<%BANNER%>

Nonlinear Control for Systems Containing Input Uncertainty Via a Lyapunov-Based Approach

Permanent Link: http://ufdc.ufl.edu/UFE0024423/00001

Material Information

Title: Nonlinear Control for Systems Containing Input Uncertainty Via a Lyapunov-Based Approach
Physical Description: 1 online resource (129 p.)
Language: english
Creator: Mackunis, William
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: adaptive, aircraft, control, lyapunov, nonlinear, robust, satellite
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: In this dissertation, a tracking control methodology is proposed for aircraft and aerospace systems for which the dynamic models contain uncertainty in the control actuation. The dissertation focuses on five problems of interest: 1) adaptive control moment gyroscope (CMG)-actuated satellite attitude control in the presence of inertia and CMG gimbal friction uncertainty; 2) adaptive neural network (NN)-based satellite attitude control in the presence of uncertain gimbal friction and satellite inertia, nonlinear disturbance torques, and nonlinear electromechanical CMG disturbances; 3) dynamic inversion control for aircraft systems containing input uncertainty and nonlinear disturbances; 4) adaptive dynamic inversion control for aircraft systems as described in 3; and 5) adaptive output feedback control for aircraft systems as described in 3 and 4.
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 William Mackunis.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024423:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024423/00001

Material Information

Title: Nonlinear Control for Systems Containing Input Uncertainty Via a Lyapunov-Based Approach
Physical Description: 1 online resource (129 p.)
Language: english
Creator: Mackunis, William
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: adaptive, aircraft, control, lyapunov, nonlinear, robust, satellite
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: In this dissertation, a tracking control methodology is proposed for aircraft and aerospace systems for which the dynamic models contain uncertainty in the control actuation. The dissertation focuses on five problems of interest: 1) adaptive control moment gyroscope (CMG)-actuated satellite attitude control in the presence of inertia and CMG gimbal friction uncertainty; 2) adaptive neural network (NN)-based satellite attitude control in the presence of uncertain gimbal friction and satellite inertia, nonlinear disturbance torques, and nonlinear electromechanical CMG disturbances; 3) dynamic inversion control for aircraft systems containing input uncertainty and nonlinear disturbances; 4) adaptive dynamic inversion control for aircraft systems as described in 3; and 5) adaptive output feedback control for aircraft systems as described in 3 and 4.
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 William Mackunis.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024423:00001


This item has the following downloads:


Full Text

PAGE 1

1

PAGE 2

2

PAGE 3

3

PAGE 4

Iwouldliketoexpresssinceregratitudetomyadvisor,Dr.WarrenE.Dixon,whoseexperienceandmotivationwereinstrumentalinmyrecentacademicsuccess.Asanadvisor,heprovidedguidanceinmyresearchandencouragementindevelopingmyownideas.Asamentor,hehelpedmeunderstandtheintricaciesofworkinginaprofessionalenvironmentandhelpeddevelopmyprofessionalskills.Ifeelfortunatetohavehadtheopportunitytoworkwithhim.IwouldalsoliketoextendmygratitudetoDr.NormanFitz-Coyforhistechnicalassistanceintheaerospaceaspectsofmyresearch.IalsoappreciatemycommitteemembersDr.CarlD.CraneIIIandDr.J.Hammerforanytimeandhelptheyprovided.Iwouldliketothankmycoworkers,family,andfriendsfortheirsupportandencouragement. 4

PAGE 5

page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTIONANDMOTIVATION ...................... 12 1.1SatelliteAttitudeControl ........................... 12 1.2AircraftControl ................................. 15 1.3ResearchPlan .................................. 19 1.3.1ContributionsofCompletedResearch ................. 19 1.3.2LimitationsofCompletedResearch .................. 20 1.3.3FutureResearchPlans ......................... 20 1.3.4ResearchSchedule ............................ 20 2ADAPTIVESATELLITEATTITUDECONTROLINTHEPRESENCEOFINERTIAANDCMGGIMBALFRICTIONUNCERTAINTIES ......... 22 2.1Introduction ................................... 22 2.2DynamicModel ................................. 22 2.3KinematicModel ................................ 25 2.4ControlObjective ................................ 26 2.5AdaptiveControlDevelopment ........................ 27 2.5.1TrackingErrorDynamics ........................ 27 2.5.2StabilityAnalysis ............................ 31 2.6AsymptoticTrackingExtension ........................ 33 2.6.1Closed-LoopErrorSystem ....................... 33 2.6.2StabilityAnalysisIgnoringStaticFriction ............... 33 2.7SimulationResults ............................... 34 2.8ConclusionsandFutureWork ......................... 37 3ADAPTIVENEURALNETWORKSATELLITEATTITUDECONTROLINTHEPRESENCEOFINERTIAANDCMGACTUATORUNCERTAINTIES 39 3.1Introduction ................................... 39 3.2DynamicModelandProperties ........................ 39 3.3KinematicModel ................................ 42 3.4ControlObjective ................................ 43 3.5FeedforwardNNEstimation .......................... 45 3.6ControlDevelopment .............................. 46 5

PAGE 6

........................ 47 3.6.2Closed-LoopErrorSystem ....................... 49 3.6.3StabilityAnalysis ............................ 52 3.7SimulationResults ............................... 54 3.8Conclusion .................................... 57 4ASYMPTOTICTRACKINGFORAIRCRAFTVIAANUNCERTAINDYNAMICINVERSIONMETHOD ............................... 59 4.1Introduction ................................... 59 4.2AircraftModelandProperties ......................... 59 4.3ControlDevelopment .............................. 63 4.3.1Open-loopErrorSystem ........................ 64 4.3.2Closed-loopErrorSystem ........................ 66 4.4StabilityAnalysis ................................ 67 4.5SimulationResults ............................... 70 4.6Conclusion .................................... 75 5ADAPTIVEDYNAMICINVERSIONFORASYMPTOTICTRACKINGOFANAIRCRAFTREFERENCEMODEL ...................... 79 5.1Introduction ................................... 79 5.2AircraftModel ................................. 79 5.3ControlDevelopment .............................. 81 5.4StabilityAnalysis ................................ 85 5.5SimulationResults ............................... 88 5.6Conclusion .................................... 94 6GLOBALADAPTIVEOUTPUTFEEDBACKMRAC .............. 98 6.1Introduction ................................... 98 6.2SystemModel .................................. 98 6.3ControlDevelopment .............................. 99 6.3.1ControlObjective ............................ 99 6.3.2Open-LoopErrorSystem ........................ 100 6.3.3Closed-LoopErrorSystem ....................... 101 6.4StabilityAnalysis ................................ 106 6.5SimulationResults ............................... 109 6.6Conclusion .................................... 114 7CONTRIBUTIONSANDFUTURERESEARCHPLANS ............ 116 7.1ContributionsofPreviousResearch ...................... 116 7.2LimitationsofPreviousResearch ....................... 116 7.3ProposedResearchPlans ............................ 117 APPENDIX:PROOFOFLEMMAS4-1AND6-1 ................... 118 6

PAGE 7

....................................... 122 BIOGRAPHICALSKETCH ................................ 129 7

PAGE 8

Table page 4-1ParametersusedintheDIcontrollersimulation. .................. 74 5-1ParametersusedintheADIcontrollersimulation. ................. 90 6-1ParametersUsedintheControllerSimulations. .................. 113 8

PAGE 9

Figure page 1-1ResearchSchedule .................................. 21 2-1TheUniversityofFloridacontrolmomentgyroscopeexperimentaltestbed. ... 24 2-2Quaterniontrackingerror. .............................. 36 2-3Controlinputgimbalangularrateresponse. .................... 37 2-4_J()vs.time. ..................................... 38 3-1Quaterniontrackingerrorofclosed-loopsystem. .................. 57 3-2Controlinputgimbalangularrates. ......................... 58 3-3Timevariationoftheinertiamatrix(i.e.,_J())duringclosed-loopoperation. .. 58 4-1PhotographoftheOspreyaircrafttestbed. ..................... 60 4-2Plotofthediscretevertical(upward)windgustusedinthecontrollersimulation. 72 4-3Pitchrateresponseachievedduringclosed-looplongitudinalcontrolleroperation. ............................................. 75 4-4Forwardvelocityresponseachievedduringclosed-looplongitudinalcontrolleroperation. ...................................... 76 4-5Rollrateresponseachievedduringclosed-looplateralcontrolleroperation. ... 77 4-6Yawrateresponseachievedduringclosed-looplateralcontrolleroperation. ... 77 4-7Controlactuationawayfromtrimusedduringclosed-looprobustdynamicinversioncontrolleroperationforthelateralsubsystem(top)andthelongitudinalsubsystem(bottom) ........................................ 78 5-1Pitchrateresponseachievedduringclosed-looplongitudinalcontrolleroperation. 91 5-2Forwardvelocityresponseachievedduringclosed-looplongitudinalcontrolleroperation. ....................................... 92 5-3Rollrateresponseachievedduringclosed-looplateralcontrolleroperation. ... 93 5-4Yawrateresponseachievedduringclosed-looplateralcontrolleroperation. ... 94 5-5Controlactuationawayfromtrimusedduringclosed-loopadaptivedynamicinversioncontrolleroperationforthelateralsubsystem(top)andthelongitudinalsubsystem(bottom). ................................. 95 9

PAGE 10

.................. 96 5-7Controlactuationawayfromtrimusedduringclosed-loopadaptivecontrolleroperationfortheangleofattacktrackingobjective.Elevatordeectionangle(top)andthrust(bottom). .............................. 97 6-1Referenceandactualforwardvelocity(topleft),pitchrate(topright),rollrate(bottomleft),andyawrate(bottomright)responsesduringclosed-looplongitudinalandlateralcontrolleroperation. ........................... 114 6-2Controlinputelevatordeection(topleft),thrust(topright),ailerondeection(bottomleft),andrudderdeection(bottomright)usedduringclosed-looplongitudinalandlateralcontrolleroperation. ........................... 115 10

PAGE 11

AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy NONLINEARCONTROLFORSYSTEMSCONTAININGINPUTUNCERTAINT YVIA ALYAPUNOV-BASEDAPPROACH By WilliamMacKunis May2009 Chair:Dr.WarrenE.DixonMajor:AerospaceEngineering Controllersareoftendesignedbasedontheassumptionthat acontrolactuationcan bedirectlyappliedtothesystem.Thisassumptionmaynotbe valid,however,forsystems containingparametricinputuncertaintyorunmodeledactu atordynamics. Inthisdissertation,atrackingcontrolmethodologyispro posedforaircaftand aerospacesystemsforwhichthecorrespondingdynamicmode lscontainuncertainty inthecontrolactuation.Thedissertationwillfocusonve problemsofinterest:1) adaptiveCMG-actuatedsatelliteattitudecontrolinthepr esenceofinertiauncertainty anduncertainCMGgimbalfriction;2)adaptiveneuralnetwo rk(NN)-basedsatellite attitudecontrolforCMG-actuatedsmall-satsinthepresen ceofuncertainsatellite inertia,nonlineardisturbancetorques,uncertainCMGgim balfriction,andnonlinear electromechanicalCMGactuatordisturbances;3)dynamici nversion(DI)controlfor aircraftsystemscontainingparametricinputuncertainty andadditive,nonlinearly parameterizable(non-LP)disturbances;4)adaptivedynam icinversion(ADI)controlfor aircraftsystemsasdescribedin3);and5)adaptiveoutputf eedbackcontrolforaircraft systemsasdescribedin3)and4). 11

PAGE 12

1 ],thespaceindustryismovingtowardsmallersatellitesandthebusesthatsupportthem.Someproposedusesofthesesmallsatellites(small-sats)includeastrophysicsresearch,surveillance,andautonomousservicing,allofwhichrequireprecisionattitudemotion.However,duetotheirsmallersizes,theattitudemotionofthesesmall-satsismoresusceptibletoexternaldisturbancesthantheirlargercounterparts.Furthermore,thesmallersizesofthesenewsmall-satslimitthemass,powerandsizebudgetsallocatedtotheirattitudecontrolsystems(ACS).ThesecontradictoryrequirementsnecessitatenovelsolutionsfortheACS.Controllersthatarebasedontheassumptionthatatorquecanbedirectlyappliedaboutthebody-xedsatelliteaxes(e.g.,[ 2 { 5 ])maynotbewellsuitedforapplicationsthatrequirehigh-precisionattitudecontrol,becausethesatellitetorquesaregeneratedbyactuatorswithadditionaldynamics.Forexample,(especiallyinsmallrigid-bodysatellites),thedesiredtorquesaretypicallygeneratedbyacluster(e.g.,[ 6 7 ])ofsinglegimbalcontrolmomentgyroscopes(CMGs)duetotheirlowmassandlowpowerconsumptionproperties.Unfortunately,thetorqueproducingcapacityofCMGscandeteriorateovertimeduetochangesinthedynamicssuchasbearingdegradationandincreasedfrictioninthegimbals.TheramicationsofCMGfrictionbuild-upincludeincreasedpowerconsumptionduetoenergydissipation.ExamplesofactualsatellitefailuresresultingfromCMGproblemsaretheHipparcossatelliteandMagellansatellite[ 8 ].Hipparcosfailedand\spundown"duetonumerousgyroscopefailures.Oneofthesefailureswasduetohighandvariabledragtorqueingyronumber4,whichledtoprematuredegradation.TheMagellansatellitewasintransittoVenusforvemonthsbeforeitbeganexhibitingerraticmotorcurrentshiftsinoneofitsgyros[ 8 ].Thecauseof 12

PAGE 13

9 { 17 ].In[ 14 ],anattitudecontrolapproachbasedontheradialbasisfunctionneuralnetwork(RBFNN)isdeveloped.Thesatellitedynamicmodelutilizedin[ 14 ]includesnoctioneectsordisturbancesinthereactionwheelactuators.AnotherNNattitudecontrollerispresentedin[ 17 ],whichutilizesNNstoapproximatetheparametricuncertaintiesandnonlinearitiespresentinthesystemdynamics.AnonlineNNisusedin[ 17 ]tore-optimizeaSingleNetworkAdaptiveCritic,orSNAC-basedoptimalcontroller,whichhasbeendesignedaprioriforthenominalsystem.In[ 9 ],aNNattitudecontrollerisdevelopedbasedonasimpliednonlinearmodeloftheSpaceStationFreedom.Thedynamicmodelforthespacestationconsideredin[ 9 ]issimpliedbyassumingsmallroll/yawattitudeerrorsandsmallproductsofinertia.Theattitudecontrollerin[ 9 ]demonstratesthecapabilityoftheNNtoadaptivelycompensateforvaryinginertiacharacteristics.TheNNcontrollerspresentedin[ 9 ]and[ 17 ]aretestedinattitudecontrolproblemsundertheassumptionthatacontroltorquecanbedirectlyappliedaboutthespacecraftbody-xedaxes.Adaptivesatelliteattitudecontrolisoftenutilizedtocopewithsystemscontainingconstantparametricuncertainty.In[ 18 ],anoutputfeedbackstructuredmodelreferenceadaptivecontroller(MRAC)isdevelopedforspacecraftrendezvousanddockingproblems.Theadaptivecontrollerin[ 18 ]accommodatesinertiauncertaintyinthemomentumwheelactuatordynamics;however,nofrictionaleectswereassumedtobepresentintheactuatormodel.Aquaternion-based,full-statefeedbackattitudetrackingcontrollerwasdesignedin[ 2 ]forarigidsatelliteinthepresenceofanunknownsatelliteinertiamatrix.Amodel-errorcontrolsynthesis(MECS)approachwasusedin[ 3 ]tocanceltheeects 13

