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Chapter utorial: The Kalman Filter

Chapter11 lter[1]haslongbeenregardedastheoptimalso lutiontomanytrackinganddatapredictiontas ks,[2]. lterisconstructedasameansquarederrormini miser,butanalternativederivationofthe lterisalsoprovidedshowinghowthe lteringistoextracttherequiredinformation fromasignal, nethegoalofthe ;yk=akxk+nk( )where;ykisthetimedependentobservedsigna l,akisagainterm, erencebetweentheestimateof^xkandxkitself istermedtheerror;f(ek)=f(xk ^xk)( )Theparticularshapeoff(ek)isdependentupo ntheapplication,howeveritisclearthatthef unctionshouldbebothpositiveandincreasemo notonically[3].Anerrorfunctionwhichexhib itsthesecharac-teristicsisthesquarederro rfunction;f(ek)=(xk ^xk)2( )133 Sinceitisnecessarytoconsidertheabilityof the ltertopredictmanydataoveraperiodoftimeam oremeaningfulmetricistheexpectedvalueoft heerrorfunction;lossfunction=E(f(ek))( )Thisresultsinthemeansquarederror(MSE)fu nction; (t)=E e2k ( ) , ningthegoalofthe lterto ndingthe^ ;max[P(yj^x)]( )AssumingthattheadditiverandomnoiseisGau ssiandistributedwithastandarddeviationof kgives;P(ykj^xk)=Kkexp (yk ak^xk)22 2k ( ) ;P(yj^x)=YkKkexp (yk ak^xk)22 2k ( )Whichleadsto;logP(yj^x)= 12Xk (yk ak^xk)2 2k +constant( ) ,whichmaybemaximisedbythevariationof^ ^ lterisde nedasbeingthat lter,fromthesetofallpossible ltertheworkofNorbertWiener[4],should niteimpulseresponse(FIR) cetosaythathissolutionusesboththeautocor relationandthecrosscorrelationoftherecei vedsignalwiththeoriginaldata,inordertode riveanimpulseresponseforthe 'sprescriptionha

w the v alue of a v ariable within a pro cess of the form; x k +1 = + w (11.10) where; x k is the state v ector of the pro cess at time k, (nx1); is the state transition matrix of the pro cess from the state at k to the state at + 1, and is assumed stationary o v er time, (nxm); w k is the asso ciated white noise pro cess with kno wn co v ...

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Transcription of Chapter utorial: The Kalman Filter

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