Active Disturbance Rejection Control for MEMS Gyroscopes

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Cleveland State University

EngagedScholarship@CSU

Electrical Engineering & Computer Science Faculty

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Electrical Engineering & Computer Science

Department

11-2009

Active Disturbance Rejection Control for Mems

Gyroscopes

Qing Zheng

Cleveland State University

Lili Dong

Cleveland State University, [email protected]

Dae Hui Lee

Zhiqiang Gao

Cleveland State University, [email protected]

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Repository Citation

Zheng, Qing; Dong, Lili; Lee, Dae Hui; and Gao, Zhiqiang, "Active Disturbance Rejection Control for Mems Gyroscopes" (2009).Electrical Engineering & Computer Science Faculty Publications. 108.

https://engagedscholarship.csuohio.edu/enece_facpub/108

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Original Citation

Qing, Z., Lili, D., Dae, H. L., & Zhiqiang, G. (2009). Active disturbance rejection control for MEMS gyroscopes. Ieee Transactions on Control Systems Technology, 17, 6, 1432-1438.

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Active Disturbance Rejection Control for MEMS Gyroscopes

Qing

Zheng,

Lili

Dong,

Dae

Hui

Lee,

and

Zhiqiang

Gao

Abstract—Anewcontrolmethodispresentedtodrivethedrive axis ofaMicro-Electro-MechanicalSystems(MEMS)gyroscope toresonanceandtoregulatetheoutputamplitudeoftheaxistoa ﬁxedlevel.Itisbasedonauniqueactivedisturbancerejectioncon trol(ADRC)strategy,whichactivelyestimatesandcompensates forinternaldynamic changesofthedriveaxis andexternaldis turbancesinrealtime.Thestabilityanalysisshowsthatboththe estimationerrorandthetrackingerrorofthedriveaxisoutputare boundedandthattheupperboundsoftheerrorsmonotonouslyde creasewiththeincreaseofthecontrollerbandwidth.Thecontrol systemissimulatedandtestedusingaﬁeld-programmable-gate array-baseddigitalimplementationonapiezoelectricvibrational gyroscope.Bothsimulationandexperimentalresultsdemonstrate thattheproposedcontrollernotonlydrivesthedriveaxistovibrate alongthedesiredtrajectorybutalsocompensatesformanufacture imperfectionsinarobustfashionthatmakestheperformanceof thegyroscopeinsensitivetoparametervariationsandnoises.Such robustness,thefactthatthecontroldesigndoesnotrequireanac curateplantmodel,andtheeaseofimplementationmakethepro posedsolutionpracticalandeconomicforindustrialapplications.

Index Terms—Active disturbance rejection control (ADRC), discreteimplementation,extendedstateobserver(ESO),ﬁeld-pro grammable gate array (FPGA), Micro-Electro-Mechanical Systems(MEMS)gyroscopes.

I. INTRODUCTION

M

ICRO-ELECTRO-MECHANICAL Systems (MEMS) gyroscope isamicro- or millimeter-scaleinertial rate sensor. Ithas beenused inautomobiles(stability controland GPS), aerospace(GPS-assisted inertial navigation), andcon sumerelectronics(cameraimagestabilizationand3-Dmouse) [1]. Compared to electromechanical gyroscopes, the MEMS gyroscopeissmallinsize,inexpensive,andenergyefﬁcient.A control systemisgenerallyused toexcitethe vibrationalong twovibratingmodes(drivingandsensingmodes)oftheMEMS gyroscopeandtoestimatetherotationrate.However,thesmall sizeoftheMEMSgyroscopeputsabigchallengeoncontroller design and microfabrication. The imprecise microfabrication anddisturbancesresultinmechanicalcouplingtermsbetween twoaxes,mechanical–thermalnoises,andparametervariations, whichconsequentlydegradetheperformanceoftheMEMSgy roscope.Therefore,aclosed-loopcontrolsystemisessentialfor improving the performanceofthe MEMS gyroscope through

alongthedrive sense

effectivelycompensatingforthemechanicalimperfectionsand thedisturbancesincontrolefforts.

Sincethe1990s,therehasbeenalimitedamountofresearch on feedback control system designs for MEMS gyroscopes. Thecontrollersintroducedin[2]–[4]didnotfullyaccountfor themechanicalcouplingtermsonthedriveaxiscausedbythe manufactureimperfections.Theadaptivecontrollersin[5]and [6]achievedperformanceimprovementinanoise-freesetting. Theadaptivecontrollerin[7]isdesignedfortheMEMSgyro scopesoperatinginanadaptivemode.However,mostreported MEMSgyroscopesoperateintheconventionalmode[8]where the movement of the mass along the drive axis is relatively largeandthemovementalongthesenseaxisisverysmall.The controllerin[8]regulatesthevibrationalongthesensingmode oftheMEMSgyroscope,whilethecontrolofthedrivingmode isdisregarded.

