SMOOTHSWITCHINGLPVCONTROLANDITSAPPLICATIONS By TianyiHe ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof MechanicalEngineeringDoctorofPhilosophy 2019 ABSTRACT SMOOTHSWITCHINGLPVCONTROLANDITSAPPLICATIONS By TianyiHe ThisdissertationstudiesthesmoothswitchingLPV(LinearParameter-Varying)systemandcontrol, aswellasitsapplicationsinmechanicalsystems,aerospacesystemstoachievethesmoothtransition betweenswitchingLPVcontrollers.Bothstate-feedbackanddynamicoutput-feedbackcasesare addressedbythesimultaneousdesignapproachofsmoothswitchingLPVcontrol,andtheproposed methodhasbeenappliedtoactivevibrationcontrolofBWB(Blended-Wing-Body)aircraft˛exible wingandtheAMB(ActiveMagneticBearing)system.Moreover,asequentialdesignapproachis developedtodesignsmoothswitchingLPVcontrollers,wherethehigh-dimensionaloptimization inthesimultaneousdesignapproachcanberelaxed. InconventionalswitchingLPVcontrol,switchingcontrollersaredesignedoneachsubregion whileguaranteeingsafeswitching,butwithoutconsideringthesmoothnessduringswitchingevents. Theabruptlyvaryingcontrolsignalcanexceedactuatorauthority;moreover,abruptchangesin systemresponsescausedbyunsmoothcontrollergainswillbeharmfultosystemcomponentsand hardware.ThesimultaneousdesignofsmoothswitchingLPVcontrolminimizesacombinedcost ofsystemoutput H 2 performanceandsmooth-switchingindexsubjectto H 2 constraintsoncontrol inputsand H 1 constraintonboundedmodeluncertainty.Thesestabilityandperformancecriteria arethenformulatedusingasetofParametricLinearMatrixInequalities(PLMIs).Besides,a tunableweightingcoe˚cientisintroducedtoprovideanoptimaltrade-o˙designbetweensystem H 2 performanceandswitchingsmoothness.SimulationresultswiththeAMBmodelandBWB aircraftwingmodelareprovidedtodemonstratethee˙ectivenessoftheproposedsmoothswitching control. Intheaboveapproach,switchingcontrollersaresynthesizedbycontrollervariablesthatsi- multaneouslysatisfyPLMIsonallsubregionsandswitchingstabilityconditionsonallswitching surfaces.Whenthenumberofsubregionsgoeslarge,simultaneousdesignapproachleadstoa high-dimensionaloptimizationproblem,withahighnumberofLMIconstraints,decisionvari- ables,onlinecomputationalload,andmemoryrequirement.Asaresult,thesedrawbacksmake simultaneousdesignpracticallyinfeasibleforhigh-ordersystemswithmanydividedsubregions. Aninnovativesequentialdesignapproachisproposedbyintroducinginterpolatedcontrollerdeci- sionvariablesandformulatingindependentPLMIconditionsoneachsubregionsuchthatsystem performancesonoverlappedsubregionsareguaranteedaswell.Inthisway,theswitchingcontroller synthesisconditionsareformulatedasindependentoptimizationproblemsandcanbewellsolved sequentially. Besides,thisdissertationalsoutilizestheLPVframeworktoinvestigateoptimalsensorplace- menttoachieveoptimalvibrationsuppressionfora˛exibleBWBairplanewing.Foragiven˛ight speedrange,vibrationbehaviorsofthewingstructureareevaluatedbytheguaranteed H 2 perfor- mancewiththe H 2 LPVcontroller.Candidatesensorlocationsareidenti˝edoneachwing,and theoptimalsensorplacementscanbefoundamongthesecandidatesensorlocationsbythegreedy algorithm.Thesearchedoptimalresultsarevalidatedbygloballysearchingthroughallpossible combinations.WiththeLPVmodelofa˛exiblewingand H 2 controllersynthesisconditions, searchresultsprovidetheoptimalsensorlocations,andbesides,thetrade-o˙betweenoptimal systemperformanceandthenumberofsensorscanalsobeobtained. Copyrightby TIANYIHE 2019 ACKNOWLEDGMENTS Manyyearslater,asIfacethisdissertation,everydetailwillprobablynotberecalled,butthat distanttimedevotedtoitwillberemembered. SincejoiningMichiganStateUniversityasaPh.D.studentin2015,I'vebeenfeelingso fortunatereceivingsomanyencouragements,supports,andcommunicationswithbeautifulminds fromthiscommunity.Therearesomanypeoplewhodeserverecognition,respect,andthanksfor thecontributionstothecompletionofthisdissertation,aswellastheirkindhelpthroughoutmy entirePh.D.study. Foremost,Iwouldliketoexpressmymostenormousgratitudetomyadvisor,ProfessorGuoming GeorgeZhu,whoismorethananexceptionalresearcherwithdistinguishedresearchtastebothin controltheoryandapplications,andalsoanuncommoneducatorwithenthusiasticpassionto communicatewithandhelpstudents,totrainstudentswithoutstandingengineeringskillsand researchethics.DuringmyPh.D.study,ProfessorZhuhasbeenopen-mindedforanynewideas, andalwaysbeenthere,o˙eringdiscussionsandsuggestions.Thankstohissupervisionand˝nancial support,I'vebeenabletogetthisresearchdoneandcompletemyPh.D.study. Besides,IwouldliketothankSeniorScientistDr.SeanSweiatNASAAmesResearchCenter, andProfessorWeihuaSuatUniversityofAlabama,fortheirguidanceandhelpontheBWB airplanevibrationcontrolproject,aswellastheirinvaluablecareeradvice.Theyhavebrought methepossibilityofconnectingcontroltheoryintoaerospaceengineeringprojects,whichhas providedmepreciseexperienceincontrolengineering. IwouldalsoliketothankProfessorHassanKhalil,ProfessorRanjanMukherjee,Professor BrianFeeny,forservingonmyguidancecommittee.Theyhavealwaysbeenreadytoo˙erkind helpandgivewiseadviceformyresearchwork.What'smore,I'vebene˝tedsomuchfromtheir lecturesandtalks,whichdrawmea'big'pictureofcontroltheoryandengineering,equipmewith mathematicaltools,anddirectmeintotherighttrackstocompletemyPh.D.degree. v ManythanksgotoallthepeersintheERClabatMichiganStateUniversity.Wehavehada goodtimetakinglectures,collaborating,anddiscussingresearchproblems,playing,andsporting. Thetimewegettogetherislimited,butthejoyisundoubtedlynot.Theirnamesare:Dr.Liang Liu,Dr.AliAl-jiboory,Dr.RuitaoSong,Dr.AqeelSalim,Dr.YifanMen,Dr.AliAlhajjar,Dr. YingxuWang,Ms.RuixueChristineLi,Mr.ShenQu,Mr.WenpengWei,Mr.AnujPal,Mr.Jian Tang,Dr.DonghaoHao,Dr.HuanLi,Dr.ChengshengMiao,Ms.DaweiHu,Mr.YuHeandMr. LingyunHua. Inparticular,Iwouldliketoexpressspecialgratitudetothespecialone,Ms.SijingLi,aswell asDeltaAirlineforconnectingusbetweenDTWandBOS.Lastbutnotleast,mydeepgratitudeis giventomyfamily.Withouttheirsupport,completionofthePh.D.degreecouldn'tbepossible. vi TABLEOFCONTENTS LISTOFTABLES ....................................... ix LISTOFFIGURES ....................................... x LISTOFALGORITHMS .................................... xiv CHAPTER1INTRODUCTION ............................... 1 1.1LPVSystemandControl...............................1 1.1.1Overview...................................1 1.1.2LPVsystemandperformance.........................4 1.1.3Performancespeci˝cationsandPLMIformulations.............4 1.1.3.1 H 1 performance..........................5 1.1.3.2 H 2 performance..........................6 1.2SwitchingLPVSystemandControl.........................7 1.2.1Overview...................................7 1.2.2SwitchingLPVsystemandperformance..................9 1.2.3Performancespeci˝cationsinswitchingLPVsystem............10 1.2.3.1 H 1 performance..........................10 1.2.3.2 H 2 performance..........................10 1.2.4Switchingstrategiesandstabilityconditions.................12 1.2.4.1Hysteresisswitching........................12 1.2.4.2Average-Dwell-Time(ADT)switching..............14 1.3MotivationsofSmoothSwitchingLPVControl...................16 1.4OrganizationofThisDissertation...........................19 CHAPTER2SMOOTHSWITCHINGLPVCONTROL .................. 21 2.1MixedICC = H 1 Control...............................21 2.1.1State-feedbackLPVcontrol.........................21 2.1.2Dynamicoutput-feedbackLPVcontrol....................26 2.1.2.1ICCcondition...........................27 2.1.2.2Synthesisconditions........................27 2.2SimultaneousDesignApproach............................32 2.2.1Continuous-timestate-feedbackcase.....................32 2.2.1.1Problemformulation........................32 2.2.1.2ControllersynthesisPLMIs....................34 2.2.2Continuous-timedynamicoutput-feedback(DOF)case...........37 2.2.2.1Problemformulation........................39 2.2.2.2ControllersynthesisPLMIs....................39 2.3SequentialDesignApproach.............................45 2.3.1Motivationsofsequentialdesign.......................45 2.3.2ControllersynthesisPLMIs..........................50 vii 2.3.2.1One-dimensionalschedulingparameter..............50 2.3.2.2Two-dimensionalschedulingparameters..............54 2.3.2.3Schedulingparameterofanydimensions.............58 2.4PLMIRelaxationMethod...............................61 2.4.1Modelingschedulingparameters.......................61 2.4.2PLMIsrelaxation...............................62 CHAPTER3APPLICATIONEXAMPLES ......................... 64 3.1ActiveMagneticBearing(AMB)Model.......................64 3.1.1Trade-o˙between trace ( W ) andICCconditions..............65 3.1.2SmoothswitchingLPVcontrolbysimultaneousdesign...........67 3.1.2.1Trade-o˙between trace ( W ) andswitchingsmoothness.....67 3.1.2.2Simulationresultsanddiscussions.................70 3.2Blended-Wing-Body(BWB)AirplaneFlexibleWing................73 3.2.1LPVmodelingofBWBairplane˛exiblewing................73 3.2.2MixedICC = H 1 LPVcontrol........................76 3.2.2.1Constraintsandperformancetrade-o˙...............76 3.2.2.2MixedICCand H 1 ControlProblem...............79 3.2.2.3Time-domainsimulationresults..................80 3.2.3SmoothswitchingLPVcontrolbysimultaneousdesign...........92 3.2.3.1Time-domainsimulationresults..................92 3.3NumericalExamplesforSequentialDesign.....................100 3.3.1Example1...................................100 3.3.2Example2...................................105 CHAPTER4OPTIMALSENSORPLACEMENT ..................... 109 4.1Introduction......................................109 4.2Problemformulationofsensorplacement......................111 4.3Simulationresultsbyglobalsearch..........................114 4.3.1Simulationresults...............................114 4.3.2Discussion...................................116 4.4Sub-modularpropertyofsensorplacementproblem.................119 4.4.1Setfunctionandsub-modularproperty....................121 4.4.2Greedyalgorithm...............................121 CHAPTER5CONCLUSIONSANDRECOMMENDEDWORK ............. 125 5.1Conclusions......................................125 5.2Recommendedwork..................................126 BIBLIOGRAPHY ........................................ 130 viii LISTOFTABLES Table3.1:Modedescriptioninreduced-ordermodel.....................74 Table3.2:Comparisonofthreedi˙erentdesignmethodsineachdesigniteration......101 Table3.3:Comparisonofthreedi˙erentdesignmethodsineachdesigniteration......108 Table4.1:Summaryofoptimalsensorcandidatecombinations...............118 ix LISTOFFIGURES Figure1.1:Classicalgain-schedulingcontrol.........................2 Figure1.2:LPVngain-schedulingcontrol..................3 Figure1.3:Closed-loopLPVsysteminLFTformwithuncertaintyblock..........5 Figure1.4:SwitchingLPVgain-schedulingcontrol......................9 Figure1.5:Three-subregionpartitionofschedulingparameterregion............9 Figure1.6:UnsmoothcontrolinputsignalfromconventionalswitchingLPVcontrol[1]..17 Figure1.7:IllustrationofmotivationsofsmoothswitchingLPVcontrol...........18 Figure1.8:Roadmapofthisdissertation...........................20 Figure2.1:Subregiondivisionillustrationofone-dimensionalschedulingparameter....48 Figure2.2:Subregiondivisionillustrationoftwo-dimensionalschedulingparameter....48 Figure3.1:Trajectoryofschedulingparameter,rotorspeed.................65 Figure3.2:Trade-o˙relationshipbetween U and trace ( W ) .................66 Figure3.3: x 1 responseunderdi˙erentICCconditions....................66 Figure3.4: x 2 responseunderdi˙erentICCconditions....................67 Figure3.5: u 1 responseunderdi˙erentICCconditions....................67 Figure3.6: u 2 responseunderdi˙erentICCconditions....................68 Figure3.7:Controlinputsunderin˝niteICCconditions...................68 Figure3.8:Trade-o˙relationshipbetweenswitchingsmoothnessand trace ( W ) under U 1 = 10 7 ......................................69 Figure3.9:Trade-o˙relationshipbetweenswitchingsmoothnessand trace ( W ) under U 2 = 10 8 ......................................69 Figure3.10:State x 1 responseby[1]andproposedmethod..................70 x Figure3.11:State x 2 responseby[1]andproposedmethod..................70 Figure3.12:Controlinput u 1 responseby[1]andproposedmethod..............71 Figure3.13:Controlinput u 2 responseby[1]andproposedmethod..............71 Figure3.14:SchematiclayoutofBWBairplanecon˝guration.................74 Figure3.15:Rootlociofopen-loopsystemwithvarying˛ightspeed.............75 Figure3.16:Three-subregionpartitionforschedulingparameter...............76 Figure3.17:Schedulingparameter(˛ightspeed)trajectory..................77 Figure3.18:Trade-o˙betweencontrollimit U andtrace(W)atdi˙erentrobustnessconditions78 Figure3.19:Bendingdisplacementatwingroot........................81 Figure3.20:Bendingdisplacementatwingtip.........................81 Figure3.21:Wingrootbendingunderdi˙erent U .......................82 Figure3.22:Wingtipbendingunderdi˙erent U ........................82 Figure3.23:Controlinput1underdi˙erent U .........................83 Figure3.24:Controlinput2underdi˙erent U .........................83 Figure3.25:Controlinput3underdi˙erent U .........................84 Figure3.26:Controlinput4underdi˙erent U .........................84 Figure3.27:Controlinput5underdi˙erent U .........................85 Figure3.28:Controlinput6underdi˙erent U .........................85 Figure3.29:Wingrootbendingunderdi˙erent 1 ......................87 Figure3.30:Wingtipbendingunderdi˙erent 1 .......................87 Figure3.31:Controlinput1underdi˙erent 1 ........................88 Figure3.32:Controlinput2underdi˙erent 1 ........................88 Figure3.33:Controlinput1underdi˙erent 1 ........................89 xi Figure3.34:Controlinput2underdi˙erent 1 ........................89 Figure3.35:Controlinput1underdi˙erent 1 ........................90 Figure3.36:Controlinput2underdi˙erent 1 ........................90 Figure3.37:Rootlociofclosed-loopsystem..........................91 Figure3.38:Closed-loopICCcostwithLPVandLTImodels.................91 Figure3.39:Schedulingparameterwithswitchingevents...................92 Figure3.40:Unforcedbendingdisplacementsatwingroot(upper)andwingtip(lower)...93 Figure3.41:Trade-o˙between trace ( W ) andsmoothnessindex I sm .............93 Figure3.42:Upper:comparisonatwingrootwithsmooth/un-smoothswitchingcontroller; Lower:comparisonatwingrootwiththreecontrolmethods...........94 Figure3.43:Upper:comparisonatwingtipwithsmooth/un-smoothswitchingcontroller; Lower:comparisonatwingtipwiththreecontrolmethods............95 Figure3.44:Upper:control1responsescomparisonwithsmooth/un-smoothswitching controller;Lower:control1responsescomparisonwiththreecontrolmethods.97 Figure3.45:Upper:control2responsescomparisonwithsmooth/un-smoothswitching controller;Lower:control2responsescomparisonwiththreecontrolmethods.97 Figure3.46:Upper:control3responsescomparisonwithsmooth/un-smoothswitching controller;Lower:control3responsescomparisonwiththreecontrolmethods.98 Figure3.47:Upper:control4responsescomparisonwithsmooth/un-smoothswitching controller;Lower:control4responsescomparisonwiththreecontrolmethods.98 Figure3.48:Upper:control5responsescomparisonwithsmooth/un-smoothswitching controller;Lower:control5responsescomparisonwiththreecontrolmethods.99 Figure3.49:Upper:control6responsescomparisonwithsmooth/un-smoothswitching controller;Lower:control6responsescomparisonwiththreecontrolmethods.99 Figure3.50: 1 obtainedbysequential(black),simultaneous(blue)andnon-switching(red) designapproach..................................101 Figure3.51: 2 obtainedbysequential(black),simultaneous(blue)andnon-switching(red) designapproach..................................102 xii Figure3.52: 3 obtainedbysequential(black),simultaneous(blue)andnon-switching(red) designapproach..................................102 Figure3.53:Closed-loopsystemstatesresponsesbysequential(black),simultaneous(blue) andnon-switching(red)designapproaches....................104 Figure3.54: 11 obtainedbysequential(black),simultaneous(blue)andnon-switching(red) designapproach..................................106 Figure3.55: 12 obtainedbysequential(black),simultaneous(blue)andnon-switching(red) designapproach..................................107 Figure3.56: 21 obtainedbysequential(black),simultaneous(blue)andnon-switching(red) designapproach..................................107 Figure3.57: 22 obtainedbysequential(black),simultaneous(blue)andnon-switching(red) designapproach..................................108 Figure4.1:Sensorlocationcandidates............................112 Figure4.2: trace ( W ) versusthenumberofavailablesensors, U =6 ............114 Figure4.3: trace ( W ) versusthenumberofavailablesensors, U =8 ............115 Figure4.4: trace ( W ) versusnumberofavailablesensors,at˛ightspeed115 m=s .....117 Figure4.5: trace ( W ) versusnumberofavailablesensors,at˛ightspeed125 m=s .....117 Figure4.6:Sensorcontributiontoeachvibrationmodeat˛ightspeed110 m=s .......119 Figure4.7:Sensorcontributiontoeachvibrationmodeat˛ightspeed113 m=s .......120 Figure4.8:Sensorcontributiontoeachvibrationmodeat˛ightspeed115 m=s .......120 Figure4.9:Submodularpropertyofsensorplacementon˛exiblewing...........123 Figure5.1:Acontrollerschemewithcompensatingoperator Q ...............127 xiii LISTOFALGORITHMS Algorithm1:Greedyalgorithmforsetfunctionoptimization.................122 Algorithm2:Greedyalgorithmforoptimalsensorplacement.................124 xiv CHAPTER1 INTRODUCTION 1.1LPVSystemandControl 1.1.1Overview LinearParameter-Varying(LPV)systemandcontrolhavegainedsigni˝cantinterestfromthecontrol communityoverthepasttwodecades[2,3,4,5,6,7,8].Themainbene˝toftheLPVsystem isthatthevaryingcharacteristicsofsystemdynamicscanbecapturedbytheLPVmodelwithits linearsystemmatricesdependentonschedulingparameter.LPVcontrollerscanbedesignedwith itsgainscheduledbasedontheschedulingparametersmeasuredinreal-time. TheLPVcontrolmethodisseenasa"modern"gain-schedulingcontrol,whichisoneofthe mostpopularande˙ectiveapproachestoaddressnonlinearsystems.Gain-schedulingcontrol hasbeenwidelyusedinawidevarietyofdynamicalsystemswithnonlinearandtime-varying dynamics.The classical gain-schedulingcontrolutilizestheideaof divideandconquer .The nonlinearsystemis˝rstlylinearizedatgriddedoperatingpointstoabundleoflinearmodels, usuallycalled parametric griddedmodel.Linearcontrollersarethendesignedbasedoneach localmodelbylinearcontroltheory,whichleadstoabundleofcorresponding˝xed-gainlinear controllers.Incontrollerimplementation,controllergainsarescheduledorswitchedaccording tooperatingpoints.AsshowninFigure1.1,onelinearcontrollerisactivewhenthesystemis operatingwithintheregionclosetoitslinearizationpoint. Theclassicalgain-schedulingtacklesthecomplicatednonlinearcontrolproblembysolvinga bundleofsimplersub-problems,however,ithasafewdrawbacksandlimitationsintheoryand applications. ‹ Classicalgain-schedulingcontroldesignslinearcontrolleratgriddedoperatingpoints,thus stability,performance,androbustnesscanonlybeguaranteedlocally,butnotgloballyinthe 1 Figure1.1:Classicalgain-schedulingcontrol entireoperatingregion.Theclosed-loopsystemperformancebylinearcontrollersdesigned atgriddedpointswilldegradewhenthecurrentoperatingpointdeviatesfromlinearization points. ‹ Classicalgain-schedulingcontrolisonlysuitablefortheslow-varyingsystem,becausethe switchingstabilitybetweenlocalcontrollerswillimposeconstraintsonswitchingsignals. ThisiswellstudiedinswitchingstabilityconditionsinswitchingLTIsystems[9,10]. ‹ Thetrade-o˙betweenthedensityofgriddingpointsandcomputationalcomplexityneedsto bewellconsidered.Ingeneral,moregriddingpointsareneededtomorepreciselydescribethe systemdynamics,whichinevitablyincreasescomputationalcomplexity.Moreover,gridded linearcontrollersaredesignedbeforehandandarerestoredinmemory,thentheyareread frommemoryincontrollerimplementation,whichmeansthatmorememoryisneededby moregriddedoperatingpoints. Toavoidthedrawbacksofclassicalgain-schedulingcontrolandretainthegain-scheduling strategy, modern LPVcontrolhasbeenproposedintheearly90s'byShamma[11]andextended bypioneeringresearchersBecker,Apkarian,GahinetandWu[12,2,3,13,14,15,16].The moderngain-schedulingcontrol-LPVcontrolcanbedescribedinFigure1.2.Inmoderngain- schedulingcontrol,controllersaredesignedwithschedulinggainsovertheentireoperatingregion. Theoperatingconditionsareconsideredasschedulingparametersandassumedtobeavailablein 2 Figure1.2:LPVngain-schedulingcontrol real-time.Thecontrollermatricesaredesignedasparameter-dependentandvarywithscheduling parameters. ThemainstreamapproachofLPVgain-schedulingcontroldesignistoformulatecontrolsyn- thesisconditionsintermsofLinearMatrixInequalities(LMIs)orParameterizedLinearMatrix Inequalities(PLMIs)[12,17,3,18,19,20,13].Numericallytractableoptimizationmethods,such asconvexoptimization,canthenbeappliedtosolveforfeasibleoroptimalLPVgain-scheduling controllers.LPVcontroldesignswithpoleplacement,guaranteed H 2 and/or H 1 performance havebeenintensivelystudiedintheliterature[21,22,23,24,20,13,15],aswellasthecaseof inexactschedulingparameters[25,26,27,28,29,30,31,32],andLPVsystemswithdelay[16]. Aslongasasolutiontotheformulatedoptimizationproblemisobtained,thenthederived parameter-dependentLPVcontrollermatriceswillachievetheguaranteedsystemperformance. Apparently,themodernLPVgain-schedulingcontrolisabletoguaranteestabilitygloballyoverthe schedulingparameterregion,toachieveguaranteedclosed-loopsystemperformance,andtoavoid repeatinglinearizationandlinearcontrollerdesign. Thischapterintroducesthenon-switchingLPVsystemandcontrol,followedbytheswitching LPVsystemandcontrol.Thesystemdescription,systemperformancespeci˝cations,andmultiple performancechannelswillbeincludedinthefollowingcontext. 