Lyapunov-Based Control of Limit Cycle Oscillations in Uncertain Aircraft Systems

MISSING IMAGE

Material Information

Title:
Lyapunov-Based Control of Limit Cycle Oscillations in Uncertain Aircraft Systems
Physical Description:
1 online resource (108 p.)
Language:
english
Creator:
Bialy, Brendan J
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering, Mechanical and Aerospace Engineering
Committee Chair:
DIXON,WARREN E
Committee Co-Chair:
KUMAR,MRINAL
Committee Members:
FITZ-COY,NORMAN G
CRISALLE,OSCAR DARDO

Subjects

Subjects / Keywords:
aeroelasticity -- control -- nonlinear
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Store-induced limit cycle oscillations (LCO) affect several fighter aircraft and is expected to remain an issue for next generation fighters. LCO arises from the interaction of aerodynamic and structural forces, however the major contributor to the phenomenon is still unclear. The major concerns regarding this phenomenon include whether or not ordnance can be safely released and the ability of the aircrew to perform mission-related tasks while in an LCO condition. The focus of this dissertation is the development of control strategies to suppress LCO in aircraft systems. The first contribution of this work (Chapter 2) is the development of a controller consisting of a continuous Robust Integral of the Sign of the Error (RISE) feedback term with a neural network (NN) feedforward term to suppress LCO behavior in an uncertain airfoil system. The second contribution of this work (Chapter 3) is the extension of the development in Chapter 2 to include actuator saturation. Suppression of LCO behavior is achieved through the implementation of an auxiliary error system that features hyperbolic functions and a saturated RISE feedback control structure. Due to the lack of clarity regarding the driving mechanism behind LCO, common practice in literature and in Chapters 2 and 3 is to replicate the symptoms of LCO by including nonlinearities in the wing structure, typically a nonlinear torsional stiffness. To improve the accuracy of the system model a partial differential equation (PDE) model of a flexible wing is derived (see Appendix F) using Hamilton's principle. Chapters 4 and 5 are focused on developing boundary control strategies for regulating the bending and twisting deformations of the derived model. The contribution of Chapter 4 is the construction of a backstepping-based boundary control strategy for a linear PDE model of an aircraft wing. The backstepping-based strategy transforms the original system to a exponentially stable system. A Lyapunov-based stability analysis is then used to to show boundedness of the wing bending dynamics. A Lyapunov-based boundary control strategy for an uncertain nonlinear PDE model of an aircraft wing is developed in Chapter 5. In this chapter, a proportional feedback term is coupled with an gradient-based adaptive update law to ensure asymptotic regulation of the flexible states.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Brendan J Bialy.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: DIXON,WARREN E.
Local:
Co-adviser: KUMAR,MRINAL.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2014
System ID:
UFE0046604:00001


This item is only available as the following downloads:


