Adaptive and Robust Controls for Uncertain Linear and Nonlinear Systems

MISSING IMAGE

Material Information

Title:
Adaptive and Robust Controls for Uncertain Linear and Nonlinear Systems
Physical Description:
1 online resource (124 p.)
Language:
english
Creator:
Son, Jungeun
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
Latchman, Haniph A
Committee Co-Chair:
Khargonekar, Pramod
Committee Members:
Mcnair, Janise Y
Fitz-Coy, Norman G

Subjects

Subjects / Keywords:
adaptive -- control -- linear -- network -- neural -- nonlinear -- synchrony
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre:
Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
The presence of various uncertainties in control systems is one of the main challenging issues to engineers since this problem in real applications gives rise to instability and degradation of tracking performance. This dissertation addresses analysis and control design problems involving linear and nonlinear systems with structured and unstructured uncertainties. In the first problem, the critical direction theory (CDT) an H-infinity design is proposed as a systematic controller synthesis methodology for systems with parametric uncertainties. In the literature, a significant conservatism issue in this study of such uncertain systems withe respect to robust control design has been reported. First, the mechanisms leading to conservatism in an earlier approach utilizing a constant overbounding uncertainty template are analyzed. Then, motivated by the desire to decrease the conservative uncertainty, a static weight approach based on the CDT is examined using a constant value to describe the uncertainties over all frequencies. This approach shows that the CDT combined with H-infinity is effective and promising for less conservative stability and controller synthesis for systems with uncertainties. The second is a study of the problem of the robustness of MIMO systems with parametric uncertainties. Two criteria that are used to assess the stability of MIMO systems are evaluated. A numerical analysis provides an empirical demonstration of the relationship between the two stability margins. The dissertation then proceeds to the study of design methods for systematic adaptive controls combined with robustness for uncertain nonlinear systems. The most common technique in designing a controller for a nonlinear system is to assume the nonlinear dynamics are exactly known, but in fact system nonlinearities are unknown in real applications. To overcome the uncertainties in nonlinear systems, a wavelet neural network (WNN) identifier with a sliding mode controller (SMC) proposed. Then, a Robust Integral of the Sign of the Error (RISE) feedback control based on WNN identifier is used to attenuate the effect of the wavelet network approximation error and exogenous disturbance. The proposed method ensures asymptotic tracking of a desired reference signal. A simulation of controlling memristor based chaotic system illustrates the effectiveness of the proposed approach and its performance in the presence of disturbances, and the result is supported through rigorous Lyapunov-based stability proofs. Then, an adaptive robust tracking control scheme using a multi-layer neural network (NN) for a class of nonlinear dynamic systems with unknown time varying states is addressed. Typical adaptive NN backstepping controllers for uncertain nonlinear systems with time-delay give rise to computation complexity caused by the the repeated derivatives of virtual controllers and nonlinear functions. Moreover, the combined techniques usually result in only uniformly ultimately bounded (UUB) stability due to the inherent NN approximation error. First, we develop a new DCAL formulation which avoids the “explosion of complexity” caused by the general NN backstepping scheme to compensate for nonlinear system uncertainties, bounded system disturbances, and unknown state time delays. Then, using a Lyapunov-Krasovskii (LK) functional, it is shown that the proposed controller renders the class of uncertain nonlinear time-delay systems asymptotically stable. Finally, the effectiveness of the proposed multiple controller design methodologies is demonstrated for the 1:1 in-phase synchrony problem with unidirectional coupled interneuron (UCI).
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 Jungeun Son.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Latchman, Haniph A.
Local:
Co-adviser: Khargonekar, Pramod.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-02-28

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2012
System ID:
UFE0044609:00001


This item is only available as the following downloads:


Full Text

PAGE 1

ADAPTIVEANDROBUSTCONTROLSFORUNCERTAINLINEARANDNONLINEARSYSTEMSByJUNGE.SONADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

PAGE 2

c2012JungE.Son 2

PAGE 3

Idedicatethisworktomyparents. 3

PAGE 4

ACKNOWLEDGMENTS Iwishtoexpressmysinceregratitudetomyadvisors,Dr.HaniphLatchmanandDr.PramodKhargonekar,fortheirsupportandguidanceduringmyPh.D.study.Theygavemealotoffreedomandexibilitytochoosemyresearchtopic.Thisworkwouldnothavebeenpossiblewithouttheirencouragementandsupportofmyeffort.AlthoughIstillhavemuchtolearn,IfeelcondentasIbeginmycareerinsystemscontrolandotherareas.IalsowishtothankDr.JaniseMcNairandDr.NormanFitz-Coyforservingonmycommittee,andforprovidingconstructiveideastofurtherdevelopmyresearch.IwouldalsoliketoextendmygratitudetoDr.SachinTalathiforhishelpinprovidingvaluablecommentsonthecomputationalneuralsynchronyinmyresearch.Hegavemetremendousknowledgeandinsightintotheareaoftheneuroscience.Iwouldalsoappreciatemyundergraduateadvisor,Dr.NakYongKo,forguidingmeinbuildingthepathofmyacademicsuccessandformotivatingmetostudyabroad.IalsowishtothankmyLISTlabcolleagues,especiallyDr.KyungwooKimandMr.YoungjoonLee.Ihavehadawonderfultimeandenjoyedthelabatmosphere.Iammostgratefultomyparents,sisters,andbrother-in-lawfortheirsupport,encouragement,understanding,andpatiencewithoutwhichitwouldhavebeenimpossibletocompletethisdissertation.Finally,Ithankmyex-girlfriendsfortheirdevotionandpatience. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTIONANDBACKGROUND ...................... 14 1.1Introduction ................................... 14 1.2RobustControlforUncertainLinearSystems ................ 14 1.2.1Single-InputSingle-OutputSystem .................. 15 1.2.2Multi-InputMulti-OutputSystem .................... 18 1.3AdaptiveandRobustControlsforUncertainNonlinearSystems ...... 22 1.3.1AdaptiveNonlinearControl ...................... 23 1.3.2AdaptiveNeuralNetworkControl ................... 27 1.3.2.1NeuralNetworkBackstepping ............... 27 1.3.2.2WaveletNeuralIdenticationControl ............ 33 1.3.2.3Controlof1:1SynchronyIn-Phase ............ 34 1.4OutlineofDissertation ............................. 35 2ASTATIC-WEIGHTINGH1APPROACHFORLINEARPARAMETRICUNCERTAINTIES 36 2.1Introduction ................................... 36 2.2ProblemFormulation .............................. 37 2.3Preliminaries .................................. 39 2.3.1ReviewOfTheCriticalDirectionTheory ............... 39 2.3.2RobustSynthesisForParametricUncertainSystemsViaH1Design 40 2.4MainResults .................................. 41 2.4.1ComplexImageOfUncertainPlants ................. 41 2.4.2ConservatismOfTheMPRWeightingApproach .......... 44 2.4.3StaticControllerDesignUsingAnExactConstant-WeightingMethod 45 2.4.4RobustStabilityConditions ...................... 47 2.5Examples .................................... 48 2.6Conclusions ................................... 51 3COMPARATIVESTUDYOFSTRUCTUREDSINGULARVALUEANDCRITICALDIRECTIONTHEORYFORUNCERTAINMIMOSYSTEMS ........... 53 3.1Introduction ................................... 53 3.2MathematicalBackground ........................... 54 3.2.1MatrixDecompositions ......................... 54 5

PAGE 6

3.2.2NyquistStabilityforSISOandMIMOTransferfunctions ....... 54 3.3NyquistRobustStabilityMargin ........................ 56 3.3.1Background ............................... 56 3.3.2CriticalDirectionTheoryforSISOSystems ............. 57 3.3.3NyquistRobustStabilityMarginforSISOSystems ......... 58 3.3.4NyquistRobustStabilityMarginforMIMOSystems ......... 59 3.4StructuredSingularValue ........................... 60 3.5ExampleofChemicalReactor ......................... 60 3.5.1TransferFunctionofChemicalReactor ................ 60 3.6Results ..................................... 62 3.7ConclusionRemarks .............................. 63 4WAVELET-BASEDIDENTIFICATIONANDTRACKINGCONTROLFORMEMRISTORBASEDCIRCUIT ................................... 64 4.1INTRODUCTION ................................ 64 4.2SystemDescriptionandPreliminaries .................... 66 4.2.1Memristorbasedchaoticsystem ................... 66 4.2.2WaveletNeuralNetworks ....................... 67 4.3RobustSystemIdenticationofWNN .................... 69 4.4RobustControllerDesignusingRISEFeedbackTerm ........... 72 4.5StabilityAnalysis ................................ 75 4.6Simulation .................................... 77 4.7Conclusion ................................... 77 5ADAPTIVENON-BACKSTEPPINGNEURALCONTROLFORACLASSOFUNCERTAINNONLINEARSYSTEMSWITHUNKNOWNTIME-DELAY .... 79 5.1Introduction ................................... 79 5.2ProblemFormulation .............................. 81 5.3ErrorDynamicSystemDevelopment ..................... 83 5.3.1FilteredErrorsystem .......................... 83 5.3.2FeedforwardNNEstimation ...................... 85 5.3.3Closed-LoopErrorSystem ....................... 85 5.4StabilityAnalysis ................................ 88 5.5Simulation .................................... 94 5.6Conclusion ................................... 95 6CONTROLOFNEURALSYNCHRONYINTHEMUTUALLYCOUPLEDNETWORK 97 6.1Introduction ................................... 97 6.2ModelNeuronsintheMCINetwork ...................... 99 6.2.1ModelDescription ........................... 99 6.2.2Measureof1:1Synchrony ...................... 100 6.3SynchronousErrorDynamicSystemDevelopment ............. 103 6.3.1ProblemFormulation .......................... 103 6.3.2Closed-LoopSystemintheMCINetwork ............... 105 6

PAGE 7

6.3.3RobustControlwithInputConstraint ................. 107 6.4StabilityAnalysis ................................ 107 6.5Simulation .................................... 109 6.5.1In-PhaseSynchronyinDifferentHeterogeneousTypes ....... 109 6.5.2MCINetworkinVaryingFeedbackSynapticCoupling ........ 112 6.6Conclusion ................................... 114 7CONCLUSIONANDFUTUREWORK ....................... 115 7.1Conclusion ................................... 115 7.2FutureWork ................................... 115 REFERENCES ....................................... 117 BIOGRAPHICALSKETCH ................................ 124 7

PAGE 8

LISTOFTABLES Table page 6-1TransitionratesfortheactivationandinactivationvariablesoftheionchannelswithX=m,h,n. ................................... 100 6-2Listofalltheparametersforthemodelconsidered ................ 100 8

PAGE 9

LISTOFFIGURES Figure page 1-1Stabilityanalysisforauncertainsystemg(s)underunitynegativefeedback .. 15 1-2Uncertaintyvaluesetsatafrequency!i:(a)convexcriticalvaluesetVc(!i),(b)non-convexcriticalvaluesetV(!i).Bothguresshowtheworst-sensitivityplantgs(j!i),locatedclosesttothepoint)]TJ /F3 11.955 Tf 9.3 0 Td[(1+j0. ................ 17 1-3E-contoursfortheunstructureduncertainsystemforfrequenciesrangingfrom0.1to10Hz ...................................... 20 1-4SchematicdiagramofthecriticaldirectiontheoryforaMIMOsystem ...... 21 1-5Three-layerperceptronneuralnetwork ....................... 28 1-6PervasivebacksteppingNNcontrolaclassofuncertainnonlinearstrict-feedbacksystems ........................................ 29 1-7Waveletneuralnetworkbasedadaptivenonlinearsystemstructure ....... 33 2-1(a)Negative-feedbackloopincludingtheuncertainsystemp(s)=p0(s)+(s)andacontrollerc(s);(b)unity-negative-feedbackofsystemg(s)=c(s)p(s);(c)mixed-sensitivityapproachtotheuncertainfeedbacksystem. ........ 38 2-2Mappingrelationshipbetweentheoriginalplantp(j!)andthetransformedsystemg(j!)=c(j!)p(j!).Theshadedareasrepresenttheuncertaintyvalue-setV(j!i)(right)andtheimage-setVp(j!i)(left).Thepointg(j!i)in(b)denesadiskofradiusc(!i),whereasitsimagep(j!i)in(a)denesadiskofradiuse(!i).Theoverboundingdiskofradiusm(!i)in(a)circumscribesthediskofradiuse(!i),andleadstomoreconservativerobust-stabilityestimates. 42 2-3Plotsofmanderadiiasfunctionsoftheuncertainty-sizeparameter".Thevaluermax=0.3950isthelimitingvalueofthestaticweightW2(s)foundviaastandardunstructuredH1approach. ........................ 49 2-4Plotsofmanderadiiasfunctionsoftheuncertainty-sizeparameter".Thevaluermax=0.4023isthelimitingvalueofthestaticweightW2(s)foundviaastandardunstructuredH1approach. ........................ 52 3-1NyquistplotofSISOsystem,g0(j!) ........................ 55 3-2NyquistplotofMIMOsystem,G0(j!) ........................ 56 3-3SchematicdiagramofthecriticaldirectiontheoryforSISOsystems ...... 58 3-4SchematicdiagramofthecriticaldirectiontheoryforMIMOsystems ...... 59 3-5E-contoursfortheunstructureduncertainsystemforfrequenciesrangingfrom0.1to7Hz ...................................... 61 9

PAGE 10

3-6Unstructureduncertaintytemplatewithscaleduncertainty,= ....... 62 3-7StructuredsingularvalueandtheNyquistrobuststabilitymarginversusfortheunstructureduncertaintytemplate ..................... 63 4-1Memristorbasedchaoticcircuit ........................... 65 4-2Waveletnetworkstructure .............................. 67 4-3TrackingresponseformemristorchaoticcircuitusingWNNbasedcontrol.(A)Flux,(B)Capacitorvoltagev1,(C)Capacitorvoltagev2,(D)InductorcurrentiL .. 78 5-1Trackingerrorfortheunstructuredunknowntime-delaysystem. ......... 93 5-2ControleffortoftherobustcontrollawwithNNadaptation. ........... 94 5-3Adaptationlawk^Vkofathree-layerNN ...................... 94 5-4Adaptationlawk^Wkofathree-layerNN. ..................... 95 6-1SchematicdiagramoftheMCInetwork ...................... 100 6-2Arnoldtongueforthedomainof1:1synchronyintheabsenceofstimulator .. 102 6-3TimedifferencesnbetweenthespikesoftwoneuronsasthefunctionofH .. 102 6-4Robustnessof1:1synchronyandin-phasesynchronyinthepresenceofcontrol 109 6-5Performancemeasureinheterogeneityofmultiplestepfunction ......... 111 6-6PerformancemeasureinZAPheterogeneity .................... 112 6-7noffeedbackcouplingstrengthintheMCInetwork ............... 113 10

PAGE 11

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyADAPTIVEANDROBUSTCONTROLSFORUNCERTAINLINEARANDNONLINEARSYSTEMSByJungE.SonAugust2012Chair:HaniphA.LatchmanCochair:PramodP.KhargonekarMajor:ElectricalandComputerEngineering Thepresenceofvariousuncertaintiesincontrolsystemsisoneofthemainchallengingissuestoengineerssincethisprobleminrealapplicationsgivesrisetoinstabilityanddegradationofperformance.Thisdissertationaddressesanalysisandcontroldesignproblemsinvolvinglinearandnonlinearsystemswithstructuredandunstructureduncertainties. Intherstproblem,thecriticaldirectiontheory(CDT)anH1designisproposedasasystematiccontrollersynthesismethodologyforalinearsystemwithparametricuncertainties.Intheliterature,asignicantconservatismissueinthisstudyofsuchuncertainsystemswithrespecttorobustcontroldesignhasbeenreported.First,themechanismsleadingtoconservatisminanearlierapproachutilizingaconstantoverboundinguncertaintytemplateareanalyzed.Then,motivatedbythedesiretodecreasetheconservativeuncertainty,astaticweightapproachbasedontheCDTisexaminedusingaconstantvaluetodescribetheuncertaintiesoverallfrequencies.ThisapproachshowsthattheCDTcombinedwithH1iseffectiveandpromisingforlessconservativestabilityandcontrollersynthesisforalinearsystemwithuncertainties. Thesecondisastudyoftheproblemoftherobustnessofmulti-inputmulti-output(MIMO)systemswithparametricuncertainties.Twocriteriathatareusedtoassessthe 11

PAGE 12

stabilityofMIMOsystemsarenumericallyevaluated.Anumericalanalysisprovidesanempiricaldemonstrationoftherelationshipbetweenthetwostabilitymargins. Thedissertationthenproceedstothestudyofdesignmethodsforsystematicadaptivecontrolscombinedwithrobustnessforuncertainnonlinearsystems.Themostcommontechniqueindesigningacontrollerforanonlinearsystemistoassumethenonlineardynamicsareexactlyknown,butinfactsystemnonlinearitiesareunknowninrealapplications.Toovercometheuncertaintiesinnonlinearsystems,awaveletneuralnetwork(WNN)identierwithaslidingmodecontroller(SMC)isproposed.Then,aRobustIntegraloftheSignoftheError(RISE)feedbackcontrolbasedontheWNNidentierisusedtoattenuatetheeffectofthewaveletnetworkapproximationerrorandexogenousdisturbance.Theproposedmethodensuresasymptotictrackingofadesiredreferencesignal.Asimulationofcontrollingmemristorbasedchaoticsystemillustratestheeffectivenessoftheproposedapproachanditsperformanceinthepresenceofdisturbances,andtheresultissupportedthroughrigorousLyapunov-basedstabilityproofs. Then,anadaptiverobusttrackingcontrolschemeusingamulti-layerneuralnetwork(NN)foraclassofnonlineardynamicsystemswithunknowntimevaryingstatesisaddressed.TypicaladaptiveNNbacksteppingcontrollersforuncertainnonlinearsystemswithtime-delaygiverisetocomputationcomplexitycausedbytherepeatedderivativesofvirtualcontrollersandnonlinearfunctions.Moreover,thecombinedtechniquesusuallyresultinonlyuniformlyultimatelybounded(UUB)stabilityduetotheinherentNNapproximationerror.Tobeginwith,wedevelopamodieddesiredcompensationadaptationlaw(DCAL)formulationwhichavoidstheexplosionofcomplexitycausedbythegeneralNNbacksteppingschemetocompensatefornonlinearsystemuncertainties,boundedsystemdisturbances,andunknownstatetimedelays.Then,usingaLyapunov-Krasovskii(LK)functional,itisshownthattheproposed 12

PAGE 13

controllerrenderstheclassofuncertainnonlineartime-delaysystemsasymptoticallystable. Finally,theeffectivenessoftheproposedmultiplecontrollerdesignmethodologiesisdemonstratedforthe1:1in-phasesynchronyinthemutuallycoupledinterneuron(MCI)network. 13

PAGE 14

CHAPTER1INTRODUCTIONANDBACKGROUND 1.1Introduction Mostpracticalsystemsarenonlinearanduncertaininnature.Tocopewiththischallengeadaptiveandrobustcontroldesigntechniquesarebeingdevelopedandarethesubjectofthisdissertation. 1.2RobustControlforUncertainLinearSystems Therobustnessanalysisintheexistenceofuncertaintyisanimportantroleforalinearsystem,namely,howdegradationoftrajectorytrackingperformanceanddestabilizationareaffectedbytheuncertainty.Moreover,diverseuncertaintiesexistinrealapplication.Therearetwomainstreamsonthistopicsuchasstructuredandunstructureduncertainties.Unstructureduncertaintyrepresentedbydiskshasbeenwelladdressedbyanalysis,andforthestructureduncertaintyH1stabilitytechniqueshavebeenextensivelydevelopedinthepastdecades.Thoughthenormboundsderivedfromunstructureduncertaintyarenecessaryandsufcient,themethodologyleadstoaconservativedesigncausedbytheoverboundingoperation. Motivatedbytheobjectiveofattenuatingtheaconservativeprobleminexactmanner,theCriticalDirectionTheory(CDT)in[ 1 2 ]wasdevelopedusingNyquiststabilitytheory.CDTisaneffectivealternativeoftheMaximumPerturbationRadius(MPR),whichintroducestheconservativeofaweightingstrategy.CDTusesthedirectionalpropertiesoftheuncertaintytemplateresultinginlessconservativestabilityconditions.MotivatedbythefactthatCDTgivesnecessaryandsufcientstabilityconditionsthatinvolveaCriticalPerturbationRadius(CPR),CDTisappliedtointroduceanexactweightingstrategythatresultsinasignicantlyimprovedstabilitymargincomparedwiththatbasedontheMPRweighting.Theeffectivecriticalperturbationradius(ECPR)weightingstrategyproposedinChapter 2 recoversthesimplicityoftheMPRweightingwhilereducingtheconservatisminducedbyuncertaintyoverbounding. 14

PAGE 15

Figure1-1. Stabilityanalysisforauncertainsystemg(s)underunitynegativefeedback 1.2.1Single-InputSingle-OutputSystem Agenerallineartimeinvariantsystem(LTI)canbeexpressedin _x(t)=Ax(t)+Bu(t)y(t)=Cx(t)+Du(t)(1) Thestatespacerepresentation( 1 )canbetransformedintothefollowingtransferfunctionform: G(s)=C(sI)]TJ /F5 11.955 Tf 11.96 0 Td[(A))]TJ /F8 7.97 Tf 6.58 0 Td[(1B+D=Cadj(sI)]TJ /F5 11.955 Tf 11.96 0 Td[(A) det(sI)]TJ /F5 11.955 Tf 11.96 0 Td[(A)B+D(1) providedthattherearenocancellationsbetweenthepolynomialofanumeratorandadenominator.Thetransformationfrom( 1 )to( 1 )enablestousefrequencydomaintechniquestoanalyzethestabilityandperformanceoftheclosed-loopsystem.CriticalDirectionTheory Considertheuncertainsingle-inputsingle-outputlinearsysteminFigure 1-1 g(s)=g0(s)+g(s)(1) whereg0(s)=c(s)p0(s)isanominalsystemandg(s)2isanunknownboundedperturbationbelongingtoaknownsetofallowableperturbations.Theclosed-loopsystemsinFigure 1-1 issaidtoberobustlystableifstabilityisensuredforall(s)2 15

PAGE 16

Theproblemunderconsiderationistheanalysisoftherobuststabilityoftheuncertainclosed-loopsystem( 1 )underunitynegativefeedback.ThedevelopmentsofNyquist-basedrobustnessanalysisassumethefollowingtwopremises.(A1)Thenominalfunctiong0(j!)isstableunderunitynegativefeedback.(A2)Theuncertainsystemg(j!)andthenominalsystemg0(jw)havethesamenumberofopen-loopunstablepoles,thenthewholefamilyofuncertainsystemsunderunitynegativefeedbackisstableifandonlyifthecriticalpoint)]TJ /F3 11.955 Tf 9.3 0 Td[(1+j0isnotinthevaluesetofg(s)forallfrequencies. Thekeyelementsofdenitionspertainingtothecriticaldirectiontheory(CDT)aresummarizedasfollows: 1. Thecriticallineisthedirectedlinewhichoriginatesatthenominalpointg0(j!)andpassesthroughthecriticalpoint)]TJ /F3 11.955 Tf 9.3 0 Td[(1+j0. 2. Thecriticaldirectiondc(jw):=)]TJ /F3 11.955 Tf 17.13 8.09 Td[(1+g0(j!) j1+g0(j!)j=ejdc(!) isaunit-lengthvectorwithoriginatg0(jw)andpointingtowardsthecriticalpoint. 3. Theuncertaintyvalueset(alsocalledtheimagesetintheliterature)V:=fg(j!):g(j!)=g0(j!)+g(j!),g(s)2(s)g indicatestheNyquist-planemappingg(jw)=g0(j!)+(j!)oftheuncertainsystem. 4. ThecriticalvaluesetVc=fgc(j!):gc(j!)=g0(j!)+c(j!)=g0(j!)+dc(j!)for2R+g isthesubsetofV(!)thatliesonthecriticalline,wherec(j!)=dc(j!)representsthesetofperturbationswithfrequencyresponselyingalongastraightlinesegmentthatjoinsthepointsg0(j!)and)]TJ /F3 11.955 Tf 9.29 0 Td[(1+j0. ThecriticalvaluesetVc(!)canbecategorizedasaconvexandanon-convexset.ProvidedthatVc(!)isaconvexsetinFigure 1-2 .(a),thestraight-linesegment go(j!i)gs(j!i)joiningthepointsgo(j!i)andgs(j!i)(asetdescribedasasinglepoint 16

PAGE 17

Figure1-2. Uncertaintyvaluesetsatafrequency!i:(a)convexcriticalvaluesetVc(!i),(b)non-convexcriticalvaluesetV(!i).Bothguresshowtheworst-sensitivityplantgs(j!i),locatedclosesttothepoint)]TJ /F3 11.955 Tf 9.29 0 Td[(1+j0. ofstraightlinesegment)denotesVc(!).Inanon-convexset,aunionofisolatedpointsandstraight-linesegments(suchastheunionofthedisjointsegments go(j!i)g1(j!i)and g2(j!i)g3(j!i)inFigure 1-2 .(b).Notethatitispossibletoencounteranuncertainsystemwithahighlynon-convexvaluesetV(!)thatneverthelessfeaturesaconvexcriticalvaluesetVc(!),asillustratedinFigure 1-2 .(a).Forthegeneralcaseofbothcriticalvaluessets,thecriticalperturbationradiusisdenedas c(!):=8><>:j1+g0(j!)j)]TJ /F6 11.955 Tf 17.93 0 Td[((!)if)]TJ /F3 11.955 Tf 11.95 0 Td[(1+j0=2V(!)j1+g0(j!)j+(!)otherwise(1) where (!)=minz2Bc(!)j1+zj(1) representstheminimaldistancefromthecriticalpoint)]TJ /F3 11.955 Tf 9.3 0 Td[(1+j0tothesetofcriticalboundaryintersectionsBc(!):=f@V(!)Tr(j!)gng0(!),wherethecriticalrayischaracterizedbyr(!)=g0(j!)+dc(j!)for2R+,and0n0istheset-exclusionoperator.Inthecasewhereg0(!)istheonlyelementof@V(!)Tr(j!),thenBc(!):=fg0(!)g.AsshowninBaabetal.[ 1 ],whenVc(!)isaconvexset,asillustratedin 17

