<%BANNER%>

Dynamic Neural Network-Based Robust Control Methods for Uncertain Nonlinear Systems

Permanent Link: http://ufdc.ufl.edu/UFE0044481/00001

Material Information

Title: Dynamic Neural Network-Based Robust Control Methods for Uncertain Nonlinear Systems
Physical Description: 1 online resource (114 p.)
Language: english
Creator: Dinh, Huyen T
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: adaptive-control -- identification-for-control -- lyapunov-methods -- neural-networks -- nonlinear-time-delay-systems -- observers -- output-feedback -- robust-adaptive-control -- time-delay -- uncertain-systems
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: Neural networks (NNs) have proven to be effective tools for identification, estimation and control of complex uncertain nonlinear systems. As a natural extension of feed forward NNs with the capability to approximate nonlinear functions, dynamic neural networks (DNNs) can be used to approximate the behavior of dynamic systems. DNNs distinguish themselves from static feed forward NNs in that they have at least one feedback loop and their representation is described by differential equations. Because of internal state feedback, DNNs are known to provide faster learning and exhibit improved computational capability in comparison to static feed forward NNs. In this dissertation, a DNN architecture is utilized to approximate uncertain nonlinear systems as a means to develop identification methods and observers for estimation and control. In Chapter 3, an identification-based control method is presented, wherein a multilayer DNN is used in conjunction with a sliding mode term to approximate the input-output behavior of a plant while simultaneously tracking a desired trajectory. This result is achieved by combining the DNN-identification strategy with a RISE (Robust Integral of the Sign of the Error) controller. In Chapters 4 and 5, a class of second-order uncertain nonlinear systems with partially unmeasurable states is considered. A DNN-based observer is developed to estimate the missing states in Chapter 4, and the DNN-based observer is developed for an output feedback (OFB) tracking control method in Chapter 5. In Chapter 6, an OFB control method is developed for uncertain nonlinear systems with time-varying input delays. In all developed approaches, weights of the DNN can be adjusted on-line: no off-line weight update phase is required. Chapter 7 concludes the proposal by summarizing the work and discussing some future problems that could be further investigated.
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 Huyen T Dinh.
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: UFE0044481:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044481/00001

Material Information

Title: Dynamic Neural Network-Based Robust Control Methods for Uncertain Nonlinear Systems
Physical Description: 1 online resource (114 p.)
Language: english
Creator: Dinh, Huyen T
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: adaptive-control -- identification-for-control -- lyapunov-methods -- neural-networks -- nonlinear-time-delay-systems -- observers -- output-feedback -- robust-adaptive-control -- time-delay -- uncertain-systems
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: Neural networks (NNs) have proven to be effective tools for identification, estimation and control of complex uncertain nonlinear systems. As a natural extension of feed forward NNs with the capability to approximate nonlinear functions, dynamic neural networks (DNNs) can be used to approximate the behavior of dynamic systems. DNNs distinguish themselves from static feed forward NNs in that they have at least one feedback loop and their representation is described by differential equations. Because of internal state feedback, DNNs are known to provide faster learning and exhibit improved computational capability in comparison to static feed forward NNs. In this dissertation, a DNN architecture is utilized to approximate uncertain nonlinear systems as a means to develop identification methods and observers for estimation and control. In Chapter 3, an identification-based control method is presented, wherein a multilayer DNN is used in conjunction with a sliding mode term to approximate the input-output behavior of a plant while simultaneously tracking a desired trajectory. This result is achieved by combining the DNN-identification strategy with a RISE (Robust Integral of the Sign of the Error) controller. In Chapters 4 and 5, a class of second-order uncertain nonlinear systems with partially unmeasurable states is considered. A DNN-based observer is developed to estimate the missing states in Chapter 4, and the DNN-based observer is developed for an output feedback (OFB) tracking control method in Chapter 5. In Chapter 6, an OFB control method is developed for uncertain nonlinear systems with time-varying input delays. In all developed approaches, weights of the DNN can be adjusted on-line: no off-line weight update phase is required. Chapter 7 concludes the proposal by summarizing the work and discussing some future problems that could be further investigated.
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 Huyen T Dinh.
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: UFE0044481:00001


This item has the following downloads:


Full Text

PAGE 1

DYNAMICNEURALNETWORK-BASEDROBUSTCONTROLMETHODSFOR UNCERTAINNONLINEARSYSTEMS By HUYENT.DINH ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012

PAGE 2

2012HuyenT.Dinh 2

PAGE 3

Tomybeautifuldaughter AnhTran ,mylovinghusband ThongTran ,andmydearparents Hop Pham and TungDinh ,fortheirunwaveringsupportandconstantencouragement 3

PAGE 4

ACKNOWLEDGMENTS Iwouldliketoexpressmydeepestgratitudetomyadvisor,Dr.WarrenE.Dixon,forhis excellentguidance,caring,patienceandsupportinthefouryearsofmydoctoralstudy.Asan advisor,heexposedmetothevastandexcitingresearchareaofnonlinearcontrolandmotivated metoworkonDynamicNeuralNetwork-basedcontrolapplications.Heencouragedmeto exploremyownideasandhelpedmegrowasanindependentresearcher.Ihighlyappreciateall hiscaringandsupportasanunderstandingbossduringmymaternitytime. Iwouldliketoextendmygratitudetomycommitteemembers,Dr.PrabirBarooah,Dr. JoshepWilson,andDr.MrinalKumar,forinsightfulcommentsandmanyvaluablesuggestions toimprovethepresentationandcontentsofthisdissertation. IwouldalsoliketothankallmycoworkersattheNonlinearControlsandRoboticsLabfor theirvariousformsofsupportduringmygraduatestudy.Theirwarmfriendshiphasenrichedmy life. Fromthebottomofmyheart,Iwouldliketothankmyparentsfortheirsupportandbeliefin me.Withouttheirsupport,encouragementandunconditionallove,Iwouldneverbeabletonish mydissertation.Also,Iwouldliketothankandapologizetomydaughter,whowasbornbefore thisdissertationwascompletedandspendmostofhertimewithmymothertoallowmetofocus. Iamdeeplysorryforthetimewespendapart.Finally,Iwouldliketothankmyhusbandforhis constantpatienceandunwaveringlove. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS....................................4 LISTOFTABLES.......................................7 LISTOFFIGURES.......................................8 LISTOFABBREVIATIONS..................................9 ABSTRACT...........................................10 CHAPTER 1INTRODUCTION....................................12 1.1MotivationandProblemStatement.........................12 1.2DissertationOutline.................................17 1.3Contributions....................................18 2BACKGROUNDONNEURALNETWORKS......................23 2.1NeuralNetworks..................................23 2.2Multi-layerFeedforwardNeuralNetworks.....................24 2.3DynamicNeuralNetworks.............................27 3DYNAMICNEURALNETWORK-BASEDROBUSTIDENTIFICATIONAND CONTROLOFACLASSOFNONLINEARSYSTEMS................30 3.1DynamicSystemandProperties..........................30 3.2RobustIdenticationusingDynamicNeuralNetworks..............31 3.3RobustTrajectoryTrackingusingRISEfeedback.................34 3.4LyapunovStabilityAnalysisforDNN-basedIdenticationandControl......38 3.5Simulation......................................42 3.6Conclusion.....................................46 4DYNAMICNEURALNETWORK-BASEDROBUSTOBSERVERSFORUNCERTAINNONLINEARSYSTEMS.............................47 4.1DynamicSystemandProperties..........................47 4.2EstimationObjective................................49 4.3RobustObserverusingDynamicNeuralNetworks.................50 4.4LyapunovStabilityAnalysis............................54 4.5ExtensionforHigh-orderUncertainNonlinearSystems..............58 4.6ExperimentandSimulationResults.........................60 4.7Conclusion.....................................65 5

PAGE 6

5GLOBALOUTPUTFEEDBACKTRACKINGCONTROLFORUNCERTAINSECONDODERNONLINEARSYSTEMS............................67 5.1DynamicSystemandProperties..........................67 5.2EstimationandControlObjectives.........................68 5.3DNN-basedRobustObserver............................69 5.4RobustAdaptiveTrackingController........................72 5.5LyapunovStabilityAnalysisforDNN-basedObservationandControl......75 5.6ExperimentResults.................................78 5.7Conclusion.....................................79 6OUTPUTFEEDBACKCONTROLFORANUNCERTAINNONLINEARSYSTEMWITHSLOWLYVARYINGINPUTDELAY...................82 6.1DynamicSystemandProperties..........................82 6.2EstimationandControlObjectives.........................83 6.3RobustDNNObserverDevelopment........................84 6.4RobustTrackingControlDevelopment.......................86 6.5LyapunovStabilityAnalysisforDNN-basedObservationandControl......88 6.6SimulationResults.................................93 6.7Conclusion.....................................95 7CONCLUSIONANDFUTUREWORKS........................97 7.1DissertationSummary................................97 7.2FutureWork.....................................99 APPENDIX ADYNAMICNEURALNETWORK-BASEDROBUSTIDENTIFICATIONAND CONTROLOFACLASSOFNONLINEARSYSTEMS................101 A.1ProofoftheInequalityinEq.3........................101 A.2ProofoftheInequalityinEq.3........................101 A.3ProofoftheInequalityinEqs.3and3.................102 A.4ProofoftheInequalityinEq.3........................103 BDYNAMICNEURALNETWORK-BASEDGLOBALOUTPUTFEEDBACKTRACKINGCONTROLFORUNCERTAINSECOND-ODERNONLINEARSYSTEMS...105 REFERENCES.........................................107 BIOGRAPHICALSKETCH..................................114 6

PAGE 7

LISTOFTABLES Table page 4-1Transient t = 0 )]TJ/F22 11.9552 Tf 11.255 0 Td [(1secandsteadystate t = 1 )]TJ/F22 11.9552 Tf 11.254 0 Td [(10secvelocityestimationerrors x t fordifferentvelocityestimationmethodsinpresenceofnoise50dB.........65 5-1Steady-stateRMSerrorsandtorquesforeachoftheanalyzedcontroldesigns......79 6-1Link1andLink2RMStrackingerrorsandRMSestimationerrors............95 6-2RMSerrorsforcasesofuncertaintyintime-varyingdelayseenbytheplantascomparedtothedelayofthecontroller.............................95 7

PAGE 8

LISTOFFIGURES Figure page 2-1Nonlinearmodelofaneuron...............................24 2-2Two-layerNN......................................25 2-3HopeldDNNcircuitstructure..............................27 3-1ThearchitectureoftheMLDNN..............................32 3-2Robustidentication-basedtrajectorytrackingcontrol..................36 3-3Link1andlink2trackingerrors..............................44 3-4Link1andLink2positionidenticationerrors......................44 3-5Controlinputsofthelink1andlink2...........................45 3-6Link1andLink2feedforwardcomponentofthecontrolinput..............45 4-1Theexperimentaltestbedconsistsofatwo-linkrobot.Thelinksaremountedontwo NSKdirect-driveswitchedreluctancemotors.......................60 4-2Velocityestimate x t usinga[1],b[2],ctheproposedmethod,anddthecenterdifferencemethodonatwo-linkexperimenttestbed.................62 4-3Thesteady-statevelocityestimate x t usinga[1],b[2],ctheproposedmethod, anddthecenterdifferencemethodonatwo-linkexperimenttestbed..........63 4-4Frequencyanalysisofvelocityestimation x t usinga[1],b[2],ctheproposed method,anddthecenterdifferencemethodonatwo-linkexperimenttestbed.....63 4-5Thesteady-statevelocityestimationerror x t usinga[1],b[2],ctheproposed method,anddthecenterdifferencemethodonsimulations,inpresenceofsensor noiseSNR60dB.Therightgurese-hindicatetherespectivefrequencyanalysis ofvelocityestimation x t .................................65 5-1Thetrackingerrors e t ofaLink1andbLink2usingclassicalPID,robustdiscontinuousOFBcontroller[1],andproposedcontroller..................79 5-2ControlinputsforLink1andLink2usinga,bclassicalPID,c,drobustdiscontinuousOFBcontroller[1],ande,ftheproposedcontroller............80 5-3Velocityestimation x t usingaDNN-basedobserverandbnumericalbackwards difference..........................................81 6-1Simulationresultswith10%magnitudeand10%offsetvarianceintime-delay.....96 8

PAGE 9

LISTOFABBREVIATIONS DNNDynamicNeuralNetwork LKLyapunov-Krasovskii LPLinear-in-parameter MLNNMulti-layerNeuralNetwork NNNeuralNetwork OFBOutputFeedback RISERobustIntegraloftheSignoftheError RMSRootMeanSquares UUBUniformlyUltimatelyBounded 9

PAGE 10

AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DYNAMICNEURALNETWORK-BASEDROBUSTCONTROLMETHODSFOR UNCERTAINNONLINEARSYSTEMS By HuyenT.Dinh August2012 Chair:WarrenE.Dixon Major:MechanicalEngineering NeuralnetworksNNshaveproventobeeffectivetoolsforidentication,estimationand controlofcomplexuncertainnonlinearsystems.AsanaturalextensionoffeedforwardNNs withthecapabilitytoapproximatenonlinearfunctions,dynamicneuralnetworksDNNscan beusedtoapproximatethebehaviorofdynamicsystems.DNNsdistinguishthemselvesfrom staticfeedforwardNNsinthattheyhaveatleastonefeedbackloopandtheirrepresentationis describedbydifferentialequations.Becauseofinternalstatefeedback,DNNsareknownto providefasterlearningandexhibitimprovedcomputationalcapabilityincomparisontostatic feedforwardNNs. Inthisdissertation,aDNNarchitectureisutilizedtoapproximateuncertainnonlinear systemsasameanstodevelopidenticationmethodsandobserversforestimationandcontrol. InChapter3,anidentication-basedcontrolmethodispresented,whereinamultilayerDNN isusedinconjunctionwithaslidingmodetermtoapproximatetheinput-outputbehaviorofa plantwhilesimultaneouslytrackingadesiredtrajectory.Thisresultisachievedbycombining theDNN-identicationstrategywithaRISERobustIntegraloftheSignoftheErrorcontroller. InChapters4and5,aclassofsecond-orderuncertainnonlinearsystemswithpartiallyunmeasurablestatesisconsidered.ADNN-basedobserverisdevelopedtoestimatethemissingstates inChapter4,andtheDNN-basedobserverisdevelopedforanoutputfeedbackOFBtracking controlmethodinChapter5.InChapter6,anOFBcontrolmethodisdevelopedforuncertain nonlinearsystemswithtime-varyinginputdelays.Inalldevelopedapproaches,weightsofthe 10

PAGE 11

DNNcanbeadjustedon-line:nooff-lineweightupdatephaseisrequired.Chapter7concludes theproposalbysummarizingtheworkanddiscussingsomefutureproblemsthatcouldbefurther investigated. 11

PAGE 12

CHAPTER1 INTRODUCTION 1.1MotivationandProblemStatement Basedontheirapproximationproperties,NNshaveproventobeeffectivetoolsforidentication,estimationandcontrolofcomplexuncertainnonlinearsystems.FeedforwardNNshave beenextensivelyusedforadaptivecontrol:NNsarecascadedintothecontrolledsystemandthe NNweightsaredirectlyadjustedthroughanadaptiveupdatelawthatisafunctionofthetracking error[35].ContrarytofeedforwardNNs,theneuronsofDNNsreceivenotonlyexternalinput e.g.,trackingerrorbutalsointernalstatefeedback.FeedforwardandfeedbackconnectionsallowinformationinDNNstopropagateintwodirections:frominputneuronstooutputsandvice versa[6].SinceDNNsexhibitdynamicbehavior,theycanbeusedasdynamicmodelstorepresentnonlinearsystems,unlikefeedforwardNNswhichcanonlyapproximatenonlinearfunctions inasystem.Fromacomputationalperspective,aDNNwithstatefeedbackmayprovidemore computationaladvantagesthanastaticfeedforwardNN[7].FunahashiandNakamura[8]and Polycarpou[9]provedthatDNNscouldapproximatetheinput-outputbehaviorofaplantwith arbitraryaccuracy. Narendra[10]proposedtheideaofusingDNNsforidenticationofnonlinearsystems, whereintheidenticationmodelsDNNshavethesamestructureastheplantbutcontain NNswithadjustableweights.TheDNN-basedlearningparadigminvolvestheidentication oftheinput-outputbehavioroftheplantandtheuseoftheresultingidenticationmodelto adjusttheparametersofthecontroller[10].In[11],recurrenthigherorderNNsareusedfor identicationofnonlinearsystems,wherethedynamicalneuronsaredistributedthroughout thenetwork.RovithakisandChristodoulou[12]usedsingularperturbationtoinvestigatethe stabilityandrobustnesspropertiesoftheDNNidentier,anddesignedastatefeedbacklawto trackareferencemodel.In[13],aHopeld-typeDNNisusedtoidentifyasingleinput/single outputSISOsystem,andtheidentierisusedinthecontrollertofeedbacklinearizethesystem, whichisthencontrolledusingaPIDcontroller.PoznyakusedaparallelHopeld-typeNNfor 12

PAGE 13

identicationandtrajectorytracking[14,15]andprovedboundedLyapunovstabilityofthe identicationandtrackingerrorsinpresenceofmodelingerrors.Renetal.[16]proposeda DNNstructureforidenticationandcontrolofnonlinearsystemsbyusingjusttheinput/output measurements.TheHopeld-typeDNNiswidelyusedbecauseofitssimplestructureand desirablestabilityproperties[7,17].However,itsstructureonlyincludesthesingle-layerNN, soitsapproximationcapabilityislimitedincomparisonwithamulti-layerDNN.Amulti-layer DNNwithstablelearninglawsisusedin[18]fornonlinearsystemidentication.However, allofthepreviousDNNmethodsarelimitedtouniformlyultimatelyboundedresults,because oftheresidualfunctionapproximationerror.Incontrast,Chapter3proposesamodiedDNN identierstructuretoprovethattheidenticationerrorisasymptoticallyregulatedwithonly typicalgradientweightupdatelaws,andacontrollerincludingaDNN-identiertermanda robustfeedbacktermRISEisusedtoensureasymptotictrackingofthesystemalongadesired trajectory.Bothasymptoticidenticationandasymptotictrackingareachievedsimultaneously whiletheDNNweightsareadaptedon-line. Typicalidentication-basedcontrolapproaches[12,16,19,20]requirethesystemstates tobecompletelymeasurable.However,fullstatefeedbackisnotalwaysavailableinmany practicalsystems.Intheabsenceofsensors,therequirementoffull-statefeedbackforthe controlleristypicallyfullledbyusing adhoc numericaldifferentiationtechniques,whichcan aggravatetheproblemofnoise,leadingtounusablestateestimates.Severalnonlinearobservers areproposedinliteraturetoestimateunmeasurablestates.Forinstance,slidingmodeobservers weredesignedforgeneralnonlinearsystemsbySlotine etal. in[21],forrobotmanipulators byCanudasdeWit etal. in[22],andformechanicalsystemssubjecttoimpactsbyMohamed etal. in[23].However,alltheseobserversrequireexactmodelknowledgetocompensatefor nonlinearitiesinthesystem.Model-basedobserversarealsoproposedin[24]and[25]which requireahigh-gaintoguaranteeconvergenceoftheestimationerrortozero.Theobservers introducedin[26]and[27]arebothappliedforLagrangiandynamicsystemstoestimatethe velocity,andasymptoticconvergencetothetruevelocityisobtained.However,thesymmetric 13

PAGE 14

positive-denitenessoftheinertiamatrixandtheskew-symmetricpropertyoftheCoriolismatrix arerequired.Modelknowledgeisrequiredin[26]andapartialdifferentialequationneeds tobesolvedtocomputetheobservers.In[27],thesystemdynamicsmustbeexpressedina non-minimalmodelandonlymassandinertiaparametersareunknowninthesystem. Thedesignofrobustobserversforuncertainnonlinearsystemsisconsideredin[1,28,29]. In[28],asecond-orderslidingmodeobserverforuncertainsystemsusingasuper-twisting algorithmisdeveloped,whereanominalmodelofthesystemisassumedtobeavailableand estimationerrorsareproventoconvergeinnite-timetoaboundedsetaroundtheorigin. In[29],theproposedobservercanguaranteethatthestateestimatesconvergeexponentially fasttotheactualstate,ifthereexistsavectorfunctionsatisfyingacomplexsetofmatching conditions.In[1],oneoftherstasymptoticvelocityobserversisdevelopedforgeneralsecondordersystems,wheretheestimationerrorisproventoasymptoticallyconverge.However, allnonlinearuncertaintiesin[1]aredampedoutbyaslidingmodetermresultinginhigh frequencystateestimates.ANNapproachthatusestheuniversalapproximationpropertyis investigatedforuseinanadaptiveobserverdesignin[30].However,estimationerrorsin[30] areonlyguaranteedtobeboundedduetofunctionreconstructioninaccuracies.Inspiredby[1] and[30],arobustadaptiveDNN-basedobserverisintroducedinChapter4,wheretheDNN isusedtoapproximatetheuncertainsystem,adynamiclterworksinjunctionwiththeDNN toreconstructtheunmeasurablestate,andaslidingmodetermisaddedtotheobserverto compensatefortheapproximationerrorandexogenousdisturbance.Asymptoticestimationis provenbyaLyapunov-basedstabilityanalysisandillustratedbyexperimentsandsimulations. InadditiontoOFBobservers,variousOFBcontrollershavealsobeendeveloped.OFB controllersusingmodel-basedobserversweredevelopedin[3133].In[31],Berghuis etal designedanobserverandacontrollerforrobotmodelsusingapassivityapproachforboth positioningandtrackingobjectivesbasedontheconditionthatthesystemdynamicsareexactly known.Do etal. in[32]consideredobserver-basedOFBcontrolforunicycle-typemobile robotstostabilizethesystemandtrackadesiredtrajectory.In[33],acontrollerbasedonan 14

PAGE 15

observer-basedintegratorbacksteppingtechniquewasproposedforarevolutemanipulatorwith knowndynamics,andasemi-globalexponentialstabilityresultforthelinkpositiontracking errorandthevelocityobservationerrorwasachieved.Adisadvantageoftheseapproachesisthe requirementofexactmodelknowledge.OFBcontrolforsystemswithparametricuncertainties hasbeendevelopedin[3437].Alinearobserverisusedin[34]toestimatetheangularvelocity ofarigidrobotarmwhichisrequiredtosatisfythelinear-in-parametersLPcondition,and uniformultimateboundednessofthetrackingandobservationerrorsisobtained.Adaptive OFBcontrolforrobotmanipulatorssatisfyingtheLPconditionwhichachievessemi-global asymptotictrackingresultsisconsideredin[3537].Thedifferencebetweentheseapproaches isthejointvelocityisestimatedbyanobserverin[35],whilealterisusedforvelocity estimationin[36]and[37].Anextensionof[36]and[37]toobtainaglobalasymptotictracking resultwasintroducedin[38].However,alimitationofsuchpreviousadaptiveOFBcontrol approachesisthatonlyLPuncertaintiesareconsidered.Asaresult,ifuncertaintiesinthesystem donotsatisfytheLPconditionorifthesystemisaffectedbydisturbances,theresultswill reducetoauniformlyultimatelyboundedUUBresult.Theconditionoflineardependence uponunknownparameterscanberelaxedbyintroducingaNNorfuzzylogicintheobserver structureasin[30,3943];however,bothestimationandtrackingerrorsareonlyguaranteed tobeboundedduetotheexistenceofreconstructionerrors.Therstsemi-globalasymptotic OFBtrackingresultforsecond-orderdynamicsystemsundertheconditionthatuncertain dynamicsarerst-orderdifferentiablewasintroducedin[1]withanovellterdesign.Allof theuncertainnonlinearitiesin[1]aredampedoutbyaslidingmodeterm,sothediscontinuous controllerrequireshigh-gain.TheOFBcontrolapproachinChapter5ismotivatedby[1]andthe observerdesigninChapter4.Inthisapproach,theDNN-basedobserverisusedtoestimatethe unmeasurablestateofthesystem;thecontroller,includingthestateestimation,NN,andsliding modetermsareusedtoyieldtrajectorytracking.Bothasymptoticestimationoftheunmeasurable stateandasymptotictrackingofthedesiredtrajectoryareachievedsimultaneously.Experiments demonstratetheperformanceofthedevelopedapproach. 15

PAGE 16

Formanypracticalsystems,timedelayisinevitable.Thetorquegeneratedbyaninternal combustionenginecanbedelayedduetofuel-airmixing,ignitiondelays,cylinderpressure forcepropagationsee,e.g.,[44,45],orcommunicationdelaysinremotecontrolapplications wheretimeisrequiredtotransmitinformationusedforfeedbackcontrolsee,e.g.,master-slave teleoperationofrobotin[4650].Unfortunately,timedelayisasourceofinstabilityandcan decreasesystemperformance. Delayinthecontrolinputi.e,actuatordelayisanissuethathasattractedsignicant attention.Variousstabilityanalysismethodsandcontroldesigntechniqueshavebeendeveloped forsystemswithinputdelays.Fornonlinearsystems,Lyapunov-KrasovskiiLKfunctionalbasedmethodscf.[5153]andLyapunov-Razumikhinmethodscf.[5456]arethemost widelyusedtoolstoinvestigatethestabilityofasystemaffectedbytimedelays.Comparedwith frequencydomainmethodsthatcheckifallrootsofthecharacteristicequationofaretardedor neutralpartialdifferentialequationhavenegativerealparts[57,58],limitingitsapplicability toonlylineartime-invariantsystemswithexactmodelknowledge,theKrasovskii-typeand Razumikhin-typeapproachescanbeappliedforuncertainnonlinearsystemswithtime-varying delays.ComparingbetweentheRazumikhin-typeandLKfunctional-typetechniquesreveals thattheRazumikhinmethodscanbeconsideredasaparticularbutmoreconservativecaseof Krasovskiimethods,whereRazumikhinmethodscanbeappliedtoarbitrarilylarge,bounded time-varyingdelays 0 t t < ; whereastheKrasovskiimethodsrequireabounded derivativeofthedelays t t j < 1 .However,theRazumikhinapproachrequiresinput-tostatestabilityofthenominalsystemwithoutdelay. Variousfull-statefeedbackcontrollershavebeendevelopedthatarebasedonLKor Razumikhinstabilitycriterionfornonlinearsystemswithinputdelays.Approachesin[5963] providecontrolmethodsforuncertainnonlinearsystemswithknownandunknownconstant time-delays.However,time-delaysarelikelytovaryinpractice.Severalmethodsfornonlinear systemswithtime-varyinginputdelayshavebeenrecentlyinvestigated.Linearizedcontrollers in[64,65]areonlyvalidwithinaregionaroundthelinearizationpoint.Acontrollerdeveloped 16

PAGE 17

in[66],whichisanextensionof[61,67],dealswithforwardcompletenonlinearsystemswith time-varyinginputdelayunderanassumptionthattheplantisasymptoticallystableinthe absenceoftheinputdelay.In[66],aninvertibleinnitedimensionalbacksteppingtransformation isusedtoyieldanasymptoticallystablesystem.AnEuler-Lagrangesystemwithaslowly varyinginputdelayisconsideredin[68],wherefullstatefeedbackisrequired.However,ifonly systemoutputisavailableforfeedback,howtodesignacontrollertohandleboththelackofthe stateandthetime-varyingdelayoftheinputisrarelyinvestigated.Studiesin[6971]addressthe OFBcontrolproblemfornonlinearsystemswithconstantinputdelaybylinearizationmethod.A spacecraft,exible-jointrobotandrigidrobotwithconstanttimedelayareconsideredin[6971], respectively,wheretheobjectivesaretodesignOFBcontrollerstostabilizethesystemsaround asetpoint.Thecontrollersarerstdesignedfordelay-freelinearizedsystems,thenrobustness tothedelayisprovenprovidedcertaindelaydependentconditionsholdtrue.Totheauthor's knowledge,anOFBcontrolmethodfornonlinearsystemswithtime-varyinginputdelayand trackingcontrolobjectivesisstillanopenproblem. 1.2DissertationOutline Chapter1servesasanintroduction.Themotivation,problemstatement,literaturereview, thecontributionsandtheproposedresearchplanofthedissertationarediscussedinthischapter. Chapter2providesabackgrounddiscussiononNNs,reviewingmulti-layerneuralnetwork MLNNandDNNstructures,theirlearninglaws,andtheirapproximationproperties. Chapter3providesamethodologyforDNN-basedidenticationandtrackingcontrol. Theidentierstructureismodiedbyaddingarobustslidingmodetermtoaccountforthe reconstructionerror,hencetheinput-outputbehavioroftheidentierisproventoasymptotically tracktheinput-outputbehaviorofthesystem.Thecontroller,includinginformationfromthe identierandtheRISEfeedbacktermisproposedtoguaranteeasymptotictrackingofthesystem tothedesiredtrajectory.Theperformanceoftheidenticationandcontrolisillustratedthrough simulations. 17

