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Optimal Control of Uncertain Euler-Lagrange Systems

Permanent Link: http://ufdc.ufl.edu/UFE0024202/00001

Material Information

Title: Optimal Control of Uncertain Euler-Lagrange Systems
Physical Description: 1 online resource (111 p.)
Language: english
Creator: Dupree, Keith
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: 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: Optimal control theory involves the design of controllers that can satisfy some tracking or regulation control objective while simultaneously minimizing some performance metric. A suplus or minuscient condition to solve an optimal control problem is to solve the Hamilton-Jacobi-Bellman (HJB) equation. For the special case of linear time-invariant systems, the solution to the HJB equation reduces to solving the algebraic Riccati equation. However, for general systems, the challenge is to find a value function that satisfies the HJB equation. Finding this value function has remained problematic because it requires the solution of a partial diregistered trademarkerential equation that can not be solved explicitly. Chapter 2 illustrates the amalgamation of optimal control techniques with a recently developed continuous robust integral of the sign of the error (RISE) feedback term. Specifically, a system in which all terms are assumed known (temporarily) is feedback linearized and a control law is developed based on the HJB optimization method for a given quadratic performance index. Under the assumption that parametric uncertainty and unknown bounded disturbances are present in the dynamics, the control law is modified to contain the RISE feedback term which is used to identify the uncertainty. A Lyapunov stability analysis is included to show that the RISE feedback term asymptotically identifies the unknown dynamics (yielding semi-global asymptotic tracking) provided upper bounds on the disturbances are known and the control gains are selected appropriately. A feedforward neural network is then added to the previous controller. The utility of combining these feedforward and feedback methods are twofold. Previous efforts indicate that modifying the RISE feedback with a feedforward term can reduce the control eregistered trademarkort and improve the transient and steady state response of the RISE controller. Moreover, combining NN feedforward controllers with RISE feedback yields asymptotic results. Simulation as well as experimental results are provided to illustrate the developed controllers. Inverse optimal control is an alternative method to solve the nonlinear optimal control problem by circumventing the need to solve the HJB equation. Adaptive inverse optimal control techniques have been developed that can handle structured (i.e., linear in the parameters (LP)) uncertainty for a particular class of nonlinear systems. In Chapter 3, an adaptive inverse optimal controller is developed to minimize a meaningful performance index while the generalized coordinates of a nonlinear Euler-Lagrange system asymptotically track a desired time-varying trajectory despite LP uncertainty. A Lyapunov analysis is provided to examine the stability of the developed optimal controller, and simulation and experimental results illustrate the performance of the controller. Output feedback based controllers are more desirable than full-state feedback controllers because the necessary sensors for full-state feedback may not always be available and using numerical diregistered trademarkerentiation to obtain velocities can be problematic if position measurements are noisy. In Chapter 4, an adaptive output feedback IOC is developed which minimizes a meaningful cost, while the generalized coordinates of a nonlinear Euler-Lagrange system asymptotically tracks a desired time-varying trajectory. The new controller contains a desired compensation adaptation law (DCAL) based feedforward term and a feedback term that is shown to be implementable using only position measurements. A Lyapunov analysis is provided to prove the stability of the developed controller and to determine a meaningful cost functional. Experimental results are included to illustrate the performance of the controller. The dissertation is concluded in Chapter 5.
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 Keith Dupree.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024202:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024202/00001

Material Information

Title: Optimal Control of Uncertain Euler-Lagrange Systems
Physical Description: 1 online resource (111 p.)
Language: english
Creator: Dupree, Keith
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: 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: Optimal control theory involves the design of controllers that can satisfy some tracking or regulation control objective while simultaneously minimizing some performance metric. A suplus or minuscient condition to solve an optimal control problem is to solve the Hamilton-Jacobi-Bellman (HJB) equation. For the special case of linear time-invariant systems, the solution to the HJB equation reduces to solving the algebraic Riccati equation. However, for general systems, the challenge is to find a value function that satisfies the HJB equation. Finding this value function has remained problematic because it requires the solution of a partial diregistered trademarkerential equation that can not be solved explicitly. Chapter 2 illustrates the amalgamation of optimal control techniques with a recently developed continuous robust integral of the sign of the error (RISE) feedback term. Specifically, a system in which all terms are assumed known (temporarily) is feedback linearized and a control law is developed based on the HJB optimization method for a given quadratic performance index. Under the assumption that parametric uncertainty and unknown bounded disturbances are present in the dynamics, the control law is modified to contain the RISE feedback term which is used to identify the uncertainty. A Lyapunov stability analysis is included to show that the RISE feedback term asymptotically identifies the unknown dynamics (yielding semi-global asymptotic tracking) provided upper bounds on the disturbances are known and the control gains are selected appropriately. A feedforward neural network is then added to the previous controller. The utility of combining these feedforward and feedback methods are twofold. Previous efforts indicate that modifying the RISE feedback with a feedforward term can reduce the control eregistered trademarkort and improve the transient and steady state response of the RISE controller. Moreover, combining NN feedforward controllers with RISE feedback yields asymptotic results. Simulation as well as experimental results are provided to illustrate the developed controllers. Inverse optimal control is an alternative method to solve the nonlinear optimal control problem by circumventing the need to solve the HJB equation. Adaptive inverse optimal control techniques have been developed that can handle structured (i.e., linear in the parameters (LP)) uncertainty for a particular class of nonlinear systems. In Chapter 3, an adaptive inverse optimal controller is developed to minimize a meaningful performance index while the generalized coordinates of a nonlinear Euler-Lagrange system asymptotically track a desired time-varying trajectory despite LP uncertainty. A Lyapunov analysis is provided to examine the stability of the developed optimal controller, and simulation and experimental results illustrate the performance of the controller. Output feedback based controllers are more desirable than full-state feedback controllers because the necessary sensors for full-state feedback may not always be available and using numerical diregistered trademarkerentiation to obtain velocities can be problematic if position measurements are noisy. In Chapter 4, an adaptive output feedback IOC is developed which minimizes a meaningful cost, while the generalized coordinates of a nonlinear Euler-Lagrange system asymptotically tracks a desired time-varying trajectory. The new controller contains a desired compensation adaptation law (DCAL) based feedforward term and a feedback term that is shown to be implementable using only position measurements. A Lyapunov analysis is provided to prove the stability of the developed controller and to determine a meaningful cost functional. Experimental results are included to illustrate the performance of the controller. The dissertation is concluded in Chapter 5.
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 Keith Dupree.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024202:00001


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Iwouldliketothankmyadvisor,Dr.WarrenDixon,forhisguidanceandsupport.HisexampletaughtmetodedicatemyselftoeverythingIattempt,andtoalwaysseektowaystoimprovemywork.Theopportunitiesheprovidedmearebothunmeasurableandinvaluable.Iwouldalsoliketothankmycommitteemembers:Dr.NormanFitz-Coy,Dr.JacobHammer,andDr.PrabirBarooah,forthetimeandhelpthattheyprovided.Finally,IwouldliketothankmyNCRlabfellowsfortheirfriendshipduringthepastfouryears.Specically,IwouldliketothankParagPatreforalwayshavingthetimetoansweraquestion,andWillMacKunisformakingthelabamorelivelyenvironment. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1MotivationandLiteratureReview ....................... 12 1.2ProblemStatement ............................... 15 1.3Contributions .................................. 18 2OPTIMALCONTROLOFUNCERTAINNONLINEARSYSTEMSUSINGANEURALNETWORKANDRISEFEEDBACK ................. 20 2.1DynamicModelandProperties ........................ 20 2.2ControlObjective ................................ 22 2.3OptimalControlDesign ............................ 22 2.4RISEFeedbackControlDevelopment ..................... 25 2.5StabilityAnalysis ................................ 27 2.6NeuralNetworkExtension ........................... 31 2.7NeuralNetworks ................................ 32 2.8StabilityAnalysis ................................ 38 2.9SimulationandExperimentalResults ..................... 42 2.9.1Simulation ................................ 42 2.9.2Experiment ............................... 44 2.9.3Discussion ................................ 54 3INVERSEOPTIMALCONTROLOFANONLINEAREULER-LAGRANGESYSTEM ....................................... 56 3.1DynamicModelandProperties ........................ 57 3.2ControlDevelopment .............................. 58 3.3StabilityAnalysis ................................ 60 3.4CostFunctionalMinimization ......................... 63 3.5SimulationandExperimentalResults ..................... 65 3.5.1Simulation ................................ 65 3.5.2Experiment ............................... 69 3.5.3Discussion ................................ 72 5

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..................... 74 4.1DynamicModelandProperties ........................ 74 4.2ControlDevelopment .............................. 75 4.3StabilityAnalysis ................................ 79 4.4CostFunctionalMinimization ......................... 82 4.5OutputFeedbackFormoftheController ................... 84 4.6ExperimentalResults .............................. 86 4.6.1Experiment ............................... 86 4.6.2Discussion ................................ 89 5CONCLUSION .................................... 90 APPENDIX ASOLUTIONOFRICCATIDIFFERENTIALEQUATION ............ 92 BSOLUTIONOFHAMILTON-JACOBI-BELLMANEQUATION ......... 93 CBOUNDON~N(t) .................................. 96 DBOUNDONL(t)-PART1 ............................. 101 EBOUNDONL(t)-PART2 ............................. 102 FREVIEWOFADAPTIVEINVERSEOPTIMALCONTROL: .......... 104 REFERENCES ....................................... 107 BIOGRAPHICALSKETCH ................................ 111 6

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Table page 2-1Tabulatedvaluesforthe10runsforthedevelopedcontrollers. .......... 50 3-1Tabulatedvaluesfortheadaptiveinverseoptimalcontroller ........... 72 4-1Tabulatedvaluesforthe10runsoftheoutputfeedbackadaptiveinverseoptimalcontroller ....................................... 87 7

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Figure page 2-1ThesimulatedtrackingerrorsfortheRISEandoptimalcontroller. ....... 45 2-2ThesimulatedtorquesfortheRISEandoptimalcontroller. ............ 45 2-3ThedierencebetweentheRISEfeedbackandthenonlineareectsandboundeddisturbances. ..................................... 46 2-4ThesimulatedtrackingerrorsfortheRISE,NN,andoptimalcontroller. .... 46 2-5ThesimulatedtorquesfortheRISE,NN,andoptimalcontroller. ........ 47 2-6ThedierencebetweentheRISEfeedbackandfeedforwardNNandthenonlineareectsandboundeddisturbances(i.e.,.^fd+(h+d)). ........... 47 2-7Theexperimentaltestbedconsistsofatwo-linkrobot.ThelinksaremountedontwoNSKdirect-driveswitchedreluctancemotors. ............... 48 2-8TrackingerrorsresultingfromimplementingtheRISEandoptimalcontroller. 51 2-9TorquesresultingfromimplementingtheRISEandoptimalcontroller. ..... 51 2-10TrackingerrorsresultingfromimplementingtheRISE,NN,andoptimalcontroller. 52 2-11TorquesresultingfromimplementingtheRISE,NN,andoptimalcontroller. .. 52 2-12Trackingerrorsresultingfromimplementingthecontrollerdevelopedinliterature. 53 2-13Torquesresultingfromimplementingthecontrollerdevelopedinliterature. ... 53 3-1Thesimulatedtrackingerrorsfortheadaptiveinverseoptimalcontroller. .... 66 3-2Thesimulatedtorquesfortheadaptiveinverseoptimalcontroller. ........ 67 3-3Unknownsystemparameterestimatesfortheadaptiveinverseoptimalcontroller. 67 3-4Thevalueoflx;^fromEquation 3{30 ..................... 68 3-5TheintegralpartofthecostfunctionalinEquation 3{29 ............. 68 3-6Trackingerrorsresultingfromimplementingtheadaptiveinverseoptimalcontroller. 70 3-7Torquesresultingfromimplementingtheadaptiveinverseoptimalcontroller. .. 70 3-8Trackingerrorsresultingfromimplementingtheadaptiveinverseoptimalcontrollerforaslowertrajectory. ................................ 71 3-9Torquesresultingfromimplementingtheadaptiveinverseoptimalcontrollerforaslowertrajectory. .................................. 71 8

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.................................. 88 4-2Torquesresultingfromimplementingtheoutputfeedbackadaptiveinverseoptimalcontroller. ....................................... 88 4-3Unknownsystemparameterestimatesfortheoutputfeedbackadaptiveinverseoptimalcontroller. .................................. 89 9

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Optimalcontroltheoryinvolvesthedesignofcontrollersthatcansatisfysometrackingorregulationcontrolobjectivewhilesimultaneouslyminimizingsomeperformancemetric.AsucientconditiontosolveanoptimalcontrolproblemistosolvetheHamilton-Jacobi-Bellman(HJB)equation.Forthespecialcaseoflineartime-invariantsystems,thesolutiontotheHJBequationreducestosolvingthealgebraicRiccatiequation.However,forgeneralsystems,thechallengeistondavaluefunctionthatsatisestheHJBequation.Findingthisvaluefunctionhasremainedproblematicbecauseitrequiresthesolutionofapartialdierentialequationthatcannotbesolvedexplicitly. Chapter2illustratestheamalgamationofoptimalcontroltechniqueswitharecentlydevelopedcontinuousrobustintegralofthesignoftheerror(RISE)feedbackterm.Specically,asysteminwhichalltermsareassumedknown(temporarily)isfeedbacklinearizedandacontrollawisdevelopedbasedontheHJBoptimizationmethodforagivenquadraticperformanceindex.Undertheassumptionthatparametricuncertaintyandunknownboundeddisturbancesarepresentinthedynamics,thecontrollawismodiedtocontaintheRISEfeedbacktermwhichisusedtoidentifytheuncertainty.ALyapunovstabilityanalysisisincludedtoshowthattheRISEfeedbacktermasymptoticallyidentiestheunknowndynamics(yieldingsemi-globalasymptotictracking)providedupperboundsonthedisturbancesareknownandthecontrolgainsareselectedappropriately.Afeedforwardneuralnetworkisthenaddedtotheprevious 10