PAGE 18

Figure 1-2 .(a),thedenition( 1 )reducesto c(!):=maxf2R+:g(j!)=g0(j!)+dc(j!)2Vc(!)g(1) aformrstinvokedinLatchmanetal.[ 2 ],wherec(!)issimplyinterpretedasthedistancebetweenthecriticalpointandthepointwheretheboundary@V(!)interceptsthecriticaldirection.Finally,theNyquistrobuststabilitymarginisdenedas kN(!)=c(!) j1+g0(j!)j(1) TheNyquiststabilitymarginisrestatedbythemainresultofBaabetal.[ 1 ]inthetheorem.Givenasystemin( 1 )withassumptions(A1)and(A2),thentheclosed-loopsystemisrobustlystableunderunityfeedbackifandonlyifkN(!)<18!(SeetheproofinBaabetal.[ 1 ].Notethatthetheoremisvalidingeneralforconvexaswellasnon-convexcriticalvaluesetsVc(!).Sincecontroldesignisoftencarriedoutundersufcient-onlyconditions,forcontrolsynthesispurposesitmaybeacceptabletoadoptthedenition( 1 )insteadof( 1 )whenworkingwithnon-convexcriticalvaluesets.ThentheresultingconditionkN(!)<18!,wherekNiscalculatedthrough( 1 ),isonlysufcientforrobuststability.WiththeNyquiststabilitytheorem,Chapter 2 proposestheeffectivecriticalperturbationradius(ECPR)weightingtechniquenotonlytoreducetheconservatismofMPRbutalsotokeepthesimplicityofit. 1.2.2Multi-InputMulti-OutputSystem Chapter 3 presentsresultsonrobusteigenvalueinclusionregionsformulti-inputmulti-output(MIMO)systemswithunstructureduncertainties.ItshowshowscalingtechniquesusedinrobuststabilityanalysismaybeadaptedtogenerateusefulrobustgeneralizedNyquistplotsandhowtheproposedeigenvalueinclusionregionsgiverisetoamechanismtocomputetheNyquistrobustnessmargin,basedonCDTforthecaseofMIMOsystems.Theresultsarecomparedwithstandardresultsintheliterature 18

PAGE 19

suchasthestructuredsingularvalueandsomesuggestionsaregivenastohowtheproposedresultscanbeadaptedforsystematiccontrollerdesign. ThisevaluationoftheNyquistrobuststabilitymargincanbeextendedtothecaseofMIMOsystems.LetG(s)2CnntransferfunctionmatrixdenotingtheMIMOperturbedplant,bedenedas G(s)=G0(s)+(s)(1) whereG0(s)isanominaltransferfunctionand(s)2Duisanunstructureduncertainty.Furthermore,G0canbedecomposedas G0=WV(1) whereisthediagonalmatrixofeigenvaluesofG0andWistheeigenvectormatrix,andV=W)]TJ /F8 7.97 Tf 6.58 0 Td[(1.Typicalclassesofuncertaintiesofinterestincludethatofunstructuredandstructureduncertainties,From( 1 )wemaywrite (s)=G(s))]TJ /F5 11.955 Tf 11.96 0 Td[(G0(s)(1) andhence,theclassofunstructureduncertaintiesDu,canbedenedasfollows: Du=f(s):k(s)k=kG(s))]TJ /F5 11.955 Tf 11.95 0 Td[(G0(s)k(s)g(1) wherekkrepresentssomeappropriatematrixnorm.Thetermunstructuredisunderstoodtomeanthattheonlyconstrainton(s)issomeupperboundonitsnormsize.WithCDTaddressedinChapter 1.2.1 ,anunstructureduncertaintydenedbythesettakestheform Du:=f(s): [(j!)]
PAGE 20

Figure1-3. E-contoursfortheunstructureduncertainsystemforfrequenciesrangingfrom0.1to10Hz individualelementsof,norindeedofanysub-matrixstructure.Furthermore,thebound, [(s)](s)isoftenunsatisfactorysincethisrepresentsasingle-inputsingle-outputboundforaMIMOsystem.Nonetheless,theattractivepropertiesofthesingularvalueanalysishasledtovariousattemptstosalvagesomeofthemultivariablestructureintheuncertaintymodel,whilepreservingtheuseofsingularvalue-typebounds.Figure 1-3 illustratesthisforfrequenciesrangingfrom0.1to10HzforatransferfunctionmatrixwithaunstructureduncertaintyinChapter 3 kN=maxici(j!) j1+i(G0(j!))j=c(j!) j1+c(G0(j!))j(1) whereiisaneigenvalue,andi=1,2.ThemaximumoveriintheaboveexpressionistheNyquistrobuststabilitymarginfortheMIMOsystemandagainwehavetheresultthatforstability,kN<1.TheseideasareillustratedinFigure 1-4 .Fortheclassoftheunstructureduncertaintydenedin( 1 ),theStructuredSingularValue(SVD)is 20

PAGE 21

Figure1-4. SchematicdiagramofthecriticaldirectiontheoryforaMIMOsystem denedas (!)= f[I+Go(j!)])]TJ /F8 7.97 Tf 6.59 0 Td[(1gr(!)(1) andtheMIMOsystemsisstableifandonlyif(!)<1.Seetheproofin[ 2 ]. OneinterestingcharacterizationofstructureduncertaintieshasbeenproposedintheblockdiagonalboundedperturbationproblemformulationofDoyle[ 3 ].Theclassofblock-structuredperturbationsisdenedtobe: DB:=f:[[]]=ijij, ijijg(1) wherethenotation[[]]indicatesthatthematrixhasseveralunstructuredblocksij,embeddedwithinitandijdenotessomeweightingfactors.Theknownspatialdistributionanddimensionsoftheuncertaintyblocksmaynowbeexploitedtotransformthesystemmodeltoanequivalentrepresentationinwhichtheuncertaintyassumesablockdiagonalform,withtheblocksijonthediagonalofthetransformedperturbation 21

PAGE 22

matrix0.AnumberofinterestingresultshavebeenreportedbyDoyle[ 3 ]andSafonov[ 4 ]usingdiagonalandblockdiagonaluncertaintyrepresentations.Chpater 3 showstherelationshipbetweenthestructuredsingularvalueandtheNyquistrobuststabilitymarginasappliedtouncertainMIMOsystems.Thetwostabilitycriteriaareevaluatedforanopen-loopunstablechemicalreactorplantthathastwoinputsandtwooutputs.Thisnumericalanalysisgivesanempiricaldemonstrationoftherelationshipbetweenthetwostabilitymarginsandfurtherresearchistargetedtowardsobtainingananalyticalexpressionforsomegeneralclassesofuncertainties. 1.3AdaptiveandRobustControlsforUncertainNonlinearSystems Inrecentadaptiveandnonlinearcontrolstudies,thebackstepping,whichisapervasivecontroltechniquetoalargerclasssuchasnthorderstrictfeedbacksystems,hasbeendevelopedin[ 5 7 ].Thiscontrolschemeforaclassofuncertainnonlinearstrict-feedbacksystemprovidesasystematiccontrolprocedurefordesigntechniqueforanasymptoticaltrackingresultundertwolimitations. First,theExactModelKnowledge(EMK)arerequiredortheuncertainnonlinearfunctionscanbelinearlyparameterizable(LP)asaddressedin[ 8 11 ].However,systemnonlinearitiesareperfectlyunknowninrealapplications,themostphysicalmodelscannotbeexpressedinLP.Toovercomethenon-LPuncertainty,learningalgorithmssuchasNNandfuzzycontrolhavebeeninvestigatedin[ 12 ]andfurtherimprovedNNresultsareshownin[ 13 15 ].Intheliterature,adaptiveNNbacksteppingcontrolsdoesprovideapossibleapproachwithouttheassumptionofaLPuncertaintyandwelldenedmatchingconditionofuncertainsystems,butthisapproachyieldsreducedperformance(uniformlyultimatelyboundedresult). Thesecondlimitationisanincreasedcomputationalburdenduetotheiterativederivativesofvirtualcontroller(recursiveapproach)andsystemnonlinearities.Observerandoutputbasedschemesin[ 16 17 ],andadaptivefuzzydecentralizedoutputfeedbackcontrolin[ 18 19 ]haverecentlyintroducedtomakethegeneralbacksteppingapproach 22

PAGE 23

lesscomputational.However,theinherentlycomplicatedcomputationstillexistduetotheneedofstabilizingunmeasuredstatesignals. Toovercomethesecondlimitation,aclassofuncertainnonlinearstrict-feedbacksystemsistransformedintoanormalformofstandardafnenonlinearsystemsmotivatedin[ 20 ]notrequiringthepervasivebacksteppingalgorithm.Then,theafnesystemisremodeledbasedonthedesiredcompensationadaptivelaw(DCAL).ThemotivationofDCALincorporatedwiththeprojectionalgorithmistoblendneuralnetwork(NN)andarobustcontroller. 1.3.1AdaptiveNonlinearControl Anonlineardynamicsystemscanberepresentedbyasetofnonlineardifferentialequationintheform _x=f(x,t)(1) wherefisan1nonlinearvectorfunction,andxisthen1statevector.Thenumberofniscalledtheorderofthesystem.Astatexisanequilibriumstate(point)ofthenonlinearsystemsifoncex(t)isequaltox,itremainsequaltoxforallfuturetime.Theequilibriumpointstatex=0issaidtobestableif,foranyR>0,thereexistr>0suchthatifkx(0)k
PAGE 24

wherex1,x2aremeasurableboundedstates.isanunknownconstantparameterandaregressionmatrix(x1)isaknownsmoothnonlinearfunction.Controllawsinthisexampleneedtofullltheregulationobjective(x1!0). Step1.Addingandsubtractastabilizingfunction1(x1,)to( 1a )anddeneabacksteppingerrorvariablez2. _z1=z2+(x1)+1(1a) z2=x2)]TJ /F6 11.955 Tf 11.96 0 Td[(1,(1b) wherez1=x1.WiththederivativeofLyapunovfunctionin( 1 )astabilizingfunctiontostabilize( 1a )andaupdatelawareobtainedas (x1,1)=)]TJ /F5 11.955 Tf 9.29 0 Td[(k1z1)]TJ /F6 11.955 Tf 11.96 0 Td[(1(x1)_1=z1(x1).(1) wherek1isaconstantgainandisanadaptationgainmatrix.Sinceisunknown,theparameterestimate1isusedtocompensateforin( 1a ).Todetermineaparameterestimatev1foranunknownparameter,acontinuouslydifferentiablepositivedeniteLyapunovfunctionV1isdenedas V1=1 2z21+1 2()]TJ /F6 11.955 Tf 11.96 0 Td[(1)2(1) whereisaconstantgain.With( 1a ),( 1b )and( 1 ),_z1becomes _z1=)]TJ /F5 11.955 Tf 9.29 0 Td[(k1z1+z2+()]TJ /F6 11.955 Tf 11.95 0 Td[(1)2(1) ThederivativeoftheLyapunovfunctionis _V1=z1_z1)]TJ /F3 11.955 Tf 13.38 8.09 Td[(1 ()]TJ /F6 11.955 Tf 11.96 0 Td[(1)_1=z1z2)]TJ /F5 11.955 Tf 11.96 0 Td[(k1z21+()]TJ /F6 11.955 Tf 11.96 0 Td[(1)(z1)]TJ /F3 11.955 Tf 13.37 8.09 Td[(1 _1)=z1z2)]TJ /F5 11.955 Tf 11.96 0 Td[(k1z21.(1) 24

PAGE 25

Step.2Thederivativeofz2becomes _z2=_x2)]TJ /F3 11.955 Tf 14.29 0 Td[(_1=u)]TJ /F6 11.955 Tf 13.15 8.09 Td[(@1 @x1_x1)]TJ /F6 11.955 Tf 13.15 8.09 Td[(@1 @1_1(1) Substituting( 1a )andtheupdatelaw( 1 )resultsin _z2=u)]TJ /F6 11.955 Tf 13.15 8.08 Td[(@1 @x1(x2+))]TJ /F6 11.955 Tf 13.15 8.08 Td[(@1 @1z1(1) Now,theaugmentedLyapunovfunctionsV2isdenedas V2(z1,z2,1)=_V1(z1,1)+1 2z22=)]TJ /F5 11.955 Tf 9.3 0 Td[(k1z21+z2z1+u)]TJ /F6 11.955 Tf 13.15 8.09 Td[(@1 @x1x2)]TJ /F6 11.955 Tf 13.15 8.09 Td[(@1 @1z1)]TJ /F6 11.955 Tf 11.96 0 Td[(@1 @x1.(1) Controlleruisdesignedtocancelouttheindenitetermsin( 1 )andtheupdatelaw( 1 )isemployedtocompensateforin( 1 )asfollows: u=)]TJ /F5 11.955 Tf 9.29 0 Td[(z1)]TJ /F5 11.955 Tf 11.96 0 Td[(k2z2+@1 @x1x2+@1 @1z1+1@1 @x1.(1) Theresultingderivativeisshownas _V2=)]TJ /F5 11.955 Tf 9.3 0 Td[(k1z21)]TJ /F5 11.955 Tf 11.96 0 Td[(k2z22)]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F6 11.955 Tf 11.96 0 Td[(1)@1 @x1z2.(1) Inthisequationwecanseethattheexistingestimatein( 1 )cannotcanceltheparametricuncertainterm.Toovercomethisproblem,anewparameterestimate2replaces1intheexpressionforu: u=)]TJ /F5 11.955 Tf 9.3 0 Td[(z1)]TJ /F5 11.955 Tf 11.95 0 Td[(k2z2+@1 @x1x2+@1 @1z1+2@1 @x1(1) andthe_z2becomes _z2=)]TJ /F5 11.955 Tf 9.3 0 Td[(k2z2)]TJ /F5 11.955 Tf 11.96 0 Td[(z1)]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F6 11.955 Tf 11.96 0 Td[(2)@1 @x1.(1) 25

PAGE 26

Tocompensateanunknownparameter,theparameterestimate2isdeterminedwiththeaugmentedLyapunovfunctionas VL(z1,z2,1,2)=V1+1 2z22+1 2()]TJ /F6 11.955 Tf 11.96 0 Td[(2)2=1 2(z21+z22)+1 2()]TJ /F6 11.955 Tf 11.96 0 Td[(1)2+()]TJ /F6 11.955 Tf 11.96 0 Td[(2)2.(1) anditstimederivativeispresentedas _V2=_V1+z2_z2)]TJ /F3 11.955 Tf 13.37 8.09 Td[(1 ()]TJ /F6 11.955 Tf 11.96 0 Td[(2)_2=z1z2)]TJ /F5 11.955 Tf 11.96 0 Td[(k1z21+z2)]TJ /F5 11.955 Tf 9.29 0 Td[(k2z2)]TJ /F5 11.955 Tf 11.96 0 Td[(z1)]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(2)@1 @x1)]TJ /F3 11.955 Tf 13.37 8.08 Td[(1 ()]TJ /F6 11.955 Tf 11.95 0 Td[(2)_2=)]TJ /F5 11.955 Tf 9.29 0 Td[(k1z21)]TJ /F5 11.955 Tf 11.96 0 Td[(k2z22)]TJ /F3 11.955 Tf 11.96 0 Td[(()]TJ /F6 11.955 Tf 11.95 0 Td[(2)@1 @x1+1 _2(1) Thesecondparametricupdatelawdenotedas_2eliminates()]TJ /F6 11.955 Tf 12.07 0 Td[(2)termandgivestheform _2=)]TJ /F6 11.955 Tf 9.3 0 Td[(@1 @x1z2,(1) whichyields _V2=)]TJ /F5 11.955 Tf 9.3 0 Td[(k1z21)]TJ /F5 11.955 Tf 11.95 0 Td[(k2z22<0.(1) Withsufcientlylargegains,k1andk2,theequations( 1 )and( 1 )alongwith( 1 )and( 1 )formtheerrorsystemrepresentationsoftheresultingclosed-loopadaptivesystem _z1=)]TJ /F5 11.955 Tf 9.3 0 Td[(k1z1+z2+()]TJ /F6 11.955 Tf 11.96 0 Td[(1)2_z2=)]TJ /F5 11.955 Tf 9.3 0 Td[(k2z2)]TJ /F5 11.955 Tf 11.96 0 Td[(z1)]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F6 11.955 Tf 11.96 0 Td[(2)@1 @x1_1=z1(x1)_2=)]TJ /F6 11.955 Tf 9.3 0 Td[(@1 @x1z2.(1) Asaresult,anadaptivestabilizingbacksteppingtechniquerequireslinearlyparameterizable(LP)uncertainties,anddespiterecentadvancedbacksteppingtechniquesaddressedin[ 8 11 ](n)]TJ /F3 11.955 Tf 12.48 0 Td[(1)thvirtualcontroldesignsarerequired( 1 )forstabilizinguncertain 26

PAGE 27

nonlinearfunctionsanditsderivativesin( 1 )giverisetocomputationcomplexity.Chapter 5 proposesanalternativetechniquetoovercomethecomputationalburdenofpervasivebacksteppingtechniquesandguaranteestheasymptoticstabilitywiththepresenceofaboundeddisturbance.Theuncertainnonlinearsystemistransformedintoastandardafnenonlinearsystemnotrequiringthepervasivebacksteppingtechniquemotivatedin[ 20 ].Then,theafnesystemisremodeledbasedonaDesiredCompensationAdaptationLaw(DCAL)technique.ADCAL-basedschemeinconjunctionwithcontinuousRISEcontrollerdeterminestheNNupdatelaw.AdetailedproposedprocedurewillbeexploitedinChapter 5 1.3.2AdaptiveNeuralNetworkControl 1.3.2.1NeuralNetworkBackstepping ThegeneralNNbacksteppingtechniqueisillustrated.Consideranonlinearstrict-feedbacksystemin[ 7 21 ]. _x1=f1(x1)+g1(x1)x2_x2=f2(x1,x2)+g2(x1,x2)x3_x3=f3(x1,x2,x3)+g3(x1,x2,x3)x4......_xm=fm(x1,x2,,xm)+gm(x1,x2,,xm)u(1) wherexi2Rn,i=1,2,3,...,maretheboundedsystemstatesandareassumedtobeallmeasurable.fi()isasmoothnonlinearfunctionanduisthecontrolinput.Comparedin[ 12 14 20 ],Gi()isassumedtobeknownforillustratingasimpliedcontrolstructure.NeuralNetworkBackground:LetRdenotetherealnumbers,Rntherealn-vectors,Rmntherealmn.Givenacontinuousnonlinearsmoothfunctionf:S!Rm,whereSisasimplyconnectedset,thereexistsidealweightsinathree-layerstructure 27

PAGE 28

Figure1-5. Three-layerperceptronneuralnetwork inFigure 1-6 ,suchthatf(x)2C(S),x(t)2S.Then,thenetoutputcanbeexpressedas y=WT(VTx).(1)andanunknownnonlinearfunctionf(x)2C(S),x(t)2ScanbeapproximatedbyaNNas f(x)=WT(x)+(x)(1) whereistheactivationfunction,and(x)isaNNfunctionalreconstructionerrorvector. Denition1. LetSbeacompactsimplyconnectedofRn,and(x):S!RN2beintegrableandbounded.Then,(x)issaidtoprovideabasisforCm(S)if 1. AconstantfunctiononScanbeexpressedas( 1 )forniteN2. 2. ThefunctionalrangeofNN( 1 )isdenseinCm(S)forcountableN2. ThetrackingerrorwillbeproventobeboundedbyatimederivativeofLyapunovfunctionandthisupper-boundingfunctioncanbedecreasedbythesufcientlybiggainofcertainconstantsincontrollers. 28

PAGE 29

Figure1-6. PervasivebacksteppingNNcontrolaclassofuncertainnonlinearstrict-feedbacksystems ControllerScheme:Controlobjectiveistomakex1totrackadesiredtrajectoryx1d.Asshownin( 1b ),thebacksteppingtechniqueintroducesvirtualcontrollersfromx2uptoxmandstabilizingfunctionsaredesignedwiththeNNapproachfromx2dtoxmd.Thepurposeofstabilizingfunctionistoreduceatrajectoryerror.Moreover,anactualcontrolleruforcesatrackingerrorbetweenxmandxmdassmallaspossible.NNsapproximatetheunknownnonlinearfunctionsfn(xn)in( 1 ). Designctitiouscontrollersforx2,x3,...,xm.Astabilizingfunctionforx2isobtainedas x2d=g)]TJ /F8 7.97 Tf 6.59 0 Td[(11(^f1+_x1d)]TJ /F5 11.955 Tf 11.95 0 Td[(k1e1)(1) wherek12R+isaconstantgain,^f1istheestimateoff1.Substituting( 1 )into_x1of( 1 )givestheerrordynamics _e1=f1)]TJ /F3 11.955 Tf 12.05 2.66 Td[(^f1)]TJ /F5 11.955 Tf 11.95 0 Td[(k1e1+g1e2,(1) wheree2=x2)]TJ /F5 11.955 Tf 12.63 0 Td[(x2d.Approximatingf1withNNsdoesnotrequireLPandregressionmatrices,buttheNNtechniqueresultsinuniformlyultimatelybounded(UUB)stabilityduetogenericapproximationerrors.Theiterativestabilizingfunctionstomaketheerror 29

PAGE 30

smallcanbegeneralizedas xmd=g)]TJ /F8 7.97 Tf 6.58 0 Td[(1m)]TJ /F8 7.97 Tf 6.59 0 Td[(1()]TJ /F3 11.955 Tf 10.52 2.66 Td[(^Fm)]TJ /F8 7.97 Tf 6.59 0 Td[(1+_x(m)]TJ /F8 7.97 Tf 6.58 0 Td[(1)d)]TJ /F5 11.955 Tf 11.96 0 Td[(km)]TJ /F8 7.97 Tf 6.59 0 Td[(1em)]TJ /F8 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(GTm)]TJ /F8 7.97 Tf 6.58 0 Td[(2em)]TJ /F8 7.97 Tf 6.59 0 Td[(2).(1) Theopen-looperrorsystemofem)]TJ /F8 7.97 Tf 6.58 0 Td[(1=xm)]TJ /F8 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(x(m)]TJ /F8 7.97 Tf 6.59 0 Td[(1)disthengovernedby _em)]TJ /F8 7.97 Tf 6.59 0 Td[(1=fm)]TJ /F8 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 12.05 2.65 Td[(^fm)]TJ /F8 7.97 Tf 6.59 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(km)]TJ /F8 7.97 Tf 6.59 0 Td[(1em)]TJ /F8 7.97 Tf 6.58 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(gTm)]TJ /F8 7.97 Tf 6.59 0 Td[(2em)]TJ /F8 7.97 Tf 6.58 0 Td[(2+gm)]TJ /F8 7.97 Tf 6.59 0 Td[(1em.(1) Thestabilizingfunctionxmdisnotdesignedtominimizeerrorem=xm)]TJ /F5 11.955 Tf 12.41 0 Td[(xmd,becausetheactualcontrollerstabilizetheem.xmdisnotahelpfultermforstabilizationandiscanceledbyu(t).ThecontrollerinconjunctionwithNNbasedvirtualcontrollerisgivenby u=g)]TJ /F8 7.97 Tf 6.59 0 Td[(1m()]TJ /F3 11.955 Tf 9.4 2.66 Td[(^fm+_xmd)]TJ /F5 11.955 Tf 11.95 0 Td[(kmem)]TJ /F5 11.955 Tf 11.95 0 Td[(gTm)]TJ /F8 7.97 Tf 6.59 0 Td[(1em)]TJ /F8 7.97 Tf 6.58 0 Td[(1),(1) andtheclosed-looperrordynamics_emgivestheform _em=fm)]TJ /F3 11.955 Tf 12.06 2.66 Td[(^fm)]TJ /F5 11.955 Tf 11.95 0 Td[(kmem)]TJ /F5 11.955 Tf 11.96 0 Td[(gTm)]TJ /F8 7.97 Tf 6.59 0 Td[(1em)]TJ /F8 7.97 Tf 6.59 0 Td[(1.(1)StabilityandAnalysis:Thegeneralizedclosed-errordynamics_e1in( 1 )upto_emin( 1 )canbeobtainedas _e1=~WT11)]TJ /F5 11.955 Tf 11.95 0 Td[(k1e1+g1e2+1_e2=~WT22)]TJ /F5 11.955 Tf 11.95 0 Td[(k2e2)]TJ /F5 11.955 Tf 11.95 0 Td[(gT1e1+g2e3+2_e3=~WT33)]TJ /F5 11.955 Tf 11.95 0 Td[(k3e3)]TJ /F5 11.955 Tf 11.95 0 Td[(gT2e2+g3e4+3..._em=~WTmm)]TJ /F5 11.955 Tf 11.96 0 Td[(kmem)]TJ /F5 11.955 Tf 11.95 0 Td[(gTm)]TJ /F8 7.97 Tf 6.58 0 Td[(1em)]TJ /F8 7.97 Tf 6.59 0 Td[(1+3(1) 30

PAGE 31

Theerrordynamics,reconstructionerrors,NNupdatelaws,adaptationgains,andactivationfunctionsaretransformedintoamatrixform =eT1eT2...eTmT,=T1T2...TmT,~Z=diagf~W1,~W2,...,~Wmg,)-278(=diagf)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(1,)]TJ /F8 7.97 Tf 12.25 -1.8 Td[(2,...,)]TJ /F7 7.97 Tf 30.19 -1.8 Td[(mg,K=diagfk1,k2,...,kmg,=[T1T2...Tm]T,H=2666666666640g10...0)]TJ /F5 11.955 Tf 9.3 0 Td[(gT10g2...00)]TJ /F5 11.955 Tf 9.29 0 Td[(gT20...0............gm)]TJ /F8 7.97 Tf 6.59 0 Td[(100...)]TJ /F5 11.955 Tf 9.3 0 Td[(gTm)]TJ /F8 7.97 Tf 6.59 0 Td[(10377777777775(1) Usingamatrixformtheerrordynamics( 1 )canbeobtainedas _=)]TJ /F5 11.955 Tf 9.3 0 Td[(K+~ZT+H+.(1) whereHhasaskew-symmetricproperty.TheNNbacksteppingtechniqueresultsinthefollowingUniformlyUltimatelyBoundedstability.Thecontrollerin( 1 )withaNNweightforaclassofuncertainnonlinearsystemssatisesthefollowinguniformlyultimatelybounded(UUB)stability. limt!1ke1(t)k!0.(1) Inthestabilityanalysis,theestimatefortheNNweightisdeterminedbytheappropriateLyapunovfunctionas _^Wi=)]TJ /F7 7.97 Tf 19.39 -1.79 Td[(iieTi)]TJ /F5 11.955 Tf 11.96 0 Td[(kc)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(ikk^Wi,i=1,2,...,m(1) where)]TJ /F7 7.97 Tf 6.77 -1.79 Td[(iisthesymmetricadaptationgainmatrixandkcisaconstantgain.Then,theclosed-errorsystemsei(t),i=1,2,...,mandNNweightestimatesareUUB. 31

PAGE 32

Byauniversalapproximationproperty,theNNcontrolholdswithgivenaccuracyiNforall( 1 )inthecompactsetUsfjkkb,whereb2R+with(0)2U.AcontinuouslydifferentiablepositivedeniteLyapunovfunctionalisdenedas V,1 2T+1 2(~ZT)]TJ /F4 7.97 Tf 6.78 4.93 Td[()]TJ /F8 7.97 Tf 6.58 0 Td[(1~Z).(1) Differentiating( 1 )andsubstituting( 1 )givestheform _V=)]TJ /F6 11.955 Tf 9.3 0 Td[(TK+kckktrh~ZT(Z)]TJ /F3 11.955 Tf 13.36 2.66 Td[(^Z)i+T.(1) ApplyingtheSchwartzinequalityto( 1 ) trh~ZT(Z)]TJ /F3 11.955 Tf 13.36 2.66 Td[(~Z)i=D~Z,ZE)-222(k~Zk2Fk~ZkFk~ZkF)-222(k~Zk2F,(1) givestheinequalityofaderivativeofLyapunovfunction _V)]TJ /F6 11.955 Tf 21.91 0 Td[(minkk2+kckkk~Zk(ZM)-222(k~ZkF)+Nkk,(1) whereministheminimumeigenvaluesofK.Completingthesquarefor( 1 )yields _V)]TJ /F13 11.955 Tf 23.91 20.45 Td[("kck~ZkF)]TJ /F5 11.955 Tf 13.15 8.09 Td[(ZM 22)]TJ /F5 11.955 Tf 11.96 0 Td[(kcZ2M 4+minkk2)]TJ /F6 11.955 Tf 11.95 0 Td[(N#hkcZM 4+Ni min.Hence,_Visnegativeoutsideacompactset.Whenmincanbeselectedlargeenough,_Visguaranteedtobenegative.ThisresultsintheUUBstability. Chapter 5 dealswithanadaptiverobusttrackingcontrolusingamulti-layerneuralnetwork(MNN)foraclassofnonlineardynamicsystemswithunknowntimevaryingstatedelays.TypicaladaptiveNNbacksteppingcontrollersforuncertainnonlinearsystemswithtime-delaygiverisetocomputationcomplexitycausedbythetherepeatedderivativesofvirtualcontrollersandnonlinearfunctions.Moreover,thecombinedtechniquesusuallyresultinonlyuniformlyultimatelybounded(UUB)stabilitycausedbytheinherentNNapproximationerror.Chapter 5 presentsacontrolschemethat 32

