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Modeling and control of magnetorheological fluid dampers using neural networks

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INSTITUTEOFPHYSICSPUBLISHINGSmartMater.Struct.14(2005)111–126

SMARTMATERIALSANDSTRUCTURES

doi:10.1088/0964-1726/14/1/011

Modelingandcontrolof

magnetorheological uiddampersusingneuralnetworks

DHWangandWHLiao1

SmartMaterialsandStructuresLaboratory,DepartmentofAutomationandComputer-AidedEngineering,TheChineseUniversityofHongKong,Shatin,NT,HongKongE-mail:whliao@cuhk.edu.hk

Received27February2003,in nalform3July2004Published7December2004

http:///SMS/14/111

Abstract

Duetotheinherentnonlinearnatureofmagnetorheological(MR) uiddampers,oneofthechallengingaspectsforutilizingthesedevicesto

achievehighsystemperformanceisthedevelopmentofaccuratemodelsandcontrolalgorithmsthatcantakeadvantageoftheiruniquecharacteristics.Inthispaper,thedirectidenti cationandinversedynamicmodelingforMR uiddampersusingfeedforwardandrecurrentneuralnetworksarestudied.Thetraineddirectidenti cationneuralnetworkmodelcanbeusedtopredictthedampingforceoftheMR uiddamperonline,onthebasisofthedynamicresponsesacrosstheMR uiddamperandthecommandvoltage,andtheinversedynamicneuralnetworkmodelcanbeusedtogeneratethecommandvoltageaccordingtothedesireddampingforcethroughsupervisedlearning.ThearchitecturesandthelearningmethodsofthedynamicneuralnetworkmodelsandinverseneuralnetworkmodelsforMR uiddampersarepresented,andsomesimulationresultsarediscussed.Finally,thetrainedneuralnetworkmodelsareappliedtopredictandcontrolthedampingforceoftheMR uiddamper.Moreover,validationmethodsfortheneuralnetworkmodelsdevelopedareproposedandusedtoevaluatetheirperformance.Validationresultswithdifferentdatasetsindicatethattheproposeddirectidenti cationdynamicmodelusingtherecurrentneuralnetworkcanbeusedtopredictthedampingforceaccuratelyandtheinverseidenti cationdynamicmodelusingtherecurrentneuralnetworkcanactasadampercontrollertogeneratethecommandvoltagewhentheMR uiddamperisusedinasemi-activemode.

(Some guresinthisarticleareincolouronlyintheelectronicversion)

1.Introduction

1.1.MR uiddampers

Magnetorheological(MR) uidsaresuspensionsthatexhibitrapid,reversible,andtunabletransitionfromafree- owingstatetoasemi-solidstateupontheapplicationofanexternalmagnetic eld.Thesematerialsdemonstratedramaticchangesintheirrheologicalbehaviorinresponsetoamagnetic eld(CarlsonandWeiss1994).MR uidshaveattracted

1Authortowhomanycorrespondenceshouldbeaddressed.

considerableinterestrecentlybecausetheycanprovideasimpleandrapidresponseinterfacebetweenelectroniccontrolsandmechanicalsystems(Kordonsky1993a,1993b).TheMR uiddampers,whichutilizetheadvantagesofMR uids,aresemi-activecontroldevicesthatarecapableofgeneratingamagnitudeofforcesuf cientforlarge-scaleapplications,whilerequiringonlyabatteryforpower(Dykeetal1996,Spenceretal1997).Additionally,thesedevicesofferhighlyreliableoperationsandtheirperformancesarerelativelyinsensitivetotemperature uctuationsorimpuritiesinthe uid.Inrecentyears,researchintoanddevelopmentof

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DHWangandWHLiao

MR uiddampersandtheirapplicationshavebeenattractivetomanyresearchers.Already,a20tonMR uiddamperprototypehasbeendevelopedandtestedinthelaboratory(Spenceretal1998)andapplicationsofMR uiddamperscanbefoundovertherangefromcivilstructuressuchasbuildingsandbridges(Dykeetal1996,Housneretal1997,Spenceretal1998)toautomobiles(Choietal2000)andrailwayvehicles(LiaoandWang2003).

DuetotheinherentnonlinearnatureofMR uiddampers,oneofthechallengingaspectsforachievingahighlevelofperformanceisdevelopmentofaccuratemodelsandcontrolalgorithmsthatcantakeadvantageoftheuniquecharacteristicsofMRdevices.

1.2.ThemodelingchallengeforMR uiddampers

Recently,bothnon-parametricandparametricmodelshavebeenproposedfordescribingthebehaviorofMR uiddampers.Parametricmodelsbasedonmechanicalidealizationshavebeenconsideredbyseveralresearchers(Spenceretal1997,Wereleyetal1998,Lietal2000).Spenceretal(1997)proposedthemodi edBouc–WenmodelfordescribingbehaviorofanMR uiddamper,inwhich14parametersneedtobedeterminedthroughcurve ttingofexperimentaldata,whichisverytime-consuming(LaiandLiao2002).Pangetal(1998)discussedfourmodelsfordescribingthebehaviorofMR uiddampers,namely:(1)theBinghamplasticmodel,(2)thebiviscousmodel,(3)thehystereticbiviscousmodel,and(4)theviscoelastic–plasticmodel;andLietal(2000)testedtheabovemodelsforthefrequencyrangeupto12Hz.TheseparametrizedmodelscanmodelthedynamicsofMR uiddamperswithinalimitedrange.