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InverseoptimalcontrolisanalternativemethodtosolvethenonlinearoptimalcontrolproblembycircumventingtheneedtosolvetheHJBequation.Adaptiveinverseoptimalcontroltechniqueshavebeendevelopedthatcanhandlestructured(i.e.,linearintheparameters(LP))uncertaintyforaparticularclassofnonlinearsystems.InChapter3,anadaptiveinverseoptimalcontrollerisdevelopedtominimizeameaningfulperformanceindexwhilethegeneralizedcoordinatesofanonlinearEuler-Lagrangesystemasymptoticallytrackadesiredtime-varyingtrajectorydespiteLPuncertainty.ALyapunovanalysisisprovidedtoexaminethestabilityofthedevelopedoptimalcontroller,andsimulationandexperimentalresultsillustratetheperformanceofthecontroller. Outputfeedbackbasedcontrollersaremoredesirablethanfull-statefeedbackcontrollersbecausethenecessarysensorsforfull-statefeedbackmaynotalwaysbeavailableandusingnumericaldierentiationtoobtainvelocitiescanbeproblematicifpositionmeasurementsarenoisy.InChapter4,anadaptiveoutputfeedbackIOCisdevelopedwhichminimizesameaningfulcost,whilethegeneralizedcoordinatesofanonlinearEuler-Lagrangesystemasymptoticallytracksadesiredtime-varyingtrajectory.Thenewcontrollercontainsadesiredcompensationadaptationlaw(DCAL)basedfeedforwardtermandafeedbacktermthatisshowntobeimplementableusingonlypositionmeasurements.ALyapunovanalysisisprovidedtoprovethestabilityofthedevelopedcontrollerandtodetermineameaningfulcostfunctional.Experimentalresultsareincludedtoillustratetheperformanceofthecontroller.ThedissertationisconcludedinChapter5. 11

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Onecommontechniqueindevelopinganoptimalcontrollerforanonlinearsystemistoassumethenonlineardynamicsareexactlyknown,feedbacklinearizethesystem,andthenapplyoptimalcontroltechniquestotheresultingsystemasin( 1 { 3 ),andothers.Forexample,dynamicfeedbacklinearizationwasusedin( 1 )todevelopacontrolLyapunovfunctiontoobtainaclassofoptimalcontrollers.AreviewoftheoptimalityofnonlineardesigntechniquesandgeneralresultsinvolvingfeedbacklinearizationaswellasJacobianlinearizationandothernonlineardesigntechniquesareprovidedin( 4 ; 5 ). Motivatedbythedesiretoeliminatetherequirementforexactknowledgeofthedynamics,( 6 )developedoneoftherstresultstoillustratetheinteractionofadaptivecontrolwithanoptimalcontroller.Specically,( 6 )rstusedexactfeedbacklinearizationtocancelthenonlineardynamicsandproduceanoptimalcontroller.Then,aself-optimizingadaptivecontrollerwasdevelopedtoyieldglobalasymptotictrackingdespitelinear-in-theparametersuncertainty.Theanalysisin( 6 )indicatedthatiftheparameterestimationerrorcouldsomehowconvergetozero,thenthecontrollerwouldconvergetotheoptimalsolution. Anothermethodtocompensateforsystemuncertaintiesistoemployneuralnetworks(NN)toapproximatetheunknowndynamics.TheuniversalapproximationpropertystatesthataNNcanidentifyafunctionuptosomefunctionreconstructionerror.Theuseof 12

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7 { 12 )toaccommodatefortheuncertaintyinthesystemandtosolvetheHJBequation.Specicallythetrackingerrorsareproventobeuniformlyultimatelybounded(UUB)andtheresultingstatespacesystem,forwhichtheHJBoptimalcontrollerisdeveloped,isonlyapproximated. ThedevelopmentinChapter2ismotivatedbythedesiretoimproveupontheUUBresultpreviouslyfoundinliterature.Specically,thischapterillustrateshowtheinclusionoftheRobustIntegraloftheSignoftheError(RISE)methodin( 13 ; 14 )canbeusedtoidentifythesystemandrejectdisturbances,whileachievingasymptotictrackingandtheconvergenceofacontroltermtotheoptimalcontroller.Chapter2alsoincludesaNNextensiontothepreviouslydesignedcontrollerasamodicationtotheresultsin( 7 { 12 )thatallowsforasymptoticstabilityandconvergencetotheoptimalcontrollerratherthantoapproximatetheoptimalcontroller. Inverseoptimalcontrol(IOC)( 15 { 21 )isanalternativemethodtosolvethenonlinearoptimalcontrolproblembycircumventingtheneedtosolvetheHJBequation.PreviousIOCsfocusonndingacontrolLyapunovfunction(CLF),whichcanbeshowntoalsobeavaluefunction,andthendevelopingacontrollerthatoptimizesameaningfulcost(i.e.,acostthatputsapositivepenaltyonthestatesandactuation).TheadvantageoftheIOCisthatthecontrollerdoesnothavetoconvergetoanoptimalsolution(likedirectoptimalcontrollers);however,thecostfunctionalcannotbechosenapriori.Thecostfunctionalisdeterminedbasedonthevaluefunction. SomeadaptiveIOCmethods( 22 { 26 )havebeendevelopedtocompensateforlinearintheparameters(LP)uncertainty.Resultssuchas( 22 )and( 23 ),developadaptiveIOCsforageneralclassofnonlinearsystemswithunknownparameters.Aninverseoptimaladaptiveattitudetrackingcontrollerisdevelopedin( 25 )forrigidspacecraftwithexternal 13

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26 ),aninverseoptimaladaptivebacksteppingtechniqueisappliedtothedesignofapitchcontrollawforasurface-to-airnonlinearmissilemodelwithaconstantinertiamatrix.TheresultsinChapter3seektoapplyadaptiveinverseoptimalcontrolmethodstoanunknownEuler-Lagrangesystemwithastatedependentinertiamatrix. Outputfeedbackbasedcontrollersaremoredesirablethanfull-statefeedbackcontrollers,becausethenecessarysensorsforfull-statefeedbackmaynotalwaysbeavailableandusingnumericaldierentiationtoobtainvelocitiescanbeproblematicifpositionmeasurementsarenoisy.Severalresearchershavedevelopedoutputfeedbackoptimalcontrollers.Theresearchersin( 27 ),developanitedimensionaldynamicoutput-feedbackcontrollerthatachieveslocalnear-optimalityandsemiglobalinverseoptimalityforaoutput-feedbacksystemwithinputdisturbances.However,thenonlinearitiesonlydependonthemeasuredoutput,andthesystemparametersareassumedtobeknown.Anoptimaltrajectorytrackingcontrolisproposedin( 28 )fornonholonomicsystemsinchainedformbyusingonlyoutputfeedbackinformation.Thenonholonomicsystemin( 28 )iswritteninsuchawaythatthestateandcontrolmatricesareknown.In( 29 ),anoutputfeedbackoptimalcontrollerisdesignedusingthecertaintyequivalenceprinciple,wherethestatesareestimated,butusedinthecontrollawasiftheywerethetruestates,resultinginanearoptimalcontroller.TheauthorsinChapter8of( 30 ),designanoutputfeedbacklinearquadraticregulator,butforalinearsystemwithknownparameters. Inspiredbyoutputfeedbackdesignmethodsdevelopedin( 31 { 34 ),anadaptiveoutputfeedbackIOCisdevelopedinChapter4.ThenewcontrollercontainsaDCALbasedfeedforwardtermandafeedbacktermthatisshowntobeimplementableusingonlypositionmeasurements.Unlikethepreviousresultsinliterature,thesystemcontainsanunknownstatedependentinertiamatrix.ALyapunovanalysisisprovidedtoprovethestabilityofthedevelopedcontrollerandtodetermineameaningfulcostfunctional. 14

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Thedissertationwilladdressthefollowingproblemsofinterest:1)RISEbasedoptimalcontrolofuncertainnonlinearsystems;2)RISEandNNbasedoptimalcontrolofuncertainnonlinearsystems;3)Adaptiveinverseoptimalcontrolofuncertainnonlinearsystems;and4)Adaptiveinverseoptimalcontrolofuncertainnonlinearsystemsusingoutputfeedback.ThecontroldevelopmentinthedissertationisprovenbyusingnonlinearLyapunovbasedmethodsandisdemonstratedbyMatlabsimulationand/orexperimentalresults. 1)RISEbasedoptimalcontrolofuncertainnonlinearsystems. 35 ),thatwaslatercoinedtheRobustIntegraloftheSignoftheError(RISE)methodin( 13 ; 14 ).TheRISEmethodisusedtoidentifythesystemandrejectdisturbances,whileachievingasymptotictrackingandtheconvergenceofacontroltermtotheoptimalcontroller.Inspiredbythepreviousworkin( 6 { 12 ; 36 ; 37 ),asysteminwhichalltermsare 15

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2)RISEandNNbasedoptimalcontrolofuncertainnonlinearsystems. 13 )indicatethatmodifyingtheRISEfeedbackwithafeedforwardtermcanreducethecontroleortandimprovethetransientandsteadystateresponseoftheRISEcontroller.Hence,thecombinedresultsshouldconvergetotheoptimalcontrollerfaster.Moreover,combiningNNfeedforwardcontrollerswithRISEfeedbackyieldsasymptoticresults( 38 ).Hence,theeortshereprovideamodicationtotheresultsin( 7 { 12 )thatallowsforasymptoticstabilityandconvergencetotheoptimalcontrollerratherthantoapproximatetheoptimalcontroller.Simulationandexperimentalresultsillustratetheperformanceofthecontroller. 3)Adaptiveinverseoptimalcontrolofuncertainnonlinearsystems 16

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_x=f(x)+F(x)+g(x)u;(1{1) forsomestatex(t)2Rn,wheref(x)2Rnisasmoothvectorvaluedfunction,F(x)2Rnp;g(x)2Rnmaresmoothmatrixvaluedfunctions,2Rpisavectorofunknownconstants,andu(t)2Rmisthecontrol.Ingeneral,theinputgainmatrixg(x)mustbeknown.Systemswithaconstantinertiamatrix,suchastheapplicationsin( 25 )and( 26 ),caneasilybetransformedintoEquation 1{1 ,unlikesystemswithanuncertainstate-dependentinertiamatrixoruncertaintyintheinputmatrix.InordertodetermineifanIOCcouldbedevelopedforamorepracticalengineeringsystem( 39 ),anadaptiveIOCisdevelopedinChapter3basedonthetheoreticalfoundationpresentedin( 22 ; 25 ; 40 ).ThedevelopedcontrollerminimizesameaningfulperformanceindexasthegeneralizedcoordinatesofanonlinearEuler-Lagrangesystemgloballyasymptoticallytrackadesiredtime-varyingtrajectorydespiteLPuncertaintyinthedynamics.TheconsideredclassofsystemsdoesnotadheretothemodelgiveninEquation 1{1 .TheuniqueabilitytoconsidertheIOCproblemforuncertainEuler-Lagrangedynamicsisduetoanoveloptimizationanalysis.Ameaningfulcostfunction(i.e.,apositivefunctionofthestatesandcontrolinput)isdevelopedandananalysisisprovidedtoprovethecostisminimizedwithouthavingtoprovetheLyapunovfunctionisaCLF.Todeveloptheoptimalcontrollerfortheuncertainsystem,theopenlooperrorsystemissegregatedtoincludetwoadaptiveterms.Oneadaptivetermisbasedonthetrackingerrorwhichcontributestothecostfunction,andtheotheradaptivetermdoesnotexplicitlydependonthetrackingerror(andthereforedoesnotexplicitlycontributetothecostfunction).ALyapunovanalysisisprovidedtoexaminethestabilityofthedevelopedcontrollerandtodeterminearespectivemeaningfulcostfunctional.Simulationaswellasexperimentalresultsareprovidedtoillustratethedevelopedcontroller. 17

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31 { 34 )anadaptiveIOCisdevelopedthatcontainsaDCALbasedfeedforwardtermandafeedbacktermthatisshowntobeimplementablewithoutvelocitymeasurements.ALyapunovanalysisisprovidedtodetermineameaningfulcostfunctionalandtoprovethestabilityofthedevelopedcontroller.Experimentalresultsareincludedtoillustratetheperformanceofthecontroller. 1. Forthersttimeever,adirectoptimalcontrollerisdevelopedthatyieldsasymptotictrackingandconvergencetoanoptimalcontroller. 2. ResultsexistinliteraturethatuseaNNwithdirectoptimalcontrolmethodsthatresultinUUBstability.ThecontributioninChapter2istoillustratehowthepreviousmethodscouldbeaugmentedwithRISEfeedbacktoobtainasymptotictracking. 3. AnadaptiveIOCisdevelopedforanonlinearEuler-Lagrangesystem.TheuseofanEuler-Lagrangesystemismotivatedbythefactthatthedynamicsmodelalargeclassofcontemporaryengineeringproblems( 39 ).Thecontrollerachievesasymptotic 18

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4. AnadaptiveIOCisdesignedforanonlinearEuler-Lagrangesystemthatonlyrequirespositionmeasurements.Thecontrollerachievesasymptotictrackingandminimizesameaningfulcost,whileusingamodelthatdoesnotconformtothestandardmodelusedinliterature.ThecontrollerconsistsofaDCALfeedforwardtermandafeedbacktermthatcanbeimplementedwithoutvelocitymeasurements. 19

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Thedevelopmentinthischapterismotivatedbythedesiretouseoptimalcontroltechniquesforuncertainnonlinearsystems.Inspiredbythepreviousworkin( 6 { 12 ; 36 ; 37 ),asysteminwhichalltermsareassumedknown(temporarily)isfeedbacklinearizedandacontrollawisdevelopedbasedontheHJBoptimizationmethodforagivenquadraticperformanceindex.ThecontrollawisthenaugmentedtocontaintheRISEfeedbacktermwhichisusedtoidentifytheparametricuncertaintyandtheunknownboundeddisturbancesthatarepresentinthedynamics.TheRISEfeedbacktermisthenshown,throughaLyapunovstabilityanalysis,toasymptoticallyidentifytheunknowndynamics(yieldingsemi-globalasymptotictracking)providedupperboundsonthedisturbancesareknownandthecontrolgainsareselectedappropriately.Duetothefactthatthisresultisasymptoticthecontrollawconvergestotheoptimalcontrollaw,ratherthantheUUBresultsinliteraturewhichonlyapproximatetheoptimalcontrollaw. Theremainderofthischapterisorganizedasfollows.InSection 2.1 ,themodelisgivenalongwithseveralofitsproperties.InSection 2.2 ,thecontrolobjectiveisstatedandanerrorsystemisformulated.InSection 2.3 ,anoptimalcontrollerisdevelopedforafeedbacklinearizedsystem.InSection 2.4 ,theRISEfeedbacktermisdeveloped.InSection 2.5 ,thestabilityofthecontrollerisproven.InSection 2.6 ,themotivationbehindincludingaNNisdiscussed.InSection 2.7 ,thepropertiesofNN'sarepresented.InSection 2.8 ,thestabilityofthecontrollerisproven.InSection 2.9 ,simulationandexperimentalresultsarepresented. 39 )formulation: 20

