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A Differential Games Approach for Analysis of Spacecraft Post-Docking Operations

Permanent Link: http://ufdc.ufl.edu/UFE0021867/00001

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Title: A Differential Games Approach for Analysis of Spacecraft Post-Docking Operations
Physical Description: 1 online resource (147 p.)
Language: english
Creator: Hiramatsu, Takashi
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

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

Notes

Abstract: An increase of responsive space assets will contribute to the growing number of spacecraft in space and in turn the growing potential for failures to occur. The number of spacecraft which has past its operational life also keeps increasing. Without proper treatment these satellites become space debris, which could lead to more failure due to collision with other spacecraft. Thus, there will be a need for effective debris abatement (i.e., repair and /or disposal of failed satellites) which will require autonomous service satellites. Such a "tow truck" concept is expected to take a crucial role for sustainable small satellite utilization in the future. Current and past investigation where autonomous docking plays an important role have all considered "cooperative" interactions between satellites. That is, either the target has the same goals as the service vehicle or the target is not actuated and passively follows the lead of the service vehicle. Cooperative scenarios are not always guaranteed, thus it is imperative that we consider docking scenarios with "noncooperative" targets. Such noncooperative scenarios involve motion which may be resistible or unpredictable (i.e., the target may have lost its control authority and thus its motion becomes resistible or it may be "adversarial" and is maneuvering to avoid capture). Maintaining a post-docked state requires a control system which minimizes the effects of the uncertain interactions due to a noncooperative behavior. In the robust control sense, such uncertainty needs to be upper-bounded in order to develop effective control strategies. For that purpose it is important to characterize this uncertain interactions. An approach to approximate this uncertainty is to model the docked state of spacecraft as a differential game with each spacecraft being the player, where the interactions are the outcome of the gameplay. Differential games is a class of game theory that describe a dynamical system with multiple control inputs with different objectives. Each input which actively affects the behavior of the system (e.g., control inputs, noise, disturbance, and other external inputs) is called a player of the game, and those players cooperate with or compete against one another to achieve their objectives. This manuscript addresses the characterization of a noncooperative behavior expected in satellite post-docking, and the corresponding controller law to achieve docking-state maintenance through modeling and solving the two-person Stackelberg differential game
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 Takashi Hiramatsu.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Fitz-Coy, Norman G.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0021867:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021867/00001

Material Information

Title: A Differential Games Approach for Analysis of Spacecraft Post-Docking Operations
Physical Description: 1 online resource (147 p.)
Language: english
Creator: Hiramatsu, Takashi
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

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

Notes

Abstract: An increase of responsive space assets will contribute to the growing number of spacecraft in space and in turn the growing potential for failures to occur. The number of spacecraft which has past its operational life also keeps increasing. Without proper treatment these satellites become space debris, which could lead to more failure due to collision with other spacecraft. Thus, there will be a need for effective debris abatement (i.e., repair and /or disposal of failed satellites) which will require autonomous service satellites. Such a "tow truck" concept is expected to take a crucial role for sustainable small satellite utilization in the future. Current and past investigation where autonomous docking plays an important role have all considered "cooperative" interactions between satellites. That is, either the target has the same goals as the service vehicle or the target is not actuated and passively follows the lead of the service vehicle. Cooperative scenarios are not always guaranteed, thus it is imperative that we consider docking scenarios with "noncooperative" targets. Such noncooperative scenarios involve motion which may be resistible or unpredictable (i.e., the target may have lost its control authority and thus its motion becomes resistible or it may be "adversarial" and is maneuvering to avoid capture). Maintaining a post-docked state requires a control system which minimizes the effects of the uncertain interactions due to a noncooperative behavior. In the robust control sense, such uncertainty needs to be upper-bounded in order to develop effective control strategies. For that purpose it is important to characterize this uncertain interactions. An approach to approximate this uncertainty is to model the docked state of spacecraft as a differential game with each spacecraft being the player, where the interactions are the outcome of the gameplay. Differential games is a class of game theory that describe a dynamical system with multiple control inputs with different objectives. Each input which actively affects the behavior of the system (e.g., control inputs, noise, disturbance, and other external inputs) is called a player of the game, and those players cooperate with or compete against one another to achieve their objectives. This manuscript addresses the characterization of a noncooperative behavior expected in satellite post-docking, and the corresponding controller law to achieve docking-state maintenance through modeling and solving the two-person Stackelberg differential game
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 Takashi Hiramatsu.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Fitz-Coy, Norman G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0021867:00001


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ADIFFERENTIALGAMESAPPROACHFORANALYSISOFSPACECRAFT POST-DOCKINGOPERATIONS By TAKASHIHIRAMATSU ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012

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c 2012TakashiHiramatsu 2

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Idedicatethistoeveryonethathelpedmewritethismanuscript. 3

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ACKNOWLEDGMENTS MybiggestappreciationgoestomyadvisorDr.NormanG.Fitz-Coyforhisgreat helpandsupport.EverytimeItalkedtohimhemotivatedmewithcriticalresponses andencouragedmewheneverIwasstuckinthemiddleofmyresearch.Ialsothankmy committeeDr.WarrenDixon,Dr.GloriaWiens,andDr.WilliamHagerfortheirsupports. Finally,IthankmycolleaguesinSpaceSystemsGroupandallotherfriends,who directlyorindirectlyhelpedmethroughouttheyearsIspentatUniversityofFlorida. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................8 LISTOFFIGURES.....................................9 ABSTRACT.........................................11 CHAPTER 1INTRODUCTION...................................13 1.1SpacecraftRendezvousandDocking.....................13 1.1.1CooperativeScenarios.........................14 1.1.2NoncooperativeScenarios.......................14 1.2SmallSatellites.................................15 1.3GameTheoreticApproach...........................15 2MATHEMATICALBACKGROUNDFORTHEAPPROACH............18 2.1DifferentialGamesandControlTheory....................18 2.1.1MinimaxStrategy............................20 2.1.2NashStrategy..............................20 2.1.3StackelbergStrategy..........................21 2.1.4Open-LoopStrategiesforTwo-PersonLinearQuadraticDifferential Games..................................23 2.2NumericalMethodstoOptimalControlProblem...............24 2.3BilevelProgramming..............................25 3TECHNICALDESCRIPTION............................27 3.1ReductionofStackelbergDifferentialGamestoOptimalControl......30 3.2ConversionofStackelbergDifferentialGamestoStackelbergStaticGames32 3.3CostateMappingofStackelbergDifferentialGames............35 3.4Conclusion...................................45 4DYNAMICSOFDOCKEDSPACECRAFT.....................47 4.1FormulationofDynamics...........................47 4.1.1RelativeMotionDynamicsofaSatellite...............47 4.1.1.1Translation..........................48 4.1.1.2Rotation............................48 4.1.2DynamicsofTwoDockedSatellites..................49 4.2Simulation....................................52 4.2.1CaseI:NonzeroLinearVelocity....................52 4.2.2CaseII:NonzeroRotationalVelocity.................58 5

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4.2.3CaseIII:NonzeroLinearandRotationalVelocities.........62 4.3Conclusion...................................66 5LINEARCONTROLLERDESIGNWITHSTACKELBERGSTRATEGY.....67 5.1Post-DockingStudywithLinearQuadraticGame..............67 5.2SimulationandResults............................69 5.3Conclusion...................................75 6SOLUTIONSTOTWO-PLAYERLINEARQUADRATICSTACKELBERGGAMES WITHTIME-VARYINGSTRUCTURE........................77 6.1GameBasedonAdditiveErrors........................78 6.1.1Open-loopStackelbergSolution....................80 6.1.2Closed-loopStackelbergSolution...................85 6.2GameBasedonMultiplicativeErrors.....................92 6.2.1Open-loopStackelbergSolution....................94 6.2.2Closed-loopStackelbergSolution...................99 6.3SimulationsandResults............................99 6.4Conclusion...................................106 7CONCLUSIONANDFUTUREWORKS......................108 APPENDIX AOPTIMALITYCONDITIONSOFTWO-PERSONSTACKELBERGDIFFERENTIAL GAMES........................................109 A.1FixedFinalTime................................109 A.1.1Follower'sStrategy...........................109 A.1.1.1Variationoftheaugmentedcostfunctional........110 A.1.1.2Optimalityconditions....................112 A.1.2Leader'sStrategy............................112 A.1.2.1Variationoftheaugmentedcostfunctional........113 A.1.2.2Optimalityconditions....................115 A.2FreeFinalTime.................................115 A.2.1Follower'sStrategy...........................116 A.2.1.1Variationoftheaugmentedcostfunctional........116 A.2.1.2Optimalityconditions....................118 A.2.2Leader'sstrategy............................119 A.2.2.1Variationoftheaugmentedcostfunctional........119 A.2.2.2Optimalityconditions....................122 A.3Linear-QuadraticDifferentialGame......................122 A.3.1FixedFinalTime............................123 A.3.2FreeFinalTime.............................123 6

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BRISESTABILITYANALYSIS............................126 B.1RiseFeedbackControlDevelopment.....................126 B.2StabilityAnalysis................................128 CCOSTATEESTIMATIONFORTHETRANSCRIBEDSTACKELBERGGAMES132 C.1TransformedOptimalityConditions......................132 C.2DiscretizationofTwo-personStackelbergDifferentialGames.......134 C.3KKTConditionsandCostateMapping....................136 REFERENCES.......................................141 BIOGRAPHICALSKETCH................................147 7

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LISTOFTABLES Table page 4-1ThesimulationparametersforCaseI.......................55 4-2ThesimulationparametersforCaseII.......................59 4-3ThesimulationparametersforCaseIII.......................63 5-1Thesimulationparametersforthelinearquadraticgame.............72 6-1ThesimulationparametersfortheStackelberg-RISEcontroller.........105 8

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LISTOFFIGURES Figure page 1-1Adesigniterationthroughsatellitepost-dockinganalysis.............17 3-1Relationshipamongoptimizationproblems....................29 3-2Directandindirectmethods.............................30 4-1Arepresentationofthepositionofasatellitewiththeinertialandthenominal referenceframes....................................47 4-2Asatellitewithabody-xedreferenceframe F i ..................48 4-3Anexeggeratedviewoftwosatellitesnearthenominalorbit...........50 4-4Twosatellitesinitiallyonthesamenominalorbit..................53 4-5Twosatellitesinitiallyradiallyaligned........................54 4-6CaseI:theinteractionforcesappliedtotheSVandtheRSO..........54 4-7CaseI:theinteractiontorquesappliedtotheSVandtheRSO.........56 4-8CaseI:thelinearmotionoftheRSOrelativetotheSV..............56 4-9CaseI:therotationalmotionoftheRSOrelativetotheSV............57 4-10CaseII:theinteractionforcesappliedtotheSVandtheRSO..........58 4-11CaseII:theinteractiontorquesappliedtotheSVandtheRSO.........60 4-12CaseII:thelinearmotionoftheRSOrelativetotheSV.............60 4-13CaseII:therotationalmotionoftheRSOrelativetotheSV...........61 4-14CaseIII:theinteractionforcesappliedtotheSVandtheRSO..........62 4-15CaseIII:theinteractiontorquesappliedtotheSVandtheRSO.........64 4-16CaseIII:thelinearmotionoftheRSOrelativetotheSV.............64 4-17CaseIII:therotationalmotionoftheRSOrelativetotheSV...........65 5-1Tworigidbodiesoncircularorbits..........................67 5-2Theresultanttrajectory................................73 5-3Thecontrolforceinputs................................74 5-4Thecontroltorqueinputs...............................75 6-1Therelationshipamongthecurrentandthedesiredorientations.........92 9

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6-2Twodockedsatellitesapproximatedastworigidbodiesconnectedviaatorsion spring.........................................100 6-3 f t and g t asrespectiveweightsonthegameandanarbitrarydisturbances.104 6-4ThesimulationresultsfortheStackelbergandRISEcontroller.........107 10

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ADIFFERENTIALGAMESAPPROACHFORANALYSISOFSPACECRAFT POST-DOCKINGOPERATIONS By TakashiHiramatsu August2012 Chair:NormanG.Fitz-Coy Major:MechanicalEngineering Anincreaseofresponsivespaceassetswillcontributetothegrowingnumberof spacecraftinspaceandinturnthegrowingpotentialforfailurestooccur.Thenumber ofspacecraftwhichhaspastitsoperationallifealsokeepsincreasing.Withoutproper treatmentthesesatellitesbecomespacedebris,whichcouldleadtomorefailuredueto collisionwithotherspacecraft.Thus,therewillbeaneedforeffectivedebrisabatement i.e.,repairand/ordisposaloffailedsatelliteswhichwillrequireautonomousservice satellites.Suchatowtruckconceptisexpectedtotakeacrucialroleforsustainable smallsatelliteutilizationinthefuture.Currentandpastinvestigationwhereautonomous dockingplaysanimportantrolehaveallconsideredcooperativeinteractionsbetween satellites.Thatis,eitherthetargethasthesamegoalsastheservicevehicleorthe targetisnotactuatedandpassivelyfollowstheleadoftheservicevehicle.Cooperative scenariosarenotalwaysguaranteed,thusitisimperativethatweconsiderdocking scenarioswithnoncooperativetargets.Suchnoncooperativescenariosinvolvemotion whichmayberesistibleorunpredictablei.e.,thetargetmayhavelostitscontrol authorityandthusitsmotionbecomesresistibleoritmaybeadversarialandis maneuveringtoavoidcapture. Maintainingapost-dockedstaterequiresacontrolsystemwhichminimizesthe effectsoftheuncertaininteractionsduetoanoncooperativebehavior.Intherobust controlsense,suchuncertaintyneedstobeupper-boundedinordertodevelop 11

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effectivecontrolstrategies.Forthatpurposeitisimportanttocharacterizethisuncertain interactions.Anapproachtoapproximatethisuncertaintyistomodelthedockedstate ofspacecraftasadifferentialgamewitheachspacecraftbeingtheplayer,wherethe interactionsaretheoutcomeofthegameplay. Differentialgamesisaclassofgametheorythatdescribeadynamicalsystem withmultiplecontrolinputswithdifferentobjectives.Eachinputwhichactivelyaffects thebehaviorofthesysteme.g.,controlinputs,noise,disturbance,andotherexternal inputsiscalledaplayerofthegame,andthoseplayerscooperatewithorcompete againstoneanothertoachievetheirobjectives. Thismanuscriptaddressesthecharacterizationofanoncooperativebehavior expectedinsatellitepost-docking,andthecorrespondingcontrollerlawtoachieve docking-statemaintenancethroughmodelingandsolvingthetwo-personStackelberg differentialgame. 12

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CHAPTER1 INTRODUCTION Thisdissertationisaimedatoutlininganapproachtoestimatinganoncooperative behaviorofthetargetspacecraftinthepost-dockingstate,thecorrespondinginteraction betweenthedockedspacecraft,andtherequiredcontrolstrategytomaintainthestate. Inthischapterthebackgroundofspacecraftdockingandmotivationsforastudyof noncooperativepost-dockingarepresented. TheSputnik1,theworldrstsatellitewaslaunchedin1957.Sincethen,morethan 6,000satelliteshavebeenlaunchedtospace,andcurrentlyabout3,600satellitesare operationalinspace.Advancingofthespacetechnologiesnecessitatedtransportation ofastronautsbetweenspacecrafts,constructionofspacestations,etc.,whichwere carriedoutthroughspacecraftrendezvousanddockingR&D. Aspacecraftdockingandrendezvousplayimportantrolesinspaceutilities[1]. SincetherstrendezvousofGeminiVI-Ain1965,rendezvoushavebeenusedincases includingtransportationofastronautsbetweenspacecraftandconstructionofaspace station.Forexample,HTV-1wasdockedwiththeinternationalspacestationISSfor refueling[2]. 1.1SpacecraftRendezvousandDocking Therearemanyspaceapplicationsinvolvingdockingofspacecraft,satellites,and objects.Manyofthecurrentandpastdockingscenariosareofcooperativenature; beforetwospacecraftaredockedtheyhavetorendezvous,whichcanbeachievedifi bothspacecraftworktogethertomatchtheirmotions,oriioneofthemisstationaryor inconstantmotionsuchthattheothercanadjusttomatchit.Dockingwithaspacecraft whichistumblingisconsiderednon-cooperative.Thusmannedmissionsareinherently cooperative,whileun-mannedmissionswithautonomousrendezvousanddockingcould benoncooperative. 13

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1.1.1CooperativeScenarios Servicingoperationsandreturnmissionsassumethetargetscooperatetoget servicesortodock.ExamplesincluderefuelingOrbitalExpress[3,4],ConeXpress[5], HTV-1KOUNOTORI[2],andHTV-2KOUNOTORI2[6],towingOrbitalmaneuvering vehicleOMV[7],andrepairing,suchastheservicingmissionstorepairHubbleSpace Telescope[811]. 1.1.2NoncooperativeScenarios Cooperativescenariosarenotguaranteedforallfuturemissionsandthelikelihood ofdockingwithnoncooperativetargetisquitehigh.Forexample,themotionofthetarget mayeitherbeunpredictableorresistiblei.e.,thetargetmotionisnotfullyundercontrol orfavorable.Cook[12]denedthenon-cooperativetargetasanyspacecraftwhichis eithernotdesignedfordockingortumblingfreelyinspace.Inthefutureitispossibleto addadversarialtargets,whichspecicallytrytoavoiddockingandrendezvous,although militaryuseofspaceresourcesisprohibitedbythecurrentspacelaw[13]. Thereareafewimportantapplicationsinthefuture.Collisionavoidanceof near-Earthasteroids,forexample,hasdrawnattentions[14,15].Dockingmaneuvers areinvolvedwhensensorsareplacedonasteroidstotracktheirmotion,orwhen actuatingorexplosivedevicesareattachedtochangethecourseoftheirmotions. Anotherexampleisspacedebris.Severalrecenteventscontributedtotherapid growthofthedebrispopulation;China'sASAToperationin2007[16],thedestruction ofUSA-193in2008[17],andthecollisionbetweenIridiumandCOSMOSin2009 [18].Liou[19]estimatedthepropagationofspacedebrisandshowedthatthedebris populationkeepsgrowingevenifnomorespacecraftislaunched;theexistingdebriswill collidewithspacecraftorotherdebristoincreaseitsnumber.Therefore,bothprevention andremovalofspacedebrisareimportant.Oneofthemotivationalexamplesisthe caseofUSA-193[17],areconnaissancesatellitewhichbecamedisabledonorbit andeventuallygotdestroyedbyamissile.Theoperationnotonlywascostlybutalso 14

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generateddebris,justliketheASAToperation.Ifthereexistsatechnologytosafelydeal withnon-cooperativetargetse.g.,aspacetowtrucktocaptureandtakethemtothe graveyardorbit,suchaproblemcanbepreventedwithsmallerdamages. Severalactivedebrisremovaltechnologieshavebeenproposed,includingthe Remoraremover TM [20],andthemicroremover[21,22]. 1.2SmallSatellites Inthepast,mission-specicbigmonolithicsatellitesweredevelopedandserved formostofthespaceapplications.Butrecentlysmallsatelliteshavedrawnattentions fortheirshort-time,low-costdevelopment,andtheirversatileapplications.Theideaof constellationofsmallsatellitesisexpectedasareplacementofsomeofthetasksthat havebeentakenbytraditionalsatellites. However,havingmoresatellitesinspaceincreasestherisksofmorespacedebris. Smallsatellitesusuallyhaveahigherorbitallifethantraditionalsatellitesduetohigher ballisticcoefcientsandlowerorbits[23].Thelongorbitallifeofsmallsatellitesalong withthefactthataftertheendoftheirservicetheybecomedebriswillincreasethe threat.Furthermore,thehigherthenumberofspacecraftgets,themorelikelythefailure couldoccur.Theutilizationofsmallsatellitesforpracticalapplicationsrequiresmany satellitesbynature,andthustherewillbeaneedforeffectivedebrisabatementi.e., removethefailedsatellitesforrepairand/ordisposalwhichrequiresabilitytoworkwith non-cooperativedebris.Andwhileseveralworkshavebeendoneonnon-cooperative docking,non-cooperativepost-dockinghasnotdrawnmuchattention. 1.3GameTheoreticApproach Recent,current,andfutureactivitiesnecessitatedevelopmentofautonomous spacecraftrendezvousanddockingtechnology,withtargetspacecrafthavingnon-cooperative characteristics.Dealingwithnon-cooperativetargetsalsoemphasizestheimportance ofpost-dockingmaintenance,whichrequiresthedesignofacontrolsystemtominimize theeffectsofuncertaininteractionsduetothenon-cooperativebehaviorofthetarget 15

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spacecraft.Intherobustcontrolsensesuchuncertaintyneedstobeupper-boundedin ordertodevelopeffectivecontrolstrategies.Thereforeitisimportanttocharacterizethe non-cooperativebehaviorandthecorrespondinginteractions. Inordertosuccessfullyachievedockingandmaintainthedockedstatebetween twospacecraft,accurateinformationoftheirdynamicbehaviorsisrequiredsothatthe correspondinginteractionscanbecontrolled.Incooperativedockingmaneuvers,where twofullyfunctionalspacecraftworktogethertodockwitheachother,sucharequirement hasbeentackledbyseveraleffortsalready.Ontheotherhand,innon-cooperative dockingmaneuversandthecorrespondingpost-dockingmaneuversitisdifcultto analyzetheinteractionbecauseonespacecraftwillnotactinaccordancewiththeother. Inthisdissertationadifferentialgamestheoreticapproachisemployedtoestimating thebehaviorofthetargetnon-concooperativespacecraftinthepost-dockingsituation, thecorrespondinginteractions,andtherequiredcontrolstrategytomaintainthedocked state.Foraspeciccasewheretwosatelliteswithknownspecicationssuchasmass, size,andpowerareconsidered,adedicatedmethodofsimulationsuchasMonteCarlo methodworkswell.However,ifthegeneralbehaviorsofarbitrarysatellitesaretobe studied,thenitisbenecialtoknowhoweachparameterofthedesignspecicationof satellitesaffectsthepost-docking. Characterizingthenon-cooperativepost-dockingbehaviorsasafunctionof thedesignparametersallowsforconsiderationofdifferentpost-dockingscenarios. Figure1-1showsaniterativedesignprocesspossiblewithgametheory 1 .Adifferential gameproblemwillbeformulatedsuchthatthetowtrucktheservicevehicle,SVand thenon-cooperativetargettheresidentspaceobject,RSOrespectivelychoosetheir actuationcommandafterdockingtoaffecttheinteractionsbetweenthem.Withasetof simulationparametersincludingthespecicationsofthesatellites,theproblemwillbe 1 MoredetailsondifferentialgametheoryisdiscussedinChapter2 16

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solvedtoyieldthepossibleinteractionsandthecontrolactuationsrequiredtoachieve them.Thatinformationcanbeusedasafeedbacktoredenethedesignspecications ofthesatellitestobebuilte.g.,iftheinteractionsarekeptsmallbuttherequiredcontrol effortsfortheSVaretoohigh,thesimulationcanbeadjustedtoweighmoretowards loweringthecontroleffortsatthecostofhigherinteractions. Figure1-1.Adesigniterationthroughsatellitepost-dockinganalysis. Theanalysisisexpectedtocontributetoanestablishmentofanewtechnology forthefuturespaceapplication.InChapter2severaltechnicalbackgroundsincluding gametheorywillbedescribed.InChapter3approachestosolvingthedifferentialgame problemwillbediscussed.InChapter4thedynamicsofthesatellitepost-dockingwillbe investigated.Thesolutionstotheparticulargame-basedcontroldesignproblemswillbe presentedinChapter5andChapter6. 17

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CHAPTER2 MATHEMATICALBACKGROUNDFORTHEAPPROACH 2.1DifferentialGamesandControlTheory Gametheoryisastudyofconictamongmultiplegroupsorindividualsplayers makingdecisionsincompetitivesituations[24].Staticgamesisatypeofgameswhere eachplayermakestheirdecisionsimultaneouslywithouttheknowledgeofthedecisions oftheothers.Dynamicgamesorsequentialgamesisanextensionofstaticgames whereeitherthedecisionofeachplayerismadeinordere.g.,atwo-levelgame whereaplayerdesignatedastheleaderchoosestheirmoveandthentheotherplayer asthefollowerchoosestheirs,basedontheleader'sactionorthegameisplayed multipletimeswiththeplayersabletolearnfromtheresultsofthepastgames.Dynamic gamesmaybeplayedwithrulesdescribedbyasetofdifferentialequationse.g.,a pursuit-evasiongameplayedbytwoaircraftsubjecttotheirrespectiveequationsof motion.Suchgameisdifferentialgames[25]. Noncooperativedifferentialgametheoryhasbeenappliedtoavarietyofcontrol problems[2639].Whilezero-sumdifferentialgameshavebeenheavilyexploited innonlinear H 1 controltheory,nonzero-sumdifferentialgameshavehadlimited applicationinfeedbackcontrol.Inparticular,Stackelbergdifferentialgames,which isbasedonahierarchicalrelationshipbetweentheplayers,havebeenutilizedin adecentralizedcontrolsystem[30],hierarchicalcontrolproblems[28,29,37],and nonclassicalcontrolproblems[31].Differentialgames,aswellasoptimalcontrol,are difculttoolstoapplybecauseofthechallengesassociatedwithdetermininganalytical solutions,withafewexceptionssuchasthelinearquadraticstructure.Onewayto incorporateoptimalcontrolanddifferentialgamestructuresistoformulateasystem composedofcontroltermstofeedbacklinearizeandadditionalcontroltermstooptimize theresidualsystem.Forexample,optimalcontrolleraredevelopedwithfeedback linearizationwithexactmodelknowledgeassumption[40]andvianeuralnetworks 18

