IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY Indirect Adaptive Attenuation of Multiple - PDF document

2IEEETRANSACTIONSONCONTROLSYSTEMSTECHNOLOGYofthevariousspikesofthedisturbance.Severalmethodshavebeenproposedbythesignalprocessingcommunityforsolvingthisissue([14]).Fromthese,theadaptivenotchlter(ANF)isparticularlyinterestingandhasbeenreviewedinanumberofarticles([15],[16],[17],[18],[19],[20],[21]).Inthispaper,theestimationapproachpresentedin[22],[23]willbeused.Combiningthefrequencyestimationprocedureandthecontroldesignprocedure,anindirectadaptiveregulationsystemforattenuationofmultipleunknownand/ortimevaryingnarrow-banddisturbancesisobtained.Inthepresentcontext,thehypothesisofconstantdynamiccharacteristicsoftheAVCsystemismade(likein[1]).Furthermore,thecorrespondingcontrolmodelissupposedtobeaccuratelyidentiedfrominput/outputdata.In[24],areviewofavailablemethodsfortherejectionofnarrow-banddisturbancesisgiven.In[4],thedirectadaptiveregulationofnarrow-banddisturbancesusingIMPandtheYoula-Kuceraparametrizationisdescribedandanalyzedandextendedin[1]formultipledisturbances.Anothermethodfornarrow-banddisturbancesrejectionbyfeedbackisbasedontheuseofadisturbanceobserver([25],[26],[27]).In[27],thedisturbanceobserveriscombinedwithamodiedIMPschemethatusesanIIRlter,whichactsasafrequencyselector,tominimizetheeffectoftheadaptivecontrolontheoutputsensitivityfunction.Anindirectmethodthatcombinesthefrequencyestimatorof[17]withfeedforwardrejectionofdisturbancesispresentedin[28].Themethodisalsocomparedtoadirectalgorithmbasedonthephase-lockedloopstructureconsideredincom-municationsystems.Thispaperisorganizedasfollows.InSectionII,themainnotationsandequationsfortheindirectadaptivesystemaregiven.ThecontrollerdesignbasedontheuseofBSFsispresentedinSectionIII.AreducedcomplexityimplementationofthismethodusingtheYoula-KuceraparametrizationisthengiveninSectionIV.Theestimationmethodusedfortrackingthevariationsofthedisturbances'frequenciesisbrieydescribedinSectionV.ThestabilityanalysisoftheproposedmethodisgiveninSectionVI.InSectionVII,anexperimentalperformanceevaluationoftheresultingindirectadaptiveregulationschemeandacomparisonwiththedirectregulationmethodusingYoula-KuceraparametrizationofthecontrollerandIMP[1]arepresented.Someconcludingre-marksaregiveninSectionVIII.II.SYSTEMDESCRIPTIONThebasicindirectadaptivecontrolblocdiagramusedisshowning.1.Theprocessoutputcanbewrittenas2y(t)=G(q�1)
u(t)+p(t);(1)whereG(q�1)=q�dB(q�1) A(q�1)(2)2Thecomplexvariablez�1willbeusedtocharacterizethesystem'sbehaviorinthefrequencydomainandthedelayoperatorq�1willbeusedforthetimedomainanalysis. Fig.1.Basicschemaforindirectadaptivecontrol.iscalledthesecondarypathofthesystemandp(t)=D(q�1) D(q�1)(t);2(0;1)isaxedconstant;(3)istheeffectofthedisturbanceonthemeasuredoutput3.Asspeciedintheintroduction,thehypothesisofconstantdynamiccharacteristicsoftheAVCsystemisconsidered(similarto[1],[4]).ThedenominatorofthesecondarypathmodelisgivenbyA(q�1)=1+a1q�1+:::+anAq�nA;(4)thenumeratorisgivenbyB(q�1)=b1q�1+:::+bnBq�nB=1+q�1B(q�1)(5)anddistheintegerdelay(numberofsamplingperiods)4.Thecontrolsignalisgivenbyu(t)=�R(q�1)