In this paper, a practical solution based on the active dis turbance rejection control (ADRC) technology is applied to the driving mode of the conventional MEMS gyroscope [9], [10].TheADRChasbeensuccessfullyemployedinmanyme chanicalsystems [11]–[14].However, theemploymentof the ADRContoMEMSisrathernew.Thebasicideaofthiscontrol strategy is to estimate the plant dynamics and disturbances using anextended stateobserver (ESO) andtoactivelycom pensateforthedisturbanceincontroleffort.Withtheaccurate estimationoftheplantdynamicsanddisturbancesbyESO,the ADRCcan successfully drive the output of the drive axis to resonance. Since the ADRCdoes not depend on an accurate modeloftheaxis,itisveryrobustagainstparametervariations, disturbances, andnoise. Another advantage of the controller is its few tuning parameters, making the controller easy to implement inthe real world. To test the effectiveness of the ADRC,aﬁeld-programmable-gate-array(FPGA)-baseddigital implementationofthecontrollerisconducted onapiezoelec trically drivenvibrationalbeam gyroscope. The experimental resultsdemonstratetheeffectivenessofthecontroller.

Thispaperisorganizedasfollows.ThedynamicsofMEMS gyroscopesisdescribedinSectionII.TheADRCapproachand itsstabilityanalysisarepresentedinSectionIII.Softwaresim ulationandhardwaretestresultsareshowninSectionIV.This paperendswithconcludingremarksinSectionV.

II. DYNAMICSOFMEMSGYROSCOPES

ThemechanicalstructureoftheMEMSgyroscopecanbeun derstoodasaproofmassattachedtoarigidframebyspringsand dampers,asshowninFig.1.Asthemassisdriventoresonance axisandtherigidframeisrotatingalongthe rotationaxis,aCoriolisaccelerationwillbeproducedalongthe axis,whichisperpendiculartobothdriveandrotation axes.TheCoriolisaccelerationisproportionaltotheamplitude oftheoutputofthedriveaxisandtheunknownrotationrate[1]. Therefore,wecanestimatetherotationratethroughsensingthe

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Fig.1. Mass–spring–damperstrutureofMEMSgyroscopes.

vibrationofthesenseaxis.Inordertoaccuratelysensethero tationrate,thevibrationalmagnitudeofthedriveaxishastobe regulatedtoaﬁxedlevel.Therefore,thecontrollerofthedrive axisismainlyusedtodrivethedriveaxistoresonanceandto regulatetheoutputamplitudeatadesiredvalue.

Disregardingthedampingcouplingtermsbetweentwoaxes andassumingthatthenaturalfrequenciesofbothaxesarethe same,thevibrationalMEMSgyroscopeismodeledas

(1) and aretheoutputsofthedriveandsenseaxes,re spectively,

where

and aretheCoriolisaccelerations, isthe rotationrate, isthenaturalfrequencyofthedriveandsense axes, and are quadrature errorscaused byspring couplings betweentwoaxes, isthe damping coefficient, isthemassof theMEMSgyroscope, isthecontrollergain, and isthe control inputforthe drive axis.In(1), theme chanicalthermalnoiseonthesenseaxisisrepresentedbythe randomforce .Theeffects ofthermalnoiseonthe drive axis arenegligibleandare ignored[15].Inthe MEMSgyro scopesrepresentedby(1), thequadratureerrorsare unknown constantsignals,thetime-varyingrotationrate isunknown, andthedampingcoefficienttypicallyhasalargevaryingrange. Inthispaper,weassumethatthesenseaxisisoperatingunder theopen-loopmode.Ourcontrolobjectiveistoforcethedrive axistooscillateatspecifiedamplitudeandresonantfrequencyin thepresenceofparameteruncertainties,mechanicalcouplings, andmechanical–thermalnoises.

III. ADRC

Inthispaper,ADRCisemployedtocontroltheMEMSgy roscopesbydealingwithmodelingerrorsandstructuraluncer tainties.In particular,an ESOprovides anestimateof thein ternaldynamicsoftheMEMSgyroscopeandtheexternaldis turbanceswhichincludetheoutputdisturbances,theunknown

time-varyingrotation rate, andthe unknown quadratureerror termsarisingfrommechanicalimperfections.Withthedynamic compensationof the estimated information, the systemis re ducedtoadoubleintegrator.Then,aPDcontrollerissufﬁcient tocontrolit.

