On Maneuvering Target Tracking with Online Observed Colored Glint Noise Parameter Estimation

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1 Wold Academy of Scence, Engneeng and Technology 6 7 On Maneuveng Taget Tacng wth Onlne Obseved Coloed Glnt Nose Paamete Estmaton M. A. Masnad-Sha, and S. A. Banan Abstact In ths pape a compehensve algothm s pesented to allevate the undesed smultaneous effects of taget maneuveng, obseved glnt nose dstbuton, and coloed nose spectum usng onlne coloed glnt nose paamete estmaton. The smulaton esults llustate a sgnfcant educton n the oot mean squae eo (RMSE) poduced by the poposed algothm compaed to the algothms that do not compensate all the above effects smultaneously. Keywods Glnt nose, IMM, Kalman Flte, Knematcs, Taget Tacng. I I. INTRODUCTION N 995, Daepou and Ba-Shalom utled the IMM algothm to mplement the glnt nose model n nonmaneuveng taget tacng []. They appled two extended Kalman fltes, one matched to the dynamc system wth Gaussan measuement nose, and the othe matched to the same dynamc system but wth a hgh vaance Laplacan nose. Late n 998, E. Daepou, and et al. n [], and K. Heyda and et al. n [3]-[4], almost concuently, extended the algothm used n [] to maneuveng tagets. They appled a layeed IMM (LIMM) algothm to mplement the taget maneuveng model as well as the glnt nose model. Although they pusued the same goal, they wee dffeent n methodology. The algothm n [] was based upon the extenson of [] wth the dynamc system state equatons n Catesan and obsevaton equatons n sphecal coodnates. But, the one n [3]-[4] was developed wth the dynamc system state equatons and obsevaton equatons both n sphecal coodnates. In both [] and [3]-[4] the fltes wee splt nto two pats, one matched to the Gaussan component and the othe matched to the Laplacan component of the glnt nose. Howeve, n the fome the extended Kalman fltes (EKF) wee exploted to deal wth the Gaussan and Laplacan nose components, but n the latte, lnea Kalman flte was Manuscpt eceved Febuay 5, 7. M. A. Masnad-Sha s wth the Dept of Electcal Engneeng, School of Engneeng, Sha Unvesty, Sha, Ian (Coespondng autho phone: ; e-mal: masnad@shau.ac.). S. A. Banan s a gaduate student n the Dept of Electcal Engneeng, School of Engneeng, Sha Unvesty, Sha, Ian (e-mal: alea_banan@yahoo.com). used fo the Gaussan nose and Masele flte wth effcent appoxmate scoe functon ([5] and [6]) was used fo the Laplacan nose to flte the components of the glnt nose. In all the afoementoned methods, the obsevaton nose spectum was assumed to be whte. Howeve, n hgh fequency measuement ada systems the successve of the measuement nose ae not uncoelated, and consequently the obseved nose spectum s not whte. In 996 Wu and Chang pesented the subject of maneuveng taget tacng wth obseved coloed nose and unnown paametes [7]. The dawbac of ths appoach was the assumpton of Gaussan nose dstbuton athe than glnt dstbuton. Eale, W. R Wu n [8] had epoted a maxmum lelhood appoach to dentfy the glnt nose paametes fom ecoded data. Howeve, he assumed the spectum of the glnt nose to be whte athe than coloed, and also gnoed maneuveng effects of the taget n nose paamete estmaton. In ths pape a compehensve algothm s developed to ecusvely povde an onlne estmate of the coloed glnt nose paametes and cope wth smultaneous effects of the followng man fou factos that may degade the optmalty of the Kalman flte n taget tacng. The fou degadng factos ae: maneuveng of the taget, glnt and coloed chaactestcs of the obsevaton nose and lac of nowledge about the nose paametes. II. PROBLEM FORMULATION A. Taget Model and System Equatons In ada taget tacng the dynamc state equatons ae expessed n ectangula coodnate, whle the obsevaton equatons ae measued n sphecal coodnate. Convetng ethe of them to the othe one wll esult n nonlnea equatons. One appoach s to convet the dynamc equatons fom ectangula to sphecal coodnate, and use appoxmate lneaed sphecal model whch encountes smultaneous soluton of thee complcated nonlnea dffeental equatons [6, 8]. Fo nstance the ange channel s used to be epesented hee. Then the coespondng appoxmate sphecal epesentaton fo the second-ode dynamc model n nonmaneuveng mode s: 98

