PAGE 1 LYAPUNOV-BASEDCONTROLOFLIMITCYCLEOSCILLATIONSINUNCERTAIN AIRCRAFTSYSTEMS By BRENDANBIALY ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2014 PAGE 2 c 2014BrendanBialy 2 PAGE 3 Tomyparents,WilliamandKellyBialy,fortheirsupportandencouragement 3 PAGE 4 ACKNOWLEDGMENTS Iwouldliketoexpressmygratitudetomyadvisor,Dr.WarrenE.Dixon,forhis guidanceandmotivationduringmyacademicpursuits.Hisinuencewascrucialtothe successfulcompletionofmydoctoralstudy.Iwouldalsoliketoextendmygratitudeto mycommitteemembers:Dr.MrinalKumar,Dr.NormanG.Fitz-Coy,andDr.OscarD. Crisalle,fortheirtimeandrecommendations.Finally,Iwouldliketothankmyfamily, coworkers,andfriendsfortheirsupportandencouragement. 4 PAGE 5 TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................7 LISTOFFIGURES.....................................8 LISTOFABBREVIATIONS................................10 ABSTRACT.........................................11 CHAPTER 1INTRODUCTION...................................13 1.1MotivationandLiteratureReview.......................13 1.2Contributions..................................19 1.2.1Chapter2:Lyapunov-BasedTrackingofStore-InducedLimitCycleOscillationsinanAeroelasticSystem..............19 1.2.2Chapter3:SaturatedRISETrackingControlofStore-InducedLimit CycleOscillations............................19 1.2.3Chapter4:BoundaryControlofLimitCycleOscillationsinaFlexibleAircraftWing:............................20 1.2.4Chapter5:AdaptiveBoundaryControlofLimitCycleOscillations inaFlexibleAircraftWing.......................20 2LYAPUNOV-BASEDTRACKINGOFSTORE-INDUCEDLIMITCYCLEOSCILLATIONSINANAEROELASTICSYSTEM..................21 2.1AeroelasticSystemModel...........................21 2.2ControlObjective................................24 2.3ControlDevelopment..............................25 2.4StabilityAnalysis................................29 2.5SimulationResults...............................32 2.6Summary....................................37 3SATURATEDRISETRACKINGCONTROLOFSTORE-INDUCEDLIMITCYCLEOSCILLATIONS................................40 3.1ControlObjective................................40 3.2ControlDevelopment..............................41 3.3StabilityAnalysis................................43 3.4SimulationResults...............................47 3.5Summary....................................52 5 PAGE 6 4BOUNDARYCONTROLOFLIMITCYCLEOSCILLATIONSINAFLEXIBLE AIRCRAFTWING..................................55 4.1AircraftWingModel..............................55 4.2BoundaryControlofWingTwist........................56 4.3BoundaryControlofWingBending......................59 4.4NumericalSimulation.............................62 4.5Summary....................................65 5ADAPTIVEBOUNDARYCONTROLOFLIMITCYCLEOSCILLATIONSINA FLEXIBLEAIRCRAFTWING............................69 5.1AircraftWingModel..............................69 5.2BoundaryControlDevelopment........................70 5.3StabilityAnalysis................................72 5.4Summary....................................80 6CONCLUSIONANDFUTUREWORK.......................81 6.1DissertationSummary.............................81 6.2LimitationsandFutureWork..........................82 APPENDIX APROOFTHAT M ISINVERTIBLECH3.....................85 BPROOFOF g> 0 CH3..............................87 CGROUPINGOFTERMSIN 1 AND 2 CH3..................88 DDEVELOPMENTOFTHEBOUNDON ~ N CH3.................89 EDETAILSONTHEDEVELOPMENTOFTHECONSTANTS c m 1 c m 2 ,AND c m 3 CH4.......................................91 FDERIVATIONOFTHEBENDINGANDTWISTINGDYNAMICSOFAFLEXIBLEWINGCH5/6.................................92 GEXPONENTIALSTABILITYOFTHETARGETSYSTEMCH5........98 HINTEGRATIONBYPARTSOFSELECTTERMSIN E c CH6..........101 REFERENCES.......................................103 BIOGRAPHICALSKETCH................................108 6 PAGE 7 LISTOFTABLES Table page 2-1AeroelasticModelParameters...........................33 2-2MonteCarloSimulationResults...........................37 3-1AeroelasticModelParameters...........................47 3-2MonteCarloSimulationResults...........................51 7 PAGE 8 LISTOFFIGURES Figure page 2-1Diagramdepictingthetwodegreeoffreedomairfoilsectionwithattachedstore. .............................................22 2-2Aeroelasticsystemfreeresponsewithoutdisturbances.............34 2-3Comparisonofthecontrolledaeroelasticsystemresponse...........35 2-4Controlsurfacedeections, t ,forthedevelopedcontrollerandprevious controller.......................................35 2-5Aeroelasticsystemstatesinthepresenceofanadditivedisturbance......36 2-6Controlsurfacedeection, t ,forthedevelopedcontrollerandpreviouscontroller.........................................36 2-7MonteCarloAOAtrajectories............................38 2-8MonteCarloverticalpositiontrajectories......................38 2-9MonteCarlocontroleffort..............................39 3-1Aeroelasticsystemopen-loopresponsewithoutdisturbances..........49 3-2StatetrajectoriesoftheRISE-basedcontrollerwithandwithoutan adhoc saturation........................................49 3-3CommandedcontroleffortfortheRISE-basedcontrollerwithandwithoutan adhoc saturation...................................50 3-4Comparisonoftheclosed-loopaeroelasticsystemresponseoftheRISE-based controllerwithan adhoc saturationandthedevelopedsaturatedcontroller...51 3-5Comparisonofthecontrolsurfacedeectionsforthedevelopedsaturated controllerand adhoc saturatedRISE-basedcontroller.............51 3-6AoAtrajectoriesforall1500MonteCarlosamples.Thedevelopedsaturated controllersuppressedtheLCObehaviorinallsamplesandthemajorityofthe samplesexhibitsimilartransientperformance...................52 3-7Verticalpositiontrajectoriesofall1500MonteCarlosamples.Thevertical positionremainedboundedforallsamplesdespitebeinganuncontrolledstate. .............................................53 3-8Controlsurfacedeectionforall1500MonteCarlosamples.Thecontroleffortforallsamplesremainwithintheactuationlimitanddemonstratesimilar steadystateperformance...............................53 8 PAGE 9 4-1ApproximationofthemodiedBesselfunctionusedinthesubsequentsimulationsection......................................60 4-2Open-looptwistdeectionoftheexibleaircraftwing...............64 4-3Open-loopbendingdeectionoftheexibleaircraftwing.............64 4-4Open-loopresponseatthewingtipoftheexibleaircraftwing..........65 4-5Closed-looptwistdeectionoftheexibleaircraftwing..............66 4-6Closed-loopbendingdeectionoftheexibleaircraftwing............66 4-7Closed-loopresponseatthewingtipoftheexibleaircraftwing.........67 4-8LiftandMomentcommandedatthewingtip....................67 9 PAGE 10 LISTOFABBREVIATIONS a.e.AlmostEverywhere AoAAngleofAttack LCOLimitCycleOscillations LPLinear-in-the-Parameters LPVLinearParameterVarying LQRLinear-QuadraticRegulator NNNeuralNetwork PDEPartialDifferentialEquation RISERobustIntegraloftheSignoftheError ROMReducedOrderModel SDREState-DependentRiccatiEquation SMCSlidingModeControl SMRACStructuredModelReferenceAdaptiveControl 10 PAGE 11 AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy LYAPUNOV-BASEDCONTROLOFLIMITCYCLEOSCILLATIONSINUNCERTAIN AIRCRAFTSYSTEMS By BrendanBialy May2014 Chair:WarrenE.Dixon Major:AerospaceEngineering Store-inducedlimitcycleoscillationsLCOaffectseveralghteraircraftandis expectedtoremainanissuefornextgenerationghters.LCOarisesfromtheinteractionofaerodynamicandstructuralforces,howevertheprimarycontributortothe phenomenonisstillunclear.Thepracticalconcernsregardingthisphenomenoninclude whetherornotordnancecanbesafelyreleasedandtheabilityoftheaircrewtoperform mission-relatedtaskswhileinanLCOcondition.Thefocusofthisdissertationisthe developmentofcontrolstrategiestosuppressLCOinaircraftsystems. TherstcontributionofthisworkChapter2isthedevelopmentofacontroller consistingofacontinuousRobustIntegraloftheSignoftheErrorRISEfeedbackterm withaneuralnetworkNNfeedforwardtermtosuppressLCObehaviorinanuncertain airfoilsystem.ThesecondcontributionofthisworkChapter3istheextensionofthe developmentinChapter2toincludeactuatorsaturation.SuppressionofLCObehavior isachievedthroughtheimplementationofanauxiliaryerrorsystemthatfeatures hyperbolicfunctionsandasaturatedRISEfeedbackcontrolstructure. DuetothelackofclarityregardingthedrivingmechanismbehindLCO,common practiceinliteratureandinChapters2and3istoreplicatethesymptomsofLCOby includingnonlinearitiesinthewingstructure,typicallyanonlineartorsionalstiffness.To improvetheaccuracyofthesystemmodelapartialdifferentialequationPDEmodel ofaexiblewingisderivedseeAppendixFusingHamilton'sprinciple.Chapters4 11 PAGE 12 and5arefocusedondevelopingboundarycontrolstrategiesforregulatingthebending andtwistingdeformationsofthederivedmodel.ThecontributionofChapter4isthe constructionofabackstepping-basedboundarycontrolstrategyforalinearPDEmodel ofanaircraftwing.Thebackstepping-basedstrategytransformstheoriginalsystem toaexponentiallystablesystem.ALyapunov-basedstabilityanalysisisthenusedto toshowboundednessofthewingbendingdynamics.ALyapunov-basedboundary controlstrategyforanuncertainnonlinearPDEmodelofanaircraftwingisdeveloped inChapter5.Inthischapter,aproportionalfeedbacktermiscoupledwithangradientbasedadaptiveupdatelawtoensureasymptoticregulationoftheexiblestates. 12 PAGE 13 CHAPTER1 INTRODUCTION 1.1MotivationandLiteratureReview Store-inducedlimitcycleoscillationsLCOcommonlyoccurandremainanissue onhighperformanceghteraircraft[1].LCObehaviorischaracterizedbyantisymmetric non-divergentperiodicmotionofthewingandlateralmotionofthefuselage.LCO motioncanbeself-inducedorinitiatedthroughthecontrolinputs;howeverthemotion isself-sustainingandpersistsuntiltheightconditionshavebeensufcientlyaltered. LCObehaviorrelatedtoutter,exceptcouplingbetweentheunsteadyaerodynamic forcesandnonlinearitiesintheaircraftstructureresultsinalimitedamplitudemotion[2]. Infact,store-inducedLCOresponsesarepresentonghteraircraftcongurations thathavebeentheoreticallypredictedtobesensitivetoutter.Classicallinearutter analysistechniqueshavebeenshowntoaccuratelypredicttheoscillationfrequencyand modalcompositionofLCObehavior;however,duetounmodelednonlinearitiesinthe system,theyfailtoadequatelypredictitsonsetvelocityoramplitude[3]. ThemajorconcernwithLCOisthepilot'sabilitytosuccessfullycompletethe missioninasafeandeffectivemanner.Specically,theLCO-inducedlateralmotionof thefuselagemaycausethepilottohavedifcultyreadingcockpitgaugesandheads-up displaysandcanleadtotheterminationofthemissionortheavoidanceofapartofthe ightenvelopecriticaltocombatsurvivability.Additionally,questionshavebeenraised abouttheeffectsofLCOonordnance[2].Thesequestionsincludewhetherornotthe ordnancecanbesafelyreleasedduringLCO,theeffectsontargetacquisitionforsmart munitions,andtheeffectsontheaccuracyofunguidedweapons. ConcernsregardingtheeffectsofLCOonmissionperformancenecessitatethe developmentofacontrolstrategythatcouldsuppressLCObehaviorinanuncertain nonlinearaircraftsystem.Severalcontrolstrategieshavebeendevelopedinrecent yearstosuppressLCObehaviorinaeroelasticsystemsthatrequireknowledgeofthe 13 PAGE 14 systemdynamics.Alinear-quadraticregulatorLQRcontrollerwithaKalmanstate estimatorwasdevelopedin[4]tostabilizeatwodegreeoffreedomairfoilsection.The unsteadyaerodynamicsweremodeledusinganapproximationofTheodorsen'stheory. Thedevelopedcontrollerwasshowntobecapableofstabilizingthesystematvelocities overtwicetheuttervelocity.However,whenthecontrolsystemwasemployedafter theonsetofLCObehavior,itwasonlyeffectiveneartheuttervelocity.Afeedbacklinearizationcontrollerwasdevelopedin[5]thatusesaquasi-steadyaerodynamicmodel andrequiresexactcancellationofthenonlinearitiesinthesystem.Anoutputfeedback LQRcontrollerwasdesignedin[6]usingalinearreducedordermodelfortheunsteady transonicaerodynamics.Danowskyetal.[7]developedanactivefeedbackcontrol systembasedonalinearreducedordermodelROMofarestrainedaeroservoelastic high-speedghteraircraft.Theeffectivenessofthedesignedcontrollerwasveried usingsimulationsofthefull-orderaircraftmodel.Alinearinput-to-outputROMofan unrestrainedaeroservoelastichigh-speedghteraircraftmodelwasdevelopedin[8] thatincludedrigidbodyaircraftdynamics.Linearcontroltechniqueswereprovento stabilizethestatesoflinearvehicledynamicswhilesuppressingaeroelasticbehavior. Acontrolsystembasedonanaerodynamicenergyconceptwasdesignedforafour controlsurfaceforwardsweptwingin[9].Theaerodynamicenergyconceptdetermines thestabilityofanaeroelasticsystembyexaminingtheworkdoneperoscillationcycle bythesystem.Thecontrollerisdesignedtoproducepositiveworkperoscillationcycle whichcorrespondstothedissipationofenergyinthesystemandthusthesystemwill remainstable.Primeetal.[10]developedanLQRcontrollerbasedonalinearparametervaryingLPVmodelbasedonfreestreamvelocityofathreedegreeoffreedom wingsection.TheLPVcontrollerauto-scheduleswithfreestreamvelocityandwas showntosuppressLCObehavioroverawiderangeofvelocities.Acomparisonof State-DependentRiccatiEquationSDREandslidingmodecontrolSMCapproaches 14 PAGE 15 forLCOsuppressioninawingsectionwithoutanexternalstorewasperformedin[11]. Bothcontrolapproachesusedlinearizeddynamicsandexactmodelknowledge. Multipleadaptivecontrollershavebeendevelopedtocompensateforuncertainties onlyinthetorsionalstiffnessmodel.Anadaptivenonlinearfeedbackcontrolstrategy wasdesignedin[12]forawingsectionwithstructuralnonlinearitiesandasingletrailing edgecontrolsurface.Thedesignassumeslinear-in-the-parametersLPstructural nonlinearitiesinthemodelofthepitchstiffnessonly,andachievespartialfeedback linearizationcontrol.Experimentalresultsusingtheadaptivecontrollerdevelopedin[12] andthemultivariablelinearcontrollerdevelopedin[4]werepresentedin[13].TheresultsshowedthattheadaptivecontrollerwascapableofsuppressingtheLCObehavior atvelocitiesupto23%higherthantheuttervelocity.Astructuredmodelreference adaptivecontrolSMRACstrategywasdevelopedin[14]tosuppresstheLCObehavior ofatypicalwingsectionwithLPuncertaintiesinthepitchstiffnessmodel.TheSMRAC strategywascomparedwithanadaptivefeedbacklinearizationmethodandwasshown tosuppressLCObehaviorathigherfreestreamvelocities.Acontrolstrategythatuses multiplecontrolsurfacesandcombinesfeedbacklinearizationviaLiealgebraicmethods andmodelreferenceadaptivecontrolwasdevelopedin[15]toimprovethecontrolof LCObehavioronatypicalwingsectionwiththesameuncertaintiesasin[12].Theproposedcontrollershowedimprovedtransientperformanceandwascapableofstabilizing thewingsectionathigherfreestreamvelocitieswhencomparedtothecontrolstrategy developedin[14]. Previouslydevelopedcontrollerseitheruselinearizedsystemdynamicsandare restrictedtospecicightregimes,requireexactknowledgeofthesystemdynamics, orconsideronlyuncertaintiesinthedynamicsthatsatisfythelinear-in-the-parameters assumption.Whenanyoftheseconditionsarenotmet,thepreviouslydevelopedcontrollerscannolongerguaranteestability.Furthermore,thesecontrollershaveneglected thefactthatthecommandedcontrolinputmayexceedtheactuationlimitsofthesystem, 15 PAGE 16 whichcanresultinunpredictableclosed-loopresponses.Chapter2proposesacontrol strategytosuppressLCOinatwodegreeoffreedomairfoilsectioninthepresence ofboundeddisturbancesusingthefullnonlinearsystemmodel.Uncertaintiesinthe systemareassumedtobepresentinthestructuralandaerodynamicmodelsandare notrequiredtosatisfytheLPcondition.Thedevelopedcontrolstrategyconsistsofa neuralnetworkNNfeedforwardtermtoapproximatetheuncertainsystemdynamics whileaRobustIntegraloftheSignoftheErrorRISEfeedbacktermensuresasymptotictrackinginthepresenceofunknownboundeddisturbances.Chapter3extendsthe resultinChapter2tocompensateforactuatorconstraints.WhileChapter3buildson theworkinChapter2,theerrorsystem,controldevelopment,andstabilityanalysisare allredesignedtoaccountforactuatorlimitations.Asymptotictrackingofadesiredangle ofattackAoAisachievedthroughtheimplementationofanauxiliaryerrorsystemthat featureshyperbolicfunctionsandacontinuousRISEfeedbackcontrolstructure[16]. Previousresearch,includingthedevelopmentinChapters2and3,focusonsuppressingLCObehaviorinanairfoilsection,whichisdescribedbyasetofordinary differentialequationsODE.However,theairfoilsectionmodelisasimplieddescriptionofwhatishappeninginreality.Toimprovethedelityoftheplantmodel,itis neccessarytoexaminetheinteractionsbetweenthestructuraldynamicsandaerodynamicsonaexiblewing.Thedynamicsofaexiblewingaredescribedbyasetof partialdifferentialequationsPDE,whichrequiresadifferentcontrolmethod.Typically, thecontrolactuatorislocatedatthespatialboundaryofthesysteme.g.,atthewingtip andsothecontroldesignmustusetheboundaryconditionstoexertcontroloverthe statesofthesystemacrosstheentirespatialdomain.Chapter4examinestheLCO problemforaexiblewingdescribedbyasetofPDEsandassociatedboundaryconditions.Hamilton'sprinciplehasbeenusedpreviouslytomodeltheexibledynamicsof 16 PAGE 17 physicalsystems,includinghelicopterrotorblades[1719]andexiblerobotmanipulators[2022],andcanbeappliedtoobtainthePDEsystemdescribingthedynamicsofa exiblewingundergoingbendingandtwistingdeformations. Twocontrolstrategieshavebeendevelopedforsystemsdescribedbyasetof PDEs.TherststrategyusesGalerkinorRayleigh-Ritzmethods[2325],oroperator