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Perfect collocation using self-sensing electromagnetic actuator: Application to vibration control of flexible structures

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ContentslistsavailableatScienceDirect

Sensors

and

Actuators

A:

Physical

jo u r n al hom e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / s n a

Short

communication

Perfect

collocation

using

self-sensing

electromagnetic

actuator:

Application

to

vibration

control

of

flexible

structures

Mohit

Verma

a,∗

,

Vicente

Lafarga

b

,

Christophe

Collette

b,c

aCSIR-StructuralEngineeringResearchCentre,CSIRCampus,Taramani,Chennai-600113,India

bUniversitéLibredeBruxelles,PrecisionMechatronicsLaboratory,BEAMSDepartment,50,F.D.RooseveltAv.,1050Brussels,Belgium cUniversityofLiège,DepartmentofAerospaceandMechanicalEngineering,AlléedelaDécouverte9,4000Liège,Belgium

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received11May2020

Receivedinrevisedform6July2020 Accepted15July2020

Availableonline17July2020

Keywords: Vibrationisolation Self-sensing Collocatedcontrol Activedamping Electromagneticactuator

a

b

s

t

r

a

c

t

Collocatedcontrolhastheadvantagesofrobustnessandguaranteedstability.Incollocatedcontrol,the

sensorandactuatorareplacedclosetoeachother.Truecollocationcanbeachievedusingself-sensingin

whichtheactuatorcanalsobeusedasasensor.Inthisarticle,wepresentaself-sensingelectromagnetic

actuatorforvibrationcontrolofflexiblestructures.Thebackelectromotiveforce(emf)generatedinthe

coilismeasuredtoevaluatethevelocityofthestructure(andhence,thedisplacement).Theposition

measurementsobtainedfromself-sensingarefoundtohavegoodcorrelationwiththoseobtainedusing

aneddycurrentsensor.Theefficacyoftheproposedtechniquehasbeenverifiedforthevibrationcontrol

ofaflexiblecantileverbeam.Self-sensingofferseconomical,simpleandrobustcontrolarchitecture.It

canbeeffectivelyusedforapplicationswheresensorsandactuatorscannotbecollocatedduetospace

andsizelimitations.

©2020ElsevierB.V.Allrightsreserved.

1. Introduction

External disturbances like ground motion or environmental disturbancescangreatlyaffecttheperformance ofthesensitive equipment.Activesystemscanbeusedtoisolatetheequipment fromtheexternaldisturbances[1].Sensorsandactuatorsinmany activesystemsareoftencollocatedinordertobuildrobustnessinto thesystem.Theopenlooptransferfunctionofcollocatedsystems hasalternatingpolesandzerosontheimaginaryaxis.Insuch sys-tems,thestabilityisguaranteedasthephaselagduetoeachpoleis compensatedbythephaseleadduetothecorrespondingzero[2]. Thecollocationofsensorsandactuatorsishowevernotpossibleat alltimes.Theremightbespacerestrictionsincertainapplications. Onemightencounterasituationwherethesizeofthesensorisnot negligibleincomparisontothemainstructure.Insuchcases,the mechanicalimpedanceofthestructureisgreatlymodifiedwhen thesensorsareinstalledonthestructure.

Self-sensingprovidesanalternativetoovercomethese prob-lems. Perfect collocation can be achieved using self-sensing in whichasingletransducerisusedasanactuatorandsensor con-currently.Itoffersasimple,robustandcost-effectivesolutionfor

∗ Correspondingauthor.

E-mailaddress:mohitverma@serc.res.in(M.Verma).

collocatedcontrol.Self-sensingreducesthecomplexityby reduc-ingthenumberofactuatorsandsensorsinthesystem.Italsoleads toreductioninthepowerrequirementsandthewiringusedin theinstrumentation.Sensorsaremorepronetofailurecompared totheactuatorsinthecontrolsystem[3].Thus,self-sensingalso leadstoarobustsystemasasingletransducerservesthedual pur-poseofsensingandactuation.Manyself-sensingtransducershave beendevelopedinthepastusingdifferenttypesofactive materi-als.Self-sensingusingpiezoelectricmaterialshavebeenusedfor measuringstress,forceandposition[4–7].Adielectricelastomer actuatorhasbeenusedforpositionsensingandforcecontrol[8,9]. Insomerecentapplications,shapememoryalloys[10]and mag-netorheologicalfluid[11]basedactuatorshavealsobeenusedfor self-sensing.Electromagneticactuatorshavebeenusedforvelocity, positionandforceestimations[12–14].