However,parametricidenti cationmethodsrequireassumptionsasregardsthestructureofthemechanicalmodelthatsimulatesbehavior.Onceamodelisselected,thevaluesofsystemparametersaredeterminedinsuchawaythattheerrorbetweentheexperimentalandthesimulatedresponsesisminimized.Theapproachcouldbedivergentifthestartingassumptionsforthestructureofthemodelare awed,orifproperconstraintsarenotappliedtotheparameters.Unrealisticparameterssuchasnegativemassorstiffnessmaybeobtained.Non-parametricmethodscouldavoidsomepitfallsoftheparametricapproachesformodeling,whicharerobustandapplicabletolinear,nonlinear,andhystereticsystems(EhrgottandMasri1992).FormodelingMR uiddampers,ChangandRoschke(1998)proposedanon-parametricmodelusingneuralnetworks,inwhichafeedforwardneuralnetwork(FNN)isused.TrainingandpredictionofthenetworkrelyoninputandoutputinformationonMR uiddampers.SchurterandRoschke(2000)investigatedthemodelingofMR uiddamperswithanadaptiveneuro-fuzzyinferencesystem.Gangetal(2001)exploredthemodelingofMR uiddamperswithanonlinearblackboxapproach.

1.3.InversedynamicmodelingandcontrolofMR uiddampers

ControlofthedampingforceofanMR uiddamperisalsoverychallengingbecauseofthefollowingtwofeatures:(1)strongnonlinearityand(2)thesemi-activerelationship112

betweenthedampingforceandthecommandvoltage.SotheforcegeneratedbytheMR uiddampercannotbecommandeddirectly;onlythevoltageappliedtothecurrentdriverfortheMR uiddampercanbedirectlycontrolled.Inordertoovercomethesedif culties,thedesireddampingforceoftheMR uiddamperisoftendeterminedat rstonlyaccordingtothestructureresponsesthroughthesystemcontroller;thenthecommandvoltageisdeterminedbythedampercontrolleronthebasisofthedesireddampingforceandthemonitoreddampingforceoftheMR uiddamper.Dykeetal(1996)proposedamethodinwhichtheHeavisidestepfunctionisusedtocalculatethecommandvoltage.TheactualdampingforceoftheMR uiddamperneedstobemonitoredonlineandthecommandvoltagevariesbetweentwovoltagestates.

Basedontheabovediscussions,neuralnetworks,whichcanbeusedtomatchnonlinearitythroughlearning,couldbeanappropriatealternativeforthemodelingandcontrolofMR uiddampers.Inthispaper,directidenti cationandinversedynamicmodelingforMR uiddampersusingfeedforwardandrecurrentneuralnetworkswillbestudied.ThearchitecturesandthelearningmethodsofthedynamicneuralnetworkmodelsandinverseneuralnetworkmodelsforMRdamperswillbepresented.Theneuralnetworkmodelsdevelopedwillbevalidatedthroughnumericalexperiments.ItshouldbenotedthatadampercontrollerfortheMRdamperusingtheinverseidenti cationdynamicmodelwiththerecurrentneuralnetwork(RNN),whichcanbeusedtogeneratethecommandvoltagewhentheMRdamperisworkinginasemi-activemode,isproposedandexploredforthe http://anizationofthispaper

Thispaperisorganizedasfollows.Themodi edBouc–WenmodelforMR uiddampersisdescribedinsection2becausethetrainingdataoftheneuralnetworkmodelsinthefollowingsectionsisgeneratedbythemodi edBouc–Wenmodel.Section3dealswithdirectidenti cationofMR uiddampersusingneuralnetworks.BoththeFNNandRNNmodelsareproposedformodelingMR uiddampers,andcharacteristicsofthesetwomodelsarediscussedandcompared.Insection4,neuralnetworksformodelingtheinversedynamicsofMR uiddampersareexplored.TheperformancesoftheneuralnetworkmodelsfortheMR uiddampersarevalidated.Section5mainlydealswiththeforcepredictionandtheforcecommandfortheMR uiddamperusingthedirectidenti cationneuralnetworkmodelsandtheinverseneuralnetworkmodelstrainedinsections3and4forMR uiddampers.Insection6,conclusionsanddiscussionsonfutureworkarepresented.

2.Themodi edBouc–WenmodeloftheMR uiddamper

ThemechanicalidealizationofanMR uiddamperdepictedin gure1hasbeenshowntoaccuratelypredictbehavioroftheMR uiddamperoverabroadrangeofinputs.ThephenomenologicalmodelproposedbySpenceretal(1997)isgovernedbythefollowingequations:

F=c1y

˙+k1(x x0)(1)

Figure1.ThedynamicmodelfortheMR uiddamper.

z˙= γ|x˙ y˙||z|n 1z β(x˙ y˙)|z|n+A(x˙ y˙)

(2)y˙=

1

c[αz+c0x˙+k00+c1

(x y)](3)α=α(u)=αa+αbu(4)c1=c1(u)=c1a+c1bu(5)c0=c0(u)=c0a+c0bu

(6)u˙= η(u v)

(7)

wherexandFarethedisplacementandtheforcegenerated

bytheMR uiddamperrespectively;yistheinternaldisplacementoftheMR uiddamper;uistheoutputofa rst-order lterandvisthecommandvoltagesenttothecurrentdriver.Inthismodel,theaccumulatorstiffnessisrepresentedbyk1;theviscousdampingsobservedatlargeandlowvelocitiesarerepresentedbyc0andc1,respectively.k0ispresenttocontrolthestiffnessatlargevelocities;x0isusedtoaccountfortheeffectoftheaccumulator.αisascalingvaluefortheBouc–Wenmodel.Thescaleandshapeofthehysteresisloopcanbeadjustedbyγ,β,A,andn.Atotalof14modelparameters,whichareshownintable1(LaiandLiao2002),areobtainedtocharacterizetheRD-1005-1MR uiddamper(manufacturedbyLordCorporation)usingexperimentaldataandaconstrainednonlinearoptimizationalgorithm.

Inordertotraintheproposedneuralnetworks,appropriatetrainingdatasetsarerequired.Thetrainingdatasetsshouldcovermostsituationsofpracticalapplicationsinordertoletthetrainedneuralnetworkmodelspredictwellwhileatthesametimetheselecteddatasetsshouldbesimpletospeedupthetrainingprocess.TheselecteddatasetstobeusedtotraintheneuralnetworkmodelsforMR uiddampersareillustratedintable2,inwhichthevalidationandtestdatasetsforthenetworktrainingarealsoshown.Intable2,thedisplacementinputisaGaussianwhitenoisesignal,thecommandvoltageinputconsistsofdifferentsignalswithindifferenttimeintervals,andtheforceisproducedbythemodi edBouc–Wenmodelaccordingtothedisplacementandcommandvoltageinputs.Thetimeintervalsofsignalsusedfortraining,validation,andtestingofneuralnetworksarelistedinthistable.

