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Lyapunov-Based Robust and Adaptive Control of Nonlinear Systems Using a Novel Feedback Structure

Permanent Link: http://ufdc.ufl.edu/UFE0024807/00001

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Title: Lyapunov-Based Robust and Adaptive Control of Nonlinear Systems Using a Novel Feedback Structure
Physical Description: 1 online resource (163 p.)
Language: english
Creator: Patre, Parag
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: adaptive, composite, control, disturbances, networks, neural, nonlinear, rise, robust, uncertain
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

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Abstract: The focus of this research is an examination of the interplay between different intelligent feedforward mechanisms with a recently developed continuous robust feedback mechanism, coined Robust Integral of the Sign of the Error (RISE), to yield asymptotic tracking in the presence of generic disturbances. This result solves a decades long open problem of how to obtain asymptotic stability of nonlinear systems with general sufficiently smooth disturbances with a continuous control method. Further, it is shown that the developed technique can be fused with other feedforward methods such as function approximation and adaptive control methods. The addition of feedforward elements adds system knowledge in the control structure, which heuristically, yields better performance and reduces control effort. This heuristic notion is supported by experimental results in this research. One key element in the development of the novel feedforward mechanisms presented in this dissertation is the modularity between the controller and update law. This modularity provides flexibility in the selection of different update laws that could be easier to implement or help to achieve faster parameter convergence and better tracking performance. The efficacy of the feedforward mechanisms is further enhanced by including a prediction error in the learning process. The prediction error, which directly relates to the actual function mismatch, is used along with the system tracking errors to develop a composite adaptation law. Each result is supported through rigorous Lyapunov-based stability proofs and experimental demonstrations.
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 Parag Patre.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Dixon, Warren E.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024807:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024807/00001

Material Information

Title: Lyapunov-Based Robust and Adaptive Control of Nonlinear Systems Using a Novel Feedback Structure
Physical Description: 1 online resource (163 p.)
Language: english
Creator: Patre, Parag
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: adaptive, composite, control, disturbances, networks, neural, nonlinear, rise, robust, uncertain
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: The focus of this research is an examination of the interplay between different intelligent feedforward mechanisms with a recently developed continuous robust feedback mechanism, coined Robust Integral of the Sign of the Error (RISE), to yield asymptotic tracking in the presence of generic disturbances. This result solves a decades long open problem of how to obtain asymptotic stability of nonlinear systems with general sufficiently smooth disturbances with a continuous control method. Further, it is shown that the developed technique can be fused with other feedforward methods such as function approximation and adaptive control methods. The addition of feedforward elements adds system knowledge in the control structure, which heuristically, yields better performance and reduces control effort. This heuristic notion is supported by experimental results in this research. One key element in the development of the novel feedforward mechanisms presented in this dissertation is the modularity between the controller and update law. This modularity provides flexibility in the selection of different update laws that could be easier to implement or help to achieve faster parameter convergence and better tracking performance. The efficacy of the feedforward mechanisms is further enhanced by including a prediction error in the learning process. The prediction error, which directly relates to the actual function mismatch, is used along with the system tracking errors to develop a composite adaptation law. Each result is supported through rigorous Lyapunov-based stability proofs and experimental demonstrations.
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 Parag Patre.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Dixon, Warren E.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024807:00001


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LYAPUNOV-BASEDROBUSTANDADAPTIVECONTROLOFNONLINEAR SYSTEMSUSINGANOVELFEEDBACKSTRUCTURE By PARAGPATRE ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1

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c r 2009ParagPatre 2

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Tomyparents,MadhukarandSurekhaPatre,andmysister,Apa rna 3

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ACKNOWLEDGMENTS Iwouldliketoexpresssinceregratitudetomyadvisor,Dr.W arrenE.Dixon,whose experienceandmotivationwereinstrumentalinsuccessful completionofmyPhD.I appreciatehispatiencetowardsmeandallowingmethefreed omtoworkindependently. Iwouldalsoliketoextendmygratitudetomycommitteemembe rsDr.Norman Fitz-Coy,Dr.RickLind,Dr.PramodKhargonekar,andDr.Fra nkLewisforthetimeand helptheyprovided. Iwouldliketothankmycoworkers,family,andfriendsforth eirsupportand encouragement. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 1.1MotivationandProblemStatement ...................... 13 1.2Contributions .................................. 18 2ASYMPTOTICTRACKINGFORSYSTEMSWITHSTRUCTUREDAND UNSTRUCTUREDUNCERTAINTIES ....................... 22 2.1Introduction ................................... 22 2.2DynamicModel ................................. 22 2.3ErrorSystemDevelopment ........................... 24 2.4StabilityAnalysis ................................ 28 2.5ExperimentalResults .............................. 32 2.6Discussion .................................... 39 2.7Conclusions ................................... 40 3ASYMPTOTICTRACKINGFORUNCERTAINDYNAMICSYSTEMSVIA AMULTILAYERNEURALNETWORKFEEDFORWARDANDRISEFEEDBACKCONTROLSTRUCTURE .............................. 43 3.1Introduction ................................... 43 3.2DynamicModel ................................. 46 3.3ControlObjective ................................ 46 3.4FeedforwardNNEstimation .......................... 46 3.5RISEFeedbackControlDevelopment ..................... 49 3.5.1Open-LoopErrorSystem ........................ 49 3.5.2Closed-LoopErrorSystem ....................... 50 3.6StabilityAnalysis ................................ 54 3.7Experiment ................................... 58 3.7.1Discussion ................................ 59 3.8Conclusions ................................... 61 4ANEWCLASSOFMODULARADAPTIVECONTROLLERS ......... 63 4.1Introduction ................................... 63 4.2DynamicSystem ................................ 65 5

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4.3ControlObjective ................................ 66 4.4ControlDevelopment .............................. 67 4.5ModularAdaptiveUpdateLawDevelopment ................. 69 4.6StabilityAnalysis ................................ 73 4.7NeuralNetworkExtensiontoNon-LPSystems ................ 77 4.7.1RISEFeedbackControlDevelopment ................. 77 4.7.2ModularTuningLawDevelopment .................. 79 4.8StabilityAnalysis ................................ 82 4.9ApplicationtoEuler-LagrangeSystems .................... 82 4.10Experiment ................................... 83 4.10.1ModularAdaptiveUpdateLaw .................... 85 4.10.2ModularNeuralNetworkUpdateLaw ................. 86 4.10.3Discussion ................................ 92 4.11Conclusion .................................... 94 5COMPOSITEADAPTIVECONTROLFORSYSTEMSWITHADDITIVE DISTURBANCES .................................. 95 5.1Introduction ................................... 95 5.2DynamicSystem ................................ 96 5.3ControlObjective ................................ 97 5.4ControlDevelopment .............................. 98 5.4.1RISE-basedSwapping .......................... 100 5.4.2CompositeAdaptation ......................... 103 5.4.3Closed-LoopPredictionErrorSystem ................. 103 5.4.4Closed-LoopTrackingErrorSystem .................. 105 5.5StabilityAnalysis ............................... 108 5.6Experiment ................................... 113 5.6.1Discussion ................................ 117 5.7Conclusion .................................... 118 6COMPOSITEADAPTATIONFORNN-BASEDCONTROLLERS ....... 119 6.1Introduction ................................... 119 6.2DynamicSystem ................................ 120 6.3ControlObjective ................................ 121 6.4ControlDevelopment .............................. 122 6.4.1Swapping ................................. 124 6.4.2CompositeAdaptation ......................... 130 6.4.3Closed-LoopErrorSystem ....................... 131 6.5StabilityAnalysis ................................ 135 6.6Experiment ................................... 139 6.6.1Discussion ................................ 140 6.7Conclusion .................................... 142 6

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7CONCLUSIONSANDFUTUREWORK ...................... 144 7.1Conclusions ................................... 144 7.2FutureWork ................................... 146 APPENDIX ......................................... 147 REFERENCES ....................................... 157 BIOGRAPHICALSKETCH ................................ 163 7

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LISTOFTABLES Table page 2-1t-test:twosamplesassumingequalvariancesforRMSerr or ........... 41 2-2t-test:twosamplesassumingequalvariancesforRMStor que ........... 42 4-1LPcase:AverageRMSvaluesfor10trials ..................... 86 4-2Non-LPcase:AverageRMSvaluesfor10trials .................. 90 8

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LISTOFFIGURES Figure page 2-1Theexperimentaltestbedconsistsofacirculardiskmou ntedonaNSKdirect-drive switchedreluctancemotor. .............................. 33 2-2Desiredtrajectoryusedfortheexperiment. ..................... 34 2-3Positiontrackingerrorwithouttheadaptivefeedforwa rdterm. .......... 36 2-4Torqueinputwithouttheadaptivefeedforwardterm. ............... 37 2-5Positiontrackingerrorforthecontrolstructurethati ncludestheadaptivefeedforward term. .......................................... 37 2-6Parameterestimatesoftheadaptivefeedforwardcompon ent:(a)^ r 1 ,(b)^ r 4 ,(c) ^ r 6 ,(d) ^ J ....................................... 38 2-7Torqueinputforthecontrolstructurethatincludesthe adaptivefeedforward term. .......................................... 38 2-8ThecontributionoftheRISEtermforthecontrolstructu rethatincludesthe adaptivefeedforwardterm. .............................. 39 2-9RMSpositiontrackingerrorsandtorquesforthetwocase s-(1)withoutthe adaptationterminthecontrolinput,(2)withtheadaptatio nterminthecontrol input. ......................................... 41 3-1TrackingerrorfortheRISEcontrollawwithnoNNadaptat ion. ......... 59 3-2ControltorquefortheRISEcontrollawwithnoNNadaptat ion. ......... 60 3-3TrackingerrorfortheproposedRISE+NNcontrollaw. .............. 60 3-4ControltorquefortheproposedRISE+NNcontrollaw. .............. 61 3-5AverageRMSerrors(degrees)andtorques(N-m).1-RISE, 2-RISE+NN(proposed). 61 4-1Theexperimentaltestbedconsistsofatwo-linkrobot.T helinksaremounted ontwoNSKdirect-driveswitchedreluctancemotors. ............... 83 4-2Linkpositiontrackingerrorwithagradient-basedadap tiveupdatelaw. ..... 87 4-3Torqueinputforthemodularadaptivecontrollerwithag radient-basedadaptive updatelaw. ...................................... 87 4-4Adaptiveestimatesforthegradientupdatelaw. .................. 88 4-5Linkpositiontrackingerrorwithaleast-squaresadapt iveupdatelaw. ...... 88 9

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4-6Torqueinputforthemodularadaptivecontrollerwithal east-squaresadaptive updatelaw. ...................................... 89 4-7Adaptiveestimatesfortheleastsquaresupdatelaw. ............... 89 4-8LinkpositiontrackingerrorforthemodularNNcontroll erwithagradient-based tuninglaw. ...................................... 91 4-9TorqueinputforthemodularNNcontrollerwithagradien t-basedtuninglaw. 91 4-10LinkpositiontrackingerrorforthemodularNNcontrol lerwithaHebbiantuning law. .......................................... 92 4-11TorqueinputforthemodularNNcontrollerwithaHebbia ntuninglaw. ..... 92 5-1BlockdiagramoftheproposedRISE-basedcompositeadap tivecontroller .... 106 5-2Actualanddesiredtrajectoriesfortheproposedcompos iteadaptivecontrollaw (RISE+CFF). ..................................... 114 5-3Trackingerrorfortheproposedcompositeadaptivecont rollaw(RISE+CFF). 114 5-4Predictionerrorfortheproposedcompositeadaptiveco ntrollaw(RISE+CFF). 115 5-5Controltorquefortheproposedcompositeadaptivecont rollaw(RISE+CFF). 115 5-6ContributionoftheRISEtermintheproposedcompositea daptivecontrollaw (RISE+CFF). ..................................... 116 5-7Adaptiveestimatesfortheproposedcompositeadaptive controllaw(RISE+CFF). 116 5-8AverageRMSerrors(degrees)andtorques(N-m).1-RISE, 2-RISE+FF,3RISE+CFF(proposed). ............................... 117 6-1BlockdiagramoftheproposedRISE-basedcompositeNNco ntroller. ...... 128 6-2Trackingerrorfortheproposedcompositeadaptivecont rollaw(RISE+CNN). 140 6-3Predictionerrorfortheproposedcompositeadaptiveco ntrollaw(RISE+CNN). 141 6-4Controltorquefortheproposedcompositeadaptivecont rollaw(RISE+CNN). 141 6-5AverageRMSerrors(degrees)andtorques(N-m).1-RISE, 2-RISE+NN,3RISE+CNN(proposed). ............................... 142 10

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy LYAPUNOV-BASEDROBUSTANDADAPTIVECONTROLOFNONLINEAR SYSTEMSUSINGANOVELFEEDBACKSTRUCTURE By ParagPatre August2009 Chair:WarrenE.DixonMajor:MechanicalEngineering Thefocusofthisresearchisanexaminationoftheinterplay betweendierent intelligentfeedforwardmechanismswitharecentlydevelo pedcontinuousrobustfeedback mechanism,coinedRobustIntegraloftheSignoftheError(R ISE),toyieldasymptotic trackinginthepresenceofgenericdisturbances.Thisresu ltsolvesadecadeslong openproblemofhowtoobtainasymptoticstabilityofnonlin earsystemswithgeneral sucientlysmoothdisturbanceswithacontinuouscontrolm ethod.Further,itisshown thatthedevelopedtechniquecanbefusedwithotherfeedfor wardmethodssuchas functionapproximationandadaptivecontrolmethods.Thea dditionoffeedforward elementsaddssystemknowledgeinthecontrolstructure,wh ichheuristically,yieldsbetter performanceandreducescontroleort.Thisheuristicnoti onissupportedbyexperimental resultsinthisresearch. Onekeyelementinthedevelopmentofthenovelfeedforwardm echanismspresented inthisdissertationisthemodularitybetweenthecontroll erandupdatelaw.This modularityprovidesrexibilityintheselectionofdieren tupdatelawsthatcouldbe easiertoimplementorhelptoachievefasterparameterconv ergenceandbettertracking performance. Theecacyofthefeedforwardmechanismsisfurtherenhance dbyincludinga predictionerrorinthelearningprocess.Thepredictioner ror,whichdirectlyrelatesto 11

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theactualfunctionmismatch,isusedalongwiththesystemt rackingerrorstodevelopa compositeadaptationlaw. EachresultissupportedthroughrigorousLyapunov-baseds tabilityproofsand experimentaldemonstrations. 12

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CHAPTER1 INTRODUCTION 1.1MotivationandProblemStatement Thecontrolofsystemswithuncertainnonlineardynamicsha sbeenadecades longmainstreamareaoffocus.Forsystemswithuncertainti esthatcanbelinear parameterized,avarietyofadaptive(e.g.,[ 1 { 3 ])feedforwardcontrollerscanbeutilized toachieveanasymptoticresult.Somerecentresultshaveal sotargetedtheapplication ofadaptivecontrollersforsystemsthatarenotlinearinth eparameters[ 4 ].Learning controllershavebeendevelopedforsystemswithperiodicd isturbances[ 5 { 7 ],andrecent researchhasfocusedontheuseofexosystems[ 8 { 11 ]tocompensatefordisturbances thatarethesolutionofalineartime-invariantsystemwith unknowncoecients.A varietyofmethodshavealsobeenproposedtocompensatefor systemswithunstructured uncertaintyincluding:variousslidingmodecontrollers( e.g.,[ 3 12 ]),robustcontrol schemes[ 13 ],andneuralnetworkandfuzzylogiccontrollers[ 14 { 18 ].Fromareviewof theseapproaches,ageneraltrendisthatcontrollersdevel opedforsystemswithmore unstructureduncertaintywillrequiremorecontroleort( i.e.,highgainorhighfrequency feedback)andyieldreducedperformance(e.g.,uniformlyu ltimatelyboundedstability). Recentlyanewrobustcontrolstrategy,coinedRobustInteg raloftheSignofthe Error(RISE)in[ 19 20 ],wasdevelopedin[ 21 22 ]thatcanaccommodateforsuciently smoothboundeddisturbances.Asignicantoutcomeofthisn ewcontrolstructureis thatasymptoticstabilityisobtaineddespiteafairlygene raluncertaindisturbance.This techniquewasusedin[ 23 ]todevelopatrackingcontrollerfornonlinearsystemsint he presenceofadditivedisturbancesandparametricuncertai ntiesundertheassumption thatthedisturbancesare C 2 withboundedtimederivatives.In[ 24 ],Xianetal.utilized thisstrategytoproposeanewoutputfeedbackdiscontinuou strackingcontrollerfora generalclassofsecond-ordernonlinearsystemswhoseunce rtaindynamicsarerst-order dierentiable.In[ 25 ],Zhangetal.combinedthehighgainfeedbackstructurewit ha 13

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highgainobserveratthesacriceofyieldingasemi-global uniformlyultimatelybounded result.Thisparticularhighgainfeedbackmethodhasalsob eenusedasanidentication technique.Forexample,themethodhasbeenappliedtoident ifyfriction(e.g.,[ 26 ]), forrangeidenticationinperspectiveandparacatadioptr icvisionsystems(e.g.,[ 27 ], [ 28 ]),andforfaultdetectionandidentication(e.g.,[ 29 ]).ThedevelopmentinChapter 2ismotivatedbythedesiretoincludesomeknowledgeofthed ynamicsinthecontrol designasameanstoimprovetheperformanceandreducetheco ntroleort.Specically, forsystemsthatincludesomedynamicsthatcanbesegregate dintostructured(i.e., linearparameterizable)andunstructureduncertainty,th ischapterillustrateshowanew controller,errorsystem,andstabilityanalysiscanbecra ftedtoincludeamodel-based adaptivefeedforwardterminconjunctionwiththehighgain RISEfeedbacktechniqueto yieldanasymptotictrackingresult. Anotherlearningmethodthathasbeenextensivelyinvestig atedbycontrolresearchers overthelastfteenyearsistheuseofneuralnetworks(NNs) asafeedforwardelement inthecontrolstructure.ThefocusonNN-basedcontrolmeth odsisspawnedfromthe ramicationofthefactthatNNsareuniversalapproximator s[ 30 ].Thatis,NNscanbe usedasablack-boxestimatorforageneralclassofsystems. Examplesinclude:nonlinear systemswithparametricuncertaintythatdonotsatisfythe linear-in-the-parameters assumptionrequiredinmostadaptivecontrolmethods;syst emswithdeadzonesor discontinuities;andsystemswithbacklash.Typically,NN -basedcontrollersyieldglobal uniformlyultimatelybounded(UUB)stabilityresults(e.g .,see[ 15 31 32 ]forexamples andreviewsofliterature)duetoresidualfunctionalrecon structioninaccuraciesand aninabilitytocompensateforsomesystemdisturbances.Mo tivatedbythedesireto eliminatetheresidualsteady-stateerrors,severalresea rchershaveobtainedasymptotic trackingresultsbycombiningtheNNfeedforwardelementwi thdiscontinuousfeedback methodssuchasvariablestructurecontrollers(VSC)(e.g. ,[ 33 34 ])orslidingmode(SM) controllers(e.g.,[ 34 35 ]).AcleverVSC-likecontrollerwasalsoproposedin[ 36 ]where 14

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thecontrollerisnotinitiallydiscontinuous,butexponen tiallybecomesdiscontinuousas anexogenouscontrolelementexponentiallyvanishes.Well knownlimitationsofVSC andSMcontrollersincludearequirementforinnitecontro lbandwidthandchattering. Unfortunately,adhocxesfortheseeectsresultinalosso fasymptoticstability(i.e., UUBtypicallyresults).Motivatedbyissuesassociatedwit hdiscontinuouscontrollers andthetypicalUUBstabilityresult,aninnovativecontinu ousNN-basedcontrollerwas recentlydevelopedin[ 37 ]toachievepartialasymptoticstabilityforaparticularc lassof systems.TheresultinChapter3ismotivatedbythequestion : CanaNNfeedforward controllerbemodiedbyacontinuousfeedbackelementtoac hieveanasymptotictracking resultforageneralclassofsystems? DespitethepervasivedevelopmentofNNcontrollers inliteratureandthewidespreaduseofNNsinindustrialapp lications,theanswerto thisfundamentalquestionhasremainedanopenproblem.Top rovideananswerto thefundamentalmotivatingquestion,theresultinChapter 3focusesonaugmentinga multi-layerNN-basedfeedforwardmethodwiththeRISEcont rolstrategy Likemostoftheresearchinadaptivecontrol,theresultsin Chapter2and3exploit Lyapunov-basedtechniques(i.e.,thecontrollerandthead aptiveupdatelawaredesigned basedonaLyapunovanalysis);however,Lyapunov-basedmet hodsrestrictthedesign oftheadaptiveupdatelaw.Forexample,manyoftheprevious adaptivecontrollersare restrictedtoutilizinggradientupdatelawstocancelcros stermsinaLyapunov-based stabilityanalysis.Gradientupdatelawscanpotentiallye xhibitslowerparameter convergencewhichcouldleadtoadegradedtransientperfor manceofthetrackingerrorin comparisontootherpossibleadaptiveupdatelaws(e.g.,le ast-squaresupdatelaw).Several resultshavebeendevelopedinliteraturethataimtoaugmen tthetypicalposition/velocity trackingerror-basedgradientupdatelawincluding:compo siteadaptiveupdatelaws [ 3 38 39 ];predictionerror-basedupdatelaws[ 1 40 { 43 ];andvariousleast-squares updatelaws[ 44 { 46 ].Theadaptiveupdatelawintheseresultsareallstilldesi gnedto cancelcrosstermsintheLyapunov-basedstabilityanalysi s.Incontrasttotheseresults, 15

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researchershavealsodevelopedaclassofmodularadaptive controllers(cf.[ 1 40 42 43 ]) whereafeedbackmechanismisusedtostabilizetheerrordyn amicsprovidedcertain conditionsaresatisedontheadaptiveupdatelaw.Forexam ple,nonlineardamping [ 41 47 ]istypicallyusedtoyieldaninput-to-statestability(IS S)resultwithrespectto theparameterestimationerrorwhereitisassumedapriorit hattheupdatelawyields boundedparameterestimates.Oftenthemodularadaptiveco ntroldevelopmentexploitsa predictionerrorintheupdatelaw(e.g.,see[ 3 40 { 43 ]),wherethepredictionerrorisoften requiredtobesquareintegrable(e.g.,[ 40 42 43 ]).Abriefsurveyofmodularadaptive controlresultsisprovidedin[ 40 ].SincetheRISEfeedbackmechanismalonecanyieldan asymptoticresultwithoutafeedforwardcomponenttocance lcrosstermsinthestability analysis,theresearchinChapter4ismotivatedbythefollo wingquestion: CantheRISE controlmethodbeusedtoyieldanewclassofmodularadaptiv econtrollers? Typicaladaptive,robustadaptive,andfunctionapproxima tionmethodsandthe onesusedinChapters2-4usetrackingerrorfeedbacktoupda tetheadaptiveestimates. Asmentionedearlier,theuseofthetrackingerrorismotiva tedbytheneedforthe adaptiveupdatelawtocancelcross-termsintheclosed-loo ptrackingerrorsystemwithin aLyapunov-basedanalysis.Asthetrackingerrorconverges ,therateoftheupdate lawalsoconverges,butdrawingconclusionsabouttheconve rgentvalue(ifany)ofthe parameterupdatelawisproblematic.Ideally,theadaptive updatelawwouldinclude someestimateoftheparameterestimationerrorasameansto provetheparameter estimatesconvergetotheactualvalues;however,theparam eterestimateerroris unknown.Thedesiretoincludesomemeasurableformofthepa rameterestimation errorintheadaptationlawresultedinthedevelopmentofad aptiveupdatelawsthat aredriven,inpart,byapredictionerror[ 3 42 48 { 50 ].Thepredictionerrorisdened asthedierencebetweenthepredictedparameterestimatev alueandtheactualsystem uncertainty.Includingfeedbackoftheestimationerrorin theadaptiveupdatelawenables improvedparameterestimation.Forexample,someclassicr esults[ 1 3 41 ]haveproven 16

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theparameterestimationerrorissquareintegrableandtha ttheparameterestimates mayconvergetotheactualuncertainparameters.Sincethep redictionerrordepends ontheunmeasurablesystemuncertainty,theswappinglemma [ 3 42 48 { 50 ]iscentral tothepredictionerrorformulation.Theswappingtechniqu e(alsodescribedasinputor torquelteringinsomeliterature)transformsadynamicpa rametricmodelintoastatic formwherestandardparameterestimationtechniquescanbe applied.In[ 1 ]and[ 41 ],a nonlinearextensionoftheswappinglemmawasderived,whic hwasusedtodevelopthe modularz-swappingandx-swappingidentiersviaaninputto-statestable(ISS)controller forsystemsinparametricstrictfeedbackform.Theadvanta gesprovidedbyprediction errorbasedadaptiveupdatelawsledtoseveralresultsthat useeithertheprediction errororacompositeofthepredictionerrorandthetracking error(cf.[ 43 51 { 56 ]andthe referenceswithin).Althoughpredictionerrorbasedadapt iveupdatelawshaveexisted forapproximatelytwodecades,nostabilityresulthasbeen developedforsystemswith additiveboundeddisturbances.Ingeneral,theinclusiono fdisturbancesreducesthe steady-stateperformanceofcontinuouscontrollerstoaun iformlyultimatelybounded (UUB)result.InadditiontoaUUBresult,theinclusionofdi sturbancesmaycause unboundedgrowthoftheparameterestimates[ 57 ]fortrackingerror-basedadaptive updatelawswithouttheuseofprojectionalgorithmsorothe rupdatelawmodications suchas -modication[ 58 ].Problemsassociatedwiththeinclusionofdisturbancesa re magniedforcontrolmethodsbasedonpredictionerror-bas edupdatelawsbecausethe formulationofthepredictionerrorrequirestheswapping( orcontrolltering)method. Applyingtheswappingapproachtodynamicswithadditivedi sturbancesisproblematic becausetheunknowndisturbancetermsalsogetlteredandi ncludedintheltered controlinput. Thisproblemmotivatesthequestionofhowcanapredictione rror-based adaptiveupdatelawbedevelopedforsystemswithadditived isturbances. Toaddressthis motivatingquestion,ageneralEuler-Lagrange-likeMIMOs ystemisconsideredinChapter 17

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5withstructuredandunstructureduncertainties,andagra dient-basedcompositeadaptive updatelawisdevelopedthatisdrivenbyboththetrackinger rorandthepredictionerror. Theswappingprocedureusedinstandardadaptivecontrolca nnotbeextendedto NNcontrollersdirectly.ThepresenceofaNNreconstructio nerrorhasimpededthe developmentofcompositeadaptationlawsforNNs.Specica lly,thereconstruction errorgetslteredandincludedinthepredictionerrordest royingthetypicalprediction errorformulation.UsingthetechniquesdevelopedinChapt er5,theresultinChapter6 presentsthersteverattempttodevelopapredictionerror -basedcompositeadaptiveNN controllerforanEuler-Lagrangesecond-orderdynamicsys temusingtheRISEfeedback. AusualconcernabouttheRISEfeedbackisthepresenceofhig h-gainandhigh-frequency componentsinthecontrolstructure.However,incontrastt oatypicalwidelyused discontinuoushigh-gainslidingmodecontroller,theRISE feedbackoersacontinuous alternative.Moreover,theproposedcontroldesignsareno tpurelyhighgainasan adaptiveelementisusedasafeedforwardcomponentthatlea rnsandincorporatesthe knowledgeofsystemdynamicsinthecontrolstructure. 1.2Contributions Thisresearchfocusesoncombiningvariousfeedforwardter mswiththeRISEfeedback methodforthecontrolofuncertainnonlineardynamicsyste ms.Thecontributionsof Chapters2-6areasfollows. Chapter2: AsymptoticTrackingforSystemswithStructuredandUnstru ctured Uncertainties :Thedevelopmentinthischapterismotivatedbythedesiret oincludesome knowledgeofthedynamicsinthecontroldesignasameanstoi mprovetheperformance andreducethecontroleort.Specically,forsystemsthat includesomedynamics thatcanbesegregatedintostructured(i.e.,linearparame terizable)andunstructured uncertainty,thischapterillustrateshowanewcontroller ,errorsystem,andstability analysiscanbecraftedtoincludeamodel-basedadaptivefe edforwardterminconjunction withtheRISEfeedbacktechniquetoyieldanasymptotictrac kingresult.Thischapter 18

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presentstherstresultthatillustrateshowtheamalgamat ionofthesecompensation methodscanbeusedtoyieldanasymptoticresult. Chapter3: AsymptoticTrackingforUncertainDynamicSystemsviaaMul tilayer NeuralNetworkFeedforwardandRISEFeedbackControlStruc ture :TheuseofaNNas afeedforwardcontrolelementtocompensatefornonlinears ystemuncertaintieshasbeen investigatedforoveradecade.TypicalNN-basedcontrolle rsyielduniformlyultimately bounded(UUB)stabilityresultsduetoresidualfunctional reconstructioninaccuracies andaninabilitytocompensateforsomesystemdisturbances .Severalresearchershave proposeddiscontinuousfeedbackcontrollers(e.g.,varia blestructureorslidingmode controllers)torejecttheresidualerrorsandyieldasympt oticresults.Theresearchin thischapterdescribeshowtheRISEfeedbacktermcanbeinco rporatedwithaNN-based feedforwardtermtoachievethersteverasymptotictracki ngresult.Toachievethis result,thetypicalstabilityanalysisfortheRISEmethodi smodiedtoenablethe incorporationoftheNN-basedfeedforwardterms,andaproj ectionalgorithmisdeveloped toguaranteeboundedNNweightestimates.Experimentalres ultsarepresentedto demonstratetheperformanceoftheproposedcontroller. Chapter4: ANewClassofModularAdaptiveControllers :Anoveladaptivenonlinear controldesignisdevelopedwhichachievesmodularitybetw eenthecontrollerandthe adaptiveupdatelaw.Modularitybetweenthecontroller/up datelawdesignprovides rexibilityintheselectionofdierentupdatelawsthatcou ldpotentiallybeeasier toimplementorusedtoobtainfasterparameterconvergence and/orbettertracking performance.Forageneralclassoflinear-in-the-paramet ers(LP)uncertainmulti-input multi-outputsystemssubjecttoadditiveboundednon-LPdi sturbances,thedeveloped controllerusesamodel-basedfeedforwardadaptivetermin conjunctionwiththe RISEfeedbackterm.Modularityintheadaptivefeedforward termismadepossible byconsideringagenericformoftheadaptiveupdatelawandi tscorrespondingparameter estimate.Thisgenericformoftheupdatelawisusedtodevel opanewclosed-looperror 19

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systemandstabilityanalysisthatdoesnotdependonnonlin eardampingtoyieldthe modularadaptivecontrolresult.Theresultisthenextende dbyconsideringuncertain dynamicsystemsthatarenotnecessarilyLP,andhaveadditi venon-LPbounded disturbances.AmultilayerNNstructureisusedinthenon-L Pextensionasafeedforward elementtocompensateforthenon-LPdynamicsinconjunctio nwiththeRISEfeedback term.ANN-basedcontrollerisdevelopedwithmodularityin NNweighttuninglawsand thecontrollaw.Anextensionisprovidedthatdescribeshow thecontroldevelopment forthegeneralclassofsystemscanbeappliedtoaclassofdy namicsystemsmodeledby Euler-Lagrangeformulation.Experimentalresultsonatwo -linkrobotareincludedto illustratetheconcept. Chapter5: CompositeAdaptiveControlforSystemswithAdditiveDistu rbances :In atypicaladaptiveupdatelaw,therateofadaptationisgene rallyafunctionofthestate feedbackerror.Ideally,theadaptiveupdatelawwouldalso includesomefeedbackofthe parameterestimationerror.Thedesiretoincludesomemeas urableformoftheparameter estimationerrorintheadaptationlawresultedinthedevel opmentofcompositeadaptive updatelawsthatarefunctionsofapredictionerrorandthes tatefeedback.Inallprevious compositeadaptivecontrollers,theformulationofthepre dictionerrorispredicatedon thecriticalassumptionthatthesystemuncertaintyisline arintheuncertainparameters (LPuncertainty).Thepresenceofadditivedisturbancesth atarenotLPwoulddestroy thepredictionerrorformulationandstabilityanalysisar gumentsinpreviousresults.In thischapter,anewpredictionerrorformulationisconstru ctedthroughtheuseofthe RISEtechnique.Thecontributionofthisdesignandassocia tedstabilityanalysisisthat thepredictionerrorcanbedevelopedevenwithdisturbance sthatdonotsatisfythe LPassumption(e.g.,additiveboundeddisturbances).Acom positeadaptivecontroller isdevelopedforageneralMIMOEuler-Lagrangesystemwithm ixedstructured(i.e., LP)andunstructureduncertainties.ALyapunov-basedstab ilityanalysisisusedto 20

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derivesucientgainconditionsunderwhichtheproposedco ntrolleryieldssemi-global asymptotictracking.Experimentalresultsarepresentedt oillustratetheapproach. Chapter6: CompositeAdaptationforNN-BasedControllers :Withthemotivation ofusingmoreinformationtoupdatetheparameterestimates ,compositeadaptation thatusesboththesystemtrackingerrorsandapredictioner rorcontainingparametric informationtodrivetheupdatelaws,hasbecomewidespread inadaptivecontrol literature.However,despiteitsobviousbenets,composi teadaptationhasnotbeen implementedinNN-basedcontrol,primarilyduetotheNNrec onstructionerrorthat destroysatypicalpredictionerrorformulationrequiredf orthecompositeadaptation. Thischapterpresentsthersteverattempttodesignacompo siteadaptationlawfor NNsbydevisinganinnovativeswappingprocedurethatusest heRISEfeedbackmethod. Semi-globalasymptotictrackingforthesystemerrorsispr oved,whileallothersignalsand controlinputareshowntobebounded. 21

