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Lyapunov-Based Control of Saturated and Time-Delayed Nonlinear Systems

Permanent Link: http://ufdc.ufl.edu/UFE0044857/00001

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Title: Lyapunov-Based Control of Saturated and Time-Delayed Nonlinear Systems
Physical Description: 1 online resource (144 p.)
Language: english
Creator: Fischer, Nic
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

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

Notes

Abstract: Time delays and actuator saturation are two phenomena which affect the performance of dynamic systems under closed-loop control. Effective compensation mechanisms can be applied to systems with actuator constraints or time delays in either the state or the control. The focus of this dissertation is the design of control strategies for nonlinear systems with combinations of parametric uncertainty, bounded disturbances, actuator saturation, time delays in the state, and/or time delays in the input. The first contribution of this work is the development of a saturated control strategy based on the Robust Integral of the Sign of the Error (RISE), capable of compensating for system uncertainties and bounded disturbances. To facilitate the design of this controller and analysis, two Lyapunov-based stability corollaries based on the LaSalle-Yoshizawa Theorem (LYT) are introduced using nonsmooth analysis techniques. Leveraging these two results, a RISE-based control design for systems with time-varying state-delays is developed. Since delays can also commonly occur in the control input, a predictor-based control strategy for systems with time-varying input delays is presented. Extending the results for time-delayed systems, a predictor-based controller for uncertain nonlinear systems subject to simultaneous time-varying unknown state and known input delays is introduced. Because errors can build over the deadtime interval when input delays are present leading to large actuator demands, a predictor-based saturated controller for uncertain nonlinear systems with constant input-delays is developed. Each of the proposed controllers provides advantages over previous literature in their ability to provide smooth, continuous control signals in the presence of exogenous bounded disturbances. Lyapunov-based stability analyses, extensions to Euler-Lagrange (EL) dynamic systems, simulations, and experiments are also provided to demonstrate the performance of each of the control designs throughout the dissertation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Nic Fischer.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Dixon, Warren E.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044857:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044857/00001

Material Information

Title: Lyapunov-Based Control of Saturated and Time-Delayed Nonlinear Systems
Physical Description: 1 online resource (144 p.)
Language: english
Creator: Fischer, Nic
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

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

Notes

Abstract: Time delays and actuator saturation are two phenomena which affect the performance of dynamic systems under closed-loop control. Effective compensation mechanisms can be applied to systems with actuator constraints or time delays in either the state or the control. The focus of this dissertation is the design of control strategies for nonlinear systems with combinations of parametric uncertainty, bounded disturbances, actuator saturation, time delays in the state, and/or time delays in the input. The first contribution of this work is the development of a saturated control strategy based on the Robust Integral of the Sign of the Error (RISE), capable of compensating for system uncertainties and bounded disturbances. To facilitate the design of this controller and analysis, two Lyapunov-based stability corollaries based on the LaSalle-Yoshizawa Theorem (LYT) are introduced using nonsmooth analysis techniques. Leveraging these two results, a RISE-based control design for systems with time-varying state-delays is developed. Since delays can also commonly occur in the control input, a predictor-based control strategy for systems with time-varying input delays is presented. Extending the results for time-delayed systems, a predictor-based controller for uncertain nonlinear systems subject to simultaneous time-varying unknown state and known input delays is introduced. Because errors can build over the deadtime interval when input delays are present leading to large actuator demands, a predictor-based saturated controller for uncertain nonlinear systems with constant input-delays is developed. Each of the proposed controllers provides advantages over previous literature in their ability to provide smooth, continuous control signals in the presence of exogenous bounded disturbances. Lyapunov-based stability analyses, extensions to Euler-Lagrange (EL) dynamic systems, simulations, and experiments are also provided to demonstrate the performance of each of the control designs throughout the dissertation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Nic Fischer.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044857:00001


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LYAPUNOV-BASEDCONTROLOFSATURATEDANDTIME-DELAYEDNONLINEAR SYSTEMS By NICFISCHER ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012

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c 2012NicFischer 2

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TomyparentsDebbieandMattFischerfortheirenduringsupport andconstantencouragement 3

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ACKNOWLEDGMENTS Iwouldliketoexpresssinceregratitudetomyadvisor,Dr.WarrenE.Dixon,whose experienceandmotivationhavebeeninstrumentalinmyacademicsuccess.Asan advisor,hehasprovidedguidanceinmyresearchandencouragementindevelopingmy ownideas.Asamentor,hehashelpedmedevelopprofessionalskillsandhasprepared meforthefuture.IwouldliketoextendmygratitudetomycommitteemembersPrabir Barooah,CarlCrane,PramodKhargonekar,andEric.M.Schwartzforthetimeand recommendationstheyprovided.Also,Iwouldliketothankmyfamily,coworkers especiallyRushikeshKamalapurkarforhiscountlesshoursofcoauthoringworkwith nonsmoothanalysis,andfriendsfortheirsupportandinspiritment. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFFIGURES.....................................7 LISTOFABBREVIATIONS................................8 ABSTRACT.........................................9 CHAPTER 1INTRODUCTION...................................11 1.1MotivationandProblemStatement......................11 1.2LiteratureReview................................15 1.3Contributions..................................23 2LASALLE-YOSHIZAWACOROLLARYFORDISCONTINUOUSSYSTEMS..27 2.1Preliminaries..................................27 2.2MainResult...................................32 2.3DesignExample1Adaptive+SlidingMode................36 2.4DesignExample2RISE...........................40 2.5Summary....................................47 3SATURATEDRISEFEEDBACKCONTROL....................48 3.1DynamicModel.................................48 3.2ControlDevelopment..............................49 3.3StabilityAnalysis................................53 3.4Euler-LagrangeExtension...........................57 3.5ExperimentalResults..............................61 3.6Summary....................................64 4RISE-BASEDCONTROLOFANUNCERTAINNONLINEARSYSTEMWITH TIME-VARYINGSTATEDELAYS..........................65 4.1DynamicModel.................................65 4.2ControlDevelopment..............................66 4.3StabilityAnalysis................................71 4.4Summary....................................75 5LYAPUNOV-BASEDCONTROLOFANUNCERTAINNONLINEARSYSTEM WITHTIME-VARYINGINPUTDELAY.......................76 5.1DynamicModel.................................76 5.2ControlDevelopment..............................77 5

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5.3StabilityAnalysis................................79 5.4Euler-LagrangeExtension...........................83 5.5SimulationResults...............................86 5.6Summary....................................87 6TIME-VARYINGINPUTANDSTATEDELAYCOMPENSATIONFORUNCERTAINNONLINEARSYSTEMS...........................89 6.1DynamicModel.................................89 6.2ControlDevelopment..............................90 6.3StabilityAnalysis................................94 6.4SimulationResults...............................98 6.5Summary....................................100 7SATURATEDCONTROLOFANUNCERTAINNONLINEARSYSTEMWITH INPUTDELAY....................................103 7.1DynamicModel.................................103 7.2ControlDevelopment..............................104 7.3StabilityAnalysis................................108 7.4Euler-LagrangeExtension...........................112 7.5SimulationResults...............................114 7.6Summary....................................115 8CONCLUSIONANDFUTUREWORK.......................118 8.1DissertationSummary.............................118 8.2LimitationsandFutureWork..........................120 APPENDIX APROOFOF P CH3................................123 BPROOFOF P CH6................................125 CPROOFOF BOUNDCH7............................127 REFERENCES.......................................130 BIOGRAPHICALSKETCH................................144 6

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LISTOFFIGURES Figure page 2-1Setclosureof K [ f ] x for x> 0 case........................30 2-2Setclosureof K [ f ] x for x =0 case........................30 3-1Trackingerrorsvs.timeforcontrollerproposedin3.............63 3-2Desiredandactualtrajectoriesvs.timeforcontrollerproposedin3....63 3-3Controltorquevs.timeforcontrollerproposedin3..............64 5-1Trackingerrorsvs.timeforcontrollerproposedin5with+50%frequency varianceininputdelay.................................88 6-1Trackingerrorsvs.timefortheproposedcontrollerin6...........100 6-2Trackingerrors,actuationeffortandtime-varyingdelaysvstimeforCase3...101 6-3Trackingerrors,actuationeffortandtime-varyingdelaysvstimeforCase5...101 7-1Trackingerrorvs.timeforproposedcontrollerin7..............116 7-2Controltorquevs.timeforproposedcontrollerin7..............116 7

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LISTOFABBREVIATIONS a.e.AlmostEverywhere DCALDesiredCompensationAdaptationLaw ELEuler-Lagrange EMKExactModelKnowledge LKLyapunov-Krasovskii LMILinearMatrixInequality LPLinear-in-the-Parameters LRLyapunov-Razumikhin LYCLaSalle-YoshizawaCorollaries LYTLaSalle-YoshizawaTheorem MVTMeanValueTheorem NNNeuralNetwork non-LPNotLinear-in-the-Parameters PDProportional-Derivative PIDProportional-Integral-Derivative RHSRight-HandSide RISERobustIntegraloftheSignoftheError RMSRootMeanSquare SARCSaturatedAdaptiveRobustControl UCUniformlyContinuous UUBUniformlyUltimatelyBounded 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy LYAPUNOV-BASEDCONTROLOFSATURATEDANDTIME-DELAYEDNONLINEAR SYSTEMS By NicFischer December2012 Chair:WarrenE.Dixon Major:MechanicalEngineering Timedelaysandactuatorsaturationaretwophenomenawhichaffecttheperformanceofdynamicsystemsunderclosed-loopcontrol.Effectivecompensationmechanismscanbeappliedtosystemswithactuatorconstraintsortimedelaysineitherthe stateorthecontrol.Thefocusofthisdissertationisthedesignofcontrolstrategiesfor nonlinearsystemswithcombinationsofparametricuncertainty,boundeddisturbances, actuatorsaturation,timedelaysinthestate,and/ortimedelaysintheinput. Therstcontributionofthisworkisthedevelopmentofasaturatedcontrolstrategy basedontheRobustIntegraloftheSignoftheErrorRISE,capableofcompensating forsystemuncertaintiesandboundeddisturbances.Tofacilitatethedesignofthis controllerandanalysis,twoLyapunov-basedstabilitycorollariesbasedontheLaSalleYoshizawaTheoremLYTareintroducedusingnonsmoothanalysistechniques. Leveragingthesetworesults,aRISE-basedcontroldesignforsystemswithtimevaryingstate-delaysisdeveloped.Sincedelayscanalsocommonlyoccurinthecontrol input,apredictor-basedcontrolstrategyforsystemswithtime-varyinginputdelaysis presented.Extendingtheresultsfortime-delayedsystems,apredictor-basedcontroller foruncertainnonlinearsystemssubjecttosimultaneoustime-varyingunknownstate andknowninputdelaysisintroduced.Becauseerrorscanbuildoverthedeadtime intervalwheninputdelaysarepresentleadingtolargeactuatordemands,apredictorbasedsaturatedcontrollerforuncertainnonlinearsystemswithconstantinput-delays 9

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isdeveloped.Eachoftheproposedcontrollersprovidesadvantagesoverprevious literatureintheirabilitytoprovidesmooth,continuouscontrolsignalsinthepresence ofexogenousboundeddisturbances.Lyapunov-basedstabilityanalyses,extensions toEuler-LagrangeELdynamicsystems,simulations,andexperimentsarealso providedtodemonstratetheperformanceofeachofthecontroldesignsthroughoutthe dissertation. 10

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CHAPTER1 INTRODUCTION 1.1MotivationandProblemStatement Thereexistnumerouscontrolsolutionsfornonlinearsystemswithadditivedisturbances.Generalcontrolliteraturesuggeststhatrobusttechniquessuchashighgain, slidingmode,orvariablestructurecontrolhavesuccessfullybeendevelopedtoaccommodateforparametricuncertaintiesanddisturbancesinnonlinearplants[17].The couplingoftheserobustmethodswithadaptivecomponentshasalsobeenshownto improvetheoverallperformanceofbothregulationandtrackingproblemsfornonlinear systems.Robustcontroltechniquesthatyieldanasymptoticresultaretypicallydiscontinuous,andoftensufferfromlimitationssuchasthedemandforinnitebandwidthor chatter.ContinuousrobustcontroldesignssuchastheRISEstrategy[8]havealsobeen developedandhavebeenshowntobeeffectiveforsystemswithboundeddisturbances. TheRISEstrategyworksbyimplicitlylearning[9]andcompensatingforsufciently smoothboundeddisturbancesandunstructuredparametricuncertaintythroughtheuse ofasufcientlylargegainmultipliedbyanintegralsignumterm.RISEtechniquesare usedthroughoutthedissertationastheypresentastate-of-the-artapproachforcontrol ofuncertainnonlinearsystems. Classicalstabilitytheoryisnotapplicableforsystemsdescribedbydiscontinuous differentialequationsbasedonthelocalLipschitzassumptioni.e.,nonsmoothsystems.Examplesofsuchsystemsinclude:systemswithfrictionmodeledasaforce proportionaltothesignofavelocity,systemswithfeedbackfromanetwork,digital systems,systemswithadiscontinuouscontrollaw,etc.Differentialinclusionsarea mathematicaltoolthatcanbeusedtodiscusstheexistenceofsolutionsfornonsmooth systems.Utilizingadifferentialinclusionframework,numerousLyapunovmethodsusing generalizednotionsofsolutionshavebeendevelopedinliteratureforbothautonomous 11

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andnonautonomoussystems.Ofthese,severalstabilitytheoremshavebeenestablishedwhichapplytononsmoothsystemsforwhichthederivativeofthecandidate Lyapunovfunctioncanbeupperboundedbyanegative-denitefunction:Lyapunov's generalizedtheoremandnite-timeconvergencein[1015]aresomeexamplesofsuch. However,forcertainclassesofcontrollerse.g.,adaptivecontrollers,outputfeedback controllers,etc.,anegative-deniteboundmaybedifcultorimpossibletoachieve, restrictingtheuseofsuchmethods. StabilitytechniquessuchastheLaSalle-YoshizawaTheoremLYTwereintroduced forcontinuoussystemstospecicallyhandlethecasewhentheLyapunovfunction derivativeisboundedbyasemi-denitefunction.Historically,someauthorshavestated theuseoftheLYTincorrectlyifthesystemcontainsdiscontinuities,thenthelocally LipschitzpropertyrequiredbythetheoremdoesnotholdorhavestatedthattheLYT canappliedusingnonsmoothtechniqueswithoutproof.ThefocusofChapter2isthe explicitdevelopmentofacorollarytotheLYTwhichcanbeusedasananalysistoolfor nonsmoothsystemswithanegative-semi-denitederivativeofthecandidateLyapunov function. Whilerobustcontroltechniqueswhethercontinuousordiscontinuoushavebeen showntobeeffectiveforthecompensationofparametricuncertaintiesandadditive disturbances,ingeneral,thesetechniquesincludingallpreviousRISEmethods donotaccountforthefactthatthecommandedinputmayrequiremoreactuation thanisphysicallypossiblebythesysteme.g.,duetolargeinitialconditionoffsets, anaggressivedesiredtrajectory,orlargeperturbations.Forexample,thetypical RISEstructureusesasufcientlylargegainmultipliedbyanintegralterm,whichcan potentiallyleadtoacomputedcontrolcommandthatexceedsactuatorcapabilities. Becausedegradedcontrolperformanceandthepotentialriskofthermalormechanical failurecanoccurwhenunmodeledactuatorconstraintsareviolated,controlschemes whichcanensureperformancewhileoperatingwithinactuatorlimitationsaremotivated. 12

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LeveragingtheoutcomesdevelopedinChapter2,Chapter3presentsasaturated RISEcontrollerwhichlimitsthecontrolauthorityatorbelowanadjustableapriori limit.Saturatedcontroldesignsareavailableinliterature;however,theintegrationofa saturationschemeintothecontinuousRISEstructurehasremainedanopenproblem due,inparttotheintegratorcompensation. Asdescribedinthesurveypapers[1619]andrelativelyrecentmonographssuch as[2025],timedelaysarepervasiveinnatureandengineeredsystems.Afewwellknownanddocumentedengineeringapplicationsinclude:digitalimplementationofa continuouscontrolsignal,regenerativechatterinmetalcuttingespeciallyprevalent inhighspeedmanufacturing,delaysintorqueproductionduetoenginecycledelays ininternalcombustionengines,chemicalprocesscontrol,rollingmills,controlover networks,activequeuemanagement,nancialmarketsespecially,computercontroller exchangesofnancialproducts,etc.Delaysarealsoinherentinmanybiological processsuchas:delayinaperson'sresponseduetodrugsandalcohol,delaysin forceproductioninmuscle,thecardiovascularcontrolsystem,etc.Systemsthatdonot compensatefordelayscanexhibitreducedperformanceandpotentialinstability. Sinceatimedelaycanbeconsideredanothertypeofdisturbancetothesystem, researchershaveinvestigatedadaptiveand/orrobusttechniquestocompensatefor theundesirableimplicationsdelayshaveonclosed-loopcontrolofnonlinearsystems. Typicaltimedelayedcontrolresultshaveusednovelprediction/compensationtechniquessuchasSmithpredictorsorArtsteinreductionmethodstohandlethedelayed termsinclosed-loopcontrol;however,methodsthatachieveasymptoticorexponential resultsutilizingclassicrobusttechniquessufferfromthesamediscontinuouslimitations e.g.,demandforinnitebandwidthand/orchatterasdelay-freecontroldesigns.Leveragingadesignapproachsimilartothatofthepreviouschapter,Chapter4presentsa RISE-basedcontroldesignfornonlinearsystemswithtime-varyingstatedelays. 13

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Whilestatedelaysareprevalentinanumberofengineeredsystems,timedelays canalsooccurinthecontrol.Examplesofsystemswithinputdelayscanbefoundin numerousapplications,fromteleoperatedroboticsystemstobiologicalprocesses. Problemsarisingfromdelaycorruptionofthecontrolinputremainunsolvedforlarge classesofpracticalsystemse.g.,uncertainnonlinearsystems.Whileseveralresults haveusedvariationsoftheSmithandArtsteinmethodstosolvetheinputdelayproblem forlinearsystemswithknownandunknowndynamics,andnonlinearsystemswith exactmodelknowledgeEMKi.e.,knownforward-completeandstrictfeedforward systems,fewresultssolvetheinputdelayproblemforuncertainnonlinearsystems. AsstatedintheBeyondthisBooksectionoftheseminalworkin[22],Krsticindicates thatapproachesdevelopedforuncertainlinearsystemsdonotextendinanobvious waytononlinearplantssincethelinearboundednessoftheplantmodelisexplicitly usedinthestabilityproofofsuchresults,andthatnewmethodsmustbedeveloped fordelay-adaptivecontrolforselectclassesofnonlinearsystemswithunknowninput delays.Methodsthatsolvetheinputdelayproblemforuncertainnonlinearsystemswith knownandunknownconstanttimedelayshavebeenstudiedin[2632].However,due touncertaintiesintheinherentnatureofrealworldsystems,itisoftenmorepracticalto considertime-varyingorstate-dependenttimedelaysinthecontrol.Chapter5presents acontrollerforuncertainnonlinearsystemswithtime-varyinginputdelays.Motivatedby thesametime-varyingdelayconsiderations,Chapter6integratestheworkofChapters 2,4and5todesignacontrollerwhichiscapableofhandlingcompositetime-varying statedelaysandtime-varyinginputdelays,whileachievingbettertransientandsteady stateperformanceandstability. Forsystemswithinputdelays,errorscanbuildoverthedelayintervalalsoleading tolargeactuatordemands,exacerbatingpotentialproblemswithactuatorsaturation. MotivatedbythesameactuatorsaturationconcernspresentedinChapter3,Chapter7 developsacontrolstrategyforuncertaininput-delayednonlinearsystemswithconstant 14

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timedelaysandactuatorsaturationconstraints.Previoustechniquesandoutcomes obtainedinChapter3areutilizedtodevelopacontinuouscontroldesignwhichallows fortheboundonthecontroltobeadjustedapriori. TheworkinthisdissertationisbasedonLyapunovstabilitytheoryacommon toolinnonlinearcontrolandpresentsseveralcontrolstrategiesforopenproblemsin nonlinearcontrolliterature.Specically,theworkfocusesonreal-worldproblemswith practicalimplementationconsiderations,integratedthroughouttheindividualtheoretical contributions. 1.2LiteratureReview AliteraturereviewofChapters2-7ispresentedbelow. Chapter2:Lasalle-YoshizawaCorollaryforDiscontinuousSystems: Peano's Theoremstatesthatforadifferentialequationgivenby x = f x;t ,if f x;t iscontinuouson R n [0 ; + 1 ,thenforeachinitialpair x 0 ;t 0 2 R n [0 ; + 1 thereexistsat leastonelocalclassicalsolution x t suchthat x t 0 = x 0 .Whenthefunction f x;t isalsoassumedtobelocallyLipschitzcontinuous,itispossibletoprovelocaluniquenessandcontinuityofsolutionswithrespecttotheinitialconditions.Incontroltheory, thisassumptionisoftentoorestrictive[33].Thus,itisoftenmoreappropriatetopose assumptionson f x;t suchthatthefunction f x;t isessentiallylocallyboundedon R n [0 ; + 1 ,thatis,foreach x 2 R n ,thefunction t f x;t ismeasurableandfor almostevery t 0 ,thefunctioniscontinuous.Thissimpleassumptionisthebasis forthebranchofmathematicsanditsextensionsintocontrolsystemsanalysiswhich includesnonsmoothcomponentsofdifferentialequations. MatrosovTheoremsprovideaframeworkforexaminingthestabilityofequilibrium pointsandsetsthroughvariousextensionswhenacandidateLyapunovfunctionhas negativesemi-denitedecay.TheclassicalMatrosovTheorem[34]isbasedonthe existenceofadifferentiable,positive-deniteandradiallyunboundedLyapunov-like functionwithanegativesemi-denitederivative,whereauxiliaryfunctionsthatsum 15

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tobepositive-denitearethenusedtoestablishstabilityorasymptoticstabilityofan equilibrium.Variousextensionsofthistheoremhavebeendevelopedcf.[3539]to encompassdiscreteandhybridsystemsandtoestablishstabilityofclosedsets.In particular,[38]seealsotherelatedworkin[35]and[36]extendedMatrosov'sTheorem todifferentialinclusions,whilealsoaddressingthestabilityofsets.Anextensionof Matrosov'sTheoremtothestabilityofsetswasalsoexaminedin[39],whereaweak versionofthetheoremisdevelopedforautonomoussystemsinthespiritofLaSalle's InvariancePrinciple. IncontrasttoMatrosovTheorems,LaSalle'sInvariancePrinciple[40]hasbeen widelyadoptedasamethod,forcontinuousautonomoustime-invariantsystems,to relaxthestrictnegative-denitenessconditiononthecandidateLyapunovfunction derivativewhilestillensuringasymptoticstabilityoftheorigin.Stabilityoftheorigin isprovenbyshowingthatboundedsolutionsconvergetothelargestinvariantsubset containedinthesetofpointswherethederivativeofthecandidateLyapunovfunctionis zero.In[41],LaSalle'sInvariancePrinciplewasmodiedtostatethatboundedsolutions convergetothelargestinvariantsubsetofthesetwhereanintegrableoutputfunction iszero.Theintegralinvariancemethodwasfurtherextendedin[42]todifferential inclusions.Asdescribedin[43],additionalextensionsoftheinvarianceprincipleto systemswithdiscontinuousright-handsidesRHSwerepresentedin[4446]for Filippovsolutionsand[47]forCarathodorysolutions. VariousextensionsofLaSalle'sInvariancePrinciplehavealsobeendeveloped forhybridsystemscf.[43,4852].Theresultsin[48]and[51]focusonswitched linearsystems,whereastheresultin[52]focusesonswitchednonlinearsystems. In[50],hybridextensionsofLaSalle'sInvariancePrinciplewereappliedforsystems whereatleastonesolutionexistsforeachinitialcondition,fordeterministicsystems, andcontinuoushybridsystems.Left-continuousandimpulsivehybridsystemsare consideredinextensionsin[49].In[43],twoinvarianceprinciplesaredevelopedfor 16

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hybridsystems:oneinvolvesaLyapunov-likefunctionthatisnonincreasingalongall trajectoriesthatremaininagivenset,andtheotherconsidersapairofauxiliaryoutput functionsthatsatisfycertainconditionsonlyalongthehybridtrajectory.Areviewof invarianceprinciplesforhybridsystemsisprovidedin[53]. Thechallengefordevelopinginvariance-likeprinciplesfornonautonomoussystems isthatitmaybeunclearhowtoevendeneasetwherethederivativeofthecandidate LyapunovfunctionisstationarysincethecandidateLyapunovfunctionisafunction ofbothstateandtime[54,55].Byaugmentingthestatevectorwithtimecf.[56, 57],anonautonomoussystemcanbeexpressedasanautonomoussystem:this techniqueallowsautonomoussystemsresultscf.[58]and[59]tobeextendedto nonautonomoussystems.Whilethestateaugmentationmethodcanbeauseful tool,ingeneral,augmentingthestatevectoryieldsanon-compactattractorwhen thetimedependenceisnotperiodic,destroyingsomeofthelatentstructureofthe originalequation;forexample,thenewequationwillnothaveanybounded,periodic, oralmostperiodicmotions.Someresultscf.[6062]haveexploredwaystoutilize theaugmentedsystem'snon-compactattractorsbyfocusingonsolutionoperator decomposition,energyequationsornewnotionsofcompactness,butthesemethods typicallyrequireadditionalregularityconditionswithrespecttotimethancaseswhen timeiskeptasadistinctvariable. TheKrasovskii-LaSalleTheorem[63]wasoriginallydevelopedforperiodicsystems, withseveralgeneralizationsalsoexistingfornotnecessarilyperiodicsystemse.g., see[45,6467].Inparticular,aKrasovskii-LaSalleExtendedInvariancePrinciple isdevelopedin[67]toprovethattheoriginofanonautonomousswitchedsystem withapiecewisecontinuousuniformlyboundedintimeRHSisgloballyasymptotically stableoruniformlygloballyasymptoticallystableforautonomoussystems.Theresult in[67]usesaLipschitzcontinuous,radiallyunbounded,positive-denitefunctionwith anegativesemi-denitederivativeconditionC1alongwithanauxiliaryLipschitz 17

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continuouspossiblyindenitefunctionwhosederivativeisupperboundedbyterms whosesumarepositive-deniteconditionC2. Alsofornonautonomoussystems,theLaSalle-YoshizawaTheoremLYTi.e.,[55, Theorem8.4]and[68,TheoremA.8],basedontheworkin[40,69,70],providesa convenientanalysistoolwhichallowsthelimitingsetwhichdoesnotneedtobeinvarianttobedenedwherethenegativesemi-deniteboundonthecandidateLyapunov derivativeisequaltozero,guaranteeingasymptoticconvergenceofthestate.Givenits utility,theLYThasbeenapplied,forexample,inadaptivecontrolandinderivingstability frompassivitypropertiessuchasfeedbackpassivationandbacksteppingdesignsof nonlinearsystems[40].AvailableproofsfortheLYTexploitBarbalat'sLemma[71], whichisofteninvokedtoshowasymptoticconvergenceforgeneralclassesofnonlinear systems.Ingeneral,adaptingtheLYTtosystemswheretheRHSisnotlocallyLipschitz hasremainedanopenproblem.However,usingBarbalat'sLemmaandtheobservation thatanabsolutelycontinuousfunctionthathasauniformlylocallyintegrablederivative isuniformlycontinuous,theresultin[71]provesasymptoticconvergenceofanoutput functionfornonlinearsystemswith L p disturbances.Theresultin[71]isdeveloped fordifferentialequationswithacontinuousright-handside,but[71,Facts1-4]provide insightsintotheapplicationofBarbalat'sLemmatodiscontinuoussystems. Chapter3:SaturatedRISEFeedbackControl: Motivatedbyissueswithactuator constraintsforrobustcontrolmethods,someeffortshavefocusedondeveloping saturatedcontrollersfortheregulationproblemcf.[7277]andthemoregeneral trackingproblemcf.[7888].In[78],theauthorsdevelopedanadaptive,full-state feedbackcontrollertoproducesemi-globalasymptotictrackingwhilecompensating forunknownparametricuncertaintiesusingmultipleembeddedhyperbolicsaturation functions.Theauthorsof[79]wereabletoextendtheProportional-Integral-Derivative PID-basedworkof[74]tothetrackingcontrolproblembyutilizingageneralclassof saturationfunctionstoachieveaglobaluniformasymptotictrackingresultforalinearly 18

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parameterizableLPsystem.Thisworkwasbasedonpriorworkin[80]and[81]which incorporatedhyperbolicsaturationfunctionsintothesaturatedProportional-Derivative PD+controlstrategydevelopedin[82].Theworksof[7981]relyongainswhich mustabidebyasaturation-avoidanceinequalityrestrictingtheabilitytoadjustthe performanceofthecontrollerorthecharacterizationofdesiredtrajectoriestoavoid saturation,bothofwhichlimitthedomainforwhichthecontrollercanoperate.Antiwindupschemeshavebeendeveloped[89]tocompensateforsaturationnonlinearities innonlinearEuler-LagrangeELsystemsusingPID-likecontrolstructures.Results in[90]and[91]achievedglobalregulationofsaturatednonlinearsystemsusinga PID-likecontrolstructureandapassivity-basedanalysis.EachofthesaturatedPD+ andPID+basedcontrolmethodsprovideanelegant,intuitivestructureforwhichto controlanuncertainsystem;however,duetotheinclusionofgravitycompensation terms,aprioriknowledgeofboththemodelstructureanditsparametersisrequired. Thisassumptionisparticularlyintrusiveintheexampleofsystemswithaddedmass suchasthatofarobotmanipulatorsystemwithunknownorvaryingpayloads.To compensateforuncertaindynamicsandtheevaluationoftheunknowngravityterm, Alvarez-Ramirez,et.al[83]includesanadditionalsaturatedintegraltermanduses energyshapinganddampinginjectionmethodstoyieldasemi-globalstabilityresult. Morerecentlyin[84],asaturatedPIDframeworkcontrollerwasproposedwhichuses sigmoidalfunctionstoachieveglobalasymptoticregulationtoaset-point;however,it isunclearhowtheresultcanbeextendedtothetrackingproblemduetothecontrol structure. Whileeachofthementionedcontributionsdevelopedsaturatedcontrollerswith asymptoticstabilityresults,theyhavenotbeenproventostabilizesystemswithboth uncertaindynamicsandadditiveunmodeleddisturbances.HongandYaoproposedthe developmentofacontinuoussaturatedadaptiverobustcontrolSARCalgorithm[85] capableofachievinganultimatelyboundedtrackingresultinthepresenceofan 19