Timehistoriesofthetrainingdatasets(table2),whichareproducedusingthemodi edBouc–Wenmodeldescribedbyequations(1)–(7)andtheparametersgivenintable1,areshownin gure2.Theinputandoutputdatasetsareproducedatatimeincrementof0.002sandcanbeaccessibleasvectorsx,v,Fandtheircombinations.

ModelingandcontrolofMR uiddampersusingneural

networks

Table1.ParametersforthemodelofMR uiddamperRD-1005-1(LaiandLiao2002).ParameterValueParameterValuec784Nsm 1

c0aα12441Nm 1

0b1803NsV 1m 1kαa38430NV 1m 103610Nm 1c14649Nsm 1

γb136c1a2059320m 21b34622NsV 1m 1βA58020m 2k840Nm 1n2

x10

0.0245m

η

190s 1

3.NeuralnetworkmodelingofMR uiddampers

Inthelate1980s,conclusiveproofsweregiventhatmultilayerfeedforwardneuralnetworksarecapableofapproximatinganycontinuousfunctiononacompactset.Sointhepastdecade,neuralnetworkshavenotonlybeensuccessfullyusedinsolvingcomplexproblemsinpatternrecognitionandtimeseriesprediction,butalsohavebeenproposedfortheidenti cationandcontrolofnonlineardynamicalsystems(NarendraandParthasarathy1990,YangandLee1997,ChangandRoschke1998).Theidenti cationprocedureforMR uiddamperscanbeasillustratedin gure3,inwhichtheinputu1(k)(consistingofthedisplacementandthecommandvoltage)issimultaneouslyconnectedtotheMR uiddamperandtheneuralnetworkmodeltobetrained.u2(k)isonlyconnectedtotheneuralnetworkmodel.Theinputu2(k)maybethemeasureddampingforceanditsdelays(fortheFNNmodel)orthepredicteddampingforceanditsdelays(forRNNmodel).TheoutputoftheneuralnetworkmodelF

the

(k)iscomparedwiththeoutputoftheMR uiddamperF(k)andthedifferencee(k)isusedtoadjusttheweightsandbiasesoftheneuralnetworkmodeluntila‘suf cientlysmall’conditionissatis ed.Thearchitectureoftheneuralnetworkmodeldeterminesthetrainingmethodsforandmeritsofusingneuralnetworks.

Inthefollowingtwosubsections,themodelsofMR uiddampersusingmultilayerFNNandRNNmodelswillbediscussed.

3.1.ModelingofMR uiddamperswithFNN

BecausetheFNNiscapableofapproximatinganycontinuousfunction,itcaneasilybeselectedastheidenti cationmodelforMR uiddampers.ChangandRoschke(1998)discussedthemodelingofanMR uiddamperusinganFNN.Forthepurposeofcomparisons,anFNNformodelinganMR uiddamperisalsoconsideredhere.Figure4showstheschemeoftheneuralnetwork,whichrepresentsthemappingF

(k+1)=NN[F(k),F(k 1),···,F(k IF+1),v(k),v(k 1),···,v(k Iv+1),x(k),x(k 1),···,x(k Ix+1)]

(8)

whereNN[·]denotesaneuralnetworkwithIF+Iv+Ix(=R)inputsandoneoutput,trainedtoapproximatetheinput–outputmappingthatdescribestheMR uiddamper.Accordingtoequation(8),thedelaysofF(k),v(k),andx(k)usedasinputtotheneuralnetworkareIF 1,Iv 1,andIx 1respectively,denotedbyblocksTDL(tappeddelayline)in gure4.When

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Displacement [cm]

–Time [s]

Voltage [V]

Time [s]

Force [kN]

–Time [s]

Figure2.Thetimehistoryoftrainingdatasetsforneuralnetworkmodels:(a)displacement;(b)commandvoltage;(c)dampingforce.

Table2.Training,validation,andthetestdataset.SignalsTrainingValidationTest

DisplacementCommandvoltageForce

ab

Timeinterval(s)

00–2020–2525–30

30–3535–4040–45

45–5050–5555–60

60–6565–70

75–8070–75

80–8585–90

90–9595–100

GWN1a

GWN2b+612606sin4πt+6

Producedbymodi edBouc–Wenmodel

Gaussianwhitenoise:frequency0–3Hz,amplitude±2.5cm.Gaussianwhitenoise:frequency0–4Hz,amplitude±6V

.

Figure3.Theschemeofidenti cationfortheMR uiddamperusingtheneuralnetworkmodel.

(k)becomessuf cientlytheerrore(k)betweenF(k)andF

small,theNN[·]isconsideredtobewelltrained.

Fortheidenti cationofanMR uiddamper,afullyconnectedthree-layerfeedforwardnetworkwith18(=S1)inputlayerneurons,18(=S2)hiddenlayerneurons,and1(=S3)outputlayerneuron,asshownin gure5,hasbeenselectedtomaptheinput–outputrelationshipfortheMR uiddamper.Thetransferfunctionsoftheneuronsoftheinputlayer(layer1)andhiddenlayer(layer2)areselectedas

the114

Figure4.Theschemeofidenti cationfortheMR uiddamperusingtheFNNmodel.

hyperbolictangentsigmoidtransferfunctionwiththeform

T1(x)=T2(x)=

2

1

1+e 2x

(9)

andthetransferfunctionfortheneuronoftheoutputlayer(layer3)isselectedasthelinearfunctionwiththeform

ModelingandcontrolofMR uiddampersusingneural

networks

x e 1x(x)

12··· e 1x(x)

n e 2x(x) e e .1 2x(x) 2x(x)n ..2···... e.

Q(x) e.

...Q(x) e.

