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Partial Differential Equation Based Control of Nonlinear Systems with Input Delays

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Title:
Partial Differential Equation Based Control of Nonlinear Systems with Input Delays
Creator:
Chakraborty, Indrasis
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
DIXON,WARREN E
Committee Co-Chair:
CRANE,CARL D,III
Committee Members:
UKEILEY,LAWRENCE S
ZHANG,LEI

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Subjects / Keywords:
pde -- time-delay
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
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Mechanical Engineering thesis, Ph.D.

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Abstract:
Partial differential equation (PDE)-based control methods are developed for a class of uncertain nonlinear systems with bounded external disturbances and known/unknown time-varying input delay. Inspired by predictor-based delay compensators, a linear transformation is used to relate the control input to a spatially and time-varying function. The transformation allows the input to be expressed in a manner that separates the control into a delayed control and non-delayed control and also facilitates the ability to compensate for the time-varying aspect of the delay with less complex gain conditions than previous robust control approaches. Unlike previous predictor-based approaches, which inherently depend on the system dynamics, an auxiliary error function is introduced to facilitate a robust control structure that does not depend on known dynamics. The designed controller features gains to compensate for the delay and delay derivative independently and further robustness is achieved since the controller does not require exact model knowledge. In Chapter 2, a tracking controller is developed for a second order system with a known time-varying input delay. A novel Lyapunov-Krasovskii functional is used in the Lyapunov based stability analysis to prove uniform ultimate boundedness of the error signals. Chapter 3 and Chapter 4 focus on the development of a tracking controller for a generalized uncertain nonlinear systems with bounded external disturbances and unknown time-varying input delay. In Chapter 3, a nonlinear mapping is used to map the non-compact time domain to a compact spatial domain, and then a neural network (NN) is used to estimate the unknown time-varying input delay. In Chapter 4, an accelerated gradient descent (AGD) based optimization method is demonstrated to estimate the unknown time-varying delay magnitude. Application of input time-delay for flexible system is examined in Chapter 5. Specifically, an aircraft wing dynamics is considered. The NN based estimation scheme developed in Chapter 3 is combined with a boundary control method, to mitigate oscillations in the aircraft wing in Chapter 5. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2017.
Local:
Adviser: DIXON,WARREN E.
Local:
Co-adviser: CRANE,CARL D,III.
Statement of Responsibility:
by Indrasis Chakraborty.

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PARTIALDIFFERENTIALEQUATIONBASEDCONTROLOFNONLINEARSYSTEMS WITHINPUTDELAYS By INDRASISCHAKRABORTY ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2017

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c 2017IndrasisChakraborty

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Totheultimatebundleofjoy,mydog,Buchuandmyancee,Aishwaryaformakingmylifeso comfortable

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ACKNOWLEDGMENTS Iwouldliketoexpresssinceregratitudetomyadvisor,Dr.WarrenE.Dixon,whose experienceandmotivationhavebeeninstrumentalinmyacademicsuccess.Asanadvisor, hehasprovidedguidanceinmyresearchandencouragementindevelopingmyownideas.As amentor,hehashelpedmedevelopprofessionalskillsandhaspreparedmeforthefuture.I wouldliketoextendmygratitudetomycommitteemembersDr.CarlCrane,Dr.Lawrence Ukeiley,andDr.ZhangLei,forthetimeandrecommendationstheyprovided.Also,Iwould liketothankmyfamily,coworkers,friendsfortheirsupportandencouragement. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS...................................4 LISTOFFIGURES.....................................7 ABSTRACT.........................................8 CHAPTER 1INTRODUCTION...................................10 1.1Motivation....................................10 1.2LiteratureReview................................11 1.3Contribution...................................14 2CONTROLOFANUNCERTAINSECOND-ORDERSYSTEMWITHKNOWNTIMEVARYINGINPUTDELAY:APDE-BASEDAPPROACH...............16 2.1DynamicModelandProperties.........................16 2.2ControlObjective................................19 2.3ControlDevelopment..............................20 2.4StabilityAnalysis.................................22 2.5ExperimentalResults...............................25 2.6ControlGainSelection..............................27 2.7Conclusion....................................29 3CONTROLOFANINPUTDELAYEDUNCERTAINNONLINEARSYSTEMWITH ANADAPTIVEDELAYESTIMATE.........................32 3.1DynamicModel.................................32 3.2ControlObjective................................35 3.3DevelopmentofErrorSignals..........................36 3.4EstimationofDelay...............................37 3.5StabilityAnalysis.................................39 3.6SimulationResults................................43 3.7ProjectionLaw..................................44 3.8ControlGainSelection..............................45 3.9Conclusion....................................46 4CONTROLOFANUNCERTAINNONLINEARSYSTEMWITHANUNKNOWN TIME-VARYINGINPUTDELAYUSINGANACCELERATEDGRADIENTDESCENT BASEDDELAYESTIMATE..............................49 4.1DynamicModel.................................49 4.2ControlObjective................................50 4.3DevelopmentofErrorSignals..........................50 5

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4.4StabilityAnalysis.................................51 4.5AcceleratedGradientDescentbasedEstimationofDelay............55 4.6SimulationResults................................60 4.7ControlGainSelection..............................61 4.8Conclusion....................................63 5BOUNDARYCONTROLOFSTOREINDUCEDOSCILLATIONSINAFLEXIBLE AIRCRAFTWINGWITHUNKNOWNTIME-VARYINGINPUTDELAY......67 5.12DEuler-BernoulliBeam............................67 5.2InstabilityinPresenceofInputTimeDelay...................71 5.3ControlDevelopment..............................71 5.4NeuralNetworkBasedDelayEstimation....................75 5.5Lyapunov-basedStabilityAnalysis........................77 5.6NumericalSimulation..............................86 5.7ControlGainSelection..............................88 5.8Conclusion....................................89 6CONCLUSIONSANDFUTUREWORKS.......................96 6.1Conclusions....................................96 6.2FutureWorks...................................98 REFERENCES........................................100 BIOGRAPHICALSKETCH.................................108 6

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LISTOFFIGURES Figure page 2-1Trackingperformanceexampletakenfromtherightlegofsubject1S1-Right. PlotAincludesthedesiredtrajectorybluesolidlineandtheactualleganglered line.PlotBillustratestheangletrackingerror.PlotCdepictstheRMStracking error.PlotDdepictsthecontrolinputcurrentamplitudeinmA..........30 2-2Trackingperformanceexampletakenfromtherightlegofsubject5S5-Right. PlotAillustratestheangletrackingerrorwhen =0 in2.PlotBdepictsthe controlinputcurrentamplitudeinmA........................31 3-1Variationoferrorsandcontrolforces.........................47 3-2Delaymismatchvariationwithtrainingiteration...................48 4-1Variationoferrorsandcontrolforces.........................63 4-2Delayestimatevs.time.................................64 4-3Delayderivativesvs.time................................65 4-4Objectivefunction E in4.............................66 5-1Schematicofthewingsection,whereE.A.denotestheelasticaxisandC.G.denotesthecenterofgravity................................68 5-2Closedloopbendingdeectionwithconstantinputdelaysusingthecontrollerin[1].90 5-3Closedlooptwistingdeectionwithconstantinputdelaysusingthecontrollerin[1].91 5-4Bendingdeectionafterapplyingthedeignedcontrollerin5...........92 5-5Twistingdeectionafterapplyingthedeignedcontrollerin5..........93 5-6Controlforcevariationvs.time.............................94 5-7ControlMomentvariationvs.time...........................95 7

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy PARTIALDIFFERENTIALEQUATIONBASEDCONTROLOFNONLINEARSYSTEMS WITHINPUTDELAYS By IndrasisChakraborty December2017 Chair:WarrenE.Dixon Major:MechanicalEngineering PartialdierentialequationPDE-basedcontrolmethodsaredevelopedforaclassof uncertainnonlinearsystemswithboundedexternaldisturbancesandknown/unknowntimevaryinginputdelay.Inspiredbypredictor-baseddelaycompensators,alineartransformationis usedtorelatethecontrolinputtoaspatiallyandtime-varyingfunction.Thetransformation allowstheinputtobeexpressedinamannerthatseparatesthecontrolintoadelayedcontrol andnon-delayedcontrolandalsofacilitatestheabilitytocompensateforthetime-varying aspectofthedelaywithlesscomplexgainconditionsthanpreviousrobustcontrolapproaches. Unlikepreviouspredictor-basedapproaches,whichinherentlydependonthesystemdynamics, anauxiliaryerrorfunctionisintroducedtofacilitatearobustcontrolstructurethatdoesnot dependonknowndynamics.Thedesignedcontrollerfeaturesgainstocompensateforthe delayanddelayderivativeindependentlyandfurtherrobustnessisachievedsincethecontroller doesnotrequireexactmodelknowledge. InChapter2,atrackingcontrollerisdevelopedforasecondordersystemwithaknown time-varyinginputdelay.AnovelLyapunov-KrasovskiifunctionalisusedintheLyapunovbasedstabilityanalysistoproveuniformultimateboundednessoftheerrorsignals.Chapter3 andChapter4focusonthedevelopmentofatrackingcontrollerforageneralizeduncertain nonlinearsystemswithboundedexternaldisturbancesandunknowntime-varyinginputdelay. InChapter3,anonlinearmappingisusedtomapthenon-compacttimedomaintoacompact spatialdomain,andthenaneuralnetworkNNisusedtoestimatetheunknowntime-varying 8

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inputdelay.InChapter4,anacceleratedgradientdescentAGDbasedoptimizationmethod isdemonstratedtoestimatetheunknowntime-varyingdelaymagnitude.Applicationofinput time-delayforexiblesystemisexaminedinChapter5.Specicallyanaircraftwingdynamics isconsidered.TheNNbasedestimationschemedevelopedinChapter3iscombinedwitha boundarycontrolmethod,tomitigateoscillationsintheaircraftwinginChapter5. 9

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CHAPTER1 INTRODUCTION 1.1Motivation Variousapplicationsexhibitadelaybeforethecontrollercanaectthesystemdynamics, i.e.,theso-calledinputdelayproblem.Inputdelayproblemsaretechnicallychallengingbecause theyheuristicallyrequirethecontrollertopredictthefutureresponseofthesystem,which isdicultfornonlinearsystems,especiallywhenthedynamicsareuncertain.Thenumberof practicalsystemsthatexperienceinputdelaysandthetheoreticalchallengesassociatedwith thisproblemhavemotivatedsignicantresearchinterestoverthelastdecade.Variousresults havebeendevelopedinliteraturefortheinputdelayproblem,andmostresultscanbeprimarily subdividedintotwocategories:knowninputdelayandunknowntimedelay.Thereareprimarily twomethodsdevelopedforanonlinearsystemsubjecttoaknowntime-varyingdelay:robust controlandpredictor-basedcontrol.Robuststrategiesuseanupperboundworsecaseeects oftheinputdelayanduseadelaydependentcontrolgaintoensurestability.Predictor-based strategiescompensatefortheinputdelay,bypredictingthesystemstateforfuturetime, whileutilizingknowledgeofmodeldynamics.Oneadvantageofrobustcontrolstrategiesover predictorbasedstrategiesisthattheformerdoesnotrequiresystemmodelknowledgeto designthecontroller.However,predictorbasedcontrolstrategiestypicallyyieldexponential stability,unliketheuniformlyultimatelyboundedtypestabilityofrobustmethods. Thesetwoapproachesclearlyhavetheiradvantages,andthismotivatesthenecessityof combiningthesetwoapproachestodeneapartialdierentialequationPDEbasedrobust controlstrategy,tocompensatetime-varyinginputdelayinanuncertainnonlineardynamics. Thisnewapproachalsosignicantlyopensanewavenue,byprovidinganinherentwayof applyingdierentparameterestimationmethodstocompensateforunknowntime-varying inputdelays.ANeuralnetworkNNbasedfunctionalapproximationmethodandanadaptive gradientdescentbasedoptimizationmethodaredemonstratedtobeabletosuccessfully estimatetheunknowntime-varyinginputdelayforanuncertainnonlineardynamics.Tofully 10

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demonstratetheeectivenessofNNbasedfunctionalapproximator,aboundarycontrolleris developedforaexibleaircraftsystem,subjectedtoanunknowntime-varyinginputdelay. 1.2LiteratureReview Variousapplicationsexhibitadelaybeforethecontrollercanaectthesystemdynamics, i.e.,theso-calledinputdelayproblem[220].Inputdelayproblemsaretechnicallychallenging becausetheyheuristicallyrequirethecontrollertopredictthefutureresponseofthesystem, whichisdicultfornonlinearsystems,especiallywhensubjecttouncertaintime-varying delays,whicharethefocusinthiswork. In[21],apioneeringstrategyisdevelopedtocompensatefortheeectsofinputdelay, followedbythenitespectrumapproachin[22],andmodelreductionin[23].Motivatedby pioneeringdevelopmentsuchas[2123],variousresultshavebeendevelopedforlinearsystems withinputdelay,includingresultssuchas[2426]foruncertainlinearsystemswithknown time-varyingdelays.However,suchresultsexplicitlyusethelinearityofthesystemtoconclude thestabilityresult. Morerecently,researchershavefocusedonnonlinearsystemswithinputdelays.For example,inresultssuchas[3,711,20,24,2732],controllersaredevelopedfornonlinear systemswithknowndynamicssubjectedtoknowntime-varyinginputdelay.Predictor-based controllers[24,2730]androbustcontrollers[3,7,8,20,31,32]aretwoprominentstrategies tocompensateforinputdelay,fornonlinearsystems.Bothstrategiesinjectdelayedand delay-freecontrolinputsintheopen-looperrorsystem.Predictor-basedmethodsmakeuse ofalineartransformationtomapthetimedependentcontrolinputtoamodiedcontrol inputwithtemporalandspatialvaryingcomponentscf.,[24,2730].Anadvantageofthis approachisthatresultinggainconditionsaregenerallylessconservative/restrictivethan robustcontrolmethods.Forexample,comparedtorobustcontrolresultssuchas[30,33,34] predictormethodsoftenhavelessrestrictiveconditionsonthemagnitudeofthedelay,and ofthefewpredictor-basedresultsthatconsidervariabledelayscf.,[27,29,30,35],boththe delayandthedelayratehavelessrestrictiveconditions.However,ingeneral,basedonthe 11

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needtopredictthestatetransition,suchstrategieshaveonlybeenappliedtolinearsystems cf.,[29,34,3638]ornonlinearsystemswithknowndynamicscf.,[27,28,30,33,35,3943]. Unlikepredictorbasedapproaches,robustapproachesdonotrequireexactmodelknowledge cf.,[3,7,8,20,31,32];although,theycanyieldcomplexsucientgainconditions,compared topredictorbasedmethods. Inthepreliminaryworkin[44,45]arobustcontrolapproachwascombinedwiththePDEbasedtransformationforuncertainnonlinearsystemswithknowntime-varyingdelays.The resultsin[44,45]wereextendedin[46].Specically,in[46]aPDE-basedmethodisdeveloped foruncertainsecondordersystemswithunknowntime-varyinginputdelay,withanadaptive delayestimate. Controllersforsystemswithunknownboundedtime-varyingdelayshavebeenstudiedin recentyears[4751].Anobserverbasedcontrollerisdesignedin[47]foranonlinearsystem withunknownboundedstateandinputdelays,whichyieldsexponentialconvergence.Similarly, foralinearsystemwithunknowntime-varyinginputdelay,aslidingmodeobserverbased designapproachisusedin[48]toyieldexponentialconvergence.In[50],foraclassofan uncertainlinearsystemswithunknowntime-varyingstatedelay,anobserverandacontroller isdeveloped.Anadaptivelearningapproachisutilizedin[51],whichshowserrorsignal boundednessforarstorderuncertainnonlinearsystemwithunknowntime-varyinginput delay,althoughthisworkisspecictorstordernonlinearsystemsandcan'tbegeneralizedfor higherordersystems. Timedelaysarepresentincommonengineeringapplicationsandcancauseinstabilityin anotherwisestablesystem,andalsosubsequentlycancausesystemperformancedegradation. Toalleviatethenegativeeectsofdelay,acontrolsignalcanbedesignedbypredictingthe futurestateofthesystem.However,anunknownandtime-varyinginputdelayaddstothe complexityoftheproblem.Forexample,whenthecontroliscommunicatedtoaplantovera network,theinputdelaycanbeunknownandtime-varyingduetotransmissionuncertainties inthecommunicationchannel.Inadditiontounknownvariationoftheinputdelay,unmodeled 12

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eectse.g.,exogenousdisturbancesmakethedelaycontrolproblemmorechallengingfor uncertainnonlinearsystems.Datkoetal.in[52]provedthataarbitrarilysmallamountoftime delay,presentintheboundaryfeedbackcontrolcandestabilizeanelasticsysteme.g.,one dimensionalEuler-Bernoullibeam. Thereexistliteraturethatdealswithcontrollerdesigntosuppressoscillationfortwodimensionalairfoilsystem.Theseliteratureincludelinear-quadraticregulator[5355],feedback linearization[56],linearreducedordermodel-basedcontrolapproaches[57,58],aNissim aerodynamicenergy-basedcontrolapproach[59],andstate-dependentRiccatiequationand slidingmodecontrolapproaches[60].Mostrecently,aRISEcontrolstructurewasusedto ensureasymptotictrackingofatwo-dimensionalairfoilsectionwithmodelinguncertaintiesin thestructuralandaerodynamicproperties[61],andthenextendedtocompensateforactuator saturation[62].Therearetwoboundarycontrolmethodologiesthathavebeendevelopedfor asystemdescribedbyasetofPDEs.TherstmethodapproximatesthePDEsystemwitha nitenumberofordinarydierentialequationsODEusingoperatortheoretictools[6366] orGalerkinandRayleigh-Ritzmethods[6769].Aboundarycontrolleristhendesignedusing theresultingreduced-ordermodel.Theprimaryconcernwithusingareduced-ordermodel forthecontroldesignisthepotentialforspilloverinstabilities[70,71],inwhichthecontroller exciteshigher-ordermodesthatwereneglectedintheapproximation.Inspecialcases,the placementofactuatorsandsensorscanguaranteetheneglectedmodesarenotexcited[72]. Specically,placingactuatorsatknownzerolocationsofthehigher-ordermodeswillalleviate spilloverinstabilities;however,thiscanconictwiththedesiretoplaceactuatorsawayfrom thezerosofthecontrolledmodes.ManyPDE-basedandODE-basedcontrolstrategieshave beendevelopedtostabilizethebendingofaexiblebeamsuchas[65,66,73,74]. Motivatedbythisdevelopment,theeectofaninputdelayhasbeenstudiedforan aircraftwing,whichissubjectedtostoreinducedlimitcycleoscillation[1].Thecontroller compensatesfortheinputdelay,themagnitudeofwhichisunknownandtime-varying.The contributionoftheworkistoconsideranexampleofaexibleelasticsystemwithinput 13

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delay.ToreplicatetheclaimofDatkoin[52],Bialyetal.in[1]designedaadaptiveboundary controllerfora2Daircraftwinginordertomitigatetheeectoflimitcycleoscillationonthe elasticwing.Thiscurrentworkextendstheboundarycontrollerdesigntocompensateforan unknowntime-varyingdelayinboundarycontrolfeedback. 1.3Contribution InChapter2,theamalgamationofpredictorbasedandrobustbasedcontrolstrategies isexamined.AsecondorderEuler-Lagrangesystemisconsidered,subjectedtoaknown time-varyinginputdelay.Theeectivenessofbothpredictorandrobustcontrolstrategiesfor atime-delayedsystem,aretakenintoconsiderationwhiledevelopingthisPDEbasedrobust controlstrategy.AlinearPDEbasedtransformationisdevelopedtomapthetime-varying controlinputtoatwovariablecontrolinput.Doingsoprovidessomeaddedexibilityby keepingthetime-varyingdelayoutsideofthecontrolinputterm.Similartoarobustcontrol strategy,thisPDEbasedmethodhastheexibilitytobeappliedtoanuncertainsystem,since itdoesnotrequirethesystemmodelforfuturesystemstateprediction,unliketypicalpredictor basedcontrolstrategies.FinallytodemonstratethetheoreticalclaimofthedevelopedPDE basedrobustcontrolmethod,anexperimentaldemonstrationofthecontrollerisshownfor knee-shankdynamicsinChapter2. Chapter3focusesonmoregeneralizeddynamicsandextendsthePDE-basedrobust controllerdevelopedforanEuler-Lagrangesystem,toanuncertain n +1 th orderdynamic system.Aspreviouslydescribed,becauseofaddedexibilityofthePDEbasedrobuststrategy, anunknowntime-varyinginputdelaycaseisconsideredforChapter3.Forestimationof anunknowntime-delay,aNeuralNetworkNNbasedfunctionapproximatorisdesignedin Chapter3.UnlikeatraditionalNNfunctionapproximator,thismodiedversioncaneectively handleaexplicittimedependentfunctionsuchasthetime-delay,byusinganonlinearmapping toconverttimeintoacompactvariablespace.Thismethodmotivatesthefurtherdevelopment ofadelaydependentcontrollerdesign,andsubsequentdemonstrationinLyapunovbased stabilityanalysis,foranunknowntime-varyinginputdelayedsystem. 14