Full Text

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. [2]R.W.BuntonandC.M.DenegriJr.,Limitcycleoscillationcharacteristicsofghter aircraft, J.Aircraft ,vol.37,pp.916,2000. [3]C.M.DenegriJr.,Limitcycleoscillationighttestresultsofaghterwithexternal stores, J.Aircraft ,vol.37,pp.761,2000. [4]J.J.BlockandT.W.Strganac,Appliedactivecontrolforanonlinearaeroelastic structure, J.Guid.Contr.Dynam. ,vol.21,pp.838,1999. [5]J.Ko,T.W.Strganac,andA.Kurdila,Stabilityandcontrolofastructurallynonlinearaeroelasticsystem, J.Guid.Contr.Dynam. ,vol.21,pp.718,1998. [6]W.ZhangandZ.Ye,Controllawdesignfortransonicaeroservoelasticity, AerospaceScienceandTechnology ,vol.11,pp.136,2007. [7]B.P.Danowsky,P.M.Thompson,C.Farhat,T.Lieu,C.Harris,andJ.Lechniak, Incorporationoffeedbackcontrolintoahigh-delityaeroservoelasticghteraircraft model, J.Aircraft ,vol.47,pp.1274,2010. [8]P.M.Thompson,B.P.Danowsky,C.Farhat,T.Lieu,J.Lechniak,andC.Harris, High-delityaeroservoelasticpredictiveanalysiscapabilityincorporatingrigidbody dynamics,in Proc.AIAAAtmosphericFlightMechanics,AIAA2011-6209 ,2011. [9]L.Cavagna,S.Ricci,andA.Scotti,Activeaeroelasticcontroloverafourcontrol surfacewingmodel, AerospaceScienceandTechnology ,vol.13,pp.374, 2009. [10]Z.Prime,B.Cazzolato,C.Doolan,andT.Strganac,Linear-parameter-varying controlofanimprovedthree-degree-of-freedomaeroelasticmodel, J.Guid.Contr. Dynam. ,vol.33,pp.615,2010. [11]M.R.ElhamiandM.F.Narab,ComparisonofSDREandSMCcontrolapproaches foruttersuppressioninanonlinearwingsection,in Proc.Am.ControlConf. 2012,pp.6148. [12]J.Ko,T.W.Strganac,andA.Kurdila,Adaptivefeedbacklinearizationforthe controlofatypicalwingsectionwithstructuralnonlinearity, NonlinearDynamics vol.18,pp.289,1999. [13]T.W.Strganac,J.Ko,D.E.Thompson,andA.Kurdila,Identicationandcontrolof limitcycleoscillationsinaeroelasticsystems, J.Guid.Contr.Dynam. ,vol.23,pp. 1127,2000. 103

PAGE 104

[14]J.Ko,T.W.Strganac,J.L.Junkins,M.R.Akella,andA.Kurdila,Structuredmodel referenceadaptivecontrolforawingsectionwithstructuralnonlinearity, J.Vib. Control ,vol.8,pp.553,2002. [15]G.PlatanitisandT.W.Strganac,Controlofanonlinearwingsectionusingleadingandtrailing-edgesurfaces, J.Guid.Contr.Dynam. ,vol.27,pp.52,2004. [16]N.Fischer,Z.Kan,R.Kamalapurkar,andW.E.Dixon,SaturatedRISEfeedback controlforaclassofsecond-ordernonlinearsystems, IEEETrans.Automat. Control ,toappear. [17]D.H.HodgesandR.A.Ormiston,Stabilityofelasticbendingandtorsionof uniformcantileverrotorbladesinhoverwithvariablestructuralcoupling,National AeronauticsandSpaceAdministration,TechnicalNoteD-8192,1976. [18]M.Y.ZiabariandB.Ghadiri,Vibrationanalysisofelasticuniformcantileverrotor bladesinunsteadyaerodynamicsmodeling, J.Aircraft ,vol.47,no.4,pp.1430 1434,2010. [19]D.H.HodgesandE.H.Dowell,Nonlinearequationsofmotionfortheelastic bendingandtorsionoftwistednonuniformrotorblades,NationalAeronauticsand SpaceAdministration,TechnicalNoteD-7818,1974. [20]X.Zhang,W.Xu,S.S.Nair,andV.Chellaboina,Pdemodelingandcontrolofa exibletwo-linkmanipulator, IEEETrans.Contr.Syst.Tech. ,vol.13,no.2,pp. 301,2005. [21]Y.Morita,F.Matsuno,Y.Kobayashi,M.Ikeda,H.Ukai,andH.Kando,Lyapunovbasedforcecontrolofaexiblearmconsideringbendingandtorsionaldeformation,in Proc.ofthe15thTriennialIFACWorldCongress ,2002. [22]J.Martins,Z.Mohamed,M.Tokhi,J.SdaCosta,andM.Botto,Approachesfor dynamicmodellingofexiblemanipulatorsystems, IEEProc.Contr.Theor.Appl. vol.150,no.4,pp.401,2003. [23]P.D.ChristodesandP.Daoutidis,Finite-dimensionalcontrolofparabolicpde systemsusingapproximateinertialmanifolds, J.Math.Anal.Appl. ,vol.216,pp. 398,1997. [24]A.Shawky,A.Ordys,andM.Grimble,End-pointcontrolofaexible-linkmanipulatorusing H 1 nonlinearcontrolviaastate-dependentriccatiequation,in 2002IEEE InternationalConferenceonControlApplications ,2002. [25]L.Meirovitch, AnalyticalMethodsinVibrations .NewYork,NY,USA:TheMacmillan Company,1967. [26]F.BucciandI.Lasiecka,Optimalboundarycontrolwithcriticalpenalizationfora PDEmodelofuid-solidinteractions, Calc.Var. ,vol.37,pp.217,2010. 104