Q(x) x1

x2

···

xn

andIisanidentitymatrix.Whenthescalarµiszero,thisis

justNewton’smethod,usingtheapproximateHessianmatrix.Whenµislarge,thisbecomesgradientdescentwithasmallstepsize.Newton’smethodisfasterandmoreaccuratenearanerrorminimum,sotheaimistoshifttowardsNewton’smethodasquicklyaspossible.Thus,µisdecreasedaftereachsuccessfulstep(reductioninperformancefunction)andisincreasedonlywhenatentativestepwouldincreasetheperformancefunction.Inthisway,theperformancefunctionwillalwaysbereducedateachiterationofthealgorithm.

Thenetworkshownin gure5canbetrainedusingtheLevenberg–Marquardtalgorithmexpressedbyequation(14)withthetrainingdatasetshownintable2.Theinputvectorisintheformofpq=[xqvqFq]T,inwhich

x(q) v(q)

xq=x(q 1)vq=v(q 1)

x(q 2) v(q 2)

F(q)

(15)Fq=F(q 1)

F(q 2)wherexq∈x,vq∈v,Fq∈F,andthetargetvectortq=Fq.Beforetrainingtheneuralnetwork,thetrainingdatasetshouldbepreprocessedbynormalizingtheinputsandtargetssothattheyhavemeansofzeroandstandarddeviationsof1.

WhetherthetrainedneuralnetworkmodelcanpredicttheresponsesofMR uiddampersshouldbevalidatedthroughsimulationand/orexperiments.Inordertovalidatethenetworksproposedinthispaper,aseriesofvalidationdatasetsarede nedintable3.PredicteddampingforcesfortheMR uiddamperusingthetrainedFNNmodelandthemodi edBouc–Wenmodelwiththevalidationset1(k=2)areshownin gure6,inwhichthedampingforceversustime,thedisplacement,andthevelocityareplotted.ItisclearthatthetrainedFNNcanaccuratelypredictthedampingforceifbothinputandoutputinformationoftheMR uiddampercanbeaccessedbecausethenetworkusesthemeasureddampingforcefortrainingandvalidation.Thatistosay,monitoringoftheforceintheMR uiddamperisneededwhentheabovetrainedFNNmodelisused,whichdetersonefromusingtheFNNmodelinpracticalsituations.

3.2.ModelingofMR uiddamperswithRNN

AlthoughtheFNNmodeldiscussedintheabovesubsectioncanaccuratelypredictthedampingforceofanMR uiddamper,theinputandoutputinformationfortheMR uiddamperneedstobeassessedduringthetrainingandpredictionstagesoftheneuralnetwork,whichrestrictstheusageoftheneuralnetworkmodel.Whenusingtheneuralnetworkmodeltopredictthedampingforceonline,itisadvantageousnottomonitorthedampingforceusingsensors,whichneedtobeimplementedbyinstallingoneforcesensorinserieswitheachMR uid

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Figure6.ThedampingforcepredictedusingtheFNNmodel.

Table3.De nitionofvalidationdatasets.

Validationset123456

ab

Displacement(cm)sin(2kπt)bsin(2kπt)bAsin(4πt)dsin(4πt)GWNeGWNe

Voltage(V)

1.5

GWNc+1.51.5

1.5+0.75sin(2kπt)b1+GWNc

1.5+0.75sin(2kπt)b

Forcea(N)Producedbymodi edBouc–Wenmodel

Timespan(s)666666

Onlyforvalidationofinversemodeling.k=0.5,1.0,1.5,···,5.0.c

Gaussianwhitenoise(frequency:0–2Hz;amplitude:±2).d

A=0.2,0.4,0.6,···,2.0.e

Gaussianwhitenoise(frequency:0–2Hz;amplitude:±2).

damper.Inthissubsection,anRNNisutilizedandtrainedtopredicttheforceoftheMR uiddamper.Inthisway,theforceintheMR uiddamperisonlyneededduringthetrainingstageoftheneuralnetworkmodel.Whenusingthetrainedneuralnetworkmodeltopredictthedampingforce,theforcesensorisnolongerneeded.

Inviewoftheabovediscussion,anRNNmodelisused,inwhichtheoutputoftheneuralnetworkmodelisdelayedandfedbacktoitsinputlayer.Figure7showstheschemeoftheRNNmodelforanMR uiddamperandthemappingoftheneuralnetworkisrepresentedas (k+1)=NN[F (k),F (k 1),···,F (k OF+1),F

v(k),v(k 1),···,v(k Iv+1),x(k),

(16)x(k 1),···,x(k Ix+1)]

whereNN[·]denotesaneuralnetworkwithIv+Ix(=R)inputsandoneoutput,trainedtoapproximatetheinput–outputmappingthatdescribestheMR uiddamper.OFisthenumberofdelaysthattheoutputofneuralnetworkfeedsbacktotheinputlayer.

Exceptthattheoutputoftheneuralnetworkisdelayedandfedbacktotheinputlayeroftheneuralnetwork,

the116

Figure7.Theschemeofidenti cationfortheMR uiddamperusingtheRNNmodel.

architectureoftheRNN,asshownin gure8,isbasicallythesameasthatoftheFNNshownin gure5.Thetransferfunctionfortheneuronoftheoutputlayerisselectedasthelinearfunctionwiththeformgivenbyequation(10)andthe

ModelingandcontrolofMR uiddampersusingneural

networks

Figure9.ThedampingforcepredictedusingtheRNNmodel(validationdataset1,k=2).

transferfunctionsfortheneuronsoftheinputandhiddenlayersareselectedasthehyperbolictangentsigmoidtransferfunctionwiththeformgivenbyequation(9),soequation(16)canberewrittenas

(k)+b1)+b2)+b3) (k+1)=T3(W3T2(W2T1(W1P+LW1FF

(17)

123

whereW,W,andWaretheweightmatricesofthethreelayers,respectively.LW1∈ S1×OFisthelayerweightmatrix.b1,b2,andb3arethebias

(k)=matricesofthosethreelayersrespectively,andF

(k)F (k 1)F (k 2)···F (k 5)]T.[F

Theinputvectorisintheformofpq=[xqvq]T,andthetargetvectortq=Fq,inwhichxqandvqaregivenbyequation(15),andFq∈F.Beforetrainingtheneuralnetwork,thetrainingdatasetispreprocessedbynormalizingtheinputsandtargetssothattheyhavemeansofzeroandstandarddeviationsof1.