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Chapter4startswiththesamegeneralizeddynamicalsystemdescribedinChapter3.The maincontributionofChapter4isthedemonstrationofanoptimizationbasedstrategy,namely Nesterov'sacceleratedgradientbasedAGDstrategy,forestimatingthepresentunknown time-varyinginputdelay.ThisapproachutilizesthePDEbasedrobustcontrollerdevelopedin Chapter3,byaddingaAGDbasedestimation,anddemonstratingthetheoreticaleectiveness fora2-linksystem,subjectedtoanunknowntime-varyinginputdelay.Simulationresultsshow theeectiveestimationoftheunknowntime-varyinginputdelay.Further,thisAGDbased estimationisextendedfora m variabledynamicalsystem,subjectedto n dierentunknown time-varyinginputdelays.While,mostofthepreviousdevelopmentinChapter4,remains unchanged,asignicantmodicationoftheestimationschemehasbeendemonstratedby mappingthe m statevariableontoa m )]TJ/F15 11.9552 Tf 12.745 0 Td [(1 orderhyperbolicmanifold,whilemappingthe unknown n inputdelayontoa n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 orderhypersphere. ThedevelopmentofcontrolstrategiesinthepreviouschaptersChapter2-Chapter4 isextendedforaexiblenonlinearsysteminChapter5.Withoutlossofgenerality,aexible aircraftwingdynamicisexamined.AboundarycontrollerisdevelopedforChapter5and adelaycompensationtermisaddedtothedevelopedadaptiveboundarycontroller.The noveltyofChapter5isthecombinationoftheNNfunctionalapproximatorbasedunknown time-varyingdelaycompensatorwithaadaptiveboundarycontroller,tosuccessfullymitigate unnecessaryoscillationsofa2Dexibleaircraftwing.AGalerkinbasedsimulationmodel demonstratesthecontrollereectiveness,throughaLyapunovstabilityanalysis. 15

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CHAPTER2 CONTROLOFANUNCERTAINSECOND-ORDERSYSTEMWITHKNOWN TIME-VARYINGINPUTDELAY:APDE-BASEDAPPROACH ThischapterdevelopsaPDE-basedapproach,whichusesthelineartransformation discussedin[30]tomapthetimedependentcontrolinputtoamodiedcontrolinputthat dependsonbothtimeandaspatialvariable.PDE-basedapproachesaregenerallyusedto solveinputdelayproblemssincetheyproducenite-dimensionalsolutions,whilesolutions toinputdelayeddierentialequationsareinnite-dimensionalingeneral.Inadditiontothe transformationofinnitetonite-dimensionality,aPDE-basedapproachgivesanadded advantageofextractingthedelaytermfromitsfunctional,facilitatingthestabilityanalysis. Additionally,theseadvantagesmayhelptoreducecontroleortandimprovethedelay compensatingterm,asdiscussedin[30].Themostprominentbenetoftheapproachinthe currentworkistheamalgamationoftheexibilityseeninotherPDE-basedapproacheswith theabilitytorobustlycompensateforuncertaindynamics.Thisisaccomplishedbymodifying therobustcontrolapproachinresultssuchas[20,32]sothatitincorporatesthePDE-based approachin[30].Anoveltrackingerrorisusedinthischaptertoinjectadelay-freecontrol termintotheclosed-looperrordynamicsbyintegratingthecontrolstatesoverthenitedelay interval.Moreover,thisrobust,PDE-baseddesignapproachallowsforthedelayderivative gain,aswellastheconventionaldelaymagnitudegain,tobetunedindependently,unlikethe previousworksin[20,32].AnovelLyapunov-Krasovskiifunctionalisdevelopedandusedinthe stabilityanalysistoyielduniformlyultimateboundednessofthetrackingerrorsignal.Tracking experimentsfortheknee-shankdynamicsaredemonstratedtoshowtheeectivenessofthe developedcontroller. 2.1DynamicModelandProperties Consideraclassofsecond-ordersystemsmodeledas x t = f x t ; x t + d t + U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t ; 16

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where x; x and x 2 R n denotethesystemstates, f : R n R n [ t 0 ; 1 R n isanuncertain nonlinearfunction, d :[ t 0 ; 1 R n isanuncertainexogenousdisturbancee.g.,unmodeled eects, U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t 2 R n representsthegeneralizeddelayedinputvector,and D 2 R isa known,bounded,time-varyingdelay.Thesubsequentdevelopmentisbasedontheassumption that x; x aremeasurable.Furthermore,thefollowingconditionsareassumedtobesatised. Alineartransformationisusedtorepresentthegeneralizedinput U t asafunctionoftwo independentvariables,i.e. p and t ,where t 2 [ t 0 ; 1 and p 2 [0 ; 1] .Thespatialvariable, p denotesdelayedanddelayfreecontrolinputsattwoendpointsofitsdomain.Thetwovariable controlinput u : R [ t 0 ; 1 R n isanalogousto U t inthesensethat[30] u p;t U )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( t + p )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t ; t t; 8 t 0 where :[ t 0 ; 1 R isaknowndelaydependentinvertiblemonotonoustimefunction.Let t t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t ,s0 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t = )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t ; t = )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t : Basedon2and2,thedelayedcontrolinputcanbeexpressedas U t )]TJ/F25 11.9552 Tf 10.486 0 Td [(D t = u ;t andthedelay-freecontrolinputcanbeexpressedas U t = u ;t Tofurtherfacilitatethesubsequentdevelopment,arelationshipbetweenthespatialand temporalvariationof u p;t ,i.e., u p p;t and u t p;t respectively,isdeveloped.From2 andfromtheauxiliaryfunction t d :[ t 0 ; 1 [0 ; 1] R denedas t d t + p )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.956 0 Td [(t u t p;t = dU t d dt d 1+ p d )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t dt )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; = @u p;t @p @p @t d 1+ p d )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t dt )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 ; = 1+ p d )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t dt )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t u p p;t ; = p;t u p p;t ; 17

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wheretheauxiliaryfunction : R ; 1 2 R n isdenedas 1 p;t 1+ p d )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t dt )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t : Since D isaknowntime-varyingfunction, )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t iscalculatedbysolving2.Tofacilitate thesubsequentstabilityanalysisthetimederivativeof )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t canbecalculatedbytakingthe timederivativeof2as t = )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t ; 1= d dt )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(dD )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t d )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t d )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t dt ; = d )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t dt )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t ; andthenrearrangingtoyield d )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t dt = 1 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t ; where D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t dD )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t d )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t .Using2,2,and2,the p variationof p;t ,i.e, p p;t canbecalculatedas p = D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.956 0 Td [(D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t 1 D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t : From2,2,and2, ;t 0 = 1 D )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t ; ;t 1 = 1 )]TJ/F26 7.9701 Tf 6.587 0 Td [(D )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t 0 ; and p p;t p = D )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t )]TJ/F26 7.9701 Tf 6.586 0 Td [(D )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t 0 .UsingAssumptions2.1and2.3, 0 0 0 1 1 1 ,| p | j p j j p j ,where 0 ; 0 ; 1 ; 1 ; | p | ; j p j2 R areknownpositiveconstants. 2 1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F25 11.9552 Tf 11.956 0 Td [(t = D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t and D t isassumedtobenon-zero. 2 Theupperboundsof D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t ,and D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t arethesameas D t and D t ,respectively. 18

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Remark 2.1 From2,singularitiescanoccurin 0 1 and p when D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t =0 and when D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t =1 .When D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t =0 thereisnodelayinthesystem.While 0 1 and p willbesingularinthiscase,thesetermswillvanishfromthecontrolandstability analysis.From2,when D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t =1 ,foranonzerodelaymagnitude D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t u p ;t =0 u ;t = u ;t p =0 0 = 1 = 1 D )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t Assumption2.1. Thenonlinearexogenousdisturbancetermanditsrsttimederivativei.e., d d existandareboundedbyknownpositiveconstants[7577]. Assumption2.2. Thedesiredtrajectory x d 2 R n isdesignedsuchthat x d ; x d ; x d existand areboundedbyknownpositiveconstants. Assumption2.3. Thetime-varyingdelay D t 2 R isboundedbyknownpositiveconstantsD and D ,asD D t D .Thesystemremainsboundedintheinterval [ t 0 ;t 0 + D ] Assumption2.4. Thedelayrate D t 2 R isboundedbyknownconstants D and D ,as D D t D 2.2ControlObjective Theobjectiveistodevelopacontinuouscontrollerwhichensuresthatthegeneralized state x oftheinput-delayedsystemin2tracksadesiredtrajectory, x d ,despiteuncertaintiesandadditivedisturbancesinthedynamics.Toquantifythecontrolobjective,atracking error,denotedby e 2 R n ,isdenedas e x d )]TJ/F25 11.9552 Tf 11.956 0 Td [(x: Tofacilitatethesubsequentanalysis,ameasurableauxiliarytrackingerror,denotedby r 2 R n isdenedas r e + e )]TJ/F25 11.9552 Tf 11.955 0 Td [(e u ; where ; 2 R areknown,positiveconstants.In2, e u 2 R n isanauxiliarysignalthat isusedtoobtainadelaydependentcontrolsignaltonegatetheeectofthedelayedinputin 19

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2,denedas e u 1 0 u p;t dp: ByusingtheLeibnitzintegralruleand2,thetimederivativeof e u canbedeterminedas e u = 1 0 u t p;t dp = 1 0 p;t d u p;t : Integrating2bypartsyields 3 e u = ;t u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( ;t u ;t )]TJ/F23 7.9701 Tf 20.788 18.664 Td [(1 0 p u p;t dp; = ;t u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( ;t u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( p e u : Since e; e areassumedtobemeasurable e u canbeobtainedfrom2and r canbe computedforalltimeandusedasfeedback. 2.3ControlDevelopment Theopen-looperrorsystemfor r isobtainedbytakingthetimederivativeof2and usingtheexpressionsin2,2,and2as r = x d )]TJ/F25 11.9552 Tf 11.956 0 Td [(f x t ; x t )]TJ/F25 11.9552 Tf 11.955 0 Td [(d t )]TJ/F25 11.9552 Tf 11.955 0 Td [(u ;t + r )]TJ/F25 11.9552 Tf 11.956 0 Td [(e + e u )]TJ/F25 11.9552 Tf 11.956 0 Td [( 1 u ;t + 0 u ;t + p e u : Theopen-looperrorsystemin2containsbothadelayedanddelay-freecontrolinput resultingfromthetimederivativeof e u .Basedonthesubsequentstabilityanalysis,thecontrol inputisdesignedas U t u ;t 1 k r; 3 p isindependentof p 20

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where k 2 R + isconstant,adjustablecontrolgain.Tofacilitatethesubsequentstabilityanalysis,thetermsin2canbesegregatedintotermsupperboundedbyastate-dependent functionandtermsupperboundedbyaknownconstantas r = ~ N + N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(e + e u )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 u ;t )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 u ;t + p e u ; wheretheterms ~ N;N d 2 R n aredenedas ~ N f x d t ; x d t )]TJ/F25 11.9552 Tf 11.955 0 Td [(f x t ; x t + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 e + e; and N d )]TJ/F25 11.9552 Tf 9.299 0 Td [(f x d t ; x d t + x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(d t : Bysubstituting2into2,theclosed-looperrorsystemfor r is r = ~ N + N d )]TJ/F25 11.9552 Tf 11.956 0 Td [(e + e u )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 1 r k )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 u ;t + p e u : Remark 2.2 UsingtheMeanValueTheoremandAssumption2.2,theexpressionin2can beupperboundedas ~ N k z k k z k ; where : R R ispositivedenite,non-decreasing,radiallyunboundedfunction,and z 2 R 3 n isavectoroferrorsignals,denedas z e T r T e T u T : Remark 2.3 UsingAssumptions2.1and2.2, N d canbeupperboundedas sup t 2 [0 ; 1 k N d k ; where 2 R isaknownpositiveconstant. 21

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2.4StabilityAnalysis Tofacilitatethesubsequentstabilityanalysis,let y 2 R 3 n +1 bedenedas y z T p Q T ; where Q 2 R isanLKfunctionaldenedas Q Q 1 0 e 2 p k u p;t k 2 dp; where 2 ; Q 2 R + areconstants.Let D beanopenandconnectedset,and S D D is denedas S D y 2 R 3 n +1 jk y k < inf )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 [ r 1 1 2 ; 1 ; where 1 and 1 2 R areknown,positiveconstants. Theorem2.1. Giventheopen-looperrorsystemin2,thecontrollerin2ensures UUBtrackinginthesensethat k e k )]TJ/F23 7.9701 Tf 7.253 -1.793 Td [(0 exp )]TJ/F25 11.9552 Tf 9.298 0 Td [()]TJ/F23 7.9701 Tf 7.253 -1.793 Td [(1 t + )]TJ/F23 7.9701 Tf 7.252 -1.793 Td [(2 ; providedthat y 2S D ; 8 2 [ t 0 ;t 0 + D max ] ,thecontrolgainssatisfysucientgain conditionsseeSection2.6,andAssumptions2.1-2.3aresatised,where )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [(0 ; )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [(1 and )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [(2 are knownpositiveconstants,where )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [(2 canbemadearbitrarilysmall. Proof. Let V : D [ t 0 ; 1 R beacontinuouslydierentiable,positive-denitefunctional denedas V 1 2 e T e + 1 2 r T r + 1 2 e T u e u + Q; where 1 k y k 2 V 2 k y k 2 .Thetimederivativeof2canbeobtainedafterapplying theLeibnizintegralruletoobtainthetimederivativeof2andutilizing2,2,and 2as V = r T ~ N + N d + e u )]TJ/F25 11.9552 Tf 11.956 0 Td [( 1 1 k r )]TJ/F25 11.9552 Tf 11.955 0 Td [(r T )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( p e u )]TJ/F25 11.9552 Tf 11.955 0 Td [(! 1 p e T u e u 22

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)]TJ/F25 11.9552 Tf 9.298 0 Td [(e T e + e T e u + 1 1 e T u r k )]TJ/F25 11.9552 Tf 11.955 0 Td [(! 1 0 e T u u ;t + Q 1 e 2 k u ;t k 2 )]TJ/F25 11.9552 Tf 9.299 0 Td [( Q 0 k u ;t k 2 )]TJ/F25 11.9552 Tf 9.298 0 Td [( Q 2 1 0 p;t e 2 p k u p;t k 2 dp )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q p 1 0 e 2 p k u p;t k 2 dp: Byusing2,2,and2,andcancelingcommontermsin2,anupperbound canbeobtainedfor2as V k r k k z k k z k + k r k + 1 j p jk e u k 2 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [( 1 jj r jj 2 k )]TJ/F25 11.9552 Tf 11.955 0 Td [( jj e jj 2 + jj e T e u jj + )]TJ/F15 11.9552 Tf 7.511 -6.529 Td [( 0 +1 jj r T u ;t jj + )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( + j p j k r T e u k + 1 1 jj e T u r k jj + Q 1 e 2 jj r jj 2 k 2 )]TJ/F25 11.9552 Tf 9.299 0 Td [( Q 0 jj u ;t jj 2 + 1 0 jj e T u u ;t jj )]TJ0 0 1 rg 0 0 1 RG0 g 0 G/F25 11.9552 Tf 9.299 0 Td [( Q 2 1 0 p;t e 2 p k u p;t k 2 dp + Q j p j 1 0 e 2 p k u p;t k 2 dp: From2and2, p;t canbelowerboundedas p;t = 0 + 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 p min f 0 ; 1 g ; for p 2 [0 ; 1] .UsingtheCauchy-Schwarzinequality, k e u k 2 1 0 k u 2 p;t k dp 1 0 1 dp; k e u k 2 1 0 k u p;t k 2 dp: AfterusingYoung'sInequalityandtheinequalitiesin2and2,2canbeupper boundedas V 1 1 2 k z k k z k 2 + 1 2 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( )]TJ/F25 11.9552 Tf 15.686 8.088 Td [( 2 k jj e jj 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( Q 2 2 min f 0 ; 1 gk e u k 2 )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( 1 k )]TJ0 g 0 G/F15 11.9552 Tf 18.758 11.243 Td [( 0 2 k )]TJ/F15 11.9552 Tf 13.15 8.088 Td [( 1 + 2 4 k r k 2 + 2 k + j p j 2 2 k + 2 2 k r k 2 + k j p j 2 2 + 1 j p j + k 2 k e u k 2 + 1 1 k 2 + 1 0 2 k k e u k 2 + Q 1 e 2 k 2 + 1 1 2 k 3 k r k 2 )]TJ/F30 11.9552 Tf 11.956 16.857 Td [( Q 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 0 2 2 + 1 0 +1 2 k u ;t k 2 )]TJ/F25 11.9552 Tf 10.494 8.087 Td [( Q 2 2 min f 0 ; 1 g 1 0 e 2 p k u p;t k 2 dp + Q j p j 1 0 e 2 p k u p;t k 2 dp: 23

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Sincebydenition k y kk z k ,thefollowingupperboundcanbeobtained V )]TJ/F30 11.9552 Tf 23.91 16.857 Td [( 1 2 )]TJ/F15 11.9552 Tf 14.954 8.088 Td [(1 1 2 k y k k z k 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 1 2 jj z jj 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q res k u ;t k 2 + 1 2 2 )]TJ/F30 11.9552 Tf 11.291 13.27 Td [(h 2 2 min f 0 ; 1 g)-222(j p j i Q; where 1 ; r ; e u ; Q res 2 R aredenedas 1 min )]TJ/F25 11.9552 Tf 15.687 8.088 Td [( 2 k ; r ; e u ; r = 1 k )]TJ/F15 11.9552 Tf 13.151 8.088 Td [( 1 + 2 4 )]TJ/F15 11.9552 Tf 18.758 11.243 Td [( 0 2 k )]TJ/F25 11.9552 Tf 13.15 8.088 Td [( Q 1 e 2 k 2 )]TJ/F25 11.9552 Tf 16.958 8.088 Td [( 2 k )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( j p j 2 2 k )]TJ1 0 0 rg 1 0 0 RG0 g 0 G/F25 11.9552 Tf 13.151 8.088 Td [( 2 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(! 1 1 2 k 3 ; e u = Q 2 2 min f 0 ; 1 g)]TJ/F25 11.9552 Tf 20.589 0 Td [(! 1 j p j)]TJ/F25 11.9552 Tf 19.128 8.087 Td [(k 2 )]TJ/F25 11.9552 Tf 13.151 8.087 Td [( j p j k 2 2 )]TJ/F25 11.9552 Tf 13.151 8.087 Td [(! 1 1 k 2 )]TJ/F25 11.9552 Tf 13.151 8.087 Td [(! 1 0 2 k ; Q res = Q 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 0 2 2 + 1 0 +1 2 : Provided y 2S D ; 8 2 [ t )]TJ/F15 11.9552 Tf 15.114 3.022 Td [( D;t ] andallthegainconditionsaresatisedsucientlysee Section2.6,theexpressionin2reducesto V )]TJ/F25 11.9552 Tf 21.918 0 Td [( 2 k y k 2 + 1 2 2 ; where 2 2 R isdenedas 2 min 1 2 ; 2 2 min f 0 ; 1 g)-222(j p j : Theinequalityin2canbefurtherupperboundedas V )]TJ/F25 11.9552 Tf 23.926 8.088 Td [( 2 2 V + 1 2 2 : Thesolutionoftheinequalityin2canbeobtainedas V t V t 0 exp )]TJ/F25 11.9552 Tf 11.307 8.088 Td [( 2 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t 0 + 2 2 2 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F25 11.9552 Tf 11.307 8.088 Td [( 2 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t 0 ; where 2 2 R canbemadearbitrarilylargetomake )]TJ/F23 7.9701 Tf 7.314 -1.793 Td [(2 ,introducedin2,arbitrarilysmall. Using2and2,theinequalityin2canbeobtained,and e t ;r t ;e u t 2L 1 Hence,from2, u t 2L 1 : 24

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2.5ExperimentalResults NeuromuscularelectricalstimulationNMESorfunctionalelectricalstimulationFES forfunctionalexercisetasksistheapplicationofelectricalcurrentacrossmusclebersto produceamusclecontraction.ThepresenceofanelectromechanicaldelayEMDinthe muscleresponsetothecontrolinputresultsinperformancedegradationinthetrackingofa humanlimbviaNMES,includingpotentiallydestabilizingeectscf.,[13,14,19,78,79]. MotivatedbythenatureoftheEMDinNMESsystems,theperformanceofthedeveloped controllerin2wasexaminedthroughaseriesofdynamictrackingexperimentsofthe knee-jointdynamics.Thenonlineardynamicsoftheknee-shankcanbemodeledasin[80]as B m q; q U t )]TJ/F25 11.9552 Tf 11.956 0 Td [(D t = J q + mgl sin q + M e + 1 exp )]TJ/F25 11.9552 Tf 9.298 0 Td [( 2 q q )]TJ/F25 11.9552 Tf 11.955 0 Td [( 3 )]TJ/F25 11.9552 Tf 9.299 0 Td [( 1 tanh )]TJ/F25 11.9552 Tf 9.298 0 Td [( 2 q + 3 q + d t ; where q; q; q 2 R denotetheangularposition,velocityandaccelerationoftheshankaboutthe knee-joint,respectively, i ; i 2 R ;i = ; 2 ; 3 areuncertainpositiveconstants, d :[0 ; 1 R isaunknownexogenousdisturbanceassumedtobebounded, B m q; q : R R denotesthe controleectivenessofthequadricepsmusclegroup,whichisauncertainfunctiondependent onthemuscle'smomentarmandthemapbetweenstimulationintensitytomuscleforce, U :[0 ; 1 R isthevoltagepotentialappliedacrossthequadricepsmusclegroupbythe electricalstimulation,and D :[0 ; 1 R denotestheEMD.Theuncertainpositiveconstants J;m;g;l 2 R symbolizetheinertiaofshankandfoot,thecombinedmassoftheshankand foot,thegravitationalacceleration,andthedistancebetweentheknee-jointandthelumped centerofmassoftheshankandfoot,respectively. Sixhealthyindividualsmale,1female,aged21-31participatedintheexperiments. Priortoparticipation,writteninformedconsentwasobtainedfromallparticipants,inaccordancewiththeinstitutionalreviewboardattheUniversityofFlorida.Theexperimental apparatus,illustratedin[81],consistedofthefollowing:1amodiedlegextensionmachine equippedwithorthoticbootstoxtheankleandsecurelyfastentheshankandthefoot,2 25