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[4143].In[44]anopen-loopStackelberggame-basedcontrollerisdevelopedbasedon theRobustIntegraloftheSignoftheErrorRISE[4547]technique. InordertodesignaSVforaspaceoperationtodealwithanon-cooperative interaction,itisnecessarytostudytheirdynamicbehaviorsaswellaspropercontrol architecture.Controllingeachindividualsatellitewhichinterfereoneanotherrequires gametheoreticconsideration.Multiobjectiveoptimizationproblems,witheachobjective possiblyconictingoneanother,havebeenstudiedintheframeworkofgametheory. ThearchitectureofdifferentialgameswasrstdeveopedbyIsaacs[25]andhasbeen appliedtovariousengineeringapplications.Gametheoreticapproachtodesignthe controllercanhandletheoptimalcontrolofmultipleobjectswithconictingobjectives; evenwhenthemotionoftheRSOisunknownandthereforenon-cooperative,itmay stillbepossibletoanalyzetheinteractionbetweentheSVandtheRSO,i.e.,howmuch forcesareappliedtothevehiclesorneededtoapply. Thesimplestformofatwo-persondifferentialgameisdenedasfollows:The systemisgivenbyadifferentialequations x = f x u 1 u 2 t x t 0 = x 0 consistingoftwoindependentcontrolinputs u 1 and u 2 .Byconventioneachcontrol inputisassignedtoa player ,suchthat u 1 isdesignedbyPlayer1and u 2 isdesigned byPlayer2.Eachplayerchoosesitscontrolstrategyinsuchawaythatitminimizesthe correspondingcostfunctionals J 1 u 1 u 2 = 1 t 0 t f x 0 x f + Z t f t 0 L 1 x u 1 u 2 t dt J 2 u 1 u 2 = 2 t 0 t f x 0 x f + Z t f t 0 L 2 x u 1 u 2 t dt Unlikeoptimalcontrolproblems,thissetisnotwell-posed.WHyInanoptimalcontrol problemtheoptimalsolution,ifexists,guaranteesthatthecostisminimizedtosatisfy alltheconstraints,whileinatwo-persondifferentialgameingeneraltheminimumcost 19

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ofplayer1and2cannotbeobtainedsimultaneously.Minimizationof J 1 ofteninterferes minimizationof J 2 andviceversa.Inordertosolvefor u 1 and u 2 ,strategiestodene thenatureoftheequilibriumsolution,inotherwordshowthegameisplayed,need tobeimposed.Formodellingthenon-cooperativeinteractions,Minimax,Nash,and Stackelbergstrategiesareconsidered. 2.1.1MinimaxStrategy InMinimaxstrategy,byassumingtheworstcaseandtryingtominimizethe damage,onecanobtainthesolutionwhichisthesafest.Minimaxconsidersazero-sum gamethecostsofallplayersadduptozerosuchthat J 1 = )]TJ/F57 11.9552 Tf 9.298 0 Td [(J 2 = J thereforeeachcontrolstrategyisexpressedas u 1 =argmin u 1 J 1 =argmin u 1 J u 2 =argmin u 2 J 2 =argmax u 2 J therefore,ifthesolutionexists,itwillbeasaddle-pointsolution u 1 =argmin u 1 max u 2 J Minimaxstrategiesareforzero-sumdifferentialgames,butevenwhenagameis nonzero-sum,itisusefultoconsiderminimaxfornoncooperativecasesbecausethe Player1maynotknowtheobjectiveofthePlayer2;if J 2 isunknown,byassuming J 2 = )]TJ/F57 11.9552 Tf 9.298 0 Td [(J 1 thePlayer1isabletoestimatethepossibleinteractionwiththePlayer2. 2.1.2NashStrategy InNashstrategyeachplayertriestooptimizetheirobjectivewithoutcaringabout oneanotherhoweveritcouldresultintheequilibriumsolutionwhichisnotoptimalto eachindividualplayere.g.,Prisoner'sDilemma.Inatwo-persongameNashstrategy eachplayertriestominimizetheircostsimultaneously,knowingthattheotherplayer 20

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doesthesame.Therefore,giventhecostofeachplayeras J 1 u 1 u 2 and J 2 u 1 u 2 ,the Nashstrategy f u 1n u 2n g shouldsatisfythefollowing: J 1 u 1 n u 2 n J 1 u 1 u 2 n J 2 u 1 n u 2 n J 2 u 1 n u 2 thatis,ifoneplayerchangestheirstrategy,itresultsinincreaseintheircost.Therefore forzero-sumgamesthesolutionisthesameasMinimaxsolution.Fornonzero-sum gamesthenecessaryconditionsforexistenceofNashsolutionhavebeendevelopedby [48]. Ingeneral,therearemorethanonesetoftheequilibriumsolutionforNashstrategy, ortheremayexistnosolutionatall.Foraspecialstructureofthegame,linearquadratic differentialgame,wherethecostfunctionalsarequadraticinthestateandthecontrol, subjecttotheconstraintwhichisadifferentialequationlinearinthestateandthe control. 2.1.3StackelbergStrategy Stackelbergstrategyassumesoneplayertheleaderhasanadvantageoverthe otherthefollowerinminimizingitscost.InStackelbergstrategy,theleadercanenforce theiraction,andtheresultantequilibriumsolutionisalwaysfavorabletotheleader.With Player2astheleader,thecostofeachplayerassociatedwiththeStackelbergstrategy isasfollows: J 1 u 1 s u 2 s J 1 u 1 u 2 s J 2 u 1 s u 2 s J 2 u 1 u 2 wherethesubscript s denotestheStackelbergstrategy.Equation2showsthat resultantcostPlayer1followerachieveisbetteroffplayingtheStackelbergstrategy iftheknowledgeofPlayer2leaderplayingtheStackelbergstrategyisavailable. StackelbergsolutionisalwaysbetterthanorequaltotheNashsolution[49],suggesting 21

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thattheStackelbergstrategycancharacterizebothhierarchicalandnon-hierarchical cases. J 2 u 1 s u 2 s J 2 u 1 n u 2 n InStackelberggames,theleaderchoosesitsstrategyrst.Itmustbenoted, however,thattheorderofchoosingthestrategydoesnotnecessarilymeanthatone playerphysicallyactsbeforetheother.Nextthefollowerchoosesitsstrategysuchthat itminimizesitscostgiventheleader'sstrategy.Thustothefollowerthegameismerely anoptimizationproblemwhereitminimizesthecostwithanystrategyprovidedbythe leader.Anoptimalcontrolinputofthefollowerisdenedby u 1 u 2 =argmin u 1 J 1 u 1 u 2 thatis,thefollower'scontrolinputisoptimaltoanarbitraryinputoftheleader.When theleaderplaysStackelbergstrategy,itassumesthatthefollowerplaysStackelberg strategy;thatthefollower'sdecisionisbasedonEq.2.Thereforetheleader choosesitsdecisionsuchthatitsolvestheoptimalcontrolproblemgiven u 1 : u 2 s =argmin u 2 J 2 u 1 u 2 u 2 OncetheStackelbergsolutionoftheleaderisobtained,thentheStackelbergsolutionof thefollowerisalsofoundas u 1 s = u 1 u 2 s =argmin u 1 J 1 u 1 u 2 s whichisbasedonthefollower'sassumptionthattheleaderknowsthefollowerfollows theleader,andthattheleadertriestooptimizeitsobjectiveaccordingly.IfEq.2can besolvedanalyticallyasafunctionof u 2 thenEq.2isautomaticallydetermined onceEq.2isfound,allowing u 1 s and u 2 s tobesolvedsequentiallyandseparately, notsimultaneously. 22

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Theconceptwasrstappliedtodifferentialgamesin[50],andlatersolvability conditionsweredevelopedin[51,52].InasimilarmannertotheNashstrategy,the uniquenessofsolutionisnotguaranteedingeneral,exceptforlinearquadraticcases. 2.1.4Open-LoopStrategiesforTwo-PersonLinearQuadraticDifferentialGames Thesolutionsoftwo-persondifferentialgamesarefoundbysolvingtheoptimality conditionsobtainedbythecalculusofvariations, 1 buttheboundary-valueproblems aresocomplicatedthatinmanycasestheydon'thaveanalyticalsolutions.However, theanalyticalsolutionexistsforasimpleproblem.Two-personlinearquadraticLQ differentialgamesisaclassofdifferentialgames,wheredynamicconstraintsgivenby Eq.2aredescribedbyasetoflineardifferentialequations x = d x dt = Ax + B 1 u 1 + B 2 u 2 x t 0 = x 0 andthecostfunctionalsgivenbyEqs.2-2aredescribedbyquadraticform J 1 u 1 u 2 = 1 2 x T f K 1 f x f + 1 2 Z t f t 0 )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(x T Q 1 x + u T 1 R 11 u 1 + u T 2 R 12 u 2 dt J 2 u 1 u 2 = 1 2 x T f K 2 f x f + 1 2 Z t f t 0 )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(x T Q 2 x + u T 1 R 21 u 1 + u T 2 R 22 u 2 dt Inthiscasethecontrolstrategiesarealsolinearinthestates x u 1 = )]TJ/F59 11.9552 Tf 9.299 0 Td [(R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T 1 K 1 x u 2 = )]TJ/F59 11.9552 Tf 9.299 0 Td [(R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T 2 K 2 x where K 1 and K 2 arethesolutionsoftheRiccatidifferentialequations,whichare associatedwiththeoptimalityconditionsfortheLQdifferentialgamesandvarywith 1 ThederivationoftheoptimalityconditionsisprovidedinAppendix. 23

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strategies.ForMinimaxstrategies, K 1 = )]TJ/F59 11.9552 Tf 9.299 0 Td [(K 1 A )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T K 1 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 1 + K 1 B 1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B 1 K 1 + K 1 B 2 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 12 B 2 K 1 K 1 t f = K 1 f K 2 = )]TJ/F59 11.9552 Tf 9.299 0 Td [(K 2 A )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T K 2 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 2 + K 2 B 1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 21 B 1 K 2 + K 2 B 2 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B 2 K 2 K 2 t f = K 2 f ForNashstrategies, K 1 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(K 1 A )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T K 1 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 1 + K 1 B 1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T 1 K 1 + K 1 B 2 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T 2 K 2 K 1 t f = K 1 f K 2 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(K 2 A )]TJ/F59 11.9552 Tf 11.956 0 Td [(A T K 2 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 2 + K 2 B 2 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T 2 K 2 + K 2 B 1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T 1 K 1 K 2 t f = K 2 f ForStackelbergstrategies,thereisanadditionalsetofdifferentialequationsthatneedto besolved. K 1 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(K 1 A + K 1 B 1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T 1 K 1 + K 1 B 2 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T 2 K 2 )]TJ/F59 11.9552 Tf 11.956 0 Td [(A T K 1 )]TJ/F59 11.9552 Tf 11.956 0 Td [(Q K 1 t f = K 1 f K 2 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(K 2 A + K 2 B 1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T 1 K 1 + K 2 B 2 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T 2 K 2 )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T K 2 )]TJ/F59 11.9552 Tf 11.956 0 Td [(Q 2 + Q 1 P K 2 t f = K 2 f P = )]TJ/F59 11.9552 Tf 9.299 0 Td [(PA + PB 1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T 1 K 1 + PB 2 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T 2 K 2 + AP )]TJ/F59 11.9552 Tf 11.955 0 Td [(B 1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 R T 21 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T 1 K 1 + B 1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T 1 K 2 P t f = 0 2.2NumericalMethodstoOptimalControlProblem Anoptimalcontrolproblemisdenedasfollows:Find u whichminimizesthecost functional J = x t 0 x t f t 0 t f + x t + Z t f t 0 L x u t dt where istheterminalconstraint, isthepathconstraint,andwithdynamicconstraint x = f x u t x t 0 = x 0 Optimalcontrolcanbeseenasaone-playerdifferentialgameandthusshares withitthesameproblemwithndingthesolution.Aclassicalapproach,theindirect 24

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method,istousecalculusofvariationstoconstructasetofdifferentialequationswhose solutionistheoptimalcontrolstrategy.Theresultantsetofdifferentialequationsisa boundary-valueproblem,anditssolutionisrarelyanalytical.Numericallysolvinga boundary-valueproblemisoftendifcultduetonecessityofinitialguessandsmall radiusofconvergence. Inoptimalcontrolthedirectmethodtranscribestheproblemintoaparameter optimizationproblemNonlinearProgramming.Thestateequationandcontrolinputs arediscretizedandapproximatedbyinterpolatingpolynomialfunctions,andthecost functionalisevaluatedbynumericalintegration.Asdiscussedin[53],therearemany waysoftranscription,butwhenanoptimalcontrolproblemistranscribedtoanonlinear programmingproblem,theyareessentiallytwodifferentproblems,anditisimportant thatthesolutiontotheconvertednonlinearprogrammingproblemisindeedthesolution totheoriginaloptimalcontrolproblem.Thiscanbecheckedbycomparingthekkt multipliersofthetranscribedNLPproblemtothecostateoftheoptimalcontrolproblem withindirectmethod. Pseudospectralmethod,whichisalsoknownasorthogonalcollocationmethod [54],convertsanoptimalcontrolproblemtoanonlinearprogrammingproblemby approximatingthedynamicswiththederivativeoforthogonalinterpolatingpolynomial functions,andtheintegralformofthecostwiththegaussquadraturenumerical integration.Forexample,LegendrepseudospectralmethodLPM[55],useslagrange interpolationatLobattocollocationpoints,whicharerootsofthederivativeofLegendre polynomial,aswellasthecorrespondingLobattoweightstoevaluatethenumerical integration.[56]showedthatthecostateapproximationwithLPMisexactatevery collocationpoint. 2.3BilevelProgramming BilevelProgrammingisaclassofproblemswheretwoparameteroptimization problemsarearrangedinsuchawaythatsomeoftheconstraintsofoneproblemis 25

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denedbythesolutiontoanotherproblem.TherelationshipbetweenStackelberg differentialgamesandbilevelprogrammingisanalogoustothatbetweenoptimal controlandnonlinearprogramming.Thecouplingofmultipleoptimizationproblems denesgames,andthehierarchicalnaturerelatesStackelberggames,inparticular tooptimisticbilevelprogrammingproblems[57].Unlikethesingle-playerparameter optimizationproblemse.g.,nonlinearprogrammingthereisnowell-established techniquestosolvingbilevelprogrammingproblemsduetoitscomplexity.[58]shows anefcientalgorithmtosolvinglinearbilevelprogrammingproblems.However,for nonlinearcases,previousworksmainlyfocusonspecicproblems[5961]. InordertofullyinvestigatecomplexnonlinearStackelbergdifferentialgames,itis necessarytohaveawellestablishednumericalmethod.InSection3.2anorthogonal collocationapproachisshowntotranscribeatwo-playerStackelbergdifferentialgame problemtoa2 N -playerStackelbergstaticgame,andinSection3.3anexampleproblem ispresented.However,thisdissertationdoesnotexplorebilevelprogrammingand insteadfocusesondesigninggame-theoreticcontrollersforspacecraftpost-docking. 26

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CHAPTER3 TECHNICALDESCRIPTION Inthissectionseveralapproachestosolvingtwo-playerStackelbergdifferential gamesarediscussed.Oneofthemconsiderstransformingtheproblemtoastatic gameproblembydiscretization.Thestaticgamesconsideredhereareparticularlythe multi-objectiveoptimizationproblemswhicharesimilartodifferentialgamesexceptthat thedifferentialconstraintsandthecostfunctionalsarereplacedbystaticconstraintsand costfunctions,respectively. Thesolutionofanoptimalcontrolproblemisobtainedasfollows[62];rstthe differentialconstraintsareaugmentedtothecostfunctionaltoformtheHamiltonian,and withcalculusofvariationsthevariationoftheHamiltonianwithrespecttoeachofthe stateandcontrolvariablesareobtained.Thesevariationsaredifferentialequationsand calledtheoptimalityconditions.Theoptimalityconditionsconsistofthedynamicsofthe statesi.e.,thesystemdynamicsortheoriginaldifferentialconstraintsandthecostates i.e.,theLagrangemultipliersusedtoaugmentthedifferentialconstraintstothecost, andcombinedthesetofdifferentialequationsisaboundary-valueproblem. Whetheradifferentialgameproblemadmitsananalyticalsolutionornotdepends ontheexistenceoftheanalyticalsolutiontotheboundaryvalueproblemdenedbythe optimalityconditions.Sincedifferentialequationsdonothaveananalyticalsolutionin manycases[63], Thesolutionofdifferentialgameswithcalculusofvariationsindirectmethod, wherethesolutionisobtainedbysolvingdifferentialequationssatisfyingtheoptimality condition,islimited;itisdifculttoensuretheexistenceofthesolutionespecially whentheproblemisnonlinear,andeventhoughtheexistenceisensured,itisstill difculttosolvetheboundaryvalueproblem.Althoughoptimalcontrolingeneral suffersthesameproblemsasdifferentialgames,thereisaclassofnumericalmethods totranscribetheoptimalcontrolproblemtoaparameteroptimizationproblemusing 27

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DirectCollocation,whichcanbesolvedwithNonlinearProgramming.Thisso-called directmethodguaranteestheexistenceofthesolutioninexchangeofpossiblelossof optimality. Threecandidatesforsolvingdifferentialgamesareconsideredhere.First,indirect methodistobeusedwhentheproblemissimpleandcantakecertainformssuch thattheanalyticalsolutioniswelldeveloped.Second,insomecasesatwo-person differentialgamecanbeconvertedtoanoptimalcontrolproblemandsolvedwithdirect methods.Thirdchoiceisinspiredbydirectmethodsforoptimalcontrol;transcribing differentialgamestostaticgamesusingorthogonalcollocationPseudospectral method. Sinceoptimalcontrolcanbeconsideredasasingle-playergame,thereisa relationshipbetweenStackelberggamesandoptimalcontrol,asshowninFig.3-1. StackelbergdifferentialgameshavesimilarstructuretoStackelbergstaticgames,as optimalcontroldoestononlinearprogramming.AStackelbergdifferentialgameproblem canbereducedtoanoptimalcontrolproblembyincludingtheoptimalityconditionsfor thefollower,andthesolutionoftheoptimalcontrolproblemprovidesthesolutionofthe Stackelbergleader.However,itdoesn'tprovidethesolutionofthefollower;itneedsto becomputedseparatelybasedonthesolutionoftheoptimalcontrolproblem,anditis difculttodiscussthatthesolutionoftheleaderandthefollowerhavethesamelevel ofaccuracy.Thesameconceptappliestothetransitionfromabilevelprogramming problemtoanonlinearprogrammingproblemThereforethosearrowsareindotted lines. Figure3-2showsmoregeneralrelationshipsamongoptimizationproblems.Not alldifferentialgamescanbereducedtooptimalcontrolproblems,thustheindirect approachtosolvingdifferentialgamesdoesnotnecessarilygothroughsolvingoptimal control.Likewise,notallstaticgamescanbereducedtononlinearprogramming. Differentialgamesandoptimalcontrolcanbesolvedindirectlyviacalculusof 28

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Figure3-1.Stackelberggamesarehierarchicaloptimizationandcanbereducedto singleoptimizationproblems.Differentialgamesandoptimalcontrolcanbe convertedthroughdiscretizationtostaticgamesandnonlinearprogramming, respectively. variations.Ifsuccessfullytranscribed,itispossibletosolveoptimalcontrolproblems anddifferentialgameproblemsasparameteroptimizationproblems,suchasnonlinear programmingandbilevelprogramming. Althoughithasn'tbeenaspopularasoptimalcontrolcasetodiscretizedifferential gameproblemstostaticones,researchershavestudiedthisdiscretizationforpursuit-evasion zero-sumgames.Ehtamo[64]performedbothdiscretizationoftheoptimality conditionstononlinearprogramminganddirectconversiontobilevelprogramming andshowedtheyledtothesamesolution.Horie[65]convertedtheoptimalityconditions andsolvedtwosetsofnonlinearprogrammingcombinedwithgeneticalgorithm.Still, noattempthasbeenmadetoapplythePseudospectralmethodtosolvenonzero-sum differentialgames,totheauthor'sknowledge. Theseprioreffortsinspireanapproachtosolvingnonzero-sumStackelberg differentialgamesofpost-dockedsatellitesbytranscribingtononlinearprogramming andstaticgamebilevelprogramming.DuetothestructureofStackelbergdifferential games,buildingconnectionsamongdifferential/staticgames,optimalcontrol,and 29

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Figure3-2.Differentialgamesandoptimalcontrolproblemscanbesolvedbyindirect methodsusingcalculusofvariations,orbydirectmethodsthroughdirect transcription. nonlinearprogrammingshouldbepossible.AlthoughtheeffortisonlytowardStackelberg games,itcouldbeextendedtoNashgamesinthefuture. 3.1ReductionofStackelbergDifferentialGamestoOptimalControl Two-persondifferentialgamescouldbeposedastwocoupledoptimalcontrol problems.Letthecontrolsetoftheplayer1andtheplayer2be u 1 and u 2 ,respectively, then u 1 solvesanoptimalcontrolproblem Minimize J 1 = 1 x 0 x f t 0 t f + Z t f t 0 L 1 x u 1 u 2 t dt Subjectto x = f x u 1 u 2 t x t 0 = x 0 30

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and u 2 solvesanotheroptimalcontrolproblem Minimize J 2 = 2 x 0 x f t 0 t f + Z t f t 0 L 2 x u 1 u 2 t dt subjectto x = f x u 1 u 2 t x t 0 = x 0 Suppose u 1 and u 2 respectivelyasthefollowerandtheleader.WhenaStackelberg strategyisplayed,atwo-playernonzero-sumdifferentialgameissolvedasfollows [66];rstthedifferentialconstraintsareaugmentedtothefollower'scostfunctionalto formthefollower'sHamiltonian,fromwhichtheoptimalityconditionsforthefollower areobtained.canbeconvertedtotheoptimalcontrolproblemoftheleader,whichcan besolvednumericallyusingcollocationmethod.OnecharacteristicoftheStackelberg strategyintwo-persondifferentialgameisthatthefolloweralwaysactsoptimallytothe leader,thereforeiftheleader'scontrolstrategyispresented,thefollower u 1 solvesa trackingproblemdenedbyEq.3assuming u 2 isaprescribedfunctionoftime.The knowledgethatthefollowertrackstheleadercanbeusedasadditionalconstraintsin solvingfortheleader'sstrategyasfollows.LettheHamiltonianofthefollowerbe H 1 denedas H 1 = L 1 x u 1 u 2 t + T 1 f x u 1 u 2 t thenthefollower'soptimalityconditionisgivenby @ H 1 @ u 1 T = 0 T 1 = )]TJ/F23 11.9552 Tf 10.494 8.088 Td [(@ H 1 @ x T 1 t f = @ 1 @ x t f Equation3relatesthefollower'scontrol u 1 andthecostate 1 .If u 1 canbe expressedexplicitlyintermsof x 1 u 2 ,and t as u 1 = u 1 x 1 u 2 t 31

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thenbyreplacing u 1 withEq.3andcombiningEq.3,thetwo-playerdifferential gameproblemisreducedtotheoptimalcontrolproblemoftheleader u 2 : Minimize J 2 = 2 x 0 x f t 0 t f + Z t f t 0 L 2 x u 1 x 1 u 2 t u 2 t dt subjectto x = f x 1 u 2 t x = x 0 T 1 = )]TJ/F23 11.9552 Tf 10.494 8.088 Td [(@ H 1 @ x T 1 t f = @ 1 @ x t f OncetheoptimalcontrolproblemisdenedbyEq.3itispossibletosolveusing existingnumericalmethodsdiscussedinSection2.2. Therearetwoissueswiththisconversion:ithefollower'scontrolstrategyis restrictedtocontinuous,whiletheleadercanadmitdiscontinuouscontrolinputs.ii directmethodofoptimalcontrolmaysacricetheoptimalityfortheexistenceofthe solution,butduetotheconversion,onlytheleader'soptimalityissacricedalthough Stackelbergleaderisbetteratoptimizingtheircostthanthefollower.Therefore,except forthecasewhenEq.3canbeanalyticallysolved,itismorereasonabletomaintain thegamestructure,bytranscribingthedifferentialgameproblemtoastaticgame problem. 3.2ConversionofStackelbergDifferentialGamestoStackelbergStaticGames IntheSection2.2itwasshownthatanoptimalcontrolproblemcanbeconverted toanoptimizationproblembydiscretizingthetimedomain,approximatingthestates withinterpolatingfunctions,andnumericallyintegratingthecostfunctionals.Differential gameproblemshavethesamestructureasoptimalcontrolproblems,asbothinvolve optimizationoffunctionsovertimesubjecttodynamicconstraints.Thus,thesame methodoftranscriptioncanbeemployedtoconvertdifferentialgameandoptimalcontrol problemstostaticgameandnonlinearprogrammingproblems,respectively. Inthissection,inthesimilarmannerthatanoptimalcontrolproblemistranscribed toanonlinearprogrammingproblem,atwo-personStackelbergdifferentialgameis 32