PAGE 14

3 ]requiresamodel-errortermtocanceltheeectsofatimedelay,whichisinherenttotheMECSdesign.Anadaptivecontrollawisdesignedin[ 19 ],whichincorporatesavelocity-generatinglterfromattitudemeasurements.Thecontrollerin[ 19 ]isshowntoachieveasymptoticconvergenceoftheattitudeandangularvelocitytrackingerrorsdespiteuncertaintyinthesatelliteinertia,butitassumesnodynamicuncertaintyinthecontroltorque.Whiletheaforementionedcontrollersperformwellforapplicationsinvolvinglargesatellites,theymaynotbewellsuitedforattitudecontrolofCMG-actuatedsmall-sats.InChapter2,amoresuitablecontroldesignforsuchsmall-satsisdeveloped.AnonlinearadaptivecontrollerisdevelopedinChapter2thatcompensatesforinertiauncertaintiesanduncertainCMGgimbalfriction.Insteadofdevelopingacontroltorquetosolvetheattitudetrackingproblem,theattitudetrackingcontrollerinChapter2isdevelopedintermsoftheCMGgimbalangularvelocity.Thedevelopmentiscomplicatedbythefactthatthecontrolinputismultipliedbyatime-varying,nonlinearuncertainmatrix.Additionalcomplicationsarisebecausethegimbalvelocitycontroltermisembeddedinsideofadiscontinousnonlinearity(i.e.,thestandardsignumfunction)resultingfromtheCMGstaticfrictioneects.Arobustcontrolmethodisusedtomitigatethedisturbanceresultingfromthestaticfriction.Inaddition,potentialsingularitiesmayexistintheJacobianthattransformsthetorqueproducedbyeachCMGtodesiredtorquesaboutthesatellitecoordinateframe[ 20 ].ThesingularityproblemiscircumventedbytheuseofaparticularJacobianpseudoinverse,coinedthesingularityrobuststeeringlaw,whichwasintroducedin[ 21 ],andhasbeenimplementedinseveralaerospacevehicles(e.g.,see[ 20 ]and[ 22 ]).Auniformlyultimatelybounded(UUB)stabilityresultisprovenviaLyapunovanalysisforthecaseinwhichbothstaticanddynamicfrictioneectsareincludedintheCMGdynamicmodel.Anasymptotictrackingextensionisthenformulatedforthecasewherestaticfrictioneectsareignored. 14

PAGE 15

23 ]),unknownelectromechanicaldisturbancesareassumedtobepresentintheCMGactuators.Someofthechallengesencounteredinthecontroldesignarethatthecontrolinput(i.e.,CMGgimbalangularrate)is:premultipliedbyanon-square,time-varying,nonlinearuncertainmatrixduetodynamicgimbalfrictionandelectromechanicaldisturbances;andisembeddedinahardnonlinearityduetostaticgimbalfriction.Furthermore,duetothesmallsizeofthesatelliteconsideredinthisdevelopment,themotionoftheCMGscausessignicanttime-variationinthesatelliteinertiacharacteristics.Thetime-variationofthesatelliteinertiamanifestsitselfasanonlineardisturbancetorqueinthesatellitedynamicmodel,whichishandledviaonlineNNapproximation.Simulationresultsareprovidedtoillustratetheecacyoftheproposedcontroldesign. 24 { 34 ].Forexample,ageneraldynamicinversionapproachispresentedin[ 27 ]forareferencetrackingproblemforaminimum-phaseandleft-invertiblelinearsystem.Adynamicinversioncontrollerisdesignedforanonminimum-phasehypersonicaircraftsystemin[ 25 ],whichutilizesanadditionalcontrollertostabilizethezerodynamics.Anite-timestabilizationdesignisproposedin[ 26 ],whichutilizesdynamicinversion.Thetechniquein[ 26 ]requiredtheinputmatrixtobefullrank.Typically,dynamicinversionmethods(e.g., 15

PAGE 16

24 25 ])assumethecorrespondingplantmodelsareexactlyknown.However,parametricuncertainty,additivedisturbances,andunmodeledplantdynamicsarealwayspresentinpracticalsystems.AdditionaldicultiesexistindesigningADIcontrollersforsystemscontaininguncertaintyintheinputmatrix.Whilerobustcontrolmethodsareoftenutilizedtocompensatefortheinversionerrorinsuchcases[ 35 { 38 ],therequiredcontroleortcanbelargeduetothehighgainorhighfrequencyfeedbacktypicallyrequiredintherobustcontroldesign.ThereremainsaneedforanADIcontroller,whichiscapableofachievingasymptotictrackingforsystemscontainingparametricuncertaintyandunknownnonlineardisturbanceswhileminimizingtherequiredcontroleort.RobustdesignmethodsareoftenutilizedinDIcontrollerstocompensateforparametricuncertaintyandinversionerror(e.g.,see[ 35 { 38 ]).InChapter4[ 35 ],abest-guessfeedforwardestimatefortheparametericuncertaintyisusedinconjunctionwitharobustcontroltermtocompensateforthecorrespondinginversionerror.In[ 36 ],astochasticrobustdynamicinversiontechniqueisappliedtoanonlinearaircraftmodelathighangleofattack.Thecontrollerin[ 36 ]isdesignedtocompensateforuncertaintiesintheaerodynamicparameters,andisapplicabletosystemsforwhichthenominalmodelisfeedbacklinearizeable.In[ 37 ],arobusttrajectorytrackingcontrollerisdesignedforanunmannedaerialvehicle(UAV)usingatwo-time-scaleddynamicinversionmethod.Thecontrollersin[ 36 ]and[ 37 ]arebasedontheassumptionthatonesubsetofthestatecomponentsevolvesmuchfasterthantheothersubset.Asliding-modecontrollerisdesignedin[ 39 ]foranagilemissilemodelcontainingaerodynamicuncertainty.Thescalarinputuncertaintyin[ 39 ]wasboundedanddampedoutthroughadiscontinuoussliding-modecontrolelement.Adiscontinuousslidingmodecontrollerwasalsodevelopedin[ 38 ]forattitudetrackingofanunpoweredyingvehiclewithanuncertaincolumndecientnon-symmetricinputmatrix.Inourpreviousworkin[ 35 ],acontinuousrobustcontrollerwasusedtoachievesemi-globalasymptotictrackingofanaircraftreferencemodelwheretheaircraftdynamicscontainedcolumndecientinput 16

PAGE 17

27 29 30 33 ]).Typically,ADImethodsMRACtechniqueswherethedesiredinput-outputbehavioroftheclosed-loopsystemisgivenviathecorrespondingdynamicsofareferencemodel[ 28 30 40 ].Therefore,thebasictaskistodesignacontrollerwhichwillensuretheminimalerrorbetweenthereferencemodelandtheplantoutputsdespiteuncertaintiesintheplantparametersandworkingconditions.In[ 41 ],afull-statefeedbackadaptivecontroldesignwaspresentedforageneralclassoffully-actuatednonlinearsystemscontainingstate-varyinginputuncertaintyandanonlineardisturbancethatislinearintheuncertainty.TheADIdesignin[ 41 ]utilizesamatrixdecompositiontechnique[ 42 43 ]toyieldaglobalasymptotictrackingresultwhentheinputuncertaintyisassumedtobesquareandpositivedenite.Asemi-globalMIMOextensionisalsoprovidedin[ 41 ]usingarobustcontrollerforthecasewhentheinputmatrixuncertaintyissquare,positivedenite,andsymmetric.Afull-statefeedbackadaptivecontrollerisdevelopedin[ 44 ],whichcompensatesforparametricuncertaintyinalinearlyparametrizeablenonlinearityandasquareinputgainmatrix.Theapproachin[ 44 ]appliesamatrixdecompositiontechniquetoavoidsingularitiesinthecontrollaw.Thetechniquein[ 44 ]isextendedtoanadaptiveoutputfeedbackcontrollerin[ 45 ]viatheuseofstateestimators.Anadaptivetrackingcontrollerisdevelopedin[ 46 ]fornonlinearrobotsystemswithkinematic,dynamicandactuatoruncertaintieswheretheinputuncertaintyisaconstantdiagonalmatrix.Inourpreviousworkin[ 47 ],anADIcontroller 17

PAGE 18

47 ]dependsontheoutputstatesandtherespectivetimederivatives.Severaleorts(e.g.,[ 31 { 33 48 { 51 ])havebeendevelopedforthemoregeneralproblemwheretheuncertainparametersortheinversionmismatchtermsdonotsatisfythelinear-in-the-parametersassumption(i.e.,non-LP).Onemethodtocompensatefornon-LPuncertaintyistoexploitaneuralnetworkasanon-linefunctionapproximationmethodasin[ 48 { 50 ];however,alloftheseresultsyielduniformlyultimatelyboundedstabilityduetotheinherentfunctionreconstructionerror.Incontrasttoneuralnetwork-basedmethodstocompensateforthenon-LPuncertainty,arobustcontrolapproachwasrecentlydevelopedin[ 52 ]thatexploitsauniqueintegralofthesignoftheerror(coinedRISEcontrolin[ 53 ])toyieldanasymptoticstabilityresult.TheRISE-basedcontrolstructurehasbeenusedforavarietyoffullyactuatedsystemsin[ 52 { 60 ].ThecontributioninChapter4istheuseoftheRISEcontrolstructuretoachieveasymptotictrackingcontrolofamodelreferencesystem,wheretheplantdynamicscontainaboundedadditivedisturbance(e.g.,potentialdisturbancesinclude:gravity,inertialcoupling,nonlineargustmodeling,etc.).ThisresultrepresentsthersteverapplicationoftheRISEmethodwherethecontrollerismultipliedbyanon-squarematrixcontainingparametricuncertainty.Toachievetheresult,thetypicalRISEcontrolstructureismodiedbyaddingarobustcontrolterm,whichisdesignedtocompensatefortheuncertaintyintheinputmatrix.TheresultisprovenviaLyapunov-basedstabilityanalysisanddemonstratedthroughnumericalsimulation.Motivatedbythedesiretodevelopanadaptivemethod,neuralnetwork(NN)-basedcontrollershavebeentypicallyusedtocompensateforunstructureduncertainty(e.g.,see[ 61 ]).OnedrawbackofNN-basedcontrolisthatasymptoticstabilityisdiculttoproveduetotheinherentfunctionalreconstructionerror.AcontributioninChapter5isthatanewrobustcontroltechniqueisusedalongwithanadaptivecontrollawtoachievean 18

PAGE 19

42 43 ]sothatthecontrollerdependsonlyontheoutputstates,andnottherespectivetimederivatives.GlobalasymptotictrackingisprovenviaaLyapunovstabilityanalysis,andahighdelitynumericalsimulationisprovidedtoshowtheperformanceofthedevelopedcontroller. 1.3.1ContributionsofCompletedResearch 19

PAGE 20

1-1 20

PAGE 21

ResearchSchedule 21

PAGE 22

20 ].ThesingularityproblemiscircumventedbytheuseofaparticularJacobianpseudoinverse,coinedthesingularityrobuststeeringlaw,whichwasintroducedin[ 21 ],andhasbeenimplementedinseveralaerospacevehicles(e.g.,see[ 20 ]and[ 22 ]).Auniformlyultimatelybounded(UUB)stabilityresultisprovenviaLyapunovanalysisforthecaseinwhichbothstaticanddynamicfrictioneectsareincludedintheCMGdynamicmodel.Anasymptotictrackingextensionisthenformulatedforthecasewherestaticfrictioneectsareignored. 62 63 ] 2{1 ),J()2R33representsthepositivedenite,symmetricsatelliteinertiamatrixthatisafunctionoftheCMGgimbalangularpositionvector(t)2R4,!(t)2R3denotestheangularvelocityofthesatellitebody-xedframeFwithrespecttoIexpressedin 22

PAGE 23

2{1 )canbelowerandupperboundedasfollows: 1 2minfJgkk2TJ1 2maxfJgkk282Rn(2{3)whereminfJg;maxfJg2RaretheminimumandmaximumeigenvaluesofJ(),respectively.ThetorquegeneratedfromtheCMGclustercanbemodeledas 7 ] _hcmg=hA()_;(2{5)whereh2RrepresentstheconstantangularmomentumofeachCMGexpressedinthegimbal-xedframe(i.e.,histhesameforallfourCMGs).In( 2{4 )and( 2{5 ),_(t)2R4 1 23

PAGE 24

_,_1_2_3_4T;(2{6)where_i(t)2RdenotestheangularvelocityoftheindividualCMGgimbals8i=1;2;3;4,sgn_(t)2R4denotesavectorformofthestandardsgn()functionwherethesgn()isappliedtoeachelementof_(t),andA()2R34denotesameasurableJacobianmatrixdenedas 2.2 .SincetheelementsofA()in( 2{7 )arecombinationsof Figure2-1. TheUniversityofFloridacontrolmomentgyroscopeexperimentaltestbed. boundedtrigonometricterms,thefollowinginequalitycanbedeveloped: 24

PAGE 25

2 ] _qv=1 2qv!+q0!(2{9) _q0=1 2qTv!:(2{10)In( 2{9 )and( 2{10 ),q(t),fq0(t);qv(t)g2RR3representstheunitquaternion[ 62 ]describingtheorientationofthebody-xedframeFwithrespecttoI,subjecttotheconstraint 2{9 )and( 2{10 ),!(t)canbeexpressedintermsofthequaternionas 2{15 )anditsrsttwotimederivativesareboundedforalltime. 25

PAGE 26

~R,RRTd=e20eTvevI3+2eveTv2e0ev;(2{16)whereR(qv;q0)andRd(qvd;q0d)weredenedin( 2{12 )and( 2{13 ),respectively,andthequaterniontrackingerrore(t),fe0(t);ev(t)g2RR3isdenedas 2{16 ),theattitudecontrolobjectivecanbestatedas ~R(ev(t);e0(t))!I3ast!1:(2{19)Basedonthetrackingerrorformulation,theangularvelocityofFwithrespecttoFdexpressedinF,denotedby~!(t)2R3,isdenedas ~!,!~R!d:(2{20)Fromthedenitionsofthequaterniontrackingerrorvariables,thefollowingconstraintcanbedeveloped[ 2 ]: 0kev(t)k10je0(t)j1;(2{22)wherekkrepresentsthestandardEuclideannorm.From( 2{21 ), 26

PAGE 27

2{16 )canbeusedtoconcludethatif( 2{23 )issatised,thenthecontrolobjectivein( 2{19 )willbeachieved. 2{24 )into( 2{20 ),theangularvelocitytrackingerrorcanbeexpressedas ~!=rev:(2{25)Motivationforthedesignofr(t)isobtainedfromthesubsequentLyapunov-basedstabilityanalysisandthefactthat( 2{14 )-( 2{18 )canbeusedtodeterminetheopen-loopquaterniontrackingerroras _ev=1 2ev+e0I~!_e0=1 2eTv~!:(2{26) 2{24 )andpremultiplyingtheresultingexpressionbyJ()as 2{1 ),( 2{4 ),( 2{5 ),( 2{24 ),and( 2{26 ),theexpressionin( 2{27 )canbewrittenas 2_Jr+Y111_hA_AFssgn_:(2{28) 27