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opticalencodersBEItechnologiestomeasuretheleganglei.e.,theanglebetweentheverticalandtheshank,3acurrent-controlled8-channelstimulatorRehaStim,HasomedGmbH, 4adataacquisitionboardQuanserQPIDewithQUARCsoftware,5adesktopcomputer runningMatlab/Simulink,and6apairof3"by5"Valutrode r surfaceelectrodesplaced proximallyanddistallyoverthequadricepsmusclegroup. 4 Surfaceelectricalstimulation wasappliedtoquadricepsmusclegroupusingasingleconventionalchannelduringknee-joint trackingexperimentswithatestingdurationof60seconds. Duringtheexperiments,electricalpulsesweredeliveredataconstantstimulationfrequencyof35Hz,thepulsewidthwasxedtoaconstantvaluei.e.,between300and400 s ,andthecontrollerin2wasusedtomodulatetheamplitude.Themainfactorsto determinethepulsewidthvalueforanindividualwerethemusclesensitivitytostimulation, trackingaccuracy,andstimulationsensitivity.Thecontrolgainswereadjustedduringpretrial teststoachievesatisfactorytrajectorytrackingwherethedesiredangulartrajectoryoftheknee jointwasselectedasasinusoidwitharangeof5 to50 andaperiodof2seconds[81].The time-varyingnatureoftheEMDwasmodeledasacontinuouslydierentiablesigmoidfunction rangingbetween [80 )]TJ/F15 11.9552 Tf 12.64 0 Td [(140] ms.ThischoiceofEMDisbasedonrecentresultsreportedin NMESstudiessuchas[19,78].Theroot-mean-squareRMStrackingerrorwascomputed overtheentiretrialasacontrolperformancemetric.Table2-1presentsthemeanRMSerror overtheentireexperimentdurationinallthetrackingtrials.Anillustrativeexampleofa completedynamictrackingtrialisshowninFigure2-1.Tofurtherillustratetheimpactof compensatingforthedelay,Figure2-2showsabrieftrialwherethecontrolgainmultiplying thedelaycompensationterm e u in2wassettozeroi.e., =0 ,whilekeepingthe othergainsthesame.TheresultingperformancedepictedinFigure2-2showsunsatisfactory performance. 4 SurfaceelectrodesforthestudywereprovidedcomplimentsofAxelgaardManufacturing Co.,Ltd. 26

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Remark 2.4 ThegainconditionsinSection2.6,althoughonlysucientconditions,were usedasaguidetodeterminethegainsduringtheexperiments.Specically,forthesetsof experiments 2 [3 : 5 ; 12] k 2 [0 : 02 ; 0 : 06] 2 [0 : 4 ; 0 : 5] ,where 2 [0 : 0008 ; 0 : 005] 1 =0 : 01 2 =2 and Q 2 [0 : 01 ; 0 : 025] ,basedontheapproximatedelaymodelfrom[19,78].For example,forsubject4S4rightlegR,thegainswereselectedas k =0 : 03 =7 : 5 and =0 : 5 ,whichsatisfythegainconditionsinSection2.6.Notethat,inadditionto gainselection,variousfactorsalsocontributetotheexperimentalresultsincluding:electrode placement,levelofmusclefatigue,sensitivitytostimulation,etc.Theexperimentswere performedinhealthynormalvolunteers,asinresultssuchas[14,78,8087],toillustratethe robustnesstouncertaintyinthedynamicsandtheinputdelay.Dierentneurologicalconditions canimpacttheresultse.g.,increased/decreasedsensitivitytostimulation,moresusceptible tofatigue,etc..Hence,furtherexperimentswouldberequiredviaclinicaltrialsinspecic populationsofindividualswithneurologicalconditionstogaugetheimpactofthedeveloped controllerforspecicrehabilitationoutcomes. 2.6ControlGainSelection ThestabilityanalysisinSection2.4requiresthat 1 in2, r in2, e u in 2, Q res in2,and 2 in2bepositiveconstants.Forsomegivenlowerand upperboundsonthedelayanddelayrate D D and D ,i.e., 0 ; 0 1 ; 1 j p j ,thissection developssucientgainconditionstoensure 1 r e u Q res and 2 > 0 .Usingthe denitionsof 1 in2and 2 in2,andusingtheinequalityin2,sucientlower boundsfor and 2 canbeobtainedas > 2 k ; 2 > 2 j p j min f 0 ; 1 g : 27

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Fromthedenitionof Q res in2,itisclearthatforanarbitrarilylarge k and and arbitrarilysmall 1 and ,that Q canbeselectedsucientlylargeas Q > k 0 0 2 2 + 1 0 +1 2 ; toensurethat Q res > 0 .Alsofromthedenitionof e u in2, Q alsoneedstosatisfy thefollowinginequality Q > 2 k! 1 j p j + k 2 + j p j k 2 2 + 1 1 k 2 + 2 1 0 k! 2 min f 0 ; 1 g ; toensure e u > 0 .Byselecting ontheorderof k 2 ontheorderof k 3 ontheorder of 1 k 4 ,and 1 ontheorderof 1 k 5 ,thelowerboundin2canbeproventobelargerthan 2.Forexample,if = k 2 = k 3 = 1 k 4 and 1 = 1 k 5 then 0 k 3 2 0 + 0 + k 5 2 0 > 1 2 min f 0 ; 1 g 2 j p j k 5 + k 4 + k 4 j p j + 1 + 0 k 10 ; forlargevaluesof k> 1 and 2 From2, Q ismultipliedbyanegativeterminthedenitionof r .Todevelopa sucientconditiontoensure r > 0 ,thelowerboundfor Q in2issubstitutedinto 2todevelopthefollowingsucientinequality: 2 1 )]TJ/F15 11.9552 Tf 15.183 11.243 Td [( 0 k 2 0 e 2 )-222(j p j 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 > 1 + 2 k 2 + 1 0 + 1 0 0 1 e 2 + + 1 1 k 2 + 0 : Basedon2,asucientconditionfortheupperboundon canbeestablishedas < 2 1 j p j 2 + 2 + 0 k 2 0 e 2 ; toensuretheparentheticaltermsontheleftsideof27arepositive.Basedon2and 2asucientlowerboundfor canbeestablishedforanarbitrarilylarge k 2 2 and 28

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arbitrarilysmall 1 and ,toensure r > 0 as > 1 + 2 k 2 + 1 0 + 1 0 0 1 e 2 + + 1 1 k 2 + 0 2 1 )]TJ/F23 7.9701 Tf 14.982 6.916 Td [( 0 2 0 e 2 )-222(j p j 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 : Toyield2andtosatisfy2,2,and2, 2 , Q and areselected sucientlylargeand and 1 areselectedsucientlysmall.Forexample,selecting = k 2 = k 3 and = 1 k 4 ,aspreviously,then2canbewrittenas k 6 1 0 + 1 0 0 1 e 2 + 0 + k 5 0 2 0 e 2 +1+ 1 + 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 + k 3 j p j + k 2 + 1 1 > 0 ; whichclearlyholdsforany k> 1 2.7Conclusion Inthiswork,arobustcontrollerwasdevelopedforanuncertainnonlinearsecond-order systemwithanadditivedisturbancesubjecttotime-varyinginputdelays.Alteredtrackingerrorsignalwasdesignedtofacilitatethecontroldesignandstabilityanalysis.Anovel Lyapunov-KrasovskiifunctionalwasusedintheLyapunov-basedstabilityanalysistoshowUUB ofthetrackingerror.Thedesignedcontrollerisanovel,continuous,robustcontrollerwhich hasexplicitdelaymagnitudeanddelayderivativedependentcontrolgainterms.Dynamic trackingexperimentsfortheknee-shankdynamicsareperformedtodemonstratetheapplicabilityandtheeectivenessofthePDErobustcontrolapproach.Motivatedbythepresent results,futureworkwillfocusonextendingtheinputdelaymethoddevelopedinthispaperto compensateforuncertaintime-varyingdelays. 29

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Table2-1.MeanRMSErrorDegreesforSubject1S1toSubject6S6forbothRightR andLeftLlegs. SubjectLegRMSError deg. S1 R3.44 L4.52 S2 R5.03 L6.75 S3 R5.86 L5.85 S4 R4.72 L3.72 S5 R5.27 L5.10 S6 R5.73 L5.26 Mean5.10 SD0.93 Figure2-1.Trackingperformanceexampletakenfromtherightlegofsubject1S1-Right. PlotAincludesthedesiredtrajectorybluesolidlineandtheactualleganglered line.PlotBillustratestheangletrackingerror.PlotCdepictstheRMStracking error.PlotDdepictsthecontrolinputcurrentamplitudeinmA. 30

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Figure2-2.Trackingperformanceexampletakenfromtherightlegofsubject5S5-Right. PlotAillustratestheangletrackingerrorwhen =0 in2.PlotBdepictsthe controlinputcurrentamplitudeinmA. 31

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CHAPTER3 CONTROLOFANINPUTDELAYEDUNCERTAINNONLINEARSYSTEMWITHAN ADAPTIVEDELAYESTIMATE Thefocusofthischapterisarobustcontrollerforanuncertainnonlinearsystemwith boundeddisturbancesandanunknowntime-varyinginputdelay.Thedevelopedcontrolleruses thelinearmappingapproachinspiredbypredictor-basedapproachessuchas[30]tomapthe timedependentcontrolinputtoamodiedcontrolinputthatdependsbothontimeanda spatialvariable.Similartopredictor-basedapproaches,themodiedinputcanbesegregated intodelayedanddelay-freecomponents.Thissegregationimpactsthestabilityanalysisina waythatallowsforarbitrarilylargedelayrates,unlikeexistingresultscf.,[3,8,31,32].While benetingfromtheaddedexibilityinthestabilityanalysisresultingfromthelinearmapping, thecontrollermaintainsrobustnesstounmodeledeects.Duetochallengesassociated withstabilityanalysis,previousapproachese.g.,[88]havereliedonaconstantestimate ofthedelay,despitethefactthatthedelayisknowntovaryintime.Motivatedbythis fact,anothercontributionofthisresultisthataneuralnetworkNNestimationscheme isintroducedtoestimatetheunknowntime-varyingdelaymagnitude.Sincetheuniversal functionalapproximationtheoremonlyholdsforcontinuousfunctionswhosedomainis compact,anonlinearmappingisintroducedtomapthenon-compacttimedomaintoa compactdomain.Lyapunov-KrasovskiifunctionalsLKareusedintheLyapunov-based analysistoprovethatthetrackingerrorsexponentiallyconvergetoasteady-stateresidualthat isafunctionofthesystemuncertaintyi.e.,uniformultimatelyboundedUUBtracking. 3.1DynamicModel ConsideraclassofnonlienarsystemsexpressedinBrunovskycanonicalform,describedas x i = x i +1 ;i =1 ;:::;n; x n +1 = f X + d t + U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t ; where x i :[0 ; 1 R m i =1 ;:::;n denotethesystemstates, X x T 1 :::x T n T : [0 ; 1 R m n f : R m n R m isanuncertainnonlinearfunction,uniformlybounded 32

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in t d :[0 ; 1 R m isanunknownexogenousdisturbancee.g.,unmodeledeects, U :[0 ; 1 R m representsthegeneralizedinputvector,and D :[0 ; 1 R isanunknown, bounded,time-varyingdelay.Alineartransformationisusedtorepresentthegeneralized input U t asafunctionoftwoindependentvariables,i.e. p and t ,where t 2 [0 ; 1 and p 2 [0 ; 1] [30].Thespatialvariable, p ,denotesdelayedanddelayfreecontrolinputsat p =0 and p =1 ,respectively.Thedynamicmodelin3canbewrittenas x n +1 1 = f X + d t + U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t ; wherethesuperscript i denotesthe i th timederivative.Thetwovariablecontrolinput u :[0 ; 1] [ t 0 ; 1 R m isanalogousto U t inthesensethat[30] u p;t U )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( t + p )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t t t; 8 t 0 ; where :[0 ; 1 R isaknowndelaydependentinvertiblemonotonictimefunction, denedas t t )]TJ/F15 11.9552 Tf 14.898 3.022 Td [(^ D t ,where ^ D t representsasubsequentlydesignedtime-varyingdelay estimate 1 ,where )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t existsforalltime.Thetransformationdenedin3,isusedto expressthedelayedcontrolinputas U t )]TJ/F15 11.9552 Tf 13.15 0 Td [(^ t = u ;t ,andthedelay-freecontrolinput as U t = u ;t .Thespatialandtimevariationof u p;t ,denotedby u p p;t and u t p;t respectively,canberelatedas u t p;t = p;t u p p;t ; wheretheauxiliaryfunction :[0 ; 1] ; 1 2 R isdenedas p;t 1+ p d )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t dt )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t = 1+ p d )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t dt )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t : 1 Theupperboundsof ^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t ,and ^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t arethesameasof ^ D t and ^ D t ,respectively. 33

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Tofacilitatethesubsequentstabilityanalysisthetimederivativeof )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t canbedetermined byrstsubstituting t = )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t inthedenitionof t andtakingthetimederivativeofthe resultingexpression t = )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F15 11.9552 Tf 14.702 3.022 Td [(^ D )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t ; 1= d dt )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(d ^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t d )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t d )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t dt ; = d )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t dt 1 )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t ; d )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t dt = 1 1 )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t ; where ^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t d ^ D )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t d )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t .Using3and3,the p variationof p;t ,i.e, p p;t canbecalculatedas p = ^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t 1 )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t 1 ^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t : Evaluating p;t at p =0 ; 1 andusing3,yields ;t 0 = 1 ^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t ; ;t 1 = 1 1 )]TJ/F15 11.9552 Tf 14.702 3.022 Td [(^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t 0 : UsingAssumptions3.1andtheprojectionlawdiscussedinSection3.7,followingboundshave beendevelopedfor 0 1 and p ,as 0 0 0 1 1 1 ,| p | j p j j p j ,where 0 0 1 1 ,| p | and j p j areknownpositiveconstants. Remark 3.1 From3-3singularitiescanoccurin 0 1 and p when ^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t =1 keepinginmindthesingularitydueto ^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t =0 isavoidedbydesigningprojectionlawas discussedinSection3.7.From3,when ^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t =1 ,foranonzerodelaymagnitude ^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t u p ;t =0 u ;t = u ;t p =0 0 = 1 = 1 ^ D )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t Assumption3.1. Theunknowntime-varyingdelay D t 2 R isupperandlowerboundedby knownpositiveconstants D andD respectively,asD D t D; 8 t 34

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Assumption3.2. Thedesiredtrajectory x d t 2 R m isdesignedsuchthat x i d t 2 R m ; 8 i = 0 ; 1 ;:::; n +2 existandareboundedbyknownpositiveconstants. 3.2ControlObjective Thecontrolobjectiveistodevelopacontrollerwhichensuresthatthestate x 1 of3 tracks x d ,despiteuncertaintiesandadditivedisturbancesinthedynamics.Toquantifythe controlobjective,atrackingerror, e 1 :[0 ; 1 R m ,isdenedas e 1 x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(x 1 : Tofacilitatethesubsequentanalysis,measurableauxiliarytrackingerrors,denotedby e i t 2 R m ;i =2 ; 3 ;:::;n ,aredenedas e 2 e 1 + e 1 ; e 3 e 2 + e 2 + e 1 ; . e n e n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + e n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + e n )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 : Ageneralexpressionfor e i t ;i =2 ; 3 ;:::;n canbewrittenas e i = i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 X j =0 a i;j e j 1 ; where a i;j 2 R ,areknowncoecients,calculatedusingthedenitionofFibonaccisequences, with a n; n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 =1 .Anauxiliarytrackingerrorsignal, r :[0 ; 1 R m ,isdenedas r e n + e n )]TJ/F25 11.9552 Tf 11.955 0 Td [(e u ; where ; 2 R areknown,positive,constantgains.In3, e u :[0 ; 1 R m isan auxiliaryerrorterm,introducedtoobtainadelay-freecontrolexpressionfortheinputinthe 35

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closed-looperrorsystemandisdenedas e u 1 0 u p;t dp: Tocalculate e u e u needstobemeasuredandintegratedoverthetimedomain [0 ;t ] .Usingthe Leibnitzintegralrule,anddierentiating3withrespecttotimegives e u as e u = 1 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 u ;t )]TJ/F25 11.9552 Tf 11.956 0 Td [( p e u : Givenaninitialconditionfor e u ,3canbeusedtocompute e u and e u 3.3DevelopmentofErrorSignals Theopen-looperrorsystemfor r t canbeobtainedbytakingthetimederivativeof 3andusing3,3,3and3as r = x n +1 d )]TJ/F25 11.9552 Tf 11.955 0 Td [(f X )]TJ/F25 11.9552 Tf 11.955 0 Td [(d t )]TJ/F25 11.9552 Tf 11.956 0 Td [(U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t + e n + p e u + n )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 X j =0 a n;j e j +2 1 )]TJ/F25 11.9552 Tf 9.299 0 Td [( 1 u ;t + 0 u ;t : Theopen-looperrorsystemin3containsbothadelayedanddelay-freecontrolinput, sincethetimederivativeof3isusedin3.Basedonthesubsequentstability analysis,thedelay-freecontrolinputisdesignedas U t = u ;t 1 k r; where k 2 R + isaconstant,adjustablecontrolgain.Tofacilitatethesubsequentstability analysis,3canbesegregatedintotermsthatcanbeupperboundedbyastate-dependent functionandtermswhichcanbeupperboundedbyaknownconstantas r = ~ N + N d )]TJ/F25 11.9552 Tf 11.956 0 Td [(e n )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 u ;t + 0 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t + p e u : 36

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In3, ~ N X;X d ;e n ; e n ;e 1 ;e 1 ;:::;e n 1 2 R m isanauxiliarytermdenedas ~ N )]TJ/F25 11.9552 Tf 9.299 0 Td [(f X + f X d + e n + e n + n )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 X j =0 a n;j e j +2 1 ; where X d t [ x T d ; 0 ;::: 0] T .UsingtheMeanValueTheorem,andAssumption3.2, ~ N X;X d ;e n ; e n ;e 1 ;e 1 ;:::;e n 1 in3canbeupperboundedas ~ N k z k k z k ; where : R R isaknownpositivedenite,non-decreasing,radiallyunboundedfunction,and z 2 R m n +2 isavectoroferrorsignals,denedas z e T 1 e T 2 :::e T n r T e T u T : Alsoin3, N d X d ;x n +1 d ;d 2 R m isanauxiliarytermdenedas N d )]TJ/F25 11.9552 Tf 9.298 0 Td [(f X d + x n +1 d )]TJ/F25 11.9552 Tf 11.955 0 Td [(d t : UsingAssumptions3.1and3.2, N d X d ;x n +1 d ;d canbeupperboundedas sup t 2 [0 ; 1 k N d k ; where 2 R + isaknownconstant.Substituting3intotheopen-looperrorsystemin 3,theclosed-looperrorsystemcanbeobtainedas r = ~ N + N d )]TJ/F25 11.9552 Tf 11.956 0 Td [(e n )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 1 r k + 0 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t + p e u : 3.4EstimationofDelay AneuralnetworkNNbasedfunctionapproximatorisusedtoestimatetheunknown delaymagnitude.Theuniversalfunctionapproximationtheoremonlyholdsoveracompact domain.Therefore,toapproximatetheunknowndelayfunction,anonlinearmappingisdened 37

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tomapthenon-compactdomaintoacompactspatialdomain.Let f L : t bedenedas f L t 1+ t ;t 2 [0 ; 1 ; 2 [0 ; 1] ; where 2 R + isauserdenedsaturationcoecient.Using3, D t canbemappedinto thedomain as D t = D )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(f )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 L D f L : Theuniversalfunctionalapproximationtheoremcanbeusedtorepresent D f L byathreelayerNNas D f L W T )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(V T + ; where V 2 R 2 L and W 2 R L +1 1 aretheunknownboundedunknownconstantidealweights fortherst-to-secondandsecond-to-thirdlayers,respectively, L isthenumberofneuronsin thehiddenlayer, 2 R L +1 isactivationfunction, isthefunctionalreconstructionerror, and =[1 ] T denotestheinputtotheNN.Basedon3,theNNestimationfor ^ t is givenby ^ D t = ^ W T ^ V T ; where ^ W and ^ V areestimatesoftheidealweights.In3, isselectedasasaturated activationfunctioni.e., logsig tanh ,inordertosimplifythedevelopmentofprojectionlaw, discussedinSection3.7.Using3and3,themismatchbetween D t and ^ D t can beobtainedusingaTaylor'sseriesapproximation,whichaftersomealgebraicmanipulation,can beexpressedas D t )]TJ/F15 11.9552 Tf 14.702 3.022 Td [(^ D t = W T )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(V T )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W T ^ V T + ; = ~ W T ^ V T + ^ W T 0 ^ V T ~ V T + W T O ~ V T 2 + + ~ W 0 ^ V T ~ V T ; 38

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where ~ W = W )]TJ/F15 11.9552 Tf 15.566 3.022 Td [(^ W 2 R L +1 1 and ~ V = V )]TJ/F15 11.9552 Tf 13.94 3.022 Td [(^ V 2 R 2 L ,aretheestimatemismatchforthe idealweightmatrices,and O representshigherorderterms.Asmentionedbefore,duetothe developmentofprojectionlawinSection3.7,theelementsof ^ W and ^ V canallbeupperand lowerboundedbyknownpositiveconstants.Hence, ~ W T 0 ^ V T ~ V T and W T O ~ V T 2 canalsobeboundedbyknownpositiveconstants,andtherefore, D t )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D t ~ W T ^ V T + ^ W T 0 ^ V T ~ V T + ; where 2 R isapositiveboundingconstant. 3.5StabilityAnalysis Tofacilitatethestabilityanalysis,let y 2 R m n +2+1 bedenedas y z T p Q T ; where Q 2 R denotesanLKfunctionaldenedas Q Q 1 0 e 2 p k u p;t k 2 dp; where Q ;! 2 2 R areknown,positiveconstants.Let D beanopenandconnectedset,and S D D isdenedas S D y 2 R 3 n +1 jk y k < inf )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 [ r 1 1 2 ; 1 ; where 1 and 1 2 R areknown,positiveconstants. Theorem3.1. Giventheopen-looperrorsystemin3,thecontrollerin3ensures UUBtrackinginthesensethat k e 1 k )]TJ/F23 7.9701 Tf 7.253 -1.793 Td [(0 exp )]TJ/F25 11.9552 Tf 9.298 0 Td [()]TJ/F23 7.9701 Tf 7.253 -1.793 Td [(1 t + )]TJ/F23 7.9701 Tf 7.252 -1.793 Td [(2 ; where )]TJ/F23 7.9701 Tf 7.253 -1.793 Td [(0 ;)]TJ/F23 7.9701 Tf 10.504 -1.793 Td [(1 and )]TJ/F23 7.9701 Tf 7.253 -1.793 Td [(2 areknownpositiveconstants,providedthat y 2S D ; 8 2 [ t 0 ;t 0 + D ] 39