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transcribedtoaStackelbergstaticprogrammingproblem.AsdiscussedinSection2.3, Stackelbergstaticgamesisasubsetofbilevelprogrammingproblems. TranscriptionwiththeLGLCollocation. .Inthissectiongeneraltranscription formulationisdevelopedfortwo-personStackelbergdifferentialgamesusingthe Legendre-Gauss-LobattoLGLcollocationpoints.Themainideaofthenumerical approachwithcollocationpointsistoapproximate x u 1 ,and u 2 aspolynomials constructedfromnitedatapoints,andndtheircoefcientssuchthattheapproximated functionssatisfytheoriginalgameproblemateachcollocationpoint[54].LGLpointsare denedas )]TJ/F20 11.9552 Tf 9.298 0 Td [(1,1,andtherootsofthederivativeofthe N th-orderLegendrepolynomialin theinterval t 2 [ )]TJ/F20 11.9552 Tf 9.299 0 Td [(1,1 ] thustotalof N +1points. Recallthegeneraltwo-playerStackelbergdifferentialgamewiththeplayer2asthe leadercanbemodeledas Minimize J 1 = 1 x t 0 t 0 x t f t f + Z t f t 0 L 1 x t u 1 t u 2 t t dt J 2 = 2 x t 0 t 0 x t f t f + Z t f t 0 L 2 x t u 1 t u 2 t t dt subjectto x = f x t u 1 t u 2 t t x t 0 = x 0 Forconvenience,let M = L 1 N = L 2 u = u 1 ,and v = u 2 .Thetimedomainneedstobe scaledto 2 [ )]TJ/F20 11.9552 Tf 9.299 0 Td [(1,1 ] : t = t 0 + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 + t f )]TJ/F57 11.9552 Tf 11.956 0 Td [(t 0 2 dt = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 d For N +1collocationpoints,thestatedynamicsbecomes N +1equalityconstraints x i = t f )]TJ/F57 11.9552 Tf 11.956 0 Td [(t 0 2 f x i u i v i i i =1,..., N x 0 = x 0 33

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andthecostfunctionalsbecome J 1 = 1 + Z t f t 0 L 1 x t u 1 t u 2 t t dt = 1 + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 Z N 0 M x u v d = 1 + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 N X i =0 w i M i and J 2 = 2 + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 N X i =0 w i N i where w i 'saretheweightsassociatedwithGauss-Lobattoquadrature,whichapproximates theintegralpartofthecostfunctional[67].Thus,theresultantbilevelprogramming problemisasfollows;rstthefollower u solvesthelowerlevelproblem Minimize J 1 = 1 + t f )]TJ/F57 11.9552 Tf 11.956 0 Td [(t 0 2 N X i =0 w i M i subjectto t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 f i )]TJ/F20 11.9552 Tf 13.515 0 Td [( x i =0, i =1,..., N however,sinceEq.3dependsalsoontheleader v ,whichhasnotbeenfoundyet, thelowerlevelproblemalonedoesnotprovidetheuniquesolutionforthefollower. Instead,thelowerlevelproblemdenesanoptimalreactionsetforthefollower,such that u isdeterminedonce v isdenedi.e., u = u v .Theleader,ontheotherhand,in 34

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solvingtheupperlevelproblem Minimize J 2 = 2 + t f )]TJ/F57 11.9552 Tf 11.956 0 Td [(t 0 2 N X i =0 w i N i subjectto t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 f i )]TJ/F20 11.9552 Tf 13.515 0 Td [( x i =0, i =1,..., N takesintoconsiderationthesolutionofEq.3.Bysubstituting u 'swiththefollower's informationfromthelowerlevelproblem,Eq.3becomesawell-posedparameter optimizationproblemoftheleader. 3.3CostateMappingofStackelbergDifferentialGames Directtranscriptionisappliedtoanexampleproblemwheretheanalyticalsolution exists,inordertoobservethevalidityoftheapproachtotwo-personStackelberg differentialgames.ThecomparisonbetweenthecostatesofaStackelbergdifferential gameproblemandthecostatesortheKKTmultipliersofthetranscribedbilevel programmingproblemisgiveninAppendixC. Example. .Consideranonzero-sumpursuit-evasiongamethatSimaanpresentedin [51]givenby x = u 1 )]TJ/F57 11.9552 Tf 11.955 0 Td [(u 2 x = x 0 J 1 = 1 2 x 2 f + 1 2 c p Z 1 0 u 2 1 dt J 2 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 x 2 f + 1 2 c e Z 1 0 u 2 2 dt where x u 1 u 2 2 R with u 2 astheleader,and c p > 0, c e > 0areknownconstants c p c e =1.Thisproblemischosensincetheanalyticalsolutionexists,sothatthe solutionwithadirectmethodcanbecompared.TheanalyticalsolutiontoEq.3 35

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providedin[51]is u 1 = )]TJ/F57 11.9552 Tf 36.352 8.117 Td [(c p c p )]TJ/F23 11.9552 Tf 11.955 0 Td [( c e +1 x 0 u 2 = )]TJ/F23 11.9552 Tf 32.836 8.087 Td [( c e c p )]TJ/F23 11.9552 Tf 11.955 0 Td [( c e +1 x 0 and x 0 = x 0 x f = 1 c p )]TJ/F23 11.9552 Tf 11.955 0 Td [( c e +1 x 0 where = 1 1+ c p NowtheproblemissolvedviaLegendrepseudospectralmethodLPM.For N =2,the scaledtimedomainis 0 = )]TJ/F20 11.9552 Tf 9.298 0 Td [(1, 1 =0, 2 =1 withthecorrespondingLobattoweights w 0 = 1 3 w 1 = 4 3 w 2 = 1 3 ThestateisapproximatedwithLagrangepolynomialoforder2: x = 2 1 2 x 0 )]TJ/F57 11.9552 Tf 11.956 0 Td [(x 1 + 1 2 x 2 + )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 x 0 + 1 2 x 2 + x 1 Thederivative x isthen dx d =2 1 2 x 0 )]TJ/F57 11.9552 Tf 11.956 0 Td [(x 1 + 1 2 x 2 )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(1 2 x 0 + 1 2 x 2 36

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Evaluatingateachdiscretizationpoints, x 0 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(3 2 x 0 +2 x 1 )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(1 2 x 2 x 1 = )]TJ/F20 11.9552 Tf 10.494 8.087 Td [(1 2 x 0 + 1 2 x 2 x 2 = 1 2 x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 x 1 + 3 2 x 2 Thesederivativesareexpressedinamatrixform X = DX where X = 2 6 6 6 6 4 x 0 x 1 x 2 3 7 7 7 7 5 X = 2 6 6 6 6 4 x 0 x 1 x 2 3 7 7 7 7 5 D = 2 6 6 6 6 4 )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(3 2 2 )]TJ/F20 7.9701 Tf 10.494 4.708 Td [(1 2 )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(1 2 0 1 2 1 2 )]TJ/F20 11.9552 Tf 9.299 0 Td [(2 3 2 3 7 7 7 7 5 Thedynamicsisdiscretizedatthreecollocationpoints x i = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 f i i =1,2,3 or DX = 1 2 F whichresultsinthreeequalityconstraints h 0 = 1 2 f 0 )]TJ/F20 11.9552 Tf 13.515 0 Td [( x 0 h 1 = 1 2 f 1 )]TJ/F20 11.9552 Tf 13.515 0 Td [( x 1 h 2 = 1 2 f 2 )]TJ/F20 11.9552 Tf 13.515 0 Td [( x 2 37

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Thecostfunctionalsareapproximatedwithquadraturerule J 1 = 1 + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 N X i =0 w i M i = 1 2 x 2 f + 1 2 1 2 c p 1 3 u 2 0 + 4 3 u 2 1 + 1 3 u 2 2 = 1 2 x 2 f + 1 12 c p u 2 0 + 1 3 c p u 2 1 + 1 12 c p u 2 2 J 2 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 x 2 f + 1 12 c e v 2 0 + 1 3 c e v 2 1 + 1 12 c e v 2 2 Theresultantbilevelprogrammingproblemisasfollows: Lowerlevelproblem: min x u J 1 = 1 2 x 2 2 + 1 12 c p u 2 0 + 1 3 c p u 2 1 + 1 12 c p u 2 2 subjectto )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(3 2 x 0 +2 x 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 x 2 = 1 2 u 0 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 v 0 )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 x 0 + 1 2 x 2 = 1 2 u 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 v 1 1 2 x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 x 1 + 3 2 x 2 = 1 2 u 2 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 v 2 Upperlevelproblem: min x v J 2 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 x 2 2 + 1 12 c e v 2 0 + 1 3 c e v 2 1 + 1 12 c e v 2 2 subjectto )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(3 2 x 0 +2 x 1 )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(1 2 x 2 = 1 2 u 0 )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(1 2 v 0 )]TJ/F20 11.9552 Tf 10.494 8.087 Td [(1 2 x 0 + 1 2 x 2 = 1 2 u 1 )]TJ/F20 11.9552 Tf 13.15 8.087 Td [(1 2 v 1 1 2 x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 x 1 + 3 2 x 2 = 1 2 u 2 )]TJ/F20 11.9552 Tf 13.15 8.087 Td [(1 2 v 2 Nowusethealgorithmin[68]tosolvethebilevelprogrammingproblemdenedby Eqs.3-3. 38

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LettheaugmentedLagrangianfor J 1 be J a 1 as J a 1 = J 1 + + N X i =0 w i i h i = J 1 + + w 0 0 h 0 + w 1 1 h 1 + w 2 2 h 2 = 1 2 x 2 2 + 1 12 c p u 2 0 + 1 3 c p u 2 1 + 1 12 c p u 2 2 + x 0 )]TJ/F57 11.9552 Tf 11.955 0 Td [(y 0 + 1 3 0 )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(3 2 x 0 +2 x 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 x 2 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 u 0 + 1 2 v 0 + 4 3 1 )]TJ/F20 11.9552 Tf 10.494 8.087 Td [(1 2 x 0 + 1 2 x 2 )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(1 2 u 1 + 1 2 v 1 + 1 3 2 1 2 x 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(2 x 1 + 3 2 x 2 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 u 2 + 1 2 v 2 andthepartialderivativeswithrespecttostatesandcontrolsare @ J a 1 @ x 1 = 2 3 0 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(2 3 2 @ J a 1 @ x 2 = x 2 )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(1 6 0 + 2 3 1 + 1 2 2 @ J a 1 @ u 0 = 1 6 c p u 0 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 6 0 @ J a 1 @ u 1 = 2 3 c p u 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(2 3 1 @ J a 1 @ u 2 = 1 6 c p u 2 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 6 2 Thisgives u 0 = c p 0 u 1 = c p 1 u 2 = c p 2 39

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thenmakesubstitutionstotheleader'saugmentedcostfunction J a 2 : J a 2 = J 2 + + N X i =0 w i i h i + N X i =0 w i i )]TJ/F23 11.9552 Tf 10.494 8.087 Td [(@ J a 1 @ x i = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 x 2 2 + 1 12 c e v 2 0 + 1 3 c e v 2 1 + 1 12 c e v 2 2 + x 0 )]TJ/F57 11.9552 Tf 11.956 0 Td [(y 0 + 1 3 0 )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 c p 0 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 v 0 + 3 2 x 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 x 1 + 1 2 x 2 + 4 3 1 )]TJ/F20 11.9552 Tf 10.494 8.087 Td [(1 2 c p 1 )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(1 2 v 1 + 1 2 x 0 )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(1 2 x 2 + 1 3 2 )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 c p 2 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 v 2 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 x 0 +2 x 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(3 2 x 2 + 1 2 3 0 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(2 3 2 + 2 x 2 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 6 0 + 2 3 1 + 1 2 2 Thepartialderivativesare @ J a 2 @ x 1 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(2 3 0 + 2 3 2 @ J a 2 @ x 2 = )]TJ/F57 11.9552 Tf 9.298 0 Td [(x 2 + 1 6 0 )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(2 3 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 2 + 2 @ J a 2 @ v 0 = 1 6 c e v 0 )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(1 6 0 @ J a 2 @ v 1 = 2 3 c e v 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(2 3 1 @ J a 2 @ v 2 = 1 6 c e v 2 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 6 2 @ J a 2 @ 0 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 6 c p 0 + 2 3 1 )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(1 6 2 @ J a 2 @ 1 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(2 3 c p 1 + 2 3 2 @ J a 2 @ 2 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 6 c p 2 )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(2 3 1 + 1 2 2 40

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Solvingfor x v ,and x 0 = x 0 x 1 = 1 2 x 0 + x 2 x 2 = 1 c p )]TJ/F23 11.9552 Tf 11.955 0 Td [( c e +1 x 0 0 = 1 = 2 = x 2 v 0 = v 1 = v 2 = c e x 2 0 = 1 = 2 = )]TJ/F23 11.9552 Tf 9.298 0 Td [( x 2 1 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 c p x 2 2 = )]TJ/F23 11.9552 Tf 9.298 0 Td [( c p x 2 where x 0 x 2 u i ,and v i correspondto x t 0 x t f u 1 t ,and u 2 t ,respectively.This showsthatsamevaluesof x u 1 ,and u 2 wereobtainedfromthetranscribedbilevel programmingproblem. Nowexaminewhetherthesolutiontothebilevelproblemistrulyanoptimalsolution totheoriginalStackelbergdifferentialgameproblem.Inthisparticularexampleitis, sincetheoptimalsolutionisalreadygivenanalyticallyin[51],whichisidenticalto thesolutiontothebilevelprogrammingproblem.Thereforethefollowingisratherto conrmthevalidityofcostatemapping.Ifthetranscriptionmaintainsthestructureofthe problem, i 'sinthebilevelprogrammingproblemcorrespondsto t inthedifferential gameproblem. 0 t 0 1 t 1 2 t f 41

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Firstcomparethevaluesof .TheLagrangemultipliersare 0 = 1 = 2 = x 2 andthecorrespondingcostateoftheoriginaldifferentialgamediscretizedateach collocationpointis 0 = 1 = N = x f anditwasshownthat x 2 = x f ,thusthemultipliersforthebilevelprogrammingproblem hastheexactvaluesasthecostateateverycollocationpoint. Nowlookattheoptimalityconditions: At = 0 : ThederivativeoftheLagrangianwithrespectto x 0 is @ L 1 @ x 0 = + 1 2 0 + 2 3 1 )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(1 6 2 =0 + 1 3 3 2 0 +2 1 )]TJ/F20 11.9552 Tf 13.15 8.087 Td [(1 2 2 =0 + 1 3 |{z} w 0 )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(3 2 0 +2 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 2 | {z } 0 + 1 3 2 3 2 0 =0 w 0 0 = )]TJ/F23 11.9552 Tf 9.299 0 Td [( 0 )]TJ/F23 11.9552 Tf 11.955 0 Td [( Notethat ontherighthandsideofEq.3isinfact @ @ x 0 .Intheoriginal Stackelbergproblem, 0 =0 0 = )]TJ/F23 11.9552 Tf 10.494 8.088 Td [(@ x 0 0 @ x 0 42

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thus,ThelefthandsideofEq.3isthecostatedynamicsat 0 ,andtherighthand sideistheinitialtransversalitycondition. At = 1 : ThederivativeoftheLagrangianwithrespectto x 1 is @ L 1 @ x 1 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(2 3 0 + 2 3 2 =0 4 3 |{z} w 1 )]TJ/F20 11.9552 Tf 10.494 8.087 Td [(1 2 0 + 1 2 2 | {z } 1 =0 w 1 1 =0 Since t =0intheoriginalproblem,Eq.3isindeedthecostatedynamicsat = 1 At = 2 : ThederivativeoftheLagrangianwithrespectto x 2 is @ L 1 @ x 2 = x 2 + 1 6 0 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(2 3 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 2 =0 x 2 + 1 3 1 2 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [(2 1 )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(3 2 2 =0 x 2 + 1 3 |{z} w 2 1 2 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 1 + 3 2 2 | {z } 2 )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 3 2 3 2 2 =0 w 2 2 = )]TJ/F57 11.9552 Tf 9.298 0 Td [(x 2 + 2 43

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Intheoriginalproblem t f =0 t f = x f ThereforeThelefthandsideofEq.3matchesthecostatedynamicsat = 2 ,and theridehandsidematchesthenaltransversalitycondition.Thatis,thetranscribed problemexactlymaps 1 t ateverycollocationpoint. Nowlookatthecostatescorrespondingtotheupperlevelproblem. 0 1 ,and 2 ofthebilevelprogrammingproblemarecomparedwith t ofthedifferentialgame problem.Similarly, 1 and 2 arecomparedwith t 0 = 1 = 2 = )]TJ/F23 11.9552 Tf 9.298 0 Td [( x 2 0 = 1 = 2 = )]TJ/F23 11.9552 Tf 9.299 0 Td [( x f 1 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 c p x 2 2 = )]TJ/F23 11.9552 Tf 9.298 0 Td [( c p x 2 0 =0 1 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 c p x f 2 = )]TJ/F23 11.9552 Tf 9.298 0 Td [( c p x f At = 1 : @ L 2 @ x 1 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(2 3 0 + 2 3 2 = 4 3 |{z} w 1 )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 0 + 1 2 2 | {z } 1 =0 44

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w 1 1 =0 At = 2 : @ L 2 @ x 2 = )]TJ/F57 11.9552 Tf 9.298 0 Td [(x 2 + 1 6 0 )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(2 3 1 )]TJ/F20 11.9552 Tf 13.15 8.087 Td [(1 2 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( 2 =0 )]TJ/F57 11.9552 Tf 9.299 0 Td [(x 2 + 1 3 1 2 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(3 2 2 = 2 )]TJ/F57 11.9552 Tf 9.298 0 Td [(x 2 + 1 3 |{z} w 2 1 2 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 1 + 3 2 2 | {z } 2 )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(2 3 3 2 2 = 2 w 2 2 = 2 + x 2 + 2 whichmatchesthecostatedynamicsandnaltransversalitycondition =0 f = )]TJ/F57 11.9552 Tf 9.298 0 Td [(x f )]TJ/F23 11.9552 Tf 11.955 0 Td [( f ThustheLagrangemultipliersintheupperlevelproblemmatchthecostatesofthe leader'ssolutioninthedifferentialgameexactlyateverycollocationpoint.Thereforeit wasconrmedthatwithLGLcollocationmethodsuccessfullyconvertedtheStackelberg differentialgameproblemtostaticone,whosesolutionisindeedoptimalateach discretizedpointintheoriginalproblem. 3.4Conclusion Twonumericalapproachestosolvingtwo-playerStackelbergdifferentialgames werepresented.Convertingthedifferentialgametoanoptimalcontrolproblem requiresthefollower'soptimalstrategytobeanalyticallydened,butforarelatively simpleproblemwell-establishedoptimalcontrolsolverscanbeutilizedtoobtainthe solution.Convertingtoastaticgameproblemretainsmorecharacteristicsoftheoriginal differentialgameproblem,butduetolackofabilevelprogrammingsolvercomparable 45

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tothoseforoptimalcontrol,solvabilityofthetranscribedproblemisnotguaranteed.In bothcases,thesolutiontotheissuesliesinadvancementofparameteroptimization techniques,whichisnotthescopeofthisdissertation. Therefore,insteadofpursuingnumericalmethodsforcomplexnonlineardifferential games,thefollowingchaptersinvestigatesimpliedproblemstofocusonStackelberg-based non-cooperativecharacteristicsinspacecraftpost-dockingandthecorresponding necessarycontrolactionsformaintainingthedocking. 46

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CHAPTER4 DYNAMICSOFDOCKEDSPACECRAFT 4.1FormulationofDynamics Thedynamicsoftwodockedsatellitesisderivedinthissection.Thisislaterusedas asetofdynamicconstraintsintheStackelbergdifferentialgame. 4.1.1RelativeMotionDynamicsofaSatellite ConsiderasatellitenearacircularorbitasshowninFig.4-1.Letthecircularorbit bethenominalorbittowhichareferenceframe F N nominalframeisattached.The originofthenominalframe F N relativetothecenteroftheEarth,whichistheorigin oftheinertialreferenceframe F I ,is R O ,andthepositionofthesatelliterelativeto F N is r i 1 Supposethatthe F N andthesatellitehaveangularvelocities O and i respectively. Figure4-1.Arepresentationofthepositionofasatellitewiththeinertialandthenominal referenceframes. 1 Thesubscript i impliesthe i th bodyalthoughonlyonebodyisconsidered.Latertwo bodiesareconsideredwithinthesameframework. 47

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4.1.1.1Translation Letthepositionofthei th satelliterelativetothecenteroftheEarth F I bedenoted as R i = R O + r i withthecorrespondingvelocityandacceleration R i = R O + r i = R O + r i + O r i R i = R O + r i = R O + r i +2 O r i + O O r i Thenthetranslationdynamicsofthesatelliterelativeto F N canbewrittenas r i = R i )]TJ/F20 11.9552 Tf 14.28 2.308 Td [( R O )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 O r i )]TJ/F28 11.9552 Tf 11.956 9.684 Td [( O O r i 4.1.1.2Rotation Fortranslationthesatelliteistreatedasaparticlewhereitspositionisthecenter ofmassofthebody.Forrotation,considerthesatelliteasarigidbody,asshownin Fig.4-2.Thesatellitehasthereferenceframexedtocenterofmassofthebody,which isalignedtotheprincipalaxis. Figure4-2.Asatellitewithabody-xedreferenceframe F i 48

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Letthemomentofinertiamatrixofthesatellitebe J i ,thentherotationalmotionof thesatelliteis J i i + i J i i = i J i i + i J i i = i wherethesuperscript denotesthematrixoperationequivalenttothevectorcross product.Introducingtheangularvelocityrelativeto F N i = O + i = O thensince O isconstant, i = i = O FromEqs.4-4therotationdynamicsofthesatelliterelativeto F N canbewritten as i = O = )]TJ/F59 11.9552 Tf 9.298 0 Td [(J )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i h )]TJ/F48 11.9552 Tf 5.48 -9.684 Td [(! i = O + O J i )]TJ/F48 11.9552 Tf 5.48 -9.684 Td [(! i = O + O i + J )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 i i 4.1.2DynamicsofTwoDockedSatellites Nowconsidertwosatellitessubjecttothelinearandtherotationalmotionsdened previously.Figure4-3showstwosatellitesnearthenominalorbit.Figure4-3shows twosatellitesdockedtogethernearthenominalorbit.FromEqs.4and4,the translationalandrotationalmotionscanbeexpressedas 49

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Figure4-3.Anexeggeratedviewoftwosatellitesnearthenominalorbit. r SV = R SV )]TJ/F20 11.9552 Tf 14.28 2.307 Td [( R O )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 O r SV )]TJ/F48 11.9552 Tf 11.955 0 Td [(! O )]TJ/F48 11.9552 Tf 5.479 -9.684 Td [(! O r SV + 1 m SV F SV r RSO = R RSO )]TJ/F20 11.9552 Tf 14.28 2.308 Td [( R O )]TJ/F20 11.9552 Tf 11.956 0 Td [(2 O r RSO )]TJ/F48 11.9552 Tf 11.955 0 Td [(! O )]TJ/F48 11.9552 Tf 5.479 -9.684 Td [(! O r RSO + 1 m RSO F RSO SV = O = )]TJ/F59 11.9552 Tf 9.298 0 Td [(J )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 SV )]TJ/F48 11.9552 Tf 10.461 -9.684 Td [(! SV = O + O J SV )]TJ/F48 11.9552 Tf 5.48 -9.684 Td [(! SV = O + O + J )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 SV M SV RSO = O = )]TJ/F59 11.9552 Tf 9.298 0 Td [(J )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 RSO )]TJ/F48 11.9552 Tf 10.461 -9.684 Td [(! RSO = O + O J RSO )]TJ/F48 11.9552 Tf 5.48 -9.684 Td [(! RSO = O + O + J )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 RSO M RSO where F SV and F RSO aretheforcesappliedtothedockingpointofeachbodydueto contact,and SV and RSO respectivelyrepresentthecorrespondingmoments,whichare duetothecomponentsof F SV and F RSO thatarenotradialtothecenterofmassofSV andRSO,respectively. TheinteractionsbetweentheSVandtheRSOaremodeledwithaspringanda damperwhichconnectthedockingpoints P and Q ontheSVandtheRSO.Thespring 50

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anddampingforceappliedto P ontheSVare F SV = F SV k + F SV c F SV k = )]TJ/F57 11.9552 Tf 9.298 0 Td [(k L R P )]TJ/F59 11.9552 Tf 11.955 0 Td [(R Q )]TJ/F57 11.9552 Tf 11.955 0 Td [(l 0 R P )]TJ/F59 11.9552 Tf 11.955 0 Td [(R Q k R P )]TJ/F59 11.9552 Tf 11.955 0 Td [(R Q k F SV c = )]TJ/F57 11.9552 Tf 9.298 0 Td [(c L )]TJ/F20 11.9552 Tf 7.804 -7.376 Td [( R P )]TJ/F20 11.9552 Tf 14.281 2.308 Td [( R Q where k L and c L arethelinearspringandthelineardampingcoefcient,respectively. Thetangentialcomponentof F SV resultsinthemomentappliedtotheSVas M SV F = SV F SV where` F 'inthesubscriptindicatesitcomesfromtheinteractionforce.Thetorque appliedto P ontheSVcanbemodeledas M SV = M SV k + M SV c + M SV F where M SV k and M SV c arefromthetorsionalspringandthetorsionaldamper,respectively, as M SV k = )]TJ/F57 11.9552 Tf 9.298 0 Td [(k T a M SV c = )]TJ/F57 11.9552 Tf 9.298 0 Td [(c T RSO = SV where and a respectivelydenotetheangleandtheaxisofrotationtoexpressthe RSO'sattituderelativetotheSV,and c T and k T aretheconstantofthetorsionaldamper andspring. Coordinatization. .Itiscustomarytodenotethetranslationalmotioninthe coordinateframeforthenominalorbit,whiletherotationalmotionisbetterdescribedin thecoordinateframeofthebodyitself.Thusthetranslationalmotionsarecoordinatized inthenominalframe,andtherotationalmotionsarecoordinatizedineachbodyframe. Thus,thetranslationalmotionandtherotationalmotionofSVandRSO,respectively, 51

