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HYBRID FILTER WITH PREDICT-ESTIMATOR AND COMPENSATOR FOR THE LINEAR TIME INVARIANT DELAYED SYSTEM

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HYBRID FILTER WITH PREDICT-ESTIMATOR AND COMPENSATOR FOR THE LINEAR TIME INVARIANT DELAYED SYSTEM

    HYBRID FILTER WITH

    PREDICT-ESTIMATOR AND

    COMPENSATOR FOR THE LINEAR

    TIME INVARIANT DELAYED SYSTEM

    ,b1.26No.5JOURNALOFELECTRONICS(CHINA)September2009

    HYBRIDFIIERWITHPREDICT.ESTIMTORANDCOMPENSTOR

    FORTHELINEARTIMEINVARIANTDELAYEDSYSTEM

    ,venChenglinGeQuanboFengXiaoliang

    (InstituteofInformationandControl,HangzhouDianziUniversity,Hangzhou310018,China)

    AbstractThispaperinvestigatestheproblemofreal

    timeestimationforonekindof1ineartime

    invariantsystemswhichsubjecttolimitedcommunicationcapacity.Thecommunication1imitations

    includesignaltransmissiondelay,theout

    of-sequencemeasurementsanddatapacketdropout,which

    appeartypicallyinanetworkenvironment.Thekerneloffilterdesignisequallytoformularizethe

    traditionalKalmanfilterasone1inearweightedsummationwhichiscomposedoft;heinitialstatees

    timateandallsequentialsampledmeasurements.Foritcanadaptaforementionedinformationlimi

    tations.thelinearweightedsummationisthendecomposedintotwostages.Oneisapredict.estimator

    composedbyallreachedmeasurements.anotherisonecompensatorconstructedbythosetimedelayed

data.Inthenetworkenvironment,thereareobviousdifierencesbetweenthenewhybridfiltera

    ndthose

    existingdelayedKalmanfilters.Forexample,thenovelfiltercanbeoptimalinthesenseofline

    ar

    minimummeansquareerrorassoonasallmeasurementsavailableandhasthelowestrunningt

    ime

    thantheseexistingdelayedfilters.Onesimulation,includingtwocases,isutilizedtoillustrate

    the

    designproceduresproposedinthispaper.

    KeywordsKalmanfilter;Signaltransmissiondelay;Measurementsummation;Predict

    estimator

    Compensator

    CLCindexTN911

    DOI10.1007/s117670090071x

    I.Introduction

    Kalmanfilterisoneofthemostimportant

    methodstoestimatethedynamicsystemstatell1.In recentyears,Kalmanfilterforthedelayedsystemis oneofresearchhotpointsfornetworkeddatafusion Uptonow,someusabledelayedKalmanfilters havebeenpresented,includingDiscardDelayed MeasurementEstimator(DDME),Discard0utof-

    SequenceMeasurementsEstimator(D0SME),Re

    FilteringEstimator(RFE),andDirectlyUpdate EstimatorfDUE)andsoon.Firstly,DDMEin

    Ref.f2]meansthatthefiltergivesupthedelayed onesoncethedelayedmeasurementsappear.That istosay,noneofdelayedmeasurementscanbeused duringtheKalmanupdateprocess.Theideaofthis Manuscriptreceiveddate:June6.2009;reviseddate:

August10,2009.

    SupportedbytheNationalNaturalScienceFoundationof ChinafNo.60804064,60772006).

    Communicationauthor:WnChenglinbornin1963 male.Professor.CollegeofAutomation,HangzhouDianzi University,Hangzhou310018,China.

    Email:wencl@hdu.edu.cn.

    methodisveryeasyanditscomputationisthe lowest.buttheintegratedperformanceistheworst amongthem.Thesecondschemeisonlytodiscard theOutOf-SequenceMeasurementsfO0SM1,

    whichmeansthatthenewestdelayedmeasurement shouldbeonlyusedtoupdatethecurrentstate estimate[.Comparatively.theperformanceof DOSMEisbetterthanDDME.Refilteringin

    Ref.41istorerunthenormalKalmanfilterwhen allofmeasurementssampledinthisintervalarrive. Itcanarchivethebestestimationaccuracy,butits timecostisveryhigh.

    Then,wehavethefollowingbriefconclusions: f11AlthoughDDMEissimple,theestimateac

    curacyistheworst.f2)DOSMEhasbetteraccu

    racythanDDME.butitstillcannotsatisfyprac

    ticalrequirements.f3)ForRefiltering,ithasthe

    bestestimateaccuracyamongthem,unfortunately, itscomputationalandtimecomplexitiesareboth veryhigh.Namely,itgivesuptherealtimeper

    formanceofKalmanfilterwhichisveryimportant inthetargettrackingandstateestimatefields.