PAGE 105

[27]C.I.Byrnes,I.G.Lauk,D.S.Gilliam,andV.I.Shubov,Outputregulationfor lineardistributedparametersystems, IEEETrans.Autom.Contr. ,vol.45,pp. 2236,2000. [28]Z.-H.Luo,Directstrainfeedbackcontrolofexiblerobotarms:Newtheoreticaland experimentalresults, IEEETrans.Autom.Contr. ,vol.38,pp.1610,1993. [29]Z.-H.LuoandB.Guo,Furthertheoreticalresultsondirectstrainfeedbackcontrol ofexiblerobotarms, IEEETrans.Autom.Contr. ,vol.40,pp.747,1995. [30]M.J.Balas,Feedbackcontrolofexiblesystems, IEEETrans.Autom.Contr. ,vol. AC-23,pp.673,1978. [31]L.MeirovitchandH.Baruh,Ontheproblemofobservationspilloverinself-adjoint distributed-parametersystems, J.Optim.TheoryApp. ,vol.39,pp.269,1983. [32]M.J.Balas,Trendsinlargespacestructurecontroltheory:Fondesthopes,wildest dreams, IEEETrans.Autom.Contr. ,vol.AC-27,pp.522,1982. [33]M.KrsticandA.Smyshlyaev, BoundarycontrolofPDEs:AcourseonBackstepping Designs .SIAM,2008. [34]M.S.deQueiroz,D.M.Dawson,S.P.Nagarkatti,andF.Zhang, Lyapunov-Based ControlofMechanicalSystems .Birkhauser,2000. [35]M.S.deQueirozandC.D.Rahn,Boundarycontrolofvibrationandnoisein distributedparametersystems:Anoverview, Mech.Syst.SignalProcess. ,vol.16, pp.19,2002. [36]R.VazquezandM.Krstic,Aclosed-formfeedbackcontrollerforstabilizationofthe linearized2-dnavier-stokespoiseuillesystem, IEEETrans.Autom.Contr. ,vol.52, pp.2298,2007. [37]M.S.deQueiroz,D.M.Dawson,M.Agarwal,andF.Zhang,Adaptivenonlinear boundarycontrolofaexiblelinkrobotarm, IEEETrans.Robot.Autom. ,vol.15, pp.779,1999. [38]M.P.FardandS.I.Sagatun,Exponentialstabilizationofatransverselyvibrating beambyboundarycontrolvialyapunov'sdirectmethod, J.Dyn.Syst.Meas. Contr. ,vol.123,pp.195,2001. [39]W.He,S.S.Ge,B.V.E.How,Y.S.Choo,andK.-S.Hong,Robustadaptive boundarycontrolofaexiblemarineriserwithvesseldynamics, Automatica vol.47,pp.722,2011. [40]A.A.Siranosian,M.Krstic,A.Smyshlyaev,andM.Bement,Gainschedulinginspiredboundarycontrolfornonlinearpartialdifferentialequations, J.Dyn.Syst. Meas.Contr. ,vol.133,p.051007,2011. 105