PAGE 28

2{28 ),Y1(ev;e0;!;;t)2R3p1isaknownandmeasurableregressionmatrix,and12Rp1isavectorofp1unknownconstants(i.e.,satelliteinertiaparameters)where 2Jev+e0I~!:(2{29)Alsoin( 2{28 ),1(r;ev;e0;t)2R34denotesanauxiliarymatrixcontainingparametricuncertaintydenedas 1_=@J @_1 2r+~R!d+ev+AFd_(2{30)thatcanbelinearlyparameterizedintermsofaknownregressionmatrixY2(ev;e0;r;!;;_;t)2R3p2andavectorofp2unknownconstants(i.e.,inertiaparametersandfrictioncoecients)22Rp2as 2{28 )arethatthecontrolinput_(t)ispremultipliedbyanonsquareknowntime-varyingmatrixplusanonsquareunknowntime-varyingmatrix,and_(t)isembeddedinsideofadiscontinuousnonlinearity(i.e.,thesignumfunction).Toaddressthefactthat_(t)ispremultipliedbyanonsquareunknowntime-varyingmatrix,anestimateoftheuncertaintyin( 2{31 ),denotedby^1(r;ev;e0;t)2R34,isdenedas ^1_,Y2^2(2{32) 2{29 ).Inthiscase,p1=6,correspondingtothe6uncertaininertiaparameters.3 2{30 ). 28

PAGE 29

2{31 )and( 2{32 ),( 2{28 )canberewrittenas 2_Jr+Y11Y2~2B_AFssgn_;(2{33)whereB(r;ev;e0;;t)2R34isdenedas ~2=2^2:(2{35)Basedontheopen-loopdynamicsin( 2{33 )andthesubsequentstabilityanalysis,thecontrolinputisdesignedas _=B+hY1^1!hcmg+kr+knrevi;(2{36)wherek;kn2Rdenotepositivecontrolgains,andB+(r;ev;e0;;t)2R33denotesapseudoinverseofB(;ev;e0;^2;t)denedas[ 20 { 22 ] 2{37 ),(t)2Rdenotesasingularityavoidanceparameter.Forexample,Nakamuraetal.[ 21 ]designed(t)as 2{36 )into( 2{33 ),theclosed-looperrorsystemforr(t)canbeobtainedas 2_Jr+Y1~1Y2~2krknr+evAFssgn_;(2{39) 29

PAGE 30

~1=1^1:(2{40)Basedon( 2{39 )andthesubsequentstabilityanalysis,theparameterestimates^1(t)and^2(t)aredesignedas 2{36 )and( 2{41 )requiresmeasurementsofangularpositionandvelocity,notacceleration. 30

PAGE 31

2{24 )and( 2{33 ),theadaptivecontrollerof( 2{36 )and( 2{41 )ensuresglobaluniformlyultimatelybounded(GUUB)attitudetrackinginthesensethat 2rTJr+1 2~T111~1+1 2~T212~2:(2{44)Theexpressionin( 2{44 )canberewrittenas 2eTvrT264evJr375+(1e0)2+1 2~T111~1+1 2~T212~2;(2{45)anditfollowsdirectlyfromtheboundsgivenin( 2{3 ),( 2{22 ),and( 2{42 )thatV(e0;ev;r;t)canbeupperandlowerboundasfollows: 2{26 ),( 2{35 ),( 2{39 ),and( 2{40 ),thetimederivativeofV(e0;ev;r;t)canbeexpressedas_V=eTvev+e0I~!+(1e0)eTv~!+rTY1~1Y2~2krknr+evAFssgn_~T111^1~T212^2: 31

PAGE 32

2{8 ),( 2{25 ),( 2{41 ),andexploitingthefactthateTvev~!=0;theexpressionin( 2{48 )canbewrittenas _V3kzk2knkrk2+krk0kFski1;(2{49)where3=minf;kg2R.Aftercompletingthesquares,( 2{49 )canbewrittenas _V(t)3kzk2+(0kFski1)2 2{46 )canbeutilizedtolowerboundkzk2as 2{50 )canbeexpressedas _V(t)3 2{52 )canbesolvedas 2{44 )and( 2{54 )canbeusedtoconcludethatr(t)2L1.Thus,from( 2{22 ),( 2{25 ),and( 2{47 ),~!(t);z(t)2L1,and( 2{24 )canbeusedtoconcludethat!(t)2L1.Equation( 2{26 )thenshowsthat_ev(t);_e0(t)2L1:Hence,( 2{29 ),( 2{32 ),( 2{34 ),and( 2{42 )canbeusedtoprovethatthecontrolinput_(t)2L1:Standardsignalchasingargumentscanthenbeutilizedtoprovethatallremainingsignalsremainboundedduringclosed-loopoperation.Theinequalitiesin( 2{46 )cannowbeusedalong 32

PAGE 33

2{53 )and( 2{54 )toconcludethat 2{43 )cannowbedirectlyobtainedfrom( 2{55 ). 2{36 )resultsinthefollowingexpressionfortheclosed-looptrackingerrorsystem: 2_Jr+Y1~1Y2~2kr+ev:(2{56) 2{56 ),theadaptivecon-trollerof( 2{36 )and( 2{41 )ensuresasymptoticattitudetrackinginthesensethat 2{3 ).Proof:ToproveTheorem2-2,thesameprocedureasintheprevioussectioncanbeusedtocalculatethetimederivativeofthefunctionV(t)denedin( 2{44 )as _V(t)3kzk2;(2{59) 33

PAGE 34

2{47 ),and3wasdenedin( 2{49 ).From( 2{59 ),_V(t)isnegativesemi-denite,andV(t)isboundedasshownin( 2{46 ).Furthermore,( 2{22 ),( 2{24 ),( 2{41 ),( 2{42 ),and( 2{56 )canbeusedtoconcludethate(t);_e(t);_r(t)2L1:Thus,_z(t)2L1;andz(t)2L2\L1.Barbalat'sLemmacannowbeusedtoconcludethatkz(t)k!0ast!1:Hence,theadaptivecontrollawgivenby( 2{36 )and( 2{41 )achievestheasymptotictrackingclaimgivenin( 2{57 )forthecaseinwhichstaticfrictionisignoredinthedynamics.Vericationoftheboundednessoftheremainingsignalsduringclosed-loopoperationissimilartothatintheprevioussection. 2.2 ).Using( 2{1 ),thedynamicequationofmotionintermsoftheCMGtestbedcanbeexpressedas 2{61 ),J02R33isdenedas 34

PAGE 35

2{61 ),BJgi()2R338i=1;2;3;4denotestheinertiamatrixoftheithgimbalasexpressedintheCMGtestbedbody-xedframe,andisdenedas 2{60 ),_hcmg2R3isdenedusing( 2{5 ),whereh=0:078.Theobjectiveistoregulateasatellite'sattitudetothedesiredquaterniondenedby 64 ]) 35

PAGE 36

2-2 and 2-3 showthesimulationresultsoftheclosed-loopsystemforthiscasewithcontrolgainsselectedas(e.g.,see( 2{36 ),( 2{37 ),( 2{38 ),and( 2{41 )):k=0:3kn=0:850=0:2=21=0:2I62=2I4:Figure 2-4 illustratesthevariationintheinertiaparametersduringclosedloopoperation.Thiseectisonlysignicantforabrieftransientperiodbeforetheadaptationlawfor^2in( 2{41 )compensatesforthedisturbance. Figure2-2. Quaterniontrackingerror. 36

PAGE 37

Controlinputgimbalangularrateresponse. 37

PAGE 38

_J()vs.time. resultsareprovidedtoshowtheecacyoftheproposedcontroller.Anasymptotictrackingextensionisalsopresentedintheabsenceofstaticfrictioninthedynamicmodel.FutureworkwilladdresstheissuesofexplicitgimbalaccelerationdependenceintheCMGtorquemodel,variationsinCMGwheelspeed,andhardstopsintheCMGgimbals. 38

PAGE 39

23 ]),unknownelectromechanicaldisturbancesareassumedtobepresentintheCMGactuators.Someofthechallengesencounteredinthecontroldesignarethatthecontrolinput(i.e.,CMGgimbalangularrate)is:premultipliedbyanon-square,time-varying,nonlinearuncertainmatrixduetodynamicgimbalfrictionandelectromechanicaldisturbances;andisembeddedinahardnonlinearityduetostaticgimbalfriction.Furthermore,duetothesmallsizeofthesatelliteconsideredinthisdevelopment,themotionoftheCMGscausessignicanttime-variationinthesatelliteinertiacharacteristics.Thetime-variationofthesatelliteinertiamanifestsitselfasanonlineardisturbancetorqueinthesatellitedynamicmodel,whichishandledviaonlineNNapproximation.Simulationresultsareprovidedtoillustratetheecacyoftheproposedcontroldesign. 62 63 ] 3{1 ),J()2R33representsthepositivedenite,symmetricsatelliteinertiamatrixthatisafunctionoftheCMGgimbalangularpositionvector(t)2R4,!(t)2R3denotestheangularvelocityofthesatellitebody-xedframeFwithrespecttoIexpressedinF,cmg(t)2R3denotesthetorquegeneratedviaaCMGclusterconsistingoffoursingle 39

PAGE 40

7 ] _hcmg=hA()_;(3{4)whereh2RrepresentstheconstantangularmomentumofeachCMGexpressedinthegimbal-xedframe(i.e.,histhesameforallfourCMGs).In( 3{3 )and( 3{4 ),_(t)2R4denotestheCMGgimbalangularvelocitycontrolinput,whichisdenedas _,_1_2_3_4T;(3{5)where_i(t)2R8i=1;2;3;4denotestheangularvelocityoftheithCMGgimbal,sgn_(t)2R4denotesavectorformofthestandardsgn()functionwherethesgn()isappliedtoeachelementof_(t),andA()2R34denotesameasurableJacobianmatrix 40

PAGE 41

3{3 ),Td;_2R4representstorquesinthegimbalaxesduetotachometerdisturbances,denedexplicitlyas[ 65 ] 3{1 )canbelowerandupperboundedas 1 2minfJgkk2TJ1 2maxfJgkk282Rn;(3{10)whereminfJg;maxfJg2RaretheminimumandmaximumeigenvaluesofJ(),respectively.Property3-2:SincetheelementsofA()in( 3{6 )arecombinationsofboundedtrigonometricterms,thefollowinginequalitycanbedeveloped: 41

PAGE 42

66 ],d(t)isassumedtobeoftheform 2 ] _qv=1 2qv!+q0!(3{13) _q0=1 2qTv!:(3{14)In( 3{13 )and( 3{14 ),q(t)representstheunitquaternion[ 62 ]describingtheorientationofthebody-xedframeFwithrespecttoI,subjecttotheconstraint 42

PAGE 43

3{13 )and( 3{14 ),!(t)canbeexpressedintermsofthequaternionas 3{19 )anditsrsttwotimederivativesareboundedforalltime. ~R,RRTd=e20eTvevI3+2eveTv2e0ev;(3{20)whereR(qv;q0)andRd(qvd;q0d)weredenedin( 3{16 )and( 3{17 ),respectively,andthequaterniontrackingerrore(t),fe0(t);ev(t)g2RR3isdenedas 3{20 ),theattitudecontrolobjectivecanbestatedas ~R(ev(t);e0(t))!I3ast!1:(3{23) 43

PAGE 44

~!,!~R!d:(3{24)Tofacilitatethesubsequentcontrollerdesign,anauxiliarycontrolsignal,denotedbyr(t)2R3,isdenedas[ 67 ] 3{25 )into( 3{24 ),theangularvelocitytrackingerrorcanbeexpressedas ~!=rev:(3{26)Motivationforthedesignofr(t)isobtainedfromthesubsequentLyapunov-basedstabilityanalysisandthatfactthat( 3{18 )-( 3{22 )canbeusedtodeterminetheopen-loopquaterniontrackingerroras _ev=1 2ev+e0I~!_e0=1 2eTv~!:(3{27)Fromthedenitionsofthequaterniontrackingerrorvariables,thefollowingconstraintcanbedeveloped[ 2 ]: 0kev(t)k10je0(t)j1;(3{29)wherekkrepresentsthestandardEuclideannorm.From( 3{28 ), 3{20 )canbeusedtoconcludethatif( 3{30 )issatised,thenthecontrolobjectivein( 3{23 )willbeachieved. 44

PAGE 45

3{1 ).ThemainfeaturethatempowersNN-basedcontrollersistheuniversalapproximationproperty.LetSbeacompactsimplyconnectedsetofRN1+1.LetCn(S)bedenedasthespacewheref:S!Rniscontinuous.Theuniversalapproximationpropertystatesthatthereexistweightsandthresholdssuchthatsomefunctionf(x)2Cn(S)canberepresentedbyathree-layerNNas[ 68 69 ] 3{31 ),V2R(N1+1)N2andW2R(N2+1)nareboundedconstantidealweightmatricesfortherst-to-secondandsecond-to-thirdlayers,respectively,whereN1isthenumberofneuronsintheinputlayer,N2isthenumberofneuronsinthehiddenlayer,andnisthenumberofneuronsinthethirdlayer.Theactivationfunctionin( 3{31 )isdenotedby():RN1+1!RN2+1,and"(x):RN1+1!Rnisthefunctionalreconstructionerror.Basedon( 3{31 ),thetypicalthree-layerNNapproximationforf(x)isgivenas[ 68 69 ] ^f(x)=^WT^VTx;(3{32)where^V(t)2R(N1+1)N2and^W(t)2R(N2+1)naresubsequentlydesignedestimatesoftheidealweightmatrices.Theestimatemismatchfortheidealweightmatrices,denotedby~V(t)2R(N1+1)N2and~W(t)2R(N2+1)n,aredenedas ~V,V^V~W,W^W;(3{33)andthemismatchforthehiddenlayeroutputerrorforagivenx(t),denotedby~(x)2RN2+1,isdenedas ~,^=VTx^VTx:(3{34) 45

PAGE 46

68 69 ] 3{35 )into( 3{34 ),thefollowingexpressioncanbeobtained: ~=^0~VTx+O~VTx2;(3{36)where^0,0^VTx:Property3-6:(BoundednessoftheIdealWeights)Theidealweightsareassumedtoexistandbeboundedbyknownpositivevaluessothat 3{1 )and( 3{3 ),whichcontainnonlineardisturbancesandparametricuncertaintyinadditiontouncertaintycausedbyactuatordynamics. 46

PAGE 47

3{25 )andpremultiplyingtheresultingexpressionbyJ()as 3{1 ),( 3{3 ),( 3{4 ),( 3{7 ),( 3{25 ),and( 3{27 ),theexpressionin( 3{40 )canbeexpressedas 2_Jr:(3{41)In( 3{41 ),theuncertainfunctionf(r;qv;q0;ev;e0;!;!d;_!d;;t)2R3isdenedas 2Jev+e0I~!+d;(3{42)where1(r;qv;q0;ev;e0;!d;_!d;;t)2R34denotesanuncertainauxiliarymatrix,whichisdenedviatheparameterization 1_=@J @_1 2r+~R!dev+AFd_+AKGEd_:(3{43)Theexpressionin( 3{43 )canbelinearlyparameterizedintermsofaknownregressionmatrixY1(r;qv;q0;ev;e0;!;!d;_!d;;_;t)2R3p1andavectorofp1unknownconstants12Rp1as 1_,Y11:(3{44)Someofthecontroldesignchallengesfortheopen-loopsystemin( 3{41 )arethatthecontrolinput_(t)ispremultipliedbyanonsquare,uncertaintime-varyingmatrix,andisembeddedinsideofadiscontinuousnonlinearity(i.e.,thesignumfunction).Toaddressthefactthat_(t)ispremultipliedbyanonsquareunknowntime-varyingmatrix,anestimateoftheuncertaintyin( 3{44 ),denotedby^1(r;qv;q0;ev;e0;!d;_!d;;t)2R34,is 47