3 1.1.2LPVsystemandperformance Considertheclosed-loopLPVsystemdescribedby cl ( ): 8 > > > > < > > > > : _ x ( t )= A cl ( ) x ( t )+ B 1 ( ) w 1 ( t )+ B 2 ( ) w 2 ( t ); z 1 ( t )= C cl; 1 ( ) x ( t )+ D 1 ( ) w 1 ( t ) z 2 ( t )= C cl; 2 ( ) x ( t ) (1.1) where ( t )= 1 ( t ) ; 2 ( t ) ;:::; q ( t ) T denotestheschedulingparametervectorof q elements, x ( t ) denotesthestate, w 1 ( t ) theexogenousinputs(forinstance,systemuncertaintyinput,sensornoises, etc.),and w 2 thedisturbanceinput; z 1 ( t ) the H 1 controlledoutput, z 2 ( t ) the H 2 performance output.Thesystemmatricesdependontheschedulingparametervector ,whichisassumedtobe measurableinreal-time.Themagnitudeandvariationalrate _ areboundedas 2 = i i ( t ) i ;i 2f 1 ; 2 ;:::;q g ; _ 2 = i _ i ( t ) i ;i 2f 1 ; 2 ;:::;q g : g (1.2) Therearetwoindependentperformancechannelsinthissystem, H 2 channelfrom w 2 to z 2 and H 1 channelfrom w 1 to z 1 .Inthenextsubsections,systemperformancesarespeci˝ed.Throughout thisdissertation,wemakeuseofthefollowingstandardde˝nitionof L 2 and L 1 normson x ( t ) 2 R n forall t 0 , k x k 2 2 := Z 1 0 x T ( t ) x ( t ) dt; k x k 2 1 :=sup t 0 x ( t ) T x ( t ) : 1.1.3Performancespeci˝cationsandPLMIformulations Itshouldbenotedthattherearetwoseparateinputandoutputpairsde˝nedin(1.1),andthey arespeci˝callydesignatedforassessingtheclosed-loopLPVsystemperformances,asshownin Figure1.3.Inthemixed H 1 = H 2 control,theLPVsystem ) achievesspeci˝c H 2 performance whilesubjectto H 1 performanceconstraints.Notethattheinterconnectionof inFigure1.3 istocapturethemodeluncertaintiesin ) ,andtherobustnessagainstmodelinguncertaintyis addressedby H 1 channel.Thede˝nitionsof H 1 , H 2 performancesaregivenbelow. 4 Figure1.3:Closed-loopLPVsysteminLFTformwithuncertaintyblock 1.1.3.1 H 1 performance The H 1 performance,de˝nedfrom w 1 ( t ) to z 1 ( t ) with L 2 inputand L 2 output,isutilizedto assesstheclosed-loopsystemrobustnessinthepresenceofmodeluncertainties.Mathematically, T 1 ( ;s ):= T z 1 w 1 ( ;s ) denotestheparameter-dependenttransferfunctionfrom w ( t ) to z 1 ( t ) and jj T 1 jj 1 the H 1 normof T 1 .Then,the H 1 performanceforthe( w 1 ( t ) , z 1 ( t ) )pairis de˝nedas L 2 gain[14,15],where jj T 1 jj 1 =sup 2 sup w 1 ;z 1 2L 2 ;w 1 6 =0 jj z 1 ( t ) jj 2 jj w 1 ( t ) jj 2 : (1.3) Physically, H 1 normisrelatedtotherobuststabilityofagivensystemwithmodelingerror.Based ontheSmallGainTheorem[33],theclosed-loopsystemsatisfyingthecondition jj T 1 jj 1 1 is well-posedandinternallystableforalluncertaintysatisfyingtheconstrain jj jj 1 < 1 1 ,where issystemuncertaindynamicsinterconnectedfrom z 1 to w 1 ,seeFigure1.3.Thefollowing Lemma1for H 1 performanceisgiven[12,2,15]. Lemma1. Supposethatthereexistsaparameterdependentpositive-de˝nitematrix P 1 ( ) ,such that (1.4) holdsforanyadmissible ( ; _ ) 2 ( j ) .Thentheclosed-loopsystem (1.1) is exponentiallystablewithguaranteedperformance jj z 1 jj 2 < jj w 1 jj 2 foragivenrobustnesslevel 5 > 0 andforalladmissibletrajectories ( ; _ ) 2 .( denotessymmetricterms.) 2 6 6 6 6 4 _ P 1 + A cl P 1 +( ) B 1 C cl; 1 P 1 ID 1 I 3 7 7 7 7 5 < 0 (1.4) 1.1.3.2 H 2 performance The H 2 performance,de˝nedfrom w 2 ( t ) to z 2 ( t ) ,isutilizedtoassesstheclosed-loopsystemoutput performance.Let T 2 ( ;s ):= T z 2 w 2 ( ;s ) betheparameter-dependenttransferfunctionfrom w 2 ( t ) to z 2 ( t ) ,andiftheclosed-loopsystemmatrix A cl isstable,thenthe H 2 normof T 2 ( ;s ) isde˝ned astheworst-case H 2 performanceonthesubregion [34,20], jj T 2 ( K ( ) ;s ) jj 2 2 =sup 2 1 2 ˇ R 1 trace T 2 ( ;j! ) T 2 ( ;j! ) d!; =sup 2 trace ( C cl; 2 ( ) P 2 ( ) C T cl; 2 ( )) : (1.5) where P 2 solvesthedi˙erentialRiccatiequation, _ P 2 = A cl P 2 + P 2 ( A cl ) T + B 2 ( B 2 ) T (1.6) withzeroinitialcondition. The H 2 normofasystemhastwointerestingphysicalinterpretationsbothstochasticallyand deterministically.Tobemorespeci˝c,stochastically, H 2 normofasystemdenotesthetraceofthe outputcovariancematrix,orinotherwords,thesummationofRMS-valueofthesystemoutputs toawhitenoiseinput;anddeterministically, H 2 normofasystemdenotesthesquaresummation of L 2 to L 1 gainsofindividualchannelsfromexogenousdisturbanceinputstosystemoutputs.In vibrationcontrol,system H 2 normcanbeusedasameasureofoutputmagnitude( L 1 norm)due toenergylimited( L 2 norm)disturbanceinputs. NotethatforLPVcontrolcase, jj T 2 ( ;s ) jj 2 dependsonvaryingschedulingparameter , leadingtoincreasedcomplexityduetounknownschedulingparametertrajectory.Toreduce complexityandkeepoptimizationasauni˝edapproachtoderivethe H 2 norm,theupperbound 6 trace ( W )=sup n trace ( C cl; 2 ( ) P 2 ( ) C T cl; 2 ( )) o forall satisfying(1.2)issoughtinstead. W isanintroducedauxiliaryvariable,whichisasymmetricmatrixwithcompatibledimensionswith outputs.Usingthisconstraint,theguaranteed H 2 performanceforalladmissiblescheduling parametercanbeformulated. ThefollowingLemma2isgiventoevaluatethe H 2 performanceforLPVsystem[20,34]. Lemma2. Forastable A cl ,ifthereexistaparameterdependentpositive-de˝nitematrix P 2 ( ) andaconstantmatrix W ,suchthat 2 6 4 _ P 2 + A cl P 2 +( ) B cl I 3 7 5 < 0 ; (1.7) and 2 6 4 WC cl; 2 P 2 P 2 3 7 5 > 0 ; (1.8) holdforall ( ; _ ) 2 ,thenthe H 2 normoftheclosed-loopsystemisboundedby trace ( W ) , i.e. trace ( C cl; 2 P 2 ( C cl; 2 ) T ) > > > < > > > > : _ x ( t )= A j cl ( ) x ( t )+ B j 1 ( ) w 1 ( t )+ B j 2 ( ) w 2 ( t ); z 1 ( t )= C j cl; 1 ( ) x ( t )+ D j 1 ( ) w 1 ( t ) z 2 ( t )= C j cl; 2 ( ) x ( t ) (1.10) 1.2.3Performancespeci˝cationsinswitchingLPVsystem 1.2.3.1 H 1 performance The H 1 performance,de˝nedfrom w 1 ( t ) to z 1 ( t ) with L 2 inputand L 2 output,isutilizedto assesstheclosed-loopsystemrobustnessinthepresenceofmodeluncertainties.Mathematically, let T 1 ( ;s ):= T z 1 w 1 ( ;s ) denotestheparameter-dependenttransferfunctionfrom w ( t ) to z 1 ( t ) and jj T 1 jj 1 the H 1 normof T 1 .Then,the H 1 performanceforthe( w 1 ( t ) , z 1 ( t ) )pair isde˝nedsimilartothatofnon-switchingLPVsystem L 2 gain[15],where jj T 1 jj 1 =sup 2 ( j ) ;j 2 N J sup w 1 ;z 1 2L 2 ;w 1 6 =0 jj z 1 ( t ) jj 2 jj w 1 ( t ) jj 2 : (1.11) ThefollowingLemma3canbeusedtoformulatethe H 1 performancefor j cl [1,32]. Lemma3. Supposethatthereexistsafamilyofparameter-dependentpositive-de˝nitematrices P j 1 ( ) suchthat (1.12) holdsforalladmissibletrajectories ( ; _ ) 2 ( j ) ,thentheclosed- loopsubsystem (1.10) isexponentiallystableonentiresubregionwithguaranteedperformance jj z 1 jj 2 < jj w 1 jj 2 foragivenrobustnesslevel > 0 .( denotessymmetricterms.) 2 6 6 6 6 4 _ P j 1 + A j cl P j 1 +( ) B j 1 C j cl; 1 P j 1 ID j 1 I 3 7 7 7 7 5 < 0 (1.12) 1.2.3.2 H 2 performance The H 2 performance,de˝nedfrom w 2 ( t ) to z 2 ( t ) ,isutilizedtoassesstheclosed-loopsystem outputperformance.Let T 2 ( ;s ):= T z 2 w 2 ( ;s ) betheparameter-dependenttransferfunction 10 from w 2 ( t ) to z 2 ( t ) ,andifeachofsubregionclosed-loopsystemmatrix A j cl isstable,thenthe H 2 normofisde˝nedastheworst-case H 2 performancesonallthesubregions ( j ) , jj T 2 jj 2 2 =sup 2 ( j ) ;j 2 N J trace ( C j cl; 2 ( ) P j 2 ( ) C j cl; 2 ( ) T ) : (1.13) where P j 2 solvesthedi˙erentialRiccatiequation, _ P j 2 = A j cl P j 2 + P j 2 ( A j cl ) T + B j 2 ( B j 2 ) T ; (1.14) withzeroinitialcondition. The H 2 normofastochasticsystemisthetraceofoutputstochasticcovariancematrix,or thesummationofRMS-valueoftheoutputstoawhitenoiseinput,whereasthe H 2 normofa deterministicsystemdenotesthesquaresummationof L 2 to L 1 gainsofindividualchannelfrom exogenousinputstosystemoutputs.Alternatively,the H 2 normcanbeinterpretedasdeterministic outputscovarianceintermsoftimecorrelation[51].Withthefollowinglemma,The H 2 normfor 2 ( j ) subregionisboundedbyitsupperbound trace ( W ) ,andcanbederivedbyminimizing trace ( W ) ,whichfallsintothetypicalmin-maxproblem. Lemma4. [20]Foranystable A j cl ,ifthereexistaparameterdependentpositive-de˝nitematrix P j 2 ( ) andaconstantmatrix W ,suchthat 2 6 4 _ P j 2 + A j cl P j 2 +( ) B j cl I 3 7 5 < 0 ; (1.15) and 2 6 4 WC j cl; 2 P j 2 P j 2 3 7 5 > 0 ; (1.16) holdforall ( ; _ ) 2 ( j ) ,thenthe H 2 normoftheclosed-looplocalsubsysteminthe j th subregionisboundedby trace ( W ) ,i.e., trace ( C j cl; 2 P j 2 ( C j cl; 2 ) T ) 0 thatsatis˝es P j ( ) A j cl ( )+( A j ) T cl ( ) P j ( )+ _ P j ( ) < j ( ) : (1.22) Onthetimeinterval t 2 [ t k ;t k +1 ) which j th controllerisactive,wehave V j ( x ( t ) ; ) e ( t t k ) V j ( x ( t k ) ; ) : (1.23) 13 Moreover,theswitchingstabilityconditiononswitchingsurface(1.21)willleadto V j ( x ( t k ) ; ) V j ( x ( t k ) ; ) .Therefore, V j ( x ( t ) ; ) e ( t t k ) V j ( x ( t k ) ; ) e ( t t k ) e ( t k t k 1 ) V j ( x ( t k 1 ) ; ) ::: e ( t t 0 ) V j ( x ( t 0 ) ; ) (1.24) sotheglobalexponentialstabilityisachieved. 1.2.4.2Average-Dwell-Time(ADT)switching TheAverage-Dwell-Time(ADT)switchingstrategyenforcesthe"slow-switching"propertyof switchingsignalssothattheclosed-loopsystemachievesglobalstabilityundertheswitching sequence.ByADTswitchingstrategy,onlyalimitednumberofswitchesareallowedwithina ˝nitetimeinterval[1,4,57]. Weassumethatswitchingsignal ˙ ( t ) renders N ˙ ( T;t ) numberofswitchingeventswithinthe timeinterval [ t;T ] .Ifthereexisttwopositivenumbers N 0 and ˝ a suchthat N ˙ ( T;t ) N 0 + T t ˝ a ; 8 T t 0 (1.25) where N 0 isthechatterboundtoavoidchatteringphenomenon.Thensu˚cientconditionforADT switchingisgiveninTheorem2andtheproofisgivenfollowingthereference[10,57]. Theorem2. Givenpositivescalar 0 and ,ifthereexistsafamilyofparameter-dependent Lyapunovmatrices P j ( ) satisfyingcondition (1.26) oneachsubregion ( ; _ ) 2 ( j ) and condition (1.27) onswitchingsurface,thentheexponentiallystabilityisachievedbyswitchingsignal withaveragedwelltime ˝ a > l 0 withintheentireschedulingparameterregion ( ; _ ) 2 . P j ( ) A j cl ( )+( A j ) T cl ( ) P j ( )+ _ P j ( )+ 0 P j ( ) < 0 (1.26) 1 P j ( ) P i ( ) j ( ) ; 2 S ( i;j ) ;i;j 2 N j ;i 6 = j (1.27) 14 Proof. Withoutlossofgenerality,weassumethatthesequenceofswitchingtimeis t 0 ;t 1 ;:::;t N . By(1.26),itiseasytoobtain _ V j ( x ( t ) ; ) < 0 V j ( x ( t ) ; ) < 0 ; (1.28) thuslocalexponentialstabilityoneachsubregionisachieved. ConsidertheLyapunovfunction W ( x ( t ) ; )= e 2 0 t V j ( x ( t ) ; ) when j th controllerisactive, thus _ W =2 0 W + e 2 0 t _ V: Thefunction W isobviouslypositiveandnon-increasingbetweenswitchingintervals.Thenatthe timeinterval [ t i ;t i +1 ) ,wearriveat W ( t i +1 )= e 2 0 t i +1 V j ( t i +1 ) ( x ( t i +1 ) ; ) 2 0 t i +1 V j ( t i ) ( x ( t i +1 ) ; ) = ( t i +1 ) ( t i ) (1.29) Sumupfrom t 0 toterminaltime T ,thenwehave W ( T ) W ( t N ) N ˙ ( t N ;t 0 ) W ( t 0 ) (1.30) Fromthede˝ned W ( t ) , e 2 0 T V j ( T ) ( x ( T ) ; ) N ˙ ( T;t 0 ) V j ( t 0 ) ( x ( t 0 ) ; ) (1.31) V j ( T ) ( x ( T ) ; ) e 2 0 T + N 0 + T ˝ a ln V j ( t 0 ) ( x ( t 0 ) ; ) = e N 0 ln e ln ˝ a 2 0 T V j ( t 0 ) ( x ( t 0 ) ; ) (1.32) Therefore,iftheswitchingsignalsatis˝esthelimitationofaverage-dwell-time ˝ a > ln 0 ,thenitis concludedthat V j ( T ) ( x ( T ) ; ) convergestozeroexponentiallyas T !1 ,whichindicatesglobal exponentialstabilityofswitchingLPVsystem. 15 1.3MotivationsofSmoothSwitchingLPVControl Asdiscussedintheaboveparagraphs,withgivenpartitionedsubregions,afamilyofLPVcon- trollersisdesignedbyconstructingParametricLinearMatrixInequalities(PLMIs)withmultiple parameter-dependentLyapunovfunctions.Controllersaredesignedbysolvingthecorresponding optimizationproblemassociatedwithswitchingstabilityconditionsandspeci˝cperformancecri- teria.ManyengineeringapplicationsofswitchingLPVcontrolhaveshownsystemperformance improvementovernon-switchingLPVcontrollers.Theseapplicationsincludeactivemagnetic bearing(AMB)system[1],F-16aircraftmodel[58],˛exibleball-screwdrives[59]andair-fuel ratiocontrolofsparkignitionengines[48]. However,inmostoftheseapplications,thedrawbackofunsmoothtransientresponsesover theswitchingsurfacescanbeobserved[1,48,32],andtheun-smoothnesscanbeattributedto sharpchangesincontrolinputsorcontrollergains.InFigure1.6,theconventionalswitchingLPV control[1]resultedintheabruptchangesofcontrolinput,markedbyredsquares.Thesespikesin controlcommandsignalsimposeheavy-dutytasksonactuators,whichwillbeharmfultohardware andsometimesexceedactuator'sauthority. Onlyafewsmoothswitchingtechniqueshavebeenproposedintheliteraturetocompensate forsharpjumps.Chen[60]consideredthehysteresisswitchingstate-feedbackLPVcontroland conductedlinearinterpolationofcontrollervariablesonswitchingsurfacestoachievesmooth switchingduringswitch-inandswitch-outontheoverlappingregion.However,thismethodcannot quantitativelyevaluateswitchingsmoothness,andonlyrelativestabilityisachieved.Hanifzadegan andNagamune[61]followedtheideaoflinearinterpolationofcontrollermatricesonswitching surfaces,andintroducedameasureofsmoothnessindexandimposedconstraintsoncontroller matrixderivativetocompensateforthedrawbacksfoundinChen[60].Thedesignofstabilizing controllerswasformulatedintoanon-convexoptimizationproblem,andaniterativedescentalgo- rithmwasthenappliedto˝ndalocalLPVcontrollerforeachsubregion.Thelinearinterpolationsof controllermatricesonswitchingsurfaceswereconductedtoobtainswitchingLPVcontrolleronthe overlappingregion.Thismethodreliesheavilyoniterativecomputationstosolvemulti-objective 16 Figure1.6:UnsmoothcontrolinputsignalfromconventionalswitchingLPVcontrol[1] non-convexproblems.Moreover,theintroducedsmoothnessindexlacksphysicalmeaning,andthe smoothnessconstraintsoncontrollermatricesareselectedthroughtrialanderror. Consideringthatexistingmethodscannote˚cientlyaddressthedesignofsmoothswitching LPVcontroller,itishighlyneededtodevelopane˚cientandsystematicapproachtoachievesmooth switchingbetweenadjacentLPVcontrollers.Intheauthors'pointofview,theleadingcauseof un-smoothcontrolinputsandsystemresponsesisduetothesuddenchangeofcontrolvariables duringswitchingevents.Theultimatereasonisthatun-smoothswitchingLPVcontroloptimizes closed-loopsystemperformanceovereachsubregion,neverthelessswitchingsmoothnessbetween adjacentcontrollersisnotconsidered.Thesystemperformanceoptimizationovereachsubregion, butignoringswitchingoftenleadstohigh-gaincontrollerswithjumpedcontrollergains.Thiscan beeasilyvalidatedbycheckingthecontrolgaindi˙erencebetweentwoneighboringsubregions overtheswitchingsurface. ThecoreideaofsmoothswitchingLPVcontrollercanbeillustratedbythecomparisonof Figures1.7aand1.7b.ThesmoothswitchingLPVcontrollerminimizesthegapbetweencontroller 17 (a)UnsmoothswitchingLPVcontrol (b)SmoothswitchingLPVcontrol Figure1.7:IllustrationofmotivationsofsmoothswitchingLPVcontrol gainsandachievessmoothswitching,whereastheconventionalLPVcontrolleronlyconsidersthe switchingstabilitybutnottheswitchingsmoothness.Twoapproaches,simultaneousdesignand sequentialdesign,areproposedinthisdissertationandthedesignstrategiesaresummarizedinthe followingparagraphs. Inthesimultaneousdesignapproach,aconvexoptimizationproblemisformulatedtodesign allswitchingcontrollersatthesametime.Anumericallytractablesmoothnessindexisintroduced intothecostfunctionbyusingthenormofdeviationofcontrollerparametersbetweenanytwo switchingsurfaces.Bymeansofminimizingthissmoothnessindex,itcanbedemonstratedthat sharpchangesincontrolstatesoroutputscanbesigni˝cantlyreduced,butatthecostofdegraded H 2 and H 1 systemperformance.Inotherwords,thereexistsatrade-o˙relationshipbetween systemperformanceandswitchingsmoothness.Intuitively,atunableweightingcoe˚cientcanbe adoptedtobalancethesystemperformanceandswitchingsmoothnessinthecostfunction.By tuningtheweightingcoe˚cient,i.e.,linesearch,anoptimaltrade-o˙canbeobtained,leadingtoa smooth-switchingLPVcontrollerwithacceptablesystemperformance. Controllersynthesisconditionsbythesimultaneousdesignapproacharenotindependenton adjacentsubregionsduetotheswitchingstabilitycondition.Whenthenumberofsubregions goeslarge,thesimultaneousdesignapproachleadstoahigh-dimensionaloptimizationproblem, withahighamountofLMI(LinearMatrixInequality)constraints,decisionvariables,online 18 computationalload,andmemoryrequirement[62,63].Asaresult,thesimultaneousdesignwould bepracticallyinfeasibleforhigh-ordersystemswithmanydividedsubregions. Toreducethecomputationalcomplexity,asequentialcontrollerdesignapproachisproposed. InterpolatedcontrollervariablesforoverlappedsubregionsandnewlyformulatedPLMIsareutilized tosynthesizeswitchingLPVcontrollersoneachsubregionindependently.Oneachoverlappedsub- region,theLyapunovmatrixisformulatedbyconvexlycombiningPDLMonadjacentsubregions. ThePLMIsfor H 1 performanceoneachsubregionisformulated,suchthattheconvexcombination ofadjacentPLMIsleadstoaguaranteed H 1 performanceoneveryoverlappedsubregion.More- over,theproposedmethodguaranteesthattheoverlappedsubregionhasintermediateperformance betweenitsneighboringsubregions.Theproposedmethoddesignsanindividualcontrollerforeach subregioninsequentialorder,insteadofsynthesizingallcontrollerssimultaneously.Byiteratively solvingthereduced-dimensionaloptimizationproblemforeachsubregion,switchingcontrollers withguaranteed H 1 performanceonallsubregionsandoverlappedsubregionscanbeobtained. 1.4OrganizationofThisDissertation Inthisdissertation,twoapproachesofsmoothswitchingLPVcontrollerdesignareproposed, includingsimultaneousdesignandsequentialdesign.AftertheintroductionofswitchingLPV systemandcontrolinChapter1,simultaneousdesignandsequentialdesignofsmoothswitching LPVcontrolarediscussed,andcontrollersynthesisconditionsaregiveninChapter2.InChapter3, afewapplicationexamplesaregiventodemonstratethee˙ectivenessofsmoothswitchingLPV controldesigns.SimultaneousdesignapproachisappliedtotheAMBmodelandBWBaircraft ˛exiblewing,thenanothertwonumericalexamplesareappliedwithsequentialdesignapproach. Also,theoptimalsensorplacementproblemusingtheLPVframeworkisinvestigatedinChapter4. Atlast,conclusionsandfutureworkarediscussedinChapter5. ThestructureofthisdissertationisshowninFigure1.8. 19 Figure1.8:Roadmapofthisdissertation 20 CHAPTER2 SMOOTHSWITCHINGLPVCONTROL AsdiscussedinChapter1,smoothswitchingLPVcontrolisneededtoremedyforunsmooth responsesinconventionalswitchingLPVcontrol.Thischapterwillgivethetheoreticalderivations andcontrollersynthesisconditionsforsmoothswitchingLPVcontrollers. BeforewegetintothesmoothswitchingLPVcontrol,themixedInputCovarianceConstraint (ICC)and H 1 LPVcontrolis˝rstlyintroduced.ThemixedICC = H 1 LPVcontrolisabletoachieve multi-objectiveperformanceofclosed-loopsystem.The H 2 performanceisoptimizedwhilethe closed-loopsystemsatisfying H 1 performanceandinputcovarianceconstraint.TheICC = H 1 controlisabletoavoidhigh-gaincontrollerbysettingupperlimitofcontrolinputcovariance. 2.1MixedICC = H 1 Control 2.1.1State-feedbackLPVcontrol Considerthefollowinga˚neLPVsystems, ): 8 > > > > < > > > > : _ x ( t )= A ( ( t )) x ( t )+ B 1 ( ( t )) w 1 ( t )+ B 2 ( ( t )) w 2 ( t )+ B u ( ( t )) u ( t ) z 1 ( t )= C 1 ( ( t )) x ( t )+ D 1 ( ( t )) w 1 ( t )+ E 1 ( ( t )) u ( t ) z 2 ( t )= C 2 ( ( t )) x ( t ) (2.1) where ( t )= 1 ( t ) ; 2 ( t ) ;:::; q ( t ) T denotestheschedulingparametervectorof q elements, x ( t ) 2 R n x denotesthestate, w 1 ( t ) 2 R n w 1 the H 1 disturbanceinputduetomodelingerror, w 2 ( t ) 2 R n w 2 the H 2 disturbanceinput, u ( t ) 2 R n u thecontrolinput, z 1 ( t ) 2 R n z 1 the H 1 controlledoutput,and z 2 ( t ) 2 R n z 2 the H 2 performanceoutput.Allsystemmatricesareassumed tohavecompatibledimensionsandina˚neparameter-dependentform.Forexample, A ( ) canbe describedby A ( ( t ))= A 0 + q X i =1 A i i ; (2.2) 21 where A 0 and A i , i =1 ; 2 ;:::;q ,areconstantmatrices.Itisassumedthattheschedulingparameters aremeasurableinreal-time,andtheirmagnitudeandvariationalratearebounded.Speci˝cally,the schedulingparametersetisformulatedas: 2 = i i ( t ) i ;i 2f 1 ; 2 ;:::;q g ; _ 2 = i _ i ( t ) i ;i 2f 1 ; 2 ;:::;q g : g (2.3) Assumeweareseekingforagain-schedulingstate-feedbackcontrollersoftheform u ( t )= K ( ( t )) x ( t ) ; (2.4) where K ( ) istheparameter-dependentcontrolgainmatrix.Notethat u ( t ) canbepartitionedas u ( t )=[ u 1 ( t ) ;u 2 ( t ) ;:::;u n u ( t )] T .Then,substituting(2.52)into(2.1)yieldstheclosed-loopLPV systemdescribedby cl ( ): 8 > > > > < > > > > : _ x ( t )= A cl ( ) x ( t )+ B 1 ( ) w 1 ( t )+ B 2 ( ) w 2 ( t ); z 1 ( t )= C cl; 1 ( ) x ( t )+ D 1 ( ) w 1 ( t ) z 2 ( t )= C 2 ( ) x ( t ) (2.5) where A cl ( )= A ( )+ B u ( ) K ( ) , C cl; 1 ( )= C 1 ( )+ E 1 ( ) K ( ) . Thecontrolinputisgivenas u ( t )= K ( ( t )) x ( t ) : Hence,thevarianceof k thcontrolinputof j thcontrollerisboundedas[64,65] cov ( u k ( ( t ))) sup 2 e k K ( ) P 2 ( ) K T ( ) e T k = U k ; (2.6) where e k isaselectionrowvectorsuchthat e k K ( ) equalstothe k throwofmatrix K ( ) ,and P 2 isgivenby(1.6).Thefollowinglemmaprovideshardconstraintonvarianceofthe k thcontrol inputforany 2 . Lemma5. TheICCconditionofthe k thcontrolinputofthestate-feedbackcontroller U k = e k K P 2 K T e T k 0 ;k =1 ; 2 ; ;n u ; (2.8) where n u isthenumberofcontrolinputs. ThefollowinglemmagivesthesynthesisconditionsformixedICC = H 1 LPVstate-feedback controller. Theorem3. Giventheinputcovarianceconstraints U k , k =1 ; 2 ; ;n u ,andapositivescalar 1 ,ifthereexistcontinuouslydi˙erentiableparameter-dependentmatrices 0