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arewrittenas N r SV = N R SV )]TJ/F57 7.9701 Tf 11.955 4.936 Td [(N R O )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 N O N r SV )]TJ/F28 11.9552 Tf 11.956 9.684 Td [( N O N O N r SV + 1 m SV N F SV N r RSO = N R RSO )]TJ/F57 7.9701 Tf 11.955 4.936 Td [(N R O )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 N O N r RSO )]TJ/F28 11.9552 Tf 11.955 9.684 Td [( N O N O N r RSO + 1 m RSO N F RSO SV SV = O = )]TJ/F57 7.9701 Tf 9.299 4.936 Td [(SV J )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 SV SV SV = O + SV O SV J SV SV SV = O + SV O + SV J )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 SV SV M SV RSO RSO = O = )]TJ/F57 7.9701 Tf 9.299 4.937 Td [(RSO J )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 RSO RSO RSO = O + RSO O RSO J RSO RSO RSO = O + SV O + RSO J )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 RSO RSO M RSO Alsotheinteractionforceapplliedtothedockingpointisequalandopposite,thusin thenominalreferenceframe N F RSO = )]TJ/F57 7.9701 Tf 9.298 4.936 Td [(N F SV Obviouslythisisnottruefortheinteractiontorques N M RSO 6 = )]TJ/F57 7.9701 Tf 9.298 4.936 Td [(N M SV dueto M SV F ,sincethemomentarms SV and RSO aredifferentregardlessofthe coordinatization. 4.2Simulation ThedynamicmodelofdockedsatellitesdenedbyEq.4isanalyzed. Threedifferentcasesareconsideredtovalidatethestabilityofthepost-dockedstate withoutexternalperturbationi.e.,withoutnon-cooperativedisturbanceoftheRSOthe interactionsnevergrowstodamagethedocking. 4.2.1CaseI:NonzeroLinearVelocity SupposetwosatellitesareonthesameorbitwheretheSVismovingaheadofthe RSO,asshowninFig.4-4.Figure4-5showsthattheyareorientedinsuchawaythat thedockingpointsandcenterofmassofeachbodyareonthesameline.Althoughthis makestheinitialorientationoftheSVandtheRSOdifferentfromthenominalorbit,the 52

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differenceisassumednegligibleduetosmallseparationrelativetolargealtitude.The initialdockingseparationisexactlytheunstretchednaturallengthofthespring. Figure4-4.Twosatellitesinitiallyonthesamenominalorbit Firstacasewiththeinitialnon-zerolinearrelativevelocitybetweentheSVand theRSOisconsidered.ThelistofthesimulationparametersisshowninTable4-1. TheresultsareplottedinFigs.4-6-4-9.AsshowninFigs.4-6and4-7,theforceand thetorqueappliedtothespacecraftquicklydecaytozero.Figure4-8showsthatthe separationbetweentheSVandtheRSOconvergesto0.5m,whichistheun-stretched naturalspringlength.TheSVandtheRSOmaintainedtherelativepositionthroughout thesimulationwithoutcausingrotationalmotion,asshowninFig.4-9. 53

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Figure4-5.Twosatellitesinitiallyradiallyaligned Figure4-6.CaseI:theinteractionforcesappliedtotheSVandtheRSO 54

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Table4-1.ThesimulationparametersforCaseI NameDescriptionValueUnit t Timeofthesimulation100s m SV Massoftheservicevehicle100kg m RSO Massofthetargetvehicle120kg I SV Momentofinertiaoftheservicevehicle diag [ 101012 ] T kg m 2 I RSO Momentofinertiaofthetargetvehicle diag [ 121215 ] T kg m 2 R O Thealtitudeofthenominalorbit7,031km k Thespringconstant10N/m c Thedampingconstant20N s/m k T Thetorsionalspringconstant10Nm/rad c T Thetorsionaldampingconstant20Nm s/rad l 0 Theunstretchedspringlength0.5m Thegravitationalparameter398,600km 3 / s 2 SV ThedockingpositionoftheSVrelativetoitscenterofmass [ 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(0.250 ] T m RSO ThedockingpositionoftheRSOrelativetoitscenterofmass [ 00.250 ] T m r SV Theinitialseparationfromthenominal [ 00.50 ] T m r RSO Theinitialseparationfromthenominal [ 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(0.50 ] T m r SV TheinitiallocalvelocityoftheSV [ 000 ] T m/s r RSO TheinitiallocalvelocityoftheRSO [ 00.20 ] T m/s SV TheinitialangularvelocityoftheSVrelativetothenominal [ 000 ] T rad/s RSO TheinitialangularvelocityoftheRSOrelativetothenominal [ 000 ] T rad/s q SV TheinitialquaternionoftheSVrelativetothenominal [ 0001 ] T q RSO TheinitialquaternionoftheRSOrelativetothenominal [ 0001 ] T 55

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Figure4-7.CaseI:theinteractiontorquesappliedtotheSVandtheRSO Figure4-8.CaseI:thelinearmotionoftheRSOrelativetotheSV 56

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Figure4-9.CaseI:therotationalmotionoftheRSOrelativetotheSV 57

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4.2.2CaseII:NonzeroRotationalVelocity Nextcaseconsidersnon-zeroinitialangularvelocitybetweentheSVandtheRSO, whilethelinearvelocityissettozero.ComparedtothecaseI,theperturbationtothe rotationalresultsinhigherrelativerotationalmotion RSO = SV ,asshowninFig.4-13,and thushigherinteractiontorqueisobservedinFig.4-11.Itisnotedthattheforceisalso signicantwithoutinitiallinearvelocityFig.4-10duetotheseparationcausedbythe misalignmentoftheSVandtheRSO.OveralltherelativemotionbetweentheSVand theRSOdecaystotheequilibrium. Figure4-10.CaseII:theinteractionforcesappliedtotheSVandtheRSO 58

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Table4-2.ThesimulationparametersforCaseII NameDescriptionValueUnit t Timeofthesimulation100s m SV Massoftheservicevehicle100kg m RSO Massofthetargetvehicle120kg I SV Momentofinertiaoftheservicevehicle diag [ 101012 ] T kg m 2 I RSO Momentofinertiaofthetargetvehicle diag [ 121215 ] T kg m 2 R O Thealtitudeofthenominalorbit7,031km k Thespringconstant10N/m c Thedampingconstant20N s/m k T Thetorsionalspringconstant10Nm/rad c T Thetorsionaldampingconstant20Nm s/rad l 0 Theunstretchedspringlength0.5m Thegravitationalparameter398,600km 3 / s 2 SV ThedockingpositionoftheSVrelativetoitscenterofmass [ 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(0.250 ] T m RSO ThedockingpositionoftheRSOrelativetoitscenterofmass [ 00.250 ] T m r SV Theinitialseparationfromthenominal [ 00.50 ] T m r RSO Theinitialseparationfromthenominal [ 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(0.50 ] T m r SV TheinitiallocalvelocityoftheSV [ 000 ] T m/s r RSO TheinitiallocalvelocityoftheRSO [ 000 ] T m/s SV TheinitialangularvelocityoftheSVrelativetothenominal [ 000 ] T rad/s RSO TheinitialangularvelocityoftheRSOrelativetothenominal [ 000.05 ] T rad/s q SV TheinitialquaternionoftheSVrelativetothenominal [ 0001 ] T q RSO TheinitialquaternionoftheRSOrelativetothenominal [ 0001 ] T 59

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Figure4-11.CaseII:theinteractiontorquesappliedtotheSVandtheRSO Figure4-12.CaseII:thelinearmotionoftheRSOrelativetotheSV 60

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Figure4-13.CaseII:therotationalmotionoftheRSOrelativetotheSV 61

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4.2.3CaseIII:NonzeroLinearandRotationalVelocities Nowthesimulationisrunwithnon-zeroinitiallinearandrotationalvelocities. ThesimulationparametersarechosenasinTable4-3,andtheresultsareshownin Figs.4-14-4-17.Figures4-14and4-15showthehighestinteractionforcesandtorques respectivelyduetothecombinationoftheinitiallinearandrotationalperturbations. However,interestinglyboththeinteractionsandtherelativelinearmotionFig.4-16 appeartoconvergetothelowerboundsthaninCaseIIFigs.4-10,-4-12.Thiscanbe interpretedasapartofthelinearrelativemotionindependentoftherotationalmotion contributedtothestabilization,whileinCaseIIthewholemotionwascoupled,making ithardertostabilizethemotiontothesmallerradiiofconvergence.Howevertheresult doesnotsuggestCaseIIIbetterthanCaseIIastheprimaryfocusshouldbeonthe upperboundsoftheinteractions,whicharecertainlyhigherinCaseIIIthantheother twocases. Figure4-14.CaseIII:theinteractionforcesappliedtotheSVandtheRSO 62

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Table4-3.ThesimulationparametersforCaseIII NameDescriptionValueUnit t Timeofthesimulation100s m SV Massoftheservicevehicle100kg m RSO Massofthetargetvehicle120kg I SV Momentofinertiaoftheservicevehicle diag [ 101012 ] T kg m 2 I RSO Momentofinertiaofthetargetvehicle diag [ 121215 ] T kg m 2 R O Thealtitudeofthenominalorbit7,031km k Thespringconstant10N/m c Thedampingconstant20N s/m k T Thetorsionalspringconstant10Nm/rad c T Thetorsionaldampingconstant20Nm s/rad l 0 Theunstretchedspringlength0.5m Thegravitationalparameter398,600km 3 / s 2 SV ThedockingpositionoftheSVrelativetoitscenterofmass [ 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(0.250 ] T m RSO ThedockingpositionoftheRSOrelativetoitscenterofmass [ 00.250 ] T m r SV Theinitialseparationfromthenominal [ 00.50 ] T m r RSO Theinitialseparationfromthenominal [ 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(0.50 ] T m r SV TheinitiallocalvelocityoftheSV [ 000 ] T m/s r RSO TheinitiallocalvelocityoftheRSO [ 00.20 ] T m/s SV TheinitialangularvelocityoftheSVrelativetothenominal [ 000 ] T rad/s RSO TheinitialangularvelocityoftheRSOrelativetothenominal [ 000.05 ] T rad/s q SV TheinitialquaternionoftheSVrelativetothenominal [ 0001 ] T q RSO TheinitialquaternionoftheRSOrelativetothenominal [ 0001 ] T 63

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Figure4-15.CaseIII:theinteractiontorquesappliedtotheSVandtheRSO Figure4-16.CaseIII:thelinearmotionoftheRSOrelativetotheSV 64

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Figure4-17.CaseIII:therotationalmotionoftheRSOrelativetotheSV 65

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4.3Conclusion Adynamicmodeloftwodockedsatelliteswasderivedwiththeaimofinvestigating theinteractionsbetweenthemthroughdifferentialgames. Thedevelopedmodelhereisshowntobestablewithoutactuation;theinteractions betweenthedockedsatellitesareboundedwithoutnon-cooperativeactuation.Itshould benotedthattherelativemotionobservedinthesimulationisduetothemodelingof theproblemsuchthattheinteractionforceandtorquearenotabsorbedbythedocking mechanismbutinsteaddirectlyaffectthemotionoftwospacecraft;iftheinteractionsare smallenoughthedockingcanbemaintainedandtheSVandtheRSOshouldbehaveas asinglebody.Thenextstepistoformulateatwo-persondifferentialgamebasedonthis dynamicsandthenwithStackelbergstrategytoanalyzetheproblem. 66

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CHAPTER5 LINEARCONTROLLERDESIGNWITHSTACKELBERGSTRATEGY Inthissectiontheinteractionsoftwodockedsatellitesarestudiedviatwo-person differentialgames.Asimpliedversionofthedynamicsobtainedinthepreviouschapter isusedtoformulatealinearquadraticLQgamewherethedynamicsislinearizedat theequilibrium.AsdiscussedinChapter2,LQdifferentialgameshavewell-dened solvabilityconditionse.g.,existenceanduniqueness[69],sothatthespacecraft post-dockingproblemcanbeinvestigatedwithoutsufferingcomputationaldifculty. 5.1Post-DockingStudywithLinearQuadraticGame Thesubsequentdiscussionfollows[70].Twosatellites,thetargetortheresident spaceobjectRSOandtheservicevehicleSV,areshowninFig.5-1.Theinteraction betweentheSVandtheRSOismodeledbyaspringanddamperconnectingthemat thepointPandQsuchthatachangeindistanceorvelocitybetweenPandQproduces theinteractionforces,whicharenotnecessarilyradialandthereforecausetorques,too. Inordertomaintaindocking,thoseforcesandtorquesneedtobeminimized.Eachbody iscontrolledbyaforceinputandatorqueinput.Itisfurtherassumedthatthethrusts andtorquesaredecoupled. Figure5-1.Tworigidbodiesoncircularorbits. 67

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Thetranslationalmotionofeachbodymovinginacircularorbitisobtainedfromthe two-bodyequationgivenby[71]as r i + k r i k 3 r i = 1 m i F i + u i i =SV,RSO where r i denotesthepositionvectorlocatedinthecenterofmassofeachbody, F i denotestheexternalforceduetothespringandthedampster,and u i isthecontrolforce input.TherotationalmotionisgovernedbyEuler'sequation J i i + i J i i = M i + i i =SV,RSO where J i istheinertiamatrix, i istheangularvelocity, M i isthemomentduetothe springandthedamper,and i isthecontroltorqueinput.Thepositionvectorofeach bodyis r SV = R SV 0 + r RSO = R RSO 0 + wherethenominalradiioftheorbitoftheSVandtheRSO,coordinatizedintheir respectiveframes,are SV R SV 0 = 00 )]TJ/F57 11.9552 Tf 9.299 0 Td [(R SV 0 T RSO R RSO 0 = 00 )]TJ/F57 11.9552 Tf 9.299 0 Td [(R RSO 0 T andthelinearperturbationsarecoordinatizedinthesamemanneras SV = 1 2 3 T RSO = 1 2 3 T Eqs.5-5arelinearizedtoyieldthelineardynamicmodel.Withtheassumption ofsmallperturbationandthattwosatellitesareonthesameorbiti.e., R SV 0 R RSO 0 68

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sincetheyaredockedandhavesmallbodycomparedtotheorbit,thecoordinateframes areapproximatedtocoincide.Therelativedistanceandattitudeerrorsaredenedas r = x y z T )]TJ/F48 11.9552 Tf 11.956 0 Td [( = 1 2 3 T )]TJ/F48 11.9552 Tf 11.955 0 Td [( Thenthefollowinglinearizeddynamicmodelisobtained: x = Ax + B 1 u 1 + B 2 u 2 andthestatevector x ofthesystemiscomposedoftherelativedistance,theattitude error,andtheirrates: x = r T T r T T T Inthefollowingnumericalanalysis,thestatesarecoordinatizedinthebodyaxisofthe SVsincethefocusofthisstudyistocontroltheSV. 5.2SimulationandResults Thesimulationforainnite-horizoncasewasrunwiththesystemparameters showninTable5-1withthecorrespondingstate-spacesystem A = 2 6 4 0 6 6 I 6 6 A 21 A 22 3 7 5 where B 1 = 2 6 4 0 6 6 B 11 3 7 5 B 2 = 2 6 4 0 6 6 B 21 3 7 5 69

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B 11 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F20 7.9701 Tf 16.231 4.707 Td [(1 m SV 000 )]TJ/F57 7.9701 Tf 10.494 7.617 Td [(R SV 0 I SV 2 0 0 )]TJ/F20 7.9701 Tf 16.231 4.707 Td [(1 m SV 0 R SV 0 I SV 1 00 00 )]TJ/F20 7.9701 Tf 16.232 4.707 Td [(1 m SV 000 000 )]TJ/F20 7.9701 Tf 15.93 4.707 Td [(1 I SV 1 00 0000 )]TJ/F20 7.9701 Tf 15.93 4.707 Td [(1 I SV 2 0 00000 )]TJ/F20 7.9701 Tf 15.931 4.707 Td [(1 I SV 3 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 B 21 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 m RSO 000 R RSO 0 I RSO 2 0 0 1 m RSO 0 )]TJ/F57 7.9701 Tf 10.494 7.617 Td [(R RSO 0 I RSO 1 00 00 1 m RSO 000 000 1 I RSO 1 00 0000 1 I RSO 2 0 00000 1 I RSO 3 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 A 21 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F20 11.9552 Tf 9.298 0 Td [(1.852 )]TJ/F20 11.9552 Tf 9.299 0 Td [(1.852 )]TJ/F20 11.9552 Tf 9.298 0 Td [(1.852 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.001 )]TJ/F20 11.9552 Tf 9.299 0 Td [(1.4820.001 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.926 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.926 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.926 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.001 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.7410.001 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.926 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.926 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.926 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.001 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.7410.001 000000 000000 000000 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 70

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A 22 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.185 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.275 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.185000 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.188 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.278 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.185000 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.185 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.185 )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.185000 00000.0010 000 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.00100 000000 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 and B 1 = 2 6 4 0 6 6 diag f)]TJ/F20 11.9552 Tf 15.276 0 Td [(0.01, )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.01, )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.01, )]TJ/F20 11.9552 Tf 9.299 0 Td [(0.0167, )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0167, )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.0167 g 3 7 5 B 2 = 2 6 4 0 6 6 diag f 0.067,0.067,0.067,0.01,0.01,0.01 g 3 7 5 Theunitsrepresentedinmatrices A ii and B i arethelineardistanceinmeters,massin kg,andangulardisplacementinradians.Theinitialconditionsare x = r T T r T T T 0 = 0.2 I 1 3 0.05 I 1 3 )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.1 I 1 6 T Thecostfunctionalsarechosenas Q 1 =diag I 1 3 0.1 I 1 3 0.5 0 1 5 Q 2 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(Q 1 71

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Table5-1.Thesimulationparametersforthelinearquadraticgame NameDescriptionValueUnit c Dampingconstant10N s/m k Springconstant20N/m l 0 Equilibriumdisplacementofthespring0.3m m SV ThemassofSV150kg m RSO ThemassofRSO200kg R SV 0 ThenominalaltitudeofSV6600.00km R RSO 0 ThenominalaltitudeofRSO6600.00km I SV TheSV'smomentofinertiadiag f 60,60,60 g kg m 2 I RSO TheRSO'smomentofinertiadiag f 100,100,100 g kg m 2 SV ThedockingpositionfromtheSV'scm[111] T m RSO ThedockingpositionfromtheRSO'scm[1.51.51.5] T m R 11 =diag 10.10.1111 R 12 = 1 6 6 R 21 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(R 11 R 22 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(R 12 Forthesakeofcomparison,theproblemisalsosolvedwiththelinearquadratic regulatorLQRcontrollerassumingtheinteractionbetweentheSVandtheRSO cooperative.IftheobjectivesofboththeSVandtheRSOaretominimizetheinteraction, acorrespondingLQRproblemcanbeconstructedas x = Ax + Bu and J = 1 2 Z 1 0 x T Qx + u T Ru dt where B = [ B 1 B 2 ] T u = u T 1 u T 2 T Q = Q 1 )]TJ/F59 11.9552 Tf 12.877 0 Td [(Q 2 ,and R =diag f R 11 R 22 g .Note thatinordertobeabletosolvetheLQRproblem, R mustbeinvertibleandsymmetric. Forsimplicity,theoff-diagnaltermsarechosentobezeroinsteadof R 12 and R 21 .The 72

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resultanttrajectoriesareplottedinFig.5-2,andthecontrolforceandtorqueinputsare comparedinFigs.5-3and5-4. Show x y ,and z ALQR BNash CRSOasleader DSVasleader Figure5-2.Theresultanttrajectory. Theresultanttrajectorydoesn'thavenotabledifferencewhicheverplayerplays StackelbergleaderorNashstrategy,orLQR.Ontheotherhand,thecontrolefforts changessignicantly.ItisobviousthatwithLQRthecontroleffortsarethesmallest becauseitisbasedontheidealsituationthatRSOisnotdisabled.Incaseswhere wecan'tcommunicatewiththeRSOanddon'tknowitsmotion,suchacooperation isimpossible,andthusLQRisnotapplicable.BoththeSVandtheRSOhavemuch 73

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ALQR BNash CStackalberg:RSOasleader DStackalberg:SVasleader Figure5-3.Thecontrolforceinputs. smallercontrolforceandtorqueinputbyplayingStackelbergleaderthanStackelberg follower.Also,itisshownthatthecontroleffortswithStackelbergfollowerisalmost thesameaswithNashstrategy.NotethatthetrajectoriesandtheSV'scontrolhistory representthepotentialupper-bound,thecontrolinputsoftheRSOinFig.5-3and Fig.5-4areimaginaryactuationtocausemotionofthedisabledRSO. Itisshownthattheresultsusinggame-theoreticcontrollersaresimilartoLQR controller,whichassumedcooperationbetweentheSVandtheRSO.Thesepreliminary resultsconrmfeasibilityofgametheoreticcontrolleraswellasthehypothesisthatit 74

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ALQR BNash CStackalberg:RSOasleader DStackelberg:SVasleader Figure5-4.Thecontroltorqueinputs. ispossibletoobtainsmalleractuationinputsfortheleaderofaStackelbergapproach whileobtainingrelativelythesametrajectoryvariables. 5.3Conclusion Itwasshownthattheinteractionbetweencontrolledservicevehicleandthe disabledtargetcanbeminimizedwiththeLQRandthegametheoreticcontrollers forapost-dockedmaneuverwherethedistanceandtheattitudeerrorarekeptsmall enoughthatlinearapproximationisvalid.AlthoughtheLQRcontrollerworkedthebest, itwasbasedonanidealcasewheretheRSOwasnotdisabled.SincetheRSObeing 75

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disabledisthecharacteristicoftheproblem,theLQRcontrollerisnotpractical.Onthe otherhand,Thegametheoreticapproach,withtheNashortheStackelbergstrategy, stillkeptthedistanceandtheattitudeerrorssmallevenwithnoncooperativebehavior ofthetargetRSO.Therefore,thefeasibilityofthegametheoreticcontrollersinthis noncooperativescenariowasvalidated. ItwasalsosuggestedthatthecontroleffortoftheSVcanbeloweredbyutilizing Stackelbergleader;suchacasecanprovidethelower-boundforthepossibleinteraction duringthespacecraftpost-dockingphase,whichisalsousefulasadesignfactor. 76

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CHAPTER6 SOLUTIONSTOTWO-PLAYERLINEARQUADRATICSTACKELBERGGAMESWITH TIME-VARYINGSTRUCTURE Two-playerlinearquadraticLQdifferentialgamesdiscussedinChapter5have thewell-denedsolutionstructure.However,itisnotsuitablefordynamicsthatare highlynonlinearsincethelineardynamicmodelcannotaccuratelydescribethesystem behaviors,thusfailingtomaketheresultantgamesolutionoptimalinreality.There areafewwaystoaddressnonlinearitiesinclassofinnitehorizonproblemsinoptimal controlanddifferentialgames.Thoseincludethestate-dependentRiccatiequation SDREtechnique,wherenonlineardynamicsisrewritteninalinear-parameterizedform fromwhichanalgebraicRiccatiequationisformedandsolved,withassumptionofthe solutiondrivingtheoptimalityconditiontothesteady-stateform.Anotherapproachis themodelpredictivecontrolMPC,alsoknownastherecedinghorizoncontrol,where theproblemisbrokenuptomultipleshortnitehorizonproblemsthatareiteratively solved.Thesemethodsgenerallyrequirenumericalmethodsinndingthesolution,and asthenonlinearityincreasessodoesthecomputationaleffort.Thereisalsoatrade-off betweenthecomputationaleffortandoptimalityofthesolution. Inthischapter,anapproachtoinvestigatethesatellitepost-dockingthatutilizethe LQstructurewhileattemptingtopreservenonlinearitiesispresented.Firstthesystem dynamicsiswrittenintheeuler-lagrangeformGivenbyRef.[72]: M q q + V m q q q + f q + g q + d = where M q 2 R n n denotestheinertiamatrix, V m q q 2 R n n denotescentripetal andCorioliseffects, f q 2 R n containsstaticanddynamicfrictionterms, g q 2 R n is thevectorofgravitationaleffects,and d 2 R n containsallunmodeleddisturbances. Insteadofdirectlyformulatinga2-playerLQdifferentialgamefromthesystemdynamics, severalerrorstatesareintroducedtoformerrordynamics,andafeedbackcontrolleris designedtoderivealinearerrormodel.Althoughsimplied,theresiduallinearsystem 77

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istimevariantwhichisassociatedwiththeoriginalsystem'snonlinearity.Formulating agameproblemwiththislinearerrordynamicsyieldstheRiccatidifferentialequations independentofthestatesi.e.,theerrors,whichallowsananalyticalsolution,ifexists, whichworkswithanydynamicswrittenintheeuler-lagrangeform. Twodifferenterrormodelsbasedonlinearandmultiplicativeerrorsarepresented. Foreachlinearerrorsystem,boththeopen-loopandtheclosed-loopRiccatiequations arederived.ThechallengeissolvingtheRiccatidifferentialequationsassociatedwith thetwo-playerlinearquadraticStackelbergdifferentialgames.Thefeedback-linearized errorsystemsaretime-variant,andthusfortheinnitehorizonstructuretheoptimality conditionsarenotreducedtothesteady-stateformi.e.,algebraicRiccatiequations,so ananalyticalsolutiontothematrixdifferentialequationsissought. 6.1GameBasedonAdditiveErrors Supposeadesiredtrajectory q d t isprovidedsuchthat q d t q d t q d t ,and ... q d t existandarebounded.Thetrackingerror e 1 ,theauxiliarytrackingerror e 2 ,andthe lteredtrackingerror r arethendenedas e 1 = q d )]TJ/F59 11.9552 Tf 11.956 0 Td [(q e 2 = e 1 + 1 e 1 r = e 2 + 2 e 2 where 1 2 2 R n n areconstantgainmatrices.Equation6canberewrittenusing Eq.6toyieldanerrormodel M e 2 = )]TJ/F59 11.9552 Tf 9.299 0 Td [(V m e 2 )]TJ/F48 11.9552 Tf 11.956 0 Td [( + M q + 1 e 1 + V m q + 1 e 1 + f + g + d ThenonlineardynamicsinEq.6arefeedbacklinearizedbychoosing suchthatall theknownnonlineartermsarecanceled.GroupthetermsinEq.6tobecanceled 78