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Proof. Let V L : D [ t 0 ; 1 R beacontinuouslydierentiable,positive-denitefunction denedas V L 1 2 n X i =1 e T i e i + 1 2 r T r + 1 2 e T u e u + Q + 1 2 tr ~ W T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 1 ~ W + 1 2 tr ~ V T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 ~ V ; where 1 k y k 2 + c L V L 2 k y k 2 + c U c L ;c U 2 R + areknownboundingconstants.Taking thetimederivativeof3andusing3-3and3,yields V L = r T ~ N + N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(e n )]TJ/F25 11.9552 Tf 11.956 0 Td [(e T n e u + e T n r + r T )]TJ/F25 11.9552 Tf 9.299 0 Td [(U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t )]TJ/F25 11.9552 Tf 11.956 0 Td [( Q 2 1 0 e 2 p k u k 2 dp + r T )]TJ/F25 11.9552 Tf 9.299 0 Td [( 1 r k + 0 u ;t + p e u )]TJ/F26 7.9701 Tf 12.61 14.944 Td [(n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 X i =1 e T i e i )]TJ/F25 11.9552 Tf 11.955 0 Td [(e T n e n + e T n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 e n + 1 e T u 1 r k )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( p e u + Q 1 e 2 k u ;t k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q 0 jj u ;t jj 2 )]TJ/F25 11.9552 Tf 9.299 0 Td [( Q p 1 0 e 2 p k u k 2 dp + tr ~ W T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 1 ~ W + tr ~ V T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 ~ V : Byusing3,3,3,andthefactsthat ~ W = )]TJ/F15 11.9552 Tf 14.012 6.176 Td [(_ ^ W and ~ V = )]TJ/F15 11.9552 Tf 12.385 6.176 Td [(_ ^ V ,anupperbound on V L canbeobtainedas V L k r k k z k k z k + k r k + )]TJ/F15 11.9552 Tf 7.512 -6.529 Td [( 0 +1 k r T u ;t k)]TJ/F26 7.9701 Tf 21.244 14.944 Td [(n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 X i =1 e T i e i )]TJ/F25 11.9552 Tf 11.955 0 Td [( k e n k 2 )]TJ0 g 0 G/F25 11.9552 Tf 10.494 8.088 Td [( 1 k r k 2 k + j r T u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(U t )]TJ/F25 11.9552 Tf 11.956 0 Td [(D t j + j p j k r T e u k + k e T n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 e n k + k e T n e u k + 1 j p jk e u k 2 + 1 1 k e T u r k k + 1 0 k e T u u ;t k )]TJ/F25 11.9552 Tf 9.298 0 Td [( Q 2 1 0 e 2 p k u k 2 dp + Q j p j 1 0 e 2 p k u k 2 dp )]TJ/F25 11.9552 Tf 11.955 0 Td [(tr ~ W T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 1 ^ W + Q 1 e 2 k r k 2 k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q 0 jj u ;t jj 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(tr ~ V T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 ^ V : Tofacilitatethesubsequentanalysis,notethat p;t = 1+ p d )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t dt )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t min f 0 ; 1 g : 40

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UsingtheCauchy-Schwarzinequality, k e u k 2 1 0 k u 2 p;t k dp 1 0 1 dp; k e u k 2 1 0 k u p;t k 2 dp: UsingtheMeanValueTheoremandtheexpressionin3,thefollowinginequalitiescanbe developed: j r T u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(U t )]TJ/F25 11.9552 Tf 11.955 0 Td [( t jj r T D t )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D t u ;t j M j r T ~ W T ^ V T j + M j r T ^ W T 0 ^ V T ~ V T j + M j r T j : Theinequalitiesin3-3,canbeusedtoupperbound3as V L 1 1 2 k z k k z k 2 + 1 2 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( 2 + M 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(tr ~ W T )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 1 ^ W )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 )]TJ/F25 11.9552 Tf 15.686 8.088 Td [( 2 k jj e n jj 2 + M j r T h ~ W T ^ V T + ^ W T 0 ^ V T ~ V T i j)]TJ/F25 11.9552 Tf 17.933 0 Td [(tr ~ V T )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 ^ V )]TJ/F26 7.9701 Tf 11.946 14.944 Td [(n )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 X i =1 e T i e i )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 k e n )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 k 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( 1 k )]TJ/F15 11.9552 Tf 18.518 11.907 Td [( j p j 2 2 k )]TJ/F25 11.9552 Tf 13.15 8.088 Td [(! 1 1 2 k 3 k r k 2 + Q 1 e 2 k 2 + 1 + 2 4 + M 2 4 k r k 2 + 2 k + 0 2 k k r k 2 )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( Q 0 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [(! 1 0 k 2 k u ;t k 2 + k 0 2 2 + k 2 k u ;t k 2 )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( Q 2 min f 0 ; 1 g 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q j p j Q )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( Q 2 min f 0 ; 1 g 2 )]TJ/F15 11.9552 Tf 18.518 11.907 Td [( j p j k 2 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(! 1 j p j k e u k 2 + 1 1 k 2 + 1 0 2 k + k 2 jj e u jj 2 : Usingthefactthat a T b = trace ba T ^ W t and ^ V t aredesignedtocancelcrosstermsas ^ W = proj 1 M ^ V T r T ; ^ W ; ^ V = 2 M r T ^ W T 0 ^ V T : 41

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Thefunction proj :;: in3denotesaLipschitzcontinuousdiscussedinSection3.7 projectionoperator,whichensuresD + l ^ D t D + u ,where l ; u 2 R aresubsequently denedpositiveconstantsseeSection3.8.Since k y kk z k ; thefollowingupperboundcan beobtained V L )]TJ/F30 11.9552 Tf 23.91 16.857 Td [( 1 2 )]TJ/F15 11.9552 Tf 14.954 8.088 Td [(1 1 2 k y k k z k 2 + 1 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( 2 + M 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 1 2 jj z jj 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q res k u ;t k 2 )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( 2 min f 0 ; 1 g 2 )-222(j p j Q; where 1 ;;; Q res 2 R aredenedas 1 min )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(1 2 )]TJ/F25 11.9552 Tf 15.686 8.087 Td [( 2 k ; 1 2 ;; ; 1 k )]TJ/F15 11.9552 Tf 18.518 11.907 Td [( j p j 2 2 k )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(! 1 1 2 k 3 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(M 2 4 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [( Q 1 e 2 k 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [( 1 + 2 4 )]TJ/F25 11.9552 Tf 16.959 8.088 Td [( 2 k )]TJ/F15 11.9552 Tf 15.182 11.242 Td [( 0 2 k ; Q 2 min f 0 ; 1 g 2 )]TJ/F15 11.9552 Tf 18.517 11.907 Td [( j p j k 2 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(! 1 j p j)]TJ/F25 11.9552 Tf 19.128 8.088 Td [(! 1 1 k 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(! 1 0 2 k )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(k 2 ; Q res Q 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 0 1 +1 2 + 0 2 2 : Provided y 2S D ; 8 2 [ t )]TJ/F15 11.9552 Tf 14.701 3.022 Td [( D;t ] ,theexpressionin3reducesto V L )]TJ/F25 11.9552 Tf 21.918 0 Td [( 2 k y k 2 + '; where '; 2 2 R aredenedas 1 2 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( 2 + M 2 ; 2 min 1 2 ; 2 min f 0 ; 1 g 2 )-222(j p j : Anupperboundcanbeobtainedfor3as V L )]TJ/F25 11.9552 Tf 23.926 8.088 Td [( 2 2 V L + c U 2 + ': Thesolutionoftheinequalityin3canbeobtainedas V L t V L exp )]TJ/F25 11.9552 Tf 11.307 8.088 Td [( 2 2 t + 2 + c U 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F25 11.9552 Tf 11.307 8.088 Td [( 2 2 t : 42

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Using3and3,theinequalityin3canbeobtained.Since e i 1 t ;r t ;e u t 2 L 1 ; 3canbeusedtoconcludethat u t 2L 1 : 3.6SimulationResults Anumericalsimulationwasperformedtoexaminetheperformanceofthecontrollerin3 17alongwiththeNNweightsupdatelawsdescribedin3and3.Forthesimulation thefollowingsecondordersystemwasused 2 6 4 U 1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t U 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t 3 7 5 = 2 6 4 p 1 +2 p 3 c 2 p 2 + p 3 c 2 p 2 + p 3 c 2 p 2 3 7 5 2 6 4 x 1 x 2 3 7 5 + 2 6 4 )]TJ/F25 11.9552 Tf 9.299 0 Td [(p 3 s 2 x 2 )]TJ/F25 11.9552 Tf 9.298 0 Td [(p 3 s 2 x 1 +_ x 2 p 3 s 2 x 1 0 3 7 5 2 6 4 x 1 x 2 3 7 5 + 2 6 4 f d 1 0 0 f d 2 3 7 5 2 6 4 x 1 x 2 3 7 5 + 2 6 4 d 1 d 2 3 7 5 : In3, d 1 d 2 representaddeddisturbancesdenedas d 1 =0 : 2sin : 5 t and d 2 = 0 : 1sin : 25 t .Additionally, p 1 =3 : 473 kg m 2 p 2 =0 : 196 kg m 2 p 3 =0 : 242 kg m 2 p 4 =0 : 238 kg m 2 p 5 =0 : 146 kg m 2 f d 1 =5 : 3 Nm sec, f d 2 =1 ; 1 Nm sec,and s 2 ;c 2 denote sin x 2 ; and cos x 2 ,respectively. Theinitialconditionsforthesystemwereselectedas x 1 ;x 2 =0 .Thedesired trajectorieswereselectedas x d 1 t =0sin : 5 t +20 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F23 7.9701 Tf 6.587 0 Td [(0 : 01 t 3 ; x d 2 t = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin t= 2+10 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F23 7.9701 Tf 6.587 0 Td [(0 : 01 t 3 : Thedynamicsdescribedin3,issimulatedfortimevaryingdelaymagnitudeof D t = 0 : 04tanh t 30 +0 : 08 ,where 0 : 08 D t 0 : 12 .Aneuralnetworkoftwohiddenlayers with 5 neuronsineachlayerisusedforthedelayestimation.Firsthiddenlayerhasasigmoid activationfunctionandthesecondhiddenlayerhasalinearactivationfunction.Updatelaws developedin3and3areusedtoupdateneuralnetworkweightsineachtraining iteration.Figure3-1showsthetrackingerrorvariationandthecontrolforcevariationfor the2-linkrobotdynamics.Usingtheauxiliarytrackingerrordata,availableindiscretetime 43

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fromthesimulation,NNweightsaretrainedusing 90 percentavailabletrainingdata,and testedonremaining 10 percentavailablesimulateddata.Inordertoenforcelearning, k -fold methodisusedwhiletrainingthedesignednetwork,fordelaymagnitude D t estimation,with k =10 .Figureshowsthepercentagevariationofdelaymismatch,withtrainingiteration, whichindicatesasuccessfultrainingwhileestimatingunknowndelaymagnitudeusingthe updatelawsin3and33. 3.7ProjectionLaw BasedonAssumption,unknowntime-varyingdelay D t belongtothecompactconvex set := f D t : D D t D g ,whereD D 2 R areknownpositiveconstants.The standardLipschitzcontinuousprojectionoperatore.g.,[89,90],introducedin3isgiven by ^ W = proj ; ^ W ^ W = 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : ; if p j low ^ D 0 or p j high ^ D 0 ; if p j low ^ D 0 and r p j low ^ D T 0 ; if p j high ^ D 0 and r p j high ^ D T 0 I )]TJ0 g 0 G/F25 11.9552 Tf 11.955 0 Td [(& ; if p j low ^ D 0 and r p j low ^ D T < 0 I )]TJ/F15 11.9552 Tf 12.602 0 Td [( & ; if p j high ^ D 0 and r p j high ^ D T > 0 where & p j low ^ D r p j low ^ D r p j low ^ D T r p j low ^ D T r p j low ^ D 2 R & p j high ^ D r p j high ^ D r p j high ^ D T r p j high ^ D T r p j high ^ D 2 R , 1 M ^ V T r T 2 R p j low ^ D ^ D T ^ D )]TJ0 g 0 G/F20 11.9552 Tf 6.586 0 Td [(D 2 2 l +2 l D 2 R p j high ^ D ^ D T ^ D )]TJ/F23 7.9701 Tf 8.531 2.015 Td [( D 2 2 u +2 u D 2 R l ; u 2 R are positiveconstants,and r isthegradientoperator.Given ^ D 2 ,theprojectionoperator mentionedin3hastheproperties,D + l ^ D t D + u and proj ; ^ W isLipschitz continuous.Lipschitzcontinuityof ^ W ,alongwiththecontinuousupdatelawof ^ V in3, guaranteetheexistenceof ^ D t ,whichgivesacontinuousestimateofunknowndelay ^ D t at alltime. 44

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3.8ControlGainSelection Controlgains,suchas 1 in3, in3, in3, Q res in3and 2 in 3,introducedinthestabilityanalysisSection3.5arerequiredtobepositiveconstants. Basedonthedesignedboundsoftime-delayestimate ^ inthissection,andsubsequently derivedboundsof 0 1 and p i.e., 0 ; 0 1 ; 1 j p j ,thisappendixdevelopssucientgain conditionstoensure 1 , Q res and 2 > 0 .Usingthedenitionsof 1 in3and 2 in3,sucientlowerboundsfor and 2 canbeobtainedas > + k 2 k ; 2 > 2 j p j min f 0 ; 1 g : Usingthedenitionof Q res in3,itisclearthatforanarbitrarilylarge k and and arbitrarilysmall 1 and ,that Q canbeselectedsucientlylargeas Q > k 0 0 1 +1 2 + 0 2 2 ; toensurethat Q res > 0 .Alsofromthedenitionof in3, Q alsoneedstosatisfythe followinginequality Q > 2 2 min f 0 ; 1 g j p j k 2 2 + 1 j p j + 1 1 k 2 + 1 0 2 k + k 2 toensure > 0 .Byselecting ontheorderof k 2 ontheorderof k 3 ontheorderof 1 k 4 and 1 ontheorderof 1 k 5 ,thelowerboundin3canbeproventobelargerthan3. Forexample,if = k 2 = k 3 = 1 k 4 and 1 = 1 k 5 then 1 0 k 5 2 + 0 k 3 2 + 0 2 > 2 2 min f 0 ; 1 g j p j k 4 2 + j p j k 5 + 1 2 k 4 + 0 2 k 10 + k 4 2 ; forlargevaluesof k> 1 and 2 From3, Q ismultipliedbyanegativeterminthedenitionof .Todevelopa sucientconditiontoensure > 0 ,thelowerboundfor Q in3issubstitutedinto 45

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3todevelopthefollowingsucientinequality: 1 k )]TJ/F15 11.9552 Tf 18.518 11.907 Td [( j p j 2 2 k )]TJ/F15 11.9552 Tf 15.183 11.242 Td [( 0 1 e 2 2 0 > 1 + 2 4 + 2 k + 0 2 k + 1 0 +1 2 0 1 e 2 k + 1 1 2 k 3 + M 2 4 : Basedon3,asucientconditionfortheupperboundon canbeestablishedas < 2 1 0 j p j 0 2 + 0 k 1 e 2 ; toensuretheparentheticaltermsontheleftsideof30arepositive.Basedon3and 3asucientlowerboundfor canbeestablishedforanarbitrarilylarge k 2 2 and arbitrarilysmall 1 and ,toensure > 0 as > 1 + 2 4 + 2 k + 0 2 k + 1 0 +1 2 0 1 e 2 k + 1 1 2 k 3 + M 2 4 1 k )]TJ/F23 7.9701 Tf 17.429 8.411 Td [( j p j 2 2 k )]TJ/F23 7.9701 Tf 14.983 6.916 Td [( 0 1 e 2 2 0 : Toobtainthesucientconditionin3andtosatisfythesucientconditionsin3, 3,and3, 2 , Q and areselectedsucientlylargeand and 1 areselected sucientlysmall.Forexample,selecting = k 2 = k 3 and = 1 k 4 ,aspreviously,sothat k 6 1 0 +1 2 0 1 e 2 + 0 2 + k 5 0 1 e 2 2 0 + 1 2 + 1 + 2 4 + M 2 4 )]TJ0 g 0 G/F25 11.9552 Tf 11.955 0 Td [( 1 + k 3 j p j 2 + 1 1 2 > 0 ; whichclearlyholdsforany k> 1 ,thenthesucientconditionin3issatised. 3.9Conclusion Foraclassofuncertainnonlinearsystemssubjecttounknowntime-varyinginputdelay, atrackingcontrollerisdesignedwherethecontrolinputvarieswithbothtimeandaspatial variable.Thedesignedcontrollerfeaturesgainstocompensateforthedelayandthedelay derivativeindependently.Duetotheseparationofthedelaytermoutsidethecontrolinput, aNN-basedestimationschemeisusedtoestimatetheunknowninputdelaymagnitude.A nonlinearmappingisusedtotransformthenon-compacttimeintervaltoacompactsetto 46

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facilitatetheuseofaNN.ALyapunov-KrasovskiifunctionalisusedintheLyapunov-based stabilityanalysistoproveuniformultimateboundednessoftheerrorsignals. Figure3-1.Variationoferrorsandcontrolforces. 47

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Figure3-2.Delaymismatchvariationwithtrainingiteration. 48

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CHAPTER4 CONTROLOFANUNCERTAINNONLINEARSYSTEMWITHANUNKNOWN TIME-VARYINGINPUTDELAYUSINGANACCELERATEDGRADIENTDESCENTBASED DELAYESTIMATE ThedevelopedrobustPDE-baseddesignapproachdoesn'trestrictthedelayratemagnitude.Thedelayderivativegain,aswellastheconventionaldelaymagnitudegain,canbe independentlyadjusted.Theamalgamofthepredictor-basedtransformation,therobustcontrol errorsystem,andthedelayestimateallowsfornewcontroldevelopmentandstabilityanalysis methodsthatcanbeappliedtouncertainnonlinearsystemswithunknowntime-varyingdelays. ThesecontributionsarebasedonNesterov'sAGDbasedstrategy[91]tosuccessfullyestimate thedelaymagnitudeatalltime.Twodierentobserverbasedmethodsaredeveloped,where onerequirestheknowledgeofhighestorderstatederivative,andtheotherusesNesterov's AGDbasedapproach.Aconstrainedoptimizationproblemisformulatedtoestimatethedelay. Subsequently,anaugmentedLagrangianbasedunconstrainedoptimizationinthedualspace isformulated,whichissolvedusingNesterov'sAGDbasedtechnique.Lyapunov-Krasovskii functionalsareusedintheLyapunov-basedanalysistoprovethetrackingerrorsexponentially convergetoasteady-stateresidualthatisafunctionofsystemuncertaintyi.e.,uniform ultimatelyboundedUUBtracking.Simulationresultsdemonstratethecontrollerperformanceforasecondordernonlinearsystemandshowsanestimationofdelayanddelayrate magnitudes. 4.1DynamicModel Consideraclassof n +1 th ordernonlinearsystemsmodeldevelopedinSection3.1. Alongwiththedynamicmodel,Assumptions3.1,andAssumptions3.2areconsideredforthis chapter.Inaddition,followingassumptionsarespecicforthischapter. Assumption4.1. Thenonlinearexogenousdisturbancetermanditsrstderivativei.e., d d existandareboundedbyknownpositiveconstants,cf.[88]. Assumption4.2. Theunknownrstandseconddelayderivatives D t ; D t 2 R areupper boundedbyknownpositiveconstants D D respectively,as D t D; D t D; 8 t 49

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Assumption4.3. Theestimate ^ D t issucientlyaccuratesuchthat ~ D t D t )]TJ/F15 11.9552 Tf 14.966 3.022 Td [(^ D t canbeupperboundedas j ~ D t j ~ D 8 t 2 R ,where ~ D 2 R isaknownpositiveconstant, cf.[88]. BasedonthedelayestimationinSection4.5andAssumptions3.1-4.2, 0 0 0 1 1 1 and j p jj p j ,where 0 ; 0 ; 1 ; 1 and j p j areknownpositiveconstants. 4.2ControlObjective Theobjectiveistodevelopacontrollerwhichensuresthatthestate x 1 t oftheinputdelayedsystemin3tracks x d t ,despiteuncertaintiesandadditivedisturbancesinthe dynamics.Toquantifythecontrolobjective,atrackingerror,denotedby e 1 t 2 R m ,is denedin3.Measurableauxiliarytrackingerrors, e i t 2 R m ;i =2 ; 3 ;:::;n ,are denedin3.Ageneralexpressionfor e i t ;i =2 ; 3 ;:::;n canbeexpressedasin 3.Anothermeasurableauxiliarytrackingerrorsignal r t 2 R m ,isdenedin3. e u :[0 ; 1 R m isdesignedasin3toobtainadelaydependentcontroltermtonegate theeectofthedelayedinputin3. 4.3DevelopmentofErrorSignals Theopen-looperrorsystemfor r t canbeobtainedbytakingthetimederivativeof r t in3,andcanbeexpressedas3.Basedonthesubsequentstabilityanalysis,the delay-freecontrolinputisdesignedas U t u ;t kr; where k 2 R isaconstant,positive,adjustablecontrolgain.Usingthedenitionof ~ N;N d 2 R m in3and3,andusingthedenitionof z 2 R n +2 m in3,theclosed-loop errorsystemfor r canbeobtainedas r = ~ N + N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(e n )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 kr + 0 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t + p e u : 50