PAGE 18

Chapter4illustratesanovelrobustadaptiveobserverdesignforsecond-orderuncertainnonlinearsystems.TheobserverisdesignedbasedonaDNNtoapproximatetheuncertainsystem, adynamicltertoprovideasurrogatefortheunmeasurablestate,andaslidingmodetermto cancelouttheapproximationerrorandexogenousdisturbance.Theasymptoticestimationresult isprovenbyLyapunov-basedstabilityanalysisandillustratedbyexperimentsandsimulations. Chapter5developsanOFBcontrolapproachforsecond-orderuncertainnonlinearsystems, wheretheDNN-basedobserverisusedtoestimatetheunmeasurablestateofthesystemand thecontrollerincludesthestateestimation,NN,andslidingmodetermstoforcethesystemto trackadesiredtrajectory.Bothasymptoticestimationoftheunmeasurablestateandasymptotic trackingofthedesiredtrajectoryareachievedsimultaneously.Experimentresultsdemonstrate theperformanceofthedevelopedapproach. Chapter6considersOFBcontrolmethodforuncertainnonlinearsystemsaffectedbytimevaryinginputdelaysandadditivedisturbances.Thedelayisassumedtobeboundedandslowly varying.TheDNN-basedobserverworksinjunctionwiththecontrollertoprovideanestimate fortheunmeasurablestate.UUBestimationoftheunmeasurablestateandUUBtrackinginthe presenceofmodeluncertainty,disturbances,andtimedelaysareprovenbyaLyapunov-based stabilityanalysis. Chapter7describesthepossibledirectionsthatcouldextendtheoutcomesoftheworkin thisdissertation. 1.3Contributions ThecontributionsinthisdissertationareprovidedinChapters3-6. Chapter3:DNN-basedrobustidenticationandcontrolofaclassofnonlinearsystems: Thefocusofthischapteristhedevelopmentofanindirectcontrolstrategyidenticationbasedcontrolforaclassofuncertainnonlinearsystems.AmodiedDNNstructureisproposed whereamulti-layerDNNiscombinedwithanidenticationerror-basedslidingmodeterm. UnlikemostDNNresultswhichconsiderasingle-layerHopeld-typeseries-parallelcongurationoftheDNN,thismethodconsidersaparallelmulti-layerDNNconguration,whichhasthe 18

PAGE 19

advantageofprovidingbetterapproximationaccuracy[15].Theadditionalslidingmodetermis usedtorobustlyaccountforexogenousboundeddisturbances,modelingerrors,andthefunction reconstructionerrorsoftheDNN.ThismodiedDNNstructureallowsidenticationofuncertain nonlinearsystemswhileensuringrobustnesstoexternaldisturbances.TheideaofrobustidenticationofnonlinearsystemswasrstproposedbyPoznyak[72],whousedaslidingmodetermin thealgebraicweightupdatelawsandensuredregulationoftheidenticationerrortozero.Huang andLewis[73]usedahighgainrobustifyingtermintheDNNstructuretoproveUUBstability ofnonlinearsystemswithtime-delay.However,theproposeduseoftheslidingmodeterminthe DNNstructureisnovelandadvantageoussinceitprovidesrobustnesstomatcheddisturbances inthesystemwithouttheneedtomodifytheweightupdatelaws.Asymptoticconvergenceof theidenticationerrorisalsoguaranteed.Theidentierisdevelopedtofacilitatethedesignof thecontrollerforthepurposeoftrajectorytracking.ThecontrollerconsistsofaDNN-identier termandarobustfeedbacktermRISE[74,75]toensureasymptotictrackingofthesystem alongadesiredtrajectory.Asymptotictrackingisacontributionoverpreviousresults,where onlyboundedstabilityofthetrackingerrorcouldbeprovenduetothepresenceofmodelingand functionreconstructionerrorsoftheDNN.TheuseofthecontinuousRISEtermispreferred overtheslidingmodeterminthefeedbackcontrollertoavoidchatteringandotherside-effects associatedwithusingadiscontinuouscontrolstrategy.Oneoftheassumptionsintheuseofthe RISEfeedbacktechniqueisthatthedisturbancetermsareboundedbyknownconstantsandtheir derivativesbeboundedbyeitheraconstantoralinearcombinationofthestates.Tosatisfythese boundednessassumptions,abounded,user-denedsamplestateisintroducedinthedesignof theweightupdatelawsfortheDNN.Noofineidenticationstageisrequired,andboththe controllerandtheidentieroperatesimultaneouslyinreal-time. Chapter4:DNN-basedrobustobserversforsecond-orderuncertainnonlinearsystems: Thechallengetoobtainasymptoticestimationstemsfromthefactthattorobustlyaccount fordisturbances,feedbackoftheunmeasurableerroranditsestimatearerequired.Typically, feedbackoftheunmeasurableerrorisderivedbytakingthederivativeofthemeasurablestate 19

PAGE 20

andmanipulatingtheresultingdynamicse.g.,thisistheapproachusedinmethodssuchas[1] and[30].However,suchanapproachprovidesalinearfeedbacktermoftheunmeasurablestate. Hence,aslidingmodetermcouldnotbesimplyaddedtotheNNstructureoftheresultin[30] toyieldanasymptoticresult,becauseitwouldrequirethesignumoftheunmeasurablestate,and itdoesnotseemclearhowthisnonlinearfunctionoftheunmeasurablestatecanbeinjectedin theclosed-looperrorsystemusingtraditionalmethods.Likewise,itisnotclearhowtosimply addaNN-basedfeedforwardestimationofthenonlinearitiesinresultssuchas[1]becauseof theneedtoinjectnonlinearfunctionsoftheunmeasurablestate.Thenovelapproachusedinthis chapteravoidsthisissuebyusingnonlinearslidingmodefeedbackofthemeasurablestate,and thenexploitingtherecurrentnatureofaDNNstructuretoinjecttermsthatcancelcrossterms associatedwiththeunmeasurablestate.Theapproachisfacilitatedbyusingthelterstructureof thecontrollerin[1]andanovelstabilityanalysis.Thestabilityanalysisisbasedontheideaof segregatingthenonlinearuncertaintiesintotermswhichcanbeupper-boundedbyconstantsand termswhichcanbeupper-boundedbystates.Thetermsupper-boundedbystatescanbecanceled bylinearfeedbackofthemeasurableerrors,whilethetermsupper-boundedbyconstantsare partiallyrejectedbythesignfeedbackofthemeasurablestateandpartiallyeliminatedbythe novelDNN-basedweightupdatelaws.Thecontributionofthischapteroverpreviousresultsis thattheobserverisdesignedforuncertainnonlinearsystems,andtheon-lineapproximationof theunmeasurableuncertainnonlinearitiesviatheDNNstructureshouldheuristicallyimprove theperformanceofmethodsthatonlyusehigh-gainfeedback.Asymptoticconvergenceof theestimatedstatestotherealstatesisprovenusingaLyapunov-basedanalysisforageneral second-ordersystem.Anextensionoftheproposedobserverforahigh-ordersystemisshown, whereas,theoutputofthe n th ordersystemisassumedtobemeasurableupto n )]TJ/F22 11.9552 Tf 11.1 0 Td [(1derivatives. Thedevelopedobservercanbeusedseparatelyfromthecontrollereveniftherelativedegree betweenthecontrolinputandtheoutputisarbitrary.Simulationandexperimentresultsona two-linkrobotmanipulatorshowtheeffectivenessoftheproposedobserverwhencomparedwith 20

PAGE 21

astandardnumericalcentraldifferentiationalgorithm,thehighgainobserverproposedin[2], andtheobserverin[1]. Chapter5:DNN-basedglobaloutputfeedbacktrackingcontrolforsecond-order uncertainnonlinearsystems :Inthischapter,aDNN-basedobserver-controllerisproposed foruncertainnonlinearsystemsaffectedbyboundedexternaldisturbances,toachieveatwofoldresult:asymptoticestimationofunmeasurablestatesandasymptotictrackingcontrol. AsymptoticestimationofunmeasurablestatesisexploitedfromDNN-basedobserverdesign introducedinChapter4;however,asymptotictrackingisnotsimplyobtainedbyreplacing theestimationstatewiththeunmeasurablestateinthecontrollaw.Thechallengeisthatthe disturbancesareagainincludedintheopen-looptrackingerrorsystem.Torobustlyaccountfor disturbances,bothlinearandnonlinearfeedbackoftheunmeasurabletrackingerrorarerequired. Thelinearfeedbackisutilizedfromthelinearfeedbackofunmeasurableestimationerror,as inChapter4.However,itisnotclearhowtoinjectthenonlinearfeedbackoftheunmeasurable trackingerrorfromthemeasurablestateandtheestimationstate.TheapproachusedinChapter5 avoidsthisissuebyusingtheslidingmodefeedbackofthemeasurabletrackingerrorcombined withthenovelstabilityanalysis.Amodiedversionofthelterintroducedin[1]isusedto estimatetheoutputderivative.Modicationinthedenitionofthelteredestimationand trackingerrorsisutilized.AcombinationofaNNfeedforwardterm,alongwithestimatedstate feedbackandslidingmodetermsaredesignedforthecontroller.TheDNNobserveradapts on-linefornonlinearuncertaintiesandshouldheuristicallyperformbetterthanarobustfeedback observer.NewweightupdatelawsfortheDNNbasedontheestimationerror,trackingerrorand lteroutputareproposed.Asymptoticregulationoftheestimationerrorandasymptotictracking areachieved. Chapter6:OutputFeedbackControlforanUncertainNonlinearSystemwith SlowlyVaryingInputDelay: ThechallengetodesignanOFBcontrolforuncertainnonlinear systemswithtime-varyinginputdelaysstemsfromtwoquestions:howtoinjectthenegative feedbackofthestatethroughthedelayedinput,andhowtoaccountforthedelayedstatewhichis 21

PAGE 22

introducedintotheclosed-loopsystembytheinput.Normally,withfull-statefeedbackmethods asin[63,68],theanswerfortherstquestionistheuseofapredictortermwhichcanprovidea freedelayinputtothesystemandthesolutionforthesecondquestionistheuseofanauxiliary LKfunctional,whichistheintegraloveradelaytimeintervalofthenormsquareofthestate, hence,thetimederivativeoftheLKfunctionalcanprovideanegativefeedbacktermofthenorm squareofthedelayedstatewhichitselfcancelsallstatedelayterms.However,withOFBcontrol, thedifcultyisthatthecorrespondingstateisunmeasurablesoitcannotindirectlyfeedbackinto thesystemviathepredictorterm.Theapproachtosolvetheissueinthischapterismotivated fromtheuseofaDNN-basedobserverin[76]and[77]toincludeanegativefeedbackofan unmeasurableestimationerrorsignalintotheclosed-loopsystemviaadynamiclter,thena controllerisdesignedbasedonthedifferenceresidualbetweentheunmeasurablestatewith theunmeasurableerrorsignal,wherethisresidualismeasurable.Hence,nallythroughthe predictorterm,theresidualwithoutdelayisaddedtotheclosed-loopsystemalongwiththeerror signaltoformthenegativefeedbackoftheunmeasurablestate.Todealwiththedelayedresidual injectedtothesystem,similarly,anauxiliaryLKfunctionalincludingthenormsquareofthe residualtermisused,thenUUBtrackingandestimationresultsareprovenbyLyapunov-based techniques. Inthischapter,anOFBcontrolforasecond-orderuncertainnonlinearsystemwithadditivedisturbancesisdevelopedtocompensateforboththeinaccessibilityofallstatesandthe time-varyingdelayoftheinput.Thedelayisassumedtobeboundedandslowlyvarying.A DNN-basedobserverwithon-lineupdatealgorithmsisusedtoprovideasurrogatefortheunmeasurablestate,apredictortermisutilizedtoinjectadelayfreecontrolintotheanalysis,and LKfunctionalsareusedtofacilitatethedesignandstabilityanalysis.Thedevelopedcontroller achievessimultaneouslyUUBestimationoftheunmeasurablestateandUUBtrackingresults, despitethelackoffullstatefeedback,thetime-varyinginputdelay,uncertainties,andexogenous disturbancesintheplantdynamics.Anumericalsimulationforatwo-linkrobotmanipulatoris providedtoexaminetheperformanceoftheproposedmethod. 22

PAGE 23

CHAPTER2 BACKGROUNDONNEURALNETWORKS 2.1NeuralNetworks Inthischapter,abriefbackgroundonarticialNNsisprovided.NNstructures,learning methods,andapproximationpropertiesareincluded.ThestructuresofbothmultilayerNNsand DNNsaredescribed. ANNisamassivelyparalleldistributedprocessorcomposedofsimpleprocessingunitsthat haveanaturalpropensityforstoringexperientialknowledgeandmakingitavailableforuse.A NNresemblesthebrainintworespects[78]: 1.Knowledgeisacquiredbythenetworkfromitsenvironmentthroughalearningprocess. 2.Interneuronconnectionstrengths,knownassynapticweights,areusedtostoretheacquired knowledge. Theprocedureforthelearningprocessiscalledalearningalgorithm.Itsfunctionistomodify thesynapticweightsofthenetworktoattainadesireddesignobjective.Theweightmodication equipsthemethodforNNdesignandimplementation.Facilitatingidentication,estimation, andcontrolforawideclassofnonlinearsystems,NNsofferseveralusefulpropertiesand capabilities: 1.N ONLINEARITY .ANNisconstructedfromaninterconnectionofnonlinearneurons,soit isitselfnonlinear.Thisisanimportantproperty,especiallyiftheunderlyingmechanismis inherentlynonlinear. 2.I NPUT OUTPUTMAPPING .Thesynapticweightsfreeparametersofthenetworkare modiedtominimizethedifferencebetweenthedesiredresponseandtheactualresponse ofthenetworkproducedbytheinputsignalinaccordancewithappropriatecriterion. Hence,theNNcanadapttoconstructthedesiredinput-outputmapping. 3.A DAPTIVITY .NNshaveabuilt-incapabilitytoadapttheirweightsbasedondesign criterion.Theon-linelearningalgorithmcanleadthenetworktoadaptinrealtime. 23

PAGE 24

Figure2-1.Nonlinearmodelofaneuron AneuronisthefundamentalunitfortheoperationofaNN.ItsmodelisshowninFig.2-1with threebasicelements: 1.Asetofsynapseslinks,witheachelementcharacterizedbyitsownweight, 2.Anadderforsummingtheinputsignalsmultipliedbytheirrespectiveweights, 3.Anonlinearactivationfunctiontransformingtheadderoutputintotheoutputoftheneuron. Inmathematicalterms,theneuroncanbedescribedas y j = s n i = 1 w ij x i + b j ; where x 1 ; x 2 ;:::; x n aretheinputsignals, w 1 j ; w 2 j ;::; w nj aretherespectivesynapticweightsof neuron j b j isthebias, s istheactivationfunction,and y j istheoutputsignaloftheneuron. Theactivationfunction s isoftenchosenashardlimit,linearthreshold,sigmoid,hyperbolic tangent,augmentedratioofsquares,orradialbasisfunctions. ThewayinwhichneuronsofaNNareinterconnecteddeterminesitsstructure.Inthe following,thestructureforamultilayerfeedforwardneuralnetworkMLNNandthestructure foraDNNareconsidered. 2.2Multi-layerFeedforwardNeuralNetworks Multi-layerfeedforwardNNsdistinguishesitselfbythepresenceofoneormorehidden layersinadditiontoinputandoutputlayers.Theneuronsineachlayerhavetheoutputofthe precedinglayerastheirinputs.Ifeachneuronineachlayerisconnectedtoeveryneuroninthe 24

PAGE 25

Figure2-2.Two-layerNN adjacentforwardlayer,thentheNNisfullyconnected.ThemostcommonstructureofMLNNs isatwo-layerNN,showninFig.2-2. Amathematicalformuladescribingatwo-layerNNisgivenby y i = L j = 1 w ij s n k = 1 v jk x k + q vj + q wi # ; i = 1 ; 2 ;::: m : TheNNcanberewritteninmatrixformas y = W T s V T x ; wheretheoutputvectoris y =[ y 1 y 2 ::: y m ] T 2 R m ; theinputvectoris x =[ 1 x 1 x 2 ::: x n ] T 2 R n + 1 theactivationvectordenedforavector x =[ x 1 x 2 ::: x L ] T is s x =[ 1 s x 1 s x 2 ::: s x L ] T 2 25

PAGE 26

R L + 1 ,andtheweightmatrices W ; V containthethresholdsintherstcolumnsas W = 2 6 6 6 6 6 6 6 4 q w 1 w 11 ::: w 1 L q w 2 w 21 ::: w 2 L . . q wm w m 1 ::: w mL 3 7 7 7 7 7 7 7 5 T 2 R L + 1 m ; V = 2 6 6 6 6 6 6 6 4 q v 1 v 11 ::: v 1 n q v 2 v 21 ::: v 2 n . . q vL v L 1 ::: w Ln 3 7 7 7 7 7 7 7 5 T 2 R n + 1 L : TheweightsofNNscanbetunedbymanytechniques.Acommonweight-tuningalgorithmisthegradientalgorithmbasedonthebackpropagatederror.Acontinuousversionof backpropagationtuningisgivenby W = G w s V T x d E T ; V = G v x d s 0 T WE T ; where G w ; G v arethedesigngains,thebackpropagatederror E = y d )]TJ/F25 11.9552 Tf 11.296 0 Td [(y ,with y d 2 R m isthe desiredNNoutputinresponsetothereferenceinput x d 2 R n ; and y 2 R m istheactualNN output.Theterm s 0 isthederivativeoftheactivationfunction s ,whichcanbecalculated easily.Forexample,iftheactivationfunctionischosenasthesigmoidfunction,theterm s 0 is equalto s 0 diag s V T x d I )]TJ/F25 11.9552 Tf 10.95 0 Td [(diag s V T x d : Approximationusingtwo-layerNNs: Let f x beageneralsmoothfunctionfrom R n to R m .Aslongas x isrestrictedtoa compactset S of R n ,thereexistNNweightsandthresholdssuchthat f x = W T s V T x + e ; forsomenumber L ofhidden-layerneurons.Theuniversalfunctionapproximationproperty holdsforalargeclassofactivationfunctionsandthefunctionalreconstructionerror e canbe madearbitrarilysmallbyincreasingthenumberofnodesinthenetworkstructure.Generally, e decreasesas L increases.Infact,foranypositivenumber e N ,thereexistweightsandanLsuch that k e k < e N ; forall x 2 S .Furtherdetailsareprovidedin[78]and[46]. 26

PAGE 27

Figure2-3.HopeldDNNcircuitstructure 2.3DynamicNeuralNetworks DNNsdistinguishthemselvesfromothertypesofNNsstaticMLNNsinthattheyhaveat leastonefeedbackloop.ThefeedbackloopsresultinanonlineardynamicalbehaviorofDNNs. ADNNstructurethatcontainsstatefeedbackmayprovidemorecomputationaladvantagesthan astaticneuralstructure,whichcontainsonlyafeedforwardneuralstructure.Ingeneral,asmall feedbacksystemisequivalenttoalargeandpossiblyinnitefeedforwardsystem[79].Avery wellknownDNNstructureistheHopeldstructure[17,80],whichcanbeimplementedby anelectroniccircuit.Acontinuous-timeHopeldDNNcontaining n unitsisdescribedbythe followingdifferentialequations[7]: Stateequation : C i dx i t dt = )]TJ/F25 11.9552 Tf 10.484 8.093 Td [(x i t R i + n j = 1 w ij y j t + s i t ; i = 1 ; 2 ;:::; n ; Outputequation : y i t = s i x i t : Thisnonlinearsystemcanbeimplementedbyananalog RC resistance-capacitancenetwork circuitasshowninFig.2-3,where u i = x i istheinputvoltageofthe i thamplier, V i = s i u i is 27

PAGE 28

theoutputofthe i thamplier,theparameter R i isdenedas 1 R i = 1 r i + n j = 1 1 R ij ; andtheweightparameter w ij as w ij = 8 > > < > > : + 1 R ij ; R ij isconnectedto V j )]TJ/F22 8.9664 Tf 13.78 4.711 Td [(1 R ij ; R ij isconnectedto )]TJ/F25 11.9552 Tf 8.691 0 Td [(V j : Thissystemcanbewritteninmatrixformas dx dt = Ax + W 1 s x + W 2 u ; where x =[ x 1 x 2 ::: x n ] T ; s x =[ s x 1 s x 2 ::: s x 2 ] T ; u =[ s 1 s 2 ::: s n ] T ; and A = 2 6 6 6 6 6 6 6 4 )]TJ/F22 8.9664 Tf 17.621 4.71 Td [(1 R 1 C 1 0 0 0 )]TJ/F22 8.9664 Tf 17.621 4.711 Td [(1 R 2 C 2 0 . . 00 ::: )]TJ/F22 8.9664 Tf 17.621 4.71 Td [(1 R n C n 3 7 7 7 7 7 7 7 5 ; W 1 = 2 6 6 6 6 6 6 6 4 w 11 C 1 w 12 C 1 w 1 n C 1 w 21 C 2 w 22 C 2 w 2 n C 2 . . w n 1 C n w n 2 C n ::: w nn C n 3 7 7 7 7 7 7 7 5 ; W 2 = 2 6 6 6 6 6 6 6 4 1 C 1 0 0 0 1 C 2 0 . . 00 ::: 1 C n 3 7 7 7 7 7 7 7 5 : ApproximationusingDNNs: 28

PAGE 29

TheapproximationcapabilityfornonlinearsystembehaviorswithDNNsisdocumentedin literature.Therstproofbasedonnaturalextensionofthefunctionapproximationproperties ofstaticNNsisshownin[81]and[8],hencetheinputofDNNsislimitedfortimebelonging toaclosedset.Thesecondproofusesasystemrepresentationoperatortoderiveconditionsfor theapproximationvaliditybyaDNN.IthasbeenextensivelyanalyzedbyI.W.Sandberg,both forcontinuousanddiscretetime[8284].AllapproachesintroducedinChapters3-6prove theapproximationcapabilityofDNNsbasedontheextensionofthefunctionapproximation propertiesofNNsandLyapunovstabilityanalysis. 29

PAGE 30

CHAPTER3 DYNAMICNEURALNETWORK-BASEDROBUSTIDENTIFICATIONANDCONTROLOF ACLASSOFNONLINEARSYSTEMS AmethodologyforDNNidentication-basedcontrolofuncertainnonlinearsystemsis proposed.Themulti-layerDNNstructureismodiedbytheadditionofaslidingmodeterm torobustlyaccountforexogenousdisturbancesandDNNreconstructionerrors.Weightupdate lawsareproposedwhichguaranteeasymptoticregulationoftheidenticationerror.Arecently developedrobustfeedbacktechniqueRISEisusedinconjunctionwiththeDNNidentier forasymptotictrackingofadesiredtrajectory.Boththeidentierandthecontrolleroperate simultaneouslyinrealtime.Numericalsimulationsforatwo-linkrobotareprovidedtoexamine thestabilityandperformanceofthedevelopedmethod. 3.1DynamicSystemandProperties Consideracontrol-afnenonlinearsystemoftheform x = f x + g x u t + d t ; where x t 2 R n isthemeasurablestatewithaniteinitialcondition x 0 = x 0 u t 2 R m isthe controlinput, f x 2 R n isanunknown C 1 function,locallyLipschitzin x g x 2 R n m ,and d t 2 R n isanexogenousdisturbance.Thefollowingassumptionsaboutthesystemin3will beutilizedinthesubsequentdevelopment. Assumption3.1. Theinputmatrixg x isknown,boundedandhasfull-rowrank. Assumption3.2. Thedisturbanced t anditsrstandsecondtimederivativesarebounded,i.e. d t ; d t ; d t 2 L : TheuniversalapproximationpropertyoftheMLNNstatesthatgivenanycontinuous function F : S R n ,where S isacompactset,thereexistidealweights q ; suchthatthe outputoftheNN, F ; q approximates F toanarbitraryaccuracy[85].Hence,theunknown nonlinearityin3canbereplacedbyaMLNN,andthesystemcanberepresentedas x = A s x + W T s V T x + e + gu + d ; 30

PAGE 31

wheretheuniversalapproximationpropertyoftheMLNNs[86,87]isusedtoapproximatethe function f x )]TJ/F25 11.9552 Tf 10.95 0 Td [(A s x as f x )]TJ/F25 11.9552 Tf 10.95 0 Td [(A s x = W T s V T x + e : In4, A s 2 R n n isHurwitz, W 2 R N n and V 2 R n N areboundedconstantidealweight matricesoftheDNNhaving N hiddenlayerneurons, s 2 R N istheactivationfunction sigmoid,hyperbolictangentetc.,and e x 2 R n isthefunctionreconstructionerror.The feedbackofthestate x t astheinputoftheMLNN W T s V T x makesthewholesysteminthe structureofamulti-layerDNN.ThefollowingassumptionsontheDNNmodelofthesystemin 3willbeutilizedforthestabilityanalysis. Assumption3.3. TheidealNNweightsareboundedbyknownpositiveconstants[46]i.e. k W k Wand k V k V : Assumption3.4. Theactivationfunction s anditsderivativeswithrespecttoitsarguments arebounded[46]. Assumption3.5. Thefunctionreconstructionerrorsanditsrstandsecondderivativesare bounded[46],as k e x k e 1 ; k e x ; x k e 2 ; k e x ; x ; x k e 3 : Sincetheinitialstate x 0 isassumedtobeboundedandthecontinuouscontroller u t is subsequentlydesignedtoguaranteethatthesystemstate x t isalwaysbounded,thefunction f x )]TJ/F25 11.9552 Tf 11.32 0 Td [(A s x canbedenedonacompactset;hencetheNNuniversalapproximationproperty holds.Withtheselectionoftheactivationfunctionasthesigmoidand/orhyperbolictangent functions,Assumption3.4issatised. 3.2RobustIdenticationusingDynamicNeuralNetworks Toidentifytheunknownnonlinearsystemin3,thefollowingMLDNNarchitectureis proposed x = A s x + W s V T x + gu + k x + b sgn x ; 31

PAGE 32

Figure3-1.ThearchitectureoftheMLDNN. where x t 2 R n isthestateoftheDNN, W t 2 R N n and V t 2 R n N aretheweightestimates, b 2 R isapositiveconstantcontrolgain,and x t 2 R n istheidenticationerrordenedas x x )]TJ/F22 11.9552 Tf 12.528 0 Td [( x : ThearchitectureoftheDNNisshowninFig.3-1. Considering d 0in3 ; [9]provedthatforsomeniteinitialconditionand u 2 U R m ; where U issomecompactset,thenforanite T > 0 ; thereexistsidealweights W ; V suchthat forall u 2 U theDNNstateandthestateoftheplantsatisfy max t 2 [ 0 ; T ] k x t )]TJ/F25 11.9552 Tf 10.95 0 Td [(x t k e x ; where e x 2 R isapositiveconstant.Acontributionofthischapteristheadditionofarobust slidingmodetermtotheclassicalDNNstructure[9,15,17],whichrobustlyaccountsforthe boundeddisturbance d t andtheNNfunctionreconstructionerror e x toguaranteeasymptotic convergenceoftheidenticationerrortozero,asseenfromthesubsequentstabilityanalysis. Theidenticationobjectiveistoprovethattheinput-outputbehavioroftheDNNapproximatestheinput-outputbehavioroftheplant.Quantitatively,theaimistoregulatethe 32

