The Unscented Kalman Filter for Nonlinear Estimation

Transcription

1 The Usceted Kala Filte fo Noliea Estiatio Eic A Wa ad Rudolh va de Mewe Oego Gaduate Istitute of Sciece & Techology 2 NW Wale Rd, Beaveto, Oego 976 eicwa@eceogiedu, vdewe@eceogiedu Abstact The Exteded Kala Filte () has becoe a stadad techique used i a ube of oliea estiatio ad achie leaig alicatios These iclude estiatig the state of a oliea dyaic syste, estiatig aaetes fo oliea syste idetificatio (eg, leaig the weights of a eual etwo), ad dual estiatio (eg, the Exectatio Maxiizatio (EM) algoith) whee both states ad aaetes ae estiated siultaeously This ae oits out the flaws i usig the, ad itoduces a ioveet, the Usceted Kala Filte (), oosed by Julie ad Uhla [] A cetal ad vital oeatio efoed i the Kala Filte is the oagatio of a Gaussia ado vaiable (GRV) though the syste dyaics I the, the state distibutio is aoxiated by a GRV, which is the oagated aalytically though the fist-ode lieaizatio of the oliea syste This ca itoduce lage eos i the tue osteio ea ad covaiace of the tasfoed GRV, which ay lead to sub-otial efoace ad soeties divegece of the filte The addesses this oble by usig a deteiistic salig aoach The state distibutio is agai aoxiated by a GRV, but is ow eeseted usig a iial set of caefully chose sale oits These sale oits coletely catue the tue ea ad covaiace of the GRV, ad whe oagated though the tue oliea syste, catues the osteio ea ad covaiace accuately to the 3d ode (Taylo seies exasio) fo ay olieaity The, i cotast, oly achieves fist-ode accuacy Reaably, the coutatioal colexity of the is the sae ode as that of the Julie ad Uhla deostated the substatial efoace gais of the i the cotext of state-estiatio fo oliea cotol Machie leaig obles wee ot cosideed We exted the use of the to a boade class of oliea estiatio obles, icludig oliea syste idetificatio, taiig of eual etwos, ad dual estiatio obles Ou eliiay esults wee eseted i [13] I this ae, the algoiths ae futhe develoed ad illustated with a ube of additioal exales This wo was sosoed by the NSF ude gat gat IRI Itoductio The has bee alied extesively to the field of oliea estiatio Geeal alicatio aeas ay be divided ito state-estiatio ad achie leaig We futhe divide achie leaig ito aaete estiatio ad dual estiatio The faewo fo these aeas ae biefly eviewed ext State-estiatio The basic faewo fo the ivolves estiatio of the state of a discete-tie oliea dyaic syste, (1) (2) whee eeset the uobseved state of the syste ad is the oly obseved sigal The ocess oise dives the dyaic syste, ad the obsevatio oise is give by Note that we ae ot assuig additivity of the oise souces The syste dyaic odel ad ae assued ow I state-estiatio, the is the stadad ethod of choice to achieve a ecusive (aoxiate) axiulielihood estiatio of the state We will eview the itself i this cotext i Sectio 2 to hel otivate the Usceted Kala Filte () Paaete Estiatio The classic achie leaig oble ivolves deteiig a oliea aig!" (3) whee is the iut, is the outut, ad the oliea a is aaeteized by the vecto " The oliea a, fo exale, ay be a feedfowad o ecuet eual etwo ( " ae the weights), with ueous alicatios i egessio, classificatio, ad dyaic odelig Leaig coesods to estiatig the aaetes " Tyically, a taiig set is ovided with sale ais cosistig of ow iut ad desied oututs, $ '& The eo of the achie is defied as ( )+*,!", ad the goal of leaig ivolves solvig fo the aaetes " ode to iiize the exected squaed eo i