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externaldisturbance.Corradini,et.alproposedadiscontinuoussaturatedslidingmode controller[86]forlinearplantmodelsinthepresenceofboundedmatcheduncertainties toachieveasemi-globaltrackingresult.In[87],twocontrolalgorithmsaredevelopedfor robuststabilizationofspacecraftinthepresenceofcontrolinputsaturation,parametric uncertainty,andexternaldisturbancesusingadiscontinuousvariablestructurecontrol design.In[88],theauthorsdevelopaSARCcontrollerausingdiscontinuousprojection methodtoachievegloballyboundedtrackingofarticialmuscles.However,whileeach ofthesesaturatedrobusttechniquesareabletoaddressuncertainnonlinearsystems withadditivedisturbances,thediscontinuousnatureoftheresultsmotivatesthedesign ofcontinuoussaturatedrobustcontroltechniques.Robustcontroldesignsutilizing nestedsaturationfunctionsforuncertainfeedforwardnonlinearsystems[9294]have guaranteedglobalasymptoticstabilitydespiteunmodeleddynamicdisturbances. Chapter4:RISE-BasedControlofanUncertainNonlinearSystemWith Time-VaryingStateDelays: Motivatedbyperformanceandstabilityproblemswith time-delayedsystems,solutionstypicallyuseappropriateLyapunov-Razumikhin LRorLyapunov-KrasovskiiLKfunctionalstoderiveboundsonthedelaysuch thattheclosed-loopsystemisstable.Numerousmethodshavebeendeveloped throughoutliteraturefortime-delayedlinearsystemsandnonlinearsystemswithknown dynamics[16,18,2123].Foruncertainnonlinearsystems,techniqueshavealsobeen developedtocompensateforbothknownandunknownconstantstate-delays[95102]. Extensionsofthesedesignstosystemswithnonlinear,boundeddisturbancesalso exist[100,102,103]. Forsomeapplications,itisoftenmorepracticaltoconsidertime-varyingorstatedependenttimedelays.Controlmethodsforuncertainnonlinearsystemswithtimevaryingstatedelayshavebeenstudiedinresultssuchas[99,104107].However, compensationoftime-varyingstate-delaysinsystemswithbothuncertaindynamics andaddedexogenousdisturbancesisexploredinonlyafewresults.Arobustintegral 20

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slidingmodetechniqueforstochasticsystemswithtime-varyingdelaysandlinearly state-boundednonlinearuncertaintiesisdevelopedin[108]butdependsonconvex optimizationroutinesandaLinearMatrixInequalityLMIfeasibilitycondition.In[109], anadaptivefuzzylogiccontrolmethodyieldingasemi-globaluniformlyultimately boundedUUBtrackingresultisillustratedforasysteminBrunovskyform.Theauthors of[110]utilizethecirclecriterionandanLMIfeasibilityconditiontodesignanonlinear observerforneural-network-basedcontrolofaclassofuncertainstochasticnonlinear strict-feedbacksystems.ThedesignproposesaneuralnetworkNNweightupdate lawthatdirectlycancelstheboundonthereconstructionerrortoyieldagloballystable result.Discontinuousmodelreferenceadaptivecontrollershavebeendesignedin[111] and[112]foruncertainnonlinearplantswithtime-varyingdelaystoachieveasymptotic stabilityresults;however,thediscontinuousnatureoftheseresultsmotivatesthedesign ofcontinuouscontroltechniques. Chapter5:Lyapunov-BasedControlofanUncertainNonlinearSystemwith Time-VaryingInputDelay: Manyoftheresultsforlinearsystemswithconstantdelays areextensionsofclassicSmithpredictors[113],Artsteinmodelreduction[114],ornite spectrumassignment[115].Duetouncertaintiesintheinherentnatureofrealworld systems,itisoftenmorepracticaltoconsidertime-varyingorstate-dependenttime delaysinthecontrol.Extensionsoflinearcontroltechniquestotime-varyinginputdelays arealsoavailable[18,116121]. Fornonlinearsystems,controllersconsideringconstant[95102]andtime-varying [99,104112,122,123]statedelayshavebeenrecentlydeveloped.However,results whichconsiderdelayedinputsarefarlessprevalent,especiallyforsystemswithmodel uncertaintiesand/ordisturbances.Examplesoftheseincludeconstantinputdelay resultsin[2632,124129]andtime-varyinginputdelayresultsbasedonLMI[130,131] andbackstepping[132134]techniques. 21

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Chapter6:Time-VaryingInputAndStateDelayCompensationforUncertain NonlinearSystemsResults: Resultswhichfocusonsimultaneousconstantstateand inputdelaysforlinearsystemsareprovidedin[135137].Resultswhichtackleboth time-varyingstateandinputdelaysinuncertainnonlinearsystemsarerare.Thereview ofliteratureinChapter5illustratedthatfewresultsevenexistfornonlinearsystemswith solelytime-varyinginputdelays.Recentlyin[134],authorsextendedthepredictor-based techniquesin[135]and[133]wereextendedtononlinearsystemswithtime-varying delaysinthestateand/ortheinpututilizingabacksteppingtransformationtoconstruct apredictor-basedcompensator.Thedevelopmentin[135]and[133]assumesthatthe disturbance-freeplantisasymptoticallystabilizableintheabsenceofdelay,andthatthe rateofchangeofthedelayisboundedby1acommonassumptionforpredictor-based work.Totheauthor'sknowledge,developmentofacontrolmethodforanuncertain nonlinearsystemwithsimultaneoustime-varyingdelayedstateandactuationwith additiveboundeddisturbancesremainsasanunsolvedproblem. Chapter7:SaturatedControlofanUncertainNonlinearSystemwithInput Delay: Saturatedcontrollersforstatedelaysystemshavebeenrigorouslystudied forbothlinearandnonlinearsystems[138142].However,themajorityofsaturated controllerspresentlyavailableforsystemswithinputdelaysarebasedonlinearplant models[141,143145]andonlyafewresultsarepresentfornonlinearsystemsespeciallythosewithuncertainties.Theauthorsof[144]proposedaparametricLyapunov equation-basedlow-gainfeedbacklawwhichguaranteesstabilityofalinearsystem withdelayedandsaturatedcontrolinput.In[146],globaluniformasymptoticstabilizationisobtainedwithboundedfeedbackofastrict-feedforwardlinearsystemwith delayinthecontrolinput.Theauthorswereabletoextendtheresulttoanuncertain butdisturbance-freestrict-feedforwardnonlinearsystemwithdelaysinthecontrolinput in[28]usingasystemofnestedsaturationfunctions.Thecontrollerrequiresanonlinear strict-feedforwarddynamicsystemwithparametricuncertainty, h t ,whichsatises 22

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thefollowingcondition: j h x i +1 ;x i +2 ;:::;x n j M )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(x 2 i +1 ;x 2 i +2 ;:::;x 2 n where M denotes apositiverealnumberwhen j x j j 1 ;j = i +1 ;:::;n .Unlikecompensation-based delaymethods,thedesignin[28]cleverlyexploitstheinherentrobustnesstodelay intheparticularstructureofthefeedbacklawandtheplant.Krsticproposedasaturatedcompensator-basedapproachin[30]whichresultsinanonlinearversionofthe SmithPredictor[113]withnestedsaturationfunctions.Thecontrollerisabletoachieve quantiableclosed-loopperformancebyusinganinnitedimensionalcompensatorfor strict-feedforwardnonlinearsystemswithnouncertainties. 1.3Contributions ThecontributionsofChapters2-7arediscussedasfollows: Chapter2:Lasalle-YoshizawaCorollariesforDiscontinuousSystems: Two generalLyapunov-basedstabilitytheoremsaredevelopedusingFilippovsolutionsfor nonautonomousnonlinearsystemswithRHSdiscontinuitiesthroughlocallyLipschitz continuousandregularLyapunovfunctionswhosetimederivativesinthesenseof Filippovcanbeboundedbynegativesemi-denitefunctions.Thechapteralsoposes asanintroductiontoFilippovsolutionsandtheiruseincontroldesignandanalysis. Applicabilityofthecorollariesisillustratedwithtwodesignexamplesincludingan adaptiveslidingmodecontrollawandastandardRISEcontrollaw. Chapter3:SaturatedRISEFeedbackControl: ThemaincontributionofChapter 3isthedevelopmentofanewRISE-basedclosed-looperrorsystemthatconsistsof asaturated,continuoustrackingcontrollerforaclassofuncertain,nonlinearsystems whichincludestime-varyingandnon-LPfunctionsandunmodeleddynamiceffects. NonsmoothanalysismethodsintroducedinChapter2areusedthroughoutthedevelopment.Thetechnicalchallengepresentedbythisobjectiveistheneedtointroduce saturationboundsontheintegralsignumtermwhilemaintainingitsfunctionalityto implicitlylearnthesystemdisturbances.Toachievetheresult,anewauxiliarylter structureisdesignedusinghyperbolicfunctionsthatworkintandemwiththeredesigned 23

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continuoussaturatedRISE-likecontrolstructure.Whilethecontrolleriscontinuous, theclosedlooperrorsystemcontaindiscontinuitieswhichareexaminedthrougha differentialinclusionframework.Theresultingcontrollerisboundedbythemagnitude ofanadjustablecontrolgain,andyieldsasymptotictracking.Theresultisextendedto generalnonlinearsystemswhichcanbedescribedbyELdynamicsandisillustrated withexperimentalresultstodemonstratethecontrolperformance. Chapter4:RISE-BasedControlofanUncertainNonlinearSystemWithTimeVaryingStateDelays: Acontinuouscontrollerisdevelopedforuncertainnonlinear systemswithanunknown,arbitrarilylarge,time-varyingstatedelay.Motivatedby previousworkin[147],acontinuousRISEcontrolstructureisaugmentedwitha three-layerNNtocompensatefortime-varyingstatedelayswhichareargumentsof uncertainnonautonomousfunctionsthatcontainnotlinear-in-the-parametersnon-LP uncertainty.Undertheassumptionthatthetimedelaycanbearbitrarilylarge,bounded andslowlyvarying,LKfunctionalsareutilizedtoprovesemi-globalasymptotictracking. Incomparisontothepreviousworkforconstantstatedelaysin[122],neweffortsinthis chapterrequiredtocompensatefortime-varyingstatedelaysinclude:strategicgrouping ofdelay-dependentanddelay-freetermsandaredesignedLKfunctional.Incomparison to[122],NNsareusedinthecurrentworktocompensateforthenon-LPdisturbances, andneweffortsarerequiredtodesigntheonlineNNupdatelawsinthepresenceofthe unknowntime-varyingdelay. Chapter5:Lyapunov-BasedControlofanUncertainNonlinearSystemwith Time-VaryingInputDelay: Lookinginsteadattimedelayswhichoccurintheinput insteadofthestate,Chapter5presentsacontrolmethodtocompensatefortimevaryinginputdelaysinuncertainnonlinearsystemswithadditivedisturbancesunder theassumptionthatthetimedelayisboundedandslowlyvarying.Inthisresult,LK functionalsandaninnovativePD-likecontrolstructurewithapredictiveintegraltermof pastcontrolvaluesareusedtofacilitatethedesignandanalysisofacontrolmethod 24

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thatcancompensatefortheinputdelay.SincetheLKfunctionalscontaintime-varying delayterms,additionalcomplexitiesareintroducedintotheanalysis.Techniquesused tocompensateforthetime-varyingdelayresultinnewsufcientcontrolconditionsthat dependonthelengthofthedelayaswellastherateofdelay.Thedevelopedcontroller achievessemi-globalUUBtrackingdespitethetime-varyinginputdelay,parametric uncertaintiesandadditiveboundeddisturbancesintheplantdynamics.Anextension togeneralEuler-Lagrangedynamicsystemsisprovidedandtheresultingcontrolleris numericallysimulatedforatwo-linkrobotmanipulatortoexaminetheperformanceofthe developedcontroller. Chapter6:Time-varyingInputAndStateDelayCompensationforUncertain NonlinearSystemsResults: MotivatedbyChapter5'sUUBresult,theprevious time-varyinginputdelayworkisextendedintwodirections:aUtilizingtechniquesfor constantinput-delayedsystemsrstintroducedin[129],time-varyinginputdelaysin anonlinearplantarenowconsidered,btheabilitytocompensateforsimultaneous unknowntime-varyingstatedelaysisadded,andcthestabilityoftheclosed-loop systemisimprovedtoasymptotictracking.Thestatedelayspresentinthesystemare robustlycompensatedforusingadesiredcompensationadaptationlawDCAL-based approach.However,thistechniqueisnotsufcienttocompensateforthesystem's inputdelays.Apredictor-likeerrorsignalbasedonpreviouscontrolvaluesprovides adelay-freeopen-loopsystem,allowingforcontroldesignexibilityandtheuseof morecomplicatedfeedbacksignalsoverthepreviousresultinChapter5.InChapter 5,complexcross-termsthatresultedfromthecontrollerinhibitedtheabilitytoachieve anasymptoticstabilityresult.Incomparison,thisresultusesarobusttechnique, termedtherobustintegralofthesignoftheerrorRISEinsteadofthepreviousPDlikecompensatorisused,allowingforcompensationofthesystemdisturbanceand eliminationoftheultimateboundonthetrackingerror.ALyapunov-basedstability analysisutilizingLyapunov-KrasovskiiLKfunctionalsdemonstratestheabilityto 25

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achievesemi-globalasymptotictrackinginthepresenceofmodeluncertainty,additive sufcientlysmoothdisturbancesandsimultaneoustime-varyingstateandinputdelays. Thestabilityanalysisconsiderstheeffectofarbitrarilysmallmeasurementnoiseand theexistenceofsolutionsfordiscontinuousdifferentialequations.Thesubsequent developmentisbasedontheassumptionthatthestatedelayisboundedandslowly varying,butunknown.ImprovingontheresultinChapter5,theassumptionthatthe inputdelaysmustbesufcientlysmallisrelaxed;instead,theinputdelaysareassumed tobeknown,boundedandslowlyvarying.Numericalsimulationscomparetheresultto thepreviousinput-delayedcontroldesigninChapter5andexaminetherobustnessof themethodtovariouscombinationsofsimultaneousinputandstatedelays. Chapter7:SaturatedControlofanUncertainNonlinearSystemwithInput Delay: Tosafeguardfromtheriskofactuatorsaturationforinput-delayedsystems, theworkpresentedinChapter7introducesanewsaturatedcontroldesignthatcan predict/compensateforinputdelaysinuncertainnonlinearsystems.Basedonthe previousnon-saturatedfeedbackworkandthedesignstructuresutilizedinChapters 3and5,acontinuoussaturatedcontrollerisdevelopedwhichallowstheboundonthe controltobeknownaprioriandtobeadjustedbychangingthefeedbackgains.The saturatedcontrollerisshowntoguaranteeUUBtrackingdespiteaknown,constantinput delay,parametricuncertaintiesandsufcientlysmoothadditivedisturbances.Efforts focusondevelopingadelaycompensatingauxiliarysignaltoobtainadelay-freeopenlooperrorsystemandtheconstructionofanLKfunctionaltocancelthetimedelayed terms.Theresultisextendedtogeneralnonlinearsystemswhichcanbedescribed byELdynamicsandisillustratedwithexperimentalresultstodemonstratethecontrol performance. 26

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CHAPTER2 LASALLE-YOSHIZAWACOROLLARYFORDISCONTINUOUSSYSTEMS Inthischapter,twogeneralizedcorollariestotheLYTarepresentedfornonautonomousnonlinearsystemsdescribedbydifferentialequationswithdiscontinuous right-handsides.Lyapunov-basedanalysismethodswhichachieveasymptoticconvergencewhenthecandidateLyapunovderivativeisupperboundedbyanegative semi-denitefunctioninthepresenceofdifferentialinclusionsarepresented.Two designexamplesillustratetheutilityofthecorollaries. 2.1Preliminaries Afunction f denedonaspace X iscalledessentiallylocallybounded,ifforany x 2 X thereexistsaneighborhood U X of x suchthat f U isaboundedsetfor almostall u 2 U .Theessentialsupremumisthepropergeneralizationofthemaximum tomeasurablefunctions,thetechnicaldifferenceisthatthevaluesofafunctionona setofmeasurezero 1 donotaffecttheessentialsupremum.Giventwometricspaces X;d X and Y;d Y thefunction f : X Y iscalledlocallyLipschitzifforany x 2 X thereexistsaneighborhood U X of x sothat f restrictedto U isLipschitzcontinuous. Asanexample,any C 1 continuousfunctionislocallyLipschitz. Considerthesystem x = f x;t where x t 2D R n denotesthestatevector, f : D [0 ; 1 R n isaLebesgue measurableandessentiallylocallybounded,uniformlyin t function,and D issome openandconnectedset.Existenceanduniquenessofthecontinuoussolution x t areprovidedundertheconditionthatthefunction f isLipschitzcontinuous[148]. 1 RecallthatforsetsintheEuclideann-space R n ,Lebesguemeasureiscommonly utilized.Forexample,anysingletonsets,countablesets,orsubsetsof R n whosedimensionislessthan n areconsideredLebesguemeasurezeroin R n 27

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However,if f containsadiscontinuityatanypointin D ,thenasolutionto2maynot existintheclassicalsense.Thus,itisnecessarytoredenetheconceptofasolution. Utilizingdifferentialinclusions,thevalueofageneralizedsolutione.g.,Filippov[149]or Krasovskii[150]solutionsatacertainpointcanbefoundbyinterpretingthebehaviorof itsderivativeatnearbypoints.Generalizedsolutionswillbeclosetothetrajectoriesof theactualsystemsincetheyarealimitofsolutionsofordinarydifferentialequationswith acontinuousright-handside[10].WhilethereexistsaFilippovsolutionforanyarbitrary initialcondition x t 0 2D ,thesolutionisgenerallynotunique[149,151]. Denition2.1.FilippovSolution [149]Afunction x :[0 ; 1 R n iscalleda solutionof2ontheinterval [0 ; 1 if x t isabsolutelycontinuousandforalmostall t 2 [0 ; 1 x 2 K [ f ] x t ;t where K [ f ] x t ;t isanuppersemi-continuous,nonempty,compactandconvex valuedmapon D ,denedas K [ f ] x t ;t > 0 N =0 cof B x t ; n N;t ; T N =0 denotestheintersectionoversets N ofLebesguemeasurezero, co denotes convexclosure,and B x t ; = f 2 R n jk x t )]TJ/F25 11.9552 Tf 11.955 0 Td [( k < g Remark 2.1 Onecanalsoformulatethesolutionsof2inotherways[152];for instance,usingKrasovskii'sdenitionofsolutions[150].Thecorollariespresentedin thisworkcanalsobeextendedtoKrasovskiisolutionssee[153],forexample.Inthe caseofKrasovskiisolutions,onewouldgetstrongerconclusionsi.e.,conclusionsfora potentiallylargersetofsolutionsatthecostofslightlystrongerassumptionse.g.,local boundednessratherthanessentiallylocalboundedness. Example2.1.DifferentialInclusionComputation 28

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Considerthedifferentialsystemgivenby x = f x;t + g x;t where f x;t = sgn x and g x;t = sin x .BasedonDenition2.1,theFilippov solutionforthesystemin2isgivenby x 2 K [ f + g ] x;t : Basedonthecalculusfor K [ ] developedin[154], K [ f + g ] x K [ f ] x;t + K [ g ] x;t Forcontinuousfunctions,thedifferentialinclusionevaluatedateverypointisequivalent tothecontinuousfunctionevaluatedatthatpoint,i.e, K [ g ] x;t = g x;t .Toexamine howthedifferentialinclusioniscomputed,rstnotethatthesetsofLebesguemeasure zeroofdiscontinuityfor f x;t includethesingletonset f 0 g When x> 0 or x< 0 ,itisstraightforwardtocomputethattheexpressionsfor K [ f ] reducetothesingletons f 1 g and f)]TJ/F15 11.9552 Tf 15.276 0 Td [(1 g ,respectively.Anillustrationofthe positivecaseisdepictedinFigure2-1where 8 only3oftheinnitesizesareshown, thefunction f evaluatedattheappropriatereducedsetisequivalentto K [ f ] x + = co f)]TJ/F15 11.9552 Tf 15.276 0 Td [(1 ; 1 g co f)]TJ/F15 11.9552 Tf 15.276 0 Td [(1 ; 1 g co f 1 g ::: .Computingtheclosedconvexhullofeachintersection reducestheinclusionto K [ f ] x + =[ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ; 1] [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ; 1] f 1 g ::: = f 1 g .Thesame argumentscanbeusedtocomputethedifferentialinclusionfor x< 0 At x =0 ,theexpressionfor K [ f ] x reducesto K [ f ]= T > 0 co [ sgn B ; )-222(f 0 g ] Since B ; > 0 ,anopenintervalcontainingtheorigin,intersectsboth ; 1 and ; 0 onsetsofpositivemeasure, K [ f ]= T > 0 co [ sgn [ x )]TJ/F25 11.9552 Tf 11.955 0 Td [(;x + ] )-222(f 0 g ]= co f)]TJ/F15 11.9552 Tf 15.276 0 Td [(1 ; 1 g =[ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ; 1] .ThisclosureisillustratedinFigure2-2. Thusitiseasytoseethatthedifferentialinclusioncanbedescribedby x 2 SGN x + sin x where SGN istheset-valuedsignfunctiondenedby SGN x =1 if x> 0 ; [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ; 1] if x =0 ,and )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 if x< 0 .Soat x 6 =0 x 2 K [ f + g ] isasingletonandat x =0 x 2 K [ f + g ] isaset. 29

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Figure2-1.Setclosureof K [ f ] x for x> 0 case. Figure2-2.Setclosureof K [ f ] x for x =0 case. Tofacilitatethemainresults,threedenitionsareprovided.Clarke'sgeneralized gradientisusedinmanyLyapunov-basedtheoremsusingnonsmoothanalysis.To introducethisidea,thedenitionofaregularfunctionasdenedbyClarke[56]is presented. Denition2.2.DirectionalDerivative [155]Givenafunction f : R m R n ,theright directionalderivativeof f at x 2 R m inthedirectionof v 2 R m isdenedas f 0 x;v =lim t 0 + f x + tv )]TJ/F25 11.9552 Tf 11.955 0 Td [(f x t : Additionally,thegeneralizeddirectionalderivativeof f at x inthedirectionof v isdened as f o x;v =lim y x sup t 0 + f y + tv )]TJ/F25 11.9552 Tf 11.955 0 Td [(f y t : 30

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Denition2.3.RegularFunction [56]Afunction f : R m R n issaidtoberegularat x 2 R m ifforall v 2 R m ,therightdirectionalderivativeof f at x inthedirectionof v exists and f 0 x;v = f o x;v 2 ThefollowingLemmaprovidesamethodforcomputingthetimederivativeofa regularfunction V usingClarke'sgeneralizedgradient[56]and K [ f ] x;t alongthe solutiontrajectoriesofthesystemin2. Denition2.4.Clarke'sGeneralizedGradient [56]Forafunction V : R n R R thatislocallyLipschitzin x;t ,denethegeneralizedgradientof V at x;t by @V x;t = co f lim r V x;t j x i ;t i x;t ; x i ;t i = 2 V g where V isthesetofmeasurezerowherethegradientof V isnotdened. Denition2.5.Locallybounded,uniformlyin t Let f : D [0 ; 1 R .Themap x f x;t islocallybounded,uniformlyin t ,ifforeachcompactset K D ,thereexists c> 0 suchthat j f x;t j c 8 x;t 2 K [0 ; 1 Lemma2.1. ChainRule[45]Let x t beaFilippovsolutionofsystem2and V : D [0 ; 1 R bealocallyLipschitz,regularfunction.Then V x t ;t isabsolutely continuous, d dt V x t ;t existsalmosteverywherea.e.,i.e.,foralmostall t 2 [0 ; 1 and V x t ;t a:e: 2 ~ V x t ;t where ~ V x;t 2 @V x;t T 2 6 4 K [ f ] x;t 1 3 7 5 : Remark 2.2 Throughoutthesubsequentdiscussion,forbrevityofnotation,leta.e.refer toalmostall t 2 [0 ; 1 2 Notethatany C 1 continuousfunctionisregularandthesumofregularfunctionsis regular[156]. 31

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2.2MainResult Forthesystemdescribedin2withacontinuousright-handside,existing Lyapunovtheorycanbeusedtoexaminethestabilityoftheclosed-loopsystemusing continuoustechniquessuchasthosedescribedin[148].However,thesetheorems mustbealteredfortheset-valuedmap ~ V x t ;t forsystemswithright-handsides whicharenotLipschitzcontinuous[10,11,45].Lyapunovanalysisfornonsmooth systemsisanalogoustotheanalysisusedforcontinuoussystems.Thedifferences arethatdifferentialequationsarereplacedwithinclusions,gradientsarereplacedwith generalizedgradients,andpointsarereplacedwithsetsthroughouttheanalysis.The followingpresentationandsubsequentproofsdemonstratehowtheLYTcanbeadapted forsuchsystems. Thefollowingauxiliarylemmafrom[154]andBarbalat'sLemmaareprovidedto facilitatetheproofsofthenonsmoothLYC. Lemma2.2. [154]Let x t beanyFilippovsolutiontothesystemin2and V : D [0 ; 1 R bealocallyLipschitz,regularfunction.If V x t ;t a:e: 0 ,then V x t ;t V x t 0 ;t 0 8 t>t 0 Proof. Forthesakeofcontradiction,letthereexistsome t>t 0 suchthat V x t ;t > V x t ;t 0 .Then, t t 0 V x ; d = V x t ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(V x t ;t 0 > 0 : Itfollowsthat V x t ;t > 0 onasetofpositivemeasure,whichcontradictsthat V x t ;t 0 ,a.e. ThefollowingLemmarecallsBarbalat'slemmafornonautonomoussystems,which willbeusedintheproofofthenonsmoothLYC. 32

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Lemma2.3. Barbalat'sLemma [148]Let : R R beauniformlycontinuousUC functionon [0 ; 1 .Supposethat lim t !1 t 0 d existsandisnite.Then, t 0 ast !1 : BasedonLemmas2.2and2.3,nonsmoothcorollariestotheLYTc.f.,[55,Theorem8.4]and[68,TheoremA.8]areprovidedinCorollary2.1and2.2. Corollary2.1. Forthesystemin2,let D R n beanopenandconnectedset containing x =0 andsuppose f isLebesguemeasurableandessentiallylocally bounded,uniformlyin t .Let V : D [0 ; 1 R belocallyLipschitzandregularsuchthat W 1 x V x;t W 2 x 8 t 0 ; 8 x 2D V x t ;t a:e: )]TJ/F25 11.9552 Tf 23.835 0 Td [(W x t where W 1 and W 2 arecontinuouspositivedenitefunctions,and W isacontinuous positivesemi-denitefunctionon D .Choose r> 0 and c> 0 suchthat B r D and c< min k x k = r W 1 x and x t isaFilippovsolutionto2where x t 0 2f x 2 B r j W 2 x c g Then x t isboundedandsatises W x t 0 ast !1 : Proof. Since B r D and c< min k x k = r W 1 x f x 2 B r j W 1 x c g isintheinteriorof B r Deneatime-dependentset t;c by t;c = f x 2 B r j V x;t c g : From2,theset t;c contains f x 2 B r j W 2 x c g since W 2 x c V x;t c: 33

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Ontheotherhand, t;c isasubsetof f x 2 B r j W 1 x c g since V x;t c W 1 x c: Thus, f x 2 B r j W 2 x c g t;c f x 2 B r j W 1 x c g B r D: Basedon2, V x t ;t a:e: 0 ,hence, V x t ;t isnon-increasingfromLemma 2.2.Forany t 0 0 andany x t 0 2 t 0 ;c ,thesolutionstartingat x t 0 ;t 0 staysin t;c forevery t t 0 .Therefore,anysolutionstartingin f x 2 B r j W 2 x c g staysin t;c ,andconsequentlyin f x 2 B r j W 1 x c g ,forallfuturetime.Hence,theFilippov solution x t isboundedsuchthat k x t k t 0 .Existenceof lim t !1 t t 0 W x d isguaranteedsincetheleft-handsideof2ismonotonicallynondecreasingbasedonthe denitionof W x in2.1andboundedabove.Since x t islocallyabsolutelycontinuousand f isessentiallylocallybounded,uniformlyin t x t isuniformlycontinuous. 3 Because W x iscontinuousin x and x isonthecompactset B r W x t isuniformly continuousin t on t 0 ; 1 ] .Therefore,byLemma2.3,itconcludesthat W x t 0 ast !1 : 3 Since x t islocallyabsolutelycontinuous, j x t 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(x t 1 j = t 2 t 1 x t dt .Fromtheassumptionthat x f x;t isessentiallylocallybounded,uniformlyin t andsince x 2 L 1 ,then, x 2L 1 .Usingthefactthatdening x t onasetofzeromeasuredoesnot change x impliesthat t 2 t 1 x t dt t 2 t 1 Mdt ,where M isaconstant.Thus, t 2 t 1 Mdt = M j t 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(t 1 j ,hence x t isuniformlycontinuous. 34

