MODELING,IDENTIFICATION,ANDCONTROLOFHYSTERETICSYSTE MSWITH APPLICATIONTOVANADIUMDIOXIDEMICROACTUATORS By JunZhang ADISSERTATION Submittedto MichiganStateUniversity inpartialfulllmentoftherequirements forthedegreeof ElectricalEngineering-DoctorofPhilosophy 2015 ABSTRACT MODELING,IDENTIFICATION,ANDCONTROLOFHYSTERETICSYSTE MSWITH APPLICATIONTOVANADIUMDIOXIDEMICROACTUATORS By JunZhang Hysteresisnonlinearityinmagneticandsmartmaterialsys temshinderstherealizationoftheir potentialinsensorsandactuators.Thegoalofthisdissert ationistoadvancemethodsformod- eling,identication,andcontrolofhystereticsystems.T hesemethodsareappliedtotheinverse compensation,self-sensingfeedbackcontrol,androbustc ontrolofvanadiumdioxide(VO 2 )mi- croactuators.Asanovelsmartmaterial,VO 2 undergoesathermallyinducedinsulator-to-metal transitionandastructuralphasetransition,exhibitingp ronouncedhysteresisinelectricalandme- chanicaldomains. Withthegoalofobtainingaccuratehysteresismodelswhile maintainingalowmodelcom- plexity,optimalcompressionsfortwopopularhysteresism odels,namelythePreisachoperator andthegeneralizedPrandtl-Ishlinskii(GPI)model,arest udied,wheretheKullback-Leiblerdiver- genceandentropy,respectively,areadoptedtoquantifyth einformationlossinmodelcompression. WhiletheoptimalcompressionofthePreisachoperatorisre alizedusingexhaustivesearch,dy- namicprogrammingisemployedtooptimallycompresstheGPI modelefciently.Bothsimulation andexperimentalresultsdemonstratethattheproposedalg orithmsyieldsuperiorperformances thantypicallyadoptedschemes. InordertoidentifythePreisachoperator,existingworkin volvesapplyingacomplicatedinput sequenceandmeasuringalargesetofoutputdata.Wepropose anefcientapproachtoidentify thePreisachoperatorthatrequiresfewermeasurements.Th eoutputofthePreisachoperatoris transformedintothefrequencydomain,generatingasparse vectorofdiscretecosinetransform (DCT)coefcients.Themodelparametersarereconstructed usingacompressivesensing-based algorithm.Theeffectivenessoftheproposedschemeisillu stratedthroughsimulationandexperi- ments. Afewnewcontributionshavebeenmadetothemodelingandcon trolofVO 2 microactuators. Inordertocapturethenon-monotoniccurvature-temperatu rehysteresisofVO 2 microactuators, physics-motivatedmodelsthatcombineamonotonichystere sisoperatorforphasetransitionin- ducedcurvatureandamemorylessoperatorfordifferential thermalexpansioninducedcurvature areproposed.Effectiveinversecompensationschemesfort heproposednon-monotonichysteresis modelsarepresented.Themodelingandinversecompensatio nschemesarevalidatedexperimen- tally. Sinceexternalsensingsystemsarenotdesirablewithmicro devices,aself-sensingmodelis developedforVO 2 microactuatorstoestimatethedeectionfromtheresistan cemeasurement.We exploitthephysicalunderstandingthateachoftheresista nceandthedeectionisdeterminedbya hystereticrelationshipwiththetemperature,whichismod eledwithaGPImodelandanextended GPImodel,respectively.Theself-sensingmodelisobtaine dbycascadingtheextendedGPImodel withtheinverseoftheGPImodel.Theperformanceoftheself -sensingschemeisexperimentally evaluatedwithproportional-integralcontrol.Finally,a n H ¥ robustcontrollerisfurtherdeveloped, whereasimplepolynomial-basedself-sensingschemeisado pted,astheemphasisisonaccom- modatingtheuncertaintiesproducedbythehysteresisnonl inearityandtheself-sensingerror.The effectivenessoftheproposedapproachisdemonstratedthr oughexperiments. ToMom,Dad,andmywifeRan iv ACKNOWLEDGMENTS Iwouldliketoexpressmywarmestgratitudetomyadvisor,Pr of.XiaoboTan,forprovidingme withtheopportunityinSmartMicrosystemsLabatMichiganS tateUniversity(MSU)topursuea Ph.D.degree.Duringmydoctoralstudy,hehasprofoundlyin spiredandinuencedmeinmany wayswithhisexpertise,enthusiasm,andvision.Iamforeve rgratefulforhisguidancewithen- couragementandpatienceinthefascinatingeldofsmartma terials.Hehasalsoofferedmewith invaluableadvicesandgeneroushelponmyjobsearchandaca demiccareerdevelopment. IwouldliketothankProf.NelsonSep´ulveda,Prof.HassanK halil,andProf.RanjanMukher- jeeforkindlyservingonmyacademiccommitteeandoffering meinsightfulsuggestionsonmy research.Inparticular,IthankProf.NelsonSep´ulveda,w holedmeintotheareaofvanadium dioxide.Hisgrouphavecontributedsignicantlyontheexp erimentalsideofmyresearchprojects includingfabrication,experimentalsetup,anddataacqui sition. IthankProf.GuomingZhuforhisrobustcontrolcourse,whic hhelpedmetoaccomplishthe workinChapter8.IalsobenetedalotfromProf.MarkIwen's courseoncompressivesensing, andthecourseonsignalcompressionbyProf.HayderRadha.T hesetwocourseshavebeenhelpful fortheworkinChapter2,Chapter3,andChapter4. IamgratefultomycolleaguesatMSUfortheirhelpanddiscus sions:Dr.EmmanuelleMerced, DavidTorres,Dr.HongLei,Dr.FeitianZhang,Dr.JianxunWa ng,Dr.AhmadT.Abdulsadda,Dr. AlexEsbrook,Dr.JianguoZhao,SanazBehbahani,JasonGree nberg,AliAbul,TongyuWang, BoSongandmanyothers.SpecialthanksareduetoEmmanuelle andDavid,fortheirconsistent supportontheexperimentalsideofmyresearchprojects.Iw ouldliketothankfriendsatMSU whoofferedmegeneroushelp:Dr.FangHou,Dr.JiyingLi,Mia oYu,Dr.BaolinYu,Dr.Zheng FanandXiaofengZhao. v Ialsowanttothankalloftheadministratorsandstaffmembe rsintheECEdepartmentfortheir assistanceduringmystudyandlifehere. IamgratefulforthenancialsupportformyresearchbytheN ationalScienceFoundation (CMMI0824830,CMMI1301243,ECCS0547131). Mostofall,Iamforemostthankfulformyfamily.Iwouldlike tothankmymother,Lizhen Zhang,myfather,YuntingZhang,andmybrother,JieZhangfo rtheireverlastingsupportinallmy endeavors.Iamdeeplyindebtedtomydearwife,RanDuan,for herconstantlove,supportand encouragement. vi TABLEOFCONTENTS LISTOFTABLES ....................................... x LISTOFFIGURES ...................................... xi Chapter1Introduction .................................. 1 1.1Modeling,Identication,andControlofHystereticSys tems.............1 1.1.1Modeling...................................1 1.1.2Identication................................. 2 1.1.3Control....................................3 1.2ModelingandControlofVanadiumDioxide(VO 2 )Microactuators.........4 1.2.1ModelingandInverseCompensation................ .....5 1.2.2Self-sensingFeedbackControl................... .....6 1.2.3RobustControl................................7 1.3ContributionandOrganization..................... .......8 1.3.1OverviewofContribution........................ ...8 1.3.2Organization.................................1 0 Chapter2Kullback-Leibler(KL)Divergence-basedOptimal Compressionofthe PreisachOperator .............................. 11 2.1ProblemFormulation.............................. ...11 2.2InformationLossMetric:KLDivergence-basedMeasure. .............12 2.3OptimalCompressionScheme........................ ....14 2.4ExperimentalResults............................. ....16 2.4.1ExperimentalCharacterization.................. ......16 2.4.2CompressionPerformance........................ ..17 Chapter3Entropy-basedOptimalCompressionoftheGeneral izedPrandtl-Ishlinskii (GPI)Model .................................. 22 3.1ProblemFormulation.............................. ...23 3.2OptimalCompressionScheme........................ ....25 3.3InformationLossMetrics:Entropy-basedMeasure..... ............27 3.4ScalingoftheWeightsfortheGPIModel............... .......29 3.5SimulationResults............................... ...30 3.5.1Case1:UniformDistributionfortheScaledWeights.. ..........32 3.5.2Case2:ScaledWeightswithOneProminentPeak....... .......33 3.5.3Case3:ScaledWeightswithTwoProminentPeaks...... .......35 3.5.4Case4:RandomDistributionfortheScaledWeights... .........35 3.5.5ComputationalTimefortheAlgorithms............. ......37 3.5.6ComparisonwithaTraditionalModelIdenticationAp proach.......38 vii 3.6ExperimentalResults............................. ....39 3.6.1CompressionPerformance........................ ..42 3.6.2ModelVerication.............................. 43 Chapter4CompressiveSensing-basedPreisachOperatorIde ntication ...... 46 4.1ProblemFormulation.............................. ...46 4.2CompressiveSensingSchemeforIdentifyingthePreisac hOperator........50 4.2.1OverviewofCompressiveSensing.................. ....50 4.2.2CompressiveSensingforthePreisachOperator...... .........52 4.3SimulationResults............................... ...57 4.4ExperimentalResults............................. ....63 4.4.1MeasurementSetup.............................. 63 4.4.2IdenticationandVerication................... .....64 Chapter5ModelingandInverseCompensationofNon-monoton icHysteresisbased onthePreisachOperator .......................... 70 5.1ExperimentalCharacterizationofVO 2 -coatedMicroactuators............71 5.1.1MaterialPreparationandExperimentalSetup....... .........71 5.1.2CharacterizationofNon-monotonicHysteresis..... ..........73 5.2Non-monotonicHysteresisModel.................... ......76 5.2.1ActuationEffectduetoPhaseTransition........... ........76 5.2.2DifferentialThermalExpansionEffect............ ........77 5.3ModelIdenticationandValidation................. .........79 5.3.1ParameterIdentication........................ ....79 5.3.2ExperimentalResults........................... ..80 5.4InverseCompensation............................. ....84 5.4.1InverseCompensationAlgorithm.................. .....89 5.4.2ExperimentalValidation........................ ...92 Chapter6ModelingandInverseCompensationofHysteresisu singanExtended GPI(EGPI)model .............................. 94 6.1EGPImodelforNon-monotonicHysteresis............. ........94 6.2InverseCompensationAlgorithm.................... .......95 6.3ExperimentalResults:Modeling.................... .......98 6.3.1Curvature-temperatureHysteresisofaVO 2 -coatedMicroactuator.....99 6.3.2Resistance-temperatureHysteresisofaVO 2 Film..............102 6.4InverseCompensationResults...................... ......104 6.4.1Simulation...................................1 04 6.4.2ExperimentalVerication....................... ....106 Chapter7ACompositeHysteresisModelinSelf-SensingFeed backControlof VO 2 -integratedMicroactuators ....................... 109 7.1ExperimentalProcedures.......................... .....110 7.1.1VO 2 -integratedActuatorFabrication....................1 10 7.1.2ExperimentalSetup............................. .111 viii 7.1.3MeasurementofHystereticBehavior............... ......113 7.2ProposedCompositeModelforSelf-sensing........... ..........115 7.2.1MainIdea...................................115 7.2.2TemperatureSurrogate g ( T ) basedonaGPIModel.............119 7.2.3EstimatedDeection ‹ D BasedonanEGPIModel..............120 7.3ModelIdenticationandVerication................ .........121 7.3.1ModelIdentication............................ .121 7.3.2ModelVerication.............................. 127 7.4Self-sensing-basedFeedbackControl............... ..........129 7.4.1StepReferenceTracking......................... ..129 7.4.2SinusoidalReferenceTracking................... .....130 7.4.3Multi-frequencyReferenceTracking.............. .......130 Chapter8RobustControlof VO 2 MicroactuatorsusingSelf-SensingFeedback .. 133 8.1ExperimentalProcedures.......................... .....133 8.1.1VO 2 Deposition................................133 8.1.2MeasurementSetup.............................. 134 8.2Self-SensingDeection........................... .....135 8.3RobustControllerDesign.......................... .....137 8.3.1ModelingofVO 2 Microactuator.......................138 8.3.2RobustControllerDesign........................ ...141 8.4ExperimentalResult.............................. ....145 8.4.1StepReferenceTracking......................... ..146 8.4.2Multi-frequencySignalsReferenceTracking....... ..........148 8.4.3NoiseRejectionforMulti-frequencySignalsReferen ceTracking......150 Chapter9ConclusionsandFutureWork ........................ 153 9.1Conclusions..................................... .153 9.2FutureWork...................................... 154 APPENDICES ......................................... 156 AppendixAReviewofthePreisachOperator............... ........157 AppendixBReviewofthePrandtl-IshlinskiiModels....... ............161 BIBLIOGRAPHY ....................................... 166 ix LISTOFTABLES Table3.1ParametersoftheGPImodelenvelopefunctions... ...........31 Table3.2Compressionperformancecomparison:theuniform case..........32 Table3.3Compressionperformancecomparison:thecaseofo nepeak........35 Table3.4Compressionperformancecomparison:thecaseoft wopeaks.......36 Table3.5Compressionperformancecomparison:thecaseofr andomdistribution..36 Table3.6Identiedparametersoftheenvelopefunctions.. .............39 Table3.7Modelingvericationerrorcomparison......... ..........43 Table6.1ParametersoftheGPImodelandtheEGPImodelforhy steresisofa VO 2 -coatedmicroactuator..........................100 Table6.2ParametersoftheGPImodelandtheEGPImodelforhy steresisofaVO 2 lm......................................102 Table6.3ParameteroftheEGPImodel.................... ....106 Table7.1IdentiedparametersoftheGPImodel........... ........122 Table7.2IdentiedparametersoftheEGPImodel.......... .........123 Table8.1Steady-statevaluesofstepexperimentsforsyste midentication......140 Table8.2Controllercomparisonforstepreferencetrackin g..............148 Table8.3Controllercomparisonformultisinusoidalrefer encetracking........150 Table8.4Controllercomparisonformultisinusoidalrefer encetrackingwithnoise..151 x LISTOFFIGURES Figure1.1SuperimposedSEMpicturesofthe300 m mVO 2 -coatedsiliconcantilever takenwhenthesubstratetemperaturewas30 C(lowercurvature)and90 C(highercurvature),respectively.................... .5 Figure1.2Non-monotonichysteresisbehaviorinaVO 2 microactuator.Thegood repeatabilityoftheactuationbehaviorisalsoshown..... .......6 Figure2.1Experimentalsetupforresistancevstemperatur emeasurementsofaVO 2 lm......................................16 Figure2.2(a)Measured log 10 ( R ) Œ T hysteresisinVO 2 .(b)IdentiedPreisach densityfunction................................17 Figure2.3Uniformdiscretization..................... .......18 Figure2.4Non-uniformdiscretization:(a)Usingmaximumo finformationlosses J ¥ ascostfunction.(b)Usingsumofinformationlosses J 1 ascostfunction. (c)UsingKLdivergence J KL ofinformationlossesascostfunction....19 Figure2.5(a)Inputtemperaturesequenceformodelvalidat ion.(b)Outputerror comparisonbetweenuniformdiscretization,maximumofinf ormation loss,sumofinformationlossandKLdivergenceofinformati onloss....20 Figure2.6(a):Desiredoutputofsinusoidalshapesequence .(b):Inversecompensa- tionerrorsbetweenuniformdiscretization,maxofinforma tionloss,sum ofinformationlossandKLdivergenceofinformationloss.. .......21 Figure3.1Schematicillustratingthecompressionofaweig htingfunction.The solid-linesegmentsaretheoriginalweightingfunction,a ndthedotted- linesegmentsarethenewweightingfunction............. ...24 Figure3.2Illustrationoftheradius-dependentoutputran geforageneralizedplay operator....................................30 Figure3.3Weightingfunction(uniformcase)oftheGPImode l:(a)Unscaled.(b) Scaled.....................................33 Figure3.4(a)Inputsequence.(b)InputvsoutputfortheGPI modelwithuniform weightfunction................................34 xi Figure3.5Scaledweightingfunction(one-peakcase)..... ............34 Figure3.6Scaledweightingfunction(two-peakcase)..... ............35 Figure3.7Scaledweightingfunction(therandomcase).... ............36 Figure3.8Comparisonofaverageoptimizationtime.Noteth elogscale........37 Figure3.9Comparisonofthenumberofinformationlosseval uations.Notethelog scale......................................38 Figure3.10(a)Athird-orderreversalinputsequence.(b)T hecorrespondingoutput sequence.(c)TheoutputpredictionerrorbetweentheEntro pySumap- proachandoutputoptimizationapproach................ ..40 Figure3.11(a)Arandominputsequence,and(b)thecorrespo ndingoutputprediction errorperformance.(c)Theoutputpredictionperformanceb asedon50 randominputsequences...........................41 Figure3.12TheperformanceofaGPImodel(30plays)inmodel ingoftheresistance- temperaturehysteresisinVO 2 ........................42 Figure3.13(a)Identiedweightsforalltheplayoperators oftheGPImodel.(b)The scaledweightsfortheGPIoperators.................... 42 Figure3.14ParametersofthecompressedGPImodel:uniform compression......43 Figure3.15ParametersofthecompressedGPImodel:(a).Ent ropySum.(b)Entropy Max......................................44 Figure3.16(a)Anewtemperatureinputsequenceformodelve rication.(b)Corre- spondingoutputsequence.(c)Theoutputpredictionerrorc omparision ofEntropySumUnscaledapproachandtheEntropySumScaleda pproach.45 Figure4.1IllustrationofadiscretizationofthePreisach densityfunction,wherethe discretizationlevel L = 4..........................47 Figure4.2ThefidampedoscillationflsequenceforPreisacho peratoridentication ( L = 30)...................................49 Figure4.3(a)Signal q showingthesparseness;(b)signal q order ismoreapprox- imatelysparsethan q ;(c)thereconstructionperformancecomparison basedonfiCSflandfiCSOrderfl........................59 xii Figure4.4(a)Density w reconstructionerrorcomparison;(b)themodelingerror comparison..................................60 Figure4.5(a)Densityreconstructionerrorwithvaryingme asurementnoisebased onfiCSfl;(b)theaverageidenticationrun-timecomparison ........60 Figure4.6(a)Arandominputsequenceformodelvalidation; (b)correspondingout- putundertherandominputsequencein(a);(c)modelestimat ionerror comparison..................................62 Figure4.7a)Sideviewschematicforthemeasurementssetup fordeectionofa microactuatorwithanintegratedheater;b)TopviewoftheV O 2 -based integratedactuatordevices.......................... 64 Figure4.8(a)Input-outputdataforidentifyingPreisacho perator( L = 30);(b)true densityfunctionidentiedbasedon467measurements..... .....66 Figure4.9Densityreconstructionerrorcomparisonbasedo n(a)LeastSquares;(b) CS;(c)CSOrder;(d)CSOrderNon-negativeapproaches..... ....67 Figure4.10TheaverageRMSEmodelingerrorcomparison.... ...........68 Figure4.11Densityreconstructionerrorbasedon(a)arand ominputsequencefor modelvalidation;(b)outputoftherandominputsequencein (a);(c) modelestimationerrors...........................69 Figure5.1Setupusedformeasuringthecantilevertipdeec tionasafunctionof temperature..................................72 Figure5.2Illustrationofthegeometricrelationshipbetw eenthecurvatureandthe tipdeectionofabentcantilever...................... .73 Figure5.3Deectionasafunctionoftimethroughheatingan dcoolingtemperature steps.Thereisnoobservablecreep..................... 74 Figure5.4Proposedmodel.(a):Measurednon-monotoniccur vature-temperature hysteresisandthatbasedontheproposedmodel.(b):Modeli ngerrorfor theentiretemperaturesequence....................... 81 Figure5.5IdentiedPreisachdensityfunctionfortheprop osedmodel.........82 Figure5.6ModelingerrorwiththesignedPreisachoperator fortheentiretempera- turesequence.................................83 xiii Figure5.7Thenegativeoftheidentieddensityvaluesfort hesignedPreisachoper- ator.Thenegativeistakenheresothatthenegativeelement softheden- sityfunctioncanbeseen(ontop);thepositiveelementsare nowipped tothebottomoftheplane,whicharenotvisiblehere....... .....83 Figure5.8Modelingerrorwithapolynomialmodelfortheent iretemperaturese- quence....................................83 Figure5.9(a):Arandomlychosentemperatureinputsequenc eformodelvalidation. (b):Errorsinpredictionsbydifferentmodelsundertheinp utsequence...84 Figure5.10Illustrationofthevariables d ( k ) 1 and d ( k ) 2 usedininversion.........87 Figure5.11(a):Open-loopinversecontrolperformancefor theproposedmodeland thepolynomialmodel.(b):Inversecompensationerrors... .......93 Figure6.1AGPImodelandanEGPImodelwithidenticalweight sofgeneralized playoperators................................95 Figure6.2Theperformanceofmodelingcurvature-temperat urehysteresisofaVO 2 - coatedmicrocantileverbasedon:(a)GPImodel.(b)EGPImod el.....101 Figure6.3Theperformanceofmodelingtheresistance-temp eraturehysteresisofa VO 2 lmbasedon:(a)GPImodel.(b)EGPImodel.............103 Figure6.4Modelvericationoftheresistance-temperatur ehysteresisinaVO 2 lm: (a)arandomtemperaturesequence.(b)correspondingresis tanceoutput. (c)ModelingcomparisonbetweentheGPImodelandEGPImodel .....105 Figure6.5Simulationvericationoftheinversealgorithm .Hysteresisrelationship: (a)Inputsequence.(b)Input-outputoftheEGPImodel..... ......107 Figure6.6Compensationofhysteresisinsimulation:(a)In put-outputoftheinverse EGPImodel.(b)Therelationshipofthedesiredoutputandth eactual outputafterhysteresiscompensation.................. ...107 Figure6.7(a).Inversecompensationperformanceinexperi ment.(b).Inversion compensationerrorfortheEGPImodel................... 108 xiv Figure7.1FabricationprocessowfortheVO 2 -integratedactuator.a)Deposition ofSiO 2 (1 m m)byPECVD;(b)depositionofVO 2 (270nm)byPLD; (c)patterning(etch)ofVO 2 byRIE;(d)depositionofSiO 2 (0.4 m m)by PECVD;(e)patterning(etch)ofSiO 2 byRIE;(f-g)depositionofTi/Au byevaporationandpatterningbylift-off;(h)RIEofSiO 2 fordevicepat- tern;(j)cantileverreleasedbyXeF 2 isotropicetchingofSi.........112 Figure7.2TheVO 2 -integratedmicroactuatorusedinthiswork,withlength42 5µm andwidth65µm...............................113 Figure7.3(a)Thehysteresisbetweentheresistanceandthe current;(b)thehystere- sisbetweenthedeectionandthecurrent................ ..114 Figure7.4(a)Thehysteresisbetweenthedeectionandther esistance;(b)theresis- tancesequence;(c)zoom-inplotofthehysteresisbetweent hedeection andtheresistance,revealinganon-nestedstructure..... ........116 Figure7.5Thehysteresisbetweenthedeectionandthecurr entundervaryinginput frequencies;(b)Thehysteresisbetweenthedeectionandt heresistance undervaryinginputfrequencies....................... 117 Figure7.6(a)ThecomparisonbetweentheGPImodelpredicti onandexperimental measurementfortheasymmetrichysteresisbetweentheresi stanceoutput andthecurrentinput;(b)thecomparisonbetweentheEGPImo delpre- dictionandexperimentalmeasurementforthenon-monotoni chysteresis betweenthedeectionoutputandthecurrentinput........ .....123 Figure7.7(a)Performanceoftheself-sensingschemeusing thecompositemodel; (b)theself-sensingerrorbasedonthecompositemodel.... .......124 Figure7.8Performancesoftheself-sensingschemesusing( a)aPreisachmodel;(b) anEGPImodel;(c)ahigh-orderpolynomialmodel.......... ...125 Figure7.9Modelaccuracyandtherunningtimecomparisonbe tweenthecomposite model,thePreisachmodel,theEGPImodel,andthepolynomia lmodel..127 Figure7.10(a)Arandomlychosencurrentinputsequencefor self-sensingmodelver- ication;(b)theexperimentaldeectionmeasurementunde rtherandom currentinputsequence;(c)errorsinpredictionsbydiffer entself-sensing approaches..................................128 Figure7.11Blockdiagramoftheclosed-loopcontrolsystem withself-sensing.....129 Figure7.12Experimentalperformanceoftrackingastepref erenceunderdifferent self-sensingschemes.............................130 xv Figure7.13(a)Asinusoidalreferencesignalfortrackingc ontroloftheVO 2 -integrated microactuator;(b)experimentaltrackingerrorsunderdif ferentself-sensing schemes...................................131 Figure7.14(a)Amulti-frequencyreferencesignalfortrac kingcontroloftheVO 2 - integratedmicroactuator;(b)experimentaltrackingerro rsunderdifferent self-sensingschemes.............................132 Figure8.1(a)VO 2 lmresistance,and(b)VO 2 -coatedmicroactuatoractualdeec- tionasafunctionoftemperaturethroughaheating-cooling cycle(20-85 C).Bothvariablesweresimultaneouslymeasured......... ....136 Figure8.2(a)VO 2 -coatedmicroactuatoractualdeectionasafunctionofVO 2 lm resistanceduringtheheating-coolingcycle.Apolynomial functionof degree9wasusedtomodelthedeection-resistancemapping .(b)Maxi- mummodelerrorobtainedfromthemajorheatingandcoolingc urves...137 Figure8.3Blockdiagramsofthe(a)simpliedphysicalclos ed-loopcontrolsystem withself-sensingand(b)closed-loopsystemaugmentedwit hweighted functions...................................138 Figure8.4(a)Frameworkof H ¥ controlforthesystemand(b)robustperformance testbyaugmentingtheuncertainty D toM..................142 Figure8.5(a)Actual,and(b)self-sensedmicroactuatorde ectionunderself-sensed, closed-loopPIDandrobustcontrolthroughaseriesofstepr eferenceinputs.147 Figure8.6(a)Actualdeectionerrorand(b)controllereff ortforthePIDandrobust controlapproachesthroughthestepreferencetrackingexp eriment.....147 Figure8.7Microactuatordeectionresponsetoamultisinu soidalreferenceinputun- derPIDandrobustcontrol..........................149 Figure8.8(a)Actualdeectionerrorand(b)controllereff ortforthePIDandrobust controlapproachesinthemultisinusoidalreferencetrack ingexperiment..149 Figure8.9(a)Microactuatordeectionresponseand(b)err orunderamultisinu- soidalreferenceinputunderPIDandrobustcontrolwithind ucedwhite noiseincurrentinputtothesystem..................... 151 Figure8.10(a)Controleffortand(b)Peltierinputforamul tisinusoidalreference inputunderPIDandrobustclosed-controlwithinducedwhit enoisein currentinputtothesystem..........................15 2 xvi Figure8.11Frequencyspectrumanalysisofthetrackingerr orsundertherobustcon- trollerandthePIDcontroller,forscenarioswithandwitho utinjected actuationnoise................................152 FigureA.1IllustrationofuniformdiscretizationofthePr eisachplane,wherethedis- cretizationlevel M = 4............................159 FigureA.2Inversecompensationofhysteresis........... ...........160 FigureB.1Input-outputrelationshipsof(a)aclassicalpl ayoperatorwithradius r ; (b)ageneralizedplayoperatorwithradius r (shownassolidcurves)....163 xvii Chapter1 Introduction Inthischapter,abriefbackgroundonmodeling,identicat ion,andcontrolofhystereticsystemsis presented.Limitationsoftheexistingworkonmodelingand identicationofhysteresticsystems arediscussed.Afterwards,themotivationsofinversecomp ensation,self-sensingfeedbackcontrol, androbustcontrolforvanadiumdioxide(VO 2 )microactuatorsarebrieydiscussed.Atlast,an overviewofthecontributionsandtheorganizationofthedi ssertationarepresented. 1.1Modeling,Identication,andControlofHystereticSys tems 1.1.1Modeling ThetermfihysteresisflwascoinedbyJamesA.Ewinginhis1881 studyofferromagnetism[1]. Hysteresisisanonlineareffectthatoccursinawiderangeo fareas,suchasbiology[2],eco- nomics[3],ferromagneticmaterials[4]andvarioussmartm aterials[5Œ8].Therehasbeenex- tensiveworkdealingwithmodelingandcontrolofsystemswi thhysteresis.Hysteresismodels canberoughlyclassiedasphysics-basedandphenomenolog y-based.JilesandAtherton[4]pro- posedaphysics-basedhysteresismodelforferromagnetics .Whilephysics-basedmodelmaybe validtoalimitedquantityofsystems,phenomenology-base dmodels,suchasthePreisachoper- ator[9Œ12],generalizedPrandtl-Ishlinskii(GPI)model[ 13Œ16],Duhemmodel[17],Bouc-Wen model[18],andMaxwellmodel[19],areoftenapplicabletoa broaderclassofsystemswithhys- 1 teresis,andthushavebeenadoptedmoreextensivelytocapt urehysteresisnonlinearity.Among them,thePreisachoperatorandtheGPImodelarewidelyadop tedandhavebeenproveneffective incapturingdifferentformsofhysteresis. ThepracticalutilizationofthePreisachoperatormostlyi nvolvesuniformdiscretizationof thedensityfunctiononthePreisachplane[9,10,12].Forex ample,TanandBarasproposedto approximatethePreisachdensityfunctionwithapiecewise constantfunction,wherethePreisach densityfunctionisdiscretizedintoagridconsistingofeq ual-sizedcells[9].Itisanticipatedand veriedthattheaccuracyofthemodelimprovesasthenumber ofdiscretizationlevelincreases [12];however,computationalcomplexityanddatastoragec ostalsoincreasewiththenumber ofdiscretizationlevel,posingchallengesinparameterid enticationandcontrolofsystemswith hysteresis.Similarly,existingworkontheGPImodelhasty picallyadoptedsomepredened playradii[13,20Œ22],themodelingperformanceofwhichco uldbefarfromoptimal.While itisgenerallytruethatthemodelingperformanceimproves withanincreasingnumberofplay operators,similarly,computationalanddatastoragecost swillalsoincreaseforthemodelandthe correspondingmodel-basedinversecompensation.Obtaini ngaccuratehysteresismodelswhile maintainingarelativelylowcalculationandstoragecosts isthusanissueofpracticalinterest. 