0 and 1 > 0 ,andmatrix W = W T 2 R n z 2 n z 2 thatminimizethefollowingcostfunction withagivenscalingmatrix Q> 0 , min trace ( QW ) (2.9) subjecttothefollowinginequalities( denotessymmetricterms), 2 6 6 6 6 4 11 12 2 ( G ( )+ G ( ) T ) B 2 ( ) T 0 n w n w I n w 3 7 7 7 7 5 < 0 ; (2.10) 2 6 4 WC 2 ( ) G ( ) G ( )+ G ( ) T P 2 ( ) 3 7 5 > 0 ; (2.11) 2 6 4 U k e k Z ( ) G ( )+ G ( ) T P 2 ( ) 3 7 5 > 0 ;k =1 ; 2 ; ;n u ; (2.12) 2 6 6 6 6 6 6 6 4 1 1 1 2 1 ( G ( )+ G ( ) T ) 1 3 1 1 3 I n z B 1 ( ) T 0 n w n x D 1 ( ) T 2 1 I n w 3 7 7 7 7 7 7 7 5 < 0 ; (2.13) 23 where 11 = A ( ) G ( )+ B u ( ) Z ( )+( A ( ) G ( )+ B u ( ) Z ( )) T @P 2 ( ) @ _ , 12 = P 2 ( ) G ( )+ 2 ( A ( ) G ( )+ B u ( ) Z ( )) T ,and e k isinputchannelselectionmatrixforcontrolinputof interest,and 1 1 = A ( ) G ( )+ B u ( ) Z ( )+( A ( ) G ( )+ B u ( ) Z ( )) T @P 1 ( ) @ _ , 1 2 = P 1 ( ) G ( )+ 1 ( A ( ) G ( )+ B u ( ) Z ( )) T ,and 1 3 = C 1 ( ) G ( )+ E 1 ( ) Z ( ) .Then, thegain-schedulingcontroller u ( t )= K ( ) x ( t ) ;K ( )= Z ( ) G 1 ( ) (2.14) exponentiallystabilizestheLPVsystem ) forany ( ; _ ) 2 withaguaranteed H 1 performancebound 1 .Inaddition,theICCcostisboundedby trace ( W ) >trace n C 2 ( ) P 2 ( ) C 2 ( ) T o ; (2.15) andtheconstraint (2.6) issatis˝ed. Proof. Forclosed-loopLPVsystem(2.5),assume A cl ( ) isstableforanypair ( ; _ ) 2 , thenthereisacontinuouslydi˙erentiableparameter-dependentpositive-de˝nitematrix P 2 ( )= P 2 ( ) T > 0 ,suchthat _ P 2 ( )+ A cl ( ) P 2 ( )+ P 2 ( ) A cl ( ) T + B 2 ( ) B 2 ( ) T =0 (2.16) where P 2 ( ) isthecontrollabilityGramianoftheLPVsystem.Inotherwords,thereisaparameter- dependentpositive-de˝nitematrix P 2 ( ) > P 2 ( ) satisfyingthefollowinginequality _ P 2 ( )+ A cl ( ) P 2 ( )+ P 2 ( ) A cl ( ) T + B 2 ( ) B 2 ( ) T < 0 : (2.17) Todecouple A cl ( ) and P 2 ( ) in(2.17),weutilize Finsler'sLemma [66]toobtainthefollowing, ( )+ X ( ) V ( )+ V T ( ) X T ( ) < 0 ; (2.18) where ( )= 2 6 6 6 6 4 _ P 2 ( ) P 2 ( )0 P 2 ( )00 00 I 3 7 7 7 7 5 ;X ( )= 2 6 6 6 6 4 G T ( )0 R T ( )0 0 I 3 7 7 7 7 5 ;V ( )= 2 6 4 A T cl ( ) I 0 B T 2 0 I 3 7 5 ; 24 and G ( ) and R ( ) areintroducedasslackvariables.Tomaintainconvexparametrizationproperty, R ( ) ischosentobe R ( )= 2 G ( ) ,where 2 > 0 isascalarthatisusedtoprovidean extradegree-of-freedomwhenperformingthelinesearchandtoreduceconservativeness.Letting Z ( )= K ( ) G ( ) yields(2.62). Now,consider(2.63).Pre-andpost-multiplying(2.63)by [ I;C 2 ] and [ I;C 2 ] T renders IC 2 2 6 4 WC 2 ( ) G ( ) G ( )+ G ( ) T P 2 ( ) 3 7 5 2 6 4 I C T 2 3 7 5 > 0 (2.19) fromwhichweobtain W>C 2 ( ) P 2 ( ) C 2 ( ) T ; (2.20) hence(2.20)leadsto(2.67).Since C 2 ( ) P 2 ( ) C 2 ( ) T >C 2 ( ) P 2 ( ) C 2 ( ) T ,asaresult,mini- mizing trace ( QW ) impliesminimizingtheupperboundoftheweightedICCcost. Similarly,pre-andpost-multiplying(2.64)by [ I;e k K ( )] and [ I;e k K ( )] T toobtain Ie k K ( ) 2 6 4 U k e k Z ( ) G ( )+ G ( ) T P 2 ( ) 3 7 5 2 6 4 I ( e k K ( )) T 3 7 5 > 0 ; (2.21) whichyields U k >e k K ( ) P ( ) K ( ) T e T k ;k =1 ; 2 ; ;n u : Thisimpliesthattheselectedcontrolinputcovarianceisupperboundedby U k . Now,for H 1 performanceinequality(2.65),weconsiderthefollowingtransformationmatrix T ( )= 2 6 6 6 6 4 IA cl ( )00 0 C cl; 1 ( ) I 0 000 I 3 7 7 7 7 5 : Pre-andpost-multiplying(2.65)by T ( ) and T ( ) T leadstothe H 1 performancecriterionbased uponthewell-known RealBoundedLemma [15]thatthe H 1 normoftheclosed-loopsystemis boundedby 1 .Thiscanbeeasilyveri˝edbyplugginginsearchvariablesandoperatingmatrix multiplication. 25 Remark1. Foreachgivenpairofsmallpositivescalarvariables 2 and 1 ,theminimizationleads toasub-optimalsolution.Fixingbothscalarvariableswouldleadtoconservativeness,however, thelinesearchofscalarvariablescanreduceconservativenesssigni˝cantly.Notethatconstraining P 1 = P 2 formulti-objectivecontroldesign,ascommonlyfoundintheliterature,couldleadto largeconservativeness.Theoptimizationprocesscanberepeatedforasetofgriddedscalarvalues tominimize trace ( QW ) .Thelinesearchprocessmayburdenthecomputationalload,butwith currentadvancedcomputationalcapacity,thisshouldnotbeanissue. 2.1.2Dynamicoutput-feedbackLPVcontrol SupposeaLPVsystemwithindependent H 2 and H 1 channels: ): 8 > > > > > > > < > > > > > > > : _ x p ( t )= A ( ( t )) x p ( t )+ B 1 ( ( t )) w ( t )+ B 2 ( ( t )) u ( t ) z 1 ( t )= C 1 ( ( t )) x p ( t )+ D 11 ( ( t )) w ( t )+ D 12 ( ( t )) u ( t ) z 2 ( t )= C 2 ( ( t )) x p ( t )+ E 2 ( ( t )) u ( t ) y ( t )= C y ( ( t )) x p ( t )+ D y ( ( t )) w ( t ) (2.22) Withoutlossofgenerality, D yu =0 .Systemmatricesrepresentina˚neformas: A ( )= A 0 + q X i =1 A i i (2.23) Eachparameterandtherateofvariationsareassumedtobeboundedasby i 2 i ; i ; _ i 2 v i ; v i .Theproposedgain-schedulingoutput-feedbackcontrollerisde˝nedas(2.24)and D K =0 sothattheclosed-loopsystemisstrictlyproperandhasmeaningful H 2 norm. 8 > < > : _ x K = A K ( ; _ ) x K + B K ( ; _ ) y u = C K ( ; _ ) x K (2.24) whichensuresinternalstabilityandaguaranteed H 1 performance jj T z 1 w jj 1 < fromdistur- bance w toperformanceoutput z 1 ,andminimizethe H 2 performance jj T z 2 w jj 2 ,whilecontrol covariance Cov ( u k ( t )) < U k ;k =1 ; 2 ; ;n u ,foralladmissibletrajectories ( ; _ ) andzero-state initialconditions. 26 Theresultedclosed-loopsystemis: cl ( ): 8 > > > > < > > > > : _ x cl ( t )= A cl ( ( t )) x cl ( t )+ B cl ( ( t )) w ( t ) z 1 ( t )= C cl; 1 ( ( t )) x cl ( t )+ D cl; 1 ( ( t )) w ( t ) z 2 ( t )= C cl; 2 ( ( t )) x cl ( t ) (2.25) x T cl =[ x T p ;x T K ] ,where isomittedinfollowingnotations: A cl = 2 6 4 AB 2 C K B K C y A K 3 7 5 ;B cl = 2 6 4 B 1 B K D y 3 7 5 (2.26) C cl; 1 = C 1 D 12 C K ;D cl; 1 = D 11 (2.27) C cl; 2 = C 2 E 2 C K ;D cl; 2 =0 (2.28) 2.1.2.1ICCcondition Thecontrolinputiscalculatedas u ( t )= C K x K = C u x cl = 0 C K 2 6 4 x p x K 3 7 5 : TheICCconditionofthe k th controlinput U k = k C u ~ P 2 C T u T k < k C u P 2 C T u T k < U k (2.29) isequivalenttoLMI[67] 2 6 4 U k k C u P 2 P 2 3 7 5 > 0 ;k =1 ; 2 ; ;n u : (2.30) 2.1.2.2Synthesisconditions Theorem4. ConsidertheLPVsystem (2.22) ,thereexistsagain-schedulingoutput-feedbackcon- troller (2.24) ,whichminimizeoutputperformancebound trace ( Q ) ,whileICCconstrained (2.29) controlinputenforcinginternalstabilityandguaranteed H 1 performanceofclosed-loopsystem,if 27 existparameter-dependentsymmetricmatrices R , S ,andaparameter-dependentstate-spacedata ( ^ A K ; ^ B K ; ^ C K ) suchthattheLMIsholdforalladmissible ( ; _ ) set. min ^ A K ; ^ B K ; ^ C K R;S trace ( Q ) (2.31) 2 6 6 6 6 6 6 6 4 AR + B 2 ^ C K +( ) _ R A T + ^ A K SA + ^ B K C y +( )+ _ S B T 1 ( SB 1 + ^ B K D y ) T I C 1 R + D 12 ^ C K C 1 D 11 I 3 7 7 7 7 7 7 7 5 < 0 (2.32) 2 6 4 RI IS 3 7 5 > 0 (2.33) 2 6 6 6 6 4 AR + B 2 ^ C K +( ) _ R A T + ^ A K SA + ^ B K C y +( )+ _ S B T 1 ( SB 1 + ^ B K D y ) T I 3 7 7 7 7 5 < 0 (2.34) 2 6 6 6 6 4 QC 1 R + D 12 ^ C K C 1 RI IS 3 7 7 7 7 5 > 0 (2.35) 2 6 6 6 6 4 U k k ^ C K 0 RI IS 3 7 7 7 7 5 > 0 ;k =1 ; 2 ; ;n u : (2.36) Iftheparameter-dependentmatricesarefoundtosatisfythePLMIconditions,thegain- schedulingoutput-feedbackcontrollercanbeobtainedbytwo-stepscheme: ‹ Solvefor N , M ,thefactorizationproblem I RS = NM T . 28 ‹ compute A K ;B K ;C K ;D K with A K = N 1 ( ^ A K S _ R N _ M T SAR ^ B K C y R SB 2 ^ C K ) M T B K = N 1 ^ B K C K = ^ C K M T (2.37) Proof. H 1 channel SupposeLyapunovmatrixfor H 1 channel,partition P = 1 1 2 = T 2 T 1 . 1 = 2 6 4 RI M T 0 3 7 5 ; 2 = 2 6 4 IS 0 N T 3 7 5 ;P = 2 6 4 RM M T U 3 7 5 ;P 1 = 2 6 4 SN N T V 3 7 5 (2.38) De˝nenonsingularcongruencematrix T 1 = diag 2 ;I;I ) ,whichmeansthatreversederivation isvalid.Pre-andpost-multiply T T 1 and T 1 onleftandrightsideof(1.4). 2 6 6 6 6 4 T 2 I I 3 7 7 7 7 5 2 6 6 6 6 4 A cl P + PA T cl _ PB cl PC T cl; 1 B T cl ID T cl; 1 C cl; 1 PD cl; 1 I 3 7 7 7 7 5 2 6 6 6 6 4 2 I I 3 7 7 7 7 5 < 0 (2.39) 2 6 6 6 6 4 T 2 A cl 1 +( ) T 2 _ P 2 B T cl 2 I C cl; 1 1 D cl; 1 I 3 7 7 7 7 5 < 0 (2.40) Bychangeofvariables 8 > > > > < > > > > : ^ A K = SAR + NB K C y R + SB 2 C K M T + NA K M T + S _ R + N _ M T ^ B K = NB K ^ C K = C K M T (2.41) Then(1.4)istransformedto 2 6 6 6 6 6 6 6 4 AR + B 2 ^ C K +( )+ _ R A T + ^ A K SA + ^ B K C y +( ) _ S B T 1 ( SB 1 + ^ B K D y ) T I CR + D 12 ^ C K CD 11 I 3 7 7 7 7 7 7 7 5 < 0 (2.42) 29 Ineqaulity(2.33)ensures P> 0 ,and I RS isnonsingular,leadingtouniquemappingofchange ofvariables. 2 6 4 RI IS 3 7 5 > 0 ) P = 1 1 2 > 0 (2.43) H 2 channel Inordertoconvexifythecontrollervariables,sameLyapunovmatrixisusedfor H 2 channel, partition P = 1 1 2 = T 2 T 1 . 1 = 2 6 4 RI M T 0 3 7 5 ; 2 = 2 6 4 IS 0 N T 3 7 5 ;P = 2 6 4 RM M T U 3 7 5 ;P 1 = 2 6 4 SN N T V 3 7 5 (2.44) De˝necongruencematrix T 2 = diag 2 ;I ) .Pre-andpost-multiply T T 2 and T 2 onleftand rightsideof(1.7). 2 6 4 T 2 I 3 7 5 2 6 4 _ P + A cl P + PA cl B cl I 3 7 5 2 6 4 2 I 3 7 5 < 0 (2.45) ) 2 6 6 6 6 4 AR + B ^ C K +( )+ _ R A T + ^ A K SA + ^ B K C y +( ) _ S B T 1 ( SB 1 + ^ B K D y ) T I 3 7 7 7 7 5 < 0 (2.46) De˝ne T 3 = diag ( I; 2 ) ,Pre-andpost-multiply T T 1 and T 1 onleftandrightsideof(1.8). 2 6 4 I T 2 3 7 5 2 6 4 QC cl 2 P P 3 7 5 2 6 4 I 2 3 7 5 > 0 (2.47) ) 2 6 6 6 6 4 QCR + E 2 ^ C K C RI IS 3 7 7 7 7 5 > 0 (2.48) ICCcondition Pre-andpost-multiply T T 3 and T 3 onleftandrightsideof(2.30). 30 2 6 4 I T 2 3 7 5 2 6 4 U k k C u P P 3 7 5 2 6 4 I 2 3 7 5 > 0 ; (2.49) ) 2 6 6 6 6 4 U k k ^ C K 0 RI IS 3 7 7 7 7 5 > 0 ;k =1 ; 2 ; ;n u : (2.50) Remark2. Toremovethe _ informationintroducedby _ R and _ M ,practicalvalidityapproach from[68]isapplied.Duetofactorizationproblemdoesn'tin˛uenceexistenceofcontrollerbut withintroducedconservativeness.Setoneofthemasconstantmatrix,thenderivativetermwill beeliminated.Forexample,set N ( )= R ( )= R 0 (constant), M T ( )=( I R 0 S ( )) .then controllermatrix ^ A K isnow A K = N 1 ( ^ A K SAR ^ B K C y R SB 2 ^ C K ) M T (2.51) R 0 = R 0 S ( )= S 0 + q X i =1 S i i Determinevariables ^ A K ; ^ B K ; ^ C K arechosenina˚neformasplantmatrix. ^ A K ( )= ^ A K 0 + q X i =1 ^ A Ki i ^ B K ( )= ^ B K 0 + q X i =1 ^ B Ki i ^ C K ( )= ^ C K 0 + q X i =1 ^ C Ki i 31 2.2SimultaneousDesignApproach 2.2.1Continuous-timestate-feedbackcase Theschedulingparameterregionisdividedinto J subregions,with J 1 overlappedregionbetween anytwoadjacentsubregions.A J numberofgain-schedulingstate-feedbackcontrollersdesigned on J subregionsforswitchingaregivenby u j ( t )= K j ( ( t )) x ( t ) ;j 2 N J = f 1 ; 2 ;:::;J g ; (2.52) where u j ( t ) ispartitionedas u j ( t )= h u j 1 ( t ) ;u j 2 ( t ) ;:::;u j n u ( t ) i T .Then,theclosed-loopLPV systeminvolvedwiththe j th controllerisnowwrittenas[69] 8 > > > > < > > > > : _ x ( t )= A j cl ( ( t )) x ( t )+ B 1 ( ( t )) w 1 ( t )+ B 2 ( ( t )) w 2 ( t ); z 1 ( t )= C j cl; 1 ( ( t )) x ( t )+ D 1 ( ( t )) w 1 ( t ) z 2 ( t )= C 2 ( ( t )) x ( t ) (2.53) where A cl ( ( t ))= A ( ( t ))+ B u ( ( t )) K j ( ( t )) , C cl; 1 ( ( t ))= C 1 ( ( t ))+ E 1 ( ( t )) K j ( ( t )) . Therearetwoseparatedinputandoutputpairsde˝nedin(2.53),andtheyarespeci˝cally designatedforassessingtheclosed-loopLPVsystemperformances,asdescribedbelow:(1) H 1 performanceisde˝nedfrom w 1 ( t ) to z 1 ( t ) with L 2 inputand L 2 outputusedtohandlemodel uncertainties;(2) H 2 performanceisde˝nedfrom w 2 ( t ) to z 2 ( t ) with L 2 inputand L 1 output(or L 2 - L 1 gains),forimprovingsystemperformance. ThecontrolobjectiveistodesignafamilyofsmoothswitchingICC = H 1 LPVcontrollers torobustlystabilizesystemin(2.1).Thiscontrolproblemcanbedividedintotwoparts:mixed ICC = H 1 controlforeachsubregionandsmoothswitchingwithhysteresisswitchingstrategy. 2.2.1.1Problemformulation ThemixedICC = H 1 controlproblemisto˝ndastate-feedbackgain-schedulingcontroller(2.52) oneachsubregionfortheLPVsystem(2.1)thatminimizestheupperboundof H 2 performance 32 cost: minsup K j ( ) jj T z 2 ;w 2 ( K j ( ) ;s ) jj 2 ;j 2 N J ; (2.54) suchthattheclosed-loopsystem(2.53)isexponentiallystable,andinaddition,thefollowing constraintsaresatis˝ed, jj T z 1 ;w 1 ( K j ( ) ;s ) jj 1 1 ; (2.55) Cov ( u k ( t )) U k ;k =1 ; 2 ;:::;n u ; (2.56) where 1 isthegiven H 1 -normboundonsystemrobustnesssubjecttomodeluncertainties,and U k isthegivenboundonthecontrolcovariance Cov ( u k ( t )) forthe k th controlinput u k ( t ) de˝ned below, Cov ( u k ( t ))= 1 2 ˇ Z 1 T u k ( K j ( ) ;j! ) T u k ( K j ( ) ;j! ) d! ; (2.57) where T u k ( K j ( ) ;s ):= T w 2 ! u k ( K j ( ) ;s ) denotesthetransferfunctionfrom w 2 ( t ) to u k ( t ) fortheclosed-loopLPVsystem(2.53).Iftheexogenousinput w 2 ( t ) isanunknowndisturbance thatbelongstoabounded L 2 set,thecovariance Cov ( u k ( t )) de˝nedin(2.56)becomesthetime correlationofcontrolsignal u k ( t ) .Then,themixedICCand H 1 controlproblemistominimizethe summationof L 2 to L 1 gainsfrom w 2 ( t ) toindividualoutputchannel z 2 ;k ( t ) for k =1 ; 2 ;:::;n z 2 subjecttothe L 2 to L 1 gainconstraintson u k ( t ) for k =1 ; 2 ;:::;n u andthe H 1 constraint.In otherwords,themixedICCand H 1 problemminimizestheweightedsumoftheworstcasepeak valuesofperformanceoutputsubjecttotheconstraintsontheworst-casepeakvaluesofcontrol inputsandthe H 1 constraint. TodesignafamilyofswitchingLPVcontrollers,hysteresisswitchingstrategyisutilizedto switchbetweenadjacentcontrollers,ensuringtheswitchingstabilityoveranytwoneighboring subregions. Forthe j th subregion,consideracontinuouslydi˙erentiableparameter-dependentmatrix P j ( )= P j ( ) T > 0 in H 1 channel,ormoreprecisely,theLyapunovmatrix f P j ( ) g j 2 N J . 33 ThentheLyapunovfunctioncanbeexpressedas, V j ( x; )= x T P j ( ) x (2.58) where x istheclosed-loopsystemstate.Ontheswitchingsurfaces S ( i;j ) ,theconditionbelow shouldbesatis˝ed, P i ( ) P j ( ) (2.59) indicatingthattheLyapunovfunctionoftheclosed-loopsystemisnon-increasingwhenswitching from ( i ) to ( j ) .TheconditionofLyapunovmatricesimpliesthat V i ( x; ) V j ( x; ) : (2.60) Then,switchingfromthe i th controllertothe j th controllerissafe[1]. Tosmoothenthepotentialsharpchangeincontrollergains,acostfunctiontobeminimizedis formulatedas F = trace ( W )+ X jj ( K i K j ) j 2 S ( i;j ) jj 2 2 ;i;j 2 N J ;i 6 = j: (2.61) where P jj ( K i K j ) j 2 S ( i;j ) jj 2 2 denotesthegaindi˙erencesonswitchingsurfaces 2 S ( i;j ) . The˝rstterm trace ( W ) inEqn.(2.61)isviewedasanindexofoutput H 2 performance,while secondtermisthemeasureofswitchingsmoothness. 0 isthetunablevariabletobalance thesetwoindexes,leadingtoatrade-o˙relationshipbetweenoutputperformanceandswitching smoothness. 2.2.1.2ControllersynthesisPLMIs ThissectionprovidesthesynthesisPLMIconditionsfortheproposedsmoothswitchingICC = H 1 controllers.Theupperboundofthe H 2 -norm,insteadofactual H 2 -norm,isminimizedinorder tomakeoptimizationnumericallytractable.Theorem5givesthePLMIsconditionsforcontroller synthesiswithguaranteed H 2 = H 1 performance.CombiningmixedICC = H 1 controllersynthesis conditionsandhysteresisswitchingconditions,Theorem6thenprovidesconditionsfordesigning switchingcontrollersdesign. 34 Theorem5. Giventheinputcovarianceconstraints U k ( k =1 ; 2 ; ;n u )andapositivescalar 1 , inthe j th subregionofschedulingparameter,ifthereexistcontinuouslydi˙erentiableparameter- dependentmatrices 0