theoretictools[2629]toapproximatethePDEsystembyanitenumberofODEs, thenacontrollerisdesignedusingthereduced-ordermodelapproximation.Themain concernofusingareduced-ordermodelinthecontroldesignisthepotentialforspillover instabilities[30,31],inwhichthecontrolstrategyexcitesthehigher-ordermodesthat wereneglectedinthereduced-ordermodel.Inspecialcases,sensorandactuator placementcanguaranteetheneglectedmodesarenotaffected[32].Specically,when thezerosofthehigher-ordermodesareknown,placingactuatorsattheselocationswill mitigatespilloverinstabilities;howeverthiscanconictwiththedesiretoplaceactuators awayfromthezerosofthecontrolledmodes. ThesecondstrategyretainsthefullPDEsystemforthecontrollerdesignand onlyrequiresmodelreductiontechniquesforimplementation.PDE-basedcontrol techniques[33,34]areoftendevelopedwiththedesiretoimplementboundarycontrol inwhichthecontrolactuationisappliedthroughtheboundaryconditions.ThePDE backsteppingmethoddescribedin[33]compensatesfordestabilizingtermsthat actacrossthesystemdomainbyconstructingastatetransformation,involvingan invertibleVolterraintegral,thatmapstheoriginalPDEsystemtoanexponentiallystable targetPDEsystem.Sincethetransformationisinvertible,stabilityofthetargetsystem translatesdirectlytostabilityoftheclosed-loopsystemthatconsistsoftheoriginal systemplusboundaryfeedbackcontrol.WhilethePDEbacksteppingmethodyields elegantsolutionstoboundarycontrolofPDEsystems,itislimitedtolinearPDEsand nonlinearPDEsinwhichthenonlinearitiesarenotdestabilizing.Theboundarycontrol methodsdescribedin[34,35]useLyapunov-baseddesignandanalysisarguments 17 PAGE 18 tocontrolPDEsystems.Thecruxofthismethodistheassumptionthatforaphysical system,iftheenergyofthesystemisbounded,thenthestatesthatcomposethe energyofthesystemarealsobounded.Basedonthisassumption,theobjectiveof theLyapunov-basedstabilityanalysisistoshowthattheenergyintheclosed-loop PDEsystemremainsboundedanddecaystozeroasymptotically.Thismethodis applicabletobothlinearandnonlinearPDEsystems;however,morecomplexsystems typicallyrequiremorecomplexcontrollersandcandidateLyapunovfunctions.Anotable difference,fromanimplementationperspective,betweenthebacksteppingmethod in[33]andtheLyapunov-basedenergyapproachin[34,35]isthesignalsthatare requiredtobemeasurable.Thebacksteppingapproachtypicallyrequiresknowledgeof thedistributedstatethroughoutthespacialdomainwhiletheLyapunov-basedenergy methodonlyrequiresmeasurementsattheboundary,howeverthesemeasurementare typicallyhigher-orderspatialderivatives.APDE-basedboundarycontrolapproachhas beenpreviouslyusedtostabilizeuidowthroughachannel[36],maneuverexible roboticarms[37],controlthebendinginanEulerbeam[3840],regulateaexiblerotor system[35,41],andtrackthenetaerodynamicforce,ormoment,ofaappingwing aircraft[42]. SeveralPDEandODEcontrollershavebeenpreviouslydevelopedtocontrol thebendinginaexiblebeam[28,29,38,40];howeverthisbodyofworkisprimarily concernedwithstructuralbeamsandroboticarmswhichdon'tencountertheclosedloopinteractionsbetweentheexibledynamicsandaerodynamicsintrinsictoexible aircraftwings.Recently,[42]usedthePDE-basedbacksteppingcontroltechnique from[33]totrackthenetaerodynamicforcesonaappingwingmicroairvehicleusing eitherroot-basedactuationortip-basedactuation.Thecontrolobjectivein[42]isnot concernedwiththeperformanceofthedistributedstatevariables,insteadtheboundary controlisdesignedtotrackaspatialintegralofthedistributedstatevariables.Thefocus ofChapter 4 isthedevelopmentofaPDE-basedcontrollertosuppressLCObehavior 18 PAGE 19 inaexibleaircraftwingdescribedbyalinearPDEviaregulationofthedistributed statevariables.Thebacksteppingtechniquein[33]isusedtoensurethewingtwist decaysexponentially,andaLyapunov-basedstabilityanalysisofthewingbending dynamicsisusedtoprovethattheoscillationsinthewingbendingdynamicsdecay asymptoticallyandthewingbendingstatereachesasteady-stateprole.Chapter5 usesLyapunov-basedboundarycontroldesignandanalysismethodsmotivatedbythe approachesin[34,35]toregulatethedistributedstatesofaexiblewingdescribedby asetofuncertainnonlinearPDEs.TheconsideredPDEmodelhasuncertaintiesthat arelinear-in-the-parametersandarecompensatedforusingagradient-basedadaptive updatelaw. 1.2Contributions ThecontributionsofChapters2-5areasfollows: 1.2.1Chapter2:Lyapunov-BasedTrackingofStore-InducedLimitCycleOscillationsinanAeroelasticSystem ThemaincontributionofChapter2isthedevelopmentofaRISE-basedcontrol strategyforthesuppressionofLCObehaviorinanuncertainnonlinearaeroelastic system.ANNfeedforwardtermisusedtocompensateforuncertaintiesinthestructuraldynamicsandaerodynamicswhileacontinuousRISEfeedbacktermensures asymptotictrackingofadesiredAoAtrajectory.Numericalsimulationsillustratetheperformanceofthedevelopedcontrolleraswellasprovidingacomparisonwithapreviously developedcontroller.Furthermore,aMonte-Carlosimulationisprovidedtodemonstrate robustnesstovariationsintheplantdynamicsandmeasurementnoise. 1.2.2Chapter3:SaturatedRISETrackingControlofStore-InducedLimitCycle Oscillations ThecontributionofChapter3istoextendtheresultinChapter2tocompensate foractuatorlimits.Toaccountforactuatorconstraints,theerrorsystemandcontrol developmentareaugmentedwithsmooth,boundedhyperbolicfunctions.Anumerical simulationdemonstratedtheunpredictableclosed-loopresponseoftheRISE-based 19 PAGE 20 controllerfromChapter2whenan adhoc saturationisappliedtothecommanded controleffort.Furthermore,thesimulationsshowthedevelopedsaturatedcontroller achievesasymptotictrackingofthedesiredAoAwithoutbreachingactuatorconstraints. 1.2.3Chapter4:BoundaryControlofLimitCycleOscillationsinaFlexible AircraftWing: ThecontributionofChapter4isthedevelopmentofaboundarycontrolstrategy forthesuppressionofLCOinaexibleaircraftwingdescribedbyasetoflinearPDEs. ThecontrolstrategyusesaPDE-basedbacksteppingtechniquetotransformtheoriginal systemtoanexponentiallystablesysteminwhichthedestabilizingtermsintheoriginal systemareshiftedtotheboundaryconditions.Aboundarycontrolisthendeveloped tocompensateforthedestabilizingterms.Thebacksteppingapproachensuresthe wingtwistdecaysexponentiallywhileaLyapunov-basedstabilityanalysisproves theoscillationsinthewingbendingaresuppressedandthewingbendingachieves asteady-stateprole.Numericalsimulationsdemonstratetheperformanceofthe proposedcontrolstrategy. 1.2.4Chapter5:AdaptiveBoundaryControlofLimitCycleOscillationsina FlexibleAircraftWing ThecontributionofChapter5isthedesignofaboundarycontrolstrategyto suppressLCOmotioninanuncertainnonlinearexibleaircraftwingmodel.The controlstrategyusesagradient-basedadaptiveupdatelawtocompensatefortheLP uncertaintiesandaLyapunov-basedanalysisisusedtoshowthattheenergyinthe systemremainsboundedandasymptoticallydecaystozero.Argumentsthatrelatethe energyinthesystemtothedistributedstatesareusedtoconcludethatthedistributed statesareregulatedasymptotically. 20 PAGE 21 CHAPTER2 LYAPUNOV-BASEDTRACKINGOFSTORE-INDUCEDLIMITCYCLEOSCILLATIONS INANAEROELASTICSYSTEM ThefocusofthischapteristodevelopacontrollertosuppressLCObehaviorina twodegreeoffreedomairfoilsectionwithanattachedstore,onecontrolsurface,andan additiveunknownnonlineardisturbancethatdoesnotsatisfytheLPassumption.The unknowndisturbancerepresentsunsteadynonlinearaerodynamiceffects.ANNisused asafeedforwardcontroltermtocompensatefortheunknownnonlineardisturbanceand aRISEfeedbackterm[4345]ensuresasymptotictrackingofadesiredstatetrajectory. 2.1AeroelasticSystemModel Thesubsequentdevelopmentandstabilityanalysisisbasedonanaeroelastic modelseeFigure2-1[46],similarto[46],givenas M q + C q + Kq = F where q h T 2 R 2 isacompositevectoroftheverticalpositionandAoAofthe wing-storesection,respectively.Itisassumedthat k q k 1 k q k 2 ,and k q k 3 where 1 ; 2 ; 3 2 R areknownpositiveconstants,whichisjustiedbythebounded oscillatorynatureofLCObehavior.In2, M 2 R 2 2 C 2 R 2 2 K 2 R 2 2 and F 2 R 2 aredenedas M 2 6 4 m 1 m 2 m 2 m 4 3 7 5 ;C 2 6 4 c h 1 c h 2 0 c 3 7 5 K 2 6 4 k h 0 0 k 3 7 5 ;F 2 6 4 )]TJ/F26 11.9552 Tf 9.299 0 Td [(L P M 3 7 5 : In2,theterms m 1 ;m 2 ;m 4 2 R aredenedas m 1 m s + m w m 2 q r x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a m w b cos + s x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a m s b cos 21 PAGE 22 Figure2-1.Diagramdepictingthetwodegreeoffreedomairfoilsectionwithattached store. )]TJ/F15 11.9552 Tf 11.291 0 Td [( r h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h m w b sin )]TJ/F15 11.9552 Tf 11.955 0 Td [( s h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h m s b sin m 4 r x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a 2 + r h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h 2 b 2 m w + s x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a 2 + s h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h 2 b 2 m s + I w + I s where m w m s b r x r h a a h s x s h I w ,and I s 2 R areunknownconstants.Specically, m w isthemassofthewingsection, m s isthemassoftheattachedstore, b isthe semichordlengthofthewing, r x r h arethedistancesfromthewingcenterofmass tothewingmidchordandthewingchordlineinpercentageofthewingsemichord,respectively, a a h arethedistancesfromtheelasticaxisofthewingtothewingmidchord andthewingchordlineinpercentageofthewingsemichord,respectively, s x s h are thedistancesfromthestorecenterofmasstothewingmidchordandwingchordline inpercentageofthewingsemichord,respectively,and I w I s arethewingandstore momentsofinertia,respectively.InEqn.2, c h 1 ;c 2 R aretheunknownconstant dampingcoefcientsoftheplungeandpitchmotion,respectively,and c h 2 2 R isdened 22 PAGE 23 as c h 2 q )]TJ/F15 11.9552 Tf 11.291 0 Td [( r x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a m w b cos )]TJ/F15 11.9552 Tf 11.955 0 Td [( s x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a m s b cos )]TJ/F15 11.9552 Tf 11.291 0 Td [( s h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h m s b sin )]TJ/F15 11.9552 Tf 11.956 0 Td [( r h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h m w b sin : In2, k h 2 R istheunknownplungestiffnesscoefcient,and k q 2 R isthe unknownnonlinearpitchstiffnesscoefcientmodeledas k q = k 1 + k 2 + k 3 2 + k 4 3 + k 5 4 where k 1 k 2 k 3 k 4 ,and k 5 2 R areconstantunknownstiffnessparameters.Also in2, L and P M 2 R aretheliftforceandpitchmomentactingonthewing-store section,respectively,andaremodeledas L = U 2 bSC l ef + C l P M = U 2 b 2 SC l 1 2 + a ef + C m where U S C l C l ,and C m 2 R areunknownconstantcoefcients.Specically, istheatmosphericdensity, U isthefreestreamvelocity, S isthewingspan, C l isthelift coefcientofthewing,and C l C m arethecontroleffectivenesscoefcientsforliftand pitchingmoment,respectively.InEqns.2and2, t 2 R isthecontrolsurface deectionangle,and ef 2 R isdenedas ef + h U + b 1 2 )]TJ/F27 7.9701 Tf 6.587 0 Td [(a U Thedynamicsin2canberewrittenas 1 q = M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 h C )]TJ/F15 11.9552 Tf 14.621 3.022 Td [(~ C q )]TJ/F15 11.9552 Tf 15.088 3.022 Td [(~ Kq i + d wheretheauxiliaryterms C )]TJ/F26 11.9552 Tf 9.299 0 Td [(C l C m T 2 R 2 d d h d T 2 R 2 denotes anunknown,nonlineardisturbancethatrepresentsunmodeled,unsteadyaerodynamic 1 SeeAppendixAfordetailsontheinvertibilityof M 23 PAGE 24 effects.Moreover,in2, ~ C 2 R 2 2 and ~ K 2 R 2 2 aredenedas ~ C 2 6 4 c h 1 + C L c h 2 + C L b )]TJ/F24 7.9701 Tf 6.675 -4.977 Td [(1 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [(a )]TJ/F26 11.9552 Tf 9.299 0 Td [(C L b )]TJ/F24 7.9701 Tf 6.675 -4.976 Td [(1 2 + a c )]TJ/F26 11.9552 Tf 11.955 0 Td [(C L b 2 )]TJ/F24 7.9701 Tf 6.675 -4.976 Td [(1 4 )]TJ/F26 11.9552 Tf 11.955 0 Td [(a 2 3 7 5 = 2 6 4 ~ C 11 ~ C 12 ~ C 21 ~ C 22 3 7 5 ~ K 2 6 4 k h C L U 0 k )]TJ/F26 11.9552 Tf 11.955 0 Td [(C L Ub )]TJ/F24 7.9701 Tf 6.675 -4.977 Td [(1 2 + a 3 7 5 = 2 6 4 ~ K 11 ~ K 12 0 ~ K 22 3 7 5 ; and C L UbSC l 2 R isanunknownconstant.Thesubsequentcontroldevelopmentis basedontheassumptionthatthenonlineardisturbancesareboundedas j d h j 1 ; d h 2 ; j d j 3 ; d 4 ; where j 2 R ; j =1 ;:::; 4 arepositive,knownconstants. 2.2ControlObjective ThecontrolobjectiveistoensuretheairfoilsectionAoA, ,tracksadesired trajectorydenedas d 2 R .TheformulationofanAoAtrackingproblemenablesthe AoAofthewingtobeoptimizedforagivenmetricandightcondition.Fortheextension tothethreedimensionalcase,thecontrolobjectiveprovidestheabilitytoalterthe wingtwistforagivenightconditiontooptimizeagivenperformancemetric,suchas aerodynamicefciency.Thesubsequentcontroldevelopmentandanalysisisbasedon theassumptionthat d ; d ; d ; ... d 2L 1 .Toquantifythecontrolobjectiveandfacilitate thecontroldesign,atrackingerror, e 1 2 R ,andtwoauxiliarytrackingerrors, e 2 ;r 2 R aredenedas e 1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( d e 2 e 1 + 1 e 1 r e 2 + 2 e 2 where 1 ; 2 2 R arepositiveconstants.Thesubsequentdevelopmentisbasedonthe assumptionthat q and q aremeasurable.Hence,theauxiliarytrackingerror, r ,isnot 24 PAGE 25 measurablesinceitdependson q .Substitutingthesystemdynamicsfrom2intothe errordynamicsin2yieldsthefollowingexpression r = f + g + d wheretheauxiliaryterms f 2 R and g 2 R aredenedas f = )]TJ/F26 11.9552 Tf 22.973 8.088 Td [(m 2 det M )]TJ/F15 11.9552 Tf 11.964 3.022 Td [(~ C 11 h )]TJ/F15 11.9552 Tf 14.621 3.022 Td [(~ C 12 )]TJ/F15 11.9552 Tf 15.088 3.022 Td [(~ K 11 h )]TJ/F15 11.9552 Tf 15.088 3.022 Td [(~ K 12 + m 1 det M )]TJ/F15 11.9552 Tf 11.964 3.022 Td [(~ C 21 h )]TJ/F15 11.9552 Tf 14.621 3.022 Td [(~ C 22 )]TJ/F15 11.9552 Tf 15.088 3.022 Td [(~ K 22 )]TJ/F15 11.9552 Tf 13.115 0 Td [( d + 1 e 1 + 2 e 2 g = m 2 det M C l + m 1 det M C m and g isinvertible 2 providedthatsufcientconditionsonthewinggeometryandstore locationaremet. 2.3ControlDevelopment Aftersomealgebraicmanipulation,theopen-looperrorsystemfor r t canbe obtainedas 1 g r = + 1 g d f d + + d where g d 2 R and f d 2 R aredenedas f d = )]TJ/F26 11.9552 Tf 22.973 8.088 Td [(m 2 q d det M q d )]TJ/F15 11.9552 Tf 11.964 3.022 Td [(~ C 11 h d )]TJ/F15 11.9552 Tf 14.621 3.022 Td [(~ C 12 q d ; q d d )]TJ/F15 11.9552 Tf 15.088 3.022 Td [(~ K 11 h d )]TJ/F15 11.9552 Tf 15.088 3.022 Td [(~ K 12 d + m 1 det M q d )]TJ/F15 11.9552 Tf 11.964 3.022 Td [(~ C 21 h d )]TJ/F15 11.9552 Tf 14.621 3.022 Td [(~ C 22 d )]TJ/F15 11.9552 Tf 15.088 3.022 Td [(~ K 22 q d d )]TJ/F15 11.9552 Tf 13.115 0 Td [( d ; g d = m 2 q d det M q d C l + m 1 det M q d C m ; where q d h d d T 2 R 2 ,and h d 2 R isadesiredtrajectoryfortheverticalposition ofthewing.Thesubsequentdevelopmentisbasedontheassumptionthatthedesired trajectories, h d and h d ,arebounded.In2,theauxiliaryfunction 2 R isdenedas 2 SeeAppendixBfordetails. 25 PAGE 26 = 1 g f )]TJ/F24 7.9701 Tf 15.321 4.707 Td [(1 g d f d .Basedontheuniversalfunctionapproximationproperty,amulti-layerNN isusedtoapproximatetheuncertaindynamics f d g d h d ; h d ; d ; d as[43] f d g d = W T )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(V T x d + x d wheretheNNinput x d 2 R 7 isdenedas x d t 1 h d h d h d d d d T .In 2, V 2 R 7 n 2 isaconstantidealweightmatrixfortherst-to-secondlayerofthe NN, W 2 R n 2 +1 isaconstantidealweightmatrixforthesecond-to-thirdlayerofthe NN, n 2 isthenumberofneuronsinthehiddenlayer, 2 R n 2 +1 denotestheactivation function,and 2 R isthefunctionreconstructionerror.Since x d isdenedinterms ofdesiredboundedterms,theinputstotheNNremainonacompactset.Sincethe desiredtrajectoriesareassumedtobebounded,then[43] j x d j 1 ; j x d ; x d j 2 ; j x d ; x d ; x d j 3 ,where 1 ;" 2 ;" 3 2 R areknownpositiveconstants. Basedontheopen-looperrorsystemin2andthesubsequentstability analysis,thecontrolsurfacedeectionangleisdesignedas = )]TJ/F31 11.9552 Tf 9.829 11.243 Td [(c f d g d )]TJ/F26 11.9552 Tf 11.955 0 Td [( where b f d g d 2 R isdenedas c f d g d ^ W T ^ V T x d and 2 R denotesthesubsequentlydenedRISEfeedbackterm.In2, ^ W 2 R n 2 +1 and ^ V 2 R 7 n 2 denoteestimatesfortheidealweightmatriceswhoseupdatelawsare denedas ^ W proj )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(1 ^ 0 ^ V T x d e 2 ^ V proj )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(2 x d ^ 0 T ^ We 2 T where )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(1 2 R n 2 +1 n 2 +1 )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(2 2 R 7 7 areconstant,positivedenitecontrolmatricesand ^ 0 d ^ V T x d d ^ V T x d : Thesmoothprojectionalgorithmin2and2isusedtoensure 26 PAGE 27 thattheidealNNweightestimates, ^ W and ^ V ,remainbounded[47].TheRISEfeedback termin2isdenedas k s 1 + k s 2 e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k s 1 + k s 2 e 2 + where 2 R istheFilippovsolutiontothefollowingdifferentialequation = k s 1 + k s 2 2 e 2 + 1 sgn e 2 ; = 0 where k s 1 ;k s 2 ; 1 2 R arepositive,constantcontrolgainsand 0 2 R isaknowninitial condition.Theexistenceofsolutionsfor 2 K [ w 1 ] canbeshownusingFilippov's theoryofdifferentialinclusions[4851]where w 1 : R R isdenedastheright-hand sideof2and K [ w 1 ] T > 0 T S m =0 cow 1 e 1 ;B )]TJ/F26 11.9552 Tf 11.955 0 Td [(S m ,where T S m =0 representsthe intersectionofallsets S m ofLebesguemeasurezero, co representsconvexclosure,and B = f 2 R jj e 2 )]TJ/F26 11.9552 Tf 11.956 0 Td [( j < g [52,53]. Theclosed-looperrorsystemisobtainedbysubstituting2into2as 1 g r = + f d g d )]TJ/F31 11.9552 Tf 12.486 11.243 Td [(c f d g d )]TJ/F26 