Inthisarticle,wepresentaself-sensingelectromagnetic actua-torforvibrationcontrolofflexiblestructures.Thepresentscheme doesnotrequireanymodificationstothedesignofthe electro-magneticactuator.Ittakesadvantageofthefactthatthevoltage dropacrossthecoilcanbemeasuredandrelatedtoitsmotion.The electromagneticactuatorisoperatedincurrentmode.Itdoesnot requireanadditionalbridgestructureformeasuringthevoltage dropacrossthecoil(whichisusuallyrequirediftheactuatoris operatedinvoltagemode).Theorganizationofthisarticleisas fol-lows:Section2presentscomparisonofcollocatedcontrolwiththe https://doi.org/10.1016/j.sna.2020.112210

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Fig.1. Locationofactuatorandsensorforcollocatedandnon-collocatedcontrol.

non-collocatedcontrol.Theconceptofself-sensingusing electro-magneticactuatorispresentedinSection3.Theproof-of-concept experimentscarriedouttovalidateself-sensingareexplainedin Section4.Theenvisionedapplicationforself-sensingispresented inSection5,whichisfollowedbyconcludingremarksinSection6.

2. Collocatedvsnon-collocatedcontrol

Inordertohighlighttheadvantagesofthecollocatedcontrol overnon-collocatedcontrol,we considera caseofacontinuous beampinnedatboththeends.UsingEuler–Bernoullibeamtheory, thedynamicsofthebeamcanbewrittenasthefollowingpartial differentialequation

2

x2



EI

2 w

x2



+m

2 w

t2 =p (1)

wherewisthedisplacement,EIrepresentsthebendingstiffness, misthemassperunitlengthandpistheexternalloadperunit length.Foracaseoffreevibrationandharmonicsolutionoftheform w(x,t)=(x)ejωt,theabovepartialdifferentialequationisreduced

tothefollowingordinarydifferentialequation d4 dx4 − m EIω 2 =0 (2)

The above equation is an eigenvalue problem whose solutions areωiandirepresentingnaturalfrequenciesandcorresponding

modeshapes,respectively.Forabeampinnedatboththeends,the naturalfrequenciesandthemodeshapesare

ω2 i =(i) 4 EI ml4 i(x) =sinixl (3)

Fig.2. Frequencyresponsesoftheopenlooptransferfunctionfromactuatorforcetothedisplacementmeasuredbythesensorincaseof—(a)collocatedcontroland(b) non-collocatedcontrol.

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Fig.4. Modelofelectromagneticactuator.

wherelisthelengthofthebeam.Thefrequencyresponsebetween theactuatorforce,u,atpointxaandsensordisplacement,y,atpoint

xbcanbewrittenas[15] G(s)= y u = ∞



i=1 i(xa)i(xb) i(s2+2iωis+ω2i) (4)

whereiisthegeneralizedmass(equaltoml/2forabeampinned

atboththeends)and i isthedampingassociatedwiththeith

mode.Theseriesexpansioncanbelimitedtoafinitenumberof modesincaseofadiscretesystem.Thesensorandactuator loca-tionforcollocatedandnon-collocatedcontrolareshowninFig.1. Incaseofcollocatedcontrolitisassumedthatboththeactuator andsensorarelocatedatxa=xs=0.5l.Fornon-collocatedcontrol,

itisassumed thattheactuatorislocatedatmidspan(x1=0.5l)

whilethesensorisplacedatxs=0.9l.Thebendingstiffness,EI,of

thebeamistakenas100Nm2,massperunitlength,m,istaken

as1kg/mand length,l,istakenas1m. Thefirsttenmodesare consideredforevaluationoftheopenlooptransferfunction.The openlooptransferfunctionsfromactuatorforcetothesensor dis-placementforcollocatedandnon-collocatedcontrolsareshownin Fig.2aandb,respectively.Itisobservedthatthecollocated sys-temhasalternatingpolesandzeros.Thephasedropduetoeach

Table1

Goodnessoffitbetweenthedisplacementsignalsobtainedfromeddycurrentsensor andself-sensing.