PAGE 48

^1_,Y1^1;(3{45)where^1(t)2Rp1isasubsequentlydesignedestimatefortheparametricuncertaintyin1(r;qv;q0;ev;e0;!d;_!d;;t).Basedon( 3{44 )and( 3{45 ),( 3{41 )canberewrittenas 2_JrY1~1AFssgn_;(3{46)whereB(r;qv;q0;ev;e0;!d;_!d;;t)2R34isdenedas ~1,1^1:(3{48)Theauxiliaryfunctionin( 3{42 )canberepresentedbyathree-layerNNas 3{49 ),theinputx(t)2R25isdenedas 3{31 ).Basedontheassumptionthattheactualanddesiredtrajectoriesarebounded,thefollowinginequalityholds: 48

PAGE 49

3{46 )andthesubsequentstabilityanalysis,thecontrolinputisdesignedas _=B+h^f!hcmg+Kvrv+evi;(3{52)whereKv2Rdenotesapositivecontrolgain,andv(t)2R3denotesarobustifyingterm,denedas[ 70 ] 3{69 ).Alsoin( 3{52 ),B+(r;qv;q0;ev;e0;!d;_!d;;t)2R43denotesthegeneralizedinverseofB(r;qv;q0;ev;e0;!d;_!d;;t),whichcouldbedenedusingtheMoore-PenrosedenitionorthesingularityrobustpseudoinversedenitioncoinedbyNakamuraetal.as(e.g.,see[ 20 { 22 ]) 3{56 ),(t)2Rdenotesasingularityavoidanceparameter.Forexample,in[ 21 ]Nakamuraetal.designed(t)as 3{52 ),thefeedforwardNNcomponent, 49

PAGE 50

^f,^WT^VTx;(3{58)wherethestatevectorx(t)2R25wasdenedin( 3{49 ).TheestimatesoftheNNweightsin( 3{58 )aregeneratedon-line(thereisnoo-linelearningphase)as[ 70 ]^W,1(^rT^0^VTxrTkrk^W) (3{59)^V,2xrT^0T^WT2krk^V; 3{59 )and( 3{60 )ensurethat^W(t)and^V(t)remainboundedprovidedx(t)remainsbounded.Thisfactwillbeexploitedinthesubsequentstabilityanalysis.Theclosed-looptrackingerrorsystemcanbedevelopedbysubstituting( 3{52 )into( 3{46 )as 2_Jr+~fY1~1Kvr+vevAFssgn_;(3{61)where~f(x)2R3representsafunctionestimationerrorvectordenedas ~f,f^f:(3{62)Basedon( 3{61 )andthesubsequentstabilityanalysis,theparameterestimate^1(t)isdesignedas 50

PAGE 51

3{63 )assumestheavailablilityofangularpositionandvelocitymeasurementsonly.Using( 3{49 ),( 3{58 )and( 3{62 ),theclosed-looperrorsystemin( 3{61 )canbeexpressedas 2_Jr+WTVTx^WT^VTx+vY1~1Kvr+"(x)evAFssgn_:(3{65)AfteraddingandsubtractingthetermsWT^and^WT~to( 3{65 ),thefollowingexpressionisobtained: 2_Jr+~WT^+^WT~+~WT~+"(x)Y1~1Kvr+vevAFssgn_(3{66)wherethenotations^and~wereintroducedin( 3{34 ).TheTaylorseriesapproximationdescribedin( 3{35 )and( 3{36 )cannowbeusedtorewrite( 3{66 )as 2_Jr+wKvr+vevY1~1+~WT^^0^VTx+^WT^0~VTx;(3{67)wherew(t)2R3isdenedas 70 ] 51

PAGE 52

3{67 ),theadaptivecontrollerof( 3{52 ),( 3{59 ),( 3{60 ),and( 3{63 )ensuresglobaluniformlyultimatelybounded(GUUB)attitudetrackinginthesensethat 2rTJr+1 2tr~WT11~W+1 2tr~VT12~V+1 2~T113~1:(3{72)Basedon( 3{10 ),( 3{29 ),( 3{48 ),( 3{59 ),( 3{60 ),and( 3{64 ),( 3{72 )canbeupperandlowerboundedas 3{26 ),( 3{27 ),( 3{67 ),andexploitingthefactthateTvev~!=0;thetimederivativeofV(t)canbeexpressedas_V(t)=eTvev+rTwKvr+vY1~1tr~WT11^W1^rT+^0^VTxrTtr~VT12^V1xrT^WT^0~T113^1: 52

PAGE 53

3{59 ),( 3{60 ),and( 3{63 ),( 3{75 )canbeexpressedas _V=eTvev+rT(wKvr+v)+krktr~ZTZ~Z:(3{76)Aftersubstituting( 3{53 )andusingthefactthattr~ZTZ~Z=h~Z;ZiF~Z2F~ZFkZkF~Z2F,( 3{76 )canbeupperboundedasfollows[ 70 ]:_V(t)kevk2Kvminkrk2knkrk2+krkkwkKZ^ZF+ZMkrk2+krk~ZFZM~ZF: 3{69 )andutilizinginequality( 3{55 ),_V(t)canbeboundedas _V(t)3kyk2knkrk2+krk;(3{78)where3,minf;Kvming,and,c0+c1~ZF+~ZFZM~ZF.Completingthesquaresin( 3{78 )yields _V(t)3kyk2+2 3{73 ),( 3{79 )canbeexpressedas _V(t)3 3{80 )canbesolvedas 3{72 ),( 3{73 ),and( 3{82 )canbeusedtoconcludethatr(t)2L1.Thus,from( 3{26 ),( 3{29 ),and( 3{74 ),~!(t);y(t)2L1,and( 3{25 )canbeusedto 53

PAGE 54

3{27 )thenshowsthat_ev(t);_e0(t)2L1:Hence,( 3{47 ),( 3{52 ),( 3{53 ),and( 3{58 )-( 3{60 )canbeusedtoprovethatthecontrolinput_(t)2L1:Standardsignalchasingargumentscanthenbeutilizedtoprovethatallremainingsignalsremainboundedduringclosed-loopoperation.Theinequalitiesin( 3{73 )cannowbeusedalongwith( 3{81 )and( 3{82 )toconcludethat 3{71 )cannowbedirectlyobtainedfrom( 3{83 ). 2.2 ).Using( 3{1 ),thedynamicequationofmotionintermsoftheCMGtestbedcanbeexpressedas 3{4 ),whereh=0:078kgm2=s,andhcmg2R3wasdenedin( 3{3 ),andtheCMGtestbedinertiamatrixJcmg()2R33isdenedusingtheparallelaxistheoremas 3{85 ),J0isdenedas 54

PAGE 55

3{85 ),BJgi()2R338i=1;2;3;4denotestheinertiamatrixoftheithgimbalexpressedintheCMGtestbedbody-xedframe,denedas 3{89 ),thecoordinatetransformationmatrixCBgi2SO(3)8i=1;2;3;4relatestheithgimbal-xedframetotheCMGclusterbody-xedframe,andgiJgi=diag4:891052:491042:79104kgm28i=1;2;3;4representstheinertiamatrixoftheithgimbalexpressedintheithgimbal-xedframe.In( 3{84 ),thefrictionmatricesFd2R44andFs2R44forthesimulatedCMGtestbedare(e.g.,see[ 64 ]) 3{6 ).Thenonlineardisturbancetermsd(t)andTd;_,Td1Td2Td3Td4Taregivenbyd=0:1266664sin10t+cos20tsin20t+cos30tsin40t+cos50t377775

PAGE 56

3-1 and 3-2 showthesimulationresultsoftheclosed-loopsystemwithcontrolgainsselectedasfollows(e.g.,see( 3{52 ),( 3{53 ),( 3{56 ),( 3{57 ),( 3{59 ),( 3{60 ),and( 3{63 )):Kv=0:2kn=0:7KZ=0:05=2:5=20=0:21=10:2I212=5I213=0:1I6Figure 3-3 illustratesthevariationintheinertiaparametersduringclosedloopoperation.Remark3-3:Thegimbalratecontrolinputsremainedboundedduringclosed-loopoperationforthecasewhensaturationlimitingwasnotincludedinthesimulation.However,theabovesimulationresultswereachievedusingagimbalratesaturationlimitof7:5=sectotestthetrackingcapabilityofthecontrollerinthepresenceofactuatorlimitations(see[ 6 ]foramoredetaileddiscussionofCMGgimbalrateranges).Thelargestangularexcursion(i.e.,max)duringthesimulationintervalisapproximately1:6. 56

PAGE 57

Quaterniontrackingerrorofclosed-loopsystem. 57

PAGE 58

Controlinputgimbalangularrates. Figure3-3. Timevariationoftheinertiamatrix(i.e.,_J())duringclosed-loopoperation. 58

PAGE 59

53 ])toachieveasymptotictrackingcontrolofamodelreferencesystem,wheretheplantdynamicscontainaboundedadditivedisturbance(e.g.,potentialdisturbancesinclude:gravity,inertialcoupling,nonlineargustmodeling,etc.).TheRISE-basedcontrolstructurehasbeenusedforavarietyoffullyactuatedsystemsin[ 52 { 60 ].TheresultinthischapterrepresentsthersteverapplicationoftheRISEmethodwherethecontrollerismultipliedbyanon-squarematrixcontainingparametricuncertainty.Toachievetheresult,thetypicalRISEcontrolstructureismodiedbyaddingarobustcontrolterm,whichisdesignedtocompensatefortheuncertaintyintheinputmatrix.TheresultisprovenviaLyapunov-basedstabilityanalysisanddemonstratedthroughnumericalsimulation. 25 29 34 71 72 ]:_x=Ax+Bu+f(x) (4{1)y=Cx; 4{1 )containparametricuncer-tainty.

PAGE 60

(4{3)_f(x;_x)=_f1(t)+_f2(x;_x); 4{1 willbeutilized,wherethestateandinputmatricesarebasedonthedynamicparametersoftheOspreyxedwingaerialvehicle(seeFigure 4-1 ).TheOspreyisacommerciallyavailable,low-costexperimentalighttestbedforinvestigatingnovelcontrolapproaches.Basedonthe Figure4-1. PhotographoftheOspreyaircrafttestbed. standardassumptionthatthelongitudinalandlateralmodesoftheaircraftaredecoupled,thestatespacemodelfortheOspreyaircrafttestbedcanberepresentedusing( 4{1 )and 60

PAGE 61

4{2 ),wherethestatematrixA2R88andinputmatrixB2R84aregivenas 4{12 ),elev2Rdenotestheelevatordeectionangle,thrust2Risthecontrolthrust,ail2Ristheailerondeectionangle,andrud2Ristherudderdeectionangle. 61

PAGE 62

4{1 )canrepresentseveralboundednonlinearities.Themorepromisingexampleofdisturbancesthatcanberepresentedbyf(x)isthenonlinearformofaselectivelyextractedportionofthestatespacematrixAlon2R44thatwouldnormallybelinearized.Thisnonlinearitywouldthenbeaddedtothenewstatespaceplantbysuperposition,resultinginthefollowingquasi-linearplantmodel: _xlon=A0lonxlon+Blonulon+f(xlon);(4{13)whereA0lon2R44isthestatespacematrixAlonwiththelinearizedportionremoved,andf(xlon)2R4denotesthenonlineardisturbancespresentinthelongitudinaldynamics.Somephysicalexamplesofthiswouldbetheselectivenonlinearitiesthatcannotbeignored,suchaswhendealingwithsupermaneuveringvehicles,wherepost-stallanglesofattackandinertiacoupling,forexample,areencountered.GiventhattheOspreyisessentiallyaverybenignmaneuveringvehicle,f(x)inthisdissertationwillrepresentlessrigorousnonlinearitiesforillustrativepurposes.Asimilartechniquecanbefollowedwiththelateraldirectionstatespacerepresentation,wherethenonlinearpartofAlatisextracted,andanewquasi-linearmodelforthelateraldynamicsisdevelopedas _xlat=A0latxlat+Blatulat+f(xlat);(4{14)whereA0lat2R44isthenewlateralstatematrixwiththelinearizedcomponentsremoved,andf(xlat)2R4denotesthenonlineardisturbancespresentinthelateraldynamics.Anotherexampleofboundednonlineardisturbances,whichcanberepresentedbyf(x)in( 4{1 ),isadiscreteverticalgust.Theformulagivenin[ 73 ],forexample,denessucha 62

PAGE 63

Hi;(4{15)whereHdenotesthedistance(between35feetand350feet)alongtheairplane'sightpathforthegusttoreachitspeakvelocity,V0istheforwardvelocityoftheaircraftwhenitentersthegust,s2[0;2H]representsthedistancepenetratedintothegust(e.g.,s=Rt2t1V(t)dt),andUdsisthedesigngustvelocityasspeciedin[ 73 ].Thisregulationisintendedtobeusedtoevaluatebothverticalandlateralgustloads,soasimilarrepresentationcanbedevelopedforthelateraldynamics.Anothersourceofboundednonlineardisturbancesthatcouldberepresentedbyf(x)istransportdelayfromcommunicationwithagroundstation. 4{2 ).Thelateralandlongitudinalreferencemodelswerechosenwiththespecicpurposeofdecouplingthelongitudinalmodevelocityandpitchrateaswellasdecouplingthe 63

PAGE 64

74 ],denotedbyr(t)2Rm;isdenedas: 4{20 )andutilizingtheexpressionsin( 4{1 ),( 4{2 ),( 4{16 ),and( 4{17 )toobtainthefollowingexpression: _r=~N+Nd+C~A(_e+e)+C~B(_u+u)+C^B(_u+u)e;(4{21)wheretheauxiliaryfunction~N(x;_x;e;_e)2Rmisdenedas ~N,C^A_e+C_f2(x)_f2(xm)+CAe+C(f2(x)f2(xm))+e;(4{22)theauxiliaryfunctionNdxm;_xm;;_isdenedasNd=CAm_xmCBm_+C^A_xmCAmxmCBm+C^Axm+C_f1+Cf1+C_f2(xm)+Cf2(xm); 64

PAGE 65

~A,A^A~B,B^B:(4{24)Tosimplifythenotationinthesubsequentdevelopment,theconstant,unknownmatrix2Rmmisdenedas ,CB;(4{25)andtheestimateandestimatemismatchforaredenedas ^,C^B~,C~B;(4{26)respectively.Thequantities~N(x;_x;e;_e)andNdxm;_xm;;_andthederivative_Ndxm;_xm;xm;;_;canbeupperboundedasfollows:~N(kzk)kzk 4{21 )andthesubsequentstabilityanalysis,thecontrolinputisdesignedasu=Zt0u()d(ks+1)^1e(t)+(ks+1)^1e(0)Zt0k^1sgn(r())d^1Zt0[(ks+1)e()+sgn(e())]d; 65