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2{1 ,M(q)2Rnndenotestheinertiamatrix,Vm(q;_q)2Rnndenotesthecentripetal-Coriolismatrix,G(q)2Rndenotesthegravityvector,F(_q)2Rndenotesfriction,d(t)2Rndenotesageneralnonlineardisturbance(e.g.,unmodeledeects),u(t)2Rnrepresentstheinputcontrolvector,andq(t),_q(t),q(t)2Rndenotetheposition,velocity,andaccelerationvectors,respectively.Thesubsequentdevelopmentisbasedontheassumptionthatq(t)and_q(t)aremeasurableandthatM(q),Vm(q;_q);G(q),F(_q)andd(t)areunknown.Moreover,thefollowingpropertieswillbeexploitedinthesubsequentdevelopment. 2{1 alongwiththesubsequenterrorsystemdevelopmentisbasedontheassumptionthatthegeneralizedcoordinates,q(t);areonlydenedfortranslationsandrotationsaboutasingleaxis. wherem12Risaknownpositiveconstant,m(q)2Risaknownpositivefunction,andkkdenotesthestandardEuclideannorm. 21

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Tofacilitatethesubsequentanalysis,lteredtrackingerrors,denotedbye2(t),r(t)2Rn,arealsodenedas where12Rnn,denotesasubsequentlydenedpositivedenite,constant,gainmatrix,and22Risapositiveconstant.Thelteredtrackingerrorr(t)isnotmeasurablesincetheexpressioninEquation 2{6 dependsonq(t). 2{4 andEquation 2{5 .Basedonthismodel,acontrollerisdevelopedthatminimizesaquadraticperformanceindexunderthe(temporary)assumptionthatthedynamicsinEquation 2{1 ,includingtheadditivedisturbance,areknown.ThedevelopmentinthissectionmotivatesthecontroldesigninSection 2.4 ,wherearobustcontrollerisdevelopedtoidentifytheunknowndynamicsandadditivedisturbance. Todevelopastate-spacemodelforthetrackingerrorsinEquation 2{4 andEquation 2{5 ,thetimederivativeofEquation 2{5 ispremultipliedbytheinertiamatrix,and 22

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2{1 andEquation 2{4 toobtain wherethenonlinearfunctionh(q,_q,qd,_qd,qd)2Rnisdenedas Underthe(temporary)assumptionthatthedynamicsinEquation 2{1 areknown,thecontrolinputcanbedesignedas whereuo(t)2Rnisanauxiliarycontrolinputthatwillbedesignedtominimizeasubsequentperformanceindex.BysubstitutingEquation 2{9 intoEquation 2{7 theclosed-looperrorsystemfore2(t)canbeobtainedas Astate-spacemodelforEquation 2{5 andEquation 2{10 cannowbedevelopedas _z=A(q;_q)z+B(q)uo;(2{11) whereA(q;_q)2R2n2n,B(q)2R2nn,andz(t)2R2naredenedasA(q;_q),2641Inn0nnM1Vm375;B(q),0nnM1T;z(t),eT1eT2T; 23

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2{11 is 2zTQz+1 2uToRuodt;(2{12) whereL(z;uo)istheLagrangian.InEquation 2{12 ,Q2R2n2nandR2Rnnarepositivedenitesymmetricmatricestoweighttheinuenceofthestatesand(partial)controleort,respectively.Furthermore,thematrixQcanbebrokenintoblocksasfollows:Q=264Q11Q12QT12Q22375: 7 ; 8 ),thefactthattheperformanceindexisonlypenalizedfortheauxiliarycontroluo(t)ispracticalsincethegravity,Coriolis,andfrictioncompensationtermsinEquation 2{8 cannotbemodiedbytheoptimaldesignphase. Tofacilitatethesubsequentdevelopment,letP(q)2R2n2nbedenedas whereK2Rnndenotesagainmatrix.If1,R,andK,introducedinEquation 2{5 ,Equation 2{12 ,andEquation 2{13 ,satisfythefollowingalgebraicrelationshipsK=KT=1 2Q12+QT12>0 (2{14)Q11=T1K+K1; 24

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2zTPz(2{17) satisestheHJBequation.Itcanthenbeconcludedthattheoptimalcontroluo(t)thatminimizesEquation 2{12 subjecttoEquation 2{11 is2 2{8 areunknown,sothecontrollergiveninEquation 2{9 cannotbeimplemented.However,ifthecontrolinputcontainssomemethodtoidentifyandcanceltheseeects,thenz(t)willconvergetothestatespacemodelinEquation 2{11 sothatuo(t)minimizestherespectiveperformanceindex.Asstatedintheintroduction,severalresultshaveexploredthisstrategyusingfunctionapproximationmethodssuchasneuralnetworks,wherethetrackingcontrolerrorsconvergetoaneighborhoodnearthestatespacemodelyieldingatypeofapproximateoptimalcontroller.Inthissection,acontrolinputisdevelopedthatexploitsRISEfeedbacktoidentifythenonlineareectsandboundeddisturbancestoenablez(t)toasymptoticallyconvergetothestatespacemodel. Todevelopthecontrolinput,theerrorsysteminEquation 2{6 ispremultipliedbyM(q)andtheexpressionsinEquation 2{1 ,Equation 2{4 ,andEquation 2{5 areutilizedtoobtain 25

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2{19 ,thecontrolinputiscomposedoftheoptimalcontroldevelopedinEquation 2{18 ,plusasubsequentlydesignedauxiliarycontrolterm(t)2Rnas Theclosed-looptrackingerrorsystemcanbedevelopedbysubstitutingEquation 2{20 intoEquation 2{19 as Tofacilitatethesubsequentstabilityanalysistheauxiliaryfunctionfd(qd;_qd;qd)2Rn;whichisdenedas isaddedandsubtractedtoEquation 2{21 toyield whereh(q,_q,qd,_qd,qd)2Rnisdenedas h,hfd:(2{24) ThetimederivativeofEquation 2{23 canbewrittenas 2_Mr+~N+NDe2R1r_(2{25) afterstrategicallygroupingspecicterms.InEquation 2{25 ,theunmeasurableauxiliaryterms~N(e1;e2;r;t),ND(t)2Rnaredenedas~N,_Vme2Vm_e21 2_Mr+h+2_Me2+2M_e2+e2+2R1e2ND,_fd+_d:

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wherey(t)2R3nisdenedas theboundingfunction(kyk)2Risapositivegloballyinvertiblenondecreasingfunction,andi2R(i=1;2)denoteknownpositiveconstants.BasedonEquation 2{25 ,thecontrolterm(t)isdesignedbasedontheRISEframework(see( 13 ; 35 ; 41 ))as whereks;12Rarepositiveconstantcontrolgains.Theclosedlooperrorsystemsforr(t)cannowbeobtainedbysubstitutingthetimederivativeofEquation 2{29 intoEquation 2{25 as 2_Mr+~N+NDe2R1r(ks+1)r1sgn(e2):(2{30) 2{20 canbeexaminedthroughthefollowingtheorem. 2{20 ensuresthatallsystemsignalsareboundedunderclosed-loopoperation,andthetrackingerrorsareregulatedinthesensethat 27

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2{31 canbeobtainedprovidedthecontrolgainksintroducedinEquation 2{29 isselectedsucientlylarge(seethesubsequentstabilityanalysis),and1,2areselectedaccordingtothesucientconditions 22>1;(2{32) wheremin(1)istheminimumeigenvalueof1;and1isselectedaccordingtothefollowingsucientcondition: where1wasintroducedinEquation 2{29 .Furthermore,u(t)convergestoanoptimalcontrollerthatminimizesEquation 2{12 subjecttoEquation 2{11 providedthegainconditionsgiveninEquation 2{14 -Equation 2{16 aresatised. 2{32 issatised. (t),[yT(t)p InEquation 2{34 ,theauxiliaryfunctionO(t)2Risdenedas wheretheauxiliaryfunctionL(t)2Risdenedas where12RisapositiveconstantchosenaccordingtothesucientconditionsinEquation 2{33 .AsillustratedinAppendixD,providedthesucientconditionsintroduced 28

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2{33 aresatised,thefollowinginequalitycanbeobtained: Hence,Equation 2{37 canbeusedtoconcludethatO(t)0. LetVL(;t):D[0;1)!Rbeacontinuouslydierentiablepositivedenitefunctiondenedas 2eT2e2+1 2rTM(q)r+O;(2{38) whichsatisesthefollowinginequalities: providedthesucientconditionsintroducedinEquation 2{33 aresatised.InEquation 2{39 ,thecontinuouspositivedenitefunctionsU1(),andU2()2RaredenedasU1(),1kk2,andU2(),2(q)kk2;where1,2(q)2Raredenedas1,1 2minf1;m1g2(q),max1 2m(q);1; 2{2 .AftertakingthetimederivativeofEquation 2{38 ,_VL(;t)canbeexpressedas_VL(;t)=2eT1_e1+eT2_e2+1 2rT_M(q)r+rTM(q)_r+_O: 2{5 ,Equation 2{6 ,Equation 2{30 ,andsubstitutinginforthetimederivativeofO(t),_V(;t)canbesimpliedasfollows:_VL(;t)2eT11e1+2eT2e1+rT~N(t) (2{40)(ks+1+minR1)krk22ke2k2: 2ke1k2+1 2ke2k2;

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2{40 canbesimpliedas_VL(;t)rT~N(t)(ks+1+minR1)krk2 2{26 ,theexpressioninEquation 2{41 canberewrittenas _VL(;t)3kyk2kskrk2(kyk)krkkyk;(2{42) where3,minf2min(1)1;21;1+min(R1)gand1and2arechosenaccordingtothesucientconditioninEquation 2{32 .AftercompletingthesquaresforthetermsinsidethebracketsinEquation 2{42 ,thefollowingexpressioncanbeobtained _VL(;t)3kyk2+2(kyk)kyk2 whereU()=ckyk2,forsomepositiveconstantc,isacontinuous,positivesemi-denitefunctionthatisdenedonthefollowingdomain:D,n2R3n+1jkk12p 2{39 andEquation 2{43 canbeusedtoshowthatVL(;t)2L1inD;hence,e1(t),e2(t),andr(t)2L1inD.Giventhate1(t),e2(t),andr(t)2L1inD,standardlinearanalysismethodscanbeusedtoprovethat_e1(t),_e2(t)2L1inDfromEquation 2{5 andEquation 2{6 .Sincee1(t),e2(t),r(t)2L1inD,thepropertythatqd(t),_qd(t),qd(t)existandareboundedcanbeusedalongwithEquation 2{4 -Equation 2{6 toconcludethatq(t),_q(t),q(t)2L1inD.Sinceq(t),_q(t)2L1inD,Property2.4canbeusedtoconcludethatM(q),Vm(q;_q),G(q),andF(_q)2L1inD.ThusfromEquation 2{1 andProperty2.5,wecanshowthatu(t)2L1inD.Giventhatr(t)2L1inD,itcanbeshownthat_(t)2L1inD.Since_q(t),q(t)2L1inD,Property2.4canbeusedtoshowthat_Vm(q;_q),_G(q),_F(q)and_M(q)2L1inD;hence,Equation 2{30 can 30

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LetSDdenoteasetdenedasfollows: TheregionofattractioninEquation 2{44 canbemadearbitrarilylargetoincludeanyinitialconditionsbyincreasingthecontrolgainks(i.e.,asemi-globaltypeofstabilityresult)( 41 ).Theorem8.4of( 42 )cannowbeinvokedtostatethat Basedonthedenitionofy(t),Equation 2{45 canbeusedtoconcludetheresultsinEquation 2{31 2{18 ),thenEquation 2{23 canbeusedtoconcludethat TheresultinEquation 2{46 indicatesthatthedynamicsinEquation 2{1 convergetothestate-spacesysteminEquation 2{11 .Hence,u(t)convergestoanoptimalcontrollerthatminimizesEquation 2{12 subjecttoEquation 2{11 providedthegainconditionsgiveninEquation 2{14 -Equation 2{16 aresatised. 13 )indicatethatmodifyingtheRISEfeedbackwithafeedforwardtermcanreducethecontroleortandimprovethetransientandsteadystateresponseoftheRISEcontroller.Hence,thecombinedresultsshouldconvergetotheoptimalcontrollerfaster.Moreover,combiningNNfeedforwardcontrollerswithRISEfeedbackyieldsasymptoticresults( 38 ).Hence,the 31

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7 { 12 )thatallowsforasymptoticstabilityandconvergencetotheoptimalcontrollerratherthantoapproximatetheoptimalcontroller. BasedonthepreviousworkdoneinthechaptertheunknownLPandnon-LPdynamicsaretemporarilyassumedtobeknownsothatacontrollercanbedevelopedforaresidualsystembasedontheHJBoptimizationmethodforagivenquadraticperformanceindex.Theoriginaluncertainnonlinearsystemisthenexamined,wheretheoptimalcontrollerisaugmentedtoincludetheRISEfeedbackandNNfeedforwardtermstoasymptoticallycanceltheuncertainties.ALyapunov-basedstabilityanalysisisincludedtoshowthattheRISEandNNcomponentsasymptoticallyidentifytheunknowndynamics(yieldingsemi-globalasymptotictracking)providedupperboundsonthedisturbancesareknownandthecontrolgainsareselectedappropriately.Moreover,thecontrollerconvergestotheoptimalcontrollerfortheapriorigivenquadraticperformanceindex. 43 ),( 44 ) InEquation 2{47 ,V2R(N1+1)N2andW2R(N2+1)nareboundedconstantidealweightmatricesfortherst-to-secondandsecond-to-thirdlayersrespectively,whereN1isthenumberofneuronsintheinputlayer,N2isthenumberofneuronsinthehiddenlayer,andnisthenumberofneuronsinthethirdlayer.Theactivationfunction4inEquation 2{47 isdenotedby():RN1+1!RN2+1,and"(x):RN1+1!Rnisthefunctionalreconstructionerror.BasedonEquation 2{47 ,thetypicalthree-layerNNapproximation 32

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43 ),( 44 ) ^f(x),^WT^VTx;(2{48) where^V(t)2R(N1+1)N2and^W(t)2R(N2+1)naresubsequentlydesignedestimatesoftheidealweightmatrices.Theestimatemismatchesfortheidealweightmatrices,denotedby~V(t)2R(N1+1)N2and~W(t)2R(N2+1)n,aredenedas~V,V^V;~W,W^W; ~,^=VTx^VTx:(2{49) OneoftheNNestimatepropertiesthatfacilitatethesubsequentdevelopmentisdescribedasfollows. wherekkFistheFrobeniusnormofamatrix,tr()isthetraceofamatrix. Todevelopthecontrolinput,theerrorsysteminEquation 2{6 ispremultipliedbyM(q)andtheexpressionsinEquation 2{1 ,Equation 2{4 ,andEquation 2{5 areutilizedtoobtain Tofacilitatethesubsequentstabilityanalysistheauxiliaryfunctionfd(qd;_qd;qd)2Rn;whichisdenedas 33