InordertosavethememoryusedforcalculatingthefullJacobianwhentheRNNmodelistrained,theJacobian

isdividedintotwoequalsubmatrices.SoJT(X)J(X)(theapproximateHessianmatrix)inequation(14)iscalculatedusingthefollowingequation:

JTTT

(18)J(X)J(X)=[J1J2]1.

J2ThedampingforceoftheMR uiddamperpredictedusingthetrainedRNNmodelwithvalidationset1(k=2)isshownin gure9.ThedampingforceoftheMR uiddamperpredictedusingthetrainedRNNmodelwithvalidationset5(refertotable3)isshownin gure10.

ItcanbeseenthatthetrainedRNNmodelcanpredictthedampingforceoftheMR http://paringtheresultsshownin gures6and9,theFNNmodelcanpredictthedampingforcemoreaccuratelythantheRNNmodel.However,thepredictionusingtheFNNmodelneedstomonitorthedampingforceoftheMR uiddamperonlineandthenfeedbacktotheinputoftheneuralnetworkmodel.Bycontrast,

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Figure10.ThedampingforcepredictedusingtheRNNmodel(validationdataset

5).

(a)(b)

Figure11.TheschemeoftheinversemodelingfortheMR uiddamperusingneuralnetworks:(a)forcetocommandvoltagemodel;(b)forcetodisplacementmodel.

thepredictionusingtheRNNneednotmonitorthedampingforceoftheMR uiddamper,whichmayimprovereliabilityofthesystemandreducethecostofimplementation.

TheforcepredictionofMR uiddampersusingthetrainedneuralnetworkmodelswillbeexploredinsection5.

4.InversedynamicmodelingofMR uiddampers

Inversemodelingusingneuralnetworksavoidstheneedforexplicitlyinvertingthefunctionofthesystem.BecausethedampingforceofanMR uiddamperisnonlinearlyrelatedwithdisplacementacrossthedamperandthecommandvoltage,theinversemodelingoftheMR uiddamperconsistsofthefollowingtwocases:

(i)thedampingforcetocommandvoltagewhenthepredictedoutputoftheneuralnetworkmodelu (k)=v( k)asshownin gure11(a);

(ii)thedampingforcetodisplacementwhenthepredicted

outputoftheneuralnetworkmodelu (k)=x (k)asshownin gure11(b).Inversemodelingofthedampingforcetocommandvoltageordisplacementinvolvestraininganeuralnetwork118

modelarrangedinaccordancewiththecon gurationshownin gure11,inwhichu1(k)hasthesamemeaningasde nedin gure3.WhentheinverserelationshipismodelledbytheRNN,thepredictedoutputu (k)fromtheRNNshouldbefedbacktoitsinputu2(k)(=u (k)),whichisdenotedbythedashedlinein gure11.AsforthemodelingwiththeFNN,u2(k)istheactualvaluethatneedstobepredictedbytheFNN.Byminimizingtheerrore(k)betweenthepredictedoutputu (k)oftheneuralnetworkmodelandthetargetinputu(k),theneuralnetworkmodelapproximatestheinversedynamicsoftheMR uiddamper.

Theresultsofcase(ii)canbeusedtopredictthedisplacementacrosstheMR uiddamperwhenthedampingforcecanbeaccessed,whichisnotthefocusinthispaper.Theresultsofcase(i)canbeusedtorealizecontrolofanMR uiddamper,whichwillbediscussedindetailinsection5.4.1.ModelinginversedynamicsofMR uiddamperswiththeFNN

ForinversemodelingofanMR uiddamperusingtheFNN,theneuralnetworkshownin gure12istrainedtoapproximatetheinput–outputbehavioroftheMR uiddamper.Forfurther

ModelingandcontrolofMR uiddampersusingneural

networks

hasbeenselectedtomaptheinput–outputrelationshipoftheMR uiddamper.Thetransferfunctionsfortheneuronsoftheoutputlayerareselectedasalinearfunctionwiththeformgivenbyequation(10)andthetransferfunctionsoftheneuronsoftheinputandhiddenlayersareselectedasahyperbolictangentsigmoidtransferfunctionwiththeformgivenbyequation(9),soequation(19)canberewrittenas

v( k+1)=T3(W3T2(W2T1(W1P+b1)+b2)+b3)

(20)

whereW1,W2,W3areweightmatrices;b1,b2,andb3arebiasmatricesofthreelayers,respectively;andT1andT2aregivenbyequation(12).

Thenetworkshownin gure13canbetrainedusingtheLevenberg–Marquardtalgorithmexpressedbyequation(14)withthetrainingdatasetshownin gure2.Theinputvectorisintheformofpq=[xqFqvq]T,andtheoutputvectoristq=vq,inwhichxq,vq,andFqaregivenbyequation(15).Beforetrainingtheneuralnetwork,thetrainingdatasetispreprocessedbynormalizingtheinputsandtargetssothattheyhavemeansofzeroandstandarddeviationsof1.

ThevalidationschemefortheinversemodelingusingFNNmodelsforMR uiddamperswithSIMULINKisshownin gure14.Inthis gure,SIMULINKblockslabelledasMRD1andMRD2arecreatedonthebasisofthemodi edBouc–Wenmodel,whichisusedtorepresenttheMR uiddamperinthevalidationprocess.TheblocklabelledasMRD NN INVERSErepresentsthetrainedFNNmodel,whichistobevalidated.Thedisplacement,thecommandvoltage,andthedampingforceproducedbyMRD1areinputstofeedintotheinverseFNNmodeltogeneratethecommandvoltagesignal,whichisthenfedintotheMRD2togetherwiththedisplacementtoproducethepredicteddampingforce.Thevalidationprocessincludescomparisonsbetweenthepredictedcommandvoltageandtheinputcommandvoltage,thedampingforcepredictedbyMRD2andthetargetdampingforcebyMRD1.Onlyonevalidationcaseispresentedhere,asshownin gure15.Thedisplacementandcommandvoltageinputaregivenbythevalidationset5.Observing gure15,notonlydoesthepredictedcommandvoltagecoincidewiththeinputcommandvoltage,butalsothedampingforcecalculatedusingthepredictedcommandvoltagecoincideswiththedampingforceobtainedwiththeinputcommandvoltage.