y(t)�S(q�1)
u(t�1);(6)withS(q�1)=1+q�1S(q�1)=1+s1q�1+:::+snSq�nS=S0(q�1)
HS(q�1);(7)R(q�1)=r0+r1q�1+:::+rnRq�nR=R0(q�1)
HR(q�1);(8)whereHS(q�1)andHR(q�1)representxed(imposed)partsinthecontrollerandS0(q�1)andR0(q�1)arecomputed.Underthehypothesisthattheplantmodelparametersareconstantandthatanaccurateidenticationexperimentcanberun,areliableestimate^p(t)ofthedisturbancesignalcanbeobtainedbyusingthefollowingdisturbanceobserver^p(t+1)=y(t+1)�q�dB(q�1) A(q�1)u(t);(9)asshowning.1.Thedisturbanceestimator(^p(t))isfollowedbyablockwhichestimatesspikes'frequenciesandcomputesinrealtimethecontrollerparameters.3Thedisturbancepassesthroughasocalled”primarypath”whichisnotrepresentedinthisgure,andp(t)isitsoutput4Asindicatedearlier,itisassumedthatareliablemodelidenticationisachievedandthereforetheestimatedmodelisassumedtobeequaltothetruemodel. 4IEEETRANSACTIONSONCONTROLSYSTEMSTECHNOLOGYIV.REDUCINGTHECOMPUTATIONALLOADOFTHEDESIGNBYUSINGTHEYOULA-KUCERAPARAMETRIZATIONThecomputationalcomplexityrelatedtotheBezoutequa-tion(18)issignicant(intheperspectiveofitsuseinadaptiveregulation).Inthissection,weshowhowthecomputationloadofthedesignprocedurecanbereducedbytheuseoftheYoula-Kuceraparametrization.Asbefore,amultipleband-stoplter,(13),shouldbecomputedbasedonthefrequenciesofthemultiplenarrow-banddisturbance.InwhatfollowsitwillbeshownthatusingaYoula-Kuceraparametrizationofthecontroller[12]asignicantreductionofthecomputationalloadwillbeobtained.SupposethatanominalcontrollerR0(z�1)=HR1(z�1)R00(z�1);(20)S0(z�1)=HS1(z�1)S00(z�1)(21)thatassuresnominalperformancesfortheclosedloopsystemintheabsenceofnarrow-banddisturbancesisavailable.ThiscontrollersatisestheBezoutequationP0(z�1)=A(z�1)S0(z�1)+q�zB(z�1)R0(z�1):(22)SincePBSF(z�1)willdenepartofthedesiredclosedlooppoles,itisreasonabletoconsideraYoula-KuceralteroftheformQ(z�1) PBSF(z�1)(whichwillautomaticallyintroducePBSF(z�1)aspartoftheclosedlooppoles).Forthispurpose,thecontrollerpolynomialsarefactorizedasR(z�1)=R0(z�1)PBSF(z�1)++A(z�1)HR1(z�1)HS1(z�1)Q(z�1);(23)S(z�1)=S0(z�1)PBSF(z�1)��z�dB(z�1)HR1(z�1)HS1(z�1)Q(z�1);(24)whereQ(z�1)isaFIRltercomputedinordertosatisfy(14)forP(z�1)=P0(z�1)PBSF(z�1),andR0(z�1),S0(z�1)aregivenby(20)and(21)respectively.Itcanbeseenfrom(23)and(24)thatthenewcontrollerpolynomialsconservethexedpartsofthenominalcontroller.Takingintoaccount(14),(16),(17),and(18),itremainstocomputeQ(z�1)suchthatS(z�1)=SBSF(z�1)HS1(z�1)S0(z�1):(25)Turningbacktoeq.(24)oneobtains6S0PBSF=SBSFHS1S0+z�dBHR1HS1Q:(26)andtakingintoconsiderationalso(21)itresultsS00PBSF=SBSFS0+q�dBHR1Q:(27)Inthelastequation,theleftsideoftheequalsignisknownandonitsrightsideonlyS0(z�1)andQ(z�1)areunknown.ThisisalsoaBezoutequationwhichcanbesolvedbyndingthesolutiontoamatrixequationofdimensionnBezYK=nB+d+nHR1+2