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CHAPTER2 ASYMPTOTICTRACKINGFORSYSTEMSWITHSTRUCTUREDAND UNSTRUCTUREDUNCERTAINTIES 2.1Introduction Thedevelopmentinthischapterismotivatedbythedesireto includesomeknowledge ofthedynamicsinthecontroldesignasameanstoimprovethe performanceandreduce thecontroleort.Specically,forsystemsthatincludeso medynamicsthatcanbe segregatedintostructured(i.e.,linearparameterizable )andunstructureduncertainty,this chapterillustrateshowanewcontrolleranderrorsystemca nbecraftedtoinclude amodel-basedadaptivefeedforwardterminconjunctionwit htheRISEfeedback techniquetoyieldanasymptotictrackingresult.Thischap terpresentstherstresult thatillustrateshowtheamalgamationofthesecompensatio nmethodscanbeused toyieldanasymptoticresult.Heuristically,theaddition ofthemodel-basedadaptive feedforwardtermshouldreducetheoverallcontroleortbe causesomeofthedisturbance hasbeenisolatedandcompensatedforbyanon-highgainfeed forwardelement.Moreover, theadditionoftheadaptivefeedforwardterminjectssomek nowledgeofthedynamics inthecontrolstructure,whichleadstoimprovedperforman ce.Experimentalresults arepresentedtoreinforcetheseheuristicnotions.Speci cally,thepresentedcontroller wasimplementedonasimplerotatingcirculardisktestbeda nddemonstratedreduced trackingerrorandcontroleort.Forthistestbed,thedyna micsthatwereincludedinthe feedforwardtermincludedtheinertiaofthelinkageassemb ly,andthefrictionpresentin thesystem. 2.2DynamicModel Theclassofnonlineardynamicsystemsconsideredinthisma nuscriptisassumedto bemodeledbythefollowingEuler-Lagrangeformulationtha tdescribesthebehaviorofa largeclassofengineeringsystems: M ( q ) q + V m ( q; q )_ q + G ( q )+ f (_ q )+ d ( t )= ( t ) : (2{1) 22

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In( 2{1 ), M ( q ) 2 R n n denotesageneralizedinertiamatrix, V m ( q; q ) 2 R n n denotesa generalizedcentripetal-Coriolismatrix, G ( q ) 2 R n denotesageneralizedgravityvector, f (_ q ) 2 R n denotesageneralizedfrictionvector, d ( t ) 2 R n denotesageneralizednonlinear disturbance(e.g.,unmodeledeects), ( t ) 2 R n representsthegeneralizedtorqueinput controlvector,and q ( t ),_ q ( t ), q ( t ) 2 R n denotethegeneralizedlinkposition,velocity,and accelerationvectors,respectively.Thefrictionterm f (_ q )in( 2{1 )isassumedtohavethe followingform[ 26 ]and[ 59 ]: f (_ q )= r 1 (tanh( r 2 q ) tanh( r 3 q ))+ r 4 tanh( r 5 q )+ r 6 q; (2{2) where r i 2 R 8 i =1 ; 2 ;:::; 6denoteunknownpositiveconstants.Thefrictionmodel in( 2{2 )hasthefollowingproperties:1)itissymmetricabouttheo rigin,2)ithasa staticcoecientoffriction,3)itexhibitstheStribecke ectwherethefrictioncoecient decreasesfromthestaticcoecientoffrictionwithincrea singslipvelocityneartheorigin, 4)itincludesaviscousdissipationterm,and5)ithasaCoul ombicfrictioncoecientin theabsenceofviscousdissipation.Toagoodapproximation ,thestaticfrictioncoecient isgivenby r 1 + r 4 ,andtheStribeckeectiscapturedbytanh( r 2 q ) tanh( r 3 q ).The Coulombicfrictioncoecientisgivenby r 4 tanh( r 5 q ),andtheviscousdissipationisgiven by r 6 q .Forfurtherdetailsregardingthefrictionmodel,see[ 26 ]and[ 59 ].Thesubsequent developmentisbasedontheassumptionthat q ( t )and_ q ( t )aremeasurableandthat M ( q ), V m ( q; q ), G ( q ), f (_ q ),and d ( t )areunknown.Theerrorsystemsystemsutilizedinthis manuscriptassumethatthegeneralizedcoordinates, q ( t ),oftheEuler-Lagrangedynamics in( 2{1 )allowadditiveandnotmultiplicativeerrors.Moreover,t hefollowingassumptions willbeexploitedinthesubsequentdevelopment:Assumption2-1:SymmetricandPositive-DeniteInertiaMa trix Theinertiamatrix M ( q )issymmetric,positivedenite,andsatisesthefollowin g inequality 8 ( t ) 2 R n : m 1 k k 2 T M ( q ) m ( q ) k k 2 (2{3) 23

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where m 1 2 R isaknownpositiveconstant, m ( q ) 2 R isaknownpositivefunction,and kk denotesthestandardEuclideannorm. Assumption2-2: If q ( t ) ; q ( t ) 2L 1 ,then V m ( q; q ), F (_ q )and G ( q )arebounded. Moreover,if q ( t ) ; q ( t ) 2L 1 ,thentherstandsecondpartialderivativesoftheelement sof M ( q ), V m ( q; q ), G ( q )withrespectto q ( t )existandarebounded,andtherstandsecond partialderivativesoftheelementsof V m ( q; q ), F (_ q )withrespectto_ q ( t )existandare bounded.Assumption2-3: Thenonlineardisturbancetermanditsrsttwotimederivat ives,i.e. d ( t ) ; d ( t ) ; d ( t )areboundedbyknownconstants. Assumption2-4: Thedesiredtrajectoryisdesignedsuchthat q ( i ) d ( t ) 2 R n ( i =0 ; 1 ;:::; 4) existandarebounded. 2.3ErrorSystemDevelopment Thecontrolobjectiveistoensurethatthesystemtracksade siredtime-varying trajectorydespitestructuredandunstructureduncertain tiesinthedynamicmodel.To quantifythisobjective,apositiontrackingerror,denote dby e 1 ( t ) 2 R n ,isdenedas e 1 = q d q: (2{4) Tofacilitatethesubsequentanalysis,lteredtrackinger rors,denotedby e 2 ( t ), r ( t ) 2 R n arealsodenedas e 2 =_ e 1 + 1 e 1 (2{5) r =_ e 2 + 2 e 2 ; (2{6) where 1 2 2 R denotepositiveconstants.Thesubsequentdevelopmentisb asedonthe assumptionthat q ( t )and_ q ( t )aremeasurable,sothelteredtrackingerror r ( t )isnot measurablesincetheexpressionin( 2{6 )dependson q ( t ). 24

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Theopen-looptrackingerrorsystemcanbedevelopedbyprem ultiplying( 2{6 )by M ( q )andutilizingtheexpressionsin( 2{1 )-( 2{5 )toobtainthefollowingexpression: M ( q ) r = Y d + S + W d + d ( t ) ( t ) ; (2{7) where Y d ( q d ; q d ; q d ) 2 R n isdenedas Y d M ( q d ) q d + V m ( q d ; q d )_ q d + G ( q d )+ r 1 (tanh( r 2 q d ) tanh( r 3 q d ))(2{8) + r 4 tanh( r 5 q d )+ r 6 q d : In( 2{8 ), 2 R p containstheconstantunknownsystemparameters, Y d ( q d ; q d ; q d ) 2 R n p isthedesiredregressionmatrixthatcontainsknownfuncti onsofthedesiredlinkposition, velocity,andacceleration, q d ( t ) ; q d ( t ) ; q d ( t ) 2 R n ,respectively,and r 2 r 3 r 5 2 R are thebestguessestimatesfor r 2 r 3 ,and r 5 ,respectively.In( 2{7 ),theauxiliaryfunction S ( q; q;q d ; q d ; q d ) 2 R n isdenedas S M ( q )( 1 e 1 + 2 e 2 )+ M ( q ) q d M ( q d ) q d + V m ( q; q )_ q V m ( q d ; q d )_ q d + G ( q ) G ( q d ) + r 6 q r 6 q d + r 4 tanh( r 5 q ) r 4 tanh( r 5 q d )+ r 1 (tanh( r 2 q ) tanh( r 3 q ))(2{9) r 1 (tanh( r 2 q d ) tanh( r 3 q d )) ; andtheauxiliaryfunction W d (_ q d ) 2 R n isdenedas W d r 4 tanh( r 5 q d ) r 4 tanh( r 5 q d )+ r 1 (tanh( r 2 q d ) tanh( r 3 q d ))(2{10) r 1 (tanh( r 2 q d ) tanh( r 3 q d )) : Basedontheexpressionin( 2{7 ),thecontroltorqueinputisdesignedas = Y d ^ + : (2{11) In( 2{11 ), ( t ) 2 R n denotestheRISEtermdenedas ( t ) ( k s +1) e 2 ( t ) ( k s +1) e 2 (0)+ Z t 0 [( k s +1) 2 e 2 ( )+ sgn ( e 2 ( ))] d; (2{12) 25

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where k s ; 2 R arepositive,constantcontrolgains,and ^ ( t ) 2 R p denotesaparameter estimatevectorgeneratedon-lineaccordingtothefollowi ngupdatelaw: ^ = Y T d r (2{13) with 2 R p p beingaknown,constant,diagonal,positive-deniteadapt ationgainmatrix. Since Y d ( t )isonlyafunctionoftheknowndesiredtimevaryingtraject ory,( 2{13 )canbe integratedbypartsasfollows: ^ ( t )= ^ (0)+ Y T d e 2 ( ) t0 Z t 0 n Y T d e 2 ( ) 2 Y T d e 2 ( ) o d (2{14) sothattheparameterestimatevector ^ ( t )implementedin( 2{11 )doesnotdependonthe unmeasurablesignal r ( t ). Remark2.1. Thecontroldesignin( 2{11 )issimilartotheresultsin[ 22 ].However, previousdesignsbasedon[ 22 ]couldonlycompensateforuncertaintyinthesystemthroug h thehighgainRISEfeedbackterm ( t ) .Throughthenewdevelopmentpresentedinthe currentresult,anadaptivefeedforwardtermcanalsobeuse dtocompensateforsystem uncertainty.Thisrexibilitypresentsasignicantadvant agebecauseitallowsmoresystem dynamicstobeincorporatedinthecontroldesign.Specica lly,ifsomeofthesystem uncertaintycanbesegregatedintoalinearparameterizabl eform,thenthemodel-based adaptivefeedforwardtermcanbeinjectedtocompensatefor theuncertaintyinsteadof justrelyingonthenon-modelbasedhighgainRISEfeedbackt erm.Heuristically,this contributionshouldimprovethetrackingperformanceandr educethecontroleort. Experimentalresultsonasimpleone-linkrobotmanipulato rprovidesomevalidationofthis heuristicidea. Theclosed-looptrackingerrorsystemcanbedevelopedbysu bstituting( 2{11 )into ( 2{7 )as M ( q ) r = Y d ~ + S + W d + d ( t ) ; (2{15) 26

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where ~ ( t ) 2 R p representstheparameterestimationerrorvectordenedas ~ = ^ : (2{16) Tofacilitatethesubsequentstabilityanalysis(andtoill ustratesomeinsightintothe structureofthedesignfor ( t )),thetimederivativeof( 2{15 )isdeterminedas M ( q )_ r = 1 2 M ( q ) r + Y d ~ + ~ N ( t )+ N d ( t ) ( t ) e 2 ; (2{17) wheretheunmeasurableauxiliaryterm ~ N ( e 1 ;e 2 ;r ) 2 R n isdenedas ~ N ( t ) Y d Y T d r + S 1 2 M ( q ) r + e 2 ; (2{18) where( 2{13 )wasused.In( 2{17 ),theunmeasurableauxiliaryterm N d ( q d ; q d ; q d ) 2 R n is denedas N d ( t ) W d +_ d : (2{19) Thetimederivativeof( 2{12 )isgivenas ( t )=( k s +1) r + sgn ( e 2 ) : (2{20) Inasimilarmannerasin[ 60 ],theMeanValueTheoremcanbeusedtodevelopthe followingupperbound 1 rrr ~ N ( t ) rrr ( k z k ) k z k ; (2{21) where z ( t ) 2 R 3 n isdenedas z ( t ) e T1 e T2 r T T : (2{22) Thefollowinginequalitiescanbedevelopedbasedontheexp ressionin( 2{19 )anditstime derivative: k N d ( t ) k N d rrr N d ( t ) rrr N d 2 ; (2{23) 1 SeeLemma 1 oftheAppendixfortheproofoftheinequalityin( 2{21 ). 27

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where N d N d 2 2 R areknownpositiveconstants. 2.4StabilityAnalysis Theorem2-1: Thecontrollergivenin( 2{11 ),( 2{12 ),and( 2{14 )ensuresthatall systemsignalsareboundedunderclosed-loopoperationand thatthepositiontracking errorisregulatedinthesensethat k e 1 ( t ) k! 0as t !1 providedthecontrolgain k s introducedin( 2{12 )isselectedsucientlylargebasedonthe initialconditionsofthesystem(seethesubsequentprooff ordetails), 1 2 areselected accordingtothefollowingsucientcondition 1 > 1 2 ; 2 > 1(2{24) and isselectedaccordingtothefollowingsucientcondition > N d + 1 2 N d 2 (2{25) where N d and N d 2 areintroducedin( 2{23 ). Proof: Let D R 3 n + p +1 beadomaincontaining y ( t )=0,where y ( t ) 2 R 3 n + p +1 is denedas y ( t ) h z T ( t ) ~ T ( t ) p P ( t ) i T (2{26) andtheauxiliaryfunction P ( t ) 2 R isdenedas P ( t ) n P i =1 j e 2 i (0) j e 2 (0) T N d (0) Z t 0 L ( ) d (2{27) wherethesubscript i =1 ; 2 ;::;n denotesthe i thelementofthevector.In( 2{27 ),the auxiliaryfunction L ( t ) 2 R isdenedas L ( t ) r T ( N d ( t ) sgn ( e 2 )) : (2{28) 28

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Thederivative P ( t ) 2 R canbeexpressedas P ( t )= L ( t )= r T ( N d ( t ) sgn ( e 2 )) : (2{29) Providedthesucientconditionintroducedin( 2{25 )issatised,thefollowinginequality canbeobtained 2 : Z t 0 L ( ) d n P i =1 j e 2 i (0) j e 2 (0) T N d (0) : (2{30) Hence,( 2{30 )canbeusedtoconcludethat P ( t ) 0. Let V ( y;t ): D [0 ; 1 ) R beacontinuouslydierentiable,positivedenitefunctio n denedas V ( y;t ) e T1 e 1 + 1 2 e T2 e 2 + 1 2 r T M ( q ) r + P + 1 2 ~ T 1 ~ (2{31) whichsatisesthefollowinginequalities: U 1 ( y ) V ( y;t ) U 2 ( y )(2{32) providedthesucientconditionintroducedin( 2{25 )issatised.In( 2{32 ),the continuous,positivedenitefunctions U 1 ( y ), U 2 ( y ) 2 R aredenedas U 1 ( y ) 1 k y k 2 U 2 ( y ) 2 ( q ) k y k 2 (2{33) where 1 2 ( q ) 2 R aredenedas 1 1 2 min 1 ;m 1 ; min 1 2 ( q ) max 1 2 m ( q ) ; 1 2 max 1 ; 1 where m 1 m ( q )areintroducedin( 2{3 )and min fg ; max fg denotetheminimumand maximumeigenvalues,respectively,oftheargument.After takingthetimederivativeof 2 Theinequalityin( 2{30 )canbeobtainedinasimilarmannerasinLemma 2 ofthe Appendix. 29

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( 2{31 ), V ( y;t )canbeexpressedas V ( y;t )= r T M ( q )_ r + 1 2 r T M ( q ) r + e T2 e 2 +2 e T1 e 1 + P ~ T 1 ^ : Remark2.2. From( 2{17 ),( 2{27 )and( 2{28 ),someofthedierentialequationsdescribingtheclosed-loopsystemforwhichthestabilityana lysisisbeingperformedhave discontinuousright-handsidesas M r = 1 2 M ( q ) r + Y d ~ + ~ N ( t )+ N d ( t ) ( k s +1) r 1 sgn ( e 2 ) e 2 (2{34) P ( t )= L ( t )= r T ( N d ( t ) sgn ( e 2 ))(2{35) Let f ( y;t ) 2 R n +1 denotetheright-handsideof( 2{34 ){( 2{35 ).Sincethesubsequent analysisrequiresthatasolutionexistsfor y = f ( y;t ) ,itisimportanttoshowtheexistence anduniquenessofthesolutionto( 2{34 ){( 2{35 ).Asdescribedin[ 13 61 ],theexistence ofFilippov'sgeneralizedsolutioncanbeestablishedfor( 2{34 ){( 2{35 ).First,notethat f ( y;t ) iscontinuousexceptintheset f ( y;t ) j e 2 =0 g .Let F ( y;t ) beacompact,convex, uppersemicontinuousset-valuedmapthatembedsthediere ntialequation y = f ( y;t ) intothedierentialinclusions y 2F ( y;t ) .FromTheorem27of[ 13 ],anabsolute continuoussolutionexiststo y 2F ( y;t ) thatisageneralizedsolutionto y = f ( y;t ) Acommonchoicefor F ( y;t ) thatsatisestheaboveconditionsistheclosedconvexhull of f ( y;t ) [ 13 61 ].Aproofthatthischoicefor F ( y;t ) isuppersemicontinuousisgiven in[ 62 ].Moreover,notethatthedierentialequationdescribing theoriginalclosed-loop system(i.e.,aftersubstituting( 2{11 )into( 2{1 ))hasacontinuousright-handside;thus, satisfyingtheconditionforexistenceofclassicalsoluti ons.Similarargumentsareusedfor alltheresultsinthisdissertation. Afterutilizing( 2{5 ),( 2{6 ),( 2{13 ),( 2{17 ),( 2{20 ),and( 2{29 ), V ( y;t )canbe simpliedas V ( y;t )= r T ~ N ( t ) ( k s +1) k r k 2 2 k e 2 k 2 2 1 k e 1 k 2 +2 e T2 e 1 : (2{36) 30

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Because e T2 ( t ) e 1 ( t )canbeupperboundedas e T2 e 1 1 2 k e 1 k 2 + 1 2 k e 2 k 2 V ( y;t )canbeupperboundedusingthesquaresofthecomponentsof z ( t )asfollows: V ( y;t ) r T ~ N ( t ) ( k s +1) k r k 2 2 k e 2 k 2 2 1 k e 1 k 2 + k e 1 k 2 + k e 2 k 2 : Byusing( 2{21 ),theexpressionin( 2{36 )canberewrittenasfollows: V ( y;t ) 3 k z k 2 k s k r k 2 ( k z k ) k r kk z k (2{37) where 3 min f 2 1 1 ; 2 1 ; 1 g ,andtheboundingfunction ( k z k ) 2 R isapositive, globallyinvertible,nondecreasingfunction;hence, 1 ,and 2 mustbechosenaccordingto thesucientconditionsin( 2{24 ).Aftercompletingthesquaresfortheparentheticterms in( 2{37 ),thefollowingexpressioncanbeobtained: V ( y;t ) 3 k z k 2 + 2 ( k z k ) k z k 2 4 k s : (2{38) Theexpressionin( 2{38 )canbefurtherupperboundedbyacontinuous,positive semi-denitefunction V ( y;t ) U ( y )= c k z k 2 8 y 2D (2{39) forsomepositiveconstant c 2 R ,where D n y 2 R 3 n + p +1 jk y k 1 2 p 3 k s o : Largervaluesof k willexpandthesizeofthedomain D .Theinequalitiesin( 2{32 )and ( 2{39 )canbeusedtoshowthat V ( y;t ) 2L 1 in D ;hence, e 1 ( t ), e 2 ( t ), r ( t ),and ~ ( t ) 2L 1 in D .Giventhat e 1 ( t ), e 2 ( t ),and r ( t ) 2L 1 in D ,standardlinearanalysismethodscan beusedtoprovethat_ e 1 ( t ),_ e 2 ( t ) 2L 1 in D from( 2{5 )and( 2{6 ).Since 2 R p contains theconstantunknownsystemparametersand ~ ( t ) 2L 1 in D ,( 2{16 )canbeusedto 31

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provethat ^ ( t ) 2L 1 in D .Since e 1 ( t ), e 2 ( t ), r ( t ) 2L 1 in D ,theassumptionthat q d ( t ), q d ( t ), q d ( t )existandareboundedcanbeusedalongwith( 2{4 )-( 2{6 )toconcludethat q ( t ),_ q ( t ), q ( t ) 2L 1 in D .Theassumptionthat q d ( t ),_ q d ( t ), q d ( t ), ... q d ( t ), .... q d ( t )existand areboundedalongwith( 2{8 )canbeusedtoshowthat Y d ( q d ; q d ; q d ), Y d ( q d ; q d ; q d ; ... q d ), and Y d ( q d ; q d ; q d ; ... q d ; .... q d ) 2L 1 in D .Since q ( t ),_ q ( t ) 2L 1 in D ,Assumption2-2canbe usedtoconcludethat M ( q ), V m ( q; q ), G ( q ),and F (_ q ) 2L 1 in D .Thusfrom( 2{1 )and Assumption2-3,wecanshowthat ( t ) 2L 1 in D .Giventhat r ( t ) 2L 1 in D ,( 2{20 )can beusedtoshowthat_ ( t ) 2L 1 in D .Since_ q ( t ), q ( t ) 2L 1 in D ,Assumption2-2canbe usedtoshowthat V m ( q; q ), G ( q ), F ( q )and M ( q ) 2L 1 in D ;hence,( 2{17 )canbeusedto showthat_ r ( t ) 2L 1 in D .Since_ e 1 ( t ),_ e 2 ( t ),_ r ( t ) 2L 1 in D ,thedenitionsfor U ( y )and z ( t )canbeusedtoprovethat U ( y )isuniformlycontinuousin D Let SD denoteasetdenedasfollows: 3 S y ( t ) 2Dj U ( y ( t )) < 1 1 2 p 3 k s 2 : (2{40) Theorem8.4of[ 63 ]cannowbeinvokedtostatethat c k z ( t ) k 2 0as t !18 y (0) 2S : (2{41) Basedonthedenitionof z ( t ),( 2{41 )canbeusedtoshowthat k e 1 ( t ) k! 0as t !18 y (0) 2S : 2.5ExperimentalResults ThetestbeddepictedinFigure 2-1 wasusedtoimplementthedevelopedcontroller. Thetestbedconsistsofacirculardiscofunknowninertiamo untedonaNSKdirect-drive 3 Theregionofattractionin( 2{40 )canbemadearbitrarilylargetoincludeanyinitial conditionsbyincreasingthecontrolgain k s (i.e.,asemi-globaltypeofstabilityresult)[ 22 ]. 32

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Figure2-1.Theexperimentaltestbedconsistsofacircular diskmountedonaNSK direct-driveswitchedreluctancemotor. switchedreluctancemotor(240 : 0NmModelYS5240-GN001).TheNSKmotoris controlledthroughpowerelectronicsoperatingintorquec ontrolmode.Themotorresolver providesrotorpositionmeasurementswitharesolutionof6 14 ; 400pulses/revolution. APentium2 : 8GHzPCoperatingunderQNXhoststhecontrolalgorithm,whi chwas implementedviaQmotor3 : 0,agraphicaluser-interface,tofacilitatereal-timegra phing, datalogging,andadjustmentofcontrolgainswithoutrecom pilingtheprogram(for furtherinformationonQmotor3 : 0,thereaderisreferredto[ 64 ]) Dataacquisitionand controlimplementationwereperformedatafrequencyof1 : 0kHzusingtheServoToGo I/Oboard.Arectangularnylonblockwasmountedonapneumat iclinearthrusterto applyanexternalfrictionloadtotherotatingdisk.Apneum aticregulatormaintaineda constantpressureof20poundspersquareinchonthecircula rdisk. Thedynamicsforthetestbedaregivenasfollows: J q + f (_ q )+ d ( t )= ( t ) ; (2{42) where J 2 R denotesthecombinedinertiaofthecirculardiskandrotora ssembly,the frictiontorque f (_ q ) 2 R isdenedin( 2{2 ),and d ( t ) 2 R denotesageneralnonlinear disturbance(e.g.,unmodeledeects).Theparameters r 2 ;r 3 ;r 5 areembeddedinside 33

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0 5 10 15 20 25 30 35 40 -50 0 50 Time [sec]Desired Trajectory [degrees] Figure2-2.Desiredtrajectoryusedfortheexperiment. thenonlinearhyperbolictangentfunctionsandhencecanno tbelinearlyparameterized. Sincetheseparameterscannotbecompensatedforbyanadapt ivealgorithm,best-guess estimates r 2 =50 ; r 3 =1 ; r 5 =50areused.Thevaluesfor r 2 ; r 3 ; r 5 arebasedon previousexperimentsconcernedwithfrictionidenticati on.Signicanterrorsinthese staticestimatescoulddegradetheperformanceofthesyste m.Thecontroltorqueinput ( t )isgivenby( 2{11 ),where Y d (_ q d ; q d ) 2 R 1 4 istheregressionmatrixdenedas Y d q d tanh( r 2 q d ) tanh( r 3 q d )tanh( r 5 q d )_ q d ; and ^ ( t ) 2 R 4 isthevectorconsistingoftheunknownparametersdenedas ^ ^ J ^ r 1 ^ r 4 ^ r 6 T : (2{43) Theparameterestimatesvectorin( 2{43 )isgeneratedon-lineusingtheadaptiveupdate lawin( 2{14 ).Thedesiredlinktrajectory(seeFigure 2-2 )wasselectedasfollows(in degrees): q d ( t )=45 : 0sin(1 : 2 t )(1 exp( 0 : 01 t 3 )) : (2{44) Forallexperiments,therotorvelocitysignalisobtainedb yapplyingastandard 34

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backwardsdierencealgorithmtothepositionsignal.Thei ntegralstructureofthe adaptivetermin( 2{14 )andtheRISEtermin( 2{12 )wascomputedon-lineviaastandard trapezoidalalgorithm.Inaddition,allthestatesandunkn ownparameterswereinitialized tozero.Thesignumfunctionforthecontrolschemein( 2{12 )wasdenedas: sgn ( e 2 ( t ))= 8>>>><>>>>: 1 e 2 > 0 : 0005 0 0 : 0005
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0 10 20 30 40 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 Position Tracking Error [degrees]Time [sec] Figure2-3.Positiontrackingerrorwithouttheadaptivefe edforwardterm. form[ 22 ]: ( t )= ( t ) : Thegainsforthecontrollerthatyieldedthebeststeady-st ateperformancewere determinedasfollows: k s =100 =115 1 =40 2 =30 : (2{45) Thepositiontrackingerrorobtainedfromthecontrolleris plottedinFigure 2-3 ,andthe torqueinputbythecontrollerisdepictedinFigure 2-4 Experiment2 Inthesecondexperiment,thecontrolinputgivenin( 2{11 )wasused.Theupdate lawdenedin( 2{14 )wasusedtoupdatetheparameterestimatesdenedin( 2{43 ).The followingcontrolgainsandbestguessestimateswereusedt oimplementthecontrollerin ( 2{11 ): k s =100 =115 1 =40 2 =30= diag f 10,1,1,10 g : 36

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0 10 20 30 40 -50 0 50 Torque [Nm]Time [sec] Figure2-4.Torqueinputwithouttheadaptivefeedforwardt erm. 0 10 20 30 40 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 Position Tracking Error [degrees]Time [sec] Figure2-5.Positiontrackingerrorforthecontrolstructu rethatincludestheadaptive feedforwardterm. Thepositiontrackingerrorobtainedfromthecontrolleris plottedinFigure 2-5 ,the parameterestimatesaredepictedinFigure 2-6 ,thecontributionoftheRISEtermis showninFigure 2-8 ,andthetorqueinputbythecontrollerisdepictedinFigure 2-7 37

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0 10 20 30 40 -5 0 5 10 15 Parameter EstimatesTime [sec](d) (c) (b) (a) Figure2-6.Parameterestimatesoftheadaptivefeedforwar dcomponent:(a)^ r 1 ,(b)^ r 4 ,(c) ^ r 6 ,(d) ^ J 0 10 20 30 40 -50 0 50 Torque [Nm]Time [sec] Figure2-7.Torqueinputforthecontrolstructurethatincl udestheadaptivefeedforward term. 38

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0 10 20 30 40 -40 -30 -20 -10 0 10 20 30 40 RISE Term [Nm]Time [sec] Figure2-8.ThecontributionoftheRISEtermforthecontrol structurethatincludesthe adaptivefeedforwardterm. 2.6Discussion Figure 2-5 illustratesthattheincorporationofamodel-basedfeedfo rwardterm eliminatesthe spikes presentinFigure 2-3 thatoccurwhenthemotorchangesdirection. ThespikesareinitiallypresentinFigure 2-5 ,butreduceinmagnitudeandvanishas theadaptiveupdateconverges.Theseguresexactlyillust ratehowtheadditionofthe adaptivefeedforwardelementinjectsmodelknowledgeinto thecontroldesigntoimprove theoverallperformance.Figure 2-8 indicatesthatthecontributionoftheRISEterm intheoveralltorquedecreaseswithtimeasthefeedforward adaptationtermbeginsto compensateforpartofthedisturbances. Boththeexperimentswererepeated10consecutivetimeswit hthesamegainvalues tochecktherepeatabilityandaccuracyoftheresults.Fore achrun,therootmean squared(RMS)valuesofthepositiontrackingerrorsandtor quesarecalculated.The averageoftheseRMSvaluesforthetwocases(withadaptatio nandwithoutadaptation) obtainedover10setsareplottedinFigure 2-9 ,wherethebarsindicatethevarianceabout themean.Anunpairedt-testassumingequalvarianceswaspe rformedusingastatistical 39

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package(MicrosoftOceExcel2003)withasignicanceleve lof =0 : 05.Theresultsof thet-testfortheRMSerror,andtheRMStorqueareshowninTa ble2-1andTable2-2, respectively.Table2-1indicatesthatthe P valueobtainedfortheone-tailedtestisless thanthesignicancelevel .Thus,themeanRMSerrorforcase2islowerthanthatof case1,andthisdierenceisstatisticallysignicant.Sim ilarly,fromTable2-2,themean RMStorqueforcase2islowerthanthatofcase1.Theresultsi ndicatethatthemean RMSvalueofthepositiontrackingerrorwhentheadaptivefe edforwardtermisusedis about43 : 5%lessthanthecasewhennoadaptationtermisused.Thisimp rovementin performancebytheproposedcontrollerwasobtainedwhileu sing17 : 6%lessinputtorque asshowninFigure 2-9 Whilethedevelopedcontrollerisacontinuouscontroller, itcanexhibitsome highfrequencycontentduetothepresenceoftheintegralsi gnfunction.However,the frequencycontentisniteunlikecurrentdiscontinuousno nlinearcontrolmethods.The experimentalresultsshowsomechatteringintheinput/out putsignals,butthemechanical systemactsalow-passlterbecausetheactuatorbandwidth islowerthanthebandwidth producedbythecontroller.Also,thecontrollerrequiresf ull-statefeedback(i.e.,both positionandvelocitymeasurementsareneeded),butasment ionedearlier,onlythe positionismeasuredandthevelocityisobtainedbyanunlt eredbackwarddierence algorithm.Theneedforvelocityfeedbackisalsoasourceof noise,especiallyforthe sub-degreeerrorsthatthecontrolleryields. 2.7Conclusions Anewclassofasymptoticcontrollersisdevelopedthatcont ainsanadaptive feedforwardtermtoaccountforlinearparameterizableunc ertaintyandahighgain feedbacktermwhichaccountsforunstructureddisturbance s.Incomparisonwithprevious resultsthatusedasimilarhighgainfeedbackcontrolstruc ture,newcontroldevelopment, errorsystemsandstabilityanalysisargumentswererequir edtoincludetheadditional adaptivefeedforwardterm.Themotivationforinjectingth eadaptivefeedforwardtermis 40

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Table2-1.t-test:twosamplesassumingequalvariancesfor RMSerror RMSError Variable1 Variable2 Mean 3 : 214 10 2 1 : 817 10 2 Variance 1 : 044 10 5 3 : 170 10 7 Observations 10 10 PooledVariance 5 : 378 10 6 HypothesizedMeanDierence 0 df 18 tStat 13 : 47 P(T < =t) one-tail 3 : 847 10 11 tCriticalone-tail 1 : 734 P(T < =t) two-tail 7 : 693 10 11 tCriticaltwo-tail 2 : 101 Figure2-9.RMSpositiontrackingerrorsandtorquesforthe twocases-(1)withoutthe adaptationterminthecontrolinput,(2)withtheadaptatio nterminthe controlinput. 41

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Table2-2.t-test:twosamplesassumingequalvariancesfor RMStorque RMSTorque Variable1 Variable2 Mean 27 : 79 22 : 90 Variance 3 : 452 1 : 474 Observations 10 10 PooledVariance 2 : 463 HypothesizedMeanDierence 0 df 18 tStat 3 : 971 P(T < =t) one-tail 8 : 208 10 7 tCriticalone-tail 1 : 734 P(T < =t) two-tail 1 : 642 10 6 tCriticaltwo-tail 2 : 101 thatimprovedtrackingperformanceandreducedcontroleo rtresultfromincludingmore knowledgeofthesystemdynamicsinthecontrolstructure.T hisheuristicideawasveried byourexperimentalresultsthatindicatereducedcontrole ortandreducedRMStracking errors. 42