Figure14.ThevalidationschemefortheinversemodelingwiththeFNNmodelfortheMR uiddamper.

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Voltage [V]

Time [s]

Force [N]

––Time [s]

Figure15.ValidationoftheinversemodelingfortheMR uiddamperusingtheneuralnetworkmodel:(a)thecommandvoltagepredictedusingtheFNNmodel;(b)theforcepredictedfromthecommandvoltageusingtheFNNmodel.

AlthoughtheinverseFNNmodelshowsgoodaccuracy,theobviousdisadvantageliesintheneedforcommandvoltageinputduringprediction;thereforethemodelcanonlybeusedwhenthecommandvoltagecanbeobtainedbeforehand.4.2.ModelinginversedynamicsoftheMR uiddamperwiththeRNN

InordertorealizetheinversemodelingoftheMR uiddamper,amultilayerRNNisselected,asshownin gure16.TheselectedRNNmodelcanberepresentedasamapping:v( k+1)=NN[v( k),v( k 1),···,v( k Ov+1),

F(k),F(k 1),···,F(k IF+1),

x(k),x(k 1),···,x(k Ix+1)]

(21)

whereNN[·]denotesaneuralnetworkwithIF+Ix(=R)inputsandoneoutput,trainedtoapproximatetheinverseinput–outputmappingthatdescribestheinversedynamicbehavioroftheMR uiddamper.Ovisthenumberofdelaysintheneuralnetworkmodel.

ThearchitectureoftheRNNusedhereisbasicallythesameasthatoftheRNNmodeldiscussedinsection3exceptthattheoutputoftheneuralnetworkisthecommandvoltage,whichisdelayedandfedbacktotheinputlayeroftheneuralnetworkasshownin gure17.TheinputoftheneuralnetworkmodelisalsocomposedofthedisplacementandtheforceoutputoftheMR uiddamper.Thetransferfunctionfortheneuronoftheoutputlayerisselectedasalinearfunction,andthetransferfunctionsfortheneuronsoftheinputandhiddenlayersarehyperbolictangentsigmoidfunctionswiththeformgivenbyequation(9),soequation(21)canberewrittenas (k)+b1)+b2)+b3)

v( k+1)=T3(W3T2(W2T1(W1P+LW1v

(22)120

Figure16.TheschemeoftheRNNformodelingtheinversedynamicsoftheMR uiddamper.

whereW1,W2,andW3aretheweightmatricesofthethreelayers,respectively.LW1∈ OF×S1isthelayerweight.b1,b2,andb3arethebiasmatricesofthosethreelayersrespectively.

(k)=[v(Andv k)v( k 1)v( k 2)···v( k 5)]T.Thenetworkshownin gure17istrainedusingtheLevenberg–Marquardtalgorithmexpressedbyequation(14)withthedatasetshownin gure2,inwhichtheinputvectorisintheformpq=[xqFq]T,andtheoutputvectoristq=vq,wherexqandvqaregivenbyequation(15),andFq∈F.Beforetrainingtheneuralnetworkmodel,thetrainingdatasetshouldbealsopreprocessedbynormalizingtheinputsandtargetssothattheyhavemeansofzeroandstandarddeviationsof1.Whenthenetworkisbeingtrained,theJacobianisalsodividedintotwoequalsubmatricestosavememoryandtheapproximateHessianmatrixiscalculatedusingequation(18).

ThevalidationschemeoftheinverseRNNmodelfortheMR uiddamperwithSIMULINKisshownin gure18,which

ModelingandcontrolofMR uiddampersusingneural

networks

Figure18.ThevalidationschemefortheinversemodelingwiththeRNNmodelfortheMR uiddamper.

isquitesimilarto gure14exceptthatthecommandvoltageisnotneededtofeedtoMRD NN INVERSEblock.MRD1andMRD2areblocksbasedonthemodi edBouc–Wenmodel.TheblocklabelledasMRD NN INVERSEiscreatedonthebasisofthetrainedneuralnetworkmodel.Fromthis gure,thedisplacementaccompaniedbytheforcegeneratedbyMRD1isfedtoMRD NN INVERSEtogeneratethepredictedcommandvoltage,thenthedisplacementaccompaniedbythepredictedcommandvoltageisfedtoMRD2togeneratethedampingforce.ThedifferencebetweenthepredictedcommandvoltageandtheinputcommandvoltageaswellasthedampingforcesproducedbyMRD2andMRD1willbevalidated.

Fourvalidationcasesarediscussedinthissubsection.The rstvalidationcaseisshownin gure19.ThedisplacementisasinusoidalsignalandthecommandvoltageisaGaussianwhitenoisesignal(thevalidationdataset2,k=3,asshownintable3).From gure19(a),thepredictedcommandvoltagecantrackthetargetcommandvoltagereasonablywell.Itisinterestingtoseefrom gure19(b)thatthedampingforceproducedbythepredictedcommandvoltage(MRD2)coincideswiththedampingforceproducedbythetargetcommandvoltage(MRD1).

ThesecondvalidationcaseusestheGaussianwhitenoisedisplacementandsinusoidalvoltageinput(validationset6,k=3)(see gure20).From gure20(a),weseethatthepredictedcommandvoltagecannottrackthetargetcommandvoltageverywell;however,thedampingforceproducedbythepredictedcommandvoltagetracksthedampingforceviathetargetcommandvoltagewell(see gure20(b)).

