Continuous-time Nonlinear Estimation Filters Using UKF-aided Gaussian Sum Representations

Transcription

1 Connuous-me onlnear Esmaon Flers Usn UKF-aded Gaussan Sum Reresenaons ura Gokce echnolo and Innovaon Fundn rorams Drecorae he Scenfc and echnolocal Research Councl of urke Ankara urke usafa Kuzuolu Elecrcal and Elecroncs Enneern Dearmen ddle Eas echncal Unvers Ankara urke Absrac An aromae nonlnear esmaon mehod for connuous-me ssems h dscree-me measuremens s develoed. he aroach evaluaes he Gaussan sum aromaon of he a ror robabl dens funcon df b solvn he Fokker-lanck equaon numercall. Aromae evaluaon of he a oseror df s acheved b usn Gaussan sums a ror df and measuremens n Baes rule. ean and covarance values of Gaussans are chosen b he hel of an Unscened Kalman Fler UKF h resec o a reon here a ror and a oseror dfs are aromaed. Wehs of he Gaussans are udaed usn he deermnscall chosen rd ons n he secfed domans. UKF here acs as a one se look ahead mechansm o deermne he hh robabl reons here a ror and a oseror dfs can resde. he a ror and a oseror dfs are aromaed around hese hh robabl reons. he develoed aroach s comared h UKF and arcle Fler n a one dmensonal nonlnear ssem. Keords nonlnear ssem; flern alorhm; connuousme ssem; Fokker-lanck equaon; Gaussan sum; numercal mehods I. IRODUCIO he eneral roblem of nonlnear flern can be handled for ssems ha are reresened eher n connuous-me or dscree-me. Connuous-me ssems are more eneral and can be aromaed b dscree-me ssems usn suable dscrezaon mehods. here es a fe eac.e. omal nonlnear flers n connuous-me for some secal cases and here are several aromae flern mehods aled o boh connuous-me and dscree-me ssems n he leraure. A dnamcal ssem ves he me evoluon of he sae and a measuremen model can be nroduced hch elds samles of he sae a a secfc me nsan. For he connuous-me case e assume a connuous-me dnamcal ssem and a dscree-me measuremen model. he sae s he sochasc rocess hose me evoluon s overned b he Io sochasc dfferenal equaon. d =f d + G dß h he nal condon =. easuremens are obaned n dscree-me as: k =h k k + v k є R n denoes he sae vecor f є R n s he drf funcon and ß є R m s a Wener rocess and he ncremens of Wener rocess dß mus be nfnel dvsble [7]. G є R nm s a sae deenden mar. he ndeenden varable reresens me. denoes he nal me nsan and k s he k-h samln me. he nal sae s assumed o be a random vecor h a knon no necessarl Gaussan dens. є R s a vecor of measuremens h є R s a funcon of he sae and me and v є R s a zero-mean he measuremen nose. I s assumed ha he random varables ß and v are muuall ndeenden. In order o smlf he noaon e ll re k for k ec. he flern roblem nvolves esman he values of he saes of he ssem b he hel of he measuremens. Snce samled saes are random varables e need o esmae her condonal robabl dens funcon ha deends on he measuremens a samln me k hch s called a oseror dens k k. Afer he deermnaon of he a oseror dens he sae esmae ma be comued. For he ven connuous-me dnamcal ssem he condonal robabl dens funcon evolves beeen o measuremens or o samln me nsans b means of he Fokker-lanck equaon eldn he a ror dens hch s udaed h Baes' rule usn measuremens a curren samln me vn a oseror dens. he recursve soluon of he flern roblem s ven n []. he Fokker- lanck equaon s ven as f GQβG f r r he Fokker-lanck equaon s n eneral a nonlnear aral dfferenal equaon hose soluon canno be eressed n erms of a fne number of arameers. Ece for a fe secal cases e.. lnear Gaussan case e canno fnd an analcal soluon of Fokker-lanck equaon for he a ror