PAGE 33

identicationerrorin3.Theclosed-loopdynamicsoftheidenticationerrorin3are obtainedbyusing3and3as x = x )]TJ/F57 8.9664 Tf 12.379 10.389 Td [( x = A s x + W T s V T x )]TJ/F22 11.9552 Tf 14.375 2.379 Td [( W T s V T x + e + d )]TJ/F53 11.9552 Tf 10.95 0 Td [(b sgn x : Addingandsubtractingtheterm W T s V T x yields x = A s x + W T s V T x )]TJ/F25 11.9552 Tf 10.352 0 Td [(W T s V T x + W T s V T x + e + d )]TJ/F53 11.9552 Tf 10.949 0 Td [(b sgn x ; where W t W )]TJ/F22 11.9552 Tf 14.375 2.379 Td [( W t 2 R N n istheestimatemismatchfortheidealNNweight. Tofacilitatethesubsequentanalysis,theterm W T s V T x isaddedandsubtractedto3 ; where x t 2 R n isasamplestateselectedsuchthat x i t 2 L ; i = 0 ; 1 ; 2 ; where i t denotesthe i th derivativewithrespecttotime.BasedonthefactthattheTaylorseriesofthe vectorfunction s V T x intheneighborhoodof V T x is s V T x = s V T x + s 0 V T x V T x + O V T x 2 ; where s 0 V T x d s x = d x j x = V T x V t V )]TJ/F22 11.9552 Tf 13.316 2.379 Td [( V t 2 R n N and O V T x 2 isthehigher orderterm,3canberepresentedas x = A s x + W T s 1 + W T s 2 + W T s 0 V T x V T x + W T O V T x 2 + W T s V T x + e + d )]TJ/F53 11.9552 Tf 10.95 0 Td [(b sgn x ; wheretheterms s 1 and s 2 aredenedas s 1 s V T x )]TJ/F53 11.9552 Tf 11.185 0 Td [(s V T x s 2 s V T x )]TJ/F53 11.9552 Tf 11.185 0 Td [(s V T x Rearrangingthetermsin3yields x = A s x + W T s 0 V T x V T x + W T s V T x + h )]TJ/F53 11.9552 Tf 10.949 0 Td [(b sgn x ; where h x ; x ; x ; W ; V ; e ; d 2 R n canbeconsideredasadisturbancetermdenedas h W T s 1 + W T s 2 + W T O V T x 2 + e + d + W T s 0 V T x V T x : 33

PAGE 34

TheweightupdatelawsfortheDNNaredesignedusingthesubsequentstabilityanalysisas W = G 1 proj s V T x x T ; V = G 2 proj x x T W T s 0 V T x ; where G 1 2 R N N and G 2 2 R n n areconstantsymmetricpositive-deniteadaptationgains,and proj isasmoothprojectionoperator[88,89]usedtoguaranteethattheweightestimates W t and V t remainbounded. Remark 3.1 Thesamplestate x t isintroducedintheweightupdatelaws3tosatisfythe assumptionsrequiredforthesubsequentlydesignedRISE-basedcontroller3.TheRISE feedbacktermrequiresthatthedisturbancetermsbeboundedbyknownconstantsandtheir derivativesbeboundedbyeitheraconstantoralinearcombinationofthestates[75].These assumptionsaresatisedifthereisaboundedsignal,withboundedderivativeslike x t in3,ratherthan x t or x t whichcannotbeproventobeboundedpriortothestability analysis. UsingAssumptions3.2,3.3-3.5,theTaylorseriesexpansionin3,andthe proj algorithmin3,thedisturbanceterm h in3canbeboundedas 1 k h k h ; where h isaknownconstant. 3.3RobustTrajectoryTrackingusingRISEfeedback Thecontrolobjectiveistoforcethesystemstate x t toasymptoticallytrackadesiredtimevaryingtrajectory x d t 2 R n ,despiteuncertaintiesandexternaldisturbancesinthesystem.The desiredtrajectory x d t isassumedtobeboundedsuchthat x i d t 2 L ; i = 0 ; 1 ; 2 : Toquantify 1 SeeAppendixA.1fordetail 34

PAGE 35

thetrackingobjective,thetrackingerror e t 2 R n isdenedas e x )]TJ/F25 11.9552 Tf 10.949 0 Td [(x d : Thelteredtrackingerror r t 2 R n for3isdenedas r = e + a e ; where a 2 R denotesapositiveconstant.Since r t containsaccelerationterms,itisunmeasurable.Substitutingthesystemdynamicsfrom3andusing3and3,thefollowing expressionisobtained r = A s + a I e + W T s V T x + e + d + gu )]TJ/F22 11.9552 Tf 12.528 0 Td [( x d + A s x d ; where I 2 R n n isanidentitymatrix.Thecontrolinput u t isnowdesignedasacompositionof theDNNtermandtheRISEfeedbacktermas u = g + x d )]TJ/F25 11.9552 Tf 10.949 0 Td [(A s x d )]TJ/F22 11.9552 Tf 14.374 2.379 Td [( W T s V T x d )]TJ/F53 11.9552 Tf 11.249 0 Td [(m ; where g x + istherightMoore-Penrosepseudoinverseofthematrix g x ,and m t 2 R n isthe RISEtermdenedasthegeneralizedsolutionto[90] m k tr + k m r + b 1 sgn e ; where k tr k m b 1 2 R areconstantpositivecontrolgainsand sgn denotesthesignumfunction denedas sgn e [ sgn e 1 sgn e 2 ... sgn e n ] T : Remark 3.2 Sincetheinputmatrix g x isassumedtobeknown,boundedandfull-rowrank Assumption3.1,therightpseudoinverse g x + iscalculatedas g x + = g x T g x g x T )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 andsatises g x g x + = I ; where I istheidentitymatrix. Thecontrollerin3andtheDNNidentierdevelopedin3operatesimultaneously inreal-time.Ablockdiagramoftheidentier-controllersystemisshowninFig.3-2. 35

PAGE 36

Figure3-2.Robustidentication-basedtrajectorytrackingcontrol. Substitutingthecontrol3into3,theclosed-loopsystembecomes r = A s + a I e + W T s V T x )]TJ/F22 11.9552 Tf 14.375 2.379 Td [( W T s V T x d + e + d )]TJ/F53 11.9552 Tf 11.249 0 Td [(m : Tofacilitatethesubsequentstabilityanalysis,thetimederivativeof3iscalculatedas r = A s + a I e + W T s 0 V T x V T x )]TJ/F57 8.9664 Tf 18.434 13.147 Td [( W T s V T x d )]TJ/F22 11.9552 Tf 14.374 2.379 Td [( W T s 0 V T x d V T x d )]TJ/F22 11.9552 Tf 14.374 2.379 Td [( W T s 0 V T x d V T x d + e + d )]TJ/F56 11.9552 Tf 10.949 0 Td [( k tr + k m r )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 1 sgn e : Rearrangingthetermsin3yields r = N + N )]TJ/F25 11.9552 Tf 10.949 0 Td [(e )]TJ/F56 11.9552 Tf 10.949 0 Td [( k tr + k m r )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 sgn e ; wheretheauxiliaryfunction N e ; r ; W ; V ; t 2 R n isdenedas N = A s + a I r )]TJ/F53 11.9552 Tf 10.95 0 Td [(a e + W T s 0 V T x V T r )]TJ/F53 11.9552 Tf 10.949 0 Td [(a e )]TJ/F57 8.9664 Tf 18.435 13.148 Td [( W T s V T x d )]TJ/F22 11.9552 Tf 14.374 2.379 Td [( W T s 0 V T x d V T x d + e ; and N x ; W ; V ; t 2 R n issegregatedintotwopartsas N = N D + N B ; 36

PAGE 37

where N D t 2 R n isdenedas N D = d + e ; and N B x ; W ; V ; t 2 R n isdenedas N B = W T s 0 V T x V T x d )]TJ/F22 11.9552 Tf 14.374 2.379 Td [( W T s 0 V T x d V T x d : Thefunction N in3canbeupperboundedas 2 N z 1 k z k ; where z x ; e ; r 2 R 3 n isdenedas z [ x T e T r T ] T ; andtheboundingfunction r 2 R isapositive,globallyinvertible,non-decreasingfunction. BasedonAssumptions3.2,3.3-3.5,and3,thefollowingboundscanbedeveloped 3 k N D k z 2 k N B k z 3 k N k z 2 + z 3 : Further,theboundsforthetimederivativesof N D and N B aredevelopedas N D z 4 ; N B z 5 + z 6 k z k ; where z i 2 R i = 1 ; 2 ;:; 6 arecomputablepositiveconstants.Tofacilitatethesubsequent stabilityanalysis, y z ; P ; Q 2 R 3 n + 2 isdenedas y [ z T p P p Q ] T : 2 SeeAppendixA.2forproof 3 SeeAppendixA.3fordetail 37

PAGE 38

In3,theauxiliaryfunction P t 2 R isdenedas P b 1 n j = 1 e j 0 )]TJ/F25 11.9552 Tf 10.949 0 Td [(e T 0 N 0 )]TJ/F25 11.9552 Tf 10.949 0 Td [(L + L 0 ; wherethesubscript j = 1 ; 2 ;::; n denotesthe j th elementof e 0 ,andtheauxiliaryfunction L z ; N 2 R isgeneratedas L r T N )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 sgn e )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 2 k z k 2 ; where b 1 ; b 2 2 R arepositiveconstantschosenaccordingtothesufcientconditions b 1 > z 2 + z 3 + z 4 a + z 5 a b 2 > z 6 : Thederivative P t 2 R canbeexpressedas P = )]TJ/F22 11.9552 Tf 10.64 2.379 Td [( L = )]TJ/F25 11.9552 Tf 9.289 0 Td [(r T N )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 1 sgn e + b 2 k z k 2 : Providedthesufcientconditionsin3aresatised,thefollowinginequalitycanbeobtained L b 1 n j = 1 e j 0 )]TJ/F25 11.9552 Tf 10.95 0 Td [(e T 0 N 0 + L 0 ; whichcanbeusedtoconcludethat P t 0 4 .Theauxiliaryfunction Q W ; V 2 R in3is denedas Q 1 2 tr W T G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 1 W + 1 2 tr V T G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 2 V : Since G 1 and G 2 areconstant,symmetric,andpositivedenitematrix, Q 0. 3.4LyapunovStabilityAnalysisforDNN-basedIdenticationandControl Theorem3.1. TheDNN-basedidentierandcontrollerproposedin3and3,respectively,andtheweightupdatelawsfortheDNNdesignedin3ensurethatallsystemsignals 4 SeeAppendixA.4forproof 38

PAGE 39

areboundedandthattheidenticationandtrackingerrorsareregulatedinthesensethat k x t k 0 ; k e t k 0 ast ; providedthecontrolgainsk tr ,k m introducedin3areselectedsufcientlylarge,thegain conditionsin3aresatised,andthefollowingsufcientgainconditionsaresatised b > h l > b 2 + z 2 1 4 k tr ; where b h, z 1 b 2 ,and l areintroducedin3,3,3,3,and3,respectively. Proof. ConsidertheLyapunovcandidatefunction V L y ; t : R 3 n + 2 [ 0 ; R ,whichisa positivedenitefunctiondenedas V L 1 2 x T x + 1 2 r T r + 1 2 e T e + P + Q ; andsatisesthefollowinginequalities: U 1 y V L y ; t U 2 y ; wherethecontinuouspositivedenitefunctions U 1 y U 2 y 2 R aredenedas U 1 y 1 2 k y k 2 ; U 2 y k y k 2 : Let y = h y ; t representtheclosed-loopdifferentialequationsin3,3,3,3 31,where h y ; t 2 R 3 n + 2 denotestheright-handsideoftheclosed-looperrorsignals. UsingFilippov'stheoryofdifferentialinclusion[9194],theexistenceofsolutionscanbe establishedfor y 2 K [ h ] y ; t ,where K [ h ] d > 0 m M = 0 coh B y ; d )]TJ/F25 11.9552 Tf 11.431 0 Td [(M ; t ; where m M = 0 denotestheintersectionofallsets M ofLebesguemeasurezero, co denotesconvexclosure,and B y ; d = w 2 R 4 n + 2 j k y )]TJ/F25 11.9552 Tf 10.95 0 Td [(w k < d : Therighthandsideofthedifferentialequation, h y ; t ; iscontinuousexceptfortheLebesguemeasurezerosetoftimes t 2 t 0 ; t f when e t = 0or x t = 0.Hence,thesetoftimeinstancesforwhich y t isnotdenedisLebesguenegligible. 39

PAGE 40

Theabsolutelycontinuoussolution y t = y t 0 + t t 0 y t dt doesnotdependonthevalueof y t onaLebesguenegligiblesetoftime-instances[95].UnderFilippov'sframework,ageneralized Lyapunovstabilitytheorycanbeusedsee[94,9698]forfurtherdetailstoestablishstrong stabilityoftheclosed-loopsystem y = h y ; t .Thegeneralizedtimederivativeof3exists almosteverywherea.e.,i.e.foralmostall t 2 t 0 ; t f ,and V L y 2 a : e : V L y where V L = x 2 V L y x T K x T e T r T 1 2 P )]TJ/F22 6.9738 Tf 8.162 3.533 Td [(1 2 P 1 2 Q )]TJ/F22 6.9738 Tf 8.163 3.533 Td [(1 2 Q T ; where V L isthegeneralizedgradientof V L y [96].Since V L y islocallyLipschitzcontinuous regularandsmoothin y ,3canbesimpliedas[97] V L = V T L K x T e T r T 1 2 P )]TJ/F22 6.9738 Tf 8.162 3.533 Td [(1 2 P 1 2 Q )]TJ/F22 6.9738 Tf 8.162 3.533 Td [(1 2 Q T = h x T e T r T 2 P 1 2 2 Q 1 2 i K x T e T r T 1 2 P )]TJ/F22 6.9738 Tf 8.162 3.532 Td [(1 2 P 1 2 Q )]TJ/F22 6.9738 Tf 8.162 3.532 Td [(1 2 Q T : Usingthecalculusfor K [ ] from[98]Theorem1,Properties2 ; 5 ; 7,andsubstitutingthe dynamicsfrom3,3,3,3and3, V L y canberewrittenas V L x T A s x + W T s 0 V T x V T x + W T s V T x + h )]TJ/F53 11.9552 Tf 10.949 0 Td [(b sgn x + r T N + N )]TJ/F25 11.9552 Tf 10.95 0 Td [(e )]TJ/F56 11.9552 Tf 10.95 0 Td [( k tr + k m r )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 sgn e + e T r )]TJ/F53 11.9552 Tf 10.949 0 Td [(a e )]TJ/F25 11.9552 Tf 10.95 0 Td [(r T N )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 1 sgn e + b 2 k z k 2 )]TJ/F25 11.9552 Tf 10.65 0 Td [(tr W T G )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 1 W )]TJ/F25 11.9552 Tf 10.65 0 Td [(tr V T G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 2 V : Usingthefactthat K [ sgn e ]= SGN e [98],suchthat SGN e i = 1if e i > 0 ; [ )]TJ/F22 11.9552 Tf 9.289 0 Td [(1 ; 1 ] if e i = 0 ; and )]TJ/F22 11.9552 Tf 9.289 0 Td [(1if e i < 0 ; thesubscript i denotesthe i th element,andsimilarly K [ sgn x ]= SGN x ,the setin3reducestothescalarinequality,sincetheRHSiscontinuousa.e.,i.e.,theRHSis continuousexceptfortheLebesguemeasurezerosetoftimeswhen e i t = 0or x i t = 0forany i = 1 ; 2 ;:::; n .Substitutingtheweightupdatelawsin3andcancelingcommonterms,the aboveexpressionissimpliedas V L a : e : x T A s x + x T h )]TJ/F53 11.9552 Tf 10.949 0 Td [(b x T sgn x + r T N )]TJ/F25 11.9552 Tf 10.949 0 Td [(r T k tr + k m r )]TJ/F53 11.9552 Tf 10.949 0 Td [(a e T e + b 2 k z k 2 : 40

PAGE 41

Takingtheupperboundof3,thefollowingexpressionisobtained V L a : e : )]TJ/F25 11.9552 Tf 23.239 0 Td [(k m k r k 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(a k e k 2 + l min f A s g k x k 2 )]TJ/F25 11.9552 Tf 10.949 0 Td [(k tr k r k 2 + h k x k )]TJ/F53 11.9552 Tf 10.95 0 Td [(b n j = 1 x j + N k r k + b 2 k z k 2 ; where l min fg istheminimumeigenvalueofamatrix.Now,usingthefactthat n j = 1 x j k x k ; and3, V L canbefurtherupperboundedas V L a : e : )]TJ/F25 11.9552 Tf 23.239 0 Td [(k m k r k 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(a k e k 2 + l min f A s g k x k 2 )]TJ/F66 11.9552 Tf 10.949 13.276 Td [(h k tr k r k 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(z 1 k z kk r k i + b 2 k z k 2 )]TJ/F66 11.9552 Tf 10.95 9.689 Td [()]TJ/F53 11.9552 Tf 5.475 -9.689 Td [(b )]TJ/F22 11.9552 Tf 11.948 2.851 Td [( h k x k : Choosing b tosatisfytheconditionin3,andcompletingthesquaresonthebracketedterms, theexpressionin3canbefurtherupperboundas V L a : e : )]TJ/F66 11.9552 Tf 24.567 16.863 Td [( l )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 2 )]TJ/F53 11.9552 Tf 15.126 8.233 Td [(z 2 1 4 k tr k z k 2 ; where l min f k m ; a ; )]TJ/F53 11.9552 Tf 9.289 0 Td [(l min f A s gg : Basedon3,wecanstatethat V L a : e : )]TJ/F25 11.9552 Tf 22.641 0 Td [(U y where U y = c k z k 2 ,forsomepositiveconstant c 2 R ,isacontinuouspositivesemi-denite function.From3and3, V L y ; t 2 L ;hence, x t ; e t ; r t ; P t ; and Q t 2 L ; since e t ; r t 2 L ; using3, e t 2 L .Moreover,since x d t ; x d t 2 L byassumption, and e t ; e t 2 L ,so x t ; x t 2 L byusing3.Since x t ; x t ; f x ; d t 2 L ,from 3, u t 2 L .Thefactthat u t 2 L and W t ; s 2 L bythe proj algorithm, indicates m t 2 L by3.Similarly,sinceboth x t ; x t 2 L so x t 2 L byusing 3;moreover,byusing3, x t 2 L ;hence, x t 2 L from3 : Using d t ; e t 2 L byAssumptions3.2,3.5, W t ; V t 2 L byusingtheupdatelaws3, W V 2 L byAssumption3.3,andtheboundednessofthefunction s and sgn ,wecanprovethat 41

PAGE 42

r t 2 L from3;then z t =[ x T e T r T ] T 2 L .Hence, U y isuniformlycontinuous.It canbeconcludedthat c k z k 2 0as t : Basedonthedenitionof z t ,boththeidenticationerror x t 0andthetrackingerror e t 0as t 3.5Simulation Thefollowingdynamicsofatwolinkrobotmanipulatorareconsideredforthesimulations: M q q + V m q ; q q + F d q + t d t = u t ; where q =[ q 1 q 2 ] T aretheangularpositions rad and q = q 1 q 2 T aretheangularvelocities rad = s ofthetwolinksrespectively. M q istheinertiamatrixand V m q ; q isthecentripetalCoriolismatrix,denedas M 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 2 6 4 )]TJ/F25 11.9552 Tf 10.185 0 Td [(p 3 s 2 q 2 )]TJ/F25 11.9552 Tf 10.185 0 Td [(p 3 s 2 q 1 + q 2 p 3 s 2 q 1 0 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 ; c 2 = cos x 2 s 2 = sin x 2 ; F d = diag f 5 : 3 ; 1 : 1 g Nm sec arethemodelsfordynamicandstaticfriction,respectively,and t d istheexternaldisturbance.Thematrix M q isassumedtobeknown,andothermatrices V m q ; q ; F d areunknown. Thesystem3isrepresentedtotheformoftheconsideredsystems3as x = A s x + Bf x )]TJ/F25 11.9552 Tf 10.949 0 Td [(BM x )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 t d + BM x )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 u ; wherethenewmeasurablestatevector x 2 R 4 denedas x [ q T q T ] T ,aconstantmatrix B [ 0 2 2 I 2 2 ] T 2 R 4 2 with I n n ; 0 n n arethe n n dimensionalidentitymatrixandzeromatrix,the 42

PAGE 43

unknownvectorfunction f x 2 R 2 isdenedas f x )]TJ/F25 11.9552 Tf 9.289 0 Td [(A 1 q )]TJ/F25 11.9552 Tf 10.694 0 Td [(A 2 q )]TJ/F25 11.9552 Tf 10.694 0 Td [(M q )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 f V m q ; q q + F d q g with A 1 ; A 2 2 R 2 2 areknownconstantmatricessuchthatthematrix A s 2 R 4 4 denedas A s 2 6 4 0 2 2 I 2 2 A 1 A 2 3 7 5 isHurwitz.TheproposedDNNidentierisintheformas x = A s x + B W s V T x + BM x )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 u + b sgn x : Theobjectiveoftwolinksistotrackdesiredtrajectoriesgivenas q 1 d = 0 : 52sin 2 t 1 )]TJ/F22 11.9552 Tf 10.949 0 Td [(exp )]TJ/F22 11.9552 Tf 9.289 0 Td [(0 : 01 t 3 rad, q 2 d = q 1 d rad. Toquantifythetrackingobjective,thetrackingerror e 1 t 2 R 2 isdenedas e 1 q )]TJ/F25 11.9552 Tf 11.058 0 Td [(q d ,where q d t [ q 1 d q 2 d ] T ,andlteredtrackingerrors,denotedby e 2 t ; r t 2 R 2 arealsodened as e 2 e 1 + a e 1 ; r e 2 + a e 2 .Therelationshipbetween r t x t ; and x d t q T d q T d T is r tr = L f x )]TJ/F22 11.9552 Tf 12.528 0 Td [( x d + a x )]TJ/F25 11.9552 Tf 10.949 0 Td [(x d g ; where L =[ a I n n I n n ] .Thecontroller u t 2 R 2 isdesignedas u M x L x d )]TJ/F25 11.9552 Tf 10.949 0 Td [(A s x d )]TJ/F22 11.9552 Tf 14.375 2.379 Td [( W T s V T x d )]TJ/F53 11.9552 Tf 11.248 0 Td [(m with m t 2 R 2 istheRISEtermdenedasthegeneralizedsolutionto m t kr + b 1 sgn e 2 : Thecontrolgainsarechosenas k = diag [ 1015 ] a = diag [ 1035255 ] b 1 = 25 ; b = diag [ 113035 ] ,and G w = I 15 15 ; G v = I 2 2 ,where I n n denotesanidentitymatrixof appropriatedimensions.TheNNsaredesignedtohave15hiddenlayerneuronsandtheNN weightsareinitializedasuniformlydistributedrandomnumbersintheinterval [ )]TJ/F22 11.9552 Tf 9.289 0 Td [(0 : 1 ; 0 : 1 ] .The initialconditionsofthesystemandtheidentierarechosenas q 0 =[ )]TJ/F22 11.9552 Tf 9.289 0 Td [(0 : 30 : 2 ] T q 0 =[ 00 ] T ; and x 0 =[ 0000 ] T ,respectively. Figures3-3and3-4showthetrackingerrorsandstateidenticationerrorsforlink1and link2duringa8ssimulationperiodrespectively.Bothtrackingandidenticationerrorshave goodtransientresponsesandconvergetozeroquickly.ThecontrolinputisshowninFig.3-5,the 43

PAGE 44

Figure3-3.Link1andlink2trackingerrors. Figure3-4.Link1andLink2positionidenticationerrors. 44

PAGE 45

Figure3-5.Controlinputsofthelink1andlink2. Figure3-6.Link1andLink2feedforwardcomponentofthecontrolinput. 45

PAGE 46

controlinputisacontinuoussignal.Fig.3-6showstheNNfeedforwardpartinthecontrolinput. Bothcontrolinput u t andtheNNfeedforwardpartarebounded. 3.6Conclusion ADNN-basedrobustidenticationandcontrolmethodforafamilyofcontrol-afne nonlinearsystemsisproposed.ThenoveluseoftheslidingmodetermintheDNNstructure guaranteesasymptoticconvergenceoftheDNNstatetothestateoftheplant.Thecontrolleris comprisedofaDNNidentiertermtoaccountforuncertainnonlinearitiesinthesystemand acontinuousRISEfeedbacktermtoaccountforexternaldisturbances.Asymptotictrajectory trackingisachieved,unlikepreviousresultsinliteraturewhereonlyboundedstabilityisobtained duetoDNNreconstructionerrors. 46

PAGE 47

CHAPTER4 DYNAMICNEURALNETWORK-BASEDROBUSTOBSERVERSFORUNCERTAIN NONLINEARSYSTEMS ADNN-basedrobustobserverforuncertainnonlinearsystemsisdevelopedinthischapter. TheobserverstructureconsistsofaDNNtoestimatethesystemdynamicson-line,adynamic ltertoestimatetheunmeasurablestateandaslidingmodefeedbacktermtoaccountfor modelingerrorsandexogenousdisturbances.Theobservedstatesareproventoasymptotically convergetothesystemstatesofsecond-ordersystemsthoughaLyapunov-basedanalysis. Similarresultsareextendedtohigher-ordersystems.Simulationsandexperimentsonatwo-link robotmanipulatorareperformedtoshowtheeffectivenessoftheproposedmethodincomparison toseveralotherstateestimationmethods. 4.1DynamicSystemandProperties ConsiderasecondordercontrolafnenonlinearsystemgiveninMIMOBrunovskyformas x 1 = x 2 ; x 2 = f x + G x u + d ; y = x 1 ; where y t 2 R n isthemeasurableoutputwithaniteinitialcondition y 0 = y 0 u t 2 R m isthe controlinput, x t =[ x 1 t T x 2 t T ] T 2 R 2 n isthestateofthesystem, f x : R 2 n R n ; G x : R 2 n R n m areunknowncontinuousfunctions,and d t 2 R n isanexternaldisturbance.The followingassumptionsaboutthesystemin4willbeutilizedintheobserverdevelopment. Assumption4.1. Thestatex t isbounded,i.e,x 1 t ; x 2 t 2 L ,andispartiallymeasurable, i.e,onlyx 1 t ismeasurable. Assumption4.2. Theunknownfunctionsf x ; G x andthecontrolinputu t areC 1 ,and u t ; u t 2 L : Assumption4.3. Thedisturbanced t isdifferentiable,andd t ; d t 2 L : 47

PAGE 48

BasedoftheuniversalapproximationpropertyofMLNNs,theunknownfunctions f x ; G x in4canbereplacedbyMLNNsas f x = W T f s f V T f 1 x 1 + V T f 2 x 2 + e f x ; g i x = W T gi s gi V T gi 1 x 1 + V T gi 2 x 2 + e gi x ; where W f 2 R L f + 1 n V f 1 ; V f 2 2 R n L f areunknownidealweightmatricesoftheMLNN having L f hiddenlayerneurons, g i x isthe i th columnofthematrix G x ; W gi 2 R L gi + 1 n ; V gi 1 ; V gi 2 2 R n L gi arealsounknownidealweightmatricesoftheMLNNhaving L gi hidden layerneurons ; i = 1 ::: m s f t 2 R L f + 1 and s gi t 2 R L gi + 1 denedas s f s f V T f 1 x 1 + V T f 2 x 2 ; s gi s gi V T gi 1 x 1 + V T gi 2 x 2 aretheactivationfunctionssigmoid,hyperbolictangent,etc.,and e f x ; e gi x 2 R n ; i = 1 ::: m arethefunctionreconstructionerrors.Using4andAssumption 4.2,thesystemin4canberepresentedas x 1 = x 2 ; x 2 = W T f s f + e f + d + m i = 1 W T gi s gi + e gi u i ; where u i t 2 R isthe i th elementofthecontrolinputvector u t .Thefollowingassumptionswill beusedintheobserverdevelopmentandstabilityanalysis. Assumption4.4. TheidealNNweightsareboundedbyknownpositiveconstants[46],i.e. W f W f ; V f 1 V f 1 ; V f 2 V f 2 W gi W gi ; V gi 1 V gi 1 ,and V gi 2 V gi 2 ,i = 1 ::: m ; where k k denotesFrobeniusnormforamatrixandEuclideannormforavector. Assumption4.5. Theactivationfunctions s f ; s gi andtheirderivativeswithrespecttoits arguments, s 0 f ; s 0 gi ; s 00 f ; s 00 gi ; i = 1 ::: m ; arebounded[46]. Assumption4.6. Thefunctionreconstructionerrors e f ; e gi ; anditsrstderivativeswith respecttotheirargumentsarebounded,withi = 1 ::: m[46]. 48