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Remark 2.3 FromDef.2.1, K [ f ] x;t isanuppersemi-continuous,nonempty,compact andconvexvaluedmap.WhileexistenceofaFilippovsolutionforanyarbitraryinitial condition x t 0 2D isprovidedbythedenition,generallyspeaking,thesolutionis non-unique[149,151]. NotethatCorollary2.1establishes2foraspecic x t .Underthestronger conditionthat 4 ~ V x;t W x 8 x 2D ,itispossibletoshowthat2holdsforall Filippovsolutionsof2.Thenextcorollaryispresentedtoillustratethispoint. Corollary2.2. Forthesystemgivenin2,let D R n beadomaincontaining x =0 andsuppose f isLebesguemeasurableandessentiallylocallybounded,uniformlyin t Let V : D [0 ; 1 R belocallyLipschitzandregularsuchthat W 1 x V x;t W 2 x ~ V x;t )]TJ/F25 11.9552 Tf 21.918 0 Td [(W x 8 t 0 ; 8 x 2D where W 1 and W 2 arecontinuouspositivedenitefunctions,and W isacontinuouspositivesemi-denitefunctionon D .Choose r> 0 and c> 0 suchthat B r D and c< min k x k = r W 1 x .Then,allFilippovsolutionsof2suchthat x t 0 2f x 2 B r j W 2 x c g areboundedandsatisfy W x t 0 ast !1 : Proof. Let x t beanyarbitraryFilippovsolutionof2.Then,fromLemma2.1, and2, V x t ;t a:e: )]TJ/F25 11.9552 Tf 25.163 0 Td [(W x t ,whichispreciselythecondition2.Sincethe 4 Theinequality ~ V x;t W x isusedtoindicatethateveryelementoftheset ~ V x;t islessthanorequaltothescalar W x 35

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selectionof x t isarbitrary,Corollary2.1canbeusedtoimplythattheresultin2 holdsforeach x t 2.3DesignExample1Adaptive+SlidingMode TheLYCandtheLaSalle-YoshizawaTheoremareusefulinitsabilitytoprovide boundednessandconvergenceofsolutions,whileprovidingacompactframework todenetheregionofattractionforwhichboundednessandconvergenceresults hold.Infact,theregionofattractionisprovidedaspartofthecorollarystructures. Inthecaseofsemi-globalandlocalresults,thesedomainsandsetsareespecially useful.ItisimportanttonotethatBarbalat'sLemmacanbeusedtoachievethesame resultsinfact,itisusedintheproofforCorollary2.1;however,theuseofBarbalat's Lemmawouldrequiretheidenticationoftheregionofattractionforwhichconvergence holdsanddoesnotprovideboundednessofthetrajectories.Forillustrativepurposes, thefollowingdesignexampletargetstheregulationofarstordernonlinearsystem. Corollary2.1and2.2canalsobedirectlyappliedtogeneral n th ordertime-varying nonlinearsystemsandtotrackingcontrolproblems. ToillustratetheutilityofCorollary2.2,considerarstordernonlineardifferential equationgivenby x = f x;t + d x;t + u t where f : R n [0 ; 1 R n isanunknown,linear-parameterizable,essentiallylocally bounded,uniformlyin t functionthatcanbeexpressedas f x;t = Y x;t when 2 R p isavectorofunknownconstantparameters,and Y : R n [0 ; 1 R n p [0 ; 1 istheregressionmatrixfor f x;t u :[0 ; 1 R n isthecontrolinput, x t 2 R n is themeasurablesystemstate,and d x;t isanessentiallylocallyboundeddisturbance whichsatises k d x;t k c 1 + c 2 k x k k x k 36

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where c 1 2 R + isapositiveconstant,and c 2 k x k : R + R + isapositive,globally invertible,state-dependentfunction.Aregulationcontrollerfor2canbedesigned as u x;t )]TJ/F25 11.9552 Tf 9.298 0 Td [(k 1 x )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 2 sgn x )]TJ/F25 11.9552 Tf 11.955 0 Td [(Y ^ where ^ x;t 2 R p istheestimateof k 1 ;k 2 2 R + aregainconstants,and sgn is dened 8 2 R n = 1 2 ::: n T as sgn sgn 1 sgn 2 :::sgn n T : Basedonthesubsequentstabilityanalysis,anadaptiveupdatelawcanbedenedas ^ =)]TJ/F25 11.9552 Tf 19.74 0 Td [(Y T x where )]TJ/F23 11.9552 Tf 10.635 0 Td [(2 R n n isapositivegainmatrix.Theclosed-loopsystemisgivenby x = Y ~ + d x;t )]TJ/F25 11.9552 Tf 11.956 0 Td [(k 1 x )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 2 sgn x where ~ 2 R p denotesthemismatch ~ )]TJ/F15 11.9552 Tf 14.006 3.155 Td [(^ .In2,itisapparentthatthe RHScontainsadiscontinuityin x t andrequirestheuseofdifferentialinclusionsto provideexistenceofsolutions.Let y x; ~ 2 R n + p denote y 2 6 4 x ~ 3 7 5 andchoosea positive-denite,locallyLipschitz,regularLyapunovcandidatefunctionas V y = 1 2 x T x + 1 2 ~ T )]TJ/F28 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.587 0 Td [(1 ~ : ThecandidateLyapunovfunctionin2satisesthefollowinginequalities: W 1 y V y W 2 y wherethecontinuouspositive-denitefunctions W 1 y ;W 2 y 2 R aredenedas W 1 y 1 k y k 2 W 2 y 2 k y k 2 and 1 ; 2 2 R + areknownconstants.Then, 37

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_ V y t ;t a:e: 2 ~ V y t ;t and ~ V = 2 @V x; ~ ;t T K 2 6 6 6 6 4 x ~ 1 3 7 7 7 7 5 x; ~ ;t : Since V y;t is C 1 in y 5 ~ V r V T K 2 6 4 x ~ 3 7 5 x; ~ h x T ; ~ T )]TJ/F28 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.587 0 Td [(1 i K 2 6 4 x ~ 3 7 5 x; ~ : Afterusing2,theexpressionin2canbewrittenas ~ V x T Y ~ + d x;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 1 x )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 2 K [ sgn x ] )]TJ/F15 11.9552 Tf 12.894 3.155 Td [(~ T )]TJ/F28 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.587 0 Td [(1 ^ where K [ sgn x ]= SGN x suchthat SGN x i =1 if x i > 0 ; [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ; 1] if x i =0 ,and )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 if x i < 0 8 i =1 ; 2 ;:::;n Remark 2.4 Onecouldalsoconsiderthediscontinuousfunctioninsteadofthedifferentialinclusioni.e.,the sgn functioncanalternativelybedenedas sgn =0 usingCaratheodorysolutions;however,thismethodlackswouldnotbeanindicator forwhathappenswhenmeasurementnoiseispresentinthesystem.Asdescribedin resultssuchas[157159],FilippovandKrasovskiisolutionsfordiscontinuousdifferential equationsareappropriateforcapturingthepossibleclosed-loopsystembehaviorin thepresenceofarbitrarilysmallmeasurementnoise.Byutilizingthesetvaluedmap SGN intheanalysis,weaccountforthepossibilitythatwhenthetruestatesatises 5 ForcontinuouslydifferentiableLyapunovcandidatefunctions,thegeneralizedgradientreducestothestandardgradient.However,thisisnotrequiredbytheCorollaryitself andonlyassistsinevaluation. 38

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x =0 sgn x ofthemeasuredstatefallswithintheset [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ; 1] .Therefore,thepresentedanalysisismorerobusttomeasurementnoisethanananalysisthatdependson sgn tobedenedasaknownsingleton. Substitutingfortheadaptiveupdatelawin2,cancelingtermsandutilizingthe boundfor d x;t in2,theexpressionin2canbeupperboundedas ~ V )]TJ/F25 11.9552 Tf 21.918 0 Td [(k 1 k x k 2 + c 1 k x k + c 2 k x k k x k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 2 k x k : Thesetin2reducestothescalarinequalityin2sinceinthecasewhen K [ sgn x ] isdenedasaset,itismultipliedby x ,i.e.,when x =0 ; 0 SGN = f 0 g Regroupingsimilarterms,theexpressionin2canbewrittenas ~ V )]TJ/F15 11.9552 Tf 23.91 0 Td [( k 1 )]TJ/F25 11.9552 Tf 11.956 0 Td [(c 2 k x k k x k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 1 k x k : Provided k 2 >c 1 and k 1 >c 2 k x k ,theexpressionin2canbeupperbounded as ~ V )]TJ/F25 11.9552 Tf 24.492 0 Td [(W y t where W y isapositivesemi-denitefunctiondenedonthe domain D y jk y k
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InvokingCorollary2.2, W y t = )]TJ/F15 11.9552 Tf 11.291 0 Td [( k 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 2 k x k k x k 2 0 ast !18 y 2S ,thus, x 0 ast !18 y 2S .Theregionofattractionin2canbemadearbitrarily largetoincludeallinitialconditionsasemi-globaltyperesultbyincreasingthegain k 1 Remark 2.5 Forsomesystemse.g.,closed-looperrorsystemswithslidingmode controllaws,itmaybepossibletoshowthatCorollary2.2ismoreeasilyapplied,as isthefocusoftherstexample.However,inothercases,itmaybedifculttosatisfy theinequalityin2.TheusefulnessofCorollary2.1isdemonstratedinthosecases whereitisdifcultorimpossibletoshowthattheinequalityin2canbesatised,but itispossibletoshowthat2canbesatisedforalmostalltime,asisthefocusofthe nextexample. 2.4DesignExample2RISE ToillustratetheutilityofCorollary2.1,considerasecondordernonlineardifferential equationgivenby x = f x;t + d t + u t where f : R n [0 ; 1 R n isanunknownessentiallylocallybounded,uniformly in t function, u :[0 ; 1 R n isthecontrolinput, x t 2 R n isthemeasurable systemstate,and d t 2 R n isanessentiallylocallyboundeddisturbance,which satises d t ; d t ; d t 2L 1 : Adesiredtrajectory,denotedby x d 2 R n ,satises x i d t 2 R n ; 8 i =0 ; 1 ;:::; 4 Toquantifythecontrolobjective,atrackingerror,denotedby e 1 ; d 2 R 6 ,is denedas e 1 x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(x; andtwoauxiliarytrackingerrorsdenotedby e 2 e 1 ; e 1 ;r e 2 ; e 2 2 R 6 ,aredenedas e 2 e 1 + 1 e 1 ; r e 2 + 2 e 2 40

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where 1 ; 2 2 R + areadjustablegains.Theauxiliarysignal r e 2 ; e 2 isintroducedto facilitatethestabilityanalysisandisnotusedinthecontroldesignsincetheexpression in2dependsontheunmeasurablestate x t Theopenlooperrorsystemcanbeexpressedas r = x d + S )]TJ/F25 11.9552 Tf 11.955 0 Td [(f x d ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(d t )]TJ/F25 11.9552 Tf 11.955 0 Td [(u t + 2 e 2 wheretheauxiliaryfunction S 2 R n isdenedas S f x d ;t )]TJ/F25 11.9552 Tf 12.1 0 Td [(f x;t + 1 e 1 + 2 e 2 .A RISE-basedcontrolstructure[8,160]canbedesignedas u k s +1 e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k s +1 e 2 + where e 2 2 R n istheFilippovsolutiontothefollowingdifferentialequation k s +1 e 2 + sgn e 2 ; =0 ; ;k s 2 R arepositive,constantcontrolgainsand sgn isdened 8 2 R m = 1 2 ::: m T as sgn sgn 1 sgn 2 :::sgn m T : Thedifferential equationgivenin2iscontinuousexceptwhen e 2 =0 : UsingFilippov'stheory ofdifferentialinclusions[149,161163],theexistenceofsolutionscanbeestablished for 2 K [ h 1 ] e 2 ,where h 1 e 2 2 R n isdenedastheright-handsideof2and K [ h 1 ] T > 0 T S m =0 coh 1 B e 2 ; )]TJ/F25 11.9552 Tf 11.955 0 Td [(S m ,where T S m =0 denotestheintersectionofall sets S m ofLebesguemeasurezeroofdiscontinuities, co denotesconvexclosure,and B e 2 ; = f & 2 R jk e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(& k < g [45,154]. Tofacilitatethesubsequentanalysis,thecontrollerin2issubstitutedinto 2andthetimederivativefoundbyutilizingaDCALapproachtoregrouptermsas r = ~ N + N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(e 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [( k s +1 r )]TJ/F25 11.9552 Tf 11.955 0 Td [(sgn e 2 where ~ N e 2 ;r;t 2 R n and N d t 2 R n aredenedas ~ N S + e 2 ; 41

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N d ... x d )]TJ/F15 11.9552 Tf 15.803 3.155 Td [(_ f x d ; x d ;t + d t : Using2-2andtheMeanValueTheorem,thefunction ~ N in2canbe upperboundedas[164,AppA] ~ N k z k k z k ; where z e 1 ;e 2 ;r 2 R 3 n isdenedas z e T 1 e T 2 r T T and : R R isapositive,globallyinvertible,nondecreasingfunction.Assumingthe disturbanceanddesiredtrajectoryaresufcientlysmooth,thefollowinginequalitiescan bedeveloped: k N d k 1 ; N d 2 where 1 ; 2 2 R + areknownconstants. Let y z;P 2 R 3 n +1 bedenedas y z T p P T wheretheauxiliaryfunction P e 2 ;t 2 R isdenedastheFilippovsolutiontothe followingdifferentialequation P = )]TJ/F25 11.9552 Tf 9.299 0 Td [(r T N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(sgn e 2 P e 2 t 0 ;t 0 = n X i =1 j e 2 i t 0 j)]TJ/F25 11.9552 Tf 17.933 0 Td [(e 2 t 0 T N d t 0 wherethesubscript i =1 ; 2 ;:::;n denotesthe i thelementofthevector.Similartothe developmentin2,existenceofsolutionsfor P canbeestablishedusingFilippov's theoryofdifferentialinclusionsfor P 2 K [ h 2 ] e 2 ;r;t ,where h 2 e 2 ;r;t 2 R isdenedas h 2 )]TJ/F25 11.9552 Tf 9.299 0 Td [(r T N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(sgn e 2 and K [ h 2 ] T > 0 T S m =0 coh 2 B e 2 ; )]TJ/F25 11.9552 Tf 11.955 0 Td [(S m ;r;t asin2. Integrating2bypartsandprovided > 1 + 2 2 P e 2 ;t 0 see[165]fordetails. 42

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Let V L : D [0 ; 1 R becontinuouslydifferentiablein y ,locallyLipschitzin t regular,anddenedas V L = 1 2 e T 1 e 1 + 1 2 e T 2 e 2 + 1 2 r T r + P whichsatisesthefollowinginequalities: U 1 y V L y;t U 2 y ; where U 1 y U 2 y 2 R arepositivedenitefunctionsdenedas U 1 1 k y k 2 and U 2 2 k y k 2 UnderFilippov'sframework,thetimederivativeof2existsalmosteverywhere a.e.,i.e.,foralmostall t 2 [0 ; 1 ,and V L y;t a:e: 2 ~ V L y;t where ~ V L = 2 @V L y;t T K e T 1 e T 2 r T 1 2 P )]TJ/F18 5.9776 Tf 7.782 3.258 Td [(1 2 P 1 T ; where @V L isthegeneralizedgradientof V L y;t [45,154,166].Since V y;t is C 1 in y ~ V L r V T L K e T 1 e T 2 r T 1 2 P )]TJ/F18 5.9776 Tf 7.782 3.259 Td [(1 2 P T ; where r V L e T 1 e T 2 r T 2 P 1 2 T : Usingthecalculusfor K [ ] from[154],substituting2,2,2,2, and2,andcancelingsimilarterms,theexpressionin2becomes ~ V L e T 1 e 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [( 1 e T 1 e 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 e T 2 e 2 + r T ~ N + r T N d )]TJ/F15 11.9552 Tf 11.955 0 Td [( k s +1 r T r )]TJ/F25 11.9552 Tf 9.299 0 Td [(r T K [ sgn e 2 ] )]TJ/F25 11.9552 Tf 11.955 0 Td [(r T N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(K [ sgn e 2 ] ; 43

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where K [ sgn e 2 ]= SGN e 2 suchthat SGN e 2 i =1 if e 2 i > 0 ; [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ; 1] if e 2 i =0 and )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 if e 2 i < 0 : 6 Utilizingthefactthatthesetin2reducestoascalarequality sincetheRHSiscontinuousa.e.,i.e,theRHSiscontinuousexceptfortheLebesgue negligiblesetoftimeswhen r T K [ sgn e 2 ] )]TJ/F25 11.9552 Tf 12.466 0 Td [(r T K [ sgn e 2 ] 6 =0 [45,167],anupper boundfor V L isgivenas V L a:e: )]TJ/F25 11.9552 Tf 24.498 0 Td [( 1 k e 1 k 2 + k e 1 kk e 2 k)]TJ/F25 11.9552 Tf 20.589 0 Td [( 2 k e 2 k 2 + k z k k r kk z k)]TJ/F15 11.9552 Tf 20.59 0 Td [( k s +1 k r k 2 : Toshowthatthenumberoftimeswhen r T K [ sgn e 2 ] )]TJ/F25 11.9552 Tf 12.608 0 Td [(r T K [ sgn e 2 ] 6 =0 is measurezero,werecalltheerrorsystemdenitionin2andintroducethefollowing lemma. Lemma2.4. Let f :[0 ; 1 R beacontinuouslydifferentiablefunctionwiththe property: f x =0 ;f 0 t 6 =0 ; then )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [(f )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 f 0 g =0 ; where denotestheLebesguemeasureon [0 ; 1 Proof. Wewillrstprovethatallthepointsintheset f )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 f 0 g areisolated.Thatis, )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(8 a 2 f )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 f 0 g 9 > 0 j \000 a )]TJ/F25 11.9552 Tf 11.955 0 Td [(;a + )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(f )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 f 0 g nf a g = : Toobtainacontradiction,thenegationofthestatementaboveis, )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(9 a 2 f )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 f 0 g j 8 > 0 \000 a )]TJ/F25 11.9552 Tf 11.955 0 Td [(;a + )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(f )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 f 0 g nf a g6 = : 6 Asinthepreviousexample,the sgn functioncanalternativelybedenedas sgn =0 ;however,thisrestrictionlacksrobustnesswithrespecttomeasurement noise. 44

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Assuming2,let b 2 a )]TJ/F25 11.9552 Tf 11.955 0 Td [(;a + f )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 f 0 g nf a g .Withoutlossofgeneralitywe canassume b>a and f 0 a > 0 : As f isdifferentiableand f a = f b =0 ,byRolle's theorem, 9 c 2 a;b suchthat f 0 c =0 : Bycontinuityof f 0 at a; 8 a > 0 9 a > 0 j 8 x 2 [0 ; 1 j x )]TJ/F25 11.9552 Tf 11.955 0 Td [(a j < a = f 0 a )]TJ/F25 11.9552 Tf 11.955 0 Td [( a 0 j 8 x 2 [0 ; 1 j x )]TJ/F25 11.9552 Tf 11.955 0 Td [(a j < a = f 0 x > 0 : Now,pick = a in2.Thusfrom b 2 a )]TJ/F25 11.9552 Tf 11.955 0 Td [( a ;a + a f )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 f 0 g nf a g weget j b )]TJ/F25 11.9552 Tf 11.955 0 Td [(a j < a whichfrom c 2 a;b implies j c )]TJ/F25 11.9552 Tf 11.955 0 Td [(a j < a whichimplies f 0 c > 0 ,which contradicts2. Thus,allthepointsintheset f )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 f 0 g areisolated,andhence, f )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 f 0 g isa discreteset.AsanydiscretesubsetofEuclideanspaceiscountable,2isobtained. Thesetoftimes n t 2 [0 ; 1 : r t T K [ sgn e 2 t ] )]TJ/F25 11.9552 Tf 11.955 0 Td [(r t T K [ sgn e 2 t ] 6 =0 o [0 ; 1 isequivalenttothesetoftimes f t : e 2 t =0 ^ r t 6 =0 g .From2,thissetcan alsoberepresentedby f t : e 2 t =0 ^ e 2 t 6 =0 g .Provided e 2 t iscontinuously differentiableitisinourcase,Lemma2.4canbeusedtoshowthatthesetoftime instances f t : e 2 t =0 ^ e 2 t 6 =0 g isisolated,andthus,measurezero.Thisimplies thattheset ismeasurezero. UtilizingYoung'sInequality,theexpressionin2canbereducedto V L a:e: )]TJ/F25 11.9552 Tf 23.834 0 Td [( k z k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(k s k r k 2 + k z k k r kk z k ; 45

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where =min 1 )]TJ/F24 7.9701 Tf 13.151 4.707 Td [(1 2 ; 2 )]TJ/F24 7.9701 Tf 13.15 4.707 Td [(1 2 ; 1 and z e 1 ;e 2 ;r wasdenedin2.Ifthegains areselectedsuchthat 1 > 1 2 and 2 > 1 2 ,andbycompletingthesquaresfor r ,the expressionin2canbeupperboundedas V L a:e: )]TJ/F25 11.9552 Tf 23.834 0 Td [( k z k 2 + 2 k z k k z k 2 4 k s )]TJ/F25 11.9552 Tf 21.918 0 Td [(U y ; where U y c k z k 2 ,forsomepositiveconstant c 2 R ; isacontinuouspositive semi-denitefunctionsuchthat D n y 2 R 3 n +1 jk y k )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 p k s o Thesizeofthedomain D canbeenlargedbyincreasingthegain k s .Theinequalitiesin 2and2canbeusedtoshowthat V L 2L 1 in D .Thus, e 1 ;e 2 ;r 2L 1 in D .Theclosed-looperrorsystemcanbeusedtoconcludethattheremainingsignals areboundedin D ,andthedenitionsfor U y and z canbeusedtoshowthat U y isuniformlycontinuousin D .Let S D D denoteasetdenedas S D y 2Dj U 2 y < 1 )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 p k s 2 : Theregionofattractionin S D canbemadearbitrarilylargetoincludeanyinitialconditionsbyincreasingthecontrolgain k s .From2,Corollary2.1canbeinvokedto showthat c k z k 2 0 ast !18 y 2S D : Basedonthedenitionof z in2, k e 1 k! 0 ast !18 y 2S D : Remark 2.6 InExample2.4,weapplyCorollary2.1when V L y;t a:e: )]TJ/F25 11.9552 Tf 25.595 0 Td [(W y .The differenceinthiscasefromExample2.3,stemsfromthefactthatitisnotpossibleto directlyshowthatallsolutionssatisfy2.Instead,itispossibletoshowthat2can besatisedintheanalysis,andLemma2.4andtheassociatedargumentscanbeused toprovethatthiscaseissatisedforalltime. 46

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2.5Summary Thischapterintroducedthemechanicsrequiredtoutilizenonsmoothanalysisin Lyapunov-basedcontroldesignandhaveextendedtheLYTtodifferentialsystemswitha discontinuousRHSusingFilippovdifferentialinclusions.Theresultpresentstheoretical toolsapplicabletononlinearsystemswithdiscontinuitiesintheplantdynamicsorinthe controlstructure.ThegeneralizedLyapunov-basedanalysismethodsaredeveloped usingdifferentialinclusionsinthesenseofFilippovtoachieveasymptoticconvergence whentheLyapunovderivativeisupperboundedbyanegativesemi-denitefunction. CaseswhentheboundontheLyapunovderivativeholdsforallpossibleFilippov solutionsarealsoconsidered.Anadaptive,slidingmodecontrolexampleandaRISE controlexampleareprovidedtoillustratetheutilityofthemainresults. 47

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CHAPTER3 SATURATEDRISEFEEDBACKCONTROL Inthischapter,asaturatedcontrollerisdevelopedforaclassofuncertain,secondorder,nonlinearsystemswhichincludestime-varyingandnon-LPfunctionswith boundeddisturbancesusingacontinuouscontrollawwithsmoothsaturationfunctions. BasedontheRISEcontrolmethodology,theproposedcontrollerisabletoutilizethe benetsofhighgaincontrolstrategieswhileguaranteeingsaturationlimitsarenot surpassed.Theboundsonthecontrolareknownaprioriandcanbeadjustedby changingthefeedbackgains.Thesaturatedcontrolleryieldsasymptotictrackingdespite uncertaintyandaddeddisturbancesinthedynamics.Experimentalresultsusinga two-linkrobotmanipulatordemonstratetheperformanceoftheproposedcontroller. 3.1DynamicModel Considerageneralclassofnonlinearsystemsofthefollowingform: x = f x; x;t + u x; x;t + d t where x t ; x t 2 R n arethegeneralizedsystemstates, u x; x;t 2 R n isthe generalizedcontrolinput, f x; x;t : R 2 n [0 ; 1 R n isanunknownnonlinear C 2 function,and d t :[0 ; 1 R n denotesageneralized,sufcientlysmooth,nonvanishingnonlineardisturbancee.g.,unmodeledeffects. Thesubsequentdevelopmentisbasedontheassumptionthat x t and x t are measurableoutputs.Additionally,thefollowingassumptionswillbeexploited. Assumption3.1. Thenonlineardisturbancetermanditsrsttwotimederivativesi.e., d t ; d t ; d t existandareboundedbyknownconstants[122,147,168]. 1 1 Manypracticaldisturbancetermsarecontinuousincludingfrictionsee[169,170], winddisturbances,wave/oceandisturbances,unmodeledsufcientlysmoothdisturbances,etc.. 48

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Assumption3.2. Thedesiredtrajectory x d t 2 R n isdesignedsuchthat x i d t 2 R n ; 8 i =0 ; 1 ;:::; 4 existandarebounded. 2 Remark 3.1 Toaidthesubsequentcontroldesignandanalysis,thevector Tanh 2 R n andthematrix Cosh 2 R n n aredenedas Tanh [ tanh 1 ;:::;tanh n ] T Cosh diag f cosh 1 ;:::;cosh n g where =[ 1 ;:::; n ] T 2 R n .Basedonthedenitionof3,thefollowinginequalities hold 8 2 R n [78]: k k 2 n X i =1 ln cosh i 1 2 tanh 2 k k ; k k > k Tanh k k Tanh k 2 tanh 2 k k ; T Tanh Tanh T Tanh : Throughoutthepaper, kk denotesthestandardEuclideannorm. 3.2ControlDevelopment Theobjectiveistodesignanamplitude-limited,continuouscontrollerwhichensures thesystemstate x t tracksadesiredtrajectory x d t .Toquantifythecontrolobjective, atrackingerrordenoted e 1 x;x d 2 R n isdenedas e 1 x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(x: Embeddingthecontrolinaboundedtrigonometrictermisanobviouswayto limitthecontrolauthoritybelowanapriorilimit;however,byinjectingtheseterms, difcultyarisesintheclosed-loopstabilityanalysis.Thischallengeisexacerbatedby thepresenceofintegralcontrolfunctionsthatareincludedtocompensateforadded 2 Manyguidanceandnavigationapplicationsutilizesmooth,high-orderdifferentiable desiredtrajectories.Curvettingmethodscanalsobeusedtogeneratesufciently smoothtime-varyingtrajectories. 49

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disturbancesasinthisresult.Motivatedbythesestabilityanalysiscomplexitiesand throughaniterativeanalysisprocedure,twomeasurablelteredtrackingerrorsare designedwhichincludeextrasmoothsaturationterms.Specically,thelteredtracking errors e 2 e 1 ; e 1 ;e f ;r e 2 ; e 2 2 R n ,aredenedas e 2 e 1 + 1 Tanh e 1 + Tanh e f ; r e 2 + 2 Tanh e 2 + 3 e 2 where 1 ; 2 ; 3 2 R denoteconstantpositivecontrolgains, Tanh wasdenedin 3,and e f e 1 ;e 2 2 R n isanauxiliarysignalwhosedynamicsaregivenby e f Cosh 2 e f f)]TJ/F25 11.9552 Tf 15.276 0 Td [( 1 e 2 + Tanh e 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 Tanh e f g and 1 ; 2 2 R areconstantpositivecontrolgains.Theauxiliarysignal r e 2 ; e 2 is introducedtofacilitatethestabilityanalysisandisnotusedinthecontroldesignsince theexpressionin3dependsontheunmeasurablegeneralizedstate x t .The structureoftheerrorsystemsandincludedauxiliarysignalsismotivatedbytheneed toinjectandcanceltermsinthesubsequentstabilityanalysis,andwillbecomeapparent inSection3.3. Anopen-looptrackingerrorcanbeobtainedbyutilizingthelteredtrackingerrorin 3andsubstitutingfrom3,3,3,and3toyield r = S )]TJ/F25 11.9552 Tf 11.955 0 Td [(f d + x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(d )]TJ/F25 11.9552 Tf 11.955 0 Td [(u t )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 e 2 wheretheauxiliaryfunction S e 1 ;e 2 ;e f ;t 2 R n isdenedas S f d )]TJ/F25 11.9552 Tf 11.956 0 Td [(f )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 Tanh e f + 1 Cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e 1 [ e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 Tanh e 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh e f ] + 2 Tanh e 2 + 3 e 2 + Tanh e 1 ; andadesiredtrajectorydependentauxiliarytermisdenedas f d = f x d ; x d ;t 2 R n 50