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2{52 toyield whereh(q,_q,qd,_qd,qd)2Rnisdenedas h,hfd:(2{55) TheexpressioninEquation 2{53 canberepresentedbyathree-layerNNas InEquation 2{56 ,theinputxd(t)2R3n+1isdenedasxd(t),[1qTd(t)_qTd(t)qTd(t)]TsothatN1=3nwhereN1wasintroducedinEquation 2{47 .BasedontheProperty2.5thatthedesiredtrajectoryisbounded,thefollowinginequalitiesholdk"(xd)k"b1k_"(xd;_xd)k"b2 Basedontheopen-looperrorsysteminEquation 2{52 ,thecontrolinputiscomposedoftheoptimalcontroldevelopedinEquation 2{18 ,athree-layerNNfeedforwardterm,plustheRISEfeedbacktermas Specically,(t)2RndenotestheRISEfeedbackcontroltermdenedinEquation 2{29 .ThefeedforwardNNcomponentinEquation 2{58 ,denotedby^fd(t)2Rn,isgeneratedas ^fd,^WT^VTxd:(2{59) 34

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2{59 aregeneratedon-line(thereisnoo-linelearningphase)as^W=proj(1^0^VT_xdeT2) (2{60)^V=proj(2_xd(^0T^We2)T) where0(^VTx)dVTx=dVTxjVTx=^VTx;and12R(N2+1)(N2+1),22R(3n+1)(3n+1)areconstant,positivedenite,symmetricmatrices.InEquation 2{60 ,proj()denotesasmoothconvexprojectionalgorithmthatensures^W(t)and^V(t)remainboundedinsideknownboundedconvexregions.SeeSection4.3in( 45 )forfurtherdetails. Theclosed-looptrackingerrorsystemisobtainedbysubstitutingEquation 2{58 intoEquation 2{52 as Tofacilitatethesubsequentstabilityanalysis,thetimederivativeofEquation 2{61 isdeterminedas UsingEquation 2{47 andEquation 2{59 ,theclosed-looperrorsysteminEquation 2{62 canbeexpressedasM_r=_Mr_Vme2Vm_e2+2_Me2+2M_e2+WT0VT_xd^WT^

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2{49 .AddingandsubtractingthetermsWT^0^VT_xd+^WT^0~VT_xdtoEquation 2{63 ,yieldsM_r=_Mr_Vme2Vm_e2+2_Me2+2M_e2+^WT^0~VT_xd 2{18 andtheNNweighttuninglawsinEquation 2{60 ,theexpressioninEquation 2{64 canberewrittenas 2_M(q)r+~N+Ne2R1r(ks+1)r1sgn(e2);(2{65) wherethefactthatthetimederivativeofEquation 2{29 isgivenas _=(ks+1)r+1sgn(e2)(2{66) wasutilized,andwheretheunmeasurableauxiliaryterms~N(e1;e2;r;t),N^W;^V;xd;t2Rnaredenedas~N,1 2_Mr+h+e2+2R1e2_Vme2Vm_e2+2_Me2+2M_e2 InEquation 2{68 ,ND(t)2Rnisdenedas whileNB^W;^V;xd2Rnisfurthersegregatedas 36

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andthetermNB2^W;^V;xd2Rnisdenedas SegregatingthetermsasinEquation 2{69 -Equation 2{72 facilitatesthedevelopmentoftheNNweightupdatelawsandthesubsequentstabilityanalysis.Forexample,thetermsinEquation 2{69 aregroupedtogetherbecausethetermsandtheirtimederivativescanbeupperboundedbyaconstantandrejectedbytheRISEfeedback,whereasthetermsgroupedinEquation 2{70 canbeupperboundedbyaconstantbuttheirderivativesarestatedependent.ThetermsinEquation 2{70 arefurthersegregatedbecauseNB1^W;^V;xdwillberejectedbytheRISEfeedback,whereasNB2^W;^V;xdwillbepartiallyrejectedbytheRISEfeedbackandpartiallycanceledbytheadaptiveupdatelawfortheNNweightestimates. Inasimilarmannerasin( 41 ),theMeanValueTheoremcanbeusedtodevelopthefollowingupperbound5 wherey(t)2R3nisdenedas andtheboundingfunction(kyk)2Risapositivegloballyinvertiblenondecreasingfunction.ThefollowinginequalitiescanbedevelopedbasedonProperty2.6,Equation 37

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,Equation 2{51 ,Equation 2{57 ,Equation 2{60 andEquation 2{70 -Equation 2{72 : InEquation 2{75 andEquation 2{76 ,i2R(i=1;2;:::;5)areknownpositiveconstants. 2{58 andEquation 2{60 canbeexaminedthroughthefollowingtheorem. 2{58 andEquation 2{60 ensuresthatallsystemsignalsareboundedunderclosed-loopoperationandthatthepositiontrackingerrorisregulatedinthesensethat TheresultinEquation 2{77 canbeachievedprovidedthecontrolgainksintroducedinEquation 2{29 isselectedsucientlylarge,and1,2areselectedaccordingtothefollowingsucientconditions: 22>2+1;(2{78) wheremin()2Rdenotestheminimumeigenvalue,andi(i=1;2)areselectedaccordingtothefollowingsucientconditions: wherei2R,i=1;2;...;5areintroducedinEquation 2{75 -Equation 2{76 ,1wasintroducedinEquation 2{29 ,and2isintroducedinEquation 2{82 .Furthermore,u(t)convergestoanoptimalcontrollerthatminimizesEquation 2{12 subjecttoEquation 2{11 providedthegainconditionsgiveninEquation 2{14 -Equation 2{16 aresatised. 38

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2{32 issatised. (t),[yT(t)p InEquation 2{80 ,theauxiliaryfunctionO(t)2Risdenedas wheree2i(0)isequaltotheithelementofe2(0)andtheauxiliaryfunctionL(t)2RisdenedasL(t),rT(NB1(t)+ND(t)1sgn(e2)) (2{82)+_eT2(t)NB2(t)2ke2(t)k2; 2{79 .ProvidedthesucientconditionsintroducedinEquation 2{79 aresatised6 Hence,Equation 2{83 canbeusedtoconcludethatO(t)0.TheauxiliaryfunctionG(t)2RinEquation 2{80 isdenedas Since1and2areconstant,symmetric,andpositivedenitematricesand2>0,itisstraightforwardthatG(t)0. 39

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2eT2e2+1 2rTM(q)r+O+G;(2{85) whichsatisesthefollowinginequalities: providedthesucientconditionsintroducedinEquation 2{79 aresatised.InEquation 2{86 ,thecontinuouspositivedenitefunctionsU1(),andU2()2RaredenedasU1(),1kk2,andU2(),2(q)kk2;where1,2(q)2Raredenedas1,1 2minf1;m1g2(q),max1 2m(q);1; 2{2 .AftertakingthetimederivativeofEquation 2{85 ,_VL(;t)canbeexpressedas_VL(;t)=2eT1_e1+eT2_e2+1 2rT_M(q)r+rTM(q)_r+_O+_G: 2{5 ,Equation 2{6 ,Equation 2{65 ,andsubstitutinginforthetimederivativeofP(t)andG(t),_V(;t)canbesimpliedas_VL(;t)=2eT11e1(ks+1)krk2rTR1r2eT2e1+2ke2(t)k2 2ke1k2+1 2ke2k2;

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2{60 ,theexpressioninEquation 2{87 canbesimpliedas_VL(;t)rT~N(t)(ks+1+minR1)krk2 2{73 ,theexpressioninEquation 2{88 canberewrittenas _VL(;t)3kyk2kskrk2(kyk)krkkyk;(2{89) where3,minf2min(1)1;212;1+min(R1)g;hence,1,and2mustbechosenaccordingtothesucientconditioninEquation 2{78 .AftercompletingthesquaresforthetermsinsidethebracketsinEquation 2{89 ,thefollowingexpressioncanbeobtained: _VL(;t)3kyk2+2(kyk)kyk2 whereU()=ckyk2,forsomepositiveconstantc,isacontinuous,positivesemi-denitefunctionthatisdenedonthefollowingdomain:D,n2R3n+2jkk12p 2{86 andEquation 2{90 canbeusedtoshowthatVL(;t)2L1inD;hence,e1(t),e2(t),andr(t)2L1inD.Giventhate1(t),e2(t),andr(t)2L1inD,standardlinearanalysismethodscanbeusedtoprovethat_e1(t),_e2(t)2L1inDfromEquation 2{5 andEquation 2{6 .Sincee1(t),e2(t),r(t)2L1inD,thepropertythatqd(t),_qd(t),qd(t)existandareboundedcanbeusedalongwithEquation 2{4 -Equation 2{6 toconcludethatq(t),_q(t),q(t)2L1inD.Sinceq(t),_q(t)2L1inD,Property2.4canbeusedtoconcludethatM(q),Vm(q;_q),G(q),andF(_q)2L1inD.ThusfromEquation 2{1 andProperty2,5,wecanshowthatu(t)2L1inD.Giventhatr(t)2L1inD,Equation 2{66 canbeusedtoshowthat_(t)2L1inD.Since_q(t),q(t)2L1inD,Property2.4canbeusedtoshowthat_Vm(q;_q),_G(q),_F(q)and_M(q)2L1inD;hence,Equation 2{65 canbeusedtoshowthat_r(t)2L1inD.Since_e1(t),_e2(t), 41

PAGE 42

LetSDdenoteasetdenedasfollows: TheregionofattractioninEquation 2{91 canbemadearbitrarilylargetoincludeanyinitialconditionsbyincreasingthecontrolgainks(i.e.,asemi-globaltypeofstabilityresult)( 41 ).Theorem8.4of( 42 )cannowbeinvokedtostatethat Basedonthedenitionofy(t),Equation 2{92 canbeusedtoshowthat TheresultinEquation 2{92 indicatesthatast!1;Equation 2{61 reducesto ^fd+=h+d:(2{94) Therefore,dynamicsinEquation 2{7 convergetothestate-spacesysteminEquation 2{11 .Hence,u(t)convergestoanoptimalcontrollerthatminimizesEquation 2{12 subjecttoEquation 2{11 providedthegainconditionsgiveninEquation 2{14 -Equation 2{16 ,Equation 2{78 ,andEquation 2{79 aresatised. 2.9.1Simulation 2{20 andEquation 2{58 anumericalsimulationwasperformed.Thesimulationisbasedonthe 42

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Thedesiredtrajectoryisgivenas 2sin(2t);(2{97) andtheinitialconditionsoftherobotwereselectedasq1(0)=q2(0)=14:3deg_q1(0)=_q2(0)=28:6deg=sec:

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2{14 ,Equation 2{15 ,andEquation 2{16 yieldedthefollowingvaluesforK;1,andRK=26444463751=2648:15:65:65:4375R=diag1 35;1 35: 2-1 andFigure 2-2 ,respectively.ToshowthattheRISEfeedbackidentiesthenonlineareectsandboundeddisturbances,aplotofthedierenceisshowninFigure 2-3 .Asthisdierencegoestozero,thedynamicsinEquation 2{1 convergetothestate-spacesysteminEquation 2{11 ,andthecontrollerbecomesoptimal. ThetrackingerrorsandthecontrolinputsfortheRISE,NN,andoptimalcontrollerareshowninFigure 2-4 andFigure 2-5 ,respectively.ToshowthattheRISEfeedbackandfeedforwardNNidentiesthenonlineareectsandboundeddisturbances,aplotofthedierenceisshowninFigure 2-6 .Asthisdierencegoestozero,thedynamicsinEquation 2{1 convergetothestate-spacesysteminEquation 2{11 ,andthecontrollerbecomesoptimal. 2{20 andEquation 2{58 anexperimentwasperformedonatwo-linkrobottestbedasdepictedinFigure 2-7 .Thetestbediscomposedofatwo-linkdirectdriverevoluterobotconsistingoftwoaluminumlinks,mountedona240:0[Nm](basejoint)and20:0[Nm](secondjoint)switched 44

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ThesimulatedtrackingerrorsfortheRISEandoptimalcontroller. Figure2-2. ThesimulatedtorquesfortheRISEandoptimalcontroller. 45

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ThedierencebetweentheRISEfeedbackandthenonlineareectsandboundeddisturbances. Figure2-4. ThesimulatedtrackingerrorsfortheRISE,NN,andoptimalcontroller. 46

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ThesimulatedtorquesfortheRISE,NN,andoptimalcontroller. Figure2-6. ThedierencebetweentheRISEfeedbackandfeedforwardNNandthenonlineareectsandboundeddisturbances(i.e.,.^fd+(h+d)). 47

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Theexperimentaltestbedconsistsofatwo-linkrobot.ThelinksaremountedontwoNSKdirect-driveswitchedreluctancemotors. reluctancemotors.Themotorsarecontrolledthroughpowerelectronicsoperatingintorquecontrolmode.Themotorresolversproviderotorpositionmeasurementswitharesolutionof614400pulses/revolution,andastandardbackwardsdierencealgorithmisusedtonumericallydeterminevelocityfromtheencoderreadings.APentium2.8GHzPCoperatingunderQNXhoststhecontrolalgorithm,whichwasimplementedviaacustomgraphicaluser-interface( 46 ),tofacilitatereal-timegraphing,datalogging,andtheabilitytoadjustcontrolgainswithoutrecompilingtheprogram.Dataacquisitionandcontrolimplementationwereperformedatafrequencyof1.0kHzusingtheServoToGoI/Oboard. Thecontrolobjectiveistotrackthedesiredtime-varyingtrajectorybyusingtheproposedcontrollaws.Toachievethiscontrolobjective,thecontrolgains2;ks;and1denedasscalarsinEquation 2{6 andEquation 2{29 ,wereimplemented(withnon-consequentialimplicationstothestabilityresult)asdiagonalgainmatrices.TheweightingmatrixesforbothcontrollerswerechosenasQ11=264402240375Q12=2644536375Q22=diag4;4;

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2{14 ,Equation 2{15 ,andEquation 2{16 yieldedthefollowingvaluesforK;1,andRK=26444463751=26415:610:610:610:4375R=diag0:25;0:25: Tocomparethedevelopedcontrollerstothecontrollersinliterature,thecontrollerin( 7 )givenby wasimplemented.InEquation 2{99 ,(t)2Rnisrobustifyingtermdenedas=kz1r 2{8 .Theneuralnetworkupdatelawisgivenby^W=1(x)eT2kz2kzk^W;