PAGE 33

Figure1-7. Waveletneuralnetworkbasedadaptivenonlinearsystemstructure usestheRobustIntegraloftheSignoftheError(RISE)asafeedbacktermandanadaptiveneuralcontrollerasafeedforwardtermbasedonthedesiredcompensationadaptivelaw(DCAL)technique.First,wedevelopanewDCALformulationwhichavoidstheexplosionofcomplexitycausedbythegeneralNNbacksteppingschemetocompensatefornonlinearsystemuncertainties,boundedsystemdisturbances,andunknownstatetimedelays.Then,usingaLyapunov-Krasovskii(LK)functional,itisshownthattheproposedcontrollerrenderstheclassofuncertainnonlineartime-delaysystemsasymptoticallystable. 1.3.2.2WaveletNeuralIdenticationControl Recently,theneuralnetwork-basedcontroltechniquehasrepresentedanalternativemethodtosolvecontrolproblems[ 22 27 ].Themostusefulpropertyofneuralnetworksistheirabilitytoapproximatearbitrarylinearornonlinearmapping 33

PAGE 34

throughlearning.Basedontheirapproximationability,theneuralnetworkshavebeenusedforapproximationofcontrolsystemdynamicsorcontrollers.Thebasicissuesinneuralnetworkfeedbackcontrolaretoprovideonlinelearningalgorithmsthatdonotrequirepreliminaryoff-linetuning.Someoftheseonlinelearningalgorithmsarebasedontheback-propagationlearningalgorithm[ 23 ],andsomearebasedontheLyapunovstabilitytheorem[ 22 24 27 ].Recently,someresearchershavedevelopedthestructureofneuralnetworkbasedonthewaveletfunctionstoconstructthewaveletneuralnetwork(WNN)[ 28 29 ].Unlikethesigmoidalfunctionsusedinconventionalneuralnetworks,waveletfunctionsarespatiallylocalized,sothatthelearningcapabilityofWNNismoreefcientthantheconventionalsigmoidalfunctionneuralnetworkforsystemidentication.ThetrainingalgorithmsforWNNtypicallyconvergeinasmallernumberofiterationsthanfortheconventionalneuralnetworks[ 28 ].TherehasbeenconsiderableinterestinexploringtheapplicationsofWNNtodealwithnonlinearityanduncertaintiesofreal-timeservocontrolsystem[ 30 33 ].TheseWNNcontrollerscombinethecapabilityofarticialneuralnetworksforlearningabilityandthecapabilityofwaveletdecompositionforidenticationability.Thus,theWNN-basedcontrolsystemswouldachievebettercontrolperformancethangeneralNN-basedcontrolsystems. 1.3.2.3Controlof1:1SynchronyIn-Phase Chapter 6 considersamutuallycoupledinterneuronnetwork(MCI)withacontrolalgorithm.InthecaseoftheMCInetwork,theadaptiveH1controllerisusedasastimulatortoinduce1:1in-phasesynchrony.Learningalgorithmsuchastheneuralnetworkalgorithm(NN)estimatessodium,delayedrectierpotassium,andleakchannelsinaHodgkin-Huxleymodel.Inaddition,theerrorof1:1in-phaseneuralsynchronyisultimatelyuniformlyboundedtozerodespitethepresenceofheterogeneityintheMCInetwork.Thepremiseonasynchronyproblemisthatacontrollerstimulatesneuronssuchthatthetimingofanimpendingspikeismodulated,butthestimulationitselfdoesnotinduceactionpotentialspikes.Hence,thesaturationfunctioncombined 34

PAGE 35

withtheadaptiveH1controllerisemployednottoexcessinputconstraintandrobustlyachieves1:1synchronyin-phase.Finally,thetheon-lineupdatelawofaNNalgorithmisderivedfromLyapunovanalysis,sothatthesynchronystabilityofuncertainMCIdynamicscanbeguaranteed. 1.4OutlineofDissertation Therestofthisdissertationisorganizedasfollows.Chapter 2 dealswithstatic-weightingH1designforparametricuncertainties.ComparativestudyofstructuredsingularvalueandcriticaldirectiontheoryarepresentedinChapter 3 .Theapplicationofthewavelet-basedidenticationandtrackingcontrolwithMemristorbasedcircuitisillustratedinChapter 4 .Chapter 5 givestheNNtrackingcontrolschemeofanadaptivenon-backsteppingdesignforaclassofuncertainnonlinearsystemswithunknowntime-delay.Asthecasestudyofanadaptivecontrolstrategy,Chapter 6 proposesaNN-basedH1controllercombinedwithaninputconstrainttorobustlysustainstable1:1in-phasesynchronyinthepresenceofunknowndeviationsinthemutuallycoupledinterneuron(MCI)networkparameters.Chapter 7 demonstratesafutureresearchplanforaoutputfeedbackandstochasticuncertainnonlinearcontrolmotivatedbyChapter 5 and 6 35

PAGE 36

CHAPTER2ASTATIC-WEIGHTINGH1APPROACHFORLINEARPARAMETRICUNCERTAINTIES 2.1Introduction Resultsonstabilityanalysisforsystemswithparametricuncertaintieshaveemergedsteadilyovertheyears,includingoutstandingresultssuchastheedgetheorem,themappingtheorem,thegeneralizedKharitonovtheorem[ 34 ],thecriticaldirectionaltheory[ 2 ],andmanyothers.However,nonconservativerobustcontrollersynthesisforparametricuncertainsystemsremainsarelativelychallengingproblemforwhichonlyfewresultsareavailable.Someresearchershavetriedtoattackthesynthesisproblemusingxed-structurecontrollerssuchasPID,orotherlow-ordercontrollers[ 35 37 ].Yet,itcanbearguedthattheseapproachessufferfrominherentshortcomingsbyrestrictingattentiontoonlyxed-structurecontrollers.Anearlyattempttosynthesizeageneralrobustcontrollerwasgivenin[ 38 ],whereuncertainplantswereoverboundedbyaconstantuncertaintydiskinthefrequencydomain.Assigningthisconstantboundastheweightingfunction,standardH1designwasthenappliedtosolveforarobustlystabilizingcontroller.Thisapproachshowsgreatpromisebecauseitisquitesimpleandmakeseffectiveuseofthewell-developedH1designmethodology.Tofacilitatecomparisonwiththealternativeresultsdevelopedinthischapter,werefertothemethodin[ 38 ]asthemaximumperturbationradius(MPR)approach.Unfortunately,theMPRweightingstrategymayintroduceconservatismsincetheoverboundingoperationguaranteesonlysufcientconditionsforrobuststability. Motivatedbythefactthatthecriticaldirectiontheory[ 1 2 ]givesnecessaryandsufcientstabilityconditionsthatinvolveacriticalperturbationradius(CPR),theconservatismoftheMPRweightingisexploredindetailinthisnote.Furthermore,thecriticaldirectiontheoryisappliedtointroduceanexactweightingstrategythatresultsinasignicantlyimprovedstabilitymargincomparedwiththatbasedontheMPRweighting.Theeffectivecriticalperturbationradius(ECPR)weightingstrategy 36

PAGE 37

proposedinthisnoterecoversthesimplicityoftheMPRweightingwhilereducingtheconservatisminducedbyuncertaintyover-bounding. 2.2ProblemFormulation ConsideraSISOlinearuncertainplantrepresentedbyafamilyofrationaltransferfunction,p(s),conguredinfeedbackwithacontrollerc(s),asshowninFigure 2-1 (a).Thefamilyofuncertainplantscanberepresentedbyp(s)=p0(s)+(s)wherep0(s)isaknownnominalSISOsystem,and(s)isaperturbationthatbelongstoagivenuncertaintydescription.Ingeneral,thesystemp0(s)isneitheropen-loopstablenorunity-negative-feedbacksystemstable.Acontrollerc(s)needstobedesignedsothattheclosed-loopsystemofFigure 2-1 (a)isrobustlystable,i.e.,stableoveralluncertainplantsp(s). Afteracandidatecontrollerc(s)isdesigned,deneg(s):=p(s)c(s)asshowninFigure 2-1 (b).Notethatthenominalsystemisg0(s):=p0(s)c(s),andthatg(s):=(s)c(s).AsiscommoninNyquist-basedrobustnessstudies,thefollowingnecessaryconditionsshouldbesatisedforrobuststabilityofsystemg(s): (A1) Thenominalsystemg0(s)=p0(s)c(s)isstableunderunity-negative-feedback. (A2) Thefamilyofuncertainsystemsg(s)anditsnominalsystemg0(s)havethesamenumberofopen-loopunstablepoles. Notethatassumptions(A1)and(A2)regardingsystemg(s)areequivalenttothefollowingtwoassumptionsintermsofsystemp(s)andcontrollerc(s): (A3) Systemsp(s)andp0(s)havethesamenumberofopen-loopunstablepolesinthesetofallowableperturbations. (A4) Thecontrollerc(s)stabilizesthenominalplantp0(s)andthereisnounstablepole-zerocancelationbetweenthecontrollerc(s)andthefamilyofp(s)forallallowableperturbations. Acontrollerthatsatises(A4)issaidtobeadmissible.Acandidatecontrollershouldberejectedasinadmissibleifanysuchforbiddenpole-zerocancelationoccurs. 37

PAGE 38

Figure2-1. (a)Negative-feedbackloopincludingtheuncertainsystemp(s)=p0(s)+(s)andacontrollerc(s);(b)unity-negative-feedbackofsystemg(s)=c(s)p(s);(c)mixed-sensitivityapproachtotheuncertainfeedbacksystem. Allplantsandcontrollersconsideredareproper.Theprimaryproblemposedinthischapteristodeneanexactweightingfunctiontorepresenttheuncertainpart(s)inthecontextofH1design,andtoutilizethisweightingfunctiontosynthesizeacontrollerthatcanrobustlystabilizethefamilyofplantsforaslargeanuncertaintyaspossible.Herethetermexactmeansthatnoconservatismisincurred.Thesecondproblemistodeterminealargestuncertaintysize,forwhichtheentirefamilyofplantsp(s)isrobustlystabilizableusingH1designmethodsinconjunctionwithstaticweights.Inotherwords,oneisaskedtocalculateaparametricstabilitymargin,whichisdenedasthesmallestuncertaintylevelthatdestabilizestheclosed-loopsystem. 38

PAGE 39

2.3Preliminaries 2.3.1ReviewOfTheCriticalDirectionTheory Latchmanetal.[ 2 ]andBaabetal.[ 1 ]proposeanecessaryandsufcientfrequency-domainrobustnessanalysistechnique,namely,thecriticaldirectiontheory,whichmakesuseofobjectsdepictedinFigure 2-2 (b).ThesolidcurveinFigure 2-2 (b)describesthenominalsystemg0(j!).Atanyfrequencypoint!i,theuncertainfamilyg(j!i)ischaracterizedthroughthevalue-setV(!i).ThefollowingobjectsrelevanttothesubsequentdiscussioncanbeidentiedinFigure 2-2 (b): 1. Thecriticallineisthedirectedlinewhichoriginatesatthenominalpointg0(j!i)andpassesthroughthecriticalpoint)]TJ /F3 11.955 Tf 9.3 0 Td[(1+j0. 2. Thecriticaldirection dc(j!i):=)]TJ /F3 11.955 Tf 13.81 8.09 Td[(1+g0(j!i) j1+g0(j!i)j=ejdc(!i),(2) isaunit-vectorontheNyquistplanewithphaseangledc(!i). 3. Thevalueset(theshadedareaofFigure 2-2 (b))V(j!i):=fg(j!i)jg(j!i)=g0(j!i)+g(j!i),g(s)=c(s)(s),(s)2g 4. Thecriticalvalue-set(thelinesegmentofFigure 2-2 (b)connectingg0(j!i)andg(j!i))Vc(j!i):=fz(j!i)2V(j!i)jz(j!i)=g0(j!i)+dc(j!i),forsome2R+g 5. Thecriticalperturbationradius(CPR) c(!i):=max2R+fjg0(j!i)+dc(j!i)2Vc(j!i)g(2) 6. TheNyquistrobuststabilitymargin kN:=sup!c(!) j1+g0(j!)j(2) Thecriticalvalue-setVc(j!)maybeconvex(i.e.,itisastraight-linesegmentorasinglepoint)evenwhenV(j!)ishighlynon-convex.Thedenition( 2 )isstrictlyvalidforthecaseofaconvexVc(j!);anextensiontothenon-convexcaseisgiven 39

PAGE 40

in[ 1 ].Forclarityofexposition,thischapterfocusesonthecaseofwhereVc(j!)isconvex.Furthermoreasdiscussedin[ 1 ]thedenition( 2 )canbeusedinallcasestoachievesufcientconditionsforrobuststability,andhenceitisparticularlyusefulforthepurposesofcontrolsynthesispursuedinthischapter. Theorem2.1. Underassumptions(A1)and(A2),andgivenaconvexcriticalvalue-setVc(!),theuncertainsysteminFigure 2-1 (a)isstableunderunitynegative-feedbackifandonlyifkN<1. Proof. Theproofisgivenin[ 2 ].NotethatonlyVc(j!)isrequiredtobeconvex,whereasV(j!)canbenonconvex. Thenumericalcalculationofc(!)canbedoneeffectivelywhenadescriptionoftheboundaryofthevaluesetisavailable[ 2 ].Inparticular,forsystemswithafneparametricuncertainties,asthosetreatedintheExamplessection,thecriticalperturbationradiuscanbecalculatedviaasystematicandnumericallytractablealgorithmbecausetheboundaryofthevaluesetconsistsofonlycirculararcsorlinesegments[ 1 39 ]. 2.3.2RobustSynthesisForParametricUncertainSystemsViaH1Design Assuggestedin[ 34 38 40 ],therobustsynthesisproblemdescribedinSection 2.2 canbecastinthewell-knownmixed-sensitivityH1designframework[ 41 ].UsingthestandardnotationshowninFigure 2-1 (c),thecontrollerisfoundasaminimizingsolutiontotheobjectivefunctional kTzw(s)k1:=kW2(s)Tuw(s)k1(2) overallthenominallystabilizingcontrollersc(s),whereTuw(s):=c(s) 1+c(s)p0(s)isthecontrollersensitivitytransferfunction,andtheweightW2(s)mustbeselectedtorepresenttheplantuncertaintiesinasexactamanneraspossible.Strictlyspeaking,thedesignconsideredhereisasimple(ratherthanmixed)sensitivityproblemsincetheremainingconventionalsensitivitytransferfunctionsW1(s)andW3(s)arezero[ 41 42 ]. 40

PAGE 41

Forthespecialcasewheretheuncertaintysetisexactlyrepresentedbythedisk-boundeddescription,=fg(s)jj(s)jrgwhererisapositivescalar,thentheselectionofastaticweight W2(s)=r(2) ensuresthatacandidatecontrollerisrobustlystabilizingifandonlyifkTzw(s)k1<1.Toaddressthecasewheretheuncertaintyisnotdisk-bounded,amaximumperturbationradius(MPR)forthesystemp(j!)ateachfrequency!isdenedin[ 38 ]asm(!):=maxj(j!)j.Thenaworst-caseuncertaintyboundisfoundas m:=max!m(!).(2) Theapproachfollowedin[ 38 ]istoadoptr=m,sothat W2(s)=m(2) servesasastaticweightforcontrollerdesign,andtheresultingrobustcontrolleristreatedasoptimalifitdrivesmtounitywherem=minstabilizingc(s)kmTuw(s)k1.Thepurposeofthischapteristoproposeanalternativestaticweightselectionmethodbasedonthecriticaldirectiontheory. 2.4MainResults 2.4.1ComplexImageOfUncertainPlants Figure 2-2 showsthecompleximagesofp(s)andg(s).Moreprecisely,Figure 2-2 (b)istheNyquistPlanewhileFigure 2-2 (a)istheplant-imageplane.Giventhatg(s)=c(s)p(s),aftermultiplicationbythecontrollerc(j!),theimageofp(j!)mayberotated,andmagniedorcontractedtoproducethevaluesetofg(j!).Atafrequency!=!i,allsystemsg(j!i)2Vc(!i)canberepresentedbyg(j!i)=jg(j!i)jejdc(!i),wheredc(!i)isthephaseangleofthecriticaldirectiondenedin( 2 ).Consideracontrollerc(j!)=jc(j!)jejc(!),withphaseanglec(!).Alltheplantimagesp(j!i)thataremappedbythecontrollertopointsg(j!i)2Vc(!i)ontheNyquistplanemustbeofthe 41

PAGE 42

form p(j!i)=jg(j!i)jejdc(!i) jc(j!i)jejc(!i)=jp(j!i)jejdc,p(!i)(2) wheredc,p(!i)=dc(!i))]TJ /F6 11.955 Tf 12.46 0 Td[(c(!i).Therelationship( 2 )is,ofcourse,denedonlyatfrequencieswherec(j!)6=0.Likewise,thecriticalpoint)]TJ /F3 11.955 Tf 9.3 0 Td[(1+j0intheNyquistplaneismappedtothepoint)]TJ /F8 7.97 Tf 18.28 4.7 Td[(1 c(j!)intheplane-imageplane.SignicantentitiesinFigure 2-2 (a)aredenedasfollows: Figure2-2. Mappingrelationshipbetweentheoriginalplantp(j!)andthetransformedsystemg(j!)=c(j!)p(j!).Theshadedareasrepresenttheuncertaintyvalue-setV(j!i)(right)andtheimage-setVp(j!i)(left).Thepointg(j!i)in(b)denesadiskofradiusc(!i),whereasitsimagep(j!i)in(a)denesadiskofradiuse(!i).Theoverboundingdiskofradiusm(!i)in(a)circumscribesthediskofradiuse(!i),andleadstomoreconservativerobust-stabilityestimates. 1. Theeffectivelineisthedirectedlinewhichoriginatesatthenominalpointp0(j!i)andpassesthroughthepoint)]TJ /F8 7.97 Tf 19.39 4.7 Td[(1 c(j!i). 2. Theeffectivedirection dc,p(j!i):=)]TJ /F3 11.955 Tf 13.82 8.08 Td[(1+p0(j!i) j1+p0(j!i)j=ejdc,p(!i),(2) isaunit-vectorontheplant-imageplanewithphaseangledc,p(!i). 42

PAGE 43

3. Theplantvalue-set(theshadedareaofFigure 2-2 (a))Vp(!i):=fp(j!i)jp(j!i)=p0(j!i)+(j!i),(s)2g 4. Theeffectiveplantvalue-set(thelinesegmentofFigure 2-2 (a)connectingp0(j!i)andp(j!i))Vc,p(!i):=fz(j!i)2Vp(!i)jz(j!i)=p0(j!i)+dc,p(j!i),forsome2R+g 5. Theeffectivecriticalperturbationradius(ECPR) e(!i):=max2R+fjp0(j!i)+dc,p(j!i)2Vc,p(!i)g(2) NotethatwhentheNyquist-planecriticalvalue-setVc(!i)isconvex,asassumedinthischapter,theplant-imageeffectivevalue-setVc,p(!i)isalsoconvex.Whenthecontrollercoincideswithzeroatanyfrequency,thepoint)]TJ /F8 7.97 Tf 18.28 4.7 Td[(1 c(j!)islocatedaninnitedistanceawayfromthenominalpointp0(j!).Inthiscase,theeffectivedirectionisundened,andsoinsuchinstanceswechoosetheworst-casedirection,whichresultsine(!)=m(!). Lemma1. Thecriticalperturbationradiusofg(j!)andtheeffectivecriticalperturbationradiusofp(j!)arerelatedthroughtheequality c(!)=e(!)jc(j!)j.(2) Proof. Withoutlossofgenerality,theequality( 2 )canbeprovedusingtheentitiesshowninFigure 2-2 .AsillustratedinFigure 2-2 (b),g(j!i)representstheintersectionoftheboundaryofthevaluesetofg(j!i)withthecriticaldirectiondenedthroughdc(j!i).From( 2 ),itfollowsthatc(!)=jg(j!))]TJ /F5 11.955 Tf 12.26 0 Td[(g0(j!)j.Applyingthemappingrelationshipg(j!i)=c(j!i)p(j!i),oneobtains c(!)=jc(j!)jjp(j!))]TJ /F5 11.955 Tf 11.96 0 Td[(p0(j!)j=e(!)jc(j!)j,(2) wheretheequatione(!)=jp(j!))]TJ /F5 11.955 Tf 11.95 0 Td[(p0(j!)jresultsfrom( 2 ). 43

PAGE 44

2.4.2ConservatismOfTheMPRWeightingApproach Theorem2.2. Underassumptions(A1)-(A4),andgivenaconvexeffectiveplantvalue-setVc,p(j!),theclosed-loopsystemofFigure 2-1 (a)isrobustlystableifandonlyif sup!e(!)c(j!) 1+c(j!)p0(j!)<1.(2) Proof. FromTheorem 2.1 ,theclosed-loopsystemofFigure 2-1 (a)isstableunderthegivenassumptionsifandonlyifkN<1.Usingthedenition( 2 ),itfollowsthatkN<1()sup!c(!) 1+g0(j!)<1. Finally,fromLemma1,thefollowingequivalentnecessaryandsufcientstabilityconditionsareobtainedkN<1()sup!e(!)c(j!) 1+c(j!)p0(j!)<1. Givenacontrollerc(s),accordingtoTheorem 2.2 onlytheuncertainplantsthatliealongtheeffectivedirectiondc,p(s)needtobeconsideredwhenaddressingnecessaryandsufcientconditionsfortherobuststabilityofthesystemofFigure 2-1 (a).Accordingly,anexactweightingfunctionW2(s)shouldtaketheform jW2(j!)j=e(!).(2) Thentheinequalitye<1isanecessaryandsufcientconditionforrobuststability,wheree:=minstabilizingc(s)ke(!)Tuw(j!)k1.Ingeneralm(!)isgreaterthane(!).Thereforeconservatismmightbeincurredwhenusing jW2(j!)j=m(!)(2) togetherwithm<1wherem=minstabilizingc(s)km(!)Tuw(j!)k1.Theexpressionkm(!)Tuw(j!)k1<1ismerelysufcientforsystemstability,nolongernecessary. 44

PAGE 45

FurthermoretheconstantMPRweightingapproach( 2 )wouldinduceevenmoreconservatismthan( 2 ).Hence,theproposition( 2 )isinherentlylessconservative.Furthermore,( 2 )wouldyieldanexactweightforthecaseswhereVc,p(!i)isconvex,avoidingconservativedesignoutcomes.Theargumentcanbeextendedtothecasefornon-convexVc,p(!i)inasimilarmannerasin[ 1 ]fornon-convexVc(!i). 2.4.3StaticControllerDesignUsingAnExactConstant-WeightingMethod ThischapterfocusesonthestaticweightingapproachesforH1design,namely,wheretheuncertainparts(s)ofsystemsp(s)arerepresentedbyapositiverealscalarr,expressedasW2(s)=r.InthiscaseTzw(s)=rTuw(s);hence,thedesigncriterionofminimizingkTzw(s)k1inSectionIII.BreducestothatofminimizingkTuw(s)k1.Amongallnominallystabilizingcontrollersdesignedusingthestaticweightingapproach,anoptimalcontrollerwillaccommodateaslargeanuncertaintyaspossible.Thatoptimalcontrollerisdenotedas cop(s):=argminstabilizingc(s)kTuw(s)k1.(2) Theorem2.3. Forthestaticweightingapproach,thecontrollerdesigncriterionminstabilizingc(s)sup!jTuw(j!)jisequivalentto maxstabilizingc(s)inf!1 c(j!)+p0(j!).(2) Proof. Considerthefollowingequivalentexpressionsfor( 2 )cop(s)=argminstabilizingc(s)sup!c(j!) 1+c(j!)p0(j!)=argmaxstabilizingc(s)sup!c(j!) 1+c(j!)p0(j!))]TJ /F8 7.97 Tf 6.58 0 Td[(1=argmaxstabilizingc(s)inf!1+c(j!)p0(j!) c(j!) Hencecop(s)=argmaxstabilizingc(s)inf!1 c(j!)+p0(j!). 45

PAGE 46

Expression( 2 )exposesanalternativeinterpretationofthecontrollerdesignprocedure,namely,asearchoverallstabilizingcontrollertondonethatmaximizestheminimaldistancebetweenp0(j!)andthepoint)]TJ /F8 7.97 Tf 18.28 4.71 Td[(1 c(j!).NotethatbothofthedesigncriteriaofTheorem 2.3 involveonlythenominalplantp0(s);hence,theproblemisindependentoftheuncertainplant(s).Thisisanaturalconsequenceofusingastaticweight. AusefulinterpretationofthestaticweightcanbederivedfromTheorem 2.3 .Givenanoptimalcontrollercop(s)designedusingastaticweighingapproach,oneonlyhastobeconcernedwiththefrequencypointcorrespondingtotheminimaldistancebetweenthepointsp0(j!)and)]TJ /F8 7.97 Tf 18.28 4.71 Td[(1 c(j!).Hencewedenetheconstanteffectiveuncertaintyboundas e:=e(!p)(2) where !p:=arginf!1 cop(j!)+p0(j!).(2) Notethattheremayexistmorethanonefrequencypointthatdenesaminimaldistancebetweenthepointsp0(j!)and)]TJ /F8 7.97 Tf 21.59 4.71 Td[(1 cop(j!);howeverthevalueofeisunique.Thealternativestaticweight( 2 )proposedinthischapteradoptsr=e,sothat W2(s)=e.(2) Notethatingeneral,emax!e(!);hence( 2 )isproposedasaselectionthatfurtherreducestheconservatismthatwouldarisethroughthelessattractivechoiceW2(s)=max!e(!). Recallthatweassignede(!)avalueofm(!)whenc(j!)=0atsomefrequency.Sincethatfrequencycannotcoincidewith!pbecauseoftheinnitedistancebetween)]TJ /F8 7.97 Tf 18.28 4.7 Td[(1 c(j!)andp0(j!),noconservatismisinduced. 46