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andcallit h : h := M q + 1 e 1 + V m q + 1 e 1 + f + g whichleavestheterm d inEq.6andresultsin M e 2 = )]TJ/F59 11.9552 Tf 9.299 0 Td [(V m e 2 )]TJ/F48 11.9552 Tf 11.956 0 Td [( + h + d Then,designthecontroltorqueinputasthesumofthenonlinearterms h andthe game-theoreticcontroller u : = h )]TJ/F59 11.9552 Tf 11.956 0 Td [(u whichyieldstheclosed-looperrorsystem M e 2 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(V m e 2 + u + d Thus,thedynamicsofthetrackingerrorsandtheauxiliaryerrors e 1 and e 2 ,respectively, canbewrittenas e 1 = )]TJ/F48 11.9552 Tf 9.298 0 Td [( 2 e 1 + e 2 e 2 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 V m e 2 + M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 u + M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 d whichcanalsobewrittenasalineartime-varyingsystem: x = Ax + Bu + B d where x = 2 6 4 e 1 e 2 3 7 5 A = 2 6 4 )]TJ/F48 11.9552 Tf 9.298 0 Td [( 1 I n n 0 n n )]TJ/F59 11.9552 Tf 9.299 0 Td [(M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 V m 3 7 5 B = 2 6 4 0 n n M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 3 7 5 79

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WithEq.6atwo-playerLQdifferentialgameisformulatedas J 1 = 1 2 Z 1 0 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(x T Qx + u T 1 R 11 u 1 + u T 2 R 12 u 2 dt J 2 = 1 2 Z 1 0 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(x T Nx + u T 1 R 21 u 1 + u T 2 R 22 u 2 dt x = Ax + Bu 1 + B d NotethatallthegameconsideredinthischapterareoftheformgivenbyEq.6. DifferenterrormodelscorrespondtodifferenttermsinsideinsideEq.6e.g.,Ittakes theformofEq.6fortheadditiveerrorcase,whichcontributetodifferentRiccati equations.Furthermore,thestrategiesemployedi.e.,open-loopvs.closed-loopaffect theshapeoftheRiccatiequationsaswell. 6.1.1Open-loopStackelbergSolution ThreeRiccatidifferentialequationsforopen-looptwo-personLQdifferentialgame areobtainedbySimaan[51]as K + KA + A T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(KBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(KBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P + Q T = 0 P + PA + A T P )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P + N T )]TJ/F59 11.9552 Tf 11.955 0 Td [(QS = 0 S + SA )]TJ/F59 11.9552 Tf 11.956 0 Td [(AS )]TJ/F59 11.9552 Tf 11.955 0 Td [(SBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(SBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P + BR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(BR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T P = 0 Anysetof K P ,and S satisfyingEqs.6-6canbethesolutionofthegame.In thispaperonesetisshownasasolution. Thefollowingpropertiesareutilizedinsimplicationoftermsandndingthe solution. Property1 Property2.2,Qu[73] M )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 V m isskew-symmetric. FromProperty1itcanbeshownthat M )]TJ/F59 11.9552 Tf 11.995 0 Td [(V m )]TJ/F59 11.9552 Tf 11.995 0 Td [(V T m M )]TJ/F20 11.9552 Tf 11.995 0 Td [(2 V T m ,and V m )]TJ/F59 11.9552 Tf 11.995 0 Td [(V T m arealso skew-symmetric.Althoughtheskew-symmetricityisnotatheorembutaproperty,the proofforProperty1isshownasfollows: 80

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Proof. In[73] V m isrelatedtothemassmatrixby V m q q q = M q q )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 q T @ M q @ q q andsince M q = @ M q @ q d q dt = @ M q @ q q V m canbewrittenas V m = @ M q @ q q )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(1 2 q T @ M q @ q Note @ M q @ q q = @ M q @ q 1 q @ M q @ q 2 q @ M q @ q n q 2 R n n q T @ M q @ q = 2 6 6 6 6 6 6 6 6 6 4 q T @ M q @ q 1 q T @ M q @ q 2 q T @ M q @ q n 3 7 7 7 7 7 7 7 7 7 5 2 R n n FromEqs.6and6, @ M q @ q q T = q T @ M q @ q FromEqs.6,6,and6, M )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 V m = @ M q @ q q )]TJ/F20 11.9552 Tf 11.956 0 Td [(2 @ M q @ q q + q T @ M q @ q = )]TJ/F28 11.9552 Tf 11.291 16.856 Td [( @ M q @ q q + q T @ M q @ q whichisthesumofasquarematrixanditstransposeandthereforeskew-symmetric. 81

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Alltheskew-symmetricmatricesappearingintheRiccatiequationscanbetreated aszeros. Assumethat K P ,and S aresymmetricblockdiagonalmatrices K = 2 6 4 K 11 0 0K 22 3 7 5 P = 2 6 4 P 11 0 0P 22 3 7 5 S = 2 6 4 S 11 0 0S 22 3 7 5 anddenote Q and N asblockmatrices Q = 2 6 4 Q 11 Q 12 Q T 12 Q 22 3 7 5 N = 2 6 4 N 11 N 12 N T 12 N 22 3 7 5 thenEq.6iswritteninmatrixformas 2 6 4 P 11 0 0 P 22 3 7 5 + 2 6 4 P 11 0 0P 22 3 7 5 2 6 4 )]TJ/F48 11.9552 Tf 9.299 0 Td [( 1 I 0 )]TJ/F59 11.9552 Tf 9.299 0 Td [(M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 V m 3 7 5 + 2 6 4 )]TJ/F48 11.9552 Tf 9.299 0 Td [( T 1 0 I )]TJ/F59 11.9552 Tf 9.298 0 Td [(V T m M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 3 7 5 2 6 4 P 11 0 0P 22 3 7 5 )]TJ/F28 11.9552 Tf 11.291 27.616 Td [(2 6 4 00 0P 22 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 K 22 3 7 5 )]TJ/F28 11.9552 Tf 11.955 27.616 Td [(2 6 4 00 0P 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P 22 3 7 5 + 2 6 4 N 11 N 12 N T 12 N 22 3 7 5 )]TJ/F28 11.9552 Tf 11.956 27.617 Td [(2 6 4 Q 11 Q 12 Q T 12 Q 22 3 7 5 2 6 4 S 11 0 0S 22 3 7 5 = 0 whichissimpliedanddecomposedtofourmatrixequationsEqs.6-6 P 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(P 11 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 P 11 + N 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 11 S 11 = 0 P 11 + N 12 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 12 S 22 = 0 P 11 + N T 12 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q T 12 S 11 = 0 P 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(P 22 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 V m )]TJ/F59 11.9552 Tf 11.955 0 Td [(V m M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(P 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K 22 )]TJ/F59 11.9552 Tf 9.299 0 Td [(P 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P 22 + N 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 22 S 22 = 0 82

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With P 22 = K 22 = M ,Eq.6becomes )]TJ/F59 11.9552 Tf 9.299 0 Td [(R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 + N 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 22 S 22 = 0 whichmakes S 22 constantmatrix.Itisthenfollowedthat P 11 and S 11 arealsoconstant. Equation6iswritteninmatrixformas 2 6 4 K 11 0 0 K 22 3 7 5 + 2 6 4 K 11 0 0K 22 3 7 5 2 6 4 )]TJ/F48 11.9552 Tf 9.298 0 Td [( 1 I 0 )]TJ/F59 11.9552 Tf 9.298 0 Td [(M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 V m 3 7 5 + 2 6 4 )]TJ/F48 11.9552 Tf 9.298 0 Td [( T 1 0 I )]TJ/F59 11.9552 Tf 9.299 0 Td [(V T m M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 3 7 5 2 6 4 K 11 0 0K 22 3 7 5 )]TJ/F28 11.9552 Tf 11.291 27.617 Td [(2 6 4 00 0K 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K 22 3 7 5 )]TJ/F28 11.9552 Tf 11.955 27.617 Td [(2 6 4 00 0K 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P 22 3 7 5 + 2 6 4 Q 11 Q 12 Q T 12 Q 22 3 7 5 = 0 whichissimpliedanddecomposedtofourmatrixequationsEqs.6-6: K 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(K 11 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 K 11 + Q 11 = 0 K 11 + Q 12 = 0 K 11 + Q T 12 = 0 K 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(K 22 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 V m )]TJ/F59 11.9552 Tf 11.955 0 Td [(V T m M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(K 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 K 22 )]TJ/F59 11.9552 Tf 9.298 0 Td [(K 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P 22 + Q 22 = 0 Substitute P 22 = M and K 22 = M intoEq.6toyield )]TJ/F59 11.9552 Tf 9.298 0 Td [(R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 + Q 22 = 0 FromEqs.6&6,andwiththeassumptionthat N 22 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(Q 22 ,asolutionfor S 22 isobtainedas S 22 = )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 I 83

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Equation6iswritteninmatrixformas 2 6 4 S 11 0 0 S 22 3 7 5 + 2 6 4 S 11 0 0S 22 3 7 5 2 6 4 )]TJ/F48 11.9552 Tf 9.298 0 Td [( 1 I 0 )]TJ/F59 11.9552 Tf 9.299 0 Td [(M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 V m 3 7 5 )]TJ/F28 11.9552 Tf 11.955 27.617 Td [(2 6 4 )]TJ/F48 11.9552 Tf 9.298 0 Td [( 1 I 0 )]TJ/F59 11.9552 Tf 9.299 0 Td [(M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 V m 3 7 5 2 6 4 S 11 0 0S 22 3 7 5 )]TJ/F28 11.9552 Tf 11.291 27.616 Td [(2 6 4 00 0S 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K 22 3 7 5 )]TJ/F28 11.9552 Tf 11.956 27.616 Td [(2 6 4 00 0S 22 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P 22 3 7 5 + 2 6 4 00 0M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 K 22 3 7 5 )]TJ/F28 11.9552 Tf 11.955 27.617 Td [(2 6 4 00 0M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P 22 3 7 5 = 0 whichissimpliedanddecomposedtothree 1 matrixequationsEqs.6-6 S 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(S 11 1 + 1 S 11 = 0 S 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(S 22 = 0 S 22 )]TJ/F59 11.9552 Tf 11.956 0 Td [(S 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 V m + M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 V m S 22 )]TJ/F59 11.9552 Tf 11.956 0 Td [(S 22 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(S 22 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P 22 + M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 K 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P 22 = 0 Substitute P 22 = K 22 = M and S 22 = )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 I intoEq.6toyield R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 +2 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 + R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 = 0 Nowtherestof K P ,and S arefound.FromEq.6, S 11 = S 22 = )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 I Equation6isautomaticallysatisedsince S 11 isscalarmultipleofidentitymatrix. FromEq.6-6, K 11 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 )]TJ/F59 11.9552 Tf 5.479 -9.683 Td [(Q 12 + Q T 12 1 Therearefourmatrixequations,butoneofthemis 0 = 0 84

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and K 11 shouldalsosatisfyEq.6canbewrittenas 1 2 )]TJ/F59 11.9552 Tf 10.461 -9.684 Td [(Q 12 + Q T 12 1 + T 1 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(Q 12 + Q T 12 + Q 11 = 0 n n whichisaLyapunovequation.Onewaytosettheparametersistochoose 1 Q 12 ,and Q T 12 thencalculate Q 11 fromthem.FromEq.6-6, P 11 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(N 12 + N T 12 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(Q 12 + Q T 12 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(N 12 + N T 12 +2 K 11 SubstituteEqs.6-6intoEq.6andsimplifytoyieldanotherconstraint 1 2 )]TJ/F59 11.9552 Tf 10.461 -9.684 Td [(N 12 + N T 12 1 + T 1 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(N 12 + N T 12 + N 11 = 0 whichisalsoaLyapunovequation. Finally,combineEqs.6and6toget Q 22 + R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 + R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 = 0 Itmustbenotedthatthewaythesolutionisobtainedisnotallowedineverycase. Theanalyticalsolutionisguaranteedattheexpenseoftheconstraintsonthegainsof thecostfunctionals;thatis,onecannotarbitrarilychoosetheobjectiveoftheplayers andutilizethecontrolstrategyobtainedhere.Itisnotaproblemforthepurposeof modelingapost-dockingscenario,however,sincethefollower'scontrolstrategyis passivelydeterminedbasedontheassumedobjectiveoftheleaderDV,ratherthanthe leaderhavingaspecicobjectivewhichneedstobesatised. 6.1.2Closed-loopStackelbergSolution Thesolutiontoadifferentialgameconsistsoftheoptimalstrategyi.e.,control inputofeachplayer,thestatetrajectorypropagatedbasedontheplayers'strategies, andthecorrespondingcosts.Ideally,thesolutionisgoodfortheentirehorizon. However,anychangesinthegame,suchaschangesintheobjectiveofeachplayerand 85

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differentbehaviorofthesystemduetouncertaintyanddisturbances,compromisethe optimalityofthegamestrategies.Thisphenomenonisknownastimeinconsistency orsubgame-imperfectioningametheory[74].Therearestrongandweaktime inconsistencies,wherethedistinctionisbasedontheinitialconditions.Thetypeof inconsistencyaddressedinthispaperisassociatedwithstrongtimeinconsistency. Toaccountfortimeinconsistency,aclosed-loopStackelbergstrategyisconsidered. Inthedifferentialgame-theoreticsense,open-loopreferstoadecisionmakingofeach playerbasedontheinitialcondition,andclosed-loopreferstotheabilityoftheplayers tochangetheirdecisionsbasedoncurrentinformation. Aclosed-loopstrategyinStackelberggameismoreadaptivetopotentialdiscrepancy betweentherealbehaviorofthesystemanditshypotheticalmodel,whichisknownas timeinconsistency.Open-loopandclosed-loopstrategiesindifferentialgamestheory arebothinfactopen-loopstrategiesinthecontroltheoreticsense.However,theycan bebothimplementedinclosed-loopsystems,justlikelinearquadraticregulatorLQR controller.Inthissectiononepossiblesolutionspresented. Stackelberggamesprovideaframeworkforsystemsthatoperateondifferentlevels withaprescribedhierarchyofdecisions.Foratwo-personStackelberggamewherethe systemisaffectedbytwodecisionmakers,theproblemiscastintwosolutionspaces: theleaderandthefollower,whereeachplayertriestominimizetheirrespectivecost functionals.Theleaderisadecisionmakerthatcanenforceitsstrategytominimizeits objectivemetricoverthefollower.Forexample,whentwoinputsaffectthebehaviorofa system,theonewithmorerapiddynamicscanbeconsideredtheleaderinStackelberg structure;sincethesystemrespondsmorerapidlytotheleader'scontrolinput,itis reasonabletoputmoreweightinoptimizingtheleader'scontrolstrategywhilemaking thefollowercompromise. AStackelbergdifferentialgameproblem,with u F asthefollowerand u L asthe leader,isformulatedbyadifferentialconstraintandthecostfunctionals J 1 z u F u L 86

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J 2 z u F u L 2 R as z = Az + Bu F + Bu L J F = 1 2 Z 1 t 0 )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(z T Qz + u T F R 11 u F + u T L R 12 u L dt J L = 1 2 Z 1 t 0 )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(z T Nz + u T F R 21 u F + u T L R 22 u L dt where Q N 2 R 2 n 2 n aresymmetricconstantmatricesdenedas Q = 2 6 4 Q 11 Q 12 Q T 12 Q 22 3 7 5 N = 2 6 4 N 11 N 12 N T 12 N 22 3 7 5 and Q ij N ij 2 R n n arepositivedeniteandsymmetricconstantmatrices 8 i j =1,2.A closed-loopsolutionissoughtbyextending[44].Unliketheopen-loopcase,thefollower assumesthattheleader'sstrategyexplicitlyaffectsthesystem.Withthegamebeingof linear-quadraticstructure,thefollowingassumptionismade. Assumption: Incomputingitsstrategy,thefollowerassumesthattheleader'sstrategy islinearinthestatessuchthat u L = F 2 z where F 2 t 2 R 2 n 2 n isassociatedwiththesuchthatthefollower'sproblemiswrittenas z = A + BF 2 z + Bu F J F = 1 2 Z 1 t 0 )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(z T )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(Q + F T 2 R 12 F 2 z + u T F R 11 u F dt TheHamiltonianofthefolloweris H F = 1 2 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(z T Q + F T 2 R 12 F 2 z + u T F R 11 u F + T 1 A + BF 2 x + Bu F 87

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wheretheoptimalcontrolstrategyandthecostateequationofthefollowerareobtained as u F = )]TJ/F59 11.9552 Tf 9.298 0 Td [(R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T 1 1 = )]TJ/F20 11.9552 Tf 9.298 0 Td [( Q + F T 2 R 12 F 2 T z )]TJ/F20 11.9552 Tf 11.955 0 Td [( A + BF 2 T 1 Substituting6and6intothedynamicsand J 2 z u F u L t yieldsanoptimal controlproblemoftheleader z = Az )]TJ/F59 11.9552 Tf 11.955 0 Td [(BR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T 1 + Bu L J L = 1 2 Z 1 t 0 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(z T Nz + u T L R 22 u L + T 1 BR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T 1 dt wheretheHamiltonianoftheleaderisconstructedas H L = 1 2 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(z T Nz + T 1 BR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T 1 + u T L R 22 u L + T 2 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(Az )]TJ/F59 11.9552 Tf 11.955 0 Td [(BR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T 1 + Bu L + T )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F20 11.9552 Tf 9.299 0 Td [( Q + F T 2 R 12 F 2 T z )]TJ/F20 11.9552 Tf 11.955 0 Td [( A + BF 2 T 1 where u L = )]TJ/F59 11.9552 Tf 9.299 0 Td [(R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T 2 2 = )]TJ/F59 11.9552 Tf 9.299 0 Td [(N T x )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T 2 + Q + F T 2 R 12 F 2 = )]TJ/F59 11.9552 Tf 9.299 0 Td [(BR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T 1 + BR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T 2 + A + BF 2 Theexpressionsderivedin6-6denethesolutiontothedifferentialgame. Thesubsequentanalysisaimsatdevelopinganexpressionforthecostatevariables 1 t 2 t t whichcanbeimplementedbythecontrollers u F t and u L t .Suppose 88

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thatthecostatesarelinearinthestate: 1 = Kz 2 = Pz = Sz where K t P t S t 2 R 2 n 2 n aretime-varyingpositivedenitediagonalmatrices. Giventheseassumedsolutions,conditionsandconstraintsaredevelopedtoensure 6-6satisfy6,6,and6.Differentiating6-6and substitutingthedynamicconstraintinEq.6alongwithEqs.6-646yields threedifferentialRiccatiequations 0 = K + KA )]TJ/F59 11.9552 Tf 11.955 0 Td [(KBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(KBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P + Q + PBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 R 12 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P + A T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T K 0 = P + PA )]TJ/F59 11.9552 Tf 11.956 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T P + N + A T P )]TJ/F59 11.9552 Tf 11.955 0 Td [(QS )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 R 12 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T PS 0 = S + SA )]TJ/F59 11.9552 Tf 11.955 0 Td [(SBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(SBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P + BR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(BR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T P )]TJ/F59 11.9552 Tf 11.955 0 Td [(AS + BR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T PS 89

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Equations6-6canbeexpressedasopen-loopRiccatiequationsplus additionalterms.FromEqs.2theopen-loopRiccatiequationsare 0 = K + KA + A T K )]TJ/F59 11.9552 Tf 11.956 0 Td [(KBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(KBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P + Q T 0 = P + PA + A T P )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 22 B T P + N T )]TJ/F59 11.9552 Tf 11.955 0 Td [(QS 0 = S + SA )]TJ/F59 11.9552 Tf 11.955 0 Td [(AS )]TJ/F59 11.9552 Tf 11.955 0 Td [(SBR )]TJ/F59 7.9701 Tf 6.587 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(SBR )]TJ/F59 7.9701 Tf 6.587 0 Td [(1 22 B T P + BR )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 R 21 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.956 0 Td [(BR )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T P Letthesubscripts CRE and ORE denotetheclosed-loop6-6and open-loop6-6Ricattiequations,respectively.Thentheclosed-loopRiccati equationscanbewrittenas f K CRE = f K ORE + PBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 R 12 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T K = 0 f P CRE = P ORE )]TJ/F59 11.9552 Tf 11.956 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 R 12 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T PS = 0 F S CRE = F S ORE + BR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T PS = 0 InEqs.6-6, K t and P t correspondto u 1 t and u 2 t respectively,while S t constraintsthetrajectoryof K t and P t .Equations6-6mustbesolved simultaneouslytoyieldtheStackelbergcontrolstrategiesfortheleaderandthefollower. 90

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If P t K t ,and S t areselectedas P = 2 6 4 P 11 0 n n 0 n n M 3 7 5 K = 2 6 4 K 11 0 n n 0 n n M 3 7 5 S = 2 6 4 S 11 0 n n 0 n n )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 I n n 3 7 5 where K 11 and P 11 satisfy K 11 = )]TJ/F20 11.9552 Tf 10.494 8.087 Td [(1 2 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(Q 12 + Q T 12 P 11 = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(N 12 + N T 12 +2 K 11 then6-6aresolvedwiththefollowingconstraintsonthecostfunctionals: 1 2 )]TJ/F59 11.9552 Tf 10.46 -9.684 Td [(Q 12 + Q T 12 1 + T 1 )]TJ/F57 11.9552 Tf 5.48 -9.684 Td [(Q 12 + Q T 12 + Q 11 = 0 1 2 )]TJ/F59 11.9552 Tf 10.461 -9.684 Td [(N 12 + N T 12 1 + T 1 )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(N 12 + N T 12 + N 11 = 0 Q 22 + R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 + R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 = 0 )]TJ/F59 11.9552 Tf 9.299 0 Td [(R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 + Q 22 + R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 R 12 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 = 0 Q 22 + N 22 = 0 FromEqs.6,6,6,6,6,and6,theclosed-loop Stackelberggame-basedcontrollersareobtainedas u F = )]TJ/F59 11.9552 Tf 9.299 0 Td [(R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 e 2 u L = )]TJ/F59 11.9552 Tf 9.299 0 Td [(R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 e 2 Notethatthesolutionhasthesameformastheopen-loopproblemexceptthatmore conservativeconstraintsareplacedontherelationshipamongthegainmatrices.In particular,constraintsinEq.6include R 12 ,whichaffectsthedecisionof u 1 t due 91

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tothedecisionof u 2 t fortheclosed-loopcase.Therefore,aclosed-loopstrategyforthe followerisbetterthananopen-loopstrategyinaddressingthetimeinconsistency. 6.2GameBasedonMultiplicativeErrors Attitudeerrorsareinherentlynonlinear.Theadditiveerrormodeldiscussed inSection6.1,whileworksforwidevarietiesofdynamicmodelswithhierarchical non-cooperativestructure,cannotfullycapturetherotationalmotionofarigidbodywhen theorientationmismatchbetweenthecurrentandthedesiredattitudesishuge.Tobe exact,thisissueisassociatedwithWiththeasymptoticcontrollerdesign,theerroris supposedtocontinuouslyconvergetoanequilibrium. Thesubsequentdevelopmentchoosestheerrorquaternionamongthemultiplicative errors.Theerrorquaternionisdenedasanadditionalrotationfromthecurrenttothe desiredorientation,asdepictedinFig.6-1. Figure6-1.Therelationshipamongthecurrentandthedesiredorientations. Letthecurrent,thedesired,andtheerrorquaternionsbedenotedas q = 2 6 4 q q 4 3 7 5 q d = 2 6 4 q d q d 4 3 7 5 q e = 2 6 4 e q e 4 3 7 5 e = e 1 = )]TJ/F57 11.9552 Tf 5.48 -9.684 Td [(d 4 1 )]TJ/F59 11.9552 Tf 11.955 0 Td [(d q )]TJ/F57 11.9552 Tf 11.955 0 Td [(q 4 d 92

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where q 4 = q 1 )-222(k q k 2 d 4 = q 1 )-222(k d k 2 thentheauxiliaryandthelteredtrackingerrorsaredenedinthesamewayasthe additiveerrorcase e 2 = e 1 + 1 e 1 r = e 2 + 2 e 2 Assuming d = d = 0 d 4 = d 4 =0, e 1 = )]TJ/F57 11.9552 Tf 5.48 -9.684 Td [(d 4 1 )]TJ/F59 11.9552 Tf 11.955 0 Td [(d q )]TJ/F20 11.9552 Tf 13.579 0.221 Td [( q 4 d = d 4 1 )]TJ/F59 11.9552 Tf 11.955 0 Td [(d + dq T q 4 q = P q d q e 1 = d 4 1 )]TJ/F59 11.9552 Tf 11.955 0 Td [(d + dq T q 4 q + d q T q 4 + dq T q T q q 3 4 q = P q d q + Q q q d q Nowdeveloptheerrorsystem. M e 2 = M e 1 + 1 e 1 = MP q + M f Q + 1 P g q Substituting q fromEq.6 q = )]TJ/F59 11.9552 Tf 9.298 0 Td [(M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 f V m q + g + d )]TJ/F48 11.9552 Tf 11.955 0 Td [( g intoEq.6yields M e 2 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(MPM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 f V m q + g + d )]TJ/F48 11.9552 Tf 11.955 0 Td [( g + M f Q + 1 P g q Let = h )]TJ/F59 11.9552 Tf 11.955 0 Td [(u where h = V m q + g + MP )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 f Q + 1 P g q )]TJ/F59 11.9552 Tf 11.955 0 Td [(V m e 2 93

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thenEq.6becomes M e 2 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(MPM )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 V m e 2 )]TJ/F59 11.9552 Tf 11.956 0 Td [(MPM )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 u )]TJ/F59 11.9552 Tf 11.955 0 Td [(MPM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 d 6.2.1Open-loopStackelbergSolution Thedynamicsofthetrackingandtheauxiliarytrackingerroris e 1 = )]TJ/F23 11.9552 Tf 9.298 0 Td [( 1 e 1 + e 2 e 2 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(PM )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 V m e 2 )]TJ/F59 11.9552 Tf 11.956 0 Td [(PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 u )]TJ/F59 11.9552 Tf 11.956 0 Td [(PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 d Thelinear-quadraticdifferentialgameisdenedas x = Ax + B 1 u + B 2 d J 1 = 1 2 Z 1 0 )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(x T Qx + u T 1 R 11 u 1 + u T 2 R 12 u 2 dt J 2 = 1 2 Z 1 0 )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(x T Nx + u T 1 R 21 u 1 + u T 2 R 22 u 2 dt where x = 2 6 4 e 1 e 2 3 7 5 A = 2 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [( 1 1 0 )]TJ/F59 11.9552 Tf 9.299 0 Td [(PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 V m 3 7 5 B 1 = B 2 = 2 6 4 0 )]TJ/F59 11.9552 Tf 9.298 0 Td [(PM )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 3 7 5 Thedifferencefromthepreviousworkistheterm B ,sotheRiccatiequationsneedtobe investigated. K + KA + A T K T )]TJ/F59 11.9552 Tf 11.955 0 Td [(KBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T K T )]TJ/F59 11.9552 Tf 11.955 0 Td [(KBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T P T + Q T = 0 P + PA + A T P T )]TJ/F59 11.9552 Tf 11.956 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T K T )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P T + N T )]TJ/F59 11.9552 Tf 11.955 0 Td [(QS = 0 S + SA )]TJ/F59 11.9552 Tf 11.955 0 Td [(AS )]TJ/F59 11.9552 Tf 11.956 0 Td [(SBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T K T )]TJ/F59 11.9552 Tf 11.955 0 Td [(SBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P + BR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(BR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T P T = 0 Assuming K = 2 6 4 K 11 0 0K 22 3 7 5 P = 2 6 4 P 11 0 0P 22 3 7 5 S = 2 6 4 S 11 0 0S 22 3 7 5 94

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Equation.6,intheblockdiagramform,willlooklike: 2 6 4 P 11 0 0 P 22 3 7 5 + 2 6 4 )]TJ/F23 11.9552 Tf 9.298 0 Td [( 1 P 11 P 11 0 )]TJ/F59 11.9552 Tf 9.298 0 Td [(P 22 PM )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 V m 3 7 5 + 2 6 4 )]TJ/F23 11.9552 Tf 9.298 0 Td [( 1 P T 11 0 P T 11 )]TJ/F59 11.9552 Tf 9.298 0 Td [(V T m M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T P T 22 3 7 5 )]TJ/F28 11.9552 Tf 11.291 27.617 Td [(2 6 4 00 0P 22 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T K T 22 3 7 5 )]TJ/F28 11.9552 Tf 11.955 27.617 Td [(2 6 4 00 0P 22 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T P T 22 3 7 5 + 2 6 4 N 11 N 12 N T 12 N 22 3 7 5 )]TJ/F28 11.9552 Tf 11.955 27.617 Td [(2 6 4 Q 11 Q 12 Q T 12 Q 22 3 7 5 2 6 4 S 11 0 0S 22 3 7 5 = 2 6 4 00 00 3 7 5 andisdecomposedtofourmatrixequations: P 11 )]TJ/F23 11.9552 Tf 11.955 0 Td [( 1 P 11 + 1 P T 11 + N 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 11 S 11 = 0 P 11 + N 12 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 12 S 22 = 0 P T 11 + N T 12 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q T 12 S 11 = 0 P 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(P 22 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 V m )]TJ/F59 11.9552 Tf 11.956 0 Td [(V T m M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P T P T 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(P 22 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P T K T 22 )]TJ/F59 11.9552 Tf 9.299 0 Td [(P 22 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T P T 22 + N 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 22 S 22 = 0 InordertosolveEq.6,weneedtochoose P 22 .Fortheadditiveerrorcase, P 22 = M worked.Forthemultiplicativecase,try P 22 = MP )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 and K 22 = MP )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 too: MP )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 + M P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 )]TJ/F59 11.9552 Tf 11.955 0 Td [(V m )]TJ/F59 11.9552 Tf 11.956 0 Td [(V T m )]TJ/F59 11.9552 Tf 11.955 0 Td [(R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 + N 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 22 S 22 = 0 Equation6has MP )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 and M P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ,makingitimpossibletoutilizetheskew-symmetric propertyi.e., M )]TJ/F59 11.9552 Tf 13.097 0 Td [(V m )]TJ/F59 11.9552 Tf 13.097 0 Td [(V T m = 0 .However,wecontinueanywayandproceedto 95

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Eq.6: 2 6 4 K 11 0 0 K 22 3 7 5 + 2 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [( 1 K 11 K 11 0 )]TJ/F59 11.9552 Tf 9.298 0 Td [(K 22 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 V m 3 7 5 + 2 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [( 1 K T 11 0 K T 11 )]TJ/F59 11.9552 Tf 9.298 0 Td [(V T m M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T K T 22 3 7 5 )]TJ/F28 11.9552 Tf 11.291 27.616 Td [(2 6 4 00 0K 22 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T K T 22 3 7 5 )]TJ/F28 11.9552 Tf 11.955 27.616 Td [(2 6 4 00 0K 22 PM )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P T P T 22 3 7 5 + 2 6 4 Q 11 Q 12 Q T 12 Q 22 3 7 5 = 2 6 4 00 00 3 7 5 or K 11 )]TJ/F23 11.9552 Tf 11.955 0 Td [( 1 K 11 )]TJ/F23 11.9552 Tf 11.955 0 Td [( 1 K T 11 + Q 11 = 0 K 11 + Q 12 = 0 K T 11 + Q T 12 = 0 K 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(K 22 PM )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 V m )]TJ/F59 11.9552 Tf 11.955 0 Td [(V T m M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P T K T 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(K 22 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P T K T 22 )]TJ/F59 11.9552 Tf 9.298 0 Td [(K 22 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T P T 22 + Q 22 = 0 With P 22 = M 22 = MP )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ,Eq.6becomes MP )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 + M P )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 )]TJ/F59 11.9552 Tf 11.955 0 Td [(V m )]TJ/F59 11.9552 Tf 11.955 0 Td [(V T m )]TJ/F59 11.9552 Tf 11.955 0 Td [(R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 )]TJ/F59 11.9552 Tf 11.956 0 Td [(R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 + Q 22 = 0 SubtractingEq.6fromEq.6yields N 22 = Q 22 1 + S 22 Assuming Q 22 = )]TJ/F59 11.9552 Tf 9.299 0 Td [(N 22 S 22 = )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 1 96

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NowlookatEq.6. 2 6 4 S 11 0 00 3 7 5 + 2 6 4 )]TJ/F23 11.9552 Tf 9.298 0 Td [( 1 S 11 S 11 0 2 PM )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 V m 3 7 5 )]TJ/F28 11.9552 Tf 11.955 27.617 Td [(2 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [( 1 S 11 )]TJ/F20 11.9552 Tf 9.299 0 Td [(2 1 0 2 PM )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 V m 3 7 5 )]TJ/F28 11.9552 Tf 11.955 27.617 Td [(2 6 4 00 0 )]TJ/F20 11.9552 Tf 9.299 0 Td [(2 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T K T 22 3 7 5 )]TJ/F28 11.9552 Tf 11.291 27.616 Td [(2 6 4 00 0 )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T P T 22 3 7 5 + 2 6 4 00 0PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T K T 22 3 7 5 )]TJ/F28 11.9552 Tf 11.291 27.617 Td [(2 6 4 00 0PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P T P T 22 3 7 5 = 2 6 4 00 00 3 7 5 whichisdecomposedto S 11 = 0 S 11 +2 1 = 0 0 = 0 2 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T K T 22 +2 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T P T 22 + PM )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 P T K T 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P T P T 22 = 0 Substituting K 22 = P 22 = MP )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ,Eq.6becomes 2 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 +2 PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 + PM )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(PM )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 = 0 or PM )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(2 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 +2 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 + R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 = 0 FromEq.6aconstraintbetween R 11 R 21 ,and R 22 areobtainedas R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 = )]TJ/F20 11.9552 Tf 9.299 0 Td [(2 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 NowgobacktoEq.6.Itispossibletomake Q 2 2time-varyingas Q 22 = Q 2 2 t = )]TJ/F20 11.9552 Tf 12.287 2.307 Td [( MP )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 )]TJ/F59 11.9552 Tf 11.955 0 Td [(M P )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 + V m + V T m + R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 + R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 97

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P 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(P 11 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 P 11 + N 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 11 S 11 = 0 P 11 + N 12 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 12 S 22 = 0 P 11 + N T 12 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q T 12 S 11 = 0 P 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(P 22 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 V m )]TJ/F59 11.9552 Tf 11.955 0 Td [(V m M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(P 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K 22 )]TJ/F59 11.9552 Tf 9.299 0 Td [(P 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P 22 + N 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(Q 22 S 22 = 0 K 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(K 11 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 K 11 + Q 11 = 0 K 11 + Q 12 = 0 K 11 + Q T 12 = 0 K 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(K 22 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 V m )]TJ/F59 11.9552 Tf 11.955 0 Td [(V T m M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(K 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 K 22 )]TJ/F59 11.9552 Tf 9.298 0 Td [(K 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P 22 + Q 22 = 0 S 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(S 11 1 + 1 S 11 = 0 S 11 )]TJ/F59 11.9552 Tf 11.955 0 Td [(S 22 = 0 S 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(S 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 V m + M )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 V m S 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(S 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K 22 )]TJ/F59 11.9552 Tf 11.956 0 Td [(S 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P 22 + M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 K 22 )]TJ/F59 11.9552 Tf 11.955 0 Td [(M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 M )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 P 22 = 0 98

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6.2.2Closed-loopStackelbergSolution Recalltheclosed-loopStackelbergRiccatidifferentialequations 0 = K + KA )]TJ/F59 11.9552 Tf 11.955 0 Td [(KBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(KBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P + Q + PBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 R 12 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P + A T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T K 0 = P + PA )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 B T P + N + A T P )]TJ/F59 11.9552 Tf 11.955 0 Td [(QS )]TJ/F59 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 R 12 R )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T PS 0 = S + SA )]TJ/F59 11.9552 Tf 11.955 0 Td [(SBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.956 0 Td [(SBR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T P + BR )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 B T K )]TJ/F59 11.9552 Tf 11.955 0 Td [(BR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 B T P )]TJ/F59 11.9552 Tf 11.955 0 Td [(AS + BR )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 22 B T PS Eachmatrixequation 2 R 2 n 2 n isdecomposedintofourmatrixequations 2 R n n 6.3SimulationsandResults Thefollowingscenarioisconsideredfortheproblemofpost-dockedstate maintenanceviaadifferentialgame-basedcontroller.AservicevehicleSVsatellite isdockedwithanothersatellitereferredtoasadisabledvehicleDV,whichisdened asasatellitewhosebehaviorisunpredictableduetoamalfunctioningactuatororloss ofcommunication.Intheeventofaperfectrendezvous,thebody-xedcoordinate framesofbothsatellitesarealigned.Bothareequippedwith3-axisattitudecontroland canaffecttherelativeorientationofonefromtheother.Theattitudemisalignmentis assumedtocauseaninteractionforce/torqueatthedockingpoint.Itisassumedthat theattitudeerrorissmallandisupper-boundedby90 2 .TheuseofmodiedRodriguez parameterMRPwillcircumventthisconstraintandallowinniteangulardisplacement. 2 Thisupper-boundisconservativeandcomesfromeliminationoftheEuler parameter inEq.6.Theattitudemisalignmentwouldbemuchsmallerfor post-docking. 99

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Figure6-2.Twodockedsatellitesapproximatedastworigidbodiesconnectedviaa torsionspring Notethatrestrictingthemotiontorotationalisjustiedaslongasthenon-cooperating behaviorofthetargetisnotadversarial.Thusthescenarioisspecictoadisabled satellite,whichiscooperativeifnotdisabled. AmathematicalmodelofthescenarioshowninFig.6-2isconstructedwherethe interactionbetweentheSVandtheDVisafunctionoforientationmismatchanda virtualtorsionspringplacedatthedockingpoint.TheinteractionaffectstheSV'sangular motionwhichisgovernedby J + J = + d + s where J isthemomentofinertiamatrix, istheSV'sangularvelocityrelativeto theDV, istheSV'scontroltorqueinput, d istheDV'scontroltorqueinputwhich isnoncooperative,and s istheinteractionmomentduetoanorientationmismatch betweentheSVandDV.Thesuperscript inEq.6denotestheskew-symmetric matrixequivalenttovectorcrossproduct.Since d isunknowntotheSV,itisconsidered tobeadisturbance. 100

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Equation6iswrittenasafunctionofaunitquaternioni.e.,theEuler parameters.Let = [ 1 2 3 ] T and bedenedas = n sin 2 =cos 2 whichrepresenttherotationofavectoraboutaunitvector n byanangle .Thespring torqueisexpressedintermsof and as s = )]TJ/F57 11.9552 Tf 9.299 0 Td [(k n = 8 > > < > > : )]TJ/F20 11.9552 Tf 9.298 0 Td [(2 k cos )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 k k if 6 = 0 3 0 3 if = 0 3 where k isthespringconstant.Forsimplicity,thespringstiffnessisassumeduniformin anydirectioni.e., k isascalar.Dampingeffectcanalsobeconsidered,butitisapplied suchthattheorientationmismatchiscorrected,soitisnotconsideredhere.Theangular velocitycanbeexpressedintermsoftheEulerparametersandtheirtimederivativesas shownin[75]as 2 6 4 0 3 7 5 = 2 6 6 6 6 6 6 6 4 1 2 3 0 3 7 7 7 7 7 7 7 5 =2 2 6 6 6 6 6 6 6 4 3 )]TJ/F23 11.9552 Tf 9.298 0 Td [( 2 )]TJ/F23 11.9552 Tf 9.299 0 Td [( 1 )]TJ/F23 11.9552 Tf 9.299 0 Td [( 3 1 )]TJ/F23 11.9552 Tf 9.299 0 Td [( 2 2 )]TJ/F23 11.9552 Tf 9.298 0 Td [( 1 )]TJ/F23 11.9552 Tf 9.299 0 Td [( 3 1 2 3 3 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 4 1 2 3 3 7 7 7 7 7 7 7 5 orinamorecompactform =2 )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [( I 3 3 )]TJ/F48 11.9552 Tf 11.955 0 Td [( )]TJ/F20 11.9552 Tf 11.955 0 Td [(2 0= T + Sincequaternionsarefour-dimensionalwhiletheproblemis3-DOF, iseliminated andthevectorcomponents istreatedasthestateofthesystem.Basedonthe 101

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previousassumptionthattheangulardisplacementnevergoesbeyond90 ,let = p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T andfromEq.6itcanbeshownthattherstandthesecondtimederivativesare = )]TJ/F20 11.9552 Tf 31.797 8.088 Td [(1 p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T T = )]TJ/F20 11.9552 Tf 27.767 8.088 Td [( T p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T )]TJ/F20 11.9552 Tf 31.92 8.088 Td [( T 2 p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 3 )]TJ/F20 11.9552 Tf 34.454 8.088 Td [(1 p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T T thentheangularvelocityandangularaccelerationcanbeexpressedas =2 p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T I 3 3 )]TJ/F48 11.9552 Tf 11.955 0 Td [( + T p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T = )]TJ/F20 11.9552 Tf 9.299 0 Td [(2 +2 h p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T I 3 3 )]TJ/F48 11.9552 Tf 11.955 0 Td [( i + 2 p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T T + 1 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T T T + T Finally,Eqs.6and6canbesubstitutedintoEq.6and rearrangedtoobtaintheEulerLagrangeform M + V m + g + d = where M =2 J p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T I 3 3 )]TJ/F48 11.9552 Tf 11.955 0 Td [( + T p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T V m =2 J 2 6 4 )]TJ/F48 11.9552 Tf 9.299 0 Td [( + T p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T + T T p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 3 3 7 5 +4 p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T I 3 3 )]TJ/F48 11.9552 Tf 11.955 0 Td [( + T p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T J p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T I 3 3 )]TJ/F48 11.9552 Tf 11.955 0 Td [( + T p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T g = )]TJ/F48 11.9552 Tf 9.298 0 Td [( s = 8 > > < > > : 2 k cos )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 p 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T k k if 6 = 0 3 0 3 if = 0 3 102

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Thefollowingparametersarechosenfordemonstration.Supposethatafterdocking SVisexpectedtostayatthesamelocationandmaintaintheorientationmisalignment toDVassmallaspossible.SupposethatinitiallytheSV'soffsetissuchthatitisrotated about[123] T byanangle =10 3 ,whichgivestheinitialcondition q = = 2 6 6 6 6 4 0.0233 0.0466 0.0699 3 7 7 7 7 5 andthedesiredtrajectoryischosentobezero. q d t = 0 3 Inthisparticularproblem,nonzerodesiredtrajectorymeansSVtriestochangeits orientationfollowingtheprescribedtrajectory.However,sincethereisnodirectcontrol overthebehaviorofDV,thereisnoguaranteeSVcantracksuchtrajectoriesalongwith DVwhileminimizingtheinteraction. Inanidealcase,thedisturbanceduetotheDV'snoncooperativebehavioris d = )]TJ/F59 11.9552 Tf 9.298 0 Td [(R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 e 2 .However,thecontroller'sperformanceneedstobetestedforarbitrary disturbance.Atestcaseiscomposedoftwostages;roughlyfortherstveseconds thedisturbanceismodeledasgamedisturbance.aftervesecondsthegame-based disturbanceisfadingandanarbitrarydisturbanceisapplied. d t = f t Gamedisturbance z }| { )]TJ/F59 11.9552 Tf 9.298 0 Td [(R )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 22 e 2 t + g t 2 6 6 6 6 4 0.1sin t +0.15cos t 0.15sin t +0.1cos t 3sin t +1.5sin t cos t 3 7 7 7 7 5 | {z } Arbitrarydisturbance 3 Alargenumberischosentodemonstratethecontroller'scapabilitytominimizethe correspondingerror. 103

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where f t and g t aresigmoid-likefunctionstosimulatesmoothon/offswitches betweenthegame-baseddisturbanceandthearbitrarydisturbanceasshownin Fig.6-3: f t = )]TJ/F20 11.9552 Tf 48.555 8.088 Td [(1 1+exp )]TJ/F57 11.9552 Tf 9.299 0 Td [(t + T +1 g t = 1 1+exp )]TJ/F57 11.9552 Tf 9.298 0 Td [(t + T where T =5sforthistest. Figure6-3. f t and g t asrespectiveweightsonthegameandanarbitrary disturbances Table6-1showsthelistofalltheparametersusedinthesimulation. TheresultswithaStackelbergstrategyisshowninFig.6-4.Givenalargeangular displacement,theSV'scontroltorque quicklyadjustsitsattitudeandminimizesthe attitudeerror,thusminimizingtheresultantinteractionwithDV.AsshowninFig.6-4F, thecontroltorqueduetoRISEfeedbackwhichisrequiredtofeedbacklinearizethe dynamicsstayslargerthanthegame-basedcontrolinputanddoesnotdecaytozero. Ontheotherhand,Fig.6-4Eshowsthatthegame-basedcontrolinput u ,whichlinearly dependsonthetrackingerrors,quicklyconvergestozero. 104

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Table6-1.ThesimulationparametersfortheStackelberg-RISEcontroller NameDescriptionValueUnit t Timeofthesimulation20s k Thespringconstant300N m/rad Initialangulardisplacement10deg q Initialstates [0.0233,0.0466,0.0699] T \000 q d Desiredtrajectory 0 3 \000 1 Tuninggainon e 1 I 3 3 1/s 2 Tuninggainon e 2 2 I 3 3 1/s Tuninggainon 5N m k s Tuninggainon 20N m s J MomentofinertiaoftheSV diag [0.5,0.25,1]kg m 2 R 11 Weighton u 1 in J 1 Stackelberg diag [1.2,1,1.15] \000 R 21 Weighton u 1 in J 2 Stackelberg diag [ )]TJ/F20 11.9552 Tf 9.298 0 Td [(0.6, )]TJ/F20 11.9552 Tf 11.955 0 Td [(1, )]TJ/F20 11.9552 Tf 11.955 0 Td [(1.4] \000 R 22 Weighton u 2 in J 2 Stackelberg diag [ )]TJ/F20 11.9552 Tf 9.299 0 Td [(3.1579, )]TJ/F20 11.9552 Tf 11.955 0 Td [(1.2821, )]TJ/F20 11.9552 Tf 11.956 0 Td [(1.1006] \000 Q Weighton x in J 1 2 6 6 6 6 6 6 4 202.51.00 )]TJ/F20 11.9552 Tf 9.299 0 Td [(4.04.01.50 2.510.02.504.0 )]TJ/F20 11.9552 Tf 9.298 0 Td [(1.03.00 1.002.5020.001.503.00 )]TJ/F20 11.9552 Tf 9.299 0 Td [(6.00 )]TJ/F20 11.9552 Tf 9.298 0 Td [(4.004.001.500.3500 4.00 )]TJ/F20 11.9552 Tf 9.299 0 Td [(1.003.0000.350 1.503.00 )]TJ/F20 11.9552 Tf 9.298 0 Td [(6.00000.33 3 7 7 7 7 7 7 5 \000 105

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6.4Conclusion Ageneralnonlineardynamicsystemcanbetransformedtolineartime-varying errormodeltoadmitdifferentialgametheoreticcontrolstrategies.Whiletheresultisnot perfectlyoptimalduetofeedbacklinearization,nonlinearitycanbestillcapturedandas aresultthesolutioncouldbelesslimitedthanalinearizedcasewhereallthenonlinear characteristicsofthesystemdynamicsareremovedintherstplace.Withrelatively smallnoncooperativedisturbanceStackelberg/RISEfeedbackcontrollawdemonstrates itscapabilityofminimizethetrackingerrorinageneralnonlineardynamicsystem.This controldesignallowsbothzero-sumandnonzero-sumconsiderationsandtherefore providesmanywaystomodelnoncooperativecharactersinasystemi.e.,ifthereis absolutelynoknowledgeaMinimaxstrategycanbeemployed,andincaseswherethe noncooperativebehaviorscanbemodeledtosomeextentaStackelbergstrategycanbe applied. 106

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AThetrackingerror e 1 t B q t torepresenttheSV'sorientation CTheSV'scontroltorqueinput t DTheangulardisplacementofSVrelativetoDV EGametheoreticcontributionto FRISEfeedbackcontributionto Figure6-4.ThesimulationresultsfortheStackelbergandRISEcontroller 107

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CHAPTER7 CONCLUSIONANDFUTUREWORKS Necessityofunderstandingthespacecraftpost-dockingwiththetargetbeing non-cooperativee.g.,disabledduetoactuatormalfunctionandcommunicationfailure wasmotivatedbythepotentialofautonomousspacecraftforvariouspurposesuchas debrisremovalandrescue,andsuchabehaviorwasinvestigatedthroughanalysisusing differentialgamestheorywithStackelbergstrategy. Inordertofullyanalyzethenon-cooperativebehaviorofthetargetspacecraft andfeedbacktheinformationapplyingtospacecraftdesignfullnonlineardifferential gamesneedtobesolved,butdifcultyofproceedingwithsuchanalysisattributesto theimmaturityofconcretesolutiontechniquesforbilevelprogramming,withStackelberg staticgamebeingitssubclass. Ontheotherhand,thefoundationhasbeendevelopedfortheattitudecontroller forspacecraftpost-docking.UtilizationoftheEuler-lagrangestructureandthe correspondinglinearizederrormodelallowsoftheexpansionofthesystemdynamics toahigherdelitymodel.Furthermore,thedevelopedcontrollerdesigncanbe implementedinrealtime,providingtheanalyticalsolutiontothetime-varyingStackelberg differentialgames. Basedontheresults,theworkstobefollowedinthefutureincludeiinvestigating numericalmethodstosolvethecorrespondingnonlineardifferentialgamesandii expandingthesystemdynamicstoahigherdelitymodele.g.,contactforcemodel. 108

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APPENDIXA OPTIMALITYCONDITIONSOFTWO-PERSONSTACKELBERGDIFFERENTIAL GAMES Inordertomakethismanuscriptrelativelyself-contained,inthissectionthe rst-orderoptimalityconditionsfortheopen-loopsolutiontotwo-personStackelberg differentialgamesdiscussedinChapter5arederivedusingcalculusofvariations. Supposeatwo-personStackelbergdifferentialgameisdenedbythedifferential constraint x = f x u 1 u 2 t x t 0 = x 0 A ,andtwocostfunctionals J 1 = 1 x f t f + Z t f t 0 L 1 x t u 1 t u 2 t t dt ,A J 2 = 2 x f t f + Z t f t 0 L 2 x t u 1 t u 2 t t dt ,A whereplayer1andplayer2areconsideredthefollowerandtheleader,respectively. A.1FixedFinalTime ConsideragamedescribedbytheEqs.A,A,andA.Itisassumedthat t f isxedand x t f = x f isfree.Assumethatthecontrolpathsareunconstrained.For anopen-loopStackelberggame,leaderannounceshiscontrolpaths u 2 .Thenthe followertriestondtheoptimalcontrolpath u 1 A.1.1Follower'sStrategy ThentheHamiltonianofthefolloweris H 1 = L 1 x t u 1 t u 2 t t + T 1 f x t u 1 t u 2 t t ,A 109