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CHAPTER3 ASYMPTOTICTRACKINGFORUNCERTAINDYNAMICSYSTEMSVIAA MULTILAYERNEURALNETWORKFEEDFORWARDANDRISEFEEDBACK CONTROLSTRUCTURE 3.1Introduction Thecontributioninthischapterismotivatedbythequestio n: CanaNNfeedforward controllerbemodiedbyacontinuousfeedbackelementtoac hieveanasymptotictracking resultforageneralclassofsystems? DespitethepervasivedevelopmentofNNcontrollers inliteratureandthewidespreaduseofNNsinindustrialapp lications,theanswertothis fundamentalquestionhasremainedanopenproblem. Toprovideananswertothefundamentalmotivatingquestion ,theresultinthis chapterfocusesonaugmentingamulti-layerNN-basedfeedf orwardmethodwitharecently developed[ 21 ]highgaincontrolstrategycoinedtheRobustIntegralofth eSignofthe Error(RISE)in[ 19 20 ].TheRISEcontrolstructureisadvantageousbecauseitisa dierentiablecontrolmethodthatcancompensateforaddit ivesystemdisturbancesand parametricuncertaintiesundertheassumptionthatthedis turbancesare C 2 withbounded timederivatives.DuetotheadvantagesoftheRISEcontrols tructurearurryofresults haverecentlybeendeveloped(e.g.,[ 22 23 25 { 27 ]). ARISEfeedbackcontrollercanbedirectlyappliedtoyielda symptoticstabilityfor theclassofsystemsdescribedinthischapter.However,the RISEmethodisahigh-gain feedbacktool,andhence,clearmotivationexists(aswitha nyotherfeedbackcontroller) tocombineafeedforwardcontrolelementwiththefeedbackc ontrollerforpotentialgains suchasimprovedtransientandsteady-stateperformance,a ndreducedcontroleort. Thatis,itiswellacceptedthatafeedforwardcomponentcan beusedtocanceloutsome dynamiceectswithoutrelyingonhigh-gainfeedback.Give nthismotivation,someresults havealreadybeendevelopedthatcombinetheRISEfeedbacke lementwithfeedforward terms.In[ 60 ],aremarkisprovidedregardingtheuseofaconstantbest-g uessfeedforward componentinconjunctionwiththeRISEmethodtoyieldaUUBr esult.In[ 19 20 ],the 43

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RISEfeedbackcontrollerwascombinedwithastandardgradi entfeedforwardtermfor systemsthatsatisfythelinear-in-the-parametersassump tion.Theexperimentalresults in[ 19 ]illustratesignicantimprovementintheroot-mean-squa redtrackingerrorwith reducedroot-mean-squaredcontroleort.However,forsys temsthatdonotsatisfythe linear-in-the-parametersassumption,motivationexists tocombinetheRISEcontroller withanewfeedforwardmethodsuchastheNN. ToblendtheNNandRISEmethods,severaltechnicalchalleng esmustbeaddressed. One(lesser)challengeisthattheNNmustbeconstructedint ermsofthedesired trajectoryinsteadoftheactualtrajectory(i.e.,aDCAL-b asedNNstructure[ 36 ])to removethedependenceonacceleration.Thedevelopmentofa DCAL-basedNNstructure ischallengingforamulti-layerNNbecausetheadaptationl awfortheweightsisrequired tobestate-dependent.Straightforwardapplicationofthe RISEmethodwouldyieldan accelerationdependentadaptationlaw.Onemethodtoresol vethisissueistousea\dirty derivative"(asintheUUBresultin[ 69 ];seealso[ 25 ]).Inlieuofadirtyderivative,the resultinthischapterusesaLyapunov-basedstabilityanal ysisapproachforthedesign ofanadaptationlawthatisonlyvelocitydependent.Incomp arisonwiththeeortsin [ 19 20 ],amoresignicantchallengearisesfromthefactthatsinc eamulti-layerNN includestherstlayerweightestimateinsideofanonlinea ractivationfunction,the previousmethods(e.g.,[ 19 20 ])cannotbeapplied.Thatis,becauseoftheunique mannerinwhichtheNNweightestimatesappear,thestabilit yanalysisandsucient conditionsdevelopedinpreviousworksareviolated.Previ ousRISEmethodshave arestriction(encapsulatedbyasucientgaincondition)t hattermsinthestability analysisthatareupperboundedbyaconstantmustalsohavet imederivativesthatare upperboundedbyaconstant(thesetermsareusuallydenoted by N d ( t )inRISEcontrol literature,see[ 60 ]).ThenormoftheNNweightestimatescanbeboundedbyacons tant (duetoaprojectionalgorithm)butthetimederivativeisst ate-dependent(i.e.,thenorm of N d ( t )canbeboundedbyaconstantbutthenormof N d ( t )isstatedependent).To 44

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addressthisissue,modiedRISEstabilityanalysistechni quesaredevelopedthatresult inmodied(butnotmorerestrictive)sucientgainconditi ons.Byaddressingthisissue throughstabilityanalysismethods,thestandardNNweight adaptationlawdoesnotneed tobemodied.Throughuniquemodicationstothestability analysisthatenablethe RISEfeedbackcontrollertobecombinedwiththeNNfeedforw ardterm,theresultinthis chapterprovidesanarmativeanswerforthersttimetothe aforementionedmotivating question. SincetheNNandtheRISEcontrolstructuresaremodelindepe ndent(blackbox) methods,theresultingcontrollerisauniversalreusablec ontroller[ 36 ]forcontinuous systems.BecauseofthemannerinwhichtheRISEtechniqueis blendedwiththe NN-basedfeedforwardmethod,thestructureoftheNNisnota lteredfromtextbook examples[ 15 ]andcanbeconsideredasomewhatmodularelementinthecont rolstructure. Hence,theNNweightsandthresholdsareautomaticallyadju stedon-line,withno o-linelearningphaserequired.Comparedtostandardadap tivecontrollers,thecurrent asymptoticresultdoesnotrequirelinearityintheparamet ersorthedevelopmentand evaluationofaregressionmatrix. Forsystemswithlinear-in-the-parametersuncertainty,a nadaptivefeedforward controllerhasthedesirablecharacteristicsthatthecont rolleriscontinuous,canbe proventoyieldglobalasymptotictracking,andincludesth especicdynamicsofthe systeminthefeedforwardpath.ContinuousfeedbackNNcont rollersdon'tincludethe specicdynamicsinaregressionmatrixandhaveadegradeds teady-statestability result(i.e.,UUBtracking);however,theycanbeappliedwh entheuncertaintyin thesystemisunmodeled,cannotbelinearlyparameterized, orthedevelopmentand implementationofaregressionmatrixisimpractical.Slid ingmodefeedbackNN controllershavetheadvantagethattheycanachieveglobal asymptotictrackingatthe expenseofimplementingadiscontinuousfeedbackcontroll er(i.e.,innitebandwidth, excitingstructuralmodes,etc.).Incomparisontotheseco ntrollers,thedevelopment 45

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inthischapterhastheadvantageofasymptotictrackingwit hacontinuosfeedback controllerforageneralclassofuncertainty;however,the seadvantagesareattheexpense ofsemi-globaltrackinginsteadofthetypicalglobaltrack ingresults. 3.2DynamicModel ThedynamicmodelanditspropertiesarethesameasinChapte r2;however,the dynamicsarenotassumedtosatisfythelinear-in-the-para metersassumption. 3.3ControlObjective Thecontrolobjectiveistoensurethatthesystemtracksade siredtime-varying trajectory,denotedby q d ( t ) 2 R n ,despiteuncertaintiesinthedynamicmodel.Toquantify thisobjective,apositiontrackingerror,denotedby e 1 ( t ) 2 R n ,isdenedas e 1 q d q: (3{1) Tofacilitatethesubsequentanalysis,lteredtrackinger rors,denotedby e 2 ( t ), r ( t ) 2 R n arealsodenedas e 2 e 1 + 1 e 1 (3{2) r e 2 + 2 e 2 (3{3) where 1 2 2 R denotepositiveconstants.Thelteredtrackingerror r ( t )isnot measurablesincetheexpressionin( 3{3 )dependson q ( t ). 3.4FeedforwardNNEstimation NN-basedestimationmethodsarewellsuitedforcontrolsys temswherethedynamic modelcontainsunstructurednonlineardisturbancesasin( 2{1 ).Themainfeaturethat empowersNN-basedcontrollersistheuniversalapproximat ionproperty.Let S bea compactsimplyconnectedsetof R N 1 +1 .Withmap f : S R n ,dene C n ( S )asthe spacewhere f iscontinuous.Thereexistweightsandthresholdssuchthat somefunction f ( x ) 2 C n ( S )canberepresentedbyathree-layerNNas[ 15 32 ] f ( x )= W T V T x + ( x )(3{4) 46

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forsomegiveninput x ( t ) 2 R N 1 +1 .In( 3{4 ), V 2 R ( N 1 +1) N 2 and W 2 R ( N 2 +1) n are boundedconstantidealweightmatricesfortherst-to-sec ondandsecond-to-thirdlayers respectively,where N 1 isthenumberofneuronsintheinputlayer, N 2 isthenumber ofneuronsinthehiddenlayer,and n isthenumberofneuronsinthethirdlayer.The activationfunction 1 in( 3{4 )isdenotedby ( ) 2 R N 2 +1 ,and ( x ) 2 R n isthefunctional reconstructionerror.Notethat,augmentingtheinputvect or x ( t )andactivationfunction ( )by\1"allowsustohavethresholdsastherstcolumnsofthe weightmatrices [ 15 32 ].Thus,anytuningof W and V thenincludestuningofthresholdsaswell.If ( x )=0,then f ( x )isinthefunctionalrangeoftheNN.Ingeneralforanyposit ive constantrealnumber N > 0, f ( x )iswithin N oftheNNrangeifthereexistnite hiddenneurons N 2 ,andconstantweightssothatforallinputsinthecompactse t,the approximationholdswith k k <" N .Forvariousactivationfunctions,resultssuchas theStone-Weierstrasstheoremindicatethatanysucientl ysmoothfunctioncanbe approximatedbyasuitablelargenetwork.Therefore,thefa ctthattheapproximation error ( x )isboundedfollowsfromthe UniversalApproximationProperty oftheNNs[ 30 ]. Basedon( 3{4 ),thetypicalthree-layerNNapproximationfor f ( x )isgivenas[ 15 32 ] ^ f ( x ) ^ W T ( ^ V T x )(3{5) where ^ V ( t ) 2 R ( N 1 +1) N 2 and ^ W ( t ) 2 R ( N 2 +1) n aresubsequentlydesignedestimatesofthe idealweightmatrices.Theestimatemismatchfortheidealw eightmatrices,denotedby ~ V ( t ) 2 R ( N 1 +1) N 2 and ~ W ( t ) 2 R ( N 2 +1) n ,aredenedas ~ V V ^ V; ~ W W ^ W 1 Avarietyofactivationfunctions(e.g.,sigmoid,hyperbol ictangentorradialbasis) couldbeusedforthecontroldevelopmentinthisdissertati on. 47

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andthemismatchforthehidden-layeroutputerrorforagive n x ( t ),denotedby~ ( x ) 2 R N 2 +1 ,isdenedas ~ ^ = ( V T x ) ( ^ V T x ) : (3{6) TheNNestimatehasseveralpropertiesthatfacilitatethes ubsequentdevelopment.These propertiesaredescribedasfollows.Assumption3-1: ( BoundednessoftheIdealWeights )Theidealweightsareassumedto existandbeboundedbyknownpositivevaluessothat k V k 2F = tr V T V = vec ( V ) T vec ( V ) V B (3{7) k W k 2F = tr W T W = vec ( W ) T vec ( W ) W B ; (3{8) where kk F istheFrobeniusnormofamatrix, tr ( )isthetraceofamatrix,andthe operator vec ( )stacksthecolumnsofamatrix A 2 R m n toformavector vec ( A ) 2 R mn as vec ( A ) A 11 A 21 :::A m 1 A 12 A 22 :::A 1 n :::A mn T : Assumption3-2: ( ConvexRegions )Basedon( 3{7 )and( 3{8 ),convexregions(e.g.,see Section4.3of[ 70 ])canbedened.Specically,theconvexregion V canbedenedas 2 V v : v T v V B ; (3{9) where V B wasgivenin( 3{7 ).Inaddition,thefollowingdenitionsconcerningthereg ion V andtheparameterestimatevector vec ^ V 2 R ( N 1 +1) N 2 (i.e.,thedynamicestimate of vec ( V ) 2 V )areprovidedasfollows: int ( V )denotestheinterioroftheregion V @ ( V )denotestheboundaryfortheregion V vec ^ V ? 2 R ( N 1 +1) N 2 isaunitvector normalto @ ( V )atthepointofintersectionoftheboundarysurface @ ( V )and vec ^ V wherethepositivedirectionfor vec ^ V ? isdenedaspointingawayfrom int ( V )(note 2 SeeLemma 3 oftheAppendixfortheproofofconvexity. 48

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that vec ^ V ? isonlydenedfor vec ^ V 2 @ ( V )), P t r ( )isthecomponentofthevector 2 R ( N 1 +1) N 2 thatistangentto @ ( V )atthepointofintersectionoftheboundarysurface @ ( V )andthevector vec ^ V ,and P ? r ( )= P t r ( ) 2 R ( N 1 +1) N 2 (3{10) isthecomponentofthevector 2 R ( N 1 +1) N 2 thatisperpendicularto @ ( V )atthepoint ofintersectionoftheboundarysurface @ ( V )andthevector vec ^ V .Similarto( 3{9 ), theconvexregion W isdenedas W v : v T v W B ; (3{11) where W B wasgivenin( 3{8 ). 3.5RISEFeedbackControlDevelopment Thecontributionofthischapteristhecontroldevelopment andstabilityanalysis thatillustrateshowtheaforementionedtextbook(e.g.,[ 15 ])NNfeedforwardestimation strategycanbefusedwithaRISEfeedbackcontrolmethodasa meanstoachievean asymptoticstabilityresultforgeneralEuler-Lagrangesy stemsdescribedby( 2{1 ).Inthis section,theopen-loopandclosed-looptrackingerrorisde velopedforthecombinedcontrol system.3.5.1Open-LoopErrorSystem Theopen-looptrackingerrorsystemcanbedevelopedbyprem ultiplying( 3{3 )by M ( q )andutilizingtheexpressionsin( 2{1 ),( 3{1 ),and( 3{2 )toobtainthefollowing expression: M ( q ) r = f d + S + d (3{12) wheretheauxiliaryfunction f d ( q d ; q d ; q d ) 2 R n isdenedas f d M ( q d ) q d + V m ( q d ; q d )_ q d + G ( q d )+ F (_ q d )(3{13) 49

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andtheauxiliaryfunction S ( q; q;q d ; q d ; q d ) 2 R n isdenedas S M ( q )( 1 e 1 + 2 e 2 )+ M ( q ) q d M ( q d ) q d + V m ( q; q )_ q V m ( q d ; q d )_ q d (3{14) + G ( q ) G ( q d )+ F (_ q ) F (_ q d ) : Theexpressionin( 3{13 )canberepresentedbyathree-layerNNas f d = W T ( V T x d )+ ( x d ) : (3{15) In( 3{15 ),theinput x d ( t ) 2 R 3 n +1 isdenedas x d ( t ) [1 q T d ( t )_ q T d ( t ) q T d ( t )] T sothat N 1 =3 n where N 1 wasintroducedin( 3{4 ).Basedontheassumptionthatthedesired trajectoryisbounded,thefollowinginequalitieshold k ( x d ) k b 1 ; k ( x d ; x d ) k b 2 ; k ( x d ; x d ; x d ) k b 3 (3{16) where b 1 ;" b 2 ;" b 3 2 R areknownpositiveconstants. 3.5.2Closed-LoopErrorSystem Basedontheopen-looperrorsystemin( 3{12 ),thecontroltorqueinputiscomposed ofathree-layerNNfeedforwardtermplustheRISEfeedbackt ermsas ^ f d + : (3{17) Specically,theRISEfeedbackcontrolterm ( t ) 2 R n isdenedas[ 22 ] ( t ) ( k s +1) e 2 ( t ) ( k s +1) e 2 (0)+ Z t 0 [( k s +1) 2 e 2 ( )+ 1 sgn ( e 2 ( ))] d (3{18) where k s ; 1 2 R arepositiveconstantcontrolgains.ThefeedforwardNNcom ponentin ( 3{17 ),denotedby ^ f d ( t ) 2 R n ,isgeneratedas ^ f d ^ W T ( ^ V T x d ) : (3{19) 50

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TheestimatesfortheNNweightsin( 3{19 )aregeneratedon-line(thereisnoo-line learningphase)usingasmoothprojectionalgorithm(e.g., seeSection4.3of[ 70 ])as ^ W = proj ( 1 )= 8>>>>><>>>>>: 1 if vec ^ W 2 int ( W ) 1 if vec ^ W 2 @ ( W )and vec ( 1 ) T vec ^ W ? 0 P t Mr ( 1 ) if vec ^ W 2 @ ( W )and vec ( 1 ) T vec ^ W ? > 0 (3{20) ^ V = proj ( 2 )= 8>>>>><>>>>>: 2 if vec ^ V 2 int ( V ) 2 if vec ^ V 2 @ ( V )and vec ( 2 ) T vec ^ V ? 0 P t Mr ( 2 ) if vec ^ V 2 @ ( V )and vec ( 2 ) T vec ^ V ? > 0 (3{21) where vec ^ W (0) 2 int ( W ) ;vec ^ V (0) 2 int ( V ) andtheauxiliaryterms 1 ( t ) 2 R ( N 2 +1) n 2 ( t ) 2 R ( N 1 +1) N 2 aredenedas 1 1 ^ 0 ^ V T x d e T2 2 2 x d (^ 0 T ^ We 2 ) T (3{22) where 1 2 R ( N 2 +1) ( N 2 +1) 2 2 R ( N 1 +1) ( N 1 +1) areconstant,positivedenite,symmetric matrices.In( 3{20 )and( 3{21 ), P t Mr ( A )= devec ( P t r ( vec ( A )))foramatrix A ,wherethe operation devec ( )isthereverseof vec ( ). Remark3.1. Theuseoftheprojectionalgorithmin( 3{20 )and( 3{21 )istoensurethat ^ W ( t ) and ^ V ( t ) remainboundedinsidetheconvexregionsdenedin( 3{9 ),and( 3{11 ). Thisfactwillbeexploitedinthesubsequentstabilityanal ysis. Theclosed-looptrackingerrorsystemcanbedevelopedbysu bstituting( 3{17 )into ( 3{12 )as M ( q ) r = f d ^ f d + S + d : (3{23) Tofacilitatethesubsequentstabilityanalysis,thetimed erivativeof( 3{23 )isdetermined as M ( q )_ r = M ( q ) r + f d ^ f d + S +_ d : (3{24) 51

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Takingthetimederivativeoftheclosed-looperrorsystemi stypicaloftheRISEstability analysis.Inourcase,thetimedierentiationalsofacilit atesthedesignofNNweight adaptationlawsinsteadofusingthetypical(asin[ 15 32 ])Taylorseriesapproximation methodtoobtainalinearformfortheestimationerror ~ V .Using( 3{15 )and( 3{19 ),the closed-looperrorsystemin( 3{24 )canbeexpressedas M ( q )_ r = M ( q ) r + W T 0 V T x d V T x d ^ W T ( ^ V T x d )(3{25) ^ W T 0 ( ^ V T x d ) ^ V T x d ^ W T 0 ( ^ V T x d ) ^ V T x d +_ + S +_ d where 0 ( ^ V T x ) d V T x =d V T x j V T x = ^ V T x .Afteraddingandsubtractingtheterms W T ^ 0 ^ V T x d + ^ W T ^ 0 ~ V T x d to( 3{25 ),thefollowingexpressioncanbeobtained: M ( q )_ r = M ( q ) r + ^ W T ^ 0 ~ V T x d + ~ W T ^ 0 ^ V T x d + W T 0 V T x d W T ^ 0 ^ V T x d (3{26) ^ W T ^ 0 ~ V T x d + S ^ W T ^ ^ W T ^ 0 ^ V T x d +_ d +_ wherethenotations^ and~ areintroducedin( 3{6 ).UsingtheNNweighttuninglawsin ( 3{20 ),( 3{21 ),theexpressionin( 3{26 )canberewrittenas M ( q )_ r = 1 2 M ( q ) r + ~ N + N e 2 ( k s +1) r 1 sgn ( e 2 )(3{27) wherethefactthatthetimederivativeof( 3{18 )isgivenas ( t )=( k s +1) r + 1 sgn ( e 2 )(3{28) wasutilized,andwheretheunmeasurableauxiliaryterms ~ N ( e 1 ;e 2 ;r;t ) ;N ( ^ W; ^ V;x d ; x d ;t ) 2 R n aredenedas ~ N ( t ) 1 2 M ( q ) r proj ( 1 ^ 0 ^ V T x d e T2 ) T ^ (3{29) ^ W T ^ 0 proj ( 2 x d (^ 0 T ^ We 2 ) T ) T x d + S + e 2 52

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and N N d + N B : (3{30) In( 3{30 ), N d ( x d ; x d ;t ) 2 R n isdenedas N d W T 0 V T x d +_ +_ d (3{31) while N B ( ^ W; ^ V;x d ; x d ;t ) 2 R n isfurthersegregatedas N B N B 1 + N B 2 (3{32) where N B 1 ( ^ W; ^ V;x d ; x d ;t ) 2 R n isdenedas N B 1 W T ^ 0 ^ V T x d ^ W T ^ 0 ~ V T x d (3{33) andtheterm N B 2 ( ^ W; ^ V;x d ; x d ;t ) 2 R n isdenedas N B 2 ^ W T ^ 0 ~ V T x d + ~ W T ^ 0 ^ V T x d : (3{34) Motivationforsegregatingthetermsin( 3{30 )isderivedfromthefactthatthe dierentcomponentsin( 3{30 )havedierentbounds.Segregatingthetermsasin ( 3{30 )-( 3{34 )facilitatesthedevelopmentoftheNNweightupdatelawsan dthe subsequentstabilityanalysis.Forexample,thetermsin( 3{31 )aregroupedtogether becausethetermsandtheirtimederivativescanbeupperbou ndedbyaconstantand rejectedbytheRISEfeedback,whereasthetermsgroupedin( 3{32 )canbeupper boundedbyaconstantbuttheirderivativesarestatedepend ent.Thestatedependency ofthetimederivativesofthetermsin( 3{32 )violatestheassumptionsgiveninprevious RISE-basedcontrollers(e.g.,[ 19 20 22 23 25 { 27 ]),andrequiresadditionalconsideration intheadaptationlawdesignandstabilityanalysis.Theter msin( 3{32 )arefurther segregatedbecause N B 1 ( ^ W; ^ V;x d )willberejectedbytheRISEfeedback,whereas N B 2 ( ^ W; ^ V;x d )willbepartiallyrejectedbytheRISEfeedbackandpartial lycanceled bytheadaptiveupdatelawfortheNNweightestimates. 53

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InasimilarmannerasinChapter2,theMeanValueTheoremcan beusedtodevelop thefollowingupperbound rrr ~ N ( t ) rrr ( k z k ) k z k (3{35) where z ( t ) 2 R 3 n isdenedas z ( t ) [ e T1 e T2 r T ] T (3{36) andtheboundingfunction ( k z k ) 2 R isapositivegloballyinvertiblenondecreasing function.Thefollowinginequalitiescanbedevelopedbase donAssumption2-3,( 3{7 ), ( 3{8 ),( 3{16 ),( 3{32 )-( 3{34 ): k N d k 1 k N B 1 k 2 k N B 2 k 3 rrr N d rrr 4 : (3{37) From( 3{30 ),( 3{32 )and( 3{37 ),thefollowingboundcanbedeveloped k N 1 kk N d k + k N 1 B kk N d k + k N 1 B a k + k N 1 B b k 1 + 2 + 3 : (3{38) Byusing( 3{20 ),( 3{21 ),thetimederivativeof N B ( ^ W; ^ V;x d )canbeboundedas rrr N B rrr 5 + 6 k e 2 k : (3{39) In( 3{37 )and( 3{39 ), i 2 R ,( i =1 ; 2 ;:::; 6)areknownpositiveconstants. 3.6StabilityAnalysis Theorem3-1: ThecombinedNNandRISEcontrollergivenin( 3{17 )-( 3{21 )ensures thatallsystemsignalsareboundedunderclosed-loopopera tionandthattheposition trackingerrorisregulatedinthesensethat k e 1 ( t ) k! 0as t !1 providedthecontrolgain k s introducedin( 3{18 )isselectedsucientlylarge(seethe subsequentproof), 1 2 areselectedaccordingtothefollowingsucientcondition 1 > 1 2 ; 2 > 2 +1 ; (3{40) 54

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and 1 and 2 areselectedaccordingtothefollowingsucientcondition s: 1 > max 1 + 2 + 3 ; 1 + 2 + 4 2 + 5 2 3 > 6 (3{41) where i 2 R i =1 ; 2 ; ... ; 5areintroducedin( 3{37 )-( 3{39 )and 2 isintroducedin( 3{44 ). Proof: Let D R 3 n +2 beadomaincontaining y ( t )=0,where y ( t ) 2 R 3 n +2 isdened as y ( t ) [ z T ( t ) p P ( t ) p Q ( t )] T : (3{42) In( 3{42 ),theauxiliaryfunction P ( t ) 2 R isdenedas P ( t ) 1 n P i =1 j e 2 i (0) j e 2 (0) T N (0) Z t 0 L ( ) d (3{43) wherethesubscript i =1 ; 2 ;::;n denotesthe i thelementofthevector,andtheauxiliary function L ( t ) 2 R isdenedas L ( t ) r T ( N B 1 ( t )+ N d ( t ) 1 sgn ( e 2 ))+_ e 2 ( t ) T N B 2 ( t ) 2 k e 2 ( t ) k 2 (3{44) where 2 2 R isapositiveconstantchosenaccordingtothesecondsucie ntconditionin ( 3{41 ).Thederivative P ( t ) 2 R canbeexpressedas P ( t )= L ( t )= r T ( N B 1 ( t )+ N d ( t ) 1 sgn ( e 2 )) e 2 ( t ) T N B 2 ( t )+ 2 k e 2 ( t ) k 2 : (3{45) Providedthesucientconditionsintroducedin( 3{41 )aresatised,thefollowing inequalitycanbeobtainedinasimilarfashionasinLemma 2 oftheAppendix Z t 0 L ( ) d 1 n P i =1 j e 2 i (0) j e 2 (0) T N (0) : (3{46) Hence,( 3{46 )canbeusedtoconcludethat P ( t ) 0.Theauxiliaryfunction Q ( t ) 2 R in ( 3{42 )isdenedas Q ( t ) 2 2 tr ( ~ W T 1 1 ~ W )+ 2 2 tr ( ~ V T 1 2 ~ V ) : (3{47) Since 1 and 2 areconstant,symmetric,andpositivedenitematricesand 2 > 0,itis straightforwardthat Q ( t ) 0. 55

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Let V L ( y;t ): D [0 ; 1 ) R beacontinuouslydierentiablepositivedenitefunction denedas V L ( y;t ) e T1 e 1 + 1 2 e T2 e 2 + 1 2 r T M ( q ) r + P + Q (3{48) whichsatisesthefollowinginequalities: U 1 ( y ) V L ( y;t ) U 2 ( y )(3{49) providedthesucientconditionsintroducedin( 3{41 )aresatised.In( 3{49 ),the continuouspositivedenitefunctions U 1 ( y ), U 2 ( y ) 2 R aredenedas U 1 ( y ) 1 k y k 2 U 2 ( y ) 2 ( q ) k y k 2 (3{50) where 1 2 ( q ) 2 R aredenedas 1 1 2 min f 1 ;m 1 g 2 ( q ) max f 1 2 m ( q ) ; 1 g where m 1 m ( q )areintroducedin( 2{3 ).Afterutilizing( 3{2 ),( 3{3 ),( 3{27 ),( 3{28 ),the timederivativeof( 3{48 )canbeexpressedas V L ( y;t )= 2 1 k e 1 k 2 +2 e T2 e 1 + r T ~ N ( t ) ( k s +1) k r k 2 2 k e 2 k 2 (3{51) + 2 k e 2 k 2 + 2 e T2 h ^ W T ^ 0 ~ V T x d + ~ W T ^ 0 ^ V T x d i + tr ( 2 ~ W T 1 1 ~ W )+ tr ( 2 ~ V T 1 2 ~ V ) : Basedonthefactthat e T2 e 1 1 2 k e 1 k 2 + 1 2 k e 2 k 2 andusing( 3{20 ),( 3{21 ),theexpressionin( 3{51 )canbesimpliedas 3 V L ( y;t ) r T ~ N ( t ) ( k s +1) k r k 2 (2 1 1) k e 1 k 2 ( 2 2 1) k e 2 k 2 : (3{52) 3 SeeLemma 4 oftheAppendixforthedetailsofobtainingtheinequalityi n( 3{52 ). 56

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Byusing( 3{35 ),theexpressionin( 3{52 )canbefurtherboundedas V L ( y;t ) 3 k z k 2 k s k r k 2 ( k z k ) k r kk z k (3{53) where 3 min f 2 1 1 ; 2 2 1 ; 1 g ;hence, 3 ispositiveif 1 ; 2 arechosenaccording tothesucientconditionsin( 3{40 ).Aftercompletingthesquaresforthesecondand thirdtermin( 3{53 ),thefollowingexpressioncanbeobtained: V L ( y;t ) 3 k z k 2 + 2 ( z ) k z k 2 4 k s : (3{54) Theexpressionin( 3{54 )canbefurtherupperboundedbyacontinuous,positive semi-denitefunction V L ( y;t ) U ( y )= c k z k 2 8 y 2D (3{55) forsomepositiveconstant c 2 R ,where D n y 2 R 3 n +2 jk y k 1 2 p 3 k s o : Largervaluesof k willexpandthesizeofthedomain D .Theinequalitiesin( 3{49 )and ( 3{55 )canbeusedtoshowthat V L ( y;t ) 2L 1 in D ;hence, e 1 ( t ), e 2 ( t ), r ( t ), P ( t ),and Q ( t ) 2L 1 in D .Giventhat e 1 ( t ), e 2 ( t ),and r ( t ) 2L 1 in D ,standardlinearanalysis methodscanbeusedtoprovethat_ e 1 ( t ),_ e 2 ( t ) 2L 1 in D from( 3{2 )and( 3{3 ).Since e 1 ( t ), e 2 ( t ), r ( t ) 2L 1 in D ,theassumptionthat q d ( t ),_ q d ( t ), q d ( t )existandarebounded canbeusedalongwith( 3{1 )-( 3{3 )toconcludethat q ( t ),_ q ( t ), q ( t ) 2L 1 in D .Since q ( t ), q ( t ) 2L 1 in D ,Assumption2-2canbeusedtoconcludethat M ( q ), V m ( q; q ), G ( q ),and F (_ q ) 2L 1 in D .Therefore,from( 2{1 )andAssumption2-3,wecanshowthat ( t ) 2L 1 in D .Giventhat r ( t ) 2L 1 in D ,( 3{28 )canbeusedtoshowthat_ ( t ) 2L 1 in D .Since q ( t ), q ( t ) 2L 1 in D ,Assumption2-2canbeusedtoshowthat V m ( q; q ), G ( q ), F ( q )and M ( q ) 2L 1 in D ;hence,( 3{24 )canbeusedtoshowthat_ r ( t ) 2L 1 in D .Since_ e 1 ( t ), 57

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_ e 2 ( t ),_ r ( t ) 2L 1 in D ,thedenitionsfor U ( y )and z ( t )canbeusedtoprovethat U ( y )is uniformlycontinuousin D Let SD denoteasetdenedasfollows: 4 S y ( t ) 2Dj U 2 ( y ( t )) < 1 1 2 p 3 k s 2 : (3{56) Theorem8.4of[ 63 ]cannowbeinvokedtostatethat c k z ( t ) k 2 0as t !18 y (0) 2S : (3{57) Basedonthedenitionof z ( t ),( 3{57 )canbeusedtoshowthat k e 1 ( t ) k! 0as t !18 y (0) 2S : (3{58) 3.7Experiment AsinChapter2,thetestbeddepictedinFigure 2-1 wasusedtoimplementthe developedcontroller,however,noexternalfrictionisapp liedtothecirculardisk.The desiredlinktrajectoryisselectedasfollows(indegrees) : q d ( t )=60 : 0sin(3 : 0 t )(1 exp( 0 : 01 t 3 )) : (3{59) Forallexperiments,therotorvelocitysignalisobtainedb yapplyingastandardbackwards dierencealgorithmtothepositionsignal.Theintegralst ructurefortheRISEtermin ( 3{18 )wascomputedon-lineviaastandardtrapezoidalalgorithm .TheNNinputvector x d ( t ) 2 R 4 isdenedas x d =[1 q d q d q d ] T : Theinitialvaluesof ^ W (0)werechosentobeazeromatrix;however,theinitialvalu esof ^ V (0)wereselectedrandomlybetween 1 : 0and1 : 0toprovideabasis[ 71 ].Adierent 4 Theregionofattractionin( 3{56 )canbemadearbitrarilylargetoincludeanyinitial conditionsbyincreasingthecontrolgain k s (i.e.,asemi-globaltypeofstabilityresult)[ 22 ]. 58