PAGE 47

2.4.4RobustStabilityConditions FromFigure 2-2 .(a),itisobviousthatrobuststabilityislostwhenatsomefrequency!itheeffectiveradiuse(!i)isequaltothedistancebetweenthepoints)]TJ /F8 7.97 Tf 19.39 4.71 Td[(1 c(j!i)andp0(j!i).Insuchcase,apointontheboundaryoftheplantvalue-setismappedbythecontrollertothecriticalpoint)]TJ /F3 11.955 Tf 9.3 0 Td[(1+j0intheNyquistplane.Consideranoptimalcontrollercop(s)designedusingthestaticweightingapproachandthefrequency!p( 2 ),anddeneue:=1 cop(j!p)+p0(j!p)=inf!1 cop(j!)+p0(j!). Theorem2.4. Letcop(s)beanoptimalcontrollerdesignedusingthestaticweightingapproach,andletebethecorrespondingconstantECPRforanuncertainty.Theclosed-loopofFigure 2-1 (a)isrobustlystableifandonlyife
PAGE 48

Therefore,itfollowsthattheclosedloopisstableifandonlyife=e(!p)<1+cop(j!p)p0(j!p) cop(j!p)=ue. Thisnishestheproof. ThenecessaryandsufcientstabilityconditioninTheorem 2.4 validatestheutilizationoftheconstantECPRasthestaticweightingfunction,asexpressedin( 2 ).UnlikethenecessaryonlyconditionofkmTuw(s)k1<1forthestaticMPRweightingapproach,theconditionkeTuw(s)k1<1isnecessaryandsufcientforsystemstability.Furthermore,Theorem 2.4 givestheupperboundueoftheconstantECPR,whichisutilizedinthenextsectiontondaparametricstabilitymargin. Recallthemeaningoftheoptimalcontroller( 2 ),onecanobtainanalternativemeaningfulexpressionofue, ue=1 minstabilizingc(s)kTuw(s)k1.(2) FurtherconnectionswiththeclassicalH1staticweightingdesignapproachcanbeestablishedbyconsideringasysteminwhichtheweightisgivenbyW2(s)=r,andthecontrollerinquestioncop(s)attainsthevalueminstabilizingc(s)kTuw(s)k1asdenedin( 2 ).NowletrmaxdenotethevalueofrsuchthatkrmaxTuw(s)k1=krmaxcop(j!p) 1+cop(j!p)p0(j!p)k1=1.Thenitfollowsthatrmax=1+cop(j!p)p0(j!p) cop(j!p):=1 minstabilizingc(s)kTuw(s)k1.Hencermax=ue. 2.5Examples TwoexamplesaregiveninthissectiontoillustratethesubstantialimprovementinstabilitymarginassessmentusingtheconstantECPRweighting. Example1. Considertheuncertainintervalplantgivenin[ 34 36 38 43 ] p(s)=5s+q1 s2+q2s+q3(2) 48

PAGE 49

withrealparameters q12[4)]TJ /F6 11.955 Tf 11.95 0 Td[(",4+"],q22[2)]TJ /F6 11.955 Tf 11.95 0 Td[(",2+"],q32[)]TJ /F3 11.955 Tf 9.3 0 Td[(15)]TJ /F6 11.955 Tf 11.95 0 Td[(",)]TJ /F3 11.955 Tf 9.3 0 Td[(15+"],(2) where">0representsthelevelofuncertainty. Usingthe-iterationmethod[ 40 ],anH1centralcontrollerthatminimizeskTuw(s)k1=c(s) 1+c(s)p0(s)1isfoundtobe cop(s)=3603.7935s+18018.9673 s2+1434.5016s)]TJ /F3 11.955 Tf 11.96 0 Td[(2312.4499.(2) Assumption(A3)issatisedfor"<7.5[ 43 ].Itisstraightforwardtoverifythatthecontroller( 2 )satisesAssumption(A4)for"<7.5 Figure2-3. Plotsofmanderadiiasfunctionsoftheuncertainty-sizeparameter".Thevaluermax=0.3950isthelimitingvalueofthestaticweightW2(s)foundviaastandardunstructuredH1approach. 49

PAGE 50

Let"maxdenotetheparametricstabilitymargin,denedasthesmallestvalueof"in( 2 )thatdestabilizestheclosedloopofFigure 2-1 (a)withthecontrollercop(s).Giventhecontroller( 2 ),kTuw(s)k1=2.5316.ForstaticweightingwhereW2(s)=r,themaximumvalueofrthatsatiseskTzw(s)k1=1is0.3950. Themethodproposedin[ 38 ]tocompute"maxusestheconstantMPRweighting( 27 ),i.e.,W2(s)=m,andthen,"andhencemisincreaseduntilm=rmax,whichyieldskmTuw(s)k1=1.AsanalyzedinSectionIV.B,thisstaticMPRweightingapproachmightincurconservatismincalculating"max.BasedonthediscussioninSectionIV.D,thischapterproducesanalternativeapproachtocomputethevalueof"maxusingtheconstantECPRweighting,i.e.,W2(s)=e.Oneincreases"andhenceeuntil"="max,forwhiche=rmax=ueandkeTuw(s)k1=1. ECPRmethodyieldsamuchlargerintervalofstabilizationasshowninFigure 2-3 .Thehorizontaldottedlinerepresentsthevaluermax=0.3950.Thedashedlineplotsthevariationofmvs.",fromwhichonegets"=2.8300whenm=rmax.Thevalue"max=2.8300isproposedin[ 38 ]astheparametricstabilitymargininthecaseoftheconstantweighingmethodusingtheMPRapproach.Thesolidlineplotsthevariationofevs.",fromwhichonegets"=5.2511whene=rmax.Heretheworstcasefrequencypointisfoundtobe!p=0.Therefore,theparametricstabilitymargincalculatedbytheMPRweightingapproach("max=2.8300)isonlyabout54percentofthatobtainedbytheECPRweightingapproach("max=5.2511). Thesamevalueofparametricstabilitymargin"max=5.2511canalsobeobtainedusingareal-analysisapproachwhentheplant( 2 )andcontroller( 2 )areknown.Anefcientalgorithmcanbeconstructedbyreferringtothelinearprogrammingalgorithmproposedin[ 44 ]. Example2. Considertheuncertainintervalplantgivenin[ 38 ] p(s)=30s+q1 s3+q2s2+q3s+q4(2) 50

PAGE 51

withrealparametersq12[10)]TJ /F6 11.955 Tf 11.95 0 Td[(",10+"],q22[)]TJ /F3 11.955 Tf 9.3 0 Td[(3)]TJ /F6 11.955 Tf 11.95 0 Td[(",)]TJ /F3 11.955 Tf 9.3 0 Td[(3+"],q32[)]TJ /F3 11.955 Tf 9.3 0 Td[(4)]TJ /F6 11.955 Tf 11.96 0 Td[(",)]TJ /F3 11.955 Tf 9.3 0 Td[(4+"],q42[12)]TJ /F6 11.955 Tf 11.95 0 Td[(",12+"]. TosatisfyAssumption(A3)itisrequiredthat"<6.0[ 38 ]. ProceedinginthesamemannerasforExample1,thefollowingcontrollerisfound cop(s)=3617.6s2+4562.3s)]TJ /F3 11.955 Tf 11.95 0 Td[(5345.9 s3+1468.3s2+18620.7s+6605.8.(2) Giventhecontroller( 2 ),kTuw(s)k1=2.4857.Thereforermax=0.4023.Thestabilitymarginisreportedin[ 38 ]tobe"max=0.8500,whichactuallycorrespondstom=0.4023inFigure 2-4 .However,byinterpretingthevaluermaxastheue,thestabilitymarginis"max=1.3500wheree=0.4023.Heretheworstcasefrequencyis!p=2.0512.TheconservatismintroducedbyMPRapproachisdramaticallyreducedbyinterpretingtheconstantweightW2astheconstanteffectivebound,insteadoftheconstantoverboundinguncertainty. 2.6Conclusions ThischapterfocusesontheH1methodswiththeconstantweightingstrategyforsystemswithparametricuncertainties.Theconservatismofearlierworkintheliteratureisanalyzedandamuchlargerparametricstabilitymarginisobtainedusingthecriticaldirectiontheory.Theanalysisinthischapterholdspromisefordeterminingevenlargerstabilitymarginsandassociatedrobustcontrollersbyexploringdynamicweightingstrategies. 51

PAGE 52

Figure2-4. Plotsofmanderadiiasfunctionsoftheuncertainty-sizeparameter".Thevaluermax=0.4023isthelimitingvalueofthestaticweightW2(s)foundviaastandardunstructuredH1approach. 52

PAGE 53

CHAPTER3COMPARATIVESTUDYOFSTRUCTUREDSINGULARVALUEANDCRITICALDIRECTIONTHEORYFORUNCERTAINMIMOSYSTEMS 3.1Introduction Therehasbeenalotofworkcompletedontherobustnatureofphysicalsystemswithregardtoassessingstability,especiallySISOsystemsbutmorerecently,MIMOsystems,especiallythosethathaveparametricuncertainties[ 1 2 45 50 ].Twocriteriathathavebeenusedtoascertainthestabilityofaparticularsystemarethestructuredsingularvalue[ 49 ]andtheNyquistrobuststabilitymargin[ 1 2 47 50 ].Bothconditionsarenecessaryandsufcientinanalyzingthestabilityofagivensystem.Aninterestingproblemthathasdevelopedistoanalyzethesesystemswithasetofuncertaintytemplates.DanielandKouvaritakisanalyzedthisproblemforMIMOsystemswithunstructuredtemplates[ 45 46 ].Theobjectiveofthischapteristorepeatthisgeneralapplicationtoamorespeciccaseofanopen-loopchemicalreactorplantwithparametricuncertaintiesusingtheunstructureduncertaintytemplate.ThetwostabilitycriteriaareappliedtothissystemtoachieveanumericalsolutiontothisparticularcaseandtoseewhetherthisgivesanyinsighttoapossibleanalyticalrelationshipbetweenthestructuredsingularvalueandtheNyquistrobuststabilitymargin.Chapter 3.2 discussessomebackgroundregardingmatrixdecompositionsandtheNyquiststabilitycriterion.Chapter 3.3 and 3.4 examinethestructuredsingularvalueandtheNyquistrobuststabilitymargin,respectively.Chapter 3.5 presentstheexampleofthechemicalreactor.Chapter 3.6 givesthesolutionsofthesestabilitycriteriaasappliedtothisspecialcase.Finally,Chapter 3.7 comparestheresultsofthisnumericalanalysisandoffersconcludingremarks. 53

PAGE 54

3.2MathematicalBackground 3.2.1MatrixDecompositions AnymatrixA2Cnmhasasingularvaluedecomposition(SVD)givenby A=XY,(3) whereXandYareunitarymatrices,i.e.XX=XX=Iand =diag(1,2,...,n).(3) Theiarecalledsingularvalues,where1isthelargestsingularvalue.Itcanbeshownforasquarematrixthat (A)i(A) (A),(3) where (A)isthesmallestsingularvalueofA, (A)isthelargestsingularvalueofA,andi(A)isanyeigenvalueofA.ItcanalsobeshownthatwhentheSVDofamatrixisexecuted,thenthefollowingmatricescanbeformed AA=Y2YAA=X2X.(3) Theaboveinactualityisaformoftheeigenvaluedecomposition(EVD).Thefollowingstatementisalsotrue 2=,(3) where=diag(1,2,...,r). 3.2.2NyquistStabilityforSISOandMIMOTransferfunctions TheNyquiststabilitycriterionstatesthattheunitynegativefeedbacksystemformedwiththeSISOtransferfunctiong(s)intheforwardpathisstableifandonlyifthenumberofcounter-clockwiseencirclementsofthe)]TJ /F3 11.955 Tf 9.3 0 Td[(1pointbyg(j!)isequaltothenumbern0of 54

PAGE 55

Figure3-1. NyquistplotofSISOsystem,g0(j!) openloopunstablepoles,i.e., N()]TJ /F3 11.955 Tf 9.3 0 Td[(1,g(j!))=)]TJ /F5 11.955 Tf 9.3 0 Td[(n0,(3) wheren0representsthenumberofopenloopunstablepolesofg(s).ThisresultisshowngraphicallyinFigure 3-1 .TheNyquiststabilitycriteriongeneralizestothecaseofsquareMIMOtransferfunctionsviatheeigenvalues.InthischapterweshallconsiderMIMOtransferfunctionswith2inputsand2outputs,thoughtheresultsapplyalsotoG(s)2CnnMIMOsystems. ForMIMOsystemswehavethestabilitycondition[ 51 ] 2Xi=1N()]TJ /F3 11.955 Tf 9.3 0 Td[(1,i[G(j!)])=)]TJ /F5 11.955 Tf 9.29 0 Td[(n0,(3) wherei[G(j!)]denotestheitheigenvalueofthefrequencyresponsetransfermatrix.Thisresultstatesthatthesystemisstableifandonlyifthenetsumofthenumberofcounter-clockwiseencirclementsofthe)]TJ /F3 11.955 Tf 9.29 0 Td[(1pointbyeigenvaluesofG(j!)isequaltothe 55

PAGE 56

Figure3-2. NyquistplotofMIMOsystem,G0(j!) number)]TJ /F5 11.955 Tf 9.3 0 Td[(n0ofopenloopunstablepoles.TheMIMONyquiststabilitycriterionisshowninFigure 3-2 3.3NyquistRobustStabilityMargin 3.3.1Background ThecriticaldirectiontheorydevelopedbyLatchmanandCrisalle[ 1 ]isusefulinanalyzingtherobuststabilityofsystemswithbothstructuredandunstructureduncertainties.Howeverinthischapter,onlytheunstructuredcaseisconsidered.Thecriticaldirectiontheoryhasanadvantageoverotherstabilitymeasures,suchasthemaximumperturbationradius,becauseitdoesnotresultinoverlyconservativebounds.Insteadofusingthemaximumperturbationradius,thecriticaldirectiontheoryusesthedirectionalpropertiesoftheuncertaintytemplatewhichyieldslessconservativestabilityconditions.TheNyquistrobuststabilitymargin,kN,isderivedfromthistheory. 56

PAGE 57

3.3.2CriticalDirectionTheoryforSISOSystems Consideralineartimeinvariantsystemg(s)=g0(s)+(s)whereg0(s)isanominaltransferfunctionand(s)2dU,anunstructureduncertaintydenedbytheset dU=f(s):k(j!)k2
PAGE 58

Figure3-3. SchematicdiagramofthecriticaldirectiontheoryforSISOsystems whichcanbeeitheraconvexornon-convexset.Ifthecriticalvaluesetisconvex,thecriticalperturbationradiusis c(!)=max2R+f:z=g0(j!)+dC(j!)2Vc(!)g(3) Thenon-convexcaseisnotdiscussedinthischapter. 3.3.3NyquistRobustStabilityMarginforSISOSystems ThecalculationoftheNyquistrobuststabilitymarginrstrequirestheknowledgeofthecriticalperturbationradius,c.Howeverbeforeccanbedetermined,theuncertaintytemplatemustberstcalculated.Usingtheuncertaintytemplateandthenominaleigenvalue,thecriticalradiusmaybedetermined.First,thecriticaldirectionandcriticallinearecomputedfromtheuncertaintytemplatesandcriticalpoint)]TJ /F3 11.955 Tf 9.3 0 Td[(1+j0.Next,thecriticaluncertaintyvaluesetisdeterminedfromtheintersectionofthecriticallineanduncertaintyvalueset.Finally,thecriticalradiusiscalculatedfromthecriticaluncertaintyvaluesetandthenominaleigenvalue. 58

PAGE 59

Figure3-4. SchematicdiagramofthecriticaldirectiontheoryforMIMOsystems Thecriticalperturbationradius,inassense,encapsulatesthesizeoftheuncertaintythatisusedtocalculatetheNyquistrobuststabilitymargin[ 1 47 50 ] kN(!)=c(!) j1+g0(j!)j.(3) TheNyquistrobuststabilitymarginistheratioofthecriticalperturbationradiustothedistancefromthenominaleigenvaluetothecriticalpoint.Itiseasytosee,asshowninFigure 3-3 ,thatifkN1,thesystemwillbeunstablebecausethecriticalpointwillbeenclosedwithintheuncertaintytemplate.SotheassociatednecessaryandsufcientstabilitycriterioniskN1. 3.3.4NyquistRobustStabilityMarginforMIMOSystems ThisevaluationoftheNyquistrobuststabilitymargincanbeextendedtothecaseofMIMOsystems.Consideranotherlineartimeinvariantsystemgivenbythefollowingexpression G(s)=G0(s)+(s),(3) 59

PAGE 60

whereG0(s)isanominaltransferfunctionand(s)2DU.AnunstructureduncertaintyDUisdenedbytheset DU=f(s): [(j!)
PAGE 61

Figure3-5. E-contoursfortheunstructureduncertainsystemforfrequenciesrangingfrom0.1to7Hz Asmentionedearlier,thischapteronlyaddressestheeffectofunstructureduncertainties.Thereforeforthisanalysistheclassoftheunstructureduncertaintyisdenedas DU=(s): [(j!)] [G(j!)]+ 10=(!)(3)Whentheuncertaintytemplateisappliedtothesystem,agraphwiththeuncertaintyforeacheigenvaluecanbeproducedcorrespondingtovariousfrequencies.TheseguresarecalledE-contours.Thesecanbedrawnbasedontheexpressionfoundin[ 45 ]whichstates (G)]TJ /F5 11.955 Tf 11.95 0 Td[(zI)=(3) wherez=i+ei.Figure 3-5 illustratesthisforfrequenciesrangingfrom0.1to7Hzforthisparticulartransferfunction. Inthischapter,theuncertaintymatrixisscaledbyascalarsuchthatthesystemuncertaintybecomes.Thisisdonesothattheeffectofthisscalingoruncertainty 61

PAGE 62

Figure3-6. Unstructureduncertaintytemplatewithscaleduncertainty,= blow-upfactorcanbeusedtoanalyzeboththestructuredsingularvalueandtheNyquistrobuststabilitymarginkNastheunstructureduncertaintyincreases.ItisaneasiertasktocomputethesevaluesatoneparticularfrequencyforthisanalysisascomparedwithcomputingtheentireE-contours.SincetheNyquistrobuststabilitymarginrequiresonlythecriticaldirection,onlyoneangleisneededforthiscomputation.Figure 3-6 showsthesescaleduncertaintiesforthissystemataparticularfrequencyaswellasthecriticallineswhichwouldintersectatthecriticalpoint,)]TJ /F3 11.955 Tf 9.3 0 Td[(1+j0. 3.6Results Thestructuredsingularvaluewascalculatedfromthescaledunstructureduncertaintytemplateandasexpected,islinearwith.TheNyquistrobuststabilitymargin,however,isnotlinearwithrespectto.Bothofthesestabilitymeasureshavetobelessthan1forthesystemtobestable.ThiscanbeseeninFigure 3-7 ,wherebothof 62

PAGE 63

Figure3-7. StructuredsingularvalueandtheNyquistrobuststabilitymarginversusfortheunstructureduncertaintytemplate thesevaluesaregraphed.Itisalsoshownthatthetwographscoincideatthemarginalvalueof1,provingthattheyarebothequivalentmeasuresofstability. 3.7ConclusionRemarks Bothcriteriaaregoodmeasuresforstabilitysincetheybothdescribetheunstablecriticalpoint.Thestructuredsingularvalueiseasiertocomputesinceitisjustalinearrelationshipfortheunstructureduncertaintycase.TheNyquistrobuststabilitymarginhastheadvantagepreservingthedesirableNyquist-likecharacterintherobuststabilityanalysis,buttherelationshipwiththescalingfactorissignicantlymorecomplicated.FurtherresearchisrequiredtoascertainapossibleanalyticalrelationshipbetweentheStructuredsingularvalueandtheNyquistrobuststabilitymarginbutthisnumericalsolutionprovidessomeinsightintogaininganunderstandingofthisproblem. 63

PAGE 64

CHAPTER4WAVELET-BASEDIDENTIFICATIONANDTRACKINGCONTROLFORMEMRISTORBASEDCIRCUIT 4.1INTRODUCTION Chaoticdynamicshasattractedextensiveinterestinvariousdisciplines.Particulary,memristorshavebecomethefocusofagreatdealofresearchincircuitengineeringafterbeingrealizedbyWilliam'sgroupHPLabsin2008[ 23 ].In1971Chua[ 22 ]claimedafourthelementaddedtothethreefundamentalpassiveelements(theresistor,thecapacitor,andtheinductor),butitwasnotuntil2008thatausefulphysicalmodelandrealizationwasprovidedforthememristor.InChapter 4 ,weconsidermemristor-basedchaoticcircuit(MCC)[ 24 25 ]basedonthefundamentalChuacircuitwiththenonlinearelementreplacedbyamemristor.Thememristorisaninterestingtopicforvariousapplicationsbecauseitexhibitsnonlineardynamicphenomenaofbifurcationsandchaos.Insomepreviousresearchonchaoticcircuitdynamics,ithasbeenassumedthattheparametersarewellknownandnodisturbancesexist[ 26 27 52 ].However,exogenousdisturbancesanddynamicuncertaintiescannotbeneglectedinpracticalapplications.Hence,alternativestrategiesinvolvingmemristor-basedchaoticsystemsarerequired.Adaptiverobustcontrolalgorithmshavebeendevelopedfordesigningfeedbackcontrollersfornonlinearsystemswithuncertaintiesasdisturbances.However,theapproachesusuallyrequirelinearparameterizations[ 7 27 ]. Neuralnetwork(NN)algorithmhasbeensuccessfullyappliedtouniversallyapproximatedynamicsystemsthroughlearning.Twokeyrolesofadaptiveneuralnetworkisrobustcontrolforperturbedanduncertainnonlinearsystemsusingfeedfowardterm[ 29 30 ]andtheotherisidenticationbasedcontrolwhichlearnstheuncertaindynamicsusinganon-lineupdatebyalearningalgorithm[ 31 ].BasedonthestructureofNNs,feedforwardNN(FNN)isastaticmappingnetworking.Therefore,dynamicmappingisnotachievablebyFNNwithoutatappeddelay[ 32 ].ComparedtoFNNs,dynamicorrecurrentNN(RNN)havebenetofaninternalfeedbackloop, 64

PAGE 65

Figure4-1. Memristorbasedchaoticcircuit whichcapturesthedynamicresponse.Therefore,itshowsbetterperformanceeventheunmodelednonlineardynamics.TheaforementionedNNschemerequiresalargenumberofiterationsforsmallapproximationerror.Ontheotherhand,theuseofwaveletneuralnetwork(WNN)takesanadvantageofcombiningthespatiallocalizationofwaveletfunctionandthelearningabilityofNNs.ItallowsmoreefcienttrainingalgorithmwithsmalliterationsthanthegeneralNNs[ 33 ].UnlikethesigmoidfunctionsusedintheconventionalNNs,thesecondlayerofWNNisawaveletform,inwhichthetranslationanddilationparametersareincluded.Thus,WNNshavebeenshowntobebetterthantheotherNNs,sinceitsstructurecanprovidemorepotentialtoenrichthemappingrelationshipbetweeninputsandoutputs[ 33 53 ]. Chapter 4 adoptsaWNNidenticationbasedcontrolinconjunctionwithslidingmodecontrol(SMC)toachieveasymptoticandrobustconvergenceidenticationerrorstozero,sinceSMCattenuatesboundeddisturbancesandtheWNNapproximationerrors.TheRISEcontrollerinconjunctionwithaWNNidentierensuresasymptotictrackingofthedesiredreferencesignalofthememristorbasedsystem. EventhoughSMCgiveszeroidenticationerror,itsdiscontinuouspropertycauseschattering.Therefore,theRISEfeedbacktermisusedtoreducechatteringbythecontinuouspropertyofRISE[ 54 55 ].Theon-lineupdatelawofaWNNisderivedfrom 65

PAGE 66

Lyapunovanalysis,sothatthestabilityofuncertainsystemcanbeguaranteed.TherestofChapter 4 isorganizedasfollows.Chapter 4.2 isadescriptionofthememristorsystemwhileChapter 4.3 providestheframefortherobustidenticationoftheWNN.Chapter 4.4 givesthedescriptionoftherobustcontrollerdesignusingRISEcontroller.StabilityanalysisandconvergenceresultsareprovedinChapter 4.5 andChapter 4.6 givessimulationresults. 4.2SystemDescriptionandPreliminaries 4.2.1Memristorbasedchaoticsystem ThememristivecircuitshowninFigure 4-1 consistsofoneinductorL,twocapacitorsC1,C2,resistorR,andonenonlinearmemristorM.InChapter 4 ,MisauxcontrolledmemristorcharacterizedbyitsincrementalmemductancefunctionW((t))[ 24 25 ]thatdescribestheux-dependentrateofchangeofchargeq((t)): W()=dq() dq()=+3(4) where(t)isux.Theq-functionisdenedbythecubicnonlinearityq()=+3,whereandareunknownconstant.Therelationshipbetweenthevoltagev(t)acrossandthecurrenti(t)throughthememristorisgivenbyi(t),dq dt=dq dd dt=W((t))v1(t),whered dt=v1(t)isthevoltageacrossthememristorasshownFigure 4-1 .Thememristorbasedchaoticcircuitwithcontrolinputui(t)anddisturbancesdi(t),i=1,2,3,4,isdescribedbythefollowingsetofequations( 4 ). 66

PAGE 67

d dt=)]TJ /F6 11.955 Tf 9.3 0 Td[(v1(t)+u1(t)+d1(t)dv1 dt=1 C1v2(t))]TJ /F5 11.955 Tf 11.95 0 Td[(v1(t) R)]TJ /F5 11.955 Tf 11.96 0 Td[(W((t))v1(t)+u2(t)+d2(t)dv2 dt=1 C2v1(t))]TJ /F5 11.955 Tf 11.95 0 Td[(v2(t) R)]TJ /F5 11.955 Tf 11.96 0 Td[(iL(t)+u3(t)+d3(t)diL dt=v2(t)+u4(t) L+d4(t)(4) whereistheintegraltimeconstant,v2(t)isthevoltageacrosscapacitorC2,andiL(t)isthecurrentthroughinductorL.Letthex(t)=[(t),v1(t),v2(t),iL(t)]Tbethestatevariableandthedesiredreferenceinputxd(t)=[xd1(t),xd2(t),xd3(t),xd4(t)]T.FortheconvenienceinSection 4.3 ,thememristivecircuitsystemistransformedtoageneralizednonlinearequationshownin( 4 ). 4.2.2WaveletNeuralNetworks Figure4-2. Waveletnetworkstructure InChapter 4 waveletneuralnetworksaredeployedasatoolfortheidenticationofunknowncircuitsystem.WNNincorporatesthetime-frequencylocalizationofwaveletsandthelearningcapabilityofgeneralneuralnetworks.Ithasbeenshownin[ 52 ]thatWNNscanapproximateanycontinuousfunctionoveracompactsetandhavehighaccuracyandfastlearningability.Followingtheconventionin[ 33 ]wedeneasequenceofsubspaceoffunctionL2denedasVj(j=)]TJ /F3 11.955 Tf 21.58 0 Td[(2,)]TJ /F3 11.955 Tf 9.3 0 Td[(1,0,1,2).ThespacecollectionfVj,j2Zgiscalledamulti-resolutionanalysisscalefunctioniftheproperties 67