PAGE 106

[41]S.P.Nagarkatti,D.M.Dawson,M.S.deQueiroz,andB.Costic,Boundarycontrol ofatwo-dimensionalexiblerotor, Int.J.AdaptControlSignalProcess. ,vol.15, pp.589,2001. [42]A.A.Paranjape,J.Guan,S.-J.Chung,andM.Krstic,Pdeboundarycontrolfor exiblearticulatedwingsonaroboticaircraft, IEEETransactionsonRobotics vol.29,no.3,pp.625,2013. [43]P.M.Patre,W.MacKunis,K.Kaiser,andW.E.Dixon,Asymptotictrackingfor uncertaindynamicsystemsviaamultilayerneuralnetworkfeedforwardandRISE feedbackcontrolstructure, IEEETrans.Automat.Control ,vol.53,no.9,pp. 2180,2008. [44]P.Patre,W.Mackunis,K.Dupree,andW.E.Dixon,Modularadaptivecontrol ofuncertainEuler-Lagrangesystemswithadditivedisturbances, IEEETrans. Automat.Control ,vol.56,no.1,pp.155,2011. [45]W.MacKunis,P.Patre,M.Kaiser,andW.E.Dixon,Asymptotictrackingforaircraft viarobustandadaptivedynamicinversionmethods, IEEETrans.ControlSyst. Technol. ,vol.18,no.6,pp.1448,2010. [46]D.E.ThompsonJr.andT.W.Strganac,Nonlinearanalysisofstore-inducedlimit cycleoscillations, NonlinearDynamics ,vol.39,pp.159,2005. [47]M.Krstic,P.V.Kokotovic,andI.Kanellakopoulos, NonlinearandAdaptiveControl Design .JohnWiley&Sons,1995. [48]J.P.AubinandH.Frankowska, Set-valuedanalysis .Birkh a user,2008. [49]A.Filippov,Differentialequationswithdiscontinuousright-handside, Am.Math. Soc.Transl. ,vol.42no.2,pp.199,1964. [50]A.F.Filippov, DifferentialEquationswithDiscontinuousRight-handSides .Kluwer AcademicPublishers,1988. [51]G.V.Smirnov, Introductiontothetheoryofdifferentialinclusions .American MathematicalSociety,2002. [52]D.ShevitzandB.Paden,Lyapunovstabilitytheoryofnonsmoothsystems, IEEE Trans.Autom.Control ,vol.39no.9,pp.1910,1994. [53]B.PadenandS.Sastry,AcalculusforcomputingFilippov'sdifferentialinclusion withapplicationtothevariablestructurecontrolofrobotmanipulators, IEEETrans. CircuitsSyst. ,vol.34no.1,pp.73,1987. [54]R.Kamalapurkar,J.Klotz,R.Downey,andW.E.Dixon.Supportinglemmas forRISE-basedcontrolmethods.arXiv:1306.3432v2. 106

PAGE 107

[55]N.Fischer,R.Kamalapurkar,andW.E.Dixon,LaSalle-Yoshizawacorollariesfor nonsmoothsystems, IEEETrans.Automat.Control ,vol.58,no.9,pp.2333, 2013. [56]W.E.Dixon,M.S.deQueiroz,D.M.Dawson,andF.Zhang,Trackingcontrolof robotmanipulatorswithboundedtorqueinputs, Robotica ,vol.17,pp.121, 1999. [57]F.H.Clarke, Optimizationandnonsmoothanalysis .SIAM,1990. [58]W.Rudin, PrinciplesofMathematicalAnalysis .McGraw-Hill,1976. [59]Z.Qu, RobustControlofNonlinearUncertainSystems .WileyInc.,New-York, 1998. 107

PAGE 108

BIOGRAPHICALSKETCH BrendanBialywasborninBinghamton,NewYork.HereceivedaBachelorof SciencedegreeinaeronauticalandmechanicalengineeringfromClarksonUniversity in2010.Aftercompetinghisundergraduatedegree,BrendandecidedtopursuedoctoralresearchundertheadvisementofDr.WarrenDixonattheUniversityofFlorida. BrendanearnedaMasterofSciencedegreeinDecemberof2012andcompletedhis Ph.D.inMayof2014,bothinaerospaceengineeringandfocusedonnonlinearcontrol ofuncertainaircraftsystems.Additionally,Brendanhasworkedasastudentresearcher atNASALangleyResearchCenterinHampton,VirginiaandattheAirForceResearch Laboratory,MunitionsDirectorateatEglinAFB,Florida. 108