Wecanrewritethedriveaxismodelin(1)as

(2) where ; referstotheexternaldisturbance(speciﬁ

cally, here); ,orsimplydonated

as ,representsboththeinternaldynamicsandtheexternaldis turbance;and

(3) ThebasicideaoftheADRCistoobtaintheestimated in realtimeby anESO andtoactivelycompensateforit inthe controllaw.TheconceptoftheADRCisintroducedasfollows.

A. ESODesign

Let , and .

Assumingthat isdifferentiable,the state-spaceformof (2) is

(4) where

with beingtheaugmentedstateand .Acontinuous ESOfor(4)isdesignedas

(5)

where istheobservergainvector.Theob servergainsarechosensuchthatthecharacteristicpolynomial isHurwitz.Fortuningsimplicity,allthe observerpolesareplacedat .Itresultsinthecharacteristic polynomialof(5)tobe

(6) istheobserverbandwidthofthedriveaxisand

.

Generally,thelargertheobserverbandwidthis,themoreac curatetheestimationwillbe.However,alargeobserverband widthwill increase noisesensitivity. Therefore,a properob serverbandwidthshouldbeselectedinacompromisebetween thetrackingperformanceandthenoisetolerance.

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B. ControlAlgorithm

Oncetheobserverisdesignedandwelltuned,itsoutputswill track ,and ,respectively.Bycancelingtheeffectof using ,theADRCactivelycompensates for inreal time. TheADRCcontrollawisgivenby

(7) where isthedesiredtrajectoryofthedriveaxisand and arethecontrollergainparametersselectedtomake

Hurwitz.Forsimplicity,let and ,where is thecontrollerbandwidth.Theclosed-loopsystemforthedrive axisbecomes

(8) Notethat,withawell-designedESO,theﬁrsttermonthe right-handside(RHS)of(8)isnegligible,andtherestoftheterms ontheRHSof(8)constituteaPDcontrollerwithafeedforward gain.

Inpractice,thecontrollerbandwidth istunedbasedonhow fastandsteadywewanttheoutputtotrackthesetpoint.Alarge controllerbandwidthgenerallyincreasestheresponsespeed,but itmaypushthesystemtoitslimit,leadingtooscillationsoreven instability.Thus,thecontroller bandwidthshouldbeadjusted basedonthecompetingrequirementsofperformanceandsta bilitymargin,togetherwithnoisesensitivity.Inaddition,alarge controllerbandwidthusuallyincreasesthemagnitudeandrate of change incontrol signaland,therefore,the operationcost. Theobserveristunedinasimilarway:adjustingitsbandwidth foratradeoffbetweentrackingperformanceandnoisesen sitivity.

Theprimary reasonfor thisparticularparameterizationand tuningmethodispracticality.Theobserverandfeedbackgains mustbeeasilytunablebymostengineers,whoareusuallyfa miliarwiththeconceptandimplicationsofbandwidth.Itisad vantageousthat engineerscoulduseacompletelynewdesign methodwithoutlosingthecriticalinsightgainedfromclassical control:frequencyresponse.

The convergence for the estimation error of ESO and the closed-looptrackingerroroftheADRCisshownhereinafter.

C. Stability

1) ConvergenceoftheESO: Let

.From(4)and(5),theobserverestimationerrordynamics canbeshownas i.e.,let rewrittenas .Then, (9)canbe (10) where

Now,letusscaletheobserverestimationerror by ,

Theorem1: Assumingthat is bounded, then there exist aconstant anda ﬁnitetime such that and .Further more, forsomepositiveinteger .

Proof: Solving(10),wecanobtain

(11) Let

(12) Since isbounded,i.e., ,where isa positiveconstant,itfollowsthat

(13)

for .Since

onehas

(14) Since isHurwitz,thereexistsaﬁnitetime suchthat (15)

forall .Hence

(16) forall .Notethat dependson .Let

(5)

Onehas Itfollowsthat

(17)

.From(13),(14),and(17),weobtain forall

(18)

forall .Let

.Itfollowsthat

(19) .From(11),onehas

forall (20) .Accordingto Let and(18)–(20),wehave (21)

forall .Q.E.D.