n�1:(28)6Theargument(z�1)hasbeendroppedtosimplifythewritingoftheequation.Asitcanbeobserved,thesizeofthenewBezoutequationisreducedincomparisonto(19)bynA+nHS1.Forsystemswithlargedimensions,thishasasignicantinuenceonthecomputationtime(inSectionVII,nA=14andnHS1=0).TakingintoaccountthatthenominalcontrollerisanuniqueandminimaldegreesolutionoftheBezoutequation(22),wendthatthelefthandsideof(27)isapolynomialofdegreenS00+2
n=2
n+nB+d+nHR1�1;(29)whichisequaltothequantitygivenin(28).Therefore,thesolutionofthesimpliedBezoutequation(27)isuniqueandofminimaldegree.Furthermore,theorderoftheQFIRlterisequalto2
n�1.Fig.2summarizestheimplementationoftheYoula-Kuceraparametrizedindirectadaptivecontroller. Fig.2.Youla-Kuceraschemaforindirectadaptivecontrol.V.FREQUENCYESTIMATIONUSINGADAPTIVENOTCHFILTERSInordertousethepresentedcontrolstrategyinthepresenceonunknownand/ortimevaryingnarrow-banddisturbances,oneneedsanestimationinrealtimeofthespikes'frequenciesinthespectrumofthedisturbance.Intheframeworkofnarrow-banddisturbancerejection,itisusuallysupposedthatthedisturbancesareinfactsinusoidalsignalswithvariablefrequencies.Asspeciedintheintroduction,itisassumedthatthenumberofnarrow-banddisturbancesisknown(similarto[1],[4],[27]).AtechniquebasedonANFswillbeusedtoestimatethefrequenciesofthesinusoidalsignalsinthedisturbance(moredetailscanbefoundin[16],[23]).ThegeneralformofanANFisHf(z�1)=Af(z�1) Af(z�1);(30)wherethepolynomialAf(z�1)issuchthatthezerosofthetransferfunctionHf(z�1)lieontheunitcircle.AnecessaryconditionforamonicpolynomialtosatisfythispropertyisthatitscoefcientshaveamirrorsymmetricformAf(z�1)=1+af1z�1+:::+afnz�n+:::++af1z�2n+1+z�2n:(31)AnotherrequirementisthatthepolesoftheANFshouldbeonthesameradiallinesasthezerosbutslightlycloserto 6IEEETRANSACTIONSONCONTROLSYSTEMSTECHNOLOGYH8)A(q�1),B(q�1)andtheintegertimedelaydareknown.H9)B(q�1)doesnothavezerosonthecircleofradiusn.H10)Theassignedclosedlooppoles^P(t;q�1)areinsidetheunitcircle,8t.Then:(1)Thesequencesu(t)andy(t)arebounded.(2)limt!1h^P(t)y(t)�^S(t)Ap(t)i=0;(48)limt!1h^P(t)u(t)+^R(t)Ap(t)i=0:(49)Notethatthequantitiesinthesquarebracketsofeqs.(48)and(49)arezerointhecaseofknownparameters.TheproofcanbefoundinAppendixA.VII.EXPERIMENTALRESULTSA.AnActiveVibrationControlSystemUsinganInertialActuatorFigures3and4representanAVCsystemusinganinertialactuatorforreducingtheresidualacceleration.Thesystemconsistsof5metallicplatesconnectedbysprings.TheplatesM1andM3areequippedwithinertialactuators.TheoneonM1servesasdisturbancegenerator(inertialactuator1ing.4),theoneonM2servesfordisturbancecompensation(inertialactuator2ing.4).Thesystemisequippedwithameasureoftheresidualacceleration(onplateM3).Thepathbetweenthedisturbance(inthiscasegeneratedbytheinertialactuatorontopofthestructure),andtheresidualaccelerationiscalledtheprimarypath.Thepathbetweentheinertialactuatorforcompensationandtheresidualaccelerationiscalledthesecondarypath(G(q�1)inSectionII)anditcharacterizesthedynamicsfromthecontrolsignaltotheresidualaccelerationmeasurement(amplier+actuator+dynamicsofthemechanicalsystem).ItsordershavebeenestimatedasbeingnAG=nBG=14.Thedisturbanceisthepositionofthemobilepartoftheinertialactuatorlocatedontopofthestructure(seegs.3and4).Theinputtothecompensatorsystemisthepositionofthemobilepartoftheinertialactuatorlocatedonthebottomofthestructure.Asamplingfrequency,fs,of800Hzhasbeenused. Fig.3.AnAVCsystemusingafeedbackcompensation-photo. Fig.4.AnAVCsystemusingafeedbackcompensation-schema. Fig.5.SecondaryandprimarypathsBodeamplitudes.B.AttenuationofMulti-sinusoidalDisturbanceAnexperimentalcomparisonoftheproposedalgorithmwiththedirectadaptivecontrollerof[1]whichusesIMPispresented.Amulti-sinusoidalsignalhasbeenusedasdisturbance(inputtotheprimarypath).Inallofthefollowingexperiments,forthedesignusingYoula-KuceraparametrizedBSF's,dhasbeenchosenequalto0:04andanattenuationof60dBhasbeenimposedonallofthespikes.Thenominalcontroller'scharacteristicpolynomial,P0(z�1),containsallthepolesofthesecondarypathmodeland15additionalrealpolesat0:42forrobustness.Furthermore,thexedpartofthecontroller'snumeratorhasbeenchosenasHR1(z�1)=1�z�2,thusopeningtheloopat0Hzand400Hz(halfthesamplingfrequency),whilethexedpartofthecontroller'sdenominatorwasHS1(z�1)=1.TheYoula-KuceraIMPdesignusesthesamecentralcontroller.Remark:fortherejectionof3sinewiththeadaptiveIMPalgorithm,3pairsofxedpoleswithdamping0:2locatedwithintherangeofvariationofthedisturbancespikes'frequencieshavebeenaddedtothenominalclosedlooppolesinordertoimproverobustnessoutsidetheattenuationband.Fortheminimalityofthesolution,thenumberofrealpolesat0.42hasbeenreducedto9.Twotypesofdisturbanceshavebeenconsidered:a)with2spikes(g.6)havingamagnitudeof0:1eachandb)with3spikes(gs.7)havingamagnitudeof0:04each,inordertoavoidsaturationofthecontrolinputwiththedirectadaptivecontrollerof[1](thesystemisnotwellsuitedforusingthismethod-theactuatordoesnothaveenoughpower).Ings.6and7,thesignalsontoprepresenttheeffectofthedisturbanceupontheresidualaccelerationinopenloopoperation,theonesinthemiddlecorrespondtotheresidualaccelerationinclosed AIRIMIT¸OAIEetal.:Indirectadaptiveattenuationofnarrow-banddisturbances9From(56)and(60),^P(t)y(t)=^S(t)�Bu(t)�A"p(t)+Ap(t)
++B^R(t)y(t)+
11(t)y(t)(76)andtakingintoaccount(58)B^R(t)y(t)+B^S(t)u(t)=0;(77)oneobtains^P(t)y(t)=^S(t)Ap(t)�^S(t)A"p(t)++
11(t)y(t)+
12(t)u(t):(78)Similarly,^P(t)u(t)=^A(t)^S(t)u(t)+^R(t)^A(t)y(t)�^R(t)Ap(t)++^R(t)A"p(t)+
22(t)u(t)(79)=�^R(t)Ap(t)+^R(t)A"p(t)++
22(t)u(t)+
21(t)y(t):(80)Whichleadtoeqs.(48)and(49).Takingintoaccountthatthevarious
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