Figure21showstheresultsforthethirdvalidationcase,forwhichthevalidationdataset1(k=3)intable3isused.From gure21(a),weseethatthepredictedcommandvoltagecannottrackaccuratelytheconstanttargetcommandvoltage(vDE=1.5V).Althoughaperiodicallyvariedcommandvoltageisproducedforthegivenconstantcommandvoltage,thedampingforcesproducedfromthepredictedcommandvoltageandtheconstantcommandvoltagecoincidewitheachother.

Thelastvalidationcaseispresentedin gure22,wherevalidationdataset5intable3isused.Validationdataset5isapartofthetrainingdataset.From gure22(a),weseethatthepredictedcommandvoltagedoesnotcompletelycoincidewiththetargetcommandvoltage,butthedampingforcesgeneratedbythepredictedcommandvoltageandthetargetcommandvoltagecompletelycoincidewitheachother(see gure22(b)).ThisisaveryimportantrequirementforusingthenetworkmodeltocontrolthedampingforceofanMR uiddamper.

Fromtheabovevalidationresults,weseethattheinversedynamicmodelingaccuracyusingtheRNNmodelisnotasgoodasthatusingtheFNNmodel.Itisfortunatethatthedampingforcegeneratedbythepredictedcommandvoltagecantrackthedampingforcegeneratedbythetargetcommandvoltagewell.ThiscansatisfytheneedsfortheinversemodelofMR uiddamperbecausetheinversemodelingnetworkismainlyusedtocontrolthedampingforceoftheMR uid

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Voltage [V]

Time [s]

Force [N]

––Time [s]

Figure19.ValidationoftheinversemodelingfortheMR uiddamperusingtheneuralnetworkmodel(validationset2,k=3):(a)thecommandvoltagepredictedusingtheRNNmodel;(b)theforcepredictedfromthecommandvoltageusingtheRNNmodel.

Voltage [V]

Time [s]

Force [N]

––Time [s]

Figure20.ValidationoftheinversemodelingfortheMR uiddamperusingtheneuralnetworkmodel(validationset6,k=3):(a)thecommandvoltagepredictedusingtheRNNmodel;(b)theforcepredictedfromthecommandvoltageusingtheRNNmodel.

damper.TrackingthedampingforceoftheMR uiddamperusingtheinversedynamicneuralnetworkmodelwillbediscussedinsection5.2.

5.Applicationsofneuralnetworkmodels

Intheabovesections,thedirectidenti cationandinversemodelingoftheMR uiddamperusingFNNandRNNmodelsareproposed.Inthissection,applicationsoftrainedneural122

networkmodelsoftheMR uiddamperforforcepredictionandgenerationproblemsarediscussed.

Inthepastdecade,manyresearchershaveexploredthefeasibilityofsemi-activevibrationcontrolofstructuresusingMR uiddampers.AlthoughMR uiddampersarehighlynonlinear,thedampingforcecanbecontrolledbywell-designedcontrolalgorithms.Atypicalsemi-activecontrolsystemschemeusinganMR uiddampertocontrolvibrationofastructureisshownin gure23.

ModelingandcontrolofMR uiddampersusingneuralnetworks

Voltage [V]

Time [s]

Force [N]

––

Time [s]

Figure21.ValidationoftheinversemodelingfortheMR uiddamperusingtheneuralnetworkmodel(validationset1,k=3):(a)thecommandvoltagepredictedusingtheRNNmodel;(b)theforcepredictedfromthecommandvoltageusingtheRNNmodel.

Voltage [V]

Time [s]

Force [N]

––Time [s]

Figure22.ValidationoftheinversemodelingfortheMR uiddamperusingtheneuralnetworkmodel(validationset5):(a)thecommandvoltagepredictedusingtheRNNmodel;(b)theforcepredictedfromthecommandvoltageusingtheRNNmodel.

Onekeyfeaturein gure23isthatthesemi-activecontrolsystemusingMR uiddampersconsistsofasystemcontrollerandadampercontroller.Thesystemcontrollergeneratesthedesireddampingforceaccordingtothedynamicresponsesoftheplantwhilethedampercontrolleradjuststhevoltagetotrackthedesireddampingforce.Theotherfeaturein gure23isthatthedampingforceoftheMR uiddampershouldbemonitoredand/orpredictedandfedtothedampercontrollertogeneratethecommandvoltageonthebasisofthedesired

dampingforcegeneratedbythesystemcontroller.Therefore,predictionofthedesireddampingforceandcommandvoltagegenerationaretwoimportantaspectsforapplyingMR uiddamperstophysicalstructures.

5.1.ForcepredictionfortheMR uiddamperusingneuralnetworkmodels

Itisverydif culttocontrolthedampingforceofanMR uiddamperbecauseofthesemi-activenatureandthenonlinear

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Figure23.Thesemi-activecontrolsystemwiththeMR uiddamper.

relationshipsbetweenthecommandvoltageandthedampingforce.Inordertoovercomethisdif culty,thedesireddampingforceisusuallyobtainedthroughasuitablecontrolalgorithm(systemcontroller).However,theforcegeneratedbytheMR uiddampercannotbecommandeddirectly;onlythevoltageappliedtothecurrentdriverfortheMR uiddampercanbedirectlycontrolled(Dykeetal1996),wheretheactualdampingforceoftheMR uiddamperneedstobemonitoredonlineandthecommandvoltagevariesbetweenlimitedvoltagelevels.TheforcesensorneedstobeinserieswiththeMR uiddamper,whichreducesreliabilityandincreasescost.WhentheMR uiddamperisusedtoattenuatesystemvibration,theparametersofthesystemshouldbemonitoredandfedbacktothecontroller.Ifthisinformationisusedtopredictthedampingforceusingthetrainedneuralnetworkmodel,thecostofthesystemcanbereducedandthereliabilityofthesystemcanbeimproved.NeuralnetworkmodelsfortheMR uiddamperproposedin gures4and7canbeusedtopredictthedampingforceoftheMR uiddamper.Theaccuracyofthepredictioncanbeimprovedthroughthoroughtraining.Referto gures6,9,and10forthepredictionperformanceoftheneuralnetworkmodels.

