IF(Ÿa)000375;Qv(Ÿa)=264IS(Ÿa)0000X(Ÿa)0375;(5.41)suchthat,G(Ÿa)Qv(Ÿa)=Qg(Ÿa);V(Ÿa)Qg(Ÿa)=Qv(Ÿa):(5.42)Inordertoguaranteethatcongruencetransformationsin(5.41)havefull-rank,blockmatricesF(Ÿa)andX(Ÿa)shouldbenon-singular.Ifthisisnotthecase,smallperturbofF(Ÿa)andX(Ÿa)withsufsmallmatricesintermsofnormscanalwaysbedonesuchthatF(Ÿa)+DF(Ÿa)andX(Ÿa)+DX(Ÿa)areinvertible.alsoQv(Ÿa)0P(a)Qv(Ÿa):=P(a)=264P11(a)?P21(a)P22(a)375>000:(5.43)Multiplying(5.40)byT1(Ÿa)fromrightandbyT1(Ÿa)0fromleftwith,T1(Ÿa)=266666664Qv(Ÿa)000000000000Qv(Ÿa)000000000000Inz000000000000Inw377777775;andconsidering(5.42)and(5.43)yields266666664Q0v(Ÿa)AQg(Ÿa)+()0P(a)???P(a)Q0v(Ÿa)Qg(Ÿa)+e(Q0v(Ÿa)AQg(Ÿa))0e(Q0v(Ÿa)Qg(Ÿa)+()0)000?CQg(Ÿa)eCQg(Ÿa)Inz?B0Qv(Ÿa)000D0g2¥Inw377777775<000;(5.44)Then,substitutingclosed-loopmatrices(5.4)andconsidering(5.41)withthefollowingrelations,Qv(Ÿa)0Qg(Ÿa)=264R(Ÿa)IT(Ÿa)S(Ÿa)375;T(Ÿa):=S(Ÿa)R(Ÿa)+X(Ÿa)F(Ÿa);83Qv(Ÿa)0AQg(Ÿa)=264A(a)R(Ÿa)+Bu(a)K3(Ÿa)A(a)K1(Ÿa)S(Ÿa)A(a)+K2(Ÿa)Cy375:=F(Ÿa);Qv(Ÿa)0B=264Bw(a)S(Ÿa)Bw(a)+K2(Ÿa)Dyw(a)375;CQg(Ÿa)=Cz(a)R(Ÿa)+Dzu(a)K3(Ÿa)Cz(a);D=Dzw(a);inequality(5.44)canbewrittenas26666666666666664F(Ÿa)+F(Ÿa)0264P11(a)P21(a)0P21(a)P22(a)375???P(a)264R(Ÿa)InT(Ÿa)S(Ÿa)375+eF(Ÿa)0e0B@264R(Ÿa)InT(Ÿa)S(Ÿa)375+()01CA??Cz(a)R(Ÿa)+Dzu(a)K3(Ÿa)Cz(a)F1(Ÿa)Inz?Bw(a)0Bw(a)0S(Ÿa)0+Dyw(a)0K2(Ÿa)0000nw2nDzw(a)0g2¥Inw37777777777777775<0004n+nz+nw;(5.45)withF1(Ÿa)=eCz(a)R(Ÿa)+Dzu(a)¯K3(Ÿa)Cz(a);whereK1(Ÿa),K2(Ÿa),andK3(Ÿa)areintermediatecontrollervariablesas264K1(Ÿa)K2(Ÿa)K3(Ÿa)000375:=264X(Ÿa)S(Ÿa)Bu(Ÿa)000I375264Ac(Ÿa)Bc(Ÿa)Cc(Ÿa)000375264F(Ÿa)000CyR(Ÿa)I375+264S(Ÿa)000375A(Ÿa)R(Ÿa)000:(5.46)SincecontrollermatricesshouldonlydependontheMeasuredShedulingParameters(MSPs),open-loopmatricesin(5.46)needtodependonthemulti-simplexvariablesŸanota.However,A(Ÿa)andBu(Ÿa)canbewrittenasA(Ÿa)=A(Ÿa)+A(a)A(a)=A(a)+DA(a;Ÿa)Bu(Ÿa)=Bu(Ÿa)+Bu(a)Bu(a)=Bu(a)+DBu(a;Ÿa);(5.47)84whereDA(a;Ÿa):=A(Ÿa)A(a),andDBu(Ÿa;a):=Bu(Ÿa)Bu(a).Hence,thissubstitutionallowsustoconstructcontrollermatricesbasedonlyontheMSPs.Therefore,substituting(5.47)into(5.46)toobtain264K1(Ÿa)K2(Ÿa)K3(Ÿa)000375=264¯K1(Ÿa)¯K2(Ÿa)¯K3(Ÿa)000375+S(5.48)whereS:=264S(Ÿa)DA(a;Ÿa)R(Ÿa)+S(Ÿa)DBu(a;Ÿa)Cc(Ÿa)F(Ÿa)000000000375;and264¯K1(Ÿa)¯K2(Ÿa)¯K3(Ÿa)000375=264X(Ÿa)S(Ÿa)Bu(a)000I375264Ac(Ÿa)Bc(Ÿa)Cc(Ÿa)000375264F(Ÿa)000CyR(Ÿa)I375+264S(Ÿa)A(a)R(Ÿa)000000000375:SubstitutingCc(Ÿa)=¯K3(Ÿa)F(Ÿa)1=K3(Ÿa)F(Ÿa)1intoSyieldsS=264S(Ÿa)DA(a;Ÿa)R(Ÿa)+S(Ÿa)DBu(a;Ÿa)K3(Ÿa)000000000375:(5.49)Substituting(5.49)into(5.48),andtheninto(5.45)andnotingthat¯K2=K2and¯K3=K3leadsto26666666666666664F(Ÿa)+()0264P11(a)P21(a)0P21(a)P22(a)375???P(a)264R(Ÿa)InT(Ÿa)S(Ÿa)375+eF(Ÿa)0e0B@264R(Ÿa)InT(Ÿa)S(Ÿa)375+()01CA??Cz(a)R(Ÿa)+Dzu(a)K3(Ÿa)Cz(a)F1(Ÿa)Inz?Bw(a)0Bw(a)0S(Ÿa)0+Dyw(a)0K2(Ÿa)0000nw2nDzw(a)0g2¥Inw37777777777777775+2640002n2n?Y2(Ÿa)0Y1(Ÿa)0002n+nz+nw2n375<0004n+nz+nw;85Notethattheaboveequationisintheformof(5.7)ofLemma5.1withY2(Ÿa)0Y1(Ÿa)=266666664S(Ÿa)DA(a;Ÿa)R(Ÿa)+S(Ÿa)DBu(a;Ÿa)K3(Ÿa)000nn000nn000nn000nzn000nzn000nwn000nwn377777775whichcanbefactorizedintoY1(Ÿa)=[DA(a;Ÿa)R(Ÿa)+DBu(a;Ÿa)K3(Ÿa)000nn];Y2(Ÿa)=S(Ÿa)0000nn000nnz000nnw:thatdirectlyleadsto(5.35)or(5.36).Ontheotherhand,solving(5.46)forthevariablesK1(Ÿa),K2(Ÿa),andK3(Ÿa)yieldsthefollow-ingrelationshipsK1(Ÿa)=X(Ÿa)Ac(Ÿa)F(Ÿa)+S(Ÿa)A(Ÿa)R(Ÿa)+S(Ÿa)Bu(Ÿa)Cc(Ÿa)F(Ÿa)+K2(Ÿa)CyR(Ÿa);K2(Ÿa)=X(Ÿa)Bc(Ÿa);K3(Ÿa)=Cc(Ÿa)F(Ÿa):ControllermatricescanbesolvedinthefollowingorderCc(Ÿa),Bc(Ÿa),andAc(Ÿa),whichleadsto(5.39).ThenexttwocorollariesarespecialcasesofTheorem5.2.TheproofsofthesecorollariesareomittedsinceitfollowssamestepsastheproofofTheorem5.2.Corollary5.3.Supposethattheinputmatrixin(5.1)isconstant,i.e.Bu(q)=Bu.Givenascalarg¥>0andasufsmallscalare>0,thereexistsagain-schedulingdynamicoutput-feedbackcontrollerKDOFintheformof(5.2)suchthattheclosed-loopsystemSCLin(5.3)isasymptoticallystablewithaguaranteedH¥performanceboundg¥satisfying(5.34),ifthereexistcontinuouslydifferentiableparameter-dependentmatrices0<P11(a)=P11(a)02Rnn,0<P22(a)=P22(a)02Rnn,P21(a)2Rnn,andparameter-dependentmatricesR(Ÿa)2Rnn,86S(Ÿa)2Rnn,T(Ÿa)2Rnn,K1(Ÿa)2Rnn,K2(Ÿa)2Rnny,K3(Ÿa)2Rnun,andpositive-parameter-dependentmatrixH(Ÿa)2Rnnsatisfying(5.35)or(5.36)withY1(Ÿa)=[R(Ÿa)000nn];Y2(Ÿa)=DA(a;Ÿa)0S(Ÿa)0+DCy(a;Ÿa)0K2(Ÿa)0000nn000nnz000nnw;DA(a;Ÿa):=A(Ÿa)A(a);DCy(a;Ÿa):=Cy(Ÿa)Cy(a);andP¥(a;Ÿa)givenby(5.38)foranypairs(Ÿa(t);Ÿa(t))2LW.Then,thematricesofthegain-schedulingdynamicoutput-feedbackcontrollerKDOFin(5.2)canbeobtainedasin(5.39)byreplacingBu(Ÿa)andCywithBuandCy(Ÿa),respectively.Corollary5.4.Supposethattheinputandmeasurementmatricesin(5.1)areconstant,i.e.Bu(q)=BuandCy(q)=Cy.Givenascalarg¥>0andasufsmallscalare>0,thereexistsagain-schedulingdynamicoutput-feedbackcontrollerKDOFintheformof(5.2)suchthattheclosed-loopsystemSCLin(5.3)isasymptoticallystablewithaguaranteedH¥performanceboundg¥satisfying(5.34),ifthereexistcontinuouslydifferentiableparameter-dependentma-trices0<P11(a)=P11(a)02Rnn,0<P22(a)=P22(a)02Rnn,P21(a)2Rnn,andparameter-dependentmatricesR(Ÿa)2Rnn,S(Ÿa)2Rnn,T(Ÿa)2Rnn,K1(Ÿa)2Rnn,K2(Ÿa)2Rnny,K3(Ÿa)2Rnun,andparameter-dependentmatrixH(Ÿa)2Rnnsatisfying(5.35)or(5.36)withY1(Ÿa)=[R(Ÿa)000nn];Y2(Ÿa)=DA(a;Ÿa)0S(Ÿa)0000nn000nnz000nnw;(5.50)orY1(Ÿa)=[DA(a;Ÿa)R(Ÿa)000nn];Y2(Ÿa)=S(Ÿa)0000nn000nnz000nnw;DA(a;Ÿa):=A(Ÿa)A(a);(5.51)87Table5.2:PossibleformulationsforCorollary5.4.FormulationAFormulationBFormulationCFormulationD(5.35),(5.50)(5.35),(5.51)(5.36),(5.50)(5.36),(5.51)andP¥(a;Ÿa)givenby(5.38)foranypairs(Ÿa(t);Ÿa(t))2LW.Then,thematricesofthegain-schedulingdynamicoutput-feedbackcontrollerKDOFin(5.2)canbeobtainedusing(5.39)byreplacingBu(Ÿa)withBu.Remark5.5.Similarly,basedon(5.35),(5.36),(5.50),and(5.51),fourdifferentformulationsofCorollary5.4canbeobtainedasillustratedinTable5.2.Similarly,itisnotclearwhichformula-tionachievesthebestperformance.Therefore,foragivendesignproblem,allformulationsshouldbetriedtoobtainthebestpossibleperformance.5.4PBMIAlgorithmDuetothemultiplicationsbetweendecisionvariables(H(Ÿa)R(Ÿa)orH(Ÿa)S(Ÿa)inthesynthesisconditions),Theorem5.1andTheorem5.2areformulatedasPBMIsintermsoftime-varyingpa-rametersinsidemulti-simplexdomain.Thistypeofsynthesisproblemcanbeviewedasaspecialtypeofnon-convexoptimizationproblem.Inotherwords,PBMIsareequivalenttodi-mensionalBMIconstraintswhichisnumericallynon-tractable.Tosolvethisproblem,numericalalgorithm(Algorithm1)isdevelopedtosolvethistypeofoptimizationproblem.ThisalgorithmassumesthepossibilityofsolvingPLMIs(forede)whichisindeedpossiblewiththeadventofpowerfultheoreticalandcomputationaltools[41,68,69].Therefore,ROLMIP[56]isusedtoimplementthePBMIalgorithm(Algorithm1)toobtainthesub-optimalcontroller.Althoughthisalgorithmdoesnotguaranteetheconvergencetotheglobaloptima,conservativenessreductionwithafewiterationscanbeexpectedsincen(org¥)ismonotonicallynon-increasingaswillbeillustratedinthenextsection.88Algorithm1:Parameter-DependentBilinearMatrixInequalityAlgorithmInitialization:Seti=0,H0(Ÿa)=In.GivenH0(Ÿa),minimizenunderthePLMIconditionstoobtainS(Ÿa)andn.SetS0(Ÿa)=S(Ÿa)andn0=n.SetimaxandTolarance.Seti=i+1.whilei<imaxORjnini1j>TolarancedoGivenSi1(Ÿa),minimizenunderthePLMIconditionstoobtainH(Ÿa)andn.SetHi(Ÿa)=H(Ÿa)andni=n.Seti=i+1.GivenHi1(Ÿa),minimizenunderthePLMIconditionstoobtainS(Ÿa)andn.SetSi(Ÿa)=S(Ÿa)andni=n.end5.5NumericalExamplesThefollowingLPVsystemhasbeenstudiedmanytimesinliterature[70,71,72,46,73,2,37,38].Itisrepresentativeexampletoshowtheadvantagesofthesynthesisconditionsdevelopedinthischapterandtomakecomparisonswiththeexistingmethods.Example5.1:Thestate-spacemodelthatrepresentsthedynamicsofthepitch-axismotionforamissilesystemisgivenby266664A(q)Bu(q)Bw(q)Cz(q)Dzu(q)Cy(q)Dyw(q)377775=2666666640:890:89q(t)10:1190:01142:6178:25q(t)0130:800111:5200:01377777775;89Figure5.1:Algorithmconvergencefordifferentboundsofmeasurementnoise(withe=0:02).withtheboundsjq(t)j1;jq(t)j1:Theuncertaintyd(t)oftheMSPisboundedbyjd(t)jz;jd(t)j10z:DOFH2control:ThisexampleisusedtosynthesizeRGSDOFcontrolwithguaranteedH2performance.ThefourformulationsofCorollary5.2aresolvedforthisexamplewithAlgorithm1andtheresultsareshowninTable5.3.Toreducenumericalburdens,fewpointslinearlygriddedoveralogarithmicscaleintheinterval[101;104]areusedforeforthisexample.AshavebeenmentionedearlierinRemark5.3,itisdiftojudgeaprioriwhichformulationachievesbestperformance,however,fromTable5.3itisclearthatFormulationCachievesthebestperformanceamongthefourformulationsforthisdesignexample.90Table5.3:ComparisonoftheguaranteedH2boundnforCorollary5.2.eisthenumbergiveninparentheses().Actualclosed-loopH2-normisgivenbythenumberbetweenthesquarebrackets[].zFormulationAFormulationBFormulationCFormulationD00.2130.2320.2140.387(0.02)[0.165](0.0013)[0.184](0.02)[0.165](0.08)[0.238]0.010.9101.0250.8950.917(0.06)[0.354](0.013)[0.558](0.05