0 and j 1 > 0 ,andsymmetricmatrix W j 2 R n z n z subjecttothefollowinginequalities( denotessymmetricterms), 2 6 6 6 6 4 11 12 j 2 ( G j ( )+ G j ( ) T ) B 2 ( ) T 0 I 3 7 7 7 7 5 < 0; (2.62) 2 6 4 W j C 2 ( ) G j ( ) G j ( )+ G j ( ) T P j 2 ( ) 3 7 5 > 0; (2.63) 2 6 4 U k e k Z j ( ) G j ( )+ G j ( ) T P j 2 ( ) 3 7 5 > 0 ;k =1 ; 2 ; ;n u (2.64) 2 6 6 6 6 6 6 6 4 1 1 1 2 j 1 ( G j ( )+ G j ( ) T ) 1 3 j 1 1 3 1 I B 1 ( ) T 0 D 1 ( ) T 1 I 3 7 7 7 7 7 7 7 5 < 0 ; (2.65) where 11 = A ( ) G j ( )+ B u ( ) Z j ( )+( A ( ) G j ( )+ B u ( ) Z j ( )) T @P j 2 ( ) @ _ , 12 = P j 2 ( ) G j ( )+ j 2 ( A ( ) G j ( )+ B u ( ) Z j ( )) T ,and e k isinputchannelselectionmatrixforcontrolinputof interest,and 1 1 = A ( ) G j ( )+ B u ( ) Z j ( )+( A ( ) G j ( )+ B u ( ) Z j ( )) T @P j 1 ( ) @ _ , 1 2 = P j 1 ( ) G j ( )+ j 1 ( A ( ) G j ( )+ B u ( ) Z j ( )) T ,and 1 3 = C 1 ( ) G j ( )+ E 1 ( ) Z j ( ) , thenthegain-schedulingcontroller u ( t )= K j ( ) x ( t ) ;K j ( )= Z j ( ) G j ( ) 1 (2.66) exponentiallystabilizestheLPVsystem ) forany ( ; _ ) 2 withaguaranteed H 1 35 performancebound 1 ,andinaddition,theICCcostisboundedby trace ( W ) >trace ( C 2 ( ) P ( ) C 2 ( ) T ) >trace ( C 2 ( ) P ( ) C 2 ( ) T )= J ICC (2.67) andtheconstraint(3.3)issatis˝ed. Proof. Theproofisomittedbecauseitissimilartothatof3. Theorem6. Foranytwoadjacentsubregions i and j ( i;j 2 N J ),ifPLMIsinTheorem5are satis˝edsimultaneouslyoverswitchingsurfaces S ( i;j ) andthefollowingPLMIsaresatis˝ed, P i 1 ( ) P j 1 ( ) ; 2 S ( i;j ) (2.68) thenswitchingmixedICC = H 1 controllerexponentiallystabilizesLPVsystem ) ,forany ( ; _ ) 2 withaguaranteed H 1 performancebound 1 ,guaranteedICCcostbound W j on j th subregion. Theproofisomittedbecauseitcanbeeasilyprovedbycombiningswitchingstabilityand Theorem5[25]. InTheorem5,controllerisformulatedas K j ( )= Z j ( ) G j ( ) 1 .Thus,parameterdependent matrices Z ( ) and G ( ) determinethecontrollergaindeviationonswitchingsurfaces.Tooptimiz- ingswitchingsmoothness,thesmoothnessindexisintroducedassumof Z ( ) and G ( ) deviations overallswitchingsurfaces,asshowninthefollowingformula. P ( jj Z i ( ) Z j ( ) jj 2 2 + jj G i ( ) G j ( ) jj 2 2 ) ; i;j 2 N J ; 2 S ( i;j ) : (2.69) Bythen,smoothswitchingLPVcontrolhasbeentransformedintoaconvexoptimization problemwithatunablecostfunction F = P tr ( W j )+ P ( jj Z i ( ) Z j ( ) jj 2 2 + jj G i ( ) G j ( ) jj 2 2 ) ; i;j 2 N J ; 2 S ( i;j ) : (2.70) whileinequalities(2.62),(2.63),(2.64),(2.65),and(2.68)aresatis˝edsimultaneously. 36 2.2.2Continuous-timedynamicoutput-feedback(DOF)case Considerthefollowinga˚neLPVsystem, _ x p ( t )= A ( ( t )) x p ( t )+ B 1 ( ( t )) w ( t )+ B 2 ( ( t )) u ( t ) z 1 ( t )= C 1 ( ( t )) x p ( t )+ D 1 ( ( t )) w ( t )+ D 2 ( ( t )) u ( t ) z 2 ( t )= C 2 ( ( t )) x p ( t ) y ( t )= C y ( ( t )) x p ( t )+ D y ( ( t )) w ( t ) (2.71) where ( t )= 1 ( t ) ; 2 ( t ) ;:::; q ( t ) T denotestheschedulingparametervectorof q elements, x p ( t ) denotesthestate, w ( t ) theexogenousinputs(forinstance,disturbanceinputs,sensornoises, etc.), u ( t ) thecontrolinput; z 1 ( t ) the H 1 controlledoutput, z 2 ( t ) the H 2 performanceoutput, and y ( t ) themeasurementoutput.Allsystemmatriceshavecompatibledimensionsandareinthe a˚neparameter-dependentform.Forexample, A ( ( t )) canbedescribedby A ( ( t ))= A 0 + q X i =1 A i i ( t ) : (2.72) Itisassumedthattheschedulingparametersaremeasurableinreal-time,andtheirmagnitudesand variationalratesareboundedas ( ; _ ) 2 : 2 = i i ( t ) i ;i 2f 1 ; 2 ;:::;q g ; _ 2 = i _ i ( t ) i ;i 2f 1 ; 2 ;:::;q g : g (2.73) Theschedulingparameterregionisdividedinto J subregions,withanoverlappingregionbetween anytwoadjacentsubregions.Again-schedulingDOFcontrolleristobedesignedforeachsubregion, andthecontrollersforadjacentsubregionsaretobeswitchedaccordingtohysteresisswitching logic.The j th subregionisdenotedby ( j ) ( j 2 N J = f 1 ; 2 ;:::;J g ),andswitchingsurfacefrom ( i ) to ( j ) isdenotedby S ( i;j ) . The j th DOFcontroller K j ( ) forthe j th subregionisgivenby K j ( ): 8 > < > : _ x K = A j K ( ) x K + B j K ( ) y u = C j K ( ) x K (2.74) 37 where x K denotesthecontrollerstateand ( A j K ;B j K ;C j K ) arecontrollervariablestobedetermined. Notethatthereisnodirectfeedthroughtermin u ,becauseastrictlyproperDOFcontrollerleadsto a˝nite H 2 normfortransferfunctions T z 2 w andinputcovariance.Thestatevectorfortheclosed- loopLPVsystemassociatedwiththe j th controllerbecomes x T cl =[ x T p ;x T K ] ,withthefollowing statespacerealization 2 6 6 6 6 4 A j cl B j cl C j cl; 1 D j cl; 1 C j cl; 2 0 3 7 7 7 7 5 = 2 6 6 6 6 6 6 6 4 AB 2 C j K B 1 B j K C y A j K B j K D y C 1 D 2 C j K D 1 C 2 E 2 C j K 0 3 7 7 7 7 7 7 7 5 : (2.75) Forsimplicity,thedependencyonschedulingparameter willbeomittedunlessnecessaryinthe restofthesis. Theproposedcontrolinput u ( t ) associatedwiththe j th controllercanbeequivalentlyrewritten as u ( t )= C j u x cl = 0 C j K 2 6 4 x p x K 3 7 5 : Hence,thevarianceofthe k th ( k 2 N n u = f 1 ; 2 ; ;n u g ) controlinputofthe j th controlleris constrainedas cov ( u k ( t )) sup 2 ( j ) ;j 2 N J e k C j u P j 2 ( C j u ) T e T k < U k ; where e k isaselectionrowvectorwith1atthe k th entryand0elsewhere,suchthat e k C j u equals tothe k th control.Wehavethefollowinglemmaprovidinghardconstraintonvarianceofcontrol inputforanyschedulingparametertrajectory 2 ( j ) . Lemma6. [67]TheICCconditiononthe k th controlinputofthe j th controller, sup 2 ( j ) e k C j u P j 2 ( C j u ) T e T k < sup 2 ( j ) e k C j u P j 2 ( C j u ) T e T k < U k ; (2.76) 38 isequivalenttothefollowingPLMItobeheldforany 2 ( j ) , 2 6 4 U k e k C j u P j 2 P j 2 3 7 5 > 0 ;k 2 N n u : (2.77) 2.2.2.1Problemformulation Foragivendividedsubregionset,thesmooth-switchingICC = H 1 DOFLPVcontrolproblemis to˝ndafamilyofgain-schedulingDOFcontrollers K j ( ) ;j 2 N J ,de˝nedin(2.74),overall subregionsfortheLPVsystem(2.71)thatminimizesthefollowingcostfunction, min A j K ( ) ;B j K ( ) ;C j K ( ) trace ( W )+ I sm ; (2.78) subjecttothefollowingconstraints jj T 1 jj 1 <; (2.79) cov ( u k ( t )) < U k ;k 2 N n u ; (2.80) where I sm denotesthesmoothnessindextobede˝nedinthenextsection, trace ( W ) istheupper boundofthesystem H 2 normoverallsubregions,and > 0 isatunableweightingcoe˚cienttobe usedtotrade-o˙betweenswitchingsmoothnessandsystemperformance.Inordertoensurethatthe controldesignproblemisconvex, I sm ischosenasthedeviationnormofcontrollerparametersover allswitchingsurfacesthatisaconvexfunctiondescribingthesmoothnessofcontrollervariables overswitchingsurfaces. 2.2.2.2ControllersynthesisPLMIs Thefollowingtheoremcontainsthemainresult.Notethatcostfunction(2.81)isalinearcom- binationoftwoconvexfunctionsofoutputperformance trace ( W ) andsmoothnessindex I sm associatedwithcontrollerparameters.Thetunableparameter > 0 isusedtobalancetheoutput performanceandsmoothnessofcontrollerparametersoverswitchingsurfaces.Alinesearchfor isneededinorderto˝ndtheoptimaltrade-o˙relationship. 39 Theorem7. ConsiderLPVsystem (2.71) .Thereexistsafamilyofgain-schedulingDOFcontrollers (2.74) thatminimizesthe -balancedcostfunction min ^ A j K ; ^ B j K ; ^ C j K X j ;Y j trace ( W )+ I sm (2.81) subjecttotheICCcondition (2.80) and H 1 constraint (2.79) ,ifthereexistsafamilyofparameter- dependentsymmetricmatrices X j and Y j ,andafamilyofparameter-dependentcontrollervari- ables ^ A j K , ^ B j K ,and ^ C j K ( j 2 N J ),suchthatPLMIs (2.83) - (2.87) holdwithagivenrobustness level > 0 foralladmissible ( ; _ ) 2 ( j ) ,andoneofthetwoconditionsin (2.88) holdson theswitchingsurfaces S ( i;j ) for > 0 with I sm givenby I sm = X i;j;i 6 = j 0 B B B B @ jj ^ A i K ^ A j K jj 2 + jj ^ B i K ^ B j K jj 2 + jj ^ C i K ^ C j K jj 2 + jj Y i Y j jj 2 + jj X i X j jj 2 1 C C C C A j 2 S ( i;j ) : (2.82) 2 6 6 6 6 6 6 6 4 M 11 A T + ^ A j K M 22 B T 1 M 32 I C 1 X j + D 2 ^ C j K C 1 D 1 I 3 7 7 7 7 7 7 7 5 < 0 (2.83) where M 11 = AX j + B 2 ^ C j K +( ) _ X j ; M 22 = Y j A + ^ B j K C y +( )+ _ Y j ; M 32 =( Y j B 1 + ^ B j K D y ) T : 2 6 4 X j I IY j 3 7 5 > 0 ; (2.84) 2 6 6 6 6 4 M 11 A T + ^ A j K M 22 B T 1 M 32 I 3 7 7 7 7 5 < 0 ; (2.85) 40 2 6 6 6 6 4 WC 1 X j + D 2 ^ C j K C 1 X j I IY j 3 7 7 7 7 5 > 0 (2.86) 2 6 6 6 6 4 U k e k ^ C j K 0 X j I IY j 3 7 7 7 7 5 > 0 ;k 2 N n u : (2.87) 8 > < > : Y i Y j X i ( Y i ) 1 X j ( Y j ) 1 or 8 > < > : X i X j Y i ( X i ) 1 Y j ( X j ) 1 (2.88) Proof. Toconvexifycontrolstrategywith H 2 and H 1 channels,let P j = P j 2 = P j 1 forthe j th subregion.SupposethattheLyapunovmatrix P j canbepartitionedas P j = 2 6 4 Y j N j ( N j ) T ? 3 7 5 ; ( P j ) 1 = 2 6 4 X j M j ( M j ) T ? 3 7 5 ; (2.89) where ? denotestheelementswhicharenotused. Furthermore,de˝nethecongruencematricesas j 1 = 2 6 4 X j I ( M j ) T 0 3 7 5 ; j 2 = 2 6 4 IY j 0( N j ) T 3 7 5 ; suchthat P j j 1 = j 2 .Forthe H 1 performancechannel,thePLMIs(2.83)canbeeasily obtainedbyfollowingtheproceduresin[68,12,13,32].De˝nenonsingularcongruencematrix T j 1 = diag j 2 ;I;I ) ,whichmeansthatreversederivationisvalid.Pre-andpost-multiply ( T j 1 ) T and T j 1 onleftandrightsideofthe H 1 performancecondition(1.12)foreachsubregion. 41 2 6 6 6 6 4 j 2 I I 3 7 7 7 7 5 T 2 6 6 6 6 4 A j cl P j + P j ( A j cl ) T _ P j B j cl I C j cl; 1 PD j cl; 1 I 3 7 7 7 7 5 2 6 6 6 6 4 j 2 I I 3 7 7 7 7 5 < 0 (2.90) By P j = j 1 j 2 ) 1 ,theaboveLMIcanbeconvertedto 2 6 6 6 6 4 j 2 ) T A j cl j 1 +( ) j 2 ) T _ P j j 2 ( B j cl ) T j 2 I C j cl; 1 j 1 D j cl; 1 I 3 7 7 7 7 5 < 0 (2.91) Introducethechangeofcontrollervariablesas 8 > > > > < > > > > : ^ A j K = N j A j K ( M j ) T + N j B j K C y X j + Y j B 2 C j K ( M j ) T + Y j AX j ^ B j K = N j B j K ^ C j K = C j K ( M j ) T ; (2.92) thenthePLMIcondition(2.83)canbeobtained. Forthe H 2 performancechannel,de˝nethecongruencematrix T j 2 = diag j 2 ;I ) .Pre-and post-multiply(1.15)by T T 2 and T 2 toobtain, ( T j 2 ) T 2 6 4 _ P j + A j cl P j +( ) B j cl I 3 7 5 T j 2 < 0 ; (2.93) whichyields(2.85)bymeansofchangeofvariablesin(2.92).De˝ne T j 3 = diag ( I; j 2 ) ,andpre- andpost-multiply(1.16)by ( T j 3 ) T and T j 3 ,weobtain 2 6 4 I j 2 3 7 5 T 2 6 4 WC j cl; 2 P j P j 3 7 5 2 6 4 I j 2 3 7 5 > 0 ; (2.94) whichyields(2.86)bymeansofchangeofvariablesin(2.92).FortheICCcondition,pre-and post-multiplying T T 3 and T 3 to(2.77)yields 42 2 6 4 I T 2 3 7 5 2 6 4 U k e k C j u P j 2 P j 2 3 7 5 2 6 4 I 2 3 7 5 > 0 ; (2.95) whichgives(2.87). Remark3. ThePLMIsformulatedinTheorem7rendersanoptimizationproblemofin˝nite dimensionsandun-de˝neddecisionvariablestructures.Tonumericallytacklethisoptimiza- tionproblem,a˚nedecisionvariablestructureisassumed,forexample, ^ A j K ( ) isexpressedas ^ A j K ( )= ^ A j K 0 + P q i =1 ^ A j Ki i : Coe˚cientcheckinmulti-simplexdomain[19,70,71]hasbeen adoptedtosuccessfullyobtaina˝nitesetofLMIsbutwithintroducedconservativeness.Other options[72,73]canalsobeapplied,forinstance,sum-of-squarerelaxation[74]andenforcing multi-convexitymethod[75]. Remark4. Ifthecontrollervariablesareobtainedbyminimizingthe -balancedcostfunction subjecttoformulatedPLMIs,thegain-schedulingDOFcontrollercanbeconstructedby˝rst solvingthefactorizationproblem I Y j X j = N j ( M j ) T for N j and M j ,andthencomputing A j K , B j K ,and C j K fromthefollowingequations 8 > > > > > > > < > > > > > > > : A j K =( N j ) 1 h ^ A j K Y j _ X j N j ( _ M j ) T Y j AX j ^ B j K C y X j Y j B 2 ^ C j K i ( M j ) T B j K = N 1 ^ B j K C j K = ^ C K ( M j ) T (2.96) Remark5. Inordertoremovethe _ dependencyintroducedby _ X j and _ Y j ,thepracticalvalidity approachpresentedin[68]isapplied.Either X j or Y j issettobeaconstantmatrixeliminatesthe derivativeterms.Forexample,wemayset X ( )= X 0 and N = I forall 2 ,then Y j = Y j ( ) 43 and ( M j ) T =( I Y j ( ) X 0 ) .Asaresult,thereconstructedcontrollervariablescanbesimpli˝ed 8 > > > > > > > < > > > > > > > : A j K =( N j ) 1 h ^ A j K Y j AX j ^ B j K C y X j Y j B 2 ^ C j K i ( M j ) T B K = N 1 ^ B j K C K = ^ C K ( M j ) T (2.97) Notethattheswitchingstabilitycondition (2.88) isnon-convexandfreezing X ( )= X 0 will convexifyitinto Y i ( ) Y j ( ) ; 2 S ( i;j ) : (2.98) Therefore,variables ( ^ A j Ki ; ^ B j Ki ; ^ C j Ki ;Y j i ;X 0 ) canbeiterativelysearchedtooptimizethecost functionwiththetuningparameter .TheoperationofPLMIsandoptimizationproblemaresolved byusingtheparserYALMIP[76]jointlywithoptimizationalgorithmSeDuMi[77]. 44 2.3SequentialDesignApproach 2.3.1Motivationsofsequentialdesign InconventionalswitchingLPVcontroldesign,switchingcontrollersaresynthesizedbycontroller variablesthatsimultaneouslysatisfyPLMIsforboth H 1 performanceonallsubregionsand switchingstabilityconditionsonallswitchingsurfaces.Inotherwords,controllersynthesis conditionsoneachsubregionarenotindependentwithadjacentonesduetotheswitchingstability conditionimposedonswitchingsurfaces.Whenthenumberofsubregionsgoeslarge,simultaneous designapproachleadstoahigh-dimensionaloptimizationproblem,withahighamountofLMI constraints,decisionvariables,onlinecomputationalload,andmemoryrequirement[78,79].Asa result,thesedrawbacksmakesimultaneousdesignpracticallyinfeasibleforhigh-ordersystemswith manydividedsubregions.Forexample,inthepolytopicsynthesisapproach,it'swellknownthat thenumberofLMIsgrowswith O (2 q ) ,where q isthedimensionofschedulingparameter[78,63]. Chen[60]consideredthehysteresisswitchingstate-feedbackLPVcontrolandconductedlinear interpolationofcontrollervariablesonswitchingsurfaces.However,onlytherelativestabilityis achievedontheoverlappingsubregionbythismethod.HanifzadeganandNagamune[61]followed theideaoflinearinterpolationofcontrollermatricesonswitchingsurfaces,andimposedconstraints oncontrollermatrixderivative.Thedesignofstabilizingcontrollerswasformulatedintoanon- convexoptimizationproblem,andaniterativedescentalgorithmwasthenappliedto˝ndalocal LPVcontrollerforeachsubregion.Theirapproachreliesheavilyoniterativecomputationstosolve multi-objectivenon-convexproblems.Moreover,theinterpolationofcontrollermatricescannot guaranteethe H 1 robustperformanceovertheoverlappedregion.Jianget.al[80]provideda systematicapproachfordevelopingswitchingLPVcontrollerbylinearlyinterpolatingcontroller variablesforaverage-dwell-timeswitchingstrategy.However,theformulatedPLMIsarenot numericallytractable,duetotheschedulingparameterterminthedenominator,whichinduces in˝nitytermonswitchingsurfaces.Bianchi[81]proposedanewdesignapproachbasedonYoula parametrizationthatclosed-loopsystemstabilityisnota˙ectedbytheinclusionofanystable 45 switchedLPVsystem.Thismakesitpossibletodesignswitchedcontrollersindependently,but thismethodcannotbeextendedtoparameter-dependentquadraticallystablesystems. Asequentialcontrollerdesignapproachisproposedinthisthesistodesignswitchingcontrollers withhighere˚ciencyandlesscomputationalburdenthansimultaneousdesign.Interpolatedcon- trollervariablesforoverlappedsubregionsandnewlyformulatedPLMIsareutilizedtosynthesize switchingLPVcontrollersoneachsubregionindependently.Oneachoverlappedsubregion,the LyapunovmatrixisformulatedbyconvexlycombiningPDLMonadjacentsubregions.ThePLMIs for H 1 performanceoneachsubregionisformulated,suchthattheconvexcombinationofadjacent PLMIsleadstoaguaranteed H 1 performanceoneveryoverlappedsubregion.Moreover,theguar- anteedsystemperformanceonoverlappedsubregionisnoworsethanitsneighboringsubregions. Inthisway,anindividualcontrollerforeachsubregioncanbedesignedinsequentialorder,instead ofsynthesizingallcontrollerssimultaneously.Byiterativelysolvingthereduced-dimensionalopti- mizationproblemoneachsubregion,switchingcontrollersforallsubregionswithguaranteed H 1 performanceonallsubregionsandoverlappedsubregionscanbeobtained. Notethat,inordertosimplifythedesignproblem,allsequentialLPVcontrollersareassumed tohaveaccesstofullstatesandthattheysharethesameparametriccontrollerform,whilecontroller gainsaredi˙erentoneachsubregion.Inthisthesis,switching H 1 LPVstate-feedbackcontrol isconsidered,wewillpresentthebasicideasoftheproposedsequentialdesignmethodbyone- dimensionalandtwo-dimensionalcases.Afterthat,theproposedapproachwillbeextendedtothe generalcaseofanydimensionalschedulingparameters.Themaincontributionsofthisworkare three-fold:(1)propositionofsequentialdesignofswitchingLPVcontrollers;(2)formulationof synthesisconditionsforsequentialdesignofswitching H 1 state-feedbackLPVcontrollers;(3) demonstrationofthebene˝tsoftheproposedsequentialdesignapproachbynumericalexamples. Considerthea˚neLPVsysteminEqn.(2.99), _ x p ( t )= A ( ( t )) x p ( t )+ B 1 ( ( t )) w ( t )+ B 2 ( ( t )) u ( t ) z ( t )= C ( ( t )) x p ( t )+ D 1 ( ( t )) w ( t )+ D 2 ( ( t )) u ( t ) (2.99) wherethesystemstateisdenotedas x p ( t ) ,theexogenousinputsdenotedas w ( t ) (forinstancedistur- 46 banceinputs,sensornoise,etc.),thecontrolinputas u ( t ) ,andthecontrolledoutputas z ( t ) .System matricesareassumedtodependonschedulingparametervector ( t )= 1 ( t ) ; 2 ( t ) ;:::; q ( t ) T andbeinthea˚neparameter-dependentform.Forexample, A ( ( t )) isdescribedas A ( ( t ))= A 0 + q X i =1 A i i ( t ) ; (2.100) where A 0 and A i , i =1 ; 2 ;:::;q ,areconstantmatrices. Thereal-timemeasurableschedulingparametersareassumedtovarywithinparameterregion whichisformulatedbyboundsofmagnitudesandvariationalrates, i 2 i = i i ( t ) i ;i =1 ; 2 ;:::;q; ; _ i 2 i = n i _ i ( t ) i ;i =1 ; 2 ;:::;q: o (2.101) NowconsidertheswitchedLPVsystem,whichconsistsof M numbersofdividedschedul- ingparameters [ 1 ; 2 ;:::; m ;:::; M ] ,and S numbersofun-dividedschedulingparameters [ M +1 ; M +2 ;:::; M + s ;:::; M + S ] .Hence, M + S = q .Foreachofdividedscheduling parameters, m isdividedinto N m numbersofsubregionswithitsvariationalrateremainedun- dividedas = 1 q ,andneighboringsubregionswillproduceoverlappedsubregions. Theentireschedulingparameterisdividedinto M m =1 N m numbersofsubregions,amongwhich the ( n 1 ;n 2 ;:::;n M ) thsubregion,denotedas ( n 1 ;n 2 ;:::;n M ) ,isformedbyCartesianproductof subregions ( n 1 ) 1 ( n 2 ) 2 ( n M ) M M +1 q .Theoverlappedsubregionformed by ( n m ) m and ( n m +1) m ,isdenotedas ([ n m ;n m +1]) m .Figure2.1andFigure2.2illustratethe divisionsscenariosofone-andtwo-dimensionalschedulingparameters. InFigure2.1,threeadjacentsubregions ( i 1) ; ( i ) ; ( i +1) producetwooverlappedsubre- gion ([ i 1 ;i ]) and ([ i;i +1]) ,withswitchingsurfaces S ([ i 1 ;i ]) and S ([ i;i +1]) de˝nedasregion boundariesofoverlappedsubregions.InFigure2.2,anyfouradjacentsubregions ( i;j ) , ( i +1 ;j ) , ( i;j +1) , ( i +1 ;j +1) producetwokindsofoverlappedsubregions.Thecentersubregiondenoted by ([ i;i +1] ; [ j;j +1]) isformedby 2 2 overlappedsubregionsintwodimensions,whereasotherover- lappedsubregionsareindividuallyformedby 2 1 overlappingsubregionsinonedimension,denoted 47 as ([ i;i +1] ;j ) , ([ i;i +1] ;j +1) , ( i; [ j;j +1]) , ( i +1 ; [ j;j +1]) .Inthecaseof M -dimensionaldivided schedulingparameter,thecenter-overlappedsubregionisformedby 2 M overlappingsubregions. Figure2.1:Subregiondivisionillustrationofone-dimensionalschedulingparameter Figure2.2:Subregiondivisionillustrationoftwo-dimensionalschedulingparameter ForthegivenswitchingLPVsystem,weareseekingforagain-schedulingstate-feedback switchingcontroller u ( t )= K i ( ) x ( t ) (2.102) stabilizingtheLPVsystem(2.99)withguaranteed H 1 performance,andcontrollergain K i ( ) is tobeswitchedaccordingtoswitchingsignal i ( t ) .Theswitchedclosed-loopsystemmatricesare derivedas 48 2 6 4 A cl;i B cl C cl;i D cl 3 7 5 = 2 6 4 A + B 2 K i B 1 C + D 2 K i D 1 3 7 5 : (2.103) The H 1 performance,de˝nedas L 2 -inducednormsfrom w ( t ) to z ( t ) ,isutilizedtoassess theclosed-loopsystemrobustnessinthepresenceofmodeluncertainties.Mathematically,let T 1 ( ;s ):= T zw ( ;s ) denotestheparameter-dependenttransferfunctionfrom w ( t ) to z ( t ) and jj T 1 jj 1 astheworst-case H 1 normof T 1 de˝nedin 2 .Then,the H 1 performanceforthe ( w ( t ) , z ( t ) )pairisde˝nedas L 2 gain[15],where jj T 1 jj 1 =sup 2 sup w 2L 2 ; jj w jj 2 6 =0 jj z ( t ) jj 2 jj w ( t ) jj 2 : (2.104) ThefollowinglemmaprovidesPLMIconditionsforsimultaneouslydesigningswitchingLPV H 1 state-feedbackcontrollerwithaverage-dwell-timeswitchinglogic[52,53,1],whichhasbeen wellprovenandwidelyusedinliterature. Lemma7. Givenscalars 0 > 0 , > 1 ,ifthereexistparameterdependentmatrices P i ( ) > 0 , Z i ( ) suchthat (2.105) holdsforalladmissibletrajectories ( ; _ ) 2 ( i ) and (2.106) holdsforanyswitchingsurface,thentheclosed-loopsystem(2.103)isexponentiallystabilizedby switchingLPVstate-feedbackcontrollergains K i ( )= Z i ( ) P 1 i ( ) foreveryswitchingsignal i ( t ) withaveragedwelltime ˝ a > ln ( ) 0 and jj z jj 2 < jj w jj 2 isachievedwithrobustnesslevel =max f i g > 0 , 2 6 6 6 6 4 _ P i + h AP i + B 2 Z i i + 0 P i B 1 CP i + D 2 Z i i ID 1 i I 3 7 7 7 7 5 < 0 ; (2.105) 1 P i +1 ( ) P i ( ) i +1 ( ) ; 2 S ([ i;i +1]) : (2.106) Remark6. ThissimultaneousdesignmethodrequiresthatPLMIconditionsforallsubregions andswitchingstabilityaresatis˝edatthesametime.Allswitchedcontrollersforsubregionsare designedsimultaneously,leadingtoaveryhigh-dimensionaloptimizationproblem,especiallyin thescenarioofmulti-dimensionalschedulingparameters. 49 2.3.2ControllersynthesisPLMIs Anovelsequentialdesignmethodisproposedtoovercomethedisadvantagesoftheconvention- aldesignmethod.Themainideaisintroducinginterpolatedcontrollerdecisionvariablesand formulatingindependentPLMIconditionsoneachsubregionsuchthatsystemperformanceson overlappedsubregionsareguaranteedaswell.Inthisway,theswitchingcontrollersynthesiscon- ditionsareformulatedasindependentoptimizationproblemsandcanbewellsolvedsequentially. One-andtwo-dimensionalschedulingparameterscenariosareprovidedasmotivationexamples foramoregeneral q -dimensionalschedulingparameterscenario. 2.3.2.1One-dimensionalschedulingparameter Considerthreeneighboringsubregions ( i 1) , ( i ) , ( i +1) asshowninFigure2.1anddesignate controllerdecisionvariablepairs P i 1 ( ) , Z i 1 ( ) , P i ( ) , Z i ( ) ,and P i +1 ( ) , Z i +1 ( ) forcon- trollersynthesis.Ontheoverlappedsubregionof ( i 1) \ ( i ) = ([ i 1 ;i ]) ,de˝netheinterpolated parameter-dependentpositivede˝nitematrix P ( i 1 ;i ) ( ) andinterpolated Z ( i 1 ;i ) ( ) as P ( i 1 ;i ) = " ( i 1 ;i ) 11 ( ) P i ( )+ " ( i 1 ;i ) 12 ( ) P i 1 ( ) ; Z ( i 1 ;i ) = " ( i 1 ;i ) 11 ( ) Z i ( )+ " ( i 1 ;i ) 12 ( ) Z i 1 ( ) ; (2.107) whereinterpolationfunction " ( ) ischosenasasigmoidfunctionas " ( i 1 ;i ) 11 ( )= e ( ) e ( ) +1 ;" ( i 1 ;i ) 12 ( )= 1 e ( ) +1 ; and ( )= [2( i ) ( i 1 i )] i 1 i = [2( i ) L ( i 1 ;i ) ] L ( i 1 ;i ) .Thevariable L ( i 1 ;i ) denotes thesizeoftheoverlappedsubregionand isatunablescalarwhichdeterminestheinterpolation functionshape. Thenthetimederivativeofinterpolatedparametricmatrixcanbewrittenas _ P ( i 1 ;i ) = n " ( i 1 ;i ) 12 ( ) _ P i 1 ( )+ " ( i 1 ;i ) 11 ( ) _ P i ( ) o + ( " ( i 1 ;i ) 12 ( ) e ( ) 2 _ ( e ( ) +1) L ( i 1 ;i ) P i 1 ( )+ " ( i 1 ;i ) 11 ( ) 2 _ ( e ( ) +1) L ( i 1 ;i ) P i ( ) ) : (2.108) 50 TheassociatedPLMIconditionsfor ( ; _ ) 2 ( i 1) and ( ; _ ) 2 ( i ) canbeaccordingly formulatedas 2 6 6 6 6 6 4 h AP i 1 + B 2 Z i 1 i _ P i 1 + 0 P i 1 + e ( ) 2 _ ( e ( ) +1) L ( i 1 ;i ) P i 1 B 1 CP i 1 + D 2 Z i 1 i 1 ID 1 i 1 I 3 7 7 7 7 7 5 < 0 (2.109) 2 6 6 6 6 6 4 h AP i + B 2 Z i i _ P i + 0 P i + 2 _ ( e ( ) +1) L ( i 1 ;i ) P i B 1 CP i + D 2 Z i i ID 1 i I 3 7 7 7 7 7 5 < 0 (2.110) suchthat " ( i 1 ;i ) 12 (2.109) + " ( i 1 ;i ) 11 (2.110)yieldingthefollowingstandardPLMIcondition, whichindicatestheguaranteed H 1 performanceforany ( ; _ ) 2 ([ i 1 ;i ]) , 2 6 6 6 6 4 D AP ( i 1 ;i ) + B 2 Z ( i 1 ;i ) E _ P ( i 1 ;i ) + 0 P ( i 1 ;i ) B 1 CP ( i 1 ;i ) + D 2 Z ( i 1 ;i ) ( i 1 ;i ) ID 1 ( i 1 ;i ) I 3 7 7 7 7 5 < 0 (2.111) where ( i 1 ;i ) = " ( i 1 ;i ) 12 i 1 + " ( i 1 ;i ) 11 i < max f i 1 ; i g : InordertoconvertEqns.