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where u 2 t ,fornow,isassumedtobeaprescribedfunctionoftime.Thenthe augmentedcostfunctionalofthefolloweris J a 1 = 1 + Z t f t 0 H 1 )]TJ/F48 11.9552 Tf 11.956 0 Td [( T 1 x dt Therstvariationof J a 1 is J a 1 = 1 + Z t f t 0 H 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x dt .A A.1.1.1Variationoftheaugmentedcostfunctional NowinvestigatethevariationEq.A.Thevariationoftheboundaryconditionis givenby 1 = @ 1 @ x t 0 x 0 + @ 1 @ t 0 t 0 + @ 1 @ x t f x f + @ 1 @ t f t f .A Sinceonlythenalstateisfree, x 0 = 0 and t 0 = t f =0,thusEq.Abecomes 1 = @ 1 @ x t f x f .A ThevariationofHamiltonianisgivenby H 1 = @ H 1 @ x x + @ H 1 @ 1 1 + @ H 1 @ u 1 u 1 + @ H 1 @ u 2 u 2 110

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Wehave Z t f t 0 H 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x dt = Z t f t 0 H 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 1 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x dt + H 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x t f t f )]TJ/F28 11.9552 Tf 11.956 9.684 Td [( H 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x t 0 = Z t f t 0 H 1 )]TJ/F23 11.9552 Tf 11.956 0 Td [( T 1 x dt )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( T 1 x t f t 0 )]TJ/F28 11.9552 Tf 11.956 16.272 Td [(Z t f t 0 T 1 x dt = )]TJ/F48 11.9552 Tf 9.299 0 Td [( T 1 t f x t f + Z t f t 0 H 1 )]TJ/F23 11.9552 Tf 11.956 0 Td [( T 1 x + T 1 x dt = )]TJ/F48 11.9552 Tf 9.299 0 Td [( T 1 t f x t f + Z t f t 0 @ H 1 @ x x + @ H 1 @ 1 1 + @ H 1 @ u 1 u 1 + @ H 1 @ u 2 u 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 1 x + T 1 x dt = )]TJ/F48 11.9552 Tf 9.299 0 Td [( T 1 t f x t f + Z t f t 0 @ H 1 @ x + T 1 x + @ H 1 @ 1 )]TJ/F20 11.9552 Tf 13.288 0.108 Td [( x T 1 + @ H 1 @ u 1 u 1 + @ H 1 @ u 2 u 2 dt Thereforethevariationoftheaugmentedcostfunctionalofthefolloweris J a 1 = 1 + Z t f t 0 H 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x dt = @ 1 @ x t f x f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f x t f + Z t f t 0 @ H 1 @ x + T 1 x + @ H 1 @ 1 )]TJ/F20 11.9552 Tf 13.288 0.107 Td [( x T 1 + @ H 1 @ u 1 u 1 + @ H 1 @ u 2 u 2 dt Since t f isxed, x t f = x f A Also,since H 1 = 1 + T 1 f @ H 1 @ 1 = f T 111

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andtherefore J a 1 = @ 1 @ x t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f x f + Z t f t 0 @ H 1 @ x + T 1 x + )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(f T )]TJ/F20 11.9552 Tf 13.288 0.108 Td [( x T 1 + @ H 1 @ u 1 u 1 + @ H 1 @ u 2 u 2 dt A A.1.1.2Optimalityconditions SettingthevariationAequaltozeroyieldsasetofdifferentialandalgebraic equations T 1 t f = @ 1 @ x t f T 1 = )]TJ/F23 11.9552 Tf 10.494 8.088 Td [(@ H 1 @ x x = f @ H 1 @ u 1 = 0 @ H 1 @ u 2 = 0 A EquationsArepresentthefollower'soptimalresponsetotheleader'scontrol u 2 Ifsuch u 1 existstosatisfyEqs.A,thenitispossibletowrite u 1 = g x t 1 t u 2 t t A whichisusedinthesubsequentdevelopmentoftheleader'sstrategy. A.1.2Leader'sStrategy Nowconsidertheleader'scontrol.IntheStackelbergstructuretheleaderassumes thatthefollower'sreactionisbasedontheleader'saction,whichisprovidedby Eqs.A.Thustheleadersolvestheoptimizationproblemwith J 2 andtheconstraints with u 1 substitutedbyEq.A: J 2 = 2 x f t f + Z t f t 0 L 2 f x t g x t 1 t u 2 t t u 2 t t g dt 112

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x = f x g x t 1 t u 2 t u 2 t t x t 0 = x 0 1 = h = )]TJ/F28 11.9552 Tf 11.291 16.857 Td [( @ H 1 @ x T 1 t f = @ 1 @ x t f T ThentheHamiltonianoftheleaderis H 2 = L 2 + T 2 f + T h A or H 2 x 1 u 2 2 t = L 2 x g x 1 u 2 t u 2 t + T 2 f x g x 1 u 2 t + T h x 1 u 2 t A withtheaugmentedcostfunctionalanditsrstvariation J a 2 = 2 + Z t f t 0 H 2 )]TJ/F48 11.9552 Tf 11.956 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 dt J a 2 = 2 + Z t f t 0 H 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.956 0 Td [( T 1 dt A A.1.2.1Variationoftheaugmentedcostfunctional NowlookinsidetheaugmentedcostfunctionalA.Inasimilarmannertothe follower'scase,thevariationoftheboundaryconditionisfoundas 2 = @ 2 @ x t f x f Thefollower'sstrategy u 1 isreplacedwith 1 and u 2 ,andthusthevariationof Hamiltonianbecomes H 2 = H 2 x 1 u 2 2 t = @ H 2 @ x x + @ H 2 @ 1 1 + @ H 2 @ u 2 u 2 + @ H 2 @ 2 2 + @ H 2 @ 113

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ThevariationoftheintegrandofEq.Ais Z t f t 0 H 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 dt = Z t f t 0 H 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 dt = Z t f t 0 H 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 1 dt )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( T 2 x t f t 0 )]TJ/F28 11.9552 Tf 11.955 16.272 Td [(Z t f t 0 T 2 x dt )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( T 1 t f t 0 )]TJ/F28 11.9552 Tf 11.955 16.272 Td [(Z t f t 0 T 1 dt = )]TJ/F48 11.9552 Tf 9.298 0 Td [( T 2 t f x t f )]TJ/F48 11.9552 Tf 11.956 0 Td [( T t f 1 t f + T t 0 1 t 0 + Z t f t 0 H 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 1 + T 2 x + T 1 dt FromEq.A,A,andA,thevariations 1 t f and x t f arerelatedby 1 t f = @ @ x t f [ 1 t f ] x t f = @ @ x t f @ 1 @ x t f T x t f = @ @ x f @ 1 @ x f T x f Thetotalvariationoftheleader'saugmentedcostisthengroupedintorespective variations: J a 2 = @ 2 @ x t f x f + Z t f t 0 H 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 dt = @ 2 @ x t f )]TJ/F48 11.9552 Tf 11.956 0 Td [( T 2 t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T t f @ @ x t f @ 1 @ x t f T x f + T t 0 1 t 0 + Z t f t 0 H 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 1 + T 2 x + T 1 dt = @ 2 @ x t f )]TJ/F48 11.9552 Tf 11.956 0 Td [( T 2 t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T t f @ @ x t f @ 1 @ x t f T x f + Z t f t 0 @ H 2 @ x + T 2 x + @ H 2 @ 1 + T 1 + @ H 2 @ u 2 u 2 + @ H 2 @ 2 )]TJ/F20 11.9552 Tf 13.289 0.108 Td [( x T 2 + @ H 2 @ )]TJ/F20 11.9552 Tf 13.974 2.05 Td [( T 1 dt A 114

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A.1.2.2Optimalityconditions SettingEq.Atozero,bysettingeachgroupoftermsassociatedwiththe variations,willyieldtheoptimalityconditionsfortheleader @ 2 @ x t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T t f @ @ x t f @ 1 @ x t f T = 0 T t 0 = 0 @ H 2 @ x + T 2 = 0 @ H 2 @ 1 + T = 0 A @ H 2 @ u 2 = 0 @ H 2 @ 2 )]TJ/F20 11.9552 Tf 13.288 0.108 Td [( x T = 0 x = f @ H 2 @ 2 )]TJ/F20 11.9552 Tf 16.439 2.049 Td [( 1 T = 0 1 = h EquationsAaresolvedfortheleader'sstrategyandthecorresponding costates.Oncetheyareobtained,theyaresubstitutedtoEqs.Asothatthe follower'sstrategycanbesolved. A.2FreeFinalTime Thecasewithnoconstraintonthenaltimei.e.,freenaltimeispresentedhere. RecallthegamedenedbyEqs.A,A,andA.Assumethat t 0 and x t 0 are xedandthat t f and x t f areexible.Inthesamewayasthexednaltimecase,the gameisdenedby x = f x t u 1 t u 2 t t x t 0 = x 0 J 1 = 1 x f t f + Z t f t 0 L 1 x t u 1 t u 2 t t dt J 2 = 2 x f t f + Z t f t 0 L 2 x t u 1 t u 2 t t dt 115

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where t 0 and x t 0 = x 0 arexedwhile t f and x t f = x f arefree.Againitisassumedthat theplayer2istheleaderandthatthecontrolpathsareunconstrained.Foranopen-loop Stackelberggame,leaderannounceshiscontrolpath u 2 A.2.1Follower'sStrategy Thefollowertriestondtheoptimalcontrolpath u 1 .Thenthefollower's Hamiltonianis H 1 = L 1 x t u 1 t t + T 1 f x t u 1 t u 2 t t where u 2 istreatedasaprescribedfunctionoftime.Thentheaugmentedcostfunctional ofthefollowerisconstructedas J a 1 = 1 + Z t f t 0 H 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x dt withitsrstvariation J a 1 = 1 + Z t f t 0 H 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x dt .A A.2.1.1Variationoftheaugmentedcostfunctional NowevaluatethevariationA. Thevariationoftheboundaryconditionisgivenby 1 = @ 1 @ x t 0 x 0 + @ 1 @ t 0 t 0 + @ 1 @ x t f x f + @ 1 @ t f t f A Sincethenaltimeandstatearefree, x 0 = 0 t 0 =0,Eq.Aissimpliedas 1 = @ 1 @ x t f x f + @ 1 @ t f t f A Thevariationofthefollower'sHamiltonianisgivenby H 1 = @ H 1 @ x x + @ H 1 @ 1 1 + @ H 1 @ u 1 u 1 + @ H 1 @ u 2 u 2 A 116

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ThevariationoftheintegralpartofEq.Aisexpanded. Z t f t 0 H 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x dt = Z t f t 0 H 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 1 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x dt + H 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x t f t f )]TJ/F28 11.9552 Tf 11.955 9.684 Td [( H 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x t 0 t 0 = Z t f t 0 H 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 1 x dt )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( T 1 x t f t 0 )]TJ/F28 11.9552 Tf 11.955 16.272 Td [(Z t f t 0 T 1 x dt + H 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x t f t f = )]TJ/F48 11.9552 Tf 9.298 0 Td [( T 1 t f x t f + H 1 t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f x t f t f + Z t f t 0 H 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 1 x + T 1 x dt A SubstitutingEq.AintoEq.Ayields Z t f t 0 H 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 x dt = )]TJ/F48 11.9552 Tf 9.299 0 Td [( T 1 t f x t f + H 1 t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f x t f t f + Z t f t 0 @ H 1 @ x x + @ H 1 @ 1 1 + @ H 1 @ u 1 u 1 + @ H 1 @ u 2 u 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [( T 1 x + T 1 x dt = )]TJ/F48 11.9552 Tf 9.298 0 Td [( T 1 t f x t f + H 1 t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f x t f t f + Z t f t 0 @ H 1 @ x + T 1 x + @ H 1 @ 1 )]TJ/F20 11.9552 Tf 13.289 0.108 Td [( x T 1 + @ H 1 @ u 1 u 1 + @ H 1 @ u 2 u 2 dt A FromEqs.AandAthevariationoftheaugmentedfollower'svariationin Eq.Abecomes J a 1 = @ 1 @ x t f x f + @ 1 @ t f t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f x t f + H 1 t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f x t f t f + Z t f t 0 @ H 1 @ x + T 1 x + @ H 1 @ 1 )]TJ/F20 11.9552 Tf 13.289 0.107 Td [( x T 1 + @ H 1 @ u 1 u 1 + @ H 1 @ u 2 u 2 dt A 117

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Since x f and x t f arerelatedby x t f = x f )]TJ/F20 11.9552 Tf 13.289 0.107 Td [( x t f t f and @ H 1 @ 1 = f T from H 1 = 1 + T 1 f ,Eq.Abecomes J a 1 = @ 1 @ x t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f x f + @ 1 @ t f + H 1 t f t f + Z t f t 0 @ H 1 @ x + T 1 x + )]TJ/F59 11.9552 Tf 5.48 -9.683 Td [(f T )]TJ/F20 11.9552 Tf 13.288 0.107 Td [( x T 1 + @ H 1 @ u 1 u 1 + @ H 1 @ u 2 u 2 dt A A.2.1.2Optimalityconditions SettingeachofthevariationsinEq.Aequaltozeroyieldsthefollower's optimalitycondition T 1 t f = @ 1 @ x t f H 1 t f = )]TJ/F23 11.9552 Tf 10.494 8.088 Td [(@ 1 @ t f T 1 = )]TJ/F23 11.9552 Tf 10.494 8.088 Td [(@ H 1 @ x x = f @ H 1 @ u 1 = 0 A Notethattheleader'scontribution u 2 isnotinvestigatedhere.EquationA representsthefollower'soptimalresponsetotheleader'scontrol u 2 .Ifsuch u 1 existsasanexplicitfunctionof x t 1 ,and u 2 ,thenitcanbewritteninthesamewayas inthexed-naltimecase u 1 = g x t 1 t u 2 t t .A 118

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A.2.2Leader'sstrategy Nowconsidertheleader'scontrol.Thecostfunctionalis J 2 = 2 x f t f + Z t f 0 L 2 x t g x t 1 t u 2 t t u 2 t t dt subjecttotheconstraints x = f x t g x t 1 t u 2 t t u 2 t t x t 0 = x 0 1 = h = )]TJ/F28 11.9552 Tf 11.291 16.857 Td [( @ H 1 @ x T 1 t f = @ 1 @ x t f T Thentheleader'sHamiltonianisdenedas H 2 x 1 u 2 2 t = L 2 x g x 1 u 2 t t + T 2 f x g x 1 u 2 t u 2 t + T h x 1 u 2 t orinacompactform H 2 = L 2 + T 2 f + T h Theaugmentedcostfunctionalandthecorrespondingrstvariationare J a 2 = 2 + Z t f t 0 H 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 dt ,A J a 2 = 2 + Z t f t 0 H 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 dt .A A.2.2.1Variationoftheaugmentedcostfunctional Nowlookinsidethevariationoftheleader'saugmentedcostfunctionalEq.A. Inasimilarmannertothefollower'scase,thevariationoftheboundaryconditionis foundas 2 = @ 2 @ x t f x f + @ 2 @ t f t f .A 119

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ThevariationofHamiltonianis H 2 = H 2 x 1 u 2 2 t = @ H 2 @ x x + @ H 2 @ 1 1 + @ H 2 @ u 2 u 2 + @ H 2 @ 2 2 + @ H 2 @ ExpandtheintegralpartofEq.A: Z t f t 0 H 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 dt = Z t f t 0 H 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 dt + H 2 )]TJ/F48 11.9552 Tf 11.956 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f t f = Z t f t 0 H 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 dt )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( T 2 x t f t 0 )]TJ/F28 11.9552 Tf 11.955 16.273 Td [(Z t f t 0 T 2 x dt )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( T 1 t f t 0 )]TJ/F28 11.9552 Tf 11.955 16.273 Td [(Z t f t 0 T 1 dt + H 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f t f = )]TJ/F48 11.9552 Tf 9.298 0 Td [( T 2 t f x t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T t f 1 t f + Z t f t 0 H 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 1 + T 2 x + T 1 dt + H 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f t f A 120

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FromEq.AandA,thetotalvariationEq.Abecomes J a 2 = @ 2 @ x t f x f + @ 2 @ t f t f + Z t f t 0 H 2 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 dt = @ 2 @ x t f x f + @ 2 @ t f t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 t f x t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T t f 1 t f + H 2 )]TJ/F48 11.9552 Tf 11.956 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f t f + Z t f t 0 H 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 2 x )]TJ/F23 11.9552 Tf 11.955 0 Td [( T 1 + T 2 x + T 1 dt = @ 2 @ x t f x f + @ 2 @ t f t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 t f x t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T t f 1 t f + H 2 )]TJ/F48 11.9552 Tf 11.956 0 Td [( T 2 x )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 1 t f t f + Z t f t 0 @ H 2 @ x + T 2 x + @ H 2 @ 1 + T 1 + @ H 2 @ u 2 u 2 + @ H 2 @ 2 )]TJ/F20 11.9552 Tf 13.288 0.107 Td [( x T 2 + @ H 2 @ )]TJ/F20 11.9552 Tf 13.974 2.05 Td [( T 1 dt A Since x t f = x f )]TJ/F20 11.9552 Tf 13.288 0.108 Td [( x t f t f 1 t f = 1 f )]TJ/F20 11.9552 Tf 13.975 2.05 Td [( 1 t f t f EquationAbecomes J a 2 = @ 2 @ x t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 t f x f + @ 2 @ t f + H 2 t f t f )]TJ/F48 11.9552 Tf 11.955 0 Td [( T t f 1 t f + Z t f t 0 @ H 2 @ x + T 2 x + @ H 2 @ 1 + T 1 + @ H 2 @ u 2 u 2 + @ H 2 @ 2 )]TJ/F20 11.9552 Tf 13.289 0.108 Td [( x T 2 + @ H 2 @ )]TJ/F20 11.9552 Tf 17.212 3.624 Td [( T 1 dt A 121

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A.2.2.2Optimalityconditions SettingEq.Athistozero,theoptimalityconditionfortheleaderisobtainedas 2 = )]TJ/F28 11.9552 Tf 11.291 16.857 Td [( @ H 2 @ x T 2 t f = @ 2 @ x t f T = )]TJ/F28 11.9552 Tf 11.291 16.857 Td [( @ H 2 @ 1 T t f = 0 A @ H 2 @ u 2 = 0 @ H 2 @ 2 )]TJ/F20 11.9552 Tf 13.288 0.107 Td [( x T = 0 = x = f @ H 2 @ )]TJ/F20 11.9552 Tf 13.974 2.049 Td [( T 1 = 0 = 1 = h A.3Linear-QuadraticDifferentialGame Consideragamedenedbythesystemwiththedynamicsandthecostfunctionals respectivelylinearandquadraticinthestatesandcontrols, x u 1 ,and u 2 ,suchas J 1 = 1 2 x T t f K 1f t x t f + Z t f t 0 x T t Q 1 t x t + u T 1 t R 11 t u 1 t + u T 2 t R 12 t u 2 t dt J 2 = 1 2 x T t f K 2f t x t f + Z t f t 0 x T t Q 2 t x t + u T 1 t R 21 t u 1 t + u T 2 t R 22 t u 2 t dt x = A t x t + B 1 t u 1 t + B 2 t u 2 t x t 0 = x 0 Fromnowon, t willbeomittedforsimplicity;thisshouldcausenoconfusionasthe subsequentderivationoftheRiccatiequationsisvalidregardlessofwhetherthese coefcientsareconstantortime-varying. 122

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A.3.1FixedFinalTime Assume t 0 x t 0 ,and x t f arexed.Thenthefollower'sHamiltonianPlayer1is constructedas H 1 = 1 2 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(x T Q 1 x + u T 1 R 11 u 1 + u T 2 R 12 u 2 + T 1 Ax + B 1 u 1 + B 2 u 2 andthecorrespondingoptimalityconditionsareobtainedfromEq.A0as T 1 = )]TJ/F23 11.9552 Tf 10.494 8.088 Td [(@ H 1 @ x = )]TJ/F28 11.9552 Tf 11.291 9.684 Td [()]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(x T Q 1 + T 1 A T 1 t f = )]TJ/F23 11.9552 Tf 15.394 8.088 Td [(@ @ x f 1 2 x T f K 1f x f = x T f K 1f @ H 1 @ u 1 = u T 1 R 11 + T 1 B 1 = 0 Aftersimplicationthefollower'sstrategyandthecostatedynamicsareobtainedas 1 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(Q T 1 x )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T 1 1 t f = K T 1f x f u 1 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T 1 1 A Nexttheleader'sHamiltonianisconstructedbyadjoiningEq.A: H 2 = L 2 + T 2 f + T h = 1 2 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(x T Q 2 x + u T 1 R 21 u 1 + u T 2 R 22 u 2 + T 2 Ax + B 1 u 1 + B 2 u 2 + T )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [()]TJ/F59 11.9552 Tf 9.298 0 Td [(Q T 1 x )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T 1 = 1 2 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(x T Q 2 x + T 1 B 1 R )]TJ/F59 7.9701 Tf 6.587 0 Td [(1 11 R 21 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T 1 1 + u T 2 R 22 u 2 + T 2 )]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(Ax )]TJ/F59 11.9552 Tf 11.955 0 Td [(B 1 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T 1 1 + B 2 u 2 + T )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [()]TJ/F59 11.9552 Tf 9.298 0 Td [(Q T 1 x )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T 1 anditsoptimalityconditionsarefoundfollowingEq.A: A.3.2FreeFinalTime Assume t 0 and x t 0 arexedand t f and x t f = x f arefree.Thefollower's Hamiltonianis H 1 = 1 2 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(x T Q 1 x + u T 1 R 11 u 1 + u T 2 R 12 u 2 + T 1 Ax + B 1 u 1 + B 2 u 2 123

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andfromEq.Atheoptimalityconditionsforthefollowerarefoundas T 1 = )]TJ/F23 11.9552 Tf 10.494 8.088 Td [(@ H 1 @ x = )]TJ/F28 11.9552 Tf 11.291 9.684 Td [()]TJ/F59 11.9552 Tf 5.48 -9.684 Td [(x T Q 1 + T 1 A T 1 t f = @ @ x f 1 2 x T f K 1f x f = x T f K 1f @ H 1 @ u 1 = u T 1 R 11 + T 1 B 1 = 0 H 1 t f = )]TJ/F23 11.9552 Tf 14.063 8.088 Td [(@ @ t f 1 2 x T f K 1f x f = 0 whichaftersimplicationyield 1 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(Q T 1 x )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T 1 1 t f = K T 1f x F u 1 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T 1 1 A Theleader'sHamiltonianisconstructedbyaugmentingEq.Aas H 2 = L 2 + T 2 f + T h = 1 2 )]TJ/F59 11.9552 Tf 5.479 -9.683 Td [(x T Q 2 x + u T 1 R 21 u 1 + u T 2 R 22 u 2 + T 2 Ax + B 1 u 1 + B 2 u 2 + T )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [()]TJ/F59 11.9552 Tf 9.298 0 Td [(Q T 1 x )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T 1 = 1 2 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(x T Q 2 x + T 1 B 1 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 R 21 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T 1 1 + u T 2 R 22 u 2 + T 2 )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(Ax )]TJ/F59 11.9552 Tf 11.955 0 Td [(B 1 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T 1 1 + B 2 u 2 + T )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F59 11.9552 Tf 9.299 0 Td [(Q T 1 x )]TJ/F59 11.9552 Tf 11.956 0 Td [(A T 1 Thentheleader'soptimalityconditionsisobtainedfromEq.Aas T 2 = )]TJ/F23 11.9552 Tf 10.494 8.088 Td [(@ H 2 @ x = )]TJ/F28 11.9552 Tf 11.291 9.684 Td [()]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(x T Q 2 + T 2 A )]TJ/F48 11.9552 Tf 11.955 0 Td [( T Q T 1 T 2 t f = )]TJ/F23 11.9552 Tf 15.394 8.088 Td [(@ @ x f 1 2 x T f K 2f x f = x T f K 2f T = )]TJ/F23 11.9552 Tf 10.494 8.088 Td [(@ H 2 @ 1 = )]TJ/F28 11.9552 Tf 11.291 9.684 Td [()]TJ/F48 11.9552 Tf 5.479 -9.684 Td [( T 1 B 1 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 R 21 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T 1 )]TJ/F48 11.9552 Tf 11.955 0 Td [( T 2 B 1 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T 1 T t f = 0 @ H 2 @ u 2 = u T 2 R 22 + T 2 B 2 = 0 124

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whicharesimpliedtoyield 2 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(Q T 2 x )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T 2 + Q 1 2 t f = K T 2f x f = )]TJ/F59 11.9552 Tf 9.298 0 Td [(B 1 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 R T 21 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T 1 1 + B 1 R )]TJ/F59 7.9701 Tf 6.587 0 Td [(1 11 B T 1 2 t f = 0 u 2 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 22 B T 2 2 A FromEqs.AandA,theoptimalityconditionsforthetwo-playerlinear quadraticStackelbergdifferentialgamewithfreenaltimearefoundas u 1 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T 1 1 u 2 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 22 B T 2 2 1 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(Q T 1 x )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T 1 1 t f = K T 1f x f 2 = )]TJ/F59 11.9552 Tf 9.298 0 Td [(Q T 2 x )]TJ/F59 11.9552 Tf 11.955 0 Td [(A T 2 + Q 1 2 t f = K T 2f x f = )]TJ/F59 11.9552 Tf 9.298 0 Td [(B 1 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 R T 21 B T 1 1 + B 1 R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 B T 1 2 t f = 0 125