)[0.372](0.05)[0.366]0.051.6131.9951.6131.636(0.02)[0.744](0.07)[0.435](0.02)[0.745](0.04)[0.534]0.12.0682.5112.0682.159(0.02)[0.888](0.07)[0.428](0.02)[0.892](0.04)[0.604]0.22.6693.7602.6693.558(0.02)[1.040](0.01)[1.022](0.02)[1.048](0.005)[1.166]0.33.1124.3963.1103.986(0.02)[1.127](0.04)[1.427](0.02)[1.139](0.005)[1.382]0.43.4785.5593.4764.467(0.02)[1.196](0.08)[1.647](0.02)[1.219](0.005)[1.463]0.53.8036.3803.8374.879(0.02)[1.233](0.08)[1.712](0.02)[1.684](0.005)[1.521]15.0526.8104.9105.970(0.02)[1.397](0.008)[2.855](0.02)[1.956](0.005)[1.910]26.8197.1276.3197.757(0.02)[1.470](0.008)[1.925](0.02)[2.221](0.005)[2.017]ThealgorithmconvergenceisshowninFigure5.1fordifferentboundsonmeasurementnoise.ItdemonstratestheeffectivenessoftheiterativeproceduredevelopedinAlgorithm1.Clearly,evenwithafewiterations,performanceimprovementcanbeachieved.Todemonstratetheadvantageofthesynthesizedcontrollers(KDOF),comparisonswithotherexistingmethodsareconducted.Figure5.2andTable5.4showtheguaranteedH2performanceboundsforthesynthesizedcontrollers(KDOF)andothercontrollersfromliterature.Aszin-creases,thecontrollerprovidedby[2]showsconsiderablesensitivitytotheuncertaintyboundswithitsH2performancedeterioratingexponentially.Themaximumuncertaintysizeforwhichmethod[2]provideafeasiblesolutionisz=0:48.Whenz>0:48nostabilizingcontrollercanbefoundusingtheconditionsin[2]whiletheconditionsofTheorem5.2providecontrollersfor91Figure5.2:ComparisonofH2guaranteedperformancevs.uncertaintysizebetweenthedevelopedconditionsandthemethodof[2].amuchwiderrangeofzwithimprovedperformancebounds.Figure5.2illustratestheH2per-formanceofbothcontrollersoverthefeasiblerangeof[2]onalog-logscale.Thiscomparisondemonstratesthegoodimprovementoftheproposedmethodoverthatof[2].AnothercomparisonbetweenKDOFversuscontrollerssynthesizedusingmethodin[3](seeFigure5.3)forarangeofuncertaintybounds.Forsmalluncertaintysize(z<0:35),theproposedmethodachievesalittleworseperformance,butaszincreases,theproposedmethodoutperformsthemethodin[3].Forinstance,whenz=2,theachievedH2performanceboundisn=6:319forKDOFwhilethecontrollerassociatedwith[3]achievesn=36:918.Furthermore,closed-loopsimulationsarecarriedoutwithActualSchedulingParameter(ASP)byq(t)=sin(0:25t)andtheirboundsjq(t)j1,andjq(t)j1.Measurementnoisewithboundsgivenbyjd(t)j0:2,andjd(t)j2areaddedintentionallytotheschedulingparameter.Thesynthesizeddynamicoutput-feedbackcontrollerachievesn=2:669withe=0:001.Figure5.4A92Figure5.3:ComparisonoftheguaranteedH2performancevs.uncertaintyboundbetweenthedevelopedconditionsandmethodof[3].Table5.4:ComparisonofguaranteedH2performancewithothermethodsfromliterature.z00.010.20.511.52Corollary5.20.2140.8952.6693.8374.9105.6916.319Methodof[2]0.4841.436230.240Methodof[3]0.4340.4591.6695.17713.94124.50036.918meansnofeasiblesolution.showstheMeasuredSchedulingParameter(MSP)andtheASP,respectively.L2disturbancesignalgivenbyw(t)=15exp(0:3t)sin(0:3t)isgeneratedasdisturbanceinputtotheclosed-loopsystem.Figure5.4BillustratessystemresponsetothedisturbanceinputcorrespondstotheMSP.Thesesimulationsshownotonlygoodrobustnesstothemeasurementnoiseinschedulingparameterbutalsorobustnessagainstexternaldisturbance.DOFH¥control:ThesameexampleisusedtosynthesizeRGSDOFcontrollerwithguaranteedH¥performance.ThefourformulationsofCorollary5.4aresolvedwithAlgorithm1andthe93Figure5.4:Simulation:A)Measuredandactualschedulingparameters,B)Disturbanceattenua-tion.resultsareshowninTable5.5.Similarly,toreducenumericalburdens,fewpointslinearlygriddedoveralogarithmicscaleintheinterval[101;104]areusedfore.Again,itisnotclearto94Table5.5:ComparisonoftheguaranteedH¥boundg¥forCorollary5.2.eisthenumbergiveninparentheses().Actualclosed-loopH¥-normisgivenbythenumberbetweenthesquarebrackets[].zFormulationAFormulationBFormulationCFormulationD00.1160.1160.1160.116(0.00207)[0.115](0.005)[0.115](0.005)[0.115](0.005)[0.114]0.010.1350.1180.1180.118(0.00207)[0.113](0.005)[0.114](0.005)[0.113](0.005)[0.114]0.050.1570.1240.1240.125(0.00207)[0.120](0.005)[0.114](0.005)[0.114](0.005)[0.113]0.10.1780.1380.1390.139(0.00207)[0.117](0.005)[0.116](0.005)[0.115](0.005)[0.115]0.20.2130.1650.1650.170(0.00207)[0.130](0.005)[0.117](0.005)[0.130](0.005)[0.120]0.30.2450.1930.1940.196(0.00207)[0.157](0.005)[0.119](0.005)[0.146](0.007)[0.143]0.40.2760.2200.2210.229(0.00207)[0.188](0.005)[0.164](0.002)[0.172](0.007)[0.169]0.50.3060.2460.2480.303(0.00207)[0.219](0.005)[0.187](0.005)[0.196](0.007)[0.207]10.4660.3950.3750.408(0.00207)[0.368](0.008)[0.288](0.005)[0.286](0.009)[0.296]20.8530.8440.7120.801(0.00207)[0.685](0.006)[0.691](0.05)[0.117](0.07)[0.189]identifywhichformulationachievesbestperformance,thereforeallformulationsshouldbetriedtoachievethebestpossibleperformance.Additionally,thesecontrollersarecomparedwithcontrollerssynthesizedviatheconditionsof[37].Figure5.5illustratesthiscomparisonthatdemonstrategoodperformanceimprovementofthecontrollerssynthesizedusingTheorem5.2overthecontrollersofmethod[37].Forsmalluncertaintysize(z<0:1),competitiveresultscanbeobtained,butaszincreases,ourcontrollersoutperformthecontrollersin[37].AnothercomparisonbetweencontrollerssynthesizedviaTheorem5.2(withe=0:001)andcontrollersbasedontheconditionsin[2]inTable5.6.Similarly,aszincreases,thesynthesizedcontrollersshowsgoodperformanceimprovementovercontrollersprovidedby[2].95Figure5.5:ComparisonofguaranteedH¥performancebetweenTheorem5.2and[3].Table5.6:ComparisonofguaranteedH¥performancewithmethodof[2].zTheorem5.2Methodof[2]0.010.1180.0630.10.1390.5690.20.1651.5770.30.1943.9220.40.22111.300.480.242339.8810.37520.712meansnofeasiblesolution.Closed-loopsimulationareconductedwithASPbyq(t)=cos(0:25t)andtheirboundsjq(t)j1,andjq(t)j1.Measurementnoisewithboundsgivenbyjd(t)j0:2,andjd(t)j2areaddedintentionallytotheschedulingparameter.Figure5.6AshowstheMSPandtheASP,respec-tively.L2disturbancesignalgivenbyw(t)=15exp(0:3t)sin(0:3t)isgeneratedasdisturbanceinputtotheclosed-loopsystem.Figure5.6Bshowssystemresponsetothedisturbanceinputcor-96Figure5.6:Simulation:A)Measuredandactualschedulingparameters,B)Disturbanceattenua-tion.respondstotheMSP.Thesesimulationsshownotonlyachievedrobustnesstomeasurementnoiseinschedulingparameterbutalsogoodrobustnesstodisturbanceattenuation.975.6SummaryNewsynthesisconditionsarederivedtosynthesizeRGSDOFcontrollerswithguaranteedH2andH¥performanceinthischapter.TheconditionsareformulatedintermsofPBMIswithscalarsearch.Thesynthesizedcontrollersguaranteenotonlyrobuststabilitybutalsoclosed-loopperformanceagainstschedulingparametersuncertainties.TheperformanceofthesynthesizedcontrollersarecomparedwithexistingdesignmethodsfromliteratureviaarealisticLPVsystemofamissilemodel.Comparisonsresultsdemonstratetheeffectivenessofthedevelopedconditions.98CHAPTER6RGSSTATICOUTPUT-FEEDBACKCONTROLThischapter,characterizesnovelsynthesisconditionsforRGSStaticOutput-Feedback(SOF)con-trolwithguaranteedperformance.BothH2andH¥performancesareinvestigated.Two-stagedesignprocedureisadoptedtosolvethiscontrolproblem.TheState-Feedback(SF)controllerisdesignedthenitisusedasinputtothesecondstagetosynthesizeRGSSOFcontroller.Numericalexamples,simulations,andcomparisonswithotherapproachesfromliteraturearein-cluded.Asummaryisgiveninthelastsection.6.1SOFSynthesisProblemConsiderthefollowingopen-loopsystem:G(q(t)):=8>>>>>><>>>>>>:x(t)=A(q(t))x(t)+Bu(q(t))u(t)+Bw(q(t))w(t)z(t)=Cz(q(t))x(t)+Dzu(q(t))u(t)y(t)=Cy(q(t))x(t)+Dyw(q(t))w(t);(6.1)wherex(t)2Rnisthestate,u(t)2Rnuisthecontrolinput,w(t)2Rnwisthedisturbanceinput,z(t)2Rnzisthecontrolledoutputandy(t)2Rnyisthemeasuredoutput.ThesystemmatriceshavethefollowingdimensionsA(q(t))2Rnn,Bu(q(t))2Rnnu,Bw(q(t))2Rnnw,Cz(q(t))2Rnzn,Dzu(q(t))2Rnznu,Cy(q(t))2Rnyn,andDyw(q(t))2Rnynw.Theaimistosynthesizeastaticoutput-feedbackgain-schedulingcontrolleroftheform,u(t)=K(Ÿq(t))y(t)(6.2)99Figure6.1:Closed-loopsystemwithRGScontrol.thatrobustlystabilizestheclosed-loopsystemx(t)=A(q;Ÿq)x(t)+B(q;Ÿq)w(t)z(t)=C(q;Ÿq)x(t)+D(q;Ÿq)w(t)A(q;Ÿq):=A(q)+Bu(q)K(Ÿq)Cy(q)B(q;Ÿq):=Bw(q)+Bu(q)K(Ÿq)Dyw(q)C(q;Ÿq):=Cz(q)+Dzu(q)K(Ÿq)Cy(q)D(q;Ÿq):=Dzw(q)+Dzu(q)K(Ÿq)Dyw(q)(6.3)andguaranteesaprescribedlevelofH2andH¥performances1.Furthermore,thesynthesizedcontrollershouldberobusttomeasurementuncertaintiesoftheschedulingparameters.More,thecontrollerutilizesthemeasured(noisy)schedulingparametersforfeedbackcon-trol.Thecontrollermatrixin(6.2)isassumedtohaveafparametrizationwithrespecttotheMSPs.Inotherwords,thismatrixK(Ÿq(t))isparameterizedasK(Ÿq(t))=K0+qåi=1Ÿqi(t)Ki:(6.4)1NotethatintheH2controlproblem,thefeedthroughmatrixoftheclosed-loopsystemshouldbezero,i.e.D(q;Ÿq)=000.100Aftomulti-simplextransformationRateofvariationmodelingSFcontrollersynthesisSOFcontrollersynthesisInverseTransformationControllerimplementationFigure6.2:Thedevelopedsynthesisapproach.Therefore,thegoalistoobtainthecontrollercoefmatricesKifori=0;1;2;;q,toim-plementtheRGScontrollerbyusingonlytheMSPsŸqi.Followingthelinesgivenin[74,75,76,77],two-stagedesignmethodhasbeenadoptedtosolvethiscontrolproblem.Gain-schedulingstate-feedbackcontrollershouldbedesignedinthestep,then,thiscontrollerisusedasinputparametermatrixatthesecondstagetosynthesizeRGSSOFcontrollerintheformof(6.2)suchthat(6.10)6.2ModelingApproachTheoverallsynthesisapproachforRGSSOFdesigncanbeillustratedbyFigure6.2.ItisaslightlyversionofFigure3.3.Aniterativeprocedureisdevelopedforthetwo-stagedesignpro-cedure.Lemma6.1.