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A robust control method for electrostatic microbeam dynamic shaping with capacitive detection

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A robust control method for electrostatic microbeam dynamic shaping with capacitive detection
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   Arobustcontrolmethodforelectrostaticmicrobeamdynamicshapingwithcapacitivedetection ChadyKharrat*,EricColinet*AlinaVoda**   *ElectronicsandInformationTechnologiesLaboratory(CEA-LETI-MINATEC),17ruedesmartyrs,38054Grenoble,France.(Tel:334-3878-2487;e-mail:[chady.kharrat;eric.colinet]@cea.fr).**AutomaticcontrolLaboratoryofGrenoble(LAG),ENSIEG-INPG,BP46,38402,SaintMartind'Hères,France.(e-mail:alina.voda@inpg.fr) Abstract: Arobustclosed-loopcontrolandobservationmethodologyforanelectrostaticdynamicshapingofamicrobeamusing  N  smallseparateelectrodesisdescribed.Afterdecomposingthedisplacementsvectoronthe n  eigenmodesusingthemodalanalysis, n  controllersaredesignedtocontrolthedynamiccoefficientsofeachmodeandthustodeliverthestressesthatmustbedistributedthroughoutthebeam.Inpreviousworks,weconsidereddirectaccesstononnoisydisplacementmeasurements.Inthispaper,weinvestigatethecapacitivemeasurementofthelocaldisplacementsdonebyeachsmallelectrode,whichgivesanoisyreadout.Robustcontrolmethodologyappliedonextendedstandardmodelpermitstodesign n  observersassociatedto n  controllersandguaranteespreciseshapetracking,freefromnoiseandrobustagainstparametersincertitude.   1.INTRODUCTIONElectrostaticallyactuatedMEMSarewidelyusedforpositioning,sensingandsignalfilteringpurposes.Itinvolvessimplecircuitryandhasamajoradvantageinthefactthatitoffersbothelectrostaticactuationaswellasintegrateddetection,withouttheneedofanadditionalpositionsensingdevice(Napoli etal .,2004).Itsmaindrawbacksarethenoninearityremainingintheequationlinkingthevoltageinputtotheforceoutputandthe“pull-in”instabilitywhichoccurswhenthevoltageappliedbetweenthefixedelectrodeandthemicrobeamcreatesanelectrostaticforcehigherthananyotherrestoringforceforanydisplacement.Capacitivesensorshavemanyadvantagessuchashighsensitivity,lowtemperaturedependence,lownoise,largedynamicrange,andpotentialmonolithicintegrationwithCMOScircuits(Chu etal .,2002).Intheotherhand,theseassociatedelectroniccircuitsforoutputreadoutandtheADCconvertersgenerateelectronicnoisethatreaches,insomecases,significantmagnitudesinrelativetothecapacitivemeasuredsignal.Also,duetotheuncertaintiesresultingfrommanufacturingprocesses,materialproperties,andmodellingassumptions,thesemicrosystemsmayexhibitsignificantvariationsintheirperformancecomparedtonominaldesigns(Min etal .,2007).Also,residualstressesandlargedeflectionsmakethenon-linearitystretchingtermssignificantfactorsgoverningthemicrobeambehaviour(Younis etal. ,2007).Inaddition,arraysofMEMSbegintoplayanimportantroleinseveralapplications.Thusmechanicalandelectrostaticcouplingsbetweenindividualmicrostructuresappearandincreasethecomplexityofthemodel.Inconsequence,andforalltheseproblems,thecontrolofsuchmicrosystemsbecomesadifficulttasktoaccomplishbecauseoftheimportantcomputationcapacityneededfortheprocessortofindtheappropriatenonlineardecouplingcommandsignalsandtoensurethestabilityandthecontrolofthewholesystem.Theflexiblestructuresaregenerallysusceptibletostructuralvibrationanddeformation,andthusvibrationsuppressionandshapecontrolareimportant.Shapecontrolrepresentsoneofthemostimportantapplicationsofsmartmaterialsandstructures.Inlightweightadaptiveoptics,piezoelectric(PZT)actuatorswerebondedinopticalmirrorstoachievedesignedsurfaceshapes(Liu