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Basedontheformof3andthroughaniterativestabilityanalysis,thecontinuouscontroller, u v ,isdesignedas 3 u 1 Tanh v where v e 1 ;e 2 2 R n isthegeneralizedFilippovsolutiontothefollowingdifferential equation v = Cosh 2 v [ 2 Tanh e 2 + 3 e 2 + sgn e 2 )]TJ/F25 11.9552 Tf 10.738 0 Td [( 1 Cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e 1 e 2 + 2 e 2 ] ;v =0 where 2 R isapositiveconstantcontrolgainand sgn isdened 8 2 R m = 1 2 ::: m T as sgn sgn 1 sgn 2 :::sgn m T 4 UsingFilippov'stheoryofdifferentialinclusions[149,161163],theexistenceofsolutionscanbeestablishedfor v 2 K [ h 1 ] e 1 ;e 2 ,where h 1 e 1 ;e 2 2 R n isdenedasthe RHSof3and K [ h 1 ] T > 0 T S m =0 coh 1 e 1 ;B e 2 ; )]TJ/F25 11.9552 Tf 11.955 0 Td [(S m ,where T S m =0 denotesthe intersectionofallsets S m ofLebesguemeasurezero, co denotesconvexclosure,and B e 2 ; = f & 2 R n jk e 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(& k < g [45,154]. Inreviewof3-3,thecontrolstrategyin3and3entailsseveral componentsincludingthedevelopmentofthelterederrorsystemsin3and3 7,whicharecomposedofsaturatedhyperbolictangentfunctionsdesignedfromthe Lyapunovanalysistocancelcrossterms.Themotivationforthedesignof3stems fromtheneedtoinjecta )]TJ/F25 11.9552 Tf 9.299 0 Td [( 1 e 2 signalintotheclosed-looperrorsystemandtocancel crosstermsintheanalysis.Basedonthestabilityanalysismethodsassociatedwith 3 Animportantfeatureofthecontrollerin3isitsapplicabilitytothecasewhere constraintsexistontheavailablecontrol.Notethatthecontrollawisupperboundedby theadjustablecontrolgain 1 as k u k p n 1 where n isthedimensionof u 4 Theinitialconditionfor v ; 0 isselectedsuchthat u =0 51

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theRISEcontrolstrategy[8,147,169,171],anextraderivativeisappliedtotheclosedlooperrorsystem.Thetimederivativeof3willincludea Cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 v term.The designof3ismotivatedbythedesiretocancelthe Cosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 v term,enablingthe remainingtermstoprovidethedesiredfeedbackandcancelnonconstructivetermsand disturbancesasdictatedbythesubsequentstabilityanalysis. Theclosed-looptrackingerrorsystemcanbedevelopedbytakingthetimederivativeof3,andusingthetimederivativeof3toyield r = ~ N + N d )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 r )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 sgn e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e 2 where ~ N e 1 ;e 2 ;r;e f 2 R n and N d x d ; x d ; x d ;t 2 R n aredenedas ~ N S + 1 1 Cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e 1 e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 2 e 2 + Tanh e 2 + e 2 ; N d ... x d )]TJ/F15 11.9552 Tf 15.804 3.155 Td [(_ f d )]TJ/F15 11.9552 Tf 15.321 3.155 Td [(_ d: Thestructureof3ismotivatedbythedesiretosegregatetermsthatcanbeupper boundedbystate-dependenttermsandtermsthatcanbeupperboundedbyconstants. ByapplyingtheMeanValueTheoremMVT,anupperboundcanbedevelopedforthe expressionin3as[164,AppA] ~ N k w k k w k wheretheboundingfunction 2 R isapositive,globallyinvertible,nondecreasing function,and w e 1 ;e 2 ;r;e f 2 R 5 n isdenedas w Tanh T e 1 ;e T 2 ;r T ;Tanh T e f T : FromAssumptions3.1and3.2,thefollowinginequalitycanbedevelopedbasedonthe expressionin3: k N d k N d 1 ; N d N d 2 52

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where N d 1 ; N d 2 2 R ; areknownpositiveconstants. 3.3StabilityAnalysis Theorem3.1. Giventhedynamicsin3,thecontrollergivenby3and3 ensuresasymptotictrackinginthesensethat k e 1 k! 0 ast !1 providedthecontrolgainsareselectedsufcientlylargebasedontheinitialconditionsofthestatesseethesubsequentstabilityanalysisandthefollowingsufcient conditions 1 > 1 2 ; 2 > 0 ; 3 > 1 2 + 2 1 2 4 ; 2 > 1 2 ; 1 > N d 1 + N d 2 3 : where 1 ; 2 ; 3 ; 1 ; 2 and wereintroducedin3-3and3,respectively, and 2 R isasubsequentlydenedadjustablepositiveconstant. Proof. Let z e 1 ;e 2 ;r;e f 2 R 4 n bedenedas z e T 1 ;e T 2 ;r T ;Tanh T e f T and y z;P 2 R 4 n +1 bedenedas y z T p P T : In3,theauxiliaryfunction P e 2 ;t 2 R isdenedasthegeneralizedFilippov solutiontothefollowingdifferentialequation P = )]TJ/F25 11.9552 Tf 9.298 0 Td [(r T N d )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 sgn e 2 ;P e 2 t 0 ;t 0 = 1 n X i =1 j e 2 i t 0 j)]TJ/F25 11.9552 Tf 17.933 0 Td [(e 2 t 0 T N d t 0 wherethesubscript i =1 ; 2 ;:::;n denotesthe i thelementofthevector.Similartothe developmentin3,existenceofsolutionsfor P e 2 ;t canbeestablishedusing Filippov'stheoryofdifferentialinclusionsfor P 2 K [ h 2 ] e 2 ;r;t ,where h 2 e 2 ;r;t 2 R is denedas h 2 )]TJ/F25 11.9552 Tf 9.299 0 Td [(r T N d )]TJ/F25 11.9552 Tf 11.956 0 Td [( 1 sgn e 2 and K [ h 2 ] T > 0 T S m =0 coh 2 B e 2 ; )]TJ/F25 11.9552 Tf 11.955 0 Td [(S m ;r;t as 53

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in3.Providedthesufcientconditionfor in3issatised, P e 2 ;t 0 See theAppendixAfordetails. Let V L y;t : D [0 ; 1 R beapositive-deniteregularfunctiondenedas V L n X i =1 ln cosh e 1 i + n X i =1 ln cosh e 2 i + 1 2 e T 2 e 2 + 1 2 r T r + 1 2 Tanh T e f Tanh e f + P where e 1 i and e 2 i denotethe i thelementofthevector e 1 x;x d and e 2 e 1 ; e 1 ;e f respectively.TheLyapunovfunctioncandidatein3satisesthefollowinginequalities: 1 y V L y;t 2 y : Basedon3and3,thecontinuouspositivedenitefunctions 1 y ; 2 y 2 R in3aredenedas 1 y 1 2 tanh 2 k y k 2 y 3 2 k y k 2 UnderFilippov'sframework,thetimederivativeof3existsalmosteverywhere, i.e.,foralmostall t 2 [ t 0 ;t f ] ,and V y;t a:e: 2 ~ V y;t where ~ V L = 2 @V L y;t T K e T 1 e T 2 r T Cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e f e T f 1 2 P )]TJ/F18 5.9776 Tf 7.782 3.259 Td [(1 2 P 1 T ; @V L isthegeneralizedgradientof V L y;t [166].Since V L y;t isaLipschitzcontinuous regularfunction, ~ V L r V T L K e T 1 e T 2 r T Cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e f e T f 1 2 P )]TJ/F18 5.9776 Tf 7.782 3.258 Td [(1 2 P T where r V L h Tanh T e 1 ; )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(Tanh T e 2 + e T 2 ;r T ;Tanh T e f ; 2 P 1 2 i T : 54

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Usingthecalculusfor K [ ] from[154],andsubstituting3-3,and3into 3,yields ~ V L r T ~ N + N d )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 r )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh e 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 K [ sgn e 2 ] + Tanh T e 1 e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 Tanh e 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh e f + Tanh T e 2 r )]TJ/F25 11.9552 Tf 11.956 0 Td [( 2 Tanh e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 3 e 2 + e T 2 r )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 Tanh e 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [( 3 e 2 + Tanh T e f )]TJ/F25 11.9552 Tf 9.299 0 Td [( 1 e 2 + Tanh e 1 + Tanh T e f )]TJ/F25 11.9552 Tf 9.298 0 Td [( 2 Tanh e f + P where K [ sgn e 2 ]= SGN e 2 [154]suchthat SGN e 2 i =1 if e 2 i > 0 ; [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ; 1] if e 2 i =0 ,and )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 if e 2 i < 0 : Substituting3,cancelingcommontermsand rearrangingtheresultingexpressionyields V L a:e: = )]TJ/F25 11.9552 Tf 9.299 0 Td [( 1 Tanh T e 1 Tanh e 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 Tanh T e 2 Tanh e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 3 e T 2 e 2 )]TJ/F25 11.9552 Tf 9.299 0 Td [( 2 Tanh T e f Tanh e f )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 r T r + r T ~ N + Tanh T e 1 e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh T e 2 3 e 2 )]TJ/F25 11.9552 Tf 9.299 0 Td [( 1 Tanh T e f e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 e T 2 Tanh e 2 wherethesetin3reducestothescalarequalityin3sincetheRHSis continuousa.e.,i.e,theRHSiscontinuousexceptfortheLebesguemeasurezerosetof timeswhen r T 1 K [ sgn e 2 ] )]TJ/F25 11.9552 Tf 12.24 0 Td [(r T 1 K [ sgn e 2 ] 6 =0 5 Utilizingthedenitionof3, 3,and3,theexpressionin3canbeupperboundedas V L a:e: )]TJ/F25 11.9552 Tf 30.476 0 Td [( 1 k Tanh e 1 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 + 3 k Tanh e 2 k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 3 k e 2 k 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [( 2 k Tanh e f k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 k r k 2 + k w kk r k + k Tanh e 1 kk e 2 k + 1 k Tanh e f kk e 2 k : 5 Thesetoftimes n t 2 [0 ; 1 : r t T 1 K [ sgn e 2 t ] )]TJ/F25 11.9552 Tf 11.955 0 Td [(r t T 1 K [ sgn e 2 t ] 6 =0 o [0 ; 1 isequivalenttothesetoftimes f t : e 2 t =0 ^ r t 6 =0 g .From3,thissetcan alsoberepresentedby f t : e 2 t =0 ^ e 2 t 6 =0 g .Provided e 2 t iscontinuouslydifferentiable,itcanbeshownthatthesetoftimeinstances f t : e 2 t =0 ^ e 2 t 6 =0 g is isolated,andthus,measurezero.Thisimpliesthattheset ismeasurezero. 55

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YoungInequalitycanbeappliedtoselecttermsin3as k Tanh e 1 kk e 2 k 1 2 k Tanh e 1 k 2 + 1 2 k e 2 k 2 1 k Tanh e f kk e 2 k 1 2 k Tanh e f k 2 + 2 1 2 4 k e 2 k 2 : Tofacilitatethesubsequentstabilityanalysis,let 1 beselectedas 1 = a + b where a ; b 2 R arepositivegainconstants.Utilizing3,completingthesquareson r andgroupingterms,theexpressionin3canbeupperboundedby V L a:e: )]TJ/F30 11.9552 Tf 32.469 16.857 Td [( 1 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 k Tanh e 1 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 + 3 k Tanh e 2 k 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( 3 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [( 2 1 2 4 k e 2 k 2 )]TJ/F30 11.9552 Tf 11.291 16.857 Td [( 2 )]TJ/F15 11.9552 Tf 15.598 8.088 Td [(1 2 k Tanh e f k 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [( a k r k 2 + 2 k w k k w k 2 4 b : Providedthesufcientconditionsin3aresatised,3and3canbeused toconcludethat V L a:e: )]TJ/F25 11.9552 Tf 23.834 0 Td [( 3 k z k )]TJ/F25 11.9552 Tf 21.918 0 Td [(U y where 3 k z k 2 R isdenedas 3 )]TJ/F26 7.9701 Tf 13.15 5.698 Td [( 2 k w k 4 b tanh 2 k z k 2 R + isdenedas = min n 1 )]TJ/F24 7.9701 Tf 13.151 4.707 Td [(1 2 ; 2 2 + 3 ; 3 )]TJ/F24 7.9701 Tf 13.15 4.707 Td [(1 2 )]TJ/F26 7.9701 Tf 13.15 6.18 Td [( 2 1 2 4 ; 2 )]TJ/F24 7.9701 Tf 15.243 4.707 Td [(1 2 ; a o ,and U y c tanh 2 k z k 8 y D isacontinuous,positivesemi-denitefunctionforsomepositiveconstant c 2 R ,where D n y 2 R 4 n +1 jk y k )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 p b o : Theinequalitiesin3and3canbeusedtoshowthat V L y;t 2L 1 in D hence, e 1 ;e 2 ;r ;Tanh e f 2L 1 in D .From3, Tanh e 1 ;Tanh e 2 2L 1 in D .Thus,from3and3, e 1 ; e 2 2L 1 in D .From3and3, u 2L 1 in D .FromAssumption3.2andbyutilizingthefactthat e 1 ; e 1 2L 1 q t ; q t 2L 1 in D .Fromtheabovestatements,3canbeusedtoshowthat r 2L 1 in D .Since f is C 2 and q t ; q t 2L 1 f q; q;t 2L 1 .Utilizingthederivative of3,Assumption3.1andthefactsthat e 2 ; e 2 ; r ;f ;u 2L 1 ,theproduct 56

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Cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e f e f 2L 1 .Thus, z 2L 1 in D anditcanbeshownthat z isuniformly continuousUCin D .Since z isUC, tanh k z k isUC.Thedenitionsof U y and z canbeusedtoprovethat U y isUCin D .Let SD denoteasetdenedas S y 2Dj 2 y < )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 2 p b 2 : Theregionofattractionin3canbemadelargerbyincreasingthecontrolgain b Forarbitrarilylargeinitialconditionsorarbitrarilylargedisturbances,thecontrolgains requiredtosatisfythesufcientgainconditionsin3maydemandaninputthatis notphysicallydeliverablebythesystemi.e.,thegain 1 mayberequiredtobelarger thanthesaturationlimitoftheactuator.Despitegaindependencyonthesystem'sinitial conditions,thisresultdoesnotsatisfythestandardsemi-globalresultbecauseunderthe considerationofinputconstraints, b cannotbearbitrarilyincreasedandconsequently theregionofattractioncannotbearbitrarilyenlargedtoincludeallinitialconditions. 6 From3,Corollary2.1canbeinvokedtoshowthat tanh k z k 0 ast 18 y 2S : Basedonthedenitionof z in3, k e 1 k! 0 ast !18 y 2S : 3.4Euler-LagrangeExtension TheresultspresentedinChapter3canbeextendedtogeneralsystemswhichcan bedescribedbyELequationsofmotion.Specically,consideranonlinearsystemofthe form M q q + V m q; q q + G q + F q + d t = u t where M q 2 R n n denotesthegeneralized,state-dependentinertia, V m q; q 2 R n n denotesthegeneralizedcentrifugalandCoriolisforces, G q 2 R n denotesthe 6 Thisoutcomeisnotsurprisingfromaphysicalperspectiveinthesensethatsuch demandsmayyieldcaseswheretheactuationisinsufcienttostabilizethesystem. 57

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generalizedgravity, F q 2 R n denotesthegeneralizedfriction, d 2 R n denotesa generalnonlineardisturbancee.g.,unmodeledeffects, q t ; q t ; q t 2 R n denotethe generalizedstatesand u t 2 R n denotesthegeneralizedcontrolforce. ThedesignoftheerrorsystemsandcontrollerfollowsimilarlytothemethodpresentedinSection3.2.Utilizingstandardpropertiesoftheinertiaandcentrifugal/Coriolis matrices,andtheassumptionsfromSection3.1,thecontroldevelopmentcanbe extendedtoachieveasimilarresultasinSection3.3. Assumption3.3. Theinertiamatrix M q issymmetricpositive-denite,andsatises thefollowinginequality 8 y t 2 R n : m k y k 2 y T My m q k y k 2 where m 2 R isaknownpositiveconstant, m q 2 R isaknownpositivefunction,and kk denotesthestandardEuclideannorm. Theerrorsystems e 1 e 2 r e f aredesignedasin3-3,respectively.Anopen-looperrorsystemsimilarto3isdevelopedas Mr = S + R )]TJ/F25 11.9552 Tf 11.955 0 Td [(u t )]TJ/F25 11.9552 Tf 11.955 0 Td [(M 1 e 2 wheretheauxiliaryfunctions S e 1 ;e 2 ;e f ;t 2 R n and R q d ; q d ; q d ;t 2 R n aredenedas S M q d + V m q + G + F )]TJ/F25 11.9552 Tf 11.955 0 Td [(S d + M 1 Cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e 1 [ e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 Tanh e 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh e f ] )]TJ/F25 11.9552 Tf 9.298 0 Td [(M 2 Tanh e f + M 2 Tanh e 2 + M 3 e 2 + MTanh e 1 ; R S d + d andadesiredtrajectorydependentauxiliaryterm, S d q d ; q d ; q d ;t 2 R n ,denedas S d M d q d + V md q d + G d + F d ; isaddedandsubtracted.In3, M d V md G d F d denote M q d 2 R n n ;V m q d ; q d 2 R n n G q d 2 R n F q d 2 R n ,respectively. 58

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Thecontrolisdesignedsimilarlytothatof3and3,howeverseveral additionaltermsmustbeincludedtohandletermsassociatedwiththeinertiamatrix.Becauseofthis,theELextensionresultrequirestheinertiamatrixtobeknown. Basedontheformof3andthroughaniterativestabilityanalysis,thecontinuous controller, u t ,isdesignedas u 1 Tanh v where v e 1 ;e 2 2 R n isthegeneralizedFilippovsolutiontothefollowingdifferential equation v = Cosh 2 v [ M 2 Tanh e 2 + M 3 e 2 + sgn e 2 )]TJ/F25 11.9552 Tf 9.299 0 Td [( 1 Cosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 e 1 e 2 )]TJ/F15 11.9552 Tf 17.591 3.022 Td [(_ Me 2 + 2 Me 2 i ; v =0 ,where 2 R isapositiveconstantcontrolgain.BecausetheRHSof v containsthe sgn function,Filippovtheorymustagainbeusedtoproveexistenceof solutions.Becausethedetailsfollowfromthepreviousdevelopment,theyareomittedin thisextension. Theclosed-looptrackingerrorsystemcanbedevelopedbyinsertingthecontrol into3,takingthetimederivative,andbyaddingandsubtracting Tanh e 2 and e 2 to yield M r = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 Mr + ~ N + N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(M 1 r )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 sgn e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e 2 where ~ N e 1 ;e 2 ;r;e f 2 R n and N d q d ; q d ; q d ;q d ;t 2 R n aredenedas ~ N )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 Mr + S + Tanh e 2 + e 2 and N d R: In3, S e 1 ;e 2 ;e f ;t 2 R n isdenedas S S )]TJ/F25 11.9552 Tf 12.513 0 Td [( 1 1 MCosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 e 1 e 2 + 1 2 Me 2 wherethelasttwotermsarefromthetimederivativeof3andcancelwithinverse 59

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termsinsideof S whicharisedueto Tanh e f termsinside S toyield S freeofdirect useofthegainparameter 1 .Remaining 1 termsin S areencapsulatedwithin Tanh functionsandthuscanbeupperboundedby1. UtilizingasimilarLyapunovcandidatefunction V L y;t : D [0 ; 1 R denedas V L n X i =1 ln cosh e 1 i + n X i =1 ln cosh e 2 i + 1 2 e T 2 e 2 + 1 2 r T Mr + 1 2 Tanh T e f Tanh e f + P; andCorollary2.1,itcanbeshownthat V L a:e: )]TJ/F25 11.9552 Tf 23.834 0 Td [( 3 k z k )]TJ/F25 11.9552 Tf 21.918 0 Td [(U y where 3 k z k 2 R isdenedas 3 )]TJ/F26 7.9701 Tf 13.15 5.699 Td [( 2 k x k 4 b tanh 2 k z k isdenedsimilartoin 3,and U y c tanh 2 k z k forsomepositiveconstant c ,isacontinuous,positive semi-denitefunctiondenedon D denedin3.Additionally, 1 isdesignedsuch that 1 a + b m where a ; b 2 R arepositivegainconstantsand x e 1 ;e 2 ;r;e f 2 R 5 n is denedthesameas w in3.From3, tanh k z k 0 ast !18 y 2S where S isdenedas S y 2Dj max 1 2 m q ; 3 2 k y k 2 < 1 2 min f 1 ;m g )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 p b 2 : Basedonthedenitionof z in3,itcanbeshownthat k e 1 t k! 0 ast !18 y 2S : AdditionaldetailsregardingtheELextensionofthischaptercanbefoundin[172]. 60

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3.5ExperimentalResults ToexaminetheperformanceofthesaturatedRISEapproach,thecontrollerin 3and3wasimplementedonaplanarmanipulatortestbed. 7 Themanipulator canbemodeledasanELsystemwiththefollowingdynamics M q q + V m q; q q + F q + d t = t where M q 2 R 2 2 V m q; q 2 R 2 2 F q 2 R 2 ,and d t 2 R 2 weredenedin3, q t ; q t ; q t 2 R 2 denotethelinkposition,velocityandaccelerationand t 2 R 2 denotesthecontroltorque. Thecontrolobjectiveistotrackadesiredlinktrajectory,selectedas q d 1 t = q d 2 t =+60 sin t 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F24 7.9701 Tf 6.586 0 Td [(0 : 01 t 3 deg .Theinitialconditionsforthemanipulator wereselectedacompleterotationawayfromtheinitialconditionsofthedesiredtrajectoryas q 1 =360 deg and q 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(180 deg .Thecontroltorquewasarbitrarilyselected tobearticiallylimitedwell-withinthecapabilitiesoftheactuatorto j 1 j 60 N )]TJ/F25 11.9552 Tf 12.4 0 Td [(m j 2 j 15 N )]TJ/F25 11.9552 Tf 12.816 0 Td [(m ,thus, 1 waschosenaccordingly.Specically,thefeedbackgains fortheproposedcontrollerwereselectedas 1 = diag ; 13 2 = diag ; 19 = diag : 8 ; 3 : 8 1 = diag : 2 ; 6 : 0 2 = diag ; 11 3 = diag ; 45 and e f ; 0 is selectedas e f ; 0=0 8 7 Themanipulatorconsistsofatwo-linkdirectdriverevoluterobotconsistingof twoaluminumlinks,mountedon240.0N-mbasejointand20.0N-msecondjoint switchedreluctancemotors.Themotorresolversproviderotorpositionmeasurements witharesolutionof614,400pulses/revolution,andastandardbackwardsdifference algorithmisusedtonumericallydeterminevelocityfromtheencoderreadings.Data acquisitionandcontrolimplementationwereperformedinreal-timeusingQNXatafrequencyof1.0kHz. 8 ItisimportanttonotethatforthegivenEuler-Lagrangesystem,theimplemented controlleris = M q u .Thus,theboundontheimplementedcontrolwillinclude theknownboundontheinertiamatrix.Forthisexperiment,theinertiamatrixcanbe boundedby k M q k 1 : 15 61

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Table3-1.Steady-stateRMSerrorandtorqueforeachoftheanalyzedcontroldesigns. RMSError1RMSError2RMSTorque1RMSTorque2 PIDw/AW[173]8.38571.409614.56301.8732 AdaptiveFSFB[174]3.82321.653814.93671.2360 SaturatedRISEProposed0.16070.288914.33631.1883 TheperformanceofthesaturatedRISEcontroldesignwascomparedagainsttwo controllersavailableinliterature:aclassicalPIDcontrollerwithconditionalintegral clampinganti-windup[173]andanadaptivefull-statefeedbackcontrollerwithbounded inputs[174].Eachcontrollerwastunedtoachievethebestpossibleperformance,given thesaturationbounds.Sinceeachcontrollerhasadifferentstructure,itisdifcultto commentonthecomparativenatureofthegainswhichwereimplemented.Starting withthesamelargeinitialconditionoffset,thetrackingerrorsofeachcontrollerare depictedinFigure3-1.Theactualtrajectorieswithrespecttothedesiredtrajectoriesare showninFigure3-2.ThecontroltorquesforeachcontrollerareshowninFigure3-3and eachremainwithintheprescribedbounds.Toquantifythesteady-stateperformance, rootmeansquareRMSerrorsarelistedinTable3-1.Thetableillustratesthatfor comparableRMStorquevalues,thesaturatedRISEcontrollerexhibitsimprovedsteadystateperformancewhencomparedtotheothercontroldesigns. 62

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Figure3-1.Trackingerrorsvs.timeforAclassicalPIDwithintegralclamping anti-windup,Badaptivefull-statefeedbackcontroller,andCtheproposed saturatedRISEcontroller. Figure3-2.Desiredandactualtrajectoriesvs.timeforAclassicalPIDwithintegral clampinganti-windup,Badaptivefull-statefeedbackcontroller,andCthe proposedsaturatedRISEcontroller. 63

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Figure3-3.Controltorquevs.timeforAclassicalPIDwithintegralclamping anti-windup,Badaptivefull-statefeedbackcontroller,andCtheproposed saturatedRISEcontroller. 3.6Summary Acontinuoussaturatedcontrollerisdevelopedforaclassofuncertainnonlinear systemswhichincludestime-varyingandnon-LPfunctionsandadditiveboundeddisturbances.Theboundonthecontrolisknownaprioriandcanbeadjustedbychanging thefeedbackgains.Thesaturatedcontrollerisshowntoguaranteeasymptotictracking usingsmoothhyperbolicfunctions.AnextensiontoELsystemsispresentedandillustratedviaexperimentalresultsusingatwo-linkrobotmanipulatortodemonstratethe performanceofthecontroldesign. 64

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CHAPTER4 RISE-BASEDCONTROLOFANUNCERTAINNONLINEARSYSTEMWITH TIME-VARYINGSTATEDELAYS Thischapterconsidersacontinuouscontroldesignforsecond-ordercontrol afnenonlinearsystemswithtime-varyingstatedelays.Buildingonpreviousworkin Chapters2and3,aNNisaugmentedwithaRISEcontrolstructuretoachievesemiglobalasymptotictrackinginthepresenceofunknown,arbitrarilylarge,time-varying delays,non-LPuncertaintyandadditiveboundeddisturbances.Byexpressingunknown functionsintermsofthedesiredtrajectoriesandthroughstrategicgroupingofdelay-free anddelay-dependentterms,LKfunctionalsareutilizedtocancelthedelayedtermsin theanalysisandobtaindelay-freeNNupdatelaws. 4.1DynamicModel Consideraclassofuncertainsecond-ordercontrolafnenonlinearsystemswithan unknowntime-varyingstatedelaydescribedby x = f x; x;t + g x t )]TJ/F25 11.9552 Tf 9.963 0 Td [( ; x t )]TJ/F25 11.9552 Tf 9.963 0 Td [( ;t + d t + u t : In4, f x; x;t : R 2 n [0 ; 1 R n isanunknownfunction, g x t )]TJ/F25 11.9552 Tf 11.955 0 Td [( ; x t )]TJ/F25 11.9552 Tf 11.955 0 Td [( ;t : R 2 n [0 ; 1 R n isanunknowntime-delayedfunction, t 2 R isanunknown,timevarying,arbitrarilylargetime-delay, d t :[0 ; 1 R n isasufcientlysmoothbounded disturbancee.g.,unmodeledeffects, u t 2 R n isthecontrolinput,and x t ; x t 2 R n aremeasurablesystemstates.Throughoutthechapter,atime-dependentdelayed functionisdenotedas t )]TJ/F25 11.9552 Tf 11.955 0 Td [( or ,and kk denotestheEuclideannormofavector. FollowingtheworkofChapter3,Assumptions3.1and3.2areutilizedforthe systemin4.Additionally,thefollowingassumptionsareapplicable: Assumption4.1. Theunknowntimedelayisboundedsuchthat 0 t 1 andthe rateofchangeofthedelayisboundedsuchthat j t j 2 < 1 where 1 ;' 2 2 R + are knownconstants. 65

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Assumption4.2. Thefunctions f ;g andtheirrstandsecondderivativeswith respecttotheirargumentsareLipschitzcontinuous. 4.2ControlDevelopment Thecontrolobjectiveistodesignacontinuouscontrollerthatwillensure x t tracks adesiredtrajectory.AsinChapter3,atrackingerrordenoted e 1 x;t 2 R n isdenedas e 1 x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(x: Tofacilitatethesubsequentanalysis,twolteredtrackingerrors,denotedby e 2 e 1 ; e 1 ;t ;r e 2 ; e 2 ;t 2 R n ,aredenedas e 2 e 1 + 1 e 1 r e 2 + 2 e 2 where 1 ; 2 2 R + areknowngainconstants.Theauxiliarysignal r e 2 ; e 2 ;t isintroducedtofacilitatethestabilityanalysisandisnotusedinthecontroldesignsincethe expressionin4dependsontheunmeasurablestate x t Anopen-looptrackingerrorcanbeobtainedbysubstitutingfor4-4toyield r = 1 e 1 + 2 e 2 + x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(d )]TJ/F25 11.9552 Tf 9.299 0 Td [(f x; x;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(g x ; x ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(u: UsingadesiredcompensationadaptationlawDCAL-baseddesignapproach[175], 4canbewrittenas r = 1 e 1 + 2 e 2 + S 1 + S d + x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(d + g x d ; x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(g x d ; x d )]TJ/F25 11.9552 Tf 11.956 0 Td [(u 66