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Tabulatedvaluesforthe10runsforthedevelopedcontrollers. RISE RISE+NN AverageMaxSteadyStateError(deg)-Link1 0.0416 0.0416 AverageMaxSteadyStateError(deg)-Link2 0.0573 0.0550 AverageRMSError(deg)-Link1 0.0128 0.0139 AverageRMSError(deg)-Link2 0.0139 0.0143 AverageRMSTorque(Nm)-Link1 9.4217 9.4000 AverageRMSTorque(Nm)-Link2 1.7375 1.6825 ErrorStandardDeviation(deg)-Link1 0.0016 0.0011 ErrorStandardDeviation(deg)-Link2 0.0019 0.0015 TorqueStandardDeviation(Nm)-Link1 0.2775 0.3092 TorqueStandardDeviation(Nm)-Link2 0.0734 0.1746 wherekz12R.Thecontrolgainsrelatingtotheoptimaltermwerekeptconstant,however,theneuralnetworkupdatelawweightandadditiongainswereselectedasfollows:1=15I15kz2=0:1kz1=diag5;1: 2{29 wascomputedon-lineviaastandardtrapezoidalalgorithm.Inaddition,allthestateswereinitializedtozero.Eachexperiment(excludingthecontrollerinEquation 2{99 )usingwasperformedtentimes,anddatafromtheexperimentsisdisplayedinTable2-1. Figure 2-8 andFigure 2-9 depictthetrackingerrorsandcontroltorquesforoneexperimentaltrialfortheRISEandoptimalcontroller.Figure 2-10 andFigure 2-11 depictthetrackingerrorsandcontroltorquesforoneexperimentaltrialfortheRISEandoptimalcontroller.Figure 2-12 andFigure 2-13 depictthetrackingerrorsandcontroltorquesforcontrollerinEquation 2{99 .TheexperimentforthecontrollergiveninEquation 2{99 wasrunforalongerthanthedevelopedcontrollers,becausemoretimewasneededtogaugethecontrolperformance. 50

PAGE 51

TrackingerrorsresultingfromimplementingtheRISEandoptimalcontroller. Figure2-9. TorquesresultingfromimplementingtheRISEandoptimalcontroller. 51

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TrackingerrorsresultingfromimplementingtheRISE,NN,andoptimalcontroller. Figure2-11. TorquesresultingfromimplementingtheRISE,NN,andoptimalcontroller. 52

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Trackingerrorsresultingfromimplementingthecontrollerdevelopedinliterature. Figure2-13. Torquesresultingfromimplementingthecontrollerdevelopedinliterature. 53

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2{11 ,whereasitisimpossibletoperfectlyfeedbacklinearizearealsystem.Forthecomparison,thecontributionofthefeedforwardNNandRISEfeedbackterminEquation 2{20 andEquation 2{58 aswellascontributionofh(q,_q,qd,_qd,qd)andd(t)inEquation 2{9 arenotconsidered,sinceonlytheinputuo(t)wasincludedinthecostfunctionalEquation 2{12 .Forthethefeedbacklinearizedsystem,assumingnouncertainty,J(uo)wascalculatedtobe40:41:FortheRISEonlycontroller,J(uo)wascalculatedtobe43:32.FortheRISEandNNcontroller,J(uo)wascalculatedtobe42:43: 7 )forthesamecost.Thecontrollerin( 7 )maybeabletoachievesimilarresultsforadierentcost,however,changingthecostlimitsthecomparisonsthatcanbemadebetweenthecontrollers.Keepingthecostthesameresultsintheoptimalpartofthecontrollerbeingthesame;theonlypartofthecontrollerdesigntochangeis 54

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55

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Inverseoptimalcontrol(IOC)( 15 { 21 )wasdevelopedasawaytodesignoptimalcontrollersfornonlinearsystemswithouthavingtosolveanHJBequation.InIOCdesign,acontrolLyapunovfunction(CLF),whichcanbeshowntoalsobeavaluefunction,isusedtodesignacontrollerwhichstabilizesasystem.Itisthenshownthatthedevelopedcontrollerminimizesameaningfulcost(i.e.,acostthatputsapositivepenaltyonthestatesandactuation).Duetothefactthatthecontrollerisdesignedbeforethecost,thecostcannotbechosenapriori.However,anadvantageoftheIOCisthatthecontrollerdoesnothavetoconvergetoanoptimalsolution(likethepreviouslydevelopedcontrollers).AdaptiveIOCmethods( 22 { 26 )havebeendevelopedforsystemsthatcontainlinearintheparameters(LP)uncertainty. PreviousIOCsfocusontheclassproblemsmodeledas _x=f(x)+F(x)+g(x)u;(3{1) forsomestatex(t)2Rn,wheref(x)2Rnisaknownsmoothvectorvaluedfunction,F(x)2Rnp;g(x)2Rnmaresmoothmatrixvaluedfunctions,2Rpisavectorofunknownconstants,andu(t)2Rmisthecontrol.Ingeneral,theinputgainmatrixg(x)mustbeknown.ClassesofsystemswherethedynamicscanbeexpressedasEquation 3{1 wereusedtodevelopinverseoptimalcontrollersbecausethatformfacilitatedthedevelopmentofacontrolLyapunovfunction.Systemswithaconstantinertiamatrix,suchastheapplicationsin( 25 )and( 26 ),caneasilybetransformedintoEquation 3{1 ,unlikesystemswithanuncertainstate-dependentinertiamatrixoruncertaintyintheinputmatrix. Basedonthetheoreticalfoundationpresentedin( 22 ; 25 ; 40 ),anadaptiveIOCisdevelopedinthisChapter.TheclassofsystemsconsideredinthisChapterareuncertainEuler-Lagrangesystems,whichdonotadheretothemodelgiveninEquation 3{1 56

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Theremainderofthischapterisorganizedasfollows.InSection 3.1 ,themodelisgivenalongwithseveralofitsproperties.InSection 3.2 ,thecontrolobjectiveisstatedandanerrorsystemisformulated.InSection 3.3 ,thestabilityofthecontrollerisproven.InSection 3.4 ,ameaningfulcostisdevelopedandshowntobeminimizedbythecontrol.InSection 3.5 ,simulationandexperimentalresultsarepresented. 39 )formulation: where,M(q);Vm(q;_q),G(q);q(t),_q(t),q(t);andu(t)aredenedinSection 2.1 ,andFd2Rnndenotestheconstant,diagonal,positive-denite,viscousfrictioncoecientmatrix.Thesubsequentdevelopmentisbasedontheassumptionthatq(t)and_q(t)aremeasurableandthatM(q),Vm(q;_q);G(q),andFdareunknown.InadditiontoProperties2.1,2.2,and2.3thefollowingpropertieswillbeexploitedinthesubsequentdevelopment. 57

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3{2 canbelinearparameterizedas where2Rpcontainstheunknownconstantsystemparameters,andthenonlinearregressionmatrixY(q;_q;q)2Rnpcontainsknownfunctionsofthelinkposition,velocity,andacceleration,q(t);_q(t);q(t)2Rn;respectively. Tofacilitatethesubsequentcontroldesignandstabilityanalysis,alteredtrackingerrordenotedbyr(t)2Rn,isdenedas where2Risapositive,constantgain.Bytakingthetimederivativeofr(t)andpremultiplingbyM(q)thefollowingopen-looperrorsystemcanbeobtained: 58

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3{2 ,Equation 3{4 ,andEquation 3{5 wereused.TheexpressioninEquation 3{6 canthenberewrittenas where InEquation 3{8 andEquation 3{9 3{8 andEquation 3{9 isnotrequiredtoachievethetrackingcontrolobjective,ratherthetermsaresegregatedtofacilitatethedevelopmentoftheoptimalcontrollaw.Althoughbothtermscontainthesameunknownparameters,Equation 3{8 explicitlydependsonthetrackingerror,whileEquation 3{9 doesnot(itisdependentthepositionanddesiredpositionbutnotdependentontheirdierence).Therefore,thetotalcontrolu(t)ismadeupoftwoparts:uf(t)basedonEquation 3{9 whichisindependentofthetrackingerrorandthereforetheoptimization,andthefeedbacklawuo(t)basedonEquation 3{8 whichislatershowntominimizeameaningfulcost(i.e.,acostthatputsapositivepenaltyonthestatesandactuation).Thecontrolisdenedas whereuf;uo2Rnand^(t)2Rpisanestimatefor:Theparameterestimate^(t)inEquation 3{10 andEquation 3{11 isgeneratedbytheadaptiveupdatelaw 59

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3{11 intoEquation 3{7 yields wheretheparameterestimationerror~(t)2Rpisdenedas ~=^:(3{14) BasedonEquation 3{13 andthesubsequentstabilityanalysis,thecontrolinputisdesignedas whereR1x;^;K12Rnnarepositivedeniteandsymmetric,and1(t);2(t)2Rnnaredenedas 1=1 2=h^M(q)^Fdi;(3{17) whereIn2Rnnisanidentitymatrix. 3{10 -Equation 3{12 ,andEquation 3{15 canbeexaminedthroughthefollowingtheorem. 3{12 andthefeedbacklawgivenbyEquation 3{15 ensuresglobalasymptotictrackingofthesysteminEquation 3{13 inthesensethatke(t)k!0kr(t)k!0ast!1:

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2~T1~;(3{18) whereV(e;r;t)2Risdenedas 2eTe+1 2rTM(q)r:(3{19) AfterusingEquation 3{13 andProperty2.3,thetimederivativeofEquation 3{18 is _Va=eT_e+rTY1+Y2~+rTu0~T1^:(3{20) AfteraddingandsubtractingrT(t)Y1(q;_q)^(t)theexpressioninEquation 3{20 canbeexpressedas _Va=eT_e+rTY1^+rTuo+~T(Y1+Y2)Tr1^:(3{21) AftersubstitutingtheadaptiveupdatelawinEquation 3{12 theexpressioninEquation 3{21 reducesto _Va=eT_e+rTY1^+rTuo:(3{22) ThetermY1(q;_q)^(t)inEquation 3{22 canbeexpressedas where^Vm(q;_q);^M(q);^Fd2Rnndenotetheestimatesforthecentripetal-Coriolismatrix,inertiamatrix,andtheviscousfrictioncoecientmatrixrespectively.Bysubstituting_e(t)fromEquation 3{5 andusingEquation 3{23 ,theexpressioninEquation 3{21 canbewrittenas_Va=rTIn+^Vm(q;_q)2^M(q)+^Fde

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3{16 andEquation 3{17 ,respectively.SubstitutingtheexpressioninEquation 3{15 foruo(t)yields _Va=eTe+p ApplyingnonlineardampingtoEquation 3{25 yields_Va= 2rTK1r1 2p 2rT(K12)TK11(K12)r: 3{26 canbereducedto _Va 2rTK1r:(3{27) TheexpressionsinEquation 3{18 ,Equation 3{19 ,andEquation 3{27 canbeusedtoshowthatVa(e;r;~;t)2L1;hence,e(t),r(t),and~(t)2L1.Giventhate(t)andr(t)2L1,standardlinearanalysismethodscanbeusedtoprovethat_e(t)2L1(andhence,e(t)isuniformlycontinuous)fromEquation 3{5 .Sincee(t)and_e(t),2L1,thepropertythatqd(t)and_qd(t)existandareboundedcanbeusedalongwithEquation 3{4 andEquation 3{5 toconcludethatq(t)and_q(t)2L1.Since~(t)2L1;theexpressioninEquation 3{14 canbeusedtoconcludethat^(t)2L1:Since^(t);q(t);and_q(t)2L1;Property3.1canbeusedtoconcludethat^M(q),^Vm(q;_q),and^G(q)2L1.Since^M(q),^Vm(q;_q);and^Fd2L1;Equation 3{16 andEquation 3{17 canbeusedtoconcluded1(t)and2(t)2L1:Since1(t),2(t);andr(t)2L1;Equation 3{15 canbeusedtoconcludethatuo(t)2L1:Sinceq(t);_q(t);e(t);and_e(t)2L1;Property3.1,Property3.2,Property3.3,Equation 3{8 ,andEquation 3{9 ,canbeusedtoconcludethatY1(t)andY2(t)2L1:SinceY2(t)and^(t)2L1,theexpressioninEquation 3{10 canbeusedtoconcludedthatuf(t)2L1:Sinceuf(t)anduo(t)2L1;theexpressionEquation 3{11 canbeusedtoconcludedthatu(t)2L1:Sinceq(t);_q(t);r(t);Y1(t),Y2(t);andu(t)2L1;Property3.1andEquation 3{7 ,canbeusedtoconcludethat_r(t)2L1(and 62

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withthescalargainconstantselectedas2,andtheadaptiveupdatelawgiveninEquation 3{12 ,minimizesthemeaningfulcostfunctional wherelx;^2Risdeterminedtobe forthesystemgiveninEquation 3{13 3{29 isconsideredtobemeaningfulifitisapositivefunctionofthecontrolandthestates.FromEquation 3{29 ,thecostfunctionisapositivefunctioniflx;^inEquation 3{30 ispositive.Toexaminethesignoflx;^,theexpressionsinEquation 3{15 ,Equation 3{22 ,andEquation 3{27 areusedtodeterminethat 2rTK1r:(3{31) Multiplyingbothsidesby2yields 2rTK1r:(3{32) 63

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3{32 canberewrittenas2heT_e+rTY1^rTR1ri+(2)rTR1r 2rTr+(2)rTR1r: 3{30 ,theexpressioninEquation 3{33 canbesimpliedas 2rTr+(2)rTR1r:(3{34) TheinequalityinEquation 3{34 indicatesthatlx;^ispositivesinceR(t)ispositivedeniteand2.ThereforeJ(t)isameaningfulcost;penalizinge(t);r(t);andtheactuation. Toshowthatuo(t)minimizesJ(t),theauxiliarysignalv(t)2Rnisdenedas SubstitutingEquation 3{30 andEquation 3{35 intoEquation 3{29 yieldsJ=limt!1~T(t)1~(t)+Zt02rTR1r2heT_e+rTY1^id 3{14 andEquation 3{35 ,theexpressioninEquation 3{36 canbewrittenasJ=limt!1~T(t)1~(t)+Zt0vTRvd+2Zt0rTR1rvTrd

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3{20 ,theexpressioninEquation 3{37 canbesimpliedasJ=limt!1~T(t)1~(t)+Zt0vTRvd2Zt0_Va+~T1^d 3{12 intoEquation 3{38 yieldsJ=limt!1~T(t)1~(t)+Zt0vTRvd dt1 2~T1~d: 3{39 ,J(t)canbeexpressedas 3{28 intoEquation 3{24 itistrivialtoshowthatuo(t)stabilizesthesystem.BasedontheanalysisinSection 3.3 ,ke(t)kandkr(t)k!0ast!1:Therefore,V(t)!0ast!1andJ(t)isminimizedifv(t)=0:Therefore,thecontrollawuo(t)=uo(t)isoptimalandminimizesthecostfunctionalEquation 3{29 3.5.1Simulation 3{11 anumericalsimulationwasperformed.Thesimulationisbasedonthedynamicsforatwo-linkrobotgiveninEquation 2{95 withnodisturbanced(t):Thedesiredtrajectoryisgivenas 2sin(2t);(3{40) 65