2 2 2 While a ube of otiizatio aoaches exist (eg, gadiet descet usig bacoagatio), the ay be used to estiate the aaetes by witig a ew state-sace eesetatio "," (4) " () whee the aaetes " coesod to a statioay ocess with idetity state tasitio atix, dive by ocess oise (the choice of vaiace deteies tacig efoace) The outut coesods to a oliea obsevatio o " The ca the be alied diectly as a efficiet secod-ode techique fo leaig the aaetes I the liea case, the elatioshi betwee the Kala Filte (KF) ad Recusive Least Squaes (RLS) is give i [3] The use of the fo taiig eual etwos has bee develoed by Sighal ad Wu [9] ad Pusoious ad Felda [8] Dual Estiatio A secial case of achie leaig aises whe the iut is uobseved, ad equies coulig both state-estiatio ad aaete estiatio Fo these dual estiatio obles, we agai coside a discete-tie oliea dyaic syste, " (6) " (7) whee both the syste states ad the set of odel aaetes " fo the dyaic syste ust be siultaeously estiated fo oly the obseved oisy sigal Aoaches to dual-estiatio ae discussed i Sectio 42 I the ext sectio we exlai the basic assutios ad flaws with the usig the I Sectio 3, we itoduce the Usceted Kala Filte () as a ethod to aed the flaws i the Fially, i Sectio 4, we eset esults of usig the fo the diffeet aeas of oliea estiatio 2 The ad its Flaws Coside the basic state-sace estiatio faewo as i Equatios 1 ad 2 Give the oisy obsevatio, a ecusive estiatio fo ca be exessed i the fo (see [6]), edictio of edictio of "! (8) This ecusio ovides the otial iiu ea-squaed eo (MMSE) estiate fo assuig the io estiate ad cuet obsevatio ae Gaussia Rado Vaiables (GRV) We eed ot assue lieaity of the odel The otial tes i this ecusio ae give by $& (' ) *,+-/1-* $& (9) /1- /1-2 (1) 3' (11) whee the otial edictio of is witte as, ad coesods to the exectatio of a oliea fuctio of the ado vaiables ad (siila iteetatio ) The otial gai te ) fo the otial edictio is exessed as a fuctio of osteio covaiace atices (with 4 * ) Note these tes also equie taig exectatios of a oliea fuctio of the io state estiates The Kala filte calculates these quatities exactly i the liea case, ad ca be viewed as a efficiet ethod fo aalytically oagatig a GRV though liea syste dyaics Fo oliea odels, howeve, the aoxiates the otial tes as: 76 (12) ) * + - / - * /1- /1-2 (13) 6 (14) whee edictios ae aoxiated as sily the fuctio of the io ea value fo estiates (o exectatio tae) 1 The covaiace ae deteied by lieaizig the dyaic equatios ( 98 ;:! =< > ;@ ), ad the deteiig the osteio covaiace atices aalytically fo the liea syste I othe wods, i the the state distibutio is aoxiated by a GRV which is the oagated aalytically though the fist-ode lieaizatio of the oliea syste The eades ae efeed to [6] fo the exlicit equatios As such, the ca be viewed as ovidig fist-ode aoxiatios to the otial tes 2 These aoxiatios, howeve, ca itoduce lage eos i the tue osteio ea ad covaiace of the tasfoed (Gaussia) ado vaiable, which ay lead to sub-otial efoace ad soeties divegece of the filte It is these flaws which will be aeded i the ext sectio usig the 3 The Usceted Kala Filte The addesses the aoxiatio issues of the The state distibutio is agai eeseted by a GRV, but is ow secified usig a iial set of caefully chose sale oits These sale oits coletely catue the tue ea ad covaiace of the GRV, ad whe oagated though the tue o-liea syste, catues the osteio ea ad covaiace accuately to the 3d ode (Taylo seies exasio) fo ay olieaity To elaboate o this, BDC EF B1G ad HIC;EF A H G, ad ae 1 The oise eas ae deoted by A usually assued to equal to zeo 2 While secod-ode vesios of the exist, thei iceased ileetatio ad coutatioal colexity ted to ohibit thei use