Rootmeansquared error(RMSE)

Sumofsquarederror (SSE)

Coefficientof determination(R2)

0.0072 1.939 0.9927

poleiscompensatedbythephasegainduetozero.Thephaseof theopenlooptransferfunctionisboundedbetween0and.This interlacingpropertyofthecollocatedsystemprovidesasymptotic stabilityevenwhenthesystemparametersaresubjectedtolarge variations.Collocatedsystemprovidesrobuststabilitytothe sys-tem.However,thispole-zerointerlacingpropertydoesnotexist inthenon-collocatedsystem.Thephaseisnotboundedbetween 0andandthesystemisnotrobustlystable.Thisfactisfurther illustratedbystudyingtherootlocusofthefeedbacksystem.The systemisassumedtohave afollowingleadcompensatorinthe feedbackloop

K(s)= 10s+1

0.005s+1 (5)

The root loci of thecollocated and non-collocatedsystems are showninFig.3aandb,respectively.Itcanbeseenthatfor col-locatedsystemtherootlocusliesentirelyinthelefthalfplane. Thismeansthatthesystemisstableforanyvalueofthegain.The rootlocusofthenon-collocatedsystemdoesnotlieentirelyinthe lefthalfplane.Thisimpliesthatforcertainvaluesofthegainthe closedlooppolesofthesystemwillmovetotherighthalfplane resultinginanunstablesystem.Duetothepole-zerointerlacing propertyandrobustlystablenature,collocatedsystemsareusually preferredwhereverpossible.

3. Self-sensingelectromagneticactuator

ThemodeloftheelectromagneticactuatorisshowninFig.4. Theactuatorcanbedrivenbyeitheravoltageorcurrentsource.The

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Fig.6. Comparisonofthefrequencyresponsesobtainedfromtheexperimentand themodel.Agoodmatchbetweentheexperimentsandthemodelisobservedup to100Hz.

Fig.7.Comparisonofthedisplacementresponseofthefreeendofcantileverbeam obtainedfromself-sensingwiththatobtainedfromeddycurrentsensor.

Fig.8. Openloopgainoftheplantwiththedesignedleadcontroller.Thedesigned controllerhasacrossoverfrequencyat36.5Hzandhasaphasemarginof60◦.

modelingoftheactuatordependsonthemodeinwhichitis oper-ated.Theconstitutiveequationsoftheelectromagnetictransducer canbewrittenas

Vm =Zei+Tem

v

F =Tmei+Zm

v

(6) whereVmisthevoltagedropacrosstheterminalsoftheactuator

coil,Zeistheelectricalimpedanceofthecoil,iisthecurrentflowing

inthecoil,Temisacouplingcoefficientrepresentingthebackemf

constantofthecoil,

v

isthevelocityofthemechanicalstructure,Fis theforceactingonthemechanicalstructureattachedtothe actua-tor,Tmeisthecouplingcoefficientrepresentingtheforcetocurrent

ratiooftheactuatorandZmisthemechanicalimpedanceofthe

attachedstructure.Theoperationoftheelectromagneticactuator withacurrentsourceensuresthatthedesiredcurrentflowsinthe coilirrespectiveofthebackemfgeneratedbythemotionofthe mechanicalstructure.Thelossesduetobackemfarecompensated bytheamplifier.UsingKirchoff’slaw,thevoltagedropacrossthe terminalsofthecoilcanbewrittenas

Vm=Ri+Ldi

dt+Tem

v

(7)

whereRandLaretheresistanceandinductanceofthecoil, respec-tively.GiventheparametersR,LandTemoftheactuator,theabove

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Fig.10.Comparisonofthefrequencyresponsesofthecontrolledanduncontrolled structureusingfeedbackfrom—(a)eddycurrentsensorand(b)self-sensing.

equationshowsthatitispossibletoevaluatethevelocityofthe

mechanicalstructurebymeasuringthevoltagedropacrossthe ter-minalsofthecoil.Thevelocityofthemechanicalstructureisrelated toVmandibythefollowingequation

v

=Vm−Ri−L

di dt

Tem (8)

TakingtheLaplacetransformoftheaboveequation,weget

v

=Vm−(R+Ls)i

Tem (9)

Fig.11.Comparisonofthetimedomaindisplacementresponseofthebeamwith andwithoutcontrolobtainedusingself-sensing.

wheresistheLaplacetransformvariable.Thedisplacementofthe mechanicalstructureisgivenby

x= 1 sTemVm−

(R+Ls)

sTem i (10)

In-housemanufacturedcurrent amplifieris usedin order to operatetheelectromagneticactuatorincurrentmode.Thecircuit diagramofthecurrentamplifierisshowninFig.A.13ofAppendix A.