PAGE 66

4{20 ),andtheconstantfeedforwardestimate^2Rmmisdenedas ^,C^B:(4{31)Tosimplifythenotationinthesubsequentstabilityanalysis,theconstantauxiliarymatrix~2Rmmisdenedas ~,^1;(4{32)where~canbeseparatedintodiagonalando-diagonalcomponentsas ~=+;(4{33)where2Rmmcontainsonlythediagonalelementsof~,and2Rmmcontainstheo-diagonalelements. 4{30 )into( 4{21 ),thefollowingclosed-looperrorsystemisobtained: _r=~N+Nd(ks+1)~rk~sgn(r)~sgn(e(t))e:(4{34)Assumption4-2:Theconstantestimate^givenin( 4{31 )isselectedsuchthatthefollowingconditionissatised: 4{35 )issatisedforawiderangeof^6=.Remark4-2:Apossibledecitofthiscontroldesignisthattheacceleration-dependenttermrappearsinthecontrolinputgivenin( 4{30 ).Thisisundesirable

PAGE 67

75 { 79 ].Further,from( 4{30 ),thesignoftheaccelerationisallthatisrequiredformeasurementinthiscontroldesign. 4{30 )ensuresthatallsystemsignalsareboundedduringclosed-loopoperationandthatthepositiontrackingerrorisregulatedinthesensethat 4{30 )isselectedsucientlylarge(seethesubsequentstabilityproof),andandkareselectedaccordingtothefollowingsucientconditions:>Nd+1 (4{37)k>p 4{28 ),"wasdenedin( 4{35 ),andandwereintroducedin( 4{33 ).BeforeprovingTheorem4-1,thefollowinglemmawillbesetforth.Lemma4-1:Tofacilitatethesubsequentstabilityanalysis,theauxiliaryfunctionP(t)2Risdenedas 67

PAGE 68

4{37 )issatised,thefollowinginequalitycanbeobtained 4{41 )canbeusedtoconcludethatP(t)0.Proof:(SeeTheorem4-1)LetDR2m+1beadomaincontainingy(t)=0;wherey(t)2R2m+1isdenedas 2eTe+1 2rTr+P;(4{43)whereV(y;t)satisestheinequality 4{37 )issatised.In( 4{44 ),thecontinuous,positivedenitefunctionsU1(y);U2(y)2Raredenedas 2kyk2U2,kyk2:(4{45)Aftertakingthederivativeof( 4{43 )andutilizing( 4{20 ),( 4{33 ),( 4{34 ),( 4{39 ),and( 4{40 ),_V(y;t)canbeexpressedas_V(y;t)=eTe+rT~N(ks+1)rTr(ks+1)rTr+p 68

PAGE 69

4{27 ),_V(y;t)canbeupperboundedas _V(y;t)eTe"krk2ks"krk2+(kzk)krkkzk+k"+p 4{38 )issatised,thebracketedtermin( 4{47 )isnegative,and_V(y;t)canbeupperboundedusingthesquaresofthecomponentsofz(t)asfollows: _V(y;t)kek2"krk2+(kzk)krkkzkks"krk2;(4{48)Completingthesquaresforthebracketedtermsin( 4{48 )yields _V(y;t)3kzk2+2(kzk)kzk2 4{27 ).Thefollowingexpressioncanbeobtainedfrom( 4{49 ): _V(y;t)U(y);(4{50)whereU(y)=ckzk2,forsomepositiveconstantc2R,isacontinuous,positivesemi-denitefunctionthatisdenedonthefollowingdomain: 4{44 )and( 4{50 )canbeusedtoshowthatV(t)2L1inD;hencee(t);r(t)2L1inD.Giventhate(t);r(t)2L1inD,standardlinearanalysismethodscanbeusedtoprovethat_e(t)2L1inDfrom( 4{20 ).Sincee(t);_e(t)2L1inD,( 4{19 )canbeusedalongwiththeassumptionthatym;_ym2L1inDtoprovethaty(t);_y(t)2L1.Giventhatr(t)2L1inD,theassumptionthat^12L1inDcanbeusedalongwithtimederivativeof( 4{30 )toshowthat_u(t)2L1inD.Further,Equation2.78of[ 80 ]canbeusedtoshowthat_u(t)canbeupperboundedas_u(t)u()+M,8t0,whereM2R+isaboundingconstant.Theorem1.1of[ 81 ]canthenbeutilizedtoprovethatu(t)2L1inD.Hence,( 4{34 )canbeusedtoshowthat_r(t)2L1inD.Since_e(t);_r(t)2L1inD,thedenitionsforU(y)andz(t)canbeusedtoprovethat 69

PAGE 70

212p 82 ]cannowbeinvokedtostatethat 4{53 )canbeusedtoshowthat 4{1 )and( 4{2 ),wherethestatematrixA,inputauthoritymatrixB,andnonlineardisturbancefunctionf(x)aregivenbythestatespacemodelfortheOspreyaircraftgivenin( 4{7 )-( 4{12 ).Thereferencemodelforthesimulationisrepresentedbythestatespacesystemgivenin( 4{16 )-( 4{18 ),withstatematricesAlonmandAlatm,inputmatricesBlonmandBlatm,andoutputmatricesClonandClatselectedas 70

PAGE 71

4{13 )and( 4{14 ),whereA0lon,A0lat,Blon,andBlataregivenas 71

PAGE 72

4{13 )and( 4{14 ),respectively,aredenedasf(xlon)=9:81sin+g(x)000T 4{15 ),whereUds=10:12m=s,H=15:24m,andV0=25m=s(cruisevelocity).Figure 4-2 showsaplotofthewindgustusedinthesimulation.Theremainderoftheadditive Figure4-2. Plotofthediscretevertical(upward)windgustusedinthecontrollersimulation. disturbancesin( 4{62 )and( 4{63 )representnonlinearitiesnotcapturedinthelinearizedstatespacemodel(e.g.,duetosmallangleassumptions).Allstatesandcontrolinputswereinitializedtozeroforthesimulation. 72

PAGE 73

^Blon=2666666640:010:1001:4000377777775^Blat=266666664001:70:050:10:2500377777775:(4{64)Remark4-3:Forthechoicesfor^Blonand^Blatgivenin( 4{64 ),theinequalityin( 4{35 )issatised.Specically,thechoicefor^Blonyields 4-3 and 4-4 showthesimulationresultsoftheclosed-looplongitudinalsystemwithcontrolgainsselectedasfollows(e.g.,see( 4{30 )and 73

PAGE 74

ParametersusedintheDIcontrollersimulation. SamplingTime 0:01sec PitchRateSensorNoise VelocitySensorNoise YawRateSensorNoise ControlThrustSaturationLimit AileronSaturationLimit RudderSaturationLimit ( 4{25 )) 4-3 showsthereferenceandactualpitchratesduringclosed-loopoperation,andFigure 4-4 showsthereferenceandactualforwardvelocityresponses.Forthelateralcontrollersimulation,theobjectivesaretotrackrollrateandyawratecommands.Figures 4-5 and 4-6 showthesimulationresultsoftheclosed-looplateral 4{38 );however,thisconditionisnotnecessaryforstability,itissucientfortheLyapunovstabilityproof. 74

PAGE 75

Pitchrateresponseachievedduringclosed-looplongitudinalcontrolleroperation. systemwithcontrolgainsselectedasfollows:=diag0:20:6ks=diag0:23=diag1:00:2k=I22:Figure 4-5 showsthereferenceandactualrollratesduringclosed-loopoperation,andFigure 4-6 showsthereferenceandactualyawrates.Thecontrolactuation(relativetotrimconditions)usedduringclosed-loopoperationfortherobustcontrollerisshowninFigure 4-7 75

PAGE 76

Forwardvelocityresponseachievedduringclosed-looplongitudinalcontrolleroperation. additive,non-LPdisturbances,wherethecontrolinputismultipliedbyanon-squarematrixcontainingparametricuncertainty.Toachievetheresult,anovelrobustcontroltechniqueiscombinedwithaRISEcontrolstructure.ALyapunov-basedstabilityanalysisisprovidedtoverifythetheoreticalresult,andnumericalsimulationresultsareprovidedtodemonstratetheecacyoftheproposedcontroller. 76

PAGE 77

Rollrateresponseachievedduringclosed-looplateralcontrolleroperation. Figure4-6. Yawrateresponseachievedduringclosed-looplateralcontrolleroperation. 77

PAGE 78

0 5 10 -4 -2 0 2 4 Aileron Angle [deg] 0 5 10 -40 -20 0 20 40 Rudder Angle [deg] 0 5 10 -50 0 50 100 150 Time [sec]Thrust [N] 0 5 10 -6 -4 -2 0 2 4 Time [sec]Elevator Angle [deg] Figure4-7.Controlactuationawayfromtrimusedduringclo sed-looprobustdynamic inversioncontrolleroperationforthelateralsubsystem( top)andthe longitudinalsubsystem(bottom) 78

PAGE 79

27 29 30 33 ]).Specicdicultiesexist,however,indesigningADIcontrollersforsystemscontaininguncertaintyintheinputmatrix.Whilerobustcontrolmethodsareoftenutilizedtocompensatefortheinversionerrorinsuchcases[ 35 { 38 ],therequiredcontroleortcanbelargeduetothehighgainorhighfrequencyfeedbacktypicallyrequiredintherobustcontroldesign.Motivatedbythedesiretodevelopanadaptivemethodasopposedtorobust,neuralnetwork(NN)-basedcontrollershavebeentypicallyusedtocompensateforunstructureduncertainty(e.g.,see[ 61 ]).OnedrawbackofNN-basedcontrolisthatasymptoticstabilityisdiculttoproveduetotheinherentfunctionalreconstructionerror.Acontributioninthischapteristheuseofanewrobustcontroltechniqueinconjunctionwithanadaptivecontrollawtoachieveanasymptotictrackingresultinthepresenceofparametricuncertaintyintheinputandstatematricesandanadditive,nonvanishingnonlineardisturbance.AnasymptotictrackingresultisprovenviaaLyapunovstabilityanalysis,andahighdelitynumericalsimulationisprovidedtoshowtheperformanceoftheproposedcontroldesign. 25 29 34 71 72 ]:_x=Ax+Bu+f(x;t) (5{1)y=Cx: 5{1 )and( 5{2 ),A2Rnndenotesastatematrixcomposedofunknownconstantelements,B2Rnmdenotesaninputmatrixcomposedofuncertainconstantelementswithm
PAGE 80

5{1 )canrepresentseveralboundednonlinearities(e.g.,errorsduetolinearization,inertialcoupling,discreteverticalgusts,etc.).Foradetaileddiscussionofnonlinearitiesthatcanberepresentedbyf(x;t)see[ 35 ].Assumption5-2:Forsomegivenoutputmatrix,thematrixproductCBisinvertibleforallelementsofBcontainedwithinsomeboundedregion.Whilethemodelin( 5{1 )and( 5{2 )isgenerictoabroadclassofaircraft,thissectiondescribeshowaspecicaircraftcanberelatedto( 5{1 ).Basedonthestandardassumptionthatthelongitudinalandlateralmodesoftheaircraftaredecoupled,thestate-spacemodelforthecommerciallyavailableOspreyxedwingaerialvehicle(seeFigure4-1)canberepresentedusing( 5{1 )and( 5{2 ),wherethestatematrixA2R88andinputmatrixB2R84aregivenas 5{3 )and( 5{4 ),Alon;Alat2R44,Blon;Blat2R42,andClon;Clat2R24denotethestatematrices,inputmatrices,andoutputmatrices,respectively,forthelongitudinalandlateralsubsystems,andthenotation0ijdenotesanijmatrixofzeros.TheOspreystate-vectorx(t)2R8isgivenas 80

PAGE 81

5{8 ),elev(t)2Rdenotestheelevatordeectionangle,thrust(t)2Risthecontrolthrust,ail(t)2Ristheailerondeectionangle,andrud(t)2Ristherudderdeectionangle. 81

PAGE 82

5{2 ).Alsoin( 5{11 ),Alonm;Alatm2Rqq,Blonm;Blatm2Rqpdenotethestatematricesandinputmatrices,respectively,forthelongitudinalandlateralsubsystems 74 ],denotedbyr(t)2Rm,isdenedas 5{3 )and( 5{4 )). 82

PAGE 83

75 { 79 ].Theopen-looptrackingerrordynamicscanbedevelopedbytakingthetimederivativeof( 5{14 )andutilizingtheexpressionsin( 5{1 ),( 5{2 ),( 5{9 ),and( 5{10 )toobtain _r=~N+Nd+YB~B+C^B(_u+u)e:(5{15)In( 5{15 ),theauxiliaryfunctions~N(x;_x;e;xm;_xm;t)2RmandNdxm;_xm;;_;t2Rmaredenedas~N,C_f(x;_x;t)_f(xm;_xm;t)+C(f(x;t)f(xm;t))+e+CA((_x+x)(_xm+xm)); 5{16 )and( 5{17 )isderivedfromthefactthatthefollowinginequalitiescanbedeveloped[ 52 53 ]:~N(kzk)kzk 5{15 ),YB(u;_u)2Rmp1denotesameasurableregressionmatrix,andB2Rp1isavectorcontainingtheunknownparametersoftheB

PAGE 84

5{15 )andthesubsequentstabilityanalysis,~B(t)2Rp1denotestheparameterestimationerrordenedas ~B,B^B;(5{22)where^B(t)2Rp1denotesasubsequentlydesignedparameterestimatevector.Theestimatematrix^B(t)2Rnmisintroducedin( 5{15 )tofacilitatethecontroldevelopment,wheretheelementsofthematrixarecomposedoftheelementsof^B(t).Basedontheexpressionin( 5{15 )andthesubsequentstabilityanalysis,thecontrolinputisdesignedas ,C^B1[(ks+1)r+sgn(e)];(5{24)where;ks2Rmmareconstant,positivedenite,diagonalcontrolgainmatrices,andisdenedin( 5{14 ).Theadaptiveestimate^B(t)(or^B(t)invectorform)in( 5{24 )isgeneratedonlineaccordingtotheadaptiveupdatelaw 5{25 ),B2Rp1p1isaconstant,positivedenite,symmetricadaptationgainmatrix,andproj()denotesaprojectionoperatorutilizedtoguaranteethattheithelementof^B(t)isboundedas 5{23 )into( 5{15 ),the 84

PAGE 85

_r=~N+Nd+YB~B(ks+1)rsgn(e)e:(5{27)Remark5-1:Theprojectionoperatorin( 5{25 )ensuresthatthematrixestimateC^B(t)isinvertibleunderthestandardassumptionthatCBisinvertible[ 25 34 83 ]forallelementsofBcontainedwithinsomeboundedregion(i.e., 5{23 )ensuresthatallsystemsignalsareboundedduringclosed-loopoperationandthatthepositiontrackingerrorisregulatedinthesensethat 5{23 )isselectedsucientlylarge(seethesubsequentstabilityproof),andthecontrolgainmatrixintroducedin( 5{24 )is

PAGE 86

5{19 ).ThefollowinglemmaisprovidedtofacilitatethemainresultinTheorem5-1.Lemma5-1:Tofacilitatethesubsequentstabilityanalysis,theauxiliaryfunctionP(t)2Risdenedas 5{34 )canbeusedtoconcludethatP(t)0.Proof:See[ 52 53 ]forproofofLemma5-1.Proof:(SeeTheorem5-1)LetDR2m+p1+1beadomaincontainingy(t)=0;wherey(t)2R2m+p1+1isdenedas 2eTe+1 2rTr+1 2~TB1B~B+P(5{36)thatsatisesthefollowinginequalities: 86