etal .,1993;Huonker etal .,1997).Preciseshapecontrolforaflexiblecircularplatemirrorwasachievedby(Philen etal .,2004)usingPZTstripswithadecoupledactuationdirection,whilehighprecisioninstaticshapecontrolofsmartstructureswasachievedbyusingtheorthotropicactuatorsin(Luo etal .,2006).TheFEM(finiteelementmethod)wasusedby(Isobe etal .,1998)foritscapabilitytoexpressthebehaviourofthewholesystembyevaluatingthestiffnessequationsbutwithreducingthenumberofdots.Recently,dynamicshapetrackinghasgainedattention;thisintegratesstructuralshapecontrolwithmotioncontrol.In(Krommer etal .,2007),dynamicdisplacementtrackingofsmartbeamsisstudiedusingdistributedself-stressesstrainsensorsandactuatorsorpiezoelectricones(Irschik2002).In(Luo etal .,2007),anefficientalgorithmfordynamicshapetrackingwithoptimumenergycontrolisproposed.Howeveralltheseworksstillrequireimportantcomputationaltimeandalgorithmsaswellasacomplexcontrollernetworkforthecontrolofeachlocaldisplacement.Inthiswork,anovelautomaticbased-mode-controlmethodisproposedtoensurethedynamicdisplacementreferencetrackingoneachpositionalongthemicrobeamwithgoodregulationdynamics,robustnessagainstparametersincertitudeandrejectionofthemeasurementnoiseinfluence.Thisisdonebycontrollingonlythemodaldynamic    h  a   l  -   0   0   3   7   1   3   1   0 ,  v  e  r  s   i  o  n   1  -   2   7   M  a  r   2   0   0   9 Author manuscript, published in "IFAC 2008, Seoul : Korea, Republic of (2008)"   coefficientsandusingdistributedelectrostaticactuatorsandcapacitivesensors.Thenonlinearitiesaswellasthecouplingsbetweenthedisplacementsineachpositionarealsotakenintoconsideration.First,thesystemisdescribed,andpracticallimitationsareexposed.Thenthedetailedmodellingofthemicrobeamandthemodalanalysisofitsbehaviourareexplained.Inthethirdpart,the“regulatorproblem”(Wonham,1985)andtheLTR(LoopTransferrecovery)controltechniqueareexposedfordesigningthemode-based-controllersandobserversguarantyingthecontrolspecifications.Inpreviouswork(Kharrat etal. ,2007),thecontrollersusedwerePIDswhowerelimitedinperformanceandforwhomnoisymeasurementshadstronginfluenceonshapeoutputs.Finally,simulationresultsofthewholecontrolledsystemonMatlabareshown,withtestsofrobustness,noiserejectionandperformances.2.SYSTEMDESCRIPTIONThestudiedsystemconsistsofacontinuousdeformablemicrobeam,clampedonbothextremitiesandsubjectedtodistributedtime-variantelectrostaticforcesgeneratedbytheapplicationofdistributedtime-variantvoltageson  N  electrodeschosenfromthetwosetsdisposedonthebothsidesofthemicrobeam.Fig.1.Thedeformablemicrobeamsurroundedwithtwosetsof  N  electrodesoneachside,subjectedtodifferentvariantdistributedvoltages.Thetwosetsarenecessarytomakepossiblethegenerationofattractiveforcesonbothdirectionsdependingonthecalculatedcontrolsignals.The1 st orderapproximativerelationlinkingtheelectrostaticforcetotheappliedvoltageonadeterminateposition x  ,is: 220 )),(( ),(. 