1.1.2Identication ThePreisachoperatorconsistsofweightedsuperpositiono fhysterons.Parameteridentication basedonthePreisachoperatorusuallyinvolvesdiscretiza tionofthePreisachdensityfunctionin oneformoranother,andoneeffectivemethodistoapproxima tethedensityfunctionwithapiece- wiseconstantfunction[9].Bothonline[12,23]andofine[ 9,24Œ26]schemescanbeadopted formodelidentication.Whenthediscretizationlevelis L ,thereare L ( L + 1 ) = 2cellswithdif- ferentdensityvalues[27].Theinputneedstoprovidesufc ientexcitationforallthedensity 2 valuesformodelidentication[12].Oneexampleofsuchinp utstakestheformofdampedoscil- lations,whichproducesnestedhysteresisloops[12].Thei nputsequenceshouldcontainatleast L ( L + 1 ) = 2elementstoidentifyallthedensities.Whenthediscretiz ationlevelischosenlarger,the correspondingPreisachoperatorcouldbettercapturethea ctualhysteresis,buttheidentication wouldrequirealargernumberofmeasurements.Forinstance ,in[24],thePreisachdensityfunc- tionwasdiscretizedinto200levels,andatleast20,100mea surementswouldneedtobetakenand processedtoidentifyallthedensityvalues.In[26],a20-l evelPreisachoperatorwasadoptedto characterizethedisplacement-temperaturehysteresisof aVO 2 -coatedmicroactuator.Inorderto capturethehysteresisunderquasi-staticcondition,arel ativelylongwaittimewasneededforeach measurementduetotheslowthermaldynamics,resultinginl ongexperimenttimeforcollecting therequireddataformodelidentication[26].Therefore, itisofgreatinteresttodesignamore efcientidenticationapproachthatrequireslessinput- outputdata. 1.1.3Control Withthefastdevelopmentofsmartmaterial-actuatedhyste reticsystems,therehasbeenanincreas- ingamountofworkincontrolschemes.Amongthem,animporta ntclassisinversecompensation. Whenexperimentsareoperateinquasi-staticcondition[6] ,theinversecompensationproblemis simpliedto:givenadesiredoutputvalue,calculateaninp utsequencesuchthatthenalvalueof theplantreachesthedesiredvalue.Sothehysteresiseffec tisapproximatelycancelledoutinthis manner.Inversionsofafewphenomenologicalmodelshavebe enreported[9,18,20,22,27,28]. TheinversionofthePreisachmodelistypicallyderivedbas edonnumericaliteration[27],ana- lyticalinversionoftheGPImodelcanoftenbederivedanaly tically[20,28],whichfacilitatesthe real-timecontrolimplementation.Notethatanalyticalin versionofaGPImodelrequiresthatall thegeneralizedplayoperatorshavethesameenvelopefunct ions,limitingitsabilityinmodeling 3 complexhysteresis. Althoughinversecompensationiseffective,itisalsohigh lycomputationallydemandingand doesnotperformrobustlyagainstdisturbances.Inorderto controlsystemswithhysteresis,various feedbackcontrolapproacheshavebeenproposed[9,12,29,3 0].Robustcontroltheoryhasbeen usedinsystemstoreduceenvironmentaldisturbancesandpl antuncertainties,butsuchstudieshave beentypicallylimitedtoconventionalsmartmaterials,su chaspiezoelectric-basedactuators[31Œ 34],wherethecontrollersweredesignedtocontrolthedee ctionofpiezoelectricmicroactuators basedonchargemeasurements. 1.2ModelingandControlofVanadiumDioxide( VO 2 )Microac- tuators VO 2 isanovelsmartmaterialthatundergoesathermallyinduced solid-to-solidphasetransition around68 C[35].Duringthetransition,thematerial'scrystallines tructurechangesfromamono- clinicphase(M 1 )atlowtemperaturestoatetragonalphase(R)athightemper atures,whichresults indrasticchangesinmultiplephysicalproperties(includ ingresistance[8],inducedmechanical stress[36],andopticaltransmittance[37])andpronounce dhysteresiswithrespecttotempera- ture.ThesecharacteristicsmakeVO 2 apromisingmultifunctionalmaterialforsensors[38],act u- ators[36,39,40],andmemoryapplications[41].Theactuat ionpotentialofVO 2 wasnotnoticed untilrecently[36].AsshowninFig.1.1,bycoatingVO 2 onamicrostructure(e.g.,asilicon cantilever),thermallyactuatedmicro-benderscanbecrea ted,whichhaveshownfullreversible actuation,largebending,andhighenergydensity[40],mak ingthemparticularlysuitableforap- plicationssuchasmicromanipulationandmicrorobotics. 4 Figure1.1:SuperimposedSEMpicturesofthe300 m mVO 2 -coatedsiliconcantilevertakenwhen thesubstratetemperaturewas30 C(lowercurvature)and90 C(highercurvature),respectively. 1.2.1ModelingandInverseCompensation TherealizationofthepotentialofVO 2 -coatedmicroactuators,however,isgreatlyhinderedby theirsophisticatednon-monotonichystereticbehavior(s howninFig.1.2)resultingfromtwocom- petingactuationeffects.Therstactuationeffectisduet otheinternalstressgeneratedduringthe phasetransition,whichisinherentlyhystereticwithresp ecttotemperature.Thesecondactuation effectisduetothedifferentialthermalexpansionoftheVO 2 layerandthesubstrate,whichcauses anoppositebendingeffect.Whilethethermalexpansioneff ectpersiststhroughoutthetempera- turerange,thestressgeneratedduringtheVO 2 'sstructuralchangesdominatesacrossthephase transition[39].Asaresult,therelationshipbetweentheb endingcurvatureandtemperatureis non-monotonic whenthetemperatureisraisedorloweredmonotonically.It iscrucialtocapture thenon-monotonichysteresisbehaviorinVO 2 microactuators. Moststudiesonmodelingandinversecompensationofhyster esisinsmartmaterialshavefo- cusedonmonotonichysteresisnonlinearities[9,10,28,42 Œ47],whereamonotonicinputcauses amonotonicoutput.Aspecialtypeofnon-monotonichystere siswithbuttery-shapedhysteresis loopswasinvestigatedbyDrincic etal. [48];however,thestudytherewasfocusedonhysteresis loopsthatcanbeconvertedtomonotonichysteresisthrough uni-modalmappings.Forthemodel consistingofaclassicalPrandtl-Ishlinskii(CPI)modela ndamemorylessfunction,theauthors 5 20304050607080010203040506070Temperature (0C)Deflection (mm) Day 1Day 2Figure1.2:Non-monotonichysteresisbehaviorinaVO 2 microactuator.Thegoodrepeatabilityof theactuationbehaviorisalsoshown. proposedaniterativeschemeforitsinversion[49],butthe convergenceoftheinversealgorithm wasnotconsidered.Inversecompensationofnon-monotonic hysteresisneedstobedevelopedfor VO 2 microactuators. 1.2.2Self-sensingFeedbackControl Forthecontrolofmicrodevices,externalsensingsystems, suchaslaserscattering[36]andin- terferometry[50],areoftenundesirableoreveninfeasibl eduetotheirsizesandcomplexity,and self-sensingprovidesacost-effectivealternative.Inse lf-sensing,thevariableofinterest(oftena mechanicalsignal)isestimatedbasedonanothervariable( typicallyanelectricalsignal)thatis mucheasiertoobtain.Existingworkonself-sensingofactu atorshasmainlyinvolvedtraditional smartmaterials,suchaspiezoelectrics[29,51,52],shape memoryalloys(SMAs)[30,53,54],and magnetorheologicaluids[55].Forexample,Ivan etal. implementedself-sensingforpiezo- electricactuators,whereboththedisplacementandtheext ernalforceatthetipofthecantilever wereestimatedbasedonthecurrentmeasurement,andaPrand tl-Ishlinskiimodelwasadopted tocompensatefortheremaininghysteresisnonlinearity[5 2].In[53],thestrainfeedbackofthe 6 SMA-actuatedexuresformotioncontrolwasestimatedfrom resistancemeasurementusinga high-orderpolynomialmodel.Polynomialmodelswerealsou tilizedtoestimatethestrainorgrip- permotionoftheSMA-basedgrippers[30,54].Althoughtheh ysteresisgapbetweenstrainand resistancecanbedecreasedbychangingthepretensionforc e,theremaininghysteresisstillposes challengesforprecisioncontrolofthesegrippers. Theself-sensingofVO 2 -basedmicroactuatorspresentsnewchallenges.Inparticu lar,there- sistancechangeisduetoaninsulator-to-metaltransition whilethemechanicalchangeisdueto astructuralphasetransition[35].Althoughstronglycorr elated,thesetwodifferentphasetran- sitionsdonotoccursimultaneouslyandthustherelationsh ipbetweendeectionandresistance ishystereticandhighlycomplicated.In[56],amemoryless Boltzmannfunctionwasutilized forself-sensingandaproportional-integralcontrollerw asimplementedbasedontheself-sensing signal.However,memorylessfunctions-basedself-sensin gschemescannotcapturetheinherent deection-resistancehysteresisandresultinlargesensi ngerrors,whichposesasignicantlimita- tiontotrackingcontrolaccuracy.Anovelcompositeself-s ensingmodelandproportional-integral controlbasedonthecompositeself-sensingwillbestudied . 1.2.3RobustControl Robustcontrolusingself-sensingfeedbackneedstobedesi gnedtoimprovethefeedbackcontrol robustnessforVO 2 microactuators.Althoughexternaldisturbancesandmodel uncertaintieswere consideredforthecontrollerdesigns,theerrorbetweenth eactualdeectionandthereferenceerror wasnotaddressedexplicitly.Therobustcontrollersin[33 ,34]weresynthesizedforsuppressionof piezoelectricstructurevibrationsbyself-sensingthera teofstrainchange,wheretrackingdesired referencesignalswasnotaconcern.Theworkdonein[33]fol lowedasimilarcontrolframework asin[31],butitwasdesignedtofollowadesireddeectionv alueofzero(inordertoreducevi- 7 brations).Althoughthecontrollerdesignin[34]accommod atesconstraintsoncontroleffort,it doesnotaccountforeffectsofmodeluncertainties,hyster esis,ordisturbances.Intermsofother hystereticmaterials,suchasSMAs,therehasbeennoreport edworkinrobustcontrolusingself- sensingfordeectioncontrol.Arobustcontrollerwillbed eveloped,whichtakesintoaccount theerrorinmodelingtemperature-deectionhysteresis,a ndenvironmentaldisturbances,inorder tominimizethetrackingerror.Unlikepreviousworkonrobu stpositioncontrolofhystereticmi- croactuatorsbasedonself-sensing[31Œ34],thecontrolle rinthisworktakesintoconsiderationthe errorbetweenthedesiredandactualdeectionvaluesinord ertopreciselycontrolthemicroactua- tor.Theperformanceoftherobustcontrollerisalsocompar edtoaproportional-integral-derivative (PID)controller. 1.3ContributionandOrganization 1.3.1OverviewofContribution First,toolsfrominformationtheory,namelyKLdivergence andentropy,areutilizedtooptimally compressthePreisachoperatorandtheGPImodelundergiven complexityconstraints.Thecom- pressedhysteresismodelsaremoreaccuratewhilemaintain ingrelativelylowcalculationandstor- agecomplexity.WhileduetotheparticularsettingofthePr eisachplane,theoptimalcompression ofthePreisachoperatorinvolvesanexhaustivesearch,the optimalcompressionoftheGPImodel isreformulatedasanoptimalcontrolproblemandsolvedwit hdynamicprogramming.Thepro- posedschemesareveriedinsimulationandexperimentalre sultsinvolvingthehysteresisbetween theresistanceandthetemperatureofaVO 2 lm. Second,identicationofthePreisachoperatorisstudiedu nderthecompressivesensingframe- workthatrequiresmuchfewermeasurements.Theproposedap proachadoptsthediscretecosine 8 transform(DCT)transformoftheoutputdatatoobtainaspar sevector.Sparservectorisfurtherob- tainedassumingtheorderofalltheoutputdataareknown.Th emodelparameterscanbeefciently reconstructedusingtheproposedscheme.Theleast-square sschemeisalsorealized,andiscom- paredwiththeproposedapproachusingthesamenumberofmea surements.Root-mean-square error(RMSE)isadoptedtoexaminetheidentiedmodelparam etersandmodelestimationperfor- mances.Theproposedidenticationapproachisshowntohav ebetteridenticationperformance thantheleast-squaresschemethroughbothsimulationande xperimentsinvolvingaVO 2 -integrated microactuator. Third,physics-motivatednon-monotonichysteresismodel sthataccountforthetwocompeting actuationmechanismsarepresented.Therstmechanismist hestressresultingfromstructural changesinVO 2 ,whichismodeledwithamonotonicPreisachoperatororaGPI model.Thesec- ondmechanismisthedifferentialthermalexpansioneffect ,whichismodeledwithamemoryless operator.Efcientinversecompensationschemesaredevel opedfortheproposednon-monotonic hysteresismodels.Forthenon-monotonicmodelbasedonthe Preisachoperator,theinversion complexityisstudied;forthenon-monotonicmodelbasedon theGPImodel,theinversionisde- velopedbasedonxed-pointiterationwithwhichtheconver genceconditionsofthealgorithmare derived.Theproposedmodelingandcompensationschemesar evalidatedexperimentally. Fourth,self-sensingfeedbackcontrolforVO 2 microactuatorsisstudied.Theproposedcom- positeself-sensingapproachexploitsthephysicalunders tandingthatboththeresistanceandthe deectionhavedifferenthystereticrelationshipswithth etemperature.Thesteadystatecurrentis usedasasurrogateforthetemperatureofVO 2 .Theself-sensingmodelisobtainedbycascadingan extendedGPI(EGPI)modelwiththeinverseofaGPImodel.The performanceoftheself-sensing schemeisevaluatedexperimentallywithproportional-int egralcontrol. Finally,an H ¥ robustcontrollerisfurtherdesignedandimplementedforp recisiondeection 9 control.Aninth-orderpolynomialisadoptedtomodelthese lf-sensingrelationshipbetweenthe deectionandtheresistance.Theuncertaintiesproducedb ythehysteresisbetweenthedeection andthetemperatureinputandtheerrorintheself-sensingm odelareaccommodatedbythepro- posedcontroller.Therobustcontrollerisdemonstratedth roughstepandmultisinusoidalreference trackingexperimentswithsimulatedwhitenoisecurrentsi gnal. 1.3.2Organization InChapter2,theoptimalcompressionofthePreisachoperat orispresented.InChapter3,we presenttheoptimalcompressionoftheGPImodel.Thecompre ssivesensing-basedPreisachoper- atoridenticationisproposedinChapter4.InChapter5,we discussthenon-monotonichysteresis modelingandinversecompensationbasedonthePreisachope rator.TheEGPImodelanditsin- versionarestudiedinChapter6.InChapter7,acompositehy steresismodelisproposedfor self-sensingfeedbackcontrolofVO 2 microactuators.RobustcontrolforVO 2 microactuatorsis presentedinChapter8.Conclusionsandfutureworkareprov idedinChapter9. 10 Chapter2 Kullback-Leibler(KL)Divergence-based OptimalCompressionofthePreisach Operator Inthischapter,anovelschemetooptimallycompressthePre isachoperatorisproposed.The KLdivergenceisutilizedtoquantifytheinformationlossi napproximatingthePreisachdensity functionaspiecewise-constantfunctions.Inparticular, theproposedcostfunctionincorporates boththelargestcellinformationlossandthetotalinforma tionloss,foragivendiscretization schemeonthePreisachplane.Exhaustivesearchisconducte dtondtheoptimaldiscretization scheme.Theproposedapproachisappliedtothemodelingoft hehystereticrelationshipbetween resistanceandtemperatureofaVO 2 lm,anditseffectivenessisfurtherexaminedinopen-loop inversecompensationexperiments.Theproposeddiscretiz ationschemeiscomparedwithtwo otherapproachesandwithuniformdiscretizaiton,andthee ffectivenessoftheproposedapproach isvalidatedinbothmodelvericationandinversecompensa tion. 2.1ProblemFormulation KLdivergence,orrelativeentropy,characterizesthedist ancebetweenprobabilitydistribution functions[57].KLdivergencehasbeenusedextensivelyins tatistics[58],patternrecognition[59], 11 andsignalprocessing[60].Tomotivatethisapproachindis cretizationofthePreisachoperator, considertheproblemofapproximatingaprobabilitydistri butionfunction(pdf) f p ( x ) inacertain regionwithauniformpdf f q ( x ) .If f p ( x ) isauniformdistribution,thenitcanbeapproximated withouterrorby f q ( x ) .However,muchofitsinformationwillbelostif f p ( x ) variesgreatlyfrom pointtopoint. ForthePreisachoperator,thenumberofdiscretizationlev elshasadirectimpactonthecom- plexityinmodelrepresentation,identication,andinver secompensation,andisthustakenasthe complexitymeasure.ThechoiceofthePreisachoperatordis cretizationisacriticalissuethatde- terminestheaccuracyinapproximatingthePreisachoperat orandthecomplexityinimplementing theinversePreisachoperator.Considertheproblemofappr oximatingsomearbitraryPreisach densityfunctionwithacell-wiseconstantfunction.Forag ivendiscretizationlevel(thusagiven levelofalgorithmiccomplexity),itisdesirabletodiscre tizethedensityfunctioninsuchaway thattheoriginaldensityfunctionhastheleastinformatio nlossapproximatingitusingaconstant withineachcell.Inotherwords,coarser(ner,resp.)disc retizationshouldbeappliedinregions withsmaller(larger,resp.)densityvariationsoastomini mizetheoverallinformationlossinthe approximationprocess. Ourproblemisthusformulatedas:giventhenumberofdiscre tizationlevel,ndtheoptimal discretizationschemethatminimizestheinformationloss inrepresentingtheoriginalPreisach operatorbytheapproximatingone. 2.2InformationLossMetric:KLDivergence-basedMeasure Forcontinuousrandomvariables,theKLdivergenceisdene das: 12 D LK ( P jj Q )= Z X f p ( x ) log f p ( x ) f q ( x ) d x ; (2.1) where f p and f q representthepdfsof P and Q ,respectively.Weusetheconventionthat0log 0 q = 0, p log p 0 = ¥ .ItcanbeshownthattheKLdivergencebetweentwopdfsisalw aysnonnegative,and iszeroifandonlyifthetwoprobabilitydistributionsarei dentical[57]. TheconceptofKLdivergencecanbeappliedtoPreisachdensi tyfunctiondiscretizationin thefollowingway:assumethatadiscretizeddensityfuncti onwithhighdelityisknown(know- ingthetrueinnite-dimensionaldensityfunctionisnotpr actical).Theoriginaldensitycanbe approximatedwithapiecewiseconstantfunctioncompatibl ewithadiscretizationgridusinga lowerdiscretizationlevel;namely,theapproximatingden sitytakesaconstantvaluewithineach discretizationcell,butthevaluevariesfromcelltocell[ 9].TheKLdivergencebetweenthe (normalized)originaldensityrestrictedtoacellandtheu niformdistributioncanthencapturethe informationlossinthatcell. Inparticular,foracertaindiscretization,thepdf p i ;j withineachcellisrstcalculatedandthe amountofinformationloss H i ;j forrepresenting p i ;j withauniformdensity q i ;j iscomputed. T i ;j isdenedastheintegralof m overcell ( i ; j ) : T i ;j = ZZ Cell ( i ;j ) m ( b ; a ) d b d a : (2.2) Thentheprobabilitydensityfunctions p i ;j and q i ;j ,overcell ( i ; j ) ,aredenedas: p i ;j ( b ; a )= m ( b ; a ) T i ;j ; (2.3) q i ;j ( b ; a )= 1 S i ;j ; (2.4) 13 where S i ;j isthetotalareaofcell ( i ; j ) .TheKLdivergence H i ;j between p i ;j and q i ;j is: H i ;j = ZZ cell ( i ;j ) p i ;j ( b ; a ) log p i ;j ( b ; a ) q i ;j ( ba ) d b d a : (2.5) Duetothenormalizationprocess,therelativeimportanceo feachcellwithrespecttoothercells isnotcapturedin H i ;j .Toaccountforthis,theinformationlossisdenedas L i ;j forcell ( i ; j ) by weighing H i ;j with T i ;j : L i ;j = H i ;j T i ;j : TheapproximatingPreisachdensityvalueforcell ( i ; j ) is T i ;j = S i ;j . 2.3OptimalCompressionScheme Itisofinteresttoinvestigatewhatisasuitablefimetricflf ormeasuringthecompressionerror. Foragivenlevelofdiscretization M ,thediscretizationvariablesaredenotedas f b k g M k = 0 ,where b 0 = v min , b M = v max arexedand b 0 b 1 b M 1 b M . D = f b k g M 1 k = 1 iscalleda discretizationstrategy.Foragiven D ,theweightedKLdivergence L i ;j canmeasurehowwell the(normalized)densityfunctionwithineachcell ( i ; j ) isapproximatedbyauniformdistribution, butatotalmeasurefortheoverallinformationlossisneede dtoevaluatetheperformanceofthe discretizationstrategy D . Insearchingthetotalmeasurecriterion,thusachievingop timalcompression,it'snotdesirable tohavecertaincellswhoseinformationlossareverylarge. Thetotalinformationlossofallcells isanotherconsideration.Thecriterionisproposedtobe 14 J KL ( D )= H D KL ( P L k Q u ) ; where P L representsthe(normalized)pdfof f L i ;j g ,and Q u representsauniformpdfwithvalue 1 = N ,where N = M ( M + 1 ) = 2isthetotalnumberofcells.Therelativeimportanceissim ilarly chosenas H andtheareaforeverycellisassumedtobe1.Thiscriteriont akesboththe total informationlossand maximum informationlossintoconsideration. Otherthanuniformdiscretization,twoothercostfunction approachesareconsideredforcom- parisonpurposes. J ¥ ( D )= max i ;j L i ;j With J ¥ ,thelargestinformationlossamongcellsisminimized,but thetotalinformationloss couldbelarge. J 1 ( D )= å i ;j L i ;j With J 1 ,thetotalinformationlossisminimized,butlargeinforma tionlossesinsomeregions arepossible,whichwouldresultinlargeerrorwhentheoper atorworksinthatparticular region. Sincetheproposedapproachconsidersboththemaximumandt otalinformationlosses,itis expectedthatitwillbestapproximatethedensityfunction ,comparedwiththeothertwocost functionapproachesandtheuniformdiscretizationscheme underthesamecomplexity. Thecompressionschemecanbeoutlinedasfollows.Givenano riginalPreisachoperator,with uniformdiscretizeddensityfunction m i ;j ; i ; j = 1 ; ; L .Discretizethedensityfunctionwitha lowerdiscretizationlevel M < L ,andnd D = f b k g M 1 k = 1 thatminimizes J ( D ) . f b k g M 1 k = 1 values arechosenfromadiscreteset.Inthisworkthediscreteseti schosentobetheinputlevelsofthe 15 originaldiscretizedplane: b k 2f v min + ( l 1 )( v max v min ) L j l = 1 ; 2 ; ; L g ; k = 1 ; ; M 1,and exhaustivesearchisconductedtondthesolutionthatmini mizes J ( D ) . 2.4ExperimentalResults 2.4.1ExperimentalCharacterization ThehysteresisbetweenresistanceandtemperatureinaVO 2 lmwasusedasanexampletoshow theeffectivenessoftheproposeddiscretizationscheme.A VO 2 layerwasdepositedbypulsed laserdeposition.Thelmwasgluedwithahighlythermalcon ductivesilverpaint,andinclose contactwithaPeltierheater.ThePeltierheaterwascontro lledwithatemperaturecontrollerwith 0.1 Cprecision.Fig.2.1showstheexperimentalsetup.Theresi stanceofthelmwasmeasured throughtwoelectricalaluminumcontactspatternedontheV O 2 lm. Figure2.1:Experimentalsetupforresistancevstemperatu remeasurementsofaVO 2 lm. Sincethemeasuredresistance ( R ) changesapproximatelytwoordersofmagnitudeduringthe phasetransition, log 10 R wastakenastheoutput,wherethenegativesignisintroduce dsothat theresultingPreisachoperatorhasanonnegativedensityf unction.Fig.2.2(a)showsthemeasured log 10 R Œ T hysteresisloopsincludingminorloops,andFig.2.2(b)sho wstheidentied,piecewise constantdensityfunction m ( b ; a ) usingthedatainFig.2.2(a),whereuniformdiscretization onthe 16 304050607080-4.5-4-3.5-3-2.5-2Temperature (0C)-log10R (log10W)(a) 3055808067.55542.53000.020.040.060.08a (0C)b (0C)Density Function (log10W(0C)-2)(b) Figure2.2:(a)Measured log 10 ( R ) Œ T hysteresisinVO 2 .(b)IdentiedPreisachdensityfunction. Preisachdensityfunctionwasadoptedwithdiscretization level M = 40.Thefullrangeoftempera- tureinput [ v 0 min ; v 0 max ] was [ 30 ; 80 ] C.Theoffsetvaluewasidentiedtobe 3 : 308log 10 W ,where log 10 W denotestheunitoftheoutput. 2.4.2CompressionPerformance Toconductthecompressionstudies,thepiecewiseconstant Preisachdensityfunctionwith(uni- form)discretizationlevel40wastakenasthefioriginalflde nsityfunction,andtheapproximation 17 toitwithdiscretizationlevel4wasfound.Fig.2.3showsth euniformdiscretizationandFig.2.4 showsdiscretizationresultsbasedon J ¥ ( D ) , J 1 ( D ) , J KL ( D ) ,respectively.Itcanbenotedfrom theguresthatinthenon-uniformdiscretizationscheme,t hediscretizationisnerinthecenter areasincethisiswheretheoriginaldensityfunctionhasla rgervariations.However,whenthecost functionischosenas J ¥ ( D ) or J 1 ( D ) ,thecentercellshaveverysmallarea,whichisactuallyun- desirable. J KL ( D ) basedcompressioncomeswithabetterapproximationofthefi originalfldensity distribution. Figure2.3:Uniformdiscretization. Inordertoexaminetheapproximatingperformanceofeachdi scretizationscheme,arandomly choseninputsequenceforthesystemshowninFig.2.5(a)was applied.Thecorrespondingre- sistanceoutputinexperimentwasthenobtainedandcompare dwiththepredictionsfromthefour low-complexitymodelsmentionedabove.Fig.2.5(b)showst hepredictionerrorsforthefourmod- els.Themodelbasedon J KL wasabletopredictmuchbettertheresistancethantheother three schemes.ThelargestoutputerrorusingtheKLschemeisbelo w0 : 23log 10 W ,whileuniformdis- cretizationapproachresultsinerrorslargerthan0 : 57log 10 W andothertwoapproachesendup withlargerthan1 : 00log 10 W .Theeffectivenessoftheproposedapproachinrepresentin gthetrue 18 (a) (b) (c) Figure2.4:Non-uniformdiscretization:(a)Usingmaximum ofinformationlosses J ¥ ascost function.(b)Usingsumofinformationlosses J 1 ascostfunction.(c)UsingKLdivergence J KL of informationlossesascostfunction. 19 0100200300400304050607080Time indexInput Temperature (0C)(a) 0100200300400-1-0.8-0.6-0.4-0.200.2Time indexError (log10W) UniformMaxSumKL(b) Figure2.5:(a)Inputtemperaturesequenceformodelvalida tion.(b)Outputerrorcomparison betweenuniformdiscretization,maximumofinformationlo ss,sumofinformationlossandKL divergenceofinformationloss. densityfunctionofPreisachoperatorisevident. Inversecontrolperformancebasedonthefourmodelswasals oexamined.Giventhedesired outputsequence,atemperatureinputthatachievesthegive noutputcanbecalculatediteratively. Thiscalculatedinputsequencewasappliedtothesystemtov erifytheperformanceoftheidentied model.Fig.2.6(a)showsthedesiredsinusoidaloutputwith decreasingamplitudes,andFig.2.6(b) showsthecorrespondinginversionerrorsbasedonthefourd ifferentmodels.Itcanbeseenthat theinversionbasedontheproposedmodeliseffective,with thelargesterrorof0 : 22log 10 W ,for thetotalrangeof [ 4 : 28 ; 2 : 34 ] log 10 W .Astimeevolvestheoutputerroralmostconvergesto zero,andsincethediscretizationlevelisonly4,theinver sionperformancecanbeconsideredvery good.Incomparison,theinversionoftheuniformdiscretiz ationapproachyieldserrorslarger 20 0204060802.533.544.5Time indexDesired output (log10W)(a) 02040608000.51Time indexError (log10W) UniformMaxSumKL(b) Figure2.6:(a):Desiredoutputofsinusoidalshapesequenc e.(b):Inversecompensationerrorsbe- tweenuniformdiscretization,maxofinformationloss,sum ofinformationlossandKLdivergence ofinformationloss. than0 : 56log 10 W ,andaslargeas1 : 00log 10 W fortheothertwoapproaches.Thisagainshowsthe advantageoftheproposedKLdivergence-basedcompression scheme. 21 Chapter3 Entropy-basedOptimalCompressionofthe GeneralizedPrandtl-Ishlinskii(GPI)Model Inthischapter,theoptimalcompressionoftheGPImodelsub jecttoagivennumberofplayopera- torsispresented.Aninformation-theoretictool,entropy ,isadoptedtocapturetheinformationloss inreplacingagroupofweightedplayoperatorswithasingle playoperator.Priortocompression, ascalingoperationontheoriginalweightsisintroducedto accommodatethefactthat,giventhe sameweightvalue,ageneralizedplaywithalargerradiusha slessimpactonthetotaloutput.The optimalcompressionalgorithmisreformulatedasanoptima lcontrolproblemandsolvedwithdy- namicprogramming,thecomputationalcomplexityofwhichi sshowntobemuchlowerthanthat ofexhaustivesearch. Extensivesimulationresultsarepresentedtoexaminethep erformanceoftheproposedap- proachinapproximatingaGPImodelconsistingofalargenum berofplayoperators,wherecases withdifferenttypesofweightdistributionsareexplored. Simulationresultsshowthat,ingeneral, theentropy-basedapproachesdeliverfarbetterperforman cethanatypicallyadoptedschemewhere theplayradiiareassigneduniformly.Theeffectivenessof theproposedapproachisfurtherver- iedinthecompressionofanexperimentallyidentiedGPIm odelforthehystereticrelationship betweentemperatureandresistanceofaVO 2 lm.Notethattheproposedoptimalcompression approachalsoworksfortheCPImodelsinceitisaspecialcas eoftheGPImodel. 