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APPENDIXB RISESTABILITYANALYSIS ThecontrolinputobtainedinChapter6isbasedontheassumptionthattheall theothercomponentsofthedynamicscanbecapturedby h asinEq.6,whichis nottrueingeneral.InthisappendixaRISEcontrollerisdevelopedsothatwhatwas previouslycapturedby h isasymptoticallytracked,enablingtheoptimalcontrol u tobe applied.ThedevelopmentoftheRISEcontrollerisalmostidenticalto[76].Designing thefeedbackcontrollerintheEuler-lagrangeformallowsthesamestabilityanalysis applicabletodifferentgametheoreticcontrollersdevelopedinChapter6. B.1RiseFeedbackControlDevelopment Ingeneral,theboundeddisturbance d t andthenonlineardynamicsgivenin Eq.6areunknown,sothecontrollergiveninEq.6cannotbeimplemented. However,ifthecontrolinputcanidentifyandcanceltheseeffects,then x t willconverge tothestatespacemodelinEq.6suchthatthat u and d minimizetherespective performanceindex J 1 and J 2 .Inthissection,acontrolinputisdevelopedthatexploits RISEfeedbacktoidentifythenonlineareffectsandboundeddisturbancesthusenabling x t toasymptoticallyconvergetothestatespacemodelinEq.6. Todevelopthecontrolinput,thelteredtrackingerrorinEq.6ispremultiplied by M q ,andtheexpressionsinEqs.6and6areutilizedtoobtain Mr = )]TJ/F59 11.9552 Tf 9.298 0 Td [(V m e 2 + h + d + 2 Me 2 )]TJ/F20 11.9552 Tf 11.955 -0.132 Td [( F + L .B Basedontheopen-looperrorsysteminEq.B,thecontrolinputiscomposedofthe gametheoreticcontrollersdevelopedinEqs.6and6,plusasubsequently designedauxiliarycontrolterm t 2 R n as F + L )]TJ/F20 11.9552 Tf 11.955 -0.131 Td [( u F + u L .B 126

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Theclosed-looptrackingerrorsystemcanbedevelopedbysubstitutingEq.Binto Eq.Bas Mr = )]TJ/F59 11.9552 Tf 9.298 0 Td [(V m e 2 + h + d + 2 Me 2 + u F + u L )]TJ/F48 11.9552 Tf 11.956 0 Td [( .B Tofacilitatethesubsequentstabilityanalysistheauxiliaryfunction f d t 2 R n ,whichis denedas f d M q d q d + V m q d q d q d + G q d + F q d ,B isaddedandsubtractedtoEq.Btoyield Mr = )]TJ/F59 11.9552 Tf 9.299 0 Td [(V m e 2 + h + f d + d + u F + u L )]TJ/F48 11.9552 Tf 11.955 0 Td [( + 2 Me 2 ,B where h 2 R n isdenedas h h )]TJ/F59 11.9552 Tf 11.955 0 Td [(f d SubstitutingEq.6intoEq.B,takingthetimederivative,andmanipulatingwith Eq.6yields M r = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(1 2 Mr + N + N D )]TJ/F59 11.9552 Tf 11.955 0 Td [(e 2 )]TJ/F28 11.9552 Tf 11.955 9.684 Td [()]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(R )]TJ/F59 7.9701 Tf 6.587 0 Td [(1 11 + R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 22 r )]TJ/F20 11.9552 Tf 14.578 0 Td [( ,B afterstrategicallygroupingspecicterms.InEq.B,theunmeasurableauxiliary terms N e 1 e 2 r t N D t 2 R n aredenedas N )]TJ/F59 11.9552 Tf 11.295 2.199 Td [( V m e 2 )]TJ/F59 11.9552 Tf 11.955 0 Td [(V m e 2 )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(1 2 Mr + h + 2 Me 2 + 2 M e 2 + e 2 + )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 + R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 22 2 e 2 N D f d + d 127

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TheMeanValueTheoremandAssumptions3,4,and5in[77]canbeusedtoupper boundtheauxiliarytermsas N t k y k k y k k N D k 1 N D 2 ,B where y t 2 R 3 n isdenedas y t e T 1 e T 2 r T T theboundingfunction k y k 2 R isapositivegloballyinvertiblenondecreasingfunction, and i 2 R i =1,2,denoteknownpositiveconstants.BasedonEq.B,thecontrol term t isdesignedasthegeneralizedsolutionto t k s r t + 1 sgn e 2 ,B where k s 1 2 R arepositiveconstantcontrolgains.Theclosed-looperrorsystemsfor r t cannowbeobtainedbysubstitutingEq.BintoEq.Bas M r = )]TJ/F20 11.9552 Tf 10.494 8.087 Td [(1 2 Mr + N + N D )]TJ/F59 11.9552 Tf 11.956 0 Td [(e 2 )]TJ/F28 11.9552 Tf 11.955 9.683 Td [()]TJ/F59 11.9552 Tf 5.479 -9.683 Td [(R )]TJ/F59 7.9701 Tf 6.587 0 Td [(1 11 + R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 22 r )]TJ/F57 11.9552 Tf 11.955 0 Td [(k s r )]TJ/F23 11.9552 Tf 11.955 0 Td [( 1 sgn e 2 B B.2StabilityAnalysis ItcanbeshownthatthecontrollergivenbyEqs.6,6,B,andB ensuresthatallsystemsignalsareboundedunderclosed-loopoperation,andthe trackingerrorareregulatedinthesensethatsee[44]forsimilardetails k e 1 t k k e 2 t k k r t k! 0 as t !1 .B0 Theboundednessoftheclosed-loopsignalsandtheresultinEq.Bcanbe obtainedprovidedthecontrolgain k s introducedinEq.Bisselectedsufciently largeseethesubsequentstabilityanalysis,and 1 and 2 areselectedaccordingto 128

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thesufcientconditions min 1 > 1 2 min 2 > 1,B where min 1 and min 2 aretheminimumeigenvaluesof 1 and 2 ,respectively. Thegain 1 isselectedaccordingtothefollowingsufcientcondition: 1 > 1 + 2 min 2 .B LetaLyapunovfunction V L t : D [0, 1 R beacontinuouslydifferentiable positivedenitefunctiondenedin[44]as V L t k e 1 k 2 + 1 2 k e 2 k 2 + 1 2 r T Mr + O .B wheretheauxiliaryfunction O t 2 R isthesolutiontosee[44]forfurtherdetails O )]TJ/F59 11.9552 Tf 9.299 0 Td [(r T N D )]TJ/F23 11.9552 Tf 11.955 0 Td [( 1 sgn e 2 O = 1 n X i =1 j e 2i j)]TJ/F59 11.9552 Tf 17.933 0 Td [(e 2 T N D B TakingthetimederivativeofEq.Byields V L =2 e T 1 e 1 + e T 2 e 2 + 1 2 r T Mr + r T M r + O UtilizingEqs.6,B,andB,theLyapunovderivativeisrewrittenas V L t )]TJ/F20 11.9552 Tf 21.918 0 Td [(2 e T 1 1 e 1 + 2e T 2 e 1 + r T N )]TJ/F28 11.9552 Tf 11.955 9.684 Td [( k s + min )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 11 + R )]TJ/F59 7.9701 Tf 6.586 0 Td [(1 22 k r k 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( min 2 k e 2 k 2 B UtilizingEqs.BandBcanbefurthersimpliedas V L )]TJ/F23 11.9552 Tf 21.918 0 Td [( 3 k y k 2 )]TJ/F28 11.9552 Tf 11.956 13.27 Td [(h k s k r k 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( k y k k r kk y k i ,B 129

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where 3 min 8 > > > > < > > > > : 2 min 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(1 min 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(1 min )]TJ/F59 11.9552 Tf 5.479 -9.684 Td [(R )]TJ/F59 7.9701 Tf 6.587 0 Td [(1 11 + R )]TJ/F59 7.9701 Tf 6.586 0 Td [(2 22 9 > > > > = > > > > ; .B CompletingthesquaresforthetermsinsidethebracketsinEq.Byields V L )]TJ/F23 11.9552 Tf 21.918 0 Td [( 3 k y k 2 + 2 k y k k y k 2 4 k s )]TJ/F57 11.9552 Tf 21.918 0 Td [(U ,B where U = c k y k 2 forsomepositiveconstant c .Thefunction U isacontinuous, positivesemi-deniteanddenedwithintheclosedset: D n 2 R 3 n +1 k k )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 2 p 3 k s o TheinequalityinEq.Bcanbeusedtoshowthat V L t 2L 1 in D ;hence, e 1 t e 2 t ,and r t 2L 1 in D .Thenstandardlinearanalysismethodscanbeusedtoprove that e 1 t e 2 2L 1 in D fromEq.6.Since e 1 t e 2 t r t 2L 1 in D ,Assumption 4isusedalongwithEq.6toconcludethat q t q t q t 2L 1 in D ,whichisthen combinedwithAssumption3toconcludethat M q V m q q G q ,and F q 2L 1 in D Thus,fromEq.6andAssumption4,itcanbeshownthat L t F t 2L 1 in D .With r t 2L 1 in D ,itcanbeshownthat t 2L 1 in D ;hence,Eq.Bcanbeusedto showthat r t 2L 1 in D .From e 1 t e 2 t r 2L 1 in D ,thedenitionsfor U y and z t canbeusedtoprovethat U y isuniformlycontinuousin D Usingsimilarargumentsasgivenin[44]itcanbeshownthat c k y t k 2 0as t !18 y 2S .B Since u F t u L t 0 as e 2 t 0 fromEq.6and6,thenEq.Bcan beusedtoconcludethat h + f d + d as r t e 2 t 0 .B 130

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EquationBindicatesthatthedynamicsinEq.6convergestothestate-space modelinEq.6.Hence, u F t and u L t convergestooptimalcontrollerstosolvethe gamedenedinEq.6,providedthegainconstraintsinEq.6aresatised. 131

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APPENDIXC COSTATEESTIMATIONFORTHETRANSCRIBEDSTACKELBERGGAMES InSection3.2atwo-playerStackelbergdifferentialgamewasdiscretizedat Legendre-Gauss-LobattoLGLpointstoformastaticStackelberggameproblemor abilevelprogrammingproblem.Similaritiesbetweenthesolutionofanoptimalcontrol problemandthetranscribednonlinearprogrammingproblemareensuredbychecking ifthesolutionofthetranscribedproblemsatisestheoptimalityconditionsofthe originalproblem.Thisprocessiscalledcostateestimationforthetranscribedsolution estimatesthecostatedynamicsoftheoriginalproblemorcostatemappingforitis donebyndingthemappingbetweenthecostatesKKTmultipliersforthenonlinear programmingproblemandthecostatesfortheoptimalcontrolproblem.Inoptimal control,[78]performedthecostateestimationforthecasewithLGLtranscriptionLPM toshowthattheKKTmultiplierateachLGLcollocationpointisequaltothecostate weightedbythecorrespondingLGLweight. InthisappendixtheLPMcostateestimationin[78]isextendedtoopen-loop two-playerStackelbergdifferentialgames,sothatthebilevelprogrammingproblem, oncesolved,willprovideanumericalsolutiontothedifferentialgameproblemsatisfying theexactoptimalityconditionsateachcollocationpoint. C.1TransformedOptimalityConditions Firstderivetheoptimalityconditionsoftheoriginaltwo-playerStackelberg differentialgameprobleminthetransformedtimedomain,whichwillbecompared 132

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withthesolutionofthetranscribedstaticproblem.Aproblemdenedby Minimize J 1 = 1 x t 0 t 0 x t f t f + Z t f t 0 M x t u t v t t dt J 2 = 2 x t 0 t 0 x t f t f + Z t f t 0 N x t u t v t t dt subjectto x = f x t u t v t t x t 0 = x 0 isredenedin -domainthrough t = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 + t f + t 0 2 whichyieldsanewproblem Minimize J 1 = 1 x )]TJ/F20 11.9552 Tf 9.299 0 Td [(1, )]TJ/F20 11.9552 Tf 9.298 0 Td [(1, x ,1 + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 Z 1 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 M x u v d J 2 = 2 x )]TJ/F20 11.9552 Tf 9.299 0 Td [(1, )]TJ/F20 11.9552 Tf 9.298 0 Td [(1, x ,1 + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 Z 1 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 N x u v d subjectto x = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 f x u v x )]TJ/F20 11.9552 Tf 9.299 0 Td [(1= x 0 ThecostatedynamicsarederivedinAppendixA. H 1 = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 )]TJ/F57 11.9552 Tf 5.48 -9.684 Td [(M + T f = )]TJ/F28 11.9552 Tf 11.291 16.857 Td [( @ H 1 @ x T = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 )]TJ/F28 11.9552 Tf 11.291 16.857 Td [( @ M @ x T )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( @ f @ x T # | {z } h .C 133

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Letting = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 h H 2 = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 )]TJ/F57 11.9552 Tf 5.48 -9.684 Td [(N + T f + T h = )]TJ/F28 11.9552 Tf 11.291 16.856 Td [( @ H 2 @ x T = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 )]TJ/F28 11.9552 Tf 11.291 16.857 Td [( @ N @ x T )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( @ f @ x T )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( @ h @ x T # = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 2 4 )]TJ/F28 11.9552 Tf 11.291 16.857 Td [( @ N @ x T )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( @ f @ x T + @ @ x @ M @ x T + @ f @ x T T 3 5 ,C = )]TJ/F28 11.9552 Tf 11.291 16.857 Td [( @ H 2 @ T = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 )]TJ/F28 11.9552 Tf 11.291 16.857 Td [( @ N @ T )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( @ f @ T )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( @ h @ T # = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 2 4 )]TJ/F28 11.9552 Tf 11.291 16.857 Td [( @ N @ T )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( @ f @ T + @ @ @ M @ T + @ f @ T T 3 5 .C EquationsC,C,andCarethendiscretizedatLGLpointsandcompared withtheoptimalityconditionsforthetranscribedproblem. C.2DiscretizationofTwo-personStackelbergDifferentialGames Thestates x t andthecontrol u = u 1 t v = u 2 t intheoriginaltimedomainare x t = x 1 t x 2 t x n t T 2 R n 1 ,C u t = u 1 t u 2 t u m t T 2 R m 1 ,C v t = v 1 t v 2 t v m t T 2 R m 1 ,C andareintimedomain t 2 [ t 0 t f ] .Thedomainistransformedto 2 [ )]TJ/F20 11.9552 Tf 9.299 0 Td [(1,1 ] tomatch theLGLpoints,through = t )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 )]TJ/F20 11.9552 Tf 11.956 0 Td [( t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 134

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suchthat t = t 0 and t = t f correspondto = 0 = )]TJ/F20 11.9552 Tf 9.298 0 Td [(1and = N =1,respectively.The transformedstatesandcontrolsarediscretizedatthe N +1LGLpoints,suchthat x 0 := x 0 = )]TJ/F20 11.9552 Tf 9.299 0 Td [(1, x 1 := x 1 x N := x N =1, u 0 := u 0 = )]TJ/F20 11.9552 Tf 9.299 0 Td [(1, u 1 := u 1 u N := u N =1, v 0 := v 0 = )]TJ/F20 11.9552 Tf 9.298 0 Td [(1, v 1 := v 1 v N := v N =1, andtheapproximatepolynomialsfor x u ,and v areformedas x x = N X l =0 l x k u u = N X l =0 l u k v v = N X l =0 l v k where indicatestheapproximatedparametertobeoptimizedinthebilevelprogramming problem,and k = 1 N N +1 L N k )]TJ/F20 11.9552 Tf 11.955 0 Td [(1 2 L N )]TJ/F23 11.9552 Tf 11.956 0 Td [( k k =1,..., N aretheLagrangepolynomialsoforder N ,and L N istheLegendrepolynomialoforder N .Theintegrandofthecostfunctionalsarediscretizedinthesamemanner: M i := M x i u i v i i N i := N x i u i v i i Theapproximatederivativeof x isobtainedbydifferentiating x .Since x k are constants,thederivativeof x k i =1,..., N canbewrittenas x k x k = N X l =0 l k x k = N X l =0 D kl x k .C 135

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Thediscretizedparameterscanbearrangedinamatrixformwhere D 2 R N +1 N +1 isa differentiationmatrixconsistingof l evaluatedateachLGLpoint: D kl = 8 > > > > > > > > > > < > > > > > > > > > > : 1 t l )]TJ/F57 11.9552 Tf 11.956 0 Td [(t k L N l L N k k 6 = l )]TJ/F57 11.9552 Tf 10.494 8.088 Td [(N N +1 4 k = l =0, N N +1 4 k = l = N 0otherwise. Finally,abilevelprogrammingproblemisformulatedasfollows: FollowerMinimize J 1 = 1 + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 N X i =0 w i M i subjectto x k = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 f C LeaderMinimize J 2 = 2 + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 N X i =0 w i N i subjectto x k = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 f C C.3KKTConditionsandCostateMapping Thefollower'saugmentedcostfunctionis J a 1 = 1 + T + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 N X i =0 w i M i + N X i =0 T i t f )]TJ/F57 11.9552 Tf 11.956 0 Td [(t 0 2 f i )]TJ/F20 11.9552 Tf 13.289 2.277 Td [( x i LookattheKKTcondition: @ J a 1 @ x k = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 w 0 @ M 0 @ x k + w 1 @ M 1 @ x k + + w N @ M N @ x k 136

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+ t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 T 0 @ f 0 @ x k + T 1 @ f 1 @ x k + + T N @ f N @ x k )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( T 0 @ x 0 @ x k + T 1 @ x 1 @ x k + + T N @ x N @ x k = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 w k @ M k @ x k + T k @ f k @ x k + w k N X i =0 D ki T i w i = 0 or N X i =0 D ki i w i + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 @ M k @ x k T + @ f k @ x k T k w k # = 0 k =1,..., N .C NowcompareEq.CwithEq.C.FirstdiscretizeEq.Candevaluateat k k canbewrittenas k = N X i =0 D ki i andsubstitutedtoEq.Ctoyield N X i =0 D ki i + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 @ M k @ x k T + @ f k @ x k T k # = 0 .C Equations.CandCarerelatedbythemapping i = w i i i =0,..., N ,C thatis,theKKTmultipliersforthediscretizedfollower'sproblemmatchthefollower's costatesintheoriginaldifferentialgameproblem,weightedbytheLGLweights. Theleader'saugmentedcostfunctionisconstructedas J a 2 = 2 + T + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 N X i =0 w i N i + N X i =0 T i t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 f i )]TJ/F20 11.9552 Tf 13.288 2.278 Td [( x i )]TJ/F57 7.9701 Tf 17.405 14.944 Td [(N X i =0 T i 8 < : N X l =0 D il l w l + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 2 4 @ M i @ x i T + @ f i @ x i T i w i 3 5 9 = ; NotethatEq.Cissimilarto )]TJ/F59 11.9552 Tf 12.231 0 Td [(h andthusaugmentedwithminussign,makingit looklike h )]TJ/F20 11.9552 Tf 14.148 2.049 Td [( whichisconsistentwiththeaugmentedcostfunctionalforthedifferential 137

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gameinAppendixA.Theleader'sKKTconditionassociatedwith x is @ J a 2 @ x k = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 w 0 @ N 0 @ x k + w 1 @ N 1 @ x k + + w N @ N N @ x k + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 T 0 @ f 0 @ x k + T 1 @ f 1 @ x k + + T N @ f N @ x k )]TJ/F28 11.9552 Tf 11.955 20.443 Td [( T 0 @ x 0 @ x k + T 1 @ x 1 @ x k + + T N @ x 1 @ x k )]TJ/F57 11.9552 Tf 13.151 8.088 Td [(t f )]TJ/F57 11.9552 Tf 11.956 0 Td [(t 0 2 T 0 @ @ x k @ M 0 @ x 0 T + T 1 @ @ x k @ M 1 @ x 1 T + + T N @ @ x k @ M N @ x N T )]TJ/F57 11.9552 Tf 13.151 8.088 Td [(t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 2 4 T 0 @ @ x k @ f 0 @ x 0 T 0 w 0 + T 1 @ @ x k @ f 1 @ x 1 T 1 w 1 + + T N @ @ x k @ f N @ x N T N w N 3 5 = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 8 < : w k @ N k @ x k + T k @ f k @ x k )]TJ/F20 11.9552 Tf 16.247 2.05 Td [( T k 2 4 @ @ x k @ M k @ x k T + @ @ x k @ f k @ x k T k w k 3 5 9 = ; + w k N X i =0 D ki T i w i = 0 whichisdividedby w k andtransposedtoyield N X i =0 D ki i w i + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 8 < : @ N K @ x k T + @ f k @ x k T k w k )]TJ/F23 11.9552 Tf 16.458 8.088 Td [(@ @ x k @ M k @ x k T + @ f k @ x k k w k # T k w k 9 = ; = 0 C CompareEq.CwithEq.C.Inasimilarmannerto canbediscretizedand writtenas k = N X i =0 D ki i 138

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thereforeEq.Cateachdiscretizedpointcanbewrittenas N X i =0 D ki i + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 8 < : @ N k @ x k T + @ f k @ x k T k )]TJ/F23 11.9552 Tf 16.458 8.088 Td [(@ @ x k @ M k @ x k T + @ f k @ x k T k # T k 9 = ; = 0 C EquationsCandCshowthattheKKTmultipliers i andthediscretized costates i arerelatedby i = w i i i =0,..., N .C Similarly,fortheKKTconditionassociatedwith @ J a 2 @ k = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 w 0 @ N 0 @ k + w 1 @ N 1 @ k + + w N @ N N @ k + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 T 0 @ f 0 @ k + T 1 @ f 1 @ k + + T N @ f N @ k )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( T 0 D 00 w 0 @ 0 @ k + D 01 w 1 @ 1 @ k + + D 0 N w N @ N @ k + T 1 D 10 w 0 @ 0 @ k + D 11 w 1 @ 1 @ k + + D 1 N w N @ N @ k + + T N D N 0 w 0 @ 0 @ k + D N 1 w 1 @ 1 @ k + + D NN w N @ N @ k )]TJ/F57 11.9552 Tf 13.15 8.088 Td [(t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 T 0 @ @ k @ M 0 @ x 0 T + T 1 @ @ k @ M 1 @ x 1 T + + T N @ @ k @ M N @ x N T )]TJ/F57 11.9552 Tf 13.15 8.088 Td [(t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 2 4 T 0 @ @ k @ f 0 @ x 0 T 0 w 0 + T 1 @ @ k @ f 1 @ x 1 T 1 w 1 + + T N @ @ k @ f N @ x N T N w N 3 5 = t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 8 < : w k @ N k @ k + T k @ f k @ k )]TJ/F20 11.9552 Tf 16.247 2.05 Td [( T k @ @ k 2 4 @ M k @ k + @ f k @ k T k w k 3 5 9 = ; + N X i =0 D ki T i w i 139

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whichwhendividedby w k andtransposedyields N X i =0 D ki i w i + t f )]TJ/F57 11.9552 Tf 11.955 0 Td [(t 0 2 8 < : @ N k @ k T + @ f k @ k T k w k )]TJ/F23 11.9552 Tf 17.144 8.088 Td [(@ @ k 2 4 @ M k @ k + @ f k @ k T k w k 3 5 T k w k 9 > = > ; = 0 C EquationCisdiscretizedattheLGLpointsandcomparedwithEq.C.With k = N X i =0 D ki i Thediscretizedcostatedynamicsbecomes N X i =0 D ki i + t f )]TJ/F57 11.9552 Tf 11.956 0 Td [(t 0 2 8 < : @ N k @ k T + @ f k @ k T k )]TJ/F23 11.9552 Tf 17.144 8.088 Td [(@ @ k @ M k @ k T + @ f k @ k T k T k 9 = ; = 0 C ComparisonbetweenEq.CandCresultsinthemappingbetween and : i = w i i i =1,..., N .C FromEqs.C,C,andC,itisconrmedthatthetranscribedbilevel programmingproblemsharesthesameoptimalityinformationastheoriginalStackelberg differentialgameproblemviaweightsassociatedwiththeLGLcollocation. 140

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[75]Kuipers,J., Quaternionsandrotationsequences:aprimerwithapplicationsto orbits,aerospace,andvirtualreality ,PrincetonUniversityPress,2002. [76]Dupree,K.,Patre,P.,Wilcox,Z.,andDixon,W.,Optimalcontrolofuncertain nonlinearsystemsusingRISEfeedback,2008,pp.2154. [77]Hiramatsu,T.,Johnson,M.,Fitz-Coy,N.G.,andDixon,W.E.,AsymptoticOptimal TrackingControlforanUncertainNonlinearEuler-LagrangeSystem:ARISE-based Closed-LoopStackelbergGameApproach, IEEECDC-ECC2011 ,2011. [78]Fahroo,F.andRoss,I.M.,CostateestimationbyaLegendrepseudospectral method, JournalofGuidance,Control,andDynamics ,Vol.24,No.2,2001, pp.270. 146

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BIOGRAPHICALSKETCH TakashiHiramatsuwasborninKawaguchi,Japan.Motivatedandencouragedby hisfamily,Takashidevelopedhisdreamofstudyingabroad,whilehewasinhighschool; and,aftergraduating,hemovedtotheUnitedStates.Hereceivedhisbachelor'sdegree inaerospaceengineeringfromtheUniversityofFloridainthespringof2005andwent ontojointhegraduateprograminmechanicalengineering,withtheUniversityofFlorida AlumniFellowship.Throughouttheyearsingraduateschool,Takashiworkedonsolving problemsrelatedtospaceapplications. Hisresearchinterestsincludeastrodynamics,nonlinearcontrol,anddifferential gametheory. 147