21),( txw  g  txuS  tx f  e −=  ε  (1)where e S  istheelectrodesurface, ),( t xu istheappliedinstantaneousvoltage, g  istheinitialgapbetweenthemicrobeamandtheelectrodeand ),( t xw  isthetransversedisplacementofthemicrobeamwhichismeasuredusingthecapacitiveprinciple.The1 st orderapproximativeequationrelatingthemeasuredcapacitancetothedisplacementis: ),(),( 0 txw  g  S txC  e −=  ε  (2)Whenonafixedinstant t ,anelectrodeisusedasanelectrostaticactuatorandissubjectedtoavoltage,theotheroneisconnectedtoacapacitiveelectronicmeasurementcircuit,whilealowvoltageisappliedonthemicrobeamforthedetectionaim.TheelectricmeasurementcircuitriesaswellastheADCconvertersaddnoisetotheoutputsignals.Smalleraretheelectrodesurfaces,smallerarethemeasuredcapacitancemagnitudeswiththesameelectricnoiseandthusthesignaltonoiseratiodecreaseswiththeincreasingnumberoftheelectrodesused.Inaddition,thecomplexityoffabrication,miniaturizationandcomputationincreasesandthecontrollerbecomesmoredifficulttointegratewhenusingclassicalcontrolmethods.Intheotherhand,thebiggeristhenumberoftheelectrodesandthemorecontinuousisthegeneratedlocalforcesalongthemicrobeamwhichallowsmoreprecisereferencetracking.Inthiswork,1000electrodesareused.3.DEFORMATIONMODELLINGThebehaviourofthedeformationofarectangularmicrobeamwithlength l ,thickness e ,width h  ,subjectedtoanexternaldistributedstrengthobeystothefollowingdifferentialequation: 2244 ),()),(( ),(  x t x w t x w T   x t x w  EI ∂∂+∂∂  ),(),(),( 22 tx f ttxw S ttxw b =∂∂+∂∂+  ρ  (3)with eh S  . = thetransversalsectionofthebeam,EtheYoung’smodulus,  I themomentofinertia,  ρ  thedensity, b thefrictioncoefficientassociatedtotheinteractionwiththesurroundingfluid, ),( t x w  thetimedependenttransversedisplacementatposition x  ,and T(w) isthestressassociatedtotheelongationofthebeam.Fig.2.Theclamped-clampedmicrobeamwithdimensions l , h  and e .FollowingGalerkinprocedureofstandardmodalanalysis, ),( t x w  canbewrittenas: ∑  = = nkkk  x w ta t x w  1 )().(),( (4)where )(  x w  k arethenmodeshapevectorsand )( ta  k thedynamicrelatedcoefficients.Asolutionofthisequation,withrespecttotheboundaryconditionsandconsideringa      h  a   l  -   0   0   3   7   1   3   1   0 ,  v  e  r  s   i  o  n   1  -   2   7   M  a  r   2   0   0   9   clamped-clampedmicrobeam,isshownin(Kharrat etal .,07)andleadstothemodalshapesspace )(  xw  k showninfig.3.Usingequation(4)in(3)andprojectingoneachvector i w  ,weget n  equationsrepresentingthe n  modesthatcanbewritteninamatrixform:  F  X  M  X  B  X  N  KX  =+++ &&& )( (5)where [ ] Tn ta ta ta  X  )()()( 21 L  = , n  IS  M  ..  ρ  = ,  = 44241 000000 n  EI K  λ λ λ  L O MML  ,  = n bbb B  000000 21 L O MML  ,   = n w  f w  f w  f F  M 21 Inthiswork, n  islimitedto5becausetheeffectofthesuperiormodesbecomenegligibleontheshape.Thestressduetotheelongationofthebeamcanbeexpressedbytheequation llS  E w T  ∆= ..)( ,where )..( 2 12 12 1 01120  X  A  X dxa  dxdw dxdw a dxdxdw l Tlll N k N lkkl −==      =∆ ∫∑∑ ∫ = = andthusonehas  X  A  X  A  X  l ES  X  N  T .)...