3 6 2 K we stat by fist exlaiig the usceted tasfoatio The usceted tasfoatio (UT) is a ethod fo calculatig the statistics of a ado vaiable which udegoes a oliea tasfoatio [] Coside oagatig a ado vaiable (diesio ) though a oliea fuctio, Assue has ea 6 ad covaiace * + To calculate the statistics of, we fo a atix of siga vectos (with coesodig weights ), accodig to the followig: 6 (1) 6 *,+ 6 * *,+ "! *$& (' "! ) $) * & whee +$ -, * $ is a scalig aaete deteies the sead of the siga oits aoud 6 ad is usually, set to a sall ositive value (eg, 1e-3) is a secoday ' scalig aaete which is usually set to, ad is used to icooate io owledge ' of the distibutio of (fo Gaussia distibutios, is otial) * * + is the th ow of the atix squae oot These siga vectos ae oagated though the oliea fuctio, / /1 (16) ad the ea ad covaiace fo ae aoxiated usig a weighted sale ea ad covaiace of the osteio siga oits, * / (17) "! $ * 6 & $ * 6 &6 (18) Note that this ethod diffes substatially fo geeal salig ethods (eg, Mote-Calo ethods such as aticle filtes [1]) which equie odes of agitude oe sale oits i a attet to oagate a accuate (ossibly o- Gaussia) distibutio of the state The decetively sile aoach tae with the UT esults i aoxiatios that ae accuate to the thid ode fo Gaussia iuts fo all olieaities Fo o-gaussia iuts, aoxiatios ae accuate to at least the secod-ode, with the accuacy of thid ad highe ode oets deteied by the choice of $ ad ' (See [4] fo a detailed discussio of the UT) A sile exale is show i Figue 1 fo a 2-diesioal syste: the left lot shows the tue ea ad covaiace oagatio usig Mote-Calo salig; the cete lots Actual (salig) Lieaized () UT 798;:=<>A@ ea tue ea covaiace tue covaiace e4fgihe L9M;N=O PQ K BDCDEGFHBJI=F RTSVU WYX siga oits UT ea Z\[^]_badc weighted sale ea ad covaiace UT covaiace tasfoed siga oits Figue 1: Exale of the UT fo ea ad covaiace oagatio a) actual, b) fist-ode lieaizatio (), c) UT show the esults usig a lieaizatio aoach as would be doe i the ; the ight lots show the efoace of the UT (ote oly siga oits ae equied) The sueio efoace of the UT is clea The Usceted Kala Filte () is a staightfowad extesio of the UT to the ecusive estiatio i Equatio 8, whee the state RV is edefied as &j the cocateatio of the oigial state ad oise vaiables: ' The UT siga oit selectio schee (Equatio 1) is alied to this ew augeted state RV to calculate the coesodig siga atix, j The equatios ae give i Algoith 3 Note that o exlicit calculatio of Jacobias o Hessias ae ecessay to ileet this algoith Futheoe, the oveall ube of coutatios ae the sae ode as the 4 Alicatios ad Results The was oigially desiged fo the state-estiatio oble, ad has bee alied i oliea cotol alicatios equiig full-state feedbac [] I these alicatios, the dyaic odel eesets a hysically based aaetic odel, ad is assued ow I this sectio, we exted the use of the to a boade class of oliea estiatio obles, with esults eseted below 41 State Estiatio I ode to illustate the fo state-estiatio, we ovide a ew alicatio exale coesodig to oisy tieseies estiatio I this exale, the is used to estiate a udelyig clea tie-seies couted by additive Gaussia white oise The tie-seies used is the Macey-Glass-3 chaotic

4 V 4 l l j l l Iitialize with:!!!!! Fo! ", Calculate siga oits: $ &(' *) &+' ( &('+,- /&21 &+' 3 Tie udate: $4&6 &+' 7 $84&(' $9&('! KJ ML ;:=< >A@ 6 &+' CBEDGFIH J ML ;:=< >A@ 4 6 &+' CBEDON=H P &6 &+' Q $46 &+' $R&+'! TS ML >A@ :=< 6 U+' BEDGFIH J! ML 4 6 &+'! Next, white Gaussia oise was added to the clea Macey- Glass seies to geeate a oisy tie-seies f hg The coesodig state-sace eesetatio is give by: l Cc " ije oq a Cc " oq oq oq oq 1 j&e Cc f ts 1 j&j&j 1u j vg (2) I the estiatio oble, the oisy-tie seies f is the oly obseved iut to eithe the o algoiths (both utilize the ow eual etwo odel) Note that fo this state-sace foulatio both the ad ae ode colexity Figue 2 shows a sub-seget of the estiates geeated by both the ad the (the oigial oisy tie-seies has a 3dB SNR) The sueio efoace of the is clealy visible x() Estiatio of Macey Glass tie seies : 1 clea oisy Measueet udate equatios: V W - WV - ;:=< >X@ [Z - W - ;:=< >X@ S ML 6 &+' YBEDON=H J L &6 &+' YBEDON=H Z - W - (' W - V W -, I V W - WV - \ S L!! 6 &(' S ML 6 &+'!X!X whee, ^]C _+! $, $ 4 $ 9 $ R!, 1 / =coosite scalig aaete, =diesio of augeted state, =ocess oise cov, =easueet oise cov, =weights B as calculated i Eq 