[62]Letu(t)=0in(6.1),foragivenpositivescalarn,ifthereexistacontinu-ouslydifferentiablematrixP(a)=P(a)02Rnnandparameter-dependentma-trixW(a)=W(a)02RnznzsuchthatthefollowingPLMIsare101264A(a;Ÿa)0P(a)+P(a)A(a;Ÿa)+P(a)?C(a;Ÿa)Inz375<000;(6.5)264W(a)?P(a)B(a;Ÿa)P(a)375>000;(6.6)trace(W(a))<n2;(6.7)theopen-loopsystemin(6.1)isasymptoticallystableforanypairs(a;a)2LWand(6.10)issatiwhereA(a;Ÿa),B(a;Ÿa),andC(a;Ÿa)aretheclosed-loopsystemmatricesde-in(6.3)withqandŸqreplacedbyaandŸausing(3.15)and(3.16).Lemma6.2.[78](ProjectionLemma)GivenasymmetricmatrixY2RnnandtwomatricesUandVofcolumndimensionsn,thereexistsanunstructuredmatrixZsatisfyingY+VZU+(VZU)0<0(6.8)ifandonlyifthefollowingprojectioninequalitieswithrespecttoZareNvYN0v<0;(6.9a)N0uYNu<0(6.9b)whereNuandNvanymatriceswhosecolumnsformabasesofthenullspacesofUandV,respectively,suchthatNvV=0andUNu=0.6.3PLMIsSynthesisConditionswithH2performanceProblem6.1.Supposethattheschedulingparametersq(t)areprovidedasŸq(t)withuncertaintyd(t)asin(3.3).SupposefurtherthatD(q(t);Ÿq(t))=0in(6.3).Convertingalltheopen-loopsystemmatricesandcontrollervariablestothemulti-simplexvariablesaandŸainsteadofqandŸq,respectively,using(3.15)and(3.16).Foragivenpositivescalarn,aRGSstatic102output-feedbackcontrollerintheformof(6.11)tostabilizetheclosed-loopsystem(6.3)foranypair(Ÿa(t);Ÿa(t))2LWsuchthatsup(Ÿa(t);Ÿa(t))2LWEEEˆZT0z(t)0z(t)dt˙<n2;(6.10)forthedisturbanceinputw(t)givenbyw(t)=w0d(t)whered(t)istheDirac'sdeltafunctionandw0isarandomvariablesatisfyingEw0w00=InwandEgdenotesthemathematicalexpectation.Twostagedesignprocedureispresentedinthissection.Inthethestage,SFcontrollerisdesigned.Then,thiscontrollerisusedinthesecondstagetosynthesizeRGSSOFcontroller.6.3.1State-FeedbackcontrolTheorem6.1.LetK(Ÿa)=000.Givenascalarn>0andasufsmallscalare>0.Ifthereexistacontinuouslydifferentiableparameter-dependentmatrix0<P(a)=P(a)02Rnn,parameter-dependentmatricesW(a)=W(a)02Rnznz,Z(a)2Rnun,G(a)2Rnnforanypair(Ÿa(t);Ÿa(t))2LWsuchthatthefollowingPLMIs266664A(a;Ÿa)G(a)+Bu(a)Z(a)+()0P(a)??P(a)G(a)+e(A(a;Ÿa)G(a)+Bu(a)Z(a))0e(G(a)+G(a)0)?B(a;Ÿa)0000nwnInw377775<0002n+nw;264G(a)+G(a)0P(a)?C(a;Ÿa)G(a)+Dzu(a)Z(a)W(a)375>000n+nz;trace(W(a))<n2;103Then,thegain-schedulingcontrollerK(a)=Z(a)G(a)1;stabilizestheclosed-loopsystemwithguaranteedH2boundnin(6.10).Proof.TheproofisgiveninChapter4.Remark6.1.Thestate-feedbackcontrollerK(Ÿa),obtainedfromTheorem6.1,isusedasinputtoTheorem6.2tosynthesizethestaticoutput-feedbackcontroller(6.2).6.3.2StaticOutput-FeedbackControlTheorem6.2.GivenK(Ÿa),sufsmallpositivescalarh,andpositivescalarn>0.Ifthereexistacontinuouslydifferentiableparameter-dependentmatrix0<P(a)=P(a)02Rnn,parameter-dependentmatricesV(a)2Rnn,F(a)2Rnn,Q(a)2Rnznz,R(Ÿa)2Rnunu,andL(Ÿa)2Rnunyforanypair(Ÿa(t);Ÿa(t))2LWsuchthatthefollowingPLMI266666664F(a)A(a;Ÿa)+()0+P(a)???P(a)F(a)0+hV(a)A(a;Ÿa)h(V(a)+V(a)0)??Bu(a)0F(a)0+L(Ÿa)Cy(a)R(Ÿa)K(Ÿa)hBu(a)0V(a)0R(Ÿa)R(Ÿa)0?Q(a)0C(a;Ÿa)000nznQ(a)0Dzu(a)InzQ(a)Q(a)0377777775<000(6.11)264W(a)?P(a)Bw(a)P(a)375>000;(6.12)trace(W(a))<n2;(6.13)withA(a;Ÿa):=A(a)+Bu(a)K(Ÿa)C(a;Ÿa):=Cz(a)+Dzu(a)K(Ÿa);then,thestaticoutput-feedbackgain-schedulingcontrollerK(Ÿa)=R(Ÿa)1L(Ÿa);(6.14)robustlystabilizestheclosed-loopsystem(6.3)and(4.6).104Proof.Inequality(6.11)holdsif(6.8)inLemma6.2iswiththefollowingV=266666664000000I000377777775;Z=R(Ÿa);U=[X(a;Ÿa)000I000];Y=266666664A(a;Ÿa)0F(a)0+F(a)A(a;Ÿa)+P(a)???P(a)F(a)0+hV(a)A(a;Ÿa)h(V(a)+V(a)0)??Bu(a)0F(a)0hBu(a)0V(a)0000nunw?Q(a)0C(a;Ÿa)000nznQ(a)0Dzu(a)Q(a)0Q(a)3777777752;(6.15)withthefollowingnullspacesofVandU,i.e.Nv=264I000000000000I000000375;Nu=266666664I000000000I000X(a;Ÿa)000000000000I377777775andA(a;Ÿa):=A(a)+Bu(a)K(Ÿa)C(a;Ÿa):=Cz(a)+Dzu(a)K(Ÿa)X(a;Ÿa):=R(Ÿa)1L(Ÿa)Cy(a)K(Ÿa):(6.16)Thus,UNu=[X(a;Ÿa)000I000]266666664I000000000I000X(a;Ÿa)000000000000I377777775=000;2Thisisbecause(InzQ(a))0(InzQ(a))>0impliesQ(a)0Q(a)<InzQ(a)Q(a)0.Thesameideawasusedin[74].105NvV=264I000000000000I000000375266666664000000I000377777775=000;VZU=266666664000000000000000000000000L(Ÿa)Cy(a)R(Ÿa)K(Ÿa)000R(Ÿa)000000000000000377777775Considernow(6.9a),NvYN0v=264A(a;Ÿa)0F(a)0+F(a)A(a;Ÿa)+P(a)?P(a)F(a)0+eV(a)A(a;Ÿa)e(V(a)+V(a)0)375<0:(6.17)Multiplying(6.17)byInA(a;Ÿa)0fromleftandbyitstransposefromrighttoobtainA(a;Ÿa)0P(a)+P(a)A(a;Ÿa)+P(a)<0:(6.18)InadditiontoP(a)>0,(6.18)representsLyapunovstabilityconditionforA(a;Ÿa).Thus,(6.9a)isvOntheotherhand,(6.9b)isN0uYNu=266664I000X(a;Ÿa)0000000I000000000000000I377775266666664Y11???P(a)F(a)0+hV(a)A(a;Ÿa)Y22??Bu(a)0F(a)0hBu(a)0V(a)0Y33?Q(a)0C(a;Ÿa)000nznQ(a)0Dzu(a)Y44377777775266666664I000000000I000X(a;Ÿa)000000000000I377777775:106withY11=A(a;Ÿa)0F(a)0+F(a)A(a;Ÿa)+P(a);Y22=h(V(a)+V(a)0);Y33=000nunw;Y44=Q(a)0Q(a):Therefore,N0uYNu=266664Y11+X(a;Ÿa)0Bu(a)0F(a)0¯Y12F(a)Bu(a)¯Y14P(a)F(a)0+hV(a)A(a;Ÿa)Y22hV(a)Bu(a)000nnzQ(a)0C(a;Ÿa)000nznQ(a)0Dzu(a)Y44377775266666664I000000000I000X(a;Ÿa)000000000000I377777775;with¯Y12=P(a)F(a)+hA(a;Ÿa)0V(a)0+hX(a;Ÿa)0Bu(a)0V(a)0;¯Y14=C(a;Ÿa)0Q(a)+X(a;Ÿa)0Dzu(a)0Q(Ÿa);leadstoN0uYNu=266664F(a)[A(a;Ÿa)+Bu(a)X(a;Ÿa)]+()0+P(a)??P(a)F(a)0+hV(a)[A(a;Ÿa)+Bu(a)X(a;Ÿa)]h(V(a)+V(a)0)?Q(a)0[C(a;Ÿa)+Dzu(a)X(a;Ÿa)]000nznQ(a)0Q(a)377775:Considering(6.16)withA(a;Ÿa):=A(a;Ÿa)+Bu(a)X(a;Ÿa)=A(a)+Bu(a)K(Ÿa)Cy(a);C(a;Ÿa):=C(a;Ÿa)+Dzu(a)X(a;Ÿa)=Cz(a)+Dzu(a)K(Ÿa)Cy(a);K(Ÿa):=R(Ÿa)1L(Ÿa);107yieldsN0uYNu=266664F(a)A(a;Ÿa)+()0+P(a)P(a)F(a)+hA(a;Ÿa)0V(a)0C(a;Ÿa)0Q(Ÿa)P(a)F(a)0+hV(a)A(a;Ÿa)h(V(a)+V(a)0)000nnzQ(a)0C(a;Ÿa)000nznQ(a)0Q(a)377775:(6.19)Multiplying(6.19)byT2fromleftandbyitstransposefromrightwithT2=264IA(a;Ÿa)0000000000(Q(a)1)0375leadsto(6.5),i.e.T2(6.19)T02=264A(a;Ÿa)0P(a)+P(a)A(a;Ÿa)+P(a)C(a;Ÿa)0C(a;Ÿa)Inz375<0:6.4SynthesisConditionswithH¥PerformanceProblem6.2.Supposethattheschedulingparametersq(t)areprovidedasŸq(t)withuncertaintyd(t).Convertingalltheopen-loopsystemmatricesandsynthesisvariablestothemulti-simplexvariablesaorŸainsteadofqandŸq,respectively,using(3.15)and(3.16).Foragivenpositivescalarg¥,aRGSstaticoutput-feedbackcontrollerintheformof(6.2)tostabilizetheclosed-loopsystem(6.3)foranypair(Ÿa(t);Ÿa(t))2LWsuchthatsup(Ÿa(t);Ÿa(t))2LWsupw2L2;w6=0kz(t)k2kw(t)k2<g¥:(6.20)Similarly,theSFcontrollerissynthesizedandthenitisusedinthesecondstagetosyn-thesizeRGSSOFcontroller.6.4.1State-FeedbackH¥controlTheorem6.3.Givenascalarg¥>0andasufsmallscalare>0.Ifthereexistacontinu-ouslydifferentiableparameter-dependentmatrix0<P(a)=P(a)02Rnn,parameter-dependent108matricesZ(a)2Rnun,G(a)2Rnnforanypair(a(t);a(t))2LWsuchthatthefollowingPLMI266666664A(a)G(a)+Bu(a)Z(a)+()0P(a)???P(a)G(a)+e(A(a)G(a)+Bu(a)Z(a))0e(G(a)+G(a)0)??C(a)G(a)+Dzu(a)Z(a)eC(a)G(Ÿa)+eDzu(a)Z(a)Inz?B(a)0000nwnD(a)0g2¥Inw377777775<000;then,thegain-schedulingcontrollerK(a)=Z(a)G(a)1;stabilizestheclosed-loopsystemwithguaranteedH¥performanceboundg¥satisfying(6.20).Proof.TheproofcanbefoundinChapter56.4.2StaticOutput-FeedbackH¥controlTheorem6.4.GivenK(a),sufsmallpositivescalarh,andpositivescalarg¥>0.Ifthereexistacontinuouslydifferentiableparameter-dependentmatrix0<P(a)=P(a)02Rnn,parameter-dependentmatricesV(Ÿa)2Rnn,F(Ÿa)2Rnn,Q(Ÿa)2Rnznz,R(Ÿa)2Rnunu,andL(Ÿa)2Rnunyforanypair(Ÿa(t);Ÿa(t))2LWsuchthatthefollowingPLMI266666666664A(a)0F(a)0+F(a)A(a)+P(a)????P(a)F(a)0+hV(a)A(a)h(V(a)+V(a)0)???Bw(a)0F(a)0hBw(a)0V(a)0g2¥Inw??Q(a)0C(a)000nznQ(a)0Dzw(a)InzQ(a)Q(a)0?Bu(a)0F(a)0+L(Ÿa)Cy(a)R(Ÿa)K(a)hBu(a)0V(a)0L(Ÿa)Dyw(a)Dzu(a)0Q(a)R(Ÿa)R(Ÿa)0377777777775<000(6.21)withA(a):=A(a)+Bu(a)K(a)C(a):=Cz(a)+Dzu(a)K(a):Then,thestaticoutput-feedbackgain-schedulingcontrollerK(Ÿa)=R(Ÿa)1L(Ÿa)(6.22)109isthestabilizingcontrollerfortheclosed-loopsystem(6.3)satisfying(6.20).Proof.Inequality(6.21)holdsif(6.8)inLemma6.2iswiththefollowingV=266666666664000000000000I377777777775;Z=R(Ÿa);U=[X(Ÿa)000Y(Ÿa)000I];Y=266666666664A(a)0F(a)0+F(a)A(a)+P(a)????P(a)F(a)0+hV(a)A(Ÿa)h(V(a)+V(a)0)???Bw(a)0F(a)0hBw(a)0V(a)0g2¥Inw??Q(a)0C(a)000nznQ(a)0Dzw(a)Q(a)0Q(a)?Bu(a)0F(a)0hBu(a)0V(a)0000nunwDzu(a)0Q(a)000nu377777777775;withthefollowingnullspacesNv=264I000000000000000I000000000375;Nu=266666666664I000000000000I000000000000I000000000000IX(Ÿa)000Y(Ÿa)000377777777775andA(a):=A(a)+Bu(a)K(Ÿa)C(a):=Cz(a)+Dzu(a)K(Ÿa)X(Ÿa):=R(Ÿa)1L(Ÿa)Cy(a)K(Ÿa)Y(Ÿa):=R(Ÿa)1L(Ÿa)Dyw(a):110Thus,VZU=266666666664000000000000000