PAGE 49

4.2EstimationObjective Theestimationobjectiveistoprovethattheestimatedstate x t convergestothesystem state x t .Tofacilitatethesubsequentanalysis,theestimationerror x t 2 R n isdenedas x x 1 )]TJ/F22 11.9552 Tf 12.528 0 Td [( x 1 : Tocompensateforthelackofdirectmeasurementsof x 2 t ,anauxiliaryestimationerroris denedas r x + a x + h ; where a 2 R isapositiveconstantcontrolgain,and h t 2 R n isanoutputofthedynamic lter[1] h = p )]TJ/F56 11.9552 Tf 10.95 0 Td [( k + a x ; p = )]TJ/F56 11.9552 Tf 9.289 0 Td [( k + 2 a p )]TJ/F22 11.9552 Tf 12.527 0 Td [( x f + k + a 2 + 1 x ; x f = p )]TJ/F53 11.9552 Tf 10.95 0 Td [(a x f )]TJ/F56 11.9552 Tf 10.949 0 Td [( k + a x ; p 0 = k + a x 0 ; x f 0 = 0 ; where x f t 2 R n isanauxiliaryoutputofthelter, p t 2 R n isusedasaninternalltervariable, and k 2 R isapositiveconstantgain.Theestimationerror r t isnotmeasurable,sincethe expressionin4dependson x t : Thesecondorderdynamicltertoestimatethesystem velocitywasrstproposedfortheOFBcontrolin[1].Thelterin4-4admitsthe estimationerror x t asitsinputandproducestwosignaloutputs x f t and h t .Theauxiliary signal p t isutilizedtoonlygeneratethesignal h t withoutinvolvingthederivativeofthe estimationerror x t whichisunmeasurable.Hence,theltercanbephysicallyimplemented.A difcultytoobtainasymptoticestimationisthatthelteredestimationerror r t isnotavailable forfeedback.Therelationbetweentwolteroutputsis h = x f + a x f ; andthisrelationship isutilizedtogeneratethefeedbackof r t .Sincetakingtimederivativeof r t ; theterm x f t 49

PAGE 50

appearsimplicitlyinside h t ; andconsequently,theunmeasurableterm x t whichcanbe replacedby r t isintroduced : 4.3RobustObserverusingDynamicNeuralNetworks ThefollowingMLDNNarchitectureisproposedtoobservethesystemin4 x 1 = x 2 ; x 2 = W T f s f + m i = 1 W T gi s gi u i + v ; where x t =[ x 1 t T x 2 t T ] T 2 R 2 n isthestateoftheDNNobserver, W f t 2 R L f + 1 n V f 1 t ; V f 2 t 2 R n L f ; W gi t 2 R L gi + 1 n ; V gi 1 t ; V gi 2 t 2 R n L gi ; i = 1 ::: m ; arethe weightestimates, s f t 2 R L f + 1 ; and s gi t 2 R L gi + 1 denedas s f s f V T f 1 x 1 + V T f 2 x 2 ; s gi s gi V T gi 1 x 1 + V T gi 2 x 2 ; and v t 2 R n isafunctiontobedeterminedtoproviderobustness toaccountforthefunctionreconstructionerrorsandexternaldisturbances.In4,thefeedforwardNNterms W f t T s f t ; W gi t T s gi t useinternalfeedbackoftheobserverstates x t hencethisobserverhasaDNNstructure.TheDNNhasarecurrentfeedbackloop,andisproven tobeabletoapproximatedynamicsystemswithanyarbitrarydegreeofaccuracy[8],[9].This propertymotivatestheDNN-basedobserverdesign.TheDNNisautomaticallytrainedtoestimatesystemdynamicsbytheweightupdatelawsbasedonthestate,weightestimates,andthe lteroutput. Takingthederivativeof4andusingthedenitions4-4yields h = )]TJ/F56 11.9552 Tf 9.289 0 Td [( k + a r )]TJ/F53 11.9552 Tf 10.949 0 Td [(ah + x )]TJ/F22 11.9552 Tf 12.528 0 Td [( x f : Theclosed-loopdynamicsofthederivativeofthelteredestimationerrorin4isdetermined from4-4and4as r = W T f s f )]TJ/F22 11.9552 Tf 14.374 2.379 Td [( W T f s f + m i = 1 [ W T gi s gi )]TJ/F22 11.9552 Tf 14.375 2.379 Td [( W T gi s gi ] u i + e f + m i = 1 e gi u i + d )]TJ/F25 11.9552 Tf 10.949 0 Td [(v + a r )]TJ/F53 11.9552 Tf 10.95 0 Td [(a x )]TJ/F53 11.9552 Tf 10.949 0 Td [(h )]TJ/F56 11.9552 Tf 10.949 0 Td [( k + a r )]TJ/F53 11.9552 Tf 10.949 0 Td [(ah + x )]TJ/F22 11.9552 Tf 12.528 0 Td [( x f : 50

PAGE 51

Therobustdisturbancerejectionterm v t isdesignedbasedonthesubsequentanalysisas v = )]TJ/F56 11.9552 Tf 9.289 0 Td [([ g k + a + 2 a ] h + g )]TJ/F53 11.9552 Tf 10.95 0 Td [(a 2 x + b 1 sgn x + x f ; where g ; b 1 2 R arepositiveconstantcontrolgains.Addingandsubtracting W T f s f V T f 1 x 1 + V T f 2 x 2 + W T f s f V T f 1 x 1 + V T f 2 x 2 + m i = 1 [ W T gi s gi V T gi 1 x 1 + V T gi 2 x 2 + W T gi s gi V T gi 1 x 1 + V T gi 2 x 2 ] u i and substituting v t from4,theexpressionin4canberewrittenas r = N + N )]TJ/F25 11.9552 Tf 10.949 0 Td [(kr )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 1 sgn x + x f + g k + a h )]TJ/F53 11.9552 Tf 10.949 0 Td [(g x ; wheretheauxiliaryfunction N x 1 ; x 2 ; x 1 ; x 2 ; x f ; W f ; V f 1 ; V f 2 ; W gi ; V gi 1 ; V gi 2 ; t 2 R n isdenedas N W T f [ s f V T f 1 x 1 + V T f 2 x 2 )]TJ/F22 11.9552 Tf 13.508 0.52 Td [( s f ]+ x )]TJ/F22 11.9552 Tf 12.527 0 Td [( x f + m i = 1 W T gi [ s gi V T gi 1 x 1 + V T gi 2 x 2 )]TJ/F22 11.9552 Tf 13.508 0.52 Td [( s gi ] u i ; and N x 1 ; x 2 ; W f ; V f 1 ; V f 2 ; W gi ; V gi 1 ; V gi 2 ; t 2 R n issegregatedintotwopartsas N N 1 + N 2 : In4, N 1 x 1 ; x 2 ; W f ; V f 1 ; V f 2 ; W gi ; V gi 1 ; V gi 2 ; t ; N 2 x 1 ; x 2 ; W f ; V f 1 ; V f 2 ; W gi ; V gi 1 ; V gi 2 ; t 2 R n are denedas N 1 W T f s 0 f [ V T f 1 x 1 + V T f 2 x 2 ]+ W T f O V T f 1 x 1 + V T f 2 x 2 2 + e f + d + m i = 1 e gi u i + m i = 1 W T gi s 0 gi [ V T gi 1 x 1 + V T gi 2 x 2 ] u i + m i = 1 W T gi O V T gi 1 x 1 + V T gi 2 x 2 2 u i ; N 2 W T f s f V T f 1 x 1 + V T f 2 x 2 + W T f s 0 f [ V T f 1 x 1 + V T f 2 x 2 ] + m i = 1 W T gi s gi V T gi 1 x 1 + V T gi 2 x 2 u i + m i = 1 W T gi s 0 gi [ V T gi 1 x 1 + V T gi 2 x 2 ] u i ; where W f t W f )]TJ/F22 11.9552 Tf 14.613 2.379 Td [( W f t 2 R L f + 1 n ; V f 1 t V f 1 )]TJ/F22 11.9552 Tf 13.311 2.379 Td [( V f 1 t 2 R n L f ; V f 2 t V f 2 )]TJ/F22 11.9552 Tf 13.31 2.379 Td [( V f 2 t 2 R n L f ; W gi t W gi )]TJ/F22 11.9552 Tf 13.916 2.379 Td [( W gi t 2 R L gi + 1 n ; V gi 1 t V gi 1 )]TJ/F22 11.9552 Tf 12.613 2.379 Td [( V gi 1 t 2 R n L gi ; V gi 2 t V gi 2 )]TJ/F22 11.9552 Tf 12.613 2.379 Td [( V gi 2 t 2 R n L gi ; i = 1 ::: m ,aretheestimatemismatchesfortheidealNNweights; O V T f 1 x 1 + V T f 2 x 2 2 t 2 R L f + 1 ; O V T gi 1 x 1 + V T gi 2 x 2 2 t 2 R L gi + 1 arethehigherordertermsintheTaylorseriesofthevector 51

PAGE 52

functions s f ; s gi intheneighborhoodof V T f 1 x 1 + V T f 2 x 2 and V T gi 1 x 1 + V T gi 2 x 2 ,respectively,as s f = s f V T f 1 x 1 + V T f 2 x 2 + s 0 f [ V T f 1 x 1 + V T f 2 x 2 ]+ O V T f 1 x 1 + V T f 2 x 2 2 ; s gi = s gi V T gi 1 x 1 + V T gi 2 x 2 + s 0 gi [ V T gi 1 x 1 + V T gi 2 x 2 ]+ O V T gi 1 x 1 + V T gi 2 x 2 2 ; wheretheterms s 0 f t ; s 0 gi t aredenedas s 0 f s 0 f V T f 1 x 1 + V T f 2 x 2 = d s f q = d q j q = V T f 1 x 1 + V T f 2 x 2 and s 0 gi s 0 gi V T gi 1 x 1 + V T gi 2 x 2 = d s gi q = d q j q = V T gi 1 x 1 + V T gi 2 x 2 : Tofacilitatethesubsequentanalysis, anauxiliaryfunction N 2 x 1 ; x 2 ; W f ; V f 1 ; V f 2 ; W gi ; V gi 1 ; V gi 2 ; t 2 R n isdenedbyreplacingterms x 1 t ; x 2 t in N 2 by x 1 t ; x 2 t ,respectively. TheweightupdatelawsfortheDNNin4aredevelopedbasedonthesubsequent stabilityanalysisas W f = proj [ G wf s f x + x f T ] ; V f 1 = proj [ G vf 1 x 1 x + x f T W T f s 0 f ] ; V f 2 = proj [ G vf 2 x 2 x + x f T W T f s 0 f ] ; W gi = proj [ G wgi s gi u i x + x f T ] ; i = 1 ::: m V gi 1 = proj [ G vgi 1 x 1 u i x + x f T W T gi s 0 gi ] ; i = 1 ::: m V gi 2 = proj [ G vgi 2 x 2 u i x + x f T W T gi s 0 gi ] ; i = 1 ::: m where G wf 2 R L f + 1 L f + 1 ; G wgi 2 R L gi + 1 L gi + 1 ; G vf 1 ; G vf 2 ; G vgi 1 ; G vgi 2 2 R n n ; areconstantsymmetricpositive-deniteadaptationgains,theterms s 0 f t ; s 0 gi t aredenedas s 0 f s 0 f V T f 1 x 1 + V T f 2 x 2 = d s f q = d q j q = V T f 1 x 1 + V T f 2 x 2 s 0 gi s 0 gi V T gi 1 x 1 + V T gi 2 x 2 = d s gi q = d q j q = V T gi 1 x 1 + V T gi 2 x 2 ; and proj isasmoothprojectionoperator[88],[89]usedtoguaranteethattheweightestimates W f t ; V f 1 t ; V f 2 t ; W gi t ; V gi 1 t ; and V gi 2 t remainbounded. Using4-4,Assumptions4.1-4.2and4.4-4.5,the proj algorithmin4andthe MeanValueTheorem,theauxiliaryfunction N in4canbeupper-boundedas N z 1 k z k ; 52

PAGE 53

where z x ; x f ; h ; r 2 R 4 n isdenedas z [ x T x T f h T r T ] T : Basedon4-4,Assumptions4.1-4.6,theTaylorseriesexpansionin4andtheweight updatelawsin4,thefollowingboundscanbedeveloped k N 1 k z 2 ; k N 2 k z 3 ; N z 4 + r k z k k z k ; N 2 z 5 k z k ; where z i 2 R ; i = 1 ::: 5 ; arecomputablepositiveconstants, r 2 R isapositive,globally invertible,non-decreasingfunction,and N 2 x ; x ; W f ; V f 1 ; V f 2 ; W gi ; V gi 1 ; V gi 2 ; u N 2 )]TJ/F22 11.9552 Tf 13.604 2.379 Td [( N 2 Tofacilitatethesubsequentstabilityanalysis,let D R 4 n + 2 beadomaincontaining y z ; P ; Q = 0,where y z ; P ; Q 2 R 4 n + 2 isdenedas y [ z T p P p Q ] T : In4,theauxiliaryfunction P t 2 R istheFilippovsolutiontothedifferentialequation P )]TJ/F25 11.9552 Tf 9.289 0 Td [(L ; P 0 b 1 n j = 1 x j 0 + x f j 0 )]TJ/F56 11.9552 Tf 10.949 0 Td [( x 0 + x f 0 T N 0 ; wherethesubscript j = 1 ; 2 ;::; n denotesthe j th elementof x 0 or x f 0 ,andtheauxiliary function L z ; N 1 ; N 2 2 R isdenedas L r T N 1 )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 1 sgn x + x f + x + x f T N 2 )]TJ 10.949 10.619 Td [(p 2 r k z k k z k 2 ; where b 1 2 R isapositiveconstantchosenaccordingtothesufcientcondition b 1 > max z 2 + z 3 ; z 2 + z 4 a ; 53

PAGE 54

where z i ; i = 2 ; 3 ; 4areintroducedin4.Providedthesufcientconditionin4 issatised,thefollowinginequalitycanbeobtained P t 0 1 .Theauxiliaryfunction Q W f ; V f 1 ; V f 2 ; W gi ; V gi 1 ; V gi 2 2 R in4isdenedas Q a 2 tr W T f G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 wf W f + a 2 tr V T f 1 G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 vf 1 V f 1 + a 2 tr V T f 2 G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 vf 2 V f 2 + a 2 m i = 1 tr W T gi G )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 wg i W gi + a 2 m i = 1 tr V T gi 1 G )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 vgi 1 V gi 1 + a 2 m i = 1 tr V T gi 2 G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 vgi 2 V gi 2 ; where tr denotesthetraceofamatrix.Sincethegains G wf ; G wgi ; G vf 1 ; G vf 2 ; G vgi 1 ; G vgi 2 are symmetric,positive-denitematrices, Q 0 : 4.4LyapunovStabilityAnalysis Theorem4.1. TheDNN-basedobserverproposedin4alongwithitsweightupdatelawsin 4ensuresasymptoticestimationinsensethat k x t k 0 and k x 2 t )]TJ/F22 11.9552 Tf 12.528 0 Td [( x 2 t k 0 ast providedthecontrolgaink = k 1 + k 2 introducedin4-4isselectedsufcientlylargebased ontheinitialconditionsofthestates 2 ,thegainconditionin4issatised,andthefollowing sufcientconditionsaresatised g > az 2 5 + 1 2 a ; k 1 > 1 2 ; and l > z 2 1 4 p 2 k 2 ; where l 1 p 2 min a g )]TJ/F53 11.9552 Tf 10.949 0 Td [(az 2 5 ; k 1 )]TJ/F22 11.9552 Tf 12.145 8.093 Td [(1 2 ; and z 1 ; z 5 areintroducedin4and4,respectively. 1 SeeAppendixA.4andBforproof 2 Seethesubsequentproof 54

PAGE 55

Proof. ConsidertheLyapunovcandidatefunction V L y : D R ; whichisaLipschitzcontinuousregularpositivedenitefunctiondenedas V L g 2 x T x + g 2 x T f x f + g 2 h T h + 1 2 r T r + P + Q ; whichsatisesthefollowinginequalities: U 1 y V L y U 2 y : In4, U 1 y ; U 2 y 2 R arecontinuouspositivedenitefunctionsdenedas U 1 y min g 2 ; 1 2 k y k 2 ; U 2 y max g 2 ; 1 k y k 2 : Thegeneralizedtimederivativeof4existsalmosteverywherea.e.,and V L y 2 a : e : V L y seeChapter3forfurtherdetailswhere V L = x 2 V L y x T K x T x T f h T r T 1 2 P )]TJ/F22 6.9738 Tf 8.162 3.532 Td [(1 2 P 1 2 Q )]TJ/F22 6.9738 Tf 8.162 3.532 Td [(1 2 Q T ; where V L isthegeneralizedgradientof V L y [96].Since V L y islocallyLipschitzcontinuous regularandsmoothin y ,4canbesimpliedas[97] V L = V T L K x T x T f h T r T 1 2 P )]TJ/F22 6.9738 Tf 8.162 3.533 Td [(1 2 P 1 2 Q )]TJ/F22 6.9738 Tf 8.162 3.533 Td [(1 2 Q T = h g x T g x T f gh T r T 2 P 1 2 2 Q 1 2 i K [ Y ] T ; where Y x T x T f h T r T 1 2 P )]TJ/F22 6.9738 Tf 8.162 3.533 Td [(1 2 P 1 2 Q )]TJ/F22 6.9738 Tf 8.163 3.533 Td [(1 2 Q : Usingthecalculusfor K [ ] from[98]Theorem1,Properties2 ; 5 ; 7,andsubstitutingthedynamicsfrom4,4-4,4,4,4and4andaddingandsubtracting 55

PAGE 56

a x + x f T N 2 andusing4 ; V L y canberewrittenas V L g x T r )]TJ/F53 11.9552 Tf 10.949 0 Td [(a x )]TJ/F53 11.9552 Tf 10.95 0 Td [(h + g x T f h )]TJ/F53 11.9552 Tf 10.949 0 Td [(a x f + gh T )]TJ/F56 11.9552 Tf 9.289 0 Td [( k + a r )]TJ/F53 11.9552 Tf 10.949 0 Td [(ah + x )]TJ/F22 11.9552 Tf 12.528 0 Td [( x f + r T N + N )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 K sgn x + x f )]TJ/F25 11.9552 Tf 10.95 0 Td [(kr + g k + a h )]TJ/F53 11.9552 Tf 10.95 0 Td [(g r T x )]TJ/F25 11.9552 Tf 10.95 0 Td [(r T )]TJ/F25 11.9552 Tf 5.476 -9.69 Td [(N 1 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 K sgn x + x f )]TJ/F56 11.9552 Tf 10.949 0 Td [( x + x f T N 2 + p 2 r k z k k z k 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(a x + x f T N 2 + a x + x f T W T f s f + W T f s 0 f [ V T f 1 x 1 + V T f 2 x 2 ] + m i = 1 W T gi s gi u i + m i = 1 W T gi s 0 gi [ V T gi 1 x 1 + V T gi 2 x 2 ] u i )]TJ/F53 11.9552 Tf 10.949 0 Td [(a tr W T f G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 wf W f )]TJ/F53 11.9552 Tf 10.95 0 Td [(a tr V T f 1 G )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 vf 1 V f 1 )]TJ/F53 11.9552 Tf 10.95 0 Td [(a tr V T f 2 G )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 vf 2 V f 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(a m i = 1 tr W T gi G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 wg W gi )]TJ/F53 11.9552 Tf 10.95 0 Td [(a m i = 1 tr V T gi 1 G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 vgi 1 V gi 1 )]TJ/F53 11.9552 Tf 10.949 0 Td [(a m i = 1 tr V T gi 2 G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 vgi 2 V gi 2 : Usingthefactthat K [ sgn x + x f ]= SGN x + x f seeChapter3forfurtherdetails,thesetin 4canreducetothescalarinequality.Substitutingtheweightupdatelawsin4and cancelingcommonterms,theaboveexpressioncanbeupperboundedas V L a : e : )]TJ/F53 11.9552 Tf 23.239 0 Td [(ag x T x )]TJ/F53 11.9552 Tf 10.95 0 Td [(ag x T f x f )]TJ/F53 11.9552 Tf 10.95 0 Td [(agh T h )]TJ/F25 11.9552 Tf 10.949 0 Td [(kr T r + a x + x f T N 2 + r T N + p 2 r k z k k z k 2 : Using4,4,thefactthat az 5 x + x f k z k a 2 z 2 5 k x k 2 + a 2 z 2 5 x f 2 + 1 2 k z k 2 ; substituting k = k 1 + k 2 ,andcompletingthesquares,theexpressionin4canbefurther boundedas V L a : e : )]TJ/F53 11.9552 Tf 23.238 0 Td [(a g )]TJ/F53 11.9552 Tf 10.949 0 Td [(az 2 5 k x k 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(a g )]TJ/F53 11.9552 Tf 10.95 0 Td [(az 2 5 x f 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(ag k h k 2 )]TJ/F25 11.9552 Tf 10.949 0 Td [(k 1 k r k 2 + 1 2 + z 2 1 4 k 2 + p 2 r k z k k z k 2 : 56

PAGE 57

Providedthesufcientconditionsin4aresatised,theaboveexpressioncanberewrittenas V a : e : )]TJ 23.239 10.619 Td [(p 2 l )]TJ/F53 11.9552 Tf 22.05 8.233 Td [(z 2 1 4 p 2 k 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(r k z k k z k 2 a : e : )]TJ/F25 11.9552 Tf 22.641 0 Td [(U y 8 y 2 D ; where l isdenedin4and U y = c k z k 2 ,forsomepositiveconstant c ,isacontinuous positivesemi-denitefunctionwhichisdenedonthedomain D y t 2 R 4 n + 2 j k y t k r )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 l )]TJ/F53 11.9552 Tf 22.05 8.233 Td [(z 2 1 4 p 2 k 2 : Thesizeofthedomain D canbeincreasedbyincreasingthegains k and a .Theinequalities in4and4showthat V y 2 L inthedomain D ;hence, x t ; x f t ; h t ; r t ; P t and Q t 2 L in D ;4-4areusedtoshowthat x t ; x f t ; h t 2 L in D .Since x 1 t ; x 2 t 2 L byAssumption4.1, x 1 t ; x 2 t 2 L in D using4.Since x t ; x f t ; h t 2 L in D ,using4, v t 2 L in D .Since W f ; W gi ; s f ; s gi ; e f ; e gi 2 L ; i = 1 ::: m ; byAssumptions4.4-4.6,thecontrolinput u t andthedisturbance d t areboundedby Assumptions4.2-4.3,and W f t ; W gi t 2 L ; i = 1 ::: m ; bytheuseofthe proj algorithm, from4, r t 2 L in D ;then z t 2 L in D ; byusing4.Hence, U y isuniformly continuousin D : Let S D denoteasetdenedas S y t 2 D j U 2 y t < e 1 r )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 l )]TJ/F53 11.9552 Tf 22.05 8.234 Td [(z 2 1 4 p 2 k 2 2 : Theregionofattractionin4canbemadearbitrarilylargetoincludeanyinitialconditionby increasingthecontrolgains k and a i.e.,asemi-globaltypeofstabilityresult,andhence c k z k 2 0as t 8 y 0 2 S : Basedonthedenitionof z t thefollowingresultcanbeproven k x t k ; k h t k ; k r t k 0as t 8 y 0 2 S : 57

PAGE 58

From4,itcanbefurthershownthat x t 0as t 8 y 0 2 S : 4.5ExtensionforHigh-orderUncertainNonlinearSystems Theproposedmethodcanbeextendedfora N th orderuncertainnonlinearsystemas x 1 = x 2 ; x N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 = x N ; x N = f x + G x u + d ; y = x 1 ; where x t = x T 1 t x T 2 t ::: x T N t T 2 R Nn isthesystemstate,thesystemoutput y t 2 R n is measurableupto N )]TJ/F22 11.9552 Tf 11.336 0 Td [(1 th derivatives,i.e. x i t ; i = 1 ; 2 ::; N )]TJ/F22 11.9552 Tf 11.336 0 Td [(1aremeasurable,and x N t is unmeasurable,andtheunknownfunctions f x ; G x ,thecontrolinput u t andthedisturbance d t areintroducedin4. Giventhesystemin4,theobserverin4canbeextendedas x 1 = x 2 ; x N )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 = x N ; x N = W T f s f + m i = 1 W T gi s gi u i + v ; where x t = x T 1 t x T 2 t ::: x T N t T 2 R Nn isthestateoftheDNNobserver, W f t V f j t ; W gi t ; V gi j t ; i = 1 ::: m ; j = 1 ::: N ,and s f t ; s gi t areintroducedin4andtherobustcontrolterm 58

PAGE 59

v t 2 R n ismodiedas v )]TJ/F56 11.9552 Tf 10.618 0 Td [([ g K + 2 a N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 ] h + g )]TJ/F53 11.9552 Tf 10.95 0 Td [(a 2 N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 x N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 + b 1 sgn x N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 + x f : In4, K ; g ; a N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 ; b 1 2 R arepositiveconstantgains,and x f t ; h t 2 R n areoutputsofthe modieddynamiclter h = p )]TJ/F25 11.9552 Tf 10.95 0 Td [(K x N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 ; p = )]TJ/F56 11.9552 Tf 9.289 0 Td [( K + a N )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 p )]TJ/F22 11.9552 Tf 12.527 0 Td [( x f + K 2 + 1 x N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 ; x f = p )]TJ/F53 11.9552 Tf 10.95 0 Td [(a N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 x f )]TJ/F25 11.9552 Tf 10.95 0 Td [(K x N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 ; p 0 = K x N )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 0 ; x f 0 = 0 : Theestimationerror x 1 t 2 R n andthefollowinglteredestimationerrors x i t 2 R n i = 1 ;:::; N )]TJ/F22 11.9552 Tf 10.95 0 Td [(1aredenedas x 1 x 1 )]TJ/F22 11.9552 Tf 12.527 0 Td [( x 1 ; x 2 x 1 + a 1 x 1 ; x i x i )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 + a i )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 x i )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 + x i )]TJ/F22 8.9664 Tf 6.967 0 Td [(2 ; i = 3 ;:::; N )]TJ/F22 11.9552 Tf 10.949 0 Td [(1 ; r x N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 + a N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 x N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 + h ; where a i 2 R i = 1 ; 2 ;:::; N )]TJ/F22 11.9552 Tf 11.436 0 Td [(2,arepositiveconstantcontrolgains.Notethatthesignals x i t i = 1 ;:::; N )]TJ/F22 11.9552 Tf 11.343 0 Td [(1aremeasurable,whereas,thelteredestimationerror r t in4is notmeasurable,sinceitdependson x N t .TheweightupdatelawsfortheDNNin4are developedas W f = proj [ G wf s f x N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 + x f T ] ; V f j = proj [ G vf j x j x N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 + x f T W T f s 0 f ] ; W gi = proj [ G wgi s gi u i x N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 + x f T ] ; V gi j = proj [ G vgi j x j u i x N )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 + x f T W T gi s 0 gi ] ; 59

PAGE 60

Figure4-1.Theexperimentaltestbedconsistsofatwo-linkrobot.Thelinksaremountedontwo NSKdirect-driveswitchedreluctancemotors. where i = 1 ::: m ; j = 1 ::: N ; G wf ; G wgi ; G vf j ; G vgi j s 0 f t ; s 0 gi t areintroducedin4. Asimilarstabilityanalysiscanbeusedtoprovethat k x i t k ; k h t k ; k r t k 0as t ; i = 1 ; 2 ;:::; N )]TJ/F22 11.9552 Tf 10.949 0 Td [(1 : From4,itcanbefurthershownthat x N 1 t )]TJ/F22 11.9552 Tf 12.527 0 Td [( x N 1 t 0as t : 4.6ExperimentandSimulationResults Experimentsandsimulationsonatwo-linkrobotmanipulatorFig.4-1areperformedto comparetheproposedmethodwithseveralotherestimationmethods.Thetestbediscomposed ofatwo-linkdirectdriverevoluterobotconsistingoftwoaluminumlinks.Themotorencoders providepositionmeasurementswitharesolutionof614400pulses/revolution.Twoaluminum linksaremountedona240Nmrstlinkanda20Nmsecondlinkswitchedreluctance motor.Dataacquisitionandcontrolimplementationwereperformedinreal-timeusingQNXata 60