4. Proofofconceptexperiment

Thevalidationoftheself-sensingelectromagneticactuatoris carriedoutonacantileverbeamclampedatoneendandattached tothevoicecoilactuatorontheotherend.Thetestsetupusedis showninFig.5.Aneddycurrentsensorisusedtotrackthemotion ofthefreeendofcantileverbeam.Self-sensingincurrentmodeis usedfortheproofofconceptexperimentduetoitssimplicity,ease ofimplementation,andavailabilityofthecurrentsource.

4.1. Identificationoftheplant

Theplant isidentified experimentally byevaluatingthe fre-quencyresponsefromtheinputcurrenttothedisplacementofthe freeendofthecantileverbeam.Awhitenoisesignalisinjectedin theactuatorandthedisplacementisrecordedusingtheeddy cur-rentsensor.Atransferfunctionoftheplant(G(s))isthenobtained usingfitfrdfunctioninMatlab[16].Theorderofthetransfer functionischosensuchthatitisabletocaptureallthemodesof theplantupto200Hz.Thetransferfunctionoftheplantis G(s)= 76.386(s2+25.82s+1494)(s2+1.803s+1.23e05)(s2+144.2s+1.268e06)

(s2+25.87s+1491)(s2+6.88s+2932)(s2+3.461s+1.528e05)(s2+144s+1.451e06) (11)

Thefrequencyresponsesobtainedfromtheexperimentand the modelarecomparedinFig.6.Thefirstbendingmodeofthebeam isat8.6Hz. Itisobservedthatthefrequencyresponseobtained fromtheexperimentsmatcheswellwiththatobtainedfromthe modelupto200Hz.

4.2. Displacementfromself-sensing

Thedisplacementofthefreeendofthebeamisevaluatedusing Eq.(10).TheparametersoftheelectromagneticactuatorR,Land Temarefoundtobe6.2,1.23mHand3.9Vs/m,respectively.The

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Fig.12. Configurationoftheproposedisolationsystemfordronecamerastabilization.

displacementobtainedusingEq.(10)ispassedthrough a high-passfilterinordertoavoidthedriftduetointegration.Thecutoff frequencyofthefilterissetto0.2Hz.Thedisplacementresponse obtainedfromself-sensingiscomparedwiththatobtainedfroman eddycurrentsensorinFig.7.Thedisplacementresponseobtained fromself-sensingandtheeddycurrentsensorarefoundtomatch wellwitheachother.Thegoodnessoffit,definedintermsofroot meansquarederror(RMSE),sumofsquarederror(SSE)and coef-ficientofdetermination(R2),betweenthedisplacementresponses

ofthebeamobtainedfromtheeddycurrentsensorandself-sensing aregiveninTable1.

4.3. Activevibrationcontrol

Adisplacementfeedbackcontrollerisdesignedtocontrolthe vibrationsofthe freeend of cantileverbeam.A lead controller is designed to have sufficient gain margin near the crossover frequency.Thedesignedcontrollerhasacrossoverfrequencyat 36.5Hzandhasaphasemarginof60◦.Thetransferfunctionofthe designedcontroller,H(s),is

H(s)=6094(s+95.54)

(s+1404) (12)

The open loop gain (G(s)H(s)) of the plant is shown in Fig. 8. Theblockdiagramforexperimentalverificationofself-sensingis showninFig.9.Abeamisexcitedbyinjectingawhitenoisesignal intheactuator.Thiswhitenoiseactsasanexternaldisturbance. Thedisplacementfeedbackfromtheeddycurrentsensorand self-sensingareusedtoreducetheeffectoftheexternaldisturbanceon themotionofthefreeendofthebeam.Thecontrolforceobtained usingthedisplacementfeedbackisthensubtractedfromthewhite noise.First,thefeedbackfromtheeddycurrentsensorisusedto seetheeffectivenessofthedesignedcontrolleronvibration reduc-tion.Thefrequencyresponsesoftheuncontrolledandcontrolled structurearecomparedinFig.10a.Itcanbeseenthatthedesigned controlleriseffectiveinreducingthevibrationofthecantilever beam.