PAGE 68

ofnestedness,density,separation,scaling,andorthonormalbasishold[ 56 ].AWNNidentierisshowninFigure 4-2 .Forsimplicity1-layerfeedfowardNNisconsideredwithinput.TheWNNoutputwithmwaveletbasisfunctions[ 33 ]canperformthemappingdenedas f(x)=lXj=1jj(!j,(x)]TJ /F5 11.955 Tf 11.95 0 Td[(cj))(4) wherex=[x1,x2,...,xn]T2Rninputvector,j(!j,(x)]TJ /F5 11.955 Tf 12.71 0 Td[(cj))2R,j=1,2,....,larethewaveletfunctions,!j=[!1j,!2j,...,!nj]T2Rnandcj=[c1,c2,...,cj]T2Rarethedilationandtranslationparameters,respectively.j2Ristheoutputlayerweightandlisthenumberneuronsinthetranslationlayer.Eachwaveletnetwork'sneuroninthetranslationlayercanberepresentedby j=hj(x)exp0BB@)]TJ /F7 7.97 Tf 16.13 27.73 Td[(nPk=1w2kj(xk)]TJ /F5 11.955 Tf 11.96 0 Td[(ckj) 21CCA(4) wheretheMexicanhatmotherwaveletfunctionisdenedashj(x)=nQk=1(1)]TJ /F5 11.955 Tf 11.96 0 Td[(w2kx2k).Forsubsequentanalysis,( 4 )isdenedinacompactvectorformas f(x,,,c)=T(Tx+c)(4) where2Rl,2Rl,2Rnl,andc2Rl.Bytheuniversalapproximationtheorem,thereexistanidealidentierf[ 52 ]denedas f=f(x)+(x)=T(x+c)+(x)(4) where^,^,and^caretheestimatesof,,andc,respectively. 68

PAGE 69

4.3RobustSystemIdenticationofWNN Ageneralnonlineardynamicsystemisdescribedasfollows: _x(t)=f(x(t))+g(x(t))u(t)+d(t),(4) wherethestatex(t)=[(t),v1(t),v2(t),iL(t)]T2Rnismeasurable,u(t)2Rnisthecontrolinput,f(x)2Rnisunknownbutboundedcontinuousfunction,locallyLipschitzinx,andg2Rnmisknown.InChapter 4 ,wetaken=m=4. Assumption1:Thedisturbanced(t)2L1anditsrstandsecondtimederivativesarebounded.i.e.d(t),_d(t),d(t)2L1 Assumption2:WNNcanapproximateanycontinuousfunctionoveracompactsetandhavehighaccuracy[ 52 ]andWNN'sapproximationerror(x)anditsrstandsecondtimederivativesarebounded.(x),_(x),(x)2L1 Assumption3:Unknownweights,translation,dilationc,itsestimates,anddesiredinputareboundedfromsomepositiverealnumber. NonlinearsystemincludingWNNindentiercanbeexpressedas _x=Asx+T(Tx+c)++gu+d,(4) whereAs2RnnisHurwitzmatrixanditisaddedandsubtractedto( 4 ).Then,unknownfunctionincludinginternalfeed-forwardloopcanbedenedasf)]TJ /F5 11.955 Tf 12.77 0 Td[(Asx=T(x+c).AMexicanhatwaveletfunctionistheselectedrenderingforWNN.ThedetailednotationisalreadydenedinSection 4.2.2 .TheWNNidentierfor( 4 )canberepresentedas _bx=Asbx+bT(bTx+^c)+gu+d+sgn~(x),(4) where2R+,bx,bT,andbcareslidingmodecontrol(SMC)gain,theestimatesofstate,weightsdilation,andtranslation,respectively. Poznyak[ 31 ]suggestedtheuseofSMCtoensureconvergenceofidenticationerrortozero.Itsdisadvantageisthatchatteringisinevitable,butthecontinuousRISE 69

PAGE 70

statefeedbackcontrollerintroducedinnextsectionmitigatesit.AWNNinconjunctionwithSMCapproximatestheinputandoutputbehavior.Denetheidenticationerroras _ex=_x)]TJ /F3 11.955 Tf 13.54 3.48 Td[(_bx=As~x+T(Tx+c))]TJ /F13 11.955 Tf 12.72 0 Td[(bT(bTx+^c)++d)]TJ /F6 11.955 Tf 11.95 0 Td[(sgn~(x),(4) ByaddingandsubtractingT(eTx+~c)andbT(bTx+^c),weobtain _ex=As~x+eT(eTx+~c)+eT(bTx+^c)+bT(eTx+~c)++d+sgn~(x),(4) Inthesubsequentscheme,theupdatelawisderivedbyanon-lineparametertuningoftheWNNidentiertoachievethecompletedsystemidentication.ATaylorseriesmethodisemployedtotransformthenonlinearfunctionintopartiallylinearformasfollows:e=26666666666664e1e2em37777777777775=266666666666664@1 @x@2 @x@m @x377777777777775jx=bxe+266666666666664@1 @c@2 @c@m @c377777777777775jc=bcec+O2 Forconvenience,theTaylorseriesexpansionissimpliedas e=ATex+BT~c+O2(4) 70

PAGE 71

whereex=x)]TJ /F13 11.955 Tf 12.49 0 Td[(bx,ec=c)]TJ /F13 11.955 Tf 12.27 .5 Td[(bc,andO2representshigherorderterms.Wecannowsubstitutethecontroller( 4 )intotheopen-looperrorsystem( 4 )toobtain _ex=As~x+eT(ATeTx+BT~c+O2)+d+)]TJ /F6 11.955 Tf 11.96 0 Td[(sgn(~x)+bT(ATeTx+BT~c+O2)+eTb=As~x+eTAT(Tx)]TJ /F13 11.955 Tf 12.18 0 Td[(bTx)+eTBT(c)]TJ /F3 11.955 Tf 12.14 0 Td[(^c)+eTO2+bT(ATeTx+BT~c+O2)+eTb+d+)]TJ /F6 11.955 Tf 11.96 0 Td[(sgn~(x)=As~x+eT(b)]TJ /F5 11.955 Tf 11.95 0 Td[(ATeTx)]TJ /F5 11.955 Tf 11.96 0 Td[(BTc)+bTATeTx+bBT~c+h)]TJ /F6 11.955 Tf 11.96 0 Td[(sgn~(x)(4) whereb=(bTx+^c).Thelumpederrorisdenedash=TO2+d+h.ThegainofSMCcontrolisselectedtodominateh,i.e.>h. ConsiderLyapunovcandidatefunctionV1foridenticationanalysis.V1isacontinuouslydifferentiableandpositivedenitefunctiondenedas V1,1 2~xT~x+1 2tr(eT)]TJ /F4 7.97 Tf 6.78 4.94 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1e)+1 2tr(eT)]TJ /F4 7.97 Tf 6.78 4.94 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1e)+1 2tr(~cT)]TJ /F4 7.97 Tf 6.78 4.94 Td[()]TJ /F8 7.97 Tf 6.58 0 Td[(1c~c)(4) where)]TJ /F10 7.97 Tf 6.77 -1.8 Td[(,)]TJ /F10 7.97 Tf 6.77 -1.8 Td[(,and)]TJ /F7 7.97 Tf 6.78 -1.8 Td[(careconstant,symmetricandpositivedenitematrix,i.e,V10.TakingthetimederivativeofV1andusingtheidenticationerrorfrom( 4 ),then_V1isobtainedas _V1=~xT(As~x+eT(b)]TJ /F5 11.955 Tf 11.96 0 Td[(ATeTx)]TJ /F5 11.955 Tf 11.95 0 Td[(BTc))+~xT(bTATeTx+bBT~c+h))]TJ /F6 11.955 Tf 11.95 0 Td[(~xTsgn~(x))]TJ /F5 11.955 Tf 11.96 0 Td[(tr(eT)]TJ /F4 7.97 Tf 6.78 4.94 Td[()]TJ /F8 7.97 Tf 6.58 0 Td[(1e))]TJ /F5 11.955 Tf 11.95 0 Td[(tr(eT)]TJ /F4 7.97 Tf 6.77 4.94 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1e))]TJ /F5 11.955 Tf 11.95 0 Td[(tr(~cT)]TJ /F4 7.97 Tf 6.77 4.94 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1c~c)(4) Selectingtheadaptationlaws,theequationissimpliedas _V1=~xTAs~x+~xTh)]TJ /F6 11.955 Tf 11.96 0 Td[(~xTsgn~(x)(4) 71

PAGE 72

Upper-bounding( 4 )gives _V1minAsk~xk2+hk~xk)]TJ /F6 11.955 Tf 20.59 0 Td[(nXj=1j~xjjminfAsgk~xk2(4) Basedon( 4 ),weclaimthat ~x!0ast!1with)]TJ /F6 11.955 Tf 11.95 0 Td[(minfAxg<0(4) TheadaptationlawoftheWNNidentierarechosenas _b=)]TJ /F10 7.97 Tf 19.4 -1.8 Td[(proj[b)]TJ /F5 11.955 Tf 11.96 0 Td[(ATbTx)]TJ /F5 11.955 Tf 11.95 0 Td[(BTc)~xT]_b=)]TJ /F7 7.97 Tf 19.4 -1.79 Td[(vproj[x~xTbTAT]_bc=)]TJ /F7 7.97 Tf 19.4 -1.79 Td[(cproj[~xTbTBT](4) 4.4RobustControllerDesignusingRISEFeedbackTerm Thecontrolproblemistondasuitablecontrollersothatthestatex(t)canasymptoticallytrackadesiredtime-varyingreferencetrajectoryxd(t).Thedesiredreferencesignalisassumedtobeboundedanditsrstandsecondtimederivativearealsoassumedtobebounded.Thetrackingerrore(t)2L1andlteredtrackingerrorr(t)2L1aredenedas e(t),x)]TJ /F5 11.955 Tf 11.96 0 Td[(xd,(4) r(t),_e+ke,(4) wherekisrealpositivenumber.x(t)ande(t)aremeasurablebutr(t)isanunmeasurablesignal.Accordingly,r(t)isreplacedintermsofmeasurablesignalsusing( 4 ,( 4 ). r=(As+kI)e+T(Tx+c)++d+gu)]TJ /F3 11.955 Tf 13.53 0 Td[(_xd+Asxd,(4) 72

PAGE 73

whereI2R+isidentitymatrix.UsingaRISEfeedbackterm[ 54 ],[ 55 ]thecontrolisdesignedas u=g)]TJ /F8 7.97 Tf 6.58 0 Td[(1(_xd)]TJ /F5 11.955 Tf 11.96 0 Td[(Asxd)]TJ /F13 11.955 Tf 12.72 0 Td[(bT(bTx+^c))]TJ /F6 11.955 Tf 11.96 0 Td[(),(4) wheretheRISEfeedbacktermisrepresentedintermsofthetimederivativeofas _,(k1+k2)r+1sgn(e)(4) wherek1,k2,12R+arecontrolgainsandsgn(e)2Rn. Substituting( 4 )into( 4 ),thentheclosed-loopltererrorsystemisobtainedas r=(As+kI)e+T(Tx+c))]TJ /F13 11.955 Tf 12.72 0 Td[(bT(bTxd+^c)++d)]TJ /F6 11.955 Tf 11.95 0 Td[(,(4) ThetimederivativeofthelterederrorsystemrequiredforthesubsequentanalysisinLyapunovanalysis,isobtainedusing( 4 )as _r=(As+kI)_e+T0(Txd+c)T_x)]TJ /F3 11.955 Tf 13.97 2.99 Td[(_bT(bTxd+^c))]TJ /F13 11.955 Tf 12.72 0 Td[(bT0(bTxd+^c)(_bTxd+_^c))]TJ /F13 11.955 Tf 12.72 0 Td[(bT0(bTxd)bT_xd+_+_d)]TJ /F3 11.955 Tf 11.95 0 Td[((k1+k2)r)]TJ /F6 11.955 Tf 11.96 0 Td[(sgn(e),(4) Substituting_x=r)]TJ /F6 11.955 Tf 12.8 0 Td[(ke+_xdandsegregatingtermsintothestatedependentauxiliaryfunctionswhichareinturnupperboundbyarealpositiveconstant,thenthetimederivativeofthelterederrorsystemisgivenby _r=eN+N)]TJ /F5 11.955 Tf 11.95 0 Td[(e)]TJ /F3 11.955 Tf 11.95 0 Td[((k1+k2)r)]TJ /F6 11.955 Tf 11.96 0 Td[(1sgn(e)(4) wherethestatedependentauxiliaryterm~N2Rnisdenedas 73

PAGE 74

eN=(As+kI)(r)]TJ /F6 11.955 Tf 11.96 0 Td[(k)_e+T0(Tx+c)T(r)]TJ /F6 11.955 Tf 11.95 0 Td[(ke))]TJ /F13 11.955 Tf 10.07 0 Td[(bT(bTxd+^c))]TJ /F13 11.955 Tf 12.72 0 Td[(bT0(bTxd+^c)(bTxd+^c)+e whereN(t)issegregatedbytwoconstanttermsofNDandNB.NDconsistsofthetimederivativeoftheWNNapproximationerroranddisturbance.BasedonAssumption3,NBfunctioncomprisedofweightsanditsestimatetakesupperboundbyaconstant.Accordingly,Ncanbeexpressedinsomepositiverealnumber,sothatwehave N=ND+NBND=_d+_NB=T0(Tx+c)T_xd)]TJ /F13 11.955 Tf 12.72 0 Td[(bT0(bTxd+^c)bT_xd.(4) Basedonthemeanvaluetheorem,theauxiliarytermof( 4 )isupperboundedas eN(kzk)kzk(4) wheretheboundingfunction2Risapositive,globally,invertible,andnondecreasingfunctionandz(t)2R3nisdenedas z(t)[~xTeTrT]T,(4) ThefollowinginequalitycanbedevelopedfromAssumption3 kNDk1,kNBk2,kNk1+2,(4) ByusingAssumption1and2,thetimederivativeofNDcanbeboundedbyapositiveconstantandNBcanbeboundedbyconstant.ButthetimederivativeofNbisnolongerconstantnumberasshownin k_NDk3,k_NBk4+5kzk.(4) 74

PAGE 75

InChapter 4 ,theRISEstatefeedbacktermisusedforthetrackingcontrolproblem,asexplainedin[ 54 ],[ 55 ]. LetDR3n+2beadomaincontainingy(t)=0,wherey(t)2R3n+2isdenedas y(t),[zTp PTp QT]T,(4) andtheauxiliaryfunction P(t),1nXi=1jei(0)j)]TJ /F5 11.955 Tf 17.93 0 Td[(e(0)TN(0))]TJ /F5 11.955 Tf 11.95 0 Td[(L(t)+L(0)(4) wherethesubscripti=1,2,...,ndenotestheithelementofthevector.TheauxiliaryfunctionL(t)2Risdenedas _P=)]TJ /F3 11.955 Tf 10.94 2.65 Td[(_L,)]TJ /F5 11.955 Tf 9.29 0 Td[(rT(N(t))]TJ /F6 11.955 Tf 11.96 0 Td[(1sgn(e)))]TJ /F6 11.955 Tf 11.96 0 Td[(2kzk2(4) where1and22Rarepositiveconstant.Refertosufcientconditionforgainselectiongivenin[ 54 ],[ 55 ].TheauxiliaryfunctionQ(t)2Risdenedas Q(t),1 2tr(eT)]TJ /F4 7.97 Tf 6.77 4.94 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1e)+1 2tr(eT)]TJ /F4 7.97 Tf 6.77 4.94 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1e)+1 2tr(~cT)]TJ /F4 7.97 Tf 6.77 4.94 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1c~c)(4) where)]TJ /F10 7.97 Tf 6.77 -1.79 Td[(,)]TJ /F10 7.97 Tf 6.77 -1.79 Td[(,and)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(careconstant,symmetricandpositivedenitematrix.Q(t)0. 4.5StabilityAnalysis ConsidertheLyapunovfunctionV(y,t):D[0,1)!R+satisfyingkyk22V2kyk2asfollows: V=1 2rTr+1 2eTe+P+V1,(4) Substitute( 4 ),( 4 ),( 4 ),and( 4 ),thenatimederivativeof( 4 )is _V=r(~N+N)]TJ /F5 11.955 Tf 11.95 0 Td[(e)]TJ /F3 11.955 Tf 11.95 0 Td[((k1+k2)r)]TJ /F6 11.955 Tf 11.96 0 Td[(1sgn(e))+e(r)]TJ /F6 11.955 Tf 11.95 0 Td[(e))]TJ /F5 11.955 Tf 11.95 0 Td[(rT(N)]TJ /F6 11.955 Tf 11.96 0 Td[(1sgn(e)+2kzk2+_V1.(4) Substituting( 4 )andcancelingcommonterms,theequationcanbesimpliedas _V=rT~N)]TJ /F5 11.955 Tf 11.96 0 Td[(rT(k1+k2)r)]TJ /F6 11.955 Tf 11.96 0 Td[(eTe+2kzk2,(4) 75

PAGE 76

Takingtheupperboundof( 4 ),theinequalityisobtainedas _V)]TJ /F5 11.955 Tf 21.92 0 Td[(k1krk2)]TJ /F6 11.955 Tf 11.96 0 Td[(kek2+mink~xk2)]TJ /F5 11.955 Tf 11.96 0 Td[(k2krk2+k~Nkkrk+2kzk2)]TJ /F5 11.955 Tf 21.92 0 Td[(k1krk2)]TJ /F6 11.955 Tf 11.96 0 Td[(kek2+mink~xk2)]TJ /F3 11.955 Tf 11.96 0 Td[([k2krk2)]TJ /F6 11.955 Tf 11.95 0 Td[((kzk)kzkkrk]+2kzk2)]TJ /F13 11.955 Tf 23.91 16.85 Td[()]TJ /F6 11.955 Tf 11.95 0 Td[(2)]TJ /F6 11.955 Tf 13.15 8.09 Td[(2(kzk) 4k2kzk2,(4) where,minfk1,,)]TJ /F6 11.955 Tf 9.3 0 Td[(fAsgg.Theresultisderivedbycompletingthesquares.2isappropriatelychosenwith( 6 ),thenthefollowinginequalityholds V)]TJ /F5 11.955 Tf 21.92 0 Td[(ckzk2,(4) wherec2R+andthedomainofckzk2canbedenedas D,ny(t)2R3n+2jky(t)k)]TJ /F8 7.97 Tf 6.59 0 Td[(1(2p k2()]TJ /F6 11.955 Tf 11.96 0 Td[(2))o.(4) UsingBarbalat'slemma,asignalchasingisperformedasfollows.V2L1isappliedto~x,e,r,P,Q2L1in( 4 )and( 4 )._e2L1in( 4 )impliesx,_x2L1asin( 4 ).Ifthedesiredsignalxd,_xd2L1byassumption,then,f,d2L1in( 4 )leadstothecontroller'sboundnessu2L1in( 4 ).Basedontheprojectionproperty,b,b,and^c2L1holdsin( 4 ).( 4 )shows^x,_^x,and_~x2L1,and_d,_2L1fromAssumption1and2.Thus,_b,_e,and_~c2L1in( 4 ).Usingtheboundnessproperty,_r2L1in( 4 )isproved.Byinduction,thetimederivativeofthelumpedstate_z2L1in( 4 ).DenoteasetSasfollow: S,ny(t)2DjU2(y(t))<[)]TJ /F8 7.97 Tf 6.58 0 Td[(12p k2()]TJ /F6 11.955 Tf 11.95 0 Td[(2)]o.(4) Bytheappropriateselectionofthegaink2,Scanincludeanyinitialconditions.Finally,weclaimthat ckzk2!0ast!1)e(t)!0ast!18y(0)2S.(4) 76

PAGE 77

Thisstatementalsoguaranteesthestabilityofidenticationerrorshownin( 4 )with8y(0)2S. 4.6Simulation ConsiderthetrackingcontrolobjectiveoftheuncertainmemristivecircuitasshowninFigure 4-1 .Theuncertainsystemisdescribedin( 4 )andtheparametersaresettoR=2000,C1=6.8nF,C2=68nF,L=18mH,=)]TJ /F3 11.955 Tf 9.29 0 Td[(0.66710)]TJ /F8 7.97 Tf 6.59 0 Td[(3,and=0.02810)]TJ /F8 7.97 Tf 6.58 0 Td[(3.Forconvenience,thedesiredinputsisdenedasxd=[xd1,xd2,xd3,xd4]T=[0.5sin(3t),2cos(2t),0.5sin(2t),cos(2t)]T.Disturbanceisd=[0.1sin(t),0.1cos(2t),0.05sin(2t),0.1cos(2t)]T.TheinputvectorofWNNisx=[x(t)]andthenumberofWNNnodeis10.Basedonthesufcientconditionin[ 55 ],thegainsaredeterminedasfollows:)]TJ /F10 7.97 Tf 6.77 -1.79 Td[(=diagf5,3,2,4g,)]TJ /F10 7.97 Tf 6.78 -1.79 Td[(=diagf2,4,1,1g,)]TJ /F7 7.97 Tf 6.77 -1.79 Td[(c=diagf3,4,1,2g.k=15,=3,1=28,2=1,As=diagf2,3,1,2g,K1=15,k2=25and=8.ThesimulationresultsofWNNformemristivechaoticcircuitareshowninFigure 4-3 .Basedonthesimulationresults,wecanclaimthatWNNidentieraccountsforuncertainnonlinearity,anddisturbancesarefastattenuatedtozerobyRISEfeedbackterm. 4.7Conclusion InChapter 4 ,weproposedawaveletneuralnetworkbasedidenticationandcontrolformemristorbasedchaoticsystem.ThemaincontributionofChapter 4 isthattheapplicationofadaptivehybridWNNidentierguaranteesglobalasymptoticconvergenceandtheRISEfeedbackterminconjunctionwithWNNidentierachievesglobalasymptotictrackingeveninthepresenceoftheresidualerrorsofWNNandconstantboundeddisturbance.Asimulationofmemristor-basedcircuitisgiventodemonstratetheresults.Moreover,itsconvergenceisfasterthangeneralneuralnetworkidentierinconjunctionwithRISEcontrolduetotheintroductionofthewaveletalgorithm. 77

PAGE 78

A B C D Figure4-3. TrackingresponseformemristorchaoticcircuitusingWNNbasedcontrol.(A)Flux,(B)Capacitorvoltagev1,(C)Capacitorvoltagev2,(D)InductorcurrentiL. 78

PAGE 79

CHAPTER5ADAPTIVENON-BACKSTEPPINGNEURALCONTROLFORACLASSOFUNCERTAINNONLINEARSYSTEMSWITHUNKNOWNTIME-DELAY 5.1Introduction Mostdynamicsystemsinrealapplicationsexhibittimedelayswhichmaketheircontrolproblemmoredifcult.Furthermore,theexistenceofunknowntimedelaysinanuncertainsystemcausesinstabilityoftheclosed-loopsystemanddegradationoftrackingperformance.Theseissueshavereceivedincreasingattentioninthepastdecades(seeforexample[ 57 59 ]andthereferencestherein).Inparticular,theLyapunov-Krasovskii(LK)functionalswhichenabletheeliminationofthetimedelaytermhavebeenwidelydevelopedforrobuststabilizationofnonlinearuncertainsystemin[ 8 10 15 60 61 ]. Adaptivestabilizingbacksteppingtechniquesfornonlineartime-delaysystemswiththelinearlyparameterizable(LP)uncertaintieshavebeenaddressedin[ 8 11 ].LearningalgorithmssuchasNNandfuzzycontrolhavebeeninvestigatedin[ 12 ]andfurtherimprovedNNresultsareshownin[ 13 15 ].Intheliterature,adaptiveNNbacksteppingcontrolsfornonlineartime-delaysystemsdoesprovideapossibleapproachwithoutassumptionsoflinearparameterizationsandwelldenedmatchingconditionofuncertainsystems.AlthoughtheadaptiveNNbacksteppingcontrolhasshowngreatpromiseasatoolfortime-delayuncertainsystems,twolimitationshavebeenreportedintheliterature. First,anincreasedcomputationalburdenisinevitableduetotheiterativederivativesofvirtualcontrollersandnonlinearfunctions.Toovercomethecomplicatedcomputations,inChapter 5 theuncertainnonlineartime-delaysystemistransformedintoastandardafnenonlinearsystemnotrequiringthepervasivebacksteppingtechniquemotivatedin[ 20 ].Then,theafnesystemisremodeledbasedonasimpliedDCALtechnique.ADCAL-basedschemeinconjunctionwithLKfunctionalsdeterminestheNNupdatelawunimpededbytime-delayphenomenon. 79

PAGE 80

Thesecondlimitationisthattypicalstabilityresultsshownin[ 12 20 61 ]areonlyuniformlyultimatelybounded(UUB)(semi-globalorglobal)duetoinherentresidualreconstructionerrors,andotherboundedsystemdisturbances.Motivatedbytheobjectiveofrobustlycompensatingfortheboundedsteadystateerror,traditionaltechniquescalledvariablestructurecontroller(VSC)andslidingmodecontrollers(SMC)havebeenproposedin[ 62 63 ].Thesetwocontrollersleadtoasymptotictrackingresults.However,bothhaveacriticalweakness,namely,thechatteringphenomenonduetothediscontinuityofcontrollers.Toovercometheissueofdiscontinuouscontrollers,acontinuousversioncalledtheRobustIntegraloftheSignoftheError(RISE)wasdevelopedin[ 62 65 ].TheRISEcontrollercanyieldasymptotictrackingresultsdespitetheexistenceofresidualerrorsoftheNNapproximationerrorandboundeddisturbances.Thetechniquewasalsoemployedtoachieveanasymptoticresultinthepresenceoftheunstructuredandstructureduncertainties[ 61 64 66 ]. Motivatedbytheaboveobservations,anuncertainnonlinearsystemwithunknownstatetime-delayisreformulatedintoanafnenonlinearsysteminthenormalform(notrequiringthecomputationcomplexityofthegeneralNNbacksteppingschemeinChapter 1.3.2.1 )withstructureduncertainty.Toavoidpotentialtime-delaysofadaptationlaws,Chapter 5 usestheDCALbasedapproachwhichleadstoanormalafnesystemintermsofdesiredtrackingsignals,andfacilitatescontrollerdesign.WiththegoalofimprovingontheUUBresults(semi-globalorglobal)ofadaptiveNNbacksteppingschemesintheliterature,Chapter 5 employstheRISEcontrollertoeliminatetheresidualapproximationerrorandadditiveboundeddisturbances.InconjunctionwiththeappropriateLKfunctional,itisshownthattheproposedcontrolschemeappliedtothisuncertainnonlinearsystemyieldsasymptotictrackingresults.TherestofChapter 5 isorganizedasfollows.Chapter 5.2 denesaclassoftheuncertainnonlinearsystemsaddressedinChapter 5 andgivesabriefsynopsisofthebacksteppingcontrolbasedapproach,whileChapter 5.3 describesthetheorybehind 80