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Thesimulatedtrackingerrorsfortheadaptiveinverseoptimalcontroller. andtheinitialconditionsoftherobotwereselectedasq1(0)=q2(0)=5:72deg_q1(0)=_q2(0)=51:56deg=sec: 3-1 andFigure 3-2 ,respectively.Figure 3-1 showsthattheerrorsasymptoticallyconvergetozero,whileFigure 3-2 showstheboundedinputtorque.TheestimatesforareshowninFigure 3-3 .Figure 3-4 indicatesthatlx;^ispositive,andFigure 3-5 indicatesthatthecostismeaningful. 66

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Thesimulatedtorquesfortheadaptiveinverseoptimalcontroller. Figure3-3. Unknownsystemparameterestimatesfortheadaptiveinverseoptimalcontroller. 67

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Thevalueoflx;^fromEquation 3{30 Figure3-5. TheintegralpartofthecostfunctionalinEquation 3{29 68

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3{11 anexperimentwasperformedonthetwo-linkrobottestbedasdescribedSection 2.9 .Thecontrolobjectiveistotrackthedesiredtime-varyingtrajectorybyusingthedevelopedadaptiveinverseoptimalcontrollaw.Toachievethiscontrolobjective,thecontrolgains;denedasascalarinEquation 3{5 wasimplemented(withnon-consequentialimplicationstothestabilityresult)asdiagonalgainmatrices.Specically,thecontrolgainswereselectedas andtheadaptationgainswereselectedas=diag([5;5;5;5;5]): Theexperimentwasrunasecondtimewithaslowerdesiredtrajectory,chosenasfollows: Forthistrajectorywassetequalto5:5:DatafromtheexperimentsisdisplayedinTable3-1. Figure 3-6 andFigure 3-7 depictthetrackingerrorsandcontroltorquesforoneexperimentaltrialfortheadaptiveinverseoptimalcontrollerforthetrajectorygiveninEquation 3{42 .Figure 3-8 andFigure 3-9 depictthetrackingerrorsandcontroltorquesforoneexperimentaltrialfortheadaptiveinverseoptimalcontrollerforthetrajectorygiveninEquation 3{43 69

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Trackingerrorsresultingfromimplementingtheadaptiveinverseoptimalcontroller. Figure3-7. Torquesresultingfromimplementingtheadaptiveinverseoptimalcontroller. 70

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Trackingerrorsresultingfromimplementingtheadaptiveinverseoptimalcontrollerforaslowertrajectory. Figure3-9. Torquesresultingfromimplementingtheadaptiveinverseoptimalcontrollerforaslowertrajectory. 71

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Tabulatedvaluesfortheadaptiveinverseoptimalcontroller TrajectoryinEquation 3{42 TrajectoryinEquation 3{43 MaxSteadyStateError(deg)-Link1 1.5213 0.5337 MaxSteadyStateError(deg)-Link2 2.0865 0.7342 RMSError(deg)-Link1 0.5553 0.3605 RMSError(deg)-Link2 0.8176 0.3954 RMSTorque(Nm)-Link1 11.4267 6.0446 RMSTorque(Nm)-Link2 1.5702 0.9610 2.9 .Inaneorttoreducetheerrorsthefrequencyofthetrajectorywasreduced,andwasincreased.Thesemodicationsresultedinsub-degreetracking,stillapproximately10timesgreaterthanthecontrollersdiscussedinSection 2.9 .Thereasonwasduetothegain.ThegaininEquation 3{5 ,tendstobeoneofthemostimportanttuninggains.ItbehavesliketheproportionalgaininaPIDcontroller.Increasingtendstoresultinfasterconvergenceandreducedsteadystateerror.Inthiscase,appearsinothertermsbesidesEquation 3{5 .InEquation 3{15 ,appearstothethirdpower.ThetermsinEquation 3{15 arethenmultipliedbyr(t)inEquation 3{15 whichresultsintothethirdpowermultipliedby_e(t)andtotheforthpowermultipliedbye(t):Soanof10;whichgenerallywouldnotbeunreasonable,wouldresultinagainof1;000multipliedbyavelocityerrorand10;000multipliedbyapositionerror.Thismakesthecontrollerverysensitivetonoiseandfasttrajectories,aswellasdiculttoimplementduetolargetorques.Todoso,thevalueofhadtobedecreased,whichresultedinpoortrackingperformance.Somesolutionstothiswouldbetoalterthecontrollerdesigntoreducethepowerof,aswellastoeliminatetheneedforvelocitymeasurementswhichwouldmitigatetheeectnoisymeasurementshasonthe 72

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73

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Inthischapter,anadaptiveoutputfeedbackinverseoptimalcontrollerisdesigned.Outputfeedbackbasedcontrollersaremoredesirablethanfull-statefeedbackcontrollers,becausethenecessarysensorsforfull-statefeedbackmaynotalwaysbeavailable,andusingnumericaldierentiationtoobtainvelocitiescanbeproblematicifpositionmeasurementsarenoisy.Usingtheerrorsystemdevelopedin( 31 { 34 ),anadaptiveoutputfeedbackIOCisdevelopedbasedonthetheoreticalfoundationpresentedin( 22 ; 25 ; 40 ).Thedevelopedcontrollerminimizesameaningfulperformanceindex(i.e.,apositivefunctionofthestatesandcontrolinput)asthegeneralizedcoordinatesofanonlinearEuler-Lagrangesystemgloballyasymptoticallytrackadesiredtime-varyingtrajectorydespiteLPuncertaintyinthedynamics.Likethepreviouslydevelopedcontroller,theconsideredclassofsystemsdoesnotadheretothemodelgiveninEquation 3{1 .ALyapunovanalysisisprovidedtoprovethestabilityofthedevelopedcontrollerandtodetermineameaningfulcostfunctional.Experimentalresultsareincludedtoillustratetheperformanceofthecontroller. Theremainderofthischapterisorganizedasfollows.InSection 4.1 ,themodelisgivenalongwithseveralofitsproperties.InSection 4.2 ,thecontrolobjectiveisstatedandanerrorsystemisformulated.InSection 4.3 ,thestabilityofthecontrollerisproven.InSection 4.4 ,ameaningfulcostisdevelopedandshowntobeminimizedbythecontrol.InSection 4.5 ,itisshownhowthecontrollercanbeimplementedusingonlypositionmeasurements.InSection 4.6 ,experimentalresultsarepresented. 3{2 .InadditiontoProperties2.1,2.2,2.3,3.1,3.2,and3.4,thefollowingpropertywillbeexploitedinthesubsequentdevelopment. 74

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31 )kM()M(v)kmkTanh(v)k; 75

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Tofacilitatethesubsequentcontroldesignandstabilityanalysis,alteredtrackingerror,denotedby(t)2Rn,isdenedas( 31 ; 47 ) where1;22Rarepositive,constantgains,andef(t)2Rnisanauxiliaryltervariabledesignedas( 31 ; 47 ) _ef=3Tanh(ef)+2Tanh(e)k1Cosh2(ef)(4{4)ef(0)=0; 47 )and( 31 )toovercometheproblemofinjectinghigherordertermsinthecontrollerandtofacilitatethedevelopmentofsucientgainconditionsusedinthesubsequentstabilityanalysis.Bytakingthetimederivativeof(t)andpremultiplingbyM(q)thefollowingopen-looperrorsystemcanbeobtained: AfterutilizingEquation 4{2 -Equation 4{4 ,theexpressioninEquation 4{5 ,canberewrittenasM(q)_=M(q)(qdq)2k1M(q)Cosh2(ef)Cosh2(ef)+1M(q)Cosh2(e)(1Tanh(e)2Tanh(ef))+2M(q)Cosh2(ef)(3Tanh(ef)+2Tanh(e)):

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3{2 forM(q)q(t)yieldsM(q)_=M(q)qdVm(q;_q)2k1M(q)+Fd_q+G(q) (4{6)+1M(q)Cosh2(e)(1Tanh(e)2Tanh(ef))+2M(q)Cosh2(ef)(3Tanh(ef)+2Tanh(e))+Vm(q;_q)(_qd+1Tanh(e)+2Tanh(ef))u: 4{6 canbeexpressedas where(e;ef;;t)2Rnand~Y(e;ef;;t)2Rnaredenedas=1M(q)Cosh2(e)(1Tanh(e)2Tanh(ef)) (4{8)+2M(q)Cosh2(ef)(3Tanh(ef)+2Tanh(e))+Vm(q;_qd+1Tanh(e)+2Tanh(ef))(1Tanh(e)+2Tanh(ef))+Vm(q;_qd)(1Tanh(e)+2Tanh(ef))Vm(q;)(_qd+1Tanh(e)+2Tanh(ef)); ~Y=M(q)qd+Vm(q;_qd)_qd+G(q)+Fd_qYd(4{9) Byexploitingthefactthatthedesiredtrajectoryisbounded,andusingProperties2.1,3.1,4.3,andthepropertiesofhyperbolicfunctions,(e;ef;;t)ofEquation 4{8 canbeupperboundedas 77

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4{9 canbeupperboundedas where22Risalsosomepositiveboundingconstantthatdependsonthemechanicalparametersandthedesiredtrajectory.ThetermsinEquation 4{9 andEquation 4{10 aredevelopedtofacilitatethedevelopmentoftheoptimalcontrollaw.Althoughbothtermscontainthesameunknownparameters,Equation 4{10 dependspurelyonthedesiredtrajectory,whileEquation 4{9 dependsontheactualcurrenttrajectory.Basedonthesegregationofthesetwoterms,thetotalcontrolu(t)ismadeupoftwoparts:uf(t)anduo(t):Thefeedforwardcontroltermuf(t)isbasedonEquation 4{10 andisindependentofthestateofthesystemandthereforetheoptimization..Thefeedbacklawuo(t)isbasedonEquation 4{9 andtheopen-looperrorsystemandislatershowntominimizeameaningfulcost(i.e.,acostthatputsapositivepenaltyonthestatesandactuation).Thetotalcontrolisdenedas whereuo;uf2Rn;and^(t)2Rpisanestimatefor:Theparameterestimate^(t)inEquation 4{13 andEquation 4{14 isgeneratedbytheadaptiveupdatelaw 78

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4{14 intoEquation 4{7 yields wheretheparameterestimateerror~(t)2Rpisdenedas ~=^:(4{17) BasedonEquation 4{16 andthesubsequentstabilityanalysis,thecontrolinputuo(t)isdesignedas 4{13 -Equation 4{15 ,andEquation 4{18 canbeexaminedthroughthefollowingtheorem. 4{15 andthefeedbacklawgivenbyEquation 4{18 ensuresglobalasymptotictrackingofthesysteminEquation 4{16 inthesensethatke(t)k!0k(t)k!0ast!1; wherem1,1,and1areconstantsdenedinEquation 2{2 ,Equation 4{11 ,andEquation 4{12 ,respectively,andk22Risacontrolgainthatmustsatisfythefollowingsucientcondition: 41;(4{20) 79

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2~T1~;(4{21) whereV(e;r;t)2Risdenedas 2TM(q);(4{22) whereei(t)2Randefi(t)2Raretheithelementsofthee(t)andef(t)vectorsrespectively.AfterusingEquation 4{16 andProperty2.3,thetimederivativeofEquation 4{21 is_Va=Tanh(e)T_e+Tanh(ef)T_ef~T1^ 4{15 theexpressioninEquation 4{23 reducesto _Va=Tanh(e)T_e+Tanh(ef)T_ef+Tu0+Th2k1M(q)++~Yi:(4{24) AftersubstitutingEquation 4{3 andEquation 4{4 ,Equation 4{24 canbeexpressedas_Va=1Tanh(e)TTanh(e)3Tanh(ef)TTanh(ef) (4{25)+TTanh(e)T2k1M(q)+Th+~Yik1Tanh(ef)TCosh2(ef)+Tuo:

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4{11 ,Equation 4{12 ,Equation 4{18 ,andEquation 4{19 theexpressioninEquation 4{25 canbewrittenas_Va1Tanh(e)TTanh(e)3Tanh(ef)TTanh(ef)T 4{26 canbewrittenas _Va1Tanh(e)TTanh(e)T3Tanh(ef)TTanh(ef)+kxk TheexpressioninEquation 4{27 canbereducedto _Va11 4k2kxk2:(4{28) Ifk2isselectedaccordingtoEquation 4{20 ,theinequalityinEquation 4{28 canbereducedto _Va2kxk2;(4{29) where22Risapositiveconstant. TheexpressionsinEquation 4{21 ,Equation 4{22 ,andEquation 4{29 canbeusedtoshowthatVa(e;r;~;t)2L1;hence,e(t),ef(t);(t),and~(t)2L1.Giventhat(t)2L1,Equation 4{3 canbeusedtoprovethat_e(t)2L1(andhence,e(t)isuniformlycontinuous).Sincee(t)and_e(t),2L1,thepropertythatqd(t)and_qd(t)existandareboundedcanbeusedalongwithEquation 4{2 andEquation 4{3 toconcludethatq(t)and_q(t)2L1.Since~(t)2L1;theexpressioninEquation 4{17 canbeusedtoconcludethat^(t)2L1:Since,bythepropertythat,qd(t);_qd(t);andqd(t)2L1;Property3.1canbeusedtoconcludethatYd(qd;_qd;qd)2L1:SinceYd(qd;_qd;qd)and^(t)2L1;Equation 4{13 canbeusedtoconcludedthatuf(t)2L1:Sinceef(t);and(t),2L1;Equation 4{18 canbeusedtoconcludethatuo(t)2L1:Sinceuf(t)anduo(t)2L1;theexpression 81