11.9552 Tf 11.956 0 Td [( + d : Tofacilitatethesubsequentstabilityanalysis,thetimederivativeof27isdetermined as 1 g r = )]TJ/F26 11.9552 Tf 12.607 8.088 Td [(d dt 1 g r +_ + d dt f d g d )]TJ/F26 11.9552 Tf 15.265 8.088 Td [(d dt c f d g d )]TJ/F15 11.9552 Tf 14.176 0 Td [(_ + d : Using2and2,theclosed-looperrorsystemin2canberewrittenas 1 g r = )]TJ/F26 11.9552 Tf 12.608 8.088 Td [(d dt 1 g r +_ + W T 0 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(V T x d V T x d )]TJ/F15 11.9552 Tf 15.368 3.022 Td [(^ W T 0 ^ V T x d ^ V T x d )]TJ/F15 11.9552 Tf 14.012 6.177 Td [(_ ^ W T ^ V T x d )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W T 0 ^ V T x d ^ V T x d +_ )]TJ/F15 11.9552 Tf 14.176 0 Td [(_ + d : Aftersomealgebraicmanipulation,2canberewrittenas 1 g r = )]TJ/F26 11.9552 Tf 12.607 8.088 Td [(d dt 1 g r + ^ W T ^ 0 ~ V T x d )]TJ/F15 11.9552 Tf 16.668 6.177 Td [(_ ^ W T ^ +_ +_ )]TJ/F15 11.9552 Tf 14.176 0 Td [(_ + d + ~ W T ^ 0 ^ V T x d 27 PAGE 28 )]TJ/F15 11.9552 Tf 12.711 3.022 Td [(^ W T ^ 0 ^ V T x d )]TJ/F26 11.9552 Tf 11.956 0 Td [(W T ^ 0 ^ V T x d + W T 0 V T x d )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W T ^ 0 ~ V T x d where 0 = 0 )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(V T x d 2 R n 2 +1 n 2 ^ =^ ^ V T x d 2 R n 2 +1 andtheparameterestimation errormatrices ~ W 2 R n 2 +1 and ~ V 2 R 7 n 2 aredenedas ~ W = W )]TJ/F15 11.9552 Tf 15.609 3.022 Td [(^ W and ~ V = V )]TJ/F15 11.9552 Tf 13.982 3.022 Td [(^ V respectively.UsingtheNNweightupdatelawsin2and2andthetime derivativeoftheRISEfeedbacktermin2,theclosed-looperrorsystemin2 canbeexpressedas 1 g r = ~ N + N d + N B )]TJ/F26 11.9552 Tf 11.955 0 Td [(e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k s 1 + k s 2 r )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 sgn e 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 d dt 1 g r where ~ N 2 R N d 2 R ,and N B 2 R aredenedas ~ N )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 d dt 1 g r +_ 1 + e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [(proj )]TJ/F24 7.9701 Tf 7.315 -1.793 Td [(1 ^ 0 ^ V T x d e 2 T ^ )]TJ/F15 11.9552 Tf 12.711 3.022 Td [(^ W T ^ 0 proj )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(2 x d ^ 0 T ^ We 2 T T x d N d W T 0 V T x d +_ +_ 2 + d N B N B 1 + N B 2 : In2,theterms N B 1 2 R and N B 2 2 R aredenedas N B 1 )]TJ/F26 11.9552 Tf 9.299 0 Td [(W T ^ 0 ^ V T x d )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W T ^ 0 ~ V T x d N B 2 ^ W T ^ 0 ~ V T x d + ~ W T ^ 0 ^ V T x d : Thetermsin2aresegregatedbasedontheirbounds.Allthetermsin2 aredependentonthedesiredtrajectories,therefore N d anditsderivativecanbe upperboundedbyaconstant,whichwillberejectedbytheRISEfeedbacktermin thecontroller.Thetermsin2aresegregatedintotermsthatwillberejectedby theRISEfeedback, N B 1 ,andtermsthatwillberejectedbyacombinationoftheRISE feedbackandNNweightestimateadaptiveupdatelaws, N B 2 .In2and2, hasbeensegregatedinto 1 and 2 where 1 denotesthecomponentsof thatare statedependentorcanbeupperboundedbythenormofthestates,and 2 denotesthe 28 PAGE 29 componentsthatcanbeupperboundedbyaconstant 3 .Thetermsin ~ N canbeupper boundedas 4 ~ N k z k where z e 1 e 2 r T 2 R 3 ,and 2 R isapositiveboundingconstant.Similar to[43],thefollowinginequalitiescanbedeveloped j N d j 1 ; N d 2 ; j N B j 3 ; N B 4 + 5 j e 2 j where i 2 R i =1 ; 2 ;:::; 5 arepositiveboundingconstants. 2.4StabilityAnalysis TofacilitatethesubsequentLyapunov-basedstabilityanalysis,let P 2 R bedened astheFilippovsolutiontothefollowingdifferentialequation P = )]TJ/F26 11.9552 Tf 9.299 0 Td [(r N B 1 + N d )]TJ/F26 11.9552 Tf 11.956 0 Td [( 1 sgn e 2 )]TJ/F15 11.9552 Tf 13.693 0 Td [(_ e 2 N B 2 + 2 e 2 2 ; P = 1 j e 2 j)]TJ/F26 11.9552 Tf 17.933 0 Td [(e 2 N d + N B : Theexistenceofsolutionsfor P t canbeestablishedinasimilarmannerasin2 byusingFilippov'stheoryofdifferentialinclusionsfor P t 2 K [ w 2 ] ,where w 2 2 R is denedastheright-handsideof2.Providedthat 1 and 2 areselectedbasedon thesufcientconditionsin2, P t 0 [43].Furthermore,let Q 2 R bedenedas Q 2 2 ~ W T )]TJ/F29 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.587 0 Td [(1 1 ~ W + 2 2 tr ~ V T )]TJ/F29 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 ~ V ; where Q 0 since )]TJ/F24 7.9701 Tf 7.315 -1.793 Td [(1 and )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(2 areconstantpositivedenitematrices,and 2 2 R + 3 SeeAppendixCfordetails 4 SeeAppendixDfordetails. 29 PAGE 30 Theorem2.1. Thecontrollergivenin 2 2 ensuresthatallclosed-loopsignals areboundedandthetrackingerrorisregulatedinthesensethat e 1 t 0 as t !1 providedthatthecontrolgainsareselectedas 1 > 1 + 2 + 1 2 3 + 1 2 4 ; 2 > 5 ; 1 > 1 2 ; 2 > 2 +1 : Proof. Let D R 5 beadomaincontaining y =0 ,where y 2 R 5 andisdenedas y e 1 e 2 r p P p Q T : Let V L y : D! R beapositivedenite,continuouslydifferentiablefunctiondenedas V L e 2 1 + 1 2 e 2 2 + 1 2 1 g r 2 + P + Q: Equation2satises U 1 V L U 2 providedthat 1 and 2 areselectedbasedon thesufcientconditionsin2.Thecontinuouspositivedenitefunctions U 1 ;U 2 2 R aredenedas U 1 1 k y k 2 U 2 2 k y k 2 where 1 ; 2 2 R aredenedas 1 1 2 min f 1 ;g l g 2 min 1 2 g m ; 1 and g l j g j g m Thetimederivativeof2existsalmosteverywherea.e,and V L 2 ~ V L where ~ V L = T 2 @V L T K e 1 e 2 r P )]TJ/F25 5.9776 Tf 6.952 2.345 Td [(1 2 P 2 Q )]TJ/F25 5.9776 Tf 6.951 2.346 Td [(1 2 Q 2 1 T ,where @V L isthegeneralizedgradient of V L .Since V L isacontinuouslydifferentiablefunction, ~ V L canbeexpressedas ~ V L = r V T L K e 1 e 2 r P )]TJ/F25 5.9776 Tf 6.952 2.345 Td [(1 2 P 2 Q )]TJ/F25 5.9776 Tf 6.951 2.345 Td [(1 2 Q 2 1 T ; where r V L = 2 e 1 e 2 1 g r 2 P 1 2 2 Q 1 2 1 2 d dt 1 g r 2 .Usingthecalculusfor K from[53],2,2,2,and2,2canbeexpressedas ~ V L 2 e 1 e 2 )]TJ/F26 11.9552 Tf 11.956 0 Td [( 1 e 1 + e 2 r )]TJ/F26 11.9552 Tf 11.956 0 Td [( 2 e 2 + r ~ N + N d + N B )]TJ/F26 11.9552 Tf 11.955 0 Td [(e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k s 1 + k s 2 r )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 K [sgn e 2 ] )]TJ/F26 11.9552 Tf 9.299 0 Td [(r N B 1 + N d )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 K [sgn e 2 ] )]TJ/F15 11.9552 Tf 13.693 0 Td [(_ e 2 N B 2 + 2 e 2 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 2 ~ W T )]TJ/F29 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.587 0 Td [(1 1 ^ W )]TJ/F26 11.9552 Tf 11.955 0 Td [( 2 tr ~ V T )]TJ/F29 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.587 0 Td [(1 2 ^ V ; 30 PAGE 31 where K [sgn e 2 ]=sgn e 2 suchthat sgn e 2 =1 if e 2 > 0 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ; 1] if e 2 =0 ,and )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 if e 2 < 0 .Thesetoftimes f t 2 [0 ; 1 : r 1 K [sgn e 2 ] )]TJ/F26 11.9552 Tf 11.955 0 Td [(r 1 K [sgn e 2 ] 6 = f 0 gg [0 ; 1 isequaltothesetoftimes f t : e 2 t =0 ^ r t 6 =0 g .FromEqn.2,thissetcanalso beexpressedas f t : e 2 t =0 ^ e 2 t 6 =0 g .Since e 2 iscontinuouslydifferentiable,itcan beshownusing[54],Lemma2thatthesetoftimeinstances f t : e 2 t =0 ^ e 2 t 6 =0 g is isolatedandmeasurezero;hence ismeasurezero.Since ismeasurezero,2 canbereducedtothefollowingscalarinequality V L a:e: 2 e 1 e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 e 2 1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 2 e 2 2 + 2 e 2 2 + r ~ N )]TJ/F26 11.9552 Tf 11.955 0 Td [(k s 1 r 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [(k s 2 r 2 + 2 e 2 h ^ W T ^ 0 ~ V T x d + ~ W T ^ 0 ^ V T x d i )]TJ/F26 11.9552 Tf 11.956 0 Td [( 2 ~ W T )]TJ/F29 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.586 0 Td [(1 1 ^ W )]TJ/F26 11.9552 Tf 11.955 0 Td [( 2 tr ~ V T )]TJ/F29 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.587 0 Td [(1 2 ^ V ; ByusingYoung'sinequalityandtheNNweightupdatelawsin2and2along withtheupperboundon ~ N givenin2,theexpressionin2canberewrittenas V L a:e: )]TJ/F15 11.9552 Tf 32.469 0 Td [( 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 e 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 e 2 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [(k s 1 r 2 + 2 4 k s 2 k z k 2 : Theexpressionin2canbefurthersimpliedas V L a:e: )]TJ/F31 11.9552 Tf 25.827 16.857 Td [( 3 )]TJ/F26 11.9552 Tf 17.935 8.088 Td [( 2 4 k s 2 k z k 2 ; where 3 =min f 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 ;k s 1 g isapositiveconstantprovidedthat 1 ; 2 are selectedaccordingto2.Theexpressionin2canbeupperboundedas V L a:e: )]TJ/F26 11.9552 Tf 23.834 0 Td [(c k z k 2 ; where c 2 R isapositiveconstantprovidedthat 3 > 4 k s 2 .Theexpressionsin2 and2canbeusedtoshowthat V L 2L 1 ,andhence, e 1 ;e 2 ;r;P;Q 2L 1 .Given that e 1 ;e 2 ;r 2L 1 ,2and2indicatethat e 1 ; e 2 2L 1 .Since e 1 ;e 2 ;r 2L 1 and d ; d ; d 2L 1 byassumption,2-2canbeusedtoshowthat ; ; 2L 1 .If ; 2L 1 ,2canbeusedtoshowthat M;C;K 2L 1 .Giventhat M 2L 1 ,2 indicatesthat g 2L 1 .Since t ; t 2L 1 in D and h t 2L 1 then,2,2, 31 PAGE 32 and2canbeusedtoshowthat F 2L 1 ;hence,withtheboundsin2itcanbe concludedfrom2thatthecontrolinput 2L 1 .Giventhat ~ N;N d ;N B ;r;e 2 ;g 2L 1 itcanbeconcludedfrom2that r 2L 1 .Since e 1 ; e 2 ; r 2L 1 ,thedenitionof z t canbeusedtoshowthat z isuniformlycontinuous.Corollary1from[55]canbeusedto showthat k z k! 0 ,andtherefore, e 1 0 as t !1 2.5SimulationResults Anumericalsimulationispresentedtoillustratetheperformanceofthedeveloped controllerandprovideacomparisonwiththecontrollerin[13].Thecontrollerfrom[13] wasselectedforcomparisonbecauseitisoneofthefewcontrollersthatconsider structuraluncertainties.However,thisisnotanequalcomparison,sincethecontroller in[13]considersuncertaintiesinthepitchstiffnessonly,whilethecontrolstrategy developedinthispaperconsidersuncertaintiesinallparametersinthestructuraland aerodynamicmodels.Forthisreason,thestructuralandaerodynamicparametersthat areassumedtobeknownin[13]aretakentobeoffby 10% fromtheactualvalues.The controllerin[13]isgivenby = 1 g 4 U 2 )]TJ/F26 11.9552 Tf 9.299 0 Td [(F L q; q )]TJ/F15 11.9552 Tf 13.581 3.022 Td [(^ T R q )]TJ/F15 11.9552 Tf 12.273 3.155 Td [( k 1 )]TJ/F15 11.9552 Tf 12.273 3.155 Td [( k 2 ; where g 4 2 R isacontroleffectivenessparameter, U 2 R denotesthefreestream velocity, F L q; q 2 R isafeedbacklinearizationtermthatrequiresexactmodel knowledgeofcertainparametersinthestructuralmodelandallparametersinthe aerodynamicmodel, ^ 2 R i denotesavectoroftheestimatesoftheuncertain parametersinthepitchstiffnessmodel, R q 2 R i representsaknownregression matrix,and k 1 ; k 2 2 R arepositivecontrolgains.Thecontrolgainswereselectedas k 1 = k 2 =60 basedonimprovingtheresultingtransientperformanceofthecontroller whilekeepingthecontroleffortwithintolerablelimits 10 deg.Theestimate, ^ ,is 32 PAGE 33 updatedviaagradientupdatelawgivenby ^ =_ R T q : ThemodelparametersforthesimulationareshowninTable2-1and2 Table2-1.AeroelasticModelParameters ParameterParameter m w 4.0kg I s 0.0050kg m 2 m s 4.0kg c h 1 2.743x10 1 kg/s r x 0.0 c 0.036kg m 2 /s r h 0.0 k h 2.200x10 3 N/m a -0.6 1.225kg/m 3 a h 0.0 U 1.20x10 1 m/s b 0.14m S 1.0m s x 0.098 C l 6.81/rad s h 1.4 C l 9.3x10 1 N/rad I w 0.043kg m 2 C m 2.3N m/rad k q =0 : 5 )]TJ/F15 11.9552 Tf 11.955 0 Td [(11 : 05 +657 : 75 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4290 3 +8644 : 85 4 : ThecontrolobjectiveistoregulatetheAOAtozerodegreesfromtheinitialcondition h =0 m, h =0 m/s, =3 : 0 deg,and =0 deg/s.FromFigure2-2it isevidentthatthesystem,undertheaboveconditions,experiencesLCObehaviorin theabsenceofacontrolstrategyandexogenousdisturbances.Thedevelopedcontrol strategywasappliedtothesystemintheabsenceofexogenousdisturbanceswiththe followinggains: 1 =2 2 =3 k s 1 + k s 2 =3 1 =0 : 1 n 2 =25 )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(1 =10 I 26 ,and )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(2 =10 I 7 where I m denotesan m m identitymatrix. Figures2-3and2-4showthestatesofthewingsectionandthecontrolsurface deection,respectively.Theguresindicatethatthedevelopedcontrollersuppresses theLCObehaviorwithcontrolsurfacedeectionsthatremainwithinreasonablelimits. Furthermore,thedevelopedcontrollerrequiresasmallercontroleffortthanthecontroller in[13]andhasbettertransientperformance.Thetwocontrollerswerealsoappliedto thesysteminthepresenceofanadditiveexogenousdisturbanceselectedas N t = 33 PAGE 34 Figure2-2.Aeroelasticsystemfreeresponsewithoutdisturbances 0 : 25cos t 0 : 25sin t T .Figures2-5and2-6showthesystemstatesandcontrol effortinthepresenceoftheadditivedisturbance,respectively.Thedevelopedcontroller iscapableofregulatingtheAOAofthewingsectioninthepresenceofexogenous disturbanceswithcontrolsurfacedeectionsthatremainwithintolerablelimits.However, thecontrollerin[13]isnotcapableofeliminatingtheeffectsofthedisturbanceinthe wingsectionverticalposition.Duetothecouplednatureoftheaeroelasticsystem dynamicsandtheavailabilityofasinglecontrolsurface,anydisturbanceintheAOAwill propagateintotheverticalpositionasanunmatcheddisturbance.Onesolutiontothis issueistoincludeanadditionalcontrolsurfaceattheleadingedgethatcouldbeusedto suppressunwantedmotionintheverticalposition. A1500sampleMonteCarlosimulationwasexecutedtodemonstratetherobustnessofthedevelopedcontrollertoplantuncertaintiesandsensornoise.Theuncertain modelparameterswereuniformlydistributedoverarangethatextendedfrom 80% to 120% ofthenominalvaluesfoundinTable2-1and2.Azeromeannoisesignal uniformlydistributedoveranintervalwasaddedtoeachmeasurement.Forthevertical 34 PAGE 35 Figure2-3.Comparisonofthecontrolledaeroelasticsystemresponse Figure2-4.Controlsurfacedeections, t ,forthedevelopedcontrollerandprevious controller 35 PAGE 36 Figure2-5.Aeroelasticsystemstatesinthepresenceofanadditivedisturbance Figure2-6.Controlsurfacedeection, t ,forthedevelopedcontrollerandprevious controller 36 PAGE 37 Table2-2.MonteCarloSimulationResults MeanStandardDeviation MaximumError2.9deg0.0038deg RMSError0.97deg0.073deg MaximumControlEffort7.5deg2.6deg displacementandvelocity,theintervalwas 2 : 5 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(3 m and 2 : 5 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(3 m = s ,respectively.FortheAOAandAOArate,theintervalwas 4 : 5 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(3 rad and 1 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 rad = s Foreachsample,themaximumoftheabsolutevalueofthetrackingerrorandcontrol surfacedeection,andtheRMSvalueofthetrackingerrorwerecalculated.Theresults, presentedinTable2-2,indicatethatthemaximumerrorandRMSerrorofthesystemdo notvarysignicantlyovertherangeoftheuncertaintiesconsidered. Figures2-7-2-9showtheaveragetrajectoryand 3 condenceboundsfor thesystemstatesandcontroleffortforthe1500MonteCarlosamples.Figure2-7 showsthattheAOAforallsamplesconvergestozeroinapproximately3.5secondsand thetightcondenceboundsindicatethatthesystemperformanceisnotsignicantly impactedbyvariationsintheuncertainparameters.ItisevidentfromFigure2-8 thattheuncontrolledverticaldisplacementdampsoutforallsamples.Figure2-9 showsthatthecontrolsurfacedeectionismoresensitivetochangesinthesystem parameters.The 3 condenceboundforthemaximumcontroleffortisapproximately threetimesthatofthenumericalresultshowninFigure2-6.Thissensitivityindicates thatinamoresevereLCO,variationsintheuncertainparameterscouldleadtoacontrol effortgreaterthantheactuatorlimits. 2.6Summary Arobustadaptivecontrolstrategyisdevelopedtosuppressstore-inducedLCO behaviorofanaeroelasticsystem.ThedevelopedcontrollerusesaNNfeedforward termtoaccountforstructuralandaerodynamicuncertaintiesandaRISEfeedback termtoguaranteeasymptotictrackingofadesiredAOAtrajectory.ALyapunov-based stabilityanalysisisusedtoproveanasymptotictrackingresult.Numericalsimulations 37 PAGE 38 Figure2-7.MonteCarloAOAtrajectories Figure2-8.MonteCarloverticalpositiontrajectories 38 PAGE 39 Figure2-9.MonteCarlocontroleffort illustrateLCOsuppressionandAOAtrackingperformanceoverarangeofuncertainty. Apotentialdrawbacktothedevelopedcontrolstrategyisthatthecontrollawdoesnot accountforactuatorlimits.AstheseverityoftheLCObehaviorincreases,thedeveloped controllercandemandalargecontrolsurfacedeection.Additionally,theMonteCarlo simulationresultsindicatedthatthemaximumcontroleffortissensitivetovariationsin theparameteruncertainties,whichcouldleadtounexpectedactuatorsaturation. 39 PAGE 40 CHAPTER3 SATURATEDRISETRACKINGCONTROLOFSTORE-INDUCEDLIMITCYCLE OSCILLATIONS ThefocusofthischapteristodevelopasaturatedcontrollertosuppressLCO behaviorinatwodegreeoffreedomairfoilsectioninthepresenceofstructuraland aerodynamicuncertaintieswithoutbreachingactuatorlimits.Asmoothsaturation functionisincludedintheclosed-looperrorsystemdesigntoensurethecommanded controleffortremainswithinactuatorlimitsandacontinuoussaturatedRISEfeedback controlstructureensuresasymptotictrackingoftheAoA[16]. 