Afterverificationofthecontroller,thesignalobtainedfrom self-sensingisusedforthefeedbackcontrol.Thenoiseintheself-sensed signaldrivestheactuatorwhich in turnforcesthestructure to followthenoise. Hence,thesignalobtainedfromtheeddy cur-rentsensor providesbetterindication of the performance. The uncontrolledandcontrolledfrequencyresponsesofthebeamwhen self-sensing is usedas feedback are compared in Fig. 10b. The feedback fromself-sensing is found toexhibit thesimilar per-formanceasthatobtainedbyusingfeedbackfromeddycurrent sensorforfrequenciesabove0.2Hz.Thecomparisonofthe con-trolledanduncontrolleddisplacementtimehistoriesofthebeam

obtainedusingself-sensingisshowninFig.11.Thisdemonstrates thatself-sensingcanbeeffectivelyusedforsensingandactuation simultaneously.Animportantadvantageofself-sensingisthatit resultsinperfectlycollocatedcontrol.

5. Envisionedapplication:Dronecamerastabilization

Dronecameraissubjectedtovibrationsfromthedronewhich hampersthequalityofthecapturedimages.Thecurrentsystem usespassiverubbermounts.Thesemounts,althougheffectivein reducing theresponsenearresonance, degradetheresponse at highfrequencies (adding dampingto thesystem). Many active mountshavebeendevelopedbuttheyworkonlyinonedirection [17,18].Thismotivatedthedevelopmentofasixdegreesof free-domisolationsystemfordronecamerawherehighqualityimages arerequired.Theisolationsystemisbasedonthecubic configu-rationoftheStewartplatform.Theproposedisolationsystemis showninFig.12.Theuniquefeaturesoftheisolationsystemare —useofnon-contactelectromagneticactuators,noflexiblejoints (clearancebetweenthecoilandmagnetisusedtoallowfor rota-tions),3Dprintedlightweightpartsandhighfrequencyroll-offas thereisnostiffnessinthelegs.Duetothecompactarchitecture andweightlimitations,itisnotpossibletohaveanarrayof sen-sorswhichcanbecollocatedwiththeactuators.Thismotivates theuseofself-sensing.Itdoesnotrequireanymodificationsinthe electromagneticactuators.Thevoltagedropacrosseachcoilcanbe measured.Themeasuredvoltagedropcanthenbeusedtoevaluate themotionineachofthelegsoftheisolator.Thisisthenusedas afeedbacktoactivelydamptheresonancesoftheisolationsystem withoutamplifyingthehighfrequencyresponse.Theproperties ofthesystemlikestiffnessmaychangewithtimeduetofatigueor otherenvironmentalfactors.Self-sensingprovidesarobustcontrol strategywhichcanbeusefulinsuchsituations.Currently,workis underwaytodevelopaprototypeofsuchisolationsystemusing self-sensingforcollocatedcontrol.

6. Concludingremarks

In many applicationsperfect collocationis not possible due tospace and size limitations. In suchcases, onemay resortto self-sensingwhereasingletransducerservesthedualpurposeof sensingandactuation.Thisbriefpresentedself-sensingusing elec-tromagneticactuator.Theactuatorisoperatedincurrentmode.The voltagedropacrossthecoilcanbedirectlymeasuredtoevaluatethe motionofthecoil.Noadditionalbridgestructuresarerequired.The measurementsfromself-sensinghavebeenvalidatedwiththose obtainedusingeddycurrentsensor.Themeasurementsfrom

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self-sensinghavealsobeenusedtoactivelycontrolthevibrationsof theflexiblecantileverbeam.Theperformanceoftheactivecontrol usingself-sensingarefoundtobeatparwiththeactivecontrol usinganeddycurrentsensor.Self-sensingprovidesarobust, cost-effectiveandsimplecontrolarchitecture.Itresultsinaperfectly collocatedsystemwiththeguaranteedstabilityastheopenloop transferfunctionhasalternatingpolesandzeros.Itisenvisioned tobeusedfortheapplicationswherealargearrayofsensorscould notbeusedduetospaceandsizelimitations.

Authors’contribution

MohitVerma: conceptualization, methodology, experiments, data curation, investigation, software, writing – original draft. VicenteLafarga:experiments,datacuration,investigation, valida-tion,writing–review&editing.ChristopheCollette: conceptualiza-tion,fundingacquisition,supervision,writing–review&editing.