( 2)( −= where  = nnnn w dxw d w dxw d w dxw d w dxw d w dxw d w dxw d  A  22212221212212221212 L L MO MMO L  representsthecouplingsbetweenallthemodes.Fig.3.Thefirstfive )(  xw  k representingthefirstfivemodalshapesofthemicrobeam.4."REGULATORPROBLEM"AND"LTR"TECHNIQUEFORMODE-BASED-CONTROLTocontrolthemicrobeamshapehavingthe  N  actuatorswiththedisplacementsreferencevectorandthedisplacementsmeasuresineachpositionalongthemicrobeam,arudimentaryideaistocalculateeachlocaltrackingerrorandtouseitascontrollerinputforthecomputationoftherequiredlocalforcetobeapplied,atacostofahugenetworkof  N  controllers.Also,onewillbedealingwiththeequation(1)directlywhichhasunknowncouplingtermswiththeotherdisplacementsandnodesiredexactdynamicscanbeimposedondisplacementsresponses.Bycontrollingthedynamicmodecoefficients )( ta  k ,thesumofthemodulatedshapevectors ∑  = = nkkk  xw ta t xw  1 )().(),( willleadtothedesiredshapewhenall krk a ta  → )( ,thereferencemodecoefficients.Anyvibratingshapereferenceisdescribedbydifferentsinusoidaldynamiccoefficientswhosemodelisaddedtothesystemstatespacemodeltoobtaintheso-calledstandardmodel.Totakeintoconsiderationapossibledisturbanceactionontheinput,amodelofaconstantdisturbance d  isalsoaddedsothatthestandardmodelbecomes: kkrkrkkkkrkrkk  f m a a d a a m bm k  m a a d a a   +  −−−=  000/100000 10000 00000 000// 00/110 2 &&&&&&&&& ω   ω  isthevibrationpulsation.Thisstandardmodelwithitstrackingerrors e andoutputs  y  canbewrittenas: u B  x x A  A  A  x x  +  =  00 12122121121 &&  [ ] kkree a a  x xC C e  −=  = 2121 , [ ]   =  = krk y y a a xxC C  y 2121 where 1  x  arethesystemstatesand 2  x  aretheexosytstemstates(disturbanceandreference).    221211 0,  A  A  A C C  ye isobservable,   ( ) 111 ,  B  A  isstabilizableand 22  A  isunstable.Findingaregulator K  thattends e towards 0  foranyinitialconditionsandkeepstheclosedloopsystemstableiswhatwasreferredtoas“regulatorproblemwithinternalstability”(RPIS)by(Wonham1985).Ifthesolutions ( 3132 ,  ×× ℜ∈ℜ∈ aa T  F  ofthefollowingequations:  =+−−=++− 0 211122211 eaeaaa C T C  F  B  A  A T T  A  (6)exist,thantheregulatorofanFSF(FullStateFeedback)–observertypewiththegains [ ] aa  F T  F  F  F  += 11 where 1  F  issuchthat 1111  F  B  A  + isstableand  K  issuchthat  KC  A   − isstable,guarantiesthestabilityoftheclosed-loopsystemandthereferencetracking.Resolvingtheseequationsinourcasegives  −−= 100010 a T  and kka bm k  F  2 1  ω  −−= .    h  a   l  -   0   0   3   7   1   3   1   0 ,  v  e  r  s   i  o  n   1  -   2   7   M  a  r   2   0   0   9   Havingtheappropriatestandardmodel(thenumberofreferenceanddisturbancestatesincludedinthestandardmodelisequalorbiggerthanthenumberofmeasuredoutput)arobustasymptoticreferencetrackingcanbeachieved(DeLarminat,1995).Forrobuststability,onehastoadjustthegains 1  F  and [ ] r xd  K  K  K  = forsystem+disturbance( sd  K  )andreference( r  K  )statesobservation.TheLTRtechniquewasusedtochoosethesegains:Consideringpossibleaccesstotheallthestates,andbychoosingtheclosed-looppolesthat i ∀ and w   ∀ , ii  po  jw  pc jw  −≥− ,where i  pc and i  po  arerespectivelytheclosed-loopandopen-looppoles,onehas 1 11 ≤−− ∏∏ == niinii  pc jw  po  jw  thustheloopsensibilty 1)(  ≤  jw S  , 1)(1 ≥+ ⇒   jw L  , w   ∀ whichmeansthatthelooptransferfunction )(  jw L  