22 3.1ProblemFormulation Notethat,foraGPImodel,itsoutputisaweightedsuperposi tionofoutputs(states)ofindividual playoperatorsandhenceislinearwithrespecttotheweight parameters.Asaresult,onecan identifytheweightparametersofineoronlinebyminimizi ngthedifferencebetweentheactual outputandtheoutputoftheestimatedmodel.Ontheotherhan d,theplayradiiofaGPImodel determinesthestatesofindividualplayoperators,butthe relationshipbetweentheradiiandthe statesarecomplexandinvolvesthepasthistoryoftheinput ,andonecannotexpresstheoutputofa GPImodeldirectlyintermsofitsplayradii.Consequently, determiningtheplayradiibasedonthe outputdifferencebetweentheoriginalmodelandtheestima tedmodelisdifcultifnotimpossible. Therefore,inthisworkweseektominimizethedifferencein fiweightdistributionfl(includingboth playradiiandtheirweights)betweentheoriginalandreduc edGPIoperators,whichwouldimply thattheoutputofthereducedmodelwillbeclosetothatofth eoriginalmodel,underallinput functions. ThenumberofplayoperatorsintheGPImodeldirectlydeterm inesthecomputationaland storagecostinhysteresismodeling,parameteridenticat ion,andinversecompensation.Therefore, itistakenasthemeasureofcomplexityforaGPImodel.Conse quently,thecompressionofahigh- delityGPImodelwithalargenumber( N )ofplayoperatorsdealswithndingasmallernumber ( M , M < N )ofplayoperatorsandthecorrespondingweightstobestrep resenttheoriginalGPI model.Thisproblemiscloselytiedtooptimalcompressiono ftheweightvector f p ( r j ) g N j = 1 inthe discretecase,whichisformulatedprecisely. Fig.3.1showsanonnegativeweightingfunction p ( r ) with N elements, p j = p ( r j ) ,0 < = r 1 < r 2 < r N < ¥ .Thecompressionoftheoriginalweightingfunctionistous eanewweighting functionwith( M , M < N )elements:‹ p j = ‹ p ( ‹ r j ) ; 0 < = ‹ r 1 ‹ r 2 ‹ r M 1 ‹ r M < ¥ toapproximate 23 theoriginalweightingfunction. 0br() pr 1r2r3r4r5r6r7r8r9r10 r11 r12 r13 rNr××× 2Nr-1Nr-××× 14 r15 r16 r××× 1b2b3bMbFigure3.1:Schematicillustratingthecompressionofawei ghtingfunction.Thesolid-lineseg- mentsaretheoriginalweightingfunction,andthedotted-l inesegmentsarethenewweighting function. Inordertocompresstheweightingfunction,thenotionofpa rtitionisrstlyintroduced.Denote D = f b k g M k = 0 asthesetofpartitionindices ( 0 = b 0 < b 1 < < b M 1 < b M = N ) ,thatpartitions theweightingfunctioninto M groups.Theoriginalweightswithindices b k 1 + 1 ; b k 1 + 2 ; ; b k belonginthe k -thgroup, k = 1 ; 2 ; ; M .Foreachgroup,theoriginalweightingfunctionisap- proximatedwithonlyoneelement(showninFig.3.1asreddot tedsegments),andthenewelement ischaracterizedas ‹ r k = b k å i = b k 1 + 1 p ( r i ) b k å j = b k 1 + 1 p ( r j ) r i ; ‹ p ( ‹ r k )= b k å i = b k 1 + 1 p ( r i ) : (3.1) Theoptimalcompressionproblemistondthecompressionst rategyD = f b k g M k = 0 thatbest approximatesthegivenweightingfunction.Tofacilitatet heformulationoftheproblem,wecon- siderafunction F k astheinformationlossmeasureinapproximatingthedistri bution p ( r i ) ; b k 1 + 1 i b k with‹ p k ( ‹ r k ) .Theoverallcompressioncostfunctioncanbechosenaseith er M å k = 1 F k or max k F k . 24 3.2OptimalCompressionScheme Whileanumberofmethods,suchasevolutionaryalgorithms[ 61]andsimulatedannealing[62], couldbeusedtosolvenonlinearoptimizationproblems,the seapproachesaretypicallycomputation- intensiveandcannotguaranteegloballyoptimalsolutions .Inthiswork,weexploitthestructure ofthecompressionproblemandreformulateitasanoptimalc ontrolproblem.Thereformulation allowsustousedynamicprogrammingtoobtainthe(globally )optimalsolution,aswellasan- alyzethecomplexityofthealgorithm.Denote x k = b k asthestate,and u k = b k b k 1 asthe controlinput, k = 1 ; ; M .Theoptimizationproblemisthenreformulatedas:ndingi nputs u =( u 1 ; u 2 ; ; u M 1 ) ,suchthatthetotalcostisminimized.Notethatsince b 0 = 0and b M = N arexed, u M willbedeterminedautomaticallyby u andthusisnotadecisionvariable.Thecom- pressioncost F k foreachgroupisclearlydeterminedby b k 1 and u k ,or F k = F k ( x k 1 ; u k ) .The dynamicprogrammingalgorithmtobepresentednextconside rstheoverallcostfunction J 1 with theformof M å i = 1 F i .Thealgorithmissimilarforthecasewhenthecostfunction isintheformof max i F i .Specically,wehave x k = x k 1 + u k ; J 1 ( x 0 ; u )= M 1 å i = 1 F i ( x i 1 ; u i )+ f ( x M 1 ) ; (3.2) where f ( x M 1 ) representsthefiterminalcostflŒtheinformationlossforth elastgroup.Theoptimal control u =( u 1 ; u 2 ; ; u M 1 ) isdenedas u = argmin u J 1 ( x 0 ; u ) : (3.3) 25 Oncetheoptimalcontrol u isfound,theoptimalcompressionstrategyisobtainedas: b 0 = 0 ; b 1 = b 0 + u 1 ; ; b k = b k 1 + u k ; ; b M = N .Thefollowingpropositionprovidesthealgo- rithmfornding u ,theproofofwhichfollowsthestandarddynamicprogrammin gprinciple[63]. Proposition1. ConsiderasequenceofM 1 optimizationproblems,withthecostfunctionsde- nedas J k ( x k 1 ; f u i g M 1 i = k )= M 1 å i = k F i ( x i 1 ; u i )+ f ( x M 1 ) ; (3.4) k = 1 ; ; M 1 ,andthecorrespondingvaluefunctionas V k ( x k 1 )= min f u i g M 1 i = k J k ( x k 1 ; f u i g M 1 i = k ) : (3.5) Thenthevaluefunctionsalongwiththeoptimalcontrolsequ ence f u i g canbeobtainedrecursively asfollows: V M 1 ( x M 2 )= min u M 1 F M 1 ( x M 2 ; u M 1 )+ f ( x M 2 + u M 1 ) ; (3.6) V k ( x k 1 )= min u k F k ( x k 1 ; u k )+ V k + 1 ( x k 1 + u k ) ; (3.7) k = M 2 ; ; 2 ; 1 ,andu k isobtainedastheminimizingu k inthecomputationofV k ( x k 1 ) ,k = 1 ; ; M 1 . Remark1. NotethattheprocedureinProposition1willgenerate f u k g asastate-dependent policy.Theoriginaloptimizationproblemhasaxedinitia lstateofx 0 = 0 ,whichresultsina specicoptimalcontrolsequencewhenappliedtothefeedba ckpolicy. 26 Thedynamicprogrammingapproachhasasignicantadvantag eovertheexhaustivesearch intermsofcomputationalcomplexity.Takethenumberofeva luationsofinformationlossin partitionedgroupsrequiredbyeachalgorithmasthemetric ofcomputationalcomplexity.For thedynamicprogrammingapproach,theterminalcostfuncti on f ( x M 1 ) requires N evaluations since x M 1 couldtakeanyvaluesof f 0 ; 1 ; ; N 1 g ,(3.6)requires N 1evaluations,and(3.7) requires N M + k evaluations,1 k M 2,resultinginatotalof M ( 2 N M + 1 ) 2 evaluations. Fortheexhaustivesearch,ontheotherhand,thereare ( N 1 ) ! ( M 1 ) ! ( N M ) ! possiblepartitionsforthe weights,andeachpartitionrequires M evaluations,resultinginatotalof ( N 1 ) ! M ( M 1 ) ! ( N M ) ! forthe numberofevaluations,whichissignicantlylargerthanth ecomplexityofthedynamicprogram algorithm. 3.3InformationLossMetrics:Entropy-basedMeasure Thediscussionssofarhaveassumedagenericfunction F k thatrepresentstheinformationlossin replacingtheweightdistributionofthe k thgroup, f p ( r i ) g b k i = b k 1 + 1 ,withasingleweight‹ p k ( ‹ r k ) . Aninformation-theoretictool,entropy,isexploitedtode netheinformationlossincompression. Entropy[64]isameasureoftheuncertaintyinarandomvaria ble,whichhasbeenusedexten- sivelyinstatistics[65]andsignalprocessing[66].Forad iscreterandomvariable G withprobabil- itymassfunction(pmf)¯ p ( r i ) , i = 1 ; 2 ; ; L ,theentropyisdenedas H ( G )= L å i = 1 ¯ p ( r i ) log ( ¯ p ( r i )) : (3.8) Theconvention0log0 = 0isadopted.Foragiven L ,theentropyof G islowestwhenthere existsa k L ,suchthat¯ p ( r k )= 1.Ontheotherhand,theuniformdistribution,where¯ p ( r i )= 1 = L , i = 1 ; 2 ; ; L ,hasthelargestentropy. 27 Intuitively,iftheweightdistributionof(multiple)play operators,whenproperlynormalized,is closetoauniformdistribution,thecompressionlossishig h;Conversely,ifthegrouphasasingle operatorwithweightdominantlylargerthanthoseoftheres toperators,thecompressionlossis expectedtobesmall.Theseconsiderationsmaketheentropy anaturalcandidateformeasuring theinformationloss.Inaddition,ifthedominantplayoper atorsarelocatedfarawayfrom‹ r k ,the compressionlossisalsohigh,motivatingtheincorporatio nofthedistancesbetweentheplayradii andtheficentroidfl‹ r k intothecostfunction.Specically,thefollowingprocedu reisproposedto computeanentropy-basedmeasurefortheinformationlossi napproximatingadiscretedistribution group p ( r i ) , i = b k 1 + 1 ; b k 1 + 2 ; ; b k : 1.Calculatethetotalweightinthegroup: T k = b k å i = b k 1 + 1 p ( r i ) : 2.Getthenormalizedpmfforthegroup: ¯ p i = p ( r i ) = T k ; i = b k 1 + 1 ; b k 1 + 2 ; ; b k : 3.Obtaintheentropyforthenormalizedpmf: H k = b k å i = b k 1 + 1 ¯ p i log¯ p i : 4.Theeffectofthedistancesbetweenplayradiiandthecent roidneedstobeincorporated;one waytodothisistodenethecostfunctionforthe k -thgroupas 28 E k = T k v u u u t b k å i = b k 1 + 1 ( ¯ p i ( r i ‹ r k )) 2 H k ; where T k isincludedtoreecttheimpactofthetotalweightforthegr oup. Notethatwhiletheremightbeotheralternatives,wewillsh owlaterinthisworkthattheproposed schemeisadequatelyeffective.Finally,forapartitionst rategy D ,theentropy-basedoverallcost functionscanbechosenas: J E SUM ( D )= M å k = 1 E k ; (3.9) J E MAX ( D )= max k E k : (3.10) Theoptimizationalgorithmsbasedonthecostfunctions(3. 9)and(3.10)aredenotedasEntropy SumandEntropyMax,respectively. 3.4ScalingoftheWeightsfortheGPIModel ForaGPImodelwithacertaininputrange,theconstituentpl ayoperatorswillhavedifferentranges ofoutputs,andthushavedifferentlevelsofimportancetot heoutputoftheGPImodeleveniftheir weightsareequal.Properfiscalingfloftheweightingfuncti onisintroducedtoaccommodatethe playradius-dependentimportance. Fig.3.2showsageneralizedplayoperatorwithradius r ,wheretheinputrangeis [ v min ; v max ] , andtheinitialcondition w ( 0 )= g L ( v min )+ r .Itcanbeeasilyseenthattheoutputrangeoftheplay operatorisdependenton r ;specically,theoutput w 2 [ g L ( v min )+ r ; g R ( v max ) r ] ,withatotal changeof g R ( v max ) g L ( v min ) 2 r .Accordingly,thefollowingschemeisintroducedtoproduc e afiscaledflweightdistributionforthecompression. 29 v() Rv-r g() Lv+r gmin vmax vwFigure3.2:Illustrationoftheradius-dependentoutputra ngeforageneralizedplayoperator. Denotetheactualweightas p ,andtheweightafterscalingas p 0 ,thenfortheplayoperator withradius r j : p 0 ( r j )= g R ( v max ) g L ( v min ) 2 r j 2 p ( r j ) : (3.11) Forageneralizedplayoperatorwhoseenvelopesareinthefo rmofhyperbolic-tangentfunc- tions,when v min ! ¥ and u max ! + ¥ ,theoutput z 2 [ 1 + r ; 1 r ] .Itcanbeseenthat0 r 1. Theplayoperatorwillnotproduceanyoutputchangeunderac yclicinputwhentheradius r > 1, sincetheoutputwillneverreachbothenvelopesduetothedi sjointrangesoftheenvelopes.Itis forthisreasonthattheradiusisalwayschosentobenolarge rthan1.Theadvantageofusingthe scaledweightsoverthenon-scaledweightsincompressionw illbefurtherdemonstrated. 3.5SimulationResults Theproposedoptimalcompressionalgorithmsaretestedins imulationforGPImodelswithdiffer- entcharacteristicsfortheirscaledweightingfunctions. Following[13,28],theenvelopefunctionsforthegenerali zedplayoperatorarechosentobe 30 Table3.1:ParametersoftheGPImodelenvelopefunctions. a R b R a L b L 3.504.51 hyperbolic-tangentfunctionsintheformof g R ( v ( t ))= tanh ( a R v ( t )+ b R ) ; (3.12) g L ( v ( t ))= tanh ( a L v ( t )+ b L ) : (3.13) Forsimplicityofdemonstration,thenon-hystereticcompo nent D ( v ( t )) issettobezero.The parametervaluesofthegeneralizedplayoperatorareshown inTable3.1.TheoriginalGPImodel consistsof N = 30playoperators,withradii r j = j = ( N + 1 ) , j = 1 ; 2 ; ; N ,andinputrangeof v 2 [ 1 ; 1 ] . ThecompressiongoalistouseanewGPImodelwith M = 6playoperatorstoapproximate theoriginalGPImodel.Althoughtheunscaledweightsaredi rectlyrelatedtotheoutput,the scaledweightdistributionisconsidered.Theoutputperfo rmanceoftheproposedapproachwill bediscussed.Fourcasesforthescaledweightdistribution oftheoriginalmodelareconsidered,1) uniform,2)onepeak,3)twopeaks,and4)random.Inaddition tothetwocompressionschemes presentedintheprevioussection(EntropySum,EntropyMax ),auniformcompressionscheme, whereeveryveconsecutiveplayoperatorsareclusteredin toonegroup,isconsideredforcom- parisonpurposes. Inordertoassesstheoutputpredictionperformanceofther educedmodel,throughoutthechap- ter,thenormalizedRMSEisadoptedtoquantifythemodeling performanceunderdifferentcom- pressionstrategies.Theerrorisobtainedasfollows:rst ,calculatetheRMSEbetweentheoutput 31 Table3.2:Compressionperformancecomparison:theunifor mcase. SchemePartitionindicesError Uniform(0,5,10,15,20,25,30)0.75% EntropySum(0,5,10,15,20,25,30)0.75% EntropyMax(0,5,10,15,20,25,30)0.75% ofeachnewGPImodelandthatoftheoriginalGPImodelundert heinputshowninFig.3.3(a); thendividetheRMSEbytheoutputrangeoftheoriginalmodel .NormalizationoftheRMSE allowsassessingandcomparingthealgorithms'performanc eacrossdifferentweightdistributions. 3.5.1Case1:UniformDistributionfortheScaledWeights First,thefollowinguniformdistributionisconsideredfo rthescaledweights: p 0 ( r j )= 0 : 5 ; j = 1 ; 2 ; ; 30 : (3.14) Fig.3.3(a)showstheactualweightdistribution(unscaled )andFig.3.3(b)showsthecorresponding scaledweightdistribution.Fig.3.4(a)showsaninputsequ enceandFig.3.4(b)showstheinput- outputrelationshipofthegivenGPImodel.Notethattheact ualweightingfunctionandtheinput- outputrelationshipwillnotbeshownforotherformsofweig htingfunctionsintheinterestof brevity;however,thehysteresisloopsinothercasesareal soveriedtobelarge. ThesimulationresultsaresummarizedinTable3.2.Itissho wnthat,giventheuniformdis- tribution,thetwoentropy-basedalgorithmsareabletocom pressthedistributionuniformly,and generatedesirableperformance. 32 00.20.40.60.8105101520WeightsPlay radius(a) 00.20.40.60.8100.20.40.6WeightsPlay radius(b) Figure3.3:Weightingfunction(uniformcase)oftheGPImod el:(a)Unscaled.(b)Scaled. 3.5.2Case2:ScaledWeightswithOneProminentPeak Inthesecondcase,thescaledweightdistributionisassume dtohaveonepeak,expressedas: p 0 ( r j )= 5 p 2 p exp ( ( j 15 ) 2 15 ) ; j = 1 ; 2 ; ; 30 : (3.15) Fig.3.5showsthescaledweightdistribution.Table3.3sho wsthecompressionperformancesbased ondifferentapproaches.Fromthesimulationresults,itis seenthatEntropySumandEntropyMax approachesareabletogenerateconsiderablybetterperfor mancethantheuniformcompression scheme.Thepeakoftheoriginalweightingfunctionisinthe middleregion;thesimulationresults showthattheentropyapproachespartitiontheweightsdens elyinthemiddleregion(withmany groupshavingonlyoneortwoelements). 33 01000200030004000-1-0.500.51IndexInput(a) -1-0.500.51-15-10-5051015InputOutput(b) Figure3.4:(a)Inputsequence.(b)InputvsoutputfortheGP Imodelwithuniformweightfunction. 00.20.40.60.8100.511.52WeightsPlay radiusFigure3.5:Scaledweightingfunction(one-peakcase). 34 Table3.3:Compressionperformancecomparison:thecaseof onepeak. SchemeCutindicesError Uniform(0,5,10,15,20,25,30)0.34% EntropySum(0,11,13,14,16,18,30)0.08% EntropyMax(0,9,12,14,16,18,30)0.14% 3.5.3Case3:ScaledWeightswithTwoProminentPeaks Inthethirdcase,thescaledweightingfunctionhastwopeak s,expressedas: p 0 ( r j )= 8 > < > : 5 p 2 p exp ( ( j 8 ) 2 8 ) ; j = 1 ; 2 ; ; 16 5 p 2 p exp ( ( j 23 ) 2 8 ) ; j = 17 ; 18 ; ; 30 : (3.16) Fig.3.6showsthescaledweightdistribution.Table3.4sho wsthecompressionperformances basedonthedifferentcompressionapproaches.Fromthesim ulationresults,bothoftheproposed approachesshowverygoodperformance,withabout40%lesse rrorcomparingtotheuniform compressionscheme. 00.20.40.60.8100.511.52WeightsPlay radiusFigure3.6:Scaledweightingfunction(two-peakcase). 3.5.4Case4:RandomDistributionfortheScaledWeights Finally,weconsiderthecasewherethescaledweightingfun ctionhasarandomdistributionas showninFig.3.7.Table3.5liststhecorrespondingsimulat ionresults.Itcanbeseenthat,under 35 Table3.4:Compressionperformancecomparison:thecaseof twopeaks. SchemePartitionindicesError Uniform(0,5,10,15,20,25,30)0.42% EntropySum(0,7,8,14,22,23,30)0.27% EntropyMax(0,6,8,11,21,23,30)0.20% Table3.5:Compressionperformancecomparison:thecaseof randomdistribution. SchemePartitionindicesError Uniform(0,5,10,15,20,25,30)0.53% EntropySum(0,5,9,13,18,24,30)0.54% EntropyMax(0,6,10,15,20,25,30)0.53% arandomdistribution,theentropyapproachescompressthe distributionalmostuniformly,with slightlybetterperformancethantheuniformcompressions cheme.Arandomdistributionissim- ilarlydifcultasauniformdistributiontocompress,sinc ethereareusuallynoparticularpatterns thatfacilitatecompression. 00.20.40.60.8100.20.40.60.81WeightsPlay radiusFigure3.7:Scaledweightingfunction(therandomcase). Fromthesimulationresults,overall,bothoftheproposeda pproachesshowgoodcompression performance.Whenthepatternofthe(scaled)weightingfun ctionisuniformorrandom,theopti- malcompressionalmostdegeneratestouniformcompression ,andthecompressionerrorislarger comparingwithothercasesthathavemorefeatures(peaks). 36 3.5.5ComputationalTimefortheAlgorithms Thecomputationaltimeofthedynamicprogramming-basedop timizationisalsocomparedwith thatusingexhaustivesearch.Duetothesimilaroptimizati onprocessunderdifferentcostfunction candidates,onlyEntropySumisconsideredinthiscomparis on.AGPImodelwith N = 30is used,whichhasascaledweightdistributionasusedinSecti on3.5.2.Thecomputationsarerunin MatlabonacomputerLenovoThinkpadT420with2.80GHzCPUan d4.00GBmemory. Inordertocomparethecomputationalefciency,thedynami cprogramming-basedalgorithm andtheexhausitivesearch-basedalgorithmarerun10times foreachsettingof M ,whichisvaried from2to7inthisstudy.Theaveragerunningtimesareshowni nFig.3.8.Itcanbeseenthat, when N isxed,thetimecostunderdynamicprogramminggrowsmuchs lowerthantheexhaustive searchwhen M isincreased.Theseresultsagreewellwiththecomplexitya nalysisinSection3.2, asshowninFig.3.9,whichplotsthenumberofinformationlo ssevaluationsforthedynamic programmingandtheexhaustivesearchmethods,respective ly.Thecomputationaladvantageof thedynamicprogrammingapproachisevident. 234567-3-2-101Number of play operatorslog10(time) (s) Exhaustive SearchDynamic ProgrammingFigure3.8:Comparisonofaverageoptimizationtime.Notet helogscale. 37 234567123456Number of play operatorslog10(Number of evaluations) Exhaustive SearchDynamic ProgrammingFigure3.9:Comparisonofthenumberofinformationlosseva luations.Notethelogscale. 3.5.6ComparisonwithaTraditionalModelIdenticationAp proach Theeffectivenessoftheproposedoptimalcompressionappr oachisfurthercomparedwithatra- ditionalmodelidenticationscheme(referredtoasfioutpu toptimizationflinthiswork),wherea modelwiththesamecomplexity(sixgeneralizedplayoperat ors)isdeterminedbyminimizingthe outputerrorunderagiventraininginput.Whiletherearein nitenumberofpossiblechoicesfor thetraininginput,athird-orderreversalinputsequence( showninFig.3.10(a))isadoptedasa representativeexample.Anextensivesearchwithinallpos sibleparameterizationsofthe6play operatorsareconductedinMatlabusingthefunction fmincon ,tomatchtheoutputoftheoriginal modelwith30plays.Thescaledweightingfunctionfortheor iginalmodelhasthesameran- domcaseasshowninFig.3.7andthecorrespondingoutputseq uenceisshowninFig.3.10(b). Fig.3.10(c)showsthecorrespondingoutputpredictionerr orundertheEntropySumapproachand theoutputoptimizationapproach.TheRMSEerrorsoftheEnt ropySumapproachandtheoutput optimizationapproachare0.165and0.088,respectively.W hilethatlatterindicatestheoutputop- timizationapproachcoulddeliverbetterperformancefora giveninputsequence,Fig.3.11shows thattheproposedapproachismorerobustinoutputpredicti onwithrespecttoinputvariability.In particular,simulationisrun50timeswithdifferent,rand omlygeneratedinputsequencesandthe correspondingoutputpredictionperformanceisrecorded. Fig.3.11(a)and(b)showsoneexample 38 Table3.6:Identiedparametersoftheenvelopefunctions. a R b R a L b L a D b D d 0.201-11.5780.162-10.2620.029-1.611-3.257 forthe50cases,whileFig.3.11(c)summarizestheerrorsta tisticsoverthe50runs. 3.6ExperimentalResults ThehysteresisbetweenresistanceandtemperatureofaVO 2 lmwasusedasanexampletoshow theeffectivenessoftheproposeddiscretizationscheme. Inordertogetdesirablemodelingperformance,theorigina lGPImodelhas N = 30playop- erators.Similarly,theirplayradiiare r j = j = N , j = 1 ; 2 ; ; N ,respectively,andtheenvelope functionsforthegeneralizedplayoperatorarechosentobe hyperbolic-tangentfunctionsinthe formofEq.(3.12)andEq.(3.13). Thenon-hystereticcomponent D ( v ( t )) ischosentobe D ( v ( t ))= tanh ( a D v ( t )+ b D )+ d : (3.17) Thefullrangeoftemperatureinputis [ 30 ; 90 ] C.ThehysteresisbehaviorshowninFig.3.12 isasymmetricandpartiallysaturated.TheGPImodelisiden tiedbasedontheapproachin[13]. TheeffectivenessoftheGPImodelisveriedinFig.3.12.Ta ble3.6andFig.3.13(a)showthe identiedparametersfortheenvelopefunctionsandthewei ghtsofthegeneralizedplayoperators, respectively.Fig.3.13(b)showstheweightafterscalingb asedontheactualweight.Theweights presentanon-uniformdistribution. 39 010203040-1-0.500.51IndexInput(a) 010203040-15-10-5051015IndexOutput(b) 010203040-0.4-0.200.20.4IndexOutput Estimation Error Entropy SumOutput Optimization(c) Figure3.10:(a)Athird-orderreversalinputsequence.(b) Thecorrespondingoutputsequence.(c) TheoutputpredictionerrorbetweentheEntropySumapproac handoutputoptimizationapproach. 40 020406080100-1-0.500.51IndexInput(a) 020406080100-0.500.5IndexOutput Estimation Error Entropy SumOutput Optimization(b) Largest ErrorRMSE Error00.20.40.60.811.2 Entropy SumOutput Optimization(c) Figure3.11:(a)Arandominputsequence,and(b)thecorresp ondingoutputpredictionerrorper- formance.(c)Theoutputpredictionperformancebasedon50 randominputsequences. 41 30405060708090-4-3.5-3-2.5Temperature (0C)-log10Z (log10W) ExperimentExtended PIFigure3.12:TheperformanceofaGPImodel(30plays)inmode lingoftheresistance-temperature hysteresisinVO 2 . 00.20.40.60.810123Weights (log10W(0C)-1)Play radius(a) 00.20.40.60.8100.050.10.150.2WeightsPlay radius(b) Figure3.13:(a)Identiedweightsforalltheplayoperator softheGPImodel.(b)Thescaled weightsfortheGPIoperators. 3.6.1CompressionPerformance Toconductthecompressionstudies,theidentiedGPImodel istakenasthefioriginalfldistribu- tion.ThenewGPImodelconsistsof M = 5playoperators.Thepartitionschemeunderuniform compressionwas f 0,6,12,18,24,30 g .Fig.3.14showstheplayradiiandweightsforthe M play operators.Uniformcompressionfailstoaccommodatethewe ightingdistribution,withanRMSE of1.10%forthemodelingerror. ThepartitionschemeunderEntropySumwas f 0,7,14,20,23,30 g ,andthatunderEntropyMax 42 00.20.40.60.8100.511.522.5Weights (log10W(0C)-1)Play radius OriginalUniformFigure3.14:ParametersofthecompressedGPImodel:unifor mcompression. Table3.7:Modelingvericationerrorcomparison. SchemeNon-scaleddistributionScaleddistribution Uniform1.76%1.45% EntropySum1.21%1.05% EntropyMax0.87%0.72% was f 0,4,10,17,22,30 g .Fig.3.15(a),(b)showthecompressedplayoperatorradiia ndweights basedonEntropySumandEntropyMax,respectively.Bothsch emesworkedmuchbetterthan theuniformcompressioncase,withRMSEvaluesof0.73%and0 .76%,respectively,whichwere about32%smallerthanthatintheuniformcase. 3.6.2ModelVerication Inordertofurthervalidatetheproposedapproach,arandom lychosentemperatureinputsequence, showninFig.3.16(a),wasappliedtotheVO 2 lm,andthecorrespondingresistanceoutputwas measuredasshowninFig.3.16(b).Predictionsoftheresist anceoutputwereobtainedbasedon thecompressedGPImodelsobtainedwithdifferentschemes. Thecorrespondingestimationerrors werecalculatedandshowninFig.3.16(c)andTable3.7. Themodelvericationexperimentsfurtherdemonstratetha ttheproposedcompressionschemes outperformtheuniformcompression.InTable3.7,themodel ingperformancewithoutconsidering 43 00.20.40.60.8100.511.522.5Weights (log10W(0C)-1)Play radius OriginalEntropy Sum(a) 00.20.40.60.8100.511.522.5Weights (log10W(0C)-1)Play radius OriginalEntropy Max(b) Figure3.15:ParametersofthecompressedGPImodel:(a).En tropySum.(b)EntropyMax. thescalingeffectisalsoincluded[67].Itisevidentthatt heperformanceimproveswithproposed scalingstrategy;theresultsimproveabout10-20%withthe scaling. 44 0100200300400304050607080Time indexInput Temperature (0C)(a) 050100150200250300-4-3.5-3-2.5Index-log10R (log10W)(b) 050100150200250300-0.1-0.0500.05IndexModel error Entropy Sum - UnscaledEntropy Sum - Scaled(c) Figure3.16:(a)Anewtemperatureinputsequenceformodelv erication.(b)Corresponding outputsequence.(c)Theoutputpredictionerrorcomparisi onofEntropySumUnscaledapproach andtheEntropySumScaledapproach. 45 Chapter4 CompressiveSensing-basedPreisach OperatorIdentication Inthischapter,identicationofthePreisachoperatoriss tudiedunderthecompressivesensing frameworkthatrequiresmuchfewermeasurements.Thepropo sedapproachadoptstheDCT transformoftheoutputdatatoobtainasparsevector.Themo delparameterscanbeefciently reconstructedusingtheproposedscheme.Sparsercoefcie ntsareobtainedassumingtheorderof alltheoutputdataareknown,andaconstraintleast-square smethodisfurtheradoptedtoensure thereconstructeddensityvectorcontainsnonegativeelem ents.Theleast-squaresschemehasbeen alsorealized,andiscomparedwiththeproposedapproach.R MSEerrorisadoptedtoexamine theidentiedmodelparametersandmodelestimationperfor mances.Theproposedidentication approachhasbeenshowntohavebetteridenticationperfor mancethantheleast-squaresscheme throughbothsimulationandexperimentsinvolvingaVO 2 -integratedmicroactuator. 4.1ProblemFormulation ConsiderthePreisachoperator G [ v ( ) ; z 0 ]( t ) u ( t )= G [ v ( ) ; z 0 ]( t )= Z P 0 m ( b ; a ) g b ;a [ v ( ) ; z 0 ( b ; a )]( t ) d b d a ; (4.1) 46 where v istheinput, z 0 denotestheinitialcondition, m 0isthedensityfunction, P = f ( b ; a ) : b a g isthe Preisachplane .Thedensityisapproximatedbyapiecewiseconstantfuncti onŒ thedensityvalue m ij isconstantwithincell ( i ; j ) , i = 1 ; 2 ; ; L ; j = 1 ; 2 ; ; L i + 1[12].An exampleofPreisachoperatordensityfunctiondiscretizat ionisshowninFig.4.1.Notethatthe cellsonthediagonalareassumedtohavethesameareaasothe rcellsinthischapter. ab11 m12 m13 m14 m21 m22 m23 m31 m32 m41 mFigure4.1:IllustrationofadiscretizationofthePreisac hdensityfunction,wherethediscretization level L = 4. TheoutputofthePreisachoperator(inthediscrete-timese tting)attime n iswrittenas Ÿ u ( n )= m 0 + L å i = 1 L + 1 i å j = 1 m ij s ij ( n ) ; (4.2) where m 0 isabiasconstant, m ij isthedensityvalueforcell ( i ; j ) ,and s ij ( n ) denotesthe signed areaofthecell ( i ; j ) ,representingtheaccumulativeeffectbroughtbyallthehy steronswithincell ( i ; j ) . Tosimplifythediscussion,writeallthemodelparametersi ntoacolumnvector w = w 1 w 2 w L ( L + 1 ) = 2 m 0 > ; 47 where w k = m ij ; k =( i 1 )( 2 L i + 2 ) = 2 + j 1.Applyaninputsequence v [ n ] ; n = 1 ; 2 ; ; N , withsufcientexcitationandthendeterminethecorrespon ding s ij [ n ] bytrackingtheevolutionof thememorycurveonthePreisachplane.Stack s ij [ n ] intoarowofamatrix: S ( n ; k )= s ij ( n ) ,and S ( n ; L ( L + 1 ) = 2 + 1 )= 1.Theoutputvectorofthemodel, Ÿ u = Ÿ u ( 1 ) Ÿ u ( 2 ) Ÿ u ( N ) > ; canbeexpressedas Ÿ u = Sw : (4.3) Assumethatthemeasuredoutputunder v [ n ] isexpressedas b = b ( 1 ) b ( 2 ) b ( N ) > : Theparameters w canbedeterminedsuchthat k Sw b k 2 isminimizedwiththenon-negative constraintimposedonalldensityvalues[25].Thisapproac hisdenotedasfiLeastSquaresfl. WhentheinputsequenceforthePreisachoperatoridentica tionischosenintheformof dampedoscillations,asshowninFig.4.2asanexamplefor L = 30,theinputsequenceisknown toproducesufcientexcitationforallthedensityvalues[ 12].Theinputlevelsarerightatthecell walls,namely,eachcellwillbeeither1or-1intermsofthes ignedarea.Thisparticularinputse- quenceisdenotedasthefidampedoscillationflinputsequenc e.Thenumberofinputvaluesequals tothenumberofmodelparameters( N = L ( L + 1 ) = 2 + 1).Itcanbeprovedthatthecorresponding S isafull-rank N N matrix.Notethatonlythefidampedoscillationflsequenceis consideredin thischapter,otherinputsequencesalsoexistsuchthatthe corresponding S isafull-rank N N 48 matrix. 