Transcription of Chapter utorial: The Kalman Filter
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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 lterth
In suc h a case the MSE serv es to pro vide the v alue of ^ x k whic h maximises the lik eliho o d of the signal y k. In the follo wing deriv ation the optimal lter is de ned ... 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 ...
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