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0 5 10 15 20 25 30 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Tracking Error [degrees]Time [sec] Figure3-1.TrackingerrorfortheRISEcontrollawwithnoNN adaptation. transientresponsecouldbeobtainediftheNNweightsarein itializeddierently.Ten hiddenlayerneuronswerechosenbasedontrialanderror(i. e., N 2 =10).Inaddition,all thestateswereinitializedtozero Thefollowingcontrolgainswereusedtoimplementthe controllerin( 3{17 )inconjunctionwiththeNNupdatelawsin( 3{20 )and( 3{22 ): k s =30 ; 1 =10 ; 1 =10 ; 2 =10 ; 1 =5 I 11 ; 2 =0 : 05 I 4 : (3{60) 3.7.1Discussion Twodierentexperimentswereconductedtodemonstratethe ecacyoftheproposed controller.Thecontrolgainswerechosentoobtainanarbit rarytrackingerroraccuracy (notnecessarilythebestperformance).Foreachcontrolle r,thegainswerenotretuned (i.e.,thecommoncontrolgainsremainthesameforbothcont rollers).Fortherst experiment,noadaptationwasusedandthecontrollerwitho nlytheRISEfeedback wasimplemented.ThetrackingerrorisshowninFigure 3-1 ,andthecontroltorqueis showninFigure 3-2 .Forthesecondexperiment,theproposedNNcontrollerwasu sedin (hereinafterdenotedasRISE+NN).Thetrackingerrorissho wninFigure 3-3 ,andthe 59

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0 5 10 15 20 25 30 -15 -10 -5 0 5 10 15 Torque [N-m]Time [sec] Figure3-2.ControltorquefortheRISEcontrollawwithnoNN adaptation. 0 5 10 15 20 25 30 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Tracking Error [degrees]Time [sec] Figure3-3.TrackingerrorfortheproposedRISE+NNcontrol law. 60

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0 5 10 15 20 25 30 -15 -10 -5 0 5 10 15 Torque [N-m]Time [sec] Figure3-4.ControltorquefortheproposedRISE+NNcontrol law. Figure3-5.AverageRMSerrors(degrees)andtorques(N-m). 1-RISE,2-RISE+NN (proposed). controltorqueisshowninFigure 3-4 .Eachexperimentwasperformedvetimesandthe averageRMSerrorandtorquevaluesareshowninFigure 3-5 ,whichindicatethatthe proposedRISE+NNcontrolleryieldsalowerRMSerrorwithas imilarcontroleort. 3.8Conclusions TheresultsinthischapterillustratehowamultilayerNNfe edforwardtermcan befusedwithaRISEfeedbackterminacontinuouscontroller toachievesemi-global 61

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asymptotictracking.Improvedweighttuninglawsareprese ntedwhichguarantee boundednessofNNweights.ToblendtheNNandRISEmethods,s everaltechnical challengeswereaddressedthroughLyapunov-basedtechniq ues.Thesechallengesinclude developingadaptiveupdatelawsfortheNNweightestimates thatdonotdependon acceleration,anddevelopingnewRISEstabilityanalysism ethodsandsucientgain conditionstoaccommodatetheincorporationoftheNNadapt iveupdatesintheRISE structure.Experimentalresultsarepresentedthatindica tereducedRMStrackingerrors whilerequiringslightlyhigherRMScontroleort. 62

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CHAPTER4 ANEWCLASSOFMODULARADAPTIVECONTROLLERS 4.1Introduction Theresultsinthischapterprovidetherstinvestigationo ftheabilitytoyield controller/updatelawmodularityusingtheRISEfeedback. First,weconsiderageneral classofmulti-inputmulti-output(MIMO)dynamicsystemsw ithstructured(i.e.,LP) andunstructureduncertaintiesanddevelopacontrollerwi thmodularitybetween thecontroller/updatelaw,whereamodel-basedadaptivefe edforwardtermisused inconjunctionwiththeRISEfeedbackterm[ 19 ].TheRISE-basedmodularadaptive approachisdierentthanpreviouswork(cf.[ 40 41 43 ])inthesensethatitdoesnot relyonnonlineardamping.TheuseoftheRISEmethodinlieuo fnonlineardamping hasseveraladvantagesthatmotivatethisinvestigationin cluding:anasymptoticmodular adaptivetrackingresultcanbeobtainedfornonlinearsyst emswithnon-LPadditive boundeddisturbances;thedualobjectivesofasymptotictr ackingandcontroller/update lawmodularityareachievedinasinglestepunlikethetwost ageanalysisrequiredinsome results(cf.,[ 40 43 ]);thedevelopmentdoesnotrequirethattheadaptiveestim atesare aprioribounded;andthedevelopmentdoesnotrequireaposi tivedeniteestimateof theinertiamatrixorasquareintegrablepredictionerrora sin[ 40 43 ].Modularityinthe adaptivefeedforwardtermismadepossiblebyconsideringa genericformoftheadaptive updatelawanditscorrespondingparameterestimate.Thege neralformoftheadaptive updatelawincludesexamplessuchasgradient,least-squar es,andetc.Thisgenericform oftheupdatelawisusedtodevelopanewclosed-looperrorsy stem,andthetypicalRISE stabilityanalysisismodiedtoaccommodatethegenericup datelaw.Newsucientgain conditionsarederivedtoproveanasymptotictrackingresu lt. TheclassofRISE-basedmodularadaptivecontrollersisthe nextendedtoinclude uncertaindynamicsystemsthatdonotsatisfytheLPassumpt ion.Neuralnetworks(NNs) havegainedpopularityasafeedforwardadaptivecontrolme thodthatcancompensate 63

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fornon-LPuncertaintyinnonlinearsystems.Alimitingfac torinpreviousNN-based feedforwardcontrolresultsisthataresidualfunctionapp roximationerrorexiststhat limitsthesteady-stateperformancetoauniformlyultimat elyboundedresult,rather thananasymptoticresult.Someresults(cf.[ 33 { 36 72 { 74 ])havebeendevelopedto augmenttheNNfeedforwardcomponentwithadiscontinuousf eedbackelementtoachieve asymptotictracking.Motivatedbythepracticallimitatio nsofdiscontinuousfeedback, amultilayerNN-basedcontrollerwasaugmentedbyRISEfeed backin[ 75 ]toyieldthe rstasymptotictrackingresultusingacontinuouscontrol ler.However,inmostNN-based controllers,theNNadaptationisgovernedbyagradientupd atelawtofacilitatethe Lyapunov-basedstabilityanalysis. SincemultilayerNNsarenonlinearintheweights,achallen geistoderiveweight tuninglawsinclosed-loopfeedbackcontrolsystemsthatyi eldstabilityaswellasbounded weights.Thedevelopmentinthecurrentchapterillustrate showtoextendtheclassof modularadaptivecontrollersforNNs.Specically,theres ultallowstheNNweighttuning lawstobedeterminedfromadevelopedgenericupdatelaw(ra therthanberestrictedto agradientupdatelaw).Wearenotawareofanymodularmultil ayerNN-basedcontroller inliteraturewithmodularityinthetuninglaws/controlle r.TheNNfeedforwardstructure adaptivelycompensatesforthenon-LPuncertaindynamics. Forthetuninglawsthat couldbeusedinthisresult,theNNweightscanbeinitialize drandomly,andnoo-line trainingisrequired. Themodularadaptivecontroldevelopmentforthegeneralcl assofmulti-input systemsisthenappliedtoaclassofdynamicsystemsmodeled bytheEuler-Lagrange formulation.TheEuler-Lagrangeformulationdescribesth ebehaviorofalargeclass ofengineeringsystems(e.g.,robotmanipulators,satelli tes,vehicularsystems).An experimentalsectionisincludedthatillustratesthatdi erentadaptationlawscanbe includedwiththefeedbackcontrollerthroughexamplesinc ludingagradientupdatelaw andaleastsquaresupdatelawfortheLPdynamicscase,andac ommongradientweight 64

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tuninglawbasedonbackpropagatederror[ 76 ]andasimpliedtuninglawbasedonthe Hebbianalgorithm[ 77 ]forthenon-LPdynamicscase. Whilethecurrentresultencompassesalargevarietyofadap tiveupdatelaws,an updatelawdesignbasedonthepredictionerrorisnotpossib lebecausetheformulationof apredictionerrorrequiresthesystemdynamicstobecomple telyLP.Futureeortscan focusondevelopingaRISE-basedadaptivecontrollerforac ompletelyLPsystemthat couldalsouseapredictionerror/torquelteringapproach .Also,oneoftheshortcomings ofcurrentworkisthatonlyasemi-globalasymptoticstabil ityisachieved,andfurther investigationisneededtoachieveaglobalstabilityresul t.Inroadstosolvetheglobal trackingproblemareprovidedin[ 78 ]underasetofassumptions. 4.2DynamicSystem ConsideraclassofMIMOnonlinearsystemsofthefollowingf orm: x ( m ) = f ( x; x;:::;x ( m 1) )+ G ( x; x;:::;x ( m 1) ) u + h ( t ) ; (4{1) where( ) ( i ) ( t )denotesthe i th derivativewithrespecttotime, x ( i ) ( t ) 2 R n ;i =0 ;:::;m 1 arethesystemstates, u ( t ) 2 R n isthecontrolinput, f ( ) 2 R n and G ( ) 2 R n n are unknownnonlinear C 2 functions,and h ( t ) 2 R n denotesageneralnonlineardisturbance (e.g.,unmodeledeects).Theoutputsofthesystemarethes ystemstates.Throughout thechapter jj denotestheabsolutevalueofthescalarargument, kk denotesthestandard Euclideannormforavectorortheinducedinnitynormforam atrix,and kk F denotes theFrobeniusnormofamatrix. Thesubsequentdevelopmentisbasedontheassumptionthata llthesystemstatesare measurable.Moreover,thefollowingpropertiesandassump tionswillbeexploitedinthe subsequentdevelopment.Assumption4-1: G ( )issymmetricpositivedenite,andsatisesthefollowing inequality 8 y ( t ) 2 R n : g k y k 2 y T G 1 ( ) y g ( x; x;:::;x ( m 1) ) k y k 2 ; (4{2) 65

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where g 2 R isaknownpositiveconstant, g ( x; x;:::;x ( m 1) ) 2 R isaknownpositive function.Assumption4-2: Thefunctions G 1 ( )and f ( )aresecondorderdierentiablesuchthat G 1 ( ), G 1 ( ) ; G 1 ( ) ;f ( ) ; f ( ) ; f ( ) 2L 1 if x ( i ) ( t ) 2L 1 ;i =0 ; 1 ;:::;m +1. Assumption4-3: Thenonlineardisturbancetermanditsrsttwotimederivat ives(i.e., h ( t ) ; h ( t ) ; h ( t ))areboundedbyknownconstants. Assumption4-4: Theunknownnonlinearities G 1 ( )and f ( )arelinearintermsof unknownconstantsystemparameters(i.e.,LP).Assumption4-5: Thedesiredtrajectory x d ( t ) 2 R n isassumedtobedesignedsuchthat x ( i ) d ( t ) 2L 1 ;i =0 ; 1 ;:::;m +2. 4.3ControlObjective Theobjectiveistodesignacontinuousmodularadaptivecon trollerwhichensures thatthesystemtracksadesiredtime-varyingtrajectory x d ( t )despiteuncertaintiesand boundeddisturbancesinthedynamicmodel.Toquantifythis objective,atrackingerror, denotedby e 1 ( t ) 2 R n ,isdenedas e 1 x d x: (4{3) Tofacilitateacompactpresentationofthesubsequentcont roldevelopmentandstability analysis,auxiliaryerrorsignalsdenotedby e i ( t ) 2 R n ;i =2 ; 3 ;:::;m aredenedas e 2 e 1 + 1 e 1 e 3 e 2 + 2 e 2 + e 1 e 4 e 3 + 3 e 3 + e 2 ... e i e i 1 + i 1 e i 1 + e i 2 (4{4) ... 66

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e m e m 1 + m 1 e m 1 + e m 2 ; where i 2 R ;i =1 ; 2 ;:::;m 1denoteconstantpositivecontrolgains.Theerrorsignals e i ( t ) ;i =2 ; 3 ;:::;m canbeexpressedintermsof e 1 ( t )anditstimederivativesas e i = i 1 X j =0 a i;j e ( j ) 1 ; (4{5) wheretheconstantcoecients a i;j 2 R canbeevaluatedbysubstituting( 4{5 )in( 4{4 ), andcomparingcoecients[ 79 ].Alteredtrackingerror[ 57 ],denotedby r ( t ) 2 R n ,isalso denedas r e m + m e m ; (4{6) where m 2 R isapositive,constantcontrolgain.Thelteredtrackinge rror r ( t )isnot measurablesincetheexpressionin( 4{6 )dependson x ( m ) ( t ). 4.4ControlDevelopment Theopen-looptrackingerrorsystemisdevelopedbypremult iplying( 4{6 )by G 1 x; x;:::;x ( m 1) andutilizingtheexpressionsin( 4{1 ),( 4{4 ),( 4{5 )as G 1 r = Y d + S G 1 d h u (4{7) wherethefactthat a m;m 1 =1,wasused.In( 4{7 ), Y d 2 R n isdenedas Y d G 1 d x ( m ) d G 1 d f d ; (4{8) where Y d ( x d ; x d ;:::;x ( m ) d ) 2 R n p isadesiredregressionmatrix,and 2 R p containsthe constantunknownsystemparameters.In( 4{8 ),thefunctions G 1 d ( x d ; x d ;:::;x ( m 1) d ) 2 R n n ;f d ( x d ; x d ;:::;x ( m 1) d ) 2 R n aredenedas G 1 d G 1 ( x d ; x d ;:::;x ( m 1) d )(4{9) f d f ( x d ; x d ;:::;x ( m 1) d ) : 67

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Alsoin( 4{7 ),theauxiliaryfunction S x; x;:::;x ( m 1) ;t 2 R n isdenedas S G 1 m 2 X j =0 a m;j e ( j +1) 1 + m e m + G 1 x ( m ) d G 1 d x ( m ) d G 1 f + G 1 d f d G 1 h + G 1 d h: (4{10) Basedontheopen-looperrorsystemin( 4{7 ),thecontrolinputiscomposedofan adaptivefeedforwardtermplustheRISEfeedbacktermas u Y d ^ + : (4{11) In( 4{11 ), ( t ) 2 R n denotestheRISEfeedbacktermdenedas[ 19 22 ] ( t ) ( k s +1) e m ( t ) ( k s +1) e m (0)+ Z t 0 [( k s +1) m e m ( )+ 1 sgn ( e m ( ))] d; (4{12) where k s ; 1 2 R arepositiveconstantcontrolgains,and m 2 R wasintroducedin( 4{6 ). In( 4{11 ), ^ ( t ) 2 R p denotesasubsequentlydesignedparameterestimatevector .The closed-looptrackingerrorsystemisdevelopedbysubstitu ting( 4{11 )into( 4{7 )as G 1 r = Y d ^ + S G 1 d h : (4{13) Tofacilitatethesubsequentmodularadaptivecontroldeve lopmentandstabilityanalysis, thetimederivativeof( 4{13 )isexpressedas G 1 r = 1 2 G 1 r + ~ N ( t )+ N B ( t ) ( k s +1) r 1 sgn ( e m ) e m ; (4{14) wherethefactthatthetimederivativeof( 4{12 )isgivenas ( t )=( k s +1) r + 1 sgn ( e m )(4{15) wasutilized.In( 4{14 ),theunmeasurable/unknownauxiliaryterms ~ N ( e 1 ;e 2 ;:::;e m ;r;t ), N B ( t ) 2 R n aredenedas ~ N ( t ) 1 2 G 1 r + S + e m + ~ N 0 (4{16) 68

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N B ( t ) N B 1 ( t )+ N B 2 ( t ) ; (4{17) where N B 1 ( t ) 2 R n isgivenby N B 1 Y d G 1 d h G 1 d h; (4{18) andthesumoftheauxiliaryterms ~ N 0 ( t ), N B 2 ( t ) 2 R n isgivenby N B 2 ( t )+ ~ N 0 = Y d ^ Y d ^ : (4{19) Specicdenitionsfor ~ N 0 ( t ), N B 2 ( t )areprovidedsubsequentlybasedonthedenition oftheadaptiveupdatelawfor ^ ( t ).Thestructureof( 4{14 )andtheintroductionofthe auxiliarytermsin( 4{16 )-( 4{19 )ismotivatedbythedesiretosegregatetermsthatcan beupperboundedbystate-dependenttermsandtermsthatcan beupperboundedby constants.Specically,dependingonhowtheadaptiveupda telawisdesigned,analysisis providedinthenextsectiontoupperbound ~ N ( t )bystate-dependenttermsand N B ( t )by aconstant.Theneedtofurthersegregate N B ( t )isthatsometermsin N B ( t )havetime derivativesthatareupperboundedbyaconstant,whileothe rtermshavetime-derivatives thatareupper-boundedbystatedependentterms.Thesegreg ationofthesetermsbased onthestructureoftheadaptiveupdatelaw(see( 4{19 ))iskeyforthedevelopmentofa stabilityanalysisforthemodularRISE-basedadaptiveupd atelaw/controller. 4.5ModularAdaptiveUpdateLawDevelopment Akeydierencebetweenthetraditionalmodularadaptiveco ntrollersthatuse nonlineardamping(cf.,[ 1 41 63 ])andthecurrentRISE-basedapproachisthatthe RISE-basedmethoddoesnotexploittheISSpropertywithres pecttotheparameter estimationerror.Thecurrentapproachdoesnotrelyonnonl ineardamping,butinstead usestheabilityoftheRISEtechniquetocompensateforsmoo thboundeddisturbances. Ingeneral,previousnonlineardamping-basedmodularadap tivecontrollersrstprovean ISSstabilityresultprovidedtheadaptiveupdatelawyield sboundedparameterestimates (e.g., ^ ( t ) 2L 1 viaaprojectionalgorithm),andthenuseadditionalanalys isalongwith 69

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assumptions(PDestimateoftheinertiamatrix,squareinte grablepredictionerror,etc.) toconcludeasymptoticconvergence.Incontrast,sincethe RISE-basedmodularadaptive controlapproachinthischapterdoesnotexploitanISSanal ysis,theassumptions regardingtheparameterestimatearemodied.Thefollowin gdevelopmentrequiressome generalboundsonthestructureoftheadaptiveupdatelaw ^ ( t )andthecorresponding parameterestimate ^ ( t )tosegregatethecomponentsoftheauxiliarytermsintrodu ced in( 4{16 )-( 4{19 ).Specically,insteadofassumingthat ^ ( t ) 2L 1 ,thesubsequent developmentisbasedonthelessrestrictiveassumptiontha ttheparameterestimate ^ ( t ) canbedescribedas ^ ( t )= f 1 ( t )+( x; x;:::;x ( m 1) ;e 1 ;e 2 ;:::;e m ;t ) : (4{20) In( 4{20 ), f 1 ( t ) 2 R p isaknownfunctionsuchthat k f 1 ( t ) k r 1 (4{21) rrr f 1 ( t ) rrr r 2 + m X i =1 r i +2 k e i k + r m +3 k r k ; where r i 2 R i =1 ; 2 ;:::;m +3areknownnon-negativeconstants(i.e.,theconstantsca n besettozerofordierentupdatelaws),and x; x;:::;x ( m 1) ;e 1 ;e 2 ;:::;e m ;t 2 R p isa knownfunctionthatsatisesthefollowingbound: k ( t ) k 1 ( k e k ) k e k ; (4{22) wheretheboundingfunction 1 ( ) 2 R isapositive,globallyinvertible,nondecreasing function,and e ( t ) 2 R nm isdenedas e ( t ) e T1 e T2 :::e Tm T : (4{23) 70

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Theestimatein( 4{20 )isassumedtobegeneratedaccordingtoanupdatelawofthe followinggeneralform ^ ( t )= g 1 ( t )+n( x; x;:::;x ( m 1) ;e 1 ;e 2 ;:::;e m ;r;t ) : (4{24) In( 4{24 ), g 1 ( t ) 2 R p isaknownfunctionsuchthat k g 1 ( t ) k 1 (4{25) k g 1 ( t ) k 2 + m X i =1 i +2 k e i k + m +3 k r k ; where i 2 R i =1 ; 2 ;:::;m +3areknownnon-negativeconstants,andn( x; x;:::;x ( m 1) ; e 1 ;e 2 ;:::;e m ;r;t ) 2 R p satisesthefollowingbound: k n( t ) k 2 ( k z k ) k z k ; (4{26) wheretheboundingfunction 2 ( ) 2 R isapositive,globallyinvertible,nondecreasing function,and z ( t ) 2 R n ( m +1) isdenedas z ( t ) e T1 e T2 :::e Tm r T T : (4{27) Remark4.1. Theupdatelawin( 4{24 )dependsontheunmeasurablesignal r ( t ) .Butit isassumedthattheupdatelawin( 4{24 )isoftheformwhichuponintegrationyieldsan estimate ^ ( t ) thatisindependentof r ( t ) .Thusthecontrollerneedsonlythemeasurable signalsforimplementation. Thestructureoftheadaptiveestimateandtheadaptiveupda telawisrexibleinthe sensethatanyofthetermsin( 4{20 )and( 4{24 )canberemovedforanyspecicupdate lawandestimate.Forexampleifalltheerror-dependentter msin( 4{20 )areremoved, thentheconditionon ^ ( t )isthesameasinthestandardnonlineardamping-based modularadaptivemethods(i.e., ^ ( t ) 2L 1 ).Inthissense,theISSpropertywith respecttotheparameterestimationerrorisautomatically provenbyconsideringthis specialcaseof ^ ( t ).Theresultsinthischapterarenotprovenforestimatesor update 71

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lawswithadditionaltermsthatarenotincludedinthegener icstructurein( 4{20 )and ( 4{24 ).Forexample,astandardgradient-basedupdatelawisofth eform( 4{24 ),butthe correspondingestimate(obtainedviaintegrationbyparts )isnotoftheform( 4{20 )dueto thepresenceofsometermsthatareboundedbytheintegralof theerrorinsteadofbeing boundedbytheerror.However,thesamegradient-basedupda telawanditscorresponding estimatecanbeusedin( 4{11 )ifasmoothprojectionalgorithmisusedthatkeepsthe estimatesbounded.Asshownin[ 19 ],thestandardgradient-basedupdatelawcanbe usedin( 4{11 )withoutaprojectionalgorithm,yetincludingthisstruct ureinthemodular adaptiveanalysisisproblematicbecausetheintegralofth eerrorcouldbeunbounded (sothisupdatelawcouldnotbeusedinnonlineardamping-ba sedmodularadaptivelaws withoutaprojectioneither).Sincethegoalinthischapter istodevelopamodularupdate law,aspecicupdatelawcannotbeusedtoinjecttermsinthe stabilityanalysistocancel thetermscontainingtheparametermismatcherror.Instead ,thetermscontainingthe parametermismatcherroraresegregateddependingonwheth ertheyarestate-dependent orboundedbyaconstant(see( 4{19 )). Basedonthedevelopmentgivenin( 4{20 )-( 4{25 ),theterms ~ N 0 ( t )and N B 2 ( t ) introducedin( 4{16 )-( 4{19 )aredenedas ~ N 0 ( t ) Y d Y d n(4{28) N B 2 ( t ) Y d f 1 Y d g 1 : (4{29) InasimilarmannerasinLemma 1 oftheAppendix,byapplyingtheMeanValue Theoremalongwiththeinequalitiesin( 4{22 )and( 4{26 )yieldsanupperboundfor theexpressionin( 4{16 )as rrr ~ N ( t ) rrr ( k z k ) k z k ; (4{30) wheretheboundingfunction ( ) 2 R isapositive,globallyinvertible,nondecreasing function,and z ( t ) 2 R n ( m +1) isdenedin( 4{27 ).Thefollowinginequalitiesare developedbasedontheexpressionsin( 4{17 ),( 4{18 ),( 4{29 ),theirtimederivatives, 72

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andtheinequalitiesin( 4{21 )and( 4{25 ): k N B ( t ) k 1 rrr N B 1 ( t ) rrr 2 (4{31) rrr N B 2 ( t ) rrr 3 + m X i =1 i +3 k e i k + m +4 k r k ; where i 2 R i =1 ; 2 ;:::;m +4areknownpositiveconstants. 4.6StabilityAnalysis Theorem4-1: Thecontrollergivenin( 4{11 ),( 4{12 ),( 4{20 ),and( 4{24 )ensures thatallsystemsignalsareboundedunderclosed-loopopera tionandthattheposition trackingerrorisregulatedinthesensethat k e 1 ( t ) k! 0as t !1 providedthecontrolgain k s introducedin( 4{12 )isselectedsucientlylarge(seethe subsequentproof), i ;i =1 ; 2 ;:::;m areselectedaccordingtothefollowingconditions i > 1 2 i +1 ;i =1 ; 2 ;:::;m 2 m 1 > 1 2 m + 1 2 (4{32) m > m +1 + 1 2 m +2 + 1 2 m 1 X i =1 i +1 + 1 2 ; and i ;i =1 ; 2 ;:::;m +2areselectedaccordingtothefollowingsucientconditi ons: 1 > 1 + 1 m 2 + 1 m 3 (4{33) i +1 > i +3 ;i =1 ; 2 ;:::;m +1 ; where 1 wasintroducedin( 4{12 ),and 2 ;:::; m +2 areintroducedin( 4{36 ). Proof: Let D R n ( m +1)+1 beadomaincontaining y ( t )=0,where y ( t ) 2 R n ( m +1)+1 is denedas y ( t ) [ z T ( t ) p P ( t )] T : (4{34) 73

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In( 4{34 ),theauxiliaryfunction P ( t ) 2 R isdenedas P ( t ) 1 n X i =1 j e mi (0) j e m (0) T N B (0) Z t 0 L ( ) d; (4{35) where e mi (0)denotesthe i thelementofthevector e m (0),andtheauxiliaryfunction L ( t ) 2 R isdenedas L ( t ) r T ( N B ( t ) 1 sgn ( e m )) m X i =1 i +1 k e i ( t ) kk e m ( t ) k m +2 k e m ( t ) kk r ( t ) k ; (4{36) where i 2 R i =2 ; 3 ;:::;m +2arepositiveconstantschosenaccordingtothesucient conditionsin( 4{33 ).Providedthesucientconditionsintroducedin( 4{33 )aresatised, thefollowinginequalityisobtained 1 : Z t 0 L ( ) d 1 n X i =1 j e mi (0) j e m (0) T N B (0) : (4{37) Hence,( 4{37 )indicatesthat P ( t ) 0. Let V L ( y;t ): D [0 ; 1 ) R beacontinuouslydierentiable,positivedenite functiondenedas V L ( y;t ) 1 2 m X i =1 e Ti e i + 1 2 r T G 1 r + P; (4{38) whichsatisesthefollowinginequalities: U 1 ( y ) V L ( y;t ) U 2 ( y )(4{39) providedthesucientconditionsintroducedin( 4{32 )-( 4{33 )aresatised.TheRayleigh-Ritz theoremwasusedtodeveloptheinequalitiesin( 4{39 ),wherethecontinuouspositive denitefunctions U 1 ( y ), U 2 ( y ) 2 R aredenedas U 1 ( y ) 1 k y k 2 and U 2 ( y ) 2 ( x; x;:::;x ( m 1) ) k y k 2 ,where 1 2 ( x; x;:::;x ( m 1) ) 2 R aredenedas 1 1 2 min 1 ;g 2 ( x; x;:::;x ( m 1) ) max f 1 2 g ( x; x;:::;x ( m 1) ) ; 1 g ; (4{40) 1 Detailsoftheboundin( 4{37 )areprovidedintheLemma 1 oftheAppendix. 74

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where g g ( x; x;:::;x ( m 1) )areintroducedin( 4{2 ).Aftertakingthetimederivativeof ( 4{38 ), V ( y;t )isexpressedas V L ( y;t )= r T G 1 r + 1 2 r T G 1 r + m X i =1 e Ti e i + P: Thederivative P ( t ) 2 R isgivenby P ( t )= L ( t )= r T ( N B ( t ) 1 sgn ( e m ))+ m X i =1 i +1 k e i ( t ) kk e m ( t ) k + m +2 k e m ( t ) kk r ( t ) k : (4{41) Afterutilizing( 4{4 ),( 4{6 ),( 4{14 ),( 4{15 ),and( 4{41 ), V ( y;t )isexpressedas V L ( y;t )= 1 2 r T G 1 r m X i =1 i e Ti e i + e Tm 1 e m 1 2 r T G 1 r + r T ~ N + r T N B r T r k s r T r 1 r T sgn ( e m ) r T ( N B 1 sgn ( e m ))+ m X i =1 i +1 k e i kk e m k + m +2 k e m kk r k : Aftercancelingsimilarterms, V ( y;t )issimpliedas V L ( y;t )= m X i =1 i e Ti e i + e Tm 1 e m r T r k s r T r + r T ~ N + m X i =1 i +1 k e i kk e m k + m +2 k e m kk r k : Basedonthefactthat a T b 1 2 ( k a k 2 + k b k 2 )forsome a;b 2 R n V L ( y;t )isupperbounded usingthesquaresofthecomponentsof z ( t )as V L ( y;t ) m 2 X i =1 ( i 1 2 i +1 ) k e i k 2 ( m 1 1 2 m 1 2 ) k e m 1 k 2 ( m m +1 1 2 m +2 1 2 m 1 X i =1 i +1 1 2 ) k e m k 2 k r k 2 (4{42) ( k s 1 2 m +2 ) k r k 2 + r T ~ N: 75

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Byusing( 4{30 ),theexpressionin( 4{42 )isrewrittenas V L ( y;t ) 3 k z k 2 ( k s m +2 2 ) k r k 2 ( k z k ) k r kk z k ; (4{43) where 3 min f 1 1 2 2 ; 2 1 2 3 ;:::; m 2 1 2 m 1 ; m 1 1 2 m 1 2 ; m m +1 1 2 m +2 1 2 m 1 X i =1 i +1 1 2 ; 1 g : Aftercompletingthesquaresforthetermsinsidethebracke tsin( 4{43 ),thefollowing expressioncanbeobtained,providedthesucientgaincond itionsin( 4{32 )and( 4{33 ) aresatised: V L ( y;t ) 3 k z k 2 + 2 ( k z k ) k z k 2 4 k s m +2 2 : (4{44) Theexpressionin( 4{44 )canbefurtherupperboundedbyacontinuous,positive semi-denitefunction V L ( y;t ) U ( y )= c k z k 2 8 y 2D (4{45) forsomepositiveconstant c 2 R ,where D ( y 2 R n ( m +1)+1 jk y k 1 2 s 3 k s m +2 2 !) : Largervaluesof k willexpandthesizeofthedomain D .Theinequalitiesin( 4{39 )and ( 4{45 )indicatethat V L ( y;t ) 2L 1 in D ;hence, e i ( t ) 2L 1 ; and r ( t ) 2L 1 in D .Given that e i ( t ) 2L 1 ; and r ( t ) 2L 1 in D ,then_ e i ( t ) 2L 1 in D from( 4{4 )and( 4{6 ).Since e i ( t ) 2L 1 ; and r ( t ) 2L 1 in D ,theassumptionthat x ( i ) d ( t )existandareboundedand ( 4{3 )-( 4{6 )indicatethat x ( i ) ( t ) 2L 1 in D .Since x ( i ) ( t ) 2L 1 in D ,( 4{20 )-( 4{25 )indicate that ^ ( t ) ; ^ ( t ) 2L 1 in D ,and G 1 ( )and f ( ) 2L 1 in D fromProperty2.Thus,from ( 4{1 )andProperty3,wecanshowthat u ( t ) 2L 1 in D .Giventhat r ( t ) 2L 1 in D ( 4{15 )indicatesthat_ ( t ) 2L 1 in D .Since x ( i ) ( t ) 2L 1 in D ,then G 1 ( )and f ( ) 2L 1 76

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in D basedonProperty2;hence,( 4{14 )indicatesthat_ r ( t ) 2L 1 in D .Since_ e i ( t ) 2L 1 and_ r ( t ) 2L 1 in D ,then U ( y )isuniformlycontinuousin D basedonthedenitionsfor U ( y )and z ( t ). Let SD denoteasetdenedas S 8<: y ( t ) 2Dj U 2 ( y ( t )) < 1 1 2 s 3 k s m +2 2 !! 2 9=; : (4{46) Theregionofattractionin( 4{46 )isarbitrarilylargeandcanincludeanyinitialcondition byincreasingthecontrolgain k s (i.e.,asemi-globalstabilityresult).ByinvokingTheore m 8.4of[ 63 ] c k z ( t ) k 2 0 ast !18 y (0) 2S : (4{47) Basedonthedenitionof z ( t ),( 4{47 )indicatesthat k e 1 ( t ) k! 0 ast !18 y (0) 2S : (4{48) 4.7NeuralNetworkExtensiontoNon-LPSystems TheclassofRISE-basedmodularadaptivecontrollersdevel opedinthepreceding sectionsareextendedtoincludeuncertaindynamicsystems thatdonotsatisfytheLP assumption(i.e.,Assumption4-4isnotsatised).NN-base destimationmethodsarewell suitedforcontrolsystemswherethedynamicmodelcontains unstructurednonlinear disturbancesasin( 4{1 ).ThemainfeaturethatempowersNN-basedcontrollersisth e universalapproximationpropertyasdescribedinSection3 .4ofChapter3. 4.7.1RISEFeedbackControlDevelopment Themodularcontroldevelopmentandstabilityanalysisisp rovidedtoillustrate howtheaforementionedtextbook(e.g.,[ 15 ])NNfeedforwardestimationstrategycanbe fusedwithaRISEfeedbackcontrolmethodasameanstoachiev easymptoticstabilityfor generalclassofMIMOsystemsdescribedby( 4{1 )whileusinggenericNNweightupdate laws.Theopen-loopandclosed-looptrackingerrorisdevel opedforthecombinedcontrol system. 77