PAGE 87

5{31 )issatised,where1,1 2min1;min1B,and2,max1 2max1B;1.Aftertakingthetimederivativeof( 5{36 )andutilizing( 5{14 ),( 5{23 ),( 5{27 ),( 5{32 ),and( 5{33 ),_V(y;t)canbeexpressedas _V(y;t)=eTe+rT~N+rTYB~B(ks+1)rTr~TB1B^B:(5{38)Afterutilizing( 5{18 )and( 5{25 ),_V(y;t)canbeupperboundedas _V(y;t)eTekrk2kskrk2(kzk)krkkzk:(5{39)Completingthesquaresforthebracketedtermsin( 5{39 )yields _V(y;t)3kzk2+2(kzk)kzk2 5{18 ).Thefollowingexpressioncanbeobtainedfrom( 5{40 ): _V(y;t)U(y);(5{41)whereU(y)=ckzk2,forsomepositiveconstantc2R,isacontinuous,positivesemi-denitefunctionthatisdenedonthefollowingdomain: 5{37 )and( 5{41 )canbeusedtoshowthatV(t)2L1inD;hence,e(t);r(t);~B(t);P(t)2L1inD.Giventhate(t);r(t)2L1inD,standardlinearanalysismethodscanbeusedtoprovethat_e(t)2L1inDfrom( 5{14 ).Sincee(t);_e(t)2L1,( 5{13 )canbeusedalongwiththeassumptionthatym;_ym2L1toprovethaty(t);_y(t)2L1.Giventhatr(t);C^B(t)12L1inD,( 5{24 )canbeusedtoshowthat(t)2L1inD.Since(t)2L1inD,Equation2.78of[ 80 ]canbeusedtoshowthatk(t)kM,8t0,whereM2R+isaboundingconstant.Thetimederivativeof( 5{23 )canthenbeusedtoprovethat_u(t)2L1inD.Giventhat_u(t)2L1inDandk(t)kM,thetimederivativeof( 5{23 )canbeusedtoupperbound_u(t)as 87

PAGE 88

81 ]canthenbeutilizedtoprovethatu(t)2L1inD.Giventhatu(t);_u(t)2L1inD,( 5{21 )canbeusedtoprovethatYB2L1inD.Hence,( 5{27 )canbeusedtoshowthat_r(t)2L1inD.Since_e(t);_r(t)2L1inD,thedenitionsforU(y)andz(t)canbeusedtoprovethatU(y)isuniformlycontinuousinD.LetSDdenoteasetdenedas 212p 82 ]cannowbeinvokedtostatethat 5{44 )canbeusedtoshowthat 5{1 )and( 5{2 ),wherethestatematrixA,inputauthoritymatrixB,andnonlineardisturbancefunctionf(x)aregivenbythestatespacemodelfortheOspreyaircraftgivenin( 5{3 )-( 5{8 ).Thereferencemodelforthesimulationisrepresentedbythestatespacesystemgivenin( 5{9 )-( 5{11 ),withstatematricesAlonmandAlatm,inputmatricesBlonmandBlatm,andoutputmatricesClonandClatselectedas 88

PAGE 89

89

PAGE 90

ParametersusedintheADIcontrollersimulation. SamplingTime 0:01sec PitchRateSensorNoise VelocitySensorNoise YawRateSensorNoise respectively.Thenonlineardisturbanceterms,denotedf(xlon)andf(xlat),aredenedasf(xlon)=9:81sin+g(x)000T 73 ],whereUds=10:12m=s,H=15:24m,andV0=25m=s(cruisevelocity).Figure3-2showsaplotofthewindgustusedinthesimulation.Theremainderoftheadditivedisturbancesin( 5{53 )and( 5{54 )representnonlinearitiesnotcapturedinthelinearizedstatespacemodel(e.g.,duetosmallangleassumptions).Allstatesandcontrolinputswereinitializedtozeroforthesimulation.Inordertodeveloparealisticsteppingstonetoanactualexperimentaldemonstrationoftheproposedaircraftcontroller,thesimulationparameterswereselectedbasedondetaileddataanalysesandspecications.ThesensornoisevaluesarebaseduponCloudCapTechnology'sPiccoloAutopilotandanalysisofdataloggedduringstraightandlevelight.ThesevaluesarealsocorroboratedwiththespecicationsgivenforCloudCapTechnology'sCristaInertialMeasurementUnit(IMU).Thesimulationparametersaresummarizedinthefollowingtable:Theobjectivesforthelongitudinalcontrollersimulationaretotrackpitchrateandforwardvelocitycommands.Figures 5-1 and 5-2 showthesimulationresultsoftheclosed-looplongitudinalsystemwithcontrolgainsselectedasfollows(e.g.,see 90

PAGE 91

5{23 )-( 5{25 )):=diag2050ks=diag7060=0:02I22B=1E5I44wherethenotationIjjdenotesthejjidentitymatrix.Figure 5-1 showsthereferenceandactualpitchratesduringclosed-loopoperation,andFigure 5-2 showsthereferenceandactualforwardvelocityresponses.Forthelateralcontrollersimulation,the Figure5-1. Pitchrateresponseachievedduringclosed-looplongitudinalcontrolleroperation. objectivesaretotrackrollrateandyawratecommands.Figures 5-3 and 5-4 showthesimulationresultsoftheclosed-looplateralsystemwithcontrolgainsselectedasfollows:=40I22ks=60I22=0:02I22B=0:01I44Figure 5-3 showsthereferenceandactualrollratesduringclosed-loopoperation,andFigure 5-4 showsthereferenceandactualyawrates. 91

PAGE 92

Forwardvelocityresponseachievedduringclosed-looplongitudinalcontrolleroperation. Thecontrolactuation(relativetotrimconditions)usedduringclosed-loopoperationfortheadaptivecontrollerisshowninFigure 5-5 .Tofurthertesttheperformanceoftheproposedcontroldesign,anadditionalsimulationwascreatedtotesttheadaptivelongitudinalcontroller'sabilitytotracksimultaneousangleofattack(providedanangleofattacksensorisavailable(e.g.,apitottube))andforwardvelocitycommands.Theangleofattackandforwardvelocityresponsesoftheclosed-loopadaptivelongitudinalsystemareshowninFigure 5-6 ,andthecontrolactuationusedduringclosedloopoperationisshowninFigure 5-7 .ThecontrolactuationvaluesshowninFigure 5-7 aremeasuredwithrespecttotrim.ThesimulationparameterssummarizedinTable1wereusedfortheangleofattacksimulation.ThemotivationforincludingtheangleofattacktrackingresultsinFigure 5-6 istoillustratethatangleofattackcontrolcanbeachievedusingthedevelopedadaptivecontroldesign(similarresultscanalsobeobtainedfortherobustcontroller),sothesudden,individual 92

PAGE 93

Rollrateresponseachievedduringclosed-looplateralcontrolleroperation. gustwasnotincludedintheangleofattacksimulation.Fortheangleofattacksimulation,theoutputmatrixis 93

PAGE 94

Yawrateresponseachievedduringclosed-looplateralcontrolleroperation. Theconstantfeedforwardestimate^Blonwasselectedas ^Blon=2666666640:0010:05001000377777775:(5{57)Notethatforthechoiceof^Blongivenin( 5{57 ),theinequalityin( 4{35 )issatised.Specically,thechoicefor^Blonyields 94

PAGE 95

0 5 10 -3 -2 -1 0 1 2 Aileron Angle [deg] 0 5 10 -20 -10 0 10 20 30 Rudder Angle [deg] 0 5 10 0 50 100 150 Time [sec]Thrust [N] 0 5 10 -4 -2 0 2 4 6 Time [sec]Elevator Angle [deg] Figure5-5.Controlactuationawayfromtrimusedduringclo sed-loopadaptivedynamic inversioncontrolleroperationforthelateralsubsystem( top)andthe longitudinalsubsystem(bottom). strategyinanADIframeworktoanonlinearsystemwithaddit ive,non-LPdisturbances, wherethecontrolinputismultipliedbyanon-squarematrix containingparametric uncertainty.ALyapunov-basedstabilityanalysisisprovi dedtoverifythetheoretical result,andnumericalsimulationresultsareprovidedtode monstratetheecacyofthe proposedcontroller. 95

PAGE 96

0 1 2 3 4 5 6 7 8 9 10 -2 -1 0 1 2 Angle of Attack [deg] 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 Forward Velocity [m/sec]Time [sec] Model Reference Actual Response Model Reference Actual Response Figure5-6.Angleofattack(top)andforwardvelocityawayf romtrim(bottom)responses fortheclosed-loopadaptivelongitudinalsystem. 96

PAGE 97

0 1 2 3 4 5 6 7 8 9 10 -8 -6 -4 -2 0 2 Elevator Deflection [deg] 0 1 2 3 4 5 6 7 8 9 10 0 50 100 150 200 Thrust [N]Time [sec] Figure5-7.Controlactuationawayfromtrimusedduringclo sed-loopadaptivecontroller operationfortheangleofattacktrackingobjective.Eleva torderectionangle (top)andthrust(bottom). 97

PAGE 98

41 ],theunknownnonlinearityinthecurrentresultdoesnotsatisfythelinear-in-the-paramtersassumption.Incomparisonwithourpreviousresultsin[ 35 47 ],thecurrentdevelopmentexploitsthematrixdecompositiontechniquein[ 42 43 ]sothatthecontrollerdependsonlyontheoutputstates,andnottherespectivetimederivatives.GlobalasymptotictrackingisprovenviaaLyapunovstabilityanalysis,andahighdelitynumericalsimulationisprovidedtoshowtheperformanceofthedevelopedcontroller. (6{1)y=Cx: 6{1 )and( 6{2 ),A2Rnndenotesastatematrixcomposedofunknownconstantelements,B2Rnmdenotesacolumndecientinputmatrixcomposedofuncertainconstantelementswithm
PAGE 99

6{2 ),areferencemodelisdenedas_xm=Amxm+Bm 6{2 ).Assumption6-1:Thenonlineardisturbancef(x;t)anditsrsttwotimederivativesareassumedtoexistandbeboundedbyknownconstants.Adiscussionofnonlinearitiesthatcanberepresentedbyf(x;t)foranaircraftisprovidedin[ 35 ]. 6.3.1ControlObjectiveThecontrolobjectiveistoensurethattheoutputsy(t)trackthetime-varyingoutputsgeneratedfromthereferencemodeloutputsin( 6{4 ).Toquantifythecontrolobjective,anoutputtrackingerror,denotedbye(t)=e1(t)e2(t)em(t)T2Rm,isdenedas 74 ],denotedbyr(t)=r1(t)r2(t)rm(t)T2Rm,isdenedas 6{5 ))areavailable.Hence,_e(t)andr(t)aredenedtosimplifythestabilityanalysis.Theunmeasurablesignalr(t)iscontainedinsideanintegralinthesubsequentadaptiveupdatelaw,butisnotrequiredforthecontrolimplementationduetointegrationbyparts. 99

PAGE 100

6{7 ).Assumption6-2:Thestatesxu(t)in( 6{7 )andthecorrespondingtimederivativescanbefurtherseparatedasxu(t)=xu(t)+xu(t) (6{8)_xu(t)=_xu(t)+_xu(t);wherexu(t);_xu(t);xu(t);_xu(t)2Rnareassumedtobeupperboundedaskxu(t)kc1kzkkxu(t)kxuk_xu(t)kc2kzkk_xu(t)k_xuwherez(t)2R2misdenedas 6{6 )andutilizingtheexpressionsin( 6{1 )-( 6{4 )toobtain _r=~N0+Nd0+CA(_x 100

PAGE 101

6{10 )aredenedas~N0,CA(_x (6{11)+CA(_xu+xu);Nd0,C_f(x;t)+f(x;t)+CA(_xu+xu)CAm(_xm+xm)CBm_+: 6{11 )and( 6{12 )isderivedfromthefactthatthefollowinginequalitiescanbedeveloped[ 52 84 ]: 6{10 )andthesubsequentstabilityanalysis,thecontrolinputisdesignedas 6{14 )into( 6{10 ),theerrordynamicscanbeexpressedas _r=~N0+Nd0+CA(_x 101

PAGE 102

41 { 43 56 ]: 6{17 ),"2(0;1)andQ2R+areknownboundingconstants,andTik2Rdenotesthe(i;k)thelementofthematrixT.Preliminaryresultsindicatethatthisassumptionismildinthesensethatthedecompositionin( 6{16 )resultsinadiagonallydominantTforawiderangeof^B6=B.Basedon( 6{16 ),theerrordynamicsin( 6{15 )are 6{18 )cannowberewrittenas 102

PAGE 103

6{20 ),theauxiliarycontrolterm0(t)isdesignedas0,(ks+Imm)e(t)+(ks+Imm)e(0) (6{22)Zt0hYA^A+(ks+Imm)e()id;andtheauxiliarycontrolterm1(t)isdesignedas 6{6 ).Theadaptiveestimate^A(t)2Rp1in( 6{22 )isgeneratedaccordingtotheadaptiveupdatelaw (xm;_xm;r),AYTA(xm;_xm)r:(6{25)Fortheadaptationlawin( 6{24 )and( 6{25 ),A2Rp1p1isaconstant,positivedenite,symmetricadaptationgainmatrix.SincethemeasurableregressionmatrixYA()containsonlythereferencetrajectoriesxmand_xm,theexpressionin( 6{24 )canbeintegratedbypartstoprovethattheadaptiveestimate^A(t)canbegeneratedusingonlymeasurementsofe(t)(i.e.,nor(t)measurements,andhence,no_x(t)measurementsarerequired). 103

PAGE 104

6{24 )ensuresthatthefollowinginequalityissatised(forfurtherdetails,see[ 85 86 ]): 6{22 )into( 6{20 ),theclosed-looperrorsystemcanbedeterminedasS1_r=~N1+T_0+Nd1(ks+Imm)r ~A,A^A:(6{28)Usingthetimederivativeof( 6{22 ),thevectorT_0canbeexpressedas T_0=26666666666664mPj=2T1j_0jmPj=3T2j_0j...T(m1)m_0m037777777777775=2640375+2640375;(6{29)wheretheauxiliarysignals,12(m1)T2Rm1and,12(m1)T2Rm1,andtheindividualelementsaredenedasi,mXj=i+1Tij(ksj+1)rj 104

PAGE 105

jk,p1Xk=1YAjk^Ak8j=1;:::;m:(6{32)Basedon( 6{30 )-( 6{32 ),thefollowinginequalitiescanbedeveloped[ 41 56 ]: 6{9 ),and;12Rareknownpositiveboundingconstants.Notethatonlydependsonthediagonalelementsi+1tomofksduetothestrictlyuppertriangularnatureofT.Afterusing( 6{31 )and( 6{32 ),thetimederivativeofcanbeexpressedas _=NB1+NB2;(6{34)whereNB1,mXj=i+1Tijp1Xk=1_YAjk^Ak 6{24 ),and( 6{26 ),thefollowinginequalitiescanbedeveloped: 6{29 ),theclosed-looperrorsystemcanbeexpressedasS1_r=~N2+Nd2+YA~A(ks+Imm)r

PAGE 106

~N2=~N1+2640375Nd2=Nd1+2640375:(6{39)Basedon( 6{19 ),( 6{33 ),and( 6{39 ),thefollowinginequalitiescanbedeveloped:~N22kzkkNd2kNd2 6{19 )and( 6{37 ). 6{14 ),( 6{22 )-( 6{24 )ensuresthattheoutputtrackingerrorisregulatedinthesensethat 6{22 )isselectedsucientlylarge(seethesubesquentproof),isselectedtosatisfythesucientcondition 20;(6{42)andthecontrolgainsand0areselectedtosatisfythefollowingsucientconditions: 6{23 ),02Risintroducedin( 6{45 ),2,Nd2,_Nd1,2,and3areintroducedin( 6{19 ),( 6{37 ),and( 6{39 ),and"isintroducedin( 6{17 ).BeforeprovingTheorem6-1,thefollowinglemmawillbesetforth. 106