Itis shownthat,inthe absenceof theplantmodel, thees timationerrorofESO(5)isboundedandthatitsupperbound monotonouslydecreaseswiththeincreaseoftheobserverband width.Theassumptionoftheboundednessof meansthat thereisalimittotherateof changeofthetotaleffortsof the internaldynamicsandtheexternaldisturbances,excludingthe control inputfor theMEMS gyroscope,or that thechange is notinstantaneous.Inthiscase, canchangeveryrapidly,and themagnitudeof canbequitelargealthoughbounded.Under standably,thisrequirestheobserverbandwidthtobesufﬁciently largeforanaccurateestimateof .Intheabsenceofthisbound ednessassumption,therateofchangein wouldbeunlimited, whichwouldmake verydifﬁculttoestimate.Fortunately,for MEMSgyroscopes,thisassumptionseemstobeareasonable onebecauseitsmechanicalconstructionandworkingenviron mentdoesnotallowtheaccelerationtochangeinstantaneously, thusforcing tobeaboundedvariable.

TheconvergenceoftheADRC,whereESOisemployed,is analyzednext.

2) ConvergenceoftheADRC: Let

and .

Theorem2: Assumingthat isbounded,thereexistacon stant anda ﬁnite time such that

,and forsomepositiveinteger

Proof: From(7),onehas

.

.Furthermore,

(22)

(23)

Let and ,then

(24) where

Solving(24),wehave

(25) Accordingto(24)andTheorem1,onehas

forall where Deﬁne .Let .Itfollowsthat . Since (27) (26) wehave (28) Since isHurwitz,thereexistsaﬁnitetime suchthat (29) forall .Notethat dependson .Let

.Itfollowsthat

(30)

forall .Let . We have

(6)

forall ,and

(32)

forall .From(27),(28),and(32),weobtain

(33)

forall .From(25),onehas

(34) Accordingto(30),(33),and(34),wehave

(35)

for ,where

.

It has been shownthat, withplant dynamics being largely unknown,thetrackingerroranditsderivativearebounded,and theirupperboundsmonotonouslydecreasewiththeincreaseof theobserverandcontrollerbandwidths.Withtheconvergence ofthetrackingerrorestablished,wenowpresentthesimulation andhardwaretestresults.

IV. SIMULATIONANDHARDWARETESTS

AnFPGA-baseddigitalimplementationoftheADRCisde signedandconductedonapiezoelectricallydrivenvibrational beam gyroscope, whichis analternativetothe MEMS gyro scope for experimental use. The control algorithm is imple mentedincustomlogicusingVHDL.

A. HardwareSetup

The block diagram of the FPGA-based digital implemen tationisshowninFig.2. Thehardwaresetupincludesacore hardwareboarddevelopedearlierasreconfigurablecontroland communication module(RCCM) [14],as shown inFig.3. It mainlyconsists oftwo analog-to-digitalconverters, eachpro ceededbyananalogprogrammablefilter,aflashmemory,and anFPGAchip.TheRCCMalsosupportstheEthernetandcon troller area network communication. In this implementation, thesinusoidalreferencesignal,theNioscoreprocessor,andthe first-input–first-output bufferare programmed into the FPGA circuit. The control algorithm, ADRC, is also programmed into FPGA using the VHDL language. One external 12-b digital-to-analog converteris employed toconvert the digital control signalfromFPGAtoanalogformbefore itentersthe gyroscope circuitry.Toclosetheloop,theoutputof thedrive axisofthegyroscopeisfirstamplifiedandthenfedbacktothe field-programmable analog array chip on the RCCM. In the

Fig.2. ControlsystemfortheMEMSgyroscope.

Fig.3. FPGA-basedRCCM.

controlsystemdesign,theADRCemploysverylargecontroller and observer gains, which are beyond the limited range of integernumbersrepresentedby32-bitbinary.Forthisreason, thesingle-precision ﬂoatingpointfromIEEEStandard754 is usedfortheFPGA-basedADRCdesign.

In terms of developmenttools,Quartus II,version3.2, for FPGAdesign,SOPCbuilderforNiosembeddedprocessorde sign,andGNUProcompilerforbuildingbothsoftwareandli brariesareemployed.

B. FPGAImplementationoftheADRC

For digital implementation, discretizing the state-space model(4)usingzero-orderholdbyignoring , we have

(36) where

and isthesamplingperiod. AdiscreteESOisdesignedas[16]

(7)

where istheestimatorgain, providesacurrentestimate of basedonthecurrentmeasurement ,and isthe predictedestimatebasedonapredictionfromtheprevioustime estimate,i.e.,

(38)

Let .Itfollowsthat

(39) Fortuningsimplicity,weplaceallthepolesofthedesireddis creteESOcharacteristicequationat ,where

Then,thedesiredcharacteristicpolynomialis

.

(40) AccordingtoAckermann’sformula[16],onehas

(41) Thediscreteimplementationofthecontrollaw(7)is

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C. SimulationResults

ThekeyparametersofthevibrationalMEMSgyroscopeare

rad/s , , and rad /s .