1 Algoith 31: Usceted Kala Filte () equatios seies The clea ties-seies is fist odeled as a oliea autoegessio Eab Yc "^d e (19) whee the odel a (aaeteized by w) was aoxiated by taiig a feedfowad eual etwo o the clea sequece The esidual eo afte covegece was tae to be the ocess oise vaiace x() oalized MSE Estiatio of Macey Glass tie seies : clea oisy Estiatio Eo : vs o Macey Glass Figue 2: Estiatio of Macey-Glass tie-seies with the ad usig a ow odel Botto gah shows coaiso of estiatio eos fo colete sequece 42 dual estiatio Recall that the dual estiatio oble cosists of siultaeously estiatig the clea state ad the odel a-

5 aetes " fo the oisy data f (see Equatio 7) As exessed ealie, a ube of algoithic aoaches exist fo this oble We eset esults fo the Dual ad Joit Develoet of a Usceted Soothe fo a EM aoach [2] was eseted i [13] As i the io state-estiatio exale, we utilize a oisy tie-seies alicatio odeled with eual etwos fo illustatio of the aoaches I the the dual exteded Kala filte [11], a seaate state-sace eesetatio is used fo the sigal ad the weights The state-sace eesetatio fo the state > is the sae as i Equatio 2 I the cotext of a tie-seies, the statesace eesetatio fo the weights is give by "," (21) f a " e vg (22) couted by additive white Gaussia oise (SNR 3dB) g A stadad MLP with hidde activatio fuctios ad a liea outut laye was used fo all the filtes i the Macey-Glass oble A -3-1 MLP was used fo the secod oble The ocess ad easueet oise vaiaces wee assued to be ow Note that i cotast to the state-estiatio exale i the evious sectio, oly the oisy tie-seies is obseved A clea efeece is eve ovided fo taiig Exale taiig cuves fo the diffeet dual ad joit Kala based estiatio ethods ae show i Figue 3 A fial estiate fo the Macey-Glass seies is also show fo the Dual The sueio efoace of the based algoiths ae clea These ioveets have bee foud to be cosistet ad statistically sigificat o a ube of additioal exeiets whee we set the iovatios covaiace * equal to * 3 Two s ca ow be u siultaeously fo sigal ad weight estiatio At evey tie-ste, the cuet estiate of the weights is used i the sigal-filte, ad the cuet estiate of the sigal-state is used i the weight-filte I the ew dual algoith, both state- ad weight-estiatio ae doe with the Note that the state-tasitio is liea i the weight filte, so the olieaity is esticted to the easueet equatio oalized MSE Chaotic AR eual etwo Dual Dual Joit Joit I the joit exteded Kala filte [7], the sigal-state ad weight vectos ae cocateated ito a sigle, joit state vecto: " ' Estiatio is doe ecusively by witig the state-sace equatios fo the joit state as: " " j " f s 1 j&jj 1 u " i je (23) g (24) ad uig a o the joit state-sace 4 to oduce siultaeous estiates of the states ad " Agai, ou aoach is to use the istead of the Dual Estiatio Exeiets We eset esults o two tie-seies to ovide a clea illustatio of the use of the ove the The fist seies is agai the Macey-Glass-3 chaotic seies with additive oise (SNR 3dB) The secod tie seies (also chaotic) coes fo a autoegessive eual etwo with ado weights dive by Gaussia ocess oise ad also 3 is usually set to a sall costat which ca be elated to the tiecostat fo RLS weight decay [3] Fo a data legth of 1, was used 4 The covaiace of is agai adated usig the RLS-weight-decay ethod x() oalized MSE 2 eoch Macey Glass chaotic tie seies eoch Dual Dual Joit Joit Estiatio of Macey Glass tie seies : Dual clea oisy Dual Figue 3: Coaative leaig cuves ad esults fo the dual estiatio exeiets

6 43 aaete estiatio As at of the dual algoith, we ileeted the fo weight estiatio This eesets a ew aaete estiatio techique that ca be alied to such obles as taiig feedfowad eual etwos fo eithe egessio o classificatio obles Recall that i this case we wite a state-sace eesetatio fo the uow weight aaetes " as give i Equatio Note that i this case both the ad ae ode ( is the ube of weights) The advatage of the ove the i this case is also ot as obvious, as the state-tasitio fuctio is liea Howeve, as oited out ealie, the obsevatio is oliea Effectively, the builds u a aoxiatio to the exected Hessia by taig oute oducts of the gadiet The, howeve, ay ovide a oe accuate estiate though diect aoxiatio of the exectatio of the Hessia Note aothe distict advatage of the occus whe eithe the achitectue o eo etic is such that diffeetiatio with esect to the aaetes is ot easily deived as ecessay i the The effectively evaluates both the Jacobia ad Hessia ecisely though its siga oit oagatio, without the eed to efo ay aalytic diffeetiatio We have efoed a ube of exeiets alied to taiig eual etwos o stadad becha data Figue 4 illustates the diffeeces i leaig cuves (aveaged ove 1 exeiets with diffeet iitial weights) fo the Macay-Robot-A dataset ad the Ieda chaotic tie seies Note the slightly faste covegece ad lowe fial MSE efoace of the weight taiig While these esults ae clealy ecouagig, futhe study is still ecessay to fully cotast diffeeces betwee ad weight taiig 1 1 Macay Robot A : Leaig cuves 1 2 ea MSE eoch Ieda chaotic tie seies : Leaig cuves ea MSE eoch Figue 4: Coaiso of leaig cuves fo the ad taiig a) Macay-Robot-A, MLP, b) Ieda tie seies, MLP Coclusios ad futue wo The has bee widely acceted as a stadad tool i the achie leaig couity I this ae we have eseted a alteative to the usig the usceted filte The cosistetly achieves a bette level of accuacy tha the at a coaable level of colexity We have deostated this efoace gai i a ube of alicatio doais, icludig state-estiatio, dual estiatio, ad aaete estiatio Futue wo icludes additioal chaacteizatio of efoace beefits, extesios to batch leaig ad o-mse cost fuctios, as well as alicatio to othe eual ad o-eual (eg, aaetic) achitectues I additio, we ae also exloig the use of the as a ethod to iove Paticle Filtes [1], as well as a extesio of the itself that avoids the liea udate assutio by usig a diect Bayesia udate [12] 6 Refeeces [1] J de Feitas, M Niaja, A Gee, ad A Doucet Sequetial ote calo ethods fo otiisatio of eual etwo odels Techical Reot CUES/F-INFENG/TR-328, Det of Egieeig, Uivesity of Cabidge, Nov 1998 [2] A Deste, N M Laid, ad D Rubi Maxiu-lielihood fo icolete data via the EM algoith Joual of the Royal Statistical Society, B39:1 38, 1977 [3] S Hayi Adative Filte Theoy Petice-Hall, Ic, 3 editio, 1996 [4] S J Julie The Scaled Usceted Tasfoatio To aea i Autoatica, Febuay 2 [] S J Julie ad J K Uhla A New Extesio of the Kala Filte to Noliea Systes I Poc of AeoSese: The 11th It Sy o Aeosace/Defece Sesig, Siulatio ad Cotols, 1997 [6] F L Lewis Otial Estiatio Joh Wiley & Sos, Ic, New Yo, 1986 [7] M B Matthews A state-sace aoach to adative oliea filteig usig ecuet eual etwos I Poceedigs IASTED Iteat Sy Atificial Itelligece Alicatio ad Neual Netwos, ages 197 2, 199 [8] G Pusoius ad L Felda Decouled Exteded Kala Filte Taiig of Feedfowad Layeed Netwos I IJCNN, volue 1, ages , 1991 [9] S Sighal ad L Wu Taiig ultilaye ecetos with the exteded Kala filte I Advaces i Neual Ifoatio Pocessig Systes 1, ages , Sa Mateo, CA, 1989 Moga Kauffa [1] R va de Mewe, J F G de Feitas, A Doucet, ad E A Wa The Usceted Paticle Filte Techical eot, Det of Egieeig, Uivesity of Cabidge, 2 I eaatio [11] E A Wa ad A T Nelso Neual dual exteded Kala filteig: alicatios i seech ehaceet ad oaual blid sigal seaatio I Poc Neual Netwos fo Sigal Pocessig Wosho IEEE, 1997 [12] E A Wa ad R va de Mewe The Usceted Bayes Filte Techical eot, CSLU, Oego Gaduate Istitute of Sciece ad Techology, 2 I eaatio (htt://cslucseogiedu/sel) [13] E A Wa, R va de Mewe, ad A T Nelso Dual Estiatio ad the Usceted Tasfoatio I S Solla, T Lee, ad K-R Mülle, editos, Advaces i Neual Ifoatio Pocessig Systes 12, ages MIT Pess, 2