2 Wold Academy of Scence, Engneeng and Technology 6 7 whee T T / = + w T () + w s the eo mean whte Gaussan nose. We get the thd-ode model as 3 T T T 6 M T = + T w () + T M whee w stands fo eo mean whte Guassan nose n maneuveng case. In both cases, the eceved obsevaton data n the ange channel s = + v (3) whee v the obsevaton Glnt nose. B. IMM Algothm and the System Model In the poposed algothm, two flte bans ae used: one s matched to the Gaussan nose and the othe matched to the Laplacan nose components of the obseved data. The Kalman flte s used to deal wth the Gaussan component, and the Masele flte wth appoxmate scoe functon s used to face the Laplacan component of the glnt nose. Wth each of the above flte bans thee ae two Kalman fltes wth dffeent dynamc systems (see Fg. ). The Kalman flte matched to the constant velocty mode deals wth the taget system n nonmaneuveng condtons and the one whch s matched to the constant acceleaton mode deals wth the taget system n maneuveng case. The same s pefomed fo the Masele flte. The IMM algothm wll combne the outputs of the fou fltes, and the esultant output s a weghted sum of all the subflte outputs. Fg. Usng Kalman flte and IMM algothm to model the maneuve and Glnt nose Fo the Laplacan mode, the flteng and updatng steps ae pefomed by the appoxmate scoe functon whch ae summaed n [[], (39)-(45)]. These equatons ae only applcable to the glnt whte obsevaton nose. Futhe modfcatons has been pefomed n ths pape to the case of glnt coloed nose ncludng decoelatng the data befoe applyng to the IMM algothm. C. Coloed Nose and Decoelatng Pocess Snce the obsevaton nose paametes ae unnown, one may not be able to use the conventonal method of whtenng pocess by ncopoatng the modal matx coespondng to the nose. The othe poblem wth usng the conventonal whtenng pocess even wth the case of nown nose paametes s the ll-condtoned nose covaance matx that one may encounte. Guu and We n [] modeled the coloed obsevaton nose as an AR model wth unnown paametes. Eventually Wu n [7] modeled the coloed nose wth Gaussan dstbuton as the followng AR pocess to pevent all the above poblems and get the beneft of less ntensve computatonal effot: ν = αν + η (4) ν epesents the coloed nose output and η the whte Gaussan nose nput of the above AR pocess. The whte nose η has eo mean and vaanceσ. The AR paamete α s the unnown coelaton paamete of the coloed nose. To decoelate the coloed nose, we geneate the followng atfcal measuement as suggested n [7]: Δ = α = H + η (5) whee H = H( I α Φ ) (6) η = αφ w + η (7) and I stands fo the dentty matx. Snce n pactce α Φ w n (7) s vey small compaed to η, the atfcal data can be teated as a measuement data fo the Kalman flte wth whte nose η η [7]. D. Nose Paamete Estmaton As the ada tacng pocess advances, the paametes α, ε, σ, and μ ae estmated ecusvely by the onlne obseved data eceved n the ada detecton mode. The estmaton pocess s based on the method dscussed n [8], whee the paametes ε, σ and μ ae estmated unde the assumpton of whte glnt obsevaton nose. Thus, t eques usng (4) to decoelate the coloed data; howeve, paamete α n (4) must be fst estmated. By ecevng a new obseved glnt nose data at th teaton, we geneate the sgnal u = + and pass t though the flte F ( ) = to obtan the new sgnal u [7]. Unde ( ρ ) nonmaneuveng mode f we select ρ =, u wll satsfy the followng AR pocess: 99

3 Wold Academy of Scence, Engneeng and Technology 6 7 u = α u + η (8) Ths pocess s smla to (4) and the paamete α can easly be estmated fom () ˆ α = (9) () whee (.) s the estmated autocoelaton functon of u and can be evaluated fom the followng fadng memoy appoach [8]: ˆ () = β ˆ () + ( β ) u ˆ ˆ () = β () + ( β ) u u () whee < β < s the fogettng facto. Choosng the best value of ρ has been dscussed and smulated