isalwaysoutsidethecircleofcentre-1andbeamequalto1intheNyquistdiagram,whichguarantiesgoodstabilitymargins.Thiscanbedonebychoosingaparameter c T  /1 − whichdefinestheaxisintheleftsideofthecomplexspace(Delarminat,2000).Afterprojectionoftheunstableopen-looppolestotheleftquadrant,theprojectionsonthe c T  /1 − axisofthosewhoareonitsright+thosewhoareonitsleftrepresenttheclosed-loopsystempoles.Inotherwords, c T  representsthedesiredresponsespeedfortheclosed-loopsystem.Oncethepolesarechosen,resolvingtheAckerman’sformulaallowstoobtainthegainsof 1  F  .Fig.4.Thepolesplacementtechniquethatassurestherobuststabilityoftheclosedloopsystem.Whenwedon’thaveaccesstoallthe“system+disturbance”states(whichisourcase),anobserverisneededwhichleadstoanew )(  jw L  n .ForanexactLTR,thislatterisequaltothetargetonebutthiscanonlybeobtainedwithaderivativeregulatorthatamplifiesthenoiseinfluenceoncontrolsignalsandoutputs.That’swhyanasymptoticLTRisadoptedandaparameter o T  /1 − ischosen,whichdefinestheaxisintheleftsideofthecomplexspace.Afterprojectionoftheunstableopen-loopzerostotheleftquadrant,theprojectionsonthe o T  /1 − axisofthosewhoareonitsleft+thosewhoareonitsright+theremainingobserverpolesseton o T  /1 − ,representtheobserverpoles.Itwasproventhatwhen 0  → o T  ,andtheobserverpolesaresetexactlyontheopen-loopzerosandtheremainingoneson ∞− andso )()(  jw L  jw L  n  → (Saberietal.,1993).Generally, o T  ischosen3,5or10timessmallerthan c T  .Oncethepolesarechosen,resolvingtheAckerman’sformulaallowstoobtainthegainsof sd  K  .Asforthereferenceobserver’spoles,theycanbechosenwithdynamicsfasterthanthosedefinedbythereferencevibrationfrequency,whichallowsprecisereconstructionofthereferenceandofitsderivative.UsingAckerman’sformula,thegainof r  K  iscalculated.Thetwoparameters c T  and o T  arechosenbytakingintoconsiderationthecontrolspecificationsandthereferencedynamics.Thesmaller c T  isandthefasteristheresponsespeedwhichcanbeusefulincaseofdisturbancerejection.Inaddition,robustnessviatheparametersuncertaintiesisimproved.But,intheotherhand,thenoiseinfluenceishigheronoutputandoncontrolsignalandbigsolicitationoftheactuatorsispredicted.Oncethetargetloopisselectedwith c T  , o T  istunedsothatweapproachthespecifiedperformancesandtherobustnessofthetargetloopbutwithouthavinganimportantderivativeaction,toobtainabetternoiserejection.Guarantyingoneofthetwoobjectivesiseasytoobtain,buthavingbothonthesametimeismuchmorecomplexandacompromisemustbedone.Thiscontrolschemeisappliedtoeachofthefivedynamiccoefficientsofthefirstfivemodes,requiringfiveregulatorsinsteadofNones.Thecalculatedcontrolsignals k  f  allowthecalculationofthedistributeddesiredelectrostaticforce ∑    = = nkkk xw t f tx f  1 )().(),( andtherequiredlocalvoltagestobeappliedontheNelectrodesarecalculatedby S tx f txw  g txu 0 ),(2 .),(),( ε  −= andareappliedonthehigherelectrodeif 0),( > tx f  ,andontheloweroneif 0),(  < tx f  .Aproblemisthat ),( t xw  arethenoisymeasurementssothecalculatedvoltages ),( t xu willnotgenerateexactlytherequiredforcesfortherealdisplacements.Thislimitstheintendedadvantagesjustwhentheserequiredforceshavebigmagnitudeswhichamplifythenoiseinthecalculatedvoltages.5.PRACTICALCASEOFSIMULATIONSANDTESTSThemicrobeammodelisimplementedinSimulink/Matlabwithalength m l  µ  35,13 = ,awidth m h   µ  2.0 = andathickness m e  µ  2.0 = .Themomentofinertiaisthen 12/. 