PAGE 107

6{17 ),andtheauxiliaryfunctionL(t)2Risdenedas 6{43 )aresatised,thefollowinginequalitycanbeobtained 6{46 )canbeusedtoconcludethatP(t)0.Proof:(SeeTheorem6-1)LetDR2m+p1+1beadomaincontainingw(t)=0,wherew(t)2R2m+p1+1isdenedas 2eTe+1 2rTS1r+1 2~TA1A~A+P;(6{48)whichispositivedeniteprovidedthesucientconditionin( 6{43 )issatised(seetheappendix).Aftertakingthetimederivativeof( 6{48 )andutilizing( 6{6 ),( 6{38 ),( 6{44 ), 107

PAGE 108

6{45 ),_V(w;t)canbeexpressedas_V(w;t)=eTe+rT~N2rT(ks+Imm)r+0kekkrk~TA1A^A+rTYA~A: 6{24 )and( 6{40 ),andusingthefactthat 2kek2+1 2krk2;(6{50)_V(w;t)canbeupperboundedas_V(w;t)1kzk2 20krk22krkkzk;where1,min1 20;1.Completingthesquaresforthebracketedtermsin( 6{51 )yields _V(w;t)122 20!kzk2:(6{52)Theinequalityin( 6{52 )canbeusedtoshowthatV(w;t)2L1;hence,e(t);r(t);~A(t);P(t)2L1.Giventhate(t);r(t)2L1,standardlinearanalysismethodscanbeusedtoprovethat_e(t)2L1from( 6{6 ).Sincee(t);_e(t)2L1,( 6{5 )canbeusedalongwiththeassumptionthatym(t);_ym(t)2L1toprovethaty(t);_y(t)2L1.SinceA2L1,theassumptionthatxm(t);_xm(t)2L1canbeusedalongwith( 6{21 )toprovethatYA(t)2L1.Giventhatr(t);^A(t);YA(t)2L1,theassumptionthatC^B12L1canbeusedalongwiththetimederivativeof( 6{22 )toshowthatC^B1_0(t)2L1.SinceC^B1_0(t)2L1andthetimederivativeof( 6{23 )canbeusedtoshowthat_1(t)2L1,Equation2.78of[ 80 ]canbeusedtoshowthatC^B1(_0(t)_1(t))canbeupperboundedasC^B1(_0(t)_1(t))R,8t0,whereR2R+isaboundingconstant.GiventhatC^B1(_0(t)_1(t))R,thetimederivativeof( 6{14 )canbeusedtoupperboundtheelements_ui(t)8i=1;:::;mof_u(t)as_ui(t)ui(t)+R. 108

PAGE 109

81 ]canthenbeutilizedtoprovethatu(t)2L1.Hence,( 6{38 )canbeusedtoshowthat_r(t)2L1.Since_e(t);_r(t)2L1,( 6{9 )canbeusedtoshowthatz(t)isuniformlycontinuous.Sincez(t)isuniformlycontinuous,V(w;t)isradiallyunbounded,and( 6{48 )and( 6{52 )canbeusedtoshowthatz(t)2L1\L2,Barbalat'sLemma[ 82 ]canbeinvokedtostatethat 6{53 )canbeusedtoshowthat 6{1 )and( 6{2 ),wherethestatematrixA,inputauthoritymatrixB,andnonlineardisturbancefunctionf(x;t)aredenedasin( 6{1 ).Thereferencemodelforthesimulationisrepresentedbythestatespacesystemgivenin( 6{3 )-( 6{4 ),wherethestatematrixAmandinputmatrixBmaredesignedwiththespecicpurposeofdecouplingthelongitudinalvelocityandpitchrateaswellasdecouplingthelateralrollrateandyawrate.Inadditiontothiscriterion,thedesignisintendedtoexhibitfavorabletransientresponsecharacteristicsandtoachievezerosteady-stateerror[ 35 47 ].Simultaneousanduncorrelatedcommandsareinputintoeachofthelongitudinalandlateralmodelsimulationstoillustratethateachmodelbehavesastwocompletelydecoupledsecondordersystems.TheoutputmatricesClonandClatareselectedas 109

PAGE 110

6{1 )and( 6{2 ),wherethestatematrixA2R88andinputmatrixB2R84aregivenas 6{56 )and( 6{57 ),Alon;Alat2R44,Blon;Blat2R42,andClon;Clat2R24denotethestatematrices,inputmatrices,andoutputmatrices,respectively,forthelongitudinalandlateralsubsystems,andthenotation0ijdenotesanijmatrixofzeros.Thestate-vectorx(t)2R8isgivenas

PAGE 111

6{61 ),elev(t)2Rdenotestheelevatordeectionangle,thrust(t)2Risthecontrolthrust,ail(t)2Ristheailerondeectionangle,andrud(t)2Ristherudderdeectionangle.ThestateandinputmatricesforthelongitudinalandlateraldynamicmodelsoftheOspreyxed-wingaircraftyingat25m/satanaltitudeof60metersaregivenas[ 35 47 ] 111

PAGE 112

73 ],andthetrigonometrictermsinf(xlon)andf(xlat)representnonlineardependenceongravity.Allstates,controlinputs,andadaptiveestimateswereinitializedtozeroforthesimulation.Thefeedforwardestimates^Blonand^Blatwereselectedas ^Blon=26666666400:20:101:5000377777775^Blat=26666666400100:50:2500377777775:(6{67) 6{67 ),Assumption3issatised.Specically,thechoicefor^Blonyieldsthefollowing: 6-1 showsthesimulationresultsoftheclosed-looplongitudinalsystemwithcontrolgainsselectedasfollows(e.g.,see( 6{14 ), 112

PAGE 113

ParametersUsedintheControllerSimulations. PitchRateSensorNoise VelocitySensorNoise RollRateSensorNoise YawRateSensorNoise ControlThrustSaturationLimit ElevatorSaturationLimit AileronSaturationLimit RudderSaturationLimit ( 6{22 ),and( 6{24 )):=diag0:310ks=diag0:134=1:2A=105I44wherethenotationIjjdenotesthejjidentitymatrix.Fig. 6-1 alsoshowsthesimulationresultsoftheclosed-looplateralsystemwithcontrolgainsselectedas=diag0:30:7ks=diag0:12:1=2:7A=106I44Fig. 6-2 showsthecontroleortusedduringclosed-loopoperation.Specically,Fig. 6-2 showstheelevatordeectionangleandthrustusedduringclosed-looplongitudinalcontrolleroperationandtheaileronandrudderdeectionangleusedduringclosed-looplateralcontrolleroperation. 113

PAGE 114

Referenceandactualforwardvelocity(topleft),pitchrate(topright),rollrate(bottomleft),andyawrate(bottomright)responsesduringclosed-looplongitudinalandlateralcontrolleroperation. 114

PAGE 115

Controlinputelevatordeection(topleft),thrust(topright),ailerondeection(bottomleft),andrudderdeection(bottomright)usedduringclosed-looplongitudinalandlateralcontrolleroperation. 115

PAGE 116

116

PAGE 117

117

PAGE 118

4{37 )issatised,thefollowinginequalitycanbeobtained:Zt0L()dke(0)kkke(0)TNd(0) (A{1)+p A{1 )canbeusedtoconcludethatP(t)0.Proof:Integratingbothsidesof( 4{40 )yields 4{20 )into( A{2 ),utilizing( 4{33 ),andrearrangingyields Integratingtherstintegralin( A{3 )usingintegrationbyparts, 118

PAGE 119

A{4 ),thefollowingboundcanbeobtained: wheremwasdenedin( 4{1 ).Thus,itisclearfrom( A{5 )thatifsatises( 4{37 ),then( A{1 )holds.Lemma6-1:Providedthecontrolgainsand0introducedin( 6{23 )and( 6{45 ),respectively,areselectedaccordingtothesucientconditionsin( 6{43 ),thefollowinginequalitycanbeobtained: A{6 )canbeusedtoconcludethatP(t)0,whereP(t)isdenedin( 6{44 ).Proof:Integratingbothsidesof( 6{45 )yieldsZt0L()d=Zt0mXi=1ri()mXj=iTij_1j() (A{7)Nd2i())0mXi=1jei()jjri()j!d;whereei(t);ri(t);Nd2i(t);_1i(t)2Rdenotetheithelementsofr(t),Nd2(t),and_1(t),respectively,andTijisintroducedin( 6{17 ).Substituting( 6{6 )into( A{7 ),rearranging, 119

PAGE 120

A{7 )canbeexpressedasZt0L()d=mXi=1ei()Nd2i()jt0Zt0mXi=1ei()@Nd2i() (A{8)+mXj=i+1Tijsgn(ej())!d+Zt0mXi=1ei()(Nd2i()(sgn(ei())+mXj=i+1Tijsgn(ej())!!dZt00mXi=1jei()jjri()jd:In( A{8 ),thefactthatmPj=iTij_1j(t)isgivenby 6{17 ).Byusing( A{10 )alongwiththefactthat 120

PAGE 121

A{8 )canbeexpressedasZt0L()d=mXi=1ei()Nd2i()jt0 6{19 ),( 6{37 ),and( 6{40 ),( A{12 )canbeupperboundedasZt0L()dZt0mXi=1jei()j(Nd2 A{14 )thatifand0satisfy( 6{43 ),then( A{6 )holds. 121

PAGE 122

[1] A.CebrowskiandJ.Raymond,\Operationallyresponsivespace:Anewdefensebusinessmodel,"Parameters,vol.Vol.35,no.2,pp.67{77,2005. [2] B.Costic,D.Dawson,M.deQueiroz,andV.Kapila,\Aquaternion-basedadaptiveattitudetrackingcontrollerwithoutvelocitymeasurements,"inProc.oftheIEEEConf.onDecisionandControl,vol.3,Sydney,Australia,Dec.2000,pp.2424{2429. [3] J.KimandJ.Crassidis,\Robustspacecraftattitudecontrolusingmodel-errorcontrolsynthesis,"inAIAAGuidance,Navigation,andControlConf.,Monterey,CA,Aug.2002. [4] H.Pan,H.Wong,andV.Kapila,\Outputfeedbackcontrolforspacecraftwithcoupledtranslationandattitudedynamics,"inProc.ofIEEEConf.onDecisionandControl,ParadiseIsland,Bahamas,Dec.2004,pp.4453{4458. [5] K.SubbaraoandM.R.Akella,\Dierentiator-freenonlinearproportional-integralcontrollersforrigid-bodyattitudestabilization,"inAAS/AIAA14thSpaceFlightMechanicsMeeting,vol.27,no.6,2004,pp.1092{1096. [6] V.Lappas,W.Steyn,andC.Underwood,\Designandtestingofacontrolmomentgyroscopeclusterforsmallsatellites,"JournalofSpacecraftandRockets,vol.42,no.4,pp.729{739,2005. [7] K.Omagari,T.Usuda,andS.Matunaga,\Researchofcontrolmomentumgyrosformicro-satellitesand3-DOFattitudedynamicssimulatorexperiments,"inProc.oftheInt'lSymposiumonArticialIntelligence,RoboticsandAutomationinSpace,Munich,Germany,2005. [8] D.M.HarlandandR.D.Lorenz,SpaceSystemsFailures.,J.Mason,Ed.Springer-Praxis,2005. [9] K.KrishnaKumar,\Adaptiveneuro-controlforspacecraftattitudecontrol,"inProc.oftheIEEEConf.onControlApplications,Aug.1994. [10] Y.Liu,J.Cao,andN.Wang,\Attitudeandvibrationcontrolofexiblespacecraftusingadaptiveinversedisturbancecanceling,"inInt'lJointConf.onNeuralNet-works,Vancouver,BC,Canada,July2006. [11] M.-T.ChoiandH.Flashner,\Neural-network-basedspacecraftattitudecontrolandmomentummanagement,"inAIAAGuidance,Navigation,andControlConf.,Denver,CO,Aug.2000. [12] ||,\Neural-network-basedspacecraftattitudecontrol,"inAIAAGuidance,Navigation,andControlConf.,Denver,CO,Aug.2000. [13] N.Sadati,A.Meghdari,andN.Dadkhah,\Optimaltrackingneuro-controllerinsatelliteattitudecontrol,"inIEEEInt'lConf.onIndustrialTechnology,Dec.2002. 122

PAGE 123

N.Sadati,N.d.Tehrani,andH.R.Bolandhemmat,\MultivariableadaptivesatelliteattitudecontrollerdesignusingRBFneuralnetwork,"inProc.oftheIEEEInt'lConf.onNetworking,SensingandControl,Taipei,Taiwan,Mar.2004. [15] N.SivaprakashandJ.Shanmugam,\Neuralnetworkbasedthreeaxissatelliteattitudecontrolusingonlymagnetictorquers,"inDigitalAvionicsSystemsConf.,Nov.2005. [16] C.W.Tan,S.Park,K.Mostov,andP.Varaiya,\Designofgyroscope-freenavigationsystems,"inIEEEInt'lConf.onIntelligentTransportationSystems,2001,pp.286{291. [17] N.Unnikrishnan,S.N.Balakrishnan,andR.Padhi,\Dynamicre-optimizationofaspacecraftattitudecontrollerinthepresenceofuncertainties,"inProc.ofIEEEInt'lSymposiumonIntelligentControl,Munich,Germany,Oct.2006. [18] P.Singla,K.Subbarao,andJ.L.Junkins,\Adaptiveoutputfeedbackcontrolforspacecraftrendezvousanddockingundermeasurementuncertainty,"JournalofGuidance,Control,andDynamics,vol.29,no.4,pp.892{902,2006. [19] H.Wong,M.deQueiroz,andV.Kapila,\Adaptivetrackingcontrolusingsynthesizedvelocityfromattitudemeasurements,"inProc.oftheAmericanControlConf.,vol.3,2000,pp.1572{1576. [20] K.A.FordandC.D.Hall,\Singulardirectionavoidancesteeringforcontrol-momentgyros,"JournalofGuidance,Control,andDynamics,vol.23,no.4,pp.648{656,2000. [21] Y.NakamuraandH.Hanafusa,\Inversekinematicsolutionswithsingularityrobustnessforrobotmanipulatorcontrol,"JournalofDynamicSystems,Mea-surement,andControl,vol.108,no.3,pp.163{171,1986. [22] N.Bedrossian,J.Paradiso,E.Bergmann,andD.Rowell,\SteeringlawdesignsforredundantSGCMGsystems,"AIAAJ.Guidance&Control,vol.13,no.6,pp.1083{1089,1991. [23] W.MacKunis,K.Dupree,N.Fitz-Coy,andW.E.Dixon,\AdaptivesatelliteattitudecontrolinthepresenceofinertiaandCMGgimbalfrictionuncertainties,"inProc.AIAAGuidance,Navigation,andControlConf.,HiltonHead,SC,Aug.2007,AIAA-2007-6432. [24] A.MoutinhoandJ.R.Azinheira,\StabilityandrobustnessanalysisoftheAURORAairshipcontrolsystemusingdynamicinversion,"inProc.ofInt'lConf.onRoboticsandAutomation,Barcelona,Spain,April2005,pp.2265{2270. [25] M.W.OppenheimerandD.B.Doman,\Controlofanunstable,nonminimumphasehypersonicvehiclemodel,"inProc.oftheIEEEAerospaceConf.,BigSky,MT,Mar.2006,pp.1{7. 123