0100200300400-1-0.500.51IndexInputFigure4.2:ThefidampedoscillationflsequenceforPreisach operatoridentication( L = 30). Whenonly M < N outputmeasurements y b areavailable y b = b ( n 1 ) b ( n 2 ) b ( n M ) > = Ab ; 1 n 1 ; n 2 ; ; n M N ,where A isan M N matrixwhose M rowsarerandomlychosenfrom N rowsofan N N identitymatrix,then y b = A Sw : (4.4) Thenumberofmeasurementsislessthanthenumberofmodelpa rameters.Thegoalofthiswork istofaithfullyidentifythePreisachoperatorweightings w basedonlimitedmeasurements y b . 49 4.2CompressiveSensingSchemeforIdentifyingthePreisac h Operator 4.2.1OverviewofCompressiveSensing CompressivesensingisanalternativetoNyquist-Shannons amplingtheoryforacquisitionand reconstructionofsparsesignals.Thecompressivesensing theory[68Œ71]statesthatanylength- N signal q thatcanbewellapproximatedusing K coefcientscanbefaithfullyrecoveredfrom M = O ( K log ( N = K )) randomlinearprojectionsofthesignal.Practically,many naturalandman- madesignalsaresparseorcompressibleinthesensethatthe yhavecompactrepresentationsin atransformeddomain,throughdiscreteFouriertransform( DFT)[72],DCT[73],anddiscrete Wavelettransform(DWT)[74],etc.Forexample,in[73],aud iosignalsweretransformedusing one-dimensionalDCT,andthesparseDCTcoefcientswerere constructedusingacompressive sensing-basedalgorithm.Thecompressivesensingtechniq uehasbeensuccessfullyappliedin signalprocessing[74Œ76],networks[77,78],machinelear ning[79],aswellassystemandcontrol [72,80].However,therehasbeenlittlework,ifanyatall,r eportedontheuseofcompressive sensinginhysteresismodelidentication. Thecompressivesensingtheory[68Œ71]statesthat,ifalen gth- N signal p is K -sparse,which meansitcontainsnomorethan K non-zeroentries,thenitispossibletofaithfullyrecover p from its M = O ( K log ( N = K )) ˝ N randomlinearprojections.Inotherwords,consider y = A F p ; (4.5) where y isan M 1vectorofobservations, A isan M N measurementmatrix, F isan N N basis transformmatrix,and p isan N 1 K -sparsesignaltoberecovered.Itisproventhatthesparse 50 signal p canberecoveredifthematrix A F satisesthefollowingrestrictedisometryproperty(RIP) condition[70], ( 1 d S ) k p k 2 2 k A F p k 2 2 ( 1 + d S ) k p k 2 2 ; (4.6) forall S -sparsesignal p ,where d S isthesmallestisometryconstantofmatrix A F .Basedon[68], p canberecoveredefcientlybysolvingthefollowing l 1 minimizationproblem, argmin k p k 1 subjectto y = A F p : (4.7) When A F isarandomlysampledGaussianmatrix,Bernoullimatrix,or Fouriermatrix,ithas showntosatisfytheRIPconditionwithveryhighprobabilit y[69].Practically,manynaturaland man-madesignalsaresparseorcompressibleinthesensetha ttheyhavecompactrepresentations underDFT[72]orDCT[73]. ThemostcommonDCTdenition[73]for1-dimensionalsignal x 1 ; x 2 ; ; x N is X d = N å l = 1 x l cos p N ( l + 1 2 ) d ; (4.8) where d = 1 ; 2 ; ; N .Theresulting N N DCTmatrix Y isorthogonal,whoseelementscanbe writtenas Y dl = cos p N ( l + 1 2 ) d : (4.9) 51 4.2.2CompressiveSensingforthePreisachOperator Anovelcompressivesensing-basedapproachforidentifyin gthePreisachoperatorbasedonpartial outputmeasurements y b isproposed.Incompressivesensing,arandommeasurementm atrixis usuallyadoptedtofaithfullyrecoverthesparsesignals.U nfortunately,compressivesensingcannot beappliedinthePreisachoperatoridenticationdirectly .First,thematrix S inPreisachoperator identicationmustfollowcertainpatternsduetothePreis achplanestructure.Second,sincethe inputsequenceforidenticationmustprovidesufcientex citation,theexibilityofdesigningthe matrix S isfurtherlimited.Finally,thedensityvector w isnotnecessarilysparseinitsoriginal domain. WhentheinputsequenceforthePreisachoperatoridentica tionischosenasfidampedoscil- lationflinputsequenceshowninFig.4.2,thecorresponding S isafull-rank N N matrix.The followingpropositionisproposed. Proposition2. ConsideraPreisachoperator(writtenintheformofEq.(4.4 )withdiscretization levelL,applythefidampedoscillationflinputsequencewith N = L ( L + 1 ) = 2 + 1 elements,by trackingtheevolutionofthememorycurveonthePreisachpl ane,thecorrespondingSisafull- rankN Nmatrix. Proof. Denote S L asthematrix S underthedampedoscillationinputsequenceforidentifyin g thePreisachoperatorwithdiscretizationlevel L .Thepropositioncanbeprovedbymathematical inductionasfollows, 1.When L = 1, rank ( S 1 )= rank 2 6 4 01 11 3 7 5 2 2 = 2;(4.10) 52 2.Assumethatunderfidampedoscillationflinputsequencewi th N = L ( L + 1 ) = 2 + 1elements, rank ( S L )= N ; 3.Underdampedoscillationinputsequencewith Z =( L + 1 )( L + 1 + 1 ) = 2 + 1 = N + L + 1 elements,byreorderingtherowsandcolumnsofcorrespondi ng S L + 1 , rank ( S L + 1 )= rank 0 B B B B B @ 2 6 6 6 6 6 4 s 1 ;1 s 1 ;N + L + 1 : : : : : : : : : s N + L + 1 ;1 s N + L + 1 ;N + L + 1 3 7 7 7 7 7 5 1 C C C C C A Z Z = rank 0 B B B B B B B B B B B B B B B B B B B B B B B B @ 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 s 1 ;1 s 1 ;N + L + 1 : : : : : : : : : s L + 1 ;1 s L + 1 ;N + L + 1 s 2 L + 1 ;1 s 2 L + 1 ;N + L + 1 s L + 2 ;1 s L + 2 ;N + L + 1 s 2 L + 2 ;1 s 2 L + 2 ;N + L + 1 : : : : : : : : : s N + L + 1 ;1 s N + L + 1 ;N + L + 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 1 C C C C C C C C C C C C C C C C C C C C C C C C A Z Z = rank 0 B B B B B B B B B @ 2 6 6 6 6 6 6 6 6 6 4 0 1 : : : : : : : : : : : : : : : 1 1 : : : ( S L ) N N 3 7 7 7 7 7 7 7 7 7 5 1 C C C C C C C C C A Z Z = Z : (4.11) 53 Thethirdequationisobtainedbyrearrangingcertaincolum nsofthepreviousmatrix,more specically,byrearrangingcolumnswithcolumnindex1 ; L + 2 ; ; 1 + ( l 1 )( 2 L l + 4 ) 2 ; ; 1 + L ( L + 3 ) 2 ,where l = 1 ; 2 ; ; L + 1,totheleftofthematrix. Sounderfidampedoscillationflinputsequence, S isafull-rankmatrix. Exploitingthefactthatmanynaturalandman-madesignalsa resparseinatransformedfre- quencydomain,write q = Y b = Y Sw ; (4.12) where Y isan N N DCTmatrix. q isthusan N 1columnvectorcontainstheDCTcoefcients of Sw .Itisfoundthat q isapproximatelysparseinthisworkwhentheallthedensiti esfollow uniformdistributionsindependently,partiallybecauseo fthedampedoscillationstructureof Sw . Since S isafull-rankmatrix,thedensityvalues w canbeexpressedas w = S 1 Y 1 q ; (4.13) andEq.(4.4)canberewrittenas y b = ASS 1 Y 1 q = A Y 1 q : (4.14) When Y ischosenastheDCTmatrix,thesparsesignal q canbereconstructedviacompres- sivesensingalgorithm.Thealgorithm l 1 -MAGICisadoptedtoefcientlyreconstructthesparse signal q usingagenericpath-followingprimal-dualmethod[68].Th edensityparameter w canbe obtainedthroughEq.(4.14)afterwards.Thisapproachisde notedasfiCSfl. 54 Theabovecompressivesensing-basedapproachonlyutilize s M input-outputdatatoreconstruct thedensityvalues w .Iftheorderof N outputdataisalsoknown,denote b order = b ( l 1 ) b ( l 2 ) b ( l N ) > ; suchthat b ( l 1 ) b ( l 2 ) b ( l N ) ,then q order = Y b order = Y S order w : (4.15) Thereconstructionschemein[68]canstillbeadoptedtoide ntifythedensity w . q order isfoundto bemoreapproximatelysparsethan q ,partiallybecause S order w ismonotonicallyincreasing,and hasmoreconcentratedfrequencycomponentsatlowfrequenc iesthanthatof Sw .Itisveriedin simulationandexperimentsthatthisapproachgeneratesbe ttermodelreconstructionandmodel estimationperformance.ThisapproachisdenotedasfiCSOrd erfl. Thedensity w ,asobtainedinEq.(4.14),isnotnecessarilynon-negative .Inordertofacilitate theinversecompensation[9,26]fordynamichystereticsys temsbasedonPreisachoperator,the densityneedstobenon-negative.Basedontheaforemention edfiCSOrderflapproach,aconstraint least-squaresapproachisfurtheradoptedasfollows(deno tedasfiCSOrderNon-negativefl) min w k Y S order w q order k 2 2 where w 0 : (4.16) Whensignal q is K -sparseandthemeasurementsarewithoutanynoise,thedens itymaybe reconstructedwithhighaccuracy.However,insimulationa ndexperiments, q isfoundtobeap- proximatelysparse.Inpracticalapplications,themeasur ementsarewithmeasurementerrorsand noises.Itisthusofpracticalimportancetostudytherecon structionperformanceforapproximately 55 sparsesignalswithmeasurementnoises. Proposition3. ConsideridentifyingthedensitywofaPreisachoperatorus ingcompressivesens- ingalgorithm[68],basedontheexpressionofEq.(4.14).Th emeasurementnoise e satises k A Y S order w y b k 2 e ,andletq s bethetruncatedsignalcorrespondingtotheslargestabso- lutevaluesofq,thenthereconstructionofdensityw obeys k w w k C 1 ;s e + C 2 ;s k q q s k 1 p s ; (4.17) whereC 1 ;s ,andC 2 ;s arepositiveconstants. Proof. k w w k 2 = k S 1 order Y 1 q S 1 order Y 1 q k 2 k S 1 order Y 1 k 2 k q q k 2 k S 1 order Y 1 k 2 C 0 1 ;s e + C 0 2 ;s k q q s k 1 p s C 1 ;s e + C 2 ;s k q q s k 1 p s ; (4.18) wherethesecondinequalityexpressioncanbeprovedbasedo n[71].Whenaspecicinputse- quenceischosen,the2-normof S 1 order Y 1 canalsobecalculated. Notethatthegeneralformulationofcompressivesensingal gorithmrequiresthatthetransfor- mationmatrix Y hasadimensionof N N .Inordertoutilizethecompressivesensingframework toidentifythePreisachoperatorusingEq.(4.15),thedime nsionofthematrix S needstobe N N , whichrequiresthattheinitialinputsequencehas N entriesaswell.Forinputsequencewitha differentnumberofentries,orthatwith N entriesbutthecorrespondingmatrix S isnotinvertable, 56 Eq.(4.15)cannotbeadopteddirectly.Moore-Penrosepseud oinverse[81]couldpotentiallybe utilizedasanapproximationoftheinverseof S .Thisworkconsiderscasesthattheinputsequence has N entriesandthecorresponding S isinvertable. 4.3SimulationResults ConsideraPreisachoperatorwithazerobias m 0 ,anddiscretizationlevel L = 30.Eachofthe 465densityvaluesisgeneratedfollowingauniformdistrib utionontheinterval[0,12].Applythe fidampedoscillationflinputsequenceandobtainthecorresp ondingoutput.Themeasuredoutput issimulatedwithaddednoisethatfollowsauniformdistrib utionontheinterval[-5,5],whichis about0.23%ofthelargestoutputvalue.Inordertoquantita tivelyexaminetherelationshipbetween thereconstructionperformanceandthenumberofmeasureme nts,thecompressivesensing-based identicationapproachiscomparedwithaconstrainedleas t-squaresscheme.Theconstrained least-squaresmethodisrealizedwiththeMatlabcommand lsqnonneg toidentifythevectorof parametersthatmeetsthesignconstraints[25].Foreachnu mberofmeasurementsusedinthe identication,simulationisrun1000times(thuswithdiff erentsetsofdensityvaluesanddifferent setsofchoseninput-outputdata),andtheperformanceisav eragedamongalloftheresults. Fig.4.3(a)showsatypicalexampleofreal q .Itisseenthatthedominantelementsonlycover thelowfrequencies.Thelargestabsolutevalueofelements of q is8,665.8,anditisfoundthatas manyas363elementsarelessthan1%ofthelargestelementso f q .Fig.4.3(b)showsatypical exampleofreal q order .Itisseenthatthedominantelementsalsocoverthelowfreq uencies.The largestabsolutevalueofelementsof q is18,703.4,anditisfoundthatasmanyas415elementsare lessthan1%ofthelargestelementsof q .Itisveriedthat q order ismoreapproximatelysparsethan q .ItisanticipatedandalsoveriedthatfiCSOrderflwouldach ievebettermodelreconstructionand 57 estimationperformancesthanfiCSfl.Fig.4.3(c)showsthere constructionperformancecomparison of q betweenfiCSflandfiCSOrderfl.ItisevidentthatthefiCSOrderfl wouldachievemuchbetter reconstructionperformanceof q order .Onaverage,theRMSEerrorof q order basedonfiCSOrderfl hasabout17.3%oftheRMSEerrorof q usingfiCSfl. Fig.4.4(a)showsthereconstructionperformancecomparis onofdensity w betweenfiLeast Squaresfl,fiCSWithoutDCTfl,fiCSfl,fiCSOrderfl,andfiCSOrderN on-negativefl,wherefiCSWith- outDCTfldirectlyusesthecompressivesensingalgorithmto identifythedensity w basedonEq. (4.12).ItisshownthatexceptthefiCSWithoutDCTfl,allofth eCS-relatedreconstructionap- proachesresultinbetterdensityreconstructionperforma ncesthanthatofthefiLeastSquaresfl. AmongtheseCS-relatedapproaches,fiCSflistheworst,andfiC SOrderNon-negativefliscon- sistentlythebest.Onaverage,theRMSEerrorsofreconstru cteddensitybasedonfiCSOrder Non-negativefl,fiCSOrderfl,andfiCSflare28.1%,34.5%,and73 .1%oftheerrorusingfiLeast Squaresfl,respectively.ThefiCSWithoutDCTflcannotfaithf ullyreconstructthedensitydueto thefactthatthedensityvectorisnotapproximatelysparse ,andthemeasurementmatrixdoesnot followtheRIPcondition,andisnotconsideredinthefollow ingofthemanuscript.Fig.4.4(b) showsthenormalizedmodelingerrorcomparison.Theerrori sobtainedasfollows:rst,calculate theRMSEbetweentheoutputoftheidentiedPreisachoperat orandthatoftheactualoutput,then dividetheRMSEbytheoutputrange.NormalizationoftheRMS Efacilitatestheassessmentof thealgorithmperformance.ItisshownthatfiCSOrderNon-ne gativeflconsistentlyproducesthe smallestmodelingerror,followedbyfiCSOrderfl,fiCSfl,andfi LeastSquaresfl.Onaverage,the RMSEerrorsbasedonfiCSOrderNon-negativefl,fiCSOrderfl,an dfiCSflare18.3%,22.6%,and 47.2%oftheerrorusingfiLeastSquaresfl,respectively. Basedon Proposition3 ,theCS-relatedapproacheshaveboundedreconstructioner rorunder noisymeasurements.Itisshownthatwhenthemagnitudeofth emeasurementnoiseincreases,the 58 0100200300400-500005000IndexTrue q (a) 0100200300400-2-1.5-1-0.50x 104IndexTrue qorder (b) 200250300350400450-50050100150Number of MeasurementsRMSE q reconstruction error CSCS Order(c) Figure4.3:(a)Signal q showingthesparseness;(b)signal q order ismoreapproximatelysparse than q ;(c)thereconstructionperformancecomparisonbasedonfiC SflandfiCSOrderfl. 59 200250300350400450010203040Number of MeasurementsError of density RMSE CSRMSE LSMAX CSMAX LS(a) 20025030035040045000.020.040.060.08Number of MeasurementsModeling error RMSE CSRMSE LSMAX CSMAX LS(b) Figure4.4:(a)Density w reconstructionerrorcomparison;(b)themodelingerrorco mparison. 20025030035040045005101520Number of MeasurementsRMSE density reconstruction error ns=0ns=10ns=20ns=30ns=40ns=50(a) 2002503003504004500.811.21.41.6Number of MeasurementsRun Time (s) CSLS(b) Figure4.5:(a)Densityreconstructionerrorwithvaryingm easurementnoisebasedonfiCSfl;(b) theaverageidenticationrun-timecomparison. 60 magnitudeofthereconstructionerrorbecomeslarger.Deno tethemeasurementnoisefollowsa uniformdistributionontheinterval [ ns 2 ; ns 2 ] ,Fig.4.5(a)showsthecorrespondingRMSEdensity reconstructionerrorwithvaryingamplitudesofmeasureme ntnoise.Fig.4.5(b)showstheaverage run-timebetweentheCS-relatedapproachesandtheleast-s quaresalgorithm.Thecomputations areruninMatlabonacomputerwithIntel(R)Core(TM)i7-260 03.40GHzCPUand4.00GB memory.Itisseenthatwhenthenumberofmeasurementsisinc reasing,fiCSflandfiCSOrderfl aremuchmoreefcientthanfiLeastSquaresfl.Theorderingof theoutputconsumesmuchless timethanthegenericpath-followingprimal-dualmethodto solvethecompressivesensing-based reconstructionalgorithm.fiCSOrderNon-negativeflisslig htlymoretime-consumingthanthe fiLeastSquaresflapproach,whileitcanensuretheidentied densityfunctioncontainsnonegative elements.Onaverage,theaverageidenticationrun-timeo ffiLeastSquaresfl,fiCSfl,fiCSOrderfl, andfiCSOrderNon-negativeflare1.18s,0.28s,0.29s,and1.9 9s,respectively. Tofurthervalidatetheproposedapproach,arandominputse quenceshowninFig.4.6(a)is used,Fig.4.6(b)showsthecorrespondingoutputcorrupted withanoisethathastheaforemen- tioneddistribution.Fig.4.6(c)showsthenormalizedmode lestimationerrorcomparisonunderthe randominput.ItisshownthatfiCSOrderNon-negativeflprodu cesthesmallestmodelingerror, followedbyfiCSOrderfl,fiCSfl,andfiLeastSquaresfl.Onaverag e,theRMSEerrorsofrecon- structeddensitybasedonfiCSOrderNon-negativefl,fiCSOrde rfl,andfiCSflare22.3%,24.4%,and 48.8%oftheerrorusingfiLeastSquaresfl,respectively.For example,whenthenumberofmea- surementsis300,theaverageRMSEerrorsusingfiLeastSquar esfl,fiCSfl,fiCSOrderfl,andfiCS OrderNon-negativeflare0.048,0.024,0.011,and0.010,res pectively. 61 050100150-1-0.500.51IndexInput(a) 050100150-1500-1000-500050010001500IndexOutput(b) 20025030035040045000.020.040.060.080.1Number of MeasurementsError of model estimation RMSE CSRMSE LSMAX CSMAX LS(c) Figure4.6:(a)Arandominputsequenceformodelvalidation ;(b)correspondingoutputunderthe randominputsequencein(a);(c)modelestimationerrorcom parison. 62 4.4ExperimentalResults VO 2 isaninterestingclassofsmartmaterialswithamyriadofmi croactuation,optical,andmem- oryapplications.Itundergoesaninsulator-to-metaltran sition(IMT)ataround68 C,during whichresistance[82],inducedmechanicalstress[26],and opticaltransmittancedemonstratepro- nouncedhysteresis.Theproposedidenticationalgorithm isveriedinexperimentsforidenti- fyingandcharacterizingthehysteresisbetweenthevoltag einputandthedeectionoutputofa VO 2 -integratedmicroactuator. 4.4.1MeasurementSetup TheexperimentalsetupusedisshowninFig.4.7(a).Themicr oactuatorusedinthissetupconsisted ofasilicondioxide(SiO 2 )microcantileverwithpatternedVO 2 lminsidethestructureandapat- ternedmetallayer(Au/Ti)ontop.TheVO 2 lmwasusedastheactiveactuationelementinthe cantilever,whilethemetallayerwasusedtoformtheheatin gelementandthetracesfortheVO 2 resistancecontacts.Themeasurementsystemwasbasedonal aserscatteringtechnique,usingan IRlaser( l =808nm)andapositionsensitivedetector(PSD)totrackthe displacementofthecan- tilever.Achargecoupledevice(CCD)camerawasusedforali gnmentandcalibrationpurposes. Adataacquisitionsystemandeldprogrammablegatearray( DAQ/FPGA)withacomputerinter- facewasusedtoautomatethecontrol/monitorofelectricsi gnals.Thepowerofthesensinglaser (222mW)wascalibratedtobetheminimumpossibletobesense dbythePSDwithoutheatingthe cantileverduetophotonabsorption.Thevoltageoutput(VD )ofthePSDwaslinearlyproportional tothepositionofthelaser.UsingimagescapturedbytheCCD camerathisvoltage(VD)was mappedtothedeectionofthecantilever.Thechipcontaini ngthemicroactuatorwasinsideaside brazepackaging(wire-bonded),whichwasconnectedtotheD AQ/FPGA.Thecurrent I H shown 63 inFig.4.7(b)wasusedtocontrolthecantilever'stemperat urebyJouleheating.Thecurrentwas generatedusingtworesistancesinseries:theheaterresis tanceandanexternalresistance,whose onlypurposewastolimitthemaximumcurrent(12.78mA)that canbeappliedtothesystem.An inputvoltagefromtheDAQ/FPGAandthecomputerinterfacew asusedtogeneratethiscurrent. b) Top View VO - integrated actuactor 2Si Ti/Au SiO 2VO 2RV+-Sensing Laser 2Electrical contacts to VO dSPACE System Computer Interface CCD PSD RV+-IHIHSensing Laser PSD VDVO - integrated actuactor 2a) Side View 2Electrical contact to VO dSPACE System Computer Interface CCD Figure4.7:a)Sideviewschematicforthemeasurementssetu pfordeectionofamicroactuator withanintegratedheater;b)TopviewoftheVO 2 -basedintegratedactuatordevices. 4.4.2IdenticationandVerication AVO 2 -integratedsiliconmicroactuatorissubjecttotwoactuat ioneffectswhenitstemperatureis varied[26].Therstisthephasetransition-inducedstres s,whichmakesthebeambendtowards theVO 2 layerwhenthemicroactuatorisheated.Thedeectiondueto phasetransitioncanbe 64 modeledbyaPreisachoperator.Thesecondeffectisthediff erentialthermalexpansion-induced stress,whichmakesthebeambendingawayfromtheVO 2 layerunderheating.Thelattereffectis modeledwithalinearterm.Asaresult,thehysteresisbetwe enthedeectionandthetemperature isnon-monotonic,andcanbemodeledas Ÿ u ( n )= m 0 + c d v ( n )+ L å i = 1 L + 1 i å j = 1 m ij s ij ( n ) : (4.19) ThemodelexpressedinEq.(4.19)canstillbeexpressedsimi larlyasEq.(4.3)assumingthe discretizationlevel L = 30,where w contains N = L ( L + 1 ) = 2 + 2 = 467elements,andcanbe writtenas w = w 1 w 2 w L ( L + 1 ) = 2 m 0 c d > .Applythefollowinginputsequencetothe system:therst466elementsarethesameasadampedoscilla tionsequenceshowninFig.4.2,the 467thelementoftheinputcanbeanyvalueotherthan0toensu rethatthecorrespondingmatrix S isinvertable. S isa467 467matrix,where S ( n ; k )= s ij ( n ) , S ( n ; L ( L + 1 ) = 2 + 1 )= 1,and S ( n ; L ( L + 1 ) = 2 + 2 )= v [ n ] .Fig.4.8(a)showsthenon-monotonichysteresisbehaviorb etweenthe voltageinputandthedeectionoutput.Fig.4.8(b)showsth ecorrespondingdensityfunction(true density)identiedbasedonthe467measurementsshowninFi g.4.8(a). Itcanbeprovedthatbyapplyingtheaforementionedinputse quence,therankofthecorre- sponding S is467.Insteadofutilizingallofthe467correspondingout putdeectionmeasure- ments,apartoftheoutputmeasurementswererandomlychose nforidentication.Foreach numberofmeasurementsusedinthecompressivesensingalgo rithmsandleast-squaresscheme, reconstructionalgorithmsarerun1000times(thuswithdif ferentsetsofchoseninput-outputdata), andtheperformanceisaveragedamongallofthesimulations . Whenthenumberofmeasurementsusedforidenticationis30 0,Fig.4.9(a)-(d)showtypi- caldensityfunctionreconstructionerrorperformances.I tiscalculatedthattheRMSEerrorusing 65 44.55-100102030Deflection output (mm)Voltage input (V)(a) 3.804.334.865.403.804.334.865.4000.511.5a (V)True Densityb (V)Density function (mm/V2)(b) Figure4.8:(a)Input-outputdataforidentifyingPreisach operator( L = 30);(b)truedensityfunc- tionidentiedbasedon467measurements. fiLeastSquaresfl,fiCSfl,fiCSOrderfl,andfiCSOrderNon-negati veflare0 : 122µm/V 2 ,0 : 116µm/V 2 , 0 : 096µm/V 2 ,and0 : 059µm/V 2 ,respectively. Fig.4.10showsthenormalizedmodelingerrorcomparison.I tisshownthatfiCSOrderNon- negativeflconsistentlyproducesthesmallestmodelingerr or,followedbyfiCSOrderfl,fiCSfl,and fiLeastSquaresfl.Onaverage,theRMSEerrorsbasedonfiCSOrd erNon-negativefl,fiCSOrderfl, andfiCSflare13.4%,25.3%,and76.8%oftheerrorusingfiLeast Squaresfl,respectively. Theeffectivenessofthecompressivesensing-basedidenti cationisfurtherexaminedbycom- paringthemodelestimationperformanceunderarandominpu tshowninFig.4.11(a).Themea- sureddeectionoutputisshowninFig.4.11(b).Fig.4.11(c )showsthemodelestimationerrors underthemodelsidentiedwiththecompressivesensingsch emesandtheleast-squaresscheme, respectively.Onaverage,theRMSEerrorsofreconstructed densitybasedonfiCSOrderNon- negativefl,fiCSOrderfl,andfiCSflare72.9%,78.4%,and84.5%o ftheerrorusingfiLeastSquaresfl, respectively.Forexample,whenthenumberofmeasurements usedforidenticationis300,the averageRMSEerrorsusingfiLeastSquaresfl,fiCSfl,fiCSOrderfl ,andfiCSOrderNon-negativeflare 66 3.804.334.865.403.804.334.865.40-1012a (V)Least-Squaresb (V)Density function error (mm/V2)(a) 3.804.334.865.403.804.334.865.40-101a (V)CSb (V)Density function error (mm/V2)(b) 3.804.334.865.403.804.334.865.40-0.500.5a (V)CS Orderb (V)Density function error (mm/V2)(c) 3.804.334.865.403.804.334.865.40-0.500.5a (V)CS Order Non-negativeb (V)Density function error (mm/V2)(d) Figure4.9:Densityreconstructionerrorcomparisonbased on(a)LeastSquares;(b)CS;(c)CS Order;(d)CSOrderNon-negativeapproaches. 67 20025030035040045000.511.5Number of MeasurementsRMSE Modeling error (mm) Least SquaresCSCS OrderCS Order Non-negativeFigure4.10:TheaverageRMSEmodelingerrorcomparison. 1 : 010µm,0 : 891µm,0 : 831µm,and0 : 778µm,respectively. 68 05010015044.55IndexVoltage input (V)(a) 050100150-100102030IndexDeflection output (mm)(b) 050100150-2-1012IndexModel estimation error (mm) LSCS(c) Figure4.11:Densityreconstructionerrorbasedon(a)aran dominputsequenceformodelvalida- tion;(b)outputoftherandominputsequencein(a);(c)mode lestimationerrors. 69 Chapter5 ModelingandInverseCompensationof Non-monotonicHysteresisbasedonthe PreisachOperator Inthischapter,thesystematicstudiesonthemodelingandi nversecompensationofnon-monotonic hysteresisexhibitedbyVO 2 -coatedmicroactuatorsarepresented.First,aphysics-mo tivatedmodel thataccountsforthetwo(opposite)actuationmechanismsi spresented.Therstmechanismisthe stressresultingfromstructuralchangesinVO 2 ,whichismodeledwithamonotonicPreisachoper- ator.Thesecondmechanismisthedifferentialthermalexpa nsioneffect.Sincethethermalexpan- sioncoefcient(TEC)ofVO 2 dependsonthephasemixtureofthematerial,alinearfuncti onof thetemperatureistakentoefcientlymodelthephasefract ionofVO 2 ,whichresultsinaquadratic operatorforthethermalexpansion-inducedactuation.The parametersofthemodelareidentied. Second,anefcientinversecompensationschemeisdevelop edfortheproposednon-monotonic hysteresismodelbyadaptingtheschemeusedin[9]foraPrei sachoperatorwithnonnegative, piecewiseconstantdensityfunction.Theeffectivenessof themodelandtheinversecompensation schemeisdemonstratedinexperiments,withcomparisontot wootherapproaches,onebasedona Preisachoperatorwithasigneddensityfunctionandtheoth erbasedonapolynomialmodel. 70 5.1ExperimentalCharacterizationof VO 2 -coatedMicroactu- ators 5.1.1MaterialPreparationandExperimentalSetup A172nmthickVO 2 layerwasdepositedbypulsedlaserdepositionona300 m mlongsilicon cantilever(MikroMaschCSC12)withwidthandthicknessof3 5 m mand1 m m,respectively.The depositionwasconductedinsideavacuumchamber.Thedepos itionfollowedasimilarprocedure asinpreviousexperiments[36],whereakryptonuorideexc imerlaser(LambdaPhysikLPX200, l =248nm)wasfocusedonarotatingmetallicvanadiumtargetw itha10Hzrepetitionrate.The backgroundpressurewas10 6 Torrandthroughoutthedepositionwaskeptat20mTorrwithg as owsof10(argon)and15(oxygen)standardcubiccentimeter perminute(sccm). Fig.1.1showstwosuperimposedscanningelectronmicrosco py(SEM)picturesoftheprepared VO 2 -coatedcantilever,whenthesubstratetemperaturewas30 Cand90 C,respectively.VO 2 is inpureM 1 andRphasesatthosetwotemperatures.Atotaltipdisplacem entchangeofabout70 m misobservedinFig.1.1,illustratingthelargebendingthe microactuatoriscapableofgenerating. Theconsiderableamountofinitialcurvatureatroomtemper atureisduetotheresidualstressafter deposition.Sincethechangeofcurvatureisofmoreinteres t,thecurvaturechangefromtheinitial curvatureistakenastheoutput. Inordertoexperimentallymeasurethetipdeectionasafun ctionoftemperature,thesetup showninFig.5.1wasused,whichwassimilartotheoneusedbe fore[36].Here,themicro- cantileverwasgluedwithahighlythermalconductivesilve rpainttoaglasssubstratethatwas directlyincontactwithaPeltierheater.Theheaterwascon trolledinclosedloopwithacommer- ciallyavailablebenchtoptemperaturecontroller(Thorla bs,TED-4015)connectedtoatemperature 71 sensor(AD592),withaprecisionintemperaturecontrolof 0 : 1 C.Acustom-madecurrentcon- trollercircuitwasusedtopoweraninfraredlaser( l =808nm)withamaximumpowerof20mW. Thelaserspotwasfocusedonthetipofthecantileverandthe reectedlaserlightwascaptured byaone-dimensionalpositionsensitivedetector(PSD)(Ha mamatsuS3270).ThePSDoutputsa voltageproportionaltothepositionofthereectedlasers potonitsactivearea.Bendingofthe VO 2 -coatedcantileverproducesanangulardisplacementonthe reectedlaserspot,whichchanges theoutputvoltageofthePSD.Thelaserintensitywaskeptat thelowestdetectablebythePSD,in ordertoobtaingoodsignal-to-noiseratiowhileminimizin gheatingofthecantileverbythelaser. TheoutputvoltagefromthePSDwasmeasuredwithananalogin putmodule(NI9201),which wasattachedtoanembeddedreal-timecontroller(NIcRIO90 75),andaLabViewprogramwas createdinordertoautomatethedeectionmeasurements. Sensing LaserPSDMicro-cantilever Peltier Heater AD592 CCDOptical Filter TED4015cRIOFigure5.1:Setupusedformeasuringthecantilevertipdee ctionasafunctionoftemperature. Themeasuredtipdeection D z wasconvertedtothecurvature k usingthegeometryillustrated inFig.5.2.Theradiusofcurvature, r = 1 = k ,isrelatedto D z via: 72 D z = sin ( d 2 ) AB = 2 r sin 2 ( L 2 r ) ; (5.1) where d = L r .Forthemicroactuatorstudiedinthiswork,thesmallangle approximationtypically used( D z ˇ L 2 2 r )willnotbevalidbecauseofthelargebending,andthetrans cendentalequation (5.1)isnumericallysolvedforthecurvature. zDdLrBAFigure5.2:Illustrationofthegeometricrelationshipbet weenthecurvatureandthetipdeection ofabentcantilever. 