000000000000000000000000000000000000000000000L(Ÿa)Cy(a)R(Ÿa)K(a)000L(Ÿa)Dyw(a)000R(Ÿa)377777777775:Considernow(6.9a),NvYN0v=264A(a)0F(a)0+F(a)A(a)+P(a)?P(a)F(a)0+eV(a)A(a)e(V(a)+V(a)0)375<0:(6.23)Multiplying(6.23)byInA(a)0fromleftanditstransposefromrighttoobtainA(a)0P(a)+P(a)A(a)+P(a)<0;whichisinadditiontoP(a)>0,representsLyapunovstabilityconditionforA(a).Thisproves(6.9a).Ontheotherhand,(6.9b)isN0uYNu=266666664I000000000X(Ÿa)0000I000000000000000I000Y(Ÿa)0000000000I000377777775266666666664A(a)0F(a)0+F(a)A(a)+P(a)P(a)F(a)+eA(a)0V(a)0F(a)Bw(Ÿa)C(a)0Q(a)F(a)Bu(a)P(a)F(a)0+hV(a)A(a)h(V(a)+V(a)0)hV(a)Bw(a)000nnzhV(a)Bu(a)Bw(a)0F(a)0hBw(a)0V(a)0g2¥InwDzw(a)0Q(a)000nwnuQ(a)0C(a)000nznQ(a)0Dzw(a)Q(a)0Q(a)Q(a)0Dzu(a)Bu(a)0F(a)0hBu(a)0V(a)0000nunwDzu(a)0Q(a)000nu377777777775266666666664I000000000000I000000000000I000000000000IX(Ÿa)000Y(Ÿa)000377777777775:111resultsN0uYNu=266666664F(a)A(a)+F(a)Bu(a)X(Ÿa)+()0+P(a)???P(a)F(a)0+hV(a)A(a)+hV(a)Bu(a)X(a)h(V(a)+V(a)0)??Bw(a)0F(a)0+Y(Ÿa)0Bu(a)0F(a)0hBw(a)0V(a)0+hY(a)0Bu(a)0V(a)0g2¥Inw?Q(a)0C(a)+Q(a)0Dzu(a)X(Ÿa)000nznQ(a)0Dzw(a)+Q(a)0Dzu(a)Y(Ÿa)Q(a)0Q(a)377777775:andN0uYNu=266666664F(a)A(a;Ÿa)+A(a;Ÿa)0F(a)0+P(a)???P(a)F(a)0+hV(a)A(a;Ÿa)h(V(a)+V(a)0)??B(a;Ÿa)0F(a)0hB(a;Ÿa)0V(a)0g2¥Inw?Q(a)0C(a;Ÿa)000nznQ(a)0D(a;Ÿa)Q(a)0Q(a)377777775:(6.24)withthefollowingclosed-loopsystemrelationshipsA(a;Ÿa)=A(a)+Bu(a)K(Ÿa)Cy(a)B(a;Ÿa)=Bw(a)+Bu(a)K(Ÿa)Dyw(a)C(a;Ÿa)=Cz(a)+Dzu(a)K(Ÿa)Cy(a)D(a;Ÿa)=Dzw(a)+Dzu(a)K(Ÿa)Dyw(a)whereK(Ÿa)isin(6.14).Inordertoshowthenegativeof(6.24),thestepsdepictedin[74]isfollowedandmayproducesomeconservativenesstotheconditions.Notethatifinequality(6.21)isnegative-thenitisalsowithQ(a)0Q(a)replacingInzQ(a)Q(a)0,since(InzQ(a))0(InzQ(a))>0impliesQ(a)0Q(a)<InzQ(a)Q(a)0.Inotherwords,N0uYNu<266666664F(a)A(a;Ÿa)+A(a;Ÿa)0F(a)0+P(a)???P(a)F(a)0+hV(a)A(a;Ÿa)h(V(a)+V(a)0)??B(a;Ÿa)0F(a)0hB(a;Ÿa)0V(a)0g2¥Inw?Q(a)0C(a;Ÿa)000nznQ(a)0D(a;Ÿa)InzQ(a)Q(a)0377777775<0(6.25)SettinghV(a):=U(a)andmultiplyingthemiddletermof(6.25)byT(Ÿa)0ontheleftandby112T(Ÿa)ontheright,withT(a;Ÿa)=266666664I000000A(a;Ÿa)B(a;Ÿa)000000I000000000Q0(a)1377777775toobtain266664IA(a;Ÿa)0000000000B(a;Ÿa)0I000000000000Q0(a)1377775266666664F(a)A(a;Ÿa)+A(a;Ÿa)0F(a)0+P(a)???P(a)F(a)0+U(a)A(a;Ÿa)U(a)U(a)0??B(a;Ÿa)0F(a)0B(a;Ÿa)0U(a)0g2¥Inw?Q(a)0C(a;Ÿa)000nznQ(a)0D(a;Ÿa)Q(a)0Q(a)0377777775266666664I000000A(a;Ÿa)B(a;Ÿa)000000I000000000Q(a)1377777775<0thatleadsto266664F(a)A(a;Ÿa)+P(a)+A(a;Ÿa)0P(a)+A(a;Ÿa)0U(a)A(a;Ÿa)P(a)F(a)A(a;Ÿa)0U(a)F(a)B(a;Ÿa)+A(a;Ÿa)0U(a)B(a;Ÿa)C(a;Ÿa)0Q(a)B(a;Ÿa)0P(a)+B(a;Ÿa)0U(a)A(a;Ÿa)B(a;Ÿa)0U(a)B(a;Ÿa)0U(a)B(a;Ÿa)g2¥InwD(a;Ÿa)0Q(a)C(a;Ÿa)000D(a;Ÿa)Q(a)377775266666664I000000A(a;Ÿa)B(a;Ÿa)000000I000000000Q(a)1377777775<0and266664A(a;Ÿa)0P(a)+P(a)A(a;Ÿa)+P(a)P(a)B(a;Ÿa)C(a;Ÿa)0B(a;Ÿa)0P(a)g2¥InwD(a;Ÿa)0C(a;Ÿa)D(a;Ÿa)Inz377775<0whichrepresentsaparameter-dependentversionofthebounded-reallemmaforLPVsystems[79].Thus,(6.9b)isvRemark6.2.OnceafeasiblesolutionexistsforTheorem6.2andTheorem6.4,thecontrollercoefKifori=0;1;;qin(6.4)canbeobtainedfromthemulti-simplexcoefof113thecontrollermatrixKj1;j2;;jq;k1;k2;;kqin(6.14)and(6.22)usingtheinversetransformation(multi-simplextoaftransformation)(3.40)and(3.41).Remark6.3.IncontrasttotheexistingliteratureofRGScontrol[2,36,80,38]thatonlyallowsthestatematrixA(q)tobeaffectedbythevaryingparameters,thesynthesisconditionsofTheorem6.2andTheorem6.4dealwiththegeneralcasethatalltheopen-loopmatricesarefunctionsofvaryingparameters.Thisisthemainadvantageofthedevelopedconditionsovertheexistingones.Remark6.4.TheconditionsofTheorem6.2andTheorem6.4associatedwithsomeconservative-nessduetoover-boundingthe(4;4)blockof(6.15)and(6.24).Therefore,IterativeStaticOutput-FeedbackDesign(ISOFD)algorithmwasdevelopedtoreduceconservativeness.ThisalgorithmisshownfortheH2performancebutitappliesforH¥performanceaswell.Initially,SFcontrollerK(a)isobtainedviaTheorem6.1orTheorem6.3.ThiscontrollerisusedinTheorem6.2orThe-orem6.4tosynthesizetheRGSSOFcontroller.EachiterationoftheISOFDalgorithmassuresafeasiblesolutionwithatleastnini1.Notethatthealgorithmconvergenceisguaranteedsincenismonotonicallydecreasingandboundedfrombelow.Remark6.5.AsaspecialcaseofTheorem6.2andTheorem6.4,robustSOFcontrollercanalsobesynthesized.More,constrainingthesynthesisvariablesR(Ÿa)andL(Ÿa)tobeparameterindependent(i.e.constantmatrices)leadstoarobustSOFcontroller.Remark6.6.Asshownabove,thesynthesisconditionsofTheorem6.2andTheorem6.4areformu-latedasPLMIs(foraeandh)intermsoftime-varyingparametersinsidethemulti-simplexdomain.ROLMIP[56]isusedtoimplementtheISOFDalgorithmtoobtaintheRGSSOFoptimalcontroller.6.5IllustrativeExamplesInthissection,twoexamplesarepresentedtodemonstratetheeffectivenessofthedevelopedap-proach.First,anacademicexampleisgiventoillustratetheachievedperformancewithdifferent114Algorithm2:IterativeStaticOutput-FeedbackDesign(ISOFD)Algorithm.Initialization:Seti=0,K0(Ÿa)=000.UsingTheorem6.1,computeinitialstate-feedbackcontrollerK0(a).SetimaxandTolarance.repeatSeti=i+1.GivenKi1(a),solvetheconditionsofTheorem6.2toobtainSOFcontrollerKi(Ÿa)withaminimalachievableboundni.GivenKi(Ÿa),solvetheconditionsofTheorem6.1toobtainKi(a).untili>imaxORjnini1j<Tolarance;boundsofmeasurementnoiselevelusingAlgorithm2.Second,thedevelopedmethodisappliedtoarealisticEVVTactuatorobtainedfromexperimentalstudy[81].ThisexampleservestovalidatethedevelopedsynthesisapproachthroughrealisticLPVmodelfromengineeringapplicationpointofview.ThesynthesisconditionsareimplementedinMATLABenvironment(R2013a).Thecom-puterusedforcontroldesignisanIntelCorei7(2.4GHz)processor,6GBRAMwithWindows10.6.5.1AcademicExampleConsiderthefollowingLPVsystem[1],266664A(q)Bu(q)Bw(q)Cz(q)Dzu(q)Cy(q)Dyw(q)377775=266666666666425:960q(t)130:032040q(t)3464q(t)20:471100011003777777777775:115Thevaryingparameterq(t)hasthefollowingbounds0q(t)1,jq(t)j1,withmeasurementuncertaintyboundjd(t)jzandjd(t)j10z.ThescalarseandhinTheorem6.1andThe-orem6.2areintroducedtoprovideadditionaldegreesoffreedominthesynthesisconditions.Toreducecomputationalburdens,thesescalarsarechosen(inalogarithmicscale)between101and105.Inthisexample,e=0:01andh=0:1arefoundtoachievebestperformancebound.Fortheiterativealgorithm,atoleranceandmaximumnumberofiterationsaresetto104andfour,respectively.Table6.1showstheguaranteedboundnfortheconditionsofTheorem6.1andTheorem6.2withtheiterativealgorithm.Inthestage,theconditionsofTheorem6.1isusedtoobtainGSstate-feedbackcontroller,andthen,thiscontrollerisfedtothesecondstagetosynthesizetheRGSSOFcontroller.Notethattheachievableperformanceusingthestageisslightlylowerthantheoneassociatedwiththesecondstage(asexpected)sinceinthecaseofstate-feedbackfullstatemeasurementisassumedtobeavailableinreal-timeforcontrollerimplementationwhileintheSOFcaseonlytheoutputfeedbackisused.Algorithm2isusedtoreduceconservativenessgraduallyasshowninTable6.1forthefouriterations.Figure6.3illustratestheconvergenceoftheiterativealgorithmwithz=0:1.Tothebestauthors'knowledgethereisnoSOFsynthesismethodthatisabletohandlethemeasurementnoisesinschedulingparameters.Comparingwithfull-orderdynamicoutput-feedbackresultsin[2],itisclearthatthemethodin[2]isverysensitivetosmallvaluesofz,whilethedevelopedsynthesisconditionspresentedinTable6.1showsgoodrobustnessagainstuncertaintiesinschedulingparameters.6.5.2EVVTActuatorTovalidatetheresultofthedevelopedapproachforpracticalengineeringapplications,theElectricVariableValveTiming(EVVT)camphasersystemisinvestigatedinthissection.TheEVVTac-tuatorconsistsoftwomaincomponents:anelectricmotorandaplanetarygearset.Theplanetarygearsetconsistsofanouterringgear,aplanetgearcarrierwithplanetgearsattached,andasun116Table6.1:H2performancewithdifferentboundsofmeasurementnoise.e=0:01,h=0:1.z00.0010.010.10.20.51Methodof[2]9.0879.159.70639.5775.406173.92324.331ststage3.1203.1313.2303.4233.7384.0534.0812ndstage3.4813.4953.4903.5123.9624.2134.311Iterations13.4633.4743.4803.5003.7323.8164.11523.4383.4583.4503.4803.6093.7753.97933.3803.4023.4243.4623.5883.6493.88843.3563.3713.3913.4373.5023.6203.820Figure6.3:AlgorithmConvergencegear.Theringgear,thatisrunningatthehalfspeedofthecrankshaft,isdrivenbythecrankshaftthroughtheenginetimingbelt.DetailsoftheEVVTmodelingworkcanbefoundin[82].Theringgear,whichisrunningatthehalfspeedofthecrankshaft,isdrivenbythecrankshaftthroughtheenginetimingbelt(seeFigure6.4).Theplanetarygearcarrierisdrivenbyanelectricmotor117Figure6.4:EVVTcam-phaseactuatorschematicdiagram.andfourplanetgearsengagingbothringandsungearsatthesametime,wherethesungearisconnectedtothecamshaft.Thespeedofthecamshaftisdeterminedbytheringgearspeedto-getherwiththeEVVTmotorspeed,whichprovidestheenginewithxiblevalveopeningtiming.Therefore,thecam-phasecanbeadjustedbycontrollingtheEVVTmotorspeedwithrespecttotheenginespeed.Aseriesofsystemexperimentaltestswereconductedatarangeofedvaluesofenginespeed(N)andbatteryvoltage(V)attheEnergyandAutomotiveResearchLaboftheMichiganStateUniversity(see[81]fortheauthors).ItwasfoundthatthemodeloftheEVVTactuatorisinthefollowingform,G(N(t);V(t);s)=q1(N(t);V(t))s(s+q2(N(t);V(t)));(6.26)whereq1(N(t);V(t))andq2(N(t);V(t))aretime-varyingcoefasfunctionsofenginespeedandbatteryvoltage.Fornotationalsimplicity,q1(t)andq2(t)areusedtorefertoq1(N(t);V(t))andq2(N(t);V(t)),respectively.Notethatq1(t)isassociatedwiththeDCgainofthetransferfunction(6.26),andq2(t)isthelocationoftheopen-looppoleofthe2ndordersystem.Inother118Table6.2:Rangeofthetime-varyingparameters.q1(t)2[0:25290:6472]q2(t)2[6:97514:540]words,theDCgainandpolelocationofthetransferfunction(6.26)aretime-varyingcoefandfunctionsofenginespeedandbatteryvoltage.Itisworthmentioningthatthevaluesofq1(t)andq2(t)wereobtainedexperimentallyoveredvaluesofbatteryvoltageandenginespeedthatcovertheentirerangeofengineoperatingconditions.Therangesofthevaryingco-efq1(t)andq2(t)aregiveninTable6.2.InordertoperformcontrollerdesignusingtheconditionsdevelopedinSection6.4,theEVVTplantmodel(6.26)needstoberealizedinthestate-spaceformasfollows:264x1x2375=264010q2(t)375264x1x2375+2640q1(t)375u;y=10264x1x2375;wherex1representsthecam-phaseangleanduistheEVVTmotorspeedcommand.Inordertoaccommodateforperformanceoutput,controlenergy,anddisturbanceinput,Cz(q)=2641000375;Dzu(q)=26401375;Bw(q)=2640:10375;areandthestate-spacerealizationoftheoverallLPVsystemcorrespondingtothematricesin(6.1)are266664A(q)Bu(q)Bw(q)Cz(q)Dzu(q)Cy(q)Dyw(q)377775=26666666666640100:10q2(t)q1(t)01000011003777777777775:119Figure6.5:Time-domainsimulationsfortheEVVTactuator(piecewise-constantschedulingsig-nals).First,theopen-loopsystemmatricesareconvertedfromtheq(t)-spaceintothea(t)-spaceusingtheprocedurepresentedinSection6.2.Then,theconditionsofTheorem6.1andTheorem6.2areusedtosynthesizeRGSSOFcontroller.Table6.3showstheachievedH2boundnforbothcontrollers(state-feedbackandstaticoutput-feedback)aswellasthenumberofLMIvariablesandLMIconstraintsassociatedwitheachtheorem.Afterafeasiblesolutionisobtained,thecontrollermatrixin(6.14)iscalculated,thantheinversetransformation(3.40)and(3.41)isusedtocalculatecontrollercoefintheoriginalparameterspace,i.e.thecoefin(6.4)toimplementthecontrollerinreal-timeusingtheMSPs.ThesecoefareobtainedasK0=0:7126;K1=0:5879;K2=0:6736:120Figure6.6:Time-domainsimulationsfortheEVVTactuator(sinusoidalschedulingsignals).Table6.3:H2performanceforstage1andstage2.e=105,h=0:01.#LMIvariables#LMIconstraintsn1ststage25600.7592ndstage684870.767ItisclearfromTable6.3thattheguaranteedperformanceusingSOFcontrollerisveryclosetotheperformanceofthestate-feedbackone.Time-domainsimulationresultsoftheEVVTactuatorareshowninFigure6.5andFigure6.6withdifferentschedulingtrajectories.Tablelook-upmethodisusedtomaptheenginespeedandbatteryvoltageintothevaryingparametersq1(t)andq2(t).Thesimulationresultsdemonstratenotonlyrobustnessofthesynthesizedcontrolleragainstuncertaintiesinschedulingparameters121butalsotheabilitytoattenuatedisturbancetorqueonthecam-phaseangle(performanceoutput).6.6SummaryPLMIsconditionsforsynthesizingRGSSOFcontrollersubjecttoinexactlyMSPsaredevelopedinthischapter.Thefoundationoftheapproachisbasedonatwo-stagedesignmethodbydesigningstate-feedbackcontrollerinthestageandusingtheresultingcontrollerforsynthesizingRGSSOFcontrollerinthesecondstage.Bothperformances,H2andH¥,areconsidered.Aniterativeprocedure(IterativeStaticOutput-FeedbackDesign(ISOFD)algorithm)isdevelopedtoreducetheupperboundsoftheH2orH¥performances.ThemainnoveltyofthedevelopedmethodisthatitcanhandlethemostgeneralLinearParameter-Varying(LPV)systems,wherethevaryingparam-etersaffectalltheopen-loopsystemmatrices.NotethatrobustSOFcontrollercanbehandledasaspecialcaseofdevelopedconditions.Twoexamplesarepresented,oneisanacademicexampleandthesecondisapracticalElectricVariableValveTiming(EVVT)actuator.Numericalresultsandsimulationsoftheseexamplesillustratetheeffectivenessofthedevelopedapproach.122CHAPTER7EXPERIMENTALVALIDATIONONEVVTACTUATORThischapterpresentsexperimentaldemonstrationofthedevelopedsynthesisconditionsfortheElectricVariableValveTiming(EVVT)actuator.Thefueleconomy,emissions,andperformanceofaninternalcombustion(IC)engineareheavilybytheintakeandexhaustvalvetim-ings.Withaconventionalvalve-trainsystem,theintakeandexhaustvalvetimingcanonlybeopti-mizedforasingleoperatingcondition.Thatis,theoptimizedvalvetimingcaneitherimprovefueleconomyandreduceemissionsatlowenginespeedsormaximizeenginepowerandtorqueout-putsathighenginespeeds.Duetothegrowingfueleconomydemandsandemissionregulations,electricvariablevalvetiming(EVVT)systems[83]weredeveloped.ThechallengingproblemofimprovingfueleconomyandreducingemissionsatlowenginespeedwhilemaintainingengineperformanceathighenginespeedcanbeaddressedwiththehelpofEVVTsystems.TheEVVTsystemisanexcellentcandidatetoverifytheeffectivenessoftheRGScontrolapproachdevelopedinthisdissertation.Sinceenginespeedandvehiclebatteryvoltagehaveimpactontheengineperformance,thesetwotime-varyingparametershavebeenselectedforfeedbackcontrolasnoisyschedulingparameters.Therefore,thegoalistodesignandimplementRGSStaticOutput-Feedback(SOF)controllerexperimentallyonEVVTactuatorsubjecttonoisyschedulingparameter.Todothis,theproceduredescribedinFig.7.1isfollowed.First,aseriesofstandardsystemtestsareconductedontheEVVTsystembenchtoobtainafamilyofLinearTime-Invariant(LTI)models.Usingen-ginespeedandbatteryvoltageastime-varyingparameters,afamilyoflinearmodelsoftheEVVTsystemareobtainedbyperformingmultipletestswhilemaintainingenginespeedandbatteryvolt-ageatedvalues.WiththefamilyoflocalLTImodels,theLPVmodeloftheEVVTsystemisformulated.SincetheobtainedLPVmodelhasafparametrizationstructureintermsofthevaryingparameters,thismodelisconvertedintoapolytopic(multi-simplex)domainsothattheH¥RGScontrolsynthesismethodpresentedinChapter5canbeappliedtoobtaintheRGS123Figure7.1:FlowchartforthedesignandimplementationofaRGScontrolleronEVVTsystem.controller.Onceacontrollerisobtained,itsperformanceisexperimentallyevaluatedonthesametestbenchthatwasusedtoperformthesystemationtests.IftheperformanceandstabilityrequirementsoftheEVVTsystemarenotanothercontrollercanbesynthesizedagainbytuningdesignparametersinthesynthesisconditions.Thisloopisperformeduntilthestabilityandsatisfactoryperformanceareobtainedonthetestbench.7.1EVVTEngineCam-PhasingActuatorTheEVVTsystemcomponentsareillustratedandthenexperimentalresultsofthesystemtestsaregivenforedvaluesofenginespeedandbatteryvoltagethatcovertheentirerangeoftheparametersvariations.Inthethirdsubsection,LPVmodelinginthestate-spacerepresentationwillbeexplainedfortheEVVTactuator.124Figure7.2:ElectricplanetarygearEVVTsystem.7.1.1ActuatorComponentsTheEVVTsystemstudiedinthischapteroperateswithtwomaincomponents:anelectricmotorandaplanetarygearset(seeFigure7.2).Theplanetarygearsetconsistsofanouterringgear,aplanetgearcarrierwithplanetgearsattached,andasungear.Theringgear,thatisrunningatthehalfspeedofthecrankshaft,isdrivenbythecrankshaftthroughtheenginetimingbelt.DetailsofthemodelingworkfortheEVVTcanbefoundin[82].Notethatforthetraditionalvalve-trainsystemtheringgearisconnectedtothecamshaftdirectly,resultinginaedvalvetiming,whilefortheEVVTsystemtheringgearandthecamshaftisconnectedthroughtheplanetarygearset.Theplanetarygearcarrierisdrivenbyanelectricmotorandfourplanetgearsengagingbothringandsungearsatthesametime,wherethesungearisconnectedtothecamshaft.ThespeedofthecamshaftisdeterminedbytheringgearspeedtogetherwiththeEVVTmotorspeed,whichprovidestheenginewithxiblevalveopeningtiming.Thecam-phaseanglecanbeexpressedastheintegrationofthespeeddifferencebetweentheEVVTmotorandringgear.Therefore,thecam-phasecanbeadjustedbycontrollingtheEVVTmotorspeedwithrespecttotheenginespeed.To125holdthecam-phaseataconstantvalue,theEVVTmotorspeedshouldmatchtheringgearspeed.Toadvancethecam-phase,theEVVTmotorspeedshouldbefasterthantheringgearspeed;andtoretardthecam-phase,theEVVTmotorspeedshouldbeslowerthantheringgearspeed.7.1.2Testbenchset-upTheEVVTexperimentaltestbenchisshowninFig.7.3.Thisbenchisusedforbothsystemidenti-testsandcontrolvalidation.Figure7.3showstheenginebody,theEVVTactuator,andthecam-phasepositionsensor.AFord5.4LV8engineheadisusedintheseexperiments.Thecylinderheadhasasinglecamshaftdrivingtwointakevalvesandoneexhaustvalve.Anelectricmotorisusedtosimulatethemotionoftheenginecrankshaft.Duetospeedlimitationofthiselectricmotor,themaximumenginespeedthatcanbeachievedonthetestbenchis1750rpm.Anencoderisin-stalledonthemotorshaftthatgeneratescrankanglesignalwith1=64crankdegreeresolutionandagatesignal(onepulseperrevolution).Bothencodersignalsareusedtoobtainenginepositionandspeed.Acampositionsensorwithfourpulsesperenginecycle,isinstalledonthebackofthecamshafttodetecttheengineTopDeadCenter(TDC)andtocalculatethecamphaseangle.Thus,thecam-phaseangleisupdatedfourtimeseveryenginecycle.AnelectricaloilpumpisusedtosupplypressurizedoilforlubricatingbothEVVTactuatorandcylinderheadassembly.Intakeandexhaustvalvesareinstalledtoprovidethecycliccamshafttorqueload.Opal-RTreal-timeengineprototypecontrollerisusedtocontroltheEVVTandcollecttheexperimentaldataforthetestbench.Thecam-phasepositionsensorsignalissampledbytheOpal-RTprototypecontrollerandthecorrespondingcam-phaseangleiscalculatedwithintheOpal-RTreal-timecontroller.TheexperimentsareperformedatMSUAutomotiveControlsLabataroomtemperatureof25.7.2SystemandLPVModelingThissectionpresentsdetailedworkofthesystemtests,constructionoftheLPVmodel,andstate-spacerepresentationoftheEVVTactuator.126Figure7.3:EngineExperimentsetup.Table7.1:FixedvaluesofenginespeedandbatteryvoltageusedinthesystemIDtests.Enginespeed=[800;1100;1400](rpm)Batteryvoltage=[9:5;11:5;14](volt)7.2.1SystemTestsofEVVTActuatorUsingthetestbenchdescribedabove,aseriesoftestsatedvaluesofenginespeedsandbatteryvoltagehavebeenconducted.Standardtests[84,85]arecarriedouttoinvestigatestepresponsedeviationsundervariousvaluesofenginespeedandvehiclebatteryvoltage.Thesetestsshowedthatwhenthevehiclebatteryvoltage,suppliedtotheEVVTmotor,dropsfrom14Vto9:5Vat800rpmenginespeed,thesettlingtimeofstepresponseisalmostdoubledduetothereducedtorqueavailablefortheEVVTmotortodrivetheringgear.However,whentheenginespeedisreducedfrom1400rpmdownto800rpmwithaedbatteryvoltageat9:5V,thesettlingtimeiscutbyhalfbecauseatlowmotorspeedtheavailablemotortorqueincreases.Thus,thesetestsshowedthattheresponsetimeoftheEVVTisstronglydependentonbothenginespeedandbatteryvoltage.Therefore,thesetwovaryingparametershavebeenselectedtobethetime-varyingparametersoftheLPVEVVTmodel.Table7.1showstheedvaluesofenginespeed(N)andbatteryvoltage(V)thathavebeenusedinthesystemtests.Therefore,toobtainafamilyoflocalLTImodels,closed-loopsystemidenttestshavebeenconductedattheedvaluesofenginespeedandbatteryvoltageshowninTable7.1andnineLTImodelsareobtained(seetheconferenceversionfortheauthors[86]).127Figure7.4:Bodeplotofthe9localLTImodels.7.2.2LPVmodelconstructionInthissubsectionthelocalLTImodelsareusedtoconstructasingleLPVmodelwithenginespeedandvehiclebatteryvoltageasthetime-varyingparameters.ItisfoundthattheLPV128Table7.2:Coefq1(t)&q2(t)BatteryvoltageEngineSpeedq1(t)q2(t)V(volt)N(rpm)9.58000.25296.97511000.33717.80914000.24416.23311.58000.38358.47011000.453310.71014000.452010.780148000.530410.26011000.539012.42014000.647214.540modeloftheEVVTactuatorisinthefollowingsecondorderform,G(N(t);V(t);s)=q1(N(t);V(t))s(s+q2(N(t);V(t)));(7.1)whereq1(N(t);V(t))andq2(N(t);V(t))aretime-varyingcoeffunctionsofenginespeedandbatteryvoltage.Fornotationalsimplicity,q1(t)andq2(t)willbeusedtorefertoq1(N(t);V(t))andq2(N(t);V(t))1,respectively.Table7.2showsthevaluesofq1andq2thathavebeenexperimentallyatthenineedoperatingconditions.Basedonthesevalues,theBodeplotcorrespondingtotheninelocalLTImodelsisshowninFig.7.4.Notethatq1(t)isassociatedwiththeDCgainofthetransferfunction(7.1),whileq2(t)isthelocationoftheopen-looppoleofthe2ndordersystem.Inotherwords,theDCgainandpolelocationofthetransferfunction(7.1)aretime-varyingcoefandfunctionsofenginespeedandbatteryvoltage.Therangeofthevaryingcoefq1(t)andq2(t)aregiveninTable7.3correspondingtotheentirerangeoftheschedulingparameters(enginespeedsandbatteryvoltage).q1andq2areplottedasfunctionofenginespeedandbatteryvoltageinFig.7.5.Fromthisitcanbeobservedthatthevehiclebatteryvoltagehassubstantialonbothcoef-q1andq2.Thehigherthebatteryvoltageis,thelargerbothq1andq2.Thisismainlyduetothefactthatmotortorqueincreasesasthebatteryvoltageincreases.Ontheotherhand,the1Sometimesthetimedependencywillbeignoredfornotations.129effectsoftheenginespeedtobothcoef(foragivenvoltage)aremixedduetothefollowingfacts.Athighvoltage(14volt),themotortorqueissaturatedandastheenginespeedincreasestheoutputpowerincreasesleadingtoincreasedvaluesofq1andq2.However,whenthebatteryvoltagedropsto(9:5volt),themotoroutputpowersaturatesathighenginespeedleadingtoslowerresponse(smallvalueofq2)andreducedDCgainq1.Tosumup,theenginespeedhaslessin-onq1andq2thanthevehiclebatteryvoltage.Itisworthmentioningthattablelook-upmethodhasbeenusedtointerpolatethevaluesofq1andq2inreal-timeimplementation.7.2.3State-SpaceRepresentationThestate-spacerepresentationofanyLPVsystemscanbegivenbyx(t)=A(q(t))x(t)+Bu(q(t))u(t)+Bw(q(t))w(t)z(t)=Cz(q(t))x(t)+Dzu(q(t))u(t)y(t)=Cyx(t)+Dyw(q(t))w(t);(7.2)wherex(t)2Rnisthestate,u(t)2Rnuisthecontrolinput,w(t)2Rnwisthedisturbanceinput,z(t)2Rnzisthecontrolledoutput,andy(t)2Rnyisthemeasuredoutput.Thesystemmatri-ceshavethefollowingcompatibledimensionsA(q(t))2Rnn,Bu(q(t))2Rnnu,Bw(q(t))2Rnnw,Cz(q(t))2Rnzn,Dzu(q(t))2Rnznu,Cy2Rnyn,andDyw(q(t))2Rnynw.q(t)isarealvectorcontainingthetime-varyingschedulingparametersasq(t)=q1(t);q2(t);;qq(t)0;whereqrepresentsthenumberofschedulingparameters.Therefore,theEVVTplantmodel(7.1)canberealizedinstate-spaceformas264x1(t)x2(t)375=264010q2(t)375264x1(t)x2(t)375+2640q1(t)375u(t);y(t)=10264x1(t)x2(t)375;130Figure7.5:Thevaryingparametersasfunctionofenginespeedandbatteryvoltage.(a)q1(N;V);(b)q2(N;V)131Table7.3:Rangeofthetime-varyingparameters.q1(t)2[0:25290:6472]q2(t)2[6:97514:540]wherex1(t)representsthecam-phaseangleandu(t)istheEVVTmotorspeedcommand.Inordertoaccommodatefordisturbanceinput,measurementnoise,andcontrolsignal,Bw(q)=2640:2000375;Dyw(q)=00:01;Cz(q)=2641000375;Dzu(q)=26400:01375:arethus,thestate-spacerealizationoftheoverallsystemcorrespondingtothematricesin(7.2)are266664A(q)Bu(q)Bw(q)Cz(q)Dzu(q)CyDyw(q)377775=26666666666640100:200q2(t)q1(t)00100000:011000:013777777777775:7.3RGSControllerDesignTheobjectiveofthecontrolproblemistoregulatethecamphaseangletoareferencephaseus-ingRGSdynamicoutput-feedbackcontrolforanytrajectoriesofenginespeedandbatteryvolt-age.More,thegoalistoguaranteetherobuststabilityandH¥performanceoftheclosed-loopsystemagainstuncertaintiesinschedulingparameters.FollowingthelinespresentedinChapter5,theRGSsynthesisapproachfortheEVVTmodelissummarizedbythefollowingpoints,1321.Settheboundsonschedulingparametersandmeasurementnoisesassociatedwitheachschedulingparameter,N2[800;1400]rpm,V2[9:5;14]volt,jd1(t)j610rpm,andjd2(t)j61:2volt.2.Settheboundsontheratesofchangeofthetime-varyingparameters,i.e.jNj6100,jVj612,jd1(t)j610d1(t),andjd2(t)j610d2(t).3.ConverttheLPVmodeloftheEVVTactuatorfromtheoriginal(afparameterspaceintomulti-simplexparameters[39,87]tothemulti-simplexdomainL.4.ModeltherateofchangeoftheschedulingparametersinaconvexsetW[45].5.SolvetheconditionsofTheorem5.2usingtheiterativealgorithmpresentedinSection5.4toobtaincontrollermatricesAc(Ÿq),Bc(Ÿq),andCc(Ÿq).Withe=0:05,theachievedupperboundontheH¥performanceisg¥=0:1312andtheobtainedcontrollermatricesattheverticesofthemulti-simplexdomainaregivenby,A(11)c=26428:99837:35612:15713:487375;A(12)c=26430:44640:08512:79515:079375;A(21)c=26431:04940:24912:82814:462375;A(22)c=26432:03242:50813:24615:817375;B(11)c=26416:2457:0632375;B(12)c=26416:9927:3396375B(21)c=26417:4357:459375;B(22)c=26417:8877:5975375;C(11)c=122:27384:5;C(12)c=19:08753:672;C(21)c=124:09394:94;C(22)c=19:33655:089;1336.Thecontrollercoefin(3.14)areobtainedbyapplyingtheinversetransformation(3.40)and(3.40).Thus,thesecoefhavebeenobtainedAc0=264122:52160:251:02658:845375;Ac1=2643:63665:31661:12231:7136375Ac2=2642:43154:98831:05472:9481375;Bc0=26468:55829:459375;Bc1=2642:08520:65373375;Bc2=2641:19920:41481375;Cc0=284:78888:21;Cc1=2:069311:863;Cc2=207:93670:68:7.4ExperimentalResultsThissectionpresentstheexperimentalimplementationoftheRGScontroller,obtainedinthepre-viouschapters,totheEVVTtestbench.Notethatthesametestbenchusedtoperformthesystemexperimentsarealsousedforcontrollervalidation.Euler'sforward(rectangular)methodthatisgivenin[88]hasbeenutilizedtodiscretizethecontrollerwithasamplingperiodof5msontheOpal-RTprototypecontroller.Forthepurposeofexperimentaldemonstration,differentoperationalconditionsareinvesti-gatedinthisexperimentwithvariousenginespeedandbatteryvoltageFigure7.6showsoneofthesetrajectoriesthatareusedintheexperimentalstudy.Itisworthmentioningthatthistra-jectoryiscorrespondingtothebestpossiblemeasurementsthatcanbeobtainedonthetestbench.ItisclearfromFigure7.6thatperfectmeasurementisverydiftoobtainexperimentally.Thissupportsthecoreideathatnoisyschedulingparameterisareasonableassumptioninpractice.Figure7.7illustratesthecam-phaseangletrackingtoareferencestepinputof40.Duringthisexperiment,enginespeedandbatteryvoltagearevaryingtocovertheentireparameterspaceand134Figure7.6:Engineexperimentaloperatingtrajectoryinparameterspace.toencompassvariousvehicleoperatingconditions.Itisclearfromthisthat,althoughsevervariationsareconsideredwithenginespeedandbatteryvoltage,fairlygoodtrackingresponseisachievedforthecam-phaseangle.Again,notethatthemeasuredbatteryvoltageisquitenoisyindicatingtheneedforgain-schedulingcontrollerstoberobustagainstthesemeasurementnoises.Thus,thecontrolledresponseisrobustagainstmeasurementnoiseassociatedwiththebatteryvoltagesignal.InFigure7.8,step-changefrom40to5andthenbackto40isappliedasareferenceinputforthecam-phaseangle.Thisdemonstratesverygoodtrackingperformanceofthesynthe-sizedcontrollersforastep-changereference.Similarly,sincethebatteryvoltagetrajectoryisnotperfect,thesynthesizedcontrollershowsrobustnesstomeasurementinaccuraciesinschedulingparameters.Furthermore,measurementnoisesareintentionallyinsertedintothemeasuredbatteryvoltageandenginespeedtoillustraterobustnessofthedesignedRGScontroller.boundsofthenoiseare135Figure7.7:Measuredenginespeedandbatteryvoltage,andcam-phasingangletrackingwithstepreferenceof40degree.givenby100rpmand1:2Volt,respectively.Intermsofthemeasurementnoiseboundsearlierin(3.4),thesenoiseboundscanbeexpressedasj¯d1j100rpm(forenginespeed)andj¯d2j1:2Volt(forbatteryvoltage).Extensiveexperimentaltestsareconductedforvariousmeasurementerrorboundsbutonlytestsfortheabovementionedboundsarepresentedheresincetheresultsofothertestsarequitesimilar.Figure7.9,Figure7.10andFigure7.11illustrateexperimentalresultsforthenoisyschedulingparameters(right-sideoftheAgain,thesedemonstraterobustnessofthesynthesizedcontroller.7.5SummaryExperimentalstudyofRobustGainScheduling(RGS)controllersynthesisfortheElectricVari-ableValveTiming(EVVT)actuatorisconsideredinthischapter.Sinceenginespeedandbatteryvoltagehavesubstantialimpactonengineperformance,fueleconomy,andemissions;thesetwo136Figure7.8:Measuredenginespeedandbatteryvoltage,andcam-phasingangletrackingwithstepchangereference.time-varyingparametersarechosenasschedulingparameterstomodeltheEVVTsysteminLPVframework.First,aseriesofsystemtestswereconductedtoobtainafamilyoflocalLinearTime-Invariant(LTI)models.Then,thesemodelsweregroupedtogetherthroughlinearinterpolationtoobtaintheLPVmodelfortheEVVTactuatorfortheentirerangeofop-erationalconditions.FollowingthetheoreticalapproachpresentedinChapter5,RGScontrollerwasdesignedtoregulatethecam-phasingangletoareferencetrajectory.Thedesignedcontrollerwasimplementedandvalidatedexperimentallyonthetestbenchutilizingthemeasured(noisy)schedulingsignals.Experimentalresultsrobustnessofthedevelopedcontrolleragainstmeasurementnoisesintheschedulingparameters.137Figure7.9:Speedandvoltagevariationswithperfectmeasurement(left)andnoisymeasurement(right)andcorrespondingcam-phaseresponses.Figure7.10:Speedandvoltagevariationswithperfectmeasurement(left)andnoisymeasurement(right)andcorrespondingcam-phaseresponses.138Figure7.11:Speedandvoltagevariationswithperfectmeasurement(left)andnoisymeasurement(right)andcorrespondingcam-phaseresponses.139CHAPTER8CONCLUSIONSANDFUTURERESEARCHInthisdissertation,aunifyingapproachforsynthesizingRobustGain-Scheduling(RGS)con-trollersofLinearParameter-Varying(LPV)systemssubjecttonoisyschedulingparameterswasdeveloped.Thischapterprovidesconcludingremarksandsuggestionsforfutureresearchdirec-tions.8.1ConclusionsSincestabilityandperformanceofdynamicalsystemsareofmostimportanceforasuccessfulcontroldesign,effortshavebeenmadebyresearcherstodevelopnewcontrolstrategiestoachievethesetwoobjectives.Inthisdissertation,ageneralapproachhasbeendevelopedtoimprovetheseobjectivesintheLPVframeworkwhenthetime-varyingparametersarepollutedbymeasurementnoise.Inthechapter,differentLPVcontrolmethodswerereviewedandtheoverallorganizationofthedissertationwasgiven.Multi-simplexmodelingapproachwasusedthroughoutthisdisser-tationtomodelthetime-varyingschedulingparametersandassociateduncertaintiesinaconvexdomain.Chapter2introducesthenecessarynotations,ofthemulti-simplexmodel-ingapproach,thehomogeneouspolynomials,andthetechnicalmachineryusedthroughoutthisdissertation.InChapter3,thecontrolproblemformulationoftheRGScontrollersynthesiswaspresented.Then,thegeneralsolutionapproachforthissynthesisproblemwasintroducedinthesamechap-ter.ThissolutionapproachincludestransformationoftheafLPVsystemsintomulti-simplexdomaint.Then,theratesofchangeoftheschedulingparametersandassociateduncertaintiesweremodeledinaconvexdomainaswell.Third,thecontrollersynthesisconditionswerederivedintermsofParametrizedLinear/BilinearMatrixInequalities(PLMIs/PBMIs).Finally,matrixco-140efcheckmethodwasutilizedtorelaxthePLMIs/PBMIsattheverticesofthemulti-simplexdomaintoobtaintheoptimalcontroller.SynthesisconditionsofRGSstate-feedbackcontrollerwithguaranteedH2andH¥perfor-manceswerepresentedinChapter4.TheseconditionswereformulatedintermsofPLMIswithscalarsearchasatuningparameter.Numericalexamplesandcomparisonswithotherexistingmethodsweregiveninthischapter.Thecomparisonresultsshowthatthedevelopedapproachachieveabetterperformanceinthepresenceofmeasurementnoises.Asaspecialcaseofthethesesynthesisconditions,robuststate-feedbackcontrollercanbesynthesizedbyconstrainingsynthesisvariablestobeparameterindependent.Chapter5introducessynthesisconditionsforRGSDynamicOutputFeedback(DOF)con-trollerwithguaranteedH2andH¥performances.TheseconditionswereformulatedintermsofPBMIswithscalarsearchforbothperformances.SincePBMIsarenon-tractable,anumericalal-gorithmwasdevelopedtosolvetheseconditions.Demonstratingexamplesandcomparisonswerepresentedtoillustratetheeffectivenessofthedevelopedapproach.SynthesisconditionsforStaticOutput-Feedback(SOF)robustgain-schedulingcontrollerweredevelopedinChapter6.Two-stagedesignmethodwasusedtosynthesizetheSOFRGScontroller.Inthestage,state-feedbackgain-schedulingcontrollerwassynthesized.Then,thiscontrollerwasusedasinputtothesecondstagetosynthesizetheSOFRGScontroller.Numericalalgorithmwasdevelopedtoreduceconservativenessiteratively.SimulationresultsoftheSOFRGScontrollerforElectricVariableValveTiming(EVVT)actuatorwaspresentedtodemonstratetheeffectivenessofthedesignedcontroller.ExperimentalvalidationofthedevelopedapproachwasdemonstratedinChapter7ontheEVVTbench.Enginespeedandbatteryvoltagewerechosenastime-varying(noisy)schedul-ingparameters.RGScontrollerhasbeendesignedandimplementedexperimentallyonthetestbenchoftheEVVTactuator.Experimentalresultsshowedverysatisfactorytrackingperformanceofthecam-phasingangle.1418.2RecommendationsforFutureResearchThissectionpresentsrecommendeddirectionsforfutureresearch.1.Inviewofthedevelopedsynthesisconditions,itwouldalsobeinterestingtostudythemixedH2=H¥synthesisproblem.SincethedevelopedconditionsutilizeslackvariableapproachandthecontrollerconstructionwasindependentonLyapunovvariables,thedevelopedsyn-thesisconditionslenditselfasanexcellentcandidatetoachievemulti-objectivecontrol.Inotherwords,eachperformanceobjectivehasitsownLyapunovmatrixallowingtoachievemultipleobjectiveswithoutintroducingadditionalconservativeness.2.IntheRGSstate-feedbackconditions,itwouldalsobeinterestingtoavoidlinesearchbyreplacingthescalar(e)withafullmatrixandsolvingtheresultingconditionsiterativelyasPBMIsproblem.Inthiscase,considerableperformanceimprovementcanbeexpectedontheexpenseofadditionalnumericalburden.3.Sincematrixcoefcheckrelaxationmethod[41]wasusedtosolvethedevelopedsyn-thesisconditions,itisinterestingtousedifferentrelaxationmethodsuchasmatrixSum-Of-Squares(SOS)[52]tosolvetheseconditionsandperformingcomparativestudyintermsoftheachievedperformanceandnumericalcomplexity.4.Aninterestingtopictostudyistoextendthedevelopedapproachtodesigngain-schedulingobserverssubjecttonoisyschedulingparameters.ByusingdualityoftheRGSstate-feedbacksynthesisconditionsitisnotdiftoderivePLMIconditionstosynthesizeobservergainthatguaranteesasymptoticconvergenceoftheestimationerrorwhileachievingrobustnessagainstschedulingparametersuncertainties.5.InthisdissertationH2andH¥performanceshavebeenconsidered.ItisveryencouragingtostudyotherperformancesuchasL2-to-L¥performanceintheRGSframework.WiththeL2-to-L¥performance,thecontrolobjectiveistominimizethecontrolenergywhile142satisfyingconstraintsontheperformanceoutput.Thisperformanceisverypow-erfulformanypracticalcontrolproblemsinautomotiveandaerospaceapplicationswhenhardconstraintsareimposedontheperformanceoutputs.6.ItwouldalsobeinterestingtoextendthedevelopedapproachtohandleRGScontrollersthatminimizetheoutputperformancewhilesatisfyingconstraintsontheavailablecontrolen-ergy.Thisisaveryinterestingprobleminpracticewhencertainconstraintsontheavailableactuatorsshouldbemetwhileachievingoptimaloutputperformance.143APPENDIX144APPENDIXFUNDAMENTALSOFLINEARMATRIXINEQUALITIESThisappendixprovidesanintroductiontotheapplicationofLinearMatrixInequalities(LMIs)incontrolsystems.BasicconceptsofLMIproblemswithessentialmanipulationtoolsareintroduced.LMIBasicsThecentralnotiontounderstandmatrixinequalitiesis.Inparticular,amatrixQistobepositiveifx0Qx>0;8x6=0(.1)Likewise,Qissaidtobepositiveifx0Qx0;8x(.2)ItiscommontowriteQ>0(Q0)toindicatingpositive(semi-)matrix.Inparticular,theinterestistopositivematricesthatarealsosymmetric,i.e.,Q=Q0.Asymmetric,positivematrixhastwokeyfeatures:itissquareandallofitseigenvaluesarepositivereal.Asymmetric,positivematrixsharestheattribute,butthelastisrelaxedtotherequirementthatallofitseigenvaluesarepositiverealorzero.AmatrixP=Qissaidtobenegative(semi-)ifQispositive(semi-)P<0(P0)isusedtoindicatenegative(semi-)ThemostgeneralformofanLMIisF(x)=F0+x1F1+x2F2++xmFm=F0+måi=1xiFi>0(.3)wherexiarereal,scalaroptimizationvariables,x=[x1x2xm]02RmandF0;Fi2Rnnaregivenconstantsymmetricmatrices.TheaboveLMIisfeasible,ifavectorxexistswhich(.3).NotethatF(x)>0describesanafrelationshipintermsofoptimizationvariablesx.145Inmostcontrolproblem,itismoreconvenienttobeformulatedasthefollowingLMI:F(X1;X2;;Xk)=F0+kåi=1GiXiHi>0(.4)whereXi2RpiqiarematrixvariablestobeobtainedandGi2RnpiandHi2Rqinareknownmatrices.Itiseasytoseethatthevectorvariablexin(.3)canbeformedbystackingthecolumnsofXiin(.4).MathematicalToolsforLMIManipulationAlthooughmanycontrolproblemcanbeformulatedasLMIproblems,asubstantialnumberoftheseneedtobemanipulatedbeforetheyareinasuitableLMIframework.Fortunately,thereareanumberofcommonmathematicaltoolsthatcanbeusedtotransformproblemsintosuitableLMIforms.Someoftheseusefultoolsaredescribedbelow.SchurComplementTheusefulnessoftheSchurcomplementistotransformquadraticmatrixinequalitiesintolinearmatrixinequalities,oratleastasastepinthisdirection.Schur'sformulastatesthatthefollowingtwostatmentsareequivalent264QSS0R375>0()8>>>>>><>>>>>>:Q>0R>0QSR1S0>0(.5)264QSS0R375>0()8>>>>>><>>>>>>:Q>0R>0RS0Q1S>0(.6)146Fornon-strictinequalities,theMoore-Penrosepseudoinverseofconstantmatrixwillbeused[89].CongruenceTransformationForagivenpositivematrixQ2Rnn,thefollowinginequalityholdsWQW0>0;(.7)fora(real)fullrankmatrixW2Rnn.Therefore,ofamatrixisinvariantunderpre-andpost-multiplicationbyafullrankrealmatrix,anditstranspose,respectively.TheprocessoftransformingQ>0into(.7)usingarealfullrankmatrixiscalledacongruencetransformation.Itisveryusefulforlinearizingnonlinearmatrixinequalitieswithasuitablechangeofvariables.Often,thecongruencetransformationmatrixWischosentobediagonal.TheS-ProcedureTheS-procedureisessentiallyamethodusedtocombineseveralquadraticinequalitiesintoasingleone(generallywithsomeconservatism).More,Itispreferredtoguaranteethatasinglequadraticfunctionofx2RnsuchthatF0(x)0;F0(x):=x0A0x+2b0x+c0whenevercertainotherquadraticfunctionsarepositiveFi(x)0;Fi(x):=x0Aix+2b0x+c0;i2f1;2;;qgToillustrate,consideri=1,ifthereexistascalarconstantt>0,suchthatFaug(x):=F0(x)+tF1(x)0;8x;s:t:F1(x)0thenF0(x)0.Inotherwords,Faug(x)0impliesthatF0(x)0iftF1(x)0becauseF0(x)Faug(x)ifF1(x)0.ExtendingtheideatoqinequalitiesconstraintF0(x)0;wheneverFi(x)0(.8)147holdifF0(x)+qåi=1tiFi(x)0;ti0:(.9)Thus,theS-procedureisamethodofverifying(.8)using(.9).ThisisquitusefulwhenF0(x)isnotaconvexfunction.Itisworthnotingthattheseconditionsaresuffori>1,itprovidessufandnecessaryconditionsonlywheni=1.Usually,ti'sareconsideredasadditionalLMIoptimizationvariables.ProjectionLemmaTheprojectionLemmaisusefulforeliminatingdecisionvariablesfromLMIs.Itisalsohasacon-vexifyingeffectoncertainnonlinearmatrixinequalities[90].Lemma.1.LetY=Y02RnnbeasymmetricmatrixandR2Rpn,Q2Rmnbegivenmatri-ces.Then,thefollowingstatementsareequivalent1.ThereexistsamatrixW2RpmsuchthatY+R0WQ+Q0W0R<0:(.10)2.TheLMIsR0?YR?<0Q0?YQ?<0hold,whereR?andP?arebasesofthenull-spaceofPandQ,respectively.3.Thereexistsscalarst1,t22RsuchthattheLMIsYt1R0R<0Yt2Q0Q<0hold.148CommonLMIcontrolProblemThissectionpresentscommoncontrolproblemsthatusesLMIformulations.LyapunovStabilityCriteriaConsiderthecontinuous-timelinearsystemx(t)=Ax(t)+Bu(t);y(t)=Cx(t)+Du(t);(.11)wherex(t)2Rnisthestatevector,u(t)2Rmthecontrol(input)vector,andy(t)2Rptheoutput(measured)vector,thesystemisasymptoticallystableifthefollowingfeasibilityproblem8><>:P>0;A0P+PA<0:(.12)StabilizationbyState-FeedbackAssumingthatthesystem(.11)isnotasymptoticallystable,state-feedbackcontrollerK,thatisu=w+Kx,maybesoughttoformthefollowingclosed-loopsystemx=(A+BK)x(t)+Bw(t);y(t)=Cx(t)+Dw(t);(.13)wherew(t)denotestheexternalinput.Inordertostabilizetheclosed-loopsystem,thestate-feedbackmatrixKrequiredtosatisfy(.12),8><>:P>0;(A+BK)0P+P(A+BF)<0(.14)or8><>:P>0;A0P+PA+K0B0P+PBK<0(.15)149Thesecondinequalityin(.15)isbilinear.Itcanbeconvertedbacktoalinearinequalitybysimplechangeofvariables.Pre-andpost-multiply(.15)byP1togetP1A0+AP1+P1K0B0+BKP1<0:(.16)LettingQ:=P1andL=KQ,QA0+AQ+L0B0+BL<0:(.17)Hence,inequality(.17)islinearwithrespecttothenewvariablesQandL.Oncethesevariablesareobtained,theoriginalvariablesPandKcanbeeasilyrecovered.H¥NormConsidertheproblemofobtainingtheH¥normforthefollowingsystemx(t)=Ax(t)+Bw(t);y(t)=Cx(t)+Dw(t):(.18)TheL2gain(g¥)isbyZ¥0y(t)0y(t)dtg2¥Z¥0w(t)0w(t)dt:(.19)Let'squadraticLyapunovfunctionV(x)=x(t)0Px(t),whereP>0,thenT:=V(t)+y(t)0y(t)g2¥w(t)0w(t)0:(.20)SubstitutingforV(t)andy(t),(.20)canbewrittenasT=x0PAx+x0A0Px+x0PBw+w0B0Px+(Cx+Dw)0(Cx+Dw)g2¥w(t)0w(t)0;(.21)thatcanalsobewritteninthefollowingform,T=x0w0264A0P+PA+C0CPB+C0DB0P+D0CD0Dg2¥I375264xw3750:(.22)150Asufconditionof(.20)is264A0P+PA+C0CPB+C0DB0P+D0CD0Dg2¥I375<0:(.23)Inequality(.23)canbewrittenindifferent(equivalent)ways,264A0P+PAPBB0Pg2¥I375+264C0D0375ICD<0;(.24)or266664A0P+PAPBC0B0Pg2¥ID0CDI377775<0:(.25)Pre-andpost-multiplythepreviousinequalityby2666641pg¥0001pg¥000pg¥377775toobtain266664A0‹P+‹PA‹PBC0B0‹Pg¥ID0CDg¥I377775<0;(.26)with‹P=P=g¥.Thereisanotherformthatisusuallyusedinsynthesisproblemthatcanbeobtainedbymultiplying(.26)fromleftandrightby266664Q000I000I377775withQ=P1,toobtain266664AQ+QA0BQC0B0g¥ID0CQDg¥I377775<0:(.27)151Thus,theboundedreallemma[91]couldbeanyoftheaboveinequalities(.25),(.26),and(.27).Inallofthepreviousinequalities,thegoalistoseekapositivematrix(PorQ)thatminimizestheH¥boundg¥.152BIBLIOGRAPHY153BIBLIOGRAPHY[1]M.Sato,Y.Ebihara,andD.Peaucelle,fiGain-ScheduledState-FeedbackControllersUsingInexactlyMeasuredSchedulingParameters:H2andH¥Problems,flinAmericanControlConference(ACC),pp.3094Œ3099,June2010.[2]J.Daafouz,J.Bernussou,andJ.Geromel,fiOnInexactLPVControlDesignofContinuousTimePolytopicSystems,flIEEETransactionsonAutomaticControl,vol.53,pp.1674Œ1678,Aug2008.[3]M.Sato,fiGain-ScheduledOutput-FeedbackControllersUsingInexactlyMeasuredSchedul-ingParametersforLinearParametricallyAfSystems,flSICEJournalofControl,Mea-surement,andSystemIntegration,vol.4,no.2,pp.145Œ152,2011.[4]D.J.LeithandW.E.Leithead,fiSurveyofGain-SchedulingAnalysisandDesign,flInterna-tionalJournalofControl,vol.73,no.11,pp.1001Œ1025,2000.[5]J.ShammaandM.Athans,fiAnalysisofGainScheduledControlforNonlinearPlants,flIEEETransactionsonAutomaticControl,vol.35,pp.898Œ907,Aug1990.[6]J.S.ShammaandM.Athans,fiGuaranteedPropertiesofGainScheduledControlforLinearParameter-VaryingPlants,flAutomatica,vol.27,no.3,pp.559Œ564,1991.[7]J.ShammaandM.Athans,fiGainScheduling:PotentialHazardsandPossibleRemedies,flIEEEControlSystemsMagazine,vol.12,pp.101Œ107,June1992.[8]H.K.Khalil,NonlinearSystems(3rdEdition).PrenticeHall,2001.[9]G.BeckerandA.Packard,fiRobustPerformanceofLinearParametricallyVaryingSystemsusingParametrically-DependentLinearFeedback,flSystems&ControlLetters,vol.23,no.3,pp.205Œ215,1994.[10]P.Apkarian,P.Gahinet,andG.Becker,fiSelf-ScheduledH¥ControlofLinearParameter-VaryingSystems:ADesignExample,flAutomatica,vol.31,no.9,pp.1251Œ1261,1995.[11]P.Gahinet,P.Apkarian,andM.Chilali,fiAfParameter-DependentLyapunovFunctionsandRealParametricUncertainty,flIEEETransactionsonAutomaticControl,vol.41,no.3,pp.436Œ442,1996.[12]F.Wu,X.H.Yang,A.Packard,andG.Becker,fiInducedL2-NormControlforLPVSystemswithBoundedParameterVariationRates,flInternationalJournalofRobustandNonlinearControl,,vol.6,pp.983Œ998,Nov.1996.[13]P.ApkarianandR.J.Adams,fiAdvancedGain-SchedulingTechniquesforUncertainSys-tems,flIEEETransactionsonControlSystemTechnology,vol.6,pp.21Œ32,1997.[14]G.I.Bara,J.Daafouz,F.Kratz,andJ.Ragot,fiParameter-DependentStateObserverDesignforAfLPVSystems,flInternationalJournalofControl,vol.74,no.16,pp.1601Œ1611,2001.154[15]V.Montagner,R.Oliveira,V.J.Leite,andP.L.D.Peres,fiLMIApproachforH¥LinearParameter-VaryingStateFeedbackControl,flIEEProceedings-ControlTheoryandApplica-tions,vol.152,no.2,pp.195Œ201,2005.[16]A.Packard,fiGainSchedulingviaLinearFractionalTransformations,flSystems&ControlLetters,vol.22,no.2,pp.79Œ92,1994.[17]C.A.DesoerandM.Vidyasagar,FeedbackSystems:Input-OutputProperties,vol.55.SIAM,1975.[18]P.ApkarianandP.Gahinet,fiAConvexCharacterizationofGain-ScheduledH¥Controllers,flAutomaticControl,IEEETransactionson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