2 df hch s used n udae se o fnd he a oseror df. Even f hese dfs are analcall avalable ma be dffcul or mossble o evaluae he nerals for normalzaon uroses or fndn he mean and covarance of he sae. As an alernave aromae mehods are used for solvn he dfferenal equaons or evaluan he nerals. Hoever some of he aromae mehods ma no eld suffcen accurac for ceran cases or ma requre ecessve comuaonal oer o acheve suffcen accurac. Generall he requred comuaonal oer does no ncrease lnearl as he dmenson of he sae varables ncreases. For he connuous-me case he Benes fler [3] and he eended Benes fler Daum fler [3] ve eac soluons for secal cases. In Benes fler he eac soluon of he Fokker- lanck equaon assumes an eonenal dens funcon characerzed n erms of sae and covarance. Daum fler eneralzes he Benes fler for he case here an analcal soluon o Fokker-lanck equaon ess. Eended Kalman Fler seudolnear Fler Second Order Flers and Eanson of Dens and Coordnae ransformaons are some of he aromae mehods for he soluon of roblem. Deals of hese flers are ven n []. As menoned n [] here are o es of aromaons suesed n he leraure eher he model s relaced b a smler one or numercal mehods are used o fnd a lobal aromaon of he a oseror dens. odel aromaon s obaned b eandn he nonlnear funcons around he oeran on a ever me se usn alor seres eansons. hs aroach elds he Eended Kalman Fler EKF. Anoher model aromaon echnque s based on he reresenaon of he sae vecor b a fne se of values. hs model class s referred o as he Hdden arkov odel H. Convered easuremen Kalman Fler CKF s also anoher echnque ha res o lnearze he nonlnear measuremen model b convern he measuremens no he sae sace. hs fler s commonl used n radar rackn alcaons. he oher alernave aroach s based on he aromaon of dfs. For nsance he dfs are aromaed usn a sum of Gaussan denses n []. Anoher df aromaon s rovded b a samln echnque knon as he ransformaon echnque o ck a mnmal se of samle ons called sma ons around he mean. hese sma ons are hen roaaed hrouh he nonlnear funcons from hch he mean and covarance of he esmae are hen recovered. UKF and Quadraure Kalman Fler are of hs form and sma ons are udaed b usn oeraons smlar o hose n he Kalman Fler. UKF s frs ublshed n [6]. An UKF formulaon for he connuous-me case s ven n [4]. Anoher df aromaon knon as he on-mass fler aromaes he a oseror dens b a se of ons on a redefned rd. An eenson of he on-mass fler s he sequenal one Carlo mehod also referred o as arcle Fler. In hs mehod he a oseror dens s also aromaed b a se of ons hoever he rd s chosen n a sochasc raher han n a deermnsc manner. A comarson of some of hese mehods can be found n [5]. he oranzaon of he aer s as follos; In Secon II a flern aroach he UKF-aded Gaussan Sum Fler for connuous-me ssems s elaned. hs mehod s based on he aromaon of dfs b Gaussan sums. Smulaon resuls relaed o hs fler are resened n Secon III. Conclusons are dran n Secon IV. II. UKF AİDED GAUSSIA SU FILER FOR COIUOUS-IE SYSES he aroach elaned n hs secon s a flern mehod based on he aromaon of boh he a ror and he a oseror dfs of he saes of he ven connuous-me ssem usn ehed Gaussan sums and nvolves he follon o ses. Evaluaon of he Gaussan sum aromaon of he a ror df b solvn he Fokker-lanck equaon numercall. Aromae evaluaon of he a oseror df b usn Gaussan sums a ror df and measuremens n Baes rule. ean and covarance values of Gaussans are deermned b he hel of an UKF hrouh hese ses. he deals of hese ses are ven belo. A. Evaluaon of he aromae a ror df For he ven ssem he Fokker-lanck equaon hch s used for fndn he a ror df s ven as follos; f f r GQβG r he rh hand sde of he equaon can be reresened as a nonlnear funcon as; here f f r GQβG r 3 can be solved b usn he Euler s mehod. A e sar h a df. We an o fnd a ror df a denoes he samln me usn 3 and. n=/ Euler ses can be used for hs urose. Hoever e canno aromae he connuous-me