5.1.2CharacterizationofNon-monotonicHysteresis Acreeptestwasconductedandnoobviouscreepwasfound.The temperaturewasvariedbya stepandthenuntilconstant,whilethedeectionwasmeasur edforthewholeprocess.AsFig.5.3 shows,thedeectionunderunchangedtemperaturevaluesva riedfrom[67.378,67.667] m mand [15.082,15.164] m m,respectively.Theseminutevariationsaremainlyattrib utedtoerrorintem- peraturecontrol(accuracy 0 : 1 C),hencecreepisnotconsidered. Asetofexperimentswereconductedtoobtainthecurvatureo ftheVO 2 microactuatorasa functionoftemperature.Thetemperaturerangewaschosent obefrom21 Cto84 C,tofully coverthephasetransitionregime.Thetemperatureprolei ntimefollowedapatternofdamped oscillations(notshown),toprovidesufcientlyrichexci tationfortheidenticationofthePreisach 73 051015-10010203040506070Time (s)Deflection (mm) HeatingCoolingFigure5.3:Deectionasafunctionoftimethroughheatinga ndcoolingtemperaturesteps.There isnoobservablecreep. hysteresismodel.Fig.5.4(a)showsthemeasurednestedhys teresisloopsbetweentheactuator curvatureandtemperature.Noticethattheapparentphaset ransitiontemperatureisshiftedfrom thetypicalvalueof68 C.Thisisattributedtotheheatingeffectbythedeection- measuringlaser; recallthattherecordedtemperaturewasonlyforthePeltie rheaterlocatedunderneaththesample. Inthiswork,theheatcontributionfromthemeasurementlas erisconsideredtobeconstant(which isareasonableassumption),andthusthetemperatureofthe Peltierheateristakenastheinput. Non-monotonichysteresiscanbeclearlyobservedinFig.5. 4(a).Asthetemperatureisin- creased,thecurvaturerstdecreasesslightly,thenincre asesabruptly,andnallydecreasesslightly againwhenthetemperatureissufcientlyhigh.Ananalogou strendholdstruewhenthetemper- atureisdecreased.Thenon-monotonicbehaviorcanbeexpla inedbytwocompetingactuation mechanisms.Ononehand,changesinthecrystallinestructu resduringtheM 1 ! Rphasetransi- tionresultinmicrocantileverbendingtowardtheVO 2 layer.Vanadiumionsarereorderedduring thephasetransition,whereoneunitcellintheM 1 phasecorrespondstotwounitcellsintheR phase.Thecrystallineplaneparalleltothesubstratechan gesfrom(011) M 1 intheM 1 phaseto 74 (110) R intheRphase.Fromthelatticeparameters[83],itiscalcul atedthatthecrystallographic planeofVO 2 thatisparalleltothecantileversurfaceforthiscase (( 011 ) M ) decreasesitsarea by1.7%(onheating),whichgeneratesastrainofapproximat ely-0.083[36],causingadrastic bendingtowardstheVO 2 layerside.Ontheotherhand,sincethethermalexpansionco efcient (TEC)ofVO 2 forboththeM 1 phaseandtheRphase[83]arelargerthanthatofsilicon[84] ,the differentialthermalexpansion-inducedstressesresulti nanoppositebendingeffectfromthatof thephasetransitioneffect. Thereareseveraladditionalinterestingobservationsfro mFig.5.4(a).First,thecurvature- temperaturerelationshipishystereticonlyintheinterme diatetemperatureregime,andbecomesa single-valuedfunctionatboththelowandhightemperature ends.Thisprovidessupportforthe twoactuationeffectsdiscussedearlier;phasetransition dominatestheintermediatetemperature region,whileattheloworhightemperatureends,VO 2 isinasinglephase(M 1 orRphase,respec- tively),andthedifferentialthermalexpansiontakesdomi nance.Second,theslopeofcurvature versustemperatureatthelow-temperatureendisdifferent fromthatatthehigh-temperatureend, suggestingthattheTECofVO 2 changeswiththematerialphase.Thisisconsistentwithwha tcan befoundinliterature[83]. Extensiveexperimentsareconductedtocharacterizethere peatabilityoftheactuationbehavior, andthehysteresisloopsmeasuredondifferentdays(withth esametemperatureinputsequence) arefoundtobenearlyidentical(showninFig.1.2). 75 5.2Non-monotonicHysteresisModel 5.2.1ActuationEffectduetoPhaseTransition Theproposedmodelforthecurvatureoutput k oftheVO 2 -coatedcantilevercomprisesthecon- tribution k P duetothephasetransitionandthecontribution k E duetothedifferentialthermal expansion.Thephasetransitioncontributionismonotonic iallyhystereticwithrespecttothetem- perature T ,thuswillbemodeledbyaPreisachoperator[27,85]withnon -negativedensityfunction m : k P ( t )= G [ T ( ) ; z 0 ]( t ) = c 0 + Z P 0 m ( b ; a ) g b ;a [ T ( ) ; z 0 ( b ; a )]( t ) d b d a ; (5.2) here c 0 issomeconstantbias, T ( ) denotesthetemperaturehistory, T ( t ) ,0 t t, P 0 iscalled thePreisachplane P 0 4 = f ( b ; a ) : T min b a T max g ,where [ T min ; T max ] denotesthetemper- aturerangeforphasetransition,andnally, g b ;a denotesthebasichystereticunit(hysteron):fora pairofthresholds ( b ; a ) andaninitialcondition z 0 ( b ; a ) 2f 1 ; 1 g ,theoutputofthehysteronis denedas: u ( t )= g b ;a [ T ( ) ; z 0 ( b ; a )]= 8 > > > > > < > > > > > : + 1if T ( t ) > a 1if T ( t ) < b u ( t ) if b T ( t ) a ; (5.3) where T ( ) isthetemperatureinputhistory T ( t ) ,0 t t ,and u ( t )= lim e > 0 ;e ! 0 u ( t e ) . Notethat P 0 canbedividedintotworegionsaccordingtotheoutputsofhy sterons,andthe boundaryofthetworegions(calledmemorycurveanddenoted y )representsequivalentlythestate 76 andthusdeterminestheoutputofthePreisachoperator.For thisreason,theinitialstatefunction z 0 canbereplacedbyaninitialmemorycurveinthePreisachpla ne. 5.2.2DifferentialThermalExpansionEffect Whenthetemperatureincreases(decreases,resp.),amater ialtypicallyexpands(shrinks,resp.). Foratwo-layerbeam,thedifferenceinthethermalexpansio nofindividuallayersresultsinbend- ing.TheVO 2 -coatedsiliconcantilevercurvature k E duetodifferentialthermalexpansionata temperature T canbederivedfollowingstandardanalysis[86]: k E = 6 ( 1 + m ) 2 ( C VO 2 C Si )( T T 0 ) h 3 ( 1 + m ) 2 +( 1 + mw )( m 2 + 1 mw ) ; (5.4) where h isthetotalthicknessofthebeam, m istheratiooftheVO 2 layerthicknesstothatofthe siliconlayer, w istheratioofthemodulusofelasticityoftheVO 2 layertothatofthesiliconlayer, C VO 2 and C Si aretheTECsoftheVO 2 andsilicon,respectively,and T 0 istheroomtemperature (20 C).In(5.4),itisdenedthat k E ispositivewhenthebeambendstowardtheVO 2 layer. Asmentionedearlier,theTECofVO 2 intheM 1 phaseisdifferentfromthatintheRphase. Sincethephasetransitionspansthroughthetemperaturera nge [ T min ; T max ] ,bothphasescoexist withinthattemperaturerange.If C M 1 and C R aretheTECsofVO 2 intheM 1 andRphases, respectively,and q ( T ) isthematerialfractionoftheRphaseataparticulartemper ature T ,then theeffectiveTECofVO 2 duringthephasetransitioncanberepresentedwithrespect to T as: C VO 2 =( 1 q ( T )) C M 1 + q ( T ) C R : (5.5) Ingeneral,theRphasefraction q ishystereticwithrespecttothetemperature T .Tomakethe problemtractable, q isapproximatedbyalinearfunctionof T ,whichissupportedbyexperimental 77 observations[87]: q = 8 > > > > > < > > > > > : 0 ; if T < T min T T min T max T min ; if T min T T max 1 ; if T > T max : (5.6) Combining(5.4)and(5.5),werewrite k E as: k E = ( k 0 ( 1 q ( T ))+ k 1 q ( T ))( T T 0 ) ; (5.7) where k 0 = 6 ( 1 + m ) 2 ( C M 1 C Si ) h 3 ( 1 + m ) 2 +( 1 + mw )( m 2 + 1 mw ) and k 1 = 6 ( 1 + m ) 2 ( C R C Si ) h 3 ( 1 + m ) 2 +( 1 + mw )( m 2 + 1 mw ) .Since C M 1 < C R [83], k 1 > k 0 .Withtheapproximation(5.6)for q ( T ) ,thedifferentialthermalexpansion- inducedcurvaturehasaquadraticdependenceon T . Byadding(5.2)and(5.7),weobtainthetotalcurvaturewith anewhysteresisoperator W : k ( t )= k P ( t )+ k E ( t )= W [ T ( ) ; z 0 ]( t ) 4 = c 0 + G [ T ( ) ; z 0 ]( t ) ( k 0 ( 1 q ( T ( t ))))( T ( t ) T 0 ) + k 1 q ( T ( t ))( T ( t ) T 0 ) : (5.8) Notethatfor W ,onlythecontribution k P ismemory-dependent,so W sharesthesamestate(or memorycurve)as G .Inparticular,theywillsharethesameinitialmemorycurv e. 78 5.3ModelIdenticationandValidation 5.3.1ParameterIdentication Theparametersofthemodel(5.8)includethedensityfuncti on m ofthePreisachoperator,andthe constants c p , k 0 ,and k 1 .FortheidenticationofaPreisachdensityfunction,adis cretizationstep istypicallyinvolved.Thediscretizationschemeadoptedi nthisworkapproximatesthedensityby apiecewiseconstantfunction,wherethedensityvalueisco nstantwithineachlatticecellbutvaries fromcelltocell[6]. Theinputrangeisdiscretizeduniformlyinto M levels,whichresultsin M ( M + 1 ) = 2cells, leadingtoatotalof K = M ( M + 1 ) = 2weightparameters.Theactualoperatingrangefortheinpu t, [ T 0 min ; T 0 max ] ,isconsideredlargerthanthephasetransitionregion [ T min ; T max ] .Inotherwords, T 0 min < T min < T max < T 0 max . Inadiscretetimesetting,thecontribution k P [ n ] attime n is: k P [ n ]= c p + M å i = 1 M + 1 i å j = 1 m ij s ij [ n ] ; (5.9) where s ij [ n ] representsthe signed areaofcell ( i ; j ) , m ij representsthedensityofcell ( i ; j ) .Note thatthesignedareaofeachcellisdenedastheareaoccupie dbyhysteronswithoutput + 1minus thatoccupiedbyhysteronswithoutput 1.Foreaseofpresentation,thecells ( i ; j ) areordered withasingleindex l = 1 ; 2 ; ; K ,andthedensityandsignedareaofthe l -thcellaredenoted(with abuseofnotation)by m l and s l ,respectively.Eq.(5.9)isrewrittenas: k P [ n ]= c 0 + K å l = 1 m l s l [ n ] : (5.10) Whentheinput T [ n ] isin [ T 0 min ; T min ] ,allhysteronsattainthevalueof 1andthePreisach 79 operatorisatthenegativesaturation, k P [ n ]= c 0 k 0 ,where k 0 = å K l = 1 m l .Similarly,positive saturationisreachedwhenthetemperature T [ n ] iswithin [ T max ; T 0 max ] . Ontheotherhand,thecontribution k E [ n ] is: k E [ n ]= ( k 0 ( 1 q ( T [ n ]))+ k 1 q ( T [ n ]))( T [ n ] T 0 ) : (5.11) Combining(5.10)and(5.11),weobtainthetotalcurvatureo utputas: k [ n ]= c p + K å l = 1 m l s l [ n ] ( k 0 ( 1 q ( T [ n ]))+ k 1 q ( T [ n ]))( T [ n ] T 0 ) : (5.12) Preisachdensityfunctionsareassumedtobenonnegative, m l 0.Inaddition, k 0 and k 1 are positivefromtheirphysicalmeanings.Finally, c p > 0sinceVO 2 -coatedmicrocantilevershave positivecurvaturebias.Aconstrainedleast-squaresmeth od,realizedwiththeMatlabcommand lsqnonneg ,isutilizedtoidentifythevectorofparameters, [ m 1 m 2 m 3 m K c p k 0 k 1 ] T , thatmeetsthesignconstraints. 5.3.2ExperimentalResults Toeffectivelyidentifythemodelparameters,theinputnee dstoprovidesufcientexcitationfor allcellsofthePreisachoperator.Onetypeofsuchinputseq uencestakestheformofdamped oscillations,whichproducesnestedhysteresisloopsandi sadoptedinthiswork.Basedonem- piricalknowledge,thetemperatureoftheclosed-loop-con trolledPeltierdevicecansettlearound asettemperaturewithinabout3s.Awaittimeof8swaschosen betweentemperaturesetpoints toensurethatthethermalsteadystatehasbeenreached.Whi letheexperimentwillremaintobe quasi-staticifthewaitingtimeislongerthan8s,itisnota dvisabletomakeitmuchshorter. 80 20304050607080050010001500Temperature (°C)Curvature change (m-1) Proposed modelExperiment(a) 050010001500200025003000-40-2002040IndexModeling error (m-1)(b) Figure5.4:Proposedmodel.(a):Measurednon-monotoniccu rvature-temperaturehysteresisand thatbasedontheproposedmodel.(b):Modelingerrorforthe entiretemperaturesequence. Thefullrangeoftemperatureinput [ T 0 min ; T 0 max ] is [ 21 ; 84 ] C.FromFig.5.4(a),thephase transitionregion [ T min ; T max ] isdeterminedtobe [ 30 ; 70 ] C.Thelevelofdiscretization M for thePreisachplaneischosentobe20.Fig.5.4(a)comparesth emeasuredhysteresisloopsand thosebasedontheidentiedmodel,andFig.5.4(b)showsthe correspondingmodelingerrorfor theentiretemperaturesequence,whichismostlyboundedby 30m 1 ,comparedwiththetotal curvaturechangerange[ 104,1846]m 1 .Fig.5.5showstheidentieddensityfunctionofthe Preisachoperator. c p , k 0 ,and k 1 havebeenidentiedtobe1026.7m 1 ,2.8m 1 K 1 ,and4.3 m 1 K 1 ,respectively. Forcomparisonpurposes,twoadditionalmodelsforthenon- monotonichysteresisareconsid- 81 405060707060504030051015b (0C)a (0C)Density Function (m-1(0C)-2)Figure5.5:IdentiedPreisachdensityfunctionforthepro posedmodel. ered:aPreisachoperatorwithasigneddensityfunction,an danonlinearsingle-valuedfunction representedbyahigh-degreepolynomial. ThesignedPreisachoperatorisidentied,wheretherearen osignconstraintsplacedonthe Preisachdensities.Thelevelofdiscretization M forthePreisachplaneisalsochosentobe20, butnowisfortheentireinputrange [ T 0 min ; T 0 max ] = [ 21 ; 84 ] C.Fig.5.6showsthemodelingerror, whichislargerthantheproposedmodel.Itisinterestingto noticethatthenegativedensityvalues ofthesignedPreisachoperatorareprimarilylocatedatthe b = a line(seeFig.5.7).Thisprovides supportfortheproposedmodel;afterall,thenon-hysteret ic,negativecomponentintheproposed modelthataccountsfordifferentialthermalexpansioncou ldberepresentedasnegativedensities locatedonthe a = b line. Forthesingle-valuednonlinearapproximation,apolynomi alofdegree12ischosen,thecoef- cientsofwhicharefoundthroughapolynomialttingbetwe encurvaturemeasurementsandthe modelpredictions.Itisnotsurprisingthatthismodelresu ltsinthelargestmodelingerroramong thethreemodelsexplored,whichisaround500m 1 (Fig.5.8).Thismodelfailstoaccountfor thehysteresiseffect. 82 050010001500200025003000-50050IndexModeling error (m-1)Figure5.6:ModelingerrorwiththesignedPreisachoperato rfortheentiretemperaturesequence. 36.7552.568.25842136.7552.568.2584-20-10010b (0C)a (0C)- Density Function (m-1(0C)-2)Figure5.7:Thenegativeoftheidentieddensityvaluesfor thesignedPreisachoperator.The negativeistakenheresothatthenegativeelementsofthede nsityfunctioncanbeseen(ontop); thepositiveelementsarenowippedtothebottomoftheplan e,whicharenotvisiblehere. 050010001500200025003000-600-400-2000200400600IndexModeling error (m-1)Figure5.8:Modelingerrorwithapolynomialmodelfortheen tiretemperaturesequence. 83 020406080-4-3-2-101234IndexInput sequence(a) 0100200300400500600-600-400-2000200400600IndexCurvature error (m-1) Proposed modelSigned PreisachPolynomial(b) Figure5.9:(a):Arandomlychosentemperatureinputsequen ceformodelvalidation.(b):Errors inpredictionsbydifferentmodelsundertheinputsequence . Inordertofurthervalidatetheproposedmodel,theVO 2 actuatorissubjectedtoarandomly chosentemperaturesequence,showninFig.5.9(a),andtheo utputcurvaturedata(notshown)is thencollected.Fig.5.9(b)showsthemodelpredictionerro rs.Themaximumpredictionerrors are108.1,108.8,and478.1m 1 fortheproposedmodel,thesignedPreisachoperator,andth e polynomialmodel,respectively.Itisclearthat,withaner roroflessthan5.6%overthecurvature rangeof1950m 1 ,theproposedmodeliseffectiveincapturingthecomplex,h ystereticbehavior oftheVO 2 actuator.WhilethesignedPreisachoperatorisabletoprov ideamodelingaccuracy comparablewiththeproposedmodel,itisnotamenabletoef cientinversionandwillnotbe pursuedfurther. 5.4InverseCompensation Therehavebeenanumberofinversion-relatedapproachesfo rhysteresiscompensationreported intheliterature.Forexample,thedirectinversePreisach operatormethod[88]aimstoidentify aPreisachoperatorastheinverseoperatorbasedonempiric aloutput-inputdata.Theempirical 84 temperaturevs.curvaturehysteresisloopsforVO 2 actuators(withthehorizontal/verticalaxesin Fig.5.4(a))exhibitsharpslopechanges,makingitdifcul ttoapproximatebyaPreisachoper- atorevenwithaprohibitivelynediscretizationscheme.I terativeLearningControl[89,90]has proveneffectiveincompensatingbothhysteresisanddynam icswithrelativelylowrequirement onmodelingaccuracy,butitonlyappliestoperiodicrefere ncesandrequiressensoryfeedbackfor thelearningprocess.Whiletheproposedinversionalgorit hmisnoncausal,onlythenextdesired outputandtheinputhistoryarerequiredtoobtaintheinput . WeproposetocompensatethehysteresiseffectinVO 2 byconstructingaproperlydened inverseofthemodelpresentedinearliersections.Thenon- monotonicnatureoftheproposed hysteresismodelpresentsseveralchallengesintheinvers ionproblem.Inthecontinuous-time setting,thereisgreatdifcultyinestablishingtheexist enceand/ortheuniquenessofacontinuous inputfunctiongivenadesired,continuousoutputfunction .Inthediscrete-timesetting,whichis thecaseofpracticalinterest,theconceptoftime-continu ityisnolongerrelevantsincethedesired outputfunctionisgivenasasequenceofvaluesfortheopera tortoachieve,andinputinterpolation betweenthesamplingtimesistypicallyusedtorealizeaqua si-continuousoutputforthephysical system.Foramonotonichysteresisoperator,foranydesire doutputvalue(withintheoutputrange) y d [ n + 1 ] atnexttimeinstant n + 1,therealwaysexistsaninputvalue v [ n + 1 ] for n + 1,suchthat, iftheinputvariesmonotonicallyfromthecurrentvalue v [ n ] to v [ n + 1 ] ,theoutput y wouldalso changemonotonicallyfromitscurrentvalue y [ n ] to y [ n + 1 ] ,whichisequalorcloseto y d [ n + 1 ] . This,however,isnolongertrueforanon-monotonichystere sisoperatorastheoneconsideredin thisworkŒOnemaynotndasingleinputvalue,monotonicint erpolationtowhichwouldresultin thedesiredoutputvalue.Therefore,a sequence ofinputvaluesneedstobefoundtoachieveagiven desiredoutputvalue;withoutproperconstraints,thelatt erproblemwouldadmitinnitelymany solutions.Aconstraintonthisproblemisimposedtominimi zetheimplementationcomplexity 85 andassureproperoutputbehavior:thenumberofelementsin thecomputedinputsequenceis minimal.Specically,fortheproposednon-monotonichyst eresismodel,theinversionproblemis formulatedasfollows:giventhecurrentinitialmemorycur ve y ( 0 ) fortheoperator W ,withthe associatedtemperatureinput T ( 0 ) ,operatoroutput k ( 0 ) = W [ T ( 0 ) ; y ( 0 ) ] ,andatargetoutputvalue ¯ k ,ndanewinputsequence ¯ T withminimalnumberofelements,suchthatthenalvalueof W [ ¯ T ; y ( 0 ) ] isequalto ¯ k . Theproposedinversionalgorithmisadaptedfromtheonein[ 9]fortheinversionofamono- tonicPreisachoperatorwithapiecewiseconstantdensityf unction.Thealgorithmin[9]exploits themonotonicityoftheoperatorandthepiecewiseconstant natureofthedensity,andndsthe desiredinputbymonotonicallyvaryingtheinputiterative ly.Assumingthattheinputisbeing increased,withthevalueofthe k -thiterationbeing T ( k ) andthecorrespondingmemorycurvebe- ing y ( k ) ,anotherincrementof d ( d min f d ( k ) 1 ; d ( k ) 2 g ) intheinputwouldresultinthefollowing changeintheoutputofthePreisachoperator: G [ T ( k ) + d ; y ( k ) ] G [ T ( k ) ; y ( k ) ]= a ( k ) 2 d 2 + a ( k ) 1 d ; (5.13) where d ( k ) 1 > 0issuchthat T ( k ) + d ( k ) 1 wouldequalthenextdiscreteinputlevel,and d ( k ) 2 > 0is suchthat T ( k ) + d ( k ) 2 woulderasethenextcornerofthememorycurve;seeFig.5.10 forillustration. In(5.13), a k 1 and a ( k ) 2 arenonnegativeconstantsassociatedwith y ( k ) andthedensityvalues. Thecoreideain[9]forinvertingthePreisachoperatoristh at,if d min f d ( k ) 1 ; d ( k ) 2 g canbe foundthatsolves a ( k ) 2 d 2 + a ( k ) 1 d = ¯ y G [ T ( k ) ; y ( k ) ] ; (5.14) where¯ y isthedesiredoutput,thentherequiredinputisobtainedas T ( k ) + d ;otherwise,let 86 T ( k + 1 ) = T ( k ) + min f d ( k ) 1 ; d ( k ) 2 g andcontinuetheiteration. max Tmin T'y() 1kd() 2kd() kT() kyabFigure5.10:Illustrationofthevariables d ( k ) 1 and d ( k ) 2 usedininversion. Sincethecurvatureoutput k startstodecrease(increase,resp.)when T > T max ( T < T min , resp.)because k P becomespositively(negatively,resp.)saturated,themax imumrangeofcurva- tureoutputcanbeachievedbyrestrictingthetemperature T totherange [ T min ; T max ] . Thecurvaturecontributionfromdifferentialthermalexpa nsionisgivenby: k E = Q [ T ] 4 = k 0 ( k 1 k 0 ) T min T max T min T 0 T k 0 T 0 + T min T max T min ( k 1 k 0 ) T 2 k 1 k 0 T max T min : (5.15) Consequently,thechangein k E foranincrement d at T ( k ) isobtainedas: Q [ T ( k ) + d ] Q [ T ( k ) ]= c 2 d 2 c ( k ) 1 d ; (5.16) 87 where c ( k ) 1 = k 0 T 0 + T min T max T min ( k 1 k 0 )+ 2 ( k 1 k 0 ) T max T min T ( k ) ,and c 2 = k 1 k 0 T max T min .Basedon T 0 T min T and k 1 > k 0 (Chapter5.2.2),itcanbeshownthatboth c ( k ) 1 and c 2 arepositive. Combining(5.13)and(5.16),weobtain: W [ T ( k ) + d ; y ( k ) ] W [ T ( k ) ; y ( k ) ]=( a ( k ) 2 c 2 ) d 2 +( a ( k ) 1 c ( k ) 1 ) d : (5.17) While a ( k ) 1 0, a ( k ) 2 0, c ( k ) 1 > 0, c 2 > 0,thesignsofboth a ( k ) 1 c ( k ) 1 and a ( k ) 2 c 2 couldbeeither positiveornegative,whichimpliesthatanincrement d inthetemperaturedoesnotnecessarily leadtoanincreaseincurvature.Analogousstatementscanb emadewhentheinputdecreases. Takeaspecialexampleoftheinputbeingincreasedtosomeva lue T 0 (withthecorresponding memorycurve y 0 )andthendecreasedbyasmall d > 0,inwhichcase a ( k ) 1 = 0,and W [ T 0 d ; y 0 ] W [ T 0 ; y 0 ]=( a ( k ) 2 c 2 ) d 2 + c 1 d .Sincethelineartermdominatesthequadratictermfor small d ,theoutput k willincreaseimmediatelyfollowingthereversingof T at T 0 .Similarly,the curvaturewilldecreaseimmediatelyfollowingthereversi ngofadecreasinginput.Interestingly, thesepredictionsareconrmedbytheexperimentaldata,as canbeseenclearlyfromFig.5.4(a). Thefollowingpropositionwillbeinstrumentalindevelopi ngtheinversionalgorithm. Proposition4. Considerthenon-monotonichysteresismodelEq.(5.8).Let k max and k min denote themaximumoutputandminimumoutputofthemodel,respecti vely.Then k max canalwaysbe achievedbyrstincreasingthetemperatureTtoT max andthendecreasingitmonotonicallyto somevalue,and k min canalwaysbeachievedbydecreasingTtoT min andthenincreasingitto somevalue. Proof. Foranytemperature T ,thememorycurveconsistingofasingleverticalsegment(i nter- sectingtheline a = b at T )dominatesanyothermemorycurvesatthesametemperature, inthe sensethatthecorrespondingset P + 4 = f ( b ; a ) : g b ;a =+ 1 g inthePreisachplaneismaximalfor 88 thegiven T .Consequently,thecorrespondingoutput k P ofthePreisachoperator(withnonnegative densityfunction)islargestunderthedominantmemorycurv e.SeeFig.5.10forillustration;and comparethetwomemorycurves y 0 (dominant)and y ( k ) forthesametemperature T ( k ) . Thecontribution k E dependsonlyonthecurrenttemperature(nomemory).Theref ore,for agiventemperature T ,themaximumcurvatureoutput k canalwaysbeachievedwithaninput sequencethatresultsinadominantmemorycurveat T .Suchadominantmemorycurveiscreated bydecreasingthetemperaturefrom T max to T .Theothercasefollowsfromasimilarargument. Thecurvatureoutput k at T max and T min isdenotedas k + and k ,respectively.Duetothepos- itive/negativesaturation, k + and k areuniquelydenedandindependentoftheinitialconditio n oftheoperator W .Thefollowingassumptionismade: Assumption1: k min < k k + k max : (5.18) TheassumptionisexpectedtoholdforallVO 2 -basedmicrobendingactuators.Inparticular,for thesampleusedinthiswork k max =1836m 1 , k + =1828m 1 , k = 20m 1 ,and k min = 75 m 1 . 5.4.1InverseCompensationAlgorithm Forbrevitypurposes,onlythecase ¯ k > k ( 0 ) isdiscussed;thecase ¯ k < k ( 0 ) istreatedinasym- metricmanner.If ¯ k > k ( 0 ) ,thediscussionisdividedintotwosub-cases:(1) ¯ k k + ,and(2) k + < ¯ k k max .Fortherstcase,aniterativeproceduremodiedfrom[9]i sdetailed;forthe secondcase,therequiredinputwillbeatwo-stepsequencea ndtheprocedureisoutlined.Speci- cally, If ¯ k k + : 89 Step1: k : = 0; Step2: 1.Determine d ( k ) 1 > 0suchthat T ( k ) + d ( k ) 1 equalsthenextdiscreteinputlevel; 2.Determinetheminimum d ( k ) 2 > 0suchthat T ( k ) + d ( k ) 2 woulderasethenextcor- nerofthememorycurve y ( k ) ,whichisgeneratedundertheiterativeinputsequence f T ( 0 ) ; T ( 1 ) ; ; T ( k ) g ; 3.Evaluatethecoefcients a ( k ) 1 , a ( k ) 2 , c ( k ) 1 , c 2 forEq.(5.13)andEq.(5.16),whosevalues aredeterminedby y ( k ) and T ( k ) andvaryfromiterationtoiteration.Solvetheequation ¯ k W [ T ( k ) ; y ( k ) ] =( a ( k ) 2 c 2 ) d 2 +( a ( k ) 1 c ( k ) 1 ) d (5.19) for d .IfEq.(5.18)hastwopositivesolutions,let d 0 ( k ) bethesmallersolution;if Eq.(5.18)hasonepositivesolution,let d ( k ) 0 equalthatsolution;ifEq.(5.18)hasno positivesolutions,let d ( k ) 0 = 1000(anumberlargerthan ( T max T min ) = M butotherwise arbitrary);Thisschemeensurestheuniquenessofthesolut ion. 4.Let d ( k ) = :min f d ( k ) 0 ; d ( k ) 1 ; d ( k ) 2 g , T ( k + 1 ) = T ( k ) + d ( k ) , k ( k + 1 ) = W [ T ( k + 1 ) ; y ( k ) ] ; 5.If d ( k ) = d ( k ) 0 ,gotoStep3;otherwiselet k : = k + 1andgobacktoStep2. Step3: ¯ T : = T ( k + 1 ) andstop. If k + < ¯ k k max :Inversioncanberealizedbyrstapplyingthetemperature T max tosaturate thePreisachoperator,andthendecreasinginputiterative lyfollowingasimilarscheme. 90 Notethattheinversionalgorithmrequiresknowingthecurr entinitialmemorycurve y ( 0 ) . AsfortheinversionofaPreisachoperator[27],thisrequir ementistypicallysatisedbysetting theinitialmemorycurveattime n = 0toa(known)congurationthatcorrespondstopositiveor negativesaturationofthePreisachoperator G ,byapplying T max or T min ,respectively.Thememory curveatanyfuturetime n > 0canthenbeinferredbasedonthecurveat n 1andthesequenceof inputvaluesappliedaftertime n 1.Thefollowingpropositionsummarizesthepropertiesoft he proposedinversionalgorithm. Proposition5. Theproposedinversionalgorithmproducestheinput ¯ Tsatisfying W [ ¯ T ; y ( 0 ) ]= ¯ k innomorethann c ( y ( 0 ) )+ Miterations,wheren c ( y ( 0 ) ) denotesthenumberofcornersin y ( 0 ) , Misthediscretizationlevel.Furthermore, ¯ Thasnomorethantwoelements. Proof. Thecasewhere ¯ k > k ( 0 ) isprovedindetail.First,considerthecase ¯ k < k + .Since theoperator W iscontinuous(i.e.,itsoutputchangescontinuouslywitht heinput),theremust exist ¯ T 2 [ T ( 0 ) ; T max ] suchthat W [ ¯ T ; y ( 0 ) ]= ¯ k .Theinversionalgorithmthensearchesforthe exactsolutionincontiguoussegmentswithin [ T ( 0 ) ; T max ] ,wherethesegmentsaredenedbythe discreteinputlevelsandthememorycurve y ( 0 ) .Thenumberofsuchsegmentsisnogreaterthan n c ( y ( 0 ) )+ M ,whichprovidestheupperboundfortheiterationsteps.The caseof k + < ¯ k k max canbeprovedfollowingsimilarandsimplerarguments.Appl yingrst T max erasesallthememory curvecorners,soitisonlyneededtosearchwithinsegments denedbythediscreteinputlevels. Themaximumiterationstepswillbe M inthiscase.Thelaststatementofthepropositioniseviden t fromthedescriptionofthealgorithm. FromProposition5,theefciencyoftheproposedinverseco mpensationalgorithminthiswork iscomparabletothatoftheinversionalgorithmforaPreisa choperator[9]. 91 5.4.2ExperimentalValidation Theperformanceoftheproposedinversionalgorithmhasbee nexaminedinopen-loopcurvature trackingexperiments.Forcomparison,thesingle-valuedp olynomialmodelhasalsobeeninverted throughalook-uptable.Thedesiredcurvatureischosentob efrom 63m 1 to1814m 1 totest theeffectivenessoftheinversecompensationalgorithmfo rawidecurvaturerange. Fig.5.11(a)showsthecurvatureoutputsobtainedunderthe twoinversionschemes.Fig. 5.11(b)showsthecorrespondinginversionerrors.Theinve rsionoftheproposedmodelisprovento beeffective,withthelargestcurvatureerrorof78m 1 ,whichisonly4.1%ofthewholecurvature range.Incomparison,theinversionofthenon-hystereticp olynomialmodelproducesamaximum errorof734.2m 1 .TheRMSEvalueisalsocalculatedtoquantifythetrackinge rror.TheRMSE oftheproposedinversionisonly26.7m 1 ,comparedto320.6m 1 forthepolynomialcase.The lastobservationfromFig.5.11(b)isthatwhenthemagnitud eofthedesiredcurvaturechanges,the proposedinversionschemecanstillmaintainasmallcurvat ureerror,whiletheinversionbasedon thepolynomialmodelproducesalargeerror. 