DeclarationofCompetingInterest

Theauthorsreportnodeclarationsofinterest.

Acknowledgements

Theauthors gratefullyacknowledgetheWallonieRegion for funding this research (WALInnov Projet Trusteye, Grant No. 1710040).The authorswouldliketothank MichelOséefor his inputsoncurrentamplifier.ThetechnicaldiscussionswithDimitri PironandAhmadPaknejadarealsoacknowledged.

AppendixA. Circuitdiagramofcurrentamplifier

Inordertousetheelectromagneticactuatorintheself-sensing mode,currentamplifierismanufacturedin-house.Thecircuit dia-gramofthecurrentamplifierisshowninFig.A.13.

AppendixB. Supplementarydata

Supplementarydataassociatedwiththisarticlecanbefound,in theonlineversion,athttps://doi.org/10.1016/j.sna.2020.112210.

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References

[1]M.Verma,A.Pece,S.Hellegouarch,J.Watchi,G.Durand,S.Chesné,C.Collette, Dynamicstabilizationofthinaperturelightcollectorspacetelescopeusing activerods,J.Astron.Telesc.Instrum.Syst.6(1)(2020)014002.

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[3]R.Lorenz,Self-SensingMotionControlforElectricalandHybridElectrical Vehicles,IEEPowerConversions&ApplicationsLecture,Midlands EngineeringCentre,Birmingham,2001.

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[5]S.Z.Mansour,R.Seethaler,Simultaneousquasi-staticdisplacementandforce self-sensingofpiezoelectricactuatorsbydetectingimpedance,Sens. ActuatorsA:Phys.274(2018)272–277.

[6]V.Babuska,R.O’donnell,Self-sensingactuatorsforprecisionstructures,IEEE, in:1998IEEEAerospaceConferenceProceedings(Cat.No.98TH8339),1, 1998,pp.179–187.

[7]J.J.Dosch,D.J.Inman,E.Garcia,Aself-sensingpiezoelectricactuatorfor collocatedcontrol,J.Intell.Mater.Syst.Struct.3(1)(1992)166–185. [8]K.Jung,K.J.Kim,H.R.Choi,Aself-sensingdielectricelastomeractuator,Sens.

ActuatorsA:Phys.143(2)(2008)343–351.

[9]R.Zhang,P.Iravani,P.Keogh,Closedloopcontrolofforceoperationinanovel self-sensingdielectricelastomeractuator,Sens.ActuatorsA:Phys.264(2017) 123–132.

[10]S.-H.Lee,S.-W.Kim,Improvedpositioncontrolofshapememoryalloy actuatorusingtheself-sensingmodel,Sens.ActuatorsA:Phys.297(2019) 111529.

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[12]B.Hanson,M.Levesley,Self-sensingapplicationsforelectromagnetic actuators,Sens.ActuatorsA:Phys.116(2)(2004)345–351.

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Biographies

MohitVermareceivedhisPh.D.inStructuralEngineeringfromAcSIR,India.After thathejoinedPrecisionMechatronicsLaboratoryatULB,BelgiumasPostdoctoral ResearchAssociate.Currently,heisworkingasSeniorScientistatCSIR-Structural EngineeringResearchCentre,India.Hisareaofinterestisvibrationcontrolof struc-tures.

VicenteLafargaisadoctoralcandidateatthePrecisionMechatronicsLaboratory atULBsince2017.Hisresearchareafocusesondevelopmentofaactiveisolation systemforspaceapplications.

ChristopheCollettereceivedaM.Sc.andPh.D.degreesinMechanical Engineer-ingfromtheUniversitéLibredeBruxellesin2003and2007.ChristopheColletteis ProfessorofMechatronicsintheFacultyofEngineeringattheUniversitéLibrede BruxellesandUniversityofLiége.HeistheDirectorofthePrecisionMechatronics Laboratory.Hismainresearchinterestsincludetheactiveandpassivecontrolof vibration,opticalinertialsensorsandfuturespacetelescopes.

Figure

Fig. 1. Location of actuator and sensor for collocated and non-collocated control.
Fig. 5. Test setup used for proof of concept experiment — (a) schematics and (b) actual setup.
Fig. 9. Block diagram for the experimental verification of self-sensing.
Fig. 10. Comparison of the frequency responses of the controlled and uncontrolled structure using feedback from — (a) eddy current sensor and (b) self-sensing.
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