3.1ControlObjective Thesubsequentcontroldevelopmentandstabilityanalysisisbasedontheaeroelasticmodeldescribedin2seeFigure2-1.Thecontrolobjectiveistoensure theairfoilsectionAoA, ,tracksadesiredtrajectorydenedas d 2 R usingalimited amplitude,continuouscontroller.AsinChapter2,itisassumedthat d ; d ; d ; ... d 2L 1 Thecontrolobjectiveisquantiedbydeningatrackingerror e 1 2 R as e 1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( d : Tofacilitatethecontroldesign,theauxiliarytrackingerrors e 2 2 R and r 2 R aredened as[16] e 2 e 1 + 1 tanh e 1 +tanh e f ; r e 2 + 2 tanh e 2 + 3 e 2 ; where 1 ; 2 ; 3 2 R arepositiveconstantcontrolgains,andtheauxiliarysignal e f 2 R is denedasthesolutiontothefollowingdifferentialequation e f cosh 2 e f )]TJ/F26 11.9552 Tf 9.298 0 Td [( 4 e 2 +tanh e 1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 5 tanh e f ;e f t 0 = e f 0 ; 40 PAGE 41 where e f 0 2 R isaknowninitialconditionand 4 ; 5 2 R arepositiveconstantcontrol gains.Thesubsequentdevelopmentisbasedontheassumptionthat q and q aremeasurable.Hence, e 1 and e 2 aremeasurable,and e f canbecomputedfrommeasurable terms,but r isnotmeasurablesinceitdependson q .Thefollowinginequalityproperties willbeusedinthesubsequentdevelopment[56]: j jj tanh j ; j tanh j 2 tanh 2 j j ; tanh tanh 2 ; j j 2 lncosh 1 2 tanh 2 j j : 3.2ControlDevelopment Substitutingthedynamicsfrom2into3andmultiplyingby det M g yields det M g r = f g + det M g d + ; wheretheauxiliaryterms f 2 R and g 2 R aredenedas f )]TJ/F26 11.9552 Tf 9.299 0 Td [(m 1 ~ C 21 h + ~ C 22 + ~ K 22 + m 2 ~ C 11 h + ~ C 12 + ~ K 11 h + ~ K 12 )]TJ/F15 11.9552 Tf 11.291 0 Td [(det M d +det M 1 cosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 e 1 e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 tanh e 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(tanh e f )]TJ/F15 11.9552 Tf 11.291 0 Td [(det M 5 tanh e f +det M tanh e 1 + 2 tanh e 2 + 3 e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 4 e 2 ; g m 2 C l + m 1 C m : Basedontheopen-looperrorsystemin3,thecontrolsurfacedeectionisdesigned as = )]TJ/F26 11.9552 Tf 9.299 0 Td [( 4 tanh v ; where v 2 R isthegeneralizedFilippovsolutiontothedifferentialequation v = cosh 2 v sgn e 2 ;v t 0 = v 0 ; where 2 R isapositiveconstantcontrolgain,and v 0 2 R isaknowninitialcondition. Theexistenceofsolutionsfor v 2 K [ w 1 ] canbeshownusingdifferentialinclusionsas 41 PAGE 42 inChapter2,where w 1 : R R isdenedastheright-handsideof3, K [ w 1 ] T > 0 T S m =0 cow 1 e 1 ;B )]TJ/F26 11.9552 Tf 11.955 0 Td [(S m ,and B = f 2 R jj e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [(" j < g .Thedesiretoinjectasmooth saturationfunctionintothecontrolstructuremotivatestheusageofthehyperbolic tangentfunctionin3.Furthermore,itisclearthatthecontrolsurfacedeection isboundedandwillnotbreachtheactuatorlimitsprovidedthatthecontrolgain 4 is selectedtobelessthanthelimit.Thedesignoftheauxiliaryterm v in3ismotivated bytheextratimederivativethatwillbeappliedtotheclosed-loopsystemobtainedby substituting3into3.Theextraderivativeintroducesa cosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 v terminthe closed-loopdynamicswhichwillbecanceledbythe cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 v termin3. Theclosed-looptrackingerrordynamicscanbeobtainedbydifferentiating3 withrespecttotimeandsubstitutingthetimederivativeof3toyield det M g r = )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 d dt det M g r + ~ N + N d + )]TJ/F15 11.9552 Tf 11.365 0 Td [(tanh e 2 )]TJ/F26 11.9552 Tf 11.365 0 Td [(e 2 )]TJ/F15 11.9552 Tf 12.56 8.087 Td [(det M g 4 r )]TJ/F26 11.9552 Tf 11.365 0 Td [( 4 sgn e 2 ; where ~ N 2 R N d 2 R ,and 2 R aredenedas ~ N )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 d dt det M g r + d dt det M g 1 cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e 1 e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 tanh e 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(tanh e f )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(2det M g 1 cosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 e 1 tanh e 1 e 2 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(det M g 2 1 cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(4 e 1 e 1 +tanh e 2 + e 2 )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(det M g 2 C l m 2 )]TJ/F26 11.9552 Tf 5.48 -9.683 Td [( 1 cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e 1 e 1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 5 tanh e f +tanh e 1 + 2 tanh e 2 + 3 e 2 )]TJ/F27 7.9701 Tf 13.219 13.493 Td [(d dt det M g 5 tanh e f )]TJ/F15 11.9552 Tf 11.955 0 Td [(tanh e 1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 2 tanh e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 3 e 2 )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(det M g )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [( 5 tanh e 1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 2 5 tanh e f )]TJ/F15 11.9552 Tf 11.955 0 Td [(cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e 1 e 1 )]TJ/F26 11.9552 Tf 11.956 0 Td [( 2 cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e 2 e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 3 e 2 + det M g 1 cosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 e 1 e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(tanh e 1 + 5 tanh e f ; N d m 2 g ~ C 11 h + ~ C 12 + ~ K 11 h + ~ K 12 )]TJ/F26 11.9552 Tf 13.151 8.088 Td [(m 1 g ~ C 21 h + ~ C 22 + ~ K 22 + ~ K 22 + m 2 g ~ C 11 h + ~ C 12 + ~ C 12 + ~ K 11 h + ~ K 12 )]TJ/F27 7.9701 Tf 15.875 13.492 Td [(d dt det M g d 42 PAGE 43 + C l m 2 g 2 m 1 ~ C 21 h + ~ C 22 + ~ K 22 )]TJ/F26 11.9552 Tf 11.955 0 Td [(m 2 ~ C 11 h + ~ C 12 + ~ K 11 h + ~ K 12 +det M d )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(det M g ... d + det M g d + d dt det M g d ; 4 e 2 det M g )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [( 1 cosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 e 1 + 5 + 3 )]TJ/F27 7.9701 Tf 15.875 13.492 Td [(d dt det M g + m 2 C l det M g 2 + det M g 2 4 tanh e 2 : UsingtheassumptionsonthedesiredtrajectoriesandboundednessoftheLCOstates, upperboundscanbedevelopedfor3and3as ~ N 0 k x k ; j N d j 1 ; N d 2 ; where 0 ; 1 ; 2 2 R areknownboundingconstants,and x 2 R 4 isdenedas x tanh e 1 e 2 r tanh e f T : 3.3StabilityAnalysis Tofacilitatethesubsequentanalysis,let z e 1 e 2 re f T 2 R 4 and y z T p P 2 R 5 where P 2 R isaFilippovsolutiontothedifferentialequation P = )]TJ/F26 11.9552 Tf 9.299 0 Td [(r N d )]TJ/F26 11.9552 Tf 11.955 0 Td [( 4 sgn e 2 ; P t 0 = 4 j e 2 t 0 j)]TJ/F26 11.9552 Tf 17.933 0 Td [(e 2 t 0 N d t 0 : Provided 4 isselectedsuchthat 4 > 1 + 2 3 P t 0 8 t 2 [0 ; 1 [16].Tofurther facilitatethestabilityanalysis,letthecontrolgain 4 beexpressedas 4 = a + b ,where a and b 2 R arepositiveconstants. Theorem3.1. Thecontrollergivenin 3 and 3 yieldsglobalasymptotictracking oftheairfoilsectionAoAinthesensethatallFilippovsolutionstothedifferential equationsin 3 3 3 ,and 3 areboundedand e 1 0 as t !1 43 PAGE 44 providedthatthecontrolgainsareselectedtosatisfythefollowingsufcientconditions 1 > 1 2 ; 3 > 2 4 +1 ; 4 > 1 + 2 3 ; 1 a > c 2 1 2 ; 5 > 2 4 2 ;> 2 0 4 1 b ; where min 1 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 ; 2 2 + 3 ; 3 )]TJ/F26 11.9552 Tf 11.956 0 Td [( 2 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; 1 a )]TJ/F26 11.9552 Tf 13.15 8.088 Td [(c 2 1 2 ; 5 )]TJ/F26 11.9552 Tf 13.151 8.088 Td [( 2 4 2 ; where c 1 and 1 2 R arepositiveboundingconstants, 1 det M g ,and c 1 det M g 1 + 3 + 5 )]TJ/F27 7.9701 Tf 15.875 13.492 Td [(d dt det M g + m 2 C l det M g 2 2 + 2 det M g 2 ; c m 1 1 + 3 + 5 + c m 2 + c m 3 C l 2 + 2 2 c 2 m 1 ; where c m 1 > det M g c m 2 > d dt det M g ,and c m 3 > m 2 det M g 2 1 Remark 3.1 Thecontrolgains 1 and 2 canbeselectedindependentlyoftheremainingcontrolgainsand 4 isselectedlessthantheactuatorlimit.After 4 isselected,the lowerboundson 3 5 ,and canbecalculated.Theselectionof a dependsonthe severityoftheLCOmotionwhichiscapturedintheboundingconstant c 1 .IftheLCO motionistoosevere,thegainconditionfor a can'tbesatisedwithoutincreasingthe saturationlimit. Proof. Let V L y : R 5 R beapositive-denite,continuouslydifferentiablefunction denedas V L lncosh e 1 +lncosh e 2 + 1 2 e 2 2 + 1 2 det M g r 2 + 1 2 tanh 2 e f + P: 1 SeeAppendixEfordetails. 44 PAGE 45 Fromtheinequalitiesin3and3, V L satisesthefollowinginequalities 1 2 min 1 ; 1tanh 2 k y k V L y 2 k y k 2 ; where 2 2 R isaknownpositiveconstant.Let y denoteaFilippovsolutiontothe closed-loopsystemdescribedby3-3,3,and3.Thetimederivativeof 3alongtheFilippovsolution y existsalmosteverywhere a:e and V L a:e 2 ~ V L where ~ V L 2 @V L T K e 1 e 2 r e f P 2 p P 1 T and @V L denotesthegeneralizedgradientof V L [57].Since V L isacontinuouslydifferentiablefunction, ~ V L canbeexpressedas ~ V L r V T L K e 1 e 2 r e f P 2 p P 1 T ; where r V T L tanh e 1 tanh e 2 + e 2 det M g r tanh e f cosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 e f 2 p P 1 2 d dt det M g r 2 : Usingthecalculusfor K from[53],3-3,3,and3,theexpressionin 3canbewrittenas ~ V L tanh e 1 e 2 )]TJ/F26 11.9552 Tf 11.956 0 Td [( 1 tanh e 1 +tanh e 2 )]TJ/F26 11.9552 Tf 9.299 0 Td [( 2 tanh e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 3 e 2 + e 2 )]TJ/F26 11.9552 Tf 9.299 0 Td [( 2 tanh e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 3 e 2 + r ~ N + )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(det M g 4 r )]TJ/F26 11.9552 Tf 11.955 0 Td [( 4 K [sgn e 2 ] +tanh e f )]TJ/F26 11.9552 Tf 9.298 0 Td [( 4 e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 5 tanh e f + r 4 K [sgn e 2 ] ; AsinChapter2,3reducestoascalarinequalitysincetheright-handsideiscontinuousexceptfortheLesbeguenegligiblesetoftimeinstanceswhen r 4 K [sgn e 2 ] )]TJ/F26 11.9552 Tf -454.366 -23.908 Td [(r 4 K [sgn e 2 ] 6 = f 0 g .Theresultingscalarinequalityisexpressedas V L a:e: )]TJ/F26 11.9552 Tf 30.476 0 Td [( 1 tanh 2 e 1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 2 tanh 2 e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 3 e 2 2 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(det M g 4 r 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 5 tanh 2 e f + r ~ N + r +tanh e 1 e 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [( 3 + 2 tanh e 2 e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 4 tanh e f e 2 : 45 PAGE 46 UsingYoung'sInequalityandtheboundsonthesystemstates,theterm r canbe upperboundedas j r j 1 2 det M g )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [( 1 cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e 1 + 5 + 3 )]TJ/F27 7.9701 Tf 15.875 13.492 Td [(d dt det M g + m 2 C l det M g 2 2 r 2 + 2 4 e 2 2 + 1 2 2 det M g 2 r 2 c 2 1 2 r 2 + 2 4 e 2 2 : ByapplyingYoung'sInequality,theinequalitiesin3and3,andtheupperbounds on ~ N and r givenin3and3,3canbeupperboundedas V L a:e: )]TJ/F26 11.9552 Tf 30.476 0 Td [( 1 tanh 2 e 1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 2 tanh 2 e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 3 e 2 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 4 r 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 5 tanh 2 e f + 0 k x kj r j + c 2 1 2 r 2 + 1 2 tanh 2 e 1 + e 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 3 + 2 tanh 2 e 2 + 1 2 2 4 tanh 2 e f + 2 4 e 2 2 : Combiningcommontermsandcompletingthesquaresontheterm )]TJ/F15 11.9552 Tf 11.291 0 Td [( 1 b r 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 0 k x kj r j yields V L a:e: )]TJ/F31 11.9552 Tf 32.468 16.857 Td [( 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 tanh 2 e 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 + 3 tanh 2 e 2 )]TJ/F31 11.9552 Tf 11.955 9.683 Td [()]TJ/F26 11.9552 Tf 5.479 -9.683 Td [( 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 2 4 e 2 2 )]TJ/F31 11.9552 Tf 11.291 16.857 Td [( 1 a )]TJ/F26 11.9552 Tf 13.151 8.088 Td [(c 2 1 2 r 2 )]TJ/F31 11.9552 Tf 11.955 16.857 Td [( 5 )]TJ/F26 11.9552 Tf 13.151 8.088 Td [( 2 4 2 tanh 2 e f + 2 0 k x k 2 4 1 b : Providedthesufcientgainconditionsin3aresatised,3andthedenition of z canbeusedtoshow V L a:e: )]TJ/F31 11.9552 Tf 25.826 16.857 Td [( )]TJ/F26 11.9552 Tf 21.58 8.088 Td [( 2 0 4 1 b tanh 2 k z k )]TJ/F26 11.9552 Tf 21.918 0 Td [(c tanh 2 k z k ; where c 2 R isapositiveconstant.Fromtheinequalitiesin3and3, V L 2L 1 ; therefore, e 1 e 2 r ,and tanh e f 2L 1 .Equations3and3canbeusedto showthat e 1 and e 2 2L 1 .From3, 2L 1 .Since e 2 ;r 2L 1 ,itcanbeconcluded from3that x 2L 1 .Equations3and3canbeusedtoshowthat r 2L 1 .Since e 2 2L 1 ,3canbeusedtoshowthat cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e f e f 2L 1 .Since 46 PAGE 47 _ e 1 ; e 2 ; r; cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e f e f 2L 1 ,thedenitionof z canbeusedtoshowthat z 2L 1 ,and hence, z isuniformlycontinuousUC.Since z isUC,thefunction )]TJ/F26 11.9552 Tf 9.299 0 Td [(c tanh 2 k z k isUC. Basedon3,Corollary1from[55]canbeusedtoprovethat tanh k z k 0 as t !1 .Fromthedenitionof z itcanbeconcludedthat e 1 0 as t !1 3.4SimulationResults Anumericalsimulationispresentedtoillustratetheperformanceofthedeveloped controllerandtoprovideacomparisonwiththecontrollerinChapter2. ThemodelparametersforthesimulationareshowninTable3-1and3. Theopen-loopsystemwassimulatedwiththefollowinginitialconditions: h =0 m, h =0 m/s, =11 : 5 deg,and =0 deg/s.ItisevidentfromFigure 3-1thattheopen-loopsystem,undertheaboveinitialconditionsandnoexogenous disturbances,experiencesLCObehavior. Table3-1.AeroelasticModelParameters ParameterParameter m w 4.0kg I s 0.0050kg m 2 m s 4.0kg c h 1 2.743x10 1 kg/s r x 0.0 c 0.036kg m 2 /s r h 0.0 k h 2.200x10 3 N/m a -0.6 1.225kg/m 3 a h 0.0 U 1.50x10 1 m/s b 0.14m S 1.0m s x 0.098 C l 6.81/rad s h 1.4 C l 9.3x10 1 N/rad I w 0.043kg m 2 C m 2.3N m/rad k q =0 : 5 )]TJ/F15 11.9552 Tf 11.955 0 Td [(11 : 05 +657 : 75 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4290 3 +8644 : 85 4 : Thecontrolobjectiveinthesubsequentnumericalsimulationsistoregulate theAoAtozerodegrees.Inaddition,anexternaldisturbance,selectedas d t = 00 : 25sin t T ,wasaddedtothenumericalsimulationandazero-meannoise signaluniformlydistributedoveranintervalwasaddedtoeachmeasurement.Forthe 47 PAGE 48 verticaldisplacementandvelocity,theintervalwas 2 : 5 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(3 mand 2 : 5 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(3 m/s,respectively.FortheAoAandAoArate,theintervalwas 4 : 5 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 radand 1 10 )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 rad/s.BasedontheidenticationperformanceoftheNN,theNNfeedforward parametersforthecontrollerdevelopedinChapter2wereselectedas n 2 =25 )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(1 =10 I 26 ,and )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(2 =10 I 7 ,where I m denotesan m m identitymatrix.TheRISE feedbackcontrolgainsforthecontrollerdevelopedinChapter2weredetermined througha1500sampleMonteCarlosimulationinwhichtheRISEfeedbackcontrol gainsforeachsamplewereselectedatrandomfromwithinaspeciedinterval.The gainsusedinthecomparisonstudywereselectedasthosethatreturnedtheminimum valueforthefollowingcostfunction J = v u u t 1 n n X i =1 2 t i ; where n isthetotalnumberoftimestepsinthenumericalsimulation.Thesetofcontrol gainsthatproducedthesmallestAoARMSerrorwere 2 =3 : 9513 k s =2 : 6112 and 1 =0 : 9966 .Figures3-2and3-3depicttheperformanceoftheunsaturated RISEcontrollerdevelopedinChapter2andthatsameRISEcontrollerwithan ad hoc saturationappliedtothecommandedcontrol.Whiletheunsaturatedcontroller suppressedtheLCObehavior,thecommandedcontroleffortbreachedtheactuator limitseveraltimes.Whenthe adhoc saturationwasappliedtothecontroller,the LCObehaviorcouldnotbesuppressedandthesystemreturnedtoanLCOstate. Thishighlightstheunpredictableresponsethatcanoccurwhenapplyingan adhoc saturationwithoutconsideringthestabilityoftheresultingclosed-loopsystem. Thedevelopedcontrolstrategywasappliedtothesystemwiththefollowinggains: 1 =0 : 8375 2 =17 : 7604 3 =33 : 9025 4 =0 : 1745 5 =15 : 4652 ,and =5 : 5539 .Note that 4 representstheactuatorlimitinradians,whichwastakentobe 10 deg.The controlgainsforthedevelopedcontrollerweredeterminedbyapplyingthesameMonte CarloapproachusedtoselectthegainsforthecontrollerinChapter2. 48 PAGE 49 Figure3-1.Aeroelasticsystemopen-loopresponsewithoutdisturbances Figure3-2.StatetrajectoriesoftheRISE-basedcontrollerwithandwithoutan adhoc saturation. 