Therefore.inordertogetthebestintegratedes

    timateperformance,thedirectlyupdateestimate WENeta1.HybridFilterwithPredictestimatorandCompensatorfortheLinearIl!坐里!!! methodispaidmoreattentionto.Themain DUEistouseeverydelayedmeasurementto updatethecurrentstateestimatein

    ideaof

    timely

    Linear

    MinimumMeanSquareErrorfLMMSE1sense. AlthoughthecurrentDUEmethodcangetgood estimateaccuracyandrealtimeperformance.its

    computationisveryhighanditdoesnothavenew informationattheupdatetime[5,6j.

    Asvnchr0nouslv,

    itisverydimculttopresenttheanalyticsolutionto optimalmultipleOOShisupdate.

    Therefore,inordertoovercomethehighcorn

    plexityanddifficultanalyticexpressionofoptimal estimatorincurrentDUE.wemakeperfectlyuseof thenatureofLMMSEfilterforakindofIIsys

    teminthispaper.Firstofal1.anovelcomputa- tionalformulaoftraditiona1Kalmanfilterises

    tablishedthatismeasurementsummation.whichis namedasWeiKhtedSummationFormofKalman FilterfWSFKF1.Namely.thetraditionalKalman filtercanberewrittenintermsoftheweighted summationoftheinitialstateestimateandthe sequentia1measurementsfortheLinearTimeIn

variantfLJTI)system.Synchronouslv,anovelHY

    bridFilterwithPredict..EstimatorandCompen. satorfHFPEC1,whichstil1belongstotheDUE frameisalsoproposedbasedontheWSFKFfor thissystemwithdelayedmeasurements.Compared withthecurrentdelayedfilters,theproposed networkedestimatorhasfoormainadvantages includingconciseanalyticexpression,highestimate accuracy,lowcomputationalcomplexityandgood realtimeperformance.

    Thispaperisorganizedasfollows:SectionII formulatestheproblem.InSectionIII,thetradi

    tiona1Kalmanfilterisrewrittenintoaneffective measurementsummationform.Theoptimalestim. ateforthemultipledelayedmeasurementssceneis consideredinSectionIV.Asimulationexampleis alsogiveninSectionV.Finally,theconclusionis made.

    II.ProblemFormulation

    Considerakindoflineartimeinvariantdy

    namicsystemgivenbylj

    f()=F(k,k1)x(k1)+w(k,k1)

    I()=H(k)x(k)+v(k)

    wherex(k)isthestatevariableoftheinterested targetattime,F(k,1)isthestatetransform

    matrixfromt

    ltotk;z(k)isthesensormeas

    urementat,andH(k)isthecorresponding measurementmatrix;w(k,k1)andv(k),in

    whichcovarianeematrixesarerespectivelyQ(k.

    k1)andR(),arebothzeromeanGaussian whitenoisesequencesandindependentfromeach other.Inaddition,Theinitialstatez(0),inwhich correspondingestimateandcovarianeearerespec

    tively(0l0)andP(0O),isindependentfromw

    (,k1)andv(k).

    III.TheWeightedSummationFormof

    KalmanFilter(WSFKF)

    Lemma1Thebasicformulasoftraditional KalmanfilterIT!

    ThetraditionalKalmanfilteris

    (lk)

    where

    E{z()l(七一1lk1),())

    (l1)+()[()一日()(I^1

    (I七一1)=F(k,k1)(1Ik1)

    P(kl七:1)=F(k,k1)P(klk1)F(k,k1)+O(k,k1)

    ()=P(I七一1)H()f()P(Ik1)H()+R(){

    P(kIk)=I()()P(I七一1)

    Eq.(2)andEq.(3)aretheKalmanfilteringre

    cursivecomputationequationswhichareusedex

    tensively.Unfortunately,itisdifficulttoestablish theaboveoptimalanalyticcomputationalequa- tionsinLMMSEwhenthedelayappearsacross (3)

    network.Thereby,weconsiderthenatureof LMMSEfilterthatisactuallythelinearweighted summationoftheinitialstateestimation(010) andallthemeasurementsz(1),z(2),,z().

    Namely,thefollowingtheoremisheld.