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wheretheauxiliaryfunctions S 1 x;x d ; x; x d ;x ; x ;x d ; x d ;t ;S d x d ; x d 2 R n are denedas S 1 )]TJ/F25 11.9552 Tf 9.299 0 Td [(f x; x;t + f x d ; x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(g x ; x ;t + g x d ; x d ; S d )]TJ/F25 11.9552 Tf 9.298 0 Td [(f x d ; x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(g x d ; x d : Thegroupingoftermsin4ismotivatedbythedesiretosegregatetermsthatcan beupperboundedbystate-dependenttermswhetherdelayedordelay-freefromthe termsthatcanbeupperboundedbyconstants. TheUniversalApproximationTheoremcanbeusedtorepresenttheauxiliary function S d byathree-layerNNas S d W T )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(V T x nn + where V t 2 R N 1 +1 N 2 and W t 2 R N 2 +1 n areboundedconstantidealweightsfor therst-to-secondandsecond-to-thirdlayers,respectively, N 1 isthenumberofneurons intheinputlayer, N 2 isthenumberofneuronsinthehiddenlayer, n isthenumberof neuronsintheoutputlayer, 2 R N 2 +1 isanactivationfunction, x nn t 2 R N 1 +1 denotestheinputtotheNNdenedonacompactsetcontainingtheknownbounded desiredtrajectoriesas x nn = 1 ;x T d ; x T d T ,and x nn 2 R n denotesthefunctional reconstructionerrors. Assumption4.3. TheidealNNweightsareassumedtoexistandbeboundedbyknown positiveconstants,i.e. k V k 2 F V B k W k 2 F W B where kk F istheFrobeniusnormfora matrix. Assumption4.4. Thefunctionalreconstructionerrors andtheirrstderivativewith respecttotheirargumentsareboundedsuchthat k x nn k b 1 k x nn ; x nn k b 2 where b 1 ;" b 2 2 R areknownpositiveconstants. Assumption4.5. Theactivationfunction anditsderivative, 0 arebounded. 67

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Remark 4.1 Assumptions4.3-4.4arestandardassumptionsinNNcontrolliterature cf.[176].Theidealweightupperboundsareassumedtobeknowntofacilitatetheuse oftheprojectionalgorithmtoensuretheweightestimatesarealwaysbounded.There arenumerousactivationsfunctionswhichsatisfyAssumption4.5,e.g.,sigmoidalor hyperbolictangentfunctions. Thecontrollerisdesignedusingathree-layerNNfeedforwardtermaugmentedbya RISEfeedbacktermas u ^ S d + : TheRISEfeedbackterm e 2 ; 2 R n isdenedas[8,160] k s +1 e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k s +1 e 2 + where e 2 2 R n isthegeneralizedFilippovsolutiontothefollowingdifferential equation k s +1 2 e 2 + sgn e 2 ; ;k s 2 R arepositive,constantcontrolgains,and sgn isdened 8 2 R n = 1 2 ::: n T as sgn sgn 1 sgn 2 :::sgn n T : 1 UsingFilippov'stheoryofdifferentialinclusions[149,161163],theexistenceof solutionscanbeestablishedfor 2 K [ h 1 ] e 2 ,where h 1 e 2 2 R n isdenedasthe RHSof in4and K [ h 1 ] T > 0 T S m =0 coh 1 B e 2 ; )]TJ/F25 11.9552 Tf 11.956 0 Td [(S m ,where T S m =0 denotesthe intersectionoverallsets S m ofLebesguemeasurezero, co denotesconvexclosure,and B e 2 ; = f & 2 R n jk e 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(& k < g [45,154]. TheNNfeedforwardterm ^ S d t 2 R n in4isdesignedas ^ S d ^ W T ^ V T x nn 1 Theinitialconditionfor v isselectedsuchthat u =0 68

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where ^ V t 2 R N 1 +1 N 2 and ^ W t 2 R N 2 +1 n areestimatesoftheidealweights. Basedonthesubsequentstabilityanalysis,theDCAL-basedweightupdatelawsforthe NNin4aregeneratedonlineas ^ W proj )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(1 ^ 0 ^ V T x nn e T 2 ^ V proj )]TJ/F24 7.9701 Tf 7.315 -1.793 Td [(2 x nn ^ 0 T ^ We 2 T ; where )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(1 2 R N 2 +1 N 2 +1 and )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(2 2 R N 1 +1 N 1 +1 arepositive-denite,constant symmetriccontrolgainmatrices,and ^ 0 2 R N 2 +1 denotesthepartialderivativeof ^ ^ V T x nn Theclosed-loopdynamicsaredevelopedbysubstituting4-4into4, takingthetimederivative,andaddingandsubtracting W T ^ 0 ^ V T x nn + ^ W T ^ 0 ~ V T x nn toyield r = 1 e 1 + 2 e 2 + S 1 + ... x d )]TJ/F15 11.9552 Tf 15.322 3.155 Td [(_ d )]TJ/F15 11.9552 Tf 11.015 0 Td [(_ g x d ; x d ; x d +_ g x d ; x d ; x d )]TJ/F15 11.9552 Tf 11.291 0 Td [( k s +1 r )]TJ/F25 11.9552 Tf 11.955 0 Td [(sgn e 2 + ^ W T ^ 0 ~ V T x nn + ~ W T ^ 0 ^ V T x nn + W T 0 V T x nn )]TJ/F25 11.9552 Tf 11.955 0 Td [(W T ^ 0 ^ V T x nn )]TJ/F15 11.9552 Tf 12.711 3.022 Td [(^ W T ^ 0 ~ V T x nn )]TJ/F15 11.9552 Tf 16.668 6.177 Td [(_ ^ W T ^ )]TJ/F15 11.9552 Tf 15.368 3.022 Td [(^ W T ^ 0 ^ V T x nn +_ whereestimatemismatchesfortheidealweights,denoted ~ V t 2 R N 1 +1 N 2 and ~ W t 2 R N 2 +1 n ,aredenedas ~ V t = V t )]TJ/F15 11.9552 Tf 13.856 3.022 Td [(^ V t and ~ W t = W t )]TJ/F15 11.9552 Tf 15.482 3.022 Td [(^ W t .Using theNNweightupdatelawsfrom4and4,theexpressionin414canbe rewrittenas r = ~ N + N + e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k s +1 r )]TJ/F25 11.9552 Tf 11.955 0 Td [(sgn e 2 69

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where ~ N ^ W; ^ V;e 1 ;e 2 ; e 1 ; e 2 ;t 2 R n and N ^ W; ^ V;t 2 R n aredenedas ~ N 1 e 1 + 2 e 2 + S 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(proj )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(1 ^ 0 ^ V T x nn e T 2 T ^ )]TJ/F15 11.9552 Tf 12.711 3.022 Td [(^ W T ^ 0 proj )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(2 x nn ^ 0 T ^ We 2 T ; N N D + N B : In4, N D x d ; x d ; x d ; ... x d ;t 2 R n isdenedas N D W T 0 V T x nn +_ + ... x d )]TJ/F15 11.9552 Tf 15.322 3.155 Td [(_ d )]TJ/F15 11.9552 Tf 11.015 0 Td [(_ g x d ; x d ; x d +_ g x d ; x d ; x d and N B ^ W; ^ V;x d ; x d ; x d ;t 2 R n isseparatedsuchthat N B N B 1 + N B 2 where N B 1 ^ W; ^ V;x d ; x d ; x d ;t ;N B 2 ^ W; ^ V;x d ; x d ; x d ;t 2 R n aredenedas N B 1 )]TJ/F25 11.9552 Tf 9.298 0 Td [(W T ^ 0 ^ V T x nn )]TJ/F15 11.9552 Tf 15.367 3.022 Td [(^ W T ^ 0 ~ V T x nn ; N B 2 ^ W T ^ 0 ~ V T x nn + ~ W T ^ 0 ^ V T x nn Separatingthetermsin4ismotivatedbythefactthatthedifferentcomponents havedifferentbounds[147]. UsingAssumptions3.1and4.1-4.5, N D from4and N B from4and theirtimederivativescanbeupperboundedas k N D k 1 ; k N B k 2 ; N D 3 ; N B 4 + 5 k e 2 k ; where i 2 R + ; 8 i =1 ;:::; 5 areknownconstants.Additionally, ~ N from4canbe upperboundedas ~ N 1 k z k k z k + 2 k z k k z k 70

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where z e 1 ;e 2 ;r 2 R 3 n denotesthevector z = e T 1 e T 2 r T T and 1 ; 2 : R R arepositive,globallyinvertible,nondecreasingfunctions.Theupperboundforthe auxiliaryfunction ~ N issegregatedintodelay-freeanddelay-dependentbounding functionstoeliminatethedelayedtermswiththeuseofanLKfunctionalinthestability analysis.Specically,let R LK z;t 2 R denoteanLKfunctionaldenedas R LK 2 k s t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( t 2 2 k z k k z k 2 d where 2 R + isanadjustableconstant,and k s and 2 wereintroducedin4and 4,respectively. 4.3StabilityAnalysis Theorem4.1. Thecontrollerproposedin4andtheweightupdatelawsdesignedin 4-4ensurethatthestatesandcontrollerareboundedandthetrackingerrors areregulatedinthesensethat k e 1 k! 0 ast !1 providedthecontrolgain k s introducedin4isselectedsufcientlylargebasedon theinitialconditionsofthestates,andtheremainingcontrolgainsareselectedbasedon thefollowingsufcientconditions 1 > 1 2 ; 2 > 2 + 1 2 ; 2 > 5 ; > 1 + 2 + 1 2 3 + 1 2 4 ; )]TJ/F25 11.9552 Tf 11.955 0 Td [(' 2 > 1 where 1 ; 2 ;; wereintroducedin4,4,4and4, 2 wasintroduced inAssumption4.1and 2 isasubsequentlydenedgainconstant. Proof. Let D R 3 n +3 beadomaincontaining y e 1 ;e 2 ;r;P;Q;R LK 2 R 3 n +3 ,denedas y z p P p Q p R LK : 71

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Similarto3,theauxiliaryfunction P e 2 ;t 2 R isdenedasthegeneralized Filippovsolutiontothefollowingdifferentialequation P )]TJ/F25 11.9552 Tf 9.298 0 Td [(r T N B 1 + N D )]TJ/F25 11.9552 Tf 11.955 0 Td [(sgn e 2 )]TJ/F15 11.9552 Tf 13.693 0 Td [(_ e T 2 N B 2 + 2 k e 2 k 2 ; P e 2 t 0 ;t 0 n X i =1 j e 2 i t 0 j)]TJ/F25 11.9552 Tf 17.933 0 Td [(e 2 t 0 T N D t 0 wherethesubscript i =1 ; 2 ;:::;n denotesthe i thelementofthevector.Similarto thedevelopmentin4,existenceofsolutionsfor P e 2 ;t canbeestablished usingFilippov'stheoryofdifferentialinclusionsfor P 2 K [ h 2 ] r; e 2 ;e 2 ;t ,where h 2 r; e 2 ;e 2 ;t 2 R isdenedastheRHSof P .Providedthesufcientconditionsin 4aresatised, P e 2 ;t 0 See[147]forproof.Additionally,theauxiliaryfunction Q ~ W; ~ V;t 2 R in4isdenedas Q 2 2 tr ~ W T )]TJ/F28 7.9701 Tf 7.314 4.937 Td [()]TJ/F24 7.9701 Tf 6.587 0 Td [(1 1 ~ W + 2 2 tr ~ V T )]TJ/F28 7.9701 Tf 7.315 4.937 Td [()]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 ~ V where Q 0 since )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(1 and )]TJ/F24 7.9701 Tf 7.314 -1.793 Td [(2 areconstant,symmetric,andpositivedenitematricesand 2 2 R + Let V y;t : D [0 ; 1 R beacontinuouslydifferentiablein y ,locallyLipschitzin t ,regularfunctiondenedas V 1 2 e T 1 e 1 + 1 2 e T 2 e 2 + 1 2 r T r + P + Q + R LK whichsatisesthefollowinginequalities 1 y V t 2 y wherethecontinuouspositive-denitefunctions 1 y ; 2 y 2 R aredenedas 1 y 1 k y k 2 2 y 2 k y k 2 and 1 ; 2 2 R + areknownconstants.UnderFilippov's framework,thetimederivativeof4existsalmosteverywhere,i.e.,foralmostall 72

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t 2 [ t 0 ;t f ] ,and V y;t a:e: 2 ~ V y;t where ~ V = 2 @V y;t T K [ % ] where % 2 R 3 n +4 isdenedas % e T 1 e T 2 r T 1 2 P )]TJ/F18 5.9776 Tf 7.782 3.259 Td [(1 2 P 1 2 Q )]TJ/F18 5.9776 Tf 7.782 3.259 Td [(1 2 Q 1 2 R )]TJ/F18 5.9776 Tf 7.782 3.259 Td [(1 2 LK R LK 1 T ; and @V isthegeneralizedgradientof V y;t [166].Since V y;t is C 1 ~ V r VK [ ] T where r V e T 1 e T 2 r T 2 P 1 2 2 Q 1 2 2 R 1 2 LK : Usingthecalculusfor K [ ] from[154],andsubstituting4-4,and4, 4,thetimederivativesof4,and4into4,yields ~ V e T 1 e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 e 1 + e T 2 r )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 e 2 + r T ~ N + N D + N B 1 + N B 2 + e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k s +1 r + r T )]TJ/F25 11.9552 Tf 9.299 0 Td [(K [ sgn e 2 ]+ 2 k e 2 k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(r T N B 1 + N D )]TJ/F25 11.9552 Tf 11.956 0 Td [(K [ sgn e 2 ] )]TJ/F15 11.9552 Tf 13.692 0 Td [(_ e T 2 N B 2 + 2 k s 2 2 k z k k z k 2 )]TJ/F25 11.9552 Tf 13.151 8.087 Td [( )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 k s 2 2 k z k k z k 2 + tr 2 ~ W T )]TJ/F28 7.9701 Tf 7.315 4.936 Td [()]TJ/F24 7.9701 Tf 6.586 0 Td [(1 1 ~ W + tr 2 ~ V T )]TJ/F28 7.9701 Tf 7.314 4.936 Td [()]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 ~ V where K [ sgn e 2 ]= SGN e 2 [154]suchthat SGN e 2 i =1 if e 2 i > 0 ; [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ; 1] if e 2 i =0 ,and )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 if e 2 i < 0 : Cancelingtermsandutilizingtheboundsfrom4and Assumption4.1,4canupperboundedas V a:e: k e 1 kk e 2 k)]TJ/F25 11.9552 Tf 20.589 0 Td [( 1 k e 1 k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 k e 2 k 2 + k r k 1 k z k k z k + k r k 2 k z k k z k )]TJ/F15 11.9552 Tf 11.291 0 Td [( k s +1 k r k 2 + 2 k e 2 k 2 + 2 k s 2 2 k z k k z k 2 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [( )]TJ/F25 11.9552 Tf 11.955 0 Td [(' 2 2 k s 2 2 k z k k z k 2 wherethesetin4reducestothescalarinequalityin4sincetheRHSis continuousa.e.,i.e,theRHSiscontinuousexceptfortheLebesguenegligiblesetof timeswhen r T K [ sgn e 2 ] )]TJ/F25 11.9552 Tf 12.558 0 Td [(r T K [ sgn e 2 ] 6 =0 .Young'sInequalitycanbeusedto showthat k e 1 kk e 2 k 1 2 k e 1 k 2 + 1 2 k e 2 k 2 and k r k 2 k z k k z k k s 2 k r k 2 + 1 2 k s 2 2 k z k k z k 2 73

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whichallowsforthefollowingupperboundfor4 V a:e: 1 2 k e 1 k 2 + 1 2 k e 2 k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 k e 1 k 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [( 2 k e 2 k 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(k s 2 k r k 2 )-222(k r k 2 + 2 k e 2 k 2 + k r k 1 k z k k z k + 1 2 k s 2 2 k z k k z k 2 + 2 k s 2 2 k z k k z k 2 )]TJ/F25 11.9552 Tf 10.494 8.088 Td [( )]TJ/F25 11.9552 Tf 11.956 0 Td [(' 2 2 k s 2 2 k z k k z k : If )]TJ/F25 11.9552 Tf 11.955 0 Td [(' 2 > 1 ,andbycompletingthesquaresfor r e 2 ; e 2 ;t ,4becomes V a:e: )]TJ/F30 11.9552 Tf 26.491 16.857 Td [( 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 k e 1 k 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 k e 2 k 2 )-222(k r k 2 + 1 2 k s 2 1 k z k k z k 2 + 2 k s 2 2 k z k k z k 2 : Regroupingsimilarterms,theexpressioncanbeupperboundedby V a:e: )]TJ/F30 11.9552 Tf 25.827 16.857 Td [( 3 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 k z k 2 k s k z k 2 where 2 k z k 2 1 k z k + 2 2 k z k and 3 min 1 )]TJ/F24 7.9701 Tf 13.151 4.707 Td [(1 2 ; 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F24 7.9701 Tf 13.151 4.707 Td [(1 2 ; 1 .The boundingfunction k z k : R R isapositive-denite,globallyinvertible,nondecreasingfunction.Theexpressionin4canbefurtherupperboundedbyacontinuous, positivesemi-denitefunction V a:e: )]TJ/F25 11.9552 Tf 23.834 0 Td [( 3 y = )]TJ/F25 11.9552 Tf 9.298 0 Td [(c k z k 2 8 y 2 D forsomepositiveconstant c 2 R + anddomain D = y 2 R 3 n +3 jk y k < )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 )]TJ/F23 11.9552 Tf 5.479 -0.392 Td [(p 2 3 k s Largervaluesof k s willexpandthesizeofthedomain D .Theinequalitiesin4 and4canbeusedtoshowthat V 2L 1 in D .Thus, e 1 ;e 2 ;r 2L 1 in D Theclosed-looperrorsystemcanbeusedtoconcludethattheremainingsignalsare boundedin D ,andthedenitionsfor 1 and z canbeusedtoshowthat 1 is uniformlycontinuousin D .Let S D D denoteasetdenedas S D y 2 D j 2 < 1 )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 p 2 3 k s 2 : 74

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Theregionofattractionin4canbemadearbitrarilylargetoincludeanyinitial conditionsbyincreasingthecontrolgain k s .From4,2.1canbeinvokedtoshow that c k z k 2 0 ast !18 y 2S D : Basedonthedenitionof z in4, k e 1 k! 0 ast !18 y 2S D : 4.4Summary Acontinuous,neuralnetworkaugmented,RISEcontrollerisutilizedforuncertain nonlinearsystemswhichincludeunknown,arbitrarilylarge,time-varyingstatedelays andadditiveboundeddisturbances.Thecontrollerassumesthetime-delayisbounded andslowlyvarying.Time-varyingLKfunctionalsareutilizedtoprovesemi-global asymptotictrackingoftheclosed-loopsysteminthepresenceoftime-varyingand non-LPfunctionsandsufcientlysmoothunmodeleddynamiceffects. 75

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CHAPTER5 LYAPUNOV-BASEDCONTROLOFANUNCERTAINNONLINEARSYSTEMWITH TIME-VARYINGINPUTDELAY Apredictor-basedcontrollerisdevelopedforuncertainsecond-ordernonlinear systemssubjecttotime-varyinginputdelayandadditiveboundeddisturbances.A Lyapunov-basedstabilityanalysisutilizingLKfunctionalsisprovidedtoprovesemiglobaluniformlyultimatelyboundedtrackingassumingtheinputdelayisknown, sufcientlysmall,andslowlyvarying.Simulationresultsdemonstratetherobustnessof thecontroldesignwithrespecttouncertaintiesinthemagnitudeandtime-variationof thedelay. 5.1DynamicModel Consideraclassofcontrolafne,second-order 1 nonlinearsystemsdescribedby x = f x; x;t + u t )]TJ/F25 11.9552 Tf 11.955 0 Td [( t + d x;t where x t ; x t 2 R n arethegeneralizedsystemstates, u t )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 R n represents thegeneralizeddelayedcontrolinputvector,where t 2 R + isaknownnon-negative time-varyingdelay, f x; x;t : R 2 n [0 ; 1 R n isanunknownnonlinear C 2 function, uniformlyboundedin t ,and d x;t 2 R n denotesasufcientlysmoothexogenous disturbancee.g.,unmodeledeffects. Thesubsequentdevelopmentisbasedontheassumptionthat x t and x t are measurableoutputs,andthetimedelayandcontrolinputvectoranditspastvaluesi.e., u t )]TJ/F25 11.9552 Tf 11.955 0 Td [( 8 2 [0 t ] aremeasurable.Throughoutthechapter,atime-dependentdelayedfunctionisdenotedas t )]TJ/F25 11.9552 Tf 11.955 0 Td [( t or Additionally,thefollowingassumptionsandpropertieswillbeexploited.Notethat theseassumptionshavebeenadjustedslightlyfromtheassumptionsintheprevious 1 Theresultinthischaptercanbeextendedto n th -ordernonlinearsystemsfollowinga similardevelopmenttothosepresentedin[122,177]. 76

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chapters.Forexample,thischapterconsidersamoregeneraldisturbancewithpossible state-dependenciesandamorerestrictiveboundonthedelay. Assumption5.1. Thenonlineardisturbancetermanditsrsttimederivativei.e., d x;t ; d x; x;t existandareboundedsuchthat k d x;t k d 1 k x k + d 2 and d x; x;t 2L 1 ,where d 1 ; d 2 2 R arenonnegativeconstants. Assumption5.2. Thetimedelayisboundedsuchthat 0 t 1 ,where 1 2 R + isasufcientlysmallseesubsequentstabilityanalysisknownconstantandtherate ofchangeofthedelayisboundedsuchthat j t j <"< 1 2 ,where 2 R + isaknown constant. Assumption5.3. Thedesiredtrajectory x d t 2 R n isdesignedtobesufcientlysmooth suchthat x d t ; x d t ; x d t 2L 1 Remark 5.1 InAssumption5.2,theslowlytime-varyingconstrainti.e., j t j <"< 1 2 is commonthoughslightlymorerestrictivetoresultswhichutilizeclassicalLKfunctionals tocompensatefortime-varyingtime-delays[18].Theinputdelayisrequiredtobeknown sincepastvaluesofthecontrolareusedinthecontrolstructure. 5.2ControlDevelopment Theobjectiveistodesignacontinuouscontrollerthatwillensurethegeneralized state x t oftheinput-delayedsystemin5tracks x d t .Toquantifythecontrol objective,atrackingerrordenotedby e x;t 2 R n ,isdenedas e x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(x: Tofacilitatethesubsequentanalysis,ameasurableauxiliarytrackingerror,denotedby r e; e;e z 2 R n ,isdenedas r e + e )]TJ/F25 11.9552 Tf 11.955 0 Td [(e z 77

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where 2 R + isaknowngainconstant,and e z t 2 R n isanauxiliarysignalcontaining thetime-delaysinthesystem,denedas e z t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( t u d : The e z t componentof5ismotivatedbythedesiretoinjectapredictor-liketerm intheerrorsystemdevelopment.Byinjectingtheintegralofthecontroleffortoverthe delayinterval,theopen-looperrorsystemfortheauxiliaryerrorcanbeexpressedin termsofadelay-freecontrolinput. Theopen-looperrorsystemcanbeobtainedtakingthetimederivativeof5and utilizingtheexpressionsin5,5and5toyield r = x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(f x; x;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(d x;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(u )]TJ/F25 11.9552 Tf 11.956 0 Td [(u + e: From5andthesubsequentstabilityanalysis,thecontrolinput u r isdesigned as[31] u = k b r where k b 2 R + isaknownconstantcontrolgain.Theclosed-looperrorsystemis obtainedutilizing5,5and5toyield r = N d + )]TJ/F25 11.9552 Tf 11.955 0 Td [(k b r )]TJ/F25 11.9552 Tf 11.955 0 Td [(k b r )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F25 11.9552 Tf 11.955 0 Td [(d x;t wheretheauxiliaryterms e;r;e z N d x d ; x d ; x d ;t 2 R n aredenedas )]TJ/F25 11.9552 Tf 9.298 0 Td [(f x; x;t + f x d ; x d ;t + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 e + e z + e; N d x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(f x d ; x d ;t : Assumptions5.1and5.3areusedtodevelopthefollowinginequalitybasedonthe expressionin5 k N d k n d 78

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where n d 2 R + isaknownconstant.Thestructureof5ismotivatedbythedesireto segregatetermsthatcanbeupperboundedbystate-dependenttermsandtermsthat canbeupperboundedbyconstants.UsingtheMVT,theexpressionin5canbe upperboundedas[164,AppA]alsosimilartothepresentationinAppendixB k k k z k k z k where k z k isapositive,globallyinvertible,nondecreasingfunction,and z e;r;e z 2 R 3 n isdenedas z e T r T e T z T : Tofacilitatethesubsequentstabilityanalysis,let y e;r;P;Q 2 R 2 n +2 bedenedas y e T r T p P p Q T where P t; ;Q r;t; 2 R denoteLKfunctionalsdenedas P t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( t t s k u k 2 d d s; Q k b 2 t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( t k r k 2 d and 2 R + isaknown,adjustableconstant.Additionally,let k b = k b 1 + k b 2 + k b 3 ,where k bi 2 R + ;i =1 ; 2 ; 3 areadjustableconstants. 5.3StabilityAnalysis Theorem5.1. Giventhedynamicsin5,thecontrollerin5ensuressemi-global uniformlyultimatelyboundedtrackinginthesensethat k e t k 0 exp )]TJ/F25 11.9552 Tf 9.298 0 Td [( 1 t + 2 where 0 ; 1 ; 2 2 R + denoteconstants,providedthetime-delayissufcientlysmall, therateofchangeofthetime-delayissufcientlyslowseeAssumption5.2andthe 79

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followingsufcientgainconditionsaresatised > 2 4 ;k b 1 > 1 3 )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(k b 2 + k b 3 +4 k 2 b ;! 2 > 2 1 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ ; 4 k b 3 > 2 2 0 B @ v u u u t 2 k y k 2 + n 2 4 k b 2 min n 1 ; 1 2 k b o 1 C A where min )]TJ/F26 7.9701 Tf 13.151 5.256 Td [( 2 4 ;k b 1 )]TJ/F26 7.9701 Tf 13.151 5.112 Td [(k b 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 2 b !; 1 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ )]TJ/F26 7.9701 Tf 15.021 4.707 Td [( 2 and ; 2 R + aresubsequentlydenedconstants. Proof. Let V L y;t : D [0 ; 1 R beacontinuouslydifferentiable,positive-denite functionalonadomain D R 2 n +2 ,denedas V L 1 2 e T e + 1 2 r T r + P + Q whichcanbeboundedas 1 2 k y k 2 V L k y k 2 : Utilizing5and5,applyingtheLeibnizRuletodeterminethetimederivativeof 5and5,andbycancelingsimilarterms,thetimederivativeof5canbe expressedas V L = )]TJ/F25 11.9552 Tf 9.298 0 Td [(e T e + e T e z + r T N d + r T )]TJ/F25 11.9552 Tf 11.955 0 Td [(r T d )]TJ/F25 11.9552 Tf 11.956 0 Td [(k b r T r )]TJ/F25 11.9552 Tf 11.955 0 Td [(k b r T r + k u k 2 )]TJ/F25 11.9552 Tf 9.298 0 Td [(! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( t k u k 2 d + k b 2 k r k 2 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [(k b )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 k r k 2 : Young'sInequalitycanbeusedtoupperboundselecttermsin5as k e kk e z k 2 4 k e k 2 + 1 2 k e z k 2 ; k r kk r k 1 2 k r k 2 + 1 2 k r k 2 : 80

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UtilizingAssumption5.2,5,5,5,5and5,520canbe expanded,regroupedandupperboundedas V L )]TJ/F25 11.9552 Tf 28.559 0 Td [( k e k 2 + 2 4 k e k 2 + 1 2 k e z k 2 + n d k r k + k z k k z kk r k + d 1 k x kk r k + d 2 k r k )]TJ/F25 11.9552 Tf 9.298 0 Td [(k b k r k 2 + k b 2 k r k 2 + k b 2 k r k 2 + k 2 b k r k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( t k u k 2 d + k b 2 k r k 2 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [(k b )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 k r k 2 : UtilizingtheCauchy-Schwarzinequality,theintegralin5canbeupperboundedas )]TJ/F25 11.9552 Tf 11.955 0 Td [(! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( t k u k 2 d )]TJ/F25 11.9552 Tf 23.114 8.088 Td [(! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 k e z k 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( t k u k 2 d: Substituting5into5yields V L )]TJ/F30 11.9552 Tf 30.552 16.857 Td [( )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 4 k e k 2 )]TJ/F15 11.9552 Tf 13.433 8.088 Td [(1 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ )]TJ/F25 11.9552 Tf 15.315 8.088 Td [( 2 k e z k 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( k b 1 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(k b 2 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [(k b 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(k 2 b k r k 2 + k b 2 k r k 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(k b )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 k r k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(k b 2 k r k 2 + n k r k)]TJ/F25 11.9552 Tf 20.589 0 Td [(k b 3 k r k 2 + 2 k z k k z kk r k )]TJ/F25 11.9552 Tf 10.494 8.088 Td [(! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( k u k 2 d where k bi 8 i =1 ; 2 ; 3 weredenedafter5, n 2 R + isdenedas n n d + d 2 and 2 k z k 2 R isapositive,globallyinvertible,nondecreasingfunctiondenedas 2 k z k k z k + d 1 k x k .BasedonAssumption5.2, k b )]TJ/F24 7.9701 Tf 7.985 0 Td [(_ 2 k r k 2 > k b 2 k r k 2 .Utilizing thisfactandcompletingthesquaresfor k r k ,theexpressionin5canbeupper boundedas V L )]TJ/F30 11.9552 Tf 30.552 16.857 Td [( )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 2 k z k 4 k b 3 k z k 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( k u k 2 d + n 2 4 k b 2 where isanauxiliaryconstantdenedinTheorem5.1.Ifthesufcientconditionsin 5aresatised,then > 0 .Theinequality[31] t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( t t s k u k 2 d d s sup s 2 [ t;t )]TJ/F26 7.9701 Tf 6.586 0 Td [( ] t s k u k 2 d = t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( t k u k 2 d 81