49 PAGE 50 Figure3-3.CommandedcontroleffortfortheRISE-basedcontrollerwithandwithoutan adhoc saturation. Thestatesandcontrolsurfacedeectionofthe adhoc saturatedcontrollerandthe developedsaturatedcontrollerareshowninFigures3-4and3-5,respectively.While differentgainselectionswillaltertheperformance,Figures3-4and3-5illustratethat thedevelopedcontrolstrategyiscapableofsupressingLCObehaviorinthepresence ofactuatorlimits.Thebenetofthedevelopedmethodisthatthesaturationlimitis includedinthestabilityanalysisguaranteeingasymptotictracking,versusthe adhoc saturationwhichyieldsanunpredictableresponse. A1500sampleMonteCarlosimulationwasalsoperformedtodemonstratethe robustnessofthedevelopedsaturatedcontrollertoplantuncertaintiesandmeasurementnoise.Themodelparameterswerevarieduniformlyoverarangethatextended from95%to105%oftheparametervalueslistedinTable3-1.Whilethedeveloped saturatedcontrollersuccessfullyregulatedtheAoAforall1500samples,thetransient performancevariedsignicantlybetweensamples. Theaveragetrajectoryand 3 condenceboundsfortheangleofattack,vertical position,andcontrolsurfacedeectionoftheMonteCarlosamplesareshownin 50 PAGE 51 Figure3-4.Comparisonoftheclosed-loopaeroelasticsystemresponseofthe RISE-basedcontrollerwithan adhoc saturationandthedeveloped saturatedcontroller. Figure3-5.Comparisonofthecontrolsurfacedeectionsforthedevelopedsaturated controllerand adhoc saturatedRISE-basedcontroller Table3-2.MonteCarloSimulationResults MeanStandardDeviation MaximumTrackingError1.272x10 1 deg3.04deg RMSTrackingError2.13deg2.53deg 51 PAGE 52 Figure3-6.AoAtrajectoriesforall1500MonteCarlosamples.Thedevelopedsaturated controllersuppressedtheLCObehaviorinallsamplesandthemajorityof thesamplesexhibitsimilartransientperformance. Figures3-6-3-8.Figure3-6indicatesthattheAoAforallsamplesconvergetozero afterapproximately7seconds,howevertheconsideredrangeofmodeluncertainties doesimpactthetransientperformanceofthecontroller.Thesensitivityintransient performancecanbeattributedtothesaturationonthecommandedcontroleffort.As notedpreviously,undercertainconditionstheseverityoftheLCOcanbecomemore thanthesaturatedcontrollercansuppressandthesystemwillreturntoanLCOstate. 3.5Summary Asaturatedcontrolstrategyisdevelopedtosuppressstore-inducedLCObehavior ofanaeroelasticsystem.ThecontrolstrategyusesasaturatedRISEcontrollerto asymptoticallytrackadesiredAoAtrajectorywithoutexceedingactuatorlimits.A Lyapunov-basedstabilityanalysisguaranteesasymptotictrackinginthepresenceof actuatorconstraints,exogenousdisturbances,andmodelinguncertainties.Simulations resultsarepresentedtoillustratetheperformanceofthedevelopedcontrolstrategy. 52 PAGE 53 Figure3-7.Verticalpositiontrajectoriesofall1500MonteCarlosamples.Thevertical positionremainedboundedforallsamplesdespitebeinganuncontrolled state. Figure3-8.Controlsurfacedeectionforall1500MonteCarlosamples.Thecontrol effortforallsamplesremainwithintheactuationlimitanddemonstrate similarsteadystateperformance. 53 PAGE 54 Anumericalsimulationwaspresentedthatdemonstratedtheunpredictableclosedloopsystemresponsewhenan adhoc saturationstrategyisappliedtothecontroller inChapter2.Acomparisonstudyrevealedthatthesaturatedcontrollerdeveloped inthispaperachievedasymptotictrackingofthedesiredAoAtrajectorywhilethe ad hoc saturationstrategywasunabletosuppresstheLCObehavior.A1500sample MonteCarlosimulationwaspresentedtodemonstratetherobustnessofthedeveloped controllertovariationsinthemodelparameters.Apotentialdrawbackofthedeveloped controlstrategyisthatundercertainconditions,theseverityoftheproducedLCOmay resultinsufcientgainconditionsthatcan'tbesatised.Thatis,ifthedisturbancesto thesystemarelargeenough,thenthesystemcouldbedestabilized.Thisisadirect resultoftheactuatorlimit;increasingtheactuatorlimitrelaxesthesufcientgain conditionsandallowsforlargerdisturbances.Furthermore,anadaptivefeedforward termcouldpotentiallybeincludedtocompensatefortheuncertaindynamics,thereby relaxingthesufcientgainconditions.However,foranycontrollerthathasrestricted controlauthority,itispossibleforsomedisturbancetodominatethecontroller'sabilityto yieldadesiredorevenstableperformance. 54 PAGE 55 CHAPTER4 BOUNDARYCONTROLOFLIMITCYCLEOSCILLATIONSINAFLEXIBLEAIRCRAFT WING Thefocusofthischapteristodevelopaboundarycontrolstrategyforsuppressing LCOmotioninanaircraftwingwhosedynamicsaredescribedbyasystemoflinearpartialdifferentialequationsPDEs.APDEbacksteppingmethodguaranteesexponential regulationofthewingtwistdynamicswhileaLyapunov-basedstabilityanalysisisused toshowboundednessofthewingbendingdynamics. 4.1AircraftWingModel Consideraexiblewingoflength l 2 R ,massperunitlengthof 2 R ,moment ofinertiaperunitlengthof I w 2 R ,andbendingandtorsionalstiffnessesof EI 2 R and GJ 2 R ,respectively,withastoreofmass m s 2 R andmomentofinertia J s 2 R attachedatthewingtip.Thebendingandtwistingdynamicsoftheexiblewingare describedbythefollowingPDEsystem 1 tt + EI! yyyy + EI! tyyyy = L w ; I w tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJ yy )]TJ/F26 11.9552 Tf 11.955 0 Td [( GJ tyy = M w ; where y;t 2 R and y;t 2 R denotethebendingandtwistingdisplacements, respectively, y 2 [0 ;l ] denotesspanwiselocationonthewing, 2 R and 2 R denoteKelvin-Voigtdampingcoefcientsinthebendingandtwistingstates,respectively, and L w = L w 2 R and M w = M w 2 R denotetheaerodynamicliftandmoment onthewing,respectively,where L w and M w 2 R denoteaerodynamicliftandmoment coefcients,respectively.In4and4,thesubscripts t and y denotepartial derivatives.Theboundaryconditionsfortip-basedcontrolare ;t = y ;t = 1 SeeAppendixFfordetailsregardingthederivationofthedynamics. 55 PAGE 56 ! yy l;t = ;t =0 and EI! yyy l;t + w EI! tyyy l;t = m s tt l;t )]TJ/F26 11.9552 Tf 11.955 0 Td [(L tip ; GJ y l;t + GJ ty l;t = )]TJ/F26 11.9552 Tf 9.299 0 Td [(J s tt l;t + M tip ; where L tip 2 R and M tip 2 R denotetheaerodynamicliftandmomentatthewingtip whichcanbeimplementedthroughapslocatedatthewingtip[42].Itisassumedin 4and4thatthecenterofmassandshearcenterarecoincidentandallmodel parametersareconstant. 4.2BoundaryControlofWingTwist Thecontrolobjectiveistoensurethatthewingtwistisregulatedinthesense that y;t 0 ; 8 y 2 [0 ;l ] as t !1 viaboundarycontrolatthewingtip.APDE backsteppingmethodwillbeusedtotransformthesystemin4intoanexponentially stabletargetsystemusinganinvertibleVolterraintegraltransformation[33].Thestate transformationisdenedas y;t y;t )]TJ/F40 11.9552 Tf 11.956 16.272 Td [( y 0 k y;x x;t dx; wherethefunction k x;y 2 R denotesthegainkernel.Theexponentiallystabletarget systemisselectedas I w tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJ yy )]TJ/F26 11.9552 Tf 11.955 0 Td [( GJ tyy + )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(cGJ )]TJ/F15 11.9552 Tf 16.291 3.022 Td [( M w + cGJ t =0 ; where c 2 R isapositiveconstantselectedtosatisfytheinequality, c> M w GJ )]TJ/F27 7.9701 Tf 13.852 4.707 Td [( 2 4 l 2 ,andthe boundaryconditionsare ;t =0 and GJ y l;t + GJ ty l;t =0 2 .Duetothefact thatthestatetransformationisinvertible,stabilityofthetargetsystemin4translates tostabilityofthesystemin4withtheboundarycontrolin4[33].Thetask 2 SeeAppendixG 56 PAGE 57 isnowtondthegainkernel k y;x thatsatises4anditsboundaryconditions. AlinearPDEandassociatedboundaryconditionsthatdescribethegainkernelare obtainedbysubstitutingthestatetransformationin4into4.Substitutingthe statetransformationintothersttermin4yields I w tt = I w tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(I w y 0 k y;x tt x;t dx = M w y;t + GJ yy y;t + GJ tyy y;t )]TJ/F40 11.9552 Tf 11.291 16.272 Td [( y 0 k y;x )]TJ/F15 11.9552 Tf 9.815 -6.662 Td [( M w x;t + GJ xx x;t + GJ txx x;t dx: Afterintegratingthelasttwotermsbyparts,4canbeexpressedas I w tt = M w y;t + GJ yy y;t + GJ tyy y;t )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJk y;y y y;t + GJk y; 0 y ;t + GJk x y;y y;t )]TJ/F26 11.9552 Tf 11.955 0 Td [( GJk y;y ty y;t + GJk y; 0 ty ;t + GJk x y;y t y;t )]TJ/F40 11.9552 Tf 11.291 16.273 Td [( y 0 k y;x M w + GJk xx y;x x;t dx )]TJ/F26 11.9552 Tf 9.299 0 Td [( GJ y 0 k xx y;x t x;t dx; where k x y;y @ @x k y;x j x = y .Similarly,expressionsforthesecondandthirdtermsin 4canbeobtainedas GJ yy = GJ yy y;t )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJ d dy k y;y y;t )]TJ/F26 11.9552 Tf 11.956 0 Td [(GJk y;y y y;t )]TJ/F26 11.9552 Tf 9.299 0 Td [(GJk y y;y y;t )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJ y 0 k yy y;x x;t dx; GJ tyy = GJ tyy y;t )]TJ/F26 11.9552 Tf 11.955 0 Td [( GJ d dy k y;y t y;t )]TJ/F26 11.9552 Tf 11.955 0 Td [( GJk y;y ty y;t )]TJ/F26 11.9552 Tf 9.298 0 Td [( GJk y y;y t y;t )]TJ/F26 11.9552 Tf 11.955 0 Td [( GJ y 0 k yy y;x t x;t dx; where d dy k y;y @ @x k y;x j x = y + @ @y k y;x j x = y and k y y;y @ @y k y;x j x = y Substitutingthestatetransformationin4intothelasttwotermsin4andutilizing 57 PAGE 58 theexpressionsin4-4yields 2 GJ d dy k y;y + cGJ y;t + GJk y; 0 y ;t + GJk y; 0 ty ;t + 2 GJ d dy k y;y + cGJ t y;t + y 0 GJk yy y;x )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJk xx y;x )]TJ/F26 11.9552 Tf 11.955 0 Td [(cGJk y;x x;t dx + y 0 GJk yy y;x )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJk xx y;x )]TJ/F26 11.9552 Tf 11.955 0 Td [(cGJk y;x t x;t dx =0 : Forthenon-trivialsolutionof y;t ,thegainkernel k y;x mustsatisfythefollowing PDE k yy y;x )]TJ/F26 11.9552 Tf 11.955 0 Td [(k xx y;x = ck y;x ; withtheboundaryconditions k y; 0=0 and 2 d dy k y;y = )]TJ/F26 11.9552 Tf 9.299 0 Td [(c .Integrationofthesecond boundaryconditionyields k y;y = )]TJ/F27 7.9701 Tf 10.778 4.708 Td [(c 2 y .ThesolutiontothegainkernelPDEin4 11conbeobtainedbyconvertingthePDEintoanintegralequationandapplyingthe methodofsuccessiveapproximations[33].Thesolutionto4is k y;x = )]TJ/F26 11.9552 Tf 9.298 0 Td [(cx I 1 p c y 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [(x 2 p c y 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [(x 2 ; where I 2 R denotesamodiedBesselfunctiondenedas I 1 X =0 )]TJ/F27 7.9701 Tf 6.675 -4.428 Td [( 2 +2 + : Theboundaryconditionat y = l canthenbeexpressedas GJ y l;t + GJ ty l;t = GJ y l;t + GJ ty l;t )]TJ/F15 11.9552 Tf 11.291 0 Td [( GJ l;t + GJ t l;t k l;l )]TJ/F26 11.9552 Tf 9.298 0 Td [(GJ l 0 k y l;x x;t + t x;t dx; 58 PAGE 59 where k l;l = )]TJ/F27 7.9701 Tf 10.777 4.707 Td [(c 2 l and k y l;x = )]TJ/F26 11.9552 Tf 9.298 0 Td [(clxI 2 p c l 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [(x 2 l 2 )]TJ/F26 11.9552 Tf 11.956 0 Td [(x 2 : Fromtheboundaryconditionofthetargetsystem, GJ y l;t + GJ ty l;t =0 theleft-handsideof4isequaltozero.From4,thersttwotermsonthe right-handsideof4canbereplacedwith )]TJ/F26 11.9552 Tf 9.299 0 Td [(J s tt l;t + M tip yielding 0= M tip )]TJ/F26 11.9552 Tf 11.956 0 Td [(J s tt l;t )]TJ/F15 11.9552 Tf 11.955 0 Td [( GJ l;t + GJ t l;t k l;l )]TJ/F26 11.9552 Tf 9.298 0 Td [(GJ l 0 k y l;x x;t + t x;t dx; whichcanbesolvedfortheboundarycontrolatthewingtip M tip = J s tt l;t + GJ l;t + GJ t l;t k l;l + GJ l 0 k y l;x x;t + t x;t dx: Duetothefactthatthestatetransformationisinvertible,stabilityofthetargetsystemin 4translatestostabilityofthesystemin4withtheboundarycontrolin4. Remark 4.1 ThemodiedBesselfunctionusedinthesolutionfor k x;y isaninnite sum,whichforimplemenationpurposesmustbeapproximatedusinganitesum. Itcanbeshownusingtheratiotest[58]that I convergesforany and 2 R Since I converges,foranysmallarbitrarynumber > 0 ,thereexists T suchthat j I ; 0 )]TJ/F26 11.9552 Tf 11.955 0 Td [(I j forall 0 T and 2 R ,where I ; 0 P 0 =0 2 +2 + .Forthe particularsystemusedinthesubsequentsimulationsection,theinput 2 0 ; p 5 and for T =10 =6 : 7 10 )]TJ/F24 7.9701 Tf 6.586 0 Td [(16 .Figure4-1showsaplotof I 1 ;10 and I 1 4.3BoundaryControlofWingBending Thecontrolobjectiveistoensurethewingbendingstate y;t remainsbounded andachievesasteadystateprole.Basedonthesystemdynamicsandboundary conditionsgivenin4and4alongwiththesubsequentstabilityanalysis,the 59 PAGE 60 Figure4-1.ApproximationofthemodiedBesselfunctionusedinthesubsequent simulationsection. boundarycontrol L tip isdesignedas L tip = )]TJ/F26 11.9552 Tf 9.299 0 Td [(! l;t )]TJ/F26 11.9552 Tf 11.955 0 Td [(K! t l;t ; where K 2 R isapositiveconstantcontrolgain. Theorem4.1. Theboundarycontrollersgivenin4and4ensurethat y;t 2 L 1 and t y;t 0 as t !1 Proof. Tofacilitatethesubsequentstabilityanalysis,let c 1 2 R bedenedas c 1 sup y 2 [0 ;l ] j y; 0 j andlet V L : R 4 R beapositive-denite,continuouslydifferentiable functiondenedas V L = 1 c 2 1 l 1 2 l 0 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(! 2 t + EI! 2 yy dy + 1 2 2 l;t + m s 2 2 t l;t ; where 2 R 4 isdenedas l 0 2 t dy 1 = 2 l 0 2 yy dy 1 = 2 l;t t l;t T .The upperandlowerboundson V L canbeexpressedas 1 k k 2 V L 2 k k 2 ,where 1 min n 2 c 2 1 l ; EI 2 c 2 1 l ; 1 2 c 2 1 l ; m s 2 c 2 1 l o 2 R and 2 max n 2 c 2 1 l ; EI 2 c 2 1 l ; 1 2 c 2 1 l ; m s 2 c 2 1 l o 2 R .Takingthetime 60 PAGE 61 derivativeof4yields V L = 1 c 2 1 l l 0 tt t dy + l 0 EI! yy tyy dy + l;t t l;t + t l;t m s tt l;t : Substitutingthebendingdynamicsfrom4intotherstintegralof4resultsin V L = 1 c 2 1 l l 0 t L w )]TJ/F26 11.9552 Tf 11.955 0 Td [(EI! yyyy dy )]TJ/F40 11.9552 Tf 11.955 16.272 Td [( l 0 EI! t tyyyy dy + l 0 EI! yy tyy dy + 1 c 2 1 l l;t t l;t + t l;t m s tt l;t : Evaluatingthesecondandthirdintegralusingintegrationbypartsandapplyingthe bendingboundaryconditionsyields )]TJ/F40 11.9552 Tf 11.955 16.273 Td [( l 0 EI! t tyyyy dy = )]TJ/F26 11.9552 Tf 9.298 0 Td [( EI! t l;t tyyy l;t )]TJ/F40 11.9552 Tf 11.955 16.273 Td [( l 0 EI! 2 tyy dy l 0 EI! yy tyy dy = )]TJ/F26 11.9552 Tf 9.299 0 Td [(EI! t l;t yyy l;t + l 0 EI! t yyyy dy: Aftersubstituting4and4into4andcancelingliketerms, V L canbe expressedas V L = 1 c 2 1 l l 0 t L w dy )]TJ/F40 11.9552 Tf 11.955 16.272 Td [( l 0 EI! 2 tyy dy + t l;t c 2 1 l l;t + m s tt l;t )]TJ/F26 11.9552 Tf 11.955 0 Td [( EI! tyyy l;t )]TJ/F26 11.9552 Tf 11.956 0 Td [(EI! yyy l;t : UsingLemmasA.12andA.13of[34],thetwointegralsin4canbeboundedas l 0 t L w dy 1 l 0 2 t dy + l 0 L 2 w 2 dy; )]TJ/F40 11.9552 Tf 11.291 16.272 Td [( l 0 EI! 2 tyy dy )]TJ/F40 11.9552 Tf 30.552 16.272 Td [( l 0 EI l 4 2 t dy; where 2 R isapositiveconstant.Substitutingtheboundaryconditionin4,the inequalitiesin4and4,andthecontrollawin4into4yields V L )]TJ/F26 11.9552 Tf 24.465 8.088 Td [(K c 2 1 l 2 t l;t )]TJ/F15 11.9552 Tf 16.985 8.088 Td [(1 c 2 1 l EI l 4 )]TJ/F15 11.9552 Tf 13.594 8.088 Td [(1 l 0 2 t dy + L 2 w c 2 1 l l 0 2 dy: 61 PAGE 62 Tofacilitatethestabilityanalysis,let z t l;t l 0 2 t dy 1 = 2 T 2 R 2 .The expressionin4canbewrittenas V L )]TJ/F26 11.9552 Tf 21.917 0 Td [( k z k + t ; where k z k = 3 k z k 2 3 min n K c 2 1 l ; 1 c 2 1 l )]TJ/F27 7.9701 Tf 6.675 -4.428 Td [( EI l 4 )]TJ/F24 7.9701 Tf 13.474 4.707 Td [(1 o = q L 2 w 3 ,and t = 1 c 2 1 l l 0 2 dy Since isexponentiallystable,thefunction 2L 1 .Duetotheselectionoftheconstant c 1 j j 1 .Corollary2.18from[59]canbeappliedtoconcludethat k k2L 1 and k z k! 0 as t !1 ;hence j y;t j2L 1 and j t y;t j! 0 as t !1 4.4NumericalSimulation Anumericalsimulationispresentedtoillustratetheperformanceofthedeveloped controller.ThesimulationsareperformedusingaGalerkin-basedmethodtoapproximatethePDEsystemwithanitenumberofODEs.Itshouldbenotedthatthecontrol designdoesnotusetheapproximation,thereforetheissueofspilloverinstabilitesis avoided.Thetwistingandbendingdeectionsarerepresentedasaweightedsumof basisfunctions y;t = a 0 t h 0 y + n X i =1 a i t h i y ; y;t = b 0 t g 0 y + p X i =1 b i t g i y ; where n and p 2 R denotethenumberofbasisfunctionsusedintheapproximationsof thewingtwistingdeectionandbendingdeection,respectively,and h 0 y h i y g 0 y and g i y 2 R arebasisfunctionsselectedtosatisfytheboundaryconditions h 0 = h i =0 ;h y 0 l =1 ;h y i l =0 ; g 0 = g i =0 ;g y 0 = g y i =0 ; g yy 0 l = g yy i l =0 ;g yyy 0 l =1 ;g yyy i l =0 : 62 PAGE 63 Substitutingtheapproximationsofthesystemstates,thePDEsystemin4and4 canbeexpressedas b tt 0 t g 0 y + p X i =1 b tt i t g i y + EI b 0 t g yyyy 0 y + p X i =1 b i t g yyyy i y + EI b t 0 t g yyyy 0 y + p X i =1 b t i t g yyyy i y = L w a 0 t h 0 y + n X i =1 a i t h i y ; I w a tt 0 t h 0 y + n X i =1 a tt i t h i y )]TJ/F26 11.9552 Tf 11.956 0 Td [(GJ a 0 t h yy 0 y + n X i =1 a i t h yy i y )]TJ/F26 11.9552 Tf 11.955 0 Td [( GJ a t 0 t h yy 0 y + n X i =1 a t i t h yy i y = M w a 0 t h 0 y + n X i =1 a i t h i y : UsingGalerkin'smethod,4and4areconvertedtoasetofODEsas B 1 b t + w B 2 b t + B 2 b t )]TJ/F26 11.9552 Tf 11.955 0 Td [(B 3 a t =0 ; I w T 1 a t )]TJ/F26 11.9552 Tf 11.955 0 Td [( T 2 a t )]TJ/F31 11.9552 Tf 11.955 9.683 Td [()]TJ/F26 11.9552 Tf 5.48 -9.683 Td [(T 2 + M w T 1 a t =0 ; where b t b 0 t b 1 t :::b p t T a t a 0 t a 1 t :::a n t T B 1 l 0 g y g T y dy B 2 EI l 0 g y g T yyyy y dy B 3 L w l 0 g y h T y dy T 1 l 0 h y h T y dy T 2 GJ l 0 h y h T yy y dy g y g 0 y g 1 y :::g p y T and h y h 0 y h 1 y :::h n y T .Theexpressionsin4and4are simulatedtoapproximatetheresponseofthePDEsystem. Theopen-loopsystemwassimulatedwiththefollowinginitialconditions: y; 0=0 mand y; 0= y 2 2 l 2 rad.ItisevidentfromFigures4-2-4-4thattheopen-loop system,undertheaboveinitialconditions,experiencesLCObehavior. Thecontrolobjectivefortheclosed-loopsystemistoregulatethetwistingand bendingdeformationsoftheexiblewing.Basedonthetransientperformanceofthe 63 PAGE 64 Figure4-2.Open-looptwistdeectionoftheexibleaircraftwing. Figure4-3.Open-loopbendingdeectionoftheexibleaircraftwing. 64 PAGE 65 Figure4-4.Open-loopresponseatthewingtipoftheexibleaircraftwing. closed-loopsystem,thecontrolgainswereselectedas c =5 and k =10 .Theexible statetrajectoriesareshowninFigures4-5-4-7.Itisevidentthatthedeveloped controlstrategyiscapableofsupressingLCObehaviorintheexibleaircraftwing. Figure4-8showstheforceandmomentcommandedbythedevelopedcontrol strategy. 