PAGE 61

frequencyof1.0kHz.Thetwo-linkrevoluterobotismodeledwiththefollowingdynamics: M x x + V m x ; x x + F d x + F s x = u t ; where x = x 1 x 2 T aretheangularpositions rad and x = x 1 x 2 T aretheangular velocities rad = s ofthetwolinksrespectively.In4, M x istheinertiamatrixand V m x ; x isthecentripetal-Coriolismatrix,denedas M 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 2 6 4 )]TJ/F25 11.9552 Tf 10.186 0 Td [(p 3 s 2 x 2 )]TJ/F25 11.9552 Tf 10.186 0 Td [(p 3 s 2 x 1 + x 2 p 3 s 2 x 1 0 3 7 5 : In4and4,parametersforsimulationarechosenasthebest-guessofthetestbedmodel as p 1 = 3 : 473 kg m 2 ; p 2 = 0 : 196 kg m 2 ; p 3 = 0 : 242 kg m 2 ; c 2 = cos x 2 s 2 = sin x 2 : F d = diag f 5 : 3 ; 1 : 1 g Nm sec and F s x = diag f 8 : 45 tanh x 1 ; 2 : 35 tanh x 2 g Nm arethemodels fordynamicandstaticfriction,respectively.Thesystemin4canberewrittenas x = f x ; x + G x ; x u + d ; where d t 2 R 2 istheadditiveexogenousdisturbanceand f x ; x 2 R 2 ,and G x ; x 2 R 2 2 are denedas f x ; x = M )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 )]TJ/F25 11.9552 Tf 8.691 0 Td [(V m )]TJ/F25 11.9552 Tf 10.95 0 Td [(F d x )]TJ/F25 11.9552 Tf 10.95 0 Td [(F s ; G x ; x = M )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 : ThecontrolinputischosenasaPDcontrollertotrackadesiredtrajectory x d t = [ 0 : 5 sin 2 t 0 : 5 cos 2 t ] T ,as u t = 20 x t )]TJ/F25 11.9552 Tf 11.489 0 Td [(x d t + 10 x t )]TJ/F22 11.9552 Tf 13.067 0 Td [( x d t ,wheretheangular velocity x t usedonlyinthecontrollawisdeterminednumericallybyastandardbackwards differencealgorithm.Theobjectiveistodesignanobserver x t toasymptoticallyestimatethe angularvelocities x t usingonlythemeasurementsofangularpositions x t .Thecontrolgains fortheexperimentarechosenas k = 7, a = 7, g = 8 ; b 1 = 6 ; and G wf = G wg 1 = G wg 2 = 3 I 8 8 ; 61

PAGE 62

Figure4-2.Velocityestimate x t usinga[1],b[2],ctheproposedmethod,anddthe centerdifferencemethodonatwo-linkexperimenttestbed. G vf = G vg 1 = G vg 2 = 3 I 2 2 ,where I n n denotesanidentitymatrixofappropriatedimensions. TheNNsaredesignedtohavesevenhiddenlayerneuronsandtheNNweightsareinitializedas uniformlydistributedrandomnumbersintheinterval [ )]TJ/F22 11.9552 Tf 9.289 0 Td [(1 ; 1 ] .Theinitialconditionsofthesystem andtheidentierarechosenas x t =[ 00 ] T x t =[ 00 ] T and x t = x t =[ 00 ] T ,respectively. Aglobalasymptoticvelocityobserverforuncertainnonlinearsystemswasdevelopedby Xianetal.[1]as x = p + K 0 x ; p = K 1 sgn x + K 2 x ; andahighgainHGobserverthatisasymptoticasthegaingoestoinnitywasdevelopedin[2] x = z h + a h 1 e h 1 x ; z h = a h 2 e h 2 x : Boththesedesignsarebasedonapurelyrobustfeedbackstrategy.Acontributionofthisworkis theadditionofafeed-forwardadaptivecomponenttocompensatefortheuncertaindynamics.To gaugethebenetofthisapproach,theproposedobserveriscomparedwiththeobserversin[1] and[2].Controlgainsfortheobserverin[1]arechosenas K 0 = 10, K 1 = 6,and K 2 = 10,and controlgainsfortheHGobserverarechosenas a 1 = 0 : 6, a 2 = 25, e 1 = 0 : 01,and e 2 = 0 : 015. 62

PAGE 63

Figure4-3.Thesteady-statevelocityestimate x t usinga[1],b[2],ctheproposedmethod, anddthecenterdifferencemethodonatwo-linkexperimenttestbed. Figure4-4.Frequencyanalysisofvelocityestimation x t usinga[1],b[2],ctheproposed method,anddthecenterdifferencemethodonatwo-linkexperimenttestbed. 63

PAGE 64

Tomakethecomparisonfeasible,thegainsofallobserversaretunedtogetthesteadystateroot meansquaresRMSofpositionestimationerrorstobeapproximatelyequal0.17forasettling timeof1second.Theexperimentresultsforthevelocityestimatorsin[1],[2],andtheproposed methodarecomparedwiththecentraldifferencealgorithm.TheresultsareshowninFigs.4-2 and4-3.Itisobservedthatthevelocityestimatesoftheproposedobserverandobserverin[2] looksimilar,butthetransientresponseoftheproposedmethodisimprovedovertheobserver in[2];moreover,bothmethodslowerfrequencycontentthantheobserverin[1]andthecentral differencemethod.Toillustratethelowerfrequencyresponseoftheproposedmethodcompared to[1]andthecentraldifferencemethod,thefrequencyanalysisplotsoftheexperimentresults areshowninFig.4-4.Fig.4-4illustratesthatthevelocityestimationusing[1]andcentral differencemethodsincludehigherfrequencysignalsthantheproposedmethodortheapproach in[2]. Giventhelackofvelocitysensorsinthetwo-linkexperimenttestbedtoverifythevelocity estimates,asimulationwasperformedusingthedynamicsin4.Toexaminetheeffectof noise,whiteGaussiannoisewithSNR60dBisaddedtothepositionmeasurements.Fig.4-5 showsthesimulationresultsforthesteady-statevelocityestimationerrorsandtherespective frequencyanalysisforthevelocityestimateoftheobserverin[1],theobserverin[2],the developedmethod,andthecentraldifferencemethod.Table4-1givesacomparisonofthe transientandsteadystateRMSvelocityestimationerrorsforthesedifferentmethods.Results ofthestandardnumericalcentraldifferentiationalgorithmaresignicantlyworsethantheother methodsinthepresenceofnoiseasseenfromFig.4-5andTable4-1.Although,simulation resultsfor[2]andthedevelopedmethodarecomparable,differencesexistinthestructureof theobserversandproofofconvergenceoftheestimates.Theobserverin[2]isapurelyrobust feedbacktechniqueandtheestimationresultisproventobeasymptoticasthegainstendto innity.Ontheotherhand,theproposedmethodisarobustadaptiveobserverwithaDNN structuretolearnthesystemuncertainties,combiningadynamiclterandarobustslidingmode structure,thusguaranteeingasymptoticconvergencewithnitegains.Further,theobserverin[1] 64

PAGE 65

Figure4-5.Thesteady-statevelocityestimationerror x t usinga[1],b[2],ctheproposed method,anddthecenterdifferencemethodonsimulations,inpresenceofsensor noiseSNR60dB.Therightgurese-hindicatetherespectivefrequencyanalysis ofvelocityestimation x t isalsoapurelyrobustfeedbackmethod,wherealluncertaintiesaredampedoutbyasliding modetermresultinginhigherfrequencyvelocityestimatesthanthedevelopedobserver,asseen frombothexperimentandsimulationresults. 4.7Conclusion AnoveldesignofanadaptiveobserverusingDNNsforuncertainnonlinearsystemsis proposed.TheDNNworksinconjunctionwithadynamiclterwithoutanyoff-linetraining Table4-1.Transient t = 0 )]TJ/F22 11.9552 Tf 10.95 0 Td [(1secandsteadystate t = 1 )]TJ/F22 11.9552 Tf 10.95 0 Td [(10secvelocityestimationerrors x t fordifferentvelocityestimationmethodsinpresenceofnoise50dB. CentraldifferenceMethodin[1]Methodin[2]Proposed TransientRMSError66.26820.17800.10400.1309 SteadyStateRMSError8.16080.05650.05380.0504 65

PAGE 66

phase.AslidingfeedbacktermisaddedtotheDNNstructuretoaccountforreconstruction errorsandexternaldisturbances.Theobservationstatesareproventoasymptoticallyconverge tothesystemstatesandasimilarobserverstructureisextendedtohigh-orderuncertainsystems. Simulationsandexperimentsshowtheimprovementoftheproposedmethodincomparisonto severalotherestimationmethods. 66

PAGE 67

CHAPTER5 GLOBALOUTPUTFEEDBACKTRACKINGCONTROLFORUNCERTAIN SECOND-ODERNONLINEARSYSTEMS ADNNobserver-basedOFBcontrollerforuncertainnonlinearsystemswithbounded disturbancesisdeveloped.Atwo-foldobjective,asymptoticestimationofunmeasurablestates andasymptotictrackingcontrol,issetup.Theasymptoticestimationoftheunmeasurablestate isachievedbyexploitingtheDNN-basedobserverinChapter4,whereinthedynamiclter andtheweightupdatelawsaremodiedforthenewobjective.Arobustcontrollerincludesa NNfeedforwardterm,alongwiththeestimatedstatefeedbackandslidingmodetermsthrough theLyapunov-basedstabilityanalysistoyieldanasymptotictrackingresult.Thedeveloped methodyieldstherstOFBtechniquesimultaneouslyachievingasymptotictrackingand asymptoticestimationofunmeasurablestatesfortheclassofuncertainnonlinearsystemswith boundeddisturbances.Experimentsonatwo-linkrobotmanipulatorareusedtoinvestigatethe performanceoftheproposedcontrolapproach. 5.1DynamicSystemandProperties Consideracontrol-afnesecondorderEuler-Lagrangelikenonlinearsystemoftheform x = f x ; x + G x u + d ; where x t 2 R n isthemeasurableoutputwithaniteinitialcondition x 0 = x 0 u t 2 R n isthecontrolinput, f x ; x 2 R n ; G x 2 R n n arecontinuousfunctions,and d t 2 R n isan exogenousdisturbance.Thefollowingassumptionsaboutthesystemin5willbeutilizedin thesubsequentdevelopment. Assumption5.1. Thetimederivativesofthesystemoutput x t ; x t areunmeasurable. Assumption5.2. Theunknownfunctionf x ; x isC 1 ,andthefunctionG x isknown,invertible andthematrixinverseG )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 x isbounded. Assumption5.3. Thedisturbanced t isdifferentiable,andd t ; d t 2 L : 67

PAGE 68

BasedontheuniversalapproximationpropertyofMLNNs,theunknownfunction f x ; x in 5canbereplacedbyaMLNN,andthesystemcanberepresentedas x = W T s V T 1 x + V T 2 x + e + Gu + d ; where W 2 R N + 1 n V 1 ; V 2 2 R n N areunknownidealweightmatricesoftheMLNNhaving N hiddenlayerneurons, s t s V T 1 x t + V T 2 x t 2 R N + 1 istheactivationfunctionsigmoid, hyperbolictangentetc.,and e x ; x 2 R n isafunctionreconstructionerror.Thefollowing assumptionswillbeusedintheDNN-basedobserverandcontrollerdevelopmentandstability analysis. Assumption5.4. TheidealNNweightsareboundedbyknownpositiveconstants[46],i.e. k W k W ; k V 1 k V 1 ; k V 2 k V 2 Assumption5.5. Theactivationfunction s anditspartialderivatives s 0 ; s 00 are bounded[46].Thisassumptionissatisedfortypicalactivationfunctionse.g.,sigmoid, hyperbolictangent. Assumption5.6. Thefunctionreconstructionerror e x ; x ; anditsrsttimederivativeare bounded[46],as k e x ; x k e 1 ; k e x ; x ; x k e 2 ; where e 1 ; e 2 areknownpositiveconstants. 5.2EstimationandControlObjectives ThecontributioninthischapteristhedevelopmentofarobustDNN-basedobserver suchthattheestimatedstatesasymptoticallyconvergetothestatesofthesystem5,anda discontinuouscontrollerenablesthesystemstatetoasymptoticallytrackadesiredtime-varying trajectory x d t 2 R n ,despiteuncertaintiesanddisturbancesinthesystem.Toquantifythese objectives,anestimationerror x t 2 R n andatrackingerror e t 2 R n aredenedas x x )]TJ/F22 11.9552 Tf 12.527 0 Td [( x ; e x )]TJ/F25 11.9552 Tf 10.949 0 Td [(x d ; where x t 2 R n isthestateoftheDNNobserverwhichisintroducedinthesubsequentdevelopment.Thedesiredtrajectory x d t anditsderivatives x i d t i = 1 ; 2 ; areassumedtoexistandbe 68

PAGE 69

bounded.Tocompensateforthelackofdirectmeasurementsof x t ,thelteredestimationerror r es t 2 R n andthelteredtrackingerror r tr t 2 R n aredenedas r es x + a x + h ; r tr e + a e + h ; where a 2 R isapositiveconstantgain,and h t 2 R n isanoutputofthedynamiclter h = p )]TJ/F56 11.9552 Tf 10.95 0 Td [( k + a x ; p = )]TJ/F56 11.9552 Tf 9.289 0 Td [( k + 2 a p )]TJ/F53 11.9552 Tf 10.949 0 Td [(n + k + a 2 + 1 x + e ; n = p )]TJ/F53 11.9552 Tf 10.95 0 Td [(an )]TJ/F56 11.9552 Tf 10.949 0 Td [( k + a x ; p 0 = k + a x 0 ; n 0 = 0 ; where n t 2 R n isanotheroutputofthelter, p t 2 R n isusedasaninternalltervariable, and k 2 R isapositiveconstantcontrolgain.Thelteredestimationerror r es t andtheltered trackingerror r tr t arenotmeasurablesincetheexpressionsin5and5dependon x t : 5.3DNN-basedRobustObserver TheMLDNNarchitectureisdevelopedtoobservethesystemin5 x = W T s + Gu )]TJ/F56 11.9552 Tf 10.949 0 Td [( k + 3 a h + b 1 sgn x + n ; where x t T x t T T 2 R 2 n arethestatesoftheDNNobserver, W t 2 R N + 1 n V 1 t ; V 2 t 2 R n N aretheweightestimates, s t s V 1 t T x t + V 2 t T x t 2 R N + 1 and b 1 2 R isa positiveconstantcontrolgain. 69

PAGE 70

TheweightupdatelawsfortheDNNin5aredevelopedbasedonthesubsequent stabilityanalysisas W = G w proj [ s d x + e + 2 n T ] ; V 1 = G v 1 proj [ x d x + e + 2 n T W T s 0 d ] ; V 2 = G v 2 proj [ x d x + e + 2 n T W T s 0 d ] ; where G w 2 R N + 1 N + 1 ; G v 1 ; G v 2 2 R n n ; areconstantsymmetricpositive-deniteadaptation gains,theterms s d t ; s 0 d t aredenedas s d t s V 1 t T x d t + V 2 t T x d t ; s 0 d t d s V = d V j V = V T 1 x d + V T 2 x d ,and proj isasmoothprojectionoperator[88],[89]usedtoguarantee thattheweightestimates W t ; V 1 t ; V 2 t remainbounded. Tofacilitatethesubsequentanalysis,5and5canbeusedtoexpressthetime derivativeof h t as h = )]TJ/F56 11.9552 Tf 9.289 0 Td [( k + a r es )]TJ/F53 11.9552 Tf 10.949 0 Td [(ah + x + e )]TJ/F53 11.9552 Tf 10.95 0 Td [(n : Theclosed-loopdynamicsofthelteredestimationerrorin5canbedeterminedbyusing 5,5,5,5and5as r es = W T s )]TJ/F22 11.9552 Tf 14.374 2.379 Td [( W T s + e + d + k + 3 a h )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 1 sgn x + n + a r es )]TJ/F53 11.9552 Tf 10.949 0 Td [(a x )]TJ/F53 11.9552 Tf 10.95 0 Td [(h )]TJ/F56 11.9552 Tf 10.949 0 Td [( k + a r es )]TJ/F53 11.9552 Tf 10.949 0 Td [(ah + x + e )]TJ/F53 11.9552 Tf 10.949 0 Td [(n : Addingandsubtracting W T s d + W T s d + W T s d where s d t s V T 1 x d t + V T 2 x d t ,the expressionin5canberewrittenas r es = N 1 + N )]TJ/F25 11.9552 Tf 10.95 0 Td [(kr es )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 sgn x + n + k + a h )]TJ/F22 11.9552 Tf 12.527 0 Td [( x ; wheretheauxiliaryfunction N 1 e ; x ; n ; r es ; r tr ; W ; V 1 ; V 2 ; t 2 R n isdenedas N 1 W T s )]TJ/F53 11.9552 Tf 10.95 0 Td [(s d )]TJ/F22 11.9552 Tf 14.374 2.379 Td [( W T s )]TJ/F22 11.9552 Tf 13.508 0.52 Td [( s d )]TJ/F56 11.9552 Tf 10.949 0 Td [( a 2 )]TJ/F22 11.9552 Tf 10.95 0 Td [(2 x )]TJ/F53 11.9552 Tf 10.95 0 Td [(n + e ; 70

PAGE 71

and N x d ; x d ; W ; V 1 ; V 2 ; t 2 R n issegregatedintotwopartsas N N D + N B : In5, N D t ; N B W ; V 1 ; V 2 ; t 2 R n aredenedas N D e + d ; and N B N B 1 + N B 2 : In5, N B 1 W ; V 1 ; V 2 ; t ; N B 2 W ; V 1 ; V 2 ; t 2 R n aredenedas N B 1 W T O V T 1 x d + V T 2 x d 2 + W T s 0 d V T 1 x d + V T 2 x d ; N B 2 W T s d + W T s 0 d V T 1 x d + V T 2 x d ; where W t W )]TJ/F22 11.9552 Tf 14.114 2.379 Td [( W t 2 R N + 1 n ; V 1 t V 1 )]TJ/F22 11.9552 Tf 12.811 2.379 Td [( V 1 t 2 R n N ; V 2 t V 2 )]TJ/F22 11.9552 Tf 12.81 2.379 Td [( V 2 t 2 R n N arethe estimatemismatchesfortheidealNNweights,and O V T 1 x d + V T 2 x d 2 2 R N + 1 isthehigherorder termintheTaylorseriesofthevectorfunctions s d intheneighborhoodof V T 1 x d + V T 2 x d as s d = s d + s 0 d V T 1 x d + V T 2 x d + O V T 1 x d + V T 2 x d 2 : Motivationforsegregatingthetermsin5,5and5isderivedfromthefactthat differenttermshavedifferentbounds.Theterm N 1 includesalltermswhichcanbeupper boundedbystates,whereas N includesalltermswhichcanbeupperboundedbyconstants. Thedifferencebetweentheterms N D and N B in5whichbothcanbeupper-boundedby constantsisthatthersttimederivativeof N D isfurtherupper-boundedbyaconstant,whereas theterm N B isstatedependent.Theterm N B isfurthersegregatedas5toaidinthe weightupdatelawdesignfortheDNNin5.Insubsequentstabilityanalysis,theterm N B 1 iscancelledbytheerrorfeedbackandtheslidingmodeterm,whiletheterm N B 2 ispartially compensatedforbytheweightupdatelawsandpartiallycancelledbytheerrorfeedbackandthe slidingmodeterm. 71

PAGE 72

Using5-5,Assumptions5.4-5.5,the proj algorithmin5andtheMeanValue Theorem,theauxiliaryfunction N 1 t in5canbeupper-boundedas N 1 z 1 k z k ; where z 1 2 R isacomputablepositiveconstant,and z x ; e ; r es ; r tr ; n ; h 2 R 6 n isdenedas z [ x T e T r T es r T tr n T h T ] T : BasedonAssumptions5.3-5.6,theTaylorseriesexpansionin5,andtheweightupdatelaws in5,thefollowingboundscanbedeveloped k N D k z 2 ; k N B 1 k z 3 ; k N B 2 k z 4 ; N D z 5 ; N B z 6 + z 7 k z k ; where z i 2 R ; i = 2 ; 3 ;:::; 7 ; arecomputablepositiveconstants. 5.4RobustAdaptiveTrackingController Thecontrolobjectiveistoforcethesystemstatetoasymptoticallytrackthedesired trajectory x d t ,despitetheuncertaintiesanddisturbancesinthesystem.Quantitatively,this objectiveistoregulatethelteredtrackingcontroller r tr t tozero.Using5,5,5 and5,theopen-loopdynamicsofthederivativeofthelteredtrackingerrorin5is expressedas r tr = W T s + G x u + e + d )]TJ/F22 11.9552 Tf 12.528 0 Td [( x d + a r tr )]TJ/F53 11.9552 Tf 10.95 0 Td [(a e )]TJ/F53 11.9552 Tf 10.95 0 Td [(h )]TJ/F56 11.9552 Tf 10.949 0 Td [( k + a r es )]TJ/F53 11.9552 Tf 10.949 0 Td [(ah + x + e )]TJ/F53 11.9552 Tf 10.949 0 Td [(n : Thecontrolinput u t isnowdesignedasacompositionoftheDNNterm,theestimatedstates x t ; x t ,andtheslidingmodetermas u t = G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 [ x d )]TJ/F22 11.9552 Tf 14.374 2.379 Td [( W T s d )]TJ/F56 11.9552 Tf 10.949 0 Td [( k + a e + a e )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 2 sgn e + n ] ; 72

PAGE 73

where b 2 2 R isapositiveconstantcontrolgainandthetrackingerrorestimate e t 2 R n is denedas e x )]TJ/F25 11.9552 Tf 10.949 0 Td [(x d : Basedonthefactthattheestimatedstatesaremeasurable,thetrackingerrorestimate e t andits derivative e t aremeasurable;moreover,thelteredtrackingerror r tr t isrelatedtotheltered estimationerror r es t viathetrackingerrorestimate e t as r tr = r es + e + a e : Hence,addingandsubtracting W T s d + W T s d andusing5-5,theclosed-looperror systembecomes r tr = N 2 + N )]TJ/F25 11.9552 Tf 10.949 0 Td [(kr tr )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 2 sgn e + n )]TJ/F25 11.9552 Tf 10.95 0 Td [(e ; wheretheauxiliaryfunction N 2 e ; x ; h ; n ; r tr ; t 2 R n isdenedas N 2 W T s )]TJ/F53 11.9552 Tf 10.95 0 Td [(s d )]TJ/F56 11.9552 Tf 10.949 0 Td [( a 2 )]TJ/F22 11.9552 Tf 10.95 0 Td [(2 e )]TJ/F53 11.9552 Tf 10.95 0 Td [(n + x )]TJ/F22 11.9552 Tf 10.949 0 Td [(2 ah ; andthefunction N isintroducedin5.Similarly,using5,5,Assumptions5.4-5.5, the proj algorithmin5,andtheMeanValueTheorem[74],theauxiliaryfunction N 2 in 5canbeupper-boundedas N 2 z 8 k z k ; where z 8 2 R isacomputablepositiveconstant. Tofacilitatethesubsequentstabilityanalysis,let y z ; P ; Q 2 R 6 n + 2 bedenedas y [ z T p P p Q ] T : 73

PAGE 74

In5,theauxiliaryfunction P r es ; r tr ; x ; e ; n ; x ; e ; n ; t 2 R istheFilippovsolutiontothe differentialequation P L ; P 0 P x 0 ; e 0 ; n 0 ; 0 = b 1 n j = 1 x j 0 + n j 0 + b 2 n j = 1 e j 0 + n j 0 )]TJ/F56 11.9552 Tf 10.95 0 Td [( x 0 + e 0 + 2 n 0 T N 0 ; wherethesubscript j = 1 ; 2 ;::; n denotesthe j th elementof x 0 e 0 or n 0 ,andtheauxiliary function L r es ; r tr ; x ; e ; n ; x ; e ; n ; t 2 R isdenedas L )]TJ/F25 11.9552 Tf 9.289 0 Td [(r T es N D + N B 1 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 sgn x + n )]TJ/F25 11.9552 Tf 10.949 0 Td [(r T tr N D + N B 1 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 2 sgn e + n )]TJ/F56 11.9552 Tf 10.95 0 Td [( x + e + 2 n T N B 2 + b 3 k z k 2 ; where b 1 ; b 2 areintroducedin5and5,and b 3 2 R isapositiveconstant.Thecontrol gains b i ; i = 1 ; 2 ; 3arechosenaccordingtothesufcientconditions b 1 ; b 2 > max z 2 + z 3 + z 4 ; z 2 + z 3 + z 5 a + z 6 a ; b 3 > 2 z 7 ; where z i i = 1 ; 2 ;:::; 7areintroducedin5and5.Providedthesufcientconditionsin 5aresatised,thefollowinginequalitycanbeobtained P 0 1 .Theauxiliaryfunction Q W ; V 1 ; V 2 2 R in5isdenedas Q t a 2 tr W T G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 w W + a 2 tr V T 1 G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 v 1 V 1 + a 2 tr V T 2 G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 v 2 V 2 ; where tr denotesthetraceofamatrix.Sincethegains G w ; G v 1 ; G v 2 aresymmetric,positivedenitematrices, Q 0 : 1 SeeAppendixBforproof 74

PAGE 75

5.5LyapunovStabilityAnalysisforDNN-basedObservationandControl Theorem5.1. TheDNN-basedobserverandcontrollerproposedin5and5,respectively,alongwiththeweightupdatelawsin5ensureasymptoticestimationandtrackingin sensethat k x t k 0 and x t 0 ast ; k e t k 0 and k e t k 0 ast ; providedthegainconditionsin5aresatised,andthecontrolgains a andk = k 1 + k 2 introducedin5-5areselectedas l min a ; k 1 > z 2 1 + z 2 8 4 k 2 + b 3 ; where z 1 ; z 8 ; b 3 areintroducedin5,5,and5,respectively. Proof. ConsidertheLyapunovcandidatefunction V L y ; t : D 0 ; R ; whichisaLipschitz continuousregularpositivedenitefunctiondenedas V L 1 2 x T x + 1 2 e T e + 1 2 n T n + 1 2 h T h + 1 2 r T es r es + 1 2 r T tr r tr + P + Q ; whichsatisesthefollowinginequalities: U 1 y V L y ; t U 2 y : In5, U 1 y ; U 2 y 2 R arecontinuouspositivedenitefunctionsdenedas U 1 y 1 2 k y k 2 ; U 2 y k y k 2 : Thegeneralizedtimederivativeof5existsalmosteverywherea.e.,and V L y 2 a : e : V L y seeChapter3forfurtherdetailswhere V L = x 2 V L y x T K x T e T n T h T r T es r T tr 1 2 P )]TJ/F22 6.9738 Tf 8.162 3.533 Td [(1 2 P 1 2 Q )]TJ/F22 6.9738 Tf 8.163 3.533 Td [(1 2 Q T ; 75