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4{14 canbeusedtoconcludedthatu(t)2L1:Sinceq(t);_q(t);e(t);ef(t);(t);Yd(qd;_qd;qd);~(t);uo(t)2L1;Property3.1andEquation 4{16 canbeusedtoconcludethat_(t)2L1(andhence(t)isuniformlycontinuous).Sincee(t);ef(t);and(t)2L1;Equation 4{4 canbeusedtoconcludethat_ef(t)2L1(andhenceef(t)isuniformlycontinuous).Fromthisitcanbeconcludedthatx(t);_x(t)2L1(andhencex(t)isuniformlycontinuous).TheexpressioninEquation 4{29 canbeusedtoconcludedthatx(t)2L2:Barbalat'sLemmacanbeusedtoconcludethatkx(t)k!0ast!1:Thereforeke(t)kandk(t)k!0ast!1: andtheadaptiveupdatelawgiveninEquation 4{15 ,minimizesthemeaningfulcostfunctional wherelx;^2Risdeterminedtobel=2hTanh(e)T_e+Tanh(ef)T_efi2Th2k1M(q)++~Yi 4{16 4{31 isconsideredtobemeaningfulifitisapositivefunctionofthecontrolandthestates.FromEquation 4{31 ,thecostfunctionisapositivefunctioniflx;^inEquation 4{32 ispositive.Toexaminethesignoflx;^,theexpressionsinEquation 4{18 ,Equation 4{24 ,andEquation 4{29 areusedto 82

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4{3 for(t)Tyields2kxk2Tanh(e)T_e+Tanh(ef)T_ef+Th2k1M(q)++~Yi+[_e+1Tanh(e)]TR1Tanh(ef)+2Tanh(ef)TR1Tanh(ef): 4{32 ,theexpressioninEquation 4{33 canbesimpliedas 22kxk2l:(4{34) TheinequalityinEquation 4{34 indicatesthatlx;^ispositive.ThereforeJ(t)isameaningfulcost;penalizinge(t);(t);ef(t);andhence,thecontrol. Toshowthatuo(t)minimizesJ(t),Equation 4{32 issubstitutedintoEquation 4{31 yieldingJ=limt!1n~T(t)1~(t) (4{35)2Zt0Tanh(e)T_e+Tanh(ef)T_ef+Th2k1M(q)++~Yid2Zt0[_e+1Tanh(e)]TR1Tanh(ef)+2Tanh(ef)TR1Tanh(ef)d:

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4{35 canbewrittenasJ=limt!1~T(t)1~(t)+2Zt0TYd~d 4{3 andEquation 4{18 ,theexpressioninEquation 4{36 canbesimpliedasJ=limt!1~T(t)1~(t)+2Zt0TYd~d 4{23 ,theexpressioninEquation 4{37 canbesimpliedto AfterusingEquation 4{15 ,theexpressioninEquation 4{38 canbewrittenas dt1 2~T1~d:(4{39) AfterintegratingEquation 4{39 ,thecostfunctionalcanbeexpressedas 4.3 ,ke(t)kandk(t)k!0ast!1:Therefore,V(t)!0ast!1;andJ(t)isminimized.Therefore,thecontrollawuo(t)isoptimalandminimizesthecostfunctionalEquation 4{31 4{18 onlyrequirespositionmeasurements,itisnotedthatthecontrolinputdoesnotactuallyrequirethecomputation 84

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Fromstandardhyperbolicidentities: cosh2(efi)=1 1tanh2(efi)=1 1y2i.(4{41) Ifyicanbecalculatedonlyfrompositionmeasurements,thentanh(efi)andcosh2(efi)canbecalculatedonlyfrompositionmeasurements.RewritingEquation 4{4 intermsofindividualelementsyields_efi=3tanh(efi)+2tanh(ei)k1cosh2(efi)i 4{40 ,andsubstitutingEquation 4{41 andEquation 4{42 yields_yi=cosh2(efi)_efi=1y2i(3yi+2tanh(ei))k1[_ei+1tanh(ei)+2yi]yi(0)=0: wherepi2Risanauxiliaryvariablegeneratedfromthefollowingdierentialequation:_pi=1(pik1ei)2[3(pik1ei)+2tanh(ei)] (4{44)k1[1tanh(ei)+2(pik1ei)]pi(0)=k1ei(0):

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4{44 ,itisobviousthatpi(t)canbecalculatedusingonlypositionmeasurements.Duetothefactthatpi(t)canbecalculatedusingonlypositionmeasurements,Equation 4{43 canbeusedtoshowthatyi(t),andthereforetanh(efi)andcosh2(efi);canbecalculatedusingonlypositionmeasurements.Duetothefactthattanh(efi)andcosh2(efi)canbecalculatedusingonlypositionmeasurements,theexpressioninEquation 4{18 canbecalculatedusingonlypositionmeasurements. ToshowthattheadaptiveupdatelawgivenbyEquation 4{15 onlyrequirespositionmeasurements,Equation 4{3 issubstitutedintoEquation 4{15 ,whichisintegratedbypartstoformthefollowingexpression:^=YTde+ 4{45 ,itisobviousthat^(t)canbecalculatedusingonlypositionmeasurements.Duetothefactthat^(t)canbecalculatedusingonlypositionmeasurements,theexpressioninEquation 4{13 canbecalculatedusingonlypositionmeasurements.DuetothefactthattheexpressionsinEquation 4{13 andEquation 4{18 canbecalculatedusingonlypositionmeasurements,thetotalcontrolgiveninEquation 4{14 canbecalculatedusingonlypositionmeasurements. 4.6.1Experiment 4{14 anexperimentwasperformedonthetwo-linkrobottestbedasdescribedSection 2.9 .Themodeleddynamicsforthetestbedarelinearinthefollowingparameters:=[p1p2p3Fd1Fd2]T: 86

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Tabulatedvaluesforthe10runsoftheoutputfeedbackadaptiveinverseoptimalcontroller AverageMaxSteadyStateError(deg)-Link1 0.0678 AverageMaxSteadyStateError(deg)-Link2 0.0973 AverageRMSError(deg)-Link1 0.0261 AverageRMSError(deg)-Link2 0.0323 AverageRMSTorque(Nm)-Link1 9.7256 AverageRMSTorque(Nm)-Link2 1.3959 ErrorStandardDeviation(deg)-Link1 0.0001 ErrorStandardDeviation(deg)-Link2 0.0030 TorqueStandardDeviation(Nm)-Link1 0.2184 TorqueStandardDeviation(Nm)-Link2 0.0446 objective,thecontrolgains1;2;3,andk1;denedasscalarsinEquation 4{3 andEquation 4{4 ,wereimplemented(withnon-consequentialimplicationstothestabilityresult)asdiagonalgainmatrices.Specically,thecontrolgainswereselectedas andtheadaptationgainswereselectedas=diag([5;5;5;5;5]): Eachexperimentwasperformedtentimes,anddatafromtheexperimentsisdisplayedinTable4-1. Figure 4-1 depictsthetrackingerrorsforoneexperimentaltrial.ThecontroltorquesandadaptiveestimatesforthesameexperimentaltrialareshowninFigures. 4-2 and 4-3 ,respectively. 87

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Trackingerrorsresultingfromimplementingtheoutputfeedbackadaptiveinverseoptimalcontroller. Figure4-2. Torquesresultingfromimplementingtheoutputfeedbackadaptiveinverseoptimalcontroller. 88

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Unknownsystemparameterestimatesfortheoutputfeedbackadaptiveinverseoptimalcontroller. 3.5 theresultsareimproved.Thecontrollerwasablekeeptheaveragemaximumsteadystate(denedasthelast10secondsoftheexperiment)errorsbelow0:07degreesfortherstlinkand0:1degreesforthesecondlink.TheaverageRMSerrorwas0:0261degreesfortherstlinkand0:0323degreesforthesecondlink.TheaverageRMStorquewas9:7256Nmfortherstlinkand1:3959Nmforthesecondlink. 89

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Inthisdissertation,optimalcontrollersaredesignedforuncertainnonlinearEuler-Lagrangesystems.Theoptimalcontrolproblemsinthisdissertationareseparatedintotwomainparts:1)directoptimalcontrol,wherethecostfunctionalwaschosenapriori;and2)inverseoptimalcontrol,whereameaningfulcostfunctionalwasdeterminedafterthecontroldesign.Thesetwodesignmethodswereapproachedusingdierentcontroltechniques. InChapter2,acontrolschemeisdevelopedforaclassofnonlinearEuler-Lagrangesystemsthatenablesthegeneralizedcoordinatestoasymptoticallytrackadesiredtime-varyingtrajectorydespitegeneraluncertaintyinthedynamicssuchasadditiveboundeddisturbancesandparametricuncertaintythatdonothavetosatisfyaLPassumption.ThemaincontributionofthisworkisthattheRISEfeedbackmethodaugmentedwithanauxiliarycontroltermisshowntominimizeaquadraticperformanceindexbasedonaHJBoptimizationscheme.Liketheinuentialworkin( 6 { 12 ; 36 ; 37 )theresultinthiseortinitiallydevelopsanoptimalcontrollerbasedonapartiallyfeedbacklinearizedstate-spacemodelassumingexactknowledgeofthedynamics.TheoptimalcontrolleristhencombinedwithafeedforwardNNandRISEfeedback.ALyapunovstabilityanalysisisincludedtoshowthattheNNandRISEidentifytheuncertainties,thereforethedynamicsasymptoticallyconvergetothestate-spacesystemthattheHJBoptimizationschemeisbasedon.Numericalsimulationsandanexperimentareincludedtosupporttheseresults. TocircumventhavingtosolveanHJBequation,anadaptiveinverseoptimalcontrollerisdevelopedinChapter3toachieveasymptotictrackingwhileminimizingameaningfulcost.Incontrasttotypicaloptimalcontrollers,inverseoptimalcontrollersdonothaveanapriorichosencost;ratherthecostiscalculatedbasedontheLyapunovfunction.Thiscontrollerconsistsofanadaptivefeedforwardtermandanfeedback 90

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AnoutputfeedbackadaptiveIOCcontrollerisdesignedinChapter4duetothefactthatoutputfeedbackcontrollersaremoredesirablethanfull-statefeedbackcontrollers.ThecontrollerisdesignedusinganewerrorsystemandaDCALfeedforwardadaptiveterm.ALyapunovstabilityanalysisisusedtoshowthatthedevelopedcontrollernotonlystabilizesasystemwheretheunknownmatricesarefunctionsofthestates,andtheinputgainmatrixisunknown,butminimizesameaningfulcost.Throughaninnovativelterdesign,theIOCisdevelopedasanoutputfeedbackcontroller;requiringonlypositionmeasurementsforimplementation.Experimentalresultsareincludedtoillustratetheimprovedperformanceofthecontrolleroverthefullstatefeedbackcontroller. Therearemanypossibleavenuesforfuturework.OnepossibledirectionistoincludedisturbancesinthecontroldesignandtoattempttosolveaHamilton-Jacobi-Issacs(HJI)equation,ratherthananHJBequation.ThesolutionofHJIequationsresultsinthesolutionofadierentialgameproblem,thataccountsfordisturbancesintheoptimization.AnotherpossibledirectionistheuseoftheRISEininverseoptimaldesign.InthecontrollersinChapter3andChapter4,anadaptivefeedforwardtermisusedtocompensateforLPuncertainty.TheuseoftheRISEmayresultininverseoptimalcontrollersthatcanhandleabroaderclassofsystems. 91

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2{5 ,Equation 2{12 ,andEquation 2{13 ,satisfythefollowingalgebraicrelationshipsK=KT=1 2Q12+QT12>0Q11=T1K+K1;R1=Q22; whereA(q;_q),B(q)andz(t)areintroducedinEquation 2{11 A{1 yields0=zT0B@264K1K0Vm375+264T1K0KVTm375 2{3 inProperty2,Equation A{2 issatisedifthefollowingconditionsaretrue: 2{14 -Equation 2{16 ,thenP(q)satisesEquation A{1 92

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2zTPz(B{1) satisestheHJBequation.Thentheoptimalcontrolu(t)thatminimizesEquation 2{12 subjecttoEquation 2{11 is wheretheHamiltonianisdenedas Toderivetheoptimalcontrollaw,thepartialderivativesofthefunctionV(z;t)needtobeevaluated.ThetimederivativeofV(z;t)canbeexpressedas dt=@V @t+@V @z_z:(B{5) ThegradientofV(z;t)withrespecttotheerrorstatez(t)is @z=zTP+1 2zTD;(B{6) where @e11z@P @e1nz00=D10:(B{7) InEquation B{7 ,D2R2n2nand02R2n1isazerovectorandthenotation@P @e1iisusedtorepresentthe2n2nmatrixwhoseelementsarepartialderivativesoftheelementsofP(q)withrespecttoe1i:

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B{4 requiresthat@H @uz;u;@V(z;t) @zB=0; Since@2H @u2=R>0; B{4 isminimizedbyu(t).SubstitutingEquation B{6 andEquation B{7 intoEquation B{8 gives 2DTz=R1BTPTz=R1e2;(B{9) wheretherelationBTDT=0D1+M0=0; AnecessaryandsucientconditionforoptimalityisthatthechosenvaluefunctionV(z;t)satisesEquation B{3 .SubstitutingEquation B{4 intoEquation B{3 yields SubstitutingEquation B{5 intoEquation B{10 yields 2zT_Pz+L(z;u)=0:(B{11) InsertingEquation 2{11 ,Equation B{9 ,andL(z;u)intoEquation B{11 yields 2zT_P+QPBR1BTPTz=0:(B{12) 94

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2zTATP+PAz;Equation B{12 canbewrittenas 1 2zT_P+ATP+PA+QPBR1BTPTz=0:(B{13) AsshowninAppendixA,P(q)satisesEquation B{13 ,thereforeV(z;t)satisestheHJBequationEquation B{3 andtheoptimalisgivenbyEquation B{9 95

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2{26 as ~N,_Vme2Vm_e21 2_Mr+h+2_Me2+2M_e2+e2+2R1e2;(C{1)ke2k^^WT^ke2kxd andtheboundingfunction(kyk)2Risapositivegloballyinvertiblenondecreasingfunction. C{1 canbeexpressedasfollows: ~N=1 2_M(q)r+_M(q)[1(e21e1)+2e2]+_M(q)qd+M(q)...qd+M(q)[1(r2e21(e21e1))+2(r2e2)]_M(qd)qd+Vm(q;_q)qdr+2e2+1e221e1_Vm(qd;_qd)_qdVm(qd;_qd)qdM(qd)...qd+_Vm(q;_q)_q+_G(q)_G(qd)+_F(_q)_F(_qd)+e2+2R1e2

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2_M(q)r+_M(q)[1(e21e1)+2e2]+_G(q)+M(q)[1(r2e21(e21e1))+2(r2e2)]+_M(q)qd+M(q)...qd+_Vm(q;_q)_q+_F(_q)+e2+2R1e2+Vm(q;_q)qdr+2e2+1e221e1:

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(C{3)+@N(q;2;qd;...qd;0;0;0)

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C{3 ,~N(t)canbeupperboundedasfollows:~N(t)@N(1;_qd;qd;...qd;0;0;0)