4.5Summary ThischapterpresentstheconstructionofaboundarycontrolstrategyforsuppressingLCObehaviorinaexibleaircraftwing.Thecontroldesignisseparatedintotwo parts:abackstepping-basedcontrolstrategyusedtodesigntheaerodynamicmoment atthewingtipandaLyapunov-basedcontrollerfortheaerodynamicliftatthewingtip. Thedevelopedcontrolstrategyensuresexponentialregulationofthewingtwistand asymptoticregulationofthewingbendingtoasteady-stateprole.Numericalsimulationsillustratetheperformanceofthedevelopedbackstepping-basedcontroldesign. Onedrawbackofthedevelopedcontrolleristhatitreliesontheassumptionthatthe distancesfromthewingelasticaxistothewingcenterofgravityandstorecenterof gravityarezero.Ifthisassumptionisdropped,thePDEdescribingthedynamicsofthe 65 PAGE 66 Figure4-5.Closed-looptwistdeectionoftheexibleaircraftwing. Figure4-6.Closed-loopbendingdeectionoftheexibleaircraftwing. 66 PAGE 67 Figure4-7.Closed-loopresponseatthewingtipoftheexibleaircraftwing. Figure4-8.LiftandMomentcommandedatthewingtip. 67 PAGE 68 wingdeformationsbecomesnonlinearwhichbecomesachallengeforthebacksteppingstrategyemployedinthischapter.Instead,anapproachsimilartothatof[34,35], inwhichaLyapunov-basedanalysisprovesthattheenergyinthesystemdecaysto zero,couldbeusedtogeneratetheaerodynamicliftandmomentatthewingtip.This strategyisconsideredinChapter5. 68 PAGE 69 CHAPTER5 ADAPTIVEBOUNDARYCONTROLOFLIMITCYCLEOSCILLATIONSINAFLEXIBLE AIRCRAFTWING Thefocusofthischapteristodevelopanadaptiveboundarycontrolstrategyfor suppressingLCOmotioninanaircraftwingwhosedynamicsaredescribedbyasystem ofnonlinearpartialdifferentialequationsPDEs.ALyapunov-basedstabilityanalysis guaranteesasymptoticregulationofthewingtwistandbendingdynamics. 5.1AircraftWingModel Consideraexiblewingoflength l 2 R ,massperunitspanof 2 R ,moment ofinertiaperunitlengthof I w 2 R ,andbendingandtorsionalstiffnessesof EI 2 R and GJ 2 R ,respectively,withastoreofmass m s 2 R andmomentofinertia J s 2 R attachedatthewingtip.Thebendingandtwistingdynamicsoftheexiblewingare describedbythefollowingPDEsystem 1 tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(x c c sin 2 t + x c c cos tt + EI! yyyy = L w ; )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c c 2 tt + x c c cos tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJ yy = M w ; where y;t 2 R and y;t 2 R denotethebendingandtwistingdisplacements, respectively, y 2 [0 ;l ] denotesspanwiselocationonthewing, x c c 2 R representsthe distancefromthewingelasticaxistothewingcenterofgravity,and L w = L w 2 R and M w = M w 2 R denotetheaerodynamicliftandmomentonthewing,respectively, where L w and M w 2 R denoteaerodynamicliftandmomentcoefcients,respectively. In5and5,thesubscripts t and y denotepartialderivatives.Theboundary conditionsfortip-basedcontrolare ;t = y ;t = yy l;t = ;t =0 and L tip = m s tt l;t )]TJ/F26 11.9552 Tf 11.955 0 Td [(m s x s c sin l;t 2 t l;t + m s x s c cos l;t tt l;t 1 SeeAppendixFfordetailsregardingthederivationofthedynamics. 69 PAGE 70 )]TJ/F26 11.9552 Tf 9.299 0 Td [(EI! yyy l;t ; M tip = )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(m s x 2 s c 2 + J s tt l;t + m s x s c cos l;t tt l;t + GJ y l;t ; where L tip 2 R and M tip 2 R denotetheaerodynamicliftandmomentatthewingtipand x s c 2 R representsthedistancefromthewingelasticaxistothestorecenterofgravity. Itisassumed,basedonRemark5.1in[34],thatthesystemhasthefollowingproperties Property1. Ifthepotentialenergyofthesystem, E P 1 2 l 0 )]TJ/F26 11.9552 Tf 5.48 -9.683 Td [(EI! 2 yy + GJ 2 y dy 2L 1 8 t 2 [0 ; 1 ,then @ n @y n y;t 2L 1 and @ m @y m y;t 2L 1 for n =2 ; 3 ; 4 and m =1 ; 2 8 t 2 [0 ; 1 and 8 y 2 [0 ;l ] Property2. Ifthekineticenergyofthesystem, E K 1 2 l 0 )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(! 2 t +2 x c c cos t t + )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c c 2 2 t dy + 1 2 m s 2 t l;t + 1 2 J s 2 t l;t ; isbounded 8 t 2 [0 ; 1 ,then @ q @t q y;t 2L 1 and @ q @t q y;t 2L 1 for q =1 ; 2 ; 3 8 t 2 [0 ; 1 and 8 y 2 [0 ;l ] 5.2BoundaryControlDevelopment Thecontrolobjectiveistoensurethewingbendingandtwistingdeformationsare regulatedinthesensethat y;t 0 and y;t 0 ; 8 y 2 [0 ;l ] as t !1 via boundarycontrolatthewingtip.Tofacilitatethesubsequentstabilityanalysis,letthe auxiliarysignal e t 2 R 2 and M 2 R 2 2 bedenedas e t l;t )]TJ/F26 11.9552 Tf 11.955 0 Td [(! yyy l;t t l;t + y l;t T ; M 2 6 4 m s m s x s c cos l;t m s x s c cos l;t m s x 2 s c 2 + J s 3 7 5 : 70 PAGE 71 Theopen-loopdynamicsoftheauxiliarysignalareobtainedbymultiplyingthetime derivativeof e by M toyield M e = 2 6 4 m s tt l;t + m s x s c cos l;t tt l;t m s x s c cos l;t tt l;t + m s x 2 s c 2 + J s tt l;t 3 7 5 + 2 6 4 m s x s c cos l;t ty l;t )]TJ/F26 11.9552 Tf 11.955 0 Td [(m s tyyy l;t m s x 2 s c 2 + J s ty l;t )]TJ/F26 11.9552 Tf 11.956 0 Td [(m s x s c cos l;t tyyy l;t 3 7 5 : Substitutingtheboundaryconditionsin5and5into5yields M e = 2 6 4 L tip M tip 3 7 5 + 2 6 4 m s x s c sin l;t 2 t l;t + EI! yyy l;t m s x 2 s c 2 + J s ty l;t )]TJ/F26 11.9552 Tf 11.955 0 Td [(m s x s c cos l;t tyyy l;t 3 7 5 + 2 6 4 )]TJ/F26 11.9552 Tf 9.298 0 Td [(m s tyyy l;t + m s x s c cos l;t ty l;t )]TJ/F26 11.9552 Tf 9.298 0 Td [(GJ y l;t 3 7 5 : Aftersomealgebraicmanipulation,5canbeexpressedas M e = U )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 Me + Y; where U LM T 2 R 2 2 R 5 isavectorofunknownparameters,and Y 2 R 2 5 is aregressionmatrixofknownquantities.Specically, and Y aredenedas m s x s cEIm s GJ m s x 2 s c 2 + J s T ; Y 2 6 4 1 2 sin l;t 2 t l;t )]TJ/F26 11.9552 Tf 11.955 0 Td [( t l;t y l;t +cos l;t ty l;t yyy l;t 1 2 sin l;t t l;t yyy l;t )]TJ/F26 11.9552 Tf 11.955 0 Td [(! t l;t )]TJ/F15 11.9552 Tf 11.955 0 Td [(cos l;t tyyy l;t 0 )]TJ/F26 11.9552 Tf 9.299 0 Td [(! tyyy l;t 00 0 y l;t ty l;t 3 7 5 : 71 PAGE 72 Basedontheopen-loopdynamicsin5,theboundarycontrolisdesignedas U = )]TJ/F26 11.9552 Tf 9.299 0 Td [(Ke )]TJ/F26 11.9552 Tf 11.956 0 Td [(Y ^ ; where K 2 R isapositiveconstantcontrolgainand ^ 2 R 5 isavectorofestimatesof theuncertainparametersin .Thevectorofparameterestimates ^ isupdatedaccording tothegradientupdatelawdenedas ^ =)]TJ/F26 11.9552 Tf 19.74 0 Td [(Y T e; where )]TJ/F23 11.9552 Tf 11.711 0 Td [(2 R 5 5 isapositiveconstantcontrolgain.Substituting5into5yields thefollowingclosed-loopdynamics M e = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 Me )]TJ/F26 11.9552 Tf 11.955 0 Td [(Ke + Y ~ ; where ~ )]TJ/F15 11.9552 Tf 12.895 3.155 Td [(^ 5.3StabilityAnalysis Tofacilitatethesubsequentstabilityanalysis,lettheauxiliaryterms E T 2 R and E c 2 R bedenedas E T 1 2 l 0 )]TJ/F26 11.9552 Tf 5.48 -9.683 Td [(! 2 t +2 x c c cos t t + )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(I w + x 2 c c 2 2 t dy + 1 2 l 0 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(EI! 2 yy + GJ 2 y dy; E c 1 l 0 y y t + x c c cos t dy + 1 l 0 y y \000 I w + x 2 c c 2 t + x c c cos t dy; where 1 2 R isapositiveweightingconstant.Theauxiliaryterm E T isanalogoustothe energyinthewing,and E c containscrosstermsusedtofacilitatethestabilityanalysis. UsingYoung'sInequality,anupperboundon E T canbeexpressedas E T 1 2 l 0 )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [( + j x c c j 2 t + )]TJ/F26 11.9552 Tf 5.48 -9.683 Td [(I w + x 2 c c 2 + j x c c j 2 t + EI! 2 yy + GJ 2 y dy 72 PAGE 73 1 2 max + j x c c j ; )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c c 2 + j x c c j ;EI;GJ E b ; where E b 2 R isdenedas E b l 0 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(! 2 t + 2 yy + 2 t + 2 y dy: Inasimilarmanner, E T canbelowerboundedas E T 1 2 min )]TJ/F26 11.9552 Tf 11.955 0 Td [( j x c c j ; )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c c 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( j x c c j ;EI;GJ E b : Providedthat j x c c j < 1 and I w >x 2 c c 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( j x c c j E T willbenon-negative. Remark 5.1 Theconditions j x c c j < 1 and I w >x 2 c c 2 )]TJ/F26 11.9552 Tf 12.349 0 Td [( j x c c j areengineeringdesign considerationsthatensurethestoreismountedsufcientlyclosetothewingcenterof mass. AfterusingYoung'sInequality,thecrossterm E c canbeupperboundedas j E c j 1 l + j x c c j l 0 2 t dy + 1 l + j x c c j l 0 2 y dy + 1 l )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c c 2 + j x c c j l 0 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [( 2 t + 2 y dy: LemmaA.12in[34]canbeappliedtothesecondintegralin5toyield j E c j 1 l + j x c c j l 0 2 t dy + 1 l 3 + j x c c j l 0 2 yy dy + 1 l )]TJ/F26 11.9552 Tf 5.48 -9.683 Td [(I w + x 2 c c 2 + j x c c j l 0 )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [( 2 t + 2 y dy 1 l max + j x c c j ;l 2 + j x c c j ; )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c c 2 + j x c c j E b : From5, E c canbelowerboundedas E c )]TJ/F26 11.9552 Tf 21.918 0 Td [( 1 l max + j x c c j ;l 2 + j x c c j ; )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c c 2 + j x c c j E b : From5and5,if 1 isselectedas 1 < min f )]TJ/F26 11.9552 Tf 11.955 0 Td [( j x c c j ; I w + x 2 c c 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( j x c c j ;EI;GJ g 2 l max f + j x c c j ;l 2 + j x c c j ; I w + x 2 c c 2 + j x c c j g ; 73 PAGE 74 then 1 E b E T + E c 2 E b wheretheconstants 1 and 2 aredenedas 1 1 2 min )]TJ/F26 11.9552 Tf 11.956 0 Td [( j x c c j ; )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c c 2 )]TJ/F26 11.9552 Tf 11.956 0 Td [( j x c c j ;EI;GJ )]TJ/F26 11.9552 Tf 9.298 0 Td [( 1 l max + j x c c j ;l 2 + j x c c j ; )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c c 2 + j x c c j ; 2 1 2 min + j x c c j ; )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c c 2 + j x c c j ;EI;GJ + 1 l max + j x c c j ;l 2 + j x c c j ; )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(I w + x 2 c c 2 + j x c c j : Remark 5.2 1 willbepositiveprovidedthatthestoreismountedsufcientlyclose tothewingcenterofmass,asmentionedinRemark5.1.If 1 ispositive,thenthe constants 1 and 2 willalsobepositive. Theorem5.1. Theboundarycontrollawin5alongwiththeadaptiveupdatelawin 5ensurethesystemstates y;t 0 and y;t 0 as t !1 providedthe followingsufcientgainconditionsaresatised: K> 1 2 max f EI + 1 EIl g ; 1 < 1 ; 1 )]TJ/F26 11.9552 Tf 11.956 0 Td [( 1 x c c )]TJ/F15 11.9552 Tf 13.336 3.022 Td [( L w > 0 ; 3 EI 2 )]TJ/F15 11.9552 Tf 14.532 11.109 Td [( L w l 3 2 > 0 ; 1 )]TJ/F26 11.9552 Tf 5.48 -9.683 Td [(I w + x 2 c c 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 x c c )]TJ/F15 11.9552 Tf 16.291 3.022 Td [( M w > 0 ; 1 GJ )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 M w l 3 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 M w l )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 L w l 3 )]TJ/F31 11.9552 Tf 11.955 9.684 Td [()]TJ/F15 11.9552 Tf 9.816 -6.662 Td [( M w + L w l 2 > 0 ; 1 EIl + EI )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F26 11.9552 Tf 11.956 0 Td [( 1 x c cl> 0 ;GJ )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 l )]TJ/F26 11.9552 Tf 5.48 -9.683 Td [(I w + x 2 c c 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 x c cl> 0 : Remark 5.3 Thesufcientgainconditionsin5-5canbesatisedbya combinationofgainselectionandengineeringdesignconsideration.Selectionofthe wingaerodynamicpropertiescanbedonetosatisfyaircraftperformancecriteriae.g., minimumtakeoffdistance,maximumrange,etc..Thestructuralpropertiesofthewing canthenbeselectedtosatisfythesufcientconditionsabove.Increasingthestiffness andmassofthewingormountingthestoreclosertothewingcenterofgravitywill satisfythesufcientconditions. 74 PAGE 75 Proof. Let V L beapositive-denite,continuouslydifferentiablefunctiondenedas V L E T + E c + 1 2 e T Me + 1 2 ~ T )]TJ/F29 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ~ : Basedon5andtheinequalitiesin5, V L canbeboundedas 1 E b + min )]TJ/F15 11.9552 Tf 9.815 -6.662 Td [( M 2 k e k 2 + min )]TJ/F29 7.9701 Tf 11.867 4.338 Td [()]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 ~ 2 V L 2 E b + max )]TJ/F15 11.9552 Tf 9.816 -6.661 Td [( M 2 k e k 2 + max )]TJ/F29 7.9701 Tf 11.866 4.338 Td [()]TJ/F24 7.9701 Tf 6.587 0 Td [(1 2 ~ 2 ; where min and max denotetheminimumandmaximumeigenvalueof ,respectively. Differentiating5andsubstituting5and5intotheresultingexpressionyields V L = E T + E c )]TJ/F26 11.9552 Tf 11.955 0 Td [(e T Ke: In5, E T isdeterminedbydifferentiating5withrespecttotimetoobtain E T = l 0 t )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(! tt + x c c cos tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(x c c sin 2 t dy + l 0 EI! yy tyy + GJ y ty dy + l 0 t \000 I w + x 2 c c 2 tt + x c c cos tt dy: Substituting5and5intotherstandthirdintegralsof527yields E T = l 0 )]TJ/F15 11.9552 Tf 6.86 -6.662 Td [( L w t + M w t dy )]TJ/F40 11.9552 Tf 11.955 16.272 Td [( l 0 EI! t yyyy dy + l 0 EI! yy tyy dy + l 0 GJ t yy dy + l 0 GJ y ty dy: Integratingbypartsthethirdandfthintegralsin5andapplyingtheboundary conditionsofthePDEsystemresultsin l 0 EI! yy tyy dy = )]TJ/F26 11.9552 Tf 9.298 0 Td [(EI! yyy l;t t l;t + l 0 EI! t yyyy dy; l 0 GJ y ty dy = GJ y l;t t l;t )]TJ/F40 11.9552 Tf 11.955 16.273 Td [( l 0 GJ t yy dy: 75 PAGE 76 Usingtheexpressionsin5and5,5canberewrittenas E T = l 0 )]TJ/F15 11.9552 Tf 6.86 -6.662 Td [( L w t + M w t dy )]TJ/F26 11.9552 Tf 11.955 0 Td [(EI! yyy l;t t l;t + GJ y l;t t l;t : Usingtheauxiliarysignaldenitionin5,5canbeexpressedas E T = l 0 )]TJ/F15 11.9552 Tf 6.86 -6.661 Td [( L w t + M w t dy + e T 2 6 4 EI 2 0 0 GJ 2 3 7 5 e )]TJ/F26 11.9552 Tf 13.15 8.087 Td [(EI 2 2 t l;t )]TJ/F26 11.9552 Tf 13.151 8.087 Td [(EI 2 2 yyy l;t )]TJ/F26 11.9552 Tf 10.494 8.088 Td [(GJ 2 2 y l;t )]TJ/F26 11.9552 Tf 13.151 8.088 Td [(GJ 2 2 t l;t : In5, E c isdeterminedbydifferentiating5withrespecttotimetoyield E c = 1 l 0 y y )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(! tt + x c c cos tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(x c c sin 2 t dy + 1 l 0 x c c cos t ty ydy + 1 l 0 t ty ydy + 1 l 0 y y \000 I w + x 2 c c 2 tt + x c c cos tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(x c c sin t t dy + 1 l 0 ty y \000 I w + x 2 c c 2 t + x c c cos t dy: Theexpressionfor E c canbesimpliedbyintegratingthesecondintegralas 1 l 0 x c c cos t ty ydy = 1 x c cl cos l;t t l;t t l;t )]TJ/F26 11.9552 Tf 9.299 0 Td [( 1 l 0 x c c cos t t dy + 1 l 0 x c c sin y t t ydy )]TJ/F26 11.9552 Tf 9.299 0 Td [( 1 l 0 x c c cos ty t ydy: Substitutingtheexpressionin5andthesystemdynamicsin5and5into 5yields E c = 1 l 0 )]TJ/F15 11.9552 Tf 6.861 -6.662 Td [( L w )]TJ/F26 11.9552 Tf 11.956 0 Td [(EI! yyyy y ydy + 1 l 0 t ty ydy + 1 x c cl cos l;t t l;t t l;t 76 PAGE 77 )]TJ/F26 11.9552 Tf 9.299 0 Td [( 1 l 0 x c c cos t t dy + 1 l 0 )]TJ/F15 11.9552 Tf 9.815 -6.661 Td [( M w + GJ yy y ydy + 1 l 0 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c c 2 t ty ydy: Afterintegratingbypartstheterms )]TJ/F26 11.9552 Tf 9.298 0 Td [( 1 l 0 EI! yyyy y ydy 1 l 0 t ty ydy 1 l 0 GJ yy y ydy ,and 1 l 0 I w + x 2 c c 2 t ty ydy from5 2 E c canbeexpressedas E c = 1 l 0 )]TJ/F15 11.9552 Tf 6.86 -6.662 Td [( L w y + M w y ydy )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 EIl! yyy l;t y l;t )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(3 2 1 EI l 0 2 yy dy + 1 2 1 2 t l;t )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 1 l 0 2 t dy )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 x c c l 0 cos t t dy + 1 x c cl cos l;t t l;t t l;t + 1 2 1 GJ 2 y l;t )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 1 GJ l 0 2 y dy + 1 2 1 l )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c c 2 2 t l;t )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 1 )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c c 2 l 0 2 t dy: UsingYoung'sInequalityandLemmaA.12from[34], E c canbeupperboundedas E c )]TJ/F15 11.9552 Tf 30.552 0 Td [( )]TJ/F26 11.9552 Tf 11.955 0 Td [(x c c 1 2 l 0 2 t dy )]TJ/F31 11.9552 Tf 11.955 16.857 Td [( 3 EI 2 )]TJ/F15 11.9552 Tf 14.532 11.11 Td [( L w l 3 2 1 l 0 2 yy dy )]TJ/F31 11.9552 Tf 11.291 9.684 Td [(\000 I w + x 2 c c 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [(x c c 1 2 l 0 2 t dy + 1 2 1 2 t l;t )]TJ/F31 11.9552 Tf 11.291 9.684 Td [()]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(GJ )]TJ/F15 11.9552 Tf 16.29 3.022 Td [( M w l 3 )]TJ/F15 11.9552 Tf 16.291 3.022 Td [( M w l )]TJ/F15 11.9552 Tf 13.336 3.022 Td [( L w l 3 1 2 l 0 2 y dy )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 EIl! yyy l;t y l;t + 1 x c cl t l;t t l;t + 1 2 1 GJ 2 y l;t + 1 2 1 l )]TJ/F26 11.9552 Tf 5.48 -9.683 Td [(I w + x 2 c c 2 2 t l;t : Using5, )]TJ/F26 11.9552 Tf 9.299 0 Td [( 1 EIl! yyy l;t y l;t canbeexpressedas )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 EIl! yyy l;t y l;t = )]TJ/F26 11.9552 Tf 10.494 8.088 Td [( 1 EIl 2 2 yyy l;t )]TJ/F26 11.9552 Tf 13.151 8.088 Td [( 1 EIl 2 2 t l;t + 1 EIl 2 e 2 1 ; where e 1 denotestherstelementofthevector e ,i.e., e 1 t l;t )]TJ/F26 11.9552 Tf 12.263 0 Td [(! yyy l;t .Using 5,5canberewrittenas E c )]TJ/F15 11.9552 Tf 30.552 0 Td [( )]TJ/F26 11.9552 Tf 11.955 0 Td [(x c c 1 2 l 0 2 t dy )]TJ/F31 11.9552 Tf 11.955 16.857 Td [( 3 EI 2 )]TJ/F15 11.9552 Tf 14.532 11.11 Td [( L w l 3 2 1 l 0 2 yy dy 2 SeeAppendixH 77 PAGE 78 )]TJ/F31 11.9552 Tf 11.291 9.683 Td [(\000 I w + x 2 c c 2 )]TJ/F26 11.9552 Tf 11.956 0 Td [(x c c 1 2 l 0 2 t dy + 1 EIl 2 e 2 1 + 1 2 1 l )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(I w + x 2 c c 2 