92 0102030405060050010001500IndexCurvature change (m-1) Desired curvatureProposed modelPolynomial(a) 0102030405060-800-600-400-2000200400IndexCurvature error (m-1) Proposed modelPolynomial(b) Figure5.11:(a):Open-loopinversecontrolperformancefo rtheproposedmodelandthepolyno- mialmodel.(b):Inversecompensationerrors. 93 Chapter6 ModelingandInverseCompensationof HysteresisusinganExtendedGPI(EGPI) model Inthischapter,anEGPImodelisproposedtocapturesophist icatedhysteresisasobservedinVO 2 . ThemodelconsistsofanonlinearmemorylessfunctionandaG PImodel,theplayoperatorsof whichhavethesameenvelopefunctions.TheEGPImodelistes tedinmodelingasymmetricand non-monotonichysteresisbetweenthecurvatureoutputand thetemperatureinputofaVO 2 -coated microactuator,demonstrating40%lessmodelingerrorthan aGPImodel.Theadvantagesofthe proposedmodelarefurtherveriedinmodelingtheasymmetr ic,partiallysaturatedhysteresisbe- tweentheresistanceoutputandthetemperatureinputofaVO 2 lm.Anovelinversionalgorithmis thenderivedbasedonthexed-pointiterationframework.T heconvergenceconditionofthepro- posedalgorithmisfurtherderived.Finally,bothsimulati onandexperimentalresultsareprovided tosupporttheeffectivenessoftheinversionalgorithm. 6.1EGPImodelforNon-monotonicHysteresis TheGPImodelcancaptureasymmetrichysteresiswithsatura tion,andithasananalyticalinversion [28]aslongastheenvelopefunctionsofallthegeneralized playoperatorsareofthesameform. 94 Weproposeaddinganonlinearmemorylessfunction D ( ) totheGPImodel: u ( t )= D ( v ( t ))+ N å j = 0 p ( r j ) F g r j [ v ]( t ) ; (6.1) ThismodeliscalledtheEGPImodel.Fig.6.1showsaGPImodel andanEGPImodelwith identicalweightsofgeneralizedplayoperators.Itisshow nthattheEGPImodelcanbettermodel complexhysteresis.Bychoosingappropriatememorylessfu nctions,theEGPImodelcanalso capturenon-monotonichysteresis. -4-2024-100-50050100InputOutput Generalized PIExtended generalized PIFigure6.1:AGPImodelandanEGPImodelwithidenticalweigh tsofgeneralizedplayoperators. 6.2InverseCompensationAlgorithm Thegoalofinversecompensationistocanceloutthehystere sisnonlinearitybyconstructingan inversehysteresismodel.Axed-pointiteration-basedin versionalgorithmforanEGPImodelis proposed. Denote u d asthedesiredoutputoftheEGPImodel.ThentheEGPImodelis expressedas u d = D ( v )+ Y [ v ] : (6.2) 95 where D ( v ) isthememorylesscomponentand Y [ v ] istheGPIhysteresismodel.Sincetheinversion of Y [ v ] isavailable[28],rewriteEq.(6.2)as v = Y 1 [ u d D ( v )] : (6.3) UnliketheinversionofGPImodel, u d D ( v ) isusedastheinputfortheinverseEGPImodel. Therightpartoftheaboveequationcanbesolvedwithaknown input v ;however, v isalsothe desiredsolution.Tosolvetheproblem,werstrecallsomeb ackgroundmaterials. Contractionmapping[91]: Let ( X ; d ) beametricspace.Amap T : X ! X iscalledacontrac- tionmappingon X ,ifthereexists q 2 [ 0 ; 1 ) suchthat: d ( T ( x ) ; T ( y )) qd ( x ; y ) forall x ; y 2 X . Proposition6. Let ( X ; d ) beanon-emptycompletemetricspacewithacontractionmapp ingT : X ! X.ThenTadmitsauniquexedpointx inX(i.e.T ( x )= x ).Furthermore,x canbefound asfollows:startwithanarbitraryelementx 0 inXanddeneasequencex n byx n = T ( x n 1 ) ,then x n ! x [91]. FromProposition6,if Y 1 [ u d D ( v )] isacontractionmappingintermsof v ,theinversioncan beobtainedbyiterating v k = Y 1 [ u d D ( v k 1 )] , k = 1 ; 2 ; ; n ; until j v n v n 1 j < s , s > 0. Thefollowingpropositionprovidesasufcientconditionf ortheconvergenceoftheinversion algorithm. Proposition7. Denote Y 1 [ u d ] astheinversionoftheGPImodel,whereu d isthedesiredoutput. Thentheoperator Y 1 [ u d D ( v )] isacontractionmappingon [ v min ; v max ] ,if min v f d g R dv ; d g L dv g p ( r 0 ) > dD dv : (6.4) 96 Proof. When u 1 > u 2 : j Y 1 [ u 1 ] Y 1 [ u 2 ] j = j g 1 R P 1 [ u 1 ] g 1 R P 1 [ u 2 ] j max u f d g 1 R du gj P 1 [ u 1 ] P 1 [ u 2 ] j max u f d g 1 R du g ‹ p ( r 0 ) j u 1 u 2 j = max u f d g 1 R du gj u 1 u 2 j = p ( r 0 ) : (6.5) ThesecondinequalityofEq.(6.5)holds,since j P 1 [ u 1 ] P 1 [ u 2 ] j = P 1 [ u 1 ] P 1 [ u 2 ] = ‹ p ( r 0 ) ( u 1 u 2 )+ N å i = 1 ‹ p ( r i )( F ‹ r i [ u 1 ]( t ) F ‹ r i [ u 2 ]( t )) ‹ p ( r 0 )( u 1 u 2 ) = ‹ p ( r 0 ) j u 1 u 2 j : (6.6) TheinequalityofEq.(6.6)holdssince‹ p ( r i ) < 0,for i 1(SeeEq.(B.17)inAppendix)and P 1 [ u 1 ] P 1 [ u 2 ] . Similarly,forthecaseof u 1 < u 2 : j Y 1 [ u 1 ] Y 1 [ u 2 ] j max u f d g 1 L du gj u 1 u 2 j = p ( r 0 ) : (6.7) 97 Combingtheabovetwocases, j Y 1 [ u d D ( v 1 )] Y 1 [ u d D ( v 2 )] j max u f d g 1 R du ; d g 1 L du g dD dv j v 1 v 2 j = p ( r 0 ) = max v f 1 = d g R dv ; 1 = d g L dv g dD dv j v 1 v 2 j = p ( r 0 ) = dD dv j v 1 v 2 j = p ( r 0 ) = min v f d g R dv ; d g L dv g : (6.8) Thus, Y 1 [ u d D ( v )] isacontractionmappingwhenthefollowinginequalityissa tised: min v f d g R dv ; d g L dv g p ( r 0 ) > dD dv : (6.9) ForCPImodel,since d g R = dv = d g L = dv = 1,sotheconvergenceconditiondegeneratesto p ( r 0 ) > dD dv : (6.10) ForeitherCPImodelorGPImodel,since dD = dv = 0,noiterationisneeded. Notethat,fromProposition7,theconvergenceofthepropos edalgorithmdependsonthepa- rametersofthehysteresismodel. 6.3ExperimentalResults:Modeling ThemodelingperformancesoftheEGPImodelsinvolvingthec urvatureandtemperaturehysteresis relationshipofaVO 2 coatedcantilever,andtheresistanceandtemperaturehyst eresisrelationship ofaVO 2 lmareshown. 98 6.3.1Curvature-temperatureHysteresisofa VO 2 -coatedMicroactuator AsdiscussedinChapter5,theVO 2 -coatedsiliconmicro-cantileversaresubjecttotwoactua tion effectswhenitstemperatureisvaried.First,thestressdu etothermallyinducedphasetransitionof VO 2 makesthebeambendtowardstheVO 2 layer,aprocessthatisinherentlyhysteretic.Second, thedifferentialthermalexpansioneffectgeneratesstres sintheoppositedirection.Asaresult,the hysteresisbetweenthebendingcurvatureandthetemperatu reisnon-monotonic. FollowingsimilartreatmentasthatinChapter5,a172nmthi ckVO 2 layerwasdepositedon asiliconcantileverwithlengthof300µm.Themicrocantile verwasgluedtoaglasssubstratethat wasdirectlyincontactwithaPeltierheater.APSDandalase rwereusedtomeasurethedeection ofthemicrocantilever.Thecurvaturewasthenobtainedbas edonthePSDmeasurement. Inordertocapturethehysteresis,theenvelopefunctionsf ortheextendedgeneralizedplay operatorarechosentobehyperbolic-tangentfunctionsint heformof g R ( v ( t ))= tanh ( a R v ( t )+ b R ) ; (6.11) g L ( v ( t ))= tanh ( a L v ( t )+ b L ) : (6.12) Thenon-hystereticcomponentisexpressedas D ( v ( t ))= p 0 sin ( w v ( t ))+ c 1 : (6.13) Thenumberofthegeneralizedplayoperatorsischosentobe N = 15,andtheplayradiiareis chosenas r = i = N ; i = 0 ; 1 ; ; N 1.TheparametersidentiedfortheGPImodelandtheEGPI modelareshowninTable6.1.TheweightsoftheGPImodelandt heEGPImodelaredifferentdue totheeffectofthenonlinearmemorylessfunction.Theiden tiedweights p ( r i ) ; i = 0 ; 1 ; ; N 1 99 Table6.1:ParametersoftheGPImodelandtheEGPImodelforh ysteresisofaVO 2 -coatedmi- croactuator. a L b L a R b R p 0 w c 1 Generalized 0.11-5.190.13-6.9400914.20 Extended 0.18-9.610.14-7.1940.10.17923.91 ofbothmodelsarenotincludedintheinterestofbrevity. Notethatgiventheenvelopefunctionsform,theidentiedE GPImodelmaynotoptimally modelthehysteresis.Thenon-hystereticcomponentisiden tiedasfollows:rstaGPImodel isadoptedtomodelthehysteresis,thenthenon-hysteretic componentischosenbasedonthe remainingmodelingerroroftheGPImodel.TheGPImodelismo deledbythesummationof weightedgeneralizedplayoperatorsandanoffset c 1 . Inordertotocoverthewholephasetransitionrange,thetem peraturerangewaschosentobe from20 Cto80 C.Inparticular,wevariedthetemperatureinrepeatedheat ing-coolingcycles withthetemperaturerangedecreasedforeachcycle.Fig.6. 2(a)and(b)showthemodelingperfor- manceofGPImodelandthatoftheEGPImodel,respectively.C omparedwiththeGPImodel,the proposedmodelcancapturetheasymmetricandnon-monotoni chysteresismoreaccurately.The RMSEandtheabsolutemaximumoftheerrorareselectedtoqua ntifythemodelingperformance. TheRMSEoftheGPImodelis38.5m 1 ,andtheRMSEoftheEGPImodelis26.4m 1 .The largesterroroftheGPImodelis148.9m 1 ,whilethatoftheEGPImodelis89.6m 1 .There- fore,theEGPImodelcancapturetheasymmetricandnon-mono tonichysteresisbehaviormore accurately,with31%and40%smallererrorintermsofRMSEan dthelargestmodelingerror, respectively. 100 20304050607080050010001500Temperature (0C)Curvature change (m-1) ExperimentGeneralized PI(a) 20304050607080050010001500Temperature (0C)Curvature change (m-1) ExperimentExtended PI(b) Figure6.2:Theperformanceofmodelingcurvature-tempera turehysteresisofaVO 2 -coatedmi- crocantileverbasedon:(a)GPImodel.(b)EGPImodel. 101 Table6.2:ParametersoftheGPImodelandtheEGPImodelforh ysteresisofaVO 2 lm. a L b L a R b R p 0 a D b D c 2 Generalized 0.14-8.50.16-9.5000-3.23 Extended 0.16-10.30.20-11.60.40.03-1.6-3.26 6.3.2Resistance-temperatureHysteresisofa VO 2 Film AVO 2 layerwasdepositedbypulsedlaserdeposition.Thelmwash eatedwithaPeltierheater. Theexperimentalsetupinthisworkwassimilartotheoneuse dinChapter2.Theresistanceof thelmwasmeasuredthroughtwoaluminiumcontactspattern edontheVO 2 lm. Similarly,thetemperatureproleintimefollowedapatter nofdampedoscillations.Itisshown inFig.6.3thatthemeasuredresistance ( Z ) changesbyapproximatelytwoordersofmagnitude. Furthermore,inordertohavenon-negativeweightsfortheh ysteresismodels, log 10 Z istakenas theoutput.ThehysteresisbehaviorshowninFig.6.3isasym metricandalsopartiallysaturated. Following[13,28],theenvelopefunctionsareselectedtob ehyperbolic-tangentfunctions.The memorylessfunctionisselectedasthesumofahyperbolic-t angentfunctionandanoffset: D ( v ( t ))= p 0 tanh ( a D v ( t )+ b D )+ c 2 : (6.14) Thenumberofplayoperatorischosentobe N = 30,andtheradiiarechosentobe r = i = N ; i = 1 ; ; N .TheGPImodelismodeledbythesummationofthesamenumbero fweightedgener- alizedplayoperatorsandanoffset c 2 .TheidentiedparametersoftheGPImodelandtheEGPI modelareshowninTable6.2.TheweightsoftheGPImodelandt heEGPImodelaredifferent duetotheeffectofthenonlinearmemorylessfunction.Thei dentiedweightsofbothmodelsare notincludedintheinterestofbrevity. Fig.6.3(a)and(b)showthemodelingperformancebasedonth eGPImodelandtheproposed 102 30405060708090-4-3.5-3-2.5Temperature (0C)-log10Z (log10W) ExperimentGeneralized PI(a) 30405060708090-4-3.5-3-2.5Temperature (0C)-log10Z (log10W) ExperimentExtended PI(b) Figure6.3:Theperformanceofmodelingtheresistance-tem peraturehysteresisofaVO 2 lm basedon:(a)GPImodel.(b)EGPImodel. model,respectively.TheRMSEandthemaximumabsoluteerro roftheGPImodelare0.031 and0.082log 10 W ,respectively,whilethecorrespondingvaluesfortheEGPI modelare0.012and 0.041log 10 W ,respectively.TheGPImodelhas158%and100%largerRMSEer rorandmaximum absoluteerror,respectively. Fig.6.4(a)and(b)showarandomtemperaturesequenceandit scorrespondingresistance output.ThemodelestimationerrorsbasedontheGPImodelan dtheEGPImodel,respectively, areshowninFig.6.4(c).TheRMSEandtheaverageabsoluteer roroftheGPImodelare0.034 103 and0.027log 10 W ,respectively,whilethecorrespondingvaluesfortheEGPI modelare0.012and 0.009log 10 W ,respectively.TheeffectivenessoftheEGPImodelincaptu ringtheasymmetricand partialsaturatedhysteresisisthusfurtherdemonstrated . 6.4InverseCompensationResults Examplesareshowntoillustratetheeffectivenessofthein versealgorithmbothinsimulationand experiments. 6.4.1Simulation ConsideranEGPIoperatorexpressedasamemorylessfunctio nandaCPImodel,i.e., g R ( v ( t ))= v ( t ) ; (6.15) g L ( v ( t ))= v ( t ) : (6.16) Thememorylesscomponentischosenas D ( v ( t ))= p 0 cos ( v ( t )) : (6.17) Notetheenvelopefunctionandmemorylesscomponentarecho senintheaboveformasan illustrativeexample.Theradiiandtheircorrespondingwe ightsoftheplayoperatorsareshownin TableIII. D ( v ( t )) ischosentobe5cos ( v ( t )) .Theconvergenceconditionforthegivenmodelis satisedsince 104 05001000150030405060708090Temperature (0C)Index(a) 050010001500-4-3.5-3-2.5Index-log10Z (log10W)(b) 050010001500-0.1-0.0500.050.1IndexError (log10W) Generalized PIExtended PI(c) Figure6.4:Modelvericationoftheresistance-temperatu rehysteresisinaVO 2 lm:(a)arandom temperaturesequence.(b)correspondingresistanceoutpu t.(c)Modelingcomparisonbetweenthe GPImodelandEGPImodel. 105 Table6.3:ParameteroftheEGPImodel i123456 r i 00.20.40.60.82 p ( r i ) 621244 min f d g R dv ; d g L dv g p ( r 0 )= 6 > 5 dD dv : (6.18) Fig.6.5(a)showsarandomlychoseninputsequence,andFig. 6.5(b)showstheinputand modeloutputrelationship.Fig.6.6(a)showstheinversion ofthemodelbasedontheproposed algorithm,andFig.6.6(b)showstheresultingrelationshi pbetweendesiredoutputandcalculated output.Thegoodlinearitydemonstratestheeffectiveness oftheinversealgorithm. Theaveragenumberofiterationsis8.38whentheconvergenc ecriterion s ischosentobe 0.0001,whichshowstheefciencyofthealgorithm.Itisfou ndinsimulationthatif s isenlarged tobe0.01,theaveragenumberofiterationdecreasesto5.31 .Itisalsoveriedthatwhenthevalue of p 0 isreduced,theinversionalgorithmmaynotconverge;onthe otherhand,if p ( r 0 ) remains sufcientlylarge,theEGPIalgorithmwillalwaysconverge . 6.4.2ExperimentalVerication Theproposedinversionalgorithmisalsotestedinexperime ntstocompensatetheresistance- temperaturehysteresisintheVO 2 lm.Itisveriedthatwhenthenumberofgeneralizedplay operatorsismorethan5,themodelingperformancewillnoti mprovesignicantlywhileincurring highercomputationalcost.Therefore,asimplerandmoreef cientmodelwith5generalizedplay operatorsandanonlineartermisutilized.Theparametersc anbefoundinTable3.6and[82]. Itisfoundthatwhen v 2 [43.2,74.2] C,theconvergencerequirementwillbesatised. 106 020406080-4-3-2-101234IndexInput sequence(a) -4-2024-60-40-200204060InputOutput(b) Figure6.5:Simulationvericationoftheinversealgorith m.Hysteresisrelationship:(a)Input sequence.(b)Input-outputoftheEGPImodel. -100-50050100-4-3-2-101234InputOuput(a) -60-40-200204060-60-40-200204060Desired ouputActual output(b) Figure6.6:Compensationofhysteresisinsimulation:(a)I nput-outputoftheinverseEGPImodel. (b)Therelationshipofthedesiredoutputandtheactualout putafterhysteresiscompensation. 107 051015202530-4-3.5-3-2.5Index DesiredProposed inversion(a) 051015202530-0.0500.050.10.15indexInversion error (log10W)(b) Figure6.7:(a).Inversecompensationperformanceinexper iment.(b).Inversioncompensation errorfortheEGPImodel. NotethatthiscoversthetypicaloperatingrangeforVO 2 .Outsidetheregion[43.2,74.2] C, theconvergencerequirementmayfail,sincewhen v ! ¥ ; d g = dv ! 0fasterthan dD = dv ,and d g = dv ˝ dD = dv ,thusmakingEq.(6.4)difculttomeet. Fig.6.7showstheinversecompensationperformanceandthe inversionerror.Theabsolute maximuminversionerrorisaround0.135log 10 W ,whichstilldemonstratestheeffectivenessof theproposedcompensationapproach. 108 Chapter7 ACompositeHysteresisModelin Self-SensingFeedbackControlof VO 2 -integratedMicroactuators Inthischapteracompositehysteresismodelisproposedfor self-sensingfeedbackcontrolofVO 2 - integratedmicroactuators.Thedeectionofthemicroactu atorisestimatedwiththeresistance measurementthroughtheproposedmodel.Tocapturethecomp licatedhysteresisbetweenthere- sistanceandthedeection,weexploitthephysicalunderst andingthatboththeresistanceandthe deectionaredeterminedbyhystereticrelationshipswith thetemperature.Sincedirecttempera- turemeasurementisnotavailable,theconceptoftemperatu resurrogate,representingtheconstant currentvalueinJouleheatingthatwouldresultinagivente mperatureatthesteadystate,isex- ploredinthemodeling.Inparticular,thehysteresisbetwe enthedeectionandthetemperaturesur- rogateandthehysteresisbetweentheresistanceandthetem peraturesurrogatearecapturedwitha generalizedPrandtl-Ishlinskii(GPI)modelandanextende dgeneralizedPrandtl-Ishlinskii(EGPI) model,respectively.Thecompositeself-sensingmodeliso btainedbycascadingtheEGPImodel withtheinverseGPImodel.Forcomparisonpurposes,twoalg orithms,basedonaPreisachmodel andanEGPImodel,respectively,arealsousedtoestimateth edeectionbasedontheresistance measurementdirectly.Theproposedself-sensingschemesi sevaluatedwithproportional-integral 109 (PI)controlofthemicroactuatorunderstepandsinusoidal references,anditssuperiorityoverthe otherschemesisdemonstratedbyexperimentalresults. 7.1ExperimentalProcedures 7.1.1 VO 2 -integratedActuatorFabrication Themicroactuatorusedinthisworkconsistedofasilicondi oxide(SiO 2 )microcantileverwith patternedVO 2 lminsidethestructure.Thefabricationprocessowforth isdeviceisshown inFig.7.1.Theprocessstartswiththedepositionof1µmlay erofSiO 2 usingplasma-enhanced chemicalvapordeposition(PECVD)attemperatureof300 Cona300µmthicksilicon(Si)wafer. ThisSiO 2 layerwasusedasthesubstratetogenerateVO 2 withhighlyorientedcrystallinestruc- ture,toachievemaximumactuationeffect[92].AVO 2 layer(270nm)wasdepositedbypulsed laserdeposition(PLD)[56]andpatternedwithreactiveion etch(RIE).ThepatternedVO 2 lm wasusedastheactiveactuationelementinthecantilever.A notherSiO 2 layer(400nm)wasde- positedbyPECVDat250 CtoisolatetheVO 2 fromthemetallayer(tobeprocessednext)and patternedwithRIEtoopenthecontacttotheVO 2 .Thelowertemperaturewasusedtomitigatethe adverseeffectsofexposingVO 2 lmstohightemperatures.Twoopeningsonthetopsideofthe SiO 2 weremadetoexposetheVO 2 inselectedregions.TwoTi(40nm)/Au(160nm)layerswere depositedbyevaporationandpatternedbylift-offtechniq ues.Therstonewastopartiallyllthe openingintheSiO 2 ,andthesecondonewastoformtheheatingelementandthetra cesforthe VO 2 resistancecontacts.CertainareasofSiO 2 withthicknessof1.4 m mwereetchedwithRIEto denethegeometryofthecantileverandexposetheSiforthe releasingstep.XeF 2 gaswasused todoanisotropicetchoftheSiandreleasethecantilever. Inthesedevices,theVO 2 lmwasfullyintegratedinthefabricationprocessowofth edevice. 110 Thegoldmetallayerusedforbothresistancemeasurementan dactuationoftheVO 2 lmwas depositedatroomtemperatureaftertheVO 2 lmwasdepositedandpatterned.Thisresultedinan directelectricalcontactbetweenthemetallayerandaunif ormVO 2 lm.InpreviousChapters, theVO 2 lmwasdepositedathightemperaturesonplatinumcontactp adsthatwereaccessedby viasthroughaSiO 2 lmthatseparatedthemetallayerandtheVO 2 lminregionsotherthanthe contactpad.ThisnotonlycreatedastepintheVO 2 lmthicknessattheelectricalcontact,butalso aVO x (xdifferentthan2)layerbetweenthelmandthecontactpad .Thus,theVO 2 lmsused inthischapterwereofbetterqualityintermsofuniformity andstoichiometry,andtheresistance measurementsonthefully-integratedVO 2 devicesincludedonlytheVO 2 thinlm. NotethattheactuatorisabimorphbenderconsistingofVO 2 andSiO 2 layers.Thermally inducedphasetransitioninVO 2 willgenerateinternalstressthatcausesdrasticbendingo fthe structuretowardtheVO 2 layer.Inaddition,differentialthermalexpansionofthet womaterials resultsinanoppositebendingeffect.Thecombinationofth etwoactuationeffectsleadstoa non-monotonichystereticbehaviorbetweenthedeectiona ndthetemperature. 7.1.2ExperimentalSetup TheexperimentalsetupwassameasFig.4.7(a).Thesystemis basedonthelaserscattering technique,usinganIRlaser( l =808nm)andapositionsensitivedetector(PSD)totrackthe dis- placementofthemicroactuator(showninFig.7.2).Acharge coupledevice(CCD)camerawas usedforalignmentandcalibrationpurposes.Notethatwhil etheCCDcamerahasalimitedpixel resolution(1 : 3 m m),theresultingcalibrationerrormainlyintroducesasca lingfactorcloseto1for therelativedisplacementmeasurement,andthusithasmini malimpactonthecharacterizationand comparisonofdifferentself-sensingschemesinthiswork. AdSPACEsystemwasusedfordata acquisitionandcontrolimplementation.Thepowerofthese nsinglaser(222mW)wascalibrated 111 a) b) c) d) e) f) g) h) j) Top View Cross Section View Si Ti/Au SiO 2VO 2Figure7.1:FabricationprocessowfortheVO 2 -integratedactuator.a)DepositionofSiO 2 (1 m m)byPECVD;(b)depositionofVO 2 (270nm)byPLD;(c)patterning(etch)ofVO 2 byRIE; (d)depositionofSiO 2 (0.4 m m)byPECVD;(e)patterning(etch)ofSiO 2 byRIE;(f-g)deposition ofTi/Aubyevaporationandpatterningbylift-off;(h)RIEo fSiO 2 fordevicepattern;(j)cantilever releasedbyXeF 2 isotropicetchingofSi. 112 tobetheminimumpossibletobesensedbythePSDwithoutheat ingthecantileverduetophoton absorption.Thevoltageoutput(VD)ofthePSDwaslinearlyp roportionaltothepositionofthe laser.WiththeimagescapturedbytheCCDcamera,thisvolta ge( V D )wasmappedtothedeec- tionofthemicroactuator.Thechipwasinsideasidebrazepa ckaging(wire-bonded),whichwas connectedtothedSPACE.Thecurrent I H showninFig.4.7(b)wasusedtocontrolthetemperature ofthemicroactuatorbyJouleheating.Thecurrentwasgener atedusingtworesistancesinseries: theheaterresistanceandanexternalresistance,whoseonl ypurposewastolimitthemaximum currentthatcanbeappliedtothesystem.TheVO 2 resistance( R V )wasmeasured insitu byusing aconstantcurrentandmonitoringthevoltageacrosstheres istanceŒthemagnitudeoftheconstant current(21 m A)waschosensothatitwouldnotheattheVO 2 considerably,butcouldbemeasured bythedSPACEsystem. 65 m425 mFigure7.2:TheVO 2 -integratedmicroactuatorusedinthiswork,withlength42 5µmandwidth 65µm. 7.1.3MeasurementofHystereticBehavior Inordertoobtainthehysteresismeasurement,asequenceof quasi-staticinputvaluesareapplied, andforeachinputvalue,thecorrespondingoutput(resista nceordeection)atthesteadystateis recorded.Inthiswork,thetermfiindexflreferstothenumber ingofthequasi-staticinputvalues aswellasthatofthecorrespondingsteady-stateoutputval ues.Fig.7.3(a)showsthecurrent inputwiththeformofdampedoscillations.Themeasurement wastakenunderaquasi-static condition,whereeachcurrentvaluewasheldfor10mssincet heheatingdynamicshadatime 113 456789x 10-30.511.52x 105Current (A)Resistance (W)(a) 456789x 10-30102030405060Current (A)Deflection (mm)(b) Figure7.3:(a)Thehysteresisbetweentheresistanceandth ecurrent;(b)thehysteresisbetween thedeectionandthecurrent. constantoflessthan2ms(seeSectionV).Fig.7.3(b)showst hecorrespondingresistanceofthe VO 2 microactuator.Thetotalresistancerangeis[11.19,231.7 5]k W ,withthecurrentranging from3.68mAto8.49mA.Thereisasymmetrichysteresisbetwe entheresistanceandthecurrent, whichshowsamonotonicbehaviorandwillbemodeledwithaGP Imodel.Fig.7.3(c)showsthe non-monotonichysteresisrelationshipbetweenthedeect ionoutputandtheinputcurrentofthe VO 2 -integratedmicroactuator,whichwillbemodeledwithanEG PImodel.Thetotaldeection rangeis[48.13,72.15] m m. Thedeectionandresistancevaluesweremeasuredsimultan eously,andFig.7.4(a)showsthe hystereticrelationshipbetweenthedeectionandtheresi stanceofthemicroactuator.Fig.7.4(b) showstheresistanceinput,whichfollowsapatternofdampe doscillations.Closerexamination (showninFig.7.4(c))ofthehysteresiscurverevealsasubt lebehaviorwherethehysteresisloops donotdemonstrateastrictfinestedflnatureunderthedamped oscillationsoftheresistance.For example,branches1and2formamajorhysteresisloop,while theminorhysteresisloopformed bybranches3and4isonlypartiallyinsideofthemajorhyste resisloopformedbybranches1and 114 2.Itcanbeshownthatsuchnon-nestedhysteresiscannotbec apturedbyatypicalsinglehysteresis model(e.g.,aPreisachoperatororaGPImodelwithnon-nega tiveweightingfunctions). Fig.7.5showsthehysteresisloopsbetweenthedeectionan dthecurrent,andbetweenthe deectionandtheresistance,atdifferentfrequenciesoft heinputcurrent.Onecanseethatthe shapeofthehysteresisloopbetweenthedeectionandthecu rrentchangesdramaticallywith thefrequency,whilethehysteresisloopbetweenthedeect ionandtheresistancehasmuchless variationwithfrequency.Thisindicatesthepromiseofusi ngresistancetoachieveself-sensingof deection. 7.2ProposedCompositeModelforSelf-sensing 7.2.1MainIdea Weusehysteresismodelsidentiedunderaquasi-staticcon dition(Fig.7.3)toderiveaself-sensing modelthatisapplicableunderdynamicconditions.Thejust icationforsuchanapproachisasfol- lows.NotethatthephasetransitioninVO 2 (includingboththemechanicalpropertychangeand theelectricalpropertychange)isinducedsolelybythetem peraturechange.Andthephasechange dynamicsisveryfast,attheorderofnanoseconds[93].Ther efore,thehysteresisbetweenthe resistanceandthetemperaturecanbeconsideredrate-inde pendentforthefrequencyrangeofin- terestinthiswork.Similarly,thehysteresisbetweenthed eectionoutputandthetemperatureis rate-independent,withinthefrequencyrangewherethestr ucturaldynamicsofthecantileverisnot excited.Consequently,withinthatsamefrequencyrange,t hehysteresisbetweenthedeection output D andtheresistance Z israte-independent,whichisakeypointbehindourpropose dap- proach.Wenotethatthe D - Z hysteresisinFig.7.5(b)showsmildrate-dependency,whic hcanbe largelyattributedtothestructuraldynamicsofthecantil ever,whichcannotbeentirelyignoredat 115 0.511.52x 1050102030405060Resistance (W)Deflection (mm)(a) 0500010000150000.511.52x 105IndexResistance (W)1234(b) 56789x 1041618202224262830Resistance (W)Deflection (mm)1234(c) Figure7.4:(a)Thehysteresisbetweenthedeectionandthe resistance;(b)theresistancese- quence;(c)zoom-inplotofthehysteresisbetweenthedeec tionandtheresistance,revealinga non-nestedstructure. 116 4681020304050Current (mA)Deflection (mm)1 Hz4681020304050Current (mA)Deflection (mm)10 Hz4681020304050Current (mA)Deflection (mm)20 Hz4681020304050Current (\mA)Deflection (mm)50 Hz(a) 00.10.21020304050Resistance (MW)Deflection (mm)1 Hz00.10.21020304050Resistance (MW)Deflection (mm)10 Hz00.10.21020304050Resistance (MW)Deflection (mm)20 Hz00.10.21020304050Resistance (MW)Deflection (mm)50 Hz(b) Figure7.5:Thehysteresisbetweenthedeectionandthecur rentundervaryinginputfrequencies; (b)Thehysteresisbetweenthedeectionandtheresistance undervaryinginputfrequencies. 117 thetestedfrequencies. Anotherkeyideaintheproposedmethodisthenotionof temperaturesurrogate ,whichis asingle-valued,strictlyincreasingfunctionofthetempe rature.Thepurposeofapplyingquasi- staticcurrentinputsduringmodelidenticationistoachi evethesteady-statetemperatureforeach valueofcurrentinput,sothattherelationshipsbetweenre sistance/deectionandtemperaturecan beestablished.SincedirectmeasurementofVO 2 temperature(whichwouldrequireadedicated sensor)isnotavailable,theappliedquasi-staticcurrent inputvalue, i ,becomesasurrogateforthe steady-statetemperature T as i isasingle-valued,increasingfunctionof T ,namely, i = g ( T ) .Since thehysteresisbetweendeection D and T andthehysteresisbetweenresistance Z and T (when structuraldynamicsofthecantileverisnotexcited)arera te-independent,soarethehysteresis between D and g ( T ) andthehysteresisbetween Z and g ( T ) .Thisiswhy, evenunderdynamic conditions ,wecaninferthesurrogatetemperature g ( T ) fromthemeasurement Z ,andthenuse g ( T ) tocalculate D . Eventhoughtheexplicitexpression g ( T ) forthetemperaturesurrogateisnotrequiredforthe implementationoftheproposedself-sensingalgorithm,fo rillustrationpurposes,weprovideone examplebasedonasimplethermalmodelofJouleheating[94] : dT ( t ) dt = d 1 ( T ( t ) T 0 )+ d 2 i 2 ( t ) ; (7.1) where d 1 and d 2 arepositiveconstantsrelatedtothedensity,volume,spec icheat,heattransfer coefcient,resistance,andsurfaceareaoftheVO 2 microactuator,and T 0 istheambienttempera- ture. Foraconstantcurrent i ,thesteady-statetemperature T under(1)canbecomputedas d 2 d 1 i 2 ( t )+ T 0 ,whichimplies 118 i = s d 1 d 2 ( T T 0 )= g ( T ) : (7.2) Notethatthefunction g ( T ) in(7.2)isindeedsingle-valuedandstrictlyincreasing,a ndthusis alegitimatesurrogatefor T .Thisnotionoftemperaturesurrogateisattheheartofourp roposed self-sensingscheme.Itisfoundinnite-elementsimulati onwithCOMSOLthatthethermaldistri- butionisapproximatelyuniformforthemajoritypartofthe cantilever,sotreatingthequasi-static currentasasurrogateofthetemperatureisacceptable. Intheproposedself-sensingscheme,thedeectionfeedbac kisestimatedbasedontheresis- tancemeasurementintwosteps:rst,thetemperaturesurro gate g ( T ) isobtainedfromtheresis- tancemeasurementbasedonainverseGPImodel;second,thed eectionestimateisobtainedfrom thetemperaturesurrogate g ( T ) basedonanEGPImodel.AbriefreviewoftheGPImodelandthe EGPImodelisprovidedbelow,andthereaderisreferredto[1 3,20,95]formoredetailsonthis subject. 