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canbeusedtoupperbound5as V L )]TJ/F30 11.9552 Tf 30.552 16.857 Td [( )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 2 k z k 4 k b 3 k z k 2 + n 2 4 k b 2 )]TJ/F25 11.9552 Tf 15.428 8.088 Td [(! 4 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( t t s k u k 2 d )]TJ/F25 11.9552 Tf 10.494 8.088 Td [(k 2 b 4 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( t k r k 2 d : Basedon5-5,anupperboundfor5canbedevelopedas V L )]TJ/F25 11.9552 Tf 21.918 0 Td [( 2 k y k 2 )]TJ/F30 11.9552 Tf 11.956 16.857 Td [( )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 2 k z k 4 k b 3 k e z k 2 + n 2 4 k b 2 where 2 k z k ;; 2 R + isdenedas 2 = inf ; )]TJ/F25 11.9552 Tf 13.15 8.088 Td [( 2 2 k z k 4 k b 3 ; k b )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 ; )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 4 : Providedthefollowinginequalityissatised, )]TJ/F25 11.9552 Tf 13.15 8.088 Td [( 2 2 k z k 4 k b 3 > 0 ; theboundin5canbeusedtoupperboundtheinequalityin5as V L )]TJ/F25 11.9552 Tf 21.917 0 Td [( 2 V L + n 2 4 k b 2 : BasedonAssumption5.2,andprovidedtheinequalityin5issatised, 2 k z k ;; canbelowerboundedbyaconstant, ,introducedinTheorem5.1;hence,thelinear differentialequationin5satises V L V L e )]TJ/F26 7.9701 Tf 6.587 0 Td [(t + n 2 4 k b 2 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F26 7.9701 Tf 6.586 0 Td [(t : From5and5, k y t k 2 2 k y k 2 + n 2 4 k b 2 : Basedon5and5, k e t k 2 + k r t k 2 + Q t 2 k y k 2 + n 2 4 k b 2 : 82

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Basedon5,5,and5, k e z t k 2 2 k b t Q t ;hencefrom5, min 1 ; 1 2 k b k z t k 2 k e t k 2 + k r t k 2 + k e z t k 2 2 k b 2 k y k 2 + n 2 4 k b 2 : From5,analsufcientconditionfor5canbeobtainedas 4 k b 3 > 2 2 0 B @ v u u u t 2 k y k 2 + n 2 4 k b 2 min n 1 ; 1 2 k b o 1 C A : Consideraset S denedas S y t 2 R 2 n +2 jk y k < s 1 2 min 1 ; 1 2 k b )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 2 p k b 3 )]TJ/F15 11.9552 Tf 20.823 8.088 Td [( n 2 4 k b 2 : From5,given y k b 3 canbeselectedsuchthat y 2S i.e.asemi-global resulttoyieldtheresultin5when issufcientlysmall. Remark 5.2 Givenaninitialcondition y k b 3 canbeselectedlargeenoughtosatisfy thesufcientconditionin5,provided issufcientlysmalli.e., j t j <" < 1 If k b 3 isselectedarbitrarilylarge,then ,introducedin5,needstobeselected arbitrarilysmallsothat k b 1 canbeselectedlargeenoughtosatisfythesecondsufcient conditiongivenin5.If isselectedarbitrarilysmall,then ,fromTheorem5.1 and5,needstobeselectedsufcientlylargetosatisfythethirdsufcientcondition givenin5,provided t satisesAssumption5.2.Theconstants and arenot presentinthecontroller.However,theramicationsofthefactthat mustbeselected largeenough,arethatthecontrolgain ,denedin5,mustbeselectedsufciently largebasedontherstinequalityin5. 5.4Euler-LagrangeExtension Althoughtheworkin[28,30,146,178]providesfundamentalcontributionstothe inputdelayproblemforfeedforwardsystems,theapplicabilityofthesemethodsto generaluncertainelectromechanicalsystemse.g.,modeledbyELdynamicsisnot clear.Atransformationisprovidedin[179]toconvertanELsystemintoafeedforward 83

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system,butthetransformationrequiresEMK.Itisnotapparenthowtotransforma feedforwardsystemintoanELsystemwhenthesystemparametersareunknownorthe dynamicsareuncertain,whichimpliesthatmethodsdevelopedforfeedforwardsystems withinputdelaysmaynotbeapplicabletouncertainELdynamics. Theresultsinthischaptercanbeextendedtogeneral,nonlinearELdynamics.To illustratethis,consideraninput-delayedEuler-Lagrangesystemoftheform M q q + V m q; q q + G q + F q + d t = u t )]TJ/F25 11.9552 Tf 11.955 0 Td [( t where M q 2 R n n denotesageneralizedinertiamatrix, V m q; q 2 R n n denotes ageneralizedcentripetal-Coriolismatrix, G q 2 R n denotesageneralizedgravity vector, F q 2 R n denotesgeneralizedfriction, d t 2 R n denotesanexogenous disturbance, u t )]TJ/F25 11.9552 Tf 11.955 0 Td [( t 2 R n representsthegeneralizeddelayedinputcontrolvector, where t 2 R isanon-negativetime-varyingdelay,and q t ; q t ; q t 2 R n denote thegeneralizedstates. Thesubsequentdevelopmentisbasedontheassumptionthat q t ; q t are measurableoutputs, M q V m q; q G q F q d t areunknown,thetime-varying inputdelayisknown.Knowledgeofpastvaluesofthecontrolinputareagainassumed. Additionally,thefollowingpropertyisused: Property1. Theinertiamatrix M q issymmetricpositive-denite,andsatisesthe followinginequality: m k k 2 T M m k k 2 ; 8 2 R n where m ; m 2 R + areknownconstantsand kk denotesthestandardEuclideannorm. TheinclusionofthisELextensioncanbeconsideredasanapplicationtothe methodpresentedinthischapter.Thedesignoftheerrorsystemsandcontrollerfollow similarlytothedevelopmentinSection5.2.Specically,5canbemodiedslightlyto accommodatetheuncertaininertiaeffectsinthedynamicsand5and5canbe designedasinSection5.2. 84

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Thelteredtrackingerror, r e;e z ;t isredenedas r e + e )]TJ/F25 11.9552 Tf 11.956 0 Td [(Be z where B 2 R n n isasymmetric,positive-deniteconstantgainmatrixthatsatisesthe followinginequality k B k 1 b where b 2 R + isaknownconstant.Theerrorbetween B and M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q isdenotedby q 2 R n n andisdenedas B )]TJ/F25 11.9552 Tf 11.955 0 Td [(M )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 andsatisesthefollowinginequality k k 1 where 2 R + isaknownconstant. Duetotheinclusionof B ,theopenlooperrorsystemwillcontainanadditivetimedelayedterm: M u )]TJ/F25 11.9552 Tf 11.955 0 Td [(u .Motivatedbytheneedtocancelthisterminthestability analysis,basedonthestructureof5andinspiredbytheworkin[180],theLK functional Q e f 2 R isredenedas Q k b m +1 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( t k r k 2 d : ALyapunov-basedstabilityanalysissimilartotheonepresentedforthegeneralsecondordernonlinearsysteminSection5.3isusedtoderivesufcientconditionsforUUB tracking.Duetotheformof5andtheinclusionof B ,TheEuler-Lagrangeextensionanalysisresultsinalteredsufcientconditions: > b 2 2 4 ;k b >sup ; 2 m +1 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ 2 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ + ; k 3 >sup ; k 2 b + 2 k b m 1 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ where 2 2 R + isaknownconstantboundonthesecondderivativeofthedelay.For moredetailsoftheextensionandacompletestabilityanalysis,see[181]. 85

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5.5SimulationResults Thecontrollerin5wassimulatedusingtwo-linkplanarrobotmanipulator dynamicstoexaminetheperformanceandrobustnesstovariationsintheinputdelay. Motivationforusingrobotdynamicsstemsfromthefactthatthedynamicscanbe expressedasanELsystemasprovidedintheextensionthatiscommontoalarge classofpracticalengineeringsystems.In5, M q 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 V m q; q 2 6 4 )]TJ/F25 11.9552 Tf 9.299 0 Td [(p 3 s 2 q 2 )]TJ/F25 11.9552 Tf 9.298 0 Td [(p 3 s 2 q 1 +_ q 2 p 3 s 2 q 1 0 3 7 5 ,and F q 2 6 4 f d 1 0 0 f d 2 3 7 5 2 6 4 q 1 q 2 3 7 5 ,where p 1 =3 : 473 kg m 2 p 2 =0 : 196 kg m 2 p 3 =0 : 242 kg m 2 f d 1 =5 : 3 Nmsec f d 2 =1 ; 1 Nmsec c 2 denotes cos q 2 ,and s 2 denotes sin q 2 .Anadditivenon-vanishingexogenous disturbancewasappliedas d 1 t =0 : 2 sin )]TJ/F26 7.9701 Tf 7.263 -4.976 Td [(t 2 ,and d 2 t =0 : 1 sin )]TJ/F26 7.9701 Tf 7.263 -4.976 Td [(t 4 .Theinitial conditionsforthemanipulatorwereselectedas q 1 ;q 2 =0 deg .Thedesiredtrajectories wereselectedas q d 1 t =20 sin : 5 t 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F24 7.9701 Tf 6.586 0 Td [(0 : 01 t 3 deg; q d 2 t =10 sin : 5 t 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(e )]TJ/F24 7.9701 Tf 6.586 0 Td [(0 : 01 t 3 deg: Toillustraterobustnesstotheinputdelay,simulationswerecompletedusing varioustime-varyingdelays.Foreachcase,theRMSerrorsareshowninTable5-1. Theresultsindicatethattheperformanceofthesystemisrelativelylesssensitivetothe delayfrequencyandmoresensitivetothedelaymagnitude.Thisoutcomeagreeswith previousinputdelayresultswherethetrackingperformancereducesaslargerconstant delaysareappliedtothesystem[31]. ResultsinTable5-1indicatethattheperformancedegradationresultingfrom thefrequencyofthedelayappearedtobeminimal.Thus,analysiswasalsoconductedtofurtherexaminetherobustnessofthecontrollerwithrespecttounknown variancesinthefrequencyandmagnitudeofthedelay.Ineachcase,theactualinput 86

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Table5-1.RMSerrorsfortime-varyingtime-delayratesandmagnitudes. Time-Delay t msRMSErrorLink1RMSErrorLink2 Fast,Small 2 sin )]TJ/F26 7.9701 Tf 7.264 -4.977 Td [(t 2 +30 : 0524 o 0 : 0363 o Fast,Large 20 sin )]TJ/F26 7.9701 Tf 7.264 -4.977 Td [(t 2 +300 : 4913 o 0 : 5687 o Slow,Small 2 sin )]TJ/F26 7.9701 Tf 9.381 -4.976 Td [(t 10 +30 : 0521 o 0 : 0341 o Slow,Large 20 sin )]TJ/F26 7.9701 Tf 9.381 -4.977 Td [(t 10 +300 : 5179 o 0 : 6970 o Table5-2.RMSerrorswhenthecontrollerisappliedwithamismatchbetweenthe assumedtimedelayandtheactualdelay.TheTime-DelayVariancecolumn indicatesthe%differenceofthemagnitudeandfrequencyoftheactualinput delayinthesystem. Time-DelayVarianceRMSErrorLink1RMSErrorLink2 -30%magnitude 0 : 0633 o 0 : 0766 o -10%magnitude 0 : 0497 o 0 : 0662 o 0%magnitude 0 : 0394 o 0 : 0605 o +10%magnitude 0 : 0495 o 0 : 0764 o +30%magnitude 0 : 0628 o 0 : 1069 o +10%frequency 0 : 0393 o 0 : 0605 o +30%frequency 0 : 0394 o 0 : 0604 o +50%frequency 0 : 0405 o 0 : 0619 o delaywasvariedfromtheassumedknowndelayusedinthecontroller.Thecontrollerwasimplementedassumingasinusoidaltime-varyinginputdelaygivenas t = )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ m v 100 m a sin t 1+ f v 100 f a + s,where m a denotesthebaselinemagnitude coefcientmsinthiscase, m v denotesthemagnitudevariance, f a denotesthebaselinefrequencycoefcientinthiscase, f v denotesthefrequencycoefcientvariance, and denotesthedelayoffsetmsinthiscase,resultinginabaselinedelaysignal withapeakmagnitudeof10ms.TheresultsinTable5-2suggestthatthecontroller isrobusttovariancesinthedelaymagnitudeandfrequency.Figure5-1illustratesthe time-delayandthetrackingerrorsassociatedwiththe+50%frequencyvariancecase. 5.6Summary Acontinuouspredictor-basedcontrollerisdevelopedforuncertainnonlinear systemswhichincludetime-varyinginputdelaysandsufcientlysmoothadditive boundeddisturbances.ThecontrollerguaranteesUUBtrackingprovidedthedelayis sufcientlysmallandslowlyvarying.Anextensionillustratesthecontroller'sapplicability 87

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Figure5-1.Trackingerrorsvs.timeforcontrollerproposedin5with+50%frequency varianceininput-delay:ATime-delayinseconds,BTrackingerrorin degrees. toawidearrayofelectromechanicalsystemsthatcanbedescribedbyELdynamics. Whilethecontroldevelopmentcanbeappliedwhenthereisuncertaintyinthesystem dynamics,thecontrollerisbasedontheassumptionthatthetime-varyingdelayis known.However,thesimulationresultsindicatesomerobustnesstouncertaintyinthe delaymagnitudeandfrequency.Variouspracticalscenariosmotivatetheneedtorelax theassumptionthatthedelayproleisknown.Futureeffortswillfocusoneliminating thisassumption,whichpresentsasignicantchallenge,sincetheinherentstructureof thepredictordependsonintegratingthecontroleffortovertheknowndelayinterval. Thestabilityanalysisalsoindicatesinaconservativemannerthroughsufcient gainconditionsanexpectedlinkbetweentheinitialconditions,thedelaymagnitude,the delayrate,andthedomainofattraction.Afavorableoutcomeofthedevelopedcontroller isthatgivenanyniteinitialconditionandniteamountofdelaythoughsufciently small,thecontrolgainscanbeselectedtoensurethetrackingerrorisregulated, assumingarbitrarilylargecontrolauthority. 88

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CHAPTER6 TIME-VARYINGINPUTANDSTATEDELAYCOMPENSATIONFORUNCERTAIN NONLINEARSYSTEMS Chapter6combinestheworkofChapters4and5byconsideringanonlinear systemwithbothtime-varyinginputandstatedelays.Acontinuous,robust,predictorbasedcontrollerisdevelopedforuncertain,second-ordernonlinearsystemssubject tosimultaneoustime-varyingunknownstateandknowninputdelaysinadditionto additiveboundeddisturbances.ADCAL-basedpredictorstructureofpreviouscontrol valuesfacilitatesadelay-freeopen-looperrorsystemandthedesignofacontroller basedontheRISEcontroltechnique.AstabilityanalysisutilizingLKfunctionals guaranteessemi-globalasymptotictrackingthanksinparttoanewerrorsystem developmentandtheinclusionoftheRISEcontrollerassumingthedelaysarebounded andslowlyvarying.NumericalsimulationsillustrateimprovedperformanceoverChapter 5'stime-varyinginputdelaycontroldesignandrobustnessofthedevelopedmethodto variouscombinationsofsimultaneousinputandstatedelays. 6.1DynamicModel Consideraclassofsecondordernonlinearsystemsofthefollowingform 1 : x = f x; x;t + g x t )]TJ/F25 11.9552 Tf 11.955 0 Td [( s t ; x t )]TJ/F25 11.9552 Tf 11.955 0 Td [( s t ;t + d t + u x; x;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( i t where x t ; x t 2 R n arethegeneralizedsystemstates, u x; x;t )]TJ/F25 11.9552 Tf 11.956 0 Td [( i 2 R n isthe generalizedcontrolinput, f x; x;t : R 2 n [0 ; 1 R n isanunknownnonlinear C 2 function, g x t )]TJ/F25 11.9552 Tf 11.955 0 Td [( s ; x t )]TJ/F25 11.9552 Tf 11.955 0 Td [( s ;t : R n R n [0 ; 1 R n isanunknownnonlinear C 2 time-delayedfunction, d t 2 R n denotesageneralized,sufcientlysmooth, 1 Theresultinthischaptercanbeextendedto n th -ordernonlinearsystemsfollowinga similardevelopmenttothosepresentedin[122,129,177]. 89

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nonvanishingnonlineardisturbancee.g.,unmodeledeffectsand i t ; s t 2 R + denotenon-negativeinputandstatedelays,respectively. Thesubsequentdevelopmentisbasedontheassumptionthat x t and x t aremeasurableoutputs.Throughoutthechapter,atime-dependentdelayedfunction isdenotedas t )]TJ/F25 11.9552 Tf 11.955 0 Td [( or ,and kk denotestheEuclideannormofavector.Asin Chapter4,Assumption3.1isusedtodescribethedisturbanceterm,andAssumption 3.2isusedtodescribethedesiredtrajectory.Additionally,thefollowingassumptionson thedelayswillbeexploited. Assumption6.1. Theinputandstatedelaysareboundedsuchthat 0 i t i 1 and 0 s t s 1 andtherateofchangeofthedelaysareboundedsuchthat j i t j i 2 < 1 and j s t j s 2 < 1 where j 2 R + 8 j = i 1 ;i 2 ;s 1 ;s 2 areknown constants.Thestatedelayisassumedtobeunknown,whiletheinputdelayisassumed tobeknown. Remark 6.1 InAssumption6.1,theslowlytime-varyingconstrainti.e., j i;s t j i 2 ;s 2 < 1 iscommontoresultswhichutilizeclassicalLKfunctionalstocompensate fortime-varyingtime-delays[18].Knowledgeofthestatedelaysinthesystemisnot requiredsincecompensationforthestatedelaysisaddressedthroughtheuseofa DCAL-basedrobustcontrolapproach.Thistechniqueisnotsufcienttoaddressthe inputdelaysinthesystem,presentingamoresignicantchallengetodevelopatechniquetocompensatefortheinputdelaysseparatelyfromthestatedelaycompensation. Theinputdelayisassumedtobeknownsincetheintervalofpastcontrolvaluesinthe predictorstructuredependsonthedelay.Thesimulationresultsillustraterobustnessto howwellthisvalueneedstobeknown. 6.2ControlDevelopment Theobjectiveistodesignacontinuouscontrollerthatwillensurethesystemstate x t ofthedelayedsystemin6tracksthedesiredstatetrajectory.Toquantifythe 90

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controlobjective,atrackingerrordenotedby e 1 x;t 2 R n ,isdenedas e 1 x d )]TJ/F25 11.9552 Tf 11.956 0 Td [(x: Tofacilitatethesubsequentanalysis,twoauxiliarytrackingerrors e 2 e 1 ; e 1 ;r e 2 ; e 2 ;e u 2 R n aredenedas[129] e 2 e 1 + 1 e 1 r e 2 + 2 e 2 + e u where 1 ; 2 2 R denoteconstantpositivecontrolgainsand e u i t ;t 2 R n denotes themismatchbetweenthedelayedcontrolinputandtheactualcontrolinput,denedas e u u t )]TJ/F25 11.9552 Tf 11.955 0 Td [( i t )]TJ/F25 11.9552 Tf 11.956 0 Td [(u t : Theauxiliarysignal e u facilitatestheabilitytoinjectadelay-freecontrolinputintothe errorsystemdevelopment.IncontrasttothedevelopmentinChapter5,thepredictorliketermin6iscontainedwithinthe r auxiliaryterminsteadofthe e 2 signal. Functionally,the e u termstillinjectsanintegralofpastcontrolvaluesintotheopenloopsysteminthisform;however,thedevelopmentinthischapterintroducesfewer cross-terms,allowingformorecontroldesignexibilityinthesubsequentanalysis.The auxiliarysignal r e 2 ; e 2 isintroducedtofacilitatethestabilityanalysisandisnotused inthecontroldesignsincetheexpressionin6dependsontheunmeasurablestate x t .Thestructureoftheerrorsystemsandincludedauxiliarysignalsismotivatedby theneedtoinjectandcanceltermsinthesubsequentstabilityanalysisandwillbecome apparentinSection6.3. Anauxiliarylterfor6,denotedby e uf e u 2 R n ,isdenedasthesolutiontothe differentialequation e uf )]TJ/F25 11.9552 Tf 9.298 0 Td [( 2 e uf + e u : 91

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Substituting6into6,yields r =_ e 2 + 2 e 2 +_ e uf + 2 e uf : Utilizinganauxiliarysignal, e 2 ;e uf 2 R n ,denedas = e 2 + e uf ; theexpressionin6canberewrittenas r =_ + 2 : Equation6isanintermediatestepinthedevelopmentoftheopenlooperrorsystem thatwillbeexplicitlyusedinthesubsequentstabilityanalysis.Substituting6-6 into6yields r = x d )]TJ/F25 11.9552 Tf 11.956 0 Td [(f x; x;t )]TJ/F25 11.9552 Tf 11.956 0 Td [(g x s ; x s ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(d t )]TJ/F25 11.9552 Tf 11.955 0 Td [(u i + 1 e 1 +_ e uf + 2 : UsingaDCAL-baseddesignapproach[175],anopen-looptrackingerrorcanbe obtainedbysubstituting6and6into6,allowingthetime-delayedcontrol inputstocancelas r = S 1 + S 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(u wheretheauxiliaryfunction S 1 x;x d ; x; x d ;x s ; x s ;x d s ; x d s ; e 1 ;e 2 ;t 2 R n and S 2 x d ; x d ; x d ;x d s ; x d s ;t 2 R n aredenedas S 1 f x d ; x d ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(f x; x;t + g x d s ; x d s ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(g x s ; x s ;t + 1 e 1 + 2 e 2 S 2 x d )]TJ/F25 11.9552 Tf 11.955 0 Td [(f x d ; x d ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(g x d s ; x d s ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(d: Basedontheformof6andthesubsequentstabilityanalysis,thecontroller, u e 2 ;v ,isdesignedas u k s +1 e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k s +1 e 2 t 0 + v 92

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where v e 2 ;e u ; 2 R n istheFilippovsolutiontothefollowingdifferentialequation v k s +1 2 e 2 + k s +1 e u + sgn where k s ; 2 R arepositiveconstantcontrolgains,and sgn isdened 8 2 R m = 1 2 ::: m T as sgn sgn 1 sgn 2 :::sgn m T 2 TheexistenceofFilippovsolutionscanbeestablishedfor v 2 K [ h 1 ] e 2 ;e u ; where h 1 e 2 ;e u ; 2 R n isdenedastheright-handsideof6,and K [ h 1 ] T > 0 T S m =0 coh 1 e 2 ;e u ;B ; )]TJ/F25 11.9552 Tf 11.955 0 Td [(S m ,where T S m =0 denotestheintersectionofall sets S m ofLebesguemeasurezero, co denotesconvexclosure,and B ; = f & 2 R n jk )]TJ/F25 11.9552 Tf 11.955 0 Td [(& k < g [45,154].Inthiscase, S m isexpressedbythesingletonset f =0 g Theclosed-looptrackingerrorsystemcanbedevelopedbytakingthetimederivativeof6andusingthetimederivativeof6toyield r = ~ N + N d )]TJ/F25 11.9552 Tf 11.956 0 Td [(e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k s +1 r )]TJ/F25 11.9552 Tf 11.955 0 Td [(sgn where ~ N S 1 ;e 2 2 R n and N d S 2 2 R n aredenedas ~ N S 1 + e 2 ; N d S 2 : Thestructureof6ismotivatedbythedesiretosegregatetermsthatcanbe upperboundedbyastate-dependentsignalandtermsthatcanbeupperbounded byconstants.BasedonAssumptions3.1and3.2,thefollowinginequalitiescanbe developedfromtheexpressionin6: k N d k N d 1 ; N d N d 2 2 Theinitialconditionfor v isselectedsuchthat u =0 93

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where N d 1 ; N d 2 2 R ; areknownpositiveconstants.ApplyingtheMVT,anupperbound canbedevelopedfortheexpressionin6as[164,AppendixA] ~ N 1 k z k k z k + 2 k z s k k z s k where z e 1 ;e 2 ;r 2 R 3 n denotesthevector z e T 1 ;e T 2 ;r T T andtheboundingterms 1 ; 2 2 R areapositive,globallyinvertible,nondecreasing functions.Theupperboundfortheauxiliaryfunction ~ N issegregatedintodelay-free anddelay-dependentboundingfunctionstoeliminatethedelayedtermswiththeuseof anLKfunctionalinthestabilityanalysis. 6.3StabilityAnalysis Theorem6.1. Giventhedynamicsin6,thecontrollergivenin6and6 ensuresasymptotictrackinginthesensethat k e 1 t k! 0 ast !1 providedthecontrolgainsareselectedbasedonthefollowingsufcientconditions 1 > 1 2 ; 2 > 1 ;> N d 1 + N d 2 2 ; 2 )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ i 2 +1 > i ; 2 k s > 2 k z k where min 1 )]TJ/F24 7.9701 Tf 13.151 4.708 Td [(1 2 ; 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; 1 2 R k z k : R R isasubsequentlydened positive-denite,globallyinvertible,nondecreasingfunction,and k z k containsthe initialconditionsofthestate. Proof. Let D R 3 n +3 beadomaincontaining y z;P;Q;R LK 2 R 3 n +3 ,denedas y z T p P p Q p R LK T : 94

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In6,theauxiliaryfunction P ;t 2 R isdenedastheFilippovsolutiontothe followingdifferentialequation P = )]TJ/F25 11.9552 Tf 9.298 0 Td [(r T N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(sgn ;P t 0 ;t 0 = n X i =1 j i t 0 j)]TJ/F25 11.9552 Tf 17.933 0 Td [( i t 0 T N d t 0 wherethesubscript i =1 ; 2 ;:::;n denotesthe i thelementofavector.Similartothe developmentin6,existenceofsolutions P ;t for6canbeestablished. Providedthesufcientconditionfor in6issatised, P ;t 0 SeeAppendixB fordetails.Additionally,let Q u; i ;t ;R LK z; s ;t 2 R denoteLKfunctionals,dened as Q t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( i t t s k u k 2 d ds R LK 2 k s t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( s t 2 2 k z k k z k 2 d where !; 2 R areknown,positive,adjustableconstants,and k s and 2 were introducedin6and6,respectively. Let V : D [0 ; 1 R beacontinuouslydifferentiablein y ,locallyLipschitzin t regularfunctiondenedas V 1 2 e T 1 e 1 + 1 2 e T 2 e 2 + 1 2 r T r + P + Q + R LK whichsatisesthefollowinginequalities: 1 y V y;t 2 y : wherethecontinuous,positive-denitefunctions 1 y ; 2 y 2 R in6aredened as 1 y 1 k y k 2 2 y 2 k y k 2 and 1 ; 2 2 R + arepositiveconstants. UnderFilippov'sframework,thetimederivativeof6existsalmosteverywhere, i.e.,foralmostall t 2 [ t 0 ;t f ] ,and V y;t a:e: 2 ~ V y;t where ~ V = 2 @V L y;t T K e T 1 e T 2 r T 1 2 P )]TJ/F18 5.9776 Tf 7.782 3.259 Td [(1 2 P 1 2 Q )]TJ/F18 5.9776 Tf 7.782 3.259 Td [(1 2 Q 1 2 R )]TJ/F18 5.9776 Tf 7.782 3.258 Td [(1 2 LK R LK 1 T ; 95

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and @V isthegeneralizedgradientof V y;t [166].Since V y;t isaLipschitzcontinuousregularfunction ~ V r V T K e T 1 e T 2 r T 1 2 P )]TJ/F18 5.9776 Tf 7.782 3.259 Td [(1 2 P 1 2 Q )]TJ/F18 5.9776 Tf 7.782 3.259 Td [(1 2 Q 1 2 R )]TJ/F18 5.9776 Tf 7.782 3.259 Td [(1 2 LK R LK T where r V h e T 1 ;e T 2 ;r T ; 2 P 1 2 ; 2 Q 1 2 ; 2 R 1 2 LK i T : Usingthecalculusfor K [ ] from[154],applyingtheLeibnizRuletodeterminethe timederivativeof6and6,andsubstituting6-6,614,and6 into6,yields ~ V e T 1 e 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 e 1 + e T 2 r )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 e 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [(e u + r T ~ N + N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(e 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k s +1 r )]TJ/F25 11.9552 Tf 11.956 0 Td [(K [ sgn ] )]TJ/F25 11.9552 Tf 9.298 0 Td [(r T N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(K [ sgn ]+ i k u k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ i t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( i t k u k 2 d + 2 k s 2 2 k z k k z k 2 )]TJ/F25 11.9552 Tf 13.15 8.088 Td [( )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ s 2 k s 2 2 k z s k k z s k 2 where K [ sgn e 2 ]= SGN e 2 suchthat SGN e 2 i =1 if e 2 i > 0 ; [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ; 1] if e 2 i =0 and )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 if e 2 i < 0 [154]. Cancelingcommonterms,theexpressionin6canbeupperboundedas ~ V a:e: )]TJ/F25 11.9552 Tf 22.506 0 Td [( 1 k e 1 k 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [( 2 k e 2 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( k s +1 k r k 2 + k e 1 kk e 2 k + k e 2 kk e u k + k r k 1 k z k k z k + k r k 2 k z s k k z s k + i k u k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ i t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( i t k u k 2 d + 2 k s 2 2 k z k k z k 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ s 2 k s 2 2 k z s k k z s k 2 wherethesetin6reducestothescalarinequalityin6sincetheRHSis continuousa.e.,i.e,theRHSiscontinuousexceptfortheLebesguenegligiblesetof timeswhen r T K [ sgn ] )]TJ/F25 11.9552 Tf 12.604 0 Td [(r T K [ sgn ] 6 =0 [45,167]. 3 Utilizingthedenitionof 3 Thesetoftimes n t 2 [0 ; 1 : r t T K [ sgn t ] )]TJ/F25 11.9552 Tf 11.956 0 Td [(r t T K [ sgn t ] 6 =0 o [0 ; 1 isequivalenttothesetoftimes f t : t =0 ^ r t 6 =0 g .From6,thissetcan 96