2 t l;t )]TJ/F31 11.9552 Tf 11.291 9.684 Td [()]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(GJ )]TJ/F15 11.9552 Tf 16.291 3.022 Td [( M w l 3 )]TJ/F15 11.9552 Tf 16.291 3.022 Td [( M w l )]TJ/F15 11.9552 Tf 13.336 3.022 Td [( L w l 3 1 2 l 0 2 y dy )]TJ/F26 11.9552 Tf 13.151 8.088 Td [( 1 EIl 2 2 yyy l;t )]TJ/F26 11.9552 Tf 13.151 8.088 Td [( 1 EIl 2 2 y l;t + 1 2 1 2 t l;t + 1 x c cl t l;t t l;t + 1 2 1 GJ 2 y l;t : Inserting5and5into5andusingYoung'sinequalityyields V L )]TJ/F15 11.9552 Tf 29.756 8.088 Td [(1 2 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [( 1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 x c c )]TJ/F15 11.9552 Tf 13.336 3.022 Td [( L w l 0 2 t dy )]TJ/F31 11.9552 Tf 11.955 16.857 Td [( 3 EI 2 )]TJ/F15 11.9552 Tf 14.532 11.11 Td [( L w l 3 2 1 l 0 2 yy dy )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [( 1 )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c c 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 x c c )]TJ/F15 11.9552 Tf 16.291 3.022 Td [( M w l 0 2 t dy )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [( 1 GJ )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 M w l 3 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 M w l )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 L w l 3 )]TJ/F31 11.9552 Tf 11.955 9.684 Td [()]TJ/F15 11.9552 Tf 9.815 -6.662 Td [( M w + L w l 2 l 0 2 y dy )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 1 EIl + EI )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F26 11.9552 Tf 11.956 0 Td [( 1 x c cl 2 t l;t )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(GJ )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 l )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c c 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 x c cl 2 t l;t )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 1 EIl + EI 2 yyy l;t )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 GJ )]TJ/F26 11.9552 Tf 11.956 0 Td [( 1 GJ 2 y l;t )]TJ/F31 11.9552 Tf 11.955 16.857 Td [( K )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 max f EI + 1 EIl;GJ g k e k 2 : Providedthesufcientconditionsin5-5aresatised,5canbeexpressedas V L )]TJ/F26 11.9552 Tf 28.56 0 Td [( 1 E b t )]TJ/F26 11.9552 Tf 11.955 0 Td [( 2 e 2 t ; where 1 2 R and 2 2 R arepositiveconstantsdenedas 1 1 2 min 1 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 x c c )]TJ/F15 11.9552 Tf 13.336 3.022 Td [( L w ; 3 EI 2 )]TJ/F15 11.9552 Tf 14.532 11.109 Td [( L w l 3 2 ; 1 )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(I w + x 2 c c 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 x c c )]TJ/F15 11.9552 Tf 16.29 3.022 Td [( M w 1 GJ )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 M w l 3 )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 M w l )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 L w l 3 )]TJ/F31 11.9552 Tf 11.955 9.684 Td [()]TJ/F15 11.9552 Tf 9.815 -6.662 Td [( M w + L w l 2 ; 2 K )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 max f EI + 1 EIl;GJ g : Itcanbeconcludedfrom5and5that V L 2L 1 ;hence E b 2L 1 e 2L 1 and ~ 2L 1 .Since E b 2L 1 ,itcanbeconcludedthat l 0 2 yy dy 2L 1 and l 0 2 y dy 2L 1 ; hencetheelasticpotentialenergyinthewing E P 2L 1 andbyProperty1 yyy l;t 2 L 1 and y l;t 2L 1 .Since e 2L 1 yyy l;t 2L 1 ,and y l;t 2L 1 ,5canbe 78 PAGE 79 usedtoshow t l;t 2L 1 and t l;t 2L 1 .Since t l;t 2L 1 t l;t 2L 1 ,and E b 2L 1 ,thekineticenergyofthesystem E K 2L 1 andbyProperty2, @ q @t q y;t 2L 1 and @ q @t q y;t 2L 1 for q =1 ; 2 ; 3 .Equations5and5canbeusedtoshowthat theboundarycontrolinput, U 2L 1 .Differentiating g t from5withrespecttotime yields g = 1 E b +2 2 e e; where E b =2 l 0 t tt + yy tyy + t tt + ty y dy: Afterintegratingbypartsthesecondandfourthtermsin5, E b canbeexpressedas E b =2 l 0 t tt + yyyy + t tt )]TJ/F26 11.9552 Tf 11.955 0 Td [( yy dy )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 t l;t yyy l;t +2 t l;t y l;t : Since t y;t tt y;t yyyy y;t t y;t tt y;t yy y;t t l;t yyy l;t t l;t and y l;t 2L 1 fromProperties1and2,5canbeusedtoconcludethat E b 2L 1 .Equations5and5canbeusedtoshowthat g 2L 1 .LemmaA.6 from[34]canbeappliedto5toconclude lim t !1 g t =0 andhence lim t !1 E b t ;e t =0 : Using5andLemmaA.12in[34]thefollowinginequalitiescanbedeveloped E b l 0 2 yy dy 1 l 3 2 y;t 0 ; E b l 0 2 y dy 1 l 2 y;t 0 : Since E b 0 as t !1 ,itcanbeconcludedfrom5and5that y;t 0 and y;t 0 as t !1 79 PAGE 80 5.4Summary ThischapterpresentstheconstructionofaboundarycontrolstrategyforsuppressingLCObehaviorinanuncertainexibleaircraftwing.Theboundarycontrolstrategy retainsthefullPDEsystem,therebyavoidingpotentialspilloverinstabilities,andensures asymptoticregulationofthedistributedstatesinthepresenceofparametricuncertainties.Apotentialdrawbacktothedevelopedmethodistheneedformeasurementsof high-orderspatialderivativesofthedistributedstatese.g., yyy l;t 80 PAGE 81 CHAPTER6 CONCLUSIONANDFUTUREWORK 6.1DissertationSummary Thefocusofthisworkistodevelopcontrolmethodsforthesuppressionoflimit cycleoscillationsLCOinaircraftsystems.ThedrivingmechanismbehindLCO behaviorremainsunknown;however,thebehaviorisprevalentonthecurrentgeneration ofghteraircraftandisexpectedtopersistonnextgenerationaircraft.Themajor concernsassociatedwithLCObehaviorareitsimpactonthesafereleaseofordnance andtheabilityofthepilottoperformnecessarymission-relatedtasks. Chapter2focusesonthedevelopmentofanadaptivecontrolstrategytosuppress LCObehaviorinanuncertaintwodegreeoffreedomairfoilsection.ThedevelopedcontrollerfeaturesaneuralnetworkNNfeedforwardtermtocompensateforuncertainties intheairfoildynamicsandarobustintegralofthesignoftheerrorRISEfeedbackterm toensureasymptotictrackingoftheairfoilangleofattack.Thesimulationresultsof Chapter2,asseeninpreviousRISE-basedcontrolstrategies,indicatethattheRISEbasedcontrollercandemandalargecontroleffortinresponetolargeinitialoffsetsor largedisturbances.InChapter3,asaturatedRISE-basedcontrollerisdevelopedin whichtheRISEcontrolstructureisenbeddedinsmoothhyberbolicfunctionstoensure actuatorcontraintsarenotbreachedwhilemaintainingasymptotictrackingwithacontinuouscontroller.Theactuatorlimitisknown apriori andcanbeadjustedviachanging thecontrolgains. Chapters4and5focusonthedevelopmentofpartialdifferentialequationPDEbasedboundarycontrolmethodsforthesuppressionofLCObehaviorinaexible aircraftwing.Chapter4usesaPDE-basedbacksteppingmethodtotransformalinear PDEsystemdescribingthedynamicsofthedistributedstatestoanexponentially stablelinearPDEsystem.Chapter5developsaboundarycontrolstrategythatusesa gradient-basedadaptiveupdatelawtocompensateforlinear-in-the-parametersLP 81 PAGE 82 uncertaintiesandaLyapunov-basedanalysistoshowthattheenergyinthesystem remainsboundedandasymptoticallydecaystozero.Thedifferencesbetweenthetwo PDE-basedcontrolstrategiesarethetypeofsystemusedinthedesignandtherequired measurementsforimplementation.ThestrategyinChapter4isdesignedforalinear PDEmodeloftheexibleaircraftwingandusesmeasurementsoftheexiblestates acrosstheentirewingspan.ThecontrollerinChapter5isdesignedforanonlinearPDE modelandrequiresmeasurementsofthehigherspatialderivativesoftheexiblestates attheactuatorlocatione.g., yyy l;t 6.2LimitationsandFutureWork Theworkinthisdissertationdevelopsnewrobustandadaptivecontrollersforthe suppressionofLCObehaviorinaircraftsystems.Inthissection,openproblemsrelated totheworkinthisdissertationarediscussed. FromChapter2: 1.ApracticallimitationinthedevelopedRISE-basedcontrolstrategyisthatasthe severityoftheLCObehaviorincreases,thedevelopedcontrollercandemanda largecontrolsurfacedeection.Additionally,theMonteCarlosimulationresults indicatedthatthemaximumcontroleffortissensitivetovariationsintheparameter uncertainties,whichcouldleadtounexpectedactuatorsaturation.Thislimitationis addressedinChapter3. FromChapter3: 1.ApotentialdrawbackofthesaturatedRISE-basedcontrolstrategyisthatunder certainconditions,theLCOproducedcouldbetoosevereresultinginsufcient gainconditionsthatcan'tbesatised.Thisisadirectresultoftheactuatorlimit; increasingtheactuatorlimitrelaxesthesufcientgainconditions.Furthermore, anadaptivefeedforwardtermcouldpotentiallybeincludedtocompensatefor theuncertaindynamics,therebyrelaxingthesufcientgainconditions.However, foranycontrollerthathasrestrictedcontrolauthority,itispossibleforsome 82 PAGE 83 disturbancetodominatethecontroller'sabilitytoyieldadesiredorevenstable performance. FromChapter4: 1.OnedrawbackofthedevelopedPDE-basedbacksteppingcontrolleristhatit reliesontheassumptionthatthedistancesfromthewingelasticaxistothewing centerofgravityandstorecenterofgravityarezero.Withoutthisassumption,the PDEdescribingthedynamicsofthewingdeformationsbecomesnonlinearwhich doesnotfacilitatetheuseofthebacksteppingstrategyemployedinthischapter. Instead,anapproachsimilartothatof[34,35],inwhichaLyapunov-basedanalysis provesthattheenergyinthesystemdecaystozero,couldbeusedtogeneratethe aerodynamicliftandmomentatthewingtip.Chapter5addressesthislimitation. 2.Duetothelackofclarityamongstresearchersastothedrivingmechanismbehind LCO,acommonpracticeinliterature,andintheworkofChapters2and3,is toreplicatethesymptomsofLCObehaviorbyincludingnonlinearitiesinthe wingstructure.Inmostcases,thisisanonlineartorsionalstiffness.Thecontrol strategiesinChapters2and3provideaframeworkthatcanbereadilyadapted tocompensateforthedrivingmechanismasitbecomesbetterunderstood. However,duetothestructureofthePDE-basedbacksteppingmethod,ifthe drivingmechanismisnonlinear,itsincorporationintothedevelopedcontrol structuremaynotbefeasible,andamethodsimilarto[34,35]mustbeemployed. FromChapter5: 1.SincethecontrollerinChapter5wasdevelopedforanonlinearPDE,itcanbe adaptedmorereadilytocompensatefortheinclusionofthedrivingmechanism behindLCObehavior.Thecontrolstructurewillrequiresmallchanges,mostly tothesufcientgainconditionstoincludetheinuenceoftheuncertainties associatedwiththedrivingmechanism;however,morecomplexsystemstypically requiremorecomplexcandidateLyapunovfunctionsi.e.,thedenitionfor E c 83 PAGE 84 willchangetoaccountforcross-termsassociatedwiththemodelofthedriving mechanism. 2.Apotentialdrawbacktothedevelopedmethodistheneedformeasurementsof high-orderspatialderivativesofthedistributedstatese.g., yyy l;t .Ashear sensorattachedatthewingtipcanbeusedtomeasure yyy l;t andtorque measurmentsatthewingtipcanbeusedtodetermine y l;t .Futureeffortsare focusedondevelopingPDE-basedoutputfeedbackboundarycontrolstrategies thatwouldeliminatetheneedforhigh-orderspatialderivativemeasurements. 84 PAGE 85 APPENDIXA PROOFTHAT M ISINVERTIBLECH3 LemmaA.1. M ,givenbytheexpressionsin2and2-2,isinvertible. Toshowthat M isinvertible,itisnecessarytoshowthat det M 6 =0 .The det M canbeexpressedas det M = m 1 m 4 )]TJ/F26 11.9552 Tf 12.166 0 Td [(m 2 2 where m 1 ;m 2 ;m 4 2 R aredenedin22.Since det M appearsin g ,whichisusedintheLyapunovfunction,thefollowing conditionisdesirable m 1 m 4 )-222(j m 2 j 2 > 0 : A From2, m 2 canbewrittenas m 2 = p cos )]TJ/F26 11.9552 Tf 12.438 0 Td [(l sin ,where p = r x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a m w b + s x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a m s b 2 R and l = s h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h m s b + r h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h m w b 2 R .Themaximumvalueof m 2 canbeexpressedas j m 2 j p p 2 + l 2 .Substitutingforthevaluesof p and l j m 2 j 2 canbe expressedas j m 2 j 2 r x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a 2 b 2 m 2 w +2 r x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a s x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a b 2 m s m w +2 s h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h r h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h b 2 m w m s + r h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h 2 b 2 m 2 w + s x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a 2 b 2 m 2 s + s h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h 2 b 2 m 2 s : A Using2and2, m 1 m 4 canbeexpressedas m 1 m 4 = r x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a 2 b 2 m 2 w + r x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a 2 b 2 m w m s + r h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h 2 b 2 m 2 w + r h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h 2 b 2 m w m s + s x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a 2 b 2 m s m w + s h )]TJ/F26 11.9552 Tf 11.956 0 Td [(a h 2 b 2 m s m w + s x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a 2 b 2 m 2 s + s h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h 2 b 2 m 2 s + I w + I s m w + m s : A Evaluatingthe det M usingAandAyields det M r x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 r x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a s x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a b 2 m w m s + s h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h 2 + s x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a 2 b 2 m w m s 85 PAGE 86 + r h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 r h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h s h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h b 2 m w m s + I w + I s m s + m w : A Aftersomealgebraicmanipulation,theexpressioninAcanberewrittenas det M [ r x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a )]TJ/F15 11.9552 Tf 11.955 0 Td [( s x )]TJ/F26 11.9552 Tf 11.956 0 Td [(a ] 2 b 2 m w m s +[ r h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h )]TJ/F15 11.9552 Tf 11.955 0 Td [( s h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h ] 2 b 2 m w m s + I w + I s m s + m w : A SincethersttwotermsinAandthemassandmomentofinertiaofthewingand storearealwayspositive, det M > 0 ;hence M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 isinvertible. 86 PAGE 87 APPENDIXB PROOFOF g> 0 CH3 LemmaB.1. Giventheexpressionin2, g> 0 ifthefollowingconditionissatised m 1 C m C l >& B Toprovethat g mustbestrictlygreaterthanzero,2isusedtowrite g as g = 1 det M [ m 2 C l + m 1 C m ] .UsingtheresultsofAppendixA, 1 det M > 0 .Therefore, for g> 0 ,theterm [ m 2 C l + m 1 C m ] mustbepositive.From2, m 2 issignindenate sofor [ m 2 C l + m 1 C m ] toremainpositive, m 1 C m >m 2 C l .From2, m 2 canbe upperboundedas j m 2 j & ,where & 2 R isaknownpositiveconstant.From2 andtheupperboundon m 2 g> 0 providedthat m 1 C m C l >& .Thissufcientcondition canbesatisedbyadjustingthegeometryofthewing-storesystem.Forexample,the left-handsidecanbeincreasedbyincreasingthecontrolsurfaceeffectivenessratio C m C l whichcanbedonebychangingthewingairfoil.Theconstant & canbemadesmallerby decreasingthedistancebetweenthewingelasticaxisandthestorecenterofgravity. 87 PAGE 88 APPENDIXC GROUPINGOFTERMSIN 1 AND 2 CH3 From2,theauxiliaryfunction 2 R isdenedas 1 g f )]TJ/F15 11.9552 Tf 15.454 8.088 Td [(1 g d f d = 1 + 2 ; where 1 2 R containsalltermsin whosetimederivativeisboundedbythenorm ofthestatesand 2 2 R containsalltermswhosetimederivativeisboundedbya constant.Theauxiliaryfunctions 1 and 2 areexplicitlydenedas 1 = det M m 1 C m + m 2 C l 1 e 1 + 2 e 2 2 = m 2 ~ C 11 h + ~ C 12 + ~ K 11 h + ~ K 12 m 1 C m + m 2 C l )]TJ/F26 11.9552 Tf 13.15 12.433 Td [(m 1 ~ C 21 h + ~ C 22 + ~ K 22 m 1 C m + m 2 C l )]TJ/F15 11.9552 Tf 28.088 8.088 Td [(det M d m 1 C m + m 2 C l : 88 PAGE 89 APPENDIXD DEVELOPMENTOFTHEBOUNDON ~ N CH3 Recallfrom2,theauxiliaryfunction ~ N isdenedas ~ N )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 d dt 1 g r +_ 1 + e 2 )]TJ/F26 11.9552 Tf 11.955 0 Td [(proj )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(1 ^ 0 ^ V T x d e 2 T ^ )]TJ/F15 11.9552 Tf 12.711 3.022 Td [(^ W T ^ 0 proj )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(2 x d ^ 0 T ^ We 2 T T x d : D Fromtheassumptiononthedesiredtrajectoriesand2and2,thelasttwo termsin ~ N canbeupperboundedas proj )]TJ/F24 7.9701 Tf 7.315 -1.793 Td [(1 ^ 0 ^ V T x d e 2 T ^ c 1 j e 2 j c 1 k z k ^ W T ^ 0 proj )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(2 x d ^ 0 T ^ We 2 T T x d c 2 j e 2 j c 2 k z k ; where c 1 ;c 2 2 R areknownpositiveconstants.Takingthetimederivativeof 1 ,dened inAppendixC,yields 1 = d dt det M m 1 C m + m 2 C l )]TJ/F15 11.9552 Tf 19.477 8.088 Td [(det M m 2 C l m 1 C m + m 2 C l 2 1 e 1 + 2 e 2 + det M m 1 C m + m 2 C l 1 e 1 + 2 e 2 : FromAppendixBandtheexpressionfor det M inAppendixA,theterms m 1 C m + m 2 C l and m 1 C m + m 2 C l 2 areboundedbelowbyaconstantwhile det M isupperboundedbyaconstant.Takingthetimederivativeof det M yields d dt det M = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 m 2 m 2 =2 m 2 m w b r h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h cos +2 m 2 m w b r x )]TJ/F26 11.9552 Tf 11.955 0 Td [(a sin +2 m 2 m s b s h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a h cos +2 m 2 m s b s h )]TJ/F26 11.9552 Tf 11.955 0 Td [(a sin : Since k q k 2 andusingtheresultinAppendixB, d dt det M c 3 ,where c 3 2 R isa knownpositiveconstant. 