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UsingtheMeanValueTheorem,andAssumption3.2,theexpressionin3canbeupper boundedas ~ N k z k k z k ; where : R R isaknownpositivedenite,non-decreasing,radiallyunboundedfunction. UsingAssumptions4.1and3.2, N d canbeupperboundedas sup t 2 [0 ; 1 k N d k ; where 2 R isaknownpositiveconstant. 4.4StabilityAnalysis Tofacilitatethesubsequentstabilityanalysis,let y t 2 R n +2 m +1 bedenedas y z T p Q T ; where Q t 2 R denotesanLKfunctionaldenedas Q Q 1 0 e 2 p k u p;t k 2 dp; where Q ;! 2 2 R areknown,positiveconstants.Let D beanopenandconnectedset,and S D D isdenedas S D n y 2 R n +2 m +1 jk y k < q min f 1 ;! 1 g max f 2 ;! 1 g inf f )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 [ p ; 1 g o ,where 1 ; 2 R areknown,positiveconstants. Theorem4.1. Giventhedynamicsin32,thecontrollerin4ensuresUUBtrackingin thesensethat k e 1 k )]TJ/F23 7.9701 Tf 7.253 -1.794 Td [(0 exp )]TJ/F25 11.9552 Tf 9.298 0 Td [()]TJ/F23 7.9701 Tf 7.253 -1.794 Td [(1 t + )]TJ/F23 7.9701 Tf 7.252 -1.794 Td [(2 ; where )]TJ/F23 7.9701 Tf 7.253 -1.794 Td [(0 q j 2 V )]TJ/F23 7.9701 Tf 13.15 5.255 Td [(2 2 j ;)]TJ/F23 7.9701 Tf 14.406 -1.794 Td [(1 )]TJ/F23 7.9701 Tf 12.099 4.707 Td [( 2 and )]TJ/F23 7.9701 Tf 7.253 -1.794 Td [(2 q 2 2 ,providedthat y 2S D ; 8 2 [ t 0 ;t 0 + D ] ,where 1 ; 2 2 R 2 max f 2 ;! 1 g ; 1 min f 1 ;! 1 g and '; 2 R are subsequentlydevelopedcontrolgains. 51

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Proof. Let V : D [ t 0 ; 1 R beacontinuouslydierentiable,positive-denitefunctional denedas V 1 2 n X i =1 e T i e i + 1 2 r T r + 1 2 e T u e u + Q; where 1 k y k 2 V 2 k y k 2 .Thetimederivativeof4canbeobtainedafterapplying Leibnizintegralruletoobtainthetimederivativeof4,andutilizing3-3and 4,as V = r T ~ N + N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(e n + r T )]TJ/F25 11.9552 Tf 9.299 0 Td [(U t )]TJ/F25 11.9552 Tf 11.956 0 Td [(D t + 1 e T u 1 kr )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( p e u + r T )]TJ/F25 11.9552 Tf 9.298 0 Td [( 1 kr + 0 u ;t + p e u )]TJ/F26 7.9701 Tf 12.61 14.944 Td [(n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 X i =1 e T i e i )]TJ/F25 11.9552 Tf 11.955 0 Td [(e T n e n + e T n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 e n )]TJ/F25 11.9552 Tf 9.299 0 Td [(e T n e u + e T n r + Q 1 e 2 k u ;t k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q 0 jj u ;t jj 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q p 1 0 e 2 p k u k 2 dp )]TJ/F30 11.9552 Tf 11.291 20.444 Td [( 0 +1+ 1 0 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.48 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 e 2 0 jj u ;t jj 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q 2 1 0 e 2 p k u k 2 dp: Byusing4,4,4,andcancelingcommontermsin4,anupperboundcanbe obtainedas V k z k k r kk z k + k r k )]TJ0 g 0 G/F25 11.9552 Tf 11.955 0 Td [( 1 k k r k 2 + )]TJ/F15 11.9552 Tf 7.511 -6.529 Td [( 0 +1 k r T u ;t k + k e T n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 e n k + k e T n e u k + j r T u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t j + j p jk r T e u k + 1 1 k k e T u r k + 1 0 k e T u u ;t k + 1 j p jk e u k 2 + Q 1 e 2 k u ;t k 2 )]TJ/F26 7.9701 Tf 11.946 14.944 Td [(n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 X i =1 e T i e i )]TJ0 g 0 G/F25 11.9552 Tf 11.955 0 Td [( k e n k 2 )]TJ/F30 11.9552 Tf 11.955 20.444 Td [(" Q )]TJ/F30 11.9552 Tf 11.955 20.444 Td [( 0 +1+ 1 0 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.479 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 e 2 0 # jj u ;t jj 2 )]TJ/F25 11.9552 Tf 9.299 0 Td [( Q 2 1 0 e 2 p k u k 2 dp + Q j p j 1 0 e 2 p k u k 2 dp: ByusingAssumption3.1,Young'sInequality,theCauchy-SchwartzInequality,thefact that p;t min f 0 ; 1 g derivedfromthedenitionof p;t in3,thefactthat k e u k 2 = k 1 0 u p;t dp k 2 1 0 jj u 2 p;t jj dp: 1 0 1 :dp 1 0 jj u 2 p;t jj dp ,thefollowing 52

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inequalitiescanbedeveloped u ;t = c u ;t D + u ;t cM D + u ;t ; k u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(U t )]TJ/F25 11.9552 Tf 11.956 0 Td [(D t k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c D + ~ D 2 M 2 ; where M 1 isapositiveconstant.Inequalitiesin4-4canbeusedtoupperbound 4as V 1 2 k 2 k z k k z k 2 + 1 2 k 2 + )]TJ/F15 11.9552 Tf 7.512 -6.529 Td [( 0 + 1 0 c 2 M 2 D 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.48 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 e 2 1 e 2 + c 2 M 2 D 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.48 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 1 e 2 1 e 2 + 1 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c D + ~ D 2 M 2 )]TJ/F26 7.9701 Tf 12.61 14.944 Td [(n )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 X i =1 e T i e i )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 k e n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 k 2 )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 2 )]TJ/F15 11.9552 Tf 14.835 11.242 Td [( 2 jj e n jj 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( 1 k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( )]TJ/F25 11.9552 Tf 11.955 0 Td [(k )]TJ/F25 11.9552 Tf 13.151 8.087 Td [( 0 2 k r k 2 )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( 1 k 4 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( j p j 2 k r k 2 )]TJ/F30 11.9552 Tf 11.956 16.857 Td [( 1 k 4 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(! 1 1 k 2 k r k 2 )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( Q 2 min f 0 ; 1 g 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q j p j Q )]TJ/F30 11.9552 Tf 11.955 20.443 Td [(" Q )]TJ/F30 11.9552 Tf 11.955 20.443 Td [( 0 +1+ 1 0 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.479 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 e 2 0 # jj u ;t jj 2 )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( Q 2 min f 0 ; 1 g 2 )]TJ/F15 11.9552 Tf 14.836 11.243 Td [( j p j 2 )]TJ/F15 11.9552 Tf 14.835 11.243 Td [( 2 jj e u jj 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( 1+ 1 1 k 2 + 1 0 2 + 1 j p j k e u k 2 : Since k y kk z k ; 4canbesimpliedtoobtain V )]TJ/F30 11.9552 Tf 23.91 16.857 Td [( 2 )]TJ/F15 11.9552 Tf 16.395 8.088 Td [(1 2 k 2 k y k k z k 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 jj z jj 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( Q 2 min f 0 ; 1 g 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q j p j Q )]TJ/F25 11.9552 Tf 9.299 0 Td [( Q res jj u ;t jj 2 + 1 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c D + ~ D 2 M 2 + )]TJ/F15 11.9552 Tf 7.511 -6.528 Td [( 0 + 1 0 c 2 M 2 D 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.479 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 e 2 1 e 2 + c 2 M 2 D 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.48 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 1 e 2 1 e 2 + 1 2 k 2 ; 1 Similarto[88],thesubsequentanalysisdoesnotassumethattheinequality u
PAGE 54

where ;;; Q res 2 R aredenedas min )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 )]TJ/F15 11.9552 Tf 14.835 11.242 Td [( 2 ; 1 2 ;; ; Q 2 min f 0 ; 1 g 2 )]TJ/F15 11.9552 Tf 14.835 11.243 Td [( j p j 2 )]TJ/F15 11.9552 Tf 14.835 11.243 Td [( 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [(! 1 1 k 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(! 1 0 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(! 1 j p j ; Q res Q )]TJ/F30 11.9552 Tf 11.955 20.443 Td [( 0 +1+ 1 0 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.48 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 e 2 0 ; 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 k )]TJ/F25 11.9552 Tf 11.955 0 Td [( )]TJ/F25 11.9552 Tf 13.151 8.847 Td [( )]TJ/F15 11.9552 Tf 7.511 -6.529 Td [( 0 + j p j 2 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [(! 1 1 k 2 : Provided y 2 D ; 8 2 [ t )]TJ/F15 11.9552 Tf 14.701 3.022 Td [( D;t ] theexpressionin4reducesto V )]TJ/F25 11.9552 Tf 21.917 0 Td [( k y k 2 + '; where ;' 2 R aredenedas min 2 ; Q 2 min f 0 ; 1 g 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q j p j ; 1 2 k 2 + 1 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c D + ~ D 2 M 2 + )]TJ/F15 11.9552 Tf 7.512 -6.529 Td [( 0 +1+ 1 0 c 2 M 2 D 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.48 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 e 2 1 e 2 : Anupperboundcanbeobtainedfor4as V )]TJ/F15 11.9552 Tf 24.829 8.088 Td [( 2 V + ': Thesolutionofthedierentialequationin4canbeobtainedas V V exp )]TJ/F15 11.9552 Tf 12.21 8.088 Td [( 2 t + 2 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(exp )]TJ/F15 11.9552 Tf 12.209 8.088 Td [( 2 t : Using4and4,followingupperboundscanbeobtainedfor e i 1 ;i =0 ; 1 ;:::;n;r and e u as k e i 1 k s 2 V exp )]TJ/F15 11.9552 Tf 12.21 8.088 Td [( 2 t + 2 2 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(exp )]TJ/F15 11.9552 Tf 12.209 8.088 Td [( 2 t ; k r k s 2 V exp )]TJ/F15 11.9552 Tf 12.21 8.087 Td [( 2 t + 2 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F15 11.9552 Tf 12.21 8.087 Td [( 2 t ; 54

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k e u k s 2 V 1 exp )]TJ/F15 11.9552 Tf 12.209 8.087 Td [( 2 t + 2 2 1 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F15 11.9552 Tf 12.21 8.087 Td [( 2 t : Since e i 1 ;r;e u 2L 1 ; from4, u 2L 1 : Basedoninequalitiesdevelopedin4-4, k e i 1 k ; k r k C e 1 ; 8 i =0 ; 1 ;:::n k e u k C e u : where r 2 V exp )]TJ/F23 7.9701 Tf 12.099 4.707 Td [( 2 t + 2 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F23 7.9701 Tf 12.099 4.707 Td [( 2 t C e 1 2 R + ; 8 t and C e u C e 1 p 1 2 R + 4.5AcceleratedGradientDescentbasedEstimationofDelay Fromtherelationshipin3thetimeandspatialvariationofthecontrolinputcanbe related.Evaluating3at p =1 ,yields u t ;t = 1 u p ;t : Theleftsideof4equalstothetimederivativeofthedelayfreecontrolinput,whichafter using4,is u t ;t = k r .Todeterminetheexpressionfor u p ;t ,thelinearvariationof u p;t over x isusedasdenedin3,whichstatesthatataxedtimeinstant, u p;t varieslinearlyin p .UsingtheMeanValueTheorem, u p ;t canbeexpressedas u p ;t = u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(u ;t 1 = u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(u ;t : Using4,4,and4,andthetimederivativeof3,thefollowingexpression canbeobtained: k e n + e n )]TJ/F25 11.9552 Tf 11.955 0 Td [( e u )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(u ;t =0 : Substitutingfor35yields k e n + e n )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 1 u ;t + k 0 u ;t + k p e u )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(u ;t =0 : 55

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Aftersubstitutingfor 0 1 and p ,theexpressionin4canbewrittenas k e n + k e n c ^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c ^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F15 11.9552 Tf 11.955 0 Td [(_ e n + e n )]TJ/F25 11.9552 Tf 11.955 0 Td [(e u )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(k 2 + k + k 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c ^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t u ;t + c ^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t ke u + u ;t =0 : Giveninitialconditionsfor e u t and ^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t ,4canbesolvedfor ^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t Remark 4.1 Toestimate ^ D t e n t needstobemeasurable,i.e.,thehighestorderstate derivativeneedstobemeasurable. Remark4.1motivatesthenecessityofremovingtherequirementofhighestorderstate derivativemeasurementbydesigninganacceleratedgradientdescentbaseddelayestimation scheme.Fromtherelationshipin3thetimeandspatialvariationofthecontrolinputcan berelated.Evaluating3at p =1 andat p =0 ,yields u t ;t = 1 u p ;t ; u t ;t = 0 u p ;t : Afterdividing4by4,andusingtherelationsin3and3,followingrelation hasbeendeveloped u t ;t u t ;t = 1 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c ^ D u p ;t u p ;t : Aftertakingapartialwithrespectto p oftherelationin3,thefollowingrelationshipis developed @ @p u p;t = @ @p U )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( t + p )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t ; u p p;t = U )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( t + p )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(t + p )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t ; u p ;t = t U t )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.956 0 Td [(t ; u p ;t = 1 )]TJ/F25 11.9552 Tf 11.956 0 Td [(c ^ D t U t )]TJ/F25 11.9552 Tf 11.956 0 Td [(c ^ D t )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t : Beforeproceedingfurtherwiththeestimation,thefollowingupperboundon r t isderived usingtherelationshipin4andsubstitutingtheinequalitiesfrom4,4,4and 56

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4as r = ~ N + N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(e n )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 kr + 0 u ;t )]TJ/F25 11.9552 Tf 11.956 0 Td [(U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t + p e u ; k r k p k ++ C e n + 1 k C r + 0 kC r + kC r + j p j C e u ; k r k C r ; where C r p k ++ C e 1 + 1 k C e 1 ++ 0 kC e 1 + kC e 1 + j p j C e u Remark 4.2 j r j r ,where 2 2 [ t;t )]TJ/F15 11.9552 Tf 14.897 3.022 Td [( D ] ; 8 t and r > 0 .Also, r issmallenoughsuch that r < 0 UsingRemark4.2,andusingTaylor'sRemainderTheorem,thefollowingupperboundis developedfor u p ;t u p ;t u p ;t u p ;t = r t r t )]TJ/F25 11.9552 Tf 11.955 0 Td [(c ^ D t r t r t )]TJ/F25 11.9552 Tf 11.956 0 Td [(c ^ D t r ; 1 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c D r C r 1+ r C r 1+ ; where 2 [ t )]TJ/F25 11.9552 Tf 11.955 0 Td [(c ^ D t ;t ] .Substituting4,theexpressionin4canbewrittenas u t ;t 1+ 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c ^ D u t ;t : Afterintegrating4andusing4andthefactthat u ;t = kr t )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D t t 0 d u ;x + t t 0 1 1 )]TJ/F15 11.9552 Tf 16.002 6.176 Td [(_ ^ D d u ;x ; r t )]TJ/F25 11.9552 Tf 11.955 0 Td [(r t 0 + r )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D 1 )]TJ/F15 11.9552 Tf 16.002 6.177 Td [(_ ^ D t t 0 )]TJ/F26 7.9701 Tf 19.061 18.664 Td [(t t 0 r )]TJ/F15 11.9552 Tf 14.702 3.022 Td [(^ D ^ D 1 )]TJ/F15 11.9552 Tf 16.002 6.177 Td [(_ ^ D 2 d # : 2 Thesubsequentanalysisdoesnotassumethattheinequality j r j r holdsforalltime. Thesubsequentanalysisonlyexploitsthefactthatprovided 2 [ t;t )]TJ/F15 11.9552 Tf 14.702 3.022 Td [( D ] ,then j r j r 57

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BasedonthedevelopmentinSection3.2,3providesacontinuous r t thatiscomposed ofmeasurablesignalsi.e., e n t e n t and e u t ,whichcanbecomputedfrom3given aninitialconditionofthedelayanditsderivativei.e., ^ D t 0 ^ D t 0 and r t 0 .Considering theupperboundof r t developedin4,adiscreteestimateof r t ,denotedby ^ r is givenby 3 ^ r =^ r t 0 ++ r )]TJ/F15 11.9552 Tf 14.702 3.022 Td [(^ D 1 )]TJ/F15 11.9552 Tf 16.002 6.177 Td [(_ ^ D )]TJ/F30 11.9552 Tf 13.15 25.704 Td [( r t 0 )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D t 0 1 )]TJ/F15 11.9552 Tf 16.001 6.177 Td [(_ ^ D t 0 )]TJ/F42 11.9552 Tf 11.291 16.272 Td [( 0 r p )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D p ^ D p 1 )]TJ/F15 11.9552 Tf 16.002 6.177 Td [(_ ^ D p 2 dp # : Thesubsequentdesignedadaptiveestimatefor ^ D t ismotivatedbythedesiretominimizethe mismatchbetweentheauxiliaryerrorsignal r andthediscreteestimate ^ r .Toquantify thisobjective,anobjectivefunctionisdenedas E = t t 0 ^ r )]TJ/F25 11.9552 Tf 11.955 0 Td [(r 2 d; where t 2 R + isthetimeofinterestwhilediscretizingfordelayestimation.From4,the followinggradientisderived @E @ ^ D =2 r )]TJ/F15 11.9552 Tf 12.479 0 Td [(^ r + )]TJ/F15 11.9552 Tf 12.319 8.087 Td [(_ r )]TJ/F15 11.9552 Tf 14.702 3.022 Td [(^ D 1 )]TJ/F15 11.9552 Tf 16.002 6.177 Td [(_ ^ D )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 d d 0 r p )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D p ^ D p )]TJ/F15 11.9552 Tf 16.002 6.177 Td [(_ ^ D p 3 dp + 0 r p )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D p ^ D p )]TJ/F15 11.9552 Tf 16.002 6.177 Td [(_ ^ D p 2 dp )]TJ/F25 11.9552 Tf 16.36 8.088 Td [(d d r )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D )]TJ/F15 11.9552 Tf 16.002 6.177 Td [(_ ^ D 2 + d 2 d 2 0 r p )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D p )]TJ/F15 11.9552 Tf 16.002 6.177 Td [(_ ^ D p 2 dp : Notethat,intheaboveequation, @ @ ^ D and @ @ ^ D areinterchangedwith byusingthe Leibnizintegralrule,sincethepartialderivativewithrespectto @ @ ^ D and @ @ ^ D ofthe integrandiscontinuousandispointwisebounded.Nesterov'sAGDtechniqueisusedto update ^ D .Sincetheupdateof ^ D ateachiterationis @ ^ D @t = @E @ ^ D see[92],theexpressionin 3 Theinequalitysignin4hasbeenreplacedbyanequalitysignfordiscreteestimateof r t ,byconsideringtheworstcasescenario. 58

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4canbeusedtoupdate ^ D .BeforeusingtheNesterov'sAGDalgorithmtoupdate ^ D developmentisprovidedtoindicatethatgivenaboundontheerror, E ,therewillalwaysbea smooth ^ D ,i.e.,theNesterov'sAGDalgorithmconverges.Inotherwords,givenanarbitrary suchthat t t 0 ^ r )]TJ/F25 11.9552 Tf 11.956 0 Td [(r 2 d< @ ^ D @t 0 Theorem4.2. Forall > 0 ,ifatsome t = T E = t t 0 ^ r )]TJ/F25 11.9552 Tf 11.955 0 Td [(r 2 d< ,thereexistsa > 0 suchthat,at t = T t t 0 @ ^ D @t 2 d< forsome T ,i.e., E 0 implies @ ^ D @t 0 in L 2 sense. Proof. Observein4that, r )]TJ/F23 7.9701 Tf 8.531 2.015 Td [(^ D 1 )]TJ/F23 7.9701 Tf 9.472 4.118 Td [(_ ^ D 0 ,where C 1 C r 1 )]TJ/F23 7.9701 Tf 8.532 3.928 Td [( D C 2 C e 1 1 )]TJ/F23 7.9701 Tf 8.531 3.929 Td [( D 2 C 3 C r D 1 )]TJ/F23 7.9701 Tf 8.531 3.928 Td [( D C 4 C e 1 D 1 )]TJ/F23 7.9701 Tf 8.532 3.929 Td [( D 3 and C 5 C e 1 1 )]TJ/F23 7.9701 Tf 8.531 3.929 Td [( D 2 .At t = T t t 0 @ ^ D @t 2 + 4 C 1 + C 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 C 4 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t 0 ; t t 0 ^ r )]TJ/F25 11.9552 Tf 11.955 0 Td [(r 2 d< 4 + C 1 + C 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 C 4 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t 0 : Nowchoose, =4 + C 1 + C 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 C 4 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t 0 toconcludetheproof. Thesubsequentdevelopmentuses ^ D .Theanalyticexpressionofthegradientderivedin 4isused.Notethat,thisapproachinvolvesderivativeandRiemannintegration.Central dierenceisusedtoapproximatethederivative.SincetheRiemannintegralfortheintegrands in4isassumedtoexist,theintegralwithupperisapproximatedwithDarbouxsumwith uniformintervallengthof dt dt isselectedbasedonthediscretizationofthetimeaxisof thedata.Toapproximate ^ D ,tominimizetheerror E isminimized,thefollowingobjective functionissolved argmin ^ D E + 1 k ^ D k 2 + 2 k ^ D k 2 subjecttobasedonAssumption3.1, D ^ D D: 59