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Similarto( 4{7 ),theopen-looptrackingerrorsystemisdevelopedbypremu ltiplying ( 4{6 )by G 1 andutilizingtheexpressionsin( 4{1 )and( 4{4 )toobtain: G 1 r = f NN + S G 1 d h u; (4{49) wheretheauxiliaryfunction f NN ( x d ; x d ;:::;x ( m ) d ) 2 R n isdenedas f NN G 1 d x ( m ) d G 1 d f d ; (4{50) where G 1 d ( x d ; x d ;:::;x ( m 1) d ) 2 R n n and f d ( x d ; x d ;:::;x ( m 1) d ) 2 R n aredenedin( 4{9 ).In ( 4{49 ),theauxiliaryfunction S x; x;:::;x ( m 1) ;t 2 R n isdenedsimilarto( 4{10 ).The expressionin( 4{50 )canberepresentedbyathree-layerNNas f NN = W T V T x d + ( x d ) : (4{51) In( 4{51 ),theinput x d ( t ) 2 R ( m +1) n +1 isdenedas x d ( t ) [1 x Td ( t )_ x Td ( t ) :::x ( m ) T d ( t )] T sothat N 1 =( m +1) n where N 1 wasintroducedin( 3{4 ).Basedontheassumption thatthedesiredtrajectoryisbounded,theinequalitiesin ( 3{16 )hold.Basedonthe open-looperrorsystemin( 4{49 ),thecontroltorqueinputiscomposedofathree-layerNN feedforwardtermplustheRISEfeedbacktermas u ^ f NN + (4{52) wheretheRISEfeedbackterm 2 R n isdenedin( 4{12 ),andthefeedforwardNN componentdenotedby ^ f NN ( t ) 2 R n ,isdenedas ^ f NN ^ W T ( ^ V T x d ) : (4{53) 78

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4.7.2ModularTuningLawDevelopment TheestimatesfortheNNweightsin( 4{53 )aregeneratedon-line(thereisnoo-line learningphase)usingasmoothprojectionalgorithmas ^ W proj ( % 1 )= 8>>>><>>>>: % 1 if vec ( ^ W ) 2 int ( W ) % 1 if vec ( ^ W ) 2 @ ( W )and vec ( % 1 ) T vec ( ^ W ) ? 0 P t Mr ( % 1 )if vec ( ^ W ) 2 @ ( W )and vec ( % 1 ) T vec ( ^ W ) ? > 0 (4{54) ^ V proj ( % 2 )= 8>>>><>>>>: % 2 if vec ( ^ V ) 2 int ( V ) % 2 if vec ( ^ V ) 2 @ ( V )and vec ( % 2 ) T vec ( ^ V ) ? 0 P t Mr ( % 2 )if vec ( ^ V ) 2 @ ( V )and vec ( % 2 ) T vec ( ^ V ) ? > 0 ; (4{55) where proj ( )istheprojectionoperatorand vec ( ^ W (0)) 2 int ( W ) ;vec ( ^ V (0)) 2 int ( V ) : In( 4{54 )and( 4{55 ),theauxiliaryterms % 1 ( t ) 2 R ( N 2 +1) n and % 2 ( t ) 2 R ( N 1 +1) N 2 denote adaptationrulesofthefollowinggeneralform: proj ( % 1 )= w 1 ( t )+ W ( x; x;:::;x ( m 1) ;e 1 ;e 2 ;:::;e m ;r;t )(4{56) proj ( % 2 )= v 1 ( t )+ V ( x; x;:::;x ( m 1) ;e 1 ;e 2 ;:::;e m ;r;t ) : In( 4{56 ), w 1 ( t ) 2 R ( N 2 +1) n and v 1 ( t ) 2 R ( N 1 +1) N 2 areknownfunctionssuchthat k w 1 ( t ) k r 1 (4{57) k w 1 ( t ) k r 2 + m X i =1 r i +2 k e i k + r m +3 k r k k v 1 ( t ) k 1 (4{58) k v 1 ( t ) k 2 + m X i =1 i +2 k e i k + m +3 k r k ; 79

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and W 2 R ( N 2 +1) n and V 2 R ( N 1 +1) N 2 satisfythefollowingbounds: k W ( t ) k m X i =1 r i + m +3 k e i k + r 2 m +4 k r k (4{59) k V ( t ) k m X i =1 i + m +3 k e i k + 2 m +4 k r k ; where r i ; i 2 R i =1 ; 2 ;:::; 2 m +4areknownnon-negativeconstants(i.e.,the constantscanbesettozerofordierentupdatelaws).In( 4{54 )and( 4{55 ), P t Mr ( A )= devec ( P t r ( vec ( A )))foramatrix A ,wheretheoperation devec ( )isthereverseof vec ( ). Theuseoftheprojectionalgorithmin( 4{54 )and( 4{55 )istoensurethat ^ W ( t )and ^ V ( t )remainboundedinsidetheconvexregionsdenedin( 3{9 )and( 3{11 ).Thisfact willbeexploitedinthesubsequentstabilityanalysis.Thu s,unlikethegeneralformfor parameterestimate ^ ( t )in( 4{20 )fortheLPcase,theNNweightestimatesarebounded byconstants.TheNNweightadaptationlawsarerestrictive comparedtotheadaptive lawsfortheparameterestimatesintheLPcase.Thisisbecau setherstlayerweight estimates ^ V ( t )areembeddedinsidethenonlinearactivationfunction ( )(i.e., ( ^ V T x d )). TypicallytheNNactivationfunctionsareboundedoverthee ntiredomain;however,their timederivativesdependontheadaptationlaw ^ V ( t ),whichcouldbestate-dependent. SimilartotheLPcase,itisassumedthatonlytheNNadaptati onrulesdependonthe unmeasurablesignal r ( t )butthecorrespondingweightestimateobtainedafterinte gration isindependentof r ( t ). Theclosed-looptrackingerrorsystemcanbedevelopedbysu bstituting( 4{52 )into ( 4{49 )as G 1 r = f NN ^ f NN + S G 1 d h : (4{60) Tofacilitatethesubsequentstabilityanalysis,thetimed erivativeof( 4{60 )isdetermined as G 1 r = G 1 r + f NN ^ f NN + S G 1 d h G 1 d h : (4{61) 80

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Using( 4{51 ),( 4{53 ),theclosed-looperrorsystemin( 4{61 )canbeexpressedas G 1 r = G 1 r + W T 0 V T x d V T x d ^ W T ( ^ V T x d )(4{62) ^ W T 0 ( ^ V T x d )( ^ V T x d + ^ V T x d )+_ + S G 1 d h G 1 d h ; where 0 ( ^ V T x ) d V T x =d V T x j V T x = ^ V T x .Afteraddingandsubtractingtheterm W T ^ 0 ^ V T x d + ^ W T ^ 0 ~ V T x d to( 4{62 ),thefollowingexpressioncanbeobtained: G 1 r = G 1 r + ^ W T ^ 0 ~ V T x d + ~ W T ^ 0 ^ V T x d + W T 0 V T x d W T ^ 0 ^ V T x d (4{63) ^ W T ^ 0 ~ V T x d ^ W T ^ ^ W T ^ 0 ^ V T x d +_ + S G 1 d h G 1 d h ; wherethenotations^ and~ areintroducedin( 3{6 ).SubstitutingtheNNweight adaptationlawsin( 4{54 ),( 4{55 )in( 4{63 )yields G 1 r = 1 2 G 1 r + ~ N + N B e m ( k s +1) r 1 sgn ( e m ) ; (4{64) where( 4{15 )wasutilized,andtheunmeasurableauxiliaryterms ~ N ( e 1 ;e 2 ;:::;e m ;r;t ), N B ( ^ W; ^ V; x d ; x d ;t ) 2 R n aredenedas ~ N 1 2 G 1 r TW ^ ^ W T ^ 0 TV x d + S + e m (4{65) N B N B 1 + N B 2 : (4{66) In( 4{66 ), N B 1 ( x d ; x d ;t ), N B 2 ( ^ W; ^ V; x d ; x d ;t ) 2 R n aregivenby N B 1 = W T 0 V T x d +_ G 1 d h G 1 d h (4{67) N B 2 = ^ W T ^ 0 ~ V T x d + ~ W T ^ 0 ^ V T x d w T 1 ^ ^ W T ^ 0 v T 1 x d (4{68) Inasimilarmannerasbefore,applicationoftheMeanValueT heoremwillyieldan upperboundfor ~ N ( t )asin( 4{30 ).Thefollowinginequalitiesaredevelopedbasedon Assumptions4-2and4-3,( 3{7 ),( 3{8 ),( 3{16 ),( 4{56 )-( 4{59 ),and( 4{66 )-( 4{68 ): k N B k 1 rrr N B 1 rrr 2 (4{69) 81

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rrr N B 2 rrr 3 + m X i =1 i +3 k e i k + m +4 k r k ; (4{70) where i 2 R i =1 ; 2 ;:::;m +4areknownpositiveconstants. 4.8StabilityAnalysis Theorem4-2: ThecombinedNNandRISEcontrollergivenin( 4{52 )-( 4{55 )ensures thatallsystemsignalsareboundedunderclosed-loopopera tionandthattheposition trackingerrorisregulatedinthesensethat k e 1 ( t ) k! 0as t !1 providedsimilargainsconditionsasinTheorem4-1aresati sed.TheproofofTheorem 4-2issimilartoTheorem4-1. 4.9ApplicationtoEuler-LagrangeSystems TheEuler-Lagrangeformulationdescribesthebehaviorofa largeclassofengineering systems.Inthissection,themodularadaptivecontroldeve lopmentforthegeneralclass ofMIMOdynamicsystemsisappliedtodynamicsystemsmodele dbytheEuler-Lagrange formulation M ( q ) q + V m ( q; q )_ q + G ( q )+ F (_ q )+ d ( t )= ( t ) : (4{71) In( 4{71 ), M ( q ) 2 R n n denotestheinertiamatrix, V m ( q; q ) 2 R n n denotesthe centripetal-Coriolismatrix, G ( q ) 2 R n denotesthegravityvector, F (_ q ) 2 R n denotes friction, d ( t ) 2 R n denotesageneralnonlineardisturbance(e.g.,unmodelede ects), ( t ) 2 R n representsthetorqueinputcontrolvector,and q ( t ),_ q ( t ), q ( t ) 2 R n denotethe linkposition,velocity,andaccelerationvectors,respec tively.Thecontroldevelopmentis basedontheassumptionthat q ( t )and_ q ( t )aremeasurableandthat M ( q ), V m ( q; q ) ;G ( q ), F (_ q )and d ( t )areunknown.Themodularadaptivecontrollerdevelopedin theearlier sectionsfortheageneralclassofMIMOcanbeeasilyapplied toEuler-Lagrangesystems. Pleasesee[ 80 ],[ 81 ]forcompletedetailsofthecontroldevelopment. 82

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Figure4-1.Theexperimentaltestbedconsistsofatwo-link robot.Thelinksaremounted ontwoNSKdirect-driveswitchedreluctancemotors. 4.10Experiment Toinvestigatetheperformanceofthemodularcontrollerde velopedinthischapter, anexperimentwasperformedonatwo-linkrobottestbedasde pictedinFig. 4-1 .The testbediscomposedofatwo-linkdirectdriverevoluterobo tconsistingoftwoaluminum links,mountedona240 : 0Nm(basejoint)and20 : 0Nm(secondjoint)switchedreluctance motors.Themotorsarecontrolledthroughpowerelectronic soperatingintorquecontrol mode.Themotorresolversproviderotorpositionmeasureme ntswitharesolutionof 614 ; 400pulses/revolution,andastandardbackwardsdierence algorithmisusedto numericallydeterminevelocityfromtheencoderreadings. APentium2 : 8GHzPC operatingunderQNXhoststhecontrolalgorithm,whichwasi mplementedviaacustom graphicaluser-interface[ 64 ],tofacilitatereal-timegraphing,datalogging,andthea bility toadjustcontrolgainswithoutrecompilingtheprogram.Da taacquisitionandcontrol implementationwereperformedatafrequencyof1 : 0kHzusingtheServoToGoI/Oboard. 83

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Thedynamicsforthetestbedare 264 1 2 375 = 264 p 1 +2 p 3 c 2 p 2 + p 3 c 2 p 2 + p 3 c 2 p 2 375 264 q 1 q 2 375 + 264 p 3 s 2 q 2 p 3 s 2 (_ q 1 +_ q 2 ) p 3 s 2 q 2 0 375 264 q 1 q 2 375 (4{72) + f (_ q )+ 264 d 1 d 2 375 ; wherethenonlinearfrictiontermisassumedtobemodeledas [ 19 26 ] f (_ q )= 264 r 1 (tanh( r 2 q 1 ) tanh( r 3 q 1 )) r 1 (tanh( r 2 q 2 ) tanh( r 3 q 2 )) 375 + 264 r 4 tanh( r 5 q 1 )+ r 6 q 1 r 4 tanh( r 5 q 2 )+ r 6 q 2 375 : (4{73) In( 4{72 )and( 4{73 ), p 1 ;p 2 ;p 3 ;r i 2 R ,( i =1 ; 2 ;:::; 6)areunknownpositiveconstant parameters, c 2 denotes cos ( q 2 ) ;s 2 denotes sin ( q 2 ),and d 1 ; d 2 2 R denotegeneral nonlineardisturbances(e.g.,unmodeledeects).Partoft hedynamicsin( 4{72 )and ( 4{73 )islinearinthefollowingparameters: =[ p 1 p 2 p 3 r 1 r 4 r 6 ] T : Theparameters r 2 ;r 3 ;r 5 areembeddedinsidethenonlinearhyperbolictangent functionsandhencecannotbelinearlyparameterized.Sinc etheseparameterscannot becompensatedforbyanadaptivealgorithm,best-guessest imates r 2 =50 ; r 3 =1 ; r 5 =50areused.Thevaluesfor r 2 ; r 3 ; r 5 arebasedonourpreviousexperiments concernedwithfrictionidentication.Signicanterrors inthesestaticestimatescould degradetheperformanceofthesystem.Anadvantageofusing theNN-basedcontroller developedinnon-LPextensionsectionisthattheNNcancomp ensateforthenon-LP dynamics.Specically,fortheNN-basedcontrollersteste dinthissection,theNNisused toestimatethefrictionmodel,andthebestguessvaluesfor r 2 ; r 3 ; r 5 arenotrequired. Thecontrolobjectiveistotrackthedesiredtime-varyingt rajectorybyusingtheproposed 84

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modularadaptivecontrollaw.Twodierentupdatelawswere chosenforboththeLP andnon-LPcases.Toachievethiscontrolobjective,thecon trolgains 1 ; 2 ;k s ,and 1 ; denedasscalarsin( 4{4 ),( 4{6 ),and( 4{12 ),wereimplemented(withnon-consequential implicationstothestabilityresult)asdiagonalgainmatr ices.Specically,thecontrol gainsforbothadaptiveupdatelawsforboththeLPandnon-LP caseswereselectedas 1 = diag f 70 ; 70 g 2 = diag f 25 ; 25 g 1 = diag f 10 ; 0 : 1 g k s = diag f 100 ; 20 g : (4{74) 4.10.1ModularAdaptiveUpdateLaw Thedesiredtrajectoriesforthisexperimentwerechosenas q d 1 = q d 2 =60sin(2 : 0 t ) 1 exp 0 : 01 t 3 : (4{75) Totestthemodularityofthecontroller,twoseparateadapt iveupdatelawsweretested includingastandardgradientupdatelawdenedas ^ = proj Y T d r ; where 2 R 6 6 isadiagonalpositive-denitegainmatrix,andaleastsqua resupdatelaw denedas ^ = proj PY T d r ; P = PY T d Y d P where P ( t ) 2 R 6 6 isatime-varyingsymmetricmatrix.Theparameterestimate swereall initializedtozero.Inpractice,theadaptiveestimateswo uldbeinitializedtoabest-guess estimateofthevalues.Initializingtheestimatestozerow asdonetotestascenarioofno parameterknowledge.Forthegradientandleastsquaresupd atelawstheadaptationgains andtheinitialvalueof P ( t )wereselectedas = P (0)= diag ([0 : 15 ; 0 : 01 ; 0 : 01 ; 0 : 01 ; 0 : 01 ; 0 : 2]) : (4{76) 85

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Table4-1.LPcase:AverageRMSvaluesfor10trials Gradient LeastSquares AverageRMSError(deg)-Link1 0 : 0203 0 : 0218 AverageRMSError(deg)-Link2 0 : 0120 0 : 0131 AverageRMSTorque(Nm)-Link1 10 : 3068 10 : 4125 AverageRMSTorque(Nm)-Link2 1 : 5770 1 : 5959 ErrorStandardDeviation(deg)-Link1 0 : 0016 0 : 0011 ErrorStandardDeviation(deg)-Link2 0 : 0021 0 : 0014 TorqueStandardDeviation(Nm)-Link1 0 : 1545 0 : 1029 TorqueStandardDeviation(Nm)-Link2 0 : 0797 0 : 0677 Theadaptationgainin( 4{76 )isgraduallyincreasedbytrialanderrortuningprocedure basedonourpreviousexperiencetoachievefasteradaptati onuntilapointwhenno signicantperformanceimprovementisnoticedwithoutcau singunnecessaryoscillations intheparameterestimates.Eachexperimentwasperformedt entimesandfollowing statisticaldataisprovidedinTable4-1. Figure 4-2 depictsthetrackingerrorsforoneexperimentaltrialwith agradient updatelaw.Thecontroltorquesandadaptiveestimatesfort hesameexperimentaltrial areshowninFigs. 4-3 and 4-4 ,respectively.Thetrackingerrorsforarepresentative experimentaltrialwithaleastsquaresupdatelawisdepict edinFigure 4-5 .Thetorques fortheleastsquaresupdatelawareshowninFigure 4-6 .Theadaptiveestimatesforthe leastsquaresupdatelawareshowninFigure 4-6 4.10.2ModularNeuralNetworkUpdateLaw Thedesiredtrajectoriesforthisexperimentwerechosenas q d 1 = q d 2 =60sin(2 : 5 t ) 1 exp 0 : 01 t 3 : (4{77) Totestthemodularityofthecontroller,twoseparateneura lnetworktuninglawswere simulatedincludingastandardgradienttuninglawbasedon backpropagatederrors[ 76 ] 86

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0 5 10 15 20 -0.2 -0.1 0 0.1 0.2 Tracking Error [degrees]Link 1 0 5 10 15 20 -0.2 -0.1 0 0.1 0.2 Time [sec]Tracking Error [degrees]Link 2 Figure4-2.Linkpositiontrackingerrorwithagradient-ba sedadaptiveupdatelaw. 0 5 10 15 20 -20 0 20 Torque [N-m]Link 1 0 5 10 15 20 -5 0 5 Time [sec]Torque [N-m]Link 2 Figure4-3.Torqueinputforthemodularadaptivecontrolle rwithagradient-based adaptiveupdatelaw. 87

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0 5 10 15 20 -0.5 0 0.5 1 1.5 2 2.5 Parameter EstimatesTime [sec] Figure4-4.Adaptiveestimatesforthegradientupdatelaw. 0 5 10 15 20 -0.2 -0.1 0 0.1 0.2 Tracking Error [degrees]Link 1 0 5 10 15 20 -0.2 -0.1 0 0.1 0.2 Time [sec]Tracking Error [degrees]Link 2 Figure4-5.Linkpositiontrackingerrorwithaleast-squar esadaptiveupdatelaw. 88

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0 5 10 15 20 -20 0 20 Torque [N-m]Link 1 0 5 10 15 20 -5 0 5 Time [sec]Torque [N-m]Link 2 Figure4-6.Torqueinputforthemodularadaptivecontrolle rwithaleast-squaresadaptive updatelaw. 0 5 10 15 20 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Parameter EstimatesTime [sec] Figure4-7.Adaptiveestimatesfortheleastsquaresupdate law. 89

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Table4-2.Non-LPcase:AverageRMSvaluesfor10trials Gradient Hebbian AverageRMSError(deg)-Link1 0 : 0221 0 : 0216 AverageRMSError(deg)-Link2 0 : 0173 0 : 0153 AverageRMSTorque(Nm)-Link1 12 : 5369 12 : 5223 AverageRMSTorque(Nm)-Link2 2 : 2129 2 : 1272 ErrorStandardDeviation(deg)-Link1 0 : 0006 0 : 0011 ErrorStandardDeviation(deg)-Link2 0 : 0009 0 : 0009 TorqueStandardDeviation(Nm)-Link1 0 : 0512 0 : 0429 TorqueStandardDeviation(Nm)-Link2 0 : 1129 0 : 0945 andaHebbiantuninglaw[ 77 ].Thegradientupdatelawisdenedas ^ W = proj F ^ 0 ^ V T x d e T2 ^ V = proj G x d ^ 0 T ^ We 2 T ; where F 2 R 11 11 and G 2 R 7 7 aregainmatrixesand ( ) 2 R 11 isasigmoidalactivation function,andtheinputvector x d ( t ) 2 R 7 isdenedas x d =[1 q d 1 q d 2 q d 1 q d 2 q d 1 q d 2 ] T : TheHebbianupdatelawisdenedas ^ W = proj F ^ e T2 ^ V = proj G x d ^ T : Theadaptationgainsforbothestimateswereselectedas F =20 I 11 G =0 : 5 I 7 ; where I 11 2 R 11 11 and I 7 2 R 7 7 areidentitymatrixes.AswiththeLPcase,theinitial valuesof ^ W (0)werechosentotobeazeromatrix;however,theinitialva luesof ^ V (0) wereselectedrandomlybetween 1 : 0and1 : 0toprovideabasis[ 71 ].Adierenttransient responsecouldbeobtainediftheNNweightsareinitialized dierently. 90

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0 5 10 15 20 -0.2 -0.1 0 0.1 0.2 Tracking Error [degrees]Link 1 0 5 10 15 20 -0.2 -0.1 0 0.1 0.2 Time [sec]Tracking Error [degrees]Link 2 Figure4-8.LinkpositiontrackingerrorforthemodularNNc ontrollerwitha gradient-basedtuninglaw. 0 5 10 15 20 -20 0 20 Torque [N-m]Link 1 0 5 10 15 20 -5 0 5 Time [sec]Torque [N-m]Link 2 Figure4-9.TorqueinputforthemodularNNcontrollerwitha gradient-basedtuninglaw. Figure 4-8 depictsthetrackingerrorsforoneexperimentaltrialwith agradient updatelaw.Thecontroltorquesforthesameexperimentaltr ialareshowninFigure 4-9 Thetrackingerrorsforarepresentativeexperimentaltria lwithaHebbianupdatelaw isdepictedinFigure 4-10 .ThetorquesfortheHebbianupdatelawareshowninFigure 4-11 91

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0 5 10 15 20 -0.2 -0.1 0 0.1 0.2 Tracking Error [degrees]Link 1 0 5 10 15 20 -0.2 -0.1 0 0.1 0.2 Time [sec]Tracking Error [degrees]Link 2 Figure4-10.LinkpositiontrackingerrorforthemodularNN controllerwithaHebbian tuninglaw. 0 5 10 15 20 -20 0 20 Torque [N-m]Link 1 0 5 10 15 20 -5 0 5 Time [sec]Torque [N-m]Link 2 Figure4-11.TorqueinputforthemodularNNcontrollerwith aHebbiantuninglaw. 4.10.3Discussion Datafromthetwosetsofexperimentsillustratethatdiere ntadaptiveupdate laws(whicharespecicpartsofthemoregeneralupdatelaws consideredinthecontrol development)canbeusedfortheproposedmodularadaptivec ontrollerbothfortheLP andnon-LPcase.Thedierentupdatelawscanbeusedwithnoc hangerequiredinthe overallstructureofthecontrollerandthestabilityisgua ranteedthroughtheLyapunov analysis.Thestabilityresultsforpreviouscontinuousmo dularadaptivecontrollerswould 92

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nothavebeenvalidforthedynamicmodeldevelopedforthisr obot.Specically,the advancedfrictionmodelderivedfrom[ 26 ]containsnon-LPdisturbancesthatcannotbe consideredinpreviousresults. InExperiment1,thesystemdynamicsareassumedtobepartia llyLP.The model-basedadaptivecontrollerin( 4{11 )isimplementedwithtwodierentupdate laws-thestandardgradient-basedandtheleast-squaresup datelaw.Ourexperimental resultsconrmthewidelyacknowledgedfactthatleast-squ aresupdatelawsyieldfaster convergenceofparameterestimatescomparedtothegradien t-basedupdatelaws.The parameterestimatesconvergemuchfasterwhenusingthelea st-squareslawascompared tothegradient-basedlaw(cf.Figure 4-4 andFigure 4-7 ).Butthisfasterconvergenceof parameterestimatesdoesnotseemtobebenecial,astheove ralltrackingperformance andcontroleortsrequiredaresimilarforbothcases(seeT able4-1). InExperiment2,thesystemdynamicsarenotassumedtobeLP. TheNN-based controllerin( 4{52 )isimplementedwithtwodierentupdatelaws,specically the gradient-basedandtheHebbianlaw.Oneoutcomeoftheexper imentalresultsisthefact thattheHebbianupdatelawsgiveatrackingperformancetha tisonparcomparedtothat ofthegradient-basedlaws(seeTable4-2).Thisresultissi gnicantbecauseingeneralitis diculttoprovestabilityandguaranteeperformanceusing theHebbianlaws[ 15 ].Onthe contrary,thegradient-basedlawsaredesignedtocancelso mecrosstermsintheLyapunov stabilityanalysis[ 32 75 ],thusfacilitatingtheanalysisandguaranteeingstabili ty.Butthe gradient-basedlawshaveacomplicatedformandrequirethe computationoftheJacobian oftheactivationfunction.Hebbianlaws,ontheotherhand, haveasimplestructureand donotrequirethecomputationoftheJacobianoftheactivat ionfunction.Thefactthat asimplerupdatelawyieldsbetterperformanceinourexperi mentsisinterestingandmay leadtoagreateruseoftheHebbian(orothersimpler)update lawsinNN-basedcontrol. 93

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4.11Conclusion ARISE-basedapproachwaspresentedtoachievemodularityi nthecontroller/update lawforageneralclassofmulti-inputsystems.Specically ,forsystemswithstructured andunstructureduncertainties,acontrollerwasemployed thatusesamodel-based feedforwardadaptiveterminconjunctionwiththeRISEfeed backterm(see[ 19 ]).The adaptivefeedforwardtermwasmademodularbyconsideringa genericformofthe adaptiveupdatelawanditscorrespondingparameterestima te.Thisgenericformof theupdatelawwasusedtodevelopanewclosed-looperrorsys tem,andthetypical RISEstabilityanalysiswasmodied.Newsucientgaincond itionswerederivedto showasymptotictrackingofthedesiredlinkposition.Thec lassofRISE-basedmodular adaptivecontrollersisthenextendedtoincludeuncertain dynamicsystemsthatdonot satisfytheLPassumption.Specically,theresultallowst heNNweighttuninglaws tobedeterminedfromadevelopedgenericupdatelaw(rather thanbeingrestrictedto agradientupdatelaw).Themodularadaptivecontroldevelo pmentisthenappliedto Euler-Lagrangedynamicsystems.Anexperimentalsectioni sincludedthatillustratesthe concept. 94

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CHAPTER5 COMPOSITEADAPTIVECONTROLFORSYSTEMSWITHADDITIVE DISTURBANCES 5.1Introduction Applyingtheswappingapproachtodynamicswithadditivedi sturbancesis problematicbecausetheunknowndisturbancetermsalsoget lteredandincludedin thelteredcontrolinput.Thisproblemmotivatesthequest ionofhowcanaprediction error-basedadaptiveupdatelawbedevelopedforsystemswi thadditivedisturbances. Toaddressthismotivatingquestion,ageneralEuler-Lagra nge-likeMIMOsystem isconsideredwithstructuredandunstructureduncertaint ies,andagradient-based compositeadaptiveupdatelawisdevelopedthatisdrivenby boththetrackingerrorand thepredictionerror.Thecontroldevelopmentisbasedonth erecentcontinuousRobust IntegraloftheSignoftheError(RISE)[ 19 ]techniquethatwasoriginallydevelopedin [ 21 ]and[ 22 ].TheRISEarchitectureisadoptedsincethismethodcanacc ommodatefor C 2 disturbancesandyieldasymptoticstability.Forexample, theRISEtechniquewasused in[ 23 ]todevelopatrackingcontrollerfornonlinearsystemsint hepresenceofadditive disturbancesandparametricuncertainties.Basedonthewe llacceptedheuristicnotion thattheadditionofsystemknowledgeinthecontrolstructu reyieldsbetterperformance andreducescontroleort,model-basedadaptiveandneural networkfeedforwardelements wereaddedtotheRISEcontrollerin[ 19 ]and[ 75 ],respectively.Incomparisontothese approachesthatusedtheRISEmethodinthefeedbackcompone ntofthecontroller,the RISEstructureisusedinboththefeedbackandfeedforwarde lementsofthecontrol structuretoenable,forthersttime,theconstructionofa predictionerrorinthe presenceofadditivedisturbances.Specically,sincethe swappingmethodwillresult indisturbancesinthepredictionerror(themainobstaclet hathaspreviouslylimitedthis development),aninnovativeuseoftheRISEstructureisals oemployedintheprediction errorupdate(i.e.,thelteredcontrolinputestimate).Ab lockdiagramindicatingthe uniqueuseoftheRISEmethodinthecontrolandthepredictio nerrorformulationis 95

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providedinFigure 5-1 .Sucientgainconditionsaredevelopedunderwhichthisun ique doubleRISEcontrollerguaranteessemi-globalasymptotic tracking.Experimentalresults arepresentedtoillustratetheperformanceoftheproposed approach. TheasymptoticstabilityfortheproposedRISE-basedcompo siteadaptivecontroller comesattheexpenseofachievingsemi-globalstabilitywhi chrequirestheinitialcondition tobewithinaspeciedregionofattractionthatcanbemadel argerbyincreasing certaingainsassubsequentlydiscussedinSection 5.5 .Developmentisalsoprovided thatprovesthepredictionerrorissquareintegrable;yet, noconclusioncanbedrawn abouttheconvergenceoftheparameterestimationerrordue tothepresenceofltered additivedisturbancesinthepredictionerror.Thepropose dmethodusesagradient-based compositeadaptivelawwithaxedadaptationgain.Futuree ortscouldalsofocuson designingacompositelawwithleast-squaresestimationwi thtime-varyingadaptationgain fortheconsideredclassofsystems. 5.2DynamicSystem ConsideraclassofMIMOnonlinearEuler-Lagrangesystemso fthefollowingform: x ( m ) = f ( x; x;:::;x ( m 1) )+ G ( x; x;:::;x ( m 2) ) u + h ( t )(5{1) where( ) ( i ) ( t )denotesthe i th derivativewithrespecttotime, x ( i ) ( t ) 2 R n ;i =0 ;:::;m 1 arethesystemstates, u ( t ) 2 R n isthecontrolinput, f x; x;:::;x ( m 1) 2 R n and G x; x;:::;x ( m 2) 2 R n n areunknownnonlinear C 2 functions,and h ( t ) 2 R n denotes ageneralnonlineardisturbance(e.g.,unmodeledeects). Throughoutthechapter, jj denotestheabsolutevalueofthescalarargument, kk denotesthestandardEuclidean normforavectorortheinducedinnitynormforamatrix. Thesubsequentdevelopmentisbasedontheassumptionthata llthesystemstates aremeasurableoutputs.Moreover,thefollowingassumptio nswillbeexploitedinthe subsequentdevelopment. 96

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Assumption5-1: G ( )issymmetricpositivedenite,andsatisesthefollowing inequality 8 y ( t ) 2 R n : g k y k 2 y T G 1 y g ( x; x;:::;x ( m 2) ) k y k 2 (5{2) where g 2 R isaknownpositiveconstant,and g ( x; x;:::;x ( m 2) ) 2 R isaknownpositive function.Assumption5-2: Thefunctions G 1 ( )and f ( )aresecondorderdierentiablesuchthat G 1 ; G 1 ; G 1 ;f; f; f 2L 1 if x ( i ) ( t ) 2L 1 ;i =0 ; 1 ;:::;m +1. Assumption5-3: Thenonlineardisturbancetermanditsrsttwotimederivat ives(i.e., h; h; h )areboundedbyknownconstants. Assumption5-4: Theunknownnonlinearities G 1 ( )and f ( )arelinearintermsof unknownconstantsystemparameters(i.e.,LP).Assumption5-5: Thedesiredtrajectory x d ( t ) 2 R n isassumedtobedesignedsuchthat x ( i ) d ( t ) 2L 1 ;i =0 ; 1 ;:::;m +2.Thedesiredtrajectory x d ( t )neednotbepersistently excitingandcanbesettoaconstantvaluefortheregulation problem. 5.3ControlObjective Theobjectiveistodesignacontinuouscompositeadaptivec ontrollerwhich ensuresthatthesystemstate x ( t )tracksadesiredtime-varyingtrajectory x d ( t )despite uncertaintiesandboundeddisturbancesinthedynamicmode l.Toquantifythisobjective, atrackingerror,denotedby e 1 ( t ) 2 R n ,isdenedas e 1 x d x: (5{3) 97