PAGE 81

theDCALbasedRISEschemeusedintheproposedcontroller.Chapter 5.4 givesthedetailedstabilityanalysiswithanappropriateLKfunctionalandprovesasymptoticstability(semi-globalsense).InChapter 5.5 ,asimulationexampleisgiventoillustratetheproposedmethodinChapter 5 andconclusionsaregiveninChapter 5.6 5.2ProblemFormulation Consideraclassofuncertainnonlinearsystemswithunknowntime-delaysdenedbythefollowingequation: _xi(t)=fi(xi(t))+gi(xi(t)]TJ /F6 11.955 Tf 11.96 0 Td[(i))+xi+1(t)_xn(t)=fn(xn(t))+gn(xn(t)]TJ /F6 11.955 Tf 11.95 0 Td[(n))+u(t)+d(t)y(t)=x1(t),i=1,...,n)]TJ /F3 11.955 Tf 11.96 0 Td[(1(5) In( 5 ),x(t),xT1xT2xTnT2Rmnistheboundedsystemstates,wherex(t)in( 5 )isassumedtobeallmeasurable.fi(),gi()2Rmareunknownsmoothnonlinearfunctions.i2R+isanarbitrarilyunknownconstanttimedelay(time-delayfunctionsaredenotedasx(t)]TJ /F6 11.955 Tf 12.32 0 Td[()orxthroughoutChapter 5 .u(t)2Rmisthecontrolinputandy2Rmisoutput. (A1) d(t)2Rmisaboundeddisturbanceanditsrstandsecondderivativesarealsobounded. (A2) Thedesiredtrajectoryisassumedtobeboundedasyd(t),y(i)d(t)2L1,wherey(i)d(t)denotestheithtimederivativefori=1,2,...,n+2. (A3) x(t)2L1ismeasurable.Then,g(x)isbounded,anditsrstandsecondpartialderivativeswithrespecttox(t)existandarebounded. Theuncertainnonlinearsystemin( 5 )isrsttransformedintothenormalafneform.Anewvariabledenotedbyv(t)2Rmnisdenedasv1,y,v2,_v1=x1+f1(x1)+ 81

PAGE 82

g1(x1)2Rm.Asshownin[ 60 ],itstimederivativeisexpressedas _v2=_x2+@f1(x1) @x1_x1+@g1(x1) @x1_x1=@f1(x1) @x1(x2+f1(x1))+f2(x2)+@f1(x1) @x1g1(x1)+@g1(x1) @x1_x1+g2(x2)+x3,a2(x2)+b2(x2)+x3,(5) wherea2(x2)=(@f1=@x1)(x2+f1)+f2(x2)andb2(x2)=g2(x2)+(@f1(x1)=@x1)g1(x1)+(@g1(x1)=@x1)_x1.Takingatimederivativeofv3=x3+a2(x2)+b2(x2)gives _v3=b3(x3)z }| {2Xj=1@a2(x2) @xjgj(x)+2Xj=1@b2(x2) @xj_xj+g3(x3)+2Xj=1@a2(x2) @xj(xj+1+fj)+f3(x3)| {z }a3(x3)+x4,a3(x3)+b3(x2,x3,_x2)+x4.(5) Bycontinuinginthismanner,wecanconcludea1=f1(x1)andb1=g1(x1).Thetransformedsystemthenbecomes vi,ai)]TJ /F8 7.97 Tf 6.58 0 Td[(1(xi)]TJ /F8 7.97 Tf 6.59 0 Td[(1)+bi)]TJ /F8 7.97 Tf 6.58 0 Td[(1(xi)]TJ /F8 7.97 Tf 6.59 0 Td[(2,xi)]TJ /F8 7.97 Tf 6.58 0 Td[(1,_xi)]TJ /F8 7.97 Tf 6.59 0 Td[(2)+xi_vi=ai(xi)+bi(xi)]TJ /F8 7.97 Tf 6.59 0 Td[(1,xi,_xi)]TJ /F8 7.97 Tf 6.59 0 Td[(1)+xi+1,(5) where ai(xi)=i)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xj=1@ai)]TJ /F8 7.97 Tf 6.59 0 Td[(1(xi)]TJ /F8 7.97 Tf 6.59 0 Td[(1) @xj(xj+1+fj)+fi(xi)bi(x,x,_x)=i)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xj=1@ai)]TJ /F8 7.97 Tf 6.59 0 Td[(1(xi)]TJ /F8 7.97 Tf 6.59 0 Td[(1) @xjgj+@bi)]TJ /F8 7.97 Tf 6.59 0 Td[(1(xi)]TJ /F16 5.978 Tf 5.76 0 Td[(1) @xj_xj+gi(xi).(5) 82

PAGE 83

Thesimpliedafnesystemtransformedfrom( 5 )canbeexpressedas _vi=vi+1_vn=a(x)+b(x)+d+uy=v1,,i=1,.....,n)]TJ /F3 11.955 Tf 11.95 0 Td[(1(5) wherea(x1,x2,...xn)2Rmisthestatedependentterm,b(x1,...xn,x1,...,xn,_x1,...,_xn)]TJ /F8 7.97 Tf 6.59 0 Td[(1)2Rmisthelumpedtime-delayterm.Bothfunctionscanbedenotedasa(x)andb(x)forconciseness. 5.3ErrorDynamicSystemDevelopment 5.3.1FilteredErrorsystem Thecontrolobjectiveisforthetransformedsystemin( 5 )totrackagivendesiredtrajectoryyd(t)2Rminthepresenceofunknowntime-delayanduncertainties.Thetrackingerrorsysteme1(t)2Rmisexpressedas e1(t),v1(t))]TJ /F5 11.955 Tf 11.95 0 Td[(yd(t),(5) wherev1=y2Rmin( 5 ).Forthesubsequentanalysis,thelterederrorsystems,e2(t),...,en(t),r(t)2Rm,areobtainedas e2(t),_e1(t)+1e1(t),ei(t),_ei)]TJ /F8 7.97 Tf 6.59 0 Td[(1(t)+i)]TJ /F8 7.97 Tf 6.59 0 Td[(1ei)]TJ /F8 7.97 Tf 6.59 0 Td[(1(t)+ei)]TJ /F8 7.97 Tf 6.58 0 Td[(2,r(t),_en(t)+nen(t),(5) where1,...,n2R+areconstantcontrolgains.Thelteredtrackingerrorr(t)isunmeasurablesinceitisafunctionof_xn(t),buttheothererrorsystemsignalsin( 5 )areallmeasurable.Thegeneralizedtrackingerrorsystemintermsofei(t)canbederivedas ei(t)=i)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xj=0ijeji,i=2,.....,n.(5) 83

PAGE 84

Using( 5 )( 5 ),thelteredopen-looperrorcanbere-writtenas r=v(n))]TJ /F5 11.955 Tf 11.95 0 Td[(y(n)d+l,(5) wherel(e1,_e1,...,e(n)]TJ /F8 7.97 Tf 6.59 0 Td[(1)1)2Rmcanbeexpressedbythemeasurableerrorsystemderivedas l=n)]TJ /F8 7.97 Tf 6.58 0 Td[(2Xj=0nje(j+1)1+ne(j)1+ne(n)]TJ /F8 7.97 Tf 6.58 0 Td[(1)1.(5) Bysubstitutingdynamicsystem( 5 )into( 5 ),weobtainthelterederrorsystemasfollows: r=a(x)+b(x,x,_x)+u+d)]TJ /F5 11.955 Tf 11.96 0 Td[(y(n)d+l.(5) Anopen-looperrorsystemcanbedevelopedbyaDCAL-basedtechniqueusing( 5 ),and( 5 ).Thesystemin( 5 )isgroupedintermsoftheindependenttime-delaytermS1(x,xd,_xd),time-delaytermS2(x,x,_x,xd,xd,_xd),andthelumpeddesiredtrajectoriesW(xd,xd,_xd,y(n)d).Aftersomealgebraicmanipulations,theerrorsystem( 5 )canberewrittenas r=d+S1+S2+W+d+u.(5) LKfunctionalscompensatefortheauxiliaryfunctionsS2inthestabilityanalysis,andtheNNfeedforwardcontrollerdealswithunknownnonlinearsystemd,where d=a(xd)+b(xd,_xd)S1=a(x))]TJ /F5 11.955 Tf 11.96 0 Td[(a(xd)S2=b(x,x,_x))]TJ /F5 11.955 Tf 11.95 0 Td[(b(xd,xd,_xd)W=)]TJ /F5 11.955 Tf 9.3 0 Td[(b(xd,_xd)+b(xd,xd,_xd))]TJ /F5 11.955 Tf 11.96 0 Td[(y(n)d(5) andwherea(xd),b(xd,_xd),b(xd)arethefunctionofthedesiredtrajectoriesofa(x),b(x,x,_x)denedin( 5 ).xd(t),hyTd_yTd(y(n)]TJ /F8 7.97 Tf 6.58 0 Td[(1)d)TiT2Rmndenotesacolumnvectorcontainingthedesiredtrajectoryanditsderivatives. 84

PAGE 85

5.3.2FeedforwardNNEstimation Givenacontinuousnonlinearsmoothfunction:S!RmwhereSisasimplyconnectedset,thereexistsidealweightsandthreshold,(W,V),suchthat(zd)2Ccanbedenedasathree-layerNNas[ 65 ].Theunknownnonlinearity(zd),in( 5 ),canbeapproximatedbyathree-layerNN (zd),WT(VTzd)+(zd).(5) In( 5 ),theinputvectorzd(t)2RN1+1isdenedaszd(t),h1yTd_yTd(y(n)]TJ /F8 7.97 Tf 6.59 0 Td[(1)d)TiT2RN1+1,(N1=mn).V2R(N1+1)N2,andW2R(N2+1)nareboundedidealweightmatrices(N1,N2arethenumberofneuronsofinputlayerandthehiddenlayer).Theactivationfunctionisdenotedas()2RN2+1and(zd)istheapproximationerror. (A4) Theactivationfunctionisboundedanditsderivativewithrespecttoaugmentsisboundedby1asthethresholds.IdealNNweightsW,VareboundedbyR+.Approximationerrors(zd)2R+arebounded,anditsrstandsecondderivativesarealsobounded. k(zd)k1,k_(zd,_zd)k2,k(zd,_zd,zd)k3.(5) 5.3.3Closed-LoopErrorSystem Usingtheopen-looperrorsystemin( 5 )andtheestimatesoftheNNsystem( 5 ),therobustcontrollerinconjunctionwithaNNfeedforwardcontrollerisgivenby u,^d+.(5)2RmisthegeneralizedRISEfeedbackcontroltermshownas _=(ks+1)r+1sgn(en),(0)=0,(5) 85

PAGE 86

whereks,2R+areknownconstantgains.ThefeedforwardNNcontrollerin( 5 ),denotedbyd(t)2Rn,givestheform ^d,^WT(^VTzd).(5) where^V(t)2R(N1+1)N2and^W(t)2R(N2+1)n.Inthestabilityanalysis,theestimatesfortheNNweightsaredeterminedbytheappropriateLyapunovfunctionas _^W,proj()]TJ /F8 7.97 Tf 11.66 -1.8 Td[(1^0^VT_zdeTn),_^V,proj()]TJ /F8 7.97 Tf 11.66 -1.79 Td[(2_zdeTn^WT^0),(5) where)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(12R(N2+1)(N2+1),)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(22R(N1+1)(N1+1)areconstant,positivedenite,symmetriccontrolgainmatrices.Substituting( 5 )into( 5 )yieldstheclosed-looperrorsystemas r=d)]TJ /F3 11.955 Tf 12.68 2.65 Td[(^d+S1+S2+W+d++l)]TJ /F5 11.955 Tf 11.95 0 Td[(u.(5) Takingthetimederivativeoftheerrorsystem( 5 ),weobtainthefollowingexpressionwhichisusefulinthesubsequentstabilityanalysis _r=)]TJ /F3 11.955 Tf 14.3 5.31 Td[(_^WT(^VTzd)+WT0(VTzd)VT)]TJ /F3 11.955 Tf 15.57 2.66 Td[(^WT0(^VTzd)_^Vzd)]TJ /F3 11.955 Tf 15.56 2.66 Td[(^WT0(^VTzd)^V_zd+_S1+_S2+_W+_d+_+_l)]TJ /F3 11.955 Tf 14.06 0 Td[(_,(5) where0(^VTzd)@(VTzd)=@(VTzd)jVTzd=^VTzd.Forsimplicity,0(^VTzd)and(^VTzd)aredenotedas^0and^,respectively.Theactivationfunction()isintroducedinAssumption4.AddingandsubtractingWT^0^VT_zd+^WT^0~VT_zdto( 5 ),weobtain _r=^WT^0~VT_zd+~WT^0VT_zd+WT0^VT_zd)]TJ /F5 11.955 Tf 11.95 0 Td[(WT0^VT_zd)]TJ /F3 11.955 Tf 15.57 2.66 Td[(^WT^0~VT_zd)]TJ /F3 11.955 Tf 16.96 5.32 Td[(_^WT)]TJ /F3 11.955 Tf 15.56 2.65 Td[(^WT^0_^VTzd+_S1+_S2+_W+_d+_+_l)]TJ /F3 11.955 Tf 14.06 0 Td[(_.(5) 86

PAGE 87

SubstitutingtheNNadaptationlaw( 5 ),cancelingcommonterms,andgroupingtermsin( 5 )-( 5 )bythedifferentupperboundingcomponents,weobtainthesimpliedclosed-errorsystems _r=~N+N)]TJ /F5 11.955 Tf 11.95 0 Td[(en+(Ks+1)r)]TJ /F6 11.955 Tf 11.96 0 Td[(sgn(en).(5) In( 5 ),Nd=(_xd,xd,t)2Rmisregroupedin( 5 )-( 5 ).Thestatedependentfunctionsincludingthetime-delayterms~N(e1,...,en,r,e1,...en,r)2Rmaresegregatedas ~N,_S1+_S2+_l+en)]TJ /F1 11.955 Tf 11.96 0 Td[(proj()]TJ /F8 7.97 Tf 11.65 -1.79 Td[(1^0^VT_zdeTn)^)]TJ /F3 11.955 Tf 15.57 2.66 Td[(^WT^0proj()]TJ /F8 7.97 Tf 11.66 -1.79 Td[(2_zdeTn^WT^0)zd,(5) ThestateindependentfunctionNB(^W,^V,zd,_zd,t)issub-segregatedas N,Nd+NBNB,NB1+NB2.(5)Nd(W,V,zd,t),NB1(^W,^V,zd,_zd,t)andNB2(^W,^V,zd,_zd,t))2Rmcanbeupperboundedbyconstants. NB1,)]TJ /F5 11.955 Tf 9.3 0 Td[(WT0^VT_zd)]TJ /F3 11.955 Tf 15.56 2.66 Td[(^WT^0~VT_zd(5) NB2,^WT^0~VT_zd+~WT^0VT_zd(5) Nd,wT0VT_zd+_W+_d+_.(5) SegregatingNB1,NB2in( 5 )-( 5 )ismotivatedbytheassumptionsoftheRISEcontrollerin[ 65 ],sincetimederivativesoftheNNestimatescannotbeexpressedastheupperboundinginequalitybyaknownpositiveconstant.Theconstantupperboundingfunctionfor_NBdoesnotholdandviolatestheassumptionsin[ 64 65 ].Hence,anadditionalcontrolschemeisrequiredtocompensateforstatedependenttermsin( 5 ).Asshownin[ 64 ],theRISEcontrollerandtheadaptiveupdatelawforNNweightestimatescompensatefor( 5 )and( 5 ),respectively.UsingtheMeanValue 87

PAGE 88

Theorem,( 5 )canbeupperboundedforthesubsequentanalysis: k~N(t)k1(kk)kk+2(kk)kk,(5) where(t)2R(n+1)misdenedas (t)=eT1eT2eTnrTT.(5) and1(kk),2(kk)2R+aregloballyinvertibleandnon-decreasingfunctions.ThedelaytermS2in( 5 )canbeexpressedbythestatedependentboundinginequality2(kk)kkasshownin[ 12 15 60 67 ].Thepurposeofsegregatingthedelaydependentterm_S2in( 5 )iscompensatedbyanLKfunctional.ThedetailedprocessisdescribedinthestabilityanalysisinChapter 5.4 .TheLKfunctionalQisdenedas Q=1 2kstZt)]TJ /F10 7.97 Tf 6.59 0 Td[(22(k()k)k()k2d,(5) whereks,2()2Rareintroducedin( 5 )and( 5 ).UsingAssumptions4,( 5 ),( 5 ),and( 5 ),thefollowinginequalitieshold kNdk1,kNBk2,k_Ndk3.(5) From( 5 ),( 5 ),and( 5 ),thetimederivativeofNBcanbeboundedas k_NBk4+5kenk.(5) wherei2R(i=1,2....,5)areknownpositiveconstants. 5.4StabilityAnalysis ThemainresultinChapter 5 maynowbestatedasfollows. Theorem5.1(Asymptoticstability). Thecombinedcontrollergivenin( 5 )-( 5 )foranuncertainnonlinearsystemwithunknowntime-delaysatisesthefollowing 88

PAGE 89

semi-globalasymptoticstabilitycondition. limt!1ke1(t)k!0(5) Thecontrolgainksin( 5 )isselectedsufcientlylarge,andn)]TJ /F8 7.97 Tf 6.58 0 Td[(1,n,1,and2areselectedaccordingtothesufcientcondition 1>1+2+3 +4 ,2>5,n)]TJ /F8 7.97 Tf 6.59 0 Td[(1>1 2,n>2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2(5)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1,nareintroducedin( 5 )anddeterminedin( 5 ).1,2areintroducedin( 518 )and( 5 ),respectively.1,2,3,4,and5areshownin( 5 ),( 5 ). Proof. Recallthatboundnessofallsignalsaregivenwiththenewlyformulatederrorsystemgivenin( 5 ),( 5 ),( 5 ),and( 5 ).Now,letDR(n+1)m+3bedenedas y(t),[T(t)p P(t)p Q(t)p R(t)]T.(5) In( 5 )(t),Q(t),andR(t)areaddressedin( 5 ),( 5 ),( 5 ),respectively.TheauxiliaryfunctionP(t)2Risthegeneralizedsolutiontotheequationgivenas P(t),1mXj=1jeni(0)j)]TJ /F5 11.955 Tf 17.94 0 Td[(en(0)TN(0))]TJ /F7 7.97 Tf 18.69 18.67 Td[(tZ0L()d,(5) wherethesubscripti=1,2,...,ndenotestheithelementofthevector,andtheauxiliaryfunctionL(t)2Risdenedas L(t),rT(NB1(t)+Nd(t))]TJ /F6 11.955 Tf 11.95 0 Td[(1sgn(en))+_en(t)TNB2(t))]TJ /F6 11.955 Tf 11.96 0 Td[(2ken(t)k2,(5) 89

PAGE 90

where22Risapositiveconstantaschosenin( 5 ).L(t)rendersthatP(t)0.ThederivativeofP(t)2Rcanbeexpressedas _P(t)=)]TJ /F3 11.955 Tf 10.94 2.66 Td[(_L(t)=)]TJ /F5 11.955 Tf 9.3 0 Td[(rT(NB1(t)+Nd(t))]TJ /F6 11.955 Tf 11.95 0 Td[(1sgn(en)))]TJ /F3 11.955 Tf 13.39 0 Td[(_en(t)TNB2(t)+2ken(t)k2.(5) Providedthatadequategainsarechosenin( 5 ),thefollowinginequalityisobtained tZ0L()d1jeni(0)j)]TJ /F5 11.955 Tf 17.93 0 Td[(en(0)TN(0).(5) Accordingly,P(t)0isprovedby( 5 ),( 5 ).TheauxiliaryfunctionR(t)2RfortheNNestimatesisdenedas R(t),n 2tr(~WT)]TJ /F4 7.97 Tf 6.78 4.95 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(11~W)+n 2tr(~VT)]TJ /F4 7.97 Tf 6.78 4.95 Td[()]TJ /F8 7.97 Tf 6.58 0 Td[(12~V).(5) In( 5 ),R(t)0because)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(1,)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(2introducedin( 5 )areconstantsymmetricandpositivedenitematricesandn>0.AcontinuouslydifferentiablepositivedeniteLyapunovfunctionalV(y,t):D[0,1)!R(includingLKfunctionalQ(t))isdenedas V,1 2eT1e1++1 2eTnen+1 2rTr+P+Q+R,(5) whichsatisesthefollowinginequalities U1(y)V(y,t)U2(y)(5) providedthesufcientconditionsintroducedin( 5 )aresatised.In( 5 ),thecontinuouspositivedenitefunctionsU1(y),U2(y)2Raredenedas U1(y),1kyk2,U2(y),2kyk2,(5) where1,22R+areknownconstants.Using( 5 ),( 5 ),( 5 ),thedifferentialequationsoftheclosed-looperrorsystemsarediscontinuousinthesetf(y,t)jen=0g. 90

PAGE 91

BasedonFilippov'sdifferentialinclusion[ 68 69 ],theexistenceofsolutionscanbeestablishedfory=f(y)2R(n+1)m+3intheright-handsideoftheclosed-errorsignals.Thegeneralizedtimederivativeofy(t)existalmosteverywhere(a.e.)and_V(y)2a.e._~V(y),denedin[ 70 ]isformulatedas _~V=\2@V(y,t)TK[_e1_e2..en1 2P)]TJ /F16 5.978 Tf 7.78 3.26 Td[(1 2_P1 2Q)]TJ /F16 5.978 Tf 7.79 3.26 Td[(1 2_Q1 2R)]TJ /F16 5.978 Tf 7.79 3.26 Td[(1 2_R]T(5) where@VisthegeneralizedgradientofVshownin[ 69 ],andK[]isdenedas K[f](y),\>0\N=0 cof(B(x,))]TJ /F5 11.955 Tf 11.96 0 Td[(N),(5) whereTN=0meanstheintersectionofallsetsNofLebesguemeasurezero, comeanstheconvexclosure,andB(x,)meansaballofradiusaroundx.Furtherdetailsareshownin[ 66 ].ALipschitzcontinuousfunctionV(y,t)canbewrittenas _~V(y,t)=5VTK[_e1 2P)]TJ /F16 5.978 Tf 7.78 3.25 Td[(1 2_P1 2Q)]TJ /F16 5.978 Tf 7.78 3.25 Td[(1 2_Q1 2R)]TJ /F16 5.978 Tf 7.78 3.25 Td[(1 2_R]T[eT2P1 22Q1 22R1 2]K[_e1 2P)]TJ /F16 5.978 Tf 7.78 3.26 Td[(1 2_P1 2Q)]TJ /F16 5.978 Tf 7.78 3.26 Td[(1 2_Q1 2R)]TJ /F16 5.978 Tf 7.79 3.26 Td[(1 2_R]T,(5) wheree=[e1e2en].Using( 5 ),( 5 ),( 5 ),and( 5 )the_~V(y,t)canbeexpressedas _~V)]TJ /F6 11.955 Tf 23.91 0 Td[(1eT1e1)]TJ /F6 11.955 Tf 11.96 0 Td[(2eT2e2)]TJ /F6 11.955 Tf 47.82 0 Td[(neTnen+eTnen)]TJ /F8 7.97 Tf 6.58 0 Td[(1+eTnr+rT(~N+Nd+NB1)]TJ /F5 11.955 Tf 11.96 0 Td[(en+^WT^0~VT_zd+~WT^0^VT_zd))]TJ /F3 11.955 Tf 11.95 0 Td[((ks+1)krk2)]TJ /F6 11.955 Tf 11.96 0 Td[(1rTK[sgn(en)])]TJ /F5 11.955 Tf 11.95 0 Td[(rT(NB1+Nd)]TJ /F6 11.955 Tf 11.95 0 Td[(1K[sgn(en)]))]TJ /F3 11.955 Tf 13.39 0 Td[(_eTnNB2+2kenk2+1 2ks(22()kk2)]TJ /F6 11.955 Tf 11.96 0 Td[(22()kk2)+tr(2~WT)]TJ /F4 7.97 Tf 6.77 4.95 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(11_~W)+tr(2~VT)]TJ /F4 7.97 Tf 6.77 4.95 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(11_~V),(5) 91

PAGE 92

whereK[sgn(en)]=SGN(en)suchthatSGN(en)=1ifen>0,[-11]ifen=0,and-1ifen<0.Basedonthefollowinginequalities 22(kk)kk2krk22()kk2 2ks+ks 2krk2eTn)]TJ /F8 7.97 Tf 6.59 0 Td[(1en1 2)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(ken)]TJ /F8 7.97 Tf 6.59 0 Td[(1k2+kenk2,(5) wegetthefollowingequation: _~V)]TJ /F7 7.97 Tf 24.75 14.94 Td[(n)]TJ /F8 7.97 Tf 6.59 0 Td[(2Xi=1keik2)]TJ /F13 11.955 Tf 11.96 16.85 Td[(n)]TJ /F8 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2ken)]TJ /F8 7.97 Tf 6.59 0 Td[(1k2)]TJ /F3 11.955 Tf 11.95 0 Td[((n)]TJ /F6 11.955 Tf 11.95 0 Td[(2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2)kenk2)-222(krk2)]TJ /F5 11.955 Tf 13.15 8.09 Td[(ks 2krk2+1(kk)kkkrk+22()kk2 2ks.(5) Completingthesquaresof( 5 ),thengives _~V(y,t))]TJ /F13 11.955 Tf 23.91 16.85 Td[(3)]TJ /F6 11.955 Tf 13.15 8.09 Td[(2(kk) 2kskk2,(5) wheretheboundingfunction,2(kk)2R,isdenedas2(kk)=21(kk)+22(kk),and3=min[1,2,...,n)]TJ /F8 7.97 Tf 6.59 0 Td[(1)]TJ /F8 7.97 Tf 13.75 4.7 Td[(1 2,n)]TJ /F6 11.955 Tf 12.56 0 Td[(2)]TJ /F8 7.97 Tf 13.76 4.7 Td[(1 2,1].Theboundingfunction(kk)isapositive,globallyinvertible,andnondecreasingfunctionunimpededbythetime-delay.Theboundingfunctionin( 5 )canbeexpressedas _~V(y,t)U(y)=)]TJ /F5 11.955 Tf 9.3 0 Td[(ckk2,c2R+(5) wherethefunctionisdenedasD,ny(t)2R(n+1)m+3jkyk)]TJ /F8 7.97 Tf 6.59 0 Td[(1p 23kso. TheinequalitiesshowthatV(y,t)2L1inD.Accordingly,e1,e2,...,en2L1arebounded,andtheprojectionalgorithmin( 5 )ensuresthat^W,^Vremainbounded.BasedontheboundingpropertyoftheNNactivationfunctionsandthesignumfunction,wecanprovethat_e1,_e2,...,_en,_risboundedinDfrom( 5 ).Hence,letSDdenotea 92

PAGE 93

setdenedas S,y(t)2DjU2(y(t))<)]TJ /F8 7.97 Tf 6.59 0 Td[(1p 23ks2.(5) Increasingthecontrolgainkscanmaketheattractiveregionlargerforincludingtheinitialcondition,andhence, ck(t)k2!0ast!18y(0)2S.(5) Finally,basedonthedenitionof(t),( 5 )ensuresthatthesemi-globalasymptoticresultforthetrackingobjectivesuchthat limt!1ke1(t)k!0,8y(0)2S.(5) Figure5-1. Trackingerrorfortheunstructuredunknowntime-delaysystem. 93

PAGE 94

Figure5-2. ControleffortoftherobustcontrollawwithNNadaptation. Figure5-3. Adaptationlawk^Vkofathree-layerNN 5.5Simulation Theuncertainnonlineartime-delaysystemisconsideredasin[ 14 61 ]. _x1=x2,_x2=f(x)+g(x)+1(x)+2(x)+d+bu,y=x1.(5) 94

