a new approach to linear filtering and prediction problems.PDF

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1、ANewApproachtoLinearFiltering1andPredictionProblemsR.E.KALMANResearchInstituteforAdvancedStudy,2Theclassicalfilteringandpredictionproblemisre-examinedusingtheBode-Baltimore,Md.Shannonrepresentationofrandomprocessesandthe“statetransition”methodofanalysisofdynamicsystems.Newre

2、sultsare:(1)Theformulationandmethodsofsolutionoftheproblemapplywithoutmodifica-tiontostationaryandnonstationarystatisticsandtogrowing-memoryandinfinite-memoryfilters.(2)Anonlineardifference(ordifferential)equationisderivedforthecovariancematrixoftheoptimalestimationerror.Fro

3、mthesolutionofthisequationtheco-efficientsofthedifference(ordifferential)equationoftheoptimallinearfilterareob-tainedwithoutfurthercalculations.(3)Thefilteringproblemisshowntobethedualofthenoise-freeregulatorproblem.Thenewmethoddevelopedhereisappliedtotwowell-knownproblems,c

4、onfirmingandextendingearlierresults.Thediscussionislargelyself-containedandproceedsfromfirstprinciples;basicconceptsofthetheoryofrandomprocessesarereviewedintheAppendix.IntroductionPresentmethodsforsolvingtheWienerproblemaresubjecttoanumberoflimitationswhichseriouslycurtailt

5、heirpracticalANIMPORTANTclassoftheoreticalandpracticalusefulness:problemsincommunicationandcontrolisofastatisticalnature.Suchproblemsare:(i)Predictionofrandomsignals;(ii)separa-(1)Theoptimalfilterisspecifiedbyitsimpulseresponse.Itistionofrandomsignalsfromrandomnoise;(iii)det

6、ectionofnotasimpletasktosynthesizethefilterfromsuchdata.signalsofknownform(pulses,sinusoids)inthepresenceof(2)Numericaldeterminationoftheoptimalimpulseresponseisrandomnoise.oftenquiteinvolvedandpoorlysuitedtomachinecomputation.Inhispioneeringwork,Wiener[1]3showedthatproblems

7、(i)Thesituationgetsrapidlyworsewithincreasingcomplexityofand(ii)leadtotheso-calledWiener-Hopfintegralequation;hetheproblem.alsogaveamethod(spectralfactorization)forthesolutionofthis(3)Importantgeneralizations(e.g.,growing-memoryfilters,integralequationinthepracticallyimporta

8、ntspecialcaseofnonstationaryprediction)requirenewderivations,frequentlyofst