89 PAGE 90 Theupperboundon 1 canbeexpressedas j 1 j d dt det M m 1 C m + m 2 C l )]TJ/F15 11.9552 Tf 19.477 8.088 Td [(det M m 2 C l m 1 C m + m 2 C l 2 j 1 e 1 + 2 e 2 j + det M m 1 C m + m 2 C l 1 e 1 + 2 e 2 : Usingtheupperboundson d dt det M m 2 ,andtheexpressionsin2and2, theupperboundon 1 canberewrittenas j 1 j c 4 j e 1 j + c 5 j e 2 j + c 6 j r j c 0 1 k z k ; where c 4 ;c 5 ;c 6 ;c 0 1 2 R areknownpositiveconstants. TherstterminDcanbeexpressedas )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 d dt 1 g r = r 2 g 2 )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 m 2 m 2 m 1 C m + m 2 C l det M 2 + r m 2 C l 2 g 2 det M : D Usingtheupperboundson m 2 and m 2 andthelowerboundson g and det M ,the expressioninDcanbeupperboundedas 1 2 d dt 1 g r c 0 2 j r j c 0 2 k z k : Theupperboundon ~ N canthenbeexpressedas ~ N 1 2 d dt 1 g r + j 1 j + j e 2 j + proj )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(1 ^ 0 ^ V T x d e 2 T ^ + ^ W T ^ 0 proj )]TJ/F24 7.9701 Tf 7.314 -1.794 Td [(2 x d ^ 0 T ^ We 2 T T x d : Therefore,usingthedevelopedupperboundsontheindividualterms, ~ N c 0 1 + c 0 2 +1+ c 1 + c 2 k z k k z k ; where 2 R isaknownpositiveconstant. 90 PAGE 91 APPENDIXE DETAILSONTHEDEVELOPMENTOFTHECONSTANTS c m 1 c m 2 ,AND c m 3 CH4 UsingtheresultsofAppendixB, g>" 1 where 1 2 R isaknownpositiveconstant. Since m 2 2 0 j det M j m 1 m 4 and det M g m 1 m 4 g " 1 d dt det M g canbeupperboundedas d dt det M g 2 1 = c m 2 : UsingtheresultinAppendixBandtheupperboundon d dt det M m 2 canbe upperboundedas m 2 3 where 3 2 R isaknownpositiveconstant.Usingtheresult inAppendixA,theterm m 2 C l det M g 2 canbeupperboundedas m 2 C l det M g 2 3 C l 4 2 1 j det M j 91 PAGE 92 APPENDIXF DERIVATIONOFTHEBENDINGANDTWISTINGDYNAMICSOFAFLEXIBLEWING CH5/6 Consideraexiblewingwithastoreattachedatthewingtipanduniformcross sectionundergoingbendingandtwistingmotions.Thewinghasspan l 2 R ,chord length c 2 R ,massperunitlengthof 2 R ,polarmomentofinertiaperunitlengthof I w 2 R ,bendingrigidity EI 2 R ,andtorsionalrigidity GJ 2 R .Theattachedstorehas mass m s 2 R andmomentofinertia J s 2 R .Denearight-handcoordinatesystemas follows:theoriginisontheshearcenterattherootofthewing,the x axispointsoutthe rearofthewing,andthe y axisextendstothewingtip.Let ! y;t 2 R denotethe bendingdeectionand y;t 2 R denotethetwistingdeformationatthespanwise location y 2 [0 ;l ] .Furthermore,itisassumedthatthecenterofgravityandaerodynamic centerofthewingcrosssectionandthecenterofgravityofthestorearenotcolinear withtheelasticaxisofthewing.Let x c c 2 R and x s c 2 R representthedistancesfrom thewingelasticaxistothewingcenterofgravityandstorecenterofgravity,respectively. Letthevectors p y;t 2 R 3 and p l t 2 R 3 denotethepositionofthecenterofgravity ofanarbitrarywingcrosssectionandthepositionofthecenterofgravityofthestore, respectively.Thesevectorsareexpressedas p y;t x c c cos y;t y! y;t + x c c sin y;t T ; p l t x s c cos l;t l! l;t + x s c sin l;t T : Thekineticenergyofthewingandstorecanbeexpressedas T wing = 2 l 0 p T t y;t p t y;t dy + I w 2 l 0 2 t y;t dy = 1 2 l 0 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(! 2 t y;t +2 x c c cos y;t t y;t t y;t + x 2 c c 2 2 t y;t dy + 1 2 l 0 I w 2 t y;t dy; T store = m s 2 p T l t t p l t t + J s 2 2 t l 92 PAGE 93 = m s 2 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(! 2 t l +2 x s c cos l t l t l + x 2 s c 2 2 t l + J s 2 2 t l ; wherethesubscript t denotesthepartialderivativewithrespectto t l l;t ,and l l;t .Thepotentialenergyinthewingcanbewrittenas U = 1 2 l 0 )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(EI! 2 yy + GJ 2 y dy; wherethesubscript y denotesthepartialderivativewithrespectto y .TheLagrangianfor thewing-storesystemisdenedas L T wing + T store )]TJ/F26 11.9552 Tf 11.955 0 Td [(U = 1 2 l 0 )]TJ/F26 11.9552 Tf 5.48 -9.683 Td [(! 2 t +2 x c c cos t t + )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(x 2 c c 2 + I w 2 t )]TJ/F26 11.9552 Tf 11.955 0 Td [(EI! 2 yy )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJ 2 y dy + m s 2 2 t l + m s x s c cos l t l t l + m s 2 x 2 s c 2 + J s 2 2 t l : Hamilton'sprincipleisgivenas t 2 t 1 W + L dt =0 ; where L denotesthevariationintheLagrangianand W denotesthevirtualwork expressedas W = l 0 L w + M w )]TJ/F26 11.9552 Tf 11.955 0 Td [( w EI! tyy yy )]TJ/F26 11.9552 Tf 11.955 0 Td [( GJ ty y dy + L tip l + M tip l ; where L w 2 R and M w 2 R representtheaerodynamicliftandmomentperunitlength, respectively, L tip 2 R and M tip 2 R denotetheaerodynamicliftandmomentatthewing tip,respectively,and w 2 R and 2 R denoteKelvin-Voigtdampingcoefcients.The variationintheLagrangiancanbewrittenas L = @ L @! t t + @ L @! yy yy + @ L @ + @ L @ t t + @ L @ y y + @ L @! t l t l + @ L @ l l + @ L @ t l t l ; 93 PAGE 94 wherethepartialderivativesareevaluatedas @ L @! t = l 0 t + x c c cos t dy; @ L @! yy = )]TJ/F40 11.9552 Tf 11.291 16.273 Td [( l 0 EI! yy dy; @ L @ = )]TJ/F26 11.9552 Tf 9.298 0 Td [(x c c l 0 sin t t dy; @ L @ t = l 0 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(x c c cos t + )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(x 2 c c 2 + I w t dy; @ L @ y = )]TJ/F40 11.9552 Tf 11.291 16.273 Td [( l 0 GJ y dy; @ L @! t l = m s t l + m s x s c cos l t l ; @ L @ l = )]TJ/F26 11.9552 Tf 9.298 0 Td [(m s x s c sin l t l t l ; @ L @ t l = m s x s c cos l t l + )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(m s x 2 s c 2 + J s t l : Substitutingtheexpressionsfor W and L intoHamilton'sprincipleyields )]TJ/F40 11.9552 Tf 11.955 16.272 Td [( t 2 t 1 l 0 EI! yy yy dydt + t 2 t 1 l 0 t + x c c cos t t dydt )]TJ/F40 11.9552 Tf 11.955 16.272 Td [( t 2 t 1 l 0 GJ y y dydt + t 2 t 1 l 0 )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(x c c cos t + )]TJ/F26 11.9552 Tf 5.48 -9.683 Td [(x 2 c c 2 + I w t t dydt )]TJ/F26 11.9552 Tf 11.955 0 Td [( w t 2 t 1 l 0 EI! tyy yy dydt )]TJ/F26 11.9552 Tf 11.955 0 Td [( t 2 t 1 l 0 GJ ty y dydt + t 2 t 1 m s t l + m s x s c cos l t l t l dt + t 2 t 1 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(m s x s c cos l t l + )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(m s x 2 s c 2 + J s t l t l dt )]TJ/F26 11.9552 Tf 11.956 0 Td [(m s x s c t 2 t 1 sin l t l t l l dt + t 2 t 1 l 0 L w + M w dydt )]TJ/F26 11.9552 Tf 11.955 0 Td [(x c c t 2 t 1 l 0 sin t t dydt + t 2 t 1 L tip l + M tip l dt =0 : F Theequationsofmotionandboundaryconditionsforthewing-storesystemareobtainedbyintegratingbypartsselecttermsfromF.Integratingbypartsthersteight 94 PAGE 95 integralsinFandrecallingthatthevariationsat t = t 1 and t = t 2 arezeroyields )]TJ/F40 11.9552 Tf 11.955 16.272 Td [( t 2 t 1 l 0 EI! yy yy dydt = )]TJ/F40 11.9552 Tf 11.291 16.272 Td [( t 2 t 1 EI! yy l y l dt + t 2 t 1 EI! yy y dt + t 2 t 1 @ @y EI! yy l l dt )]TJ/F40 11.9552 Tf 11.955 16.272 Td [( t 2 t 1 @ @y EI! yy dt )]TJ/F40 11.9552 Tf 11.291 16.272 Td [( t 2 t 1 l 0 @ 2 @y 2 EI! yy !dydt; F )]TJ/F40 11.9552 Tf 11.955 16.272 Td [( t 2 t 1 l 0 GJ y y dydt = )]TJ/F40 11.9552 Tf 11.291 16.272 Td [( t 2 t 1 GJ y l l dt + t 2 t 1 GJ y dt + t 2 t 1 l 0 @ @y GJ y dydt; F t 2 t 1 l 0 t + x c c cos t t dydt = )]TJ/F40 11.9552 Tf 11.291 16.273 Td [( t 2 t 1 l 0 )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(! tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(x c c sin 2 t !dydt )]TJ/F40 11.9552 Tf 11.291 16.272 Td [( t 2 t 1 l 0 x c c cos tt !dydt F t 2 t 1 l 0 x c c cos t t dydt + t 2 t 1 l 0 )]TJ/F26 11.9552 Tf 5.48 -9.683 Td [(x 2 c c 2 + I w t t dydt = t 2 t 1 l 0 x c c sin t t dydt )]TJ/F40 11.9552 Tf 11.291 16.273 Td [( t 2 t 1 l 0 x c c cos tt dydt; )]TJ/F40 11.9552 Tf 11.291 16.272 Td [( t 2 t 1 l 0 )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(x 2 c c 2 + I w tt dydt F )]TJ/F26 11.9552 Tf 11.955 0 Td [( w t 2 t 1 l 0 EI! tyy yy dydt = )]TJ/F26 11.9552 Tf 9.299 0 Td [( w t 2 t 1 EI! tyy l y l dt + w t 2 t 1 EI! tyy y dt + w t 2 t 1 @ @y EI! tyy l l dt )]TJ/F26 11.9552 Tf 9.299 0 Td [( w t 2 t 1 @ @y EI! tyy dt 95 PAGE 96 )]TJ/F26 11.9552 Tf 9.299 0 Td [( w t 2 t 1 l 0 @ 2 @y 2 EI! tyy !dydt; F )]TJ/F26 11.9552 Tf 11.955 0 Td [( t 2 t 1 l 0 GJ ty y dydt = )]TJ/F26 11.9552 Tf 9.298 0 Td [( t 2 t 1 GJ ty l l dt + t 2 t 1 GJ ty dt + t 2 t 1 l 0 @ @y GJ ty dydt; F t 2 t 1 m s t l t l dt t 2 t 1 m s x s c sin l 2 t l l dt + t 2 t 1 m s x s c cos l t l t l dt = )]TJ/F40 11.9552 Tf 11.291 16.273 Td [( t 2 t 1 m s x s c cos l tt l l dt )]TJ/F40 11.9552 Tf 11.291 16.273 Td [( t 2 t 1 m s tt l l dt; F t 2 t 1 m s x s c cos l t l t l dt + t 2 t 1 )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(m s x 2 s c 2 + J s t l t l dt = t 2 t 1 m s x s c sin l t l t l l dt )]TJ/F40 11.9552 Tf 11.291 16.273 Td [( t 2 t 1 m s x s c cos l tt l l dt )]TJ/F40 11.9552 Tf 11.291 16.273 Td [( t 2 t 1 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(m s x 2 s c 2 + J s tt l l dt; F SubstitutingF-FintoFyieldsthefollowingPDEsystemandboundary conditions L w = tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(x c c sin 2 t + x c c cos tt + @ 2 @y 2 EI! yy + w @ 2 @y 2 EI! tyy ; F M w = )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c c 2 tt + x c c cos tt )]TJ/F26 11.9552 Tf 16.219 8.088 Td [(@ @y GJ y )]TJ/F26 11.9552 Tf 9.298 0 Td [( @ @y GJ ty ; F = y = yy l = =0 ; F 96 PAGE 97 L tip = m s tt l )]TJ/F26 11.9552 Tf 11.955 0 Td [(m s x s c sin l 2 t l + m s x s c cos l tt l )]TJ/F26 11.9552 Tf 13.562 8.088 Td [(@ @y EI! yy l )]TJ/F26 11.9552 Tf 11.955 0 Td [( w EI! tyy l ; F M tip = )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(m s x 2 s c 2 + J s tt l + m s x s c cos l tt l + GJ y l + GJ ty l F ThedevelopmentinChapter4isbasedontheassumptionsthat EI and GJ are constantsand x c = x s =0 .Undertheseassumptions,F,F,F,and Fbecome L w = tt + EI! yyyy + w EI! tyyyy ; F M w = I w tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJ yy )]TJ/F26 11.9552 Tf 11.955 0 Td [( GJ tyy ; F EI! yyy l + w EI! tyyy l = m s tt l )]TJ/F26 11.9552 Tf 11.955 0 Td [(L tip ; GJ y l + GJ ty l = )]TJ/F26 11.9552 Tf 9.298 0 Td [(J s tt l + M tip : ThedevelopmentinChapter5isbasedontheassumptionsthat EI and GJ are constantsandtheKelvin-Voigtdampingcoefcientsarezero.Undertheseassumptions, F,F,F,andFbecome L w = tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(x c c sin 2 t + x c c cos tt + EI! yyyy ; F M w = )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(I w + x 2 c c 2 tt + x c c cos tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJ yy F L tip = m s tt l )]TJ/F26 11.9552 Tf 11.955 0 Td [(m s x s c sin l 2 t l + m s x s c cos l tt l )]TJ/F26 11.9552 Tf 11.955 0 Td [(EI! yyy l ; M tip = )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(m s x 2 s c 2 + J s tt l + m s x s c cos l tt l + GJ y l 97 PAGE 98 APPENDIXG EXPONENTIALSTABILITYOFTHETARGETSYSTEMCH5 ThetargetsysteminChapter5isgivenas I w tt )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJ yy )]TJ/F26 11.9552 Tf 11.955 0 Td [( GJ tyy + )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(cGJ )]TJ/F15 11.9552 Tf 16.291 3.022 Td [( M w + cGJ t =0 ; G where c 2 R isaconstantcontrolgainandtheboundaryconditionsare ;t =0 and GJ y l;t + GJ ty l;t =0 .SinceGisalinearPDE,itssolutionisassumedto beoftheform y;t = g t h y .SubstitutingtheassumedsolutionintoGyields I w h y g tt t )]TJ/F26 11.9552 Tf 11.955 0 Td [(GJg t h yy y )]TJ/F26 11.9552 Tf 11.955 0 Td [( GJg t t h yy t + )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(cGJ )]TJ/F15 11.9552 Tf 16.291 3.022 Td [( M w g t h y + cGJg t t h y =0 : Gatheringtheliketermsonoppositesidesoftheequationresultsin I w g tt t + )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(cGJ )]TJ/F15 11.9552 Tf 16.291 3.022 Td [( M w g t + cGJg t t )]TJ/F26 11.9552 Tf 9.299 0 Td [(GJg t )]TJ/F26 11.9552 Tf 11.955 0 Td [( GJg t t = )]TJ/F26 11.9552 Tf 10.494 8.088 Td [(h yy y h y : G TheequalityinGcanonlyholdiftheright-handsideandleft-handsideareequal toaconstant .Examiningtheright-handsideofGresultsinthefollowingordinary differentialequationfor h y h yy y + h y =0 G withtheboundaryconditions h =0 and h y l =0 .Thecaseswhere < 0 and =0 leaddirectlytothetrivialsolutioni.e., h y =0 .Anon-trivialsolutiontothecasewhere > 0 existsandisexpressedas h y = a 1 cos p x + a 2 sin p x where a 1 and a 2 2 R areconstantsdeterminedthroughtheapplicationoftheboundaryconditions.Applying theboundaryconditionsyields a 1 =0 and p l = n +1 2 ,where n =0 ; 1 ; 2 ;::: The generalsolutiontoGcanbewrittenas h y = 1 X n =0 A n sin n +1 2 l x ; where A n 2 R isaconstantassociatedwiththe n thparticularsolution. 98 PAGE 99 Examiningtheleft-handsideofGyields I w g tt t + GJ c + g t t + )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( c + GJ )]TJ/F15 11.9552 Tf 16.291 3.022 Td [( M w g t =0 ; whose n thpairofeigenvalues n satisfythefollowingquadraticexpression I w 2 n + GJ c + n n + c + n GJ )]TJ/F15 11.9552 Tf 16.291 3.022 Td [( M w =0 ; where n = n +1 2 2 4 l 2 .The n thpairofeigenvaluescanbeexpressedas n = )]TJ/F26 11.9552 Tf 9.298 0 Td [( GJ c + n q 2 GJ 2 c + n 2 +4 I w M w )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 I w GJ c + n 2 I w : G Forthecaseinwhich 2 GJ 2 c + n 2 +4 I w M w )]TJ/F15 11.9552 Tf 12.749 0 Td [(4 I w GJ c + n =0 ,theresulting eigenvaluesare n = )]TJ/F27 7.9701 Tf 10.494 6.274 Td [( GJ c + n 2 I w .Inthecasewhere 2 GJ 2 c + n 2 +4 I w M w )]TJ/F15 11.9552 Tf -420.366 -23.908 Td [(4 I w GJ c + n < 0 ,theeigenvalueswillbecomplexwith Re n = )]TJ/F27 7.9701 Tf 10.494 6.274 Td [( GJ c + n 2 I w where Re n denotestherealpartof n .Lastly,when 2 GJ 2 c + n 2 +4 I w M w )]TJ/F15 11.9552 Tf -425.193 -23.908 Td [(4 I w GJ c + n > 0 ,theresultingeignevalueswillberealanddistinct.Sincethesquare rootterminGispositive,bothrealeigenvalueswillbenegativeifthefollowing inequalityissatised, )]TJ/F26 11.9552 Tf 9.298 0 Td [( GJ c + n + q 2 GJ 2 c + n 2 +4 I w M w )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 I w GJ c + n < 0 : Aftersomealgebraicmanipulationandrecallingthat n = n +1 2 2 4 l 2 ,thesufcient conditionabovecanbeexpressedas c> M w GJ )]TJ/F15 11.9552 Tf 13.15 8.088 Td [( n +1 2 2 4 l 2 : G As n !1 ,theright-handsideofGgetssmaller;hence,iftheinequalityissatised for n =0 ,itwillbesatisedforall n .Substituting n =0 intoGyieldsthefollowing sufcientcondition c> M w GJ )]TJ/F26 11.9552 Tf 14.417 8.088 Td [( 2 4 l 2 : 99 PAGE 100 Sincealleigenvalueshavenegativerealparts,thetargetsysteminGisexponentiallystable. 100 PAGE 101 APPENDIXH INTEGRATIONBYPARTSOFSELECTTERMSIN E C CH6 Thedevelopmentofanupperboundfor E c reliesontheintegrationbypartsoftheterms )]TJ/F26 11.9552 Tf 9.298 0 Td [( 1 l 0 EI! yyyy y ydy 1 l 0 t ty ydy 1 l 0 GJ yy y ydy ,and 1 l 0 I w + x 2 c c 2 t ty ydy from5.Integrationoftherstterm, )]TJ/F26 11.9552 Tf 9.299 0 Td [( 1 l 0 EI! yyyy y ydy yields )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 l 0 EI! yyyy y ydy = )]TJ/F26 11.9552 Tf 9.298 0 Td [( 1 EIl! yyy l;t y l;t + 1 EI l 0 yyy y dy + 1 EI l 0 yyy yy ydy; )]TJ/F26 11.9552 Tf 9.298 0 Td [( 1 l 0 EI! yyyy y ydy = )]TJ/F26 11.9552 Tf 9.298 0 Td [( 1 EIl! yyy l;t y l;t )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 EI l 0 2 yy dy + 1 EI l 0 yyy yy ydy; H )]TJ/F26 11.9552 Tf 9.298 0 Td [( 1 l 0 EI! yyyy y ydy = )]TJ/F26 11.9552 Tf 9.298 0 Td [( 1 EIl! yyy l;t y l;t )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 EI l 0 2 yy dy )]TJ/F26 11.9552 Tf 9.298 0 Td [( 1 EI l 0 yyy yy ydy: H AfteraddingHtoHandcombiningliketerms, )]TJ/F26 11.9552 Tf 9.298 0 Td [( 1 l 0 EI! yyyy y ydy canbe expressedas )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 l 0 EI! yyyy y ydy = )]TJ/F26 11.9552 Tf 9.299 0 Td [( 1 EIl! yyy l;t y l;t )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(3 2 1 EI l 0 2 yy dy: H Theterms 1 l 0 t ty ydy 1 l 0 GJ yy y ydy ,and 1 l 0 I w + x 2 c c 2 t ty ydy areevaluatedas 1 l 0 t ty ydy = 1 l! 2 t l;t )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 l 0 2 t dy )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 l 0 t ty ydy; H 1 l 0 GJ yy y ydy = 1 GJl 2 y l;t )]TJ/F26 11.9552 Tf 11.956 0 Td [( 1 GJ l 0 2 y dy )]TJ/F26 11.9552 Tf 9.299 0 Td [( 1 l 0 GJ yy y ydy; H 1 l 0 )]TJ/F26 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c c 2 t ty ydy = 1 )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c c 2 l 2 t l;t )]TJ/F26 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c c 2 l 0 2 t dy 101 PAGE 102 )]TJ/F26 11.9552 Tf 9.299 0 Td [( 1 l 0 )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(I w + x 2 c c 2 t ty ydy; H whichaftersomealgebraicmanipulationarerewrittenas 1 l 0 t ty ydy = 1 2 1 l! 2 t l;t )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 2 1 l 0 2 t dy; H 1 l 0 GJ yy y ydy = 1 2 1 GJl 2 y l;t )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 1 GJ l 0 2 y dy; H 1 l 0 )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(I w + x 2 c c 2 t ty ydy = 1 2 1 )]TJ/F26 11.9552 Tf 5.479 -9.683 Td [(I w + x 2 c c 2 l 2 t l;t )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 1 )]TJ/F26 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c c 2 l 0 2 t dy: H 102 PAGE 103 REFERENCES [1]P.S.Beran,T.W.Strganac,K.Kim,andC.Nichkawde,Studiesofstore-induced limitcycleoscillationsusingamodelwithfullsystemnonlinearities, Nonlinear Dynamics ,vol.37,pp.323,2004. 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