PAGE 76

where V L isthegeneralizedgradientof V L y [96].Since V L y isalocallyLipschitzcontinuous regularfunctionthatissmoothin y ,5canbesimpliedas[97] V L = V T K x T e T n T h T r T es r T tr 1 2 P )]TJ/F22 6.9738 Tf 8.162 3.533 Td [(1 2 P 1 2 Q )]TJ/F22 6.9738 Tf 8.163 3.533 Td [(1 2 Q T = h x T e T n T h T r T es r T tr 2 P 1 2 2 Q 1 2 i K [ Y ] T ; where Y x T e T n T h T r T es r T tr 1 2 P )]TJ/F22 6.9738 Tf 8.162 3.532 Td [(1 2 P 1 2 Q )]TJ/F22 6.9738 Tf 8.163 3.532 Td [(1 2 Q : Usingthecalculusfor K [ ] from[98]Theorem1,Properties2 ; 5 ; 7,andsubstitutingthe dynamicsfrom5-5,5,5,5,5,5and5 ; V L y canbe rewrittenas V L x T r es )]TJ/F53 11.9552 Tf 10.95 0 Td [(a x )]TJ/F53 11.9552 Tf 10.949 0 Td [(h + e T r tr )]TJ/F53 11.9552 Tf 10.949 0 Td [(a e )]TJ/F53 11.9552 Tf 10.949 0 Td [(h + h T [ )]TJ/F56 11.9552 Tf 9.289 0 Td [( k + a r es )]TJ/F53 11.9552 Tf 10.949 0 Td [(ah + x + e )]TJ/F53 11.9552 Tf 10.949 0 Td [(n ] + n T h )]TJ/F53 11.9552 Tf 10.95 0 Td [(an + r T es N 1 + N )]TJ/F25 11.9552 Tf 10.949 0 Td [(kr es )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 1 K [ sgn x + n ]+ k + a h )]TJ/F22 11.9552 Tf 12.527 0 Td [( x + r T tr N 2 + N )]TJ/F25 11.9552 Tf 10.95 0 Td [(kr tr )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 2 K [ sgn e + n ] )]TJ/F25 11.9552 Tf 10.949 0 Td [(e )]TJ/F25 11.9552 Tf 10.949 0 Td [(r T es f N D + N B 1 )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 1 K [ sgn x + n ] g )]TJ/F25 11.9552 Tf 10.949 0 Td [(r T tr f N D + N B 1 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 2 K [ sgn e + n ] g + b 3 k z k 2 )]TJ/F56 11.9552 Tf 10.95 0 Td [( x + e + 2 n T N B 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(a tr W T G )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 w W )]TJ/F53 11.9552 Tf 10.949 0 Td [(a tr V T 1 G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 v 1 V 1 )]TJ/F53 11.9552 Tf 10.949 0 Td [(a tr V T 2 G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 v 2 V 2 : Usingthefactthat K [ sgn e + n ]= SGN e + n and K [ sgn x + n ]= SGN x + n seeChapter 3forfurtherdetails,thesetin5canreducetothescalarinequality.Substitutingtheweight updatelawsin5andcancelingcommonterms,theaboveexpressioncanbeupperbounded as V L a : e : )]TJ/F53 11.9552 Tf 23.239 0 Td [(a x T x )]TJ/F53 11.9552 Tf 10.95 0 Td [(a e T e )]TJ/F53 11.9552 Tf 10.95 0 Td [(an T n )]TJ/F53 11.9552 Tf 10.95 0 Td [(ah T h )]TJ/F25 11.9552 Tf 10.95 0 Td [(kr T es r es )]TJ/F25 11.9552 Tf 10.949 0 Td [(kr T tr r tr + r T es N 1 + r T tr N 2 + b 3 k z k 2 : 76

PAGE 77

Using5and5,substituting k = k 1 + k 2 ; andcompletingthesquares,theexpressionin 5canbefurtherboundedas V L a : e : )]TJ/F53 11.9552 Tf 23.239 0 Td [(a k x k 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(a k e k 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(a k n k 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(a k h k 2 )]TJ/F25 11.9552 Tf 10.949 0 Td [(k 1 k r es k 2 )]TJ/F25 11.9552 Tf 10.95 0 Td [(k 1 k r tr k 2 + z 2 1 + z 2 8 4 k 2 + b 3 k z k 2 a : e : )]TJ/F56 11.9552 Tf 23.239 0 Td [( l )]TJ/F53 11.9552 Tf 12.145 8.373 Td [(z 2 1 + z 2 8 4 k 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 3 k z k 2 a : e : )]TJ/F25 11.9552 Tf 22.641 0 Td [(U y ; where U y = c k z k 2 ,forsomepositiveconstant c ,isacontinuouspositivesemi-denitefunction, and l isdenedin5.Theinequalitiesin5and5showthat V L y 2 L ;hence, x t ; e t ; n t ; h t ; r es t ; r tr t ; P t and Q t 2 L .Using5and5,itcanbeshown that x t ; e t 2 L : Basedontheassumptionthat x d t ; x d t 2 L ; and e t ; e t 2 L x t ; x t 2 L by5;moreover,using5and x t ; x t 2 L x t ; x t 2 L : Basedon Assumptions5.2and5.5,theprojectionalgorithmin5,theboundednessofthe sgn and s functions,and x d t ; x d t ; x t ; x t 2 L ,thecontrolinput u t isboundedfrom5. Similarly, n t h t ; r es t ; r tr t 2 L byusing5,5,5,5;hence z t 2 L ; using5;hence, U y isuniformlycontinuous.Itcanbeconcludedthat c k z k 2 0as t ; andusingthedenitionof z t in5,thefollowingresultcanbeshown k x k ; k e k 0as t ; k r es k ; k r tr k ; k n k ; k h k 0as t : Using5and5,andstandardlinearanalysis,itcanbefurthershownthat x ; k e k 0as t : 77

PAGE 78

5.6ExperimentResults TheperformanceoftheproposedOFBcontrolmethodistestedonatwo-linkrobot manipulatordepictedinFig.4-1,wherethedynamicsgivenby4.Thedesiredtrajectoryfor eachlinkofthemanipulatorisgivenasindegrees x 1 d = 30sin 1 : 5 t 1 )]TJ/F22 11.9552 Tf 10.949 0 Td [(exp )]TJ/F22 11.9552 Tf 9.289 0 Td [(0 : 01 t 3 x 2 d = 30sin 2 : 0 t 1 )]TJ/F22 11.9552 Tf 10.949 0 Td [(exp )]TJ/F22 11.9552 Tf 9.289 0 Td [(0 : 05 t 3 Thecontrolgainsarechosenas k = diag 25 ; 90 a = diag 22 ; 30 b 1 = 0 : 2 ; b 2 = 0 : 2 and, G w = 0 : 2 I 8 8 ; G v 1 = G v 2 = 0 : 2 I 2 2 ,where I n n denotesanidentitymatrixofappropriate dimensions.TheNNswasimplementedwith7hiddenlayerneuronsandtheNNweights areinitializedasuniformlydistributedrandomnumbersintheinterval [ 0 : 1 ; 0 : 3 ] .Theinitial conditionsofthesystemandtheobserverwereselectedas x t =[ 00 ] T x t =[ 00 ] T ; and x t = x t =[ 00 ] T ,respectively. TheperformanceoftheproposedOFBcontrolleriscomparedwithtwocontrollers:a classicalPIDcontroller,andthediscontinuousOFBcontrollerin[1].Astandardbackwards differencealgorithmisusedtonumericallydeterminevelocityfromtheencoderreadingsto implementthePIDcontroller.Controlgainsforthediscontinuouscontrollerin[1]wereselected as K 1 = 0 : 2, K 2 = diag 410 ; 38 ,andcontrolgainsforthePIDcontrollerwereselectedas K d = diag 120 ; 30 ; K p = diag 750 ; 90 ; and K i = diag 650 ; 100 : Thetrackingerrorsand controltorquesforallcontrollersareillustratedinFigs.5-1and5-2,respectively.Table5-1 showstheRMSerrorsandRMStorquesatsteady-stateforallmethods.Thedevelopedcontroller isshowntoexhibitlowertrackingerrorswithlesscontrolauthoritythanthecomparative controllers.Moreover,theDNN-basedobserveryieldsabettervelocityestimationincomparison withthehighfrequencycontentresultsfromabackwardsdifferencemethodasdepictedinFig. 5-3.Hence,theexperimentsillustratethatusingthevelocityestimationfromaDNN-based observer,whichadaptivelycompensatesforunknownuncertaintiesinthesystem,resultsin 78

PAGE 79

Figure5-1.Thetrackingerrors e t ofaLink1andbLink2usingclassicalPID,robust discontinuousOFBcontroller[1],andproposedcontroller. Table5-1.Steady-stateRMSerrorsandtorquesforeachoftheanalyzedcontroldesigns. SSRMSError1SSRMSError2SSRMSTorque1SSRMSTorque2 ClassicalPID0.45380.27006.58052.4133 RobustOFB[1]0.35520.29478.65091.2585 Proposed0.17430.17406.34840.6944 improvedcontrolperformancewithlowerfrequencycontentincomparisontothecompared methods. 5.7Conclusion ADNNobserver-basedOFBcontrolofaclassofsecond-ordernonlinearuncertainsystems isdeveloped.TheDNN-basedobserverworksinconjunctionwithadynamicltertoestimate theunmeasurablestate.TheDNNisupdatedon-linebyweightupdatelawsbasedonthe estimationerror,trackingerror,andlteroutput.ThecontrollerisacombinationoftheNN feedforwardterm,andtheestimatedstatefeedbackandslidingmodeterms.Globalasymptotic estimationoftheunmeasurablestateandglobalasymptotictrackingresultsareachieved, 79

PAGE 80

Figure5-2.ControlinputsforLink1andLink2usinga,bclassicalPID,c,drobust discontinuousOFBcontroller[1],ande,ftheproposedcontroller. 80

PAGE 81

Figure5-3.Velocityestimation x t usingaDNN-basedobserverandbnumericalbackwards difference. simultaneously.Resultsfromanexperimentusingatwo-linkrobotmanipulatordemonstratethe performanceoftheproposedOFBcontroller. 81

PAGE 82

CHAPTER6 OUTPUTFEEDBACKCONTROLFORANUNCERTAINNONLINEARSYSTEMWITH SLOWLYVARYINGINPUTDELAY OFBcontrolforanonlinearsystemwithtime-varyingactuatordelayisachallenging problembecauseofboththeneedtocompensateforthelackofthesystemstateandtheneed todevelopsomeformofpredictionofthenonlineardynamics.Inthischapter,anOFBtracking controllerisdevelopedforageneralsecond-ordersystemwithtime-varyinginputdelay, uncertainties,andadditiveboundeddisturbances.Thedevelopedcontrollerisamodied PDcontrollerworkinginassociationwithanintegralcomponent.ThePDcomponentsare formulatedusingthedifferencebetweenadesiredtrajectoryandanestimatedstateacquiredfrom aDNNbasedobservertocompensatetheinaccessibilityofthetruesystemstate.Theintegral componentisapredictor-likefeedbacktermtocompensatefortheinputdelay.Astability analysisusingLyapunov-KrasovskiifunctionalsisprovidedtoproveUUBtrackingandUUB estimationoftheunavailablestate.Asimulationofatwo-linkrobotmanipulatorisprovidedto illustratetheeffectivenessoftheproposedcontrolstrategy. 6.1DynamicSystemandProperties Consideracontrol-afnesecondordernonlinearsystemoftheform x = f x ; x + G x u t )]TJ/F53 11.9552 Tf 10.949 0 Td [(t t + d t ; where x t 2 R n isameasurableoutputwithaniteinitialcondition x 0 = x 0 u t )]TJ/F53 11.9552 Tf 10.949 0 Td [(t t 2 R n representsageneralizeddelayedcontrolinput,where t t 2 R isanon-negativetime-varying delay, f x ; x 2 R n ; G x 2 R n n areunknowncontinuousfunctions,and d t 2 R n isan exogenousdisturbance.Thesubsequentdevelopmentisbasedontheassumptionsthatthestate x t ismeasurable,thetime-varyinginputdelay t t isknown,andthecontrolinputvector anditspastvaluesi.e., u t )]TJ/F53 11.9552 Tf 10.949 0 Td [(q 8 q 2 [ 0 ; t t ] aremeasurable.Throughoutthechapter,a timedependentdelayedfunctionisdenotedas x t )]TJ/F53 11.9552 Tf 10.949 0 Td [(t t or x t .Additionally,thefollowing assumptionswillbeexploited. Assumption6.1. Theunknownfunctionf x ; x isC 1 82

PAGE 83

Assumption6.2. Thetimedelayisboundedsuchthat 0 t t j 1 ,and k t t k < 1 3 where j 1 2 R + isaknownconstant. BasedontheuniversalapproximationpropertyofMLNNs,theunknownfunction f x ; x in 6canbereplacedbyaMLNN,andthesystemcanberepresentedas x = W T s V T 1 x + V T 2 x + e + Gu t + d ; where W 2 R N + 1 n V 1 ; V 2 2 R n N areunknownidealweightmatricesoftheMLNNhaving N hiddenlayerneurons, s t s V T 1 x t + V T 2 x t 2 R N + 1 isanactivationfunctionsigmoid, hyperbolictangent,etc.,and e x ; x 2 R n isafunctionreconstructionerror.Assumptions5.3-5.6 willbealsoexploitedintheDNN-basedobserverandcontrollerdevelopment,andthestability analysis. 6.2EstimationandControlObjectives AcontributionofthischapteristhedevelopmentofacontinuousDNN-basedobserverto estimatetheunmeasurablestate x t oftheinput-delayedsystemin6.Basedonthisestimate, acontinuouscontrollerisdesignedsothatthesystemstate x t tracksadesiredtime-varying trajectory x d t 2 R n ,despiteuncertaintiesanddisturbancesinthesystem.Toquantifythese objectives,anestimationerror x t 2 R n andatrackingerror e t 2 R n aredenedas x x )]TJ/F22 11.9552 Tf 12.527 0 Td [( x ; e x )]TJ/F25 11.9552 Tf 10.949 0 Td [(x d ; where x t 2 R n isastateoftheDNNobserverwhichisintroducedinthesubsequentdevelopment.Thedesiredtrajectory x d t anditsderivatives x i d t i = 1 ; 2 ; areassumedtoexistandbe bounded.Tocompensateforthelackofdirectmeasurementsof x t ,alteredestimationerror r es t 2 R n andalteredtrackingerror r tr t 2 R n aredenedas r es x + a x + h ; r tr e + a e + Be z + h ; 83

PAGE 84

where a 2 R + isapositiveconstantgain,and e z t 2 R n isanauxiliarytime-delayedsignal denedas e z t t )]TJ/F53 8.9664 Tf 6.967 0 Td [(t t u q d q : Theterm e z t isapredictor-liketerminsensethat e z t transformstheinputdelayedsystem intoaninputdelayfreesystemseethesubsequentanalysis.In6, B 2 R n n isaknown, symmetric,positive-denite,constantgainmatrixthatsatisesthefollowinginequality b 1 k B k b 2 ; where b 1 ; b 2 2 R + areknownconstants.Theerrorbetween B and G x isdenotedby c x 2 R n n ,whichisdenedas c G )]TJ/F25 11.9552 Tf 10.95 0 Td [(B ; andsatisesthefollowingassumption k c k c ; where c 2 R + isaknownconstant.In6and6, h t 2 R n isanoutputofthedynamic lterin5.Thelteredestimationerror r es t andthelteredtrackingerror r tr t arenot measurablesincetheexpressionsin6and6dependon x t : 6.3RobustDNNObserverDevelopment ThefollowingMLDNNarchitectureisproposedtoobservethesystemin6 x = W T s + Bu t )]TJ/F56 11.9552 Tf 10.95 0 Td [( k + 3 a h ; where x t T x t T T 2 R 2 n arethestatesoftheDNNobserver, W t 2 R N + 1 n V 1 t ; V 2 t 2 R n N areweightestimates,and s t s V 1 t T x t + V 2 t T x t 2 R N + 1 84

PAGE 85

TheweightupdatelawsfortheDNNin6aredevelopedbasedonthesubsequent stabilityanalysisas W = G w proj [ s a x + h T ] ; V 1 = G v 1 proj [ x a x + h T W T s 0 ] ; V 2 = G v 2 proj [ x a x + h T W T s 0 ] ; where G w 2 R N + 1 N + 1 ; G v 1 ; G v 2 2 R n n areconstantsymmetricpositive-deniteadaptation gains,theterm s 0 t isdenedas s 0 d s V = d V j V = V T 1 x + V T 2 x ,and proj isasmoothprojection operatorcf.[88],[89]usedtoguaranteethattheweightestimates W t ; V 1 t ; V 2 t remain bounded. Tofacilitatethesubsequentanalysis,5and6canbeusedtoexpressthetime derivativeof h t as h = )]TJ/F56 11.9552 Tf 9.289 0 Td [( k + a r es )]TJ/F53 11.9552 Tf 10.95 0 Td [(ah + x + e )]TJ/F53 11.9552 Tf 10.95 0 Td [(n : Theclosed-loopdynamicsofthelteredestimationerrorin6canbedeterminedbyusing 6,6,6,6,6and6as r es = W T s )]TJ/F22 11.9552 Tf 14.374 2.379 Td [( W T s + e + d + c u t + k + 3 a h + + a r es )]TJ/F53 11.9552 Tf 10.949 0 Td [(a x )]TJ/F53 11.9552 Tf 10.95 0 Td [(h )]TJ/F56 11.9552 Tf 10.949 0 Td [( k + a r es )]TJ/F53 11.9552 Tf 10.949 0 Td [(ah + x + e )]TJ/F53 11.9552 Tf 10.949 0 Td [(n : Aftersomealgebraicmanipulation,theclosed-loopdynamicsofthelteredestimationerror r es t canbefurtherexpressedas r es = N 1 + N 2 )]TJ/F25 11.9552 Tf 10.95 0 Td [(kr es + c u t + k + a h )]TJ/F22 11.9552 Tf 12.527 0 Td [( x ; wheretheauxiliaryfunction N 1 x ; x ; W ; V 1 ; V 2 2 R n is N 1 W T s + W T s 0 [ V T 1 x + V T 2 x ] ; 85

PAGE 86

and N 2 e ; x ; n ; r es ; r tr ; e z ; W ; V 1 ; V 2 ; t 2 R n is N 2 W T )]TJ/F53 11.9552 Tf 5.476 -9.69 Td [(s )]TJ/F53 11.9552 Tf 10.949 0 Td [(s )]TJ/F25 11.9552 Tf 4.877 -9.69 Td [(V T 1 x + V T 2 x + W T s 0 [ V T 1 x + V T 2 x ] + W T O V T 1 x + V T 2 x 2 )]TJ/F56 11.9552 Tf 10.949 0 Td [( a 2 )]TJ/F22 11.9552 Tf 10.95 0 Td [(2 x )]TJ/F53 11.9552 Tf 10.949 0 Td [(n + e + e + d ; where W t W )]TJ/F22 11.9552 Tf 14.512 2.379 Td [( W t 2 R N + 1 n ; V 1 t V 1 )]TJ/F22 11.9552 Tf 13.208 2.379 Td [( V 1 t 2 R n N ; V 2 t V 2 )]TJ/F22 11.9552 Tf 13.208 2.379 Td [( V 2 t 2 R n N are estimatemismatchesfortheidealNNweights,and O V T 1 x + V T 2 x 2 2 R N + 1 representsahigher ordertermintheTaylorseriesofthevectorfunction s )]TJ/F25 11.9552 Tf 4.877 -9.69 Td [(V T 1 x t + V T 2 x t intheneighborhoodof V T 1 x + V T 2 x as s )]TJ/F25 11.9552 Tf 4.877 -9.689 Td [(V T 1 x + V T 2 x = s + s 0 [ V T 1 x + V T 2 x ]+ O V T 1 x + V T 2 x 2 : Using6-6,Assumptions5.3-5.6,the proj algorithmin6,theTaylorseries expansionin6andtheMeanValueTheorem,theauxiliaryfunctions N 1 in6and N 2 in6canbeupper-boundedas k N 1 k z 1 k z k + z 2 ; k N 2 k z 3 k z k + z 4 ; where z i 2 R + ; i = 1 ;:::; 4arecomputablepositiveconstants,and z x ; e ; r es ; r tr ; n ; h ; e z 2 R 7 n is denedas z [ x T e T r T es r T tr n T h T e T z ] T : 6.4RobustTrackingControlDevelopment Thecontrolobjectiveistoforcethesystemstatetotrackthedesiredtrajectory x d t ,despite theuncertainties,disturbances,andtime-delaysinthesystem.Quantitatively,thisobjectiveisto regulatethetrackingerror e t tozero.Using6,6,6-6and6,theopen-loop dynamicsofthelteredtrackingerrorin6canbeexpressedas r tr = W T s + Gu t + e + d )]TJ/F22 11.9552 Tf 12.528 0 Td [( x d + a r tr )]TJ/F53 11.9552 Tf 10.95 0 Td [(a e )]TJ/F53 11.9552 Tf 10.949 0 Td [(h )]TJ/F25 11.9552 Tf 10.949 0 Td [(Be z + B e z )]TJ/F56 11.9552 Tf 10.95 0 Td [( k + a r es )]TJ/F53 11.9552 Tf 10.95 0 Td [(ah + x + e )]TJ/F53 11.9552 Tf 10.95 0 Td [(n : 86

PAGE 87

Basedontheerrorsystemformulationin6,thetimederivativeof e z t canbecalculatedas e z = u )]TJ/F25 11.9552 Tf 10.95 0 Td [(u t + u t t : Hence,theopen-looperrorsystemin6containsadelay-freecontrolinput.Basedon6 andthesubsequentstabilityanalysis,thecontrolinputisdesignedas u t = )]TJ/F25 11.9552 Tf 9.289 0 Td [(B )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 k + a e + a e + Be z ; wherethetrackingerrorestimate e t 2 R n isdenedas e x )]TJ/F25 11.9552 Tf 10.949 0 Td [(x d : Basedonthefactthattheestimatedstates x t ; x t aremeasurable,thetrackingerrorestimates e t anditsderivative e t aremeasurable;moreover,thelteredtrackingerror r tr t isrelatedto thelteredestimationerror r es t viathetrackingerrorestimate e t as r tr = r es + e + a e + Be z : Therelationin6showsthateventhoughboththelteredtrackingerror r tr t andthe lteredestimationerror r es t areunmeasurable,thedifferencebetween r tr t and r es t is measurable.TheDNNobserverprovidesnegativefeedbackofthelteredestimationerror r es t toguaranteetheconvergenceoftheestimatedstates,andthecontrollerin6compensates forthedifferencebetween r es t and r tr t toobtainthenegativefeedbackofthelteredtracking error r tr t ;hencetheconvergenceofthetrackingerrorcanbeachieved. Using6-6,theclosed-looperrorsystembecomes r tr = N 3 )]TJ/F25 11.9552 Tf 10.949 0 Td [(kr tr + c u t + Bu t t )]TJ/F25 11.9552 Tf 10.95 0 Td [(e ; wheretheauxiliaryfunction N 3 e ; x ; h ; n ; e z ; r tr ; t 2 R n isdened N 3 W T s )]TJ/F56 11.9552 Tf 10.949 0 Td [( a 2 )]TJ/F22 11.9552 Tf 10.95 0 Td [(2 e )]TJ/F53 11.9552 Tf 10.95 0 Td [(n + x )]TJ/F22 11.9552 Tf 10.95 0 Td [(2 ah )]TJ/F53 11.9552 Tf 10.949 0 Td [(a Be z + e + d )]TJ/F22 11.9552 Tf 12.528 0 Td [( x d : 87

PAGE 88

Similarly,using6,6,Assumptions5.3-5.6,thecondition6andthe proj algorithm in6,theauxiliaryfunction N 3 in6canbeupper-boundedas k N 3 k z 5 k z k + z 6 ; where z 5 ; z 6 2 R + arecomputablepositiveconstants. Tofacilitatethesubsequentstabilityanalysis,let y z ; P ; Q ; R 2 R 7 n + 3 bedenedas y [ z T p P p Q p R ] T ; where P u ; t ; t ; Q r es ; r tr ; t ; t ; t 2 R denotepositive-deniteLKfunctionalsdenedas P w t t )]TJ/F53 8.9664 Tf 6.966 0 Td [(t t t s k u q k 2 d q ds ; Q 6 c + b 1 k + a 4 b 1 t t )]TJ/F53 8.9664 Tf 6.967 0 Td [(t t k r es q )]TJ/F25 11.9552 Tf 10.949 0 Td [(r tr q k 2 d q ; and w 2 R + isaknownconstant.Additionally,theauxiliaryfunction R W ; V 1 ; V 2 2 R in6 isdenedas R 1 2 tr W T G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 w W + 1 2 tr V T 1 G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 v 1 V 1 + 1 2 tr V T 2 G )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 v 2 V 2 ; where tr denotesthetraceofamatrix.Sincethegains G w ; G v 1 ; G v 2 aresymmetric,positivedenitematrices, R 0 : UsingAssumption5.4andthe proj algorithmin6, R can beupperboundedas R t R ; where R 2 R + isaknownconstant.Moreover,theupdatelawsin6aredesignedsuchthat R + a x + h T N 1 = 0 : 6.5LyapunovStabilityAnalysisforDNN-basedObservationandControl Theorem6.1. TheDNN-basedobserverandcontrollerproposedin6and6,respectively,alongwiththeweightupdatelawsin6ensureuniformlyultimatelybounded 88

PAGE 89

estimationandtrackinginsensethat x t e 1 exp )]TJ/F53 11.9552 Tf 9.289 0 Td [(e 2 t + e 3 ; k e t k e 4 exp )]TJ/F53 11.9552 Tf 9.289 0 Td [(e 5 t + e 6 ; where e i 2 R + ; i = 1 ; 2 ;:::; 6 areknownconstants,provided t t and t t aresufcientlysmall, theapproximationmatrixBareselectedsufcientlyclosetoG x ,andthefollowingsufcient conditionsaresatised w > sup t ; t 2 t 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 1 2 y 2 + 2 z 1 z 7 ; a > b 2 2 y 2 2 + 2 z 1 z 7 ; k 1 > sup t ; t k + a j t j + 1 2 + 7 c 2 b 1 + 2 wt k + a + 2 z 1 z 7 ; where y 2 R + isaknowngainconstant,andk 1 2 R + isintroducedin6. Proof. ConsidertheLyapunovcandidatefunction V L y ; t : D 0 ; R ; whichisaLipschitz continuouspositive-denitefunctionaldenedas V L 1 2 x T x + 1 2 e T e + 1 2 n T n + 1 2 h T h + 1 2 r T es r es + 1 2 r T tr r tr + P + Q + R ; whichsatisesthefollowinginequalities: U 1 y V L y ; t U 2 y : In6, U 1 y ; U 2 y 2 R arecontinuouspositive-denitefunctionsdenedas U 1 y 1 2 k y k 2 ; U 2 y k y k 2 : 89

PAGE 90

Using6-6,6,6,and6,andbyapplyingtheLeibnizRuletodetermine thetimederivativeof6and6,thetimederivativeof6canbecalculatedas V L = x T r es )]TJ/F53 11.9552 Tf 10.95 0 Td [(a x )]TJ/F53 11.9552 Tf 10.949 0 Td [(h + e T r tr )]TJ/F53 11.9552 Tf 10.95 0 Td [(a e )]TJ/F53 11.9552 Tf 10.949 0 Td [(h )]TJ/F25 11.9552 Tf 10.949 0 Td [(Be z + n T h )]TJ/F53 11.9552 Tf 10.949 0 Td [(an + h T )]TJ/F56 11.9552 Tf 9.289 0 Td [( k + a r es )]TJ/F53 11.9552 Tf 10.95 0 Td [(ah + x + e )]TJ/F53 11.9552 Tf 10.949 0 Td [(n + r T tr N 3 )]TJ/F25 11.9552 Tf 10.95 0 Td [(kr tr + c u t + Bu t t )]TJ/F25 11.9552 Tf 10.95 0 Td [(e + r T es N 1 + N 2 )]TJ/F25 11.9552 Tf 10.95 0 Td [(kr es + c u t + k + a h )]TJ/F22 11.9552 Tf 12.527 0 Td [( x + R + wt k u k 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t t t )]TJ/F53 8.9664 Tf 6.967 0 Td [(t t k u q k 2 d q + 6 c + b 1 k + a 4 b 1 k r es )]TJ/F25 11.9552 Tf 10.95 0 Td [(r tr k 2 )]TJ/F56 11.9552 Tf 10.95 0 Td [( 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 6 c + b 1 k + a 4 b 1 k r es t )]TJ/F25 11.9552 Tf 10.949 0 Td [(r tr t k 2 : Using6,cancelingcommonterms,andutilizingtherelationshipbetweenthecontroller u t withtheunmeasurableerrors r es t ; r tr t as u t = k + a B )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 r es )]TJ/F25 11.9552 Tf 10.95 0 Td [(r tr ; theexpressionin6canbeexpandedandregroupedas V L = )]TJ/F53 11.9552 Tf 9.289 0 Td [(a x T x )]TJ/F53 11.9552 Tf 10.949 0 Td [(a e T e )]TJ/F25 11.9552 Tf 10.949 0 Td [(e T Be z )]TJ/F53 11.9552 Tf 10.949 0 Td [(an T n )]TJ/F53 11.9552 Tf 10.95 0 Td [(ah T h )]TJ/F25 11.9552 Tf 10.949 0 Td [(kr T es r es )]TJ/F25 11.9552 Tf 10.949 0 Td [(kr T tr r tr + x T N 1 + r T es N 2 + k + a r es + r tr T c B )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 r es t )]TJ/F25 11.9552 Tf 10.949 0 Td [(r tr t + t k + a r T tr r es t )]TJ/F25 11.9552 Tf 10.949 0 Td [(r tr t + r T tr N 3 + wt k + a 2 k r es )]TJ/F25 11.9552 Tf 10.95 0 Td [(r tr k 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t t t )]TJ/F53 8.9664 Tf 6.966 0 Td [(t t k u q k 2 d q + 6 c + b 1 k + a 4 b 1 k r es )]TJ/F25 11.9552 Tf 10.95 0 Td [(r tr k 2 )]TJ/F56 11.9552 Tf 10.949 0 Td [( 1 )]TJ/F22 11.9552 Tf 12.676 0.52 Td [( t 6 c + b 1 k + a 4 b 1 k r es t )]TJ/F25 11.9552 Tf 10.949 0 Td [(r tr t k 2 : Young'sinequalitycanbeusedtoupperboundselecttermsin6as k e kk B kk e z k b 2 2 y 2 2 k e k 2 + 1 2 y 2 k e z k 2 ; k r tr kk r es t )]TJ/F25 11.9552 Tf 10.949 0 Td [(r tr t k 1 2 k r tr k 2 + 1 2 k r es t )]TJ/F25 11.9552 Tf 10.949 0 Td [(r tr t k 2 ; k r es )]TJ/F25 11.9552 Tf 10.949 0 Td [(r tr k 2 2 k r tr k 2 + 2 k r es k 2 : 90