3 eh  I  = .Thegapbetweentheelectrodesis m  g   µ  5.0 = ,Sotheparametersofthemodelare: 4 .. kk  I E k  λ  = , Q m k b kk .  = and h em  ..  ρ  = with 9 10.169 =  E  , 3 10.232.2  =  ρ  and 4000  = Q  forallthemodes. k λ  arethecalculatedeignevaluesofthemodesand    h  a   l  -   0   0   3   7   1   3   1   0 ,  v  e  r  s   i  o  n   1  -   2   7   M  a  r   2   0   0   9   onehas [ ] L  27.1713.1499.1085.773.4)( = l k λ  . Theresonancefrequenciesofthe5modesgofrom10Mhzto134Mhz.Thechosenreferencecoefficientsdefineacertainshape,vibratingwithafrequency  f  =10Mhz.Thecouplingsbetweenthedisplacementsrepresentedbythecouplingsbetweenthedifferentmodesaretakenintoconsideration.Theelectricalnoiseismodelledasawhitenoiseofspectralpowerequalto  HzaF  /10 5 − forasamplingfrequencyof1GHz.Aconstantdisturbanceisaddedtotheelectrostaticforces.Inadditiontoallthesespecifications,amodificationof20%onallthemicrobeam'sparameterswereconsideredtotesttherobustnessofthecontrolschemeevenincaseofnoisymeasurementanddisturbingenvironmentwithuncertainsystems.Dependingontheirinitialdynamicsandparameters,different c T  and o T  arechosenforthedifferentmodesandareshownintable1. Table1.Controlandobservationparameters ModenumTcTo1  st mode2.10  -8 Tc/42  nd mode1.10  -8 Tc/53  rd mode8.10  -9 Tc/44  th mode7.10  -9 Tc/45  th mode5.10  -9 Tc/3 Theresultsobtainedfortheshapetrackingwithoutdisturbanceapplicationareshowninthefollowingfigures:Fig.5.The5dynamiccoefficients a  k ofthenoisymeasuredshapeusedasregulatorsinputs(ontop)comparedtotherealshapecoefficientsoftheclosed-loopsystem(onbottom).Whenaconstantdisturbanceisaddedtotheactuatorsforcesasaninstantaneousstep(whichcanrepresentamechanicalshockoracurtacceleration),thedesignedcontrollersexhibitgoodperformance(forthesamevaluesof c T  , o T  )andresultsareshowninfig.9.  Fig.6.Thecomparisonbetweenthereferenceshape(bluecurve)andtheoutputshape(redcurve)ofthemicrobeamonafixedinstant t  f (ontop)andthecorrespondentdesiredforcedistributionoftheactuators(onbottom).Fig.7.Themodaltrackingerrorscoefficientsforthe5modes.Onecannoticethaterrorsdon'texceedthe10%ofthereferencecoefficients.6.CONCLUSIONThispaperdetailsthedesignofafullyintegrablecontrolloopforamicrobeamdynamicshapingbyelectrostaticactuationdonewithtwosetsof1000   smallelectrodesdisposedonbothsidesofthemicrobeam.Firstofall,thedetailedmodellingofthestructureusingmodalanalysisispresented.Thenthecontrolproblemofeachofthefirstfivemodeswasformulatedasa“regulatorproblemwithinternalstability”forextendedsystemsmodelscontainingthemodesdynamicsmodelsaswellasthe“exosystems”,consideringaknownreferencetrajectoryafterprojectiononthemodalspaceofdimension5,andaconstantdisturbancetoberejected.Thenthecontrollersaredesignedbyplacingthepolesoftheclosed-loopsystemssuchthatgoodrobustness,stabilityandperformanceareexpectedandobserversaredesignedaccordinglytothe“LTR”techniquetoapproachtheclosedtargetloops.ThisprocedurewassuccessfullysimulatedonMatlabandcontrolspecificationsareobtained.    h  a   l  -   0   0   3   7   1   3   1   0 ,  v  e  r  s   i  o  n   1  -   2   7   M  a  r   2   0   0   9
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