(2.109)and(2.110)intonumericallytractableones,theboundsthat, _ ( t ) , 1 e ( ) +1 < 1 ,and e ( ) e ( ) +1 < 1 areusedtomodifythecontrollersynthesisPLMI conditionswithupperboundconstant ˙ ( i 1 ;i ) = 2 L ( i 1 ;i ) > 0 , 2 6 6 6 6 4 h AP i 1 + B 2 Z i 1 i _ P i 1 +( 0 + ˙ ( i 1 ;i ) ) P i 1 B 1 CP i 1 + D 2 Z i 1 i 1 ID 1 i 1 I 3 7 7 7 7 5 < 0 ; (2.112) 2 6 6 6 6 4 h AP i + B 2 Z i i _ P i +( 0 + ˙ ( i 1 ;i ) ) P i B 1 CP i + D 2 Z i i ID 1 i I 3 7 7 7 7 5 < 0 : (2.113) 51 Ifthereexistfeasiblecontrollerdecisionmatrices P i 1 , Z i 1 and P i 1 , Z i 1 ,andscalars i 1 ; i suchthatPLMIconditions(2.112)and(2.113)arevalidonsubregion ( i 1) and ( i ) ,respectively,thenthestandardPLMIconditionsasofEqn.(2.105)willbealsovalid onthesesubregions,leadingtotheguaranteed H 1 performanceoneachsubregion.Furthermore, conditions(2.112)and(2.113)leadtoconditions(2.109)and(2.110),andhence,resultin(2.111) ontheoverlappedsubregion ([ i 1 ;i ]) ,leadingtoguaranteed H 1 performance ( i 1 ;i ) noworse thanthatofneighboringsubregions. Nowdesignatecontrollerdecisionmatrices P i +1 ( ) and Z i +1 ( ) forsubregion ( i +1) ,andon theoverlappedsubregionof ( i ) \ ( i +1) = ([ i;i +1]) ,de˝netheinterpolatedparameter-dependent matrix P ( i;i +1) ( ) and Z ( i;i +1) ( ) as P ( i;i +1) ( )= " ( i;i +1) 11 ( ) P i +1 ( )+ " ( i;i +1) 12 ( ) P i ( ) ; Z ( i;i +1) ( )= " ( i;i +1) 11 ( ) Z i +1 ( )+ " ( i;i +1) 12 ( ) Z i ( ) ; (2.114) wheresimilarlysigmoidfunctionischosenforinterpolationas " ( i;i +1) 11 ( )= e ( ) e ( ) +1 ;" ( i;i +1) 12 ( )= 1 e ( ) +1 ; ( )= [2( i +1 ) L ( i;i +1) ] L ( i;i +1) ; where L ( i;i +1) = i i +1 .ThenthePLMIconditionsforcontrollersynthesisonsubregion ( i ) and ( i +1) areformulatedsimilarlywith ˙ ( i;i +1) = 2 L ( i;i +1) > 0 , 2 6 6 6 6 4 h AP i + B 2 Z i i _ P i +( 0 + ˙ ( i;i +1) ) P i B 1 CP i + D 2 Z i i ID 1 i I 3 7 7 7 7 5 < 0 ; (2.115) 2 6 6 6 6 4 h AP i +1 + B 2 Z i +1 i _ P i +1 +( 0 + ˙ ( i;i +1) ) P i +1 B 1 CP i +1 + D 2 Z i +1 i +1 ID 1 i +1 I 3 7 7 7 7 5 < 0 : (2.116) Ifthereexistfeasiblecontrollerdecisionvariables P i +1 , Z i +1 andscalar i +1 suchthatPLMI condition(2.116)isvalidonsubregion ( i +1) ,thenthestandardPLMIconditionsasofEqn.(2.105) 52 willbevalidonsubregion ( i +1) .Inotherwords,controllergain K i +1 = Z i +1 P 1 i +1 guaranteesthe H 1 performance i +1 onsubregion ( i +1) .Atthesametime,theinterpolatedcontrollervariables P ( i;i +1) ;Z ( i;i +1) willsatisfythefollowingPLMIconditionobtainedby " ( i;i +1) 12 (2.115) + " ( i;i +1) 11 (2.116), 2 6 6 6 6 4 D AP ( i;i +1) + B 2 Z ( i;i +1) E _ P ( i;i +1) + 0 P ( i;i +1) B 1 CP ( i;i +1) + D 12 Z ( i;i +1) ( i;i +1) ID 11 ( i;i +1) I 3 7 7 7 7 5 < 0 (2.117) where ( i;i +1) = " ( i;i +1) 11 i +1 + " ( i;i +1) 12 i < max f i ; i +1 g : Inotherwords,controller K ( i;i +1) = Z ( i;i +1) P 1 ( i;i +1) alsoguarantees H 1 performance max f i ; i +1 g overtheoverlappedsubregion ([ i;i +1]) . Notethat ˙ ( i 1 ;i ) and ˙ ( i;i +1) dependonthesizeofoverlappedsubregions,thustheymaynot beidentical.Inordertoidentifythecommoncontroller K i ( ) on ( i ) ,themaximumvalueoftwo variables ˙ ( i ) =max f ˙ ( i 1 ;i ) ;˙ ( i;i +1) g isusedtoreplacethecoe˚cientsofintroducedterms inEqns.(2.113)and(2.115). Toensureswitchingstability,theminimumdwelltimeforswitchingsignalcanbecalculated as ˝ a = ln 0 , =max 8 > < > : 1+ 1 e ; 1+ i +1 i 1 e +1 ; 1+ i i +1 1 e +1 9 > = > ; ,suchthat(2.106)issatis˝edon switchingsurfaces. i and i denotethemaximumandminimumeigenvaluesofmatrix P i ( ) at switchingsurfaces.Iftheinterpolationvariable ischosenlargeenough,then iscloseto1, andtheminimumdwelltimeiscloseto0.Inotherwords,theaveragedwelltimesignalisalmost arbitrary.Atthispoint,wearereadytoobtainthefollowingtheorem. Theorem8. Withgiven 0 andgivenschedulingparametersubregions,ifthereexistparameter- dependentpositive-de˝nitematrices P i ( ) ,parameter-dependentmatrices Z i ( ) ,andpositives- calars i ,satisfyingthePLMIs (2.118) forany ( ; _ ) 2 ( i ) ,thentheswitchingcontrollergain K i ( )= Z i ( ) P 1 i ( ) guaranteestheclosed-loopsystem H 1 performance i ,andtheinterpo- latedcontrollerswithitsadjacentcontrollersbyEqn. (2.114) alsoguaranteesameperformance forswitchingsignalswithaveragedwelltime ˝ a largerthan ˝ a whichcanbecloseto0.. 53 2 6 6 6 6 4 h AP i + B 2 Z i i _ P i +( 0 + ˙ ( i ) ) P i B 1 CP i + D 2 Z i i ID 1 i I 3 7 7 7 7 5 < 0 ; (2.118) Remark7. Theconstant ˙ ( i ) inthePLMIconditionillustratetheintroducedrelativestabilityof theclosed-loopsystem,whichisknowninliteratureas ˙ -stability[60].Itisdeterminedbythe sizesofoverlappedsubregions min n L ( i;i +1) ;L ( i 1 ;i ) o andinterpolationrate ofsigmoidfunc- tion.Theintroducedrelativestability,togetherwithinterpolationofcontrollerdecisionvariables, provideindependentsynthesisconditionsforeachindividualsubregion,butwithintroduceddesign conservativeness. 2.3.2.2Two-dimensionalschedulingparameters Supposethatentireschedulingparameterregionisdividedinto N 1 N 2 subregions,andconsidera subregion ( i;j ) , i 2 N 1 ;j 2 N 2 ,aswellasitsadjacentsubregions ( i +1 ;j ) ; ( i;j +1) ; ( i +1 ;j +1) . AsillustratedbyshadowsinFigure2.2,theoverlappedsubregionsarecategorizedintotwotype- s:single-overlappedsubregion(slashshadow)anddouble-overlappedsubregion(crossshadow). Thedouble-overlappedsubregionis˝rstlyfocusedandassociatePLMIconditionswillbede- rived.Designateparameter-dependentcontrollervariablesforeachsubregionis P ( i;j ) ;Z ( i;j ) , P ( i +1 ;j ) ;Z ( i +1 ;j ) , P ( i;j +1) ;Z ( i;j +1) , P ( i +1 ;j +1) ;Z ( i +1 ;j +1) ,thenthecontrollerdecisionvari- ablesonthedouble-overlappedsubregion =( 1 ; 2 ) 2 ([ i;i +1] ; [ j;j +1]) areinterpolatedas P = " 11 ( 1 ; 2 ) P ( i;j ) + " 21 ( 1 ; 2 ) P ( i +1 ;j ) + " 12 ( 1 ; 2 ) P ( i;j +1) + " 22 ( 1 ; 2 ) P ( i +1 ;j +1) ; Z = " 11 ( 1 ; 2 ) Z ( i;j ) + " 21 ( 1 ; 2 ) Z ( i +1 ;j ) + " 12 ( 1 ; 2 ) Z ( i;j +1) + " 22 ( 1 ; 2 ) Z ( i +1 ;j +1) ; (2.119) " 11 ( 1 ; 2 )= " e ( 1 ) e ( 1 ) +1 #" e ( 2 ) e ( 2 ) +1 # ;" 21 ( 1 ; 2 )= 1 e ( 1 ) +1 " e ( 2 ) e ( 2 ) +1 # ; " 12 ( 1 ; 2 )= " e ( 1 ) e ( 1 ) +1 # 1 e ( 2 ) +1 ;" 22 ( 1 ; 2 )= 1 e ( 1 ) +1 1 e ( 2 ) +1 : 54 _ P = " 11 2 4 2 1 _ 1 ( e ( 1 ) +1) L ( i;i +1) 1 + 2 2 _ 2 ( e ( 2 ) +1) L ( j;j +1) 2 3 5 P ( i;j ) + " 11 _ P ( i;j ) + " 21 2 4 e ( 1 ) 2 1 _ 1 ( e ( 1 ) +1) L ( i;i +1) 1 + 2 2 _ 2 ( e ( 2 ) +1) L ( j;j +1) 2 3 5 P ( i +1 ;j ) + " 21 _ P ( i +1 ;j ) + " 12 2 4 2 1 _ 1 ( e ( 1 ) +1) L ( i;i +1) 1 + e ( 2 ) 2 2 _ 2 ( e ( 2 ) +1) L ( j;j +1) 2 3 5 P ( i;j +1) + " 12 _ P ( i;j +1) + " 22 2 4 e ( 1 ) 2 1 _ 1 ( e ( 1 ) +1) L ( i;i +1) 1 + e ( 2 ) 2 2 _ 2 ( e ( 2 ) +1) L ( j;j +1) 2 3 5 P ( i +1 ;j +1) + " 22 _ P ( i +1 ;j +1) (2.120) It'sobviousthat, " 11 + " 12 + " 21 + " 22 =1 .Moreover, ( m )= m [2( m i +1 m ) L ( i;i +1) m ] L ( i;i +1) m ;m = 1 ; 2 ,where L ( i;i +1) m = i m i +1 m denotesthesizeofoverlappedsubregionin m direction, m determinestheinterpolationratein m direction. WiththeexpressionoftimederivativeofinterpolatedparametricmatrixinEqn.(2.120),the coe˚cientsoftheseadditionaltermsareboundedas 2 1 _ 1 ( e ( 1 ) +1) L ( i;i +1) 1 + 2 2 _ 2 ( e ( 2 ) +1) L ( j;j +1) 2 < 2 1 1 L ( i;i +1) 1 + 2 2 2 L ( j;j +1) 2 = ˙ ( i;i +1) 1 + ˙ ( j;j +1) 2 : Theotherthreecoe˚cientsarealsoboundedby ˙ ( i;i +1) 1 + ˙ ( j;j +1) 2 = ˙ ([ i;i +1] ; [ j;j +1]) ,abbreviated as ˙ infollowingformula. 2 6 6 6 6 4 D AP ( i;j ) + B 2 Z ( i;j ) E _ P ( i;j ) +( 0 + ˙ ) P ( i;j ) B 1 CP ( i;j ) + D 2 Z ( i;j ) ( i;j ) ID 1 ( i;j ) I 3 7 7 7 7 5 < 0 (2.121) 2 6 6 6 6 4 D AP ( i +1 ;j ) + B 2 Z ( i +1 ;j ) E _ P ( i +1 ;j ) +( 0 + ˙ ) P ( i +1 ;j ) B 1 CP ( i +1 ;j ) + D 2 Z ( i +1 ;j ) ( i +1 ;j ) ID 1 ( i +1 ;j ) I 3 7 7 7 7 5 < 0 (2.122) 55 2 6 6 6 6 4 D AP ( i;j +1) + B 2 Z ( i;j +1) E _ P ( i;j +1) +( 0 + ˙ ) P ( i;j +1) B 1 CP ( i;j +1) + D 2 Z ( i;j +1) ( i;j +1) ID 1 ( i;j +1) I 3 7 7 7 7 5 < 0 (2.123) 2 6 6 6 6 4 D AP ( i +1 ;j +1) + B 2 Z ( i +1 ;j +1) E _ P ( i +1 ;j +1) +( 0 + ˙ ) P ( i +1 ;j +1) B 1 CP ( i +1 ;j +1) + D 2 Z ( i +1 ;j +1) ( i +1 ;j +1) ID 1 ( i +1 ;j +1) I 3 7 7 7 7 5 < 0 (2.124) 2 6 6 6 6 4 h AP + B 2 Z i _ P + 0 P B 1 CP + D 2 Z ID 1 I 3 7 7 7 7 5 < 0 (2.125) IfthePLMIconditionsonthesubregions ( i;j ) , ( i +1 ;j ) , ( i;j +1) , ( i +1 ;j +1) areproposedin Eqns.(2.121),(2.122),(2.123)and(2.124),then H 1 performanceoneachindividualsubregionis guaranteedwithassociated -level.Meanwhile, " 11 (2.121) + " 21 (2.122) + " 12 (2.123) + " 22 (2.124)yieldsPLMI(2.125),where = " 11 ( i;j ) + " 21 ( i +1 ;j ) + " 12 ( i;j +1) + " 22 ( i +1 ;j +1) < max f ( i;j ) ; ( i +1 ;j ) ; ( i;j +1) ; ( i +1 ;j +1) g ; (2.126) whichindicatesthat H 1 performanceondouble-overlappedsubregionisguaranteedwithfour adjacentsubregionsfortwo-dimensionalschedulingparametercases. Forthesesingle-overlappedsubregion,denotedbyslashshadowsinFigure2.2,the H 1 perfor- mancecanalsobeachievedifEqns.(2.121),(2.122),(2.123)and(2.124)aresatis˝ed,whichcan beeasilyvalidatedbyeliminatingeither 1 or 2 inEqn.(2.119)andconvertitintoEqn.(2.107) usedinone-dimensionalschedulingparametercase. Whendesigning K ( i;j ) insequentialorder,allitsfouroverlappedsubregionwithadjacent subregionsshouldbeconsidered,inotherwords, ˙ -relativestabilityneedstobesatis˝edunderthe 56 mostconservativecondition.Hence,wehavethelargest ˙ -relativestabilityindex ˙ ( i;j ) =max f ˙ ( i 1 ;i ) 1 ;˙ ( i;i +1) 1 g +max f ˙ ( j 1 ;j ) 2 ;˙ ( j;j +1) 2 g : Theswitchingstabilityconditionbetweenanyadjacentsubregionscanbecalculatedaccording totheaverage-dwell-timeswitchingconditions[52,10,53].Toensureswitchingstability,the minimumaveragedwelltimecanbecalculatedas ˝ a = ln 0 , =max 8 > > > < > > > : 1+ " ( i +1 ;j ) + ( i;j +1) ( i;j ) ! e + ( i +1 ;j +1) ( i;j ) 1 # ( e +1) 2 ; 1+ 2 e +1 e 2 : 9 > > > = > > > ; ( i;j ) and ( i;j ) denotethemaximumandminimumeigenvaluesofmatrix P ( i;j ) ( ) overswitching surfaces.Iftheinterpolationcoe˚cient ischosenlargeenough, ˇ 1 andtheminimumdwell time ˝ a isverycloseto0,whichindicatesthatswitchingsignalcanbealmostarbitrary. Bythispoint,it'sobvioustoconcludethefollowingtheoremofdesigningswitchingstate- feedbackLPVcontrollerfortwo-dimensionalschedulingparametersystem.Theproofcanbe easilyprovedbythederivationprocedure. Theorem9. Withgiven 0 andgivenschedulingparametersubregions,ifthereexistparameter- dependentpositive-de˝nitematrices P ( i;j ) ( ) ,parameter-dependentmatrices Z ( i;j ) ( ) ,andpos- itivescalars ( i;j ) ,satisfyingthePLMIs (2.127) forany ( ; _ ) 2 ( i;j ) ,thentheswitching controllergain K ( i;j ) ( )= Z ( i;j ) ( ) P 1 ( i;j ) ( ) guaranteestheclosed-loopsystem H 1 performance ( i;j ) ,andtheinterpolatedcontrollerswithitsadjacentcontrollersbyEqn. (2.119) alsoguarantee sameperformanceontheoverlappedsubregionswithitsadjacentsubregionsforswitchingsignals withaveragedwelltime ˝ a largerthan ˝ a whichiscloseto0.. 2 6 6 6 6 4 D AP ( i;j ) + B 2 Z ( i;j ) E _ P ( i;j ) +( 0 + ˙ ( i;j ) ) P ( i;j ) B 1 CP ( i;j ) + D 2 Z ( i;j ) ( i;j ) ID 1 ( i;j ) I 3 7 7 7 7 5 < 0 (2.127) 57 2.3.2.3Schedulingparameterofanydimensions Considerthegeneralscenariothat ( t )= 1 ( t ) ; 2 ( t ) ;:::; q ( t ) T with M numbersofdivided schedulingparameters [ 1 ; 2 ;:::; m ;:::; M ] .Theoverlappedsubregionformedby 2 M neigh- boringsubregionsisdenotedas ([ n 1 ;n 1 +1] ; ; [ n M ;n M +1]) ,andtheassociatedLyapunovmatrix P ( ) andcontrollervariable Z ( ) arede˝nedas(2.128)bytheconvexcombinationofLyapunov matricesonneighboringoverlappingsubregions.Notethatsubregionnumberingisabbreviated, forexample P ( n 1 + i 1 1 ;n 2 + i 2 1 ;:::;n M + i M 1) isabbreviatedby P ( i 1 ;i 2 ;:::;i M ) . P = 2 P i 1 =1 2 P i 2 =1 2 P i M =1 n " 1 i 1 ( 1 ) " 2 i 2 ( 2 ) :::" Mi M ( M ) P ( i 1 ;i 2 ;:::;i M ) ( 1 ; 2 ;:::; q ) o = 2 P i 1 =1 2 P i 2 =1 2 P i M =1 ( M Q m =1 " mi m ( m ) P ( i 1 ;i 2 ;:::;i M ) ( 1 ; 2 ;:::; q ) ) (2.128) where " m 1 ( m )= e ( m ) e ( m ) +1 ;" m 2 ( m )= 1 e ( m ) +1 ;;m =1 ; 2 ; ;M: Moreover, ( m )= m [2( m n m +1 m ) L ( n m ;n m +1) m ] L ( n m ;n m +1) m ,where L ( n m ;n m +1) m = n m m n m +1 m denotesthesizeofoverlappedsubregionin m direction,and m determinestheinterpola- tionratein m direction.Obviously,wehavetheequationthatsummationofallcoe˚cientsequals to1, 2 X i 1 =1 2 X i 2 =1 2 X i M =1 n " 1 i 1 ( 1 ) " 2 i 2 ( 2 ) " Mi M ( M ) o =1 : (2.129) Thuswehavetheboundsforthederivativeofinterpolationcoe˚cientas _ " mi m ( m )= " mi m (1 " mi m ) ( 1) i m +1 2 m _ m L ( n m ;n m +1) m <" mi m 2 m m L ( n m ;n m +1) m : (2.130) 58 ThetimederivativeofLyapunovmatrix P ( 1 ; 2 ;:::; q ) canbederivedas, _ P = 2 P i 1 =1 2 P i 2 =1 2 P i M =1 n " 1 i 1 " 2 i 2 :::" Mi M _ P ( i 1 ;i 2 ;:::;i M ) +_ " 1 i 1 " 2 i 2 :::" Mi M P ( i 1 ;i 2 ;:::;i M ) + " 1 i 1 _ " 2 i 2 :::" Mi M P ( i 1 ;i 2 ;:::;i M ) + + " 1 i 1 " 2 i 2 ::: _ " Mi M P ( i 1 ;i 2 ;:::;i M ) o < 2 P i 1 =1 2 P i 2 =1 2 P i M =1 ( M Q m =1 " mi m ( m ) _ P ( i 1 ;i 2 ;:::;i M ) + M Q m =1 " mi m ( m ) 2 4 2 1 1 L ( n 1 ;n 1 +1) 1 + + 2 M M L ( n M ;n M +1) M 3 5 | {z } = M P m =1 0 @ 2 m m L ( n m ;n m +1) m 1 A = M P m =1 ˙ ( n m ;n m +1) m P ( i 1 ;i 2 ;:::;i M ) g = 2 P i 1 =1 2 P i 2 =1 2 P i M =1 ( M Q m =1 " mi m " _ P ( i 1 ;i 2 ;:::;i M ) + M P m =1 ˙ ( n m ;n m +1) m P ( i 1 ;i 2 ;:::;i M ) #) (2.131) Forthesubregion ( n 1 + i 1 1 ; ;n M + i M 1) ,PLMIfor H 1 performanceisformulatedas 2 6 6 6 6 4 e ( i 1 ; ;i M ) B 1 CP ( i 1 ; ;i M ) + D 2 Z ( i 1 ; ;i M ) ( i 1 ; ;i M ) ID 1 ( i 1 ; ;i M ) I 3 7 7 7 7 5 < 0 (2.132) where e ( i 1 ; ;i M ) = D AP ( i 1 ; ;i M ) + B 2 Z ( i 1 ; ;i M ) E _ P ( i 1 ; ;i M ) +( 0 + M X m =1 ˙ ( n m ;n m +1) m ) P ( i 1 ; ;i M ) ; suchthattheconvexcombinationofPLMIconditions(2.132)onall 2 M overlappingsubregions 2 X i 1 =1 2 X i 2 =1 2 X i M =1 8 < : M Y m =1 " mi m (2.132) 9 = ; yieldsthePLMIconditionon M -overlappedsubregion ([ n 1 ;n 1 +1] ; ; [ n M ;n M +1]) 2 6 6 6 6 4 h AP + B 2 Z i _ P + 0 P B 1 CP + D 2 Z ID 1 I 3 7 7 7 7 5 < 0 (2.133) 59 whichindicatesthatthe H 1 performance = 2 P i 1 =1 2 P i 2 =1 2 P i M =1 ( M Q m =1 " mi m ( i 1 ;i 2 ; ;i M ) ) is achievedbyinterpolatingcontrollervariables,andit'sobviousthat < max n ( i 1 ;i 2 ; ;i M ) o for any i 1 ;i 2 ; i M =1 ; 2 . The H 1 performanceontherestlessthan M -overlappedsubregioncanalsobeachieved,if Eqn.(2.132)issatis˝edoneachindividualsubregion,whichcanbeeasilyvalidatedbyeliminat- ingtermsrelatedtotheun-overlappedschedulingparameter.Whendesigning K ( n 1 ; ;n M ) on ( n 1 ; ;n M ) inasequentialorder,alloverlappedsubregionsproducedbythissubregionwithits adjacentsubregionsshouldbeconsidered,inotherwords, ˙ -relativestabilityindexneedstobe replacedby ˙ ( n 1 ; ;n m ) = M X m =1 max f ˙ ( n m 1 ;n m ) m ;˙ ( n m ;n m +1) m g : Toensureswitchingstability,theminimumaveragedwelltimecanbecalculatedbytheeigen- valuesof P ( ) overswitchingsurfaces.Iftheinterpolationcoe˚cient ischosenlargeenough, theminimumdwelltimeisverycloseto0,whichindicatesthatswitchingsignalcanbealmost arbitrary. Bythispoint,it'sobvioustoprovidethefollowingtheoremofdesigningswitchingstate- feedbackLPVcontrollerfortwo-dimensionalschedulingparametersystem.Theproofisprovided bytheabovederivationprocedures. Theorem10. Withgiven 0 andgivenschedulingparametersubregions,ifthereexistparameter- dependentpositive-de˝nitematrices P ( ) ,parameter-dependentmatrices Z ( ) ,andpositives- calars ( n 1 ; ;n M ) ,satisfyingthePLMIs (2.134) forany ( ; _ ) 2 ( n 1 ; ;n M ) ,thenthe switchingcontrollergain K ( n 1 ; ;n M ) ( )= Z ( ) P 1 ( ) guaranteestheclosed-loopsystem H 1 performance ( n 1 ; ;n M ) ,andtheinterpolatedcontrollerswithitsadjacentcontrollersbyE- qn. (2.119) alsoguaranteesameperformanceontheoverlappedsubregionswithitsadjacent 60 subregionsforswitchingsignalswithaveragedwelltime ˝ a largerthan ˝ a whichiscloseto0.. 2 6 6 6 6 4 h AP + B 2 Z i _ P + ˙ ( n 1 ; ;n m ) P B 1 CP + D 2 Z ( n 1 ; ;n M ) ID 1 ( n 1 ; ;n M ) I 3 7 7 7 7 5 < 0 (2.134) 2.4PLMIRelaxationMethod 2.4.1Modelingschedulingparameters Theschedulingparametervectorconsideredintheopen-loopsystem(2.1)isde˝nedinana˚ne manifold,sowe˝rstneedtomapthatintoamulti-simplexmanifoldforsubsequentconvexanalysis. FollowingtheaprocedurepresentedinLacerdaetal.[82]andOliveiraetal.[70],theoriginal parameterdomaincanbeconvertedintoaconvexmulti-simplexdomain.Notethatamulti-simplex domainisde˝nedastheCartesianproductofmultipleunit-simplexes.Thus,thescheduling parameter i ( t ) canbeconvertedintotheunit-simplexvariable i ( t ) usingthefollowingformula, i; 1 = i ( t ) i i i ; i; 2 =1 i; 1 = i i ( t ) i i ;i =1 ; 2 ;:::;q: (2.135) Asaresult,wehave i =( i; 1 ; i; 2 ) 2 i; 2 ,wherethetwodimensionalunit-simplex i; 2 for i isde˝nedas i; 2 := f i 2 R 2 : 2 X k =1 i;k =1 ; i;k 0 g : Hence,theunit-simplexvariable i 2 i; 2 iscreated.Similarly,thetimederivativeofthe schedulingparametercanalsobeconvertedintoaunit-simplexvariablebyutilizingthefollowing condition, _ i; 1 ( t )+_ i; 2 ( t )=0 : (2.136) Hence,theratesofconvexparametersareboundedasfollows, i i i _ i;k i i i ;i =1 ; 2 ;:::;q ; k =1 ; 2 : (2.137) 61 Notethatthetimederivativeofparameter i liesinthespacemodeledbytheconvexcombination ofthecolumnsofthematrix H i 2 R 2 2 givenby H i = 2 6 4 i i i ; i i i i i i ; i i i 3 7 5 ;i =1 ; 2 ;:::;q; (2.138) and _ i canbeestablishedusingaunit-simplexofdimension2as i; 2 := f ˚ i 2 R 2 : ˚ i = 2 X k =1 i;k H k i ; i 2 i; 2 g ;i =1 ; 2 ;:::;q; (2.139) where i =( i; 1 ; i; 2 ) and H k i denotesthe k th columnofmatrix H i .Therefore,theunit-simplex variable _ i 2 i; 2 iscreated.Furthermore,theschedulingparameters ( ; _ ) withgivenboundscan thenbeconvertedintomulti-simplexdomainfromCartesianproductofmultipleunit-simplexesas follows, ( ; _ ) 2 := q Y i =1 i; 2 q Y i =1 i; 2 : Byutilizingtheschedulingparametertransformationpresentedabove,theLPVsystem ) de- scribedin(2.1),whichisana˚nefunctionofparameter ,cannowbetransformedintoanLPV systemrepresentation ) thatisafunctionof inmulti-simplexdomain.Forsimplicity,we assumethat ) takesthesameformas ) inthatallthesystemmatricesarenowfunctionsof .Subsequently,theLPVcontrollerdesign,tobepresentedinthenextsection,willbebasedon theconvexschedulingparameter .However,inactualcontrolimplementation,thedesignedLPV controllerinmulti-simplex domainwillneedtobemappedbacktothecontrollerinthea˚ne domain[70]. 2.4.2PLMIsrelaxation ThePLMIsformulatedinTheorem5,7,8,9,and10rendersanoptimizationproblemofin˝nite dimensionsandun-de˝neddecisionvariablestructures.Tonumericallytacklethisoptimization problem,a˚nedecisionvariablestructureisassumed,forexample, ^ A j K ( ) isexpressedas ^ A j K ( )= ^ A j K 0 + P q i =1 ^ A j Ki i : Coe˚cientcheckinmulti-simplexdomain[19,70,71]hasbeenadoptedto 62 successfullyobtaina˝nitesetofLMIsbutwithintroducedconservativeness.Otheroptions[72,73] canalsobeapplied,forinstance,sum-of-squarerelaxation[74]andenforcingmulti-convexity method[75]. Therefore,controllerdecisionvariablescanbeiterativelysearchedtooptimizethecostfunction. TheoperationofPLMIsandoptimizationproblemaresolvedbyusingtheparserYALMIP[76] jointlywithoptimizationalgorithmSeDuMi[77]. 