In (1), ,where and are the

amplitudeandangularfrequencyoftherate,respectively.The actualamplitudeoftherotationrateisassumedtobe0.1rad/s, and isassumedtobe50Hz.Thereferencesignalforthe driveaxisis ,where rad/s.Typically, theidealoutputamplitudeofthedriveaxisis mVfor thepiezoelectricallydrivenvibrationalgyroscope.Thisvoltage output is linearly proportional tothe displacement output of the gyroscope inmicrometers.Weuse in“simula tionunits”torepresentthemagnitudeofthedriveaxisoutput inthe simulation.Theamplitude ofthe control signalislim ited to mV. In the simulation, the mechanical–thermal noiseisapplied,andthePSD ofmechanical–thermalnoiseis

N s. The design parameteris

.Thecontrollerbandwidthis rad/s, andtheobserverbandwidthis rad/s.Thesam

plingperiodis s.

Theoutputofthedriveaxisunderthecontrolofthediscrete ADRCisshowninFig.4.Afterapproximately2.2ms,thefre quencyofthedriveaxisisdriventotheresonantfrequency , as expected.Thetrackingerrorbetweenthereferencesignal and theoutputofthedriveaxisisshowninFig.5.Thesteady-state peak errorbetweenthe referenceandthedrive axisoutput is around0.17%ofthedesiredamplitude.Figs.4and5showthe excellenttrackingperformanceoftheADRC.

TofurtherinvestigatetherobustnessoftheADRCagainstpa rametervariations,the systemparametersare changedas

fol-Fig.4. OutputofthedriveaxiswiththeADRC.

Fig.5. Trackingerrorofthedriveaxis.

lows.Themagnitudeofthequadratureerrorterm,thedamping coefﬁcient,andthefrequencyoftherotationrateareincreased

by ten times, i.e., rad s , and

Hz,respectively.Withtheseplantparametervari ations,the trackingerrorofthe driveaxisisshowninFig.6. Notethat thetuningparametersintheADRCarethesameas thatusedinFigs.4and5.Itcanbeseenthatthefrequencyof thedriveaxisisdriventotheresonantfrequency afterapprox imately1.8ms,andthesteady-statepeakerrorbetweentheref erenceandthedriveaxisoutputisaround0.17%ofthedesired amplitude.Withthelarge-scaleplantparametervariations,the performanceoftheADRCisalmostthesame.Thisshowsthe strongrobustnessoftheADRCagainstparametervariations.

D. HardwareTestResults

Ifthesamplingperiod ischosenthesameastheoneused inthesimulation,itwillbetoofasttoimplementintotheFPGA

(8)

Fig.6. Trackingerrorofthedriveaxiswithparametervariations.

bandwidth rad/s, the observer bandwidth rad/s,andthesamplingperiod isreduced to s.TheoutputofthedriveaxisisshowninFig.7.

Fig.7. DriveaxisoutputoftheFPGAimplementation.

board sincethe FPGAboardclockspeed is50MHz,andthe digitalADRCrequiresseveralclockcyclestoprocess.Forthis reason, the tuning parameters are adjusted as the controller

FromFig.7,itcanbeseenthatthefrequencyofthedriveaxisis driventotheresonantfrequency afterapproximately18ms. Atthesteadystate,theoutputmatchesthereferenceverywell. The steady-state peak error between the reference and the drive axis output is around 0.93% of the desired amplitude. TheseshowthegoodperformanceoftheADRCintheFPGA implementation.

V. CONCLUSION

Inthispaper,anovelconcept,namely,activedisturbancere jection,issuccessfullyappliedtosolvetheproblemsinMEMS gyroscopesthatstemfrommanufacturingimperfections.Such imperfectionsmanifest themselvesasuncertaindynamicsand unknowndisturbancesthataredifﬁculttodealwithusing ex istingdesignmethodsthatarelargelydependentonanaccurate mathematical model. The proposed ADRC design proves to beagoodﬁt forthreereasons:1)Itrequiresminimalapriori

informationoftheplant(justtheorderoftheplantanditshigh frequencygain);2) it activelyestimatesandcompensates for theunknowndynamicsanddisturbances;and3)thecontroller is easy to be implemented and tuned as compared to other methods. The resultsobtained from simulation andhardware testsdemonstratedtheeffectivenessoftheADRC.Thestability analysis solidiﬁed the theoretical foundationof the proposed approach and provided much insight for the users, and the rationaleforthesuccessofbothsimulationandhardwaretests.

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