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6,6and6,andYoung'sInequalitytoshowthat e T 1 e 2 1 2 k e 1 k 2 + 1 2 k e 2 k 2 e T 2 e u 1 2 k e 2 k 2 + 1 2 k e u k 2 and k r k 2 k z s k k z s k k s 2 k r k 2 + 1 2 k s 2 2 k z s k k z s k 2 ,the expressionin6canbeupperboundedas ~ V a:e: )]TJ/F30 11.9552 Tf 24.499 16.857 Td [( 1 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 k e 1 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 k e 2 k 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( k s +1 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(k s 2 k r k 2 + 1 2 k e u k 2 + i k u k 2 )]TJ/F25 11.9552 Tf 9.298 0 Td [(! )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ i t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( i t k u k 2 d + 2 k s 2 2 k z k k z k 2 )]TJ/F25 11.9552 Tf 13.15 8.087 Td [( )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ s 2 k s 2 2 k z s k k z s k 2 + k r k 1 k z k k z k + 1 2 k s 2 2 k z s k k z s k 2 : If )]TJ/F25 11.9552 Tf 11.955 0 Td [(' s 2 > 1 ,bycompletingthesquaresfor k r k andbyutilizingthefactthat k u t k 2 t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( i t k u k 2 d; k e u k 2 i t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( i t k u k 2 d; theexpressionin6canbeupperboundedas ~ V a:e: )]TJ/F30 11.9552 Tf 24.498 16.857 Td [( 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 k e 1 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 k e 2 k 2 )-222(k r k 2 + 2 1 k z k k z k 2 2 k s + 2 2 k z k k z k 2 2 k s )]TJ/F30 11.9552 Tf 11.956 13.27 Td [( )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ i )]TJ/F25 11.9552 Tf 11.955 0 Td [(! i )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( i 2 t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( i t k u k 2 d: Iftheconditionsin6aresatised,theexpressionin6reducesto ~ V a:e: )]TJ/F30 11.9552 Tf 25.827 16.857 Td [( )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 k z k 2 k s k z k 2 )]TJ/F25 11.9552 Tf 21.918 0 Td [( 3 y = )]TJ/F25 11.9552 Tf 9.298 0 Td [(c k z k 2 8 y 2 D forsomepositiveconstant c 2 R + anddomain D = y 2 R 3 n +3 jk y k < )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 )]TJ/F23 11.9552 Tf 5.479 -0.392 Td [(p 2 k s where wasintroducedin6,andtheboundingfunction k z k from6is denedas 2 k z k 2 1 k z k + 2 2 k z k .Largervaluesof k s willexpandthesizeof thedomain D .Theinequalitiesin6and6canbeusedtoshowthat V 2L 1 in D .Thus, e 1 ;e 2 ;r 2L 1 in D .Theclosed-looperrorsystemcanbeusedto alsoberepresentedby f t : t =0 ^ t 6 =0 g .Provided t iscontinuouslydifferentiable,itcanbeshownthatthesetoftimeinstances f t : t =0 ^ t 6 =0 g isisolated, andthus,measurezero.Thisimpliesthattheset ismeasurezero. 97

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concludethattheremainingsignalsareboundedin D ,andthedenitionsfor 3 and z canbeusedtoshowthat 3 isuniformlycontinuousin D .Let S D D denotea setdenedas S D y 2 D j 2 y < 1 )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 p 2 k s 2 : Theregionofattractionin6canbemadearbitrarilylargetoincludeanyinitial conditionsbyincreasingthecontrolgain k s .From6,[182,Corollary1]canbe invokedtoshowthat c k z k 2 0 ast !18 y 2S D : Basedonthedenitionof z in 6, k e 1 k! 0 ast !18 y 2S D : 6.4SimulationResults Thecontrollerin6wassimulatedtoexaminetheperformanceandrobustnesstovariationsinboththestateandinputdelay.Specicallythedynamics from6areutilizedwhere n =2 ;f x; x;t 2 6 4 )]TJ/F25 11.9552 Tf 9.298 0 Td [(p 4 s 2 p 5 s 2 x 2 3 7 5 g x s ; x s ;t 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 s x 2 s 3 7 5 + 2 6 4 f d 1 0 0 f d 2 3 7 5 2 6 4 x 1 s x 2 s 3 7 5 x; x; x 2 R 2 denote thestateposition,velocity,andacceleration, d t 2 R 2 denotesanadditiveexternal disturbance, u x; x;t )]TJ/F25 11.9552 Tf 11.955 0 Td [( i t 2 R 2 denotesthedelayedcontrolinputand s t ; i t 2 R denotetheunknownnon-negativetime-varyingstatedelayandtheknownnon-negative time-varyinginputdelay,respectively.Additionally, p 1 =3 : 473 p 2 =0 : 196 p 3 =0 : 242 p 4 =0 : 238 p 5 =0 : 146 f d 1 =5 : 3 f d 2 =1 ; 1 ,and s 2 ;s 2 s denote sin x 2 t and sin x 2 t )]TJ/F25 11.9552 Tf 11.955 0 Td [( s Remark 6.2 Thesystemisassumedtohavedelay-freesensorfeedback.Thisis evidentinthedynamicmodelpresentedin6,asthestatedelayonlyappearswithin thedelayedfunction g x t )]TJ/F25 11.9552 Tf 11.955 0 Td [( s t ; x t )]TJ/F25 11.9552 Tf 11.955 0 Td [( s t ;t .Scenarioswheredelaysarefound intheoutputarenotconsideredinthiswork.Thedynamicsin6canbetransformed intoanEL-likesystemtoresembleaclassofsystemswhichdescribesalargenumber 98

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ofphysicalapplicationssee[129]foradditionaldetailsonextensionstoEuler-Lagrange dynamics. Anadditive,non-vanishing,exogenousdisturbancewasappliedas d 1 =0 : 2 sin )]TJ/F26 7.9701 Tf 7.263 -4.977 Td [(t 2 and d 2 =0 : 1 sin )]TJ/F26 7.9701 Tf 7.263 -4.977 Td [(t 4 .Theinitialconditionsforthesystemwereselectedas x 1 ;x 2 =0 Thedesiredtrajectorieswereselectedas x d 1 t = sin : 5 t +20 1 )]TJ/F25 11.9552 Tf 11.956 0 Td [(e )]TJ/F24 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/F24 7.9701 Tf 6.587 0 Td [(0 : 01 t 3 : Toillustraterobustnesstothedelays,severalsimulationswerecompletedusing varioustime-varyingdelays.First,tocomparetheproposedcontrollertotheprevious input-delayedworkin[123],thecontrollerin6issimulatedwithnostatedelay. Figure6-1illustratesthecomparativeresultsofthetwocontrollers,assuming i = )]TJ/F15 11.9552 Tf 9.298 0 Td [(10 sin )]TJ/F26 7.9701 Tf 7.263 -4.976 Td [(t 3 +30 and s =0 sincetheresultin[123]didnotconsiderstatedelays. Notably,thetheproposedcontrollerachievesbettertrackingperformancecomparedto thePD-likecontrollerinChapter5. Next,robustnessoftheproposedcontrollerwasexaminedforthecaseswhenboth stateandinputdelaysarepresentinthesystem.Varioustimedelaycombinationswere consideredandforeachcase,theRMSerrorsareshowninTable6-1.Toillustrate thendings,Figure6-2depictsthetrackingerrors,actuationeffortandtime-varying delaysforCase3.Additionally,Case5isprovidedinFigure6-3.Theresultsindicate thattheperformanceofthesystemisrelativelylesssensitivetothedelayfrequency andmoresensitivetothedelaymagnitude.Thisoutcomeagreeswithpreviousinput delayresultswherethetrackingperformancereducesaslargerdelaysareappliedtothe system[31,181].Ingeneral,simulationresultsillustratethattheproposedcontrolleris abletoachievebettertrackingperformanceaswellashandlelargerinputdelayseven withaddedsimultaneousstatedelaysthantheprevioustime-varyinginput-delayed workin[181].Additionally,convergenceandperformanceareachievedinmoredelay 99

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Figure6-1.TrackingerrorsfortheAproposedcontrollerandBthePD-likecontroller in[181]whenconsideringaninputdelayof i = )]TJ/F15 11.9552 Tf 9.298 0 Td [(10 sin )]TJ/F26 7.9701 Tf 7.263 -4.977 Td [(t 3 +30 andno statedelay. Table6-1.RMSerrorsfortime-varyingtime-delayratesandmagnitudes. State-DelayInput-Delay s t ms i t msError x 1 Error x 2 Case1Slow,SmallSlow,Large 5 sin )]TJ/F10 6.9738 Tf 6.243 -4.147 Td [(t 8 +10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(50 sin )]TJ/F10 6.9738 Tf 8.228 -4.147 Td [(t 10 +1001 : 46 o 1 : 47 o Case2Fast,SmallFast,Small 10 sin )]TJ/F10 6.9738 Tf 6.242 -4.147 Td [(t 2 +40 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 sin )]TJ/F10 6.9738 Tf 6.243 -4.147 Td [(t 3 +300 : 29 o 0 : 30 o Case3Slow,LargeFast,Large 10 sin )]TJ/F10 6.9738 Tf 8.228 -4.147 Td [(t 10 +40 )]TJ/F8 9.9626 Tf 7.749 0 Td [(50 sin )]TJ/F10 6.9738 Tf 6.243 -4.147 Td [(t 2 +1001 : 32 o 1 : 39 o Case4Slow,LargeSlow,Large 10 sin )]TJ/F10 6.9738 Tf 8.228 -4.148 Td [(t 10 +4050 sin )]TJ/F10 6.9738 Tf 8.229 -4.148 Td [(t 10 +1001 : 52 o 2 : 02 o Case5Fast,LargeFast,Small 50 sin t +8005 sin )]TJ/F10 6.9738 Tf 6.243 -4.147 Td [(t 2 +300 : 16 o 0 : 32 o cases,indicatingaddedrobustnesstodelaysinthesystem.Asdepictedintheexample casesforthegivendynamics,thecontrollerismorerobusttolargermagnitudedelaysin thestatethanintheinput,asindicatedinthestabilityanalysisandisapparentinCase 5.Thisisnotsurprisingbasedonthesufcientconditionin6. 6.5Summary Thischapterpresentsacontinuouspredictor-basedcontrollerforuncertainnonlinearsystemswhichincludesimultaneoustime-varyingstateandinputdelaysaswellas sufcientlysmoothadditiveboundeddisturbances.ThecontrollerutilizesaDCAL-based designapproachtoassistincompensationoftheunknownstatedelayscoupledwith 100

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Figure6-2.Trackingerrors,actuationeffortandtime-varyingdelaysvstimeforCase3. Figure6-3.Trackingerrors,actuationeffortandtime-varyingdelaysvstimeforCase5. 101

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anerrorsystemstructurethatprovidesadelay-freeopen-looperrorsystem.TheRISEbasedcontrollerandLKfunctionalsguaranteesemi-globalasymptotictrackingprovided therateofthedelaysissufcientlyslow,butdoesnotrestricttheboundonthedelaysto besufcientlysmall.Thecontroldevelopmentcanbeappliedwhenthereisuncertainty inthesystemdynamicsandwhenthestatedelayisunknown;however,thecontroller isbasedontheassumptionthatthetime-varyinginputdelayisknown.Numericalsimulationscomparetheresulttoaprevioustime-varyinginput-delaycontroldesignand examinetherobustnessofthemethodtovariouscombinationsofsimultaneousinput andstatedelays.Thesimulationresultsillustraterobustnesstotheuncertaintyinthe magnitudeandfrequencyoftheinputdelaysandstatedelays.Theseresultspointtothe possibilitythatdifferentcontroloranalysismethodscouldbedevelopedtoeliminatethe assumptionthattheinputdelayisknown. 102

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CHAPTER7 SATURATEDCONTROLOFANUNCERTAINNONLINEARSYSTEMWITHINPUT DELAY LeveragingtheworkofChapters3and5,thischapterexaminessaturatedcontrolof ageneralclassofuncertainnonlinearsystemswithtime-delayedactuationandadditive boundeddisturbances.Theboundonthecontrolisknownaprioriandcanbeadjusted bychangingthefeedbackgains.ALyapunov-basedstabilityanalysisutilizingLK functionalsisprovidedtoproveUUBtrackingdespiteuncertaintiesinthedynamics.The resultisextendedtogeneralnonlinearsystemswhichcanbedescribedbyELdynamics andisillustratedwithsimulationresultstodemonstratethecontrolperformance. 7.1DynamicModel Consideraclassofnonlinearsystemsdescribedby x = f x; x;t + u t )]TJ/F25 11.9552 Tf 11.955 0 Td [( + d t where x t ; x t 2 R n arethegeneralizedsystemstates, u t )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 R n represents thegeneralizeddelayedcontrolinputvector,where 2 R + isaconstanttimedelay, f x; x;t : R 2 n [0 ; 1 R n isanunknownnonlinear C 2 function,and d t :[0 ; 1 R n denotesasufcientlysmoothexogenousdisturbancee.g.,unmodeledeffects. Thesubsequentdevelopmentisbasedontheassumptionthat x t and x t are measurableoutputs,thetimedelayconstant, ,isknown,andthecontrolinputvector u t anditspastvaluesi.e., u t )]TJ/F25 11.9552 Tf 11.955 0 Td [( 8 2 [0 ] aremeasurable.Throughoutthe chapter,atimedependentdelayedfunctionisdenotedas t )]TJ/F25 11.9552 Tf 11.955 0 Td [( or .Additionally, Assumption5.3andthefollowingassumptionsareused. Assumption7.1. Thedisturbancetermanditsrsttimederivativeareboundedby knownconstants,i.e., k d t k c 1 ; d t c 2 where c 1 ;c 2 2 R + 103

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Remark 7.1 Thenonlinearfunction f x; x;t is C 2 andtheMVTcanbeusedtoprove thatitsatisesthefollowinginequality k f x; x;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(f x d ; x d ;t k k k k k where x; x;x d ; x d 2 R 2 n isdenedas =[ x )]TJ/F25 11.9552 Tf 11.955 0 Td [(x d ; x )]TJ/F15 11.9552 Tf 13.98 0 Td [(_ x d ] T .Dening f x; x;t inthis wayislessrestrictivethanclaimingthefunction f x; x;t satisestheglobalLipschitz conditionwhichwouldyieldalinearboundinthestates,i.e., k k = AsinChapter3,toaidthesubsequentcontroldesignandanalysis,thevector Tanh 2 R n andthematrix Cosh 2 R n n aredenedasfollows Tanh [ tanh 1 ;:::;tanh n ] T ; Cosh diag f cosh 1 ;:::;cosh n g where =[ 1 ;:::; n ] T 2 R n and diag fg representsadiagonalmatrix.Basedonthe denitionsin7and7,thefollowinginequalitieshold 8 2 R n [183]: k k 2 n X i =1 ln cosh i 1 2 tanh 2 k k ; k k > k Tanh k ; k Tanh k 2 tanh 2 k k ; T Tanh Tanh T Tanh ; k k tanh k k k k +1 : 7.2ControlDevelopment Thecontrolobjectiveistodesignanamplitude-limited,continuouscontrollerthat willensurethegeneralizedstate x t oftheinput-delayedsystemin7tracks x d t despiteuncertaintiesandadditiveboundeddisturbancesinthedynamicmodel.To quantifythecontrolobjective,atrackingerror,denotedby e x;t 2 R n ,isdenedas e x d )]TJ/F25 11.9552 Tf 11.956 0 Td [(x: Embeddingthecontrolinaboundedtrigonometricterme.g., tanh isanobvious waytolimitthecontrolauthoritybelowanapriorilimit;however,difcultyarisesinthe 104

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closed-loopstabilityanalysiswithrespecttothedelaypresentinthecontrol.Motivated bythesestabilityanalysiscomplexitiesandthroughaniterativeanalysisprocedure, ameasurablelteredtrackingerrorisdesignedwhichincludesadditionalsmooth saturationtermsandaniteintegralofpastcontrolvalues.Specically,theltered trackingerror r e; e;e f ;e z ;t 2 R n isdenedas r e x;t + Tanh e + Tanh e f )]TJ/F25 11.9552 Tf 11.955 0 Td [(e z t where 2 R + isaknownadjustablegainconstant, e f e;r;t 2 R n isthesolutionofthe auxiliaryerrorlterdynamicsgivenby e f Cosh 2 e f )]TJ/F25 11.9552 Tf 9.298 0 Td [(kr + Tanh e )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh e f where e f =0 and k; 2 R + areconstantcontrolgains,and e z t 2 R n denotesthe niteintegralofpastcontrolvalues,denedas e z t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( u d : Fromthedenitionin7,theniteintegralcanbeupperboundedas k e z k z ,where z 2 R + isaknownboundingconstantprovidedthecontrolisbounded. Theopen-looperrorsystemcanbeobtainedbytakingthetimederivativeof7 andutilizingtheexpressionsin7and7toyield r = x d t )]TJ/F25 11.9552 Tf 11.955 0 Td [(f x; x;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(u t )]TJ/F25 11.9552 Tf 11.955 0 Td [(d t + Cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(2 e e x;t + Cosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 e f e f e;e f ;r;t : From7andthesubsequentstabilityanalysis,thecontrolinput, u e;e f ;t ,isdesignedas u )]TJ/F25 11.9552 Tf 9.298 0 Td [(kTanh e f +2 Tanh e 105

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where k wasintroducedin7. 1 Animportantfeatureofthecontrollergivenby5isitsapplicabilitytothecase whereconstraintsexistontheavailableactuatorcommands.Notethatthecontrollawis boundedbytheadjustablecontrolgain k since k u k k +2 p n Inreviewof7-7,thestrategyemployedtodevelopthecontrollerin7 entailsseveralcomponents.Onecomponentisthedevelopmentofthelterederror systemin7and7,whichiscomposedofsaturatedhyperbolictangentfunctions designedfromtheLyapunovanalysistocancelcrossterms.Thelterederrorsystem alsoincludesapredictorterm7,whichutilizespastvaluesofthecontrol.The motivationforthedesignof7stemsfromtheneedtoinjecta )]TJ/F25 11.9552 Tf 9.298 0 Td [(kr signalintothe closed-looperrorsystem,sincesuchtermscannotbedirectlyinjectedthroughthe saturatedcontroller,andtocancelcrosstermsintheanalysis.Thesaturatedcontrol structuremotivatestheneedforhyperbolictangentfunctionsintheLyapunovanalysis toyield )-167(k Tanh e f k 2 terms.Thetimederivativeofthehyperbolictangentfunction willyielda Cosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 e f term.Thedesignof7ismotivatedbythedesiretocancel the Cosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 e f term,enablingtheremainingtermstoprovidethedesiredfeedbackand cancelnonconstructivetermsasdictatedbythesubsequentstabilityanalysis. 1 Toimplementthecontrollerin7,thetrackingerror e andintegralofpastcontrolvalues e z shouldbeevaluatedrst.Thesignal e z isconsideredtobe0until t = .Thelteredtrackingerror r canbeevaluatedusingeithertheinitialcondition for e f e f =0 asstatedafter7orthecomputedvalueaftertherstiteration. Theauxiliarysignal e f canbesolvedonlinebyevaluating e f ateachtimestepusingthecomputedvaluesfor e and r andthepreviousvaluefor e f .Sinceeach ofthetermsontherighthandsideof7aremeasurable,thesolution e f t canbe foundusinganyofthenumerousnumericalintegrationtechniquesavailableinliterature. Onceeachoftheauxiliaryerrorsignalshavebeencomputed,7canbeimplemented. 106

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Theclosed-looperrorsystemisobtainedbyutilizing7,7,and7to yield r = S x d ; x d ; x d ;t + e; e;e f ;t + kTanh e f )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh e )]TJ/F25 11.9552 Tf 11.955 0 Td [(kr e; e;e f ;e z ;t wheretheauxiliaryterms S x d ; x d ; x d ;t 2 R n and e; e;e f ;t 2 R n aredenedas S x d t )]TJ/F25 11.9552 Tf 11.955 0 Td [(f x d ; x d ;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(d t ; )]TJ/F25 11.9552 Tf 9.299 0 Td [(f x; x;t + f x d ; x d ;t + Cosh )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 e e x;t )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh e f : Thestructureof7ismotivatedbythedesiretosegregatetermsthatcanbeupper boundedbystate-dependenttermsandtermsthatcanbeupperboundedbyconstants. UsingAssumptions5.3and6.1,thefollowinginequalitycanbedevelopedbasedonthe expressionin7 k S k s where s 2 R + isaknownconstant.UsingtheMVT,7and7,theexpressionin 7canbeupperboundedseeAppendixCfordetailsas k k k z k k z k wheretheboundingfunction : R 4 n +1 R isapositive,globallyinvertible,nondecreasingfunction,and z e;e f ;r;e z ;P 2 R 4 n +1 isdenedas z e T Tanh T e f r T e T z p P T : In7, P t 2 R + denotesanLKfunctionaldenedas P t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( t s k u k 2 d d s where 2 R + isaknownconstant. 107

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7.3StabilityAnalysis Theorem7.1. Giventhedynamicsin7,thecontrollerin7ensuresuniformly ultimatelyboundedtrackingprovidedtheadjustablecontrolgains ;;k areselected accordingtothefollowingsufcientconditions > 2 4 +2 k +1 ;>k! k +2 ;! 2 > 2 ; 4 k 2 2 cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 e 2 2 +1 2 where 2 R + isanknown,adjustable,positiveconstant, 2 R isdenedas max d; k z k ,and d 2 R isasubsequentlydened,positiveconstantthatdenesthe radiusofaballcontainingthepositiontrackingerrors. Proof. Let V L z;t : D [0 ; 1 R beacontinuouslydifferentiable,positive-denite functionalonadomain D R 4 n +1 ,denedas V L 1 2 r T r + n X i =1 ln cosh e i + 1 2 Tanh T e f Tanh e f + P; whichcanbeboundedusing7as 1 k z k V L 2 k z k : wherethestrictlyincreasingnon-negativefunctions 1 ; 2 : R 4 n +1 R aredened as 1 k z k 1 2 ln cosh k z k ; 2 k z k k z k 2 : Afterutilizing7,7,7andbycancelingsimilarterms,thetimederivative of7canbeexpressedas V L = r T + r T S )]TJ/F25 11.9552 Tf 11.955 0 Td [(kr T r )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh T e Tanh e )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh T e f Tanh e f + Tanh T e e z + k u k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(! t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( k u k 2 d 108

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wheretheLeibnizRulewasappliedtodeterminethetimederivativeof7.Using 7,7,7,and7,7canbeupperboundedby V L )]TJ/F25 11.9552 Tf 21.918 0 Td [(k k r k 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [( k Tanh e k 2 )]TJ/F25 11.9552 Tf 11.956 0 Td [( k Tanh e f k 2 + k r k k z k k z k + k r k s + k Tanh e kk e z k + k 2 k Tanh e f k 2 +4 k Tanh e k 2 +4 k! k Tanh e f kk Tanh e k)]TJ/F25 11.9552 Tf 20.589 0 Td [(! t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( k u k 2 d : Young'sInequalitycanbeusedtoupperboundselecttermsin7as k Tanh e kk e z k 2 4 k Tanh e k 2 + 1 2 k e z k 2 ; k Tanh e f kk Tanh e k 1 2 k Tanh e f k 2 + 1 2 k Tanh e k 2 where isaknownconstant.UtilizingtheCauchy-SchwarzInequality,thelastintegral in7canbeupperboundedas )]TJ/F25 11.9552 Tf 11.955 0 Td [(! t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( k u k 2 d )]TJ/F25 11.9552 Tf 25.39 8.088 Td [(! 2 k e z k 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(! 2 t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( k u k 2 d: Using7and7,7canbeupperboundedas V L )]TJ/F25 11.9552 Tf 21.917 0 Td [(k 1 k r k 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 k 2 +1 k Tanh e k 2 )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k! )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 2 k Tanh e f k 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( 2 )]TJ/F15 11.9552 Tf 16.61 8.088 Td [(1 2 k e z k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 2 k r k 2 + k z k k z kk r k)]TJ/F25 11.9552 Tf 20.589 0 Td [(k 3 k r k 2 + s k r k)]TJ/F25 11.9552 Tf 21.784 8.087 Td [(! 2 t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( k u k 2 d where k ,introducedin7and7,issplitintoadjustableconstants k 1 ;k 2 ;k 3 2 R + as k k 1 + k 2 + k 3 : Aftercompletingthesquares,theexpressionin7canbeupper boundedas V L )]TJ/F25 11.9552 Tf 21.918 0 Td [(k 1 k r k 2 )]TJ/F30 11.9552 Tf 9.963 16.857 Td [( )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 k 2 +1 k Tanh e k 2 )]TJ/F30 11.9552 Tf 9.963 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k! )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 2 k Tanh e f k 2 )]TJ/F30 11.9552 Tf 11.956 16.857 Td [( 2 )]TJ/F15 11.9552 Tf 16.609 8.088 Td [(1 2 k e z k 2 + 2 k z k 4 k 2 k z k 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(! 2 t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( k u k 2 d + s 2 4 k 3 : 109

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Theinequality t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( t s k u k 2 d d s sup s 2 [ t;t )]TJ/F26 7.9701 Tf 6.586 0 Td [( ] t s k u k 2 d = t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( k u k 2 d canbeusedtoupperbound7as V L )]TJ/F25 11.9552 Tf 21.918 0 Td [(k 1 k r k 2 )]TJ/F30 11.9552 Tf 9.963 16.857 Td [( )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 k 2 +1 k Tanh e k 2 )]TJ/F30 11.9552 Tf 9.963 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k! )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 2 k Tanh e f k 2 )]TJ/F30 11.9552 Tf 11.955 16.857 Td [( 2 )]TJ/F15 11.9552 Tf 16.61 8.088 Td [(1 2 k e z k 2 + 2 k z k 4 k 2 k z k 2 )]TJ/F25 11.9552 Tf 15.428 8.088 Td [(! 2 t t )]TJ/F26 7.9701 Tf 6.586 0 Td [( t s k u k 2 d d s + s 2 4 k 3 : Let y e;e f ;e z ;r;P 2 R 4 n +1 bedenedas y h Tanh T e Tanh T e f e T z r T p P i T : Byusing7and7,7canbeupperboundedas V L )]TJ/F25 11.9552 Tf 21.918 0 Td [( k y k 2 + 2 k z k 4 k 2 k z k 2 + s 2 4 k 3 wheretheauxiliaryconstant 2 R + isdenedas min k 1 ; )]TJ/F25 11.9552 Tf 13.151 8.088 Td [( 2 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 k 2 +1 ; )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k! )]TJ/F25 11.9552 Tf 11.955 0 Td [(k 2 !; 2 )]TJ/F15 11.9552 Tf 16.61 8.088 Td [(1 2 ; 1 2 : Ifthesufcientconditionsin7aresatised,then > 0 .Providedthefollowing inequalityissatised 2 k z k 4 k 2 k z k 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( k y k 2 0 ; 7canbeexpressedas V L )]TJ/F25 11.9552 Tf 21.918 0 Td [( 2 k y k 2 + s 2 4 k 3 where 2 2 R + issomeconstant.Fromthedenitionsin7and7andutilizing thefactthat k y k 2 tanh 2 k z k from7,theexpressionin7issatisedif k z k tanh k z k 2 4 k 2 2 k z k : 110

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Usingthepropertiesin7,asufcientconditionfor7is k z k +1 2 4 k 2 2 k z k : Therstlowerboundon V L z;t from7canbeusedtostatethat k z k cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 exp V L ; hence,asufcientconditionfor7canbeobtainedas 2 )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [(cosh )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 exp V L +1 2 4 k 2 : Ifthecondition7issatised,thenfrom7,theexpressionin7canbe rewrittenas V L )]TJ/F25 11.9552 Tf 21.918 0 Td [( 3 k z k + s 2 4 k 3 wherethestrictlyincreasingnon-negativefunction 3 : R 4 n +1 R isdenedas 3 k z k 2 tanh 2 k z k .Given7,and7, z aswellas e and r viathe denitionin7andstandardlinearanalysisisUUB[184]inthesensethat k e t kk z t k < d; 8 t T )]TJ/F15 11.9552 Tf 7.546 -6.529 Td [( d; k z k providedthesufcientconditionsin7andtheinequalityin7aresatised. In7, d 2 R isapositiveconstantthatdenestheradiusofaballcontainingthe positiontrackingerrors,selectedaccordingto[184] d> )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 1 2 )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 3 s 2 4 k 3 ; and T )]TJ/F15 11.9552 Tf 7.545 -6.529 Td [( d; k z k 2 R isapositiveconstantthatdenotestheultimatetimetoreachthe ball[184] T 8 > > < > > : 0 k z k )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [( )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 1 )]TJ/F15 11.9552 Tf 13.025 -6.529 Td [( d 2 k z k )]TJ/F26 7.9701 Tf 6.586 0 Td [( 1 )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 2 1 d 3 )]TJ/F18 5.9776 Tf 5.757 0 Td [(1 2 1 d )]TJ/F18 5.9776 Tf 10.145 3.259 Td [( s 2 4 k 3 k z k > )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 2 1 )]TJ/F15 11.9552 Tf 13.025 -6.529 Td [( d : 111