3 bu e can onl evaluae a he samle ons. amel e can fnd for =. In order o fnd an aromae eresson for e emloed a Gaussan sum aromaon. eans covarances and ehs of Gaussans are calculaed a ever se of Euler s mehod. eans and covarances are deermned usn he redcon se of an UKF. Wehs are udaed usn deermnsc rd ons chosen from a reon covern he revous and curren means of he Gaussans. he follon ses summarze mean covarance and eh calculaon n a snle Euler se.. he redcon se of UKF s aled o he mean and covarance values of nal df o fnd he redced mean and covarance values. For hs urose assumn ha he nal df s Gaussan he unscened ransform s aled usn. he nonlnear funcon used n he ransformaon s f here = +. eans of Gaussans are deermned on a unform rd around. he dh of he rd s deermned usn a suabl scaled value of. Covarances of Gaussans are assned o he same value such ha. hs scaln can be done as follos; Under he assumon ha he mean of he sae varable s and for a Gaussan sum aromaon of he a ror df s e here > le he dh of he rd be defned as he mar lda c here l and c s he scaln facor. For c l can be chosen as zero and he covarances of Gaussans can be aken as as here <ν<. For <c< he covarances of Gaussans can be aken as here ν. eans of Gaussans are unforml deermned accordn o he daonal elemens of he dh of he rd. 3. o ses of rd ons are defned. One se nvolves he mean values of Gaussans for ha se. he oher se nvolves ons chosen from a unform rd around. here 4. he a ror df for he curren se s modeled usn Gaussan sums as here e Also e can fnd he values of a ror df a a ven on usn he Euler dscrezed Fokker-lanck Equaon. B equan 5 and 6 a he rd ons e have he follon ssem. B A

4 A. B he leas squares soluon of hs ssem can be ven as A A A B If A s nearl snular mamum values alon each ro of A mar ma be used. In hs case he ar of A + mar ll be daonal for dsnc values avodn he nverson roblems. A he above rocedure s reeaed a ever se. B. Evaluaon of he aromae a oseror df he lkelhood and he aromaed a ror df are combned n Baes rule eldn he unnormalzed a oseror df as; For a nonlnear measuremen model he lkelhood ll no be a smle eresson and he unnormalzed a oseror df hch s calculaed usn he lkelhood canno be used drecl o fnd s mean and covarance. So he unnormalzed a oseror df can be reresened aromael h ehed Gaussans as; K! e ean and covarance values of Gaussans are found n a manner smlar o he a ror df. Here mean and covarance of a ror df are used n he udae se of UKF. Grd ons are chosen as he mean values of Gaussans n hs case. he resuln mar here s an smmerc osve defne mar ven as n [] he reon used o choose he mean values of Gaussans deermnes he search sace here he a ror and a oseror dfs are red o be aromaed. Covarance values mus no be oo hh and number of Gaussans used n he aromaons s also moran for a ood reresenaon. If he number of Gaussans ncreases here can be an overfn. For aromaons nvolvn a fe Gaussans dfs ma no f ell. he number of Gaussans s roblem deenden and ma be chosen emrcall. III. SIULAIO RESULS he develoed aroach s mlemened n a one dmensonal nonlnear ssem and he resuls are comared h hose obaned b solvn he same roblem h he UKF and Sequenal Imorance Samln SIR arcle Fler. Connuous-dscree UKF of [4] s used and also nose free dnamcal model s used n he SIR arcle Fler for me evoluon of arcles before addn nose. he smulaons are carred ou on a comuer h Inel core 3 CU and.8 Gb RA usn ALAB. c&oc command s used for he comuaon me and normalzed accordn o he UKF hle fndn he comuaonal load. Consder he follon one dmensonal connuous-me nonlnear ssem. 5 d 8cos. d d here s a Wener rocess h Gaussan ncremens hose nens s aken as. easuremens are aken b usn he follon dscree me model a a samln erod of =s. k 3 k v here v k s a Gaussan measuremen nose. n= Euler dscrezaon ses beeen o samles are used. Smulaed daa ere eneraed usn he ses Euler- aruama mehod [8] a ever second and smulaon lass for s. For he ven roblem o confuraons are used. he frs one uses Gaussans hle aroman he a ror df and 5 Gaussans hle aroman he a oseror df. he second one uses Gaussans hle aroman he a ror df and Gaussans hle aroman he a oseror df. he resuls of he 3 one Carlo runs are ven n able I n erms of he ean of ean Absolue Error AE and Comuaon Load. AE here =s and =3 runs. ABLE I. Fler o k rue es COARISO OF AE AD COUAIO LOAD Flers ean of ean Absolue Error AE Averae Comuaon Load for One erod Unless UKF SIR arcle Fler h arcles SIR arcle Fler h 5 arcles UKF-aded Gaussan Sum Fler h +5 Gaussans UKF-aded Gaussan Sum Fler h + Gaussans