PAGE 91

Using6,6,andAssumption6.2,6canbeupperboundedas V L )]TJ/F53 11.9552 Tf 21.234 0 Td [(a k x k 2 )]TJ/F66 11.9552 Tf 10.95 16.863 Td [( a )]TJ/F25 11.9552 Tf 12.145 8.233 Td [(b 2 2 y 2 2 k e k 2 + 1 2 y 2 k e z k 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(a k n k 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(a k h k 2 )]TJ/F25 11.9552 Tf 10.95 0 Td [(k k r es k 2 )]TJ/F25 11.9552 Tf 10.949 0 Td [(k k r tr k 2 + x k N 1 k + k r es kk N 2 k + k r tr kk N 3 k + c k + a 2 b 1 k r es k 2 + k r tr k 2 + 2 k r es t )]TJ/F25 11.9552 Tf 10.949 0 Td [(r tr t k 2 + j t j k + a 2 k r tr k 2 + k r es t )]TJ/F25 11.9552 Tf 10.949 0 Td [(r tr t k 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t t t )]TJ/F53 8.9664 Tf 6.967 0 Td [(t t k u q k 2 d q + 2 wt k + a 2 + 6 c + b 1 k + a 2 b 1 k r tr k 2 + k r es k 2 )]TJ/F56 11.9552 Tf 10.949 0 Td [( 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t k + a 3 c 2 b 1 + 1 4 k r es t )]TJ/F25 11.9552 Tf 10.95 0 Td [(r tr t k 2 : UtilizingAssumption6.2yields j t j k + a 2 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t k + a 4 ; c k + a b 1 3 c k + a 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 2 b 1 ; andusingtheCauchy-Schwartzinequality,and6,theintegraltermin6canbeupper boundedas )]TJ/F53 11.9552 Tf 10.95 0 Td [(w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t t t )]TJ/F53 8.9664 Tf 6.966 0 Td [(t k u q k 2 d q )]TJ/F53 11.9552 Tf 22.431 8.093 Td [(w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 2 t k e z k 2 )]TJ/F53 11.9552 Tf 12.145 8.093 Td [(w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 2 t t )]TJ/F53 8.9664 Tf 6.967 0 Td [(t k u q k 2 d q : Utilizing6,6,6,6,andthefactthat x z 7 k z k ; where z 7 2 R + isdened as z 7 max 1 ; a ,theinequalityin6canbeexpressedas V L )]TJ/F53 11.9552 Tf 21.235 0 Td [(a k x k 2 )]TJ/F66 11.9552 Tf 10.949 16.863 Td [( a )]TJ/F25 11.9552 Tf 12.145 8.233 Td [(b 2 2 y 2 2 k e k 2 )]TJ/F66 11.9552 Tf 10.949 16.863 Td [( w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 2 t )]TJ/F22 11.9552 Tf 19.233 8.093 Td [(1 2 y 2 k e z k 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(a k n k 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(a k h k 2 )]TJ/F25 11.9552 Tf 10.949 0 Td [(k k r es k 2 )]TJ/F25 11.9552 Tf 10.949 0 Td [(k k r tr k 2 + z 7 z 1 k z k + z 2 k z k + z 3 k z k + z 4 k r es k + z 5 k z k + z 6 k r tr k + j t j k + a 2 k r tr k 2 )]TJ/F53 11.9552 Tf 12.145 8.094 Td [(w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 2 t t )]TJ/F53 8.9664 Tf 6.967 0 Td [(t t k u q k 2 d q + 2 wt k + a 2 + 7 c + b 1 k + a 2 b 1 k r tr k 2 + k r es k 2 : 91

PAGE 92

Let k ,introducedin5,bedenedas k k 1 + k 2 + k 3 ; where k 1 ; k 2 ; k 3 2 R + arepositivecontrolgains,andlettheauxiliaryconstants b ; l 2 R + be denedas b 1 2 min a )]TJ/F25 11.9552 Tf 12.145 8.233 Td [(b 2 2 y 2 2 ; inf t ; t w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 2 t )]TJ/F22 11.9552 Tf 19.233 8.094 Td [(1 2 y 2 ; inf t ; t k 1 )]TJ/F56 11.9552 Tf 10.95 0 Td [( k + a j t j + 1 2 + 7 c 2 b 1 + 2 wt k + a ; l )]TJ/F53 11.9552 Tf 5.475 -9.69 Td [(z 2 4 + z 2 6 4 k 3 + z 2 2 z 2 7 4 b ; where z 2 ; z 4 ; z 6 areintroducedin6and6.Using6,6,andcompletingthe squares,theexpressionin6canbefurtherupperboundedas V L )]TJ/F22 11.9552 Tf 21.235 0 Td [(2 b k z k 2 + z 7 z 1 k z k + z 2 k z k + )]TJ/F53 11.9552 Tf 5.475 -9.69 Td [(z 2 3 + z 2 5 k z k 2 4 k 2 + )]TJ/F53 11.9552 Tf 5.475 -9.69 Td [(z 2 4 + z 2 6 4 k 3 )]TJ/F53 11.9552 Tf 12.145 8.094 Td [(w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 2 t t )]TJ/F53 8.9664 Tf 6.967 0 Td [(t t k u q k 2 d q : Usingtheinequality[63] t t )]TJ/F53 8.9664 Tf 6.967 0 Td [(t t t s k u q k 2 d q ds t sup s 2 [ t )]TJ/F53 8.9664 Tf 6.967 0 Td [(t ; t ] t s k u q k 2 d q = t t t )]TJ/F53 8.9664 Tf 6.967 0 Td [(t t k u q k 2 d q ; andcompletingthesquares,theexpressionin6canbeupperboundedas V L )]TJ/F66 11.9552 Tf 22.563 20.45 Td [( b )]TJ/F66 11.9552 Tf 12.145 18.53 Td [()]TJ/F53 11.9552 Tf 5.476 -9.689 Td [(z 2 3 + z 2 5 4 k 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(z 1 z 7 k z k 2 + l )]TJ/F53 11.9552 Tf 12.145 8.093 Td [(w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 4 t t )]TJ/F53 8.9664 Tf 6.967 0 Td [(t t k u q k 2 d q )]TJ/F53 11.9552 Tf 12.145 8.093 Td [(w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 4 t t t )]TJ/F53 8.9664 Tf 6.967 0 Td [(t t t s k u q k 2 d q ds : Byfurtherutilizing6-6and6,theinequalityin6canbewrittenas V L )]TJ/F66 11.9552 Tf 29.869 20.449 Td [( b )]TJ/F66 11.9552 Tf 12.145 18.53 Td [()]TJ/F53 11.9552 Tf 5.476 -9.69 Td [(z 2 3 + z 2 5 4 k 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(z 1 z 7 k z k 2 + l )]TJ/F25 11.9552 Tf 10.949 0 Td [(R + R )]TJ/F53 11.9552 Tf 12.145 8.093 Td [(w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t k + a 2 4 b 2 2 Q )]TJ/F56 11.9552 Tf 12.145 8.093 Td [( 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 4 t P )]TJ/F53 11.9552 Tf 28.54 0 Td [(b 2 k y k 2 + l + R ; 92

PAGE 93

where b 2 2 R + isdenedas b 2 min b )]TJ/F66 11.9552 Tf 12.145 18.53 Td [()]TJ/F53 11.9552 Tf 5.475 -9.69 Td [(z 2 3 + z 2 5 4 k 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(z 1 z 7 ; inf t ; t w 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t k + a 2 4 b 2 2 ; inf t ; t 1 )]TJ/F22 11.9552 Tf 12.677 0.52 Td [( t 4 t ; 1 # : Using6,theinequalityin6canbewrittenas V L )]TJ/F53 11.9552 Tf 21.234 0 Td [(b 2 V L + l + R ; wherethelineardifferentialinequalityin6canbesolvedas V L y ; t e )]TJ/F53 8.9664 Tf 6.967 0 Td [(b 2 t V L 0 + b )]TJ/F22 8.9664 Tf 6.967 0 Td [(1 2 l + R h 1 )]TJ/F25 11.9552 Tf 10.949 0 Td [(e )]TJ/F53 8.9664 Tf 6.967 0 Td [(b 2 t i ; 8 t 0 : Basedon6and6,itcanbeconcludedthat e t ; x t ; r es t ; r tr t ; h t ; n t 2 L henceusingthedenitionof r es t in6andtherelationbetween u t r es t and r tr t in 6, x t u t 2 L 6.6SimulationResults Thefollowingdynamicsofatwolinkrobotmanipulatorareconsideredforthesimulations: M x x + V m x ; x x + F d x + t d t = u t t ; where x = x 1 x 2 T aretheangularpositions rad and x = x 1 x 2 T aretheangular velocities rad = s ofthetwolinks,respectively, M x istheinertiamatrixand V m x ; x isthe centripetal-Coriolismatrix,denedas M = 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 = 2 6 4 )]TJ/F25 11.9552 Tf 10.186 0 Td [(p 3 s 2 x 2 )]TJ/F25 11.9552 Tf 10.185 0 Td [(p 3 s 2 x 1 + x 2 p 3 s 2 x 1 0 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 ; c 2 = cos x 2 s 2 = sin x 2 ; and F d = diag f 5 : 3 ; 1 : 1 g Nm sec denotesfrictioncoefcients.Anadditiveexogenousdisturbance isappliedas t d t = 0 : 2 sin )]TJ/F25 8.9664 Tf 7.25 -4.979 Td [(t 2 0 : 1 sin )]TJ/F25 8.9664 Tf 7.249 -4.979 Td [(t 4 T .ThedesiredtrajectoriesforLinks1and2forall 93

PAGE 94

simulationsareselectedas x d 1 t = 40 sin 1 : 5 t 1 )]TJ/F25 11.9552 Tf 10.95 0 Td [(e )]TJ/F22 8.9664 Tf 6.967 0 Td [(0 : 01 t 3 deg ; x d 2 t = 20 sin 1 : 5 t 1 )]TJ/F25 11.9552 Tf 10.95 0 Td [(e )]TJ/F22 8.9664 Tf 6.967 0 Td [(0 : 01 t 3 deg : Theinitialconditionsofthesystemandtheobserverarechosenas x t =[ 00 ] T x t =[ 00 ] T and x t = x t =[ 00 ] T ,respectively.Thecontrollerin6assumesthattheinertiamatrix M x isunknown,hence,abestguessestimateoftheinertiamatrixisselectedas B = 2 6 4 3 : 50 : 2 0 : 20 : 2 3 7 5 )]TJ/F22 8.9664 Tf 6.966 0 Td [(1 : Toillustrateperformanceofthedevelopedmethod,simulationsareexecutedusingvarioustimevaryingdelays.Thetimedelaysareselectedassinusoidalfunctionswithincreasingmagnitudes, increasingvaryingspeedsandincreasingdisplacementoffsets.Foreachcase,Link1andLink2 RMStrackingerrorsandRMSestimationerrorsareshowninTable6-1,respectively.Theresults clearlyshowthatthesystemperformanceisbetterwithsmallandslowlyvaryingtime-delays. Inthetheoreticalanalysis,thetime-delayisassumedtobeexactlyknown.However,to examinetherobustnessofthedevelopedcontrollerwithrespectivetothetime-delayparameters, theinputdelayenteringtheplantisvariedfromthedelayusedinthecontrollerfeedback. Thefeedbackdelayofthecontrolleriskeptasthesinusoidfunctionwithaoffsetof10 ms as t t = 2 sin )]TJ/F25 8.9664 Tf 9.491 -4.979 Td [(t 10 + 10 ms : Table6-2presentssimulationresultswheremagnitudesand/oroffsets ofplantdelaysarevariedfromtheconrespondingparametersofthecontrollerdelay.Theresults suggestthatthecontrollerisrobusttovariancesindelaymagnitudeandoffset.However,the smallervariancesresultinbetterperformance.Figure6-1illustratesthetime-delay,tracking, andestimationerrorsandcontroltorquesassociatedwiththe+10%magnitudeand+10%offset variancecase.Thedevelopedapproachhasbeenprovenfortheexactknowledgeofthetime delay,butthesimulationresultsalsoillustratesomerobustnesswithregardtouncertaintiesin 94

PAGE 95

Table6-1.Link1andLink2RMStrackingerrorsandRMSestimationerrors. RMSTrackingdegRMSEstimationdeg/s Time-Delay t t msLink1Link2Link1Link2 2 sin )]TJ/F25 8.9664 Tf 9.491 -4.98 Td [(t 10 + 50.13460.18200.03660.1257 2 sin )]TJ/F25 8.9664 Tf 9.491 -4.979 Td [(t 10 + 100.22780.30240.04910.2050 5 sin )]TJ/F25 8.9664 Tf 7.249 -4.979 Td [(t 2 + 100.23510.34630.05460.2350 5 sin )]TJ/F25 8.9664 Tf 7.249 -4.98 Td [(t 2 + 200.91340.93500.20970.6255 Table6-2.RMSerrorsforcasesofuncertaintyintime-varyingdelayseenbytheplantas comparedtothedelayofthecontroller. RMSTrackingdegRMSEstimationdeg/s Time-DelayVarianceinPlantLink1Link2Link1Link2 -30%magnitude0.22370.25670.04700.1722 -10%magnitude0.22780.27100.04910.1818 0%magnitude0.22780.30240.04910.2050 10%magnitude0.23220.30400.05160.2053 30%magnitude0.23420.31190.05310.2105 10%offset0.24290.34890.05770.2350 30%offset0.27830.48150.07050.3288 10%magnitude,10%offset0.26410.30230.06260.2021 thetimedelay.FuturestudieswillconsiderthedevelopmentofOFBcontrollersforuncertain nonlinearsystemswithunknowntime-varyinginputdelays. 6.7Conclusion AcontinuousOFBcontrollerisdevelopedforuncertainsecond-ordernonlinearsystems affectedbytime-varyinginputdelaysandadditiveboundeddisturbances.Thedelayisassumed tobeboundedandslowlyvarying.ADNN-basedobserverworksinjunctionwiththecontroller toprovideanestimateoftheunmeasurablestate.ALyapunov-basedstabilityanalysisutilizing LKfunctionalsisusedtoprovesimultaneouslyUUBestimationoftheunmeasurablestateand UUBtrackinginthepresenceofmodeluncertainty,disturbancesandtimedelays.Numerical simulationsdemonstratetheperformanceoftheproposedmethod. 95

PAGE 96

Figure6-1.Simulationresultswith10%magnitudeand10%offsetvarianceintime-delay 96

PAGE 97

CHAPTER7 CONCLUSIONANDFUTUREWORKS Thischapterconcludesthedissertationbydiscussingthemaincontributionsdevelopedin eachchapter.Limitationsandimplementationissuesoftheworkarediscussedtoopenpossible futureresearchdirections. 7.1DissertationSummary ThisworkfocusesonvariousapplicationsofDNNstocontrolcontinuous-timenonlinear systems.TheuniversalapproximationpropertyofDNNs,equippedwiththeabilitytoapproximatedynamicsystems,enablenewopportunitiestoembedDNNsincontrolstructures.The structuraldesignandupdatelawsforDNNsdependoneachparticularapplication.InChapter3, aDNNisdesignedasanidentier,butinChapters4-6,anobserverdesignbasedonDNNshas beenused.AllDNNsaretrainedonlinetoapproximatesystemuncertaintiesbyweightupdate lawsdevelopedbasedonthestabilityanalysis.Fordisturbancerejectionpurposes,DNNscan bemodiedwiththeadditionofsomepurerobustterms.Identication/estimationandtracking controlobjectivesareconsideredineachchapter. ThefocusofChapter3istodevelopanidenticationbasedadaptivetrackingcontroller foraclassofcontinuous-timeuncertainnonlinearsystemswithadditiveboundeddisturbances. Thisworkovercomesthelimitationofpreviousworkswherecontrollersareeitherdiscretetimeand/oryieldaUUBstabilityresultduetothepresenceofdisturbancesandunknown approximationerrors.ADNNisusedtoapproximatethenonlinearuncertaindynamics,asliding modeincludedintheDNNstructureaccountsforthedisturbancesandreconstructionerrors toobtainanasymptoticidenticationresult.Inaddition,theDNNidentieristrainedonline. TheasymptotictrackingresultismadepossiblebycombiningacontinuousRISEfeedback termwithaNNfeedforwardterm.Asimulationdemonstratestheperformanceoftheproposed identierandcontroller.Althoughtheproposedmethodguaranteesasymptoticidenticationand asymptotictracking,alimitationofthecontrolleristhattheinputgainmatrixisrequiredtobe exactlyknown,andsystemstatesarecompletelymeasurable. 97

PAGE 98

ThedevelopmentoftheobserverinChapter4ismotivatedbytheneedtoestimateinaccessiblesystemstateswhenfullstatefeedbackisnotavailable.Incontrasttopurelyrobust feedbackmethodsinliterature,aDNN-basedrobustadaptiveapproachisdeveloped.TheobserverstructureconsistsofaDNNtoestimatethesystemdynamicson-line,adynamiclterto estimatetheunmeasurablestateandaslidingmodefeedbacktermtoaccountformodelingerrors andexogenousdisturbances.Theobservedstatesareproventoasymptoticallyconvergetothe systemstatesthoughLyapunov-basedanalysis.Simulationsandexperimentsonatwo-linkrobot manipulatorareperformedtoshowtheeffectivenessoftheproposedmethodincomparisonto severalotherstateestimationmethods.ThedevelopedobserverinChapter4motivatedtheOFB controldevelopmentillustratedinChapters5and6. InChapter5,aDNN-basedobserver-controllerisdevelopedforuncertainnonlinearsystems affectedbyboundedexternaldisturbances,toachieveatwo-foldresult:asymptoticestimation ofunmeasurablestatesandasymptotictrackingcontrol.AcombinationofaNNfeedforward term,alongwithestimatedstatefeedbackandslidingmodetermsaredesignedforthecontroller. ThismethodisanadaptiverobustOFBmethodwhichisshownbyexperimentstoreducehighfrequencycontentsignalsandimprovetrackingresultsincomparisonwithcomparedpurely robustmethods.Limitationsofthemethod,however,aretherequirementoftheknowledgeofthe inputgainmatrixandthediscontinuityofthecontroller. AnOFBcontrolmethodforuncertainnonlinearsystemswithexogenousdisturbancesand time-varyinginputdelaysarepresentedinChapter6.Todevelopthisapproach,boththeneed tocompensateforthelackofthesystemstateandtheneedtodevelopsomeformofprediction ofthenonlineardynamicsarerequiredsimultaneously.Thedelayisassumedtobeboundedand slowlyvarying.ADNN-basedobserverisusedtoprovideasurrogatefortheinaccessiblestate,a predictorisutilizedtoinjectadelayfreecontrolintotheanalysis,andaLyapunov-basedstability analysisfacilitatedbyLKfunctionalsisusedtoproveUUBestimationoftheunmeasurable stateandUUBtrackingresults.Acontinuouscontrollerisdeveloped,andtherequirementof knowledgeoftheinputgainmatrixinChapters3and5isrelaxedinthischapter. 98

PAGE 99

7.2FutureWork ThisworkillustratesthatDNNscanbesuccessfullyappliedtofeedbackcontrol.While thedevelopedmethodsarefairlygeneralandapplicabletoawiderangeofsystems,several limitationsstillexist.Thissectiondiscussestheopentheoreticalproblems,implementation issues,andfutureresearchdirections. 1.Inallnonlinearsystemsconsideredinthisdissertation,disturbancesareassumedtobe boundedandsufcientlysmooth.ApracticallymotivatedproblemishowtoapplyDNNs todesignnonlinearidentiers,observers,controllersfornonlinearsystemsaffectedby stochasticdisturbances.Theparallelsofthedevelopedresultstostochasticnonlinear systemsshouldbepursued. 2.InChapter6,time-varyingdelaysareassumedtobebounded,continuous,slowlyvarying andexactlyknown.Futureeffortsshouldtrytorelaxtheseassumptionsbyconsideringthe casewheretimedelaysarerandom,unknown,and/orthedelayappearsbothinthecontrol inputandsystemstates. 3.InChapter5,thedevelopedOFBcontrollerisadiscontinuouscontrollerwhichcancause chatteringandrequiresinnitecontrolbandwidth.Howtodesignacontinuouscontroller toobtainasymptoticresultsforuncertainnonlinearsystemsaffectedbyexogenous boundeddisturbancesandthelackoffull-statefeedbackremainsanopenproblem. 4.InChapters3and5,theinputgainmatrixisassumedtobeexactlyknowntoobtain anasymptoticerrorconvergence.Undersuitableconditions,istheasymptoticerror convergenceachievablewithouttheknowledgeoftheinputgainmatrix? 5.Tothebestofauthor'sknowledge,allcontrollersinliteratureforuncertainnonlinear systemswithatime-varyinginputdelayonlyobtainUUBresults.Apotentialfull-state feedbackorOFBcontrollerforthesesystemstoachieveanasymptoticresultisstillan openproblem. 6.InChapter4,anextensionoftheDNN-basedobserverfor n th ordernonlinearsystemsisintroduced.Inthismethod,however,fullaccesstosystemstatesexceptforthe 99

PAGE 100

highest-orderstateisrequired.Relaxingthisassumptionwhilestillobtainingasymptotic estimationremainsanopenproblem. 7.Thegainconditionin6canbesatisedonlyifthetimedelayissufcientlysmall andslowlyvaryingandtheapproximationoftheinputgainmatrixissufcientlygood.Is thesystemstillstableandisthetrackingobjectiveachievedifthedelayedtimeislongand fastchangingornoenoughknowledgetomakeasufcientgoodguessfortheinputgain matrix?Cantheapproximationmatrixadaptivelyapproximatefortheunknowninputgain matrix?Allofthesequestionscouldbeexploredinfutureefforts. 100

PAGE 101

APPENDIXA DYNAMICNEURALNETWORK-BASEDROBUSTIDENTIFICATIONANDCONTROLOF ACLASSOFNONLINEARSYSTEMS A.1ProofoftheInequalityinEq.3 UsingEq.3andthetriangleinequalityinEq.3yields k h k k W k k s 1 k + k s 2 k + k W k s V T x )]TJ/F53 11.9552 Tf 10.95 0 Td [(s V T x )]TJ/F53 11.9552 Tf 10.949 0 Td [(s 0 V T x V T x + k e k + k d k + W s 0 V T x V k x k k W k )]TJ/F57 11.9552 Tf 5.475 -9.642 Td [(k s 1 k + k s 2 k + s V T x )]TJ/F53 11.9552 Tf 10.949 0 Td [(s V T x + s 0 V T x V T x A + k e k + k d k + )]TJ/F57 11.9552 Tf 5.476 -9.642 Td [(k W k + W s 0 V T x )]TJ/F57 11.9552 Tf 5.476 -9.642 Td [(k V k + V k x k h whereAssumptions3.2,3.3-3.5,thepropertiesofthesamplestate x t ,theprojectionbounds ontheweightestimatesinEq.3areused.Thebound h 2 R iscomputedbyusingtheupper boundsofalltermsinEq.A. A.2ProofoftheInequalityinEq.3 UsingtheupdatelawsdesigninginEq.3andthetriangleinequalityinEq.3 yields N k A s + a I k k r k + a k e k + k W k s V T x k V k k r k + a k e k + W s V T x d + W s 0 V T x d V k x d k + k e k k A s + a I k k r k + a k e k + k W k s V T x k V k k r k + a k e k + k e k A + k x k k G 1 k s V T x s V T x d + k G 2 k W 2 s 0 V T x d s 0 V T x k x d kk x k 101

PAGE 102

Usingthedenitionof z t inEq.3andthefactthat k x k k z k ; k e k k z k ; k r k k z k ,the expressioninAcanberewrittenas N )]TJ/F53 11.9552 Tf 10.449 -9.69 Td [(l max A s + a I + k W k s V T x k V k a + 1 + 1 k z k + k z k k G 1 k s V T x s V T x d + k G 2 k W 2 s 0 V T x d s 0 V T x k x d kk x k z 1 k z k A whereAssumptions3.3-3.5,thepropertiesofthesamplestate x t ,theprojectionboundson theweightestimatesinEq.3areused.Thebound z 1 2 R iscomputedbyusingtheupper boundsofalltermsinEq.A. A.3ProofoftheInequalityinEqs.3and3 Usingthetriangleinequalityforthefollowingequationyields N D = d + e ; k N D k d + k e k z 2 ; N D d + k e k z 4 ; whereAssumptions3.2and3.5areused.Thebound z 2 ; z 4 2 R arecomputedbyusingthe upperboundsoftherstandsecondderivativesofthedisturbanceandthereconstructionerror. Similarly,theterm N B t anditsderivativecanbeupper-boundedasfollow N B = W T s 0 V T x V T x d )]TJ/F22 11.9552 Tf 14.374 2.379 Td [( W T s 0 V T x d V T x d ; k N B k k W k s 0 V T x k V kk x d k + W s 0 V T x d V k x d k z 3 ; N B = W T s 0 V T x V T x d + W T s 0 V T x V T x d )]TJ/F22 11.9552 Tf 14.374 4.998 Td [( W T s 0 V T x d V T x d )]TJ/F22 11.9552 Tf 14.375 2.379 Td [( W T s 0 V T x d V T x d )]TJ/F22 11.9552 Tf 14.375 2.379 Td [( W T s 0 V T x d V T x d )]TJ/F22 11.9552 Tf 14.375 2.379 Td [( W T s 0 V T x d V T x d 102

PAGE 103

N B k W k s 0 V T x k V kk x d k + k W k s 0 V T x k V kk x d k + W s 0 V T x d V k x d k + W s 0 V T x d V k x d k + W s 0 V T x d V k x d k + W s 0 V T x d V k x d k A UsingtheEq.3,thefollowingupper-boundsareobtained W k G 1 k s V T x k x k c 1 k z k ; V k G 2 kk x kk x k W s 0 V T x c 2 k z k ; and s 0 V T x s 00 V T x k V kk x k = s 00 V T x k V kk r )]TJ/F53 11.9552 Tf 10.95 0 Td [(a e + x d k c 3 + c 4 k z k ; s 0 V T x d s 00 V T x d V k x d k + V k x d k c 5 + c 6 k z k ; where c i 2 R i = 1 ; 2 ;:; 6 arecomputablepositiveconstants.Finally,theinequalityAcan berewrittenas N B z 5 + z 6 k z k ; wherethebounds z 5 ; z 6 2 R arecomputedbasedontheconstants c i i = 1 ; 2 ;:; 6 ,andthe upper-boundsofallothertermsintherightsideofA. A.4ProofoftheInequalityinEq.3 Integrating3andusing3yields L t = t 0 r T N D + N B )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 1 sgn e )]TJ/F53 11.9552 Tf 9.289 0 Td [(b 2 k z k 2 d t + L 0 : Usingthefactthat r = e + a e yields L t = t 0 e T N D + N B d t )]TJ/F50 11.9552 Tf 10.949 16.273 Td [( t 0 e T b 1 sgn e d t + t 0 a e T N D + N B )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 1 sgn e d t )]TJ/F50 11.9552 Tf 10.95 16.273 Td [( t 0 b 2 k z k 2 d t + L 0 : 103