63 CHAPTER3 APPLICATIONEXAMPLES 3.1ActiveMagneticBearing(AMB)Model Anactivemagneticbearing(AMB)systemborrowedfromLuandWu[1,83,84,85]isrevisited todemonstratethee˙ectivenessoftheproposedsmooth-switchingLPVcontroldesign.In[1], statesandcontrolinputsexperiencesharpjumpoverswitchingsurfaces,andthesesharpjumpswill bee˙ectivelysmoothenedbyapplyingtheproposedmethod. TheAMBsystemisformulatedintoanLPVmodelwithrotorspeedastheschedulingparameter .Intheautomaticbalancingdesign,themeasuredrotordisplacementsareassumedtobeexposed tosensornoises,andthegain-schedulingcontrollerisdesiredtosuppressthedisplacementsofrotor centerlines.While H 1 channelsarekeptthesameasthesein[1]forattainingguaranteedrobust stability,theoutputsof H 2 channelsarechosentobe [ x 1 ;x 2 ] T =[ l;l ] T ,thedisplacements ofrotorcenterline.Inthisway,thesmooth-switchingmixedICC = H 1 LPVcontrollerwillbe designedtosuppressrotordisplacementssubjecttomeasurementnoise,withconstrainedcontrol inputsandboundedmodelinguncertainty.Themainbene˝tsoftheproposedmethodover[1] arethree-fold.First, -balancedoptimal H 2 performanceisachievedwithsmoothresponsesover switchingsurfaces.Second,thecontrolinputconstraintisenforcedduringthecontroldesign. Last,thetrade-o˙amongsystem H 2 performance,ICCcondition,andswitchingsmoothnessis establishedandprovidesinsightsastohowtotunethecontrollerstoattainabalancedsystem performance. Theweightingfunctionsusedinthisstudyarethesameasthosein[1,86].Thatis, W z = 10( s +8) s +0 : 001 I 2 , W u = 0 : 01( s +100) s +100000 I 2 ,and W n =0 : 001 I 2 .Therotorspeedisassumedtovary withintherange 2 [315 ; 1100] rad=s andvariationalrate _ 2 [ 100 ; 100] rad=s 2 .Thescheduling parameterisdividedintotwooverlappingsubregions;namely, 2 [315 ; 720] and 2 [700 ; 1100] , anditstrajectoryisde˝nedinFigure3.1.Switchingeventshappenwhen =720 rad=s (at 64 t 1 =2 : 9 s )and =700 rad=s (at t 2 =6 : 5 s ).Sameas[1],thetwodimensionalmeasurement noisesarechosenasstepinputswiththesamemagnitudeof 0 : 001 m butwiththeoppositesign. Figure3.1:Trajectoryofschedulingparameter,rotorspeed 3.1.1Trade-o˙between trace ( W ) andICCconditions Tostudythein˛uenceofICCconditionswhenoptimizingthe H 2 performance,di˙erentupper boundsofcontrolinputsareconsideredinPLMIs.Whenthecostfunction(2.81)withoutsmooth- nessindexisminimizedtoobtaintheoptimal H 2 performance,thetrade-o˙relationshipofICC conditions U and H 2 performanceupperbound trace ( W ) canbefoundinFigure3.2.Itcanbe observedthatincreasingICCbound U leadstodecreasing trace ( W ) ,indicatingthatlargercontrol authoritywillresultinimprovedsystemperformance.Moreover,when U isgreaterthan 10 8 , furtherreducing trace ( W ) requiresmuchlargercontrolauthority.Hence, U =10 8 isselectedas the optimal trade-o˙point,consideringbothcontrole˙ortandachievableperformance. Thedisplacementandcontrolinputresponsesunderdi˙erent U and˝xed =36 areinvestigat- ed.AsshowninFigures3.3and3.4,withlargercontrolauthority,thedisplacementsaresuppressed toamuchsmallerlevel.Inthecaseofin˝niteICCcondition,thegraysolidcurveprovidesthebest performance,and U =10 8 producesslightlybetterresponsesthantheduplicatedresultsfollowing theprocedurein[1].Furthermore,theresponsesexperiencesmallerjumps,becauseofthelackof 65 feed-forwardterm D K inthemixedICC = H 1 control.InFigures3.5-3.7,largercontrolconstraint leadstolargercontrole˙ortinordertoachievebetterperformance.In˝nitelylarge U willproduce controlinputmagnitudelargerthan6000N,inordertoachievethebest H 2 performanceasshown inFigures3.3and3.4.Fromthetime-domainsimulationresults,consideringbothcontrole˙ort U andachievablesystem H 2 performance,theselectionof optimal trade-o˙ICCconstraint U =10 8 canbecross-validatedwithFigure3.2. Figure3.2:Trade-o˙relationshipbetween U and trace ( W ) Figure3.3: x 1 responseunderdi˙erentICCconditions 66 Figure3.4: x 2 responseunderdi˙erentICCconditions Figure3.5: u 1 responseunderdi˙erentICCconditions 3.1.2SmoothswitchingLPVcontrolbysimultaneousdesign 3.1.2.1Trade-o˙between trace ( W ) andswitchingsmoothness Inthissubsection,thesmoothnessindexisconsideredinthecostfunctioninordertoattainan optimaltrade-o˙relationshipbetweensystemperformanceandswitchingsmoothness.Withthe ˝xedrobustnesslevel =36 ,weightingfactor istunedtobalancethesystemperformanceand 67 Figure3.6: u 2 responseunderdi˙erentICCconditions Figure3.7:Controlinputsunderin˝niteICCconditions switchingsmoothnesson S (1 ; 2) and S (2 ; 1) .Twodi˙erentICCconditions U 1 =10 7 and U 2 =10 8 areconsideredinthisstudy.FromFigures3.8and3.9,onecanseethatincreasedweightingfactor leadstodecreased trace ( W ) orimprovedoutputperformance.Notethatincreased I sm leadsto decreasedcontrollerswitchingperformance.Theseresultsclearlyshowthetrade-o˙relationship betweenperformanceandswitchingsmoothness.Onechoiceof optimal trade-o˙pointisthat magnitude trace ( W ) issmall,andthesmoothnessindexisnotyetincreasedsigni˝cantly,such 68 thatsystemperformanceisclosetothebestachievablelevelwhiletheswitchingsmoothnessis acceptable.Thechosenweightingfactorfortwocasesare: ( 1 ; 2 )=(10 ; 1) . Figure3.8:Trade-o˙relationshipbetweenswitchingsmoothnessand trace ( W ) under U 1 = 10 7 Figure3.9:Trade-o˙relationshipbetweenswitchingsmoothnessand trace ( W ) under U 2 = 10 8 69 3.1.2.2Simulationresultsanddiscussions Figure3.10:State x 1 responseby[1]andproposedmethod Figure3.11:State x 2 responseby[1]andproposedmethod Afteranoptimaltrade-o˙pointischosen,thetime-domainsimulationsunderdi˙erentICC 70 Figure3.12:Controlinput u 1 responseby[1]andproposedmethod Figure3.13:Controlinput u 2 responseby[1]andproposedmethod conditionsareconductedwithdesignedcontrollers 1 .AsshowninFigures3.10and3.11,dashed- linesaretheun-smoothstateresponsesduplicatedusingthemethodin[1],whilesolid-and dotted-linesrepresenttheseresponsesobtainedbytheproposedmethodundertwodi˙erentICC conditions. 1 Thedesignedswitchingcontrollermatricesareavailableonline,https://github.com/ hetianyi1992/smooth_switching_LPV. 71 Theunsmoothstateresponsesfrom[1]experiencesharpjumpsonswitchingsurfacesat t 1 = 2 : 9 s and t 2 =6 : 5 s .However,withtheproposedmethod,thesharpjumpsofstateresponsesare successfullysmoothenedbyminimizingthe -balancedcostfunction(2.81),whichdemonstrates thee˙ectivenessoftheproposedsmooth-switchingcontroldesign. Bycomparingstateresponsesunderdi˙erentICCconditions,itiseasyto˝ndthatrotor displacementscanbefurthersuppressedwhenlargercontrolauthorityismadeavailable.With tuned U 2 =10 8 ; 2 =1 ,theproposedmethodnotonlyleadstoasmooth-switchingcontroller,but alsoreducesthepeakmagnitudeofrotordisplacementsovertheun-smoothresponsesduplicatedby followingtheprocedurein[1].Thatis,thewell-tunedICCconditionsandsmoothnessweighting coe˚cientleadtosigni˝cantlyimprovedswitchingsmoothness,whilesystemperformanceisnot degradedcomparedtoconventionalLPVcontrol. Figures3.12and3.13showtheunsmoothcontrolresponsesduplicatedfrom[1]andsmooth controlinputsunderdi˙erentICCconditions.Unsmoothcontrolinputsexperiencesharpjumps atswitchinginstants,whilecontrolinputsaresmoothenedusingtheproposedsmooth-switching controllers.Bycomparingcontrolinputresponses,itcanbefoundthatICCconditionsin˛uence thepeakmagnitudesofcontrolinputs.WithdeterminedICCconditions,awell-tunedweighting coe˚cientenforcessmoothswitchingwithoutsacri˝cingsystemperformance. Besidesthedemonstratedswitchingsmoothness,thisstudyalsoprovidesvaluableinsights regardinghowtotunethemodel-basedcontrollergain.Notethattuningcontrolgainplaysan essentialroleinimplementingmodel-basedcontrollersforpracticalapplications.Duetosystem modelingerror,highgaincontrollersoftenleadtoinstabilityordegradedsystemperformance, whilelowgaincontrollersmightnotimprovesystemperformancemuch.Therefore,theabilityto designacontrollerwithanadequategainisessentialinpractice,andtheproposedmethodmakesit possibletodesigncontrollerswithdi˙erentgainsbymodifyingICCconditions.TheICCcondition tuningalongwiththelinesearchofsmoothnessweightingcoe˚cientmakesitpossibletobalanced switchingsmoothnessandsystemperformanceinpractice,whichisverybene˝cialforpractical applications. 72 3.2Blended-Wing-Body(BWB)AirplaneFlexibleWing 3.2.1LPVmodelingofBWBairplane˛exiblewing BeforethesmoothswitchingLPVcontroldesignisappliedtotheBWBairplane˛exiblewing model,inthissection,weconsidertheLPVmodelingofBWB˛exiblewing;seeFigure3.14 foraschematicillustration.AssumethattheBWBairplaneis˛yingata˝xedaltitudebutwith varying˛ightspeed.ThemainbodyofBWBisgriddedintosixbeamelements,andeachwing isgriddedintofourbeamelements.Theinnerthreeelementsateachwingareselectedascontrol surfaces,labeledasU1-U6inFigure3.14,andwingbendingdisplacementsaretobesuppressed byactivatingthecontrolsurfaces.Inordertomodulatethevibrationalbehaviorsofentireairplane wings,atotalof18bendingdisplacementsareselectedassystemoutputs.Forexample,outputs1 and9arethenodaldisplacementsattheright-wingrootandrightwingtipinFigure3.14. TheLPVmodelingprocedurecanbedescribedasfollows: ‹ AbundleofLTIfull-ordermodels(FOMs)arederivedbylinearizingnonlinearaero-elastic modelateachgridded˛ightspeed[87]; ‹ FOMsarethentransformedintomodalcoordinatesandallsystemmodesareproperlyaligned totrackmodevariationsfromone˛ightspeedtothenext; ‹ Model-reductionisconductedtokeepthemostsigni˝cantmodesovertheentiregridded ˛ightenvelop[88]; ‹ Linearinterpolationoverthealignedreduced-ordermodelstoattainthea˚neLPVmodel. Theinterpolationofalignedmodesisabletocapturethevariationofsystem'scoupled aerodynamicmodewithvarying˛ightspeed,whichcannotbeachievedbydirectinterpolation ofLTIsystemmatrices[88]. Inthisstudy,theschedulingparameterischosentobetheairplane˛ightspeed,anditranges from110to130m/s.Abundleofreduced-orderLTImodelsarederivedatvarying˛ightspeeds 73 Table3.1:Modedescriptioninreduced-ordermodel ModeID Rigid-bodycomponent Flexiblecomponent Note M1 Plungingandpitching Firstsymmetricout-of-planebending Bending/torsioncoupling M2 Plungingandpitching Secondsymmetricout-of-planebending Bending/torsioncoupling M3 Plungingandpitching Firstsymmetricin-planebending Bending/torsioncoupling M4 Roll Secondanti-symmetricout-of-planebending Bending/torsioncoupling M5 - Firstanti-symmetricin-planebending Bending/torsioncoupling M6 - - Aerodynamicdominantmode andatanincrementof0.5m/stocapturemodelvariation.Sixdominantmodesarekeptinthe reduced-orderLTImodels,asmarkedbyM1-M6inFigure3.15.Physicalmeaningsofthese modesaresummarizedinTable3.1.Notethatallthebending/torsioncouplinge˙ectscomefrom thebacksweptofthewing,andthewingstructuralrigidityitselfhasnoinherentbending/torsion coupling.Thevibrationmodesstaystablewhen˛ightspeedisbelow115m/s,andmodeM1 becomesunstablebeyond115m/sasshowninFigure3.15. Figure3.14:SchematiclayoutofBWBairplanecon˝guration Thea˚neLPVmodelisobtainedbylinearlyinterpolatingthe˝rstandlasteigenvaluesofeach mode.Asshownintheclose-upviewofFigure3.15,thesolidlineshowsthelinearinterpolationof theeigenvalues,wherecrossesdenotethelociofactualeigenvaluesasfunctionof˛ightspeed.As aresult,intheinterpolateda˚neLPVmodel,systemdampingcoe˚cientisapproximatedwhile systemstabilityremainsunchangedovertheentire˛ightenvelope.Similarly,allothersystem matricesarealsoobtainedbyfollowingthesamelinearinterpolationprocess.Theresulteda˚ne LPVmodelconsistsof12states(6modes),6controlinputs(controlsurfacesde˛ectionangles) and18performanceoutputs(wingbendingdisplacements). 74 Figure3.15:Rootlociofopen-loopsystemwithvarying˛ightspeed Therearetwomaincontroldesigngoals.Oneistorobustlystabilizetheclosed-loopsystem underboundedmodelingerrorandtheotheristosuppresswingbendingdisplacements,excitedby thegustdisturbance,usingcontrolsurfacesonthewing.Asaresult,twoindependent H 1 and H 2 inputchannelsareusedalongwithtwoindependent H 1 and H 2 outputchannelsforthesystem describedinEqn.(2.1),wheremodelingerrorismodeledassystemdisturbanceinput w 1 excited bythesystemoutput z 1 throughuncertainty andtheclosed-looprobuststabilityisachieved bysatisfyingthedesired H 1 performance;thegustdisturbanceistreatedasdisturbanceinput w 2 withassociated H 2 performanceoutput z 2 tobeoptimizedforsuppressingbendingdisplacement z 2 causedbythegustdisturbance.Inaddition,ICCconstraintsareimposedoncontrolinputs orde˛ectionanglesofcontrolsurfaces,sothattheyarehard-constrainedtooperatewithintheir limits.InordertoapplyswitchingLPVcontrol,theswitchingLPVmodelisdevelopedbydividing 75 theschedulingparameterrangeintomultipleoverlappingsubregions,asshowninFigure3.16.In thenextsubsection,agenericLPVmodelwith H 1 and H 2 channelswillbeconsideredandthe associatedsystemperformancesde˝ned. Figure3.16:Three-subregionpartitionforschedulingparameter 3.2.2MixedICC = H 1 LPVcontrol The H 2 outputsofinterestarebendingdisplacements,whilethe H 1 outputsincludebending displacementsandcontrolinputs.Theweightingmatrix Q ischosentobeidentitymatrix,that is,alloutputsareweightedequally.Theschedulingparameterischosenasabiasedsinusoidal function, ( t )=110+20sin( t= 20) m/s,asshowninFigure3.17.Therefore,withinthetime intervalof [0 ; 20 ˇ ] second,theschedulingparameterisboundedas110m/s 130m/s,and itsrateboundedas-1m/s 2 _ 1m/s 2 .Ingeneral,theschedulingparametertrajectoryshould satisfytheboundaryconditionsforboth and _ ,andbeatleastpiece-wisedi˙erentiable.Itis commonlyacceptedthatthevariationoftheschedulingparametersmustbe "slow" comparedto thesystemdynamics,becausedesigninganLPVcontrollerforfast-varyingschedulingparameters isachallenge[89]. 3.2.2.1Constraintsandperformancetrade-o˙ InthemixedICCand H 1 (orrobustICC)LPVcontrolproblem,bothcontrolinputconstraints androbustnessrequirementwouldsigni˝cantlyimpacttheoptimalsolutiontothePLMIs.Hence, atrade-o˙studyisconductedtobetterunderstandthecharacteristicsofLPVmodels.Figure3.18 76 Figure3.17:Schedulingparameter(˛ightspeed)trajectory showsthecompletetrade-o˙betweenthecontrole˙ort U ,therobustnesslevels 1 ,andtheoutput performance trace ( W ) .Foragivenrobustnesslevel,thetrade-o˙contourillustratesthatlarger controlinputconstraintleadstosmalleroutputcovariance,hencebetter H 2 performanceforthe closed-loopsystem.Inaddition,withsmallcontrole˙ort,outputperformancewillbedegraded, resultinginalargeoutputcovariance.Anincreaseincontrole˙ortleadstonotableimprovement onsystem H 2 performancewithwiderrangeofadmissiblerobustnesslevels.Thisdemonstrates thatlargercontrolinputcane˙ectivelycompensatefortherobustnessconstraints. Furthermore,basedontheSmallGainTheorem[33],theclosed-loopsystemsatisfyingthe condition jj T 1 jj 1 1 iswell-posedandinternallystableforalluncertaintysatisfying jj jj 1 < 1 1 ,where canbeconsideredasaninterconnectionfrom z 1 to w 1 ,asshowninFigure1.3. InFigure3.18,witha˝xed U ,itisobviousthatwithmorestringentrequirementonrobust performance,i.e.smaller 1 ,theoutputperformancedegradeswithincreasein trace ( W ) ,leading toworsen H 2 performance.Notethat,while 1 decreasesincrementally, trace ( W ) increases or H 2 performancedegradesmuchdrastically.Thiscanbeexplainedbythereciprocalrelation betweenuncertainty and 1 . 77 Figure3.18:Trade-o˙betweencontrollimit U andtrace(W)atdi˙erentrobustnessconditions Thetrendathigherorlowerrobustnesslevelrevealsanimportantimplicationforcontroller design.Atlowerrobustnesslevel,forinstance 1 =2 ,theachievable H 2 performanceremains almostunchangedwhen U> 0 : 01 .Thisindicatesthattherobust H 1 performancerequirement isnotthedominantfactorforcontroldesignandthe H 2 performancecanbeachievedwitha relativelysmallcontrole˙ort.However,athigherrobustnesslevel,forinstance 1 =0 : 5 ,the H 1 performancebecomescriticalforcontroldesign.Asaresult,inordertoachieveaspeci˝c H 2 performance,morecontrole˙ortisrequired.Itisalsoobservedthattheachievable H 2 performance degradeswithincreasedrobustnesslevel.Basedontheabove-mentionedtrade-o˙s,theconstraints forthecontroldesignarechosentobe U =0 : 02 and 1 =1 ,whichensureagoodrobustness margintohandlemodelingerrorwithgoodbalancebetween H 2 performanceandcontrole˙ort. 78 3.2.2.2MixedICCand H 1 ControlProblem ThemixedICCand H 1 controlproblemisto˝ndastate-feedbackgain-schedulingcontroller(2.52) fortheLPVsystem(2.1),whileminimizingtheupperboundof H 2 performancecost[90,35] min K ( ) trace ( W ) ; (3.1) suchthat: ‹ theclosed-loopsystem(2.5)isexponentiallystable, ‹ thefollowingconstraintsofrobustnesslevelandcontrolinputcovariancearesatis˝ed, jj T 1 ( K ( ) ;s ) jj 1 1 ; (3.2) Cov ( u k ( t )) U k ;k =1 ; 2 ;:::;n u ; (3.3) where 1 > 0 isthegiven H 1 -normboundonsystemrobustness,and U k thegivenboundonthe controlcovariance Cov ( u k ( t )) forthe k th controlinput u k ( t ) de˝nedbelow, Cov ( u k ( t ))= 1 2 ˇ Z 1 T u ( K ( ) ;j! ) T u ( K ( ) ;j! ) d! ; (3.4) and T u ( K ( ) ;s ):= T w 2 ! u ( K ( ) ;s ) denotesthetransferfunctionfrom w 2 ( t ) to u ( t ) forthe LPVsystem(2.5).Notethat,fordeterministicsignal,covarianceisde˝nedintermsoftime correlation[51,91,7,92]. Asaresult,theproposedmixedICCand H 1 controlproblemhasinterestinginterpretationsin stochasticanddeterministicperspectives.Thestochasticinterpretationassumesthattheexogenous input w 2 ( t ) isanuncorrelatedzero-meanwhitenoisewithunitintensity.Then,themixedICCand H 1 controlproblemistominimizetheoutputcovariance(orRMS-value)whilesatisfyingmultiple controlinputcovarianceconstraintsand H 1 robustperformancecriterion.Thecontrolinput covarianceconstraintscanbeconsideredasconstraintsonthevariancesofthecontrolactuation.In otherwords,theproposedcontrolprovidesthebestoutput H 2 performancewiththegivencontrol 79 H 2 performanceandrobust H 1 constraints.Ontheotherhand,thedeterministicinterpretation assumesthattheexogenousinput w 2 ( t ) isanunknowndisturbancethatbelongstoabounded L 2 set.Then,themixedICCand H 1 controlproblemistominimizethesquaresummationof L 2 to L 1 gainsfrom w 2 ( t ) toindividualoutputchannel z 2 ;k ( t ) for k =1 ; 2 ;:::;n z 2 ,subjecttothe L 2 to L 1 gainconstraints(3.3)on u k ( t ) for k =1 ; 2 ;:::;n u andthe H 1 constraint(3.2).In otherwords,theproposedcontrolproblemistominimizetheweightedsumoftheworstcasepeak valuesofperformanceoutputsubjecttotheconstraintsonworst-casepeakvaluesofcontrolinputs andthe H 1 constraint.Itshouldbenotedthatthe L 2 - L 1 gainfrom w 2 ( t ) to z 2 ( t ) isde˝nedin Whiteetal.[91]asfollows, ˙ 1 2 ˇ Z 1 T 2 ( K ( ) ;j! ) T 2 ( K ( ) ;j! ) d! =sup w 2 2L 2 ;z 2 2L 1 ; jj w 2 jj 2 6 =0 jj z 2 ( t ) jj 2 1 jj w 2 ( t ) jj 2 2 (3.5) where ˙ [ ] denotesthemaximumsingularvalueoperator. 3.2.2.3Time-domainsimulationresults Giventherangeof and _ ,thecontrolinputconstraints,andtherobustnesslevel,theLPVmodelof theBWBairplaneissimulatedwhenitissubjectedtoasharp-edgedgustdisturbancefor5seconds. Figures3.19and3.20showthewingroot(output1)andwingtip(output12)bendingdisplacement oftherightwingforopen-loopcase,andascanbeseentheresultsareunstable.Therefore,a state-feedbackLPVcontrollerintheformofEqn.(2.52)isdesignedtostabilizewingelementsand suppressthebendingdisplacement. UsingTheorem3,astate-feedbackLPVcontrollercanbedesignwithscheduledcontrolgain matrixofdimension 6 12 ,mapping12statesto6controlinputs.NotethattheLPVmodelis developedinthemodalcoordinate,themeasuredorobservedstatesinoriginalcoordinateneedto betransformedtothemodalcoordinate.Inpracticalimplementation,schedulingparameter(˛ight speed)willbeonlinemeasuredineachsamplingtime,andcontrolinputsofaltering˛apangles canbecalculatedfromcorrespondingcontrollergainmatrixandmeasuredorobservedstates. 80 Figure3.19:Bendingdisplacementatwingroot Figure3.20:Bendingdisplacementatwingtip Todemonstratethee˙ectofcontrolinputconstraintsandrobustnesslevelsto H 2 performance, multiplesimulationsareperformedforcomparison.Whenrobustnesslevel 1 =1 is˝xed,each controlinputisidenticallyconstrainedbyvariousupperbounds U .Figures3.21and3.22showthe bendingdisplacementatwingrootandwingtipfor U =0 : 01 ; 0 : 02 ; 0 : 04 .Ascanbeseen,during thegustdisturbance,theoutputsareconvergedandbounded.Inaddition,withlargercontrol inputs,theoutputresponseshavesmallerovershootandfasterconvergentrate,indicatingthat H 2 81 Figure3.21:Wingrootbendingunderdi˙erent U Figure3.22:Wingtipbendingunderdi˙erent U outputperformanceareimproved.AsshowninFigures3.23-3.28,thecontrolinputsU1-U6are increasedbymorethantwicewhenupperboundsbecomedoubled.Thiscomparisonindicatesthat theselectionof U =0 : 02 o˙ersagoodbalancebetweentheperformanceandthecontrole˙ort, whichproducesanupperboundof u =0 : 14 rad ˇ 8 . 