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Tofacilitateacompactpresentationofthesubsequentcont roldevelopmentandstability analysis,auxiliaryerrorsignalsdenotedby e i ( t ) 2 R n ;i =2 ; 3 ;:::;m aredenedas e 2 e 1 + 1 e 1 e 3 e 2 + 2 e 2 + e 1 e 4 e 3 + 3 e 3 + e 2 ... e i e i 1 + i 1 e i 1 + e i 2 (5{4) ... e m e m 1 + m 1 e m 1 + e m 2 where i 2 R ;i =1 ; 2 ;:::;m 1denoteconstantpositivecontrolgains.Theerrorsignals e i ( t ) ;i =2 ; 3 ;:::;m canbeexpressedintermsof e 1 ( t )anditstimederivativesas e i = i 1 X j =0 b i;j e ( j ) 1 b i;i 1 =1(5{5) wheretheconstantcoecients b i;j 2 R canbeevaluatedbysubstituting( 5{5 )in( 5{4 ), andcomparingcoecients.Alteredtrackingerror[ 57 ],denotedby r ( t ) 2 R n ,isalso denedas r e m + m e m (5{6) where m 2 R isapositive,constantcontrolgain.Thelteredtrackinge rror r ( t )isnot measurablesincetheexpressionin( 5{6 )dependson x ( m ) 5.4ControlDevelopment Todeveloptheopen-looptrackingerrorsystem,theltered trackingerrorin( 5{6 )is premultipliedby G 1 ( )toyield G 1 r = G 1 e m + G 1 m e m : (5{7) 98

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Substituting( 5{5 )into( 5{7 )for_ e m ( t )yields G 1 r = G 1 m 1 X j =0 b m;j e ( j +1) 1 + G 1 m e m : (5{8) Byseparatingthelasttermfromthesummation,( 5{8 )canalsobeexpressedas G 1 r = G 1 b m;m 1 e ( m ) 1 + G 1 m 2 X j =0 b m;j e ( j +1) 1 + G 1 m e m : (5{9) Usingthefactthat b m;m 1 =1andmakingsubstitutionsfrom( 5{1 )and( 5{3 ),the expressionin( 5{9 )isrewrittenas G 1 r = G 1 x ( m ) d G 1 f G 1 h u + G 1 m 2 X j =0 b m;j e ( j +1) 1 + G 1 m e m whichcanberearrangedas G 1 r = Y d + S 1 G 1 d h u: (5{10) In( 5{10 ),theauxiliaryfunction S 1 x; x;:::;x ( m 1) ;t 2 R n isdenedas S 1 G 1 m 2 X j =0 b m;j e ( j +1) 1 + m e m + G 1 x ( m ) d G 1 d x ( m ) d G 1 f + G 1 d f d G 1 h + G 1 d h: (5{11) Alsoin( 5{10 ), Y d 2 R n isdenedas Y d G 1 d x ( m ) d G 1 d f d (5{12) where Y d ( x d ; x d ;:::;x ( m ) d ) 2 R n p isadesiredregressionmatrix,and 2 R p containsthe constantunknownsystemparameters.In( 5{12 ),thefunctions G 1 d ( x d ; x d ;:::;x ( m 2) d ) 2 R n n and f d ( x d ; x d ;:::;x ( m 1) d ) 2 R n aredenedas G 1 d G 1 ( x d ; x d ;:::;x ( m 2) d )(5{13) f d f ( x d ; x d ;:::;x ( m 1) d ) : 99

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Theopen-looperrorsystemin( 5{10 )istypicallywritteninaformsimilarto G 1 e m = Y d + S 1 G 1 m e m G 1 d h u (5{14) whichcanbeobtainedbysubstituting( 5{6 )into( 5{10 )forthelteredtrackingerror. Although( 5{10 )and( 5{14 )areequivalent,theatypicalformin( 5{10 )isusedtofacilitate thesubsequentclosed-looperrorsystemdevelopmentandst abilityanalysis.Specically, thesubsequentRISEcontrolmethodisdesignedbasedonthet imederivativeofthe controlinput.Thedesignofthelteredtrackingerrorin( 5{6 )isnotnecessary,butit simpliesthesubsequentdevelopmentbyallowingtheclose d-looperrorsystemtobe expressedintermsof_ r ( t )ratherthan e m ( t ). 5.4.1RISE-basedSwapping Ameasurableformofthepredictionerror ( t ) 2 R n isdenedasthedierence betweenthelteredcontrolinput u f ( t ) 2 R n andtheestimatedlteredcontrolinput ^ u f ( t ) 2 R n as u f ^ u f (5{15) wherethelteredcontrolinput u f ( t ) 2 R n isgeneratedby[ 3 ] u f + !u f = !u u f (0)=0(5{16) where 2 R isaknownpositiveconstant,and^ u f ( t ) 2 R n issubsequentlydesigned.The dierentialequationin( 5{16 )canbedirectlysolvedtoyield u f = v u (5{17) where isusedtodenotethestandardconvolutionoperation,andth escalarfunction v ( t ) isdenedas v !e !t : (5{18) 100

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Using( 5{1 ),theexpressionin( 5{17 )canberewrittenas u f = v G 1 x ( m ) G 1 f G 1 h : (5{19) Sincethesystemdynamicsin( 5{1 )includenon-LPboundeddisturbances h ( t ),they alsogetlteredandincludedinthelteredcontrolinputin ( 5{19 ).Tocompensatefor theeectsofthesedisturbances,thetypicalpredictioner rorformulationismodiedto includeaRISE-likestructureinthedesignoftheestimated lteredcontrolinput.With thismotivation,thestructureoftheopen-loopprediction errorsystemisengineeredto facilitatetheRISE-baseddesignoftheestimatedlteredc ontrolinput. Addingandsubtractingtheterm G 1 d x ( m ) d + G 1 d f d + G 1 d h totheexpressionin( 5{19 ) yields u f = v ( G 1 d x ( m ) d + G 1 d f d + G 1 x ( m ) G 1 d x ( m ) d G 1 f G 1 d f d G 1 h + G 1 d h G 1 d h ) : (5{20) Using( 5{12 ),theexpressionin( 5{20 )issimpliedas u f = v Y d + S S d G 1 d h (5{21) where S ( x; x;:::;x ( m ) ) ;S d ( x d ; x d ;:::;x ( m ) d ) 2 R n aredenedas S G 1 x ( m ) G 1 f G 1 h (5{22) S d G 1 d x ( m ) d G 1 d f d G 1 d h: (5{23) Theexpressionin( 5{21 )isfurthersimpliedas u f = Y df + v S v S d + h f (5{24) wherethelteredregressormatrix Y df ( x d ; x d ;:::;x ( m ) d ) 2 R n p isdenedas Y df v Y d (5{25) 101

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andthedisturbance h f ( t ) 2 R n isdenedas h f v G 1 d h: Theterm v S ( x; x;:::;x ( m ) ) 2 R n in( 5{24 )dependson x ( m ) .Usingthefollowingproperty ofconvolution[ 57 ]: g 1 g 2 =_ g 1 g 2 + g 1 (0) g 2 g 1 g 2 (0)(5{26) anexpressionindependentof x ( m ) canbeobtained.Consider v S = v G 1 x ( m ) G 1 f G 1 h whichcanberewrittenas v S = v ( d dt ( G 1 x ( m 1) ) G 1 x ( m 1) G 1 f G 1 h ) : (5{27) Applyingthepropertyin( 5{26 )tothersttermof( 5{27 )yields v S = S f + W (5{28) wherethestate-dependenttermsareincludedintheauxilia ryfunction S f ( x; x;:::;x ( m 1) ) 2 R n ; denedas S f v G 1 x ( m 1) + v (0) G 1 x ( m 1) v G 1 x ( m 1) v G 1 f v G 1 h (5{29) andthetermsthatdependontheinitialstatesareincludedi n W ( t ) 2 R n ,denedas W vG 1 x (0) ; x (0) ;:::;x ( m 2) (0) x ( m 1) (0) : (5{30) Similarly,followingtheprocedurein( 5{27 )-( 5{30 ),theexpression v S d in( 5{24 )is evaluatedas v S d = S df + W d (5{31) 102

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where S df ( x d ; x d ;:::;x ( m 1) d ) 2 R n isdenedas S df v ( G 1 d x ( m 1) d )+ v (0) G 1 d x ( m 1) d v G 1 d x ( m 1) d v G 1 d f d v G 1 d h (5{32) and W d ( t ) 2 R n isdenedas W d vG 1 d ( x d (0) ; x d (0) ;:::;x ( m 2) d (0)) x ( m 1) d (0) : (5{33) Substituting( 5{28 )-( 5{33 )into( 5{24 ),andthensubstitutingtheresultingexpressioninto ( 5{15 )yields = Y df + S f S df + W W d + h f ^ u f : (5{34) 5.4.2CompositeAdaptation Thecompositeadaptationfortheadaptiveestimates ^ ( t ) 2 R p in( 5{47 )isgivenby ^ = Y T d r + Y T df (5{35) where 2 R p p isapositivedenite,symmetric,constantgainmatrixandt heltered regressormatrix Y df ( x d ; x d ;:::;x ( m ) d ) 2 R n p isdenedin( 5{25 ). Remark5.1. Theparameterestimateupdatelawin( 5{35 )dependsontheunmeasurable signal r ( t ) ,buttheparameterestimatesareindependentof r ( t ) ascanbeshownby directlysolving( 5{35 )as ^ ( t )= ^ (0)+ Y T d ( ) e m ( ) t0 + Z t 0 Y T df ( ) ( ) d Z t 0 n Y T d ( ) e m ( ) m Y T d ( ) e m ( ) o d: 5.4.3Closed-LoopPredictionErrorSystem Basedon( 5{36 )andthesubsequentanalysis,thelteredcontrolinputest imateis designedas ^ u f = Y df ^ + 2 ; (5{36) 103

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where 2 ( t ) 2 R n isaRISE-liketermdenedas 2 ( t ) Z t 0 [ k 2 ( )+ 2 sgn ( ( ))] d; (5{37) where k 2 ; 2 2 R denoteconstantpositivecontrolgains.Inatypicalpredic tionerror formulation,theestimatedlteredcontrolinputisdesign edtoincludejusttherst term Y df ^ in( 5{36 ).Butaspreviouslydiscussed,thepresenceofnon-LPdistu rbances inthesystemmodelresultsinltereddisturbancesintheun measurableformofthe predictionerrorin( 5{34 ).Hence,theestimatedlteredcontrolinputisaugmented withanadditionalRISE-liketerm 2 ( t )tocanceltheeectsofdisturbancesinthe predictionerrorasillustratedinFigure 5-1 andthesubsequentdesignandstability analysis.Substituting( 5{36 )into( 5{34 )yieldsthefollowingclosed-looppredictionerror system: = Y df ~ + S f S df + W W d + h f 2 (5{38) where ~ ( t ) 2 R p denotestheparameterestimatemismatchdenedas ~ ^ : (5{39) Tofacilitatethesubsequentcompositeadaptivecontrolde velopmentandstabilityanalysis, thetimederivativeof( 5{38 )isexpressedas = Y df ~ Y df Y T df + ~ N 2 + N 2 B k 2 2 sgn ( ) ; (5{40) where( 5{35 )andthefactthat 2 = k 2 + 2 sgn ( )(5{41) wereutilized.In( 5{40 ),theunmeasurable/unknownauxiliaryterm ~ N 2 ( e 1 ;e 2 ;:::;e m ;r;t ) 2 R n isdenedas ~ N 2 S f S df Y df Y T d r; (5{42) 104

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wheretheupdatelawin( 5{35 )wasutilized,andtheterm N 2 B ( t ) 2 R n isdenedas N 2 B W W d + h f : (5{43) InasimilarmannerasinLemma 1 oftheAppendix,byapplyingtheMeanValue Theoremcanbeusedtodevelopthefollowingupperboundfort heexpressionin( 5{42 ): rrr ~ N 2 ( t ) rrr 2 ( k z k ) k z k ; (5{44) wheretheboundingfunction 2 ( ) 2 R isapositive,globallyinvertible,nondecreasing function,and z ( t ) 2 R n ( m +1) isdenedas z ( t ) e T1 e T2 :::e Tm r T T : (5{45) UsingAssumption5-3,andthefactthat v ( t )isalinear,strictlyproper,exponentially stabletransferfunction,thefollowinginequalitycanbed evelopedbasedontheexpression in( 5{43 )withasimilarapproachasinLemma2of[ 42 ]: k N 2 B ( t ) k ; (5{46) where 2 R isaknownpositiveconstant. 5.4.4Closed-LoopTrackingErrorSystem Basedontheopen-looperrorsystemin( 5{10 ),thecontrolinputiscomposedofan adaptivefeedforwardtermplustheRISEfeedbacktermas u Y d ^ + 1 (5{47) where 1 ( t ) 2 R n denotestheRISEfeedbacktermdenedas 1 ( t ) ( k 1 +1) e m ( t ) ( k 1 +1) e m (0)+ Z t 0 f ( k 1 +1) m e m ( )+ 1 sgn ( e m ( )) g d (5{48) where k 1 ; 1 2 R arepositiveconstantcontrolgains,and m 2 R wasintroducedin( 5{6 ). In( 5{47 ), ^ ( t ) 2 R p denotesaparameterestimatevectorforunknownsystempara meters 105

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Figure5-1.BlockdiagramoftheproposedRISE-basedcompos iteadaptivecontroller 2 R p ,generatedbyasubsequentlydesignedgradient-basedcomp ositeadaptiveupdate law[ 38 39 82 ]. Theclosed-looptrackingerrorsystemcanbedevelopedbysu bstituting( 5{47 )into ( 5{10 )as G 1 r = Y d ~ + S 1 G 1 d h 1 : (5{49) Tofacilitatethesubsequentcompositeadaptivecontrolde velopmentandstabilityanalysis, thetimederivativeof( 5{49 )isexpressedas G 1 r = 1 2 G 1 r + Y d ~ Y d Y T df + ~ N 1 + N 1 B ( k 1 +1) r 1 sgn ( e m ) e m (5{50) where( 5{35 )andthefactthatthetimederivativeof( 5{48 )isgivenas 1 =( k 1 +1) r + 1 sgn ( e m )(5{51) wasutilized.In( 5{50 ),theunmeasurable/unknownauxiliaryterms ~ N 1 ( e 1 ;e 2 ;:::;e m ;r;t ) and N 1 B ( t ) 2 R n aredenedas ~ N 1 1 2 G 1 r + S 1 + e m Y d Y T d r (5{52) 106

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where( 5{35 )wasused,and N 1 B G 1 d h G 1 d h: (5{53) Thestructureof( 5{50 )andtheintroductionoftheauxiliarytermsin( 5{52 )and( 5{53 )is motivatedbythedesiretosegregatetermsthatcanbeupperb oundedbystate-dependent termsandtermsthatcanbeupperboundedbyconstants.Inasi milarfashionasin ( 5{44 ),thefollowingupperboundcanbedevelopedfortheexpress ionin( 5{52 ): rrr ~ N 1 ( t ) rrr 1 ( k z k ) k z k (5{54) wheretheboundingfunction 1 ( ) 2 R isapositive,globallyinvertible,nondecreasing function,and z ( t ) 2 R n ( m +1) wasdenedin( 5{45 ).UsingAssumptions5-2and5-3,the followinginequalitiescanbedevelopedbasedontheexpres sionin( 5{53 )anditstime derivative: k N 1 B ( t ) k 1 rrr N 1 B ( t ) rrr 2 (5{55) where i 2 R i =1 ; 2areknownpositiveconstants. TheRISEcontrollerin( 5{47 )and( 5{48 ),andslidingmodecontrollers(SMCs) (e.g.,seetheclassicresultsin[ 3 12 ])aretheonlymethodsthathavebeenprovento yieldanasymptotictrackingresultforanopen-looperrors ystemsuchas( 5{10 )where additiveboundeddisturbancesarepresentthatareupperbo undedbyaconstant.Other approacheslackamechanismtocancelthedisturbanceterms (thesetermsaretypically eliminatedthroughnonlineardampingandyieldauniformly ultimatelybounded(UUB) result).SMCisadiscontinuouscontrolmethodthatrequire sinnitecontrolbandwidth andaknownupperboundonthedisturbanceterm.Continuousm odicationsofSMC reducethestabilityresulttoUUB.TheRISEcontrolmethodi n( 5{47 )and( 5{48 )is continuous/dierentiable(i.e.,nitebandwidth)andreq uiresaknownupperboundonthe disturbanceandthetimederivativeofthedisturbance.Des pitewhichfeedbackcontrol methodisselected,thecontrolchallengeisthatthedistur banceterm h ( t )in( 5{1 )willbe includedintheswapping(ortorqueltering)methodinthep redictionerrorformulation 107

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asshownin( 5{34 ).Thistechnicalobstaclehaspreventedthedevelopmentof anyprevious compositeadaptivecontrollerforadynamicsystemwithadd eddisturbancessuchas h ( t ). Thecontributionofthecurrentresultisthedevelopmentof theRISE-basedswapping methodtoenablethedevelopmentofcompositeadaptivecont rollersforsystemssuchas ( 5{1 ).Specically,theRISE-basedswappingtechniqueprovide sameanstocancelthe ltereddisturbancetermsin( 5{34 ). 5.5StabilityAnalysis Theorem5-1: Thecontrollergivenin( 5{47 )and( 5{48 )inconjunctionwiththe compositeadaptiveupdatelawin( 5{35 ),wherethepredictionerrorisgeneratedfrom ( 5{15 ),( 5{16 ),( 5{36 ),and( 5{37 ),ensuresthatallsystemsignalsareboundedunder closed-loopoperationandthatthepositiontrackingerror andthepredictionerrorare regulatedinthesensethat k e 1 ( t ) k! 0and k ( t ) k! 0as t !1 providedthecontrolgains k 1 and k 2 introducedin( 5{48 )and( 5{37 )areselected sucientlylargebasedontheinitialconditionsofthesyst em(seethesubsequentproof), andthefollowingconditionsaresatised: m 1 > 1 2 ; m > 1 2 ; (5{56) 1 > 1 + 1 m 2 ; 2 >; (5{57) wherethegains m 1 and m wereintroducedin( 5{4 ), 1 wasintroducedin( 5{48 ), 2 wasintroducedin( 5{37 ), 1 and 2 wereintroducedin( 5{55 ),and wasintroducedin ( 5{46 ). Proof: Let D R n ( m +2)+ p +2 beadomaincontaining y ( t )=0,where y ( t ) 2 R n ( m +2)+ p +2 isdenedas y [ z T T p P 1 p P 2 ~ T ] T : (5{58) 108

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In( 5{58 ),theauxiliaryfunction P 1 ( t ) 2 R isdenedas P 1 ( t ) 1 n X i =1 j e mi (0) j e m (0) T N 1 B (0) Z t 0 L 1 ( ) d; (5{59) where e mi (0) 2 R denotesthe i thelementofthevector e m (0),andtheauxiliaryfunction L 1 ( t ) 2 R isdenedas L 1 r T ( N 1 B 1 sgn ( e m )) ; (5{60) where 1 2 R isapositiveconstantchosenaccordingtothesucientcond itionin( 5{57 ). Providedthesucientconditionintroducedin( 5{57 )issatised,thefollowinginequality isobtained[ 22 ]: Z t 0 L 1 ( ) d 1 n X i =1 j e mi (0) j e m (0) T N 1 B (0) : (5{61) Hence,( 5{61 )canbeusedtoconcludethat P 1 ( t ) 0.Alsoin( 5{58 ),theauxiliary function P 2 ( t ) 2 R isdenedas P 2 ( t ) Z t 0 L 2 ( ) d; (5{62) wheretheauxiliaryfunction L 2 ( t ) 2 R isdenedas L 2 T ( N 2 B 2 sgn ( )) ; (5{63) where 2 2 R isapositiveconstantchosenaccordingtothesucientcond itionin( 5{57 ). Providedthesucientconditionintroducedin( 5{57 )issatised,then P 2 ( t ) 0. Let V L ( y;t ): D [0 ; 1 ) R beacontinuouslydierentiable,positivedenite functiondenedas V L ( y;t ) 1 2 m X i =1 e Ti e i + 1 2 r T G 1 r + 1 2 T + P 1 + P 2 + 1 2 ~ T 1 ~ (5{64) whichsatisestheinequalities U 1 ( y ) V L ( y;t ) U 2 ( y )(5{65) 109

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providedthesucientconditionsintroducedin( 5{57 )aresatised.In( 5{65 ),the continuouspositivedenitefunctions U 1 ( y ), U 2 ( y ) 2 R aredenedas U 1 ( y ) 1 k y k 2 and U 2 ( y ) 2 ( x; x;:::;x ( m 2) ) k y k 2 ,where 1 2 ( x; x;:::;x ( m 2) ) 2 R aredenedas 1 1 2 min 1 ;g ; min 1 (5{66) 2 max f 1 2 g ( x; x;:::;x ( m 2) ) ; 1 2 max 1 ; 1 g where g g ( x; x;:::;x ( m 2) )areintroducedin( 5{2 ),and min fg and max fg denotethe minimumandmaximumeigenvalueofthearguments,respectiv ely. Afterusing( 5{4 ),( 5{6 ),( 5{35 ),( 5{40 ),( 5{50 ),( 5{59 ),( 5{60 ),( 5{62 )and( 5{63 ),the timederivativeof( 5{64 )canbeexpressedas V L ( y;t )= m X i =1 i e Ti e i + e Tm 1 e m r T r k 1 r T r + r T Y d ~ + r T ~ N 1 + r T N 1 B r T Y d Y T df 1 r T sgn ( e m )+ T Y df ~ (5{67) + T ~ N 2 + T N 2 B k 2 T T Y df Y T df 2 T sgn ( ) r T ( N 1 B 1 sgn ( e m )) T N 2 B + T 2 sgn ( ) ~ T 1 ( Y T d r + Y T df ) : Aftercancelingthesimilartermsandusingthefactthat a T b 1 2 ( k a k 2 + k b k 2 )forsome a;b 2 R n ,theexpressionin( 5{67 )isupperboundedas V L ( y;t ) m X i =1 i e Ti e i + 1 2 k e m 1 k 2 + 1 2 k e m k 2 k r k 2 k 1 k r k 2 + r T ~ N 1 r T Y d Y T df + T ~ N 2 k 2 T T Y df Y T df ": Usingthefollowingupperbounds: rrr Y d Y T df rrr c 1 ; rrr Y df Y T df rrr c 2 ; 110

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where c 1 ;c 2 2 R arepositiveconstants, V L ( y;t )isupperboundedusingthesquaresofthe componentsof z ( t )as V L ( y;t ) 3 k z k 2 k 1 k r k 2 + k r k rrr ~ N 1 rrr + c 1 k kk r k + k k rrr ~ N 2 rrr ( k 2 c 2 ) k k 2 ; (5{68) where 3 min f 1 ; 2 ;:::; m 2 ; m 1 1 2 ; m 1 2 ; 1 g : Letting k 2 = k 2 a + k 2 b where k 2 a ;k 2 b 2 R arepositiveconstants,andusingtheinequalitiesin( 5{54 )and( 5{44 ), theexpressionin( 5{68 )isupperboundedas V L ( y;t ) 3 k z k 2 k 2 b k k 2 k 1 k r k 2 1 ( k z k ) k r kk z k (5{69) ( k 2 a c 2 ) k k 2 ( 2 ( k z k )+ c 1 ) k kk z k : Completingthesquaresforthetermsinsidethebracketsin( 5{69 )yields V L ( y;t ) 3 k z k 2 k 2 b k k 2 + 21 ( k z k ) k z k 2 4 k 1 + ( 2 ( k z k )+ c 1 ) 2 k z k 2 4( k 2 a c 2 ) 3 k z k 2 + 2 ( k z k ) k z k 2 4 k k 2 b k k 2 ; (5{70) where k 2 R isdenedas k k 1 ( k 2 a c 2 ) max f k 1 ; ( k 2 a c 2 ) g ;k 2 a >c 2 (5{71) and ( ) 2 R isapositive,globallyinvertible,nondecreasingfunctio ndenedas 2 ( k z k ) 21 ( k z k )+( 2 ( k z k )+ c 1 ) 2 : 111

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Theexpressionin( 5{70 )canbefurtherupperboundedbyacontinuous,positive semi-denitefunction V L ( y;t ) U ( y )= c rrr z T T T rrr 2 8 y 2D (5{72) forsomepositiveconstant c ,where D n y ( t ) 2 R n ( m +2)+ p +2 jk y k 1 2 p 3 k o : Largervaluesof k willexpandthesizeofthedomain D .Theinequalitiesin( 5{65 ) and( 5{70 )canbeusedtoshowthat V L ( y;t ) 2L 1 in D ;hence, e i ( t ) 2L 1 and ( t ) ;r ( t ) ; ~ ( t ) 2L 1 in D .Giventhat e i ( t ) 2L 1 and r ( t ) 2L 1 in D ,standardlinear analysismethodscanbeusedtoprovethat_ e i ( t ) 2L 1 in D from( 5{4 )and( 5{6 ).Since e i ( t ) 2L 1 ; and r ( t ) 2L 1 in D ,Assumption5-5canbeusedalongwith( 5{3 )-( 5{6 )to concludethat x ( i ) ( t ) 2L 1 ;i =0 ; 1 ;:::;m in D .Since ~ ( t ) 2L 1 in D ,( 5{39 )canbeused toprovethat ^ ( t ) 2L 1 in D .Since x ( i ) ( t ) 2L 1 ;i =0 ; 1 ;:::;m in D ,Assumption5-2can beusedtoconcludethat G 1 ( )and f ( ) 2L 1 in D .Thus,from( 5{1 )andAssumption 5-3,wecanshowthat u ( t ) 2L 1 in D .Therefore, u f ( t ) 2L 1 in D ,andhence,from ( 5{15 ),^ u f ( t ) 2L 1 in D .Giventhat r ( t ) 2L 1 in D ,( 5{51 )canbeusedtoshowthat 1 ( t ) 2L 1 in D ,andsince G 1 ( )and f ( ) 2L 1 in D ,( 5{50 )canbeusedtoshowthat r ( t ) 2L 1 in D ,and( 5{40 )canbeusedtoshowthat_ ( t ) 2L 1 in D .Since_ e i ( t ) 2L 1 r ( t ),and_ ( t ) 2L 1 in D ,thedenitionsfor U ( y )and z ( t )canbeusedtoprovethat U ( y ) isuniformlycontinuousin D Let SD denoteasetdenedas S y ( t ) 2Dj U 2 ( y ( t )) < 1 1 2 p 3 k 2 : (5{73) Theregionofattractionin( 5{73 )canbemadearbitrarilylargetoincludeanyinitial conditionsbyincreasingthecontrolgain k (i.e.,asemi-globalstabilityresult).Theorem 112

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8.4of[ 63 ]cannowbeinvokedtostatethat c rrr z T T T rrr 2 0as t !18 y (0) 2S : (5{74) Basedonthedenitionof z ( t ),( 5{74 )canbeusedtoshowthat k e 1 ( t ) k! 0as t !18 y (0) 2S (5{75) k ( t ) k! 0as t !18 y (0) 2S 5.6Experiment AsinChapter2,thetestbeddepictedinFigure 2-1 wasusedtoimplementthe developedcontroller.Thedesiredlinktrajectoryisselec tedasfollows(indegrees): q d ( t )=60 : 0sin(1 : 2 t )(1 exp( 0 : 01 t 3 )) : (5{76) Forallexperiments,therotorvelocitysignalisobtainedb yapplyingastandardbackwards dierencealgorithmtothepositionsignal.Theintegralst ructurefortheRISEterm in( 5{48 )wascomputedon-lineviaastandardtrapezoidalalgorithm .Theparameter estimateswereallinitializedtozero.Inpractice,theada ptiveestimateswouldbe initializedtoabest-guessestimateofthevalues.Initial izingtheestimatestozerowas donetotestascenarioofnoparameterknowledge.Inadditio n,allthestateswere initializedtozero Thefollowingcontrolgainsandbestguessestimateswereus edto implementthecontrollerin( 5{47 )and( 5{48 )inconjunctionwiththecompositeadaptive updatelawin( 5{35 ),wherethepredictionerrorisgeneratedfrom( 5{15 ),( 5{16 ),( 5{36 ), and( 5{37 ): k 1 =70 ; 1 =50 ; 1 =20 ; 2 =10 ; = diag f 4,0 : 9,0 : 9,2 g k 2 =400 ; 2 =1 ;! =10 ; r 2 =50 ; r 3 =1 ; r 5 =50 : (5{77) 113

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0 5 10 15 20 25 30 35 40 -80 -60 -40 -20 0 20 40 60 80 Actual and Desired Positions [degrees]Time [sec] Actual Desired Figure5-2.Actualanddesiredtrajectoriesforthepropose dcompositeadaptivecontrol law(RISE+CFF). 0 5 10 15 20 25 30 35 40 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Tracking Error [degrees]Time [sec] Figure5-3.Trackingerrorfortheproposedcompositeadapt ivecontrollaw(RISE+CFF). 114

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0 5 10 15 20 25 30 35 40 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Prediction Error [N-m]Time [sec] Figure5-4.Predictionerrorfortheproposedcompositeada ptivecontrollaw (RISE+CFF). 0 5 10 15 20 25 30 35 40 -60 -40 -20 0 20 40 60 Torque [N-m]Time [sec] Figure5-5.Controltorquefortheproposedcompositeadapt ivecontrollaw(RISE+CFF). 115

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0 5 10 15 20 25 30 35 40 -60 -40 -20 0 20 40 60 [N-m]Time [sec] Figure5-6.ContributionoftheRISEtermintheproposedcom positeadaptivecontrollaw (RISE+CFF). 0 5 10 15 20 25 30 35 40 -2 0 2 4 6 8 10 12 14 16 18 Parameter EstimatesTime [sec] Figure5-7.Adaptiveestimatesfortheproposedcompositea daptivecontrollaw (RISE+CFF). 116

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Figure5-8.AverageRMSerrors(degrees)andtorques(N-m). 1-RISE,2-RISE+FF,3RISE+CFF(proposed). 5.6.1Discussion Threedierentexperimentswereconductedtodemonstratet heecacyofthe proposedcontroller.Foreachcontroller,thegainswereno tretuned(i.e.,thecommon controlgainsremainthesameforallcontrollers).First,n oadaptationwasusedandthe controllerwithonlytheRISEfeedbackwasimplemented.For thesecondexperiment, thepredictionerrorcomponentoftheupdatelawin( 5{35 )wasremoved,resultingina standardgradient-basedupdatelaw(hereinafterdenoteda sRISE+FF).Forthethird experiment,theproposedcompositeadaptivecontrollerin ( 5{47 )-( 5{48 )(hereinafter denotedasRISE+CFF)wasimplemented.Figure 5-2 depictstheactualposition comparedwiththedesiredtrajectoryfortheRISE+CFFcontr oller,whilethetracking errorandthepredictionerrorareshowninFigure 5-3 andFigure 5-4 ,respectively.The controltorqueisshowninFigure 5-5 ,andthecontributionoftheRISEtermintheoverall torqueisdepictedinFigure 5-6 .ThecontributionofthefeedbackRISEtermdecreasesas theadaptiveestimatesconvergeasshowninFigure 5-7 .Eachexperimentwasperformed vetimesandtheaverageRMSerrorandtorquevaluesareshow ninFigure 5-8 ,which indicatethattheproposedRISE+CFFcontrolleryieldsthel owestRMSerrorwitha similarcontroleort. 117

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5.7Conclusion Anovelapproachforthedesignofagradient-basedcomposit eadaptivecontrollerwas proposedforgenericMIMOsystemssubjectedtoboundeddist urbances.Amodel-based feedforwardadaptivecomponentwasusedinconjunctionwit htheRISEfeedback,where theadaptiveestimatesweregeneratedusingacompositeupd atelawdrivenbyboth thetrackingandpredictionerrorwiththemotivationofusi ngmoreinformationinthe adaptiveupdatelaw.Toaccountfortheeectsofnon-LPdist urbances,thetypical predictionerrorformulationwasmodiedtoincludeasecon dRISE-liketerminthe estimatedlteredcontrolinputdesign.UsingaLyapunovst abilityanalysis,sucient gainconditionswerederivedunderwhichtheproposedcontr olleryieldssemi-global asymptoticstability.Thecurrentdevelopment,aswellasa llpreviousRISEcontrollers, requirefull-statefeedback.Thedevelopmentofanoutputf eedbackresultremainsanopen problem.Experimentsonarotatingdiskwithexternallyapp liedfrictionindicatethatthe proposedmethodyieldsbettertrackingperformancewithas imilarcontroleort. 118

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CHAPTER6 COMPOSITEADAPTATIONFORNN-BASEDCONTROLLERS 6.1Introduction Thischapterpresentsthersteverattempttodevelopapred ictionerror-based compositeadaptiveNNcontrollerforanEuler-Lagrangesec ond-orderdynamicsystem usingtherecentcontinuousRobustIntegraloftheSignofth eError(RISE)[ 19 ]technique thatwasoriginallydevelopedin[ 21 ]and[ 22 ].TheRISEarchitectureisadoptedsince thismethodcanaccommodatefor C 2 disturbancesandyieldasymptoticstability. TheRISEtechniquewasusedin[ 75 ]toprovethersteverasymptoticresultfora NN-basedcontrollerusingacontinuousfeedback.Inthisch apter,theRISEfeedbackis usedinconjunctionwithaNNfeedfowardelementsimilarto[ 75 ],however,unlikethe typicaltrackingerror-basedgradientupdatelawusedin[ 75 ],theresultinthischapter usesacompositeupdatelawdrivenbyboththetrackingandth epredictionerror.In ordertocompensatefortheeectofNNreconstructionerror ,aninnovativeuseofthe RISEstructureisalsoemployedinthepredictionerrorupda te(i.e.,thelteredcontrol inputestimate).Ablockdiagramindicatingtheuniqueuseo ftheRISEmethodin thecontrolandthepredictionerrorformulationisprovide dinFigure 6-1 .Sucient gainconditionsarederivedusingaLyapunov-basedstabili tyanalysisunderwhichthis uniquedouble-RISEcontrolstrategyyieldsasemi-globala symptoticstabilityforthe systemtrackingerrorsandthepredictionerror,whileallo thersignalsandthecontrol inputareshowntobebounded.Sinceamulti-layerNNinclude stherstlayerweight estimateinsideofanonlinearactivationfunction,provin gthattheNNweightestimates areboundedisachallengingtask.Aprojectionalgorithmis usedtoguaranteethe boundednessoftheweightestimates.However,ifinsteadas ingle-layerNNisused, projectionisnotrequiredandtheweightestimatescanbesh ownboundedviathestability analysis.Thecontroldevelopmentinthischaptercanbeeas ilysimpliedforasingle-layer NNbychoosingxedrstlayerweights. 119