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From7and7,analsufcientconditionfor7,givenin719,can beexpressedintermsofeithertheinitialconditionsofthesystemortheultimate bound. Remark 7.2 Basedon7,thesizeoftheultimateboundin7canbemade smallerbyselecting k 3 larger.Forarbitrarilylargedelaysorarbitrarilylargeinitial conditions,thecontrolgainsrequiredtosatisfythesufcientgainconditionsin7 maydemandtorquethatisnotphysicallydeliverablebythesystemi.e.,thegain k may berequiredtobelargerthanthesaturationlimitoftheactuator.Thegainconditionin 7isdirectlyinuencedbytheboundgivenin7,whichresultsfromthebounds derivedinRemark7.1.Forexample,if f isgloballyLipschitz,thentheupperbound on reducestoaconstanttimesthestateandalocalconditiononthestate z canbe determinedas k z k p 4 k 2 = )]TJ/F15 11.9552 Tf 12.422 0 Td [(1 ,whichcanbeenlargedbyincreasing k 2 upto apointbasedontheactuatorconstraints.Giventhecurrent,moregeneralboundfor inRemark7.1,asimpliedclosed-forminitialconditionboundcannotbederived. However,givenanupperboundonthedisturbance,anupperboundonthetimedelay, andtheinitialconditions,7and7canbeusedtodeterminethesufcientgain k 2 ,ifpossible,basedontheactuatorlimit.Thisresultdoesnotsatisfythestandard semi-globalresultbecauseundertheconsiderationofinputconstraints, k cannotbe arbitrarilyincreasedandconsequentlycannotsatisfyallinitialconditions.Thisoutcome isnotsurprisingfromaphysicalperspectiveinthesensethatsuchdemandsmayyield caseswheretheactuationisinsufcienttostabilizethesystem. 7.4Euler-LagrangeExtension SimilartothedevelopmentinSection5.4ofChapter5,thecontrollerpresentedin 7canbeextendedtononlinearELsystemsoftheform M q q + V m q; q q + G q + F q + d t = u t )]TJ/F25 11.9552 Tf 11.955 0 Td [( 112

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where M q 2 R n n denotesthegeneralizedinertia, V m q; q 2 R n n denotesthe generalizedcentrifugalandCoriolisforces, G q 2 R n denotesthegeneralizedgravity, F q 2 R n denotesthegeneralizedfrictionand q t ; q t ; q t 2 R n denotethe generalizedstates.Utilizingstandardpropertiesoftheinertiaandcentrifugal/Coriolis matrices,thecontroldevelopmentcanbeextendedtoachievethesameresultasin Section7.3. Thedesignoftheerrorsystemsandcontrollerfollowsimilarlytothemethod presentedpreviously.Specically,7canbemodiedslightlytoaccommodatethe uncertaininertiaeffectsinthedynamicsand7,7and7canbedesigned asinSection7.2. Asinthedevelopmentof7,thelteredtrackingerror, r e;e f ;e z ;t isredened as r e + Tanh e + Tanh e f )]TJ/F25 11.9552 Tf 11.955 0 Td [(Be z where e f isdenedasin7and B 2 R n n isasymmetric,positive-deniteconstant gainmatrixthatsatisesthefollowinginequality k B k 1 b where b 2 R + isaknown constant.Theerrorbetween B and M )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q isdenotedby q 2 R n n andisdenedas B )]TJ/F25 11.9552 Tf 11.955 0 Td [(M )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 andsatisesthefollowinginequality k k 1 where 2 R + isaknownconstant. Duetotheinclusionof B ,theopen-looperrorsystemwillcontainanadditive time-delayedterm: M u )]TJ/F25 11.9552 Tf 11.955 0 Td [(u .Motivatedbytheneedtocancelthisterminthe stabilityanalysis,basedonthestructureof7andinspiredbytheworkin[180],two additionalLKfunctionalsareaddedtotheLyapunovfunctionalcandidate, V L z;t 2 R denedas V L 1 2 r T Mr + n X i =1 ln cosh e i + 1 2 Tanh T e f Tanh e f + P + Q + R 113

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where P t isdenedasin7and Q e f ;R e 2 R denoteLKfunctionalsdened as Q k m 2 t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( k Tanh e f k 2 d ;R m 2 t t )]TJ/F26 7.9701 Tf 6.587 0 Td [( k Tanh e k 2 d : ALyapunov-basedstabilityanalysissimilartotheonepresentedforthegeneral second-ordernonlinearsysteminSection7.3isusedtoderivesufcientconditionsfor UUBtracking.TheELsystemin7requiresasufcientconditionon B inaddition togainconditionssimilartothosegivenin7and7,givenby k 1 > k 2 + k 3 2 m 1 )]TJ/F24 7.9701 Tf 13.151 5.256 Td [(2 m m where m;m 2 R aretheknownconstantupperandlowerboundontheinertiamatrix. Thesufcientgainconditionsindicatethat k 1 canbeselectedsufcientlylargeprovided 1 )]TJ/F24 7.9701 Tf 13.399 5.256 Td [(2 m m > 0 .Thisconditionindicatesthattheconstantapproximationmatrix B mustbe chosensufcientlycloseto M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q sothat k B )]TJ/F25 11.9552 Tf 11.956 0 Td [(M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q k 1 < m 2 m .Additionaldetails regardingtheELextensionofthischaptercanbefoundin[185]. 7.5SimulationResults UtilizingtheextensionfromSection7.4,thecontrollerwassimulatedforatwo-link planarmanipulator.TheELdynamicsofthemanipulatoraregivenas 2 6 4 1 2 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 q 1 q 2 3 7 5 + 2 6 4 )]TJ/F25 11.9552 Tf 9.298 0 Td [(p 3 s 2 q 2 )]TJ/F25 11.9552 Tf 9.298 0 Td [(p 3 s 2 q 1 +_ q 2 p 3 s 2 q 1 0 3 7 5 2 6 4 q 1 q 2 3 7 5 + 2 6 4 f d 1 0 0 f d 2 3 7 5 2 6 4 q 1 q 2 3 7 5 + 2 6 4 d 1 d 2 3 7 5 where p 1 =3 : 473 kg m 2 p 2 =0 : 196 kg m 2 p 3 =0 : 242 kg m 2 f d 1 =5 : 3 Nmsec f d 2 =1 ; 1 Nmsec c 2 denotes cos q 2 ,and s 2 denotes sin q 2 .Thedisturbanceterms wereselectedas d 1 =0 : 5 sin )]TJ/F26 7.9701 Tf 7.263 -4.976 Td [(t 5 ,and d 2 =0 : 1 sin )]TJ/F26 7.9701 Tf 7.263 -4.977 Td [(t 5 .Thedesiredtrajectoriesforlinks 114

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1and2forallsimulationswereselectedas q d 1 t =1 : 5 sin t= 2 rad;q d 2 t =0 : 5 sin t= 4 rad: Theinitialconditionsforthemanipulatorwereselectedasstationarywithasignicant offsetfromtheinitialconditionsofthedesiredtrajectoryas q 1 q 2 T = 12 T rad: Forcomparison,thesimulationwascompletedusingvariousvaluesofinputdelay, rangingfrom100msto1s.Foreachcaseitisdesiredfortheactuationtorquetobe limitedto 1 20 N; 2 10 N: Becausethecontrollerassumesthattheinertiamatrixis unknown,abestguessestimateoftheconstantmatrix B isselectedas B = 2 6 4 4 : 00 : 4 0 : 40 : 2 3 7 5 : Additionalresultsshowthattheperformance/robustnessofthedevelopedsaturated controllerwithrespecttothemismatchbetween B and M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q indicatinganinsignificantamountofvariationintheperformanceevenwheneachelementof M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q is overestimatedbyasmuchas300%,animprovementovertheresultshownin[31]. Figure7-1illustratesthetrackingerrorsassociatedwitheachoftheinputdelay cases.Asthedelaymagnitudeisincreased,theperformancedegradesandthetracking errorboundincreases.Figure7-2showsthatevenwithalargeinputdelayinthe system,theproposedcontrollerisabletoensurethatthecontroltorquedoesnot exceedtheactuatorlimitsasspeciedbythecontrollergainswhileensuringthe boundednessofthetrackingerror. 7.6Summary Acontinuoussaturatedcontrollerisdevelopedforuncertainnonlinearsystems whichincludeinputdelaysandsufcientlysmoothadditiveboundeddisturbances.The boundonthecontrolisknownaprioriandcanbeadjustedbychangingthefeedback gains.ThesaturatedcontrollerisshowntoguaranteeUUBtrackingprovidedthedelay 115

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Figure7-1.Trackingerrorvs.timeforproposedcontrollerin7.A100msinput delay,B500msinputdelay,C1sinputdelay. Figure7-2.Controltorquevs.timeforproposedcontrollerin7.A100msinput delay,B500msinputdelay,C1sinputdelay. 116

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issufcientlysmall.TheresultisextendedtogeneralELsystemsandsimulationresults areperformedonatwo-linkroboticmanipulatortodemonstratetheeffectivenessofthe controldesign. 117

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CHAPTER8 CONCLUSIONANDFUTUREWORK 8.1DissertationSummary Thefocusofthisworkistodevelopcontrolmethodsforuncertainnonlinearsystems withrealworldconsiderationsincludingtime-delaysandactuatorsaturation.The workcoversawidevarietyofsystemswithpracticalconsiderationsandhasfacilitated theintroductionofanimportantLyapunov-basedstabilityCorollary,suitableforuse innumerousapplicationsaswellasmanytheoreticalstudiesinnonlinearcontrol designandanalysis.Becauserealworldsystemsareaffectedbynonlinearbehaviors thatareoftennotconsidered,theworkinthisdissertationaimstocompensatefor thesephenomenawithpracticalcontroldesignsthatcanbeimplementedinrelevant engineeredsystems. Chapter2focusesonintroducingthemechanicsrequiredtoutilizenonsmooth analysisinLyapunov-basedcontroldesignandextendingtheLYTtodifferentialsystems withadiscontinuousRHSusinggeneralizedsolutionsinthesenseofFilippov.The resultpresentstheoreticaltoolsapplicabletononlinearsystemswithdiscontinuitiesin theplantdynamicsorinthecontrolstructure.GeneralizedLyapunov-basedanalysis methodsaredevelopedusingdifferentialinclusionstoachieveasymptoticconvergence ofthestatewhentheLyapunovderivativeisupperboundedbyanegativesemi-denite function.Semi-globalslidingmodecontrolandRISEcontrolexamplesillustratetheuse oftheCorollariesincontroldesignandanalysis. RISE-basedcontroltechniqueshavebeenshowntoeffectivelysuppressadditive boundeddisturbancesandparametricuncertaintiesinnonlinearsystems;however, thetechnique'shigh-gainnatureoftenlimitsitsapplicabilityinsystemswhereactuatorlimitationsexistsincestandardRISEtechniquescanpotentiallydemandlarge actuationeffortswhenlargeinitialoffsetsordisturbancesarepresent.InChapter3,a continuoussaturatedcontrollerisdevelopedforaclassofuncertainnonlinearsystems 118

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whichincludestime-varyingandnon-LPfunctionsandadditiveboundeddisturbances, achievingsemi-globalasymptotictracking.Theboundonthecontrolisknownapriori andcanbeadjustedbychangingthefeedbackgains.Embeddingthe sgn ofanerror signalinsideandintegraltermallowsforacontinuouscontroldesignwithouttherisk ofinniteactuationdemandorchatter,ascommonlyfoundinstandardslidingmode controldesigns.However,withtheinclusionofan sgn functionandbecausemost Lyapunov-basedstabilityanalysesarebasedonexistenceofsolutionswhencontinuous differentialequationsareutilized,theworkinChapter2ismotivated. Thenextfourchaptersfocusonnonlinearsystemswithtimedelays.Chapter4 extendsourRISE-basedcontroltechniquestosystemswithtime-varyingstatedelays. Whilethegeneralizedstatedelayproblemfornonlinearsystemshasbeenstudied rigorously,ourcontinuouscontroller,whichachievesasymptotictrackinginthepresence ofparametricuncertaintiesandboundeddisturbances,isoneoftherstofitskind.The developmentinvolvesaDCAL-basedapproachtoseparatedelayedandnon-delayed termsandaneuralnetworktocompensatefornon-LPuncertainties. Themorechallengingopentimedelayproblemsinliteraturefocusontheinput delayproblemfornonlinearsystems:thefocusofChapters5-7.Chapter5beginsthis workbydevelopingacontinuouspredictor-basedcontrollerforuncertainnonlinear systemswhichincludetime-varyinginputdelaysandsufcientlysmoothadditive boundeddisturbances,guaranteeingUUBtracking,providedthedelayissufciently smallandslowlyvarying.Theerrorsystemdevelopmentincludesanovelpredictorbasedintegralofpastcontrolvalues,facilitatingtheuseofadelay-freecontrolsignal whichcanbedesigned.Chapter6goesastepfurtherbycombiningtheinputdelay problemfromChapter5andthestatedelayproblemfromChapter6todevelopaRISEbasedcontrolmethodcapableofhandlingbothtime-varyingstateandinputdelays andtheinclusionofplantuncertaintiesandboundeddisturbances.Anewerrorsystem developmentwhichintroducesauxiliarylteredsignalsallowsadditionalexibilityin 119

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controldesignoverthepreviousUUBapproach.Now,thepredictor-basedcontrollerand LKfunctionaldesignguaranteesemi-globalasymptotictrackingprovidedtherateofthe delaysissufcientlyslow,butdoesnotrestricttheboundonthedelaystobesufciently small. Thenalchaptercirclesbacktoanalyzetheeffectofactuatorsaturationforinputdelayedsystems.Sinceerrorscanbuildovertheactuatordead-time,thisworkishighly motivated.Specically,acontinuoussaturatedcontrollerisdevelopedforuncertain nonlinearsystemswhichincludeconstant,knowninputdelaysandsufcientlysmooth additiveboundeddisturbances.Theboundonthecontrolisknownaprioriandcanbe adjustedbychangingthefeedbackgainswhilethecontrollerisshowntoguarantee UUBtrackingprovidedthedelayissufcientlysmall. 8.2LimitationsandFutureWork TheworkinthisdissertationopensnewdoorsforbothRISE-basedcontrolmethods andcontroldesignsfortime-delayednonlinearsystems.Inthissection,openproblems relatedtotheworkinthisdissertationarediscussed. FromChapter2: 1.TheLYCutilizesexistenceofsolutionsinthesenseofFilippov.However,similar resultsinliteraturethatdevelopstabilitytechniquesutilizingdifferentialinclusions havealsobeenshownforothertypesofsolutiondenitions.Onesuchexample areKrasovskiisolutions,asutilizedinresultssuchas[153].Extendingthework ofChapter2togeneralsolutiondenitionsnotrestrictedtoFilippovisalikely achievablegoal.Ingeneral,utilizingKrasovskiisolutionsinplaceofFilippov solutionsfacilitatesstrongeroverallstabilityresultsatthecostofrequiringmore restrictivesystemassumptions. FromChapter3: 1.Thestabilityresultachievedinthischapterissemi-global,meaningthatthecontrol gainsmustbeselectedaccordingtotheinitialconditionsofthesystem.However, 120

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inpracticebothsimulationandexperiments,thecontrollerexhibitsglobal-like performance.Utilizingtechniquessimilarto[174],itmaybepossibletoexploitthe Tanh propertiesinthecontroldevelopmenttomanipulatetheboundon to achieveaglobalstabilityresult. 2.IntheELextension,theinertiamatrix M q isrequiredtobeknown.Thisisdue tothefactthat M )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 q pre-multipliesthecontrolinputthroughouttheopenand closedloopanalysis.Itmaybepossibletomodifyingthecontroldesignortheerror systemdevelopmenttoeliminatethisassumption. FromChapter4: 1.AllLKfunctional-basedanalysistechniquesfortime-varying-delayedsystems introducearestrictiononthederivativeofthedelay.Thisisintuitivelyduetothe factthatthederivativeoftheLKfunctionalresultsina )]TJ/F15 11.9552 Tf 13.864 0 Td [(_ .Alternativestability techniquessuchasRazumikhinmethodsmaybecapableofeliminatingthe assumptionontherateofthedelay,butfutureeffortshaveyettoconrmthis. FromChapter5: 1.AsinChapter4,theuseofLKfunctionalsintroducesrestrictivesufcientconditionsontherateofthedelay. 2.Duetothefactthatthestabilityresultdependsonasufcientconditionrelatedto theinitialconditionsofthesystem,itisdifculttopredicttheadmissiblevaluesof delaythatthesystemcantolerate. 3.Thedelayisassumedtobesufcientlysmallandslowlyvarying.Thislimitsthe numberofapplicationsforwhichthecontrollercanbeapplied.Relaxingthese assumptionstoauncertaindelays,barbitrarilylargedelays,andcarbitrarily fastdelaysarefuturegoalsofthiswork,inadditiontoreducingthesteadystate ultimatelyboundedtracking. 4.Theconditionon B intheELextensionthatrequires B tobeselectedsufciently closeto M )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 q isratherrestrictiveiftheinertiamatrixisentirelyunknown.While 121

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theseparametersareoftenmeasurableofine,incorporatingamorerobust methodofhandlingtheinertiauncertaintiesismotivated. FromChapter6: 1.ThemodiederrorsystemdevelopmentallowsustointegrateaRISE-based controllawintheclosed-loopsystem,allowingustoachieveasymptotictracking. However,thecontrollerisstillrestrictedbyknowledgeoftheinputdelaysinceit isusedintheerrorsystemandtheboundonthedelayrates.Itmaybepossible toextendtheresulttoincludeuncertaininputdelaysthestatedelaysarealready assumeduncertainandtorelaxtheassumptiononthedelayrates. 2.Asin3,thetheEuler-Lagrangeextensionofthisresultrequiresknowledgeofthe inertiamatrix. 3.Thesecondorderdynamicspresentedin6donotfacilitateatrivialextension toEuler-Lagrangedynamics.Thisextensionwouldintroducesignalsthathave bothstateandinputdelayspresent.Considerationsforthesecompositedelay termsrequireadditionalfocus. FromChapter7: 1.Thestabilityanalysisinthisworkisalsodependentonasufcientcondition relatedtotheinitialconditionsofthesystem.Accuratepredictionoftheadmissible valuesofthedelayforagivensystemareobtained,however;theyarelimitedby theassumptionthatthedelayisconstantandknown. 2.UtilizingtheworkinChapter3and6,designofasaturatedcontrollerforuncertain nonlinearsystemswithasymptotictrackingmaynowbepossibleusingRISE techniques. 122

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APPENDIXA PROOFOF P CH3 LemmaA.1. Giventhedifferentialequationin3, P e 2 ;t 0 if satises 1 > N d 1 + N d 2 3 : A Proof. Forsakeofnotation,deneanauxiliarysignal e 2 ;r;t 2 R n astheintegralof termsfoundin P e 2 ;t in3 e 2 ;r;t = t 0 r T N d )]TJ/F25 11.9552 Tf 9.962 0 Td [( 1 sgn e 2 d : Byusing3,integratingbyparts,andregroupingyields e 2 ;r;t = t t 0 2 Tanh T e 2 [ N d )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 sgn e 2 ] d + t t 0 3 e T 2 [ N d )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 sgn e 2 ] d )]TJ/F42 11.9552 Tf 11.291 16.273 Td [( t t 0 3 e T 2 1 3 @N d @ d + e T 2 N d t A )]TJ/F25 11.9552 Tf 9.298 0 Td [(e T 2 t 0 N d t 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 n X i =1 j e 2 i t j + 1 n X i =1 j e 2 i t 0 j : From3and3,theexpressioninAcanbeupperboundedby e 2 ;r;t t t 0 2 k Tanh e 2 k [ N d 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 ] d + t t 0 3 k e 2 k N d 1 + N d 2 3 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 d + k e 2 t k [ N d 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 ]+ 1 n X i =1 j e 2 i t 0 j )]TJ/F25 11.9552 Tf 9.298 0 Td [(e T 2 t 0 N d t 0 : A Thus,fromA,if satisesA,then e 2 ;r;t 1 n X i =1 j e 2 i t 0 j)]TJ/F25 11.9552 Tf 17.932 0 Td [(e T 2 t 0 N d t 0 = P e 2 t 0 ;t 0 : A 123

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Integratingbothsidesof3yields P e 2 ;t = P e 2 t 0 ;t 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( e 2 ;r;t ; whichindicates P e 2 ;t 0 fromA. 124

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APPENDIXB PROOFOF P CH6 LemmaB.1. Giventhedifferentialequationin6, P ;t 0 if satises > N d 1 + N d 2 2 : B Proof. Forsakeofnotationbrevity,deneanauxiliarysignal r;;t 2 R n asthe integralofthetermsfoundin P r;;t in6as r;;t = t t 0 r T N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(sgn d: B Utilizingtheexpressionfor6intermsof6,Bcanbeexpandedas r;;t = t t 0 2 T [ N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(sgn ] d + t t 0 @ T @ N d d )]TJ/F42 11.9552 Tf 11.955 16.273 Td [( t t 0 @ T @ sgn d: B IntegratingthelasttwointegralsinBbypartsyields r;;t = t t 0 2 T [ N d )]TJ/F25 11.9552 Tf 11.955 0 Td [(sgn ] d + T t N d t )]TJ/F25 11.9552 Tf 11.955 0 Td [( T t 0 N d t 0 )]TJ/F42 11.9552 Tf 11.955 16.272 Td [( t t 0 2 T 1 2 @N d @ d )]TJ/F25 11.9552 Tf 9.298 0 Td [( n X i =1 j i t j + n X i =1 j i t 0 j : B Basedon6,theexpressioninBcanbeupperboundedas r;;t t t 0 2 k k N d 1 + N d 1 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( d + T t [ N d 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( ]+ n X i =1 j i t 0 j)]TJ/F25 11.9552 Tf 17.933 0 Td [( T t 0 N d t 0 : B FromB,if satisesthesufcientconditioninB,then r;;t n X i =1 j i t 0 j)]TJ/F25 11.9552 Tf 17.933 0 Td [( T t 0 N d t 0 = P t 0 ;t 0 : B 125

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Integratingbothsidesof P t 0 ;t 0 in6yields P ;t = P t 0 ;t 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( r;;t whichindicatesthat P ;t 0 fromB. 126

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APPENDIXC PROOFOF BOUNDCH7 LemmaC.1. TheMVTcanbeusedtodeveloptheupperboundin7 k k k z k k z k wheretheboundingfunction : R 4 n +1 R isapositive,globallyinvertible,nondecreasingfunction,and z e;e f ;r;e z ;P 2 R 4 n +1 isdenedas z e T Tanh T e f r T e T z p P T : C Proof. TheproofofLemmaC.1followsfromthatof[164,AppA].Theauxiliaryerror in7canbewrittenasthesumoferrorspertainingtoeachofitsargumentsas follows: = x; x;e;r;Tanh e f ;e z )]TJ/F25 11.9552 Tf 11.955 0 Td [( x d ; x d ; 0 ; 0 ; 0 ; 0 = x; x d ; 0 ; 0 ; 0 ; 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( x d ; x d ; 0 ; 0 ; 0 ; 0 + x; x; 0 ; 0 ; 0 ; 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( x; x d ; 0 ; 0 ; 0 ; 0 + x; x;e; 0 ; 0 ; 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( x; x; 0 ; 0 ; 0 ; 0 + x; x;e;r; 0 ; 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( x; x;e; 0 ; 0 ; 0 + x; x;e;r;Tanh e f ; 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( x; x;e;r; 0 ; 0 + x; x;e;r;Tanh e f ;e z )]TJ/F25 11.9552 Tf 11.955 0 Td [( x; x;e;r;Tanh e f ; 0 : ApplyingtheMVTtofurtherdescribe = @ 1 ; x d ; 0 ; 0 ; 0 ; 0 @ 1 j 1 = v 1 x )]TJ/F25 11.9552 Tf 11.955 0 Td [(x d + @ x; 2 ; 0 ; 0 ; 0 ; 0 @ 2 j 2 = v 2 x )]TJ/F15 11.9552 Tf 13.981 0 Td [(_ x d + @ x; x; 3 ; 0 ; 0 ; 0 @ 3 j 3 = v 3 e )]TJ/F15 11.9552 Tf 11.955 0 Td [(0+ @ x; x;e; 4 ; 0 ; 0 @ 4 j 4 = v 4 r )]TJ/F15 11.9552 Tf 11.956 0 Td [(0 @ x; x;e;r; 4 ; 0 @ 5 j 5 = v 5 Tanh e f )]TJ/F15 11.9552 Tf 11.955 0 Td [(0 + @ x; x;e;r;Tanh e f ; 5 @ 6 j 6 = v 6 e z )]TJ/F15 11.9552 Tf 11.955 0 Td [(0 C 127

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where v 1 = x )]TJ/F25 11.9552 Tf 11.956 0 Td [(c 1 x )]TJ/F25 11.9552 Tf 11.956 0 Td [(x d ;v 2 =_ x )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 2 x )]TJ/F15 11.9552 Tf 13.98 0 Td [(_ x d ;v 3 = e )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 3 ; v 4 = r )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 4 ;v 5 = e f )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 5 ;v 6 = e z )]TJ/F25 11.9552 Tf 11.955 0 Td [(c 6 and c i 2 ; 1 2 R ;i =[1 ; 6] areunknownconstants.FromC, canbeupper boundedas k k = @ 1 ; x d ; 0 ; 0 ; 0 @ 1 j 1 = v 1 k e k + @ x; 2 ; 0 ; 0 ; 0 @ 2 j 2 = v 2 k r )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh e + Tanh e f )]TJ/F25 11.9552 Tf 11.955 0 Td [(e z k + @ x; x; 3 ; 0 ; 0 @ 3 j 3 = v 3 k e k + @ x; x;e; 4 ; 0 ; 0 @ 4 j 4 = v 4 k r k + @ x; x;e;r; 4 ; 0 @ 5 j 5 = v 5 k Tanh e f k + @ x; x;e;r;Tanh e f ; 5 @ 6 j 6 = v 6 k e z k : C Wecanupperboundthepartialderivativesas @ 1 ; x d ; 0 ; 0 ; 0 @ 1 j 1 = v 1 1 e @ x; 2 ; 0 ; 0 ; 0 @ 2 j 2 = v 2 2 r;e;Tanh e f ;e z @ x; x; 3 ; 0 ; 0 @ 3 j 3 = v 3 3 e @ x; x;e; 4 ; 0 ; 0 @ 4 j 4 = v 4 4 r @ x; x;e;r; 4 ; 0 @ 5 j 5 = v 5 5 Tanh e f @ x; x;e;r;e f ; 5 @ 6 j 6 = v 6 6 e z where i 2 R ;i =[1 ; 6] arepositive,nondecreasingfunctions.Theboundon can befurthersimplied k k 1 e k e k + 2 r;e;Tanh e f ;e z k r )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh e + Tanh e f )]TJ/F25 11.9552 Tf 11.956 0 Td [(e z k + 3 e k e k + 4 r k r k + 5 Tanh e f k Tanh e f k + 6 e z k e z k : 128

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Since k Tanh e kk e k ; theupperbound k r )]TJ/F25 11.9552 Tf 11.955 0 Td [(Tanh e + Tanh e f )]TJ/F25 11.9552 Tf 11.955 0 Td [(e z kk r k + k e k + k Tanh e f k + k e z k ; canbeusedtobound as k k 1 e + 3 e + 2 r;e;Tanh e f ;e z k e k + 2 r;e;Tanh e f ;e z + 4 r k r k + 2 r;e;Tanh e f ;e z + 5 Tanh e f k Tanh e f k + 2 r;e;Tanh e f ;e z + 6 e z k e z k : UsingthedenitionfromC, canbeexpressedintermsof z as k k 1 e + 3 e + 2 r;e;Tanh e f ;e z k z k + 2 r;e;Tanh e f ;e z + 4 r k z k + 2 r;e;Tanh e f ;e z + 5 Tanh e f k z k + 2 r;e;Tanh e f ;e z + 6 e z k z k ; k k 1 + +3 2 + 3 + 4 + 5 + 6 k z k : Therefore, k k ^ k z k k z k where ^ k z k issomepositive,nondecreasingfunction.Anypositivenondecreasing functioncanbeupperboundedbyapositivestrictlyincreasingfunction.Thus,the conditionsforglobalinvertibilityhold.Finally, k k ^ k z k k z k k z k k z k where ^ k z k isapositivegloballyinvertible,nondecreasingfunction.Notethatsince p P ispositivebydenition,itsconservativematterintheboundingof doesnotplaya factor. 129

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BIOGRAPHICALSKETCH NicFischerwasborninSt.Petersburg,Florida.HereceivedaBachelorofScience degreeinmechanicalengineeringfromtheUniversityofFloridain2008.Hejoined theNonlinearControlsandRoboticsNCRresearchgroupin2006withthehopesof continuinghisresearchintograduateschool.Aftercompetinghisbachelor'sdegree, NicdecidedtopursuedoctoralresearchinundertheadvisementofDr.WarrenDixon attheUniversityofFlorida.Focusingonnonlinearcontroltheoryandapplications,Nic earnedaMasterofSciencedegreeinDecemberof2010andcompletedhisPh.D.in Decemberof2012,bothinmechanicalengineering.Asagraduateresearcher,Nicwas awardedtheOutstandingGraduateResearchAwardfortheDepartmentofMechanical andAerospaceEngineeringforhisworkinnonlinearcontroltheoryin2012.Healso receivedthePh.D.GatorEngineeringAttributeofProfessionalExcellenceawardin2012 fromtheUFCollegeofEngineering.NichasledseveralstudentprojectsincludingUF's SubjuGatorautonomousunderwatervehicle.Additionally,Nicworkedasamechanical engineeringinternatHoneywellInternationalin2006. 144