A NewParametrizing Technique fortheDerivation ofUnbiased Minimum-Variance Filters
Chien‐Shu Hsieh · 2006
Inthis paper, theproblem ofdesigning anunbi-problem. Simulation results given in(6)showedthatthe asedminimum-variance filter forsystems withunknown inputsoptimal estimator filter proposed byDarouach etal.(4) whichaffect boththesystem modelandthemeasurements is mayexhibit afiltering performance degradation problem. addressed. A newparametrizing technique forthederivation Ina recent paper(7), we havealsoproposed anoptimal ofunbiased minimum-variance filters ispresented. ThederivedIniaurecent pap er to givea p ropos en o parametrized unbiased minimum-variance filter serves asa minimum-variance filter togiveapotential refinement to unified filter structure toderive existing unbiased minimum-remedy theperformance degradation problem. variance filters, e.g., theoptimal estimator filter (4) andthewell- On theother hand,itisknownthatstochastic model knownKalman filter. Furthermore, theproposed parametrizing serve asa useful tooltorepresent systems withnoises, methodology alsosuggests amethodtoderive otherunbiased . . . F minimum-variance filters. A numerical example isincluded in dilsturbances, anduncertainties. Forthese systems, theop- order toillustrate theproposed method. timal state estimates canbeobtained fromthewell-known Kalmanfilter. Themostcommon approach tosolve the I.INTRODUCTION considered problem usingKalmanfiltering istotreat the Thispaper considers theproblem ofestimating thestateunknowninputs asastochastic process withknownwide- ofalinear time-varying discrete-time system inthepresencesensedescription. OtherthanKalmanfiltering algorithm, an ofunknown inputs. Tobefree oftheunknown-input model,optimal FIRfiltering algorithm (8)hadalsobeenproposed there arerobust filters (e.g., (1), (2), (3), (5)) that give optimaltosolve theaforementioned general unknown-input filtering state estimations. Themainadvantage ofthese robust filters problem. Inthis approach theunknowninputs aremodeled isthat their performances arenotaffected bythevalues ofthe asrandom-walk processes similar toKalmanfiltering ap- unknowninputs, whichisanimportant consideration when proaches. Theyaretreated asauxiliary states todevelop an theunknown-input modelishighly non-Gaussian orhasun- augmented system. Then, theoptimal FIRfiltering algorithm knownstatistics. Unfortunately, mostoftheaforementioned forthisaugmented system isderived toobtain estimates of filters arefocused onthelimiting casewheretheunknowntheunknowninputs utilizing measurements inthecurrent input onlyaffect thesystem modelwiththeassumption thatwindow. Itwasstated in(8)thatthisoptimal FIRfilter is thefollowing rankconditions aresatisfied knowntoberobust totemporary modeling uncertainties in mostcases.Unfortunately, this result only applies totime- ranrk(H(t +1)F(t)) =rank(F(t)) = q