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AusualconcernabouttheRISEfeedbackisthepresenceofhig h-gainandhigh-frequency componentsinthecontrolstructure.However,incontrastt oatypicalwidelyused discontinuoushigh-gainslidingmodecontroller,theRISE feedbackoersacontinuous alternative.Moreover,theproposedcontrollerisnotpure lyhighgainasamulti-layerNN isusedasafeedforwardcomponentthatlearnsandincorpora testheknowledgeofsystem dynamicsinthecontrolstructure. 6.2DynamicSystem Consideraclassofsecondordernonlinearsystemsofthefol lowingform: x = f ( x; x )+ G ( x ) u (6{1) where x ( t ) ; x ( t ) 2 R n arethesystemstates, u ( t ) 2 R n isthecontrolinput, f ( x; x ) 2 R n and G ( x ) 2 R n n areunknownnonlinear C 2 functions.Thecontroldevelopmentforthe dynamicsystemin( 6{1 )canbeeasilyextendedtoasecond-orderEuler-Lagrangesy stem ofthefollowingform: M ( q ) q + V m ( q; q )_ q + G ( q )+ F (_ q )= ( t ) where M ( q ) 2 R n n denotestheinertiamatrix, V m ( q; q ) 2 R n n denotesthecentripetal-Coriolis matrix, G ( q ) 2 R n denotesthegravityvector, F (_ q ) 2 R n denotesfriction, ( t ) 2 R n representsthetorqueinputcontrolvector,and q ( t ),_ q ( t ), q ( t ) 2 R n denotethelink position,velocity,andaccelerationvectors,respective ly.Throughoutthechapter, jj denotestheabsolutevalueofthescalarargument, kk denotesthestandardEuclidean normforavectorortheinducedinnitynormforamatrix,and kk F denotesthe Frobeniusnormofamatrix. Thefollowingpropertiesandassumptionswillbeexploited inthesubsequent development. 120

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Assumption6-1: G ( )issymmetricpositivedenite,andsatisesthefollowing inequality 8 ( t ) 2 R n : g k k 2 T G 1 g ( x ) k k 2 (6{2) where g 2 R isaknownpositiveconstant,and g ( x ) 2 R isaknownpositivefunction. Assumption6-2: Thefunctions G 1 ( )and f ( )aresecondorderdierentiablesuchthat G 1 ( ) ; G 1 ( ) ; G 1 ( ) ;f ( ) ; f ( ) ; f ( ) 2L 1 if x ( i ) ( t ) 2L 1 ;i =0 ; 1 ; 2 ; 3,where( ) ( i ) ( t ) denotesthe i th derivativewithrespecttotime. Assumption6-3: Thedesiredtrajectory x d ( t ) 2 R n isdesignedsuchthat x ( i ) d ( t ) 2L 1 ; i =0 ; 1 ;:::; 4withknownbounds. 6.3ControlObjective Theobjectiveistodesignacontinuous composite adaptiveNNcontrollerwhich ensuresthatthesystemstate x ( t )tracksadesiredtime-varyingtrajectory x d ( t )despite uncertaintiesinthedynamicmodel.Toquantifythisobject ive,atrackingerror,denoted by e 1 ( t ) 2 R n ,isdenedas e 1 x d x: (6{3) Tofacilitatethesubsequentanalysis,lteredtrackinger rors,denotedby e 2 ( t ), r ( t ) 2 R n arealsodenedas e 2 e 1 + 1 e 1 (6{4) r e 2 + 2 e 2 (6{5) where 1 2 2 R denotepositiveconstants.Thesubsequentdevelopmentisb asedonthe assumptionthatthesystemstates x ( t ) ; x ( t )aremeasurable.Hence,thelteredtracking error r ( t )isnotmeasurablesincetheexpressionin( 6{5 )dependson x ( t ). 121

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6.4ControlDevelopment Theopen-looptrackingerrorsystemisdevelopedbypremult iplying( 6{5 )by G 1 ( ) andutilizingtheexpressionsin( 6{1 ),( 6{3 ),( 6{4 )as G 1 r = + S 1 u: (6{6) In( 6{6 ), ( x d ; x d ; x d ) 2 R n isdenedas G 1 d x d G 1 d f d (6{7) wherethefunctions G 1 d ( x d ) 2 R n n and f d ( x d ; x d ) 2 R n aredenedas G 1 d G 1 ( x d ) ;f d f ( x d ; x d ) : (6{8) Alsoin( 6{6 ),theauxiliaryfunction S 1 ( x; x;t ) 2 R n isdenedas S 1 G 1 2 e 2 + G 1 x d G 1 d x d G 1 f + G 1 d f d + G 1 1 e 1 : (6{9) Theunknowndynamicsin( 6{7 )canberepresentedbyathree-layerNN[ 15 32 ]usingthe universalapproximationpropertyasdescribedinSection3 .4ofChapter3. = W T ( V T x d )+ ( x d ) : (6{10) Basedon( 6{10 ),thetypicalthree-layerNNapproximationfor ( x d )isgivenas[ 15 32 ] ^ ^ W T ^ V T x d (6{11) where ^ V ( t ) 2 R ( N 1 +1) N 2 and ^ W ( t ) 2 R ( N 2 +1) n aresubsequentlydesignedestimatesofthe idealweightmatrices.Theestimatemismatchfortheidealw eightmatrices,denotedby ~ V ( t ) 2 R ( N 1 +1) N 2 and ~ W ( t ) 2 R ( N 2 +1) n ,aredenedas ~ V V ^ V; ~ W W ^ W 122

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andthemismatchforthehidden-layeroutputerrorforagive n x d ( t ),denotedby~ ( x d ) 2 R N 2 +1 ,isdenedas ~ ^ = V T x d ^ V T x d : (6{12) Property6-1: ( TaylorSeriesApproximation )TheTaylorseriesexpansionfor V T x d foragiven x d ( t )maybewrittenas[ 15 32 ] V T x d = ^ V T x d + 0 ^ V T x d ~ V T x d + O ~ V T x d 2 (6{13) where 0 ^ V T x d d V T x d =d V T x d j V T x d = ^ V T x d ,and O ^ V T x d 2 denotesthe higherorderterms.Aftersubstituting( 6{13 )into( 6{12 ),thefollowingexpressioncanbe obtained ~ =^ 0 ~ V T x d + O ~ V T x d 2 (6{14) where^ 0 0 ^ V T x d Basedontheopen-looperrorsystemin( 6{6 ),thecontrolinputiscomposedofaNN estimatetermplustheRISEfeedbacktermas[ 75 ] u ^ + 1 (6{15) where ^ ( t ) 2 R n denotesarstever,subsequentlydesigned,prediction-er rorbased NNfeedfowardterm.In( 6{15 ), 1 ( t ) 2 R n denotestheRISEfeedbacktermdenedas [ 21 22 75 ] 1 ( t ) ( k 1 +1) e 2 ( t ) ( k 1 +1) e 2 (0)+ Z t 0 f ( k 1 +1) 2 e 2 ( )+ 1 sgn ( e 2 ( )) g d (6{16) where k 1 ; 1 2 R arepositiveconstantcontrolgains, 2 2 R wasintroducedin( 6{5 ),and sgn ( )denotesthesignumfunctiondenedas sgn ( e 2 ) [ sgn ( e 2 1 ) sgn ( e 2 2 )... sgn ( e 2 i )... sgn ( e 2 n )] T : 123

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Theclosed-looptrackingerrorsystemcanbedevelopedbysu bstituting( 6{15 )into( 6{6 ) as G 1 r = ^ + S 1 1 : (6{17) Tofacilitatethesubsequentcompositeadaptivecontrolde velopmentandstabilityanalysis, ( 6{10 )and( 6{11 )areusedtoexpressthetimederivativeof( 6{17 )as G 1 r = G 1 r + W T 0 V T x d V T x d ^ W T ( ^ V T x d ) ^ W T 0 ( ^ V T x d ) ^ V T x d (6{18) ^ W T 0 ( ^ V T x d ) ^ V T x d +_ + S 1 1 : Afteraddingandsubtractingtheterms W T ^ 0 ^ V T x d + ^ W T ^ 0 ~ V T x d to( 6{18 ),thefollowing expressioncanbeobtained: G 1 r = G 1 r + ^ W T ^ 0 ~ V T x d + ~ W T ^ 0 ^ V T x d + W T 0 V T x d W T ^ 0 ^ V T x d (6{19) ^ W T ^ 0 ~ V T x d + S 1 ^ W T ^ ^ W T ^ 0 ^ V T x d +_ 1 wherethenotations^ ^ V T x d and^ 0 ^ V T x d areintroducedin( 6{12 )and( 6{14 ), respectively.6.4.1Swapping Inthissection,theswappingprocedureisusedtogeneratea measurableformof apredictionerrorthatrelatestothefunctionmismatcherr or(i.e., ( t ) ^ ( t )).A measurableformofthepredictionerror ( t ) 2 R n isdenedasthedierencebetweena lteredcontrolinput u f ( t ) 2 R n andanestimatedlteredcontrolinput^ u f ( t ) 2 R n as u f ^ u f (6{20) wherethelteredcontrolinputisgeneratedfromthestable rstorderdierential equation u f + !u f = !u u f (0)=0(6{21) 124

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where 2 R isaknownpositiveconstant.Thedierentialequationin( 6{21 )canbe expressedasaconvolutionas u f = v u (6{22) where isusedtodenotethestandardconvolutionoperation,andth escalarfunction v ( t ) isdenedas v !e !t : (6{23) Using( 6{1 ),theexpressionin( 6{22 )canberewrittenas u f = v G 1 x G 1 f : (6{24) TheconstructionofaNN-basedcontrollertoapproximateth eunknownsystemdynamics in( 6{24 )willinherentlyresultinaresidualfunctionreconstruct ionerror ( x d ).The presenceofthereconstructionerrorhasbeenthetechnical obstaclethathasprevented thedevelopmentofcompositeadaptationlawsforNNs.Tocom pensatefortheeects ofthereconstructionerror,thetypicalpredictionerrorf ormulationismodiedto includeaRISE-likestructureinthedesignoftheestimated lteredcontrolinput.With thismotivation,theopen-looppredictionerrorsystemise ngineeredtofacilitatethe RISE-baseddesignoftheestimatedlteredcontrolinput. Addingandsubtractingtheterm v G 1 d x d + G 1 d f d totheexpressionin( 6{24 ), andusing( 6{7 )yields u f = v ( + S S d )(6{25) where S ( x; x; x ) ;S d ( x d ; x d ; x d ) 2 R n aredenedasS G 1 x G 1 f (6{26) S d G 1 d x d G 1 d f d : (6{27) Theexpressionin( 6{25 )isfurthersimpliedas u f = v + v S v S d : (6{28) 125

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Theterm v S ( x; x; x ) 2 R n in( 6{28 )dependson x ( t ).Usingthefollowingpropertyof convolution[ 57 ]: g 1 g 2 =_ g 1 g 2 + g 1 (0) g 2 g 1 g 2 (0)(6{29) anexpressionindependentof x ( t )canbeobtainedas v S = S f + D (6{30) wherethestate-dependenttermsareincludedintheauxilia ryfunction S f ( x; x ) 2 R n ; denedas S f v G 1 x + v (0) G 1 x v G 1 x v G 1 f (6{31) andthetermsthatdependontheinitialstatesareincludedi n D ( t ) 2 R n ,denedas D vG 1 ( x (0))_ x (0) : (6{32) Similarly,theexpression v S d ( x d ; x d ; x d )in( 6{28 )isevaluatedas v S d = S df + D d (6{33) where S df ( x d ; x d ) 2 R n isdenedas S df v ( G 1 d x d )+ v (0) G 1 d x d v G 1 d x d v G 1 d f d (6{34) and D d ( t ) 2 R n isdenedas D d vG 1 d ( x d (0))_ x d (0) : (6{35) Substituting( 6{30 )-( 6{35 )into( 6{28 ),andthensubstitutingtheresultingexpressioninto ( 6{20 )yields = v + S f S df + D D d ^ u f : (6{36) 126

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Basedon( 6{36 )andthesubsequentanalysis,thelteredcontrolinputest imateis designedas ^ u f = ^ f + 2 (6{37) wherethelteredNNestimate ^ f ( t ) 2 R n isgeneratedbyfromthestablerstorder dierentialequation ^ f + ^ f = ^ ; ^ f (0)=0 whichcanbeexpressedasaconvolution ^ f = v ^ : In( 6{37 ), 2 ( t ) 2 R n isaRISE-liketermdenedas 2 ( t ) Z t 0 [ k 2 ( )+ 2 sgn ( ( ))] d (6{38) where k 2 ; 2 2 R denoteconstantpositivecontrolgains.Inatypicalpredic tionerror formulation,theestimatedlteredcontrolinputisdesign edtoincludejusttherstterm ^ f ( t )in( 6{37 ).Butasdiscussedearlier,duetothepresenceoftheNNreco nstruction error,theunmeasurableformofthepredictionerrorin( 6{36 )alsoincludestheltered reconstructionerror.Hence,theestimatedlteredcontro linputisaugmentedwith anadditionalRISE-liketerm 2 ( t )tocanceltheeectsofreconstructionerrorinthe predictionerrormeasurementasillustratedinFigure 6-1 andthesubsequentdesignand stabilityanalysis. Substituting( 6{37 )into( 6{36 )yieldsthefollowingclosed-looppredictionerror system: = v ^ + S f S df + D D d 2 : (6{39) Tofacilitatethesubsequentcompositeadaptivecontrolde velopmentandstabilityanalysis, thetimederivativeof( 6{39 )isexpressedas =_ v ^ + ^ + S f S df + D D d 2 (6{40) 127

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Figure6-1.BlockdiagramoftheproposedRISE-basedcompos iteNNcontroller. wheretheproperty d dt ( f g )=( f g )( t )+ f (0) g ( t )wasused.Substituting( 6{10 )and ( 6{11 )into( 6{40 )yields =_ v W T ( V T x d )+ ^ W T ( ^ V T x d ) + W T ( V T x d )+ ^ W T ( ^ V T x d ) (6{41) + S f S df + D D d 2 : Addingandsubtracting_ v ( t ) ( ^ W T + ~ W T ^ )+ ( ^ W T + ~ W T ^ )to( 6{41 )andrearranging thetermsyields =_ v ^ W T ~ + ~ W T ^ + ~ W T ~ + + ^ W T ~ + ~ W T ^ + ~ W T ~ + (6{42) + S f S df + D D d 2 : ByusingtheTaylorseriesexpansionin( 6{13 ),( 6{42 )canbeexpressedas =_ v ~ W T ^ + ^ W T ^ 0 ~ V T x d +_ v ^ W T O ~ V T x d 2 + ~ W T ~ + (6{43) + ~ W T ^ + ^ W T ^ 0 ~ V T x d + ^ W T O ~ V T x d 2 + ~ W T ~ + + S f S df + D D d 2 : 128

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Usingthecommutativityanddistributivitypropertyofthe convolution,andrearranging thetermsin( 6{43 )yields = ~ W T (^ v + ^ )+ ^ W T ^ 0 ~ V T ( x d v + x d )+ ~ N 2 + N 2 B k 2 2 sgn ( )(6{44) wherethefactthat 2 = k 2 + 2 sgn ( )(6{45) wasutilized.In( 6{44 ),theunmeasurable/unknownauxiliaryterm ~ N 2 ( e 1 ;e 2 ;r;t ) 2 R n is denedas ~ N 2 S f S df (6{46) andtheterm N 2 B ( t ) 2 R n isdenedas N 2 B D D d +_ v ^ W T O ~ V T x d 2 + ~ W T ~ + + ^ W T O ~ V T x d 2 + ~ W T ~ + : (6{47) InasimilarmannerasinLemma 1 oftheAppendix,byapplyingtheMeanValue Theoremcanbeusedtodevelopthefollowingupperboundfort heexpressionin( 6{46 ): rrr ~ N 2 ( t ) rrr 2 ( k z k ) k z k (6{48) wheretheboundingfunction 2 ( ) 2 R isapositive,globallyinvertible,nondecreasing function,and z ( t ) 2 R 3 n isdenedas z ( t ) e T1 e T2 r T T : (6{49) UsingAssumption6-3,andthefactthat v ( t )isalinear,strictlyproper,exponentially stabletransferfunction,thefollowinginequalitycanbed evelopedbasedontheexpression in( 6{47 )withasimilarapproachasinLemma2of[ 42 ]: k N 2 B ( t ) k (6{50) where 2 R isaknownpositiveconstant. 129

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6.4.2CompositeAdaptation ThecompositeadaptationfortheNNweightestimatesisgive nby ^ W 1 proj ( 2 ^ 0 ^ V T x d e T2 + ^ f T )(6{51) ^ V 2 proj ( 2 x d e T2 ^ W T ^ 0 + x df T ^ W T ^ 0 )(6{52) where 1 2 R ( N 2 +1) ( N 2 +1) 2 2 R ( N 1 +1) ( N 1 +1) areconstant,positivedenite,symmetric controlgainmatrices, proj ( )denotesasmoothprojectionoperator(see[ 70 75 ])that isusedtoensurethat ^ W ( t )and ^ V ( t )remaininsidetheboundedconvexregion.Also, ( t ) 2 R n denotesthemeasurablepredictionerrordenedin( 6{20 ),andthescalar function v ( t )wasdenedin( 6{23 ).Thelteredactivationfunction^ f ( t ) 2 R N 2 +1 is generatedfromthestablerstorderdierentialequation ^ f + ^ f = ^ ,^ f (0)=0(6{53) whilethelteredNNinputvector x df ( t ) 2 R 3 n +1 isgeneratedby x df + x df = x d x df (0)=0 : (6{54) InatypicalNNweightadaptationlaw,onlythesystemtracki ngerrorsareused toupdatetheweightsandnoinformationabouttheactualest imatemismatch(i.e., ( t ) ^ ( t ))isutilizedasitisunmeasurable,andhencecannotbeused incontrol implementation.Theproposedmethodusestheswappingproc edureinSection 6.4.1 togenerateameasurablepredictionerror ( t )thatcontainsinformationrelatedtothe estimatemismatcherror. 130

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TheprojectionusedintheNNweightadaptationlawsin( 6{51 )and( 6{52 )canbe decomposedintotwotermsas 1 ^ W = W + We 2 ^ V = V + Ve 2 (6{55) suchthattheauxiliaryfunctions W (^ f ; ) ; We 2 ^ V; x d ; x d ;e 2 2 R ( N 2 +1) n and V x df ; ^ W; ^ V; ; Ve 2 ^ W; ^ V; x d ; x d ;e 2 2 R ( N 1 +1) N 2 satisfythefollowingbounds rr W rr b 1 k k rr We 2 rr b 2 k e 2 k (6{56) rr V rr b 01 k k rr Ve 2 rr b 02 k e 2 k where b 1 ;b 2 ;b 01 ; and b 02 2 R areknownpositiveconstants.Tofacilitatethesubsequent stabilityanalysis,thefollowinginequalityisdeveloped basedon( 6{56 )andthefactthat theNNweightestimatesareboundedbythesmoothprojection algorithm: rrr W ^ + ^ W T ^ 0 V x d rrr c 1 k k (6{57) where c 1 2 R isapositiveconstant. 6.4.3Closed-LoopErrorSystem Substitutingfor ^ W ( t )and ^ V ( t )from( 6{55 ),theexpressionin( 6{19 )canbe rewrittenas G 1 r = 1 2 G 1 r W ^ ^ W T ^ 0 V x d + ~ N 1 + N 1 ( k 1 +1) r 1 sgn ( e 2 ) e 2 (6{58) wherethefactthatthetimederivativeof( 6{16 )isgivenas 1 =( k 1 +1) r + 1 sgn ( e 2 )(6{59) 1 SeeLemma 5 oftheAppendixfortheproofofthedecompositionin( 6{55 )andthe inequalitiesin( 6{56 ). 131

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wasutilized.In( 6{58 ),theunmeasurable/unknownauxiliaryterms ~ N 1 ( e 1 ;e 2 ;r;t )and N 1 ( t ) 2 R n aredenedas ~ N 1 1 2 G 1 r + S 1 + e 2 We 2 ^ ^ W T ^ 0 Ve 2 x d (6{60) N 1 N d + N 1 B : (6{61) In( 6{61 ), N d ( x d ; x d ;t ) 2 R n isdenedas N d W T 0 V T x d +_ (6{62) while N 1 B ( ^ W; ^ V; x d ; x d ;t ) 2 R n isfurthersegregatedas N 1 B N 1 B a + N 1 B b (6{63) where N 1 B a ( ^ W; ^ V; x d ; x d ;t ) 2 R n isdenedas N 1 B a W T ^ 0 ^ V T x d ^ W T ^ 0 ~ V T x d (6{64) andtheterm N 1 B b ( ^ W; ^ V; x d ; x d ;t ) 2 R n isdenedas N 1 B b ^ W T ^ 0 ~ V T x d + ~ W T ^ 0 ^ V T x d : (6{65) Motivationforsegregatingthetermsin( 6{61 )isderivedfromthefactthatthedierent componentsin( 6{61 )havedierentbounds.Segregatingthetermsasin( 6{61 )-( 6{65 ) facilitatesthedevelopmentoftheNNweightupdatelawsand thesubsequentstability analysis.Forexample,thetermsin( 6{62 )aregroupedtogetherbecausethetermsand theirtimederivativescanbeupperboundedbyaconstantand rejectedbytheRISE feedback,whereasthetermsgroupedin( 6{63 )canbeupperboundedbyaconstantbut theirderivativesarestatedependent.Thestatedependenc yofthetimederivativesof thetermsin( 6{63 )violatestheassumptionsgiveninpreviousRISE-basedcon trollers (e.g.,[ 19 20 22 23 25 { 27 ]),andrequiresadditionalconsiderationintheadaptatio n lawdesignandstabilityanalysis.Thetermsin( 6{63 )arefurthersegregatedbecause 132

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N 1 B a ( ^ W; ^ V; x d ; x d )willberejectedbytheRISEfeedback,whereas N 1 B b ( ^ W; ^ V; x d ; x d )will bepartiallyrejectedbytheRISEfeedbackandpartiallycan celedbytheadaptiveupdate lawfortheNNweightestimates.Inasimilarmannerasin( 6{48 ),thefollowingupper boundisdevelopedfortheexpressionin( 6{60 ): rrr ~ N 1 ( t ) rrr 1 ( k z k ) k z k (6{66) wheretheboundingfunction 1 ( ) 2 R isapositive,globallyinvertible,nondecreasing function,and z ( t ) 2 R 3 n wasdenedin( 6{49 ).Thefollowinginequalitiescanbe developedbasedonAssumption6-3,( 3{7 ),( 3{8 ),( 3{16 ),( 6{63 )-( 6{65 ): k N d k 1 k N 1 B a k 2 k N 1 B b k 3 rrr N d rrr 4 : (6{67) From( 6{61 ),( 6{63 )and( 6{67 ),thefollowingboundcanbedeveloped k N 1 kk N d k + k N 1 B kk N d k + k N 1 B a k + k N 1 B b k 1 + 2 + 3 : (6{68) Byusing( 6{51 )and( 6{52 ),thetimederivativeof N 1 B ( ^ W; ^ V;x d )canbeboundedas rrr N 1 B rrr 5 + 6 k e 2 k + 7 k k : (6{69) In( 6{67 )and( 6{69 ), i 2 R ,( i =1 ; 2 ;:::; 7)areknownpositiveconstants. Forthesubsequentstabilityanalysis,let D R 4 n +3 beadomaincontaining y ( t )=0, where y ( t ) 2 R 4 n +3 isdenedas y [ z T T p P 1 p P 2 p Q ] T : (6{70) In( 6{70 ),theauxiliaryfunction P 1 ( t ) 2 R isdenedas P 1 ( t ) 1 n P i =1 j e 2 i (0) j e 2 (0) T N 1 (0) Z t 0 L 1 ( ) d (6{71) 133

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where e 2 i (0) 2 R denotesthe i thelementofthevector e 2 (0),andtheauxiliaryfunction L 1 ( t ) 2 R isdenedas L 1 r T ( N 1 B a + N d 1 sgn ( e 2 ))+_ e T2 N 1 B b 3 k e 2 k 2 4 k k 2 (6{72) where 1 ; 3 ; 4 2 R arepositiveconstantschosenaccordingtothesucientcon ditions 1 > max 1 + 2 + 3 ; 1 + 2 + 4 2 + 5 2 3 > 6 + 7 2 4 > 7 2 (6{73) where 1 7 wereintroducedin( 6{67 )and( 6{69 ).Providedthesucientconditions introducedin( 6{73 )aresatised,thefollowinginequalityisobtained[ 22 ],[ 20 ]: Z t 0 L 1 ( ) d 1 n P i =1 j e 2 i (0) j e 2 (0) T N 1 (0) : (6{74) Hence,( 6{74 )canbeusedtoconcludethat P 1 ( t ) 0.Alsoin( 6{70 ),theauxiliary function P 2 ( t ) 2 R isdenedas P 2 ( t ) Z t 0 L 2 ( ) d (6{75) wheretheauxiliaryfunction L 2 ( t ) 2 R isdenedas L 2 T ( N 2 B 2 sgn ( ))(6{76) where 2 2 R isapositiveconstantchosenaccordingtothesucientcond ition 2 > (6{77) where wasintroducedin( 6{50 ).Providedthesucientconditionintroducedin( 6{77 )is satised,then P 2 ( t ) 0.Theauxiliaryfunction Q ( t ) 2 R in( 6{70 )isdenedas Q ( t ) 1 2 tr ~ W T 1 1 ~ W + 1 2 tr ~ V T 1 2 ~ V : (6{78) Since 1 and 2 areconstant,symmetric,andpositivedenitematrices,it isstraightforward that Q ( t ) 0. 134

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6.5StabilityAnalysis Theorem6-1: Thecontrollergivenin( 6{15 )and( 6{16 )inconjunctionwith thecompositeNNadaptationlawsin( 6{51 )and( 6{52 ),wherethepredictionerror isgeneratedfrom( 6{20 ),( 6{21 ),( 6{37 ),and( 6{38 ),ensuresthatallsystemsignals areboundedunderclosed-loopoperationandthatthepositi ontrackingerrorandthe predictionerrorareregulatedinthesensethat k e 1 ( t ) k! 0and k ( t ) k! 0as t !1 providedthecontrolgains k 1 and k 2 introducedin( 6{16 )and( 6{38 )areselected sucientlylargebasedontheinitialconditionsofthesyst emsstates(seethesubsequent semi-globalstabilityproof),thesucientconditionsin( 6{73 )and( 6{77 )aresatised,and thefollowingconditionsaresatised: 1 > 1 2 ; 2 > 3 + 1 2 (6{79) wherethegains 1 and 2 wereintroducedin( 6{4 )and( 6{5 ). Proof: Let V L ( y;t ): D [0 ; 1 ) R beacontinuouslydierentiable,positivedenite functiondenedas V L 1 2 e T1 e 1 + 1 2 e T2 e 2 + 1 2 r T G 1 r + 1 2 T + P 1 + P 2 + Q (6{80) whichsatisestheinequalities U 1 ( y ) V L ( y;t ) U 2 ( y )(6{81) providedthesucientconditionsintroducedin( 6{73 )and( 6{77 )aresatised.In( 6{81 ), thecontinuouspositivedenitefunctions U 1 ( y ), U 2 ( y ) 2 R aredenedas U 1 ( y ) 1 k y k 2 135

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and U 2 ( y ) 2 ( x ) k y k 2 ,where 1 2 ( x ) 2 R aredenedas 1 1 2 min 1 ;g (6{82) 2 max f 1 2 g ( x ) ; 1 g where g g ( x )areintroducedin( 6{2 ).Using( 6{4 ),( 6{5 ),( 6{44 ),( 6{58 ),( 6{71 ),( 6{72 ), ( 6{75 )and( 6{76 ),thetimederivativeof( 6{80 )canbeexpressedas V L = e T1 ( e 2 1 e 1 )+ e T2 ( r 2 e 2 )+ r T ( 1 2 G 1 r W ^ ^ W T ^ 0 V x d + ~ N 1 (6{83) + N 1 ( k 1 +1) r 1 sgn ( e 2 ) e 2 )+ 1 2 r T G 1 r + T ( ~ W T ^ f + ^ W T ^ 0 ~ V T x df + ~ N 2 + N 2 B k 2 2 sgn ( )) r T ( N 1 B a + N d 1 sgn ( e 2 )) e T2 N 1 B b + 3 k e 2 k 2 + 4 k k 2 T ( N 2 B 2 sgn ( )) tr ( ~ W T 1 1 ^ W ) tr ( ~ V T 1 2 ^ V ) : Expandingthetermsin( 6{83 )andusing( 6{61 )and( 6{63 )yields V L = e T1 e 2 1 e T1 e 1 + e T2 r 2 e T2 e 2 + r T ~ N 1 1 2 r T G 1 r r T W ^ r T ^ W T ^ 0 V x d (6{84) + r T ( N d + N 1 B a )+(_ e 2 + 2 e 2 ) T N 1 B b ( k 1 +1) r T r 1 r T sgn ( e 2 ) r T e 2 + T ~ W T ^ f + T ^ W T ^ 0 ~ V T x df T N 2 B + T ~ N 2 + T N 2 B k 2 T 2 T sgn ( ) r T ( N 1 B a + N d 1 sgn ( e 2 )) e 2 ( t ) T N 1 B b + 3 k e 2 k 2 + 4 k k 2 + 2 T sgn ( )) + 1 2 r T G 1 r tr ( ~ W T 1 1 ^ W ) tr ( ~ V T 1 2 ^ V ) : Cancelingthecommontermsin( 6{84 ),andsubstitutingfor N 1 B b from( 6{65 ),andusing thefactthat a T b = tr ba T V L ( y;t )isexpressedas V L = e T1 e 2 1 e T1 e 1 2 e T2 e 2 k 2 T r T W ^ r T ^ W T ^ 0 V x d + r T ~ N 1 (6{85) ( k 1 +1) r T r + 3 k e 2 k 2 + 4 k k 2 + T ~ N 2 + tr 2 ~ W T ^ 0 ^ V T x d e T2 + tr ~ W T ^ f T + tr 2 ~ V T x d (^ 0 T ^ We 2 ) T + tr ~ V T x df ^ 0 T ^ W T tr ( ~ W T 1 1 ^ W ) tr ( ~ V T 1 2 ^ V ) : 136

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Substitutingtheupdatelawsfrom( 6{51 )and( 6{52 )in( 6{85 ),cancelingthesimilar terms,andusingthefactthat e T1 e 2 1 2 ( k e 1 k 2 + k e 2 k 2 ),theexpressionin( 6{85 )isupper boundedas V L 1 1 2 k e 1 k 2 2 1 2 3 k e 2 k 2 k r k 2 + c 1 k kk r k + k r k rrr ~ N 1 rrr k 1 k r k 2 + k k rrr ~ N 2 rrr ( k 2 4 ) k k 2 where( 6{57 )wasused.Usingthesquaresofthecomponentsof z ( t ) ; V L ( y;t )isupper boundedas V L 3 k z k 2 k 1 k r k 2 + k r k rrr ~ N 1 rrr + c 1 k kk z k + k k rrr ~ N 2 rrr k 2 k k 2 (6{86) where 3 min f 1 1 2 ; 2 1 2 3 ; 1 g : Letting k 2 = k 2 a + k 2 b where k 2 a ;k 2 b 2 R arepositiveconstants,andusingtheinequalitiesin( 6{48 )and( 6{66 ), theexpressionin( 6{86 )isupperboundedas V L 3 k z k 2 ( k 2 b 4 ) k k 2 k 1 k r k 2 1 ( k z k ) k r kk z k (6{87) k 2 a k k 2 ( 2 ( k z k )+ c 1 ) k kk z k : Completingthesquaresforthetermsinsidethebracketsin( 6{87 )yields V L 3 k z k 2 ( k 2 b 4 ) k k 2 + 21 ( k z k ) k z k 2 4 k 1 + ( 2 ( k z k )+ c 1 ) 2 k z k 2 4 k 2 a 3 k z k 2 + 2 ( k z k ) k z k 2 4 k ( k 2 b 4 ) k k 2 (6{88) where k 2 R isdenedas k min f k 1 ;k 2 a g (6{89) 137