5.2.ForcecommandoftheMR uiddamperusingneuralnetworkmodels

5.2.1.TheexistingforcecontrollerwiththeHeavisidestepfunction.Thedif cultyforgeneratingthedesireddampingforceoftheMR uiddamperhasbeendiscussed.WhenthedesiredandactualdampingforcesoftheMR uiddamperareknown,Dykeetal(1996)proposedamethodthatusesaHeavisidestepfunctionasfollows:

v=VmaxH{(FDE FAC)FAC}

(23)

5.2.2.Theforcecontrollerusingneuralnetworkmodels.Inthissubsection,theneuralnetworkmodelfortheinversedynamicsoftheMR uiddamperisusedtogeneratethecommandvoltageaccordingtothedesireddampingforce.Theschemeofthesystemisillustratedin gure25,inwhichFDE(k)representsthedesiredforcethattheMR uiddampershouldgenerate,andFAC(k)representstheactualdampingforcethattheMR uiddamperactuallyproduces.Theblockdenotedas‘NeuralNetwork’representstheinverseneuralnetworkmodeloftheMR uiddamper,whichistrainedinsection4.Infact,theinverseneuralnetworkmodeloftheMR uiddamperin gure25actsasadampercontrollerasshownin gure23.IfaninverseFNNmodelisused,theactualdampingforceneedstobemonitoredbyaforcetransducerandfedbacktotheinputoftheFNNmodel,whichisshownbythedashedlinein gure25.TheSIMULINKblockdiagramusingtheinverserecurrentneuralnetworkmodelisshownin gure24(b).5.2.3.Discussionsonthedampercontrollers.Inordertovalidatethedampercontrollers,thedesireddampingforcegivenbyequation(24)isconsideredtobeproducedunderthedisplacementgivenbyequation(25):

FDE=1200(sin4πt)

π

x=sin4πt .

2

(24)(25)

whereVmaxisthevoltagetothecurrentdriverassociatedwithsaturationofthemagnetic eldintheMR uiddamper,andH(·)istheHeavisidestepfunction.FDEandFACarethedesireddampingforcegeneratedbythesystemcontrollerandtheactualdampingforceoftheMR uiddamper.

In gure24(a),thedampercontrollerusingtheHeavisidestepfunctionisshownandthevalidationresultsarediscussedinsection5.2.3.124

Thevalidationresultsforthedampercontrollersareshownin gure26.In gure26(a),thecontrolleddampingforcesobtainedusingtheinverseneuralnetworkmodelandHeavisidestepfunctionarecomparedtothedesireddampingforce.ItcanbeseenthatthedampingforcegeneratedbytheHeavisidestepfunctioncontrollercoincideswiththedesireddampingforce,whichindicatesthatthedampingforcecanbecompletelycontrollableunderthiscondition.Alsothedampingforceproducedbytheneuralnetworkcontrollercantrackthedesireddampingforceonthewhole,butthedifferencebetweenthecontrolledanddesireddampingforcesislargerthanthatwiththeHeavisidestepfunctioncontroller.

In gure26(b),thecommandvoltagesproducedbythecorrespondingdampercontrollersarealsoshown.The

ModelingandcontrolofMR uiddampersusingneuralnetworks

(a)

Figure24.TheforcecontrollerfortheMR uiddamper:(a)theschemeusingtheHeavisidestepfunction;

(b)theschemeusingtheinverseneuralnetworkmodel.

Figure25.Theschemeofthecontrollerfortrackingthedesireddampingforceviatheinversedynamicneuralnetworkmodel.

commandvoltageproducedbytheHeavisidestepfunctioncontrollerisadiscretepulsewithchangingtimewidth,whichneedsafastdynamicresponseofthecurrentdriverfortheMR uiddamper.Thecommandvoltageproducedbytheneuralnetworkmodelisacontinuouslyvaryingvoltage,whichisbene cialtothecurrentdriverfortheMR uiddamper.

6.Conclusionsandfuturework

Inthispaper,directidenti cationandinversedynamicmodelingofanMR uiddamperusingfeedforwardandrecurrentneuralnetworksarestudied.Thetraineddirectidenti cationneuralnetworkmodelcanbeusedtopredictthedampingforceoftheMR uiddamperonline,onthebasisofthedynamicresponsesacrosstheMR uiddamperandthecommandvoltage.Theinversedynamicneuralnetworkmodelcanbeusedtogeneratethecommandvoltageonthebasisof

thedesireddampingforce.ThearchitecturesandthetrainingmethodsforthedirectandinverseneuralnetworkmodelsfortheMR uiddamperarepresented,andsomesimulationresultsarediscussed.Finally,thetrainedneuralnetworkmodelsareappliedtopredictandcontrolthedampingforceofanMR uiddamper.

TheneuralnetworkmodelsdevelopedfortheMR uiddamperarevalidatedandtheirperformancesareevaluated.ValidationresultsindicatethattheproposeddirectdynamicmodelusinganRNNcanbeusedtopredictthedampingforceaccuratelyonlineandtheinversedynamicmodelusinganRNNcanbeusedtogeneratethecommandvoltagewhentheMR uiddamperisusedinthesemi-activemode,whichprovidesanewmethodforthedampercontrolleroftheMR uiddamper.Notonlyaretheresultssatisfactory,butthisapproachalsosimpli esthecontrollerarchitecture.Furthermore,theRNNmodelfortheMR uiddampergeneratesasmoothercommandvoltagethanthatwiththeHeavisidestepfunction.

TheresultspresentedinthispaperarestillpreliminaryformodelingandcontroloftheMR uiddamperusingneuralnetworks;moreresearchworkisneededinordertoimplementneuralnetworksforpracticalapplicationswithMR uiddampers.Inordertoassurestabilityofthecontrolledsystem,theon-linelearningoftheneuralnetworkandinverseneuralnetworkmodelsthatareusedtoadapttotheuniquecharacteristicsoftheMR uiddamperunderdifferentconditionsareworthfurtherinvestigation.Moreover,experimentalvalidationsusingneuralnetworkchipsshouldalsobeperformed.

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