82 Figure3.23:Controlinput1underdi˙erent U Figure3.24:Controlinput2underdi˙erent U 83 Figure3.25:Controlinput3underdi˙erent U Figure3.26:Controlinput4underdi˙erent U 84 Figure3.27:Controlinput5underdi˙erent U Figure3.28:Controlinput6underdi˙erent U 85 All6inputsarecomparedtoshowhowthecontrollawallocates6independentinputsto suppressairplanewingdisplacements.Itcanbeobservedthatinputs1and3aredistributedby similarcontrolauthority.Theequaldistributionofcontrolauthorityalsohappensoncontrolinputs 2and4,controlinputs5and6. When U =0 : 02 is˝xed,therobustnesslevel 1 isvariedtostudyitsin˛uenceonoutput performance.AsshowninFigures3.29and3.30,thebendingdisplacementatwingrootand wingtipareimprovedwhen 1 increasesfrom0.5to1.However,theresponsesremainalmost unchangedwhen 1 increasesfrom1to2.Thisphenomenonmatcheswellwiththeearliertrade-o˙ studyshowninFigure3.18.Figures3.31and3.32showthecontrolinputswhentherobustnesslevel isgreaterthan1,ascanbeseenthat 1 isnolongerthedominantfactorforoutputperformance. After U ischosen,theLPVcontrollerisdesignedandappliedtoactualgriddedLTImodelsto validateitsfeasibility.Figure3.37showstherootlocioftheclosed-loopsystemwithvarying˛ight speed.Asshown,theproposedLPVcontrollerstabilizesthegriddedLTImodelssubjecttoinput constraints,whileminimizingtheoutput H 2 performance.However,inane˙orttoreducecontrol energy,somemodesarekeptunchangedbytheproposedcontroller.ComparingFigures3.15 and3.37,themodes( M 1 ;M 2 ;M 4 ),whichdominatein z directionalbendingmotion,havebeen signi˝cantlyshifted,whileothermodes( M 3 ;M 5 ;M 6 )arekeptunchanged.Inaddition,Figure3.38 showstheICCcostor H 2 normoftheclosed-loopsystemwiththeLPVcontrollerappliedtothe interpolatedLPVsystemandactualgriddedLTImodels,respectively.Theirmagnitudesarevery closeandupperboundedby trace ( W ) .WhencombiningwithFigure3.37,Figure3.38e˙ectively validatesthattheproposedinterpolationofLTImodelsandLPVcontrollerdesignisfeasiblefor vibrationcontroloftheBWBairplane. 86 Figure3.29:Wingrootbendingunderdi˙erent 1 Figure3.30:Wingtipbendingunderdi˙erent 1 87 Figure3.31:Controlinput1underdi˙erent 1 Figure3.32:Controlinput2underdi˙erent 1 88 Figure3.33:Controlinput1underdi˙erent 1 Figure3.34:Controlinput2underdi˙erent 1 89 Figure3.35:Controlinput1underdi˙erent 1 Figure3.36:Controlinput2underdi˙erent 1 90 Figure3.37:Rootlociofclosed-loopsystem Figure3.38:Closed-loopICCcostwithLPVandLTImodels 91 3.2.3SmoothswitchingLPVcontrolbysimultaneousdesign 3.2.3.1Time-domainsimulationresults ThescenariothataBWBairplaneexperiencesasharpgustdisturbanceisconsideredinthisstudy. Thegustdisturbanceisassumedtoinduceaconstantshiftangle w 2 onallcontrolsurfacesfor t 2 [0 ; 9] second,andweassumethat w 2 =0 : 005 rad ˇ 0 : 28 .AsshowninFigure3.39,two switchingeventshappenat t = T 1 =3 s and t = T 2 =8 s .Therefore,withinthetimeintervalof [0 ; 10] second,theschedulingparameterisboundedby110m/s 130m/s,anditsratebounded by-1m/s 2 _ 1m/s 2 .Notethatwhentheopen-loopsystemissubjecttogustdisturbance, bendingdisplacementsareunstable,asshowninFigure3.40.Afamilyofsmooth-switchingmixed ICC = H 1 LPVDOFcontrollersaretobedesignedusingTheorem7forstabilityaswellasachieving abalanced H 2 performanceandswitchingsmoothness,withguaranteed H 1 robustperformance (at =10 ). Figure3.39:Schedulingparameterwithswitchingevents Thetrade-o˙relationshipisexploredbylinesearchofweightingcoe˚cients underdi˙erent ICCconstraints: U 1 =8 ; U 2 =12 and U 3 =20 .AsshowninFigure3.41,theswitching smoothnessindexcanbereducedbydecreasingtheweightingcoe˚cient ,whichresultsinan 92 Figure3.40:Unforcedbendingdisplacementsatwingroot(upper)andwingtip(lower) Figure3.41:Trade-o˙between trace ( W ) andsmoothnessindex I sm increased H 2 performanceindex trace ( W ) ordegraded H 2 performance.Thisillustratesthatsys- temperformanceissacri˝cedinordertoenforceswitchingsmoothness.Especially,when < 10 2 , thesystemperformanceindexincreasessigni˝cantlyforallthreeICCconstraints,indicatingthat 93 systemperformanceisdegradingmuchdrasticallyinordertoachievesmootherresponses.Thus, anoptimalweightingcoe˚cientischosentobe =10 2 toattainsmoothswitchingwithacceptable systemperformance.Todemonstratethee˙ectivenessoftheproposedmethod,extensivesimula- tionsareconductedbyconsideringthreedi˙erentcontrollers:1)non-switchingLPVcontroller,2) un-smoothswitchingLPVcontroller,and3)theproposedsmooth-switchingLPVcontroller.And thesecontrollersareappliedtotheBWB˛exiblewingmodelforvibrationsuppression. Figures3.42and3.43showthebendingdisplacementatwingroot(output1)andwingtip (output12),respectively,whileFigures3.44-3.49showthecontrolallocationofde˛ectionangles ofsix˛apsaccordingtothreedi˙erentcontrolstrategies. Figure3.42:Upper:comparisonatwingrootwithsmooth/un-smoothswitchingcontroller;Lower: comparisonatwingrootwiththreecontrolmethods 94 Figure3.43:Upper:comparisonatwingtipwithsmooth/un-smoothswitchingcontroller;Lower: comparisonatwingtipwiththreecontrolmethods Intheuppersub-˝gureofFigure3.43,smooth(blue)andun-smooth(red)responsesofbending displacementatwingrootareshown.Atswitchingevent T 1 =3 s ,controller1isswitched tocontroller2,andthesuddenchangesofun-smoothcontrollerscauseabruptjumpsforallthree di˙erentICCconditions.Ontheotherhand,thesmooth-switchingLPVcontrollersenforcesmooth outputresponses,withslightlyincreasedbendingdisplacementasaminorpenaltyonsystem performance.Similarbehaviorscanbeobservedattheswitchingevent T 2 =8 s .Anothertrade-o˙ relationshipcanbeobservedfromoutputresponses.Di˙erentICCconstraintswillin˛uencethe optimalachievablesystemperformance.Withlargercontrolinput,thebendingdisplacementscan besuppressedevenfurther,however,when U> 12 ,muchmorecontrole˙ortwillbeconsumed tofurtherimprovesystemperformance,asseenfromcontrolresponsesinFigures3.44-3.49. Therefore,thehardconstraintoncontrolinputischosenas U =12 ,inordertoachieveacceptable performanceandenergysaving. Thelowersub-˝gureofFigure3.43showsthecomparisonofwingtipresponseswiththree di˙erentcontrollers.Asshown,allthreecontrolmethodsareabletostabilizeandsuppressbending 95 displacementsfortheentire˛ightspeedenvelope.Itcanbefurtherobservedthatbothsmoothand un-smoothswitchingLPVcontrollersproduceasmallermagnitudeofbendingdisplacementsthan thenon-switchingLPVcontroller,andthisisachievedbyrelaxingthePLMIconservativenessand enforcingtheoptimalperformanceoneachsubregion.However,un-smoothswitchingLPVleads toundesirablejumponthebendingdisplacementatwingtip,whichise˙ectivelysmoothenedby theproposedsmooth-switchingLPVcontroller. Theresponsesofcontrolinputalsodemonstratethee˙ectivenessoftheproposedcontrol method.Intheuppersub-˝guresofFigures3.44-3.49,theun-smoothcontroldesignresultsin controlinputsexhibitingsharpjumpattheswitchingevents,buttheproposedsmooth-switching LPVcontrollerse˙ectivelyremovethesejumps.Especiallyatswitchingevent T 2 =8 s ,un-smooth switchingcontrollercommandsthecontrolsurfacestode˛ectinoppositedirectionswithinavery shorttime,whichimposesaseverecapacityburdenontheactuator.Smooth-switchingcontroller, ontheotherhand,allocatesthede˛ectionanglesofcontrolsurfaceswithsmoothcontrolcommands whenswitchingoccurs.Inthelowersub-˝gures,controlcommandsofthreecontrolmethodsare compared.UnlikeswitchingLPVcontrol,non-switchingLPVcontrolresultsinaconservativecon- trolinputofaverysmallmagnitudeduetotheconservativenessintroducedinPLMIs.Un-smooth switchingLPVcontrolisabletorelaxconservativenessandassignslightlylargercontrolenergy, leadingtoimprovedvibrationsuppressionofbendingdisplacements.However,byminimizing controlgaindi˙erencesintheoptimizationcostfunction,smooth-switchingLPVcontrolcanresult inmuchsmootherresponseswithslightdegradationonsystemperformance,whichisstillbetter thantheperformanceoftheun-smoothswitchingLPVcontrol. 96 Figure3.44:Upper:control1responsescomparisonwithsmooth/un-smoothswitchingcontroller; Lower:control1responsescomparisonwiththreecontrolmethods Figure3.45:Upper:control2responsescomparisonwithsmooth/un-smoothswitchingcontroller; Lower:control2responsescomparisonwiththreecontrolmethods 97 Figure3.46:Upper:control3responsescomparisonwithsmooth/un-smoothswitchingcontroller; Lower:control3responsescomparisonwiththreecontrolmethods Figure3.47:Upper:control4responsescomparisonwithsmooth/un-smoothswitchingcontroller; Lower:control4responsescomparisonwiththreecontrolmethods 98 Figure3.48:Upper:control5responsescomparisonwithsmooth/un-smoothswitchingcontroller; Lower:control5responsescomparisonwiththreecontrolmethods Figure3.49:Upper:control6responsescomparisonwithsmooth/un-smoothswitchingcontroller; Lower:control6responsescomparisonwiththreecontrolmethods 99 3.3NumericalExamplesforSequentialDesign Inordertodemonstratethefeasibilityoftheproposedmethod,twoexampleswillbegiven andresultswillbecomparedwithsimultaneousdesign,aswellasnon-switchingLPVdesign. Furthermore,theinterpolationrate andschedulingparametervariationalratearevariedtostudy theirultimatein˛uencetoclosed-loop H 1 performance . TheformulatedPLMIsareofin˝nite-dimension,andtheycanbetransformedinto˝nite- dimensionalbymeansofvariousrelaxationmethods.Tonumericallytacklethisoptimization problem,coe˚cientcheckinmulti-simplexdomainbyPolyatheorem[19]isapplied.Gridding technique[68]orotherrelaxationmethods[73]canalsobepotentiallyusedtotacklethisproblem. SomesoftwareisavailabletomanipulatethePLMIsandhandletheconvexoptimization.Inthis study,ThePLMIsaresolvedbyusingtheparserROLMIP[93]andYALMIP[76],whichwork jointlywithoptimizationtoolSEDUMI[77].Computationisoperatedusingacomputerwith Intelcorei7-4770TCPU@2.50GHzand16GRAM,andcomputationtimesofthreedesign approachesareobtainedbyrunning tic and toc commandsinMATLAB,andtheyarecomparedto showcomputationale˙orts. 3.3.1Example1 TheLPVmodelinreference[94]isrevisitedtoillustratethefeasibilityoftheproposedsequential designapproachofsmoothswitchingLPVcontrollers.ConsidertheLPVmodelwitha˚ne dependencyofone-dimensionalschedulingparameter , A ( )= 2 6 4 25 : 9 60 1 20 40 34 64 3 7 5 ;B u = 2 6 4 3 2 3 7 5 ; B w = 2 6 4 0 : 03 0 : 47 3 7 5 ;C = 2 6 4 11 00 3 7 5 ;D w = 2 6 4 0 0 3 7 5 ;D u = 2 6 4 0 1 3 7 5 : Thetime-varyingschedulingparameter ( t ) isboundedas 0 ( t ) 1 ,anditsvariational rateisboundedas v _ ( t ) v .Thedomainof isassumedtobepartitionedasthree 100 Table3.2:Comparisonofthreedi˙erentdesignmethodsineachdesigniteration non-switching sequential simultaneous No.LMIs 8 8 27 No.variables 14 14 42 tic/toctime(s) 0.23 0.21 0.39 overlappingsubregionsof range, [0 ; 0 : 4] , [0 : 3 ; 0 : 7] , [0 : 6 ; 1] ,andvariationalrateboundiskeptas notdivided.Controllerdecisionvariables P ( ) and Z ( ) areassumedtobeinthea˚neformas P ( )= P 0 + P 1 ,and Z ( )= Z 0 + Z 1 .Controllerdecisionvariables P 0 ;P 1 ;Z 0 ;Z 1 aresought tominimizethe H 1 performanceindex ,whilePLMIsformulatedbydi˙erentdesignapproaches aresatis˝ed.Inthenon-switchingLPVcontroldesign,asingle onentireschedulingparameter regionisminimized.Inthesimultaneousdesignapproach, 1 ; 2 and 3 areassociatedwiththree subregions,and max f 1 ; 2 ; 3 g isminimizedinobjectivefunction.However, 1 ; 2 and 3 are minimizedsequentiallyoneachsubregionbysequentialdesignapproach. Figure3.50: 1 obtainedbysequential(black),simultaneous(blue)andnon-switching(red)design approach 101 Figure3.51: 2 obtainedbysequential(black),simultaneous(blue)andnon-switching(red)design approach Figure3.52: 3 obtainedbysequential(black),simultaneous(blue)andnon-switching(red)design approach 102 Withgiven 0 =2 ,interpolationrate andvariationalrate arevariedtogetinsightofhow theyin˛uencethe H 1 performance 1 ; 2 and 3 plottedinFigures.3.50-3.52.Thenon-switching designresultisplottedasredsurface,while 1 , 2 and 3 obtainedbysimultaneousandsequential designsareplottedinblueandblacksurfaces,respectively. Itcanbeobservedthat,bothswitchingdesignmethodsleadtoanimprovedclosed-loopsystem H 1 performanceovernon-switchingLPVcontrol.Inmostareaofshownregion,sequential designapproachresultsinasmaller inmagnitudethansimultaneousdesignapproachonthree subregions,indicatingdesignconservativenesscanberelaxedinthesecases.Inthesituation thatlargevariationalrate andaggressiveinterpolationrate ,switchingsmoothnessbetween adjacentcontrollerswillbeimproved,butconservativeconstraintsofadditionalrelative ˙ -stability willbeintroduced.Asaconsequence, H 1 performancebysequentialdesignisworsethanthat ofsimultaneousdesign.Thus,thereexistsatrade-o˙relationshipbetweensystemperformance andswitchingsmoothnessrepresentedbyinterpolationcoe˚cient.Performancedegradationisa sacri˝cetoguaranteetherobustperformancebyinterpolatedcontrollervariables.Inotherwords, thelimitationofthismethodisthattuningworkmaybeneededifoptimizingsystemperformance istheobjectiveratherthanreducingdesigncomplexity. Thetime-domainresponsesofthreedi˙erentdesignapproacheshavebeensimulatedandcom- paredinFigure3.53.Systemdisturbanceissetas w ( t )=0 : 5 for t 2 [0 ; 4 : 5] secondand w ( t )=0 for t> 4 : 5 second.Schedulingparametertrajectoryissetas ( t )=0 : 3+0 : 1 t: Inthesequen- tialdesignapproach,interpolationrateandvariationalratearechosenas =2 and =0 : 02 , respectively.Itiseasytoobservethatswitchingcontrollersbythesequentialdesignleadtostate responseswithsmallersignalnormsthanthesefromsimultaneousdesignapproachandnonswitch- ingcontrol.ThisconclusionmatcheswellwiththeseresultsinFigures3.50-3.52thatsequentially designedswitchingcontrollerleadstosmaller H 1 norm.Moreover,simultaneousdesignresults injumpsatswitchinginstantsof t =1 and t =4 second,whereastheproposedsequentialdesign leadstosmoothresponsesbecausecontrollergainsareinterpolatedoveroverlappedsubregions. After t =4 : 5 secondwhensystemdisturbancedisappears,statesareregulatedto 0 byallthree 103 controllers. Table3.2summarizesthenumberofrelaxedLMIsandcontrollerdecisionvariablesandcom- putationtimebythreedi˙erentdesignapproach.Non-switchingdesignapproachdealswithfewer LMIsandsearchforminimainasmallerspaceofvariables,thuslesscomputationaltimeisutilized intheoptimization.However,theoptimized H 1 performanceisworsethanbothswitchingLPV controldesignapproaches.Sequentialdesigniteratestheoptimizationoneachindividualsubregion sequentially,thusineachdesigniteration,sequentialdesigndealswithsameamountsofLMIsand variableswithnon-switchingdesign,butwithinsmallersizeofsubregion.Thesimultaneousdesign approachisimposedwithallLMIsandvariables,thushasthelargestcomputationalcomplexity. Notethatinthisexample,totalsolvingtimeofsequentialdesignisslightlylargerthansimultaneous designapproach,anditcanbepossiblyreasonedthatoptimizationproblemformulatedbylow-order systemcanstillbewellhandledbythesimultaneousdesignapproach. Figure3.53:Closed-loopsystemstatesresponsesbysequential(black),simultaneous(blue)and non-switching(red)designapproaches 104 3.3.2Example2 The A ( ) matrixinExample1ismodi˝edintotwo-dimensionala˚nedependencyof 1 and 2 ,whileothersystemmatricesareunchanged.Thedomainsoftwoschedulingparametersare 1 =[0 ; 10] , 2 =[0 ; 7] ,andtheyarerespectivelydividedinto [0 ; 7] ; [5 ; 10] and [0 ; 5] ; [3 ; 7] .With given 0 =0 : 1 anddivisionofschedulingparameterdomain,threedi˙erentdesignapproachesare conductedagaintocompareoptimized H 1 performanceandcomputationale˙orts. A ( )= 2 6 4 20 2 1 16 128 6 2 3 7 5 SimilartoExample1,controllerdecisionvariables P ( ) and Z ( ) areassumedtobeinthe a˚neformas P ( )= P 0 + P 1 1 + P 2 2 ,and Z ( )= Z 0 + Z 1 1 + Z 2 2 .Oneachsubregion, controllerdecisionvariables P 0 ;P 1 ;P 2 ;Z 0 ;Z 1 ;Z 2 aresoughttominimizethe H 1 performance index ,whilePLMIsformulatedbydi˙erentdesignapproachesaresatis˝ed. Theoptimizedsystemperformanceindexesonsubregionsbysequentialdesign,simultaneous designandnon-switchingdesignareplottedbyblack,blueandredsurfacesinFigures.3.54- 3.57.Non-switchingcontroldesignminimizes H 1 performance overentiresubregion.In thesimultaneousdesignapproach, 11 ; 12 ; 21 and 22 areassociatedwithfoursubregions, and max f 11 ; 12 ; 21 ; 22 g isminimizedinobjectivefunction.However,theyareminimized sequentiallyoneachsubregionbysequentialdesignapproach.Itcanbeseenthatinmostcases, sequentialdesignapproachobtainssmaller magnitudes,inotherwords,bettersystemperformance thansimultaneousdesignandnon-switchingLPVcontroldesign. Fromsimulationresults,theconservativenessofhigh-dimensionaloptimizationinsimultaneous designcanberelaxedbyiteratinglow-dimensionaloptimizationinthesequentialdesignapproach. However,inthescenarioofaggressiveinterpolationrate andlargevariationalrate ,sequential designprovidesveryconservative ˙ -stabilityandhenceworsesystemperformanceisobtainedby sequentialdesignapproach.Thissurfacegivesinsightofhow ˙ -stabilitywilltrade-o˙with H 1 performanceinswitchingLPVcontrol,andgiveshintonhowtofurthertunesubregiondivision 105 andinterpolationrate. Table3.3summarizesthenumberofLMIsandcontrollermatricesvariablesandcomputational timebythreedi˙erentdesignapproach.BothswitchingLPVcontroldesignapproachessacri˝ce moresolvingtimetoobtainbettersystemperformances.Inthesimultaneousdesign,muchmore constraintsthansequentialdesignareimposed,thusmoresolvingtimearetakentoobtainan optimalsolution.However,theresultingsystemperformanceiscontrarilyworsethanthatof sequentialdesignapproachifinterpolationcoe˚cientisproperlychosen.Inthisexamplewith4 subregionsformedbytwo-dimensionalschedulingparameters,sequentialdesignapproachexceeds simultaneousdesignapproachintermsofcomputationale˙ortsandachievedsystemperformance. Figure3.54: 11 obtainedbysequential(black),simultaneous(blue)andnon-switching(red)design approach 106 Figure3.55: 12 obtainedbysequential(black),simultaneous(blue)andnon-switching(red)design approach Figure3.56: 21 obtainedbysequential(black),simultaneous(blue)andnon-switching(red)design approach 107 Table3.3:Comparisonofthreedi˙erentdesignmethodsineachdesigniteration non-switching sequential simultaneous No.LMIs 40 40 192 No.variables 21 21 84 tic/toctime(s) 0.40 0.47 3.48 Figure3.57: 22 obtainedbysequential(black),simultaneous(blue)andnon-switching(red)design approach 108 CHAPTER4 OPTIMALSENSORPLACEMENT 4.1Introduction Alightand˛exibleairplanewingisbene˝tedfromstructure˛exibility,anditfeatureswithhigh aerodynamicandfuele˚ciency.However,thestructure˛exibilityposesagreatchallengetocontrol systemdesignforactivevibrationsuppression.Structuralcontrolcommunityhasmadevarious attemptsondevelopinge˙ectivecontroltechniquesinordertosuppressvibration,avoidstructural failureandenlarge˛ightstabilitymargin.Amongcontrolsystemcomponents,positioningand selectionofsensorsplayaroleofgreatimportancebuthavenotbeenpaidenoughattention. Sensorplacementneedstobeintegratedintobothmodelingandcontroldesign,andwillultimately in˛uencestateobservability,aswellastheachievableclosed-loopsystemperformance. Asalarge-scalestructuralsystem,a˛exiblewingexhibitscoupledaero-structuredynamicsat various˛ightconditions[87,88].Multiplenodalpointsalongthewingspanareoftenselectedto getinsightonoverallstructuralbehavior.Moreover,multiplesensorsareneededtobeinstalledat di˙erentlocationstoprovidefeedbackinformationforactivevibrationcontrol.Sensorpositioning onalarge-scale˛exiblewingstructureisevenmorecomplex.Sparsedensityofmeasuredlocations cannotcaptureallvibrationalmodesandverylikelyleadtonofeasiblesolutionforanoutput feedback-basedcontroller.Onthecontrary,toodenseofsensorplacementwillincreasebothmodel andcontrollerdimensions,whichdramaticallyincreasescomputationalcomplexityandpotentially limitsachievablesystemperformance.Thesepracticaldemandscallforasystematicmethodto compute,evaluateanddetermineoptimalnumberofsensorsandtheirplacement. Inthisdissertation,weinvestigatetheproblemthatwithagivenrangeofvarying˛ightspeed, howtodeterminesensorpositionwithinalimitednumberoffeasiblelocationstoachieveoptimal vibrationsuppression[95].Knownaspartofinput/outputselectionproblem,sensorpositioning togetherwithactuatorpositioninghavebeenwidelystudiedin˛exiblestructures[96,97,98]. 109 Readersaresuggestedtothereference[97]formoredetailedsurvey.Thesemethodsaremostly basedonquantitativemeasuresforstatecontrollability,observabilityore˚ciencyofmanipulation, estimation.Thesemeasurescanbeconnectedwellwiththeenergystoredbystructuralsystem, suppliedbyactuatorsorbeingsuppliedtosensors.However,thesemethodsaremostlyestablished forlineartime-invariant(LTI)systems,whichhavestaticvibrationfrequencyanddamping,and hencestaticvibrationnodes.ForaBWBairplane˛exiblewing,ithasbeendemonstratedthat ˛exiblemodeswillvaryunderdi˙erent˛ightconditions.Hence,theLTIframeworkisnotcapable ofcapturingthemodedynamics,thusanewframeworkthatisabletohandlevaryingmodesis needed. Linearparameter-varying(LPV)modelingandcontrolhavebeendemonstratedasane˙ective alternativeforactivevibrationsuppressionforaBWBairplane˛exiblewings[88,87,62,35,99]. TheLPVmodelisabletocapturemodedynamicswithvarying˛ightconditionanddepictvarying input-outputcharacteristicsbetween˛apde˛ectionangles(controlsurfaces)andwingbending displacements(controlledoutputs).TheLPVcontrollerthenschedulesthecontrolgainsaccording tothemeasuredreal-time˛ightconditiontoachievespeci˝csystemperformance.Bythisway, thecontrollersynthesisiswellde˝nedasanoptimizationproblem,withperformance-associated indexasobjectivefunctionandasetofPLMIs(ParametricLinearMatrixInequalities)derived fromspeci˝csystemperformancerequirements. TheLPVframeworkisadoptedinthisstudytonumericallyanalyzehowasensorselection anditslocationin˛uencetheclosed-loopsystemperformance.Tothebestknowledgeofauthors, suchanattempthasneverbeenmadeinstructuralcontrolliterature.With H 2 LPVcontrollers, di˙erentcombinationsofsensorlocationsareevaluatedintermsoftheguaranteedclosed-loop systemperformance.Becausethecontrolinput,e.g.