5 Fure ves he ean of Absolue ErrorAE afer 3 one Carlo runs. 5 AE rue es 5 For he one dmensonal ssem aromael one hundred arcles are enouh for he omal accurac [9]. Bu he number of arcles ncreases eensvel n hher dmensons. For o samle runs he ar of he unnormalzed aromaed dfs n he reon here he means of Gaussans are chosen are ven n F. and F. 3 for he o confuraons a he end of las smulaon me a =s. Almos eac dfs are found usn he hsoram of arcles of a SIR arcle Fler nvolvn arcles. As can be seen from F. and F. 3 for he frs confuraon onl a lmed ar of he df s aromaed n a reon close o he mean. Snce he number of rd ons s small a smaller search reon s chosen. Bu snce aromaon s carred ou n an nformaonall meannful reon urns ou ha he error levels are acceable. IV. COSLUSIOS An aromae connuous-me nonlnear esmaon mehod usn UKF-aded Gaussan sum reresenaons s derved. he mehod res o aromae a ror and a oseror dfs n he flern ses usn ehed Gaussan sum reresenaons. he mehod has he abl o aromae he almos eac df n a ven search area. Lookn a he smulaon resuls he mehod ves smaller error values comared h he UKF h an acceable comuaonal load and a b hher error levels han arcle Flers usn hh number of arcles for he ven ssem. he error levels and comuaonal loads of he aroach n hher dmensonal ssems and he erformance of he aroach under non- Gaussan noses are nended o be suded as a fuure ork a b c F.. AE afer 3 one Carlo runs F.. Unnormalzed aromae dfs a he end of =s for he frs samle one Carlo run: a Almos eac df; b Aromae df h UKF-aded Gaussan Sum Fler h + Gaussans; c Aromae df h UKF-aded Gaussan Sum Fler h +5 Gaussans.

6 a REFERECES [] L.L. Banasch A Comarave Sud of onlnear rackn Alorhms A dsseraon submed o he Sss Federal Insue of echnolo Zurch 99 []. Schön On Comuaonal ehods for onlnear Esmaon A dsseraon submed o he Dearmen of Elecrcal Enneern of Lnkön Unvers 3. [3] F. Daum "Eac Fne-Dmensonal onlnear Flers" IEEE ransacons on Auomac Conrol vol. AC-3 no Jul 984. [4] S. Smo On Unscened Kalman Flern for Sae Esmaon of Connuous me onlnear Ssems IEEE ransacons on Auomac Conrol vol Se. 7. [5] F. Daum onlnear Flers: Beond he Kalman Fler IEEE Aerosace and Elecronc Ssems aazne vol Au. 5. [6] S.J. Juler and J.K. Uhlmann A e Eenson of he Kalman Fler o onlnear Ssems Inroc. of AeroSense: he h In. Sm. on Aerosace/Defence Sensn Smulaon and Conrols 997. [7] A. Cubko and B. Solona Generalze Wener rocess and Kolmoorov s Equaon for dffuson nduced b non-gaussan nose source Flucaon and ose Leers vol [8] Desmond J. Hham An Alorhmc Inroducon o umercal Smulaon of Sochasc Dfferenal Equaons SIA REVIEW Vol. 43o [9] F. Daum and J. Huan Curse of Dmensonal and arcle Flers roceedns of he IEEE Aerosace Conference vol arch 3. []. Snla. Snh.D. Sco Adave Gaussan Sum Fler for onlnear Baesan Esmaon IEEE ransacons on Auomac Conrol vol Se.. []. onllo Choosn bass funcons and shae arameers for radal bass funcon mehods. SIA Underraduae Research Onlne Se b c F. 3. Unnormalzed aromae dfs a he end of =s for he second samle one Carlo run: a Almos eac df; b Aromae df h UKF-aded Gaussan Sum Fler h + Gaussans; c Aromae df h UKF-aded Gaussan Sum Fler h +5 Gaussans.