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Also, 1 2 aretheregularizerstoensurethatthe ^ D istwicedierentiable.Theregularizer 1 isadjustedtoenforcetheinequalityon ^ D ,as ^ D D basedonAssumption4.2.Similarly, 2 isadjustedtoenforcetheinequalityon ^ D ,as ^ D D basedonAssumption4.2.For T discretetimesteps,thentheequivalentKarush-Kuhn-TuckerKKTconditionssee[93]are givenbelow C ^ D @E @ ^ D + 1 @ k ^ D k @ ^ D + 2 @ k ^ D k @ ^ D + T X i =1 i @ @ ^ D ^ D i )]TJ/F15 11.9552 Tf 14.701 3.022 Td [( D + T X j =1 j @ @ ^ D )]TJ/F15 11.9552 Tf 12.044 3.022 Td [(^ D j + D =0 i ^ D i )]TJ/F15 11.9552 Tf 14.701 3.022 Td [( D =0 ; 8 i i 0 ; 8 i i )]TJ/F15 11.9552 Tf 12.044 3.022 Td [(^ D i + D =0 ; 8 i i 0 ; 8 i: Nesterov'sAGDisusedtosolvetheoptimizationbyformulatinganaugmentedLagrangianof theaboveconstrainedproblemtoget argmin ^ D F 1 2 C ^ D 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 1 C ^ D + 2 2 T X i =1 i ^ D i )]TJ/F25 11.9552 Tf 11.955 0 Td [(b 2 )]TJ/F26 7.9701 Tf 16.872 14.944 Td [(T X i =1 i 2 i ^ D i )]TJ/F25 11.9552 Tf 11.955 0 Td [(b + 3 2 T X j =1 j )]TJ/F15 11.9552 Tf 12.045 3.022 Td [(^ D j + a 2 )]TJ/F26 7.9701 Tf 17.536 14.944 Td [(T X j =1 j 3 j )]TJ/F15 11.9552 Tf 12.044 3.022 Td [(^ D j + a subjectto, i 0 ; 8 i i 0 : 8 i Here, 1 2 3 aretakenasuserspeciedparameters,andinAlgorithm1 j i arelearned. Now,Nesterov'sAGDwillusetosolvetheobjectivefunction F in4.Thealgorithmis givenbelow. 4.6SimulationResults Anumericalsimulationwasperformedonthe2-linkrobotdescribedin3.In 3, D t istheactualdelayinjectedinthesimulation,whichisoftheform D t = 60

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Algorithm1 Thealgorithmtolearn ^ D Require: r ,for x 2 [ t 0 ; t ] in dt incrementalsteps, 1 ; 2 1 ; 2 > 0 a
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designedtobeboundedbetweengivenboundsasmentionedinAssumption3.1.Giventhefact thatD ^ D D ,andsubsequentlyderivedboundsof 0 1 and p i.e., 0 ; 0 1 ; 1 j p j thissectiondevelopssucientgainconditionstoensure , Q res and > 0 .Usingthe denitionsof in4sucientlowerboundsfor canbeobtainedas > +1 2 : Fromthedenitionof in4, 2 alsoneedstosatisfythefollowinginequality 2 > 2 j p j min f 0 ; 1 g : Nowcombiningthedenitionsof in4and Q res in4,alowerboundover Q can beobtainedas Q > j p j + +2+ 1 1 k + 1 0 +2 1 j p j 2 min f 0 ; 1 g Q 1 ; Q > 0 +1+ 1 0 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.48 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 e 2 0 Q 2 ; Q > max f Q 1 ; Q 2 g : Tomake > 0 fromthedenitionof in4,thefollowinglowerboundof k canbe obtainedas k> 2 + )]TJ/F15 11.9552 Tf 7.511 -6.529 Td [( 0 + j p j 2 h 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F26 7.9701 Tf 13.151 4.813 Td [(! 1 1 2 i ; where 1 mustsatisfy 1 < 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 ; tomakethedenominatorpositivewhichgivesrisetothefollowinginequalityover both and > > 1 1 : Fromthedenitionof in4, isselectedsucientlysmall. 62

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4.8Conclusion Arobustcontrollerisdevelopedforaclassofuncertainnonlinearsystemswithanadditive disturbancesubjecttounknowntimevaryinginputdelaywithoutdelayrateconstraints.A lteredtrackingerrorsignalisdesignedtofacilitatethecontroldesignandstabilityanalysis. AnovelLyapunov-KrasovskiifunctionalisusedintheLyapunov-basedstabilityanalysisto provideUUBofthetrackingerror.Anobserverbasedmethodisdevelopedforunknown delayestimation,whichuseshighestorderstatederivativemeasurement,andcancause potentialdrawbackinpracticalapplications.Inordertoremovenecessityofhighestorderstate derivativemeasurement,Nesterov'sAGDbasedestimationisusedtoprovideatime-varying estimateofthedelay.Simulationresultsshowtheperformanceofthecontrolleralongwiththe estimationofthedelayanddelayratesmagnitude. Figure4-1.Variationoferrorsandcontrolforces. 63

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Figure4-2.Delayestimatevs.time. 64

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Figure4-3.Delayderivativesvs.time. 65

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Figure4-4.Objectivefunction E in4. 66

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CHAPTER5 BOUNDARYCONTROLOFSTOREINDUCEDOSCILLATIONSINAFLEXIBLEAIRCRAFT WINGWITHUNKNOWNTIME-VARYINGINPUTDELAY Thischapterpresentsarobustcontrollerforanelasticwingsubjectedtostoreinduced oscillationwithunknowntime-varyinginputdelayinboundarycontrolfeedback.2Delastic aircraftwingisdescribedbyuncertaincouplednonlinearPDEsviaregulationofthestate variablesasin[1].AnadaptiveboundarycontrolleraddedwithaPDEbasedrobust controllerisdesignedtoensurethedistributedstatesoftheexiblewingareregulated exponentiallytoaresidualballofgivenradius.UnlikeKrstic'sworkin[94],uncertainnonlinear PDEcannotbetransformedintoanexponentiallystabletargetsystemusingVolteraIntegral method.Asaresult,thecontrollerisdevelopedthroughaLyapunov-basedanalysis.In theLyapunovanalysis,wingenergytermsareusedalongwithanovelLyapunovKrasvoskii function,introducedinthePDEbaseddelayworkasin[44,45,95].Thedevelopedcontroller usesthelinearmappingapproachinspiredbypredictor-basedapproachessuchas[30]tomap thetimedependentcontrolinputtoamodiedcontrolinputthatdependsbothontimeand aspatialvariable.Similartopredictor-basedapproaches,themodiedinputcanbesegregated intodelayedanddelay-freecomponents.Thissegregationimpactsthestabilityanalysisina waythatallowsforarbitrarilylargedelayrates,unlikeexistingresultscf.,[3,8,31,32]. AnothercontributionofthisresultisthataneuralnetworkNNestimationscheme isintroducedtoestimatetheunknowndelaymagnitude.Sincetheuniversalfunctional approximationtheoremonlyholdsforcontinuousfunctionswhosedomainiscompact,a nonlinearmappingisintroducedtomapthenon-compacttimedomaintoacompactdomain. Simulationresultsdemonstratethecontrollereectivenesstodampouttheoscillationdespite thepresenceofunknowntime-varyinginputdelayinboundaryfeedback. 5.12DEuler-BernoulliBeam Flexibleaircraftwingcanbemodeledbya2DcantileverEuler-Bernoullibeammodelas showninFigure5-1oflength l 2 R ,chordlength c 2 R ,massperunitspanof 2 R momentofinertiaperunitlengthof I w 2 R ,andbendingandtorsionalstinessesof EI 2 R 67

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and GJ 2 R ,respectively,withastoreofmass m s 2 R andmomentofinertia J s 2 R attachedatthefreeendofthebeam. Figure5-1.Schematicofthewingsection,whereE.A.denotestheelasticaxisandC.G. denotesthecenterofgravity.. Similarto[1],Hamiltonianmechanicsisusedtodevelopbending andtwisting dynamicsofthecantileverbeamas L t = tt y;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c sin y;t 2 t y;t + x c cos y;t tt y;t + EI! yyyy y;t ; M t = )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c tt y;t + x c cos y;t tt y;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(GJ yy y;t ; where : R [0 ; 1 R and : R [0 ; 1 R denotethebendingandtwisting displacements,respectively, y 2 [0 ;l ] denotesspanwiselocationonthewing, x c 2 R represents thedistancefromthewingelasticaxistothewingcenterofgravity, L :[0 ; 1 R and M :[0 ; 1 R denoteaerodynamicliftandmomentperunitlength,respectively. Throughoutthispaper, : t and : y denotepartialderivativesofcorrespondingvariablewith respecttotimeandthespanwisepositionalongawing,respectively.Inadditiontothederived dynamics,boundarycontrolconditionsforthe2Dcantileverbeamaredevelopedas ;t = y ;t = yy l;t = ;t =0 ; L tip t )]TJ/F25 11.9552 Tf 11.956 0 Td [(D t = m s tt l;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(m s x s sin l;t 2 t l;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(EI! yyy l;t 68

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+ m s x s cos l;t tt l;t ; M tip t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t = )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(m s x 2 s + J s tt l;t + GJ y l;t + m s x s cos l;t tt l;t ; where L tip : [0, 1 R and M tip : [0, 1 R denotetheaerodynamicliftandmoment atthewingtip, D :[0 ; 1 R denotesunknowntime-varyinginputdelay,associatedwith thetimetakenforthecontrolforcestogetappliedonthesystem,and x s 2 R isthedistance fromthewingelasticaxistothestorecenterofgravity.Asin[44],alineartransformationis usedtotransformtime-varyingcontrolinputtoacontrolinputoftwoindependentvariables, i.e., p and t ,where t 2 [0 ; 1 and p 2 [0 ; 1] .Thistransformationproducescontrolinputas atwovariablefunctioni.e., p and t ,whereevaluatingat p =0 and p =1 givesdelayedand delay-freecontrolinput,respectively.Thelineartransformationisoftheform u p;t U )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( t + p )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t t t; 8 t 0 ; where u :[0 ; 1] [0 ; 1 R 2 U [ L tip M tip ] T 2 R 2 :[0 ; 1 R isaknown delaydependentinvertiblemonotonoustimefunction,denedas t t )]TJ/F15 11.9552 Tf 15.497 3.022 Td [(^ D t ,where ^ D t 2 R representsaknowntime-varyingsubsequentlydesigneddelayestimate, )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t exists atalltime.Transformationdenedin5,isusedtoexpressthedelayedcontrolinputas U t )]TJ/F15 11.9552 Tf 15.073 3.022 Td [(^ D t = u ;t ,andthedelay-freecontrolinputas U t = u ;t .Similarasin[44], thespatialandtimevariationof u p;t ,denotedby u p p;t and u t p;t respectively,canbe relatedas u t p;t = p;t u p p;t ; andtheauxiliaryfunction :[0 ; 1] ; 1 2 R isdenedas p;t 1+ p d )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t dt )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t ; 69

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where ;t 0 = 1 ^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t ; ;t 1 = 1 1 )]TJ/F15 11.9552 Tf 16.002 6.177 Td [(_ ^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t 0 ; @ @p p;t p = ^ D )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t 1 )]TJ/F15 11.9552 Tf 16.002 6.177 Td [(_ ^ D )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t 0 : Inrestofthepaper, and indicatemaximumandminimumvalueofthebracketedvariable, respectively.FirsttwoassumptionsarestatedbasedonRemark5.1in[96]. Property5.1. BasedonRemark5.1in[96],thepotentialenergyofthesystem, E P t 1 2 l 0 )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(EI! 2 yy + GJ 2 y dy isassumedtobebounded 8 t 2 [0 ; 1 ,and @ n @y n and @ m @y m are assumedtobebounded,uniformlyin y 8 t 2 [0 ; 1 for n =2 ; 3 ; 4 and m =1 ; 2 Property5.2. Similarly,thekineticenergyofthesystem E K t 1 2 l 0 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(! 2 t +2 x c cos t t dy + 1 2 m s 2 t l;t + 1 2 J s 2 t l;t + 1 2 l 0 \000 I w + x 2 c 2 t dy; isassumedtobebounded 8 t 2 [0 ; 1 ,and @ q @t q and @ q @t q areassumedtobebounded,uniformly in t 8 y 2 [0 ;l ] for q =1 ; 2 ; 3 Assumption5.1. Thesubsequentcontroldevelopmentisbasedontheassumptionthat l;: tyyy l;: t l;: yyy l;: y l;: ,and ty l;: aremeasurable. Assumption5.2. Theunknowntime-varyinginputdelay D t 2 R isboundedbyknown positiveconstants, D andD respectively,asD D t D Remark 5.1 Inpractice,timevariationofbothwingtipbendingandtwistingdeection canbemeasuredbytransducers.Spatialvariationofbendingdeectioncanbemeasured bystraingaugesasmentionedin[73]orshearsensorsasdiscussedin[97],basedon theorderofdierentiation.Timevariationsofthesesensormeasurementscanbeobtained throughnumericalmethods.Suchmeasurementsandnumericalmethodscanintroducenoise, 70

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andmotivationexistsforadditionalresearchtoeliminatethesehigher-ordermeasurements. AdvancesinberopticsensingbothLongPeriodFiberGratingsandFiberBraggGrating canalsobeusedtomeasurethedeformationofthewing.Forexample,beropticstraindata fromagroundloadtestofafull-scaleaircraftwingcanbeusedtomeasurethedeectionof thewingandcorrugatedlong-periodbergratingcanbeusedtomeasurestrain,bendingand torsionofthewingasin. 5.2InstabilityinPresenceofInputTimeDelay Anadaptivecontrollerisdevelopedin[1],tostabilizethestoreinducedoscillationinan aircraftwing.Inordertomotivatetheinputdelayproblem,constantinputdelaysofmagnitude between 0 )]TJ/F15 11.9552 Tf 12.109 0 Td [(900 ms,withanincrementof 1 msareinjectedinthesystemdynamicsdescribed in5-5,toshowimpactofinjecteddelayonthestabilityofthesystem.Simulation resultsvalidatetheclaim,thatevenaslightpresenceofinputdelaycandestabilizeastable system.Figure5-2,showsthedesignedcontrollerin[1],isnotsucienttohandlethepresence ofinputdelayinthesystem,incaseofbendingdeectionofthewing.Similarresultisnoticed fortwistingdeection,asshowninFigure5-3.Figure5-2and5-3motivatethenecessityofa controllerthatcancompensatethepresenceofinputdelayinthesystem.Although,forthis simulationdemonstration,delaymagnitudewasknown,itisneededtogeneralizethesystem byincorporatingdelayofunknownmagnitude.Motivatedbythisnecessity,inthefollowing section,anadaptivecontrollerisdevelopedforthesystemdescribedin5-5,subjected toanunknowntime-varyingdelay. 5.3ControlDevelopment Controlobjectiveistoensurethatinpresenceoftimedelay,bothbendingandtwisting deectiongotozerothroughoutthewholewingspan,astimeprogresses,i.e., y;t 0 and y;t 0 8 y 2 [0 ;l ] as t !1 .Tofacilitatesubsequentstabilityanalysis,anauxiliaryerror signal, e :[0 ; 1 R 2 andmassmatrix, M :[0 ; 1 R 2 2 denedas e t 2 6 4 t l;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(! yyy l;t t l;t + y l;t 3 7 5 ; 71

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M 2 6 4 m s m s x s cos l;t m s x s cos l;t m s x 2 s + J s 3 7 5 : Inordertoincludeboundarycontrolequations5-5intheopen-loopdynamics,another auxiliaryerrorsignal, r :[0 ; 1 R 2 denedas r t e t + e u t ; where 2 R isknown,diagonal,positivedenite,constantgain.In5, e u :[0 ; 1 R 2 isanauxiliaryerrorterm,introducedtoobtainadelay-freecontrolexpressionfortheinputin theclosedlooperrorsystemandcanbeexpressedas e u t 1 0 u p;t dp: Beforeproceedingfurtherwiththeerrorsignaldevelopment,usingLeibnizruleanddenitions of 0 1 and p from5-5,timederivativeof e u t iscalculatedas e u t = 1 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 u ;t )]TJ/F25 11.9552 Tf 11.956 0 Td [( p e u t : Theopen-loopdynamicsfortheerrorsignal r t ,canbeexpressedas r t = 2 6 4 tt l;t tt l;t 3 7 5 + 2 6 4 )]TJ/F25 11.9552 Tf 9.298 0 Td [(! tyyy l;t ty l;t 3 7 5 + e u t : Beforeproceedingfurther,5and5arerearrangedtofacilitateopen-looperrorsignal development,andcanbeexpressedas M inv t U t )]TJ/F25 11.9552 Tf 11.956 0 Td [(D t = I 2 2 2 6 4 tt l;t tt l;t 3 7 5 + M inv t 2 6 4 )]TJ/F25 11.9552 Tf 9.298 0 Td [(m s x s sin l;t 2 t l;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(EI! yyy l;t GJ y l;t 3 7 5 ; 72

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where I 2 2 is 2 2 identitymatrixand M inv :[0 ; 1 R 2 2 canbeexpressedas M inv t 2 6 4 m s x 2 s + J s m 2 s x 2 s sin 2 l;t + m s J s x s cos l;t m s x 2 s sin 2 l;t + J s x s cos l;t m s x 2 s sin 2 l;t + J s 1 m s x 2 s sin 2 l;t + J s 3 7 5 = M M ; where M :[0 ; 1 R 2 2 isdenedas M 2 6 4 m s x 2 s + J s m s x s cos l;t m s x s cos l;t m s 3 7 5 and M :[0 ; 1 R isdenedas M t m 2 s x 2 s sin 2 l;t + m s J s Proposition5.1. As M inv 2 R 2 2 ,itisnecessaryandsucienttoprovethatboth det M inv and trace M inv arepositive,inordertoshowthat M inv denedin5ispositive denite 8 t 2 [0 ; 1 det M inv = 1 det M ,fromthedenitionof M in5, det M = m 2 s x 2 s sin 2 l;t + m s J s > 0 ,whichgives det M inv = 1 m 2 s x 2 s sin 2 l;t + m s J s > 0 .Also, trace M inv = m s x 2 s + J s m 2 s x 2 s sin 2 l;t + m s J s + 1 m s x 2 s sin 2 l;t + J s = m s x 2 s + J s + m s m 2 s x 2 s sin 2 l;t + m s J s > 0 .Thismeans M inv ispositivedenite 8 t 2 [0 ; 1 Proposition5.2. Asetofwingandstoreparameterssatisfyingtheseconditionsarelisted in[98].Basedonthesampledatavaluesin[98], j m s x 2 s jj J s j ,so j m s x 2 s sin 2 l;t J s j < 1 M t )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 canbeexpandedasabinomialserieswhile j m s x 2 s sin 2 l;t J s j < 1 M t )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 = m 2 s x 2 s sin 2 l;t + m s J s )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 = 1 m s J s 1+ m s x 2 s sin 2 l;t J s )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 1 m s J s 1 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(m s x 2 s sin 2 l;t J s + 1 2 m s x 2 s sin 2 l;t J s 2 # = 1 m s J s )]TJ/F25 11.9552 Tf 13.15 8.088 Td [(x 2 s sin 2 l;t J 2 s + 1 2 m s x 4 s sin 4 l;t J 3 s : Aftersubstituting,5and5in5,yields r t = M M U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t + [ 1 u ;t )]TJ/F25 11.9552 Tf 11.956 0 Td [( 0 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( p e u t ]+ Y t ; 73

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where Y :[0 ; 1 R 2 27 isaregressionmatrixofknowntime-varyingquantitiesand 2 R 27 isavectorofunknownparameters,denedas Y t 2 6 4 Y 1 1 :::Y 1 6 :::Y 1 12 00 ::: 0 Y 2 1 :::Y 2 6 0 ::: 0 Y 2 22 :::Y 2 27 3 7 5 ; 1 2 3 ::: 27 T ; where 1 = J s m s x 2 s 2 = m 2 s x 4 s J 2 s 3 = m 3 s x 6 s 2 J 3 s 4 = J 2 s 5 = m s x 2 s J s 6 = m 2 s x 4 s 2 J 2 s 7 = J s m s x 3 s 8 = m 2 s x 5 s J 2 s 9 = m 3 s x 7 s 2 J 3 s 10 = J s x 2 s EI 11 = x 4 s m s EI J 2 s 12 = m 2 s x 6 s EI 2 J 3 s 13 = x s J 2 s 14 = m s x 3 s J s 15 = m 2 s x 5 s 2 J 2 s 16 = J 2 s EI m s 17 = x 2 s EI J s 18 = m s x 4 s EI 2 J 2 s 19 = J s x s GJ 20 = x 3 s GJm s J 2 s 21 = m 2 s x 5 s GJ 2 J 3 s 22 = J s GJ 23 = x 2 s m s GJ J 2 s 24 = m 2 s x 4 s GJ 2 J 3 s 25 = J s x s EI 26 = x 3 s m s EI J 2 s 27 = m 2 s x 5 s EI 2 J 3 s and Y 1 1 = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin l;t tyyy l;t Y 1 2 =sin 4 l;t tyyy l;t Y 1 3 = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 6 l;t tyyy l;t Y 1 4 = )]TJ/F25 11.9552 Tf 9.299 0 Td [(! tyyy l;t Y 1 5 =sin 2 l;t tyyy l;t Y 1 6 = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 4 l;t tyyy l;t Y 1 7 =sin l;t 2 t l;t Y 1 8 = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 3 l;t 2 t l;t Y 1 9 =sin 5 l;t 2 t l;t Y 1 10 = yyy l;t Y 1 11 = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 2 l;t yyy l;t Y 1 12 =sin 4 l;t yyy l;t Y 1 13 =sin l;t 2 t l;t Y 1 14 = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 3 l;t 2 t l;t Y 1 15 =sin 5 l;t 2 t l;t Y 1 16 = yyy l;t Y 1 17 = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 2 l;t yyy l;t Y 1 18 =sin 4 l;t yyy l;t Y 1 19 = )]TJ/F25 11.9552 Tf 9.299 0 Td [( y l;t cos l;t Y 1 20 =sin 2 l;t y l;t cos l;t Y 1 21 = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 4 l;t y l;t cos l;t Y 2 1 =sin l;t cos l;t 2 t l;t +sin 2 l;t ty l;t Y 2 2 = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 4 l;t ty l;t )]TJ/F15 11.9552 Tf -445.887 -23.908 Td [(sin 3 l;t cos l;t 2 t l;t Y 2 3 =sin 6 l;t ty l;t +sin 5 l;t cos l;t 2 t l;t Y 2 4 = ty l;t Y 2 5 = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 2 l;t ty l;t Y 2 6 =sin 4 l;t ty l;t Y 2 22 = )]TJ/F25 11.9552 Tf 9.299 0 Td [( y l;t Y 2 23 =sin 2 l;t y l;t Y 2 24 = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 4 l;t y l;t Y 2 25 = yyy l;t cos l;t Y 2 26 = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 2 l;t yyy l;t cos l;t Y 2 27 =sin 4 l;t yyy l;t cos l;t Remark 5.2 Projectionalgorithmandadaptationlawareusedtoestimatetheunknown parametersin .Duetouseofprojectionalgorithmintheestimationof M canbeupper andlowerboundedby M and M respectively.Similarlyusingthesameargument, M t can beupperandlowerboundedby M up and M low respectively. 74