˛apangle,isphysicallylimited,ICC(Input CovarianceConstraint)isappliedtocontrollersynthesisconditions.Underthisconstraint,the worst-case H 2 performancewithingivenrangeof˛ightspeedistreatedastheevaluationindex forachievablesystemperformance,whichguaranteestheperformanceunderanypossible˛ight conditionwithinthe˛ightenvelope. 110 4.2Problemformulationofsensorplacement RevisittheLPVmodeloftheBWBairplane˛exiblewing.forsimplicity,weconsiderthe H 2 performanceofsensorplacement,thusonly H 2 performanceofsignalpair ( w;z ) isconsidered. _ x p ( t )= A ( ( t )) x p ( t )+ B 1 ( ( t )) w ( t )+ B 2 ( ( t )) u ( t ) z ( t )= C ( ( t )) x p ( t ) y ( t )= C y ( ( t )) x p ( t )+ v ( t ) (4.1) wheretheexternaldisturbance w ( t ) andmeasurementnoise v ( t ) areassumedtobezero-mean, Gaussianwhitenoise,butnotnecessarilystationary.Theyarealsoassumedasindependentas E n w ( s ) w T ( t ) o = W ( t ) ( t s ) ;E n v ( s ) v T ( t ) o = V ( t ) ( t s ) (4.2) AsshowninFigure.4.1,thesensorlocationcandidatesaremarkedbytrianglesinred.Itis assumedthatbendingdisplacementsin z -directioncanbemeasuredbyavailablesensors.Allof theseequallyspacedlocationstogetherareselectedasperformanceoutputstoevaluateclosed-loop H 2 systemperformance.Someorallthesecandidatelocationsmaybeselectedtoinstallbending displacementsensors.Forthegiven˛ightspeedrange,thequestionthathowmanycandidatesand whatcandidatesgroupwillleadtotheoptimalsystemperformance,arisesastheobjectiveofthis study. Notethat y ( t )=[ y 1 ( t ) ; y 2 ( t ) ; ; y m ( t ) ; ; y M ( t )] ,where M =9 isthetotalnumberof sensorlocationcandidates.Allavailablemeasurementoutput y ( t ) canbederivedfrom C y ( ) x p , where C y ( )= 2 6 6 6 6 6 6 6 6 6 6 4 C 1 ( ) . . . C m ( ) . . . C M ( ) 3 7 7 7 7 7 7 7 7 7 7 5 : 111 Figure4.1:Sensorlocationcandidates Theselectedsubsetofmeasuredoutput y ( t ) isobtainedbystackingthechosenmeasurement y m , then N M measuredoutputsareobtainedby C y as C y = 2 6 6 6 6 4 C y 1 ( ) . . . C yN ( ) 3 7 7 7 7 5 : Therefore,thereisatotalof C N M = M ! N !( M N )! combinationstochoose N sensorlocationsfrom M candidatelocations. Supposetheprojectionoperator P N 2 R N M mapstheselectedsensorsubsetfromtheentire setofavailablesensors,wherethe m th columnis1fortheselected m th sensorandthecolumnis 0ifassociatesensorisnotchosen.Thentheselectedsensoroutputcanbeexpressedby y = P N C y ( ) x p ( t )+ P N v ( t ) : (4.3) Theprojectedmeasurementnoisehasthevarianceas E n [ P N v ( s )][ P N v ( t )] T o = P N V ( t ) P T N ( t s ) 112 Usingtheselectedsensormeasurement,thedynamicoutput-feedback(DOF)LPVcontroller K ( ) isexpressedas K ( ): 8 > < > : _ x K = A K ( ) x K + B K ( ) y u = C K ( ) x K + D K ( ) y (4.4) Recallthede˝nitionof H 2 performanceofLPVsystem,de˝nedfrom w =[ w ( t ); v ( t )] to z ( t ) , isutilizedtoassesstheclosed-loopperformanceagainstexternaldisturbance.Let T 2 ( ;s ):= T z w ( ;s ) betheparameter-dependenttransferfunctionfrom w ( t ) to z ( t ) ,andifthesystempair ( A cl ;B cl ;C cl ; 0 )isstable,the H 2 norm jj T 2 jj 2 2 canbeobtainedbyminimizing traceW while subjecttothefollowingPLMIsovertheregion ( ; _ ) 2 , 2 6 4 _ P 2 + A cl P 2 +( ) B cl I 3 7 5 < 0 ; (4.5) 2 6 4 WC cl; 2 P 2 P 2 3 7 5 > 0 ; (4.6) Theoptimalsensorplacementbyselectingfromavailablesensorsetisactuallydecidingthe projectionoperator.Bythisstep,wearereadytogivetheproblemformulationofoptimalsensor placement.The H 2 performanceofclosed-loopLPVsystemminimizedbytheprojectionoperator P N indynamicoutput-feedbackLPVcontrol. min P N min trace ( W ) (4.7) subjectto(4.5)and(4.6). Thisoptimizationproblemiswell-knownasanNPhardproblem[100,101],whichinvolves hybridoptimizationofintegervariable(binaryoptimization)andrealmatrixvariables(control design). 113 4.3Simulationresultsbyglobalsearch 4.3.1Simulationresults Followingthesame H 2 LPVcontroldesignprocedure,di˙erentcombinationsofsensorlocation candidatesareexploredtocalculatetheachievableguaranteedsystemperformance.Theobservabil- ityofsystempair ( A ( ) ;C y ( )) is˝rstlycheckedatgriddedpointsovertheschedulingparameter range.Thosesensorcombinationswhichcannotbeobservedatallgriddedpointsareconsidered asunobservable,andhencetheyareremovedfromcontrollersynthesis. Figure4.2: trace ( W ) versusthenumberofavailablesensors, U =6 Figures.4.2and4.3showthe H 2 systemperformance trace ( W ) ( y axis)whenlimitednumber ofsensors( x axis)areused.Usingonesensorandsomecombinationsoftwosensorsare determinedunobservable,thusfeasible H 2 LPVDOFcontrollercannotbedesigned.When N 3 numberofsensorlocationsareavailable,thesystemmatricespairsarecheckedasobservablefor entire˛ightspeedrange.Asaresult,thereisnodatapointsshownat N =1 ,andonlyafewdata pointscanbeseenat N =2 ,whereas M ! N !( M N )! datapointsareobtainedforothercases. Notethatweightingmatrix Q ischosenas 100 I forevaluatingallbendingdisplacements equally.Figures4.2and4.3arecorrespondingtotwodi˙erentICCconditionsforallcontrol 114 Figure4.3: trace ( W ) versusthenumberofavailablesensors, U =8 surfaces U =6 ; 8 : Itisapparentthat,whenbendingdisplacementsensorsareinstalledatmore candidatelocations,theupperboundofachievablesystemperformance trace ( W ) becomessmaller, whichindicatesthatimprovedsystemperformancecanbeachieved.Moreover,thesolidlineand dashlineareplottedbyconnectingthebestandworstperformanceofeachcombinationgroupwith availablesensornumber.Thevariancetraceofsystemperformanceisshrinkingandconverging asavailablesensornumberincreases.Whensensornumberisplentyenough,systemstatescan bewellrecoveredandvibrationbehaviorswithinentire˛ightenvelopcanbewellhandled.This issimplyduetothatwithmoresensorsmoreusefulinformationcanbeaccessibleforfeedback control. Furthermore,whenlargecontrolauthorityisallowed,theachievablesystemperformance trace ( W ) issuppressedfurther,indicatingbettersystemperformancecanbeachieved.The combinationcandidateswiththebestsystemperformanceforany N numberofsensorcandidates grouparesummarizedinTable4.1.Notethat,ifonlyonesensorisused,thereisnofeasiblesensor duetotheunobservability. 115 4.3.2Discussion AsshowninFigures.4.4and4.5,LTI H 2 controllerswithICCcondition U =8 arealsodesigned atgridded˛ightspeed =115 ; 125 m/s,andasetofsensorlocationcombinationsareglobally searchedto˝ndoptimalsensorplacement.Theoptimalgroupofsensorlocationsisfoundvarying withdi˙erent˛ightspeed.Thus,sensorpositioningdeterminedbyfollowingconventionalLTI approachcannotproduceoptimalsystemperformancewithinthe˛ightspeedrange.Forexample, when N =7 numberofsensorsareusedat =115 m/s,theoptimalsensorgroupisfoundas f 1 ; 2 ; 4 ; 5 ; 6 ; 7 ; 8 g .However,when˛exiblewingis˛yingat =125 m/s,theoptimalsensor locationcombinationisfoundas f 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 9 g .Thisvalidatesthefactthattheoptimalsensor combinationsobtainedunderdi˙erentspeci˝c˛yingconditionscanvary,andhencemaynotbe optimalfortheentire˛ightenvelope.Onthecontrary,theLPVapproachconsiderstheoptimal sensorcombinationovertheentire˛ightenvelope,andlookingfortheoptimalsensorpositioning intermsofthebestguaranteedsystemperformance.Thisisthemainadvantageoftheproposed LPVapproachovertheconventionalLTIapproach. 116 Figure4.4: trace ( W ) versusnumberofavailablesensors,at˛ightspeed115 m=s Figure4.5: trace ( W ) versusnumberofavailablesensors,at˛ightspeed125 m=s 117 Table4.1:Summaryofoptimalsensorcandidatecombinations LPVapproach LTIapproach U =6 U =8 U =8 , =115 U =8 , =125 N sensor trace(W) sensor trace(W) sensor trace(W) sensor trace(W) 1 2 f 3 ; 9 g 0 : 7009 f 3 ; 8 g 0 : 5569 f 4 ; 5 g 0 : 1223 f 2 ; 3 g 0 : 2866 3 f 3 ; 4 ; 5 g 0 : 2416 f 3 ; 7 ; 9 g 0 : 1519 f 2 ; 4 ; 5 g 0 : 0496 f 6 ; 7 ; 9 g 0 : 0814 4 f 3 ; 5 ; 7 ; 9 g 0 : 2202 f 3 ; 4 ; 7 ; 8 g 0 : 1338 f 3 ; 5 ; 7 ; 9 g 0 : 0470 f 5 ; 7 ; 8 ; 9 g 0 : 0760 5 f 1 ; 3 ; 4 ; 6 ; 8 g 0 : 2165 f 1 ; 2 ; 3 ; 4 ; 6 g 0 : 1314 f 1 ; 3 ; 5 ; 6 ; 9 g 0 : 0467 f 4 ; 5 ; 7 ; 8 ; 9 g 0 : 0745 6 f 1 ; 2 ; 3 ; 4 ; 6 ; 9 g 0 : 2148 f 1 ; 2 ; 3 ; 4 ; 6 ; 8 g 0 : 1604 f 1 ; 4 ; 5 ; 6 ; 7 ; 9 g 0 : 0468 f 1 ; 2 ; 3 ; 6 ; 7 ; 9 g 0 : 0750 7 f 1 ; 2 ; 3 ; 4 ; 5 ; 7 ; 8 g 0 : 2148 f 1 ; 2 ; 3 ; 4 ; 5 ; 7 ; 9 g 0 : 1306 f 1 ; 2 ; 4 ; 5 ; 6 ; 7 ; 8 g 0 : 0465 f 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 9 g 0 : 0749 8 f 1 ; 2 ; 3 ; 4 ; 6 ; 7 ; 8 ; 9 g 0 : 2147 f 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 8 ; 9 g 0 : 1301 f 1 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 g 0 : 0473 f 1 ; 2 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 g 0 : 0753 9 f 1 9 g 0 : 2147 f 1 9 g 0 : 1313 f 1 9 g 0 : 0483 f 1 9 g 0 : 0754 118 4.4Sub-modularpropertyofsensorplacementproblem Atdi˙erent˝xed˛ightcondition,thecontributionofeachsensortoeachvibrationmodecan becalculatedbasedon[102].TheresultsareshownintheFigures4.6-4.8. Itiseasytoobservethatthecontributionofsensortoeachmodefollowsanincreasingtrend, moreover,thecontributionofeachsensorwillvaryunderdi˙erent˛ightconditions.Eventhough theGawronski'sapproximationmethod[102]canonlybeappliedtostableLTIsystems,theresults indicatethatthesensorplacementhasthesub-modularproperty. Inthesearchingofoptimalsensorplacement,globalsearchisnotane˚cientapproach,dueto theNPhardnatureofthehybridoptimization.However,thesub-modularpropertyoftheoptimal sensorplacementproblemisuncoveredandwillbeshowninthissection. Thesub-modularitywillbe˝rstlyreviewedandbasicgreedyalgorithmwillbeintroduced[100, 101] Figure4.6:Sensorcontributiontoeachvibrationmodeat˛ightspeed110 m=s 119 Figure4.7:Sensorcontributiontoeachvibrationmodeat˛ightspeed113 m=s Figure4.8:Sensorcontributiontoeachvibrationmodeat˛ightspeed115 m=s 120 4.4.1Setfunctionandsub-modularproperty De˝nition1 (setfunction) . Let S bea˝nitesetandasetfunctionover S assignsavaluetoevery subsetof S ,i.e. f ( S ):2 S ! R . De˝nition2 (submodularity) . Let S bea˝nitesetand 2 S denotepowerset.Asetfunction f :2 S ! R issaidtobesubmodularifandonlyif f ( A [ B )+ f ( A \ B ) f ( A )+ f ( B ) ; 8 A;B S: (4.8) For˝niteset S ,thisisequivalentto f ( A + j ) f ( A ) f ( B + j ) f ( B ) ; 8 A B S; 8 j 2 S n B: (4.9) Inotherwords,thefunction f satis˝esthediminishingincrementproperty.Thesubmodular function f ismonotoneif f ( A ) f ( B ) ; 8 A B .Ifasetfunctionissubmodular,thenthe contributionofanynewelement s tothesetfunctionvaluedecreaseswhenthesetgetsbigger. Basedonthede˝nitionofsubmodularity,wecanconcludethatiftheoptimalsensorplacement problemwithLPVDOFcontrolisasubmodularfunction,thenperformanceincrementbyadding onesensorwilldecreasewiththesetsize.Thisindicatesthatgreedyalgorithmhasthepotentialto e˚cientlysolvethesensorplacementproblem. 4.4.2Greedyalgorithm GreedyalgorithmutilizesaseriesofoptimallocalstepstoconducttheoptimizationofNPhard problem.Insteadofdirectlysearchingforaglobalsolution,greedyalgorithmsearchestowards minimumstepbystep.Ithasbeenprovedthatgreedyalgorithmhaspolynomialcomplexityand achievestoasub-optimalsolutionwithin (1 [1 =e ]) oftheoptimum[103]. max j S k f ( S ) (4.10) Theoptimizationofthesetfunction f ( S ) overtheset S withthesizelimit j S j k isformulated as(4.10),thenthebasicgreedyalgorithmisgivenasAlgorithm1. 121 Algorithm1: Greedyalgorithmforsetfunctionoptimization Result: K s Step1.Initialization: s =1 , K s = fg , S s = S while s k do Step2.Determinegreedilythenextelementfromresidualset S s : k s =argmax j 2 S s f ( S ) Step3.Updatetheresidualset S s andselectedset K s S s +1 = S s n k s ;K s +1 = K s [ k s ;s s +1 end Optimalsetvariable K s issearchedwithinset S tomaximizethesetfunction f ( S ) ,andthe maximumsetsizeis k ,residualset S s andselectedset K s areupdatedineverysteptomaximize theupdatedsetfunction.Inthisalgorithm,Step2searchesamongtheresidualsetfortheelement tobeaddedintoselectedset,whichwillleadtothemaximumvalueofsetfunction f ( S ) . Inthesensorplacementproblem,Algorithm1iscustomizedtoAlgorithm2toselect N number ofsensorsfromset S of M numberofsensors. Inthismodi˝edalgorithm,theperformanceindex trace ( W ) of H 2 performancewithLPV controlisminimizedbythesetvariable K s andLPVcontroller K ( ) .LPVcontrollerisdesigned followingthedesigntechniquediscussedinChapter1.Itisnotedthattheobservabilityofsensor subsetneedstobecheckedinStep2,sothatthereexistsaLPVcontrollerstabilizingthe˛exible wingmodel. Thegreedyalgorithmiswell-knowntohavepolynomialtimee˚ciency.Ateachstep,the algorithmscansamongtheresidualsensorsubset S s andconductsetfunctionevaluation.After that,theelementresultinginmaximumsetfunctionvalueisaddedtoselectedset,whichonlyneeds polynomialnumberofoperationstogetlocaloptimalsensorset. TheFigure4.9showsthesubmodularpropertyofsensorplacement.Whensearchingonly 122 onesinglesensor,thereisnofeasiblesensortoachieveobservability,thusnovalueofminimized trace ( W ) islabeled.Inthecaseoftwosensors,thesensorcombinationof 8 ; 9 leadstotheoptimal trace ( W ) ofclosed-loopsystem,thusthesetwosensorsaretheoptimalselection.Whenmorethan twosensorsarechosen,thenitiscleartoobservethedescending trace ( W ) valuefromlefttoright. Thisindicatesthatthesensorsclosetowingtipleadtobetterclosed-loopsystemperformance. Moreover,thedecrementofoptimal trace ( W ) ,representingimprovementof H 2 performance, decreaseswiththenumberofselectedsensors,whichdemonstratesthesubmodularpropertyof optimalsensorplacementon˛exiblewing. Figure4.9:Submodularpropertyofsensorplacementon˛exiblewing 123 Algorithm2: Greedyalgorithmforoptimalsensorplacement Result: K s Step1.Initializationofsetvariable: s =1 , K s = fg , S s = S while s N do Step2.Checkobservabilityofsensorsubset i =1 ; while i j S s j do if K s \ j sensorsetisunobservable then S s = S s n j else continue; end end Step3.Determinegreedilythenextelementfromresidualset S s : k s =argmin K ( ) ;j 2 S s trace ( W ) Step4.Updatetheresidualset S s andselectedset K s S s +1 = S s n k s ;K s +1 = K s [ k s ;s s +1 end 124 CHAPTER5 CONCLUSIONSANDRECOMMENDEDWORK 5.1Conclusions Inthisdissertation,simultaneousdesignandsequentialdesignofsmoothswitchingLPVcontrol designhavebeenproposed,andoptimalsensorplacementontheBWBaircraft˛exiblewinghas beensoughtintheLPVframework.Themaincontributionscanbegroupedintothefollowing items. ‹ Thesimultaneousdesignapproachforsmooth-switching ICC= H 1 state-feedbackanddy- namicoutput-feedbackLPVcontrolhasbeenseparatelyexploredandPLMIs(Parametric LinearMatrixInequalities)forcontrollersynthesishavebeenaccordinglyderived.Toobtain smoothswitching,smoothnessandsystemperformanceindexeswereincorporatedintothe costfunctionandweightedbyatunablecoe˚cient,introducinganothertunabletrade-o˙ betweensystemperformanceandswitchingsmoothness.Bytuningthecoe˚cient,optimal balanceofswitchingsmoothnessandsystemperformancecanbeattained. ‹ ThesequentialdesignapproachdesignstheLPVcontrollersindependentlyandusessigmoid interpolationofadjacentcontrollersonoverlappedsubregion.The H 1 LPVstate-feedback caseisstudiedandcontrollersynthesisconditionsarederived.Furthermore,thee˙ectiveness ofsequentialdesignandreducedcomputationalcomplexitythansimultaneousdesignare demonstratedbytwonumericalexamples. ‹ TheproposedLPVcontrollershavebeenappliedtoanactivemagneticbearingsystemand vibrationsuppressionofaBWB˛exibleairplanewing.Thesimulationresultsdemonstrated thattheproposedsmooth-switchingLPV ICC= H 1 controllersareabletobalanceswitching smoothnessandsystemperformancesubjecttoconstraintsoncontrolinputsandsystem uncertainty.Inaddition,theresultsshowthattheproposedmethodimprovestheswitching 125 smoothnesssigni˝cantlycomparedwiththeresultsfromtheearlierstudywithoutconsidering switchingsmoothness.SimulationresultsofBWBairplanewinghaveshowedthatthe proposeddesignmethodisabletosigni˝cantlyreducethesharpjumpsinsystemcontrolsand responsesduringswitchingevents.Furthermore,theproposedtunableweightingcoe˚cient providestrade-o˙betweensystemperformanceandsmoothnessofresponse,andtheICC constraintsoncontrolinputscanalsobeusedtotunetheachievableperformance.These o˙ergreatadvantagesinpracticalimplementation. ‹ Inaddition,theLPVapproachisutilizedtodetermineoptimalsensorpositionforaBWB airplane˛exiblewing.Again-scheduling H 2 LPVcontrol,subjectto ICC hardconstraints, isdesignedforagivenschedulingparameterregion.Theoptimalcandidateforsensor allocationsisobtainedbysearchingforthebestguaranteed H 2 systemperformancewithin the˛ightspeedregion.Byglobalsearchandgreedyalgorithm,theoptimalcandidatecan beobtainedforanygivennumberofsensors,andthetrade-o˙betweenoptimalperformance andsensornumbercanalsobeobtained. 5.2Recommendedwork Withtheresultsshowninthisdissertation,therearestillafewpotentialdirectionstoworkon, inboththeoryandapplicationparts. ‹ Potentialdirectionsintheory SmoothswitchingcontrollersynthesiswithYoulaParameterization. RecalltheYoulaParameterization,let K ( s )= V 1 ( s ) U ( s ) and G ( s )= M 1 ( s ) N ( s ) betheleftco-primefactorizationofcontroller K ( s ) andnominalplant P ( s ) .ThentheY- oulaparameterizationofallstabilizablecontrollers ^ K =( V ( s ) Q ( s ) N ( s )) 1 ( U ( s )+ Q ( s ) M ( s )) forany Q 2 RH 1 suchthat det ( V ( 1 ) Q ( 1 ) N ( 1 )) 6 =0 .Acontroller schemebasedonYoulaparameterizationproposedin[104]isshowninFigure5.1. 126 Figure5.1:Acontrollerschemewithcompensatingoperator Q Thustheswitchingcontrollerhasthepotentialtobedividedintotwoparts:nominal controller U ( s ) , V 1 ( s ) andoperator Q ( s ) .Thenswitchedcontrollergaincanbe includedinto Q ,whichwillgreatlysimplifytheswitchingstabilityconditions,because aslongasswitchingoperator Q 2 RH 1 ,thecontroller ^ K isstabilizableforthegiven plant. Explorationofsub-modularpropertyinKalmanFilterdesignforLTI/LTV/LPVsystem InconventionalKalman˝lterdesign,weutilizethegivensetofsensorsandonlyfocus ontheestimatorgaintoachieveoptimalstateestimation.Thesensorplacementusually involves heuristic methodandwilllimitestimationperformance.However,inengineer- ingpractice,thesensorplacementisessentialandshouldbedecidedbeforeKalman ˝lterdesign.Ifthesetwodecisionvariablescanbeintegratedintooneoptimization problem,thestateestimationcanbeimprovedsigni˝cantly. Considerastochasticsystem _ x = A ( t ) x + B ( t ) u + F ( t ) w y = C ( t ) x + v (5.1) Assumethedisturbance w ( t ) andnoise v ( t ) arezero-mean,Gaussianwhitenoise,but notnecessarilystationary.Theyarealsoassumedasindependent. E n w ( s ) w T ( t ) o = W ( t ) ( t s ) ;E n v ( s ) v T ( t ) o = V ( t ) ( t s ) (5.2) 127 Theoptimalestimationproblemisformulatedasminimizingthemeansquareerrorof ^ x ( t ) withtruestatevalue x ( t ) E n ( x ( t ) ^ x ( t ))( x ( t ) ^ x ( t )) T o (5.3) Theorem11 (Kalman-Bucy,1961) . Theoptimalestimatorhastheformofalinear observer _ ^ x = A ( t )^ x + B ( t ) u + L ( t )[ y C ( t )^ x ] (5.4) where L ( t )= P ( t ) C T ( t ) V 1 and P ( t )= E n ( x ( t ) ^ x ( t ))( x ( t ) ^ x ( t )) T o satis˝es _ P = AP + PA T PC T V 1 ( t ) CP + FW ( t ) F T P (0)= E n x (0) x T (0) o ConsiderasensorselectionproblemforKalman-Bucy˝lterthat,select C y subset consistingof s rowelementsinset C =[ C T 1 ;C T 2 ;:::;C T m ] T ,andestimatorgain L ( t ) suchthatestimationerrorisminimized,withgiven P (0)= E n x (0) x T (0) o andgiven V = diag ( V m ) and W . TheOSPisformulatedasselectsubset S M suchthat min s 2 S M trace ( P ( t )) (5.5) and P ( t ) subjecttodi˙erentialequation _ P = AP + PA T PC T y V 1 S ( t ) C y P + FW ( t ) F T (5.6) Supposethecovarianceofsensorsareknown,thenselectingproperlythesensorsetto achieveoptimalstateestimationbyKalman˝lterisaveryinterestingresearchtopic. Ifthesub-modularpropertyormoremildpropertycanbediscovered,thenthesensor placementandoptimalKalman˝ltergaindesigncouldbetackled. ‹ Potentialdirectionsinapplications 128 Applicationsofsequentialdesignin˛ightcontrol.TheBWBairplane˛exiblewing modelisalwaysofhighorder,eventhoughmodelreductionisconducted.Thisisa promisingapplication˝ledthatcouldutilizesequentialdesignapproachofswitching controllers. Investigationofsub-modularpropertyof˛exiblestructureelements.Inthisdissertation, thesubmodularpropertyisdiscoveredbasedonthemodelofonespeci˝c˛exiblewing. 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