PAGE 95

Figure5-4. Adaptationlawk^Wkofathree-layerNN. In( 5 ),g(x)and2(x)areunknowntime-delayfunctions.Forthesimulation,functionsandparametersarechosenasf(x)=0.5sin(x1(t)),g(x)=0.2x21(t)]TJ /F6 11.955 Tf 12.33 0 Td[(1)+2cos(x2(t)),1(x)=sin(5x2(t)),2(x)=0.5x2(t)]TJ /F6 11.955 Tf 11.67 0 Td[(2)sin(2x1(t)]TJ /F6 11.955 Tf 11.67 0 Td[(1)),d=0.1sin(t),b=1.Thedesiredtrajectoryisxd=0.5[sin(t)+sin(0.5t)].Thegainsanddelaytimearechosenas1=10s,2=10s,ks=15,1=8,2=7,1=5,2=4,)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(1=1,and)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(2=1.ThesimulationresultsinFigure 5-1 demonstratethattheproposedcontrollerachievesthecontrolobjectivedespitelongtimedelays. 5.6Conclusion Chapter 5 dealswithanalternativetechniquetoovercomethecomputationalburdenofatypicalpervasiveNNbacksteppingcontrolandtoimprovethestabilityresultforuncertainnonlinearsystemswithunknowntime-delay.ThekeyelementsoftheproposedapproachisthenewlydenedsysteminDCAL-basedtechniquecombinedwithtransformingfromanuncertainnonlinearsystem,andovercomesthecomputationalexplosioncausedbygeneralvirtualcontroltechniques.AnappropriateLKfunctionalisemployedforcompensatingfortheunknowntime-delay,andaRISE-basedrobustcontrollerinconjunctionwithanadaptiveNNcontrollercompensateforthelumped 95

PAGE 96

constantboundedtermwhichresultsinsemi-asymptoticalresult.TheissueofexistenceofsolutionsishandledusingtheFilippov'stechnique. 96

PAGE 97

CHAPTER6CONTROLOFNEURALSYNCHRONYINTHEMUTUALLYCOUPLEDNETWORK 6.1Introduction Synchronousoscillationhasbeenobservedinaspecicormultipleregionofthebrain,andithasbeensuggestedthatsynchronousactivitywouldleadthecriticalmechanismrelatedtothecognitionandsensoryinformationprocessingwithmacroscopicallyobservablebrainrhythm[ 71 ],[ 72 ].Neuralsynchronystudiesrelatedtocognitionandinformationprocessusingnon-invasiverecordingtechniquessuchaselectroencephalography(EEG)haveanalyzedthatneuralsynchronyrequireslarge-scaleintegrationofdistributedneuralactivity.Inaddition,thereisgrowingevidencefromliteraturetosupportthehypothesisthatneuralsynchronyisacriticalcomponentofthetemporalcodingstrategiesforinformationprocessinginthebrain[ 73 ].Asaresult,anumberofscienticstudieshavefocusedonthemechanismsofneuralsynchronyinthebraincorticalnetworks.Computationalstudieshavedemonstratedthatsynapticallycoupledinterneuronscansustainin-phasesynchronyintheirmembranepotentialspikes,whichinduceshypothesisthatinterneuronsplayasignicantroleinthegenerationofsynchronousbrainactivity.Intheliterature,numerousevidences[ 74 ],concludesthatspikinginterneuronnetworkcanproducesynchrony,anditisverysensitivetoheterogeneityinnetworkparameters. Overthelastdecade,thevariouscontrolbasedtheoreticalinvestigationshaveappearedtodevelopthesynchronymainlyinunidirectionalcoupledinterneuron(UCI)networks.Thesecanbecategorizedintotwoclasses.Therstclassisrelatedtoneuronalsynchronywithexplicitfeedbackcontrolbasedonexactmodelknowledge(EMK)withknownsystemparameters[ 75 ],andthesecondapproachisrelatedtoanobserverbasedadaptivecontrolproblemfortheuncertainnetworkidentication[ 76 ],[ 77 ].Adaptivecontrollerswithobserver,andlearningalgorithmssuchastheneuralnetwork(NN),aswellas,fuzzytechniqueshavebeenproposedtoachievethesynchronyof 97

PAGE 98

UCInetwork[ 78 ].Theapproachhascenteredontheformofthecontrolinputforcompensatingtheparameteruncertaintiesfordynamicalinterneurondynamics,andachievesrobustsynchronyoftheneuralnetworksinthepresenceofheterogeneity.Althoughtheadaptivecontrolhasshowngreatpromiseasatoolforin-phasesynchrony,thecriticalmistakehasbeenreportedinthebiologicalviewthatcontrolinputhastobelowerthanthethresholdnottoarticiallyinduceanactionpotentialspike,butmerelyperturbsthetimingofanimpendingactionpotential. Motivatedbytheinputconstraintnottoarticiallyinduceneuron'sintrinsicalring,thewell-knownneuralnetworkalgorithmbasedH1designframeworkisproposedtoestimateunknownparametersandbroadentherobustrangeofthe1:1in-phasesynchronyinthepresenceofheterogeneityoftheMCInetwork.Then,asaturationtechniqueisemployedtolimitinputamplitudeasaswitchingcontroller.Finally,anumericalanalysisispresentedtoinvestigatetherobustnessofin-phasesynchronyversustimevaryingheterogeneityandsynapticcouplingstrength. ThelayoutofChapter 6 isorganizedasfollows:Chapter 6.2 describesthemathematicalmodelforthefastspikinginterneuron,thesynapseandthechannelkinetics.Chapter 6.3 providestheframeforthefunctionapproximationandgivesthedescriptionofaninputconstraintcontrollerdesigncombinedwiththeneuralnetwork(NN)basedH1technique,whileChapter 6.4 providesaperformanceindexanda1:1in-phasesynchronyanalysis.Finally,Chapter 6.5 comparestheresultsofthenumericalanalysisinthenominalandtheproposedcontrolsystem,andconcludingremarksaregivenin 6.6 98

PAGE 99

6.2ModelNeuronsintheMCINetwork 6.2.1ModelDescription Thedynamicequationforthemodelneuronisgivenas: CdV(t) dt=IDC+gNam31h(t)(ENa)]TJ /F5 11.955 Tf 11.95 0 Td[(V(t))+gKn4(t)(EK)]TJ /F5 11.955 Tf 11.95 0 Td[(V(t))+gL(EL)]TJ /F5 11.955 Tf 11.95 0 Td[(V(t))+IS(t),(6) whereC=1=cm2denotescapacitance,V(t)2Rdenotesthemembranepotential. IDCdenotestheexternalDCcurrentsuchthattheneuronspikeswithintrinsicperiodT0(IDC)/1=p IDC)]TJ /F5 11.955 Tf 11.96 0 Td[(I0,whereI0isthebifurcationpointfortheneurontotransitiontoregularspikingmode.AlsoIS(t)=gsS(t)(EI)]TJ /F5 11.955 Tf 12.35 0 Td[(V(t))istheinhibitorysynapticcurrentfromthepre-synapticneuron(Vpre).gsrepresentsthesynapticstrengthofconnection.Er(r=Na,K,L)2Rarereversalpotentialofthesodium,potassiumionchannels,andtheleakchannel,respectively.gr(r=Na,K,L)representstheconductanceofsodium,potassium,andtheleakchannel,respectively.Thesteadystateactivationforsodiumcurrentisgivenbym1=m=(m+m).Theinactivationvariableforsodiumchannelh(t)andtheactivationvariableforpotassiumcurrentn(t)satisesthefollowingrstorderkineticequation: dX(t) dt=[X(V(t))(1)]TJ /F5 11.955 Tf 11.95 0 Td[(X(t)))]TJ /F6 11.955 Tf 11.95 0 Td[(X(V(t))X(t)],(6) where=5,andXandXaregivenbyTable 6-1 ThequantityS(t)denotesthefractionofboundreceptorsandsatisesthefollowingrstorderkineticequation, _S(t)=S0(Vpre(t)(t)))]TJ /F5 11.955 Tf 11.96 0 Td[(S(t) ^(SI)]TJ /F5 11.955 Tf 11.96 0 Td[(S0(Vpre(t)(t)))(6) where(t)=Pi(t)]TJ /F5 11.955 Tf 11.96 0 Td[(ti)((ti+R))]TJ /F5 11.955 Tf 11.96 0 Td[(t)(X)istheheavisidefunctionsatisfying(X)=1ifX>0else(X)=0,andtiisthetimeoftheithpre-synapticneuronalspike(Vpre(t)).ThekineticequationforS(t)involvestwotimeconstants,thedocking 99

PAGE 100

timefortheneurotransmitterR=^(SI)]TJ /F3 11.955 Tf 12.26 0 Td[(1)andforneurotransmitterbindingD=^SI.Finally,S0isthesigmoidalfunctiongivenbyS0()=0.5(1+tanh(120()]TJ /F3 11.955 Tf 11.95 0 Td[(0.1))). Table6-1. TransitionratesfortheactivationandinactivationvariablesoftheionchannelswithX=m,h,n. XX m=0.1(V(t)+35) e)]TJ /F16 5.978 Tf 5.75 0 Td[(0.1(V(t)+35))]TJ /F8 7.97 Tf 6.58 0 Td[(0.1m=4e)]TJ /F8 7.97 Tf 6.58 0 Td[((V(t)+60)=18h=0.07e)]TJ /F8 7.97 Tf 6.58 0 Td[(((V(t)+58)=20h=1 e)]TJ /F16 5.978 Tf 5.76 0 Td[(0.1(V(t)+28)+1m=0.01(V(t)+34) e)]TJ /F16 5.978 Tf 5.75 0 Td[(0.1(V(t)+34))]TJ /F8 7.97 Tf 6.58 0 Td[(1n=0.125 e)]TJ /F16 5.978 Tf 5.76 0 Td[((V(t)+44)+88 Table6-2. Listofalltheparametersforthemodelconsidered NeuronmodelENa55mVEK)]TJ /F3 11.955 Tf 9.29 0 Td[(90mVEL)]TJ /F3 11.955 Tf 9.29 0 Td[(65mVgNa35mS=cm2gK9mS=cm2gL0.1mS=cm2SynapticmodelEI)]TJ /F3 11.955 Tf 9.3 0 Td[(82mVR0.1msD2ms 6.2.2Measureof1:1Synchrony Figure6-1. SchematicdiagramoftheMCInetwork HerewedescribetheintrinsicpropertiesoftheMCInetworkintermsofitsabilitytosustain1:1synchronyinabsenceofexternalstimulator(controller).Theschematic 100

PAGE 101

diagramoftheMCInetworkinthepresenceoffeedbacksynapticcouplingstrength(gf)isshowninFigure 6-1 .TheMCInetworkcomprisesanintrinsicallyring(periodT10)interneuron1interactingwithanintrinsicallyringinterneuron2(periodT20)throughafast-GABAAmediatedinhibitorysynapse.TheDCcurrentsinneuron1and2aredifferentandarerelatedviatheheterogeneityparameterHas: I2DC=8>><>>:I1DC1+H(t) 100I1DC,H(t)>0(6) suchthatT10>T20inH(t)<0andT10,whereheterogeneityH(t)canbeastaticoratimevaryingvariable.Foralltheresultspresentedbelowunlessotherwisementioned,I1DC=0.5A=cm2,correspondingtointrinsicperiodT1032.8ms.Thecontrolobjectiveistodeterminethestableregionofphasedlockedstateof1:1synchronybetweentwomutuallycoupledneurons.Lett1i(i=1,2,)andt2j(j=1,2,)representsthespiketimesforneurons1and2,respectively.ThemeanperiodhTxi=limt!1txm m,(X=1,2)andthetimedifferencen=t2n)]TJ /F5 11.955 Tf 12.38 0 Td[(t1n0betweenthenearestspiketimeofthetwoneurons,wheremisthespiketimeofneuron1suchthatt2n)]TJ /F8 7.97 Tf 6.59 0 Td[(1
PAGE 102

Figure6-2. Arnoldtongueforthedomainof1:1synchronyintheabsenceofstimulator Figure6-3. TimedifferencesnbetweenthespikesoftwoneuronsasthefunctionofH 102

PAGE 103

thetwocoupledoscillatorscansynchronizeincreases,resultinginatongueshapedtwodimensionaldomain.Forthecaseofgs=0.08,Fig. 6-3 showsthescatterplotofthephasedifferencen,betweenthetwocoupledneuronsasfunctionofheterogeneity.Thetwoneuronsphaselockin1:1synchronywithatimedifferencenexistsintherangeofvalues()]TJ /F3 11.955 Tf 9.29 0 Td[(5H5).Blackdotsdenotethetimedifferencebetweenthespiketimesoftwoneurons,andthespikingratio10hTAi hTBibetweenthemeanringperiodsoftwoneuronsisshownasreddots.ForthecaseofH=)]TJ /F3 11.955 Tf 9.29 0 Td[(5(I2DC=0.475A=cm2),weseethatthetimedifferenceatn=2.3ms. 6.3SynchronousErrorDynamicSystemDevelopment 6.3.1ProblemFormulation Therstneuronmodelinthemutuallycoupledinterneuronisexpressedas _V1=)]TJ /F3 11.955 Tf 17.78 8.09 Td[(1 CmhI1DC+gNam31h1(ENa)]TJ /F5 11.955 Tf 11.96 0 Td[(V1)+gKn41(EK)]TJ /F5 11.955 Tf 11.96 0 Td[(V1)+gL(EL)]TJ /F5 11.955 Tf 11.95 0 Td[(V1)+gsS1(t)(El)]TJ /F5 11.955 Tf 11.96 0 Td[(V1)i_S1=S0(V2(t)(t)))]TJ /F5 11.955 Tf 11.96 0 Td[(S1(t) ^(SI)]TJ /F5 11.955 Tf 11.95 0 Td[(S0(V2(t)))_X1=[X(V1)(1)]TJ /F5 11.955 Tf 11.95 0 Td[(X(t)))]TJ /F6 11.955 Tf 11.95 0 Td[(X(V1)X],X=n,h(6) andthesecondneuronas _V2=)]TJ /F3 11.955 Tf 17.78 8.09 Td[(1 CmhI1DC1+H 100+gNam32h2(ENa)]TJ /F5 11.955 Tf 11.95 0 Td[(V2)+gKn42(EK)]TJ /F5 11.955 Tf 11.96 0 Td[(V2)+gL(EL)]TJ /F5 11.955 Tf 11.95 0 Td[(V2)+gsS2(El)]TJ /F5 11.955 Tf 11.95 0 Td[(V2)+ui_S2=S0(V1(t)(t)))]TJ /F5 11.955 Tf 11.95 0 Td[(S2 ^(SI)]TJ /F5 11.955 Tf 11.96 0 Td[(S0(V1(t)))_X2=[X(V2)(1)]TJ /F5 11.955 Tf 11.95 0 Td[(X(t)))]TJ /F6 11.955 Tf 11.95 0 Td[(X(V2)X(t)],X=n,h(6) wheretheaddedtermuisthecontrolinputfor1:1in-phasesynchrony.Statevectorisdenotedasx1=[V1,S1,n1,h1]T,x2=[V2,S2,n2,h2]T.ThecontrolobjectivedenedaseV=V2)]TJ /F5 11.955 Tf 12 0 Td[(V1withe=[eV,eS,en,eh]T2R4istorobustlysynchronizetheactionpotential 103

PAGE 104

spikesoftwoneuronmodelsinthepresenceofuncertainties.Toquantifytheobjective,errordynamicscanbeexpressedas _eV=)]TJ /F3 11.955 Tf 17.78 8.09 Td[(1 CmhI1DCH 100+gNam32h2(ENa)]TJ /F5 11.955 Tf 11.95 0 Td[(V2)+gKn42(t)(EK)]TJ /F5 11.955 Tf 11.96 0 Td[(V2)+gL(EL)]TJ /F5 11.955 Tf 11.96 0 Td[(V2)+gsS2(El)]TJ /F5 11.955 Tf 11.95 0 Td[(V2))]TJ /F5 11.955 Tf 11.96 0 Td[(gNam31h1(ENa)]TJ /F5 11.955 Tf 11.96 0 Td[(V1))]TJ /F5 11.955 Tf 11.95 0 Td[(gKn41(t)(EK)]TJ /F5 11.955 Tf 11.95 0 Td[(V1))]TJ /F5 11.955 Tf 11.96 0 Td[(gL(EL)]TJ /F5 11.955 Tf 11.96 0 Td[(V1))]TJ /F5 11.955 Tf 11.95 0 Td[(gsS1(t)(El)]TJ /F5 11.955 Tf 11.95 0 Td[(V1)i+u=f(x1,x2)+I1DCH 100+u_eS=S0(V1(t)(t)))]TJ /F5 11.955 Tf 11.95 0 Td[(S2 ^(SI)]TJ /F5 11.955 Tf 11.96 0 Td[(S0(V1(t))))]TJ /F5 11.955 Tf 13.15 8.09 Td[(S0(V2(t)(t)))]TJ /F5 11.955 Tf 11.96 0 Td[(S1(t) ^(SI)]TJ /F5 11.955 Tf 11.95 0 Td[(S0(V2(t)))_eX=hX(V2)(1)]TJ /F5 11.955 Tf 11.95 0 Td[(X(t)))]TJ /F6 11.955 Tf 11.95 0 Td[(X(V2)X(t))]TJ /F6 11.955 Tf 11.95 0 Td[(X(V1)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(X(t)))]TJ /F6 11.955 Tf 11.96 0 Td[(X(V1)X(t)i,X=n,h(6) where f(x1,x2)=)]TJ /F3 11.955 Tf 15.13 8.09 Td[(1 CmhgNam32h2(ENa)]TJ /F5 11.955 Tf 11.95 0 Td[(V2)+gKn42(t)(EK)]TJ /F5 11.955 Tf 11.96 0 Td[(V2)+gL(EL)]TJ /F5 11.955 Tf 11.95 0 Td[(V2)+gsS2(El)]TJ /F5 11.955 Tf 11.95 0 Td[(V2))]TJ /F5 11.955 Tf 11.96 0 Td[(gNam31h1(ENa)]TJ /F5 11.955 Tf 11.96 0 Td[(V1))]TJ /F5 11.955 Tf 11.95 0 Td[(gKn41(t)(EK)]TJ /F5 11.955 Tf 11.95 0 Td[(V1))]TJ /F5 11.955 Tf 11.96 0 Td[(gL(EL)]TJ /F5 11.955 Tf 11.96 0 Td[(V1))]TJ /F5 11.955 Tf 11.95 0 Td[(gsS1(t)(El)]TJ /F5 11.955 Tf 11.96 0 Td[(V1)i(6)Assumption1:HeterogeneityH(t)2Rcanbeastaticortimevaryingboundedvariable.Assumption2:Potassium,leakchannelsgates(m,n)andsynapse(S)aremeasurable,theirparametersareunknown.Assumption3:MembranevoltageV(t)2L1ismeasurable.Then,f(x1,x2)isbounded Ifthenonlinearfunctionf(x1,x2):R4!Risexactlyknown,thecontrolleruisdenedas u=)]TJ /F5 11.955 Tf 9.3 0 Td[(keV)]TJ /F5 11.955 Tf 11.96 0 Td[(I1DCH 100)]TJ /F5 11.955 Tf 11.95 0 Td[(f(x1,x2),(6) wherethefeedbackgainkispositive.Then,theclosed-looperrordynamicsin( 6 )isobtainedas_eV=)]TJ /F5 11.955 Tf 9.3 0 Td[(keVwhichasymptoticallyconvergestozeroast!1.Moreover, 104

PAGE 105

theerrordynamicsofsynapse(eS)andchannelgates(eX)isaninternalstatevariablevector,whichalsoasymptoticallyconvergestozeroaslimt!1keVk=0.Finally,weensurethat1:1in-phasesynchronyisachieved. However,theparametersofsynapse(S)andchannelgates(n,h)inthenonlinearfunctionf(x1,x2),andheterogeneityHaregenerallyassumedtobeunknown.Therefore,thecontrollerin( 6 )wouldnotbeimplementable.InnextsectionthecompositeNNbasedH1controllerisproposed. 6.3.2Closed-LoopSystemintheMCINetwork Theidealcontrollerin( 6 )showsthattheerrordynamicsasymptoticallyconvergestozero.Thedynamicsofsynchronizationerrorcanbetransformedintothecanonicalform[ 79 ]as _e=Ae+Bf(x)+I1DCH 100+u(6) wheref(x)=f(x2))]TJ /F5 11.955 Tf 11.96 0 Td[(f(x1)2R4isboundedsmoothnonlinearfunction,and A=2666666640100001000010000377777775,B=2666666640001377777775(6) areconstantmatrices.HeterogeneityHcanbeaunknownxednumberortimevaryingfunction.Usingthecanonicalerrorsystemin( 6 ),therobustcontrolcombinedwithaNNfeedforwardtermisgivenby u=)]TJ /F5 11.955 Tf 9.29 0 Td[(kTee)]TJ /F3 11.955 Tf 12.06 2.66 Td[(^f)]TJ /F5 11.955 Tf 11.95 0 Td[(u1(6) wherekTe2R41isaknownconstantgain.u12RistheH1controltermshownas u1=1 22BTPe.(6) 105

PAGE 106

wherePisthesolutionoftheRiccatiequationshownintheanalysissection.TheNNcontorllerin( 6 ),denotedby^f(x),isdenedas ^f(x),^WT(x),(6) whereWisassumedtobetheidealconstantweightvector.TheupdatelawfortheNNweightsin( 6 )ischosenas _^W=)]TJ /F6 11.955 Tf 19.39 0 Td[((x)BTPe(6) where)]TJ /F1 11.955 Tf 10.09 0 Td[(isaconstantpositivedenite,symmetricmatrice. Bysubstituting( 6 )into( 6 )theclosed-loopsynchronoussystemiswrittenas _e=hA)]TJ /F5 11.955 Tf 11.96 0 Td[(BkTie+Bf(x))]TJ /F3 11.955 Tf 12.05 2.66 Td[(^f(x))]TJ /F5 11.955 Tf 11.96 0 Td[(u1+I1DCH 100,(6) wherethefeedbackkischosensuchthatA)]TJ /F5 11.955 Tf 13.11 0 Td[(BkTisHurwitz.DetailsfortheNNparametersandpropertyhaveshowninChapter 5.3.2 .Rearrangingthetermsyields _e= Ae+Bh^WT(x))]TJ /F5 11.955 Tf 11.96 0 Td[(u1+!i.(6) where A=A)]TJ /F5 11.955 Tf 11.96 0 Td[(BkT.Thereconstructionerror(x)andtheheterogeneityHarecollectedintothelumpeduncertainterm!.Thetermisassumedtobe!2L2[0,tf],8tf2[0,1).Withthelumpeduncertainty!6=0thefollowingtrackingperformanceisinvokedas 1 2Ztf0eT(t)Qe(t)dt1 2eT(0)Pe(0)+1 2~WT(0))]TJ /F4 7.97 Tf 22.81 4.94 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1~W(0)+1 22Ztf0!2(t)dt(6) whereQ,P,and)]TJ /F1 11.955 Tf 10.1 0 Td[(areconstant,symmetric,andpositivedenitematrix.isanattenuationlevel.Iftheinitialconditionsofweightingmatricesstartswith~W(0),thentheperformanceindexisdenedas sup!2L2[0,tf]kekQ k!k,(6) 106

PAGE 107

Thesynchronizationerroreisattenuatedlowerthanthedesiredtermregardlessoftheuncertaintylevel!. 6.3.3RobustControlwithInputConstraint Theobjectiveistodesignanestimationbasedcontrollerthatcaninducestable1:1in-phasesynchronyintheMCInetworkforthebroadrangeofheterogeneityH.Thecontrollerwithlowconstantgainsin( 6 ),( 6 )doesnotguaranteethatthestimulationislowerthanthethresholdvalueumax,whicharticiallyinducethemembraneactionpotential.Accordingly,thesaturationtechniquecombinedwith( 6 )asaswitchingcontrollerisproposedas usat=8>><>>:u,ifjujumaxumaxsgn(u),ifjuj>umax(6) wherethethresholdvaluefor( 6 )umaxis0.25anduistheHcontrollerin( 6 ).Neurondoesnotspikeatlowerthanheterogeneity(H)]TJ /F3 11.955 Tf 22.14 0 Td[(50),andhigherthan(H60)whichaffectsthefeedbacksynapticcouplingstrength,gfinFig. 6-1 toinhibittheotherneuron'sring.Inbothcasesofheterogeneityabovethestimulatorofusatisunabletoinduceneuronstospike.Namely,theintensityoflightstimulationislowenoughnottoinducespikesbutmerelyperturbimpendingactionpotentialspike.Thecontrollerin( 6 )leadsthe1:1in-phasesynchronyinthebroaderrangeofheterogeneitythantheresultin[ 80 ]. 6.4StabilityAnalysis Thestabilityanalysisofthischaptermaynowbestatedasfollows. Theorem6.1. ThecombinedcontrollerandtheNNupdatelawgivenin( 6 ),( 618 ),( 6 )fortheinterneuronsystemintheexistenceofheterogeneitysatisesthe 107

PAGE 108

followingsynchronouscondition: 1 2Ztf0eT(t)Qe(t)dt1 2eT(0)Pe(0)+1 2~WT(0))]TJ /F4 7.97 Tf 22.81 4.93 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1~W(0)+1 22Ztf0!2(t)dt(6) Proof. LettheLyapunovfunctionVbeacontinuouslydifferentiablepositivedenitefunctiondenedas V=1 2eTPe+1 2~WT)]TJ /F4 7.97 Tf 6.78 4.94 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1~W.(6) Takingtimederivativesof( 6 )andutilizing( 6 ),( 6 )canbesimpliedas _V=1 2eTP_e+_eTPe)]TJ /F3 11.955 Tf 15.56 2.66 Td[(~WT)]TJ /F4 7.97 Tf 6.77 4.93 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1_^W=1 2eT(ATP+PA)e+)]TJ /F3 11.955 Tf 15.98 8.09 Td[(1 22BTPe+!BTPe)]TJ /F3 11.955 Tf 15.57 2.66 Td[(~WT)]TJ /F4 7.97 Tf 6.78 4.94 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1_^W)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F6 11.955 Tf 6.77 0 Td[((x)BTPe(6) Using( 6 )andtherelationshipwithATP+PA=)]TJ /F5 11.955 Tf 9.3 0 Td[(Q,theexpressionin( 6 )canbesimpliedas _V=)]TJ /F3 11.955 Tf 10.49 8.08 Td[(1 2eTQe+1 22(BTPe)2+!BTPe=)]TJ /F3 11.955 Tf 10.49 8.08 Td[(1 2eTQe+1 22!2)]TJ /F3 11.955 Tf 13.15 8.08 Td[(1 21 BTPe)]TJ /F6 11.955 Tf 11.95 0 Td[(!2)]TJ /F3 11.955 Tf 23.11 8.09 Td[(1 2eTQe+1 22!2.(6) Byintegratingbothsidesof( 6 )fromt=[0,tf],theequationisrewrittenas V(tf))]TJ /F5 11.955 Tf 11.95 0 Td[(V(0))]TJ /F3 11.955 Tf 23.11 8.08 Td[(1 2Ztf0eTQedt+1 22Ztf0!2dt(6) TheinequalityV(tf)0provesTheorem1as 1 2Ztf0eT(t)Qe(t)dtV(0)+1 22Ztf0!2dt1 2eT(0)Pe(0)+1 2~WT(0))]TJ /F4 7.97 Tf 22.81 4.94 Td[()]TJ /F8 7.97 Tf 6.59 0 Td[(1~W(0)+1 22Ztf0!2(t)dt(6) 108