PAGE 124

S.Onori,P.Dorato,S.Galeani,andC.Abdallah,\Finitetimestabilitydesignviafeedbacklinearization,"inProc.ofConf.onDecisionandControl,andtheEuropeanControlConf.,Seville,Spain,Dec.2005,pp.4915{4920. [27] Z.Szabo,P.Gaspar,andJ.Bokor,\TrackingdesignforWienersystemsbasedondynamicinversion,"inProc.ofInt'lConf.onControlApplications,Munich,Germany,Oct.2006,pp.1386{1391. [28] J.Chen,D.Li,X.Jiang,andX.Sun,\Adaptivefeedbacklinearizationcontrolofaexiblespacecraft,"inProc.ofConf.onIntelligentSystemsDesignandApplications,Jinan,China,Oct.2006,pp.225{230. [29] A.D.NgoandD.B.Doman,\Dynamicinversion-basedadaptive/recongurablecontroloftheX-33onascent,"inProc.ofIEEEAerospaceConference,BigSky,MT,Mar.2006,pp.2683{2697. [30] M.D.TandaleandJ.Valasek,\Adaptivedynamicinversioncontrolofalinearscalarplantwithconstrainedcontrolinputs,"inProc.ofAmericanControlConf.,Portland,OR,June2005,pp.2064{2069. [31] N.Hovakimyan,E.Lavretsky,andA.Sasane,\Dynamicinversionfornonane-in-controlsystemsviatime-scaleseparation:PartI,"inProc.ofAmeri-canControlConf.,Portland,OR,June2005,pp.3542{3547. [32] N.Hovakimyan,E.Lavretsky,andC.Cao,\Dynamicinversionofmulti-inputnonanesystemsviatime-scaleseparation,"inProc.ofAmericanControlConf.,Minneapolis,MN,June2006,pp.3594{3599. [33] E.LavretskyandN.Hovakimyan,\Adaptivedynamicinversionfornonane-in-controlsystemsviatime-scaleseparation:partII,"inProc.ofAmer-icanControlConf.,Portland,OR,June2005,pp.3548{3553. [34] J.BungtonandA.Sparks,\ComparisonofdynamicinversionandLPVtaillessightcontrollawdesigns,"inProc.ofAmericanControlConf.,vol.2,Philadelphia,PA,June1998,pp.1145{1149. [35] W.MacKunis,M.K.Kaiser,P.M.Patre,andW.E.Dixon,\Asymptotictrackingforaircraftviaanuncertaindynamicinversionmethod,"inProc.AmericanControlConf.,Seattle,WA,June2008,pp.3482{3487. [36] Q.WangandR.F.Stengel,\Robustnonlinearightcontrolofahigh-performanceaircraft,"IEEETransactionsonControlSystemsTechnology,vol.13,no.1,pp.15{26,Jan.2005. [37] T.Yamasaki,H.Sakaida,K.Enomoto,H.Takano,andY.Baba,\Robusttrajectory-trackingmethodforUAVguidanceusingproportionalnavigation,"inInt'lConf.onControl,AutomationandSystems,Seoul,Korea,Oct.2007,pp.1404{1409. 124

PAGE 125

Z.Liu,F.Zhou,andJ.Zhou,\Flightcontrolofunpoweredyingvehiclebasedonrobustdynamicinversion,"inChineseControlConference,Heilongjiang,China,Aug.2006,pp.693{698. [39] J.Cheng,H.Li,andY.Zhang,\Robustlow-costslidingmodeoverloadcontrolforuncertainagilemissilemodel,"inProc.ofWorldCongressonIntelligentControlandAutomation,Dalian,China,June2006,pp.2185{2188. [40] X.-.J.Liu,F.Lara-Rosano,andC.W.Chan,\Model-referenceadaptivecontrolbasedonneurofuzzynetworks,"IEEETransactionsonSystems,Man,andCybernet-icsC,vol.34,no.3,pp.302{309,Aug.2004. [41] X.Zhang,D.Dawson,M.deQueiroz,andB.Xian,\AdaptivecontrolforaclassofMIMOnonlinearsystemswithnon-symmetricinputmatrix,"inProc.InternationalConf.onControlApplications,Taipei,Taiwan,Sep.2004,pp.1324{1329. [42] A.S.Morse,\Againmatrixdecompositionandsomeofitsproperties,"SystemsandControlLetters,vol.21,no.1,pp.1{10,July1993. [43] R.R.Costa,L.Hsu,A.K.Imai,andP.Kokotovic,\Lyapunov-basedadaptivecontrolofMIMOsystems,"Automatica,vol.39,no.7,pp.1251{1257,July2003. [44] A.Behal,V.M.Rao,andP.Marzocca,\Adaptivecontrolforanonlinearwingsectionwithmultipleaps,"JournalofGuidance,Control,andDynamics,vol.29,no.3,pp.744{748,2006. [45] K.K.Reddy,J.Chen,A.Behal,andP.Marzocca,\Multi-input/multi-outputadaptiveoutputfeedbackcontroldesignforaeroelasticvibrationsuppression,"JournalofGuidance,Control,andDynamics,vol.30,no.4,pp.1040{1048,2007. [46] C.C.Cheah,C.Liu,andJ.J.E.Slotine,\AdaptiveJacobiantrackingcontrolofrobotswithuncertaintiesinkinematic,dynamicandactuatormodels,"IEEETransactionsonAutomaticControl,vol.51,no.6,pp.1024{1029,June2006. [47] W.MacKunis,M.K.Kaiser,P.M.Patre,andW.E.Dixon,\Adaptivedynamicinversionforasymptotictrackingofanaircraftreferencemodel,"inProc.AIAAGuidance,Navigation,andControlConf.,Honolulu,HI,2008,AIAA-2008-6792. [48] A.CaliseandR.Rysdyk,\Nonlinearadaptiveightcontrolusingneuralnetworks,"IEEEControlSystemMagazine,vol.18,no.6,pp.14{25,Dec.1998. [49] J.Leitner,A.Calise,andJ.V.R.Prasad,\Analysisofadaptiveneuralnetworksforhelicopterightcontrols,"JournalofGuidance,ControlandDynamics,vol.20,no.5,pp.972{979,Sept.1997. [50] Y.Shin,\Neuralnetworkbasedadaptivecontrolfornonlineardynamicregimes,"Ph.D.dissertation,GeorgiaTechnicalInstitute,November2005. 125

PAGE 126

E.LavretskyandN.Hovakimyan,\Adaptivecompensationofcontroldependentmodelinguncertaintiesusingtime-scaleseparation,"inProc.ofConf.onDeci-sionandControl,andtheEuropeanControlConf.,Seville,Spain,Dec.2005,pp.2230{2235. [52] B.Xian,D.M.Dawson,M.S.deQueiroz,andJ.Chen,\Acontinuousasymptotictrackingcontrolstrategyforuncertainnonlinearsystems,"IEEETransactionsonAutomaticControl,vol.49,no.7,pp.1206{1211,July2004. [53] P.M.Patre,W.MacKunis,C.Makkar,andW.E.Dixon,\Asymptotictrackingforsystemswithstructuredandunstructureduncertainties,"inProc.ofConf.onDecisionandControl,SanDiego,CA,Dec.2006,pp.441{446. [54] Z.Cai,M.S.deQueiroz,andD.M.Dawson,\Robustadaptiveasymptotictrackingofnonlinearsystemswithadditivedisturbance,"IEEETransactionsonAutomaticControl,vol.51,no.3,pp.524{529,Mar.2006. [55] B.Xian,M.S.deQueiroz,andD.M.Dawson,AContinuousControlMechanismforUncertainNonlinearSystemsinOptimalControl,Stabilization,andNonsmoothAnalysis.Heidelberg,Germany:Springer-Verlag,2004. [56] X.Zhang,A.Behal,D.M.Dawson,andB.Xian,\OutputfeedbackcontrolforaclassofuncertainMIMOnonlinearsystemswithnon-symmetricinputgainmatrix,"inProc.Conf.onDecisionandControl,andtheEuropeanControlConf.,Seville,Spain,Dec.2005,pp.7762{7767. [57] M.L.McIntyre,W.E.Dixon,D.M.Dawson,andI.D.Walker,\Faultidenticationforrobotmanipulators,"IEEETransactionsonRoboticsandAutomation,vol.21,no.5,pp.1028{1034,Oct.2005. [58] S.Gupta,D.Aiken,G.Hu,andW.E.Dixon,\Lyapunov-basedrangeandmotionidenticationforanonaneperspectivedynamicsystem,"inProc.AmericanControlConf.,Minneapolis,MN,June2006,pp.4471{4476. [59] W.E.Dixon,Y.Fang,D.M.Dawson,andT.J.Flynn,\Rangeidenticationforperspectivevisionsystems,"IEEETransactionsonAutomaticControl,vol.48,no.12,pp.2232{2238,Dec.2003. [60] A.Behal,D.M.Dawson,W.E.Dixon,andY.Fang,\Trackingandregulationcontrolofanunderactuatedsurfacevesselwithnonintegrabledynamics,"inProc.ofConf.onDecisionandControl,vol.3,Sydney,Australia,Dec.2000,pp.2150{2155. [61] N.Hovakimyan,F.Nardi,A.Calise,andN.Kim,\Adaptiveoutputfeedbackcontrolofuncertainnonlinearsystemsusingsingle-hidden-layerneuralnetworks,"IEEETransactionsonNeuralNetworks,vol.13,no.6,pp.1420{1431,Nov.2002. [62] P.Hughes,SpacecraftAttitudeDynamics.NewYork:Wiley,1994. 126

PAGE 127

T.Kane,P.Likins,andD.Levinson,SpacecraftDynamics.McGraw-Hill,NewYork,1983. [64] F.Leve,A.Tatsch,andN.Fitz-Coy,\Ascalablecontrolmomentgyrodesignforattitudecontrolofmicro-,nano-,andpico-classsatellites,"inAASGuidanceandControlConf.,Breckenridge,CO,2007. [65] C.J.Heiberg,D.Bailey,andB.Wie,\Precisionspacecraftpointingusingsingle-gimbalcontrolmomentgyroscopeswithdisturbance,"JournalofGuidance,Control,andDynamics,vol.23,no.1,Jan.2000. [66] S.DiGennaro,\Adaptiverobuststabilizationofrigidspacecraftinpresenceofdisturbances,"inProc.ofConf.onDecision&Control,NewOrleans,LA,Dec.1995,pp.1147{1152. [67] W.E.Dixon,A.Behal,D.M.Dawson,andS.P.Nagarkatti,NonlinearControlofEngineeringSystems:aLyapunov-BasedApproach.Boston:Birkhuser,2003. [68] F.L.Lewis,\Nonlinearnetworkstructuresforfeedbackcontrol,"AsianJournalofControl,vol.1,no.4,pp.205{228,1999. [69] F.L.Lewis,J.Campos,andR.Selmic,\Neuro-fuzzycontrolofindustrialsystemswithactuatornonlinearities,"SIAM,2002. [70] F.L.Lewis,A.Yesildirek,andK.Liu,\Multi-layerneuralnetworkcontrollerwithguaranteedtrackingperformance,"IEEETransactionsonNeuralNetworks,vol.7,no.2,Mar.1996. [71] L.Duan,W.Lu,F.Mora-Camino,andT.Miquel,\Flight-pathtrackingcontrolofatransportationaircraft:Comparisonoftwononlineardesignapproaches,"inProc.ofDigitalAvionicsSystemsConf.,Portland,OR,Oct.2006,pp.1{9. [72] I.Szaszi,B.Kulcsar,G.J.Balas,andJ.Bokor,\DesignofFDIlterforanaircraftcontrolsystem,"inProc.ofAmer.ControlConf.,Anchorage,AK,May2002,pp.4232{4237. [73] DepartmentofTransportation,\AirworthinessStandards:Transportcategoryairplanes,"inFederalAviationRegulations-Part25,Washington,DC,1996. [74] F.L.Lewis,C.T.Abdallah,andD.M.Dawson,ControlofRobotManipulators.NewYork,NY:MacMillan,1993. [75] B.S.Davis,T.Denison,andJ.Kaung,\Amonolithichigh-gSOI-MEMSaccelerometerformeasuringprojectilelaunchandightaccelerations,"inProc.ofConf.onSensors,Vienna,Austria,Oct.2004,pp.296{299. [76] V.Janardhan,D.Schmitz,andS.N.Balakrishnan,\DevelopmentandimplementationofnewnonlinearcontrolconceptsforaUA,"inProc.ofDigitalAvionicsSystemsConf.,SaltLakeCity,UT,Oct.2004,pp.12.E.5{121{10. 127

PAGE 128

T.WagnerandJ.Valasek,\Digitalautolandcontrollawsusingquantitativefeedbacktheoryanddirectdigitaldesign,"JournalofGuidance,ControlandDynamics,vol.30,no.5,pp.1399{1413,Sept.2007. [78] M.Bodson,\Multivariableadaptivealgorithmsforrecongurableightcontrol,"inProc.ofConf.onDecisionandcontrol,LakeBuenaVista,FL,Dec.1994,pp.12.E.5{121{10. [79] B.J.Bacon,A.J.Ostro,andS.M.Joshi,\RecongurableNDIcontrollerusinginertialsensorfailuredetection&isolation,"IEEETransactionsonAerospaceandElectronicSystems,vol.37,no.4,pp.1373{1383,Oct.2001. [80] G.Tao,AdaptiveControlDesignandAnalysis,S.Haykin,Ed.Wiley-Interscience,2003. [81] D.Dawson,M.Bridges,andZ.Qu,NonlinearControlofRoboticSystemsforEnvironmentalWasteandRestoration.EnglewoodClis,NewJersey:PrenticeHallPTR,1995. [82] H.K.Khalil,NonlinearSystems,3rded.UpperSaddleRiver,NJ:PrenticeHall,2002. [83] D.EnnsandT.Keviczky,\Dynamicinversionbasedightcontrolforautonomousrmaxhelicopter,"inProc.ofAmericanControlConf.,Minneapolis,MN,June2006,pp.3916{3923. [84] P.M.Patre,W.MacKunis,C.Makkar,andW.E.Dixon,\Asymptotictrackingforsystemswithstructuredandunstructureduncertainties,"IEEETransactionsonControlSystemsTechnology,vol.16,no.2,pp.373{379,2008. [85] M.Bridges,D.M.Dawson,andC.Abdallah,\Controlofrigid-linkexible-jointrobots:Asurveyofbacksteppingapproaches,"Jnl.RoboticSystems,vol.12,pp.199{216,1995. [86] R.LozanoandB.Brogliato,\Adaptivecontrolofrobotmanipulatorswithexiblejoints,"IEEETransactionsonAutomaticControl,vol.37,no.2,pp.174{181,Feb.1992. 128

PAGE 129

BIOGRAPHICALSKETCH WilliamMacKunisreceivedhisbachelor'sdegreeinelectri calengineeringfromFlorida AtlanticUniversity(FAU)inMayof2000.Hethenwentonton ishhismaster'sdegree fromFAUin2003.Specializingincontrols,hismaster'sthe sisresearchwasbasedon autonomous,multi-agentcontrolofmobilerobots. InMayof2009,WilliamcompletedhisPh.D.inaerospaceengi neeringinthe DepartmentofMechanicalandAerospaceEngineeringattheU niversityofFlorida underthesupervisionofDr.WarrenDixon. 127