Tm hiu v lc Kalman (Kalman filter)Rudolf Kalman(sinh nm 1930) l mt nh ton hc M gc Hungary c bit n nhiu nht v cc cng trnh v ci gi l lc Kalman (Kalman filter). V cc cng trnh ny m nm 2009 Kalman c Obama trao tng National Medal of Science ca M.Lc Kalman l mt phng php thut ton lc nhiu ra khi thng tin, v n c dng rt nhiu trong cc lnh vc iu khin, hng khng, qun s, v tr, v.v, v d nh c lng v iu khin qu o ca tn la, ca phi thuyn. N cn c dng trong rt nhiu lnh vc khc, t nhn dng ting ni cho n marketing !Ngoi Kalman, cn c Thiele v Swerling ngh ra thut ton tng t trc , Bucy tham ra vo pht trin l thuyt, v Stratonovich Nga cn pht trin mt l thuyt thut ton phi tuyn m rng hn t trc . Bi vy lc ny cn c gi l Kalman-Bucy-Stratonovich filter.Lc Kalman nhm c lng gi tr ch thc ca mt ci g , bng cch d on gi tr ca n v tnh tin cy (hay bt nh) ca d on , ng thi o c gi tr (nhng b sai s v c cc nhiu), sau ly mt trung bnh c trng gia gi tr d on v gi tr o c c, lm gi tr c lng. C th coi n l mt trng hp ca suy din c iu kin kiu bayes (bayesian inference) ?M hnh ton hc:(y l mt m hnh tuyn tnh, cc trng thi c vit bi cc vector cn cc bin i c vit bi cc ma trn).Gil vector gi tr thc s ca mt ci g (v d nh v tr ca tn la) ti thi im th. Ta s gi sbin i theo qui lut sau:

trong l ma trn thay i trng thi (state transition matrix),l vector iu khin,l ma trn iu khin, cnl nhiu ngu nhin, vi gi s l n c phn b Gaussian (phn b normal nhiu chiu), trong l k hiu ca ma trn hip phng sai tng ng.Ti mi thi im th, c mt o c (measurement) trng thicho kt qu l

trong l ma trn ca m hnh quan st, cnl nhiu trong lc o c, v ta gi s nhiu ny cng tun theo mt phn b Gaussian.Cc m trnc coi l bit. Ta gi s thm l cc nhiul mt b bin ngu nhin c lp v cng c lp vi trng thi ban u.Cu hi t ra l lm sao c lng c cc trng thi $x_k$ t cc quan st?Nu khng h c nhiu, th ta ch cn t. Nhng v c nhiu nn khng tnh c chnh xcm ch c th ng lng n. Kalman filter l mt hm c lng quy (recursive estimator) cho php lm vic ny.Thut ton c lng nh sau: C th chia n thnh 2 bc, bc d on ban u (predict) v bc iu chnh sau (update)Predict:,

yl k hiu d on gi tr cada trn thng tin v gi tr ti thi im, cn $\hat{x}_{k|k}$ l c lng gi tr casau khi s dng mi thng tin ti thi im. Ma trndng ch (c lng) ma trn hip phng sai ca c lng ca.Update: lch so vi quan st (measurement residual):Thng d hip phng sai (residual covariance):Kalman ti u:c lng c iu chnh (updated estimate):Hip phng sai cho c lng mi (updated estimate covariance):