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C{5 canbefurtherupperboundedasfollows:~N(t)[1(e1)+12(e1;e2)+5(e1;e2)]ke1k+[2(e1;e2)+6(e1;e2)]ke2k+7(e1;e2;r)krk: C{2 ,~N(t)canbeexpressedintermsofy(t)asfollows:~N(t)[1(e1)+12(e1;e2)+5(e1;e2)]ky(t)k+[2(e1;e2)+6(e1;e2)]ky(t)k+7(e1;e2;r)ky(t)k: 2{67 istrivialduetothefactthatproj()ke2k,andallothertermsareboundedbyassumptionordesign. 100

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2{36 .IfthefollowingsucientconditionsinEquation 2{33 then 2{36 yields SubstitutingEquation 2{6 intoEquation D{2 ,Zt0L()d=Zt0de2() D{3 byparts,theexpressioninEquation D{3 canbewrittenas D{4 as (D{5)+Zt02ke2()k1+1 D{5 canbeobtainedif1satisesEquation 2{33 101

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2{82 .IfthefollowingsucientconditionsinEquation 2{79 then 2{82 yields (E{2)+r()T(NB1()+ND()1sgn(e2)))d: 2{6 intoEquation E{2 E{3 byparts,andbyusingthefactthatdN()

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E{3 canbewrittenasZt0L()d=Zt02e2()T(NB1(t)+ND(t)1sgn(e2))dt E{4 asZt0L()d1ke2(0)ke2(0)N(0) (E{5)+Zt02ke2()k1+2+1 E{5 canbeobtainedif1and2satisfyEquation 2{79 103

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Givenasystemoftheform _x=f(x)+g(x)u;(F{1) wherex(t)2Rndenotesthestatevector,u(t)2Rm;denotesthecontrolvector,f(x)2Rnisasmoothvectorvaluedfunctionandg(x)2Rnmisasmoothmatrixvaluedfunction,theoptimalcontrolproblemistodeterminethecontrolu(t)2Rmwhichminimizesacost subjecttothedynamicconstraintsinEquation F{1 ,whereL(z;u)2RistheLagrangian.Anecessaryandsucientconditionforanoptimalsolutiontoexist,istheexistenceofafunctionV(x;t)2R,calledthevaluefunction,whichsatisestheHamilton-Jacobi-Bellman(HJB)equation0=@V(x;t) _x=f(x)+F(x)+g(x)u;(F{3) wheref(x)2RndenotesaknownsmoothvectorfunctionandF(x)2Rnp;g(x)2Rnmareknown,smoothmatrixvaluedfunctions,2Rpisavectorofunknownconstants,andu(t)2Rmdenotesthecontrolvector.Apositivedenite,radiallyunboundedfunctionVa(x;)2RiscalledanadaptivecontrolLyapunovfunctionforEquation F{3 ifitisa 104

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_x=f(x)+F(x)+@Va where2Rppispositivedenite.ThefunctionVa(x;)isacontrolLyapunovfunctionforEquation F{4 ifthereexistsasmoothcontrollawu(;x);withu(;0)=0;whichsatises @x"f(x)+F(x)+@Va IfthereexistsacontrolLyapunovfunctionforEquation F{4 (whichmeansanadaptivecontrolLyapunovfunctionforEquation F{3 ),andafeedbackcontrollawoftheform whereR(x;)2RmmisapositivedeniteandsymmetricmatrixthatstabilizesEquation F{4 ,thenthefeedbackcontrollawu=Rx;^1@Va minimizesthecostfunctional wherelx;^2Risdenedas 105

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F{6 canbefoundforthesysteminEquation F{3 ,thenthatcontrollawminimizesthecostfunctionalinEquation F{8 106

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[1] R.FreemanandP.Kokotovic,\Optimalnonlinearcontrollersforfeedbacklinearizablesystems,"inProceedingsoftheAmericanControlsConference,Seattle,WA,1995,pp.2722{2726. [2] Q.Lu,Y.Sun,Z.Xu,andT.Mochizuki,\Decentralizednonlinearoptimalexcitationcontrol,"IEEETransactionsonPowerSystems,vol.11,no.4,pp.1957{1962,1996. [3] M.Sekoguchi,H.Konishi,M.Goto,A.Yokoyama,andQ.Lu,\NonlinearoptimalcontrolappliedtoSTATCOMforpowersystemstabilization,"inProceedingsoftheIEEE/PESTransmissionandDistributionConferenceandExhibition,Yokohama,Japan,2002,pp.342{347. [4] V.NevisticandJ.A.Primbs,\Constrainednonlinearoptimalcontrol:aconverseHJBapproach,"CaliforniaInstituteofTechnology,Pasadena,CA91125,Tech.Rep.CIT-CDS96-021,1996. [5] J.A.PrimbsandV.Nevistic,\Optimalityofnonlineardesigntechniques:AconverseHJBapproach,"CaliforniaInstituteofTechnology,Pasadena,CA91125,Tech.Rep.CIT-CDS96-022,1996. [6] R.Johansson,\Quadraticoptimizationofmotioncoordinationandcontrol,"IEEETransactionsonAutomaticControl,vol.35,no.11,pp.1197{1208,1990. [7] Y.Kim,F.Lewis,andD.Dawson,\Intelligentoptimalcontrolofroboticmanipulatorusingneuralnetworks,"Automatica,vol.36,no.9,pp.1355{1364,2000. [8] Y.KimandF.Lewis,\OptimaldesignofCMACneural-networkcontrollerforrobotmanipulators,"IEEETransactionsonSystems,Man,andCybernetics:PartC,vol.30,no.1,pp.22{31,Feb.2000. [9] M.Abu-KhalafandF.Lewis,\NearlyoptimalHJBsolutionforconstrainedinputsystemsusinganeuralnetworkleast-squaresapproach,"inProceedingsoftheIEEEConferenceonDecisionandControl,LasVegas,NV,2002,pp.943{948. [10] T.ChengandF.Lewis,\Fixed-naltimeconstrainedoptimalcontrolofnonlinearsystemsusingneuralnetworkHJBapproach,"inProceedingsoftheIEEEConferenceonDecisionandControl,SanDiego,CA,2006,pp.3016{3021. [11] ||,\Neuralnetworksolutionfornite-horizonH-innityconstrainedoptimalcontrolofnonlinearsystems,"inProceedingsoftheIEEEInternationalConferenceonControlandAutomation,Guangzhou,China,2007,pp.1966{1972. [12] T.Cheng,F.Lewis,andM.Abu-Khalaf,\Aneuralnetworksolutionforxed-naltimeoptimalcontrolofnonlinearsystems,"Automatica,vol.43,no.3,pp.482{490,2007. 107

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P.M.Patre,W.MacKunis,C.Makkar,andW.E.Dixon,\Asymptotictrackingforsystemswithstructuredandunstructureduncertainties,"IEEETransactionsonControlSystemsTechnology,vol.16,no.2,pp.373{379,2008. [14] ||,\Asymptotictrackingforsystemswithstructuredandunstructureduncertainties,"inProceedingsoftheIEEEConferenceonDecisionandControl,SanDiego,CA,2006,pp.441{446. [15] M.KrsticandH.Deng,StabilizationofNonlinearUncertainSystems.NewYork:Springer,1998. [16] M.KrsticandP.Tsiotras,\Inverseoptimalityresultsfortheattitudemotionofarigidspacecraft,"inProceedingsoftheAmericanControlsConference,Albuquerque,NM,1997,pp.1884{1888. [17] M.KrsticandZ.-H.Li,\Inverseoptimaldesignofinput-to-statestabilizingnonlinearcontrollers,"inProceedingsoftheIEEEConferenceonDecisionandControl,SanDiego,CA,1997,pp.3479{3484. [18] N.Kidane,Y.Yamashita,H.Nakamura,andH.Nishitani,\Inverseoptimizationforanonlinearsystemwithaninputconstraint,"inSICE2004AnnualConference,Sapporo,Japan,2004,pp.1210{1213. [19] T.Fukao,\Inverseoptimaltrackingcontrolofanonholonomicmobilerobot,"inProceedingsoftheIEEE/RSJInternationalConferenceonIntelligentRobotsandSystems,Sendai,Japan,2004,pp.1475{1480. [20] R.FreemanandJ.Primbs,\Controllyapunovfunctions:newideasfromanoldsource,"inProceedingsoftheIEEEConferenceonDecisionandControl,Kobe,Japan,1996,pp.3926{3931. [21] P.Gurl,\Non-linearmissileguidancesynthesisusingcontrollyapunovfunctions,"ProceedingsoftheInstitutionofMechanicalEngineers,PartG:JournalofAerospaceEngineering,vol.219,no.2,pp.77{88,2005. [22] Z.LiandM.Krstic,\Optimaldesignofadaptivetrackingcontrollersfornonlinearsystems,"inProceedingsoftheAmericanControlsConference,Albuquerque,NewMexico,1997,pp.1191{1197. [23] X.-S.CaiandZ.-Z.Han,\Inverseoptimalcontrolofnonlinearsystemswithstructuraluncertainty,"IEEProceedingsofControlTheoryandApplications,vol.152,no.1,pp.79{84,2005. [24] J.L.Fausz,V.-S.Chellaboina,andW.Haddad,\Inverseoptimaladaptivecontrolfornonlinearuncertainsystemswithexogenousdisturbances,"inProceedingsoftheIEEEConferenceonDecisionandControl,SanDiego,CA,1997,pp.2654{2659. 108

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W.Luo,Y.Chu,andK.Ling,\Inverseoptimaladaptivecontrolforattitudetrackingofspacecraft,"IEEETransactionsonAutomaticControl,vol.50,no.11,pp.1639{1654,Nov.2005. [26] L.Sonneveldt,E.VanOort,Q.P.Chu,andJ.A.Mulder,\Comparisonofinverseoptimalandtuningfunctionsdesignsforadaptivemissilecontrol,"JournalofGuidance,Control,andDynamics,vol.31,no.4,pp.1176{1182,2008. [27] K.Ezal,P.V.Kokotovic,A.R.Teel,andT.Basar,\Disturbanceattenuatingoutput-feedbackcontrolofnonlinearsystemswithlocaloptimality,"Automatica,vol.37,no.6,pp.805{817,2001. [28] Z.Qu,J.Wang,C.E.Plaisted,andR.A.Hull,\Output-feedbacknear-optimalcontrolofchainedsystems,"inProceedingsoftheAmericanControlsConference,Minneapolis,MN,2006,pp.4975{4980. [29] T.PerezandH.Haimovich,ConstrainedControlandEstimation.Berlin:Springer,2005,ch.OutputFeedbackOptimalControlwithConstraints,pp.279{294. [30] F.L.LewisandV.L.Syrmos,OptimalControl.NewYork:JohnWileyandSons,1995. [31] F.Zhang,D.M.Dawson,M.S.deQueiroz,andW.E.Dixon,\Globaladaptiveoutputfeedbacktrackingcontrolofrobotmanipulators,"IEEETransactionsonAutomaticControl,vol.45,no.6,pp.1203{1208,2000. [32] W.E.Dixon,E.Zergeroglu,andD.M.Dawson,\Globalrobustoutputfeedbacktrackingcontrolofrobotmanipulators,"Robotica,vol.22,no.4,pp.351{357,2004. [33] E.Zergeroglu,W.E.Dixon,D.Haste,andD.M.Dawson,\Acompositeadaptiveoutputfeedbacktrackingcontrollerforroboticmanipulators,"Robotica,vol.17,pp.591{600,1999. [34] T.Burg,D.M.Dawson,andP.Vedagarbha,\Aredesigneddcalcontrollerwithoutvelocitymeasurements:Theoryanddemonstration,"Robotica,vol.15,no.4,pp.337{346,1997. [35] Z.QuandJ.Xu,\Model-basedlearningcontrolsandtheircomparisonsusingLyapunovdirectmethod,"AsianJournalofControl,vol.4,no.1,pp.99{110,2002. [36] M.Abu-Khalaf,J.Huang,andF.Lewis,NonlinearH-2/H-InnityConstrainedFeedbackControl.London:Springer,2006. [37] F.L.Lewis,OptimalControl.NewYork:JohnWileyandSons,1986. [38] P.M.Patre,W.MacKunis,K.Kaiser,andW.E.Dixon,\Asymptotictrackingforuncertaindynamicsystemsviaamultilayerneuralnetworkfeedforwardandrise 109

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[39] R.Ortega,A.Lora,P.J.Nicklasson,andH.J.Sira-Ramirez,Passivity-basedControlofEuler-LagrangeSystems:Mechanical,ElectricalandElectromechanicalApplica-tions.London:Springer,1998. [40] M.KrsticandP.V.Kokotovic,\Controllyapunovfunctionsforadaptivenonlinearstabilization,"SystemsandControlLetters,vol.26,no.1,pp.17{23,1995. [41] B.Xian,D.M.Dawson,M.S.deQueiroz,andJ.Chen,\Acontinuousasymptotictrackingcontrolstrategyforuncertainnonlinearsystems,"IEEETransactionsonAutomaticControl,vol.49,no.7,pp.1206{1211,2004. [42] H.K.Khalil,NonlinearSystems.NewJersey:Prentice-Hall,2002. [43] F.L.Lewis,\Nonlinearnetworkstructuresforfeedbackcontrol,"AsianJournalofControl,vol.1,no.4,pp.205{228,1999. [44] F.Lewis,J.Campos,andR.Selmic,Neuro-FuzzyControlofIndustrialSystemswithActuatorNonlinearities.Philadelphia:SIAM,2002. [45] W.E.Dixon,A.Behal,D.M.Dawson,andS.P.Nagarkatti,NonlinearControlofEngineeringSystems:aLyapunov-BasedApproach.Boston:Birkhuser,2003. [46] M.S.Loer,N.P.Costescu,andD.M.Dawson,\Qmotor3.0andtheQmotorrobotictoolkit:apc-basedcontrolplatform,"IEEEControlSystemsMagazine,vol.22,no.3,pp.12{26,2002. [47] W.E.Dixon,E.Zergeroglu,D.M.Dawson,andM.W.Hannan,\Globaladaptivepartialstatefeedbacktrackingcontrolofrigid-linkexible-jointrobots,"inPro-ceedingsoftheIEEE/ASMEInternationalConferenceonAdvancedIntelligentMechatronics,Atlanta,GA,1999,pp.281{286. 110

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KeithDupreewasbornin1982inPortJeerson,NewYork.HereceivedhisBachelorofSciencedegreeinaerospaceengineeringwithaminorinphysicsinMayof2005,hisMastersofScienceinmechanicalengineeringinMayof2007,andhisMasterofScienceinelectricalandcomputerengineeringinAugustof2008,andhisDoctorofPhilosophyinmechaincalengineeringinMayof2009;allfromtheUniversityofFlorida.Hisresearchinterestsincludenonlinearcontrols,imageprocessing,stateestimation,andoptimalcontrol. 111