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Basedonthesubsequentstabilityanalysis,delay-freecontrolinputisdesignedas U t = u ;t )]TJ/F25 11.9552 Tf 13.804 8.088 Td [(K 1 r t )]TJ/F15 11.9552 Tf 18.944 8.088 Td [(1 1 Y t ^ t ; where K 2 R isapositiveconstantcontrolgain, ^ :[0 ; 1 R 27 isatime-varying estimateofunknownparametersasin .Gradientupdatelawisusedtoupdatetheestimate ofunknownparameters,denedas ^ t =)]TJ/F25 11.9552 Tf 27.613 0 Td [(Y t T r t ; where )]TJ/F22 11.9552 Tf 13.463 0 Td [(2 R 27 27 isapositivedenitecontrolgain.Theclosed-looperrordynamicsis developedbysubstitutingdelay-freecontrolinput5intoopen-looperrordynamicsin 5,andcanbeexpressedas r t = M M U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t )]TJ/F25 11.9552 Tf 11.956 0 Td [(Kr )]TJ/F25 11.9552 Tf 11.956 0 Td [( 0 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( p e u t + Y t ~ ; where ~ :[0 ; 1 ,denedas ~ t = )]TJ/F15 11.9552 Tf 13.069 3.154 Td [(^ t ,isdierencebetweenactualandestimationof unknownparametervalues. 5.4NeuralNetworkBasedDelayEstimation AneuralnetworkNNbasedfunctionapproximatorisusedtoestimatetheunknown delaymagnitude.Theuniversalfunctionapproximationtheoremonlyholdsoveracompact domain.Therefore,toapproximatetheunknowndelayfunction,anonlinearmappingisdened tomapthenon-compactdomaintoacompactspatialdomain.Let f L : t bedenedas f L t 1+ t ;t 2 [0 ; 1 ; 2 [0 ; 1] ; where 2 R + isauserdenedsaturationcoecient.Using5, D t canbemappedinto thedomain as D t = D )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(f )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 L D f L : 75

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Theuniversalfunctionalapproximationtheoremcanbeusedtorepresent D f L byathreelayerNNas D f L W T )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(V T + ; where W 2 R L +1 1 and V 2 R 3 L aretheboundedconstantidealweightsfortherst-tosecondandsecond-to-thirdlayers,respectively, L isthenumberofneuronsinthehiddenlayer, 2 R L +1 isanactivationfunction, isthefunctionalreconstructionerror,and =[1 ] T denotestheinputtotheNN.Basedon5,theNNestimationfor ^ D t isgivenby ^ D t = ^ W T ^ V T ; where ^ W and ^ V areestimatesoftheidealweights.Using5and5,themismatch between D t and ^ D t canbeobtainedusingaTaylor'sseriesapproximation,whichafter somealgebraicmanipulation,canbeexpressedas D t )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D t = W T )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(V T )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W T ^ V T + ; = ~ W T ^ V T + ^ W T 0 ^ V T ~ V T + W T O ~ V T 2 + + ~ W 0 ^ V T ~ V T ; where ~ W = W )]TJ/F15 11.9552 Tf 15.892 3.022 Td [(^ W 2 R L +1 1 and ~ V = V )]TJ/F15 11.9552 Tf 14.265 3.022 Td [(^ V 2 R 2 L ,aretheestimatemismatchfor theidealweightmatrices,and O representshigherorderterms.Inthesubsequentdevelopment acontinuouslydierentialprojectionalgorithmasshowninSection3.7,isusedtodesign theadaptiveupdatelawsfor ^ W and ^ V .Asaresult,theelementsof ^ W and ^ V canallbe upperandlowerboundedbyknownpositiveconstants.Hence, ~ W T 0 ^ V T ~ V T and W T O ~ V T 2 canalsobeboundedbyknownpositiveconstants,andtherefore, D t )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D t ~ W T ^ V T + ^ W T 0 ^ V T ~ V T + ; where 2 R isapositiveboundingconstant. 76

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5.5Lyapunov-basedStabilityAnalysis Tofacilitatethesubsequentstabilityanalysis,energyterms E T :[0 ; 1 R E C : [0 ; 1 R E B :[0 ; 1 R aredenedas E T t 1 2 l 0 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(! 2 t +2 x c cos t t + EI! 2 yy dy + 1 2 l 0 \000 I w + x 2 c 2 t + GJ 2 y dy; E C t l 0 y y t + x c cos t dy + l 0 y y )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c t dy + l 0 y yx c cos t dy; E B t l 0 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(! 2 t + 2 yy + 2 t + 2 y dy; where 2 R isapositivecontrolgain.Young'sInequalityisusedtoupperandlowerbound E T t E C t andcanbeexpressedas E T t 1 2 max n + j x c j )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(I w + x 2 c + j x c j ;EI;GJ o E B t ; E T t 1 2 min n )]TJ/F25 11.9552 Tf 11.955 0 Td [( j x c j ; )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(I w + x 2 c )]TJ/F25 11.9552 Tf 11.955 0 Td [( j x c j ;EI;GJ o E B t ; E C t l max n + j x c j ;l 2 + j x c j ; )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c + j x c j o E b t ; E C t l max n + j x c j ;l 2 + j x c j ; )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c + j x c j o E b t : Remark 5.3 Providedthat j x c j < 1 and I w >x 2 c )]TJ/F25 11.9552 Tf 12.593 0 Td [( j x c j E T willbenon-negative.The conditions j x c j < 1 and I w >x 2 c )]TJ/F25 11.9552 Tf 12.059 0 Td [( j x c j areengineeringdesignconsiderationsthatensurethe storeismountedsucientlyclosetothewingcenterofmass[98]. From5and5,if isselectedas < 1 2 l 2 ; where 1 min )]TJ/F25 11.9552 Tf 11.955 0 Td [( j x c j ; )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(I w + x 2 c )]TJ/F25 11.9552 Tf 11.955 0 Td [( j x c j ;EI;GJ ; 2 max + j x c j ;l 2 + j x c j ; )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c + j x c j then 1 E B t E T t + E C t 2 E B t 77

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wherethepositiveconstants 1 and 2 aredenedas 1 1 2 1 )]TJ/F25 11.9552 Tf 11.956 0 Td [(l 2 ; 2 1 2 2 + l 2 : Beforeproceedingfurtherwiththeanalysis,anLKfunctional Q :[0 ; 1 R denedas Q t Q 1 0 e 2 p u T p;t u p;t dp; where 2 ; Q 2 R areknown,positiveconstants. Theorem5.1. Giventheopen-looperrorsystemin5,thecontrollerin5along withtheadaptivelawin5,ensuresthesystemstates !; areUUB 8 y 2 [0 ;l ] as t !1 providedthefollowingsucientgainconditionsaresatised: l 0 ; 3 EI 2 )]TJ/F15 11.9552 Tf 14.532 11.11 Td [( L w l 3 2 > 0 ; )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c )]TJ/F15 11.9552 Tf 16.291 3.022 Td [( M w > 0 ; GJ )]TJ/F25 11.9552 Tf 11.955 0 Td [( M w l 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [( M w l )]TJ/F25 11.9552 Tf 11.956 0 Td [( L w l 3 )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F15 11.9552 Tf 9.815 -6.662 Td [( M w + L w l 2 > 0 ; EIl + EI )]TJ/F25 11.9552 Tf 11.955 0 Td [( )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c l> 0 ; GJ )]TJ/F25 11.9552 Tf 11.955 0 Td [(l )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c l> 0 ; andallthegainconditionsaresatisedinSection5.7. Remark 5.4 Thesucientgainconditionsin5-5canbesatisedbyacombination ofgainselectionsandengineeringdesignconsiderations.Selectionofthewingaerodynamic propertiescanbedonetosatisfyaircraftperformancecriteriae.g.,minimumtakeodistance, maximumrange,etc..Thestructuralpropertiesofthewingcanthenbeselectedtosatisfy thesucientconditions.Increasingthestinessandmassofthewingormountingthestore closertothewingcenterofmasswillsatisfythesucientconditions.Asetofwingandstore parameterssatisfyingtheseconditionsarelistedin[98]. 78

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Proof. Let V L :[0 ; 1 R + ,continuouslydierentiablefunctiondenedas V L t E T t + E C t + 1 2 r t T r t + 1 2 e u t T e u t + Q t + ~ t T )]TJ/F28 7.9701 Tf 7.314 4.338 Td [()]TJ/F23 7.9701 Tf 6.587 0 Td [(1 ~ t 2 + 1 2 tr ~ W t T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 1 ~ W t + 1 2 tr ~ V t T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 ~ V t : Fromthedenitionoftheerrorsignalsand Q ,followinginequalityisdevelopedandcanbe expressedas L k y k 2 + c L 1 2 tr ~ W t T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 1 ~ W t + 1 2 tr ~ V t T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 ~ V t + 1 2 r t T r t + 1 2 e u t T e u t + Q t U k y k 2 + c U ; where y :[0 ; 1 R 7 ,denedas y t [ e t T r t T e u t T Q ] T and c L ;c U 2 R + areknownboundingconstants.Usinginequalitiesin5,andtherelationthat V L canbe boundedas V L t 1 E B t + min [)]TJ/F28 7.9701 Tf 10.566 4.338 Td [()]TJ/F23 7.9701 Tf 6.586 0 Td [(1 ] 2 ~ t 2 + L k y k 2 + c L ; V L t 2 E B t + max [)]TJ/F28 7.9701 Tf 10.565 4.339 Td [()]TJ/F23 7.9701 Tf 6.587 0 Td [(1 ] 2 ~ t 2 + U k y k 2 + c U ; where min and max denotetheminimumandmaximumeigenvaluesofrespectiveitem. Takingthetimederivativeof5andusing5,5andtheupdatelawin5, yields V L = r t T M M U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t + Y t ~ + E T t + E C t )]TJ/F15 11.9552 Tf 12.894 3.155 Td [(~ t T )]TJ/F28 7.9701 Tf 7.314 4.936 Td [()]TJ/F23 7.9701 Tf 6.586 0 Td [(1 ^ t + r t T )]TJ/F25 11.9552 Tf 9.299 0 Td [(Kr t )]TJ/F25 11.9552 Tf 11.956 0 Td [( 0 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( p e u t + Q 1 e 2 u T ;t u ;t + 1 e u t T 1 u ;t )]TJ/F25 11.9552 Tf 11.956 0 Td [( 0 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( p e u t )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q 2 1 0 e 2 p u T p;t u p;t dp + h M M low )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 i + 1 p 2 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.48 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 12.496 3.155 Td [( 1 e 2 + M 0 u T ;t u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q 0 u T ;t u ;t + Q p 1 0 e 2 p u T p;t u p;t dp + tr ~ W t T )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 1 ~ W t + tr ~ V t T )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 ~ V t : 79

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Byusing5,Young'sinequality,andthefactsthat ~ W = )]TJ/F15 11.9552 Tf 14.012 6.176 Td [(_ ^ W and ~ V = )]TJ/F15 11.9552 Tf 12.386 6.176 Td [(_ ^ V ,anupper boundon V L canbeobtainedas V L r t T M M U t )]TJ/F25 11.9552 Tf 11.956 0 Td [(D t + 1 p 2 k r t k 2 + E T t + E C t )]TJ/F25 11.9552 Tf 9.298 0 Td [(K k r t k 2 + r t T )]TJ/F25 11.9552 Tf 9.298 0 Td [( 0 u ;t + p 2 1 k e u t k 2 + Q 1 e 2 k u ;t k 2 + 1 e u t T 1 u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 u ;t + 1 j p jk e u t k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q 2 1 0 e 2 p u T p;t u p;t dp )]TJ/F30 11.9552 Tf 11.291 24.03 Td [(0 @ Q )]TJ/F30 11.9552 Tf 11.955 24.03 Td [(2 4 h M M low )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 i + 1 p 2 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.48 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 12.496 3.155 Td [( 1 e 2 + M 3 5 1 A 0 k u ;t k 2 + Q p 1 0 e 2 p u T p;t u p;t dp )]TJ/F25 11.9552 Tf 11.956 0 Td [(tr ~ W t T )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 1 ^ W t )]TJ/F25 11.9552 Tf 11.955 0 Td [(tr ~ V t T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 ^ V t : In5, E T isdeterminedbydierentiating5withrespecttotimetoobtain E T t = l 0 t )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(! tt + x c cos tt )]TJ/F25 11.9552 Tf 11.956 0 Td [(x c sin 2 t dy + l 0 EI! yy tyy + GJ y ty dy + l 0 \000 I w + x 2 c tt + x c cos tt t dy: Substituting5and5intothersttwointegralsof5yields E T t = l 0 L! t + M t dy )]TJ/F42 11.9552 Tf 11.955 16.273 Td [( l 0 EI! t yyyy dy + l 0 EI! yy tyy dy + l 0 GJ t yy dy + l 0 GJ y ty dy: Integratingthethirdandfthintegralsin55bypartsandapplyingtheboundaryconditionsofthePDEsystemgives l 0 EI! yy tyy dy = )]TJ/F25 11.9552 Tf 9.298 0 Td [(EI! yyy l;t t l;t + l 0 EI! t yyyy dy; l 0 GJ y ty dy = GJ y l;t t l;t )]TJ/F42 11.9552 Tf 11.955 16.272 Td [( l 0 GJ t yy dy: 80

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Usingtheexpressionsin5and5andusingtheerrorsignaldenitionin5, 5canberewrittenas E T t = l 0 L! t + M t dy + e t T 2 6 4 EI 2 0 0 k GJ 2 3 7 5 e t )]TJ/F25 11.9552 Tf 13.151 8.087 Td [(EI 2 )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(! 2 t l;t + 2 yyy l;t )]TJ0 0 1 rg 0 0 1 RG0 g 0 G/F26 7.9701 Tf 10.494 6.274 Td [(k GJ 2 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( 2 y l;t + 2 t l;t : AfterintegratingandusingYoung'sInequalityandLemmaA.12from[96], E C canbeupper boundedas E C t )]TJ/F15 11.9552 Tf 23.91 0 Td [( )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c 2 l 0 2 t dy + EIl 2 e 2 1 + 1 2 l )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c 2 t l;t )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(EIl 2 2 y l;t )]TJ/F30 11.9552 Tf 11.291 16.856 Td [( 3 EI 2 )]TJ/F25 11.9552 Tf 13.15 8.087 Td [(Ll 3 2 l 0 2 yy dy )]TJ/F30 11.9552 Tf 11.955 9.683 Td [()]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(I w + x 2 c )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c 2 l 0 2 t dy + 1 2 GJl 2 y l;t )]TJ/F30 11.9552 Tf 11.291 9.683 Td [()]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(GJ )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ml 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ml )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ll 3 2 l 0 2 y dy )]TJ/F25 11.9552 Tf 13.151 8.087 Td [(EIl 2 2 yyy l;t + 1 2 l! 2 t l;t + x c l t l;t t l;t ; where e 1 denotestherstelementofthevector e ,i.e., e 1 t t l;t )]TJ/F25 11.9552 Tf 12.248 0 Td [(! yyy l;t .Before proceedingfurther,notethat p;t = 1+ p d )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 t dt )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F25 11.9552 Tf 11.955 0 Td [(t ;p 2 [0 ; 1] = 0 + p 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(p 0 ; = 0 + 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 p; min f 0 ; 1 g : UsingtheCauchy-Schwarzinequality, k e u k 2 1 0 k u T p;t u p;t k dp: 1 0 1 dp; k e u k 2 1 0 k u 2 p;t k dp: 81

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UsingMeanValueTheorem,followinginequalitieshavebeendeveloped U t = U c ^ D t + U t )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D t ; u ;t M ^ D + u ;t : u T ;t u ;t u T ;t u ;t +2 M ^ Du ;t + 2 M ^ D 2 ; k u ;t k 2 + M k u ;t k 2 + )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( M + 2 M ^ D 2 : UsingMeanValueTheoremandtheexpressionin5,thefollowingequalitycanbe developedas M M low j r T U t )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D t )]TJ/F25 11.9552 Tf 11.955 0 Td [(U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t j M M M low j r T j M M M low j r T ~ W T ^ V T j + M M M low j r T ^ W T 0 ^ V T ~ V T j + M M M low j r T D t )]TJ/F15 11.9552 Tf 14.702 3.022 Td [(^ D t j : Theexpressionsin5,5,5,5andRemark5.2,canbeusedtoupper bound5as V L M M low r t T U t )]TJ/F25 11.9552 Tf 11.955 0 Td [(D t )]TJ/F25 11.9552 Tf 11.955 0 Td [(U t )]TJ/F15 11.9552 Tf 14.701 3.022 Td [(^ D t + p 2 1 k e u t k 2 + 1 p 2 k r t k 2 )]TJ/F25 11.9552 Tf 9.299 0 Td [(K k r t k 2 + M M low )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 r t T u ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(! 1 j p jk e u t k 2 + 1 e u t T h 1 M ^ D + p u ;t i + l 0 L! t + M t dy + e t T 2 6 4 EI 2 0 0 k GJ 2 3 7 5 e t )]TJ/F25 11.9552 Tf 10.494 8.088 Td [(EI 2 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(! 2 t l;t + 2 yyy l;t )]TJ0 0 1 rg 0 0 1 RG0 g 0 G/F26 7.9701 Tf 10.494 6.274 Td [(k GJ 2 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( 2 y l;t + 2 t l;t )]TJ/F15 11.9552 Tf 11.291 0 Td [( )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c 2 l 0 2 t dy + EIl 2 e 2 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q 0 k u ;t k 2 )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( 3 EI 2 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [(Ll 3 2 l 0 2 yy dy )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c 2 l 0 2 t dy + 1 2 l )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c 2 t l;t )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(EIl 2 2 y l;t )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(GJ )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ml 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ml )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ll 3 2 l 0 2 y dy 82

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)]TJ/F25 11.9552 Tf 10.494 8.087 Td [(EIl 2 2 yyy l;t + 1 2 GJl 2 y l;t + 1 2 l! 2 t l;t + x c l t l;t t l;t + Q 1 e 2 h + M k u ;t k 2 + )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( M + 2 M ^ D 2 i + h M M low )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 i + 1 p 2 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.479 -9.683 Td [( 0 )]TJ/F15 11.9552 Tf 12.495 3.155 Td [( 1 e 2 + M 0 k u ;t k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q 2 1 0 e 2 p u T p;t u p;t dp + Q p 1 0 e 2 p u T p;t u p;t dp )]TJ/F25 11.9552 Tf 11.956 0 Td [(tr ~ W t T )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 1 ^ W t )]TJ/F25 11.9552 Tf 11.955 0 Td [(tr ~ V t T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 ^ V t : Young'sinequalityisusedtoupperbound5as V L M M low r T U t )]TJ/F25 11.9552 Tf 11.956 0 Td [(D t )]TJ/F25 11.9552 Tf 11.955 0 Td [(U t )]TJ/F15 11.9552 Tf 14.702 3.022 Td [(^ D t + 2 1 p 2 k e u t k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q 0 k u ;t k 2 )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( K 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( M M low )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 2 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 1 p 2 k r t k 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( K )]TJ/F25 11.9552 Tf 11.955 0 Td [(! 1 j p j)]TJ/F25 11.9552 Tf 19.128 8.088 Td [( p 2 1 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(! 1 1 M 2 k e u t k 2 + l 0 L! t + M t dy )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c 2 l 0 2 t dy )]TJ/F25 11.9552 Tf 10.494 8.088 Td [(EI 2 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(! 2 t l;t + 2 yyy l;t )]TJ0 0 1 rg 0 0 1 RG0 g 0 G/F26 7.9701 Tf 10.494 6.274 Td [(k GJ 2 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( 2 y l;t + 2 t l;t )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( 3 EI 2 )]TJ/F25 11.9552 Tf 13.15 8.087 Td [(Ll 3 2 l 0 2 yy dy )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c 2 l 0 2 t dy + 1 2 l )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(I w + x 2 c 2 t l;t )]TJ/F25 11.9552 Tf 13.15 8.088 Td [(EIl 2 2 y l;t + 1 2 l! 2 t l;t + x c l t l;t t l;t )]TJ/F30 11.9552 Tf 11.291 9.683 Td [()]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(GJ )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ml 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ml )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ll 3 2 l 0 2 y dy )]TJ/F25 11.9552 Tf 13.151 8.087 Td [(EIl 2 2 yyy l;t + 1 2 GJl 2 y l;t )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( K )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 2 max f EI + EIl;k GJ g k e t k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q 2 1 0 e 2 p u T p;t u p;t dp + Q 1 e 2 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( M + 2 M ^ D 2 + 1 1 M ^ D 2 2 + Q p 1 0 e 2 p u T p;t u p;t dp + h M M low )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 i + 1 p 2 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.479 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 12.495 3.155 Td [( 1 e 2 + M 0 k u ;t k 2 )]TJ/F25 11.9552 Tf 9.298 0 Td [(tr ~ W t T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 1 ^ W t )]TJ/F25 11.9552 Tf 11.956 0 Td [(tr ~ V t T )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 ^ V t : Usingthefactthat a T b = trace ba T ^ W t and ^ V t aredesignedtocancelcrosstermsas ^ W = proj 1 M M M low ^ V T r T ; ^ V = 2 M M M low r T ^ W T 0 ^ V T : 83