PAGE 104

Integratingtherstintegralbypartsandintegratingthesecondintegral,yields L t = e T N )]TJ/F25 11.9552 Tf 10.949 0 Td [(e T 0 N 0 )]TJ/F50 11.9552 Tf 10.949 16.272 Td [( t 0 e T )]TJ/F22 11.9552 Tf 8.129 -7.311 Td [( N B + N D d t + b 1 n j = 1 e j 0 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 n j = 1 e j t + t 0 a e T N D + N B )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 sgn e d t )]TJ/F50 11.9552 Tf 10.949 16.272 Td [( t 0 b 2 k z k 2 d t + L 0 : Usingthefactthat k e k n j = 1 e j t ; thefollowingupperboundisobtained L t k e kk N k )]TJ/F25 11.9552 Tf 10.949 0 Td [(e T 0 N 0 + b 1 n j = 1 e j 0 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 k e k + t 0 k e k )]TJ 5.476 0.473 Td [( N B + N D d t + t 0 a k e k k N D k + k N B k )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 d t )]TJ/F50 11.9552 Tf 10.949 16.273 Td [( t 0 b 2 k z k 2 d t + L 0 : Usingtheboundsin3and3,andrearrangingterms,thefollowingexpressionis obtained L t b 1 n j = 1 e j 0 )]TJ/F25 11.9552 Tf 10.949 0 Td [(e T 0 N 0 + L 0 )]TJ/F56 11.9552 Tf 10.95 0 Td [( b 1 )]TJ/F53 11.9552 Tf 10.95 0 Td [(z 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(z 3 k e k )]TJ/F50 11.9552 Tf 10.949 16.272 Td [( t 0 a k e k b 1 )]TJ/F53 11.9552 Tf 10.95 0 Td [(z 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(z 3 )]TJ/F53 11.9552 Tf 12.145 8.093 Td [(z 4 a )]TJ/F53 11.9552 Tf 12.145 8.093 Td [(z 5 a d t )]TJ/F50 11.9552 Tf 10.95 16.272 Td [( t 0 b 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(z 6 k z k 2 d t : Ifthesufcientconditionsin3aresatised,thenthefollowinginequalityholds L t b 1 n j = 1 e j 0 )]TJ/F25 11.9552 Tf 10.949 0 Td [(e T 0 N 0 + L 0 : 104

PAGE 105

APPENDIXB DYNAMICNEURALNETWORK-BASEDGLOBALOUTPUTFEEDBACKTRACKING CONTROLFORUNCERTAINSECOND-ODERNONLINEARSYSTEMS ProofoftheInequalityin5 Integrating5andusing5yields H = t 0 Ld t = t 0 )]TJ/F25 11.9552 Tf 5.475 -9.689 Td [(r T es N D + N B 1 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 sgn x + n + r T tr N D + N B 1 )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 2 sgn e + n + x + e + 2 n T N B 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 3 k z k 2 d t : Usingthefactthat r es = x + n + a x + n and r tr = e + n + a e + n yields H = t 0 x + e + 2 n T N D + N B 1 + N B 2 d t )]TJ/F50 11.9552 Tf 10.949 16.273 Td [( t 0 x + n T b 1 sgn x + n d t )]TJ/F50 11.9552 Tf 10.949 16.273 Td [( t 0 e + n T b 2 sgn e + n d t + t 0 a x + n T N D + N B 1 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 sgn x + n d t + t 0 a e + n T N D + N B 1 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 2 sgn e + n d t )]TJ/F50 11.9552 Tf 10.949 16.273 Td [( t 0 b 3 k z k 2 d t : Integratingtherstintegralbyparts,andintegratingthesecondandthirdintegralsyields H = x + e + 2 n T N )]TJ/F56 11.9552 Tf 10.949 0 Td [([ x 0 + e 0 + 2 n 0 ] T N 0 + b 1 n j = 1 x j 0 + n j 0 )]TJ/F66 11.9552 Tf 10.95 10.162 Td [( x j + n j )]TJ/F50 11.9552 Tf 10.95 16.273 Td [( t 0 x + e + 2 n T )]TJ/F22 11.9552 Tf 8.129 -7.311 Td [( N D + N B d t + b 2 n j = 1 e j 0 + n j 0 )]TJ/F66 11.9552 Tf 10.95 10.162 Td [( e j + n j )]TJ/F50 11.9552 Tf 10.95 16.273 Td [( t 0 b 3 k z k 2 d t + t 0 a e + n T [ N D + N B 1 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 2 sgn e + n ] d t : 105

PAGE 106

Usingthefactthat k x + n k n j = 1 x j + n j and k e + n k n j = 1 e j + n j ; thefollowingupperbound isobtained H k x + e + 2 n kk N k )]TJ/F56 11.9552 Tf 10.949 0 Td [( x 0 + e 0 + 2 n 0 T N 0 + b 1 n j = 1 x j 0 + n j 0 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 k x + n k + b 2 n j = 1 e j 0 + n j 0 )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 2 k e + n k + t 0 a k x + n k N D + N B 1 + N a )]TJ/F53 11.9552 Tf 10.949 0 Td [(b 1 d t + t 0 a k e + n k N D + N B 1 + N a )]TJ/F53 11.9552 Tf 10.95 0 Td [(b 2 d t )]TJ/F50 11.9552 Tf 10.949 16.272 Td [( t 0 b 3 k z k 2 d t : Usingtheboundsin5,basedonthefactthat k x + e + 2 n k k x + n k + k e + n k ,and rearrangingterms,thefollowingexpressionisobtained H b 1 n j = 1 x j 0 + n j 0 + b 2 n j = 1 e j 0 + n j 0 )]TJ/F56 11.9552 Tf 10.949 0 Td [( x 0 + e 0 + 2 n 0 T N 0 )]TJ/F56 11.9552 Tf 10.95 0 Td [( b 1 )]TJ/F53 11.9552 Tf 10.949 0 Td [(z 2 )]TJ/F53 11.9552 Tf 10.95 0 Td [(z 3 )]TJ/F53 11.9552 Tf 10.95 0 Td [(z 4 k x + n k )]TJ/F56 11.9552 Tf 10.949 0 Td [( b 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(z 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(z 3 )]TJ/F53 11.9552 Tf 10.949 0 Td [(z 4 k e + n k )]TJ/F50 11.9552 Tf 10.95 16.272 Td [( t 0 a k x + n k b 1 )]TJ/F53 11.9552 Tf 10.949 0 Td [(z 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(z 3 )]TJ/F53 11.9552 Tf 12.145 8.094 Td [(z 5 a )]TJ/F53 11.9552 Tf 12.145 8.093 Td [(z 6 a d t )]TJ/F50 11.9552 Tf 10.95 16.272 Td [( t 0 a k e + n k b 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(z 2 )]TJ/F53 11.9552 Tf 10.949 0 Td [(z 3 )]TJ/F53 11.9552 Tf 12.144 8.094 Td [(z 5 a )]TJ/F53 11.9552 Tf 12.145 8.094 Td [(z 6 a d t )]TJ/F50 11.9552 Tf 10.95 16.272 Td [( t 0 b 3 )]TJ/F22 11.9552 Tf 10.949 0 Td [(2 z 7 k z k 2 d t : Ifthesufcientconditionsin5aresatised,thenthefollowinginequalityholds H b 1 n j = 1 x j 0 + n j 0 + b 2 n j = 1 e j 0 + n j 0 )]TJ/F56 11.9552 Tf 10.949 0 Td [( x 0 + e 0 + 2 n 0 T N 0 ; H = t 0 Ld t P 0 : B Hence,using5andB,itcanbeshownthat P t 0 : 106

PAGE 107

REFERENCES [1]B.Xian,M.S.deQueiroz,D.M.Dawson,andM.McIntyre,Adiscontinuousoutput feedbackcontrollerandvelocityobserverfornonlinearmechanicalsystems, Automatica vol.40,no.4,pp.695,2004. [2]L.VasiljevicandH.Khalil,Errorboundsindifferentiationofnoisysignalsbyhigh-gain observers, Systems&ControlLetters ,vol.57,no.10,pp.856,2008. [3]D.WhiteandD.Sofge, Handbookofintelligentcontrol:neural,fuzzy,andadaptive approaches .VanNostrandReinholdCompany,1992. [4]W.T.Miller,R.Sutton,andP.Werbos, NeuralNetworksforControl .MITPress,1990. [5]F.L.Lewis,Nonlinearnetworkstructuresforfeedbackcontrol, AsianJ.Control ,vol.1, no.4,pp.205,1999. [6]D.PhamandX.Liu,Dynamicsystemidenticationusingpartiallyrecurrentneural networks, J.Syst.Eng. ,vol.2,no.2,pp.90,1992. [7]M.Gupta,L.Jin,andN.Homma, Staticanddynamicneuralnetworks:fromfundamentals toadvancedtheory .Wiley-IEEEPress,2003. [8]K.FunahashiandY.Nakamura,Approximationofdynamicsystemsbycontinuous-time recurrentneuralnetworks, NeuralNetworks ,vol.6,pp.801,1993. [9]M.PolycarpouandP.Ioannou,Identicationandcontrolofnonlinearsystemsusingneural networkmodels:Designandstabilityanalysis, SystemsReport91-09-01,Universityof SouthernCalifornia ,1991. [10]K.NarendraandK.Parthasarathy,Identicationandcontrolofdynamicalsystemsusing neuralnetworks, IEEETrans.NeuralNetworks ,vol.1,no.1,pp.4,1990. [11]E.Kosmatopoulos,M.Polycarpou,M.Christodoulou,andP.Ioannou,High-orderneural networkstructuresforidenticationofdynamicalsystems, IEEETrans.NeuralNetworks vol.6,no.2,pp.422431,1995. [12]G.A.RovithakisandM.A.Christodoulou,Adaptivecontrolofunknownplantsusing dynamicalneuralnetworks, IEEETrans.Syst.ManCybern. ,vol.24,pp.400,1994. [13]A.Delgado,C.Kambhampati,andK.Warwick,Dynamicrecurrentneuralnetwork forsystemidenticationandcontrol, IEEProc.Contr.Theor.Appl. ,vol.142,no.4,pp. 307,1995. [14]A.S.Poznyak,W.Yu,E.N.Sanchez,andJ.P.Perez,Nonlinearadaptivetrajectory trackingusingdynamicneuralnetworks, IEEETrans.NeuralNetworks ,vol.10,no.6,pp. 1402,Nov1999. 107

PAGE 108

[15]A.Poznyak,E.Sanchez,andW.Yu, Differentialneuralnetworksforrobustnonlinear control:identication,stateestimationandtrajectorytracking .WorldScienticPubCo Inc,2001. [16]X.Ren,A.Rad,P.Chan,andW.Lo,Identicationandcontrolofcontinuous-time nonlinearsystemsviadynamicneuralnetworks, IEEETrans.Ind.Electron. ,vol.50,no.3, pp.478,2003. [17]J.Hopeld,Neuronswithgradedresponsehavecollectivecomputationalpropertieslike thoseoftwo-stateneurons, Proc.Nat.Acad.Sci.U.S.A. ,vol.81,no.10,p.3088,1984. [18]X.LiandW.Yu,Dynamicsystemidenticationviarecurrentmultilayerperceptrons, Inf. Sci.Inf.Comput.Sci. ,vol.147,pp.45,2002. [19]H.Dinh,S.Bhasin,andW.E.Dixon,Dynamicneuralnetwork-basedrobustidentication andcontrolofaclassofnonlinearsystems,in Proc.IEEEConf.Decis.Control ,Atlanta, GA,2010,pp.5536. [20]S.Bhasin,M.Johnson,andW.E.Dixon,Amodel-freerobustpolicyiterationalgorithm foroptimalcontrolofnonlinearsystems,in Proc.IEEEConf.Decis.Control ,Atlanta,GA, 2010,pp.3060. [21]J.-J.Slotine,J.Hedrick,andE.A.Misawa,Onslidingobserversfornonlinearsystems,in AmericanControlConf. ,1986,pp.1794. [22]C.CanudasDeWitandJ.-J.Slotine,Slidingobserversforrobotmanipulators, Automatica ,vol.27,pp.859,1991. [23]M.Mohamed,T.Karim,andB.Safya,Slidingmodeobserverfornonlinearmechanical systemssubjecttononsmoothimpacts,in Int.Multi-Conf.Syst.SignalsDevices ,2010. [24]K.W.LeeandH.K.Khalil,Adaptiveoutputfeedbackcontrolofrobotmanipulatorsusing high-gainobserver, INT.J.CONTROL ,vol.67,pp.869,1997. [25]E.ShinandK.Lee,Robustoutputfeedbackcontrolofrobotmanipulatorsusinghigh-gain observer,in IEEEInt.Conf.onControlandAppl. ,1999. [26]A.Astol,R.Ortega,andA.Venkatraman,Agloballyexponentiallyconvergentimmersionandinvariancespeedobserverforndegreesoffreedommechanicalsystems,in Proc. Chin.ControlConf. ,2009. [27]N.LotandM.Namvar,Globaladaptiveestimationofjointvelocitiesinroboticmanipulators, IETControlTheoryAppl. ,vol.4,pp.2672,2010. [28]J.Davila,L.Fridman,andA.Levant,Second-ordersliding-modeobserverformechanical systems, IEEETrans.Autom.Control ,vol.50,pp.1785,2005. [29]D.Dawson,Z.Qu,andJ.Carroll,Onthestateobservationandoutputfeedbackproblems fornonlinearuncertaindynamicsystems, Syst.Contr.Lett. ,vol.18,pp.217,1992. 108

PAGE 109

[30]Y.H.KimandF.L.Lewis,Neuralnetworkoutputfeedbackcontrolofrobotmanipulators, IEEETrans.Robot.Autom. ,vol.15,pp.301,1999. [31]H.BerghuisandH.Nijmeijer,Apassivityapproachtocontroller-observerdesignfor robots, IEEETrans.Robot.Autom. ,vol.9,pp.740,1993. [32]K.D.Do,Z.Jiang,andJ.Pan,Aglobaloutput-feedbackcontrollerforsimultaneous trackingandstabilizationforunicycle-typemobilerobots, IEEETrans.Robot.Autom. vol.20,pp.589,2004. [33]S.Y.Lim,D.M.Dawson,andK.Anderson,Re-examiningthenicosia-tomeirobot observer-controllerforabacksteppingperspective, IEEETrans.ControlSyst.Technol. vol.4,pp.304,1996. [34]M.A.ArteagaandR.Kelly,Robotcontrolwithoutvelocitymeasurements:Newtheory andexperimentresults, IEEETrans.Robot.Autom. ,vol.20,pp.297,2004. [35]K.KanekoandR.Horowitz,Repetitiveandadaptivecontrolofrobotmanipulatorswith velocityestimation, IEEETrans.Robot.Autom. ,vol.12,pp.204,1997. [36]T.Burg,D.Dawson,J.Hu,andM.deQueiroz,Anadaptivepartialstatefeedback controllerforrledrobotmanipulators, IEEETrans.Autom.Control ,vol.41,pp.1024 1031,1996. [37]T.Burg,D.M.Dawson,andP.Vedagarbha,Aredesigneddcalcontrollerwithoutvelocity measurements:Theoryanddemonstration, Robotica ,vol.15,no.4,pp.337,1997. [38]F.Zhang,D.M.Dawson,M.S.deQueiroz,andW.E.Dixon,Globaladaptiveoutput feedbacktrackingcontrolofrobotmanipulators, IEEETrans.Automat.Control ,vol.45, pp.1203,2000. [39]S.SeshagiriandH.Khalil,Outputfeedbackcontrolofnonlinearsystemsusingrbfneural networks, IEEETrans.NeuralNetw. ,vol.11,pp.69,2000. [40]J.Y.ChoiandJ.A.Farrell,Adaptiveobserverbacksteppingcontrolusingneuralnetwork, IEEETrans.NeuralNetw. ,vol.12,pp.1103,2001. [41]A.J.Calise,N.Hovakimyan,andM.Idan,Adaptiveoutputfeedbackcontrolofnonlinear systemsusingneuralnetworks, Automatica ,vol.37,no.8,pp.1201,2001. [42]N.Hovakimyan,F.Nardi,A.Calise,andN.Kim,Adaptiveoutputfeedbackcontrolof uncertainnonlinearsystemsusingsingle-hidden-layerneuralnetworks, IEEETrans. NeuralNetworks ,vol.13,no.6,pp.1420,Nov.2002. [43]S.IslamandP.Liu,Robustadaptivefuzzyoutputfeedbackcontrolsystemforrobot manipulators, IEEE/ASMETrans.Mechatron. ,vol.16,pp.288,2011. [44]J.A.CookandB.K.Powell,Modelingofaninternalcombustionengineforcontrol analysis, IEEEControlSyst.Mag ,pp.20,1988. 109

PAGE 110

[45]M.KaoandJ.J.Moskwa,Turbochargeddieselenginemodelingfornonlinearengine controlandstateestimation, ASMEJ.Dyn.Syst.,Meas.,Control ,vol.117,pp.20, 1995. [46]F.L.Lewis,R.Selmic,andJ.Campos, Neuro-FuzzyControlofIndustrialSystemswith ActuatorNonlinearities .Philadelphia,PA,USA:SocietyforIndustrialandApplied Mathematics,2002. [47]G.NiemeyerandJ.-J.Slotine,Telemanipulationwithtimedelays, Int.J.Robot.Res. vol.23,pp.873,2004. [48]A.Aziminejad,M.Tavakoli,R.V.Patel,andM.Moallem,Stabilityandperformancein delayedbilateralteleoperation:Theoryandexperiments, ControlEng.Pract. ,vol.16,pp. 1329,2008. [49]Y.Gu,C.Zhang,andK.T.Chong,Adaptivepassivecontrolwithvaryingtimedelay, Simul.Model.Pract.Theory ,vol.18,pp.1,2010. [50]F.Hashemzadeh,I.Hassanzadeh,M.Tavakoli,andG.Alizadeh,Adaptivecontrolfor statesynchronizationofnonlinearhapticteleroboticsystemswithasymmetricvaryingtime delays, J.Intell.Robot.Syst. ,2012. [51]N.N.Krasovskii, Stabilityofmotion .StanfordUniversityPress,1963. [52]X.LiandC.deSouza,Delay-dependentrobuststabilityandstabilizationofuncertain lineardelaysystems:alinearmatrixinequalityapproach, IEEETrans.Autom.Control vol.42,no.8,pp.1144,Aug1997. [53]V.B.Kolmanovskii,S.-I.Niculescu,andJ.-P.Richard,Ontheliapunov-krasovskii functionalsforstabilityanalysisoflineardelaysystems, Int.J.Control ,vol.72,pp.374 384,1999. [54]B.S.Razumikhin,Onthestabilityofsystemswithadelay, Prikl.Mat.Meh ,vol.20,pp. 500,1956. [55],Applicationofliapunov'smethodtoproblemsinthestabilityofsystemswitha delay, Automat.iTelemeh ,vol.21,pp.740,1960. [56]M.Jankovic,ControlLyapunov-Razumikhinfunctionsandrobuststabilizationoftime delaysystems, IEEETrans.Autom.Control ,vol.46,no.7,pp.1048,2001. [57]R.BellmanandK.L.Cooke, Differential-DifferenceEquations .AcademicPress,1963. [58]L.E.ElsgoltsandS.B.Norkin, Introductiontothetheoryandapplicationofdifferential equationswithdeviatingarguments .AcademicPress,1973,vol.105. [59]F.Mazenc,S.Mondie,R.Francisco,P.Conge,I.Lorraine,andF.Metz,Globalasymptotic stabilizationoffeedforwardsystemswithdelayintheinput, IEEETrans.Autom.Control vol.49,,pp.844,2004. 110

PAGE 111

[60]B.Chen,X.Liu,andS.Tong,Robustfuzzycontrolofnonlinearsystemswithinputdelay, Chaos,Solitons&Fractals ,vol.37,no.3,pp.894,2008. [61]M.Krstic,Inputdelaycompensationforforwardcompleteandstrict-feedforward nonlinearsystems, IEEETrans.Autom.Control ,vol.55,pp.287,feb.2010. [62]B.Castillo-Toledo,S.DiGennaro,andG.Castro,Stabilityanalisysforaclassofsampled nonlinearsystemswithtime-delay,in Proc.IEEEConf.Decis.Control ,2010,pp.1575 [63]N.Sharma,S.Bhasin,Q.Wang,andW.E.Dixon,Predictor-basedcontrolforanuncertain Euler-Lagrangesystemwithinputdelay, Automatica ,vol.47,no.11,pp.2332, 2011. [64]L.Guo,Hinn;outputfeedbackcontrolfordelaysystemswithnonlinearandparametric uncertainties, Proc.IEEControlTheoryAppl. ,vol.149,no.3,pp.226,2002. [65]W.Li,Y.Dong,andX.Wang,Robusthinn;controlofuncertainnonlinearsystems withstateandinputtime-varyingdelays,in Proc.Chin.ControlDecis.Conf. ,2010,pp. 317. [66]N.Bekiaris-LiberisandM.Krstic,Compensationoftime-varyinginputdelayfornonlinear systems,in Mediterr.Conf.ControlandAutom. ,Corfu,Greece,2011. [67]M.Krstic,Lyapunovstabilityoflinearpredictorfeedbackfortime-varyinginputdelay, IEEETrans.Autom.Control ,vol.55,pp.554,2010. [68]N.Fischer,R.Kamalapurkar,N.Fitz-Coy,andW.E.Dixon,Lyapunov-basedcontrolof anuncertainEuler-Lagrangesystemwithtime-varyinginputdelay,in AmericanControl Conference ,Montral,Canada,June2012,pp.3919. [69]A.AilonandM.Gil,Stabilityanalysisofarigidrobotwithoutput-basedcontrollerand timedelay, Systems&ControlLetters ,vol.40,no.1,pp.31,2000. [70]A.Ailon,R.Segev,andS.Arogeti,Asimplevelocity-freecontrollerforattituderegulation ofaspacecraftwithdelayedfeedback, IEEETrans.Autom.Control ,vol.49,no.1,pp. 125,2004. [71]A.Ailon,Asymptoticstabilityinaexible-jointrobotwithmodeluncertaintyandmultiple timedelaysinfeedback, J.FranklinInst. ,vol.341,no.6,pp.519,2004. [72]A.Poznyak,W.Yu,E.Sanchez,andH.Sira-Ramirez,Robustidenticationbydynamic neuralnetworksusingslidingmodelearning, Appl.Math.Comput.Sci. ,vol.8,pp.135 144,1998. [73]J.HuangandF.Lewis,Neural-networkpredictivecontrolfornonlineardynamicsystems withtime-delay, IEEETrans.NeuralNetworks ,vol.14,no.2,pp.377,2003. 111

PAGE 112

[74]B.Xian,D.M.Dawson,M.S.deQueiroz,andJ.Chen,Acontinuousasymptotictracking controlstrategyforuncertainnonlinearsystems, IEEETrans.Autom.Control ,vol.49,pp. 1206,2004. [75]P.M.Patre,W.MacKunis,K.Kaiser,andW.E.Dixon,Asymptotictrackingforuncertain dynamicsystemsviaamultilayerneuralnetworkfeedforwardandRISEfeedbackcontrol structure, IEEETrans.Automat.Control ,vol.53,no.9,pp.2180,2008. [76]H.T.Dinh,R.Kamalapurkar,S.Bhasin,andW.E.Dixon,Dynamicneuralnetwork-based robustobserversforsecond-orderuncertainnonlinearsystems,in Proc.IEEEConf.Decis. Control ,Orlando,FL,2011,pp.7543. [77]H.T.Dinh,S.Bhasin,D.Kim,andW.E.Dixon,Dynamicneuralnetwork-basedglobal outputfeedbacktrackingcontrolforuncertainsecond-ordernonlinearsystems,in American ControlConference ,Montral,Canada,June2012,pp.6418. [78]S.Haykin, NeuralNetworksandLearningMachines ,3rded.UpperSaddleRiver,NJ: PrenticeHall,2008. [79]D.R.HushandB.G.Horne,Progressinsupervisedneuralnetworks, IEEESignal Process.Mag. ,vol.10,no.1,pp.8,1993. [80]J.Hopeld,Neuralnetworksandphysicalsystemswithemergentcollectivecomputational abilities,in Proc.Nat.Acad.Sci.USA ,vol.79,1982,pp.2554. [81]E.Sontag,Neuralnetsassystemmodelsandcontrollers,in Proc.7thYaleworkshopon AdaptiveandLearningSystems ,1992,pp.73. [82]I.Sandberg,Approximationtheoremsfordiscrete-timesystems, IEEETrans.onCircuits andSyst. ,vol.38,pp.564,1991. [83],Uniformapproximationandthecirclecriterion, IEEETrans.onAutom.Control vol.38,pp.1450,1993. [84],Uniformapproximationofmultidimensionalmyopicmaps, IEEETrans.onCircuits andSyst. ,vol.44,pp.477,1997. [85]K.Hornick,Approximationcapabilitiesofmultilayerfeedforwardnetworks, Neural Networks ,vol.4,pp.251,1991. [86]G.Cybenko,Approximationbysuperpositionsofasigmoidalfunction, Math.Control SignalsSyst. ,vol.2,pp.303,1989. [87]A.Barron,Universalapproximationboundsforsuperpositionsofasigmoidalfunction, IEEETrans.Inf.Theory ,vol.39,no.3,pp.930,1993. [88]W.E.Dixon,A.Behal,D.M.Dawson,andS.Nagarkatti, NonlinearControlofEngineering Systems:ALyapunov-BasedApproach .Birkhuser:Boston,2003. 112

PAGE 113

[89]M.Krstic,P.V.Kokotovic,andI.Kanellakopoulos, NonlinearandAdaptiveControlDesign JohnWiley&Sons,1995. [90]P.M.Patre,W.Mackunis,C.Makkar,andW.E.Dixon,Asymptotictrackingforsystems withstructuredandunstructureduncertainties, IEEETrans.ControlSyst.Technol. ,vol.16, pp.373,2008. [91]A.Filippov,Differentialequationswithdiscontinuousright-handside, Am.Math.Soc. Transl. ,vol.42no.2,pp.199,1964. [92]A.F.Filippov, DifferentialEquationswithDiscontinuousRight-handSides .Kluwer AcademicPublishers,1988. [93]G.V.Smirnov, Introductiontothetheoryofdifferentialinclusions .AmericanMathematical Society,2002. [94]J.P.AubinandH.Frankowska, Set-valuedanalysis .Birkhuser,2008. [95]R.LeineandN.vandeWouw,Non-smoothdynamicalsystems,in Stabilityand ConvergenceofMechanicalSystemswithUnilateralConstraints ,ser.LectureNotesin AppliedandComputationalMechanics.SpringerBerlin/Heidelberg,2008,vol.36,pp. 59. [96]F.H.Clarke, Optimizationandnonsmoothanalysis .SIAM,1990. [97]D.ShevitzandB.Paden,Lyapunovstabilitytheoryofnonsmoothsystems, IEEETrans. Autom.Control ,vol.39no.9,pp.1910,1994. [98]B.PadenandS.Sastry,AcalculusforcomputingFilippov'sdifferentialinclusionwith applicationtothevariablestructurecontrolofrobotmanipulators, IEEETrans.Circuits Syst. ,vol.34no.1,pp.73,1987. 113

PAGE 114

BIOGRAPHICALSKETCH HuyenT.DinhwasborninApril,1984inThaiNguyen,Vietnam.Shereceivedher BachelorofEngineeringdegreeinMechatronicEngineeringin2006fromHanoiUniversityof TechnologyHUT,Vietnam.SheworkedasalecturerinMechanicalEngineeringdepartmentof UniversityofTransportandCommunications,VietnamfromSeptember2006.Shethenjoined theNonlinearControlsandRoboticsNCRresearchgroupattheUniversityofFloridaUF topursueherdoctoralresearchundertheadvisementofDr.WarrenDixonsinceAugust2008 andcompletedherPh.D.inAugust2012.FromSeptember2012onwardshewillbeanassistant professorintheUniversityofTransportandCommunications,Vietnam. 114