/,L

    668JOURNALOFELECTRONICS(CHINA),Vo1.26No.5,September2009

    Theorem1Thestateestimateshownas

    canbeequallyformularizedasthefollowing k

    (1)=?A(k,)+B(k,1)(0l0)2=1

    where

    a(i)=IK(i)H(i);b(i)=a(i)F(i,i1)

    i=1..1

    B(k,)=6(+iJ);

    j=i

    A(k,i)=B(k,i+1)×K(i)

    A(k,k)=K(k)

    Eq.(2)

    form

    (4)

    (5)

    Inaddition,byuseoftheauxiliaryvariables definedinEq.(5),thestateestimateofthesystem showninEq.(1)attimetkcanbeexpressedasthe followingrecursiveform

    (1):

    where

    +B(k,1)(010)

    j)z(j)

    (七一LIkL)=E{(七一L)Iz(0),

    (1),,z(k)),0<<k

    IntheLTIsystem,Eq.(1)inthevariableK(i)is thegainofKalmanfilterwhenallthemeasure

    mentsarriveattheestimatecenterintimeandit

    canbecomputedoffline,soaretheauxiliary variablesinEq.(5).

    IV.HybridFilterwithPredict-Esti-

    matorandCompensator(HFPEC)

    Becauseofthedelay'srandomicitywhendata transmitsacrossnetworks,themeasurementsotten appearthephenomenonwherethedatasampled earlyarriveattheestimatorfaterthantheone sampledlate,thatisoutof-Sequencemeasurement. Inordertooptimallyestimatethestateofthe systemwithdelayedmeasurementsrealtimelyand recursively,anoveldelayedestimatornamed HFPECisproposedinthissection,whichisbased ontheWSFKFgivenintheprevioussectionand includestheprocessofpredict..estimationandde.. 1ayedmeasurementinformationcompensation. AssumingapositiveintegerLandthecon- sideredmomenttk,thesceneformultiplestep

    measurementsdelayisdescribedasfollows. (1)Assumetheestimateofx(kL)attand

    thecorrespondingcovariancematrixareasfollows (七一IL)=E{()l(0),

    (1),,z())

    P(Ll七一)=E{[(k-L)-5~(kl)]

    [(七一)(l七一))

    (8)

    (2)TherearerdelayedonesintheLmeas

    urementssampledbythesensorinthepe

    riod(tktk],whicharemarkedas(),z(k2),

    ,z(g)accordingtothemomentt.(i=1,2,,

    r1theygettotheestimator.Andallofthemcan arriveattheestimatecenterbeforet. f31Intermsofthetimetheyarriveatthe estimator,wecanmarkthesequenceoftherde

    layedmeasurementsas(1),(2),,z().

    Thesubscriptsequence{ml,m2,,mr}isa

    permutationof{1,2,,r}..isthetimesubscript whenthemeasurement()arriveattheesti- mator.Thedelayedscenecanbeillustratedbythe followingfigure.

    f4)Inordertobrieflyintroducethefiltering process,assumer=2,and()sampledatt butarriveat1,z(h)sampledatthbutarrive atkm2,whereth<t,<tk.Accordingtothe timesubscripts,theperiod(tL,t]isdividedinto

    fivepartsasthefollowfigureshown. Forthestateinthedelayedsceneshownby Fig.2,thenovelestimatorHFPECbasedonthe processofinformationcompensationandpre

    dict..estimationispresentedbythenextfiveparts. ,

    ::

    k1,,,

    ILj:L::U,,Ul

    蝇川2

    IMeasurementwhicharriveatestimatorintime ,

    .

    Statepoint:kisthesamplednlomentofdelayedmeasurement 'Finalpoint:%,,isthearrivaltimeofdelayedmeasurenlent Fig.1Thesceneofrdelayedmeasurements +

    L

    

    B

    +

    

    =

    WENef0f.HybrFilterwithPredict.estimato

    randCompensatorfortheL

    

    inearTimeInvariantDelayedSystem669

    ^

    /

    I/,

    ,/,0,,^

    llL

    I?lsIllCIlI}lltwhich;I1'rRPatestilnaIorintime I,It,point:isthesampiedi~1Olllell1.ofdelayedtlteasttrement

    rinMlJfJint:isthearrRaltimeofdelay('dilleasIlrelllent

    Fig.2Thesceneoftwodelayedmeasurements Part1(theperiod(tt])Assumethatallthe measurementsz()(七一L<i<k1)arriveatthe

    estimatorintime,theestimationofthestateattis (ji)=E{z(OI(010),z(1),,z(k-L),

    z(k-+1),?,()),=L+I,,墨一1

    AccordingtothestatisticalpropertyandTheorem 1intheSectionIII,theestimationofthestate

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