7.2.2TemperatureSurrogate g ( T ) basedonaGPIModel Inordertocapturetheasymmetrichysteresisbehaviorbetw eentemperaturesurrogate g ( T ) and theresistanceoutput Z ,aGPImodelisadopted. Z ( t )= N 1 å j = 0 p 1 ( r j ) F g 1 r j [ g ( T )]( t )+ c 1 ; (7.3) where c 1 denotesthebias. g ( T ) canbeexpressedasthefollowinginversionmodel 119 g ( T )= 8 > > > > > < > > > > > : g 1 R P 1 ( Z ( t ) c 1 ) ; if Z ( t ) > Z ( t ) g 1 L P 1 ( Z ( t ) c 1 ) ; if Z ( t ) < Z ( t ) ; ‹ i ( t ) ; if Z ( t )= Z ( t ) (7.4) 7.2.3EstimatedDeection ‹ D BasedonanEGPIModel AnEGPImodelisadoptedtocapturethenon-monotonichyster esisbehaviorbetweenthetemper- aturesurrogate g ( T ) ,whichcanbecalculatedintheprevioussubsection,andthe deectionoutput D : D ( t )= N 2 å j = 0 p 2 ( r j ) F g 2 r j [ g ( T )]( t )+ c 2 g ( T )+ c 3 ; (7.5) where c 2 isaconstantrelatedtothethermalexpansioncoefcientso fthemicroactuatorstructure, and c 3 denotesaconstantbias.Theformofthemodelischosenbased on[95].In[95],thenon- monotonichysteresisbetweentemperatureanddeectionaV O 2 microactuatorwasmodeledby thesummationofaGPImodelandamemorylessfunction. Notethatinactualoperationsoftheactuator,thecurrenti nputisnotquasi-staticingeneral anddoesnothaveaxedrelationshipwiththetemperature.T herefore,eventhoughthecurrent inputisreadilyavailable(asacontrolsignal),onecannot simplyusetheknowncurrentvalueto estimatethedeection.However,thejointuseofEq.(7.4)a ndEq.(7.5)willbeabletoproducethe deectionestimateevenunderdynamicconditions,sinceth eschemeoperatesbyrstestimating thetemperaturestate g ( T ) .Thediscussionisalsosupportedbycomparingthehysteres isloops ofthedeectionandthecurrent,andthatofthedeectionan dtheresistance,wheretherate- dependencyismuchmilder(Fig.7.5(b)). 120 7.3ModelIdenticationandVerication 7.3.1ModelIdentication Toeffectivelyidentifythemodelparameters,theinputnee dstoprovidesufcientexcitationfor individualelementsofthehysteresismodels.Inthisworka ninputwiththeformofdampedos- cillationsisused,whichproducesnestedhysteresisloops fortheresistance-currentanddeection- currentrelationships.Forcomparisonpurposes,anEGPImo del,asinglePreisachoperator,and ahigh-orderpolynomialmodelareadoptedtodirectlymodel therelationshipbetweenthedeec- tionandtheresistance.Theperformanceofeachself-sensi ngschemeismeasuredbytheaverage andmaximumabsolutepredictionerrors.Thecalculationco mplexityisalsoexaminedusingthe averagetimeofeachself-sensingcalculation. ThenumbersofthegeneralizedplayoperatorsintheGPImode l N 1 +1andintheEGPImodel N 2 + 1arebothchosentobe6,theradiiarechosenas r =( i 1 ) = 6 ; i = 1 ; 2 ; ; 6.Thenumbers ofplayoperatorsoftheGPImodelandtheEGPImodelarechose nsuchthattheidentiedmodel couldprovideadequateaccuracywithreasonablecomputati ontime.Whenthenumberofplay operatorsischosentobe6,theaveragemodelingerrorisles sthan1 m m,overthetotaldeection range[1.62,58.65] m m.Increasingthenumberfurtherdoesnotseemtoproduceapp reciable improvementinmodelingaccuracy.TheparametersoftheGPI modelincludeplayradii,envelope functions,andweights.Whenthenumberofplaysislarger,t henumberofmodelparametersalso islarger,posingdifcultiesinmodelidentication.Inpr actice,itiscommontopre-denesome oftheparametersandtheidentiedmodelcouldstillaccura telycapturethehysteresisbehavior. Forexample,inChapter6,and[13],playradiiwerepre-den edinasimilarwayasadoptedinthis chapter.Inordertocapturethehysteresisbetweenthetemp eraturesurrogateandtheresistance,the envelopefunctionsforthegeneralizedplayoperatorarech osentobehyperbolic-tangentfunctions 121 Table7.1:IdentiedparametersoftheGPImodel. c 1 -132556.3 a L 519.7 a R 624.9 b L -2.984 b R -3.656 p 1 (0.415,0.648,0.325,0,0.081,0.618) intheformof: g R ( v ( t ))= tanh ( a R v ( t )+ b R ) ; (7.6) g L ( v ( t ))= tanh ( a L v ( t )+ b L ) : (7.7) Hyperbolic-tangentfunctionscouldeffectivelycapturet hecomplicatedasymmetrichysteresiswith outputsaturationinVO 2 microactuators.Thehyperbolic-tangentfunctionshaveal sobeenadopted tomodelothertypesofhysteresisbehaviorsand[13]andthe ireffectivenesshavebeenveried. Themodelparametersareidentiedthroughminimizationof anerror-squaredfunctionbe- tweentheactualdeectionandthemodelusingtheMatlabopt imizationtoolbox[13].Table7.1 andTable7.2showtheparametersoftheGPImodelandtheEGPI model,respectively.Notethat fortheGPImodel,allthegeneralizedplayoperatorshaveth esameenvelopefunctions. Fig.7.7(a)showstheperformanceoftheproposedself-sens ingscheme,andFig.7.7(b)shows thepredictionerror.Theaverageandmaximumabsoluteerro rswiththecompositemodelare0.95 m mand4.01 m m,respectively,overthetotaldeectionrange[1.62,58.6 5] m m.Theaveragetime foreachself-sensingcalculationis0.16ms.Thecomputati onswereruninMatlabonacomputer LenovoThinkpadT420with2.80GHzCPUand4.00GBmemory. 122 Table7.2:IdentiedparametersoftheEGPImodel. c 2 -8874.3 c 3 143.29 a L (982.5,1754.8,2169.2,838.1,1864.7,1623.2) a R (1202.6,1376.2,1243.0,1173.7,788.8,788.8) b L (-6.49,-11.13,-11.56,-6.24,-6.46,-15.52) b R (-8.67,-12.15,-10.12,-7.42,-10.83,-11.21) p 2 (27.24,5.35,3.18,12.84,1.00,26.68) 456789x 10-30.511.52x 105Current (A)Resistance (W) ExperimentGeneralized PI model(a) 456789x 10-30102030405060Current (A)Deflection (mm) ExperimentExtended PI model(b) Figure7.6:(a)ThecomparisonbetweentheGPImodelpredict ionandexperimentalmeasurement fortheasymmetrichysteresisbetweentheresistanceoutpu tandthecurrentinput;(b)thecompar- isonbetweentheEGPImodelpredictionandexperimentalmea surementforthenon-monotonic hysteresisbetweenthedeectionoutputandthecurrentinp ut. Fig.7.7(a)showstheperformanceoftheproposedself-sens ingscheme,andFig.7.7(b)shows thepredictionerror.Theaverageandmaximumabsoluteerro rswiththecompositemodelare0.95 m mand4.01 m m,respectively,overthetotaldeectionrange[1.62,58.6 5] m m.Theaveragetime foreachself-sensingcalculationis0.16ms.Thecomputati onswereruninMatlabonacomputer LenovoThinkpadT420with2.80GHzCPUand4.00GBmemory. ThePreisachmodelisapopularandeffectivehysteresismod el[9,12,85].APreisachmodel consistsofweightedsuperpositionofdelayedrelays.Prac ticalparameteridenticationinvolves 123 00.511.52x 1050102030405060Resistance (W)Deflection (mm) ExperimentComposite model(a) 050001000015000-4-2024IndexModeling error (mm) Composite model(b) Figure7.7:(a)Performanceoftheself-sensingschemeusin gthecompositemodel;(b)theself- sensingerrorbasedonthecompositemodel. discretizationofthePreisachdensityfunctioninonewayo ranother,andoneeffectivemethodisto approximatethedensitywithapiecewiseconstantfunction [9].Anon-monotonichysteresismodel thatcombinesamonotonicPreisachmodelwithamemorylesso peratorisadoptedtodirectly modelthehysteresisbetweenresistanceanddeection.The numberofdiscretizationlevelofthe modelischosentobe10.Fig.7.8(a)showsthemodelingperfo rmanceofthePreisachmodel. Theaverageandmaximumabsoluteerrorsare1.19 m mand6.68 m m,respectively.Theaverage timeneededforeachself-sensingcalculationis0.68ms.Th erefore,thePreisachmodelresults inmuchlongercalculationtimeandproducinglessaccurate modelingperformanceascompared totheproposedapproach.Moreover,themodelingperforman ceshowsthatthePreisachmodel cannotcapturethenon-nestedhysteresisloops. AnEGPImodelthatcombinesaGPImodelwithamemorylessoper atorisalsoadoptedfor modelingcomparison.Thenumberofgeneralizedplayoperat orsofthemodelischosentobe6. Fig.7.8(b)showsthemodelingperformanceoftheEGPImodel .Theaverageandmaximumabso- luteerrorsare1.26 m mand4.65 m m,respectively.Theaveragetimeneededforeachself-sens ing calculationis0.11ms.Therefore,theEGPImodelresultsin comparablecalculationtimebutpro- 124 0.511.52x 1050102030405060Resistance (W)Deflection (m m) ExperimentPreisach model(a) 0.511.52x 1050102030405060Resistance (W)Deflection (mm) ExperimentEGPI model(b) 0.511.52x 1050102030405060Resistance (W)Deflection (m m) ExperimentPolynomial model(c) Figure7.8:Performancesoftheself-sensingschemesusing (a)aPreisachmodel;(b)anEGPI model;(c)ahigh-orderpolynomialmodel. 125 duceslessaccuratemodelingperformanceascomparedtothe proposedapproach.Moreover,the modelingperformanceshowsthattheEGPImodelcannotcaptu rethenon-nestedhysteresisloops. DuetothesimilarmodelingaccuracyasthePresiachmodel,t heEGPImodelisnotadoptedfor modelvericationorcontrolexperiments. Aeighth-orderpolynomialmodelisidentiedtoapproximat ethecomplicatedhysteresisrela- tionshipbetweenthedeectionandtheresistance,asshown inFig.7.8(c).Itcanbeseenthata polynomialmodelfailstocapturethehysteresisrelations hip,andtheaverageandmaximumabso- luteerrorsinself-sensingare2.66 m mand6.48 m m,respectively,whiletheaveragetimeneeded foreachself-sensingestimationisonly0.004ms.Although thepolynomialmodeltakesmuch lesstimethanthecompositemodel,itsmodelingperformanc eismuchworsethantheproposed approach. Wehaveshownthat,forsomechosendesigns,theproposedcom positehysteresismodel-based self-sensingschemeoutperformsthePreisachmodel-based schemesinbothprecisionandef- ciency,andoutperformstheEGPIandpolynomialmodel-base dschemesinprecisionwithhigher computationalcomplexity.Ontheotherhand,itisknowntha ttheerrorperformanceofeach schemedependsonthecomplexityofeachmodel.Hereamorein -depthcomparisonisprovided byvaryingthecomplexityofeachscheme.Fig.7.9comparest heself-sensingperformanceand computationaltimeofeachmodelwhenthefilevelflofeachisv ariedfrom6to10.Heretheterm filevelflreferstothenumberofgeneralizedplayoperatorsf ortheGPImodelandtheEGPImodel, thediscretizationlevelforthePreisachmodel,andthedeg reeofthepolynominalmodel,respec- tively.Itcanbeseenthatthecompositemodelconsistently hasthelowestmodelingerroramong thefourschemes.Furthermore,itscomputationalcomplexi tyisonlyslightlyhigherthanthatofthe EGPImodel-basedschemes.Additionally,unlikeothersche mes,theproposedmodelcancapture thesubtledeection-resistancehysteresisbehaviorwher ethehysteresisloopsdonotdemonstrate 126 astrictfinestedflnatureunderthedampedoscillationsofth eresistance. 10-310-210-110001234Running time (ms)Modeling error (mm) Preisach modelPolynomial modelEGPI modelComposite modelFigure7.9:Modelaccuracyandtherunningtimecomparisonb etweenthecompositemodel,the Preisachmodel,theEGPImodel,andthepolynomialmodel. 7.3.2ModelVerication Inordertofurthervalidatetheproposedapproach,theVO 2 -integratedmicroactuatorissubjectedto arandomlychosencurrentinputsequence,showninFig.7.10 (a),undereachofthethreeschemes. Foreachindex,thecurrentisheldfor10ms.Herethenumbers ofgeneralizedplayoperators intheGPImodelandtheEGPImodelfortheproposedschemeis6 ,thediscretizationlevelfor thePreisachmodelis10,andtheorderofthepolynomialmode lis8.Fig.7.10(b)showsthe experimentalmeasurementofthedeectionandFig.7.10(c) showstheself-sensingerrorsunder eachscheme.Theaverageabsoluteerrorsare1.10 m m,1.45 m m,and2.35 m m,respectively,under thecompositemodel,thePreisachmodelandthepolynomialm odel.Themaximumabsoluteerrors are2.88 m m,4.55 m m,and5.91 m m,respectively,underthecompositemodel,thePreisachmo del andthepolynomialmodel.Theeffectivenessoftheproposed modelisthusfurtherveried. 127 0204060801001206789x 10-3IndexCurrent (A)(a) 0204060801001200204060IndexDeflection (mm)(b) 020406080100120-6-4-20246IndexDeflection error (mm) PolynomialPreisachComposite(c) Figure7.10:(a)Arandomlychosencurrentinputsequencefo rself-sensingmodelverication;(b) theexperimentaldeectionmeasurementundertherandomcu rrentinputsequence;(c)errorsin predictionsbydifferentself-sensingapproaches. 128 7.4Self-sensing-basedFeedbackControl Theblockdiagramforthephysicalclosed-loopsystemissho wninFig.7.11.Theinputofthe controlleristhedeectionerrorbetweenthereferenceand theself-senseddeection.Theoutputof thecontrolleristhecurrent.Theheatingdynamicsismodel edasarst-ordersystemandthetime constantisidentiedtobe1.8msbasedonaseriesofstepres ponseexperiments.Proportional- integraltuningisconductedinsimulation.Thefollowingp roportional-integralparameterswas chosentoensuredesirablestepresponseandfastsinusoida l-trackingperformance: K p = 1 : 4 10 3 , K I = 2 : 03 10 5 . Dynamics c-[Ax;0](t) +-++kk1/s PI +-Reference 2() ×10.0018s1 +Hysteresis I Deflection Sensed deflection Hysteresis II Self-sensing Resistance Figure7.11:Blockdiagramoftheclosed-loopcontrolsyste mwithself-sensing. 7.4.1StepReferenceTracking Astepreference-trackingexperimenthasbeenrstconduct ed.Eachreferencesetpointhasdura- tionof1s.Fig.7.12showstheexperimentalperformanceint ermsofthereferenceandtheactual deectionmeasuredbytheexternalPSD.Notethatalthought heself-senseddeectionisusedin thefeedbackcontrol,theactualdeectionisofmorereleva nce.Theaverageabsolutetracking errorsunderthecompositemodel,thePreisachmodel,andth epolynomialare1.11 m m,2.21 m m, and3.12 m m,respectively.Thestepreference-trackingexperimentd emonstratestheeffectiveness oftheproposedcompositeself-sensingmodel. 129 012345671015202530354045Time (s)Deflection (mm) PolynomialPreisachCompositeSetpointFigure7.12:Experimentalperformanceoftrackingastepre ferenceunderdifferentself-sensing schemes. 7.4.2SinusoidalReferenceTracking Thenextexperimentinvolvesthetrackingofasinusoidalsi gnal.Thereferencesignal,shownin Fig.7.13(a),hasfrequencyof0.1Hz.Fig.7.13(b)showsthe trackingperformanceofthethree self-sensingapproaches.Itiscalculatedthatthecontrol lersbasedonthecompositeself-sensing approach,thePreisachmodel,andthepolynomialmodelresu ltinaverageabsoluteerrorsof0.69 m m,1.28 m m,and2.79 m m,respectively.Theresultsshowtheeffectivenessofthep roposed self-sensingapproachforfeedbackcontrol. 7.4.3Multi-frequencyReferenceTracking Experimentsontrackingmulti-frequencysignalshavebeen furtherconducted.Thereferencesig- nalischosenas4sin ( 2 p t ) 6sin ( 2 p 10 t )+ 30,whichisshowninFig.7.14(a).Fig.7.14(b)shows thetrackingperformanceofthethreeself-sensingapproac hes.Itiscalculatedthatthecontroller basedonthecompositeself-sensingapproachresultsinana verageabsoluteerrorof2.43 m m, whichis20.1%and44.1%lessthanthoseunderthePreisachmo delandthepolynomialmodel- basedschemes,respectively.Theresultsshowthatthecont rollersresultinlargertrackingerror 130 0102030405020304050Time (s)Deflection (mm) Reference(a) 01020304050-4-20246Time (s)Deflection error (mm) PolynomialPreisachComposite(b) Figure7.13:(a)Asinusoidalreferencesignalfortracking controloftheVO 2 -integratedmicroac- tuator;(b)experimentaltrackingerrorsunderdifferents elf-sensingschemes. 131 0123452025303540Time (s)Deflection (mm) Reference(a) 012345-505Time (s)Deflection error (mm) PolynomialPreisachComposite(b) Figure7.14:(a)Amulti-frequencyreferencesignalfortra ckingcontroloftheVO 2 -integrated microactuator;(b)experimentaltrackingerrorsunderdif ferentself-sensingschemes. comparingwiththetrackingofalower-frequencysignal(Fi g.7.13),whichislikelyduetothe mildfrequency-dependenceofthedeection-resistancehy steresis(Fig.7.5)thatisnotcapturedin thesemodels. 132 Chapter8 RobustControlof VO 2 Microactuators usingSelf-SensingFeedback Inthischapter,self-sensing-basedrobustcontrolofVO 2 microactuatorsisstudied.Althoughthe resistancechangeisduetoaninsulator-to-metal-transit ion(IMT)andthemechanicalchangeis duetoastructural-phase-transition(SPT),thesetwomech anismsarestronglycoupled.Thus, self-sensingisachievedbymappingdeectiontoresistanc ewithahigh-orderpolynomial.Byem- ployingthistechnique,notonlytheimpactofhysteresisca nbereducedduetothehighlycoupled deectionandresistancechangesinVO 2 ,butthemeasurementsetupisalsogreatlysimplied. Therobustcontrollertakesintoaccounttheerrorinmodeli ngtemperature-deectionhysteresis, andenvironmentaldisturbances.Thecontrollertakesinto considerationtheerrorbetweenthede- siredandactualdeectionvaluesinordertopreciselycont rolthemicroactuator.Theperformance oftherobustcontrollerisalsocomparedtoaPIDcontroller . 8.1ExperimentalProcedures 8.1.1 VO 2 Deposition TheVO 2 thinlmwasdeposited,throughpulselaserdeposition,ona chipcontainingaSimicro- cantileverwithlength,width,andthicknessof300,35,and 1 m m,respectively.Themicroactuator 133 chipwasattachedtoaSitestpieceandwasplacedinavacuumc hamberwithamixedgaspressure ofargon(40%)andoxygen(60%)at20mTorrandmaintainedthr ougha30mindeposition.A ceramicheater,controlledat600 C,wasusedtoheatthesampleduringdeposition.Although thetemperatureatthesamplewasnotdirectlymeasured,aca librationdonebeforethedeposition approximatesthetemperatureat550 C.Thesamplewasalsorotatedthroughoutthedeposition toensureuniformtemperatureandthicknessdistribution. Akryptonuorideexcimerlaserwas focusedonarotatingvanadiumtarget5cmapartfromthesamp lewithanintensityof350mJand arepetitionrateof10Hz.Afterdeposition,theVO 2 thicknesswasmeasuredtobe172nm.To determinethequalityoftheVO 2 ,theresistanceofthelmontopofthetestchipwasmeasured as afunctionoftemperaturethroughaheating-coolingcycle( 20-85 C)(videinfra).Adropoftwo ordersofmagnitudeinlmresistanceisobserved,whichiss imilartoresistancechangesreported intheliteratureforstoichiometricpolycrystallineVO 2 onSisubstrates[96]. 8.1.2MeasurementSetup ThemeasurementsetupissimilarasFig.5.1.TheVO 2 -coatedSimicroactuator(shownincross- sectionalview)wasattachedtothesametestpieceusedduri ngdeposition,whichwasalsoSi coatedwiththesameVO 2 .Thistestpiecewasneededinordertocreatetheelectrical connections totheVO 2 andmeasureitsresistance.Thesecontactswerelocatednex ttothemicroactuatorchip andfabricatedbyevaporatingaluminumthroughacustom-ma demetalmask.Avoltagedivisor (notshownintheschematic)wasusedinordertomeasurether esistanceoftheVO 2 lm. Tomeasurethedeectionofthedevice,asensinglaser( l =808nm,0.5mW)wasfocused onthetipofthemicroactuatorandthereectedlightwasthe nfocusedontheactiveareaofa PSD.Acharged-coupleddevicecamerawasusedtoaidintheal ignmentofthelaser.ThePSD outputwasavoltageproportionaltothedeectionofthemic roactuator,whichwascalibratedby 134 sideviewimagesofthecantileveratdifferentdeectionva lues.Inparticular,thecalibrationof thePSDreadingwasdonebyrstassigningtheinitialdeect ionofthecantileverat20 Cas 0 m m.Thenthesamplewasheatedto85 C,whichresultedinthemaximumdeectionofthe cantilever,70 m m,asmeasuredfromthesideviewimagesofthecantilever.Th etotalvoltage changeinthePSDoutputfrom20to85 Cwas7.8V,resultinginameasurementsensitivity of0.111V/ m m.WenotethatthePSDusedinthisworkhadaresolutionof0.2 m mforthe laserspotdisplacement,whiletherangeforthelaserspotd isplacementwas13.875mmwhen thetemperaturevariedfrom20to85 C.ThismeansthatthePSDreadingwashighlyaccurate sincetheoutputresolutionwas1 : 4 10 5 oftheoperationalrange.APeltierheaterwasusedto controlthetemperatureofthesample.Thetemperatureatth eheaterwasmeasuredwithaplatinum temperaturesensor.Adataacquisitioncardandeldprogra mmablegatearray(DAQ/FPGA)was usedtoaccessthePSDoutput,theresistanceoftheVO 2 lm,andthetemperaturesensoroutput. TheDAQ/FPGAsystemwasprogrammedtoeither:1)controlthe temperatureofthePeltierin closedloopinordertomeasurethedeectionofthemicroact uatorandtheVO 2 resistanceof thetestpiecesimultaneously,or2)control,usingPIDorro bustcontroller,thedeectionofthe microactuatorbyself-sensingthedeectionthroughresis tance.Forbothcases,theDAQ/FPGA controlledthemagnitudeofthecurrentsignalsenttothePe ltierheater.Allthevariableswere controlledandobservedinacomputerconnectedwiththeDAQ /FPGAsystem. 8.2Self-SensingDeection Fig.8.1showsthemajorheating-coolingcycleofthemicroa ctuatordeectionandlmresistance asafunctionoftemperature.Thedeectioninthisworkisde nedasthetipdisplacementchange relativetotheinitialposition.Atotaldeectionof70 m mandaresistancedropoftwoorders 135 ofmagnitudeweremeasuredduringtheVO 2 transitionthroughatemperaturespanof15 C. Bothvariablesweresimultaneouslymeasured,andbymappin gdeectionwithresistance,itwas observedthatthehysteresisbetweenthedeectionandther esistancewasinsignicant,enabling theuseoftheresistanceofthelminthetestpiecetoestima tethedeectionwithouttheneedof physicallymeasuringitsvalue. Figure8.1:(a)VO 2 lmresistance,and(b)VO 2 -coatedmicroactuatoractualdeectionasafunc- tionoftemperaturethroughaheating-coolingcycle(20-85 C).Bothvariablesweresimultane- ouslymeasured. Thedeection-resistancemappingisshowninFig.8.2(a),w hichalsoincludesaninth-degree polynomialusedtoestimatethedeectionintheexperiment s.Thismodelwasobtainedfrom ttingtheaverageoftheheatingandcoolingcurvesandwasu sedasthedeectionsensingmech- anismintheclosed-loopdeectioncontrolexperimentsdon einthiswork.Themaximumerrors betweentheheating/coolingcurvesandtheself-sensingmo delareshowninFig.8.2(b).Forawide rangeoftheresistance,thedeectionestimationerrorwas lowerthan2 m mwhereasslightlylarger estimationerrorwasfoundatthetwoends.Itisobservedtha tsomehysteresisremains.Thisis believedtobeduetotheslightlydifferentenergyrequirem entsbetweentheIMTandtheSPT[87]. Thishypothesisissupportedbythefactthatthisdifferenc einenergyrequirementshasbeenfound 136 tobemorepronouncedattheonsetofthephasetransition,wh ichwouldcorrespondtothehigher resistance-lowdeectionregioninFig.8.2(a).Hereinaft er,theestimatedandmeasureddeection valueswillbeaddressedasself-sensedandactualdeectio ns,respectively. etal. -coatedmicroactuatoractual Fig.3.(a)VO-coatedmicroactuatoractualde"ectionasafunctionofVO Figure8.2:(a)VO 2 -coatedmicroactuatoractualdeectionasafunctionofVO 2 lmresistance duringtheheating-coolingcycle.Apolynomialfunctionof degree9wasusedtomodelthe deection-resistancemapping.(b)Maximummodelerrorobt ainedfromthemajorheatingand coolingcurves. Twotypesofcontrollerswillbeconsideredandcompared:1) aPIDcontroller,whichonly considerstheerrorbetweenthecontrolledvariable(inthi scaseself-senseddeection)withthe desiredreferencesignal,and2)arobustcontroller,which asidefromconsideringtheerrorfrom thecontrolledvariable,alsoaccommodatestheerrorbroug htbytheself-sensingmodel,noises, andsystemuncertainties. 8.3RobustControllerDesign Following H ¥ designtechniques[97],an H ¥ controllerwasdesignedtoaccommodateperturba- tions,noises,andmodeluncertaintiesindeectiontracki ng. 137 8.3.1Modelingof VO 2 Microactuator Theblockdiagramforthesimpliedphysicalclosed-loopsy stemwithself-sensingisshownin Fig.8.3(a)andthemodeledclosed-loopsystemaugmentedwi thweighteduncertaintiesisshown inFig.8.3(b).Thevariable y ref isdenedasthedesireddeectionoutput.Theinputofcontr oller K ( s ) isthedeectionerrordenedasthedifferencebetween y ref andtheself-senseddeection y self .Thecontrolleroutputisthecurrent I c ,whichiscorruptedby I d thataccountforenviron- mentaldisturbances d . A ( s ) denotesthetransferfunctionforthePeltierheaterwithte mperature T asitsoutput.Thehysteresisbetweentemperature T andactualdeectionoftheVO 2 cantilever y ismodeledbythesummationofalinearrelationship K c andanoise n 1 .TheVO 2 lmresis- tanceisdenedas R .Theself-sensingerroristakenintoconsiderationthroug hanoise n 2 .The functions W u , W d , W n 1 , W n 2 ,and W e areweightingfunctionsthatrepresenttheimportanceofth e correspondingsignals.Forexample, W u reectsthecontroleffortconstraintsand W e accountsfor actualdeectionperformance.ThevariablesŸ u andŸ e areweightedinputandweighteddeection error,respectively. Figure8.3:Blockdiagramsofthe(a)simpliedphysicalclo sed-loopcontrolsystemwithself- sensingand(b)closed-loopsystemaugmentedwithweighted functions. 138 ThetemperaturedynamicsduetothePeltierheateraremodel edbyarst-ordersystemanda timedelayrepresentedbyarst-orderPad´eapproximation [98] A ( s )= t d 2 s + 1 t d 2 s + 1 ! A 0 t s + 1 (8.1) where t isthetimeconstantassociatedwiththesystemtransient, A 0 isthegain,and t d is thetimeconstantassociatedwiththesystemdelay.Inthisw ork,thedeectiontransferfunction wasassumedtohavenodynamicssincethetimeconstantsasso ciatedwithheattransferthrough thecantileveranddragproducedbyairaremuchlowerthanth atofthePeltierheaterdynamics. Hence,thePeltierheaterdynamicresponsewasconsideredt hedominantdynamicinthesystem understudy. Aseriesofopen-loopstepinputexperimentswereconducted toidentifythesystemparameters andtheresultsareshowninTable8.1.Theseweredonebymanu allycontrollingthecurrent throughthePeltierandmeasuringthetemperaturetransien ts.Theinitialcurrentvalueforeach experimentwaszero,whichcorrespondedto25 C(roomtemperature).Onlyheatingstepswere consideredinthisparameteridenticationandaparameter d wasusedasanuncertaintyparameter duetothedifferencesfromheatingandcoolingwith d 2 [ 1 ; 1 ] .Fromthemeasureddata,the timeconstant t andthegain A 0 oftheplantmodelwerecalculatedtobe25(1+0.2 d )and50 C/A,respectively.The0.2factorthatmultiplies d ischosentocovertherangeofmeasuredtime constants,whichspanfrom20to30s.Thetimedelay t d ,whichisdenedhereasthetime intervalbetweenachangeintheinputcurrenttothesystema ndthetemperatureresponsetothat signal(deadtime),wasexperimentallymeasuredtobe0.375 s. Thereexistsaconsiderableamountofhystereticnonlinear itybetweentemperatureandde- ectionoftheVO 2 microactuator.Althoughthereareseveralmodelsthatcapt urethehysteresis 139 Table8.1:Steady-statevaluesofstepexperimentsforsyst emidentication. CurrentI ( A ) Temperature T ( C)Timeconstant t (s) -0.2(heating)39.720.8 -0.4(cooling)49.227.4 -0.6(heating)59.122.6 -0.8(cooling)70.128.2 memoryeffectsinmicroactuators,duetothecomplexitycon siderationinonlineprocessing,the hysteresisnonlinearityinthisworkwasapproximatedasfo llows: y = K e T + n 1 W n 1 (8.2) where K c istherateofchangeindeectionasafunctionoftemperatur eacrossthetransition,which wasidentiedtobe5.9 m m/ C.Thesecondterminthesumrepresentsthehysteresismodel ing error.Theself-senseddeectionwasthenmodeledby y self = K e T + n 1 W n 1 + n 2 WW n 2 (8.3) wherethethirdterminthesumrepresentstheself-sensinge rrorobtainedwiththehigh-orderpoly- nomialinFig.8.3(a).Byconsideringthesemodelingerrors andtheremainingweightingfunctions inFig.8.3(b),arobustcontrolframeworkcanbedesignedth ataccommodatesenvironmentaldis- turbances,modelingerrors,anduncertaintieswhileeffec tivelycontrollingtheactualdeectionof theVO 2 -basedmicroactuators. 