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Thefunction proj :;: in5denotesaprojectionoperator,thatisLipschitzcontinuous asdiscussedinSection3.7,whichensuresD + l ^ D t D + u ,where l u 2 R are subsequentlydenedpositiveconstantsseeSection5.7.Usingtheinequalitiesin5, 5canbeupperboundedas V L )]TJ/F25 11.9552 Tf 21.918 0 Td [( r k r t k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( e u k e u t k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( e k e t k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q res 0 k u ;t k 2 + l 0 L! t + M t dy )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c 2 l 0 2 t dy )]TJ/F25 11.9552 Tf 10.494 8.088 Td [(EI 2 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(! 2 t l;t + 2 yyy l;t )]TJ0 0 1 rg 0 0 1 RG0 g 0 G/F26 7.9701 Tf 10.494 6.274 Td [(k GJ 2 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( 2 y l;t + 2 t l;t )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( 3 EI 2 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [(Ll 3 2 l 0 2 yy dy )]TJ/F25 11.9552 Tf 11.955 0 Td [( Q 1 Q + res )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c 2 l 0 2 t dy + 1 2 l )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(I w + x 2 c 2 t l;t )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(EIl 2 2 y l;t )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(GJ )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ml 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ml )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ll 3 2 l 0 2 y dy )]TJ/F25 11.9552 Tf 10.494 8.088 Td [(EIl 2 2 yyy l;t + 1 2 GJl 2 y l;t + 1 2 l! 2 t l;t + x c l t l;t t l;t ; where r ; e ; e u ; Q 1 ; Q res ; res 2 R aredenedas r K 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( M M low )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 2 2 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [( 1 p 2 ; e K )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 max f EI + EIl;k GJ g ; e u K )]TJ/F25 11.9552 Tf 11.955 0 Td [(! 1 j p j)]TJ/F25 11.9552 Tf 19.128 8.087 Td [( p 2 1 )]TJ/F25 11.9552 Tf 13.151 8.087 Td [(! 1 1 M 2 )]TJ/F25 11.9552 Tf 13.15 8.087 Td [( 2 1 p 2 ; Q 1 Q )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(! 2 min 0 ; 1 )]TJ/F15 11.9552 Tf 17.323 3.819 Td [( j p j ; Q res Q )]TJ/F30 11.9552 Tf 11.955 24.03 Td [(2 4 h M M low )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 i + 1 p 2 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.479 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 12.496 3.155 Td [( 1 e 2 + M 3 5 ; res = Q 1 e 2 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( M + 2 M ^ D 2 + 1 1 M ^ D 2 2 : Usingthedenitionof z [ r T e T e T u ] T and y [ z T p Q ] T ,andprovidedallthegain conditionsaresatisedsucientlyseeRemark5.4andSection5.7,followingupperbound of V L isdevelopedandcanbeexpressedas V L )]TJ/F25 11.9552 Tf 21.917 0 Td [( 1 k y k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 E B + res )]TJ/F25 11.9552 Tf 9.298 0 Td [(g t + res ; 84

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where 1 ; 2 2 R aredenedas 1 min f r ; e ; e u ; Q g ; 2 1 2 min n )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c )]TJ/F25 11.9552 Tf 11.956 0 Td [(L; 3 EI )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ll 3 ; )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(I w + x 2 c )]TJ/F25 11.9552 Tf 11.955 0 Td [(x c )]TJ/F25 11.9552 Tf 11.956 0 Td [(M; )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(GJ )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ml 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ml )]TJ/F25 11.9552 Tf 11.955 0 Td [(Ll 3 )]TJ/F15 11.9552 Tf 11.956 0 Td [( M + L l 2 o : From5and5, V L 2L 1 ;hence, E B 2L 1 e;r;e u 2L 1 ,and ~ 2L 1 Since E B 2L 1 ,itcanbeconcludedthat l 0 2 yy dy 2L 1 and l 0 2 y dy 2L 1 ;hence,the elasticpotentialenergyinthewing E P 2L 1 andbyProperty5.1, yyy l; 2L 1 and y l; 2L 1 .Since e 2L 1 yyy l; 2L 1 ,and y l; 2L 1 ,5canbeusedto show t l; 2L 1 and t l; 2L 1 .Since t l; 2L 1 t l; 2L 1 ,and E B 2L 1 thekineticenergyofthesystem E K 2L 1 andbyProperty5.2, @ q @t q and @ q @t q arebounded, uniformlyin t 8 y 2 [0 ;l ] for q =1 ; 2 ; 3 .Equations5and5andthefactthat e u 2 L 1 canbeusedtoshowthattheboundarycontrolinput, U t 2L 1 .Dierentiating g from 5withrespecttotimeyields g t = 2 E B t +2 1 k y t k T k y t k ; where E B t =2 l 0 t tt + yy tyy + t tt + ty y dy: Afterintegratingbypartsthesecondandfourthtermsin5, E b canbeexpressedas E B t =2 l 0 t tt + yyyy + t tt )]TJ/F25 11.9552 Tf 11.955 0 Td [( yy dy )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 t l;t yyy l;t +2 t l;t y l;t : Sinceallsystemsignalsarebounded,5canbeusedtoconcludethat E B 2L 1 Equations5and5canbeusedtoshowthat g 2L 1 .Giventhat V L t isanonnegativefunctionintimeand V L t )]TJ/F25 11.9552 Tf 22.224 0 Td [(g t + res ,where g t isanon-negativefunctionand 85

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_ g t 2 L 1 ,LemmaA.6in[96]andLemma4.3in[99]canbeusedtoshowthat E B t ;y t areUUB.Using5andLemmaA.12in[96]thefollowinginequalitiescanbedeveloped E B t l 0 2 yy dy 1 l 3 2 0 ; E B t l 0 2 y dy 1 l 2 0 : Since y t isUUBastimeapproaches,thatisalltheauxiliaryerrortermsi.e., e t r t and e u t ,itcanbeconcludedfrom5and5that !; areUUBas t !1 8 y 2 [0 ;l ] 5.6NumericalSimulation Anumericalsimulationispresentedtoillustratetheperformanceofthedeveloped controller.ToapproximatethesimultaneousnonlinearsystemofPDEsthatdescribethe bendingandtwistingofaircraftwingwithanitenumberofODEs,aGalerkin-basedmethod isused.Thetwistingandbendingdeectionsofthewingarerepresentedasaweightedsumof basisfunctionsasgivenby y;t = a 0 t h 0 y + n X i =1 a i t h i y ; y;t = b 0 t g 0 y + p X i =1 b i t g i y ; where n =5 ;p =4 ,denotethenumberofbasisfunctionsusedintheapproximationsofthe wingtwistingandbendingdeection,respectively.Equation5isastandardtrailsolution forGalerkin'sweightedresidualmethod.Selectingthetrialsolutioninthiswayensuresthat thesolutionsatisesthePDEs,byusingprincipleoforthogonalitybetweenthebasisfunctions andanyarbitraryfunction.Asetoflinearlyindependentfunctions f h i y g n i =0 and f g i y g p i =0 is usedsatisfyingthefollowingboundaryconditions. h 0 = h i =0 ;h y 0 l =1 ;h y i l =0 ; 86

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g 0 = g i =0 ;g y 0 = g y i =0 ; g yy 0 l = g yy i l =0 ;g yyy 0 l =1 ;g yyy i l =0 : Firsttheapproximationofthetwistingandbendingdeectiongivenin5issubstitutedin thesystemofPDEsin5and5,andthenTaylor'sapproximationuptotwotermsis usedtoapproximatesineandcosineterms,andtheresultingequationscanbewrittenasaset ofcouplednonlinearODEs G 1 b +_ a 2 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(G 21 a + G 22 a 3 + a )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(G 31 + G 32 a 2 + G 4 b + G 5 a =0 ; H 1 a + H 21 b + H 22 ba 2 + H 3 a + H 4 a =0 : In5and5 b t b 0 t b 1 t :::b p t T a t a 0 t a 1 t :::a n t T G 1 l 0 g y g T y dy G 21 )]TJ/F25 11.9552 Tf 9.299 0 Td [(x c l 0 g y h y h y 2 T dy G 22 x c 3! l 0 g y h y 2 h y 3 T dy G 31 x c l 0 g y h T y dy G 32 )]TJ/F26 7.9701 Tf 10.494 5.255 Td [(x c 2! l 0 g y h y h y 2 T dy G 4 EI l 0 g y g T yyyy y dy G 5 )]TJ/F15 11.9552 Tf 10.68 3.022 Td [( L w l 0 g y h T y dy H 1 I w + x 2 c l 0 h y h T y dy H 21 x c l 0 h y g T y dy H 22 )]TJ/F26 7.9701 Tf 10.494 5.256 Td [(x c 2! l 0 h y g y h y 2 T dy H 3 )]TJ/F25 11.9552 Tf 9.298 0 Td [(GJ h y h T yy y dy H 4 )]TJ/F15 11.9552 Tf 13.634 3.022 Td [( M w l 0 h y h T y dy ThecouplednonlinearODEsaresimulatedwiththefollowinginitialconditions: y; 0= 0 mand y; 0= y 2 2 l 2 rad.Theperformanceofthecontrollerdesignedin5alongwith theupdatelawsin5,inthesimulation,demonstrated.AsindicatedinFigures5-2and53,thecoupledelasticsystembecomeunstableinthepresenceoftimedelay.Thus,thecontrol objectiveistoregulatethetwistingandbendingdeectioninpresenceofunknowntimevaryinginputdelayinthesystem.Inordertoestimatethetime-varyingdelayandcompensate forthat,NNbasedupdatelawshavebeenusedforthissimulationasin5-5.Figures 5-4and5-5showthatthedesignedcontrollersucientlymitigatesthedelayinducedbending andtwistingdeectionsrespectively,alongthelengthofthebeamastimeprogresses.Figures 5-6and5-7illustratethetimevariationoftheappliedcontrolforceandmoment,respectively. 87

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5.7ControlGainSelection Controlgains,suchas r in5, e in5, e u in5, Q 1 in5, Q res in 5,and 1 in5,introducedinthestabilityanalysisSection5.5requiretobepositiveconstants.Basedonthedesignedboundsoftime-delayestimate ^ D ,andsubsequently derivedboundsof 0 1 and p i.e., 0 ; 0 1 ; 1 j p j ,thissectiondevelopssucientgain conditionstoensure r e e u Q 1 and Q res > 0 .Usingthedenitionsof r in5and e in5,sucientlowerboundsfor K canbeobtainedas K> M M low )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 2 + 1 p ; K> 1 2 max f EI + EIl;k GJ g ; K> max M M low )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 2 + 1 p ; 1 2 max f EI + EIl;k GJ g : Usingthedenitionof e u in5,followingupperboundsof 1 canbeobtainedas 1 < K )]TJ/F26 7.9701 Tf 13.15 5.865 Td [( p 2 1 j p j + 1 M 2 + 2 p 2 : Inordertoensurethatthenumeratorofinequalityin5,stayspositive,followingupper boundof canbeobtainedas < 2 K 1 p : Alsofromthedenitionof Q 1 in5, 2 needstosatisfythefollowinginequality 2 > j p j min 0 ; 1 : Inordertosatisfy Q res in5, Q hastobeselectedsucientlylargetoensure Q > 2 4 h M M low )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 i + 1 p 2 2 )]TJ0 g 0 G/F25 11.9552 Tf 5.48 -9.684 Td [( 0 )]TJ/F15 11.9552 Tf 12.496 3.155 Td [( 1 e 2 + M 3 5 : In5, 1 hastobeselectedsucientlytosatisfy 88

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< M 0 M low : From5,itisquiteclearthatbyselecting K 1 , Q ,and 2 sucientlyasin55, 1 canbemadepositive.BasedonRemark5.4, 2 > 0 hasbeensatised. 5.8Conclusion Thispaperpresentsanovelapproachofdevelopingaboundarycontrolstrategyadded withdelaycompensation,formitigatingstoreinducedoscillationsinaexibleaircraftwing, subjectedtounknowntime-varyinginputdelay.Thedesignedcontrollerguaranteestoprovide aUUBtypestabilityasshowninstabilityanalysis,unlikeregularboundarycontrollerin [1],whichensuresasymptoticstabilitywithoutdelaypresenceinthesystem.Themain contributionsofthisworkistwofold.First,thedesignedcontrolleristherstofthiskind whichensuresthestabilityofacoupledPDEbasedelasticsysteminpresenceofunknown time-varyinginputdelay.Second,NNbasedupdatelawshavebeendevelopedtomodelthe unknowndelayinthesystem,whichusesanonlinearmappingtotransformthetimedomain toacompactdomain,inordertoutilizetheuniversalfunctionapproximationtheorem.A potentialdrawbacktothedevelopedmethodistheneedformeasurementsofhigh-orderspatial derivativesofthedistributedstatese.g., yyy l;t ,asshownin5.Futureeortsare focusedondevelopingPDE-basedoutputfeedbackboundarycontrolstrategiesthatwould eliminatetheneedforhigh-orderspatialderivativemeasurements.Finally,numericalsimulation demonstratestheperformanceofthedesignedcontrolleralongwiththeadaptiveupdatelaws. 89

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Figure5-2.Closedloopbendingdeectionwithconstantinputdelaysusingthecontroller in[1]. 90

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Figure5-3.Closedlooptwistingdeectionwithconstantinputdelaysusingthecontroller in[1]. 91

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Figure5-4.Bendingdeectionafterapplyingthedeignedcontrollerin5. 92

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Figure5-5.Twistingdeectionafterapplyingthedeignedcontrollerin5. 93

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Figure5-6.Controlforcevariationvs.time. 94

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Figure5-7.ControlMomentvariationvs.time. 95

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CHAPTER6 CONCLUSIONSANDFUTUREWORKS 6.1Conclusions Inputdelayedsystemsaresubjectofinterestofmanyresearchersforpastfewdecades. Althoughthereexiststwodierentkindofcontrolstrategiesinexistingliterature,namely robuststrategyandprediction-basedstrategy,amalgamationofbothtwostrategiesare notstudied.Robuststrategyhasadvantageofnotusingsystemmodelfordevelopingthe controller,alongwitheectiveapplicationtosystemwithexogenousdisturbances.On theotherhand,althoughpredictor-basedstrategyusesmodelknowledgeforsystemstate prediction,itgivesamuchsimplercontrolgainconditionsfromstabilityanalysis,unlikerobust strategy.Theseadvantagesofbothtwostrategies,motivatethedevelopmentnecessityofa noblecontrolstrategyfornonlinearunknowndynamicalsystem,subjectedtoinputdelay. InChapter2thisproposedamalgamationofpredictor-basedandrobuststrategyis demonstratedbydevelopingapartialdierentialequationbasedcontroller,forasecondorder uncertainnonlinearsystemsubjectedtoknowntime-varyinginputdelay.Thisnoblecontrol approachutilizesanonlineardelaydependenttransformation,totransformtraditionalcontrol inputtoamodiedcontrolinput,whichdependsonbothtimeandadummyspatialvariable. Introductionofthisnewspatialvariablenotonlysimpliesthecontrolgainconditions,as demonstratedinthestabilityanalysis,butalsotakesouttime-varyingdelaytermoutofthe controlinput,whichisadvantageouswhiledesigningestimatoroftheunknowntimedelay, asshowninChapter3andChapter4.Finally,applicationofthedevelopedcontrolleris experimentallydemonstratedforaseriesofdynamictrackingexperimentsoftheknee-joint dynamics.Thedynamictrackingexperimentsshowsuccessfulimplementationofthedeveloped controlleronsixdierenthealthyindividuals. Chapter3extendstheconceptofpartialdierentialequationbasedcontroller,introducedinChapter2,foracascadinguncertaindynamicalsystemofBrunovskycanonicalform, 96

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subjectedtoanunknowninputdelay.Inordertoapplythespatialandtimevaryingtransformationsforthecontroller,anestimateofdelayisneeded.Aneuralnetworkbasedestimation strategyisdevelopedfordelayestimation,whichdependsonanonlinearmappingtotransform timeintoacompactdomain,andusesuniversalfunctionalapproximationtheoremforthe estimation.Simulationisperformedforatwolinkrobotdynamics,subjectedtoaunknown time-varyingdelay.Simulationresultsshowtheperformanceofthecontroller,andalsoan estimateoftheunknowndelaymagnitudeusingdesignedneuralnetwork. Chapter3,demonstratesaneuralnetworkbasedestimationforunknowndelaymagnitude, althoughtheperformanceoftheestimationdependshighlyonthechoiceofactivationfunctionsandtrainingdataset.Aswithallthedatadriventechniques,availabilityofsucientdata setismuchneeded,butinordertoeliminatethehighdependencyofestimationperformance onchoiceofactivationfunction,anoptimizationbasedstrategyisdemonstratedinChapter 4.Nesterov'sacceleratedgradientdescentbasedalgorithmisusedforthedelayestimation, whichusestwopreviousdiscretetimestepsinformationforestimatingthecurrenttimestep, insteadofoneasintraditionalgradientdescent.Stabilityanalysisalsoincorporatesdeveloped acceleratedgradientdescentbasedmethodfortheestimation,andshowsanUUBstabilityof thenonlinearsystem.Simulationisperformedonatwolinkrobotdynamics,andsimulation resultsshowasucientlysmoothdelayestimation,andadecayintheobjectivefunction. InChapter5,previouslydevelopeddelayestimationusingneuralnetworkisappliedfora exibleaircraftwingsubjectedtoanunknowntime-varyinginputdelay.Aboundarycontrol strategywithadelaycompensationtermisdevelopedtomitigatethelimitcycleoscillation oftheaircraftwing.Thedelaycompensationtermutilizestheneuralnetworkbaseddelay estimationstrategydevelopedin3.Simulationresultjustiesthenecessityofaddingadelay compensationtermtotheboundarycontroller,bydemonstratingtheeectofinputdelay onaexibleaircraftsystemwhichiscontrolledjustbyanadaptiveboundarycontroller.For thedevelopeddelaycompensatedboundarycontroller,samesystemissimulatedandthe 97

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performanceofthedevelopedcontrollersignicantlyimprovescomparetothecaseofnodelay compensationinthecontroller. 6.2FutureWorks Image-basedcontrolsystemsrelyonfeedbackfromasingleormultiplecamerastoachieve desiredguidance,navigation,andcontrolobjectives.Therawimagesneedtobeprocessed, eitherbyacentralprocessingunitorbyspecializedimageprocessors,toextractandmatch desiredfeaturesandpatterns.Whilededicatedsystemsandgraphicalprocessingunitsprovide signicantcomputationalresources,potentialgainsinprocessingtimehavebeenosetby thedesiretoprocesshigherresolutionimagery.Therefore,image-basedcontrolsystemsare inherentlysusceptibletotimedelaysresultingfromimageextractionandprocessing.The delayinimageprocessingtoobtainthenecessarycontrolsignalcanberegardedasatimevaryinginputdelay.Additionally,whenthecameraisnotco-locatedwiththesystemtobe controlled,i.e.,whenusingano-boardcamera,anetworkcommunicationchannelwiredor wirelessisusedtostreamimagesfromthecameratothecontroller.Theuncertaintiesinthe communicationchannelposeanotherchallengetonetworkedimagingsystemsasthestate receivedatthecontrollerisdelayed,andthedelaycouldbeunknown.Thedevelopedneural networkbaseddelayestimationmethodcanbeapplied,alongwiththepartialdierential equationbasedcontroller,forimage-basedvisualservocontrolproblems. Moreover,anotherpossibleapplicationofthedevelopedcontrolleralongwithdelay estimationstrategycanbeswitchedsystems.Switchedsystemsarehybriddynamicalsystem, consistsofswitchingbetweendierentsubsystems,andhavestrongengineeringapplications. Similartolinear/nonlinearsystemssubjectedtotimedelay,switchedtime-delaysystemsare studiedextensivelyinexistingliterature.Althoughthereexistsseveralliteratureforboth continuous,anddiscreteswitchedsystem,subjectedtoknowntime-varyingdelays,continuous uncertainnonlinearswitchedsystemsubjectedtounknowntime-varyingstatedelayremains anopenproblem,basedonauthor'sbestknowledge.Thismotivatesthenecessityofusingthe 98

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NNfunctionalapproximatorapproachtoestimateunknowntime-varyingstatedelay,foran uncertaincontinuousswitchedsystem. 99

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BIOGRAPHICALSKETCH Indrareceivedhisbachelor'sandmaster'sfromJadavpurUniversity,IndiaandIIT Kharagpur,India,respectively,bothinmechanicalengineering.Heworkedforoneandhalf yearsasanEdisonengineerinGeneralElectricGEEnergy,afterhismaster'sdegree.Heis pursuingmaster'sandPhDinmechanicalengineering,undertheadvisementofProf.Warren E.Dixon,alongwithmaster'sinAppliedMathematics,atUniversityofFlorida.Hehasworked asasummerinternatUF-REEF,duringMay2015toAugust2015andMay2016toAugust 2016.HeiscurrentlyworkingasaPhDinternatPacicNorthwestNationalLaboratory PNNL,startingfromJune2017. 108