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and ( ) 2 R isapositive,globallyinvertible,nondecreasingfunctio ndenedas 2 ( k z k ) 21 ( k z k )+( 2 ( k z k )+ c 1 ) 2 : Theexpressionin( 6{88 )canbefurtherupperboundedbyacontinuous,positive semi-denitefunction V L ( y;t ) U ( y )= c rrr z T T T rrr 2 8 y 2D (6{90) forsomepositiveconstant c 2 R ,where D n y ( t ) 2 R 4 n +3 jk y k 1 2 p 3 k o : Largervaluesof k willexpandthesizeofthedomain D .Theinequalitiesin( 6{81 ) and( 6{90 )canbeusedtoshowthat V L ( y;t ) 2L 1 in D ;hence, e 1 ( t ) ;e 2 ( t ) ;r ( t ) ; and ( t ) 2L 1 in D .Giventhat e 1 ( t ) ;e 2 ( t ) ; and r ( t ) 2L 1 in D ,standardlinearanalysis methodscanbeusedtoprovethat_ e 1 ( t ) ; and_ e 2 ( t ) 2L 1 in D from( 6{4 )and( 6{5 ).Since e 1 ( t ) ; e 2 ( t ) ; and r ( t ) 2L 1 in D ,Assumption6-3canbeusedalongwith( 6{3 )-( 6{5 )to concludethat x ( i ) ( t ) 2L 1 ; in D .Since x ( i ) ( t ) 2L 1 ; in D ,Assumption6-2canbeusedto concludethat G 1 ( )and f ( ) 2L 1 in D .Thus,from( 6{1 )wecanshowthat u ( t ) 2L 1 in D .Therefore, u f ( t ) 2L 1 in D ,andhence,from( 6{20 ),^ u f ( t ) 2L 1 in D .Giventhat r ( t ) 2L 1 in D ,( 6{59 )canbeusedtoshowthat_ 1 ( t ) 2L 1 in D ,andsince G 1 ( )and f ( ) 2L 1 in D ,( 6{58 )canbeusedtoshowthat_ r ( t ) 2L 1 in D ,and( 6{44 )canbeused toshowthat_ ( t ) 2L 1 in D .Since_ e 1 ( t ) ; e 2 ( t ),_ r ( t ),and_ ( t ) 2L 1 in D ,thedenitions for U ( y )and z ( t )canbeusedtoprovethat U ( y )isuniformlycontinuousin D Let SD denoteasetdenedas S y ( t ) 2Dj U 2 ( y ( t )) < 1 1 2 p 3 k 2 : (6{91) Theregionofattractionin( 6{91 )canbemadearbitrarilylargetoincludeanyinitial conditionsbyincreasingthecontrolgain k (i.e.,asemi-globalstabilityresult).Theorem 138

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8.4of[ 63 ]cannowbeinvokedtostatethat c rrr z T T T rrr 2 0as t !18 y (0) 2S : (6{92) Basedonthedenitionof z ( t ),( 6{92 )canbeusedtoshowthat k e 1 ( t ) k! 0as t !18 y (0) 2S k ( t ) k! 0as t !18 y (0) 2S 6.6Experiment AsinChapter2,thetestbeddepictedinFigure 2-1 wasusedtoimplementthe developedcontroller.Noexternalfrictionisappliedtoth ecirculardisk.Thedesiredlink trajectoryisselectedasfollows(indegrees): q d ( t )=60 : 0sin(3 : 0 t )(1 exp( 0 : 01 t 3 )) : (6{93) Forallexperiments,therotorvelocitysignalisobtainedb yapplyingastandardbackwards dierencealgorithmtothepositionsignal.Theintegralst ructurefortheRISEtermin ( 6{16 )wascomputedon-lineviaastandardtrapezoidalalgorithm .TheNNinputvector x d ( t ) 2 R 4 isdenedas x d =[1 q d q d q d ] T : Theinitialvaluesof ^ W (0)werechosentobeazeromatrix;however,theinitialvalu esof ^ V (0)wereselectedrandomlybetween 1 : 0and1 : 0toprovideabasis[ 71 ].Adierent transientresponsecouldbeobtainediftheNNweightsarein itializeddierently.Ten hiddenlayerneuronswerechosenbasedontrialanderror.In addition,allthestateswere initializedtozero Thefollowingcontrolgainswereusedtoimplementthecontr ollerin ( 6{15 )inconjunctionwiththecompositeadaptiveupdatelawsin( 6{51 )and( 6{52 ), wherethepredictionerrorisgeneratedfrom( 6{20 ),( 6{21 ),( 6{37 ),and( 6{38 ): 139

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0 5 10 15 20 25 30 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Tracking Error [degrees]Time [sec] Figure6-2.Trackingerrorfortheproposedcompositeadapt ivecontrollaw(RISE+CNN). k 1 =30 ; 1 =10 ; 1 =10 ; 2 =10 ; 1 =5 I 11 ; 2 =0 : 05 I 4 k 2 =30 ; 2 =10 ;! =2 : (6{94) 6.6.1Discussion Threedierentexperimentswereconductedtodemonstratet heecacyofthe proposedcontroller.Thecontrolgainswerechosentoobtai nanarbitrarytracking erroraccuracy(notnecessarilythebestperformance),and foreachcontroller,thegains werenotretuned(i.e.,thecommoncontrolgainsremainthes ameforallcontrollers). First,noadaptationwasusedandthecontrollerwithonlyth eRISEfeedbackwas implemented.Forthesecondexperiment,thepredictionerr orcomponentoftheupdate lawsin( 6{51 )and( 6{52 )wasremoved,resultinginastandardgradient-basedNNupd ate law(hereinafterdenotedasRISE+NN).Forthethirdexperim ent,theproposedcomposite adaptivecontrollerin( 6{15 )(hereinafterdenotedasRISE+CNN)wasimplemented. ThetrackingerrorandthepredictionerrorareshowninFigu re 6-2 andFigure 6-3 140

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0 5 10 15 20 25 30 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Prediction Error [N-m]Time [sec] Figure6-3.Predictionerrorfortheproposedcompositeada ptivecontrollaw (RISE+CNN). 0 5 10 15 20 25 30 -15 -10 -5 0 5 10 15 Torque [N-m]Time [sec] Figure6-4.Controltorquefortheproposedcompositeadapt ivecontrollaw(RISE+CNN). 141

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Figure6-5.AverageRMSerrors(degrees)andtorques(N-m). 1-RISE,2-RISE+NN,3RISE+CNN(proposed). respectively.ThecontroltorqueisshowninFigure 6-4 .Eachexperimentwasperformed vetimesandtheaverageRMSerrorandtorquevaluesareshow ninFigure 6-5 ,which indicatethattheproposedRISE+CNNcontrolleryieldsthel owestRMSerrorwitha similarcontroleort. 6.7Conclusion Arstevergradient-basedcompositeNNcontrollerisdevel opedfornonlinear uncertainsystems.ANNfeedforwardcomponentisusedincon junctionwiththeRISE feedback,wheretheNNweightestimatesaregeneratedusing acompositeupdatelaw drivenbyboththetrackingandthepredictionerrorwiththe motivationofusing moreinformationintheNNupdatelaw.TheconstructionofaN N-basedcontrollerto approximatetheunknownsystemdynamicsinherentlyresult sinaresidualfunction reconstructionerror.Thepresenceofthereconstructione rrorhasbeenthetechnical obstaclethathaspreventedthedevelopmentofcompositead aptationlawsforNNs. Tocompensatefortheeectsofthereconstructionerror,th etypicalpredictionerror formulationismodiedtoincludeaRISE-likestructureint hedesignoftheestimated lteredcontrolinput.UsingaLyapunovstabilityanalysis ,sucientgainconditions arederivedunderwhichtheproposedcontrolleryieldssemi -globalasymptoticstability. 142

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Experimentsonarotatingdiskindicatethattheproposedme thodyieldsbettertracking performancewithasimilarcontroleort. 143

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CHAPTER7 CONCLUSIONSANDFUTUREWORK 7.1Conclusions Anewclassofasymptoticcontrollersisdevelopedthatcont ainsanadaptive feedforwardtermtoaccountforlinearparameterizableunc ertaintyandahighgainRISE feedbacktermwhichaccountsforunstructureddisturbance s.Incomparisonwithprevious resultsthatusedasimilarhighgainfeedbackcontrolstruc ture,newcontroldevelopment wasrequiredtoincludetheadditionaladaptivefeedforwar dterm.Themotivationfor injectingtheadaptivefeedforwardtermisthatimprovedtr ackingperformanceand reducedcontroleortresultfromincludingmoreknowledge ofthesystemdynamicsinthe controlstructure.Thisheuristicideawasveriedbyourex perimentalresultsthatindicate reducedcontroleortandreducedRMStrackingerrors. TheresearchfurtherillustrateshowamultilayerNNfeedfo rwardtermcanbefused withaRISEfeedbackterminacontinuouscontrollertoachie vesemi-globalasymptotic tracking.Improvedweighttuninglawsarepresentedwhichg uaranteeboundednessofNN weights.ToblendtheNNandRISEmethods,severaltechnical challengeswereaddressed throughLyapunov-basedtechniques.Thesechallengesincl udedevelopingadaptiveupdate lawsfortheNNweightestimatesthatdonotdependonacceler ation,anddeveloping newRISEstabilityanalysismethodsandsucientgaincondi tionstoaccommodatethe incorporationoftheNNadaptiveupdatesintheRISEstructu re.Experimentalresultsare presentedthatindicatereducedRMStrackingerrors. Further,aRISE-basedapproachwaspresentedtoachievemod ularityinthe controller/updatelawforageneralclassofmulti-inputsy stems.Specically,forsystems withstructuredandunstructureduncertainties,acontrol lerwasemployedthatusesa model-basedfeedforwardadaptiveterminconjunctionwith theRISEfeedbackterm. Theadaptivefeedforwardtermwasmademodularbyconsideri ngagenericformof theadaptiveupdatelawanditscorrespondingparameterest imate.Thisgenericform 144

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oftheupdatelawwasusedtodevelopanewclosed-looperrors ystem,andthetypical RISEstabilityanalysiswasmodied.Newsucientgaincond itionswerederivedto showasymptotictrackingofthedesiredlinkposition.Thec lassofRISE-basedmodular adaptivecontrollersisthenextendedtoincludeuncertain dynamicsystemsthatdonot satisfytheLPassumption.Specically,theresultallowst heNNweighttuninglaws tobedeterminedfromadevelopedgenericupdatelaw(rather thanbeingrestrictedto agradientupdatelaw).Themodularadaptivecontroldevelo pmentisthenappliedto Euler-Lagrangedynamicsystems.Anexperimentalsectioni sincludedthatillustratesthe concept. Inaddition,anovelapproachforthedesignofagradient-ba sedcompositeadaptive controllerwasproposedforgenericMIMOsystemssubjected toboundeddisturbances. Amodel-basedfeedforwardadaptivecomponentwasusedinco njunctionwiththeRISE feedback,wheretheadaptiveestimatesweregeneratedusin gacompositeupdatelaw drivenbyboththetrackingandpredictionerrorwiththemot ivationofusingmore informationintheadaptiveupdatelaw.Toaccountforthee ectsofnon-LPdisturbances, thetypicalpredictionerrorformulationwasmodiedtoinc ludeasecondRISE-liketerm intheestimatedlteredcontrolinputdesign.UsingaLyapu novstabilityanalysis, sucientgainconditionswerederivedunderwhichthepropo sedcontrolleryields semi-globalasymptoticstability.Experimentsonarotati ngdiskwithexternallyapplied frictionindicatethattheproposedmethodyieldsbettertr ackingperformancewitha similarcontroleort. Thistechniqueisthenextendedtodeveloparstevergradie nt-basedcomposite NNcontrollerfornonlinearuncertainsystems.ANNfeedfor wardcomponentisused inconjunctionwiththeRISEfeedback,wheretheNNweightes timatesaregenerated usingacompositeupdatelawdrivenbyboththetrackingandt hepredictionerrorwith themotivationofusingmoreinformationintheNNupdatelaw .Theconstructionofa NN-basedcontrollertoapproximatetheunknownsystemdyna micsinherentlyresults 145

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inaresidualfunctionreconstructionerror.Thepresenceo fthereconstructionerrorhas beenthetechnicalobstaclethathaspreventedthedevelopm entofcompositeadaptation lawsforNNs.Tocompensatefortheeectsofthereconstruct ionerror,thetypical predictionerrorformulationismodiedtoincludeaRISE-l ikestructureinthedesignof theestimatedlteredcontrolinput.UsingaLyapunovstabi lityanalysis,sucientgain conditionsarederivedunderwhichtheproposedcontroller yieldssemi-globalasymptotic stability. 7.2FutureWork Theresultsinthisdissertation,aswellasallpreviousRIS Econtrollers,require full-statefeedback.Thedevelopmentofanoutputfeedback resultremainsanopen problem. Also,oneoftheshortcomingsofcurrentworkisthatonlyase mi-globalasymptotic stabilityisachieved,andfurtherinvestigationisneeded toachieveaglobalstability result.Inroadstosolvetheglobaltrackingproblemarepro videdin[ 78 ]underaset ofassumptions. WhilethemodularadaptiveresultinChapter4encompassesa largevarietyof adaptiveupdatelaws,anupdatelawdesignbasedonthepredi ctionerrorisnot possiblebecausetheformulationofapredictionerrorrequ iresthesystemdynamics tobecompletelyLP.Futureeortscanfocusondevelopingaa daptivecontroller thatcouldalsouseapredictionerrorinthemodularadaptiv eupdatelaw. ThecompositeadaptationpresentedinChapters5and6utili zethegradient-based updatelaws.Futureeortscouldfocusondevelopingcompos iteadaptationthatuse least-squaresestimation. 146

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APPENDIX Lemma1. TheMeanValueTheoremcanbeusedtodeveloptheupperboundi n( 2{21 ) rrr ~ N ( t ) rrr ( k z k ) k z k where z ( t ) 2 R 3 n isdenedas z ( t ) e T1 e T2 r T T : (A{1) Proof: Theauxiliaryerror ~ N ( t )in( 2{18 )canbewrittenasthesumoferrors pertainingtoeachofitsargumentsasfollows: ~ N ( t )= N ( q; q; q d ; ... q d ;e 1 ;e 2 ;r ) N ( q d ; q d ; q d ; ... q d ; 0 ; 0 ; 0) = N ( q; q d ; q d ; ... q d ; 0 ; 0 ; 0) N ( q d ; q d ; q d ; ... q d ; 0 ; 0 ; 0) + N ( q; q; q d ; ... q d ; 0 ; 0 ; 0) N ( q; q d ; q d ; ... q d ; 0 ; 0 ; 0) + N ( q; q; q d ; ... q d ; 0 ; 0 ; 0) N ( q; q; q d ; ... q d ; 0 ; 0 ; 0) + N ( q; q; q d ; ... q d ; 0 ; 0 ; 0) N ( q; q; q d ; ... q d ; 0 ; 0 ; 0) + N ( q; q; q d ; ... q d ;e 1 ; 0 ; 0) N ( q; q; q d ; ... q d ; 0 ; 0 ; 0) + N ( q; q; q d ; ... q d ;e 1 ;e 2 ; 0) N ( q; q; q d ; ... q d ;e 1 ; 0 ; 0) + N ( q; q; q d ; ... q d ;e 1 ;e 2 ;r ) N ( q; q; q d ; ... q d ;e 1 ;e 2 ; 0) : ApplyingtheMeanValueTheoremtofurtherdescribe ~ N ( t ) ; ~ N ( t )= @N ( 1 ; q d ; q d ; ... q d ; 0 ; 0 ; 0) @ 1 j 1 = v 1 ( q q d )+ @N ( q; 2 ; q d ; ... q d ; 0 ; 0 ; 0) @ 2 j 2 = v 2 (_ q q d ) + @N ( q; q; 3 ; ... q d ; 0 ; 0 ; 0) @ 3 j 3 = v 3 ( q d q d )+ @N ( q; q; q d ; 4 ; 0 ; 0 ; 0) @ 4 j 4 = v 4 ( ... q d ... q d ) (A{2) + @N ( q; q; q d ; ... q d ; 5 ; 0 ; 0) @ 5 j 5 = v 5 ( e 1 0)+ @N ( q; q; q d ; ... q d ;e 1 ; 6 ; 0) @ 6 j 6 = v 6 ( e 2 0) + @N ( q; q; q d ; ... q d ;e 1 ;e 2 ; 7 ) @ 7 j 7 = v 7 ( r 0) 147

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where v 1 2 ( q d ;q ) ;v 2 2 (_ q d ; q ) ;v 3 2 ( q d ; q d ) ;v 4 2 ( ... q d ; ... q d ) ;v 5 2 (0 ;e 1 ) ; v 6 2 (0 ;e 2 ) ;v 7 2 (0 ;r ) : Fromequation( A{2 ), ~ N ( t )canbeupperboundedasfollows: rrr ~ N ( t ) rrr rrrr @N ( 1 ; q d ; q d ; ... q d ; 0 ; 0 ; 0) @ 1 j 1 = v 1 rrrr k e 1 k (A{3) + rrrr @N ( q; 2 ; q d ; ... q d ; 0 ; 0 ; 0) @ 2 j 2 = v 2 rrrr k e 2 a 1 e 1 k + rrrr @N ( q; q; q d ; ... q d ; 5 ; 0 ; 0) @ 5 j 5 = v 5 rrrr k e 1 k + rrrr @N ( q; q; q d ; ... q d ;e 1 ; 6 ; 0) @ 6 j 6 = v 6 rrrr k e 2 k + rrrr @N ( q; q; q d ; ... q d ;e 1 ;e 2 ; 7 ) @ 7 j 7 = v 7 rrrr k r k : Bynotingthat v 1 = q c 1 ( q q d ) ;v 2 =_ q c 2 (_ q q d ) ;v 5 = e 1 (1 c 5 ) ;v 6 = e 2 (1 c 6 ) ; v 7 = r (1 c 7 ) where c i 2 (0 ; 1) 2 R i =1 ; 2 ; 5 ; 6 ; 7areunknownconstants,wecanupperboundthe partialderivativesasfollows: rrrr @N ( 1 ; q d ; q d ; ... q d ; 0 ; 0 ; 0) @ 1 j 1 = v 1 rrrr 1 ( e 1 ) rrrr @N ( q; 2 ; q d ; ... q d ; 0 ; 0 ; 0) @ 2 j 2 = v 2 rrrr 2 ( e 1 ;e 2 ) rrrr @N ( q; q; q d ; ... q d ; 5 ; 0 ; 0) @ 5 j 5 = v 5 rrrr 5 ( e 1 ;e 2 ) rrrr @N ( q; q; q d ; ... q d ;e 1 ; 6 ; 0) @ 6 j 6 = v 6 rrrr 6 ( e 1 ;e 2 ) rrrr @N ( q; q; q d ; ... q d ;e 1 ;e 2 ; 7 ) @ 7 j 7 = v 7 rrrr 7 ( e 1 ;e 2 ;r ) : 148

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Theboundon ~ N ( t )canbefurthersimplied: rrr ~ N ( t ) rrr 1 ( e 1 ) k e 1 k + 2 ( e 1 ;e 2 ) k e 2 1 e 1 k + 5 ( e 1 ;e 2 ) k e 1 k + 6 ( e 1 ;e 2 ) k e 2 k + 7 ( e 1 ;e 2 ;r ) k r k : Usingtheupperbound k e 2 1 e 1 kk e 2 k + 1 k e 1 k : ~ N ( t )canbefurtherupperboundedasfollows: rrr ~ N ( t ) rrr ( 1 ( e 1 )+ 1 2 ( e 1 ;e 2 )+ 5 ( e 1 ;e 2 )) k e 1 k +( 2 ( e 1 ;e 2 )+ 6 ( e 1 ;e 2 )) k e 2 k + 7 ( e 1 ;e 2 ;r ) k r k : Usingthedenitionof z ( t ) 2 R 3 n in( A{1 ), ~ N ( t )canbeexpressedintermsof z ( t )as follows: rrr ~ N ( t ) rrr ( 1 ( e 1 )+ 1 2 ( e 1 ;e 2 )+ 5 ( e 1 ;e 2 )) k z ( t ) k +( 2 ( e 1 ;e 2 )+ 6 ( e 1 ;e 2 )) k z ( t ) k + 7 ( e 1 ;e 2 ;r ) k z ( t ) k ( 1 ( e 1 )+ 1 2 ( e 1 ;e 2 )+ 5 ( e 1 ;e 2 )+ 2 ( e 1 ;e 2 )+ 6 ( e 1 ;e 2 )+ 7 ( e 1 ;e 2 ;r )) k z ( t ) k : Therefore, rrr ~ N ( t ) rrr ( k z k ) k z k where ( k z k )issomepositivegloballyinvertiblenondecreasingfunct ion. Lemma2. Letthefunction L ( t ) 2 R bedenedas L ( t ) r T ( N B ( t ) 1 sgn ( e m )) m X i =1 i +1 k e i ( t ) kk e m ( t ) k m +2 k e m ( t ) kk r ( t ) k : (A{4) Ifthefollowingsucientconditionsaresatised: 1 > 1 + 1 m 2 + 1 m 3 (A{5) i +1 > i +3 ;i =1 ; 2 ;:::;m +1 ; 149

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then Z t 0 L ( ) d 1 n X i =1 j e mi (0) j e m (0) T N B (0) : (A{6) Proof: Integratingbothsidesoftheequationin( A{6 ) Z t 0 L ( ) d = Z t 0 ( r ( ) T ( N B ( ) 1 sgn ( e m ( )) m X i =1 i +1 k e i ( ) kk e m ( ) k (A{7) m +2 k e m ( ) kk r ( ) k ) d: Substituting( 4{6 )into( A{7 ), Z t 0 L ( ) d = Z t 0 de m ( ) d T N B ( ) d + Z t 0 m e m ( ) T ( N B ( ) 1 sgn ( e m ( ))) d Z t 0 de m ( ) d T 1 sgn ( e m ( )) d Z t 0 m X i =1 i +1 k e i ( ) kk e m ( ) k d (A{8) Z t 0 m +2 k e m ( ) kk r ( ) k d: Integratingtherstintegralin( A{8 )bypartsyields Z t 0 L ( ) d = e m ( ) T N B ( ) t0 Z t 0 e m ( ) T dN B ( ) d d (A{9) + Z t 0 m e m ( ) T ( N B ( ) 1 sgn ( e m ( ))) d Z t 0 de m ( ) d T 1 sgn ( e m ( )) d Z t 0 m X i =1 i +1 k e i ( ) kk e m ( ) k d Z t 0 m +2 k e m ( ) kk r ( ) k d: Byusingthefactthat dN B ( ) d = d ( N B 1 ( )+ N B 2 ( )) d = dN B 1 ( ) d + dN B 2 ( ) d ; 150

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theexpressionin( A{9 )isrewrittenas Z t 0 L ( ) d = Z t 0 m e m ( ) T ( N B ( ) 1 m dN B 1 ( ) d 1 sgn ( e m ( ))) d (A{10) Z t 0 e m ( ) T dN B 2 ( ) d d Z t 0 m X i =1 i +1 k e i ( ) kk e m ( ) k d Z t 0 m +2 k e m ( ) kk r ( ) k d 1 n X i =1 j e mi ( t ) j + 1 n X i =1 j e mi (0) j + e m ( t ) T N B ( t ) e m (0) T N B (0) : Rearrangingtermsin( A{10 ),yields Z t 0 L ( ) d Z t 0 m k e m ( ) k ( k N B ( ) k + 1 m rrrr dN B 1 ( ) d rrrr 1 ) d + Z t 0 k e m ( ) k rrrr dN B 2 ( ) d rrrr d Z t 0 m X i =1 i +1 k e i ( ) kk e m ( ) k d Z t 0 m +2 k e m ( ) kk r ( ) k d (A{11) 1 k e m ( t ) k + 1 n X i =1 j e mi (0) j + k e m ( t ) kk N B ( t ) k e m (0) T N B (0) wherethefactthat n X i =1 j e mi ( t ) jk e m ( t ) k wasused.Usingtheinequalities( 4{31 )in( A{11 )yields Z t 0 L ( ) d Z t 0 m k e m ( ) k ( 1 + 1 m 2 + 1 m 3 1 ) d (A{12) + Z t 0 k e m ( ) k ( m X i =1 k e i ( ) k ( i +3 i +1 )) d + Z t 0 k e m ( ) kk r ( ) k ( m +4 m +2 ) d + k e m ( t ) k ( 1 1 )+ 1 n X i =1 j e mi (0) j e m (0) T N B (0) : From( A{12 ),ifthesucientconditionsin( A{5 )aresatised,then( A{6 )holds. Lemma3. Theregions V and W denedin( 3{9 )and( 3{11 ),respectively,areconvex [ 83 ]. 151

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Proof: Let vec ( V 1 )and vec ( V 2 ) 2 V ; therefore vec ( V 1 ) T vec ( V 1 ) V B and vec ( V 2 ) T vec ( V 2 ) V B : Consider vec ( V )= vec ( V 1 )+(1 ) vec ( V 2 ) ; 2 [0 ; 1] : (A{13) Takingthetransposeof( A{13 )andmultiplyingbyitselfyields vec ( V ) T vec ( V )= 2 vec ( V 1 ) T vec ( V 1 )+(1 ) 2 vec ( V 2 ) T vec ( V 2 ) +2 (1 ) vec ( V 1 ) T vec ( V 2 ) : Using( 3{7 ),thefollowinginequalitycanbedeveloped: vec ( V ) T vec ( V ) 2 V B +(1 ) 2 V B +2 (1 ) V B V B : Therefore, V isconvex.Theconvexityof W followsthesamedevelopment. Lemma4. GiventhecompositeNNandRISEcontrollerin( 3{17 )-( 3{21 ),if vec ( ^ W (0)) 2 int ( W ) and vec ( ^ V (0)) 2 int ( V ) ,then ^ W ( t ) and ^ V ( t ) neverleavetheregions W and V describedinAssumption3-2,respectively,andanupperbou ndfortheexpressiongiven in( 3{51 )canbeformulatedasfollows V L ( y;t ) r T ~ N ( t ) ( k s +1) k r k 2 (A{14) (2 1 1) k e 1 k 2 ( 2 2 1) k e 2 k 2 : Proof: Thefollowingninecasesmustbeconsidered(seeAppendixB. 3.2of[ 70 ]). Case1: vec ( ^ W ( t )) 2 int ( W ), vec ( ^ V ( t )) 2 int ( V ) Inthiscase,( 3{51 )canbeexpressedas V L ( y;t )= 2 1 k e 1 k 2 +2 e T2 e 1 + r T ~ N ( t ) ( k s +1) k r k 2 2 k e 2 k 2 + 2 k e 2 k 2 (A{15) + 2 e T2 h ~ W T (^ )+ ^ W T ^ 0 ~ V T x d i tr ( 2 ~ W T ^ e T2 ) tr ( 2 ~ V T x d (^ 0 T ^ We 2 ) T ) ; 152

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Basedonthefactthat a T b = tr ba T 8 a;b 2 R n ( A{15 )canbereducedto( A{14 ).Inaddition,thedirectioninwhichtheestimates vec ( ^ W )and vec ( ^ V )areupdatedforCase1isirrelevant,sincetheworstcasesc enariois that vec ( ^ W )and vec ( ^ V )willmovetowardstheboundariesoftheconvexregionsdeno ted by @ ( W )and @ ( V ),respectively,whichisaddressedsubsequently. Case2: vec ( ^ W ( t )) 2 int ( W ), vec ( ^ V ) 2 @ ( V )and vec ( 2 ) T vec ( ^ V ) ? 0 Inthiscase,( 3{51 )canbeexpressedas( A{15 ),whichcanbereducedto( A{14 ). Inaddition,thevector vec ( 2 )hasazeroornonzerocomponentperpendicularto theboundary @ ( V )at vec ( ^ V )thatpointsinthedirectiontowardsthe int ( V ). Geometrically,thismeansthat vec ( ^ V )isupdatedsuchthatiteithermovestowards the int ( V )orremainsontheboundary.Hence, vec ( ^ V )neverleaves V Case3: vec ( ^ W ( t )) 2 int ( W ), vec ( ^ V ) 2 @ ( V )and vec ( 2 ) T vec ( ^ V ) ? > 0 V L ( y;t )= 2 1 k e 1 k 2 +2 e T2 e 1 + r T ~ N ( t ) ( k s +1) k r k 2 2 k e 2 k 2 + 2 k e 2 k 2 + 2 e T2 h ~ W T (^ )+ ^ W T ^ 0 ~ V T x d i tr 2 ~ W T ^ e T2 tr 2 ~ V T 1 2 devec P t r ( vec ( 2 )) ; V L ( y;t )= 2 1 k e 1 k 2 +2 e T2 e 1 + r T ~ N ( t ) ( k s +1) k r k 2 2 k e 2 k 2 + 2 k e 2 k 2 + tr 2 ~ V T 1 2 2 tr 2 ~ V T 1 2 devec P t r ( vec ( 2 )) ; V L ( y;t )= 2 1 k e 1 k 2 +2 e T2 e 1 + r T ~ N ( t ) ( k s +1) k r k 2 2 k e 2 k 2 + 2 k e 2 k 2 + tr 2 ~ V T 1 2 2 devec P t r ( vec ( 2 )) ; 153

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Using( 3{10 ),weget V L ( y;t )= 2 1 k e 1 k 2 +2 e T2 e 1 + r T ~ N ( t ) ( k s +1) k r k 2 2 k e 2 k 2 + 2 k e 2 k 2 + tr 2 ~ V T 1 2 devec P ? r ( vec ( 2 )) ; V L ( y;t )= 2 1 k e 1 k 2 +2 e T2 e 1 + r T ~ N ( t ) ( k s +1) k r k 2 2 k e 2 k 2 + 2 k e 2 k 2 + 2 vec ( ~ V ) T diag 1 2 ; 1 2 ;:::N 2 times (A{16) P ? r ( vec ( 2 )) ; where N 2 wasdescribedin( 3{4 ).Because vec ( ^ V ) 2 @ ( V ),and vec ( V )mustlieeitheron theboundaryorintheinteriorof V ,thentheconvexityof V impliesthat vec ( ~ V )will eitherpointtangentto @ ( V )ortowards int ( V ).Thatis, vec ( ~ V )willhaveacomponent inthedirectionof vec ( ^ V ) ? thatiseitherzeroornegative.Inaddition,since P ? r ( vec ( 2 )) pointsawayfrom int ( V ),thefollowinginequalitycanbedetermined 2 vec ( ~ V ) T diag 1 2 ; 1 2 ;:::N 2 times P ? r ( vec ( 2 )) 0 (A{17) Theinequalitygivenin( A{17 )cannowbeusedtosimplifytheexpressiongivenin( A{16 ) as( A{14 ).Furthermore,since ^ V = devec ( P t r ( vec ( 2 ))),theparameterestimate vec ( ^ V )is ensuredtobeupdatedsuchthatitmovestangentto @ ( V ).Hence, vec ( ^ V )neverleaves V Case4: vec ( ^ W ) 2 @ ( W )and vec ( 1 ) T vec ( ^ W ) ? 0, vec ( ^ V ) 2 int ( V ) SimilartoCase2.Case5: vec ( ^ W ) 2 @ ( W )and vec ( 1 ) T vec ( ^ W ) ? > 0, vec ( ^ V ) 2 int ( V ) SimilartoCase3.Case6: vec ( ^ W ) 2 @ ( W )and vec ( 1 ) T vec ( ^ W ) ? 0, vec ( ^ V ) 2 @ ( V )and vec ( 2 ) T vec ( ^ V ) ? 0 SimilartoCase2. 154

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Case7: vec ( ^ W ) 2 @ ( W )and vec ( 1 ) T vec ( ^ W ) ? > 0, vec ( ^ V ) 2 @ ( V )and vec ( 2 ) T vec ( ^ V ) ? 0 SimilartotheargumentsinCase2andCase3.Case8: vec ( ^ W ) 2 @ ( W )and vec ( 1 ) T vec ( ^ W ) ? 0, vec ( ^ V ) 2 @ ( V )and vec ( 2 ) T vec ( ^ V ) ? > 0 SimilartoCase2andCase3.Case9: vec ( ^ W ) 2 @ ( W )and vec ( 1 ) T vec ( ^ W ) ? > 0, vec ( ^ V ) 2 @ ( V )and vec ( 2 ) T vec ( ^ V ) ? > 0 SimilartoCase3. Lemma5. Proofofthedecompositionin( 6{55 )andtheinequalitiesin( 6{56 ) Proof: Thesmoothprojectionusedin( 6{51 )isoftheform ^ W proj ( % 1 )= 8>>>><>>>>: % 1 if vec ( ^ W ) 2 int ( W ) % 1 if vec ( ^ W ) 2 @ ( W )and vec ( % 1 ) T vec ( ^ W ) ? 0 P t Mr ( % 1 )if vec ( ^ W ) 2 @ ( W )and vec ( % 1 ) T vec ( ^ W ) ? > 0 where % 1 = 1 2 ^ 0 ^ V T x d e T2 + 1 ^ f T : Forthersttwocasesoftheprojection,wehave proj ( % 1 )= % 1 ,andtherefore, ^ W = W + We 2 suchthat W = 1 ^ f T We 2 = 1 2 ^ 0 ^ V T x d e T2 whichusingthefactthattheNNweightestimatesandinputve ctorarebounded,canbe upperboundedas rr W rr b 1 k k rr We 2 rr b 2 k e 2 k : 155

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Forthethirdcaseoftheprojection,wehave ^ W = P t Mr 1 2 ^ 0 ^ V T x d e T2 + 1 ^ f T : Sincethetangentialcomponentofthesumisthesameasthesu mofthetangential components,weget ^ W = P t Mr 1 2 ^ 0 ^ V T x d e T2 + P t Mr 1 ^ f T : Therefore,choose W = P t Mr 1 ^ f T We 2 = P t Mr 1 2 ^ 0 ^ V T x d e T2 andthefollowinginequalitieshold rr W rr b 1 k k rr We 2 rr b 2 k e 2 k : 156

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BIOGRAPHICALSKETCH ParagPatrewasborninNagpur,India.HereceivedhisBachel orofTechnology degreeinMechanicalEngineeringfromtheIndianInstitute ofTechnologyMadras,India in2004.AfterthisheworkedforayearasaGraduateEngineer TraineeinLarsen& ToubroLimited,Asia'slargestengineeringconglomerate. HethenjoinedtheNonlinear ControlsandRobotics(NCR)researchgroupoftheDepartmen tofMechanicaland AerospaceEngineering,UniversityofFloridaintheFallof 2005topursuehisdoctoral researchundertheadvisementofDr.WarrenE.Dixon. 163