PAGE 109

Thus,( 6 )ensuresthattheuniformlyultimatelybounded(UUB)resultforthesynchronyobjectivesuchthat limt!1keV(t)k!ast!1.(6) 6.5Simulation Inthissection,numericalanalysisisinvestigatedforthemutuallycoupledsynchronyviaasaturationcontrollercombinedwiththeNNbasedH1approachin( 6 )and( 6 ).Weillustratetheperformanceandtherobustnessoftheproposedtechniquefor1:1in-phasesynchronyintheexistenceofheterogeneity.Controlparametersarechosenask=2,)-277(=2,Q=8,=0.1andP=2tocomparetheresultsin[ 81 ]. 6.5.1In-PhaseSynchronyinDifferentHeterogeneousTypes AArnoldtongueinthepresenceofcontrol BTimedifferencenoftwoneuronalspikingatgs=0.2 Figure6-4. Robustnessof1:1synchronyandin-phasesynchronyinthepresenceofcontrol Inordertoquantifytherobustnessandperformanceof1:1in-phasesynchrony,theArnoldtongueinFig. 6-4 .Arepresentstheregionof1:1synchronouslocking.ComparedwiththesynchronousregionofthenominalMCInetworkinFig. 6-2 ,theplot 109

PAGE 110

demonstratesthatthestimulationenhances1:1synchronyinthenetworkeveninhighstaticheterogeneity()]TJ /F3 11.955 Tf 9.29 0 Td[(49
PAGE 111

AHeterogeneityofmultiplestepfunction BControlinactionunderstepheterogeneity CSynchronouserror DTwoneuronalmembranepotential Figure6-5. Performancemeasureinheterogeneityofmultiplestepfunction Fig. 6-6 .AshowsthattheZAPsignaloscillatesbetween50whosefrequencyincreaseastimegoesonandcontrolisturnedonat500msinFig. 6-6 .B.Incomparisonwithstaticheterogeneityat50inFig. 6-5 ,weseethatthereexist1:1synchronybetweenacoupledneuronbetween0-800ms,butthisresultissimply1:1,butnotin-phasesynchrony.Thecontrolactionbetween500850msislimitedbytheboundaryofinputconstraintintheregionofcomparativelylowfrequency.Asfrequencyincreases,thesynchronousstateof1:1in-phaselockingintheMCInetworkinteractingthroughacouplingsynapseispresentedbetween850-1000ms.Theresultisoneofthe 111

PAGE 112

contributionthataLyapunovbasedadaptivecontrolwithinputconstraintsexhibitsin-phasesynchrony.Inliteratures[ 80 ],[ 82 ],aspikingtimingapproachhasexperiencedthelimitationofneuronalsynchronyinhighfrequencydisturbances.Insummaryahighfrequencydisturbanceintheproposedcontroltechniqueisahelpfulelementtokeeptheneuron'sringeveninthehighamplitudeofheterogeneity(H<)]TJ /F3 11.955 Tf 9.3 0 Td[(50andH>50). AZAPheterogeneity BControlinactionunderZAPheterogeneity CPhaseportrait DTwoneuronalmembranepotential Figure6-6. PerformancemeasureinZAPheterogeneity 6.5.2MCINetworkinVaryingFeedbackSynapticCoupling Inthissection,theeffectoffeedbacksynapticcouplingintheMCInetworkonthecontrolperformanceisaddressed.AsnotedinFig. 6-1 ,thefeedbackcouplingparameter 112

PAGE 113

isdenedasgf=S 100gs.S=0iscalledtheunidirectionalcouplinginterneuron(UCI)network,whichhasthebroaderregionofin-phasesynchronythantheMCInetwork.Toevaluatetheinuenceofpercentagecouplingstrength,thespecicvalueofheterogeneityandcouplingstrengtharechosenasH=)]TJ /F3 11.955 Tf 9.3 0 Td[(47andgs=0.2,whichensuresin-phasesynchronyasshowninFig. 6-4 .A.InFig. 6-7 .B,weshowtheplotofthetimedifferencenasthestrengthofthesynapticinteractioninthecouplednetwork.Inthecaseoffeedbackcouplingstrength(P<135),therobustcontrollerguaranteesrobust1:1in-phasesynchronyintheMCInetwork.However,thecontrollercannotsustain1:1in-phasesynchronyinP>135,butinducesthehigherordersynchrony,i.e.,2=1.Tocomparetheperformanceoffeedbackcouplingstrengthintheexistenceofcontroller,weconsiderthespeciccaseofsynchronyregionin 6-7 .Awithparameters:H=)]TJ /F3 11.955 Tf 9.3 0 Td[(5,gs=0.08.ByobservingtwoninFig. 6-7 therobustcontrollerextendstheregionofPinprovidingtherobustnessofin-phasesynchronyintheMCInetwork.Here,wehavenumericallyinvestigatedrobustnessandenhancementofneural1:1synchronybytheproposedcontroller. ANominalMCInetwork BMCInetworkinthepresenceofstimulation Figure6-7. noffeedbackcouplingstrengthintheMCInetwork 113

PAGE 114

6.6Conclusion Chapter 6 dealswiththecontrolproblemof1:1in-phaseneuralsynchronyinthemutuallycouplednetworkofheterogeneouslyringinterneuroninteractingwithsynapse(MCInetwork).Thecontributionofthischapterisprofound.Firstly,thereisnopriorworkaboutthe1:1in-phaseneuralsynchronyanalysisinthepresenceoftheinputconstraintcombinedwiththeadaptiveNNrobustH1control.Thetechniqueisabletosustain1:1in-phaseneuralsynchronyintheMCInetworkeveninthepresenceoflargeunknownvariationsofthenetworkheterogeneity.Whilerobustneuralsynchronyresultsagainsttheuncertaintieshavebeenreportedin[ 76 ],[ 77 ],inputamplitudeatthetransientresponseisabletoarticiallyinducethespikes.Intheproposedapproach,therobustcontroltechniquemerelystimulatestheimpendingactionpotentialspikesandbroadenstherobustnessof1:1in-phasesynchronydespitethelargevariationofheterogeneityandthesynapticcouplingstrengthintheMCInetwork. 114

PAGE 115

CHAPTER7CONCLUSIONANDFUTUREWORK 7.1Conclusion Systematiccontrolmethodologiesforlinearandnonlinearsystemswithstructuredandunstructureduncertaintiesareproposed.Thisstudyshowsthatthecriticaldirectiontheory(CDT)iseffectiveandpromisingforlessconservativestabilityanalysisandsynthesisforlinearsystemswithstructuredandunstructureduncertainties.Inaddition,theresearchinthenonlinearcaseillustrateshowaneuralnetworkfeedforwardtermcanbecombinedwithafeedbackcontrollertoensureasymptotictrackingofadesiredreferencesignalinthepresenceofunknownuncertaintiesbasedontheLyapunovanalysis.However,therearetworequirementsinthisdissertationtoclaimrobuststability.Namely,statesignalsinordinarydifferentialequationsaremeasurableandunknowndisturbancesarebounded.Thedevelopmenttotheseproblemsisannouncedinthefuturework. 7.2FutureWork Therequirementinthisdissertationisthatfull-statesignalsaremeasurable.However,inrealapplicationsonlyoutputsignalmaybeavailable.Anobserver-basedandanoutputfeedbackapproachunderanadaptivenon-backsteppingtechniqueforaclassofuncertainnonlinearstrict-feedbacksystems,stillremainsanopenproblem.Thedevelopedcontrollerwithrespecttotheoutputfeedback(OFB)positiontrackingcontrolyieldedsemi-globalasymptoticalresultin[ 83 84 ]underthelimitationthatdynamicsystemmodelsareexactlyknown.Otherwise,thestabilityresultisuniformlyultimatelybounded(UUB).Withthetransformederrorsystemin( 5 ),futureeffortcanfocusondevelopinganadaptiveoutputfeedbackcontrollerundertheassumptionthatsystemparametersareunknown. Also,oneoftheshortcomingsofcurrentworkisthatdisturbancesandtheirderivativesinordinarydynamicequations(ODE)areassumedtobeupperbounded 115

PAGE 116

byknownconstants.Furtherinvestigationisneededtofocusondevelopinganadaptivecontrollerthatcouldalsosolvethecontrolproblemundertheassumptionofunrestricteddisturbances.Inthestochasticformulationoftheseproblems,theexogenousdisturbancescanbemodeledusingstochasticprocess,e.g.,Wienerprocesses.Asaresult,theODEmodelcanbereplacedbyastochasticdifferentialequation(SDE)model.Thesestochasticversionsoftheproblemsconsideredinthisdissertationremaillargelyopen. 116

PAGE 117

REFERENCES [1] C.T.Baab,J.C.Cockburn,H.A.Latchman,andO.D.Crisalle,Generalizationofthenyquistrobuststabilitymarginanditsapplicationtosystemswithrealafneparametricuncertainties,InternationalJournalofRobustandNonlinearControl,vol.11,no.15,pp.1415,2001. [2] H.A.Latchman,O.D.Crisalle,andV.R.Basker,Thenyquistrobuststabilitymargin-newmetricforthestabilityofuncertainsystems,InternationalJournalofRobustandNonlinearControl,vol.7,no.2,pp.211,1997. [3] J.Doyle,Analysisoffeedbacksystemswithstructureduncertainties,ControlTheoryandApplications,IEEProceedingsD,vol.129,pp.242,November1982. [4] M.G.Safonov,D.J.N.Limebeer,andR.Y.Chiang,SimplifyingtheH1theoryvialoop-shifting,matrix-pencilanddescriptorconcepts,InternationalJournalofControl,vol.50,no.6,pp.2467,1989. [5] I.Kanellakopoulos,P.Kokotovic,andA.Morse,Systematicdesignofadaptivecontrollersforfeedbacklinearizablesystems,AutomaticControl,IEEETransactionson,vol.36,pp.1241,Nov.1991. [6] P.Kokotovic,Thejoyoffeedback:nonlinearandadaptive,ControlSystems,IEEE,vol.12,pp.7,June1992. [7] M.Krstic,I.Kanellakopoulos,andP.Kokotovic,Nonlinearandadaptivecontroldesign.Adaptiveandlearningsystemsforsignalprocessing,communications,andcontrol,Wiley,1995. [8] S.K.Nguang,Robuststabilizationofaclassoftime-delaynonlinearsystems,AutomaticControl,IEEETransactionson,vol.45,pp.756,Apr.2000. [9] Z.-P.Jiang,Decentralizedandadaptivenonlineartrackingoflarge-scalesystemsviaoutputfeedback,AutomaticControl,IEEETransactionson,vol.45,pp.2122,Nov.2000. [10] C.Hua,X.Guan,andP.Shi,Robustbacksteppingcontrolforaclassoftimedelayedsystems,AutomaticControl,IEEETransactionson,vol.50,pp.894899,June2005. [11] X.JiaoandT.Shen,Adaptivefeedbackcontrolofnonlineartime-delaysystems:thelasalle-razumikhin-basedapproach,AutomaticControl,IEEETransactionson,vol.50,pp.19091913,Nov.2005. [12] S.Ge,F.Hong,andT.H.Lee,Adaptiveneuralnetworkcontrolofnonlinearsystemswithunknowntimedelays,AutomaticControl,IEEETransactionson,vol.48,pp.20042010,Nov.2003. 117

PAGE 118

[13] M.Wang,B.Chen,andP.Shi,Adaptiveneuralcontrolforaclassofperturbedstrict-feedbacknonlineartime-delaysystems,Systems,Man,andCybernetics,PartB:Cybernetics,IEEETransactionson,vol.38,pp.721,June2008. [14] S.Ge,F.Hong,andT.H.Lee,Adaptiveneuralcontrolofnonlineartime-delaysystemswithunknownvirtualcontrolcoefcients,Systems,Man,andCybernetics,PartB:Cybernetics,IEEETransactionson,vol.34,pp.499,Feb.2004. [15] F.Hong,S.Ge,andT.H.Lee,Practicaladaptiveneuralcontrolofnonlinearsystemswithunknowntimedelays,Systems,Man,andCybernetics,PartB:Cybernetics,IEEETransactionson,vol.35,pp.849,Aug.2005. [16] J.ZhouandC.Wen,Decentralizedbacksteppingadaptiveoutputtrackingofinterconnectednonlinearsystems,AutomaticControl,IEEETransactionson,vol.53,pp.2378,Nov.2008. [17] S.JainandF.Khorrami,Decentralizedadaptiveoutputfeedbackdesignforlarge-scalenonlinearsystems,AutomaticControl,IEEETransactionson,vol.42,pp.729,May1997. [18] a.G.C.S.Tong,S.Li,Adaptivefuzzydecentralizedcontrolforaclassoflarge-scalenonlinearsystems,Systems,Man,andCybernetics,PartB:Cyber-netics,IEEETransactionson,vol.34,pp.770775,Feb.2004. [19] Z.Huaguang,L.Cai,andZ.Bien,Afuzzybasisfunctionvector-basedmultivariableadaptivecontrollerfornonlinearsystems,Systems,Man,andCy-bernetics,PartB:Cybernetics,IEEETransactionson,vol.30,pp.210,Feb.2000. [20] J.-H.Park,S.-H.Kim,andC.-J.Moon,Adaptiveneuralcontrolforstrict-feedbacknonlinearsystemswithoutbackstepping,NeuralNetworks,IEEETransactionson,vol.20,pp.1204,July2009. [21] C.KwanandF.Lewis,Robustbacksteppingcontrolofnonlinearsystemsusingneuralnetworks,Systems,ManandCybernetics,PartA:SystemsandHumans,IEEETransactionson,vol.30,no.6,pp.753,2000. [22] L.Chua,Memristor-themissingcircuitelement,CircuitTheory,IEEETransactionson,vol.18,pp.507519,Sep1971. [23] D.B.Strukov,G.S.Snider,D.R.Stewart,andR.S.Williams,Themissingmemristorfound.,Nature,vol.453,no.7191,pp.80,2008. [24] B.MuthuswamyandP.Kokate,Memristor-basedchaoticcircuits,IeteTechnicalReview,vol.26,no.6,p.417,2009. [25] B.Muthuswamy,Implementingmemristorbasedchaoticcircuits,InternationalJournalOfBifurcationAndChaos,vol.20,no.05,p.1335,2010. 118

PAGE 119

[26] K.Tanaka,T.Ikeda,andH.O.WAng,Auniedapproachtocontrollingchaosviaanlmi-basedfuzzycontrolsystemdesign,IeeeTransactionsOnCircuitsAndSystemsIFundamentalTheoryAndApplications,vol.45,no.10,pp.1021,1998. [27] H.O.WangandE.Y.Abed,Bifuractioncontrolofachaoticsystem,Automatica,vol.31,no.9,pp.1213,1995. [28] J.Xu,G.Chen,andL.Shieh,Digitalredesignforcontrollingthechaoticchua'scircuit,AerospaceandElectronicSystems,IEEETransactionson,vol.32,pp.1488,Oct.1996. [29] F.-C.Chen,Adaptivecontrolofnonlinearsystemsusingneuralnetworks,Interna-tionalJournalofControl,vol.55,no.6,pp.1299,1992. [30] M.T.Hagan,H.B.Demuth,andO.D.Jesus,Anintroductiontotheuseofneuralnetworksincontrolsystems,InternationalJournalofRobustandNonlinearControl,vol.12,no.11,pp.959,2002. [31] A.S.PoznyakandE.N.Sanchez,Identicationandcontrolofunknownchaoticsystemsviadynamicneuralnetworks,IeeeTransactionsOnCircuitsAndSystemsIFundamentalTheoryAndApplications,vol.46,no.12,pp.1491,1999. [32] J.-H.Park,S.-H.Huh,S.-H.Kim,S.-J.Seo,andG.-T.Park,Directadaptivecontrollerfornonafnenonlinearsystemsusingself-structuringneuralnetworks.,IEEETransactionsonNeuralNetworks,vol.16,no.2,pp.414,2005. [33] B.Delyon,A.Juditsky,andA.Benveniste,Accuracyanalysisforwaveletapproximations.,IEEETransactionsonNeuralNetworks,vol.6,no.2,pp.332,1995. [34] S.P.Bhattacharyya,H.Chapellat,andL.H.Keel,RobustControl:TheParametricApproach.UpperSaddleRiver,NJ,USA:PrenticeHallPTR,1sted.,1995. [35] T.Cho,S.Choi,andY.Kim,RobustdesignofPIDcontrollerforintervalplants,inSICEAnnual,1999.38thAnnualConferenceProceedingsofthe,pp.933,Aug.1999. [36] D.HenrionandO.Bachelier,Low-orderrobustcontrollersynthesisforintervalplants,InternationalJournalofControl,vol.74,pp.1,1998. [37] M.-T.Ho,A.Datta,andS.Bhattackaryya,DesignofP,PIandPIDcontrollersforintervalplants,inAmericanControlConference,1998.Proceedingsofthe1998,vol.4,pp.2496vol.4,Jun1998. [38] S.Bhattacharya,L.Keel,andS.Bhattacharyya,RobuststabilizersynthesisforintervalplantsusingH1methods,inDecisionandControl,1993.,Proceedingsofthe32ndIEEEConferenceon,pp.3003vol.4,Dec.1993. 119

PAGE 120

[39] M.Fu,Computingthefrequencyresponseoflinearsystemswithparametricperturbation,Syst.ControlLett.,vol.15,pp.45,July1990. [40] R.ChiangandM.Safonov,Robustcontroltoolbox:forusewithMATLAB.MathWorksInc.,1992. [41] K.Zhou,J.Doyle,andK.Glover,RobustandOptimalControl.NewJersey:Prentice-Hall,1996. [42] R.S.Sanchez-PenaandM.Sznaier,RobustSystemsTheoryandApplications.NewYork,NY,USA:JohnWiley&Sons,Inc.,1sted.,1998. [43] K.B.DattaandV.V.Patel,H1-basedsynthesisforarobustcontrollerofintervalplants,Automatica,vol.32,pp.1575,Nov.1996. [44] H.Latchman,O.Crisalle,T.Baab,andB.Ji,Theexactcalculationofrealstabilityradiiofsystemswithafneparametricuncertainties,inSystemTheory,2002.ProceedingsoftheThirty-FourthSoutheasternSymposiumon,pp.279286,2002. [45] R.W.DanielandB.Kouvaritakis,Analysisanddesignoflinearmultivariablefeedbacksystemsinthepresenceofadditiveperturbations,InternationalJournalofControl,vol.39,no.3,pp.551,1984. [46] R.W.DANIELandB.KOUVARITAKIS,Thechoiceanduseofnormalapproximationstotransfer-functionmatricesofmultivariablecontrolsystems,InternationalJournalofControl,vol.37,no.5,pp.1121,1983. [47] H.Latchman,K.-N.Hodge,O.Crisalle,W.Edmonson,andK.Yen,ThecriticalperturbationradiusweightsinH1synthesisforintervalplants,inControlAppli-cations,1998.Proceedingsofthe1998IEEEInternationalConferenceon,vol.2,pp.975vol.2,1998. [48] H.LatchmanandO.Crisalle,Exactrobustnessanalysisforhighlystructuredfrequency-domainuncertainties,inAmericanControlConference,1995.Proceed-ingsofthe,vol.6,pp.3982vol.6,Jun1995. [49] P.Young,Robustnessanalysisforfull-structureduncertainties,inDecisionandControl,1996.,Proceedingsofthe35thIEEE,vol.3,pp.3464vol.3,Dec.1996. [50] V.Basker,H.Latchman,H.Mahon,andO.Crisalle,Anewperspectiveoncomputingrobuststabilitymarginsforcomplexparametricuncertainties,inAmer-icanControlConference,1998.Proceedingsofthe1998,vol.4,pp.2299vol.4,June1998. [51] A.MacFarlane,Complexvariablemethodsforlinearmultivariablefeedbacksys-tems.Taylor&Francis,1980. 120

PAGE 121

[52] J.XuandD.Ho,Adaptivewaveletnetworksfornonlinearsystemidentication,inAmericanControlConference,1999.Proceedingsofthe1999,vol.5,pp.3472vol.5,1999. [53] J.Zhang,G.Walter,Y.Miao,andW.N.W.Lee,Waveletneuralnetworksforfunctionlearning,SignalProcessing,IEEETransactionson,vol.43,pp.1485,Jun.1995. [54] W.Dixon,Nonlinearcontrolofengineeringsystems:aLyapunov-basedapproach.Controlengineering,Birkhauser,2003. [55] Dinh,Dynamicneuralnetwork-basedrobustidenticationandcontrolofaclassofnonlinearsystems,inDecisionandControl(CDC),201049thIEEEConferenceon,pp.5536,Dec.2010. [56] C.K.Chui,Anintroductiontowavelets,Construction,vol.2,pp.1,1992. [57] J.-P.Richard,Time-delaysystems:anoverviewofsomerecentadvancesandopenproblems,Automatica,vol.39,no.10,pp.1667,2003. [58] K.Gu,V.Kharitonov,andJ.Chen,Stabilityoftime-delaysystems,Automatica,vol.41,no.12,pp.1458,2003. [59] S.Niculescu,Delayeffectsonstability:arobustcontrolapproach.Lecturenotesincontrolandinformationsciences,Springer,2001. [60] H.Yousef,M.Hamdy,andM.Shaq,Adaptivefuzzycontrolforstrict-feedbacknonlineartime-delaysystemswithoutbacksteppingscheme,inDecisionandControl(CDC),201049thIEEEConferenceon,pp.3700,Dec.2010. [61] N.Sharma,S.Bhasin,Q.Wang,andW.Dixon,Rise-basedadaptivecontrolofanuncertainnonlinearsystemwithunknownstatedelays,inDecisionandControl(CDC),201049thIEEEConferenceon,pp.1773,Dec.2010. [62] D.Meddah,A.Benallegue,andA.Cherif,Aneuralnetworkrobustcontrollerforaclassofnonlinearmimosystems,inRoboticsandAutomation,1997.Proceedings.,1997IEEEInternationalConferenceon,vol.3,pp.2645vol.3,Apr1997. [63] M.-B.ChengandC.-C.Tsai,Hybridrobusttrackingcontrolforamobilemanipulatorviasliding-modeneuralnetwork,inMechatronics,2005.ICM'05.IEEEInternationalConferenceon,pp.537,July2005. [64] P.Patre,W.MacKunis,K.Kaiser,andW.Dixon,Asymptotictrackingforuncertaindynamicsystemsviaamultilayernnfeedforwardandrisefeedbackcontrolstructure,inAmericanControlConference,2007.ACC'07,pp.5989,July2007. 121

PAGE 122

[65] P.Patre,W.MacKunis,C.Makkar,andW.Dixon,Asymptotictrackingforsystemswithstructuredandunstructureduncertainties,inDecisionandControl,200645thIEEEConferenceon,pp.441,Dec.2006. [66] F.Clarke,Optimizationandnonsmoothanalysis.Classicsinappliedmathematics,SIAM,1990. [67] D.Ho,J.Li,andY.Niu,Adaptiveneuralcontrolforaclassofnonlinearlyparametrictime-delaysystems,NeuralNetworks,IEEETransactionson,vol.16,pp.625,May2005. [68] C.Henry,Differentialequationswithdiscontinuousright-handsideforplanningprocedures,JournalofEconomicTheory,vol.4,pp.545,June1972. [69] D.ShevitzandB.Paden,Lyapunovstabilitytheoryofnonsmoothsystems,inDecisionandControl,1993.,Proceedingsofthe32ndIEEEConferenceon,pp.416vol.1,Dec1993. [70] A.F.Filippov,Differentialequationswithdiscontinuousright-handside,AmericanMathemticalSocietyTrans.,vol.42,pp.199,1964. [71] G.Buzsaki,RhythmsoftheBrain.OxfordUniversityPress,2006. [72] P.J.UhlhaasandW.Singer,Neuralsynchronyinbraindisorders:Relevanceforcognitivedysfunctionsandpathophysiology,Neuron,vol.52,no.1,pp.155168,2006. [73] P.J.Uhlhaas,F.Roux,E.Rodriguez,A.Rotarska-Jagiela,andW.Singer,Neuralsynchronyandthedevelopmentofcorticalnetworks,TrendsinCognitiveSciences,vol.14,no.2,pp.7280,2010. [74] H.A.Swadlow,I.N.Beloozerova,andM.G.Sirota,Sharp,localsynchronyamongputativefeed-forwardinhibitoryinterneuronsofrabbitsomatosensorycortex.,JNeurophysiol,vol.79,pp.567,Feb.1998. [75] khalil,NonlinearSystems.PrenticeHall,Jan.2002. [76] O.Cornejo-PerezandR.Femat,Unidirectionalsynchronizationofhodgkin-huxleyneurons,Chaos,Solitons&Fractals,vol.25,no.1,pp.4353,2005. [77] L.Tian,D.Li,andX.Sun,Nonlinear-estimator-basedrobustsynchronizationofhodgkin-huxleyneurons,Neurocomputing,vol.72,pp.186196,2008. [78] J.Wang,T.Zhang,andY.Che,Chaoscontrolandsynchronizationoftwoneuronsexposedtoelfexternalelectriceld,Chaos,Solitons&Fractals,vol.34,no.3,pp.839850,2007. [79] A.Isidori,NonlinearControlSystemsII.London,UK,UK:Springer-Verlag,2000. 122

PAGE 123

[80] S.S.Talathi,P.R.Carney,andP.P.Khargonekar,Controlofneuralsynchronyusingchannelrhodopsin-2:acomputationalstudy,J.Comput.Neurosci.,vol.31,pp.87,Aug.2011. [81] J.Wang,Y.-Q.Che,S.-S.Zhou,andB.Deng,Unidirectionalsynchronizationofhodgkin-huxleyneuronsexposedtoelfelectriceld,Chaos,Solitons&Fractals,vol.39,no.3,pp.13351345,2009. [82] S.S.Talathi,D.-U.Hwang,andW.L.Ditto,Spiketimingdependentplasticitypromotessynchronyofinhibitorynetworksinthepresenceofheterogeneity.,JournalofComputationalNeuroscience,vol.25,no.2,pp.262,2008. [83] S.NicosiaandP.Tomei,Robotcontrolbyusingonlyjointpositionmeasurements,AutomaticControl,IEEETransactionson,vol.35,pp.1058,Sep.1990. [84] S.Lim,D.Dawson,andK.Anderson,Re-examiningthenicosia-tomeirobotobserver-controllerfromabacksteppingperspective,ControlSystemsTechnology,IEEETransactionson,vol.4,pp.304,May1996. 123

PAGE 124

BIOGRAPHICALSKETCH JungEunSonobtainedthebachelor'sdegreeincontrolandinstrumentationengineeringattheChosunUniversity,Gwangju,SouthKoreain2001andthemaster'sdegreeintheelectricalandcomputerengineeringattheUniversityofFlorida,Gainesville,Florida,USAin2004,respectively.HethenjoinedtheLaboratoryforInformationSystemsandTelecommunications(LIST)researchgrouptopursuehisdoctoralresearchundertheadvisementofDr.HaniphA.LatchmanandDr.PramodP.Khargonekar.HewillbejoiningasapostdoctoralfellowinDr.SachinS.Talathi'slaboratoryinthecollegeofmedicineattheUniversityofFlorida. 124