140 8.3.2RobustControllerDesign Fig.8.4(a)showsthesystemframeworkafteralinearfracti onaltransformation(LFT),whichfa- cilitatesthe H ¥ controllerdesignprocess.Thetransferfunctions D ( s ) , P ( s ) ,and K ( s ) denotethe uncertainty,interconnectionmatrix,andcontroller,res pectively.Theinterconnectionmatrixis denotedasfollows: P ( s )= 2 6 4 P 11 ( s ) P 12 ( s ) P 21 ( s ) P 22 ( s ) 3 7 5 wherethelowerLFTisdenedas F l ( P ; D l )= P 11 + P 12 D l ( I P 22 D l ) 1 P 21 and,similarly,the upperLFTas F u ( P ; D u )= P 22 + P 21 D u ( I P 11 D u ) 1 P 12 withcompatibledimensions.The H ¥ controldesignobjectiveistondacontroller K ( s ) ,suchthat,given g > 0,itminimizesthe H ¥ normofthetransferfunctionfromtheinput W = dn 1 n 2 y ref > ,whichincludesthenoises, disturbance,andreferencesignal,totheoutput z = Ÿ e Ÿ u > ,whichincludesthecontroleffortand trackingerror,bysolving k F l ( F u ( P ( s ) ; 0 ) ; K ( s )) k ¥ < g (8.4) where F u ( P ( s ) ; 0 ) representsthenominalmodeland kk denotesthe H ¥ norm. Choosingappropriateweightsisverycrucialinrobustcont roldesign.Themainguidelines inthisworkareasfollows:1)thecontroleffortweight W u anddisturbancerejectionweight W d areveryimportantacrossawidefrequencyrangeinordertod ealwithdisturbanceswitharbitrary frequencies;2)theactualdeectionperformanceweight W e isalsogivenimportance,especially atlowfrequenciessincethefrequencyofthedesireddeect ionsignalisrelativelylowduetothe relativebigtimeconstantofthetemperaturedynamics;and 3)thenoiseweightings W n 1 and W n 2 141 Figure8.4:(a)Frameworkof H ¥ controlforthesystemand(b)robustperformancetestbyaug - mentingtheuncertainty D toM. aremoreimportantathigherfrequenciessincenoiseswillu suallyhavehigherfrequenciesthan thoseofthereferencesignals.Withtheseguidelines,thet ransferfunctionsfortheweightsare chosenasfollows: W u = 0 : 12 ( s + 1 ) ( s 10 + 1 ) ; (8.5) W d = 0 : 1 ( s 2 + 1 ) ( s + 1 ) ; (8.6) W e = 0 : 09 1 ( s 10 + 1 ) ; (8.7) W n 1 = 0 : 002 ( s + 1 ) ( s 100 + 1 ) ; (8.8) W n 2 = 0 : 002 ( s + 1 ) ( s 100 + 1 ) : (8.9) Basedonthemodelparametersandtheweightingfunctionsfo rthemodelinFig.8.3(b),the systemcanbeexpressedinthefollowingstate-spacerepres entation, P ( s )= C ( sI A ) 1 B + D , where 142 A = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 : 04005000 50 0 10050000 0 : 240 100100 000 5 : 330010 : 67 0000 10000 00000 1000 000000 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; B = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 : 008002 : 5050 0000010 : 8 0 : 048 0 : 20010 000 5 : 330 10 : 67 019 : 800 00 0019 : 8000 000 0 : 0500 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; C = 2 6 6 6 6 6 6 6 6 6 4 1000000 000 : 90000 0 100000 0 : 24000110 3 7 7 7 7 7 7 7 7 7 5 ; 143 D = 2 6 6 6 6 6 6 6 6 6 4 0 : 200000 000000 000001 : 2 0 : 048 0 : 2 0 : 2010 3 7 7 7 7 7 7 7 7 7 5 : Thetransferfunctionofthecontrolleriscalculatedbased onthealgebraicRiccatiequation K ( s )= 918 ( s + 1 : 1 )( s + 5 : 3 )( s + 8 : 2 )( s + 100 ) ( s + 40830 )( s + 89 )( s + 1 )( s + 4 : 4 1 : 6 i ) : (8.10) InordertoimplementtheresultingcontrollerinEq.(8.10) itischangedtoitsdiscretez- transformwithasamplingtimeof125ms.Thissamplingtimew asmorethananorderofmagni- tudefasterthantheclosed-loopcontrolresponseofthesys tem,whichensuredafullyreconstructed signal.Thevariable g opt istheoptimumoverall g suchthatthecontrollerisadmissible,anditwas calculatedtobe0.118. Inordertoverifytherobustnessoftheclosed-loopsystem, therearetwospecicationsto test:robuststabilityandrobustperformance.Black-Nich olsdiagramhasbeenutilizedtoanalyze therobuststabilityin[99],whereassmallgaintheory[100 ]and m synthesis[97,101]havebeen utilizedtoanalysisrobuststabilityandrobustperforman ce.Thelatterisauniedapproachfor analyzingrobuststabilityandrobustperformancewithmul tiplesourcesofuncertainties,which isadvantageousoverthesmallgaintheoryapproach.Thus, m synthesisisadoptedinthiswork, whichisrepresentedinFig.8.4(b). Totestforsystemrobuststability,denote M ( s )= F l ( P ( s ) ; K ( s ))= 2 6 4 M 11 ( s ) M 12 ( s ) M 21 ( s ) M 22 ( s ) 3 7 5 ; 144 if k M 11 k ¥ < 1 b and b > 0arealwayssatisedforall D ( s ) with k D k ¥ < b ,thenthesystemis robustlystable.ForthesystemshowninFig.8.3(b),itisve riedthat k M 11 k ¥ = 0 : 26,which makesthesystemrobustlystablesince k D k ¥ = k d k ¥ < 1.Totestforrobustperformance,assume M ( s ) have q 1 + q 2 inputsand p 1 + p 2 outputs, M 11 ( s ) has q 1 inputsand p 1 outputs,anddenote D p = 8 > < > : 2 6 4 D 0 0 D f 3 7 5 : D 2 D ; D f 2 C q 2 p 2 9 > = > ; where D p = d 2 [ 1 ; 1 ] inthisworkand D f isshowninFig.8.4.Astructuredsingularvaluecan bedenedasfollows: m D p ( M )= 1 min [ s ( D ) : D 2 D det ( I M D )= 0 ] (8.11) where s ( D ) isthelargestsingularvalueof D . Ifforall D ( s ) with k D k ¥ < b and b > 0,sup w 2 R m D p ( M ( jw )) 1 = b isalwayssatised,then thesystemhasrobustperformance.BasedonD-Kiterations, sup w 2 R m D p ( M ( jw )) isfoundtobe 0.32,forthecasepresentedhere.Hence,therobustperform anceofthedesignedrobustcontroller isveried. 8.4ExperimentalResult Experimentalresultsareprovidedusingthe H ¥ controllerforstepandmultisinusoidalreference inputswithandwithoutaddednoisetothecurrentgenerator .Itsperformanceiscomparedto thatofaPIDcontrollerinordertoshowitsrobustnesstonoi sesandperturbations,aswellasthe controleffortadvantagesoverthePIDcontroller.Wehavec hosenaPIDcontrollerinsteadof aproportional-derivativecontrollerforthecomparison, because,forthedynamicsshowninthe 145 system,aPDcontrollerwouldresultinnonzerosteady-stat eerrorforstepreferencesevenunder idealconditions.TheRMSEhasbeenselectedtoquantifythe trackingerrorinalltheexperiments, althoughtheaveragesteady-stateerrorhasbeenalsocalcu latedforexperimentswithstepreference inputs.Standarddeviation(SD)wasusedtomeasurethecont roleffortinthemultisinusoidal referenceinputexperimentwithandwithoutnoise. TheparametersofthePIDweretunedinsimulationbasedonth esamenominalmodelshown inFig.8.3(b).Sincemostofthepotentialapplicationsfor thepresentedmicrobenderswillrequire high-precisionandfastresponse,thePIDcontrollerwasde signedtohaveanovershootoflessthan 2%,whichwouldensureaccuracyduringthetransients,andt ohaveasettlingtimeofsmaller than10s,whichwouldensurerelativelyfastresponsegiven thatthetimeconstantofthePeltier dynamicswasapproximately25s.Theresultingcontrollerp arameterswere:proportionalgain K p = 0 : 059,integralgain K i = 0 : 004,andderivativegain K d = 0 : 0136.Theobtainedtransfer functionforthecontrollerwastransformedtoitsdiscrete counterpartinthez-transformwiththe samplingtimeof125msforimplementation,similartothero bustcontrollercase. 8.4.1StepReferenceTracking Experimentswithstepreferenceinputsweredesignedsotha tthemicroactuatorfollowedasetof threedifferentsetpoints,eachwithdurationof15s,progr ammedintheDAQ/FPGA.Thegoalof theseexperimentswastostudythetransientbehaviorandst eady-stateerroroftherobustcontroller andcomparethosetotheperformanceofPIDcontroller.Fig. 8.5(a)and(b)showstheexperimental performanceintermsoftheactualdeectionandself-sense ddeection.Althoughthecontrolled variableistheself-senseddeectionandabettersteady-s tateperformanceisobservedinFig.8.5(b) forthePID,Fig.8.5(a)showsthattheactualsteady-stated eectionundertherobustcontroller isclosertothesetpointforeverystepvalue,whereasithas ahigherdifferenceunderthePID 146 controller.Theactualsteady-statedeectionerrorsandc ontroleffortsareshowninFig.8.6(a)and (b). analyzetherobuststabilityin[39],whereassmallgainthe- synthesis[34],[41]havebeenutilizedtoanalysis robuststabilityandrobustperformance.Thelatterisauni!ed approachforanalyzingrobuststabilityandrobustperformance withmultiplesourcesofuncertainties,whichisadvantageous synthesisisadopted (12) arealwayssatis!edforall ,thenthesystemisrobustlystable.Forthe systemshowninFig.4(b),usingthecontrollerin(11),itisve r- ,whichmakesthesystem in- A.TrackingofStepReferenceInput Figure8.5:(a)Actual,and(b)self-sensedmicroactuatord eectionunderself-sensed,closed-loop PIDandrobustcontrolthroughaseriesofstepreferenceinp uts. etal. Fig.7.(a)Actualde!ectionerrorand(b)controllereffortforthePIDand B.TrackingofMultisinusoidalReferenceInput Fig.8.Microactuatorde!ectionresponsetoamultisinusoidalreferenceinput underPIDandrobustcontrol. C.NoiseRejectionforMultisinusoidalReferenceTracking Figure8.6:(a)Actualdeectionerrorand(b)controlleref fortforthePIDandrobustcontrol approachesthroughthestepreferencetrackingexperiment . Table8.2comparestheRMSEoftheactualdeectionandtheav erage(overthreesetpoints) steady-statedeectionerror.Althoughthelargesttracki ngerrorforbothcontrollersissimilar, theRMSEandaveragesteady-stateerrorundertherobustcon trollerare3.66%and36%,respec- tively,lessthanthoseofthePID.Thisprovesthatthedesig nedrobustcontrolleroutperformsthe 147 Table8.2:Controllercomparisonforstepreferencetracki ng. Approach RMSE( m m)Averagesteady-stateerror( m m) PID6.281.11 Robust6.050.62 PIDcontrollerineffectivelyandrobustlyreducingtheste ady-stateerrorofthemicroactuator's actualdeectionbyconsideringtheself-sensingmodeling error.Inpractice,theactualdeection performanceisofrelevance,thusonlytheactualdeection isprovided. Theadvantagesoftherobustcontrollerintermsofsettling timeoverthePIDcontrollercanbe noticedfromFig.8.5(a),althoughthereishigherovershoo twiththerobustcontroller.Acloser examinationofthecontrollerefforts,quantiedhereasth eamountofcurrentchange,revealsthat therobustcontrollerperformsslightlyhigherworkthanth ePIDforlongertimeduration(seeFig. 8.6(b)),whichexplainsthetransientdifferences. 8.4.2Multi-frequencySignalsReferenceTracking Experimentsinvolvingmultisinusoidalreferenceinputsw erecarriedouttostudytheperformance ofthemicroactuatorundercontinuousinputchanges.Forth isexperiment,thesumofthreediffer- entsinusoidalwaveformswithfrequenciesof0.001,0.005, and0.01Hz,maximumamplitudeof 20 m mandanoffsetof35 m mwaschosenastheinputsignal.Fig.8.7showstheactualde ection ofthemicroactuatorasafunctionoftimewithPIDandrobust control.Fromtheobserveddata,it isseenthattherobustcontrollerperformanceisbettertha nthatofthePID.Thisismoreevident bylookingatthetrackingerrorsandcontroleffortsundert hetwocontrollers,whichareshownin Fig.8.8(a)and(b),respectively. ThevaluesforRMSEandSDcalculatedforthisexperimentare summarizedinTable8.3.It 148 etal. B.TrackingofMultisinusoidalReferenceInput Fig.8.Microactuatorde!ectionresponsetoamultisinusoidalreferenceinput C.NoiseRejectionforMultisinusoidalReferenceTracking Figure8.7:Microactuatordeectionresponsetoamultisin usoidalreferenceinputunderPIDand robustcontrol. etal. Fig.7.(a)Actualde!ectionerrorand(b)controllereffortforthePIDand robustcontrolapproachesthroughthestepreferencetrackingexperiment. andaveragesteady-stateerrorundertherobustcontrollerare 3.66%and36%,respectively,lessthanthoseofthePID.This provesthatthedesignedrobustcontrolleroutperformsthePI D controllerineffectivelyandrobustlyreducingthesteady-stat e errorofthemicroactuator™sactualde!ectionbyconsideringthe self-sensingmodelingerror.Inpractice,theactualde!ection performanceisofrelevance,thusonlytheactualde!ectionis Theadvantagesoftherobustcontrollerintermsofsettling timeoverthePIDcontrollercanbenoticedfromFig.6(a), althoughthereishigherovershootwiththerobustcontroller.A B.TrackingofMultisinusoidalReferenceInput Fig.9.(a)Actualde!ectionerrorand(b)controllereffortforthePIDand C.NoiseRejectionforMultisinusoidalReferenceTracking Figure8.8:(a)Actualdeectionerrorand(b)controlleref fortforthePIDandrobustcontrol approachesinthemultisinusoidalreferencetrackingexpe riment. 149 Table8.3:Controllercomparisonformultisinusoidalrefe rencetracking. Approach Largesterror( m m)RMSE( m m)ControleffortSD(mA) PID3.001.2449.1 Robust3.071.0248.2 canbecalculatedthattherobustcontrollerhasaround18%l esstrackingRMSEand1.8%less controleffortthanthePIDcontroller.Theeffectivenesso ftherobustcontrollerinreducingthe steady-stateerroroftheactualdeectionisagainveried experimentally. 8.4.3NoiseRejectionforMulti-frequencySignalsReferen ceTracking Inordertostudytherobustnessofthe H ¥ controllertoenvironmentaldisturbances,modeledin Fig.8.3(b)as I d ,awhitenoisesignalwithmaximumvalueof 0.01Aandband-limitof8Hz wasaddedtothecontrollereffort.Thesameinputsignaluse dinthemultisinusoidalreference inputexperimentwithoutnoisewasadoptedforthisstudy(s eeFig.8.7).Thewhitenoiseampli- tudecorrespondedto25%ofthetotalcurrentchangeobserve dinFig.8.8(b),whichrepresented anoverestimateofreallifenoisesignalsduetocurrentvar iations.Fig.8.9(a)showstheactualde- ectionsofthemicroactuatorwiththenoisyinputunderrob ustandPIDcontrol.Itisseenfrom theobserveddatathattheclosed-loopdeectionsystemund erthe H ¥ controllerperformsrobustly againstnoisedisturbancesbetterthanwiththePID.Thisdi fferenceinperformanceisevidentin Fig.8.9(b)wherethelargesterrorbetweenactualdeectio nandreferenceinputwithrobustcontrol is3.07 m mandforPIDis4.28 m m.Thistranslatestoa28%decreaseinthelargesterrorfort he robustcontroller.WhilethelargesttrackingerrorforPID controllerundernoisytrackingis43% largerthanthenoiselesstrackingcase,therobustcontrol lerendswithnotevidentdeteriorationin thelargesttrackingerror.Fig.8.10(a)showsthecontrole ffortappliedbybothcontrollersinthese 150 Table8.4:Controllercomparisonformultisinusoidalrefe rencetrackingwithnoise. Approach Largesterror( m m)RMSE( m m)ControleffortSD(mA) PID4.281.4733.6 Robust3.070.9631.6 experimentsand,forclarity,aseparateplotinFig.8.10(b )showsthePeltierinput,whichincludes thecontrollereffortandthewhitenoise.Table8.4showsth eRMSEandcontroleffortSDvalues forthisexperiment.Forthesystemwithrobustcontrol,the RMSEis34.7%lessthanwithPID, andthecontroleffortSDis6%less.Althoughtherobustcont rolRMSEinthisexperimentis closetotheoneobtainedwithoutnoise(seeTable8.3),anin creaseof18.6%isobservedforthe systemunderPIDcontrol.Thisveriestherobustperforman ceandstabilityofthecontrollerto compensatefordeectioncontrol,notonlyformodelingerr ors,butalsoagainstenvironmental disturbances. Fig.10.(a)Microactuatorde!ectionresponseand(b)errorunderamultisi- Figure8.9:(a)Microactuatordeectionresponseand(b)er rorunderamultisinusoidalreference inputunderPIDandrobustcontrolwithinducedwhitenoisei ncurrentinputtothesystem. InadditiontotheRMSEandthemaximumtrackingerrors,weha vefurtherconductedfast FouriertransformofthetrackingerrorsunderthePIDand H ¥ controllers,forthescenarioswith andwithoutinjectedactuationnoises.AsshowninFig.8.11 ,thetrackingerrorunderthe H ¥ 151 Fig.11.(a)Controleffortand(b)Peltierinputforamultisinusoidalreference Fig.12.Frequencyspectrumanalysisofthetrackingerrorsund erthero- bustcontrollerandthePIDcontroller,forscenarioswithandwithoutinjected actuationnoise. experiment.Forthesystemwithrobustcontrol,theRMSEis 34.7%lessthanwithPID,andthecontroleffortSDis6%less. AlthoughtherobustcontrolRMSEinthisexperimentisclose totheoneobtainedwithoutnoise(seeTableIII),anincrease of18.6%isobservedforthesystemunderPIDcontrol.This veri"estherobustperformanceandstabilityofthecontrollerto compensateforde!ectioncontrol,notonlyformodelingerrors, butalsoagainstenvironmentaldisturbances. wehavefurtherconductedfastFouriertransformofthetracking errorsunderthePIDandH andwithoutinjectedactuationnoises.AsshowninFig.12,the trackingerrorundertheH thePIDcontrollerforallfrequencycomponents.Thisistruefor Figure8.10:(a)Controleffortand(b)Peltierinputforamu ltisinusoidalreferenceinputunderPID androbustclosed-controlwithinducedwhitenoiseincurre ntinputtothesystem. controllerislowerthanthatunderthePIDcontrollerforal lfrequencycomponents.Thisistruefor boththecaseswithandwithoutnoise. Fig.10.(a)Microactuatorde!ectionresponseand(b)errorunderamultisi- nusoidalreferenceinputunderPIDandrobustcontrolwithinducedwhitenoise Fig.12.Frequencyspectrumanalysisofthetrackingerrorsund erthero- Figure8.11:Frequencyspectrumanalysisofthetrackinger rorsundertherobustcontrollerandthe PIDcontroller,forscenarioswithandwithoutinjectedact uationnoise. 152 Chapter9 ConclusionsandFutureWork 9.1Conclusions Inthiswork,themodeling,identication,andcontrolofhy stereticsystemshavebeenexplored.A fewnewcontributionshavebeenmadetothemodelingandcont rolofVO 2 microactuators. First,toolsfrominformationtheoryareutilizedtooptima llycompressthePreisachoperator andtheGPImodelunderagivencomplexityconstraint.Theco mpressedhysteresismodelsachieve highdelitywhilemaintainingrelativelylowcalculation andstoragecomplexity.Whileduetothe particularsettingofthePreisachplane,theoptimalcompr essionofthePreisachoperatorinvolves anexhaustivesearch,theoptimalcompressionoftheGPImod elisreformulatedasanoptimal controlproblemandsolvedwithdynamicprogramming.Thepr oposedschemesareveriedwith simulationresultsaswellasexperimentalresults,wheret hehysteresisbetweenVO 2 resistance andtemperatureismodeled. Second,identicationofthePreisachoperatorisstudiedu nderthecompressivesensingframe- workthatrequiresfewermeasurements.Theproposedapproa chadoptstheDCTtransformof theoutputdatatoobtainasparsevector,wheretheorderofa lltheoutputdataisassumedtobe known.Themodelparameterscanbeefcientlyreconstructe dusingtheproposedscheme.The least-squaresschemeisalsoimplementedasacomparison.T heproposedidenticationapproach isshowntohavebetterperformancethantheleast-squaress chemethroughbothsimulationand experimentsinvolvingaVO 2 -integratedmicroactuator. 153 Third,aphysics-motivatednon-monotonichysteresismode lthataccountsforthetwocompet- ingactuationmechanismsispresented.Therstmechanismi sthestressresultingfromstructural changesinVO 2 ,whichismodeledwithamonotonicPreisachoperatororaGPI model.Thesecond mechanismisthedifferentialthermalexpansioneffect.Ef cientinversecompensationschemesare developedfortheproposednon-monotonichysteresismodel s. Fourth,self-sensingfeedbackcontrolforVO 2 microactuatorisstudied.Theproposedcom- positeself-sensingapproachexploitsthephysicalunders tandingthatboththeresistanceandthe deectionhavedifferenthystereticrelationshipswithth etemperature.Aconceptoftemperature surrogateisexploitedinthealgorithm.Theself-sensings chemeisvalidatedexperimentallyin feedbackcontrol,whereaproportional-integralcontroll erisused. Finally,an H ¥ robustcontrollerisdesignedandimplementedforprecisio ndeectioncontrol, wheretheuncertaintiesproducedbythehysteresisbetween thedeectionandthetemperatureinput andtheerrorintheself-sensingmodelareaccommodated.He rewetakeasimplerself-sensing approachthatmodelsthedeectionasapolynomialfunction oftheresistance.Theproposed robustcontrolapproachisexperimentallydemonstratedto beabletomitigatetheimpactofthe self-sensingerrorandotherdisturbances. 9.2FutureWork First,forthecompressivesensing-basedidenticationof thePreisachoperator,thetransforma- tionofthedensitywillbefurtherstudiedtogeneratespars ersignals.TheuseofMoore-Penrose pseudoinverse[81]willbestudiedforcaseswheretheinput sequencehasadifferentnumberof entriesfromthenumberofmodelparametersandwherethemat rix S isnotinvertable.Compres- sivesensing-basedidenticationwillalsobeexploredfor otherhysteresismodels,suchastheGPI 154 model. Second,wewillexaminetheperformanceofthecompositesel f-sensingschemeinadvanced controlofVO 2 microactuators.Forexample,robustcontrolmethodswillb eexploredtominimize theimpactoftheself-sensingerroronthetrackingperform ance.Inaddition,thecurrentcompos- iteself-sensingmodelisrate-independent.Whileithasde monstratedgoodperformanceoverall intrackingcontrol,themildrate-dependencesuggeststha tarate-dependent(forexample,accom- modatingthestructuraldynamicsoftheactuator),composi tehysteresismodelcouldofferfurther enhancedtrackingperformanceathigherfrequencies. 155 APPENDICES 156 AppendixA ReviewofthePreisachOperator ForamoredetailedtreatmentofPreisachoperator,readers arereferredto[9,11,12]. PreisachOperator APreisachoperatorconsistsofaweightedsuperpositionof acontinuumofbasichystereticele- ments,calledPreisach hysterons .AgenericPreisachhysteron, g b ;a ,isadelayedrelaycharacter- izedbyapairofthresholds ( b ; a ) .Theoutputevolveswiththeinput v ,whereaninitialcondition z 0 ( b ; a ) 2f 1 ; 1 g isneededtofullydescribethebehaviorof g b ;a : g b ;a [ v ( ) ; z 0 ( b ; a )]= 8 > > > > > < > > > > > : + 1if v > a 1if v < b z 0 ( b ; a ) if b v a ; (A.1) where v ( ) denotestheinputhistory v ( t ) ,0 t t . TheoutputofaPreisachoperator G ,withinput v andinitialcondition z 0 = f z 0 ( b ; a ) ; b a g canthenberepresentedas: u ( t )= G [ v ; z 0 ]( t )= Z P 0 m ( b ; a ) g b ;a [ v ; z 0 ( b ; a )]( t ) d b d a ; (A.2) where m isameasurabledensityfunctiontypicallyassumedtobenon negative.Eachpoint ( b ; a ) in 157 the Preisachplane ,denedas P = f ( b ; a ) : b a g ,isidentiedwiththehysteron g b ;a .Because oftheinputrangeconstraintsorphysicalsaturation(e.g. ,completephasetransitionbeyondcertain inputrangeforVO 2 ),itoftensufcestoconsider m withnitesupport f ( b ; a ) : v min b a v max g in P .ThestateofthePreisachoperator,namely,theoutputsofa llhysteronscanbecaptured by memorycurve ,astaircase-structuredlinein P separatinghysteronswithoutput + 1fromthose withoutput 1. DiscretizationofPreisachOperator Thedensityfunction m ofhysteronsistheparameterofthePreisachoperator.Forp arameter identication,adiscretizationstepistypicallyinvolve dtoobtainanitenumberofparameter values.ForthePreisachoperator,oneschemeistoapproxim atetheoperatorwithanitenumber ofhysteronslocatedatthecenterofuniformlyspacedlatti cecellsinthePreisachplane[12].This isequivalenttoapproximatingtheweightingfunctionbyas umofimpulsefunctionslocatedat thecellcenters,whichresultsinadiscontinuousoutputun deracontinuousinput.Analternative scheme,stillbasedonuniformdiscretizationofthePreisa chplane,approximatesthedensitybya piecewiseconstantfunctionŒthedensityvalueisconstant withineachlatticecellbutcouldvary fromcelltocell[7].Fig.A.1showsanexampleofuniformdis cretizationofPreisachplane.Under thisscheme,thePreisachoperatorhas M ( M + 1 ) = 2densityparameters,where M isthelevelof uniformdiscretizationalong a (orequivalently, b )directioninthePreisachplane.Thisscheme producesacontinuousoutputunderacontinuousinput;furt hermore,efcientschemesforthe identication[25]andinversion[9]areavailable. TheoutputofthePreisachmodel(inthediscrete-timesetti ng)attime n canbeexpressedas: 158 ba(1,1) (1,2) (1,3) (1,4) (2,1) (2,2) (2,3) (3,1) (3,2) (4,1) max vmin vmax v1b2b3bmin vFigureA.1:IllustrationofuniformdiscretizationoftheP reisachplane,wherethediscretization level M = 4. Ÿ u ( n )= u c + M å i = 1 M + 1 i å j = 1 m ij s ij [ n ] ; (A.3) where u c representsaconstantcontributionfromhysteronsoutside theactivePreisachplane, m ij isthedensityvalueforcell ( i ; j ) ,and s ij [ n ] representsthe signed areaofthecell ( i ; j ) ,namely, itsareaoccupiedbyhysteronswithoutput + 1minusthatoccupiedbyhysteronswithoutput 1. Thecalculationcomplexityis O ( M 2 ) ,andwhen M islarge,thecalculationandstoragecostcanbe prohibitive. Inversion Apredominantclassofcontrolapproachesinvolveapproxim atecancellationofthehysteresisef- fectthroughinversion[6,9,10].Byconstructinganapprox imate(right)inversetothePreisach operator(Fig.A.2),thehysteresiseffectcanbe(mostly)c ancelled. 159 G‹Gref yuyFigureA.2:Inversecompensationofhysteresis. Theinversionschemeusedin[9]exploitsthepiecewisecons tantstructureofthedensityfunc- tionandthepiecewisemonotonitypropertyoftheoperator. Whiletheschemewasinitiallydevel- opedforaPreisachoperatorwithuniformdiscretizationof thePreisachplane,itiseasilymodied toaccommodateanoperatorwithnonuniformdiscretization ,withoutincreasingcomputational complexity. 160 AppendixB ReviewofthePrandtl-IshlinskiiModels AbriefoverviewoftheclassicalPrandtl-Ishlinskii(CPI) modelandgeneralizedPrandtl-Ishlinskii (GPI)modelsareprovided.Readersarereferredto[13Œ15,2 2]formoredetails. ClassicalPrandtl-Ishlinskii(CPI)Model TheCPImodelconsistsofaweightedsuperpositionofbasicp lay(orstop)operators.TheCPI modelislimitedtomodelingsymmetricandnon-saturatedhy steresis.AsillustratedinFig.B.1(a), theplayoperatorischaracterizedbyitsradius r .Foragiveninputfunction v ( t ) ,theoutput w ( t ) of aplayoperatorwithradius r andinitialcondition w ( t ) isdenedas w ( t )= F r [ v ]( t )= f r ( v ( t ) ; F r [ v ]( t )) ; (B.1) where f r ( v ( t ) ; w ( t ))= 8 > > > > > < > > > > > : max ( v ( t ) r ; w ( t )) ; if v ( t ) > v ( t ) min ( v ( t )+ r ; w ( t )) ; if v ( t ) < v ( t ) w ( t ) ; if v ( t )= v ( t ) ; (B.2) and t = lim e > 0 ;e ! 0 t e . 161 TheoutputofaCPImodelisexpressedasanintegralinthefol lowingform: y P ( t )= Z R p 0 p ( r ) F r [ v ]( t ) dr ; (B.3) where p ( r ) istheweightingfunctionofthePrandtl-Ishlinskiimodel, whichisusuallychosentobe non-negative,and R p representsthemaximumplayradius. Forpracticalimplementation,theCPImodelisrepresented asaweightedsummationofanite numberofplayoperatorsasfollows: y P ( t )= p ( r 0 ) v + N å j = 1 p ( r j ) F r j [ v ]( t ) ; (B.4) where r j > 0istheplayradiusofthe j -thplayoperator, p ( r j ) isthecorrespondingweight,and N denotesthenumberofplayoperators. GeneralizedPrandtl-Ishlinskii(GPI)Model TheGPIhysteresismodelcancapturecomplexhysteresisloo pswithbothasymmetryandoutput saturation[13Œ15,28]. Followingasimilartreatmentasin[13,28],ageneralizedp layoperatorwithradius r isdened by(seeFig.B.1(b)) w ( t )= F g r [ v ]( t )= f g r ( v ( t ) ; F g r [ v ]( t )) ; (B.5) 162 vwrr-(a) i() Ri-r g() Li+r gmin imax iw(b) FigureB.1:Input-outputrelationshipsof(a)aclassicalp layoperatorwithradius r ;(b)ageneral- izedplayoperatorwithradius r (shownassolidcurves). where f g r ( t ; w ( t )) isdenedas f g r ( v ( t ) ; w ( t ))= 8 > > > > > > < > > > > > > : max ( g L ( v ( t )) r ; w ( t )) ; if v ( t ) > v ( t ) min ( g R ( v ( t ))+ r ; w ( t )) ; if v ( t ) < v ( t ) ; w ( t ) ; if v ( t )= v ( t ) (B.6) where g L ( ) ,and g R ( ) aretwoenvelopefunctionsthatarestrictlyincreasing.Th eenvelopefunc- tionsdescribethepropertiesoftheplayoperators.Forany radius r 0andinput v ( t ) ,thecondition g L ( v ( t ))+ r g R ( v ( t )) r needstobesatisedinordertomeettheorderpreservationp ropertyof hysteresisbehavior[27]. TheoutputofaGPImodelcanbeexpressedintheintegralform as y P ( t )= Z R p 0 p ( r ) F g r [ v ]( t ) dr : (B.7) SimilartotheCPIcase,adiscrete-versionoftheGPImodelc anbewrittenas y P g ( t )= N å j = 0 p ( r j ) F g r j [ v ]( t ) : (B.8) 163 When g L ( v ( t ))= g R ( v ( t )) ,theGPImodelcanbeutilizedtomodelsymmetrichysteresis ;when D ( ) islinearand g L ( v ( t ))= g R ( v ( t ))= v ( t ) ,theGPImodeldegeneratestoaCPImodel. Inversion Denote Y astheGPImodel,whichcanbewrittenas y d = Y [ v ]( t )= N å j = 0 p ( r j ) F g r [ v ]( t ) ; (B.9) where y d isthedesiredoutputoftheGPImodel,anddenote ‹ Y 1 asitsapproximateinverse.Then ideally, y = Y ‹ Y 1 [ y d ]( t ) ˇ y d ; (B.10) issatised,where y istheactualoutputoftheGPImodel Y ,and y d isthedesiredoutputofthe generalizedmodel.Notethatininversecompensation, y d isusedastheinputfortheinversemodel ‹ Y 1 .fi fldenotesthecompositionoffunctionsoroperators.Onecan write y ( t )= Y [ v ]( t )= 8 > > > > > < > > > > > : Y [ v ]( t )= P g R ( v ( t )) ; if v ( t ) > v ( t ) Y [ v ]( t )= P g L ( v ( t )) ; if v ( t ) < v ( t ) : y ( t ) if v ( t )= v ( t ) (B.11) where P denotestheclassicalPImodel. Duetotheinvertibilityoftheenvelopefunctions g L and g R ,Eq.(B.11)canbeexpressedas 164 v ( t )= 8 > > > > > < > > > > > : g 1 R P 1 y ( t ) ; if y ( t ) > y ( t ) g 1 L P 1 y ( t ) ; if y ( t ) < y ( t ) : v ( t ) ; if y ( t )= y ( t ) (B.12) TheinverseoftheGPImodeliswrittenas[28] Y 1 [ y d ]( t )= 8 > > > > > < > > > > > : g 1 R P 1 [ y d ]( t ) ; if y d ( t ) > y d ( t ) g 1 L P 1 [ y d ]( t ) ; if y d ( t ) < y d ( t ) ; Y 1 [ y d ]( t ) if y d ( t )= y d ( t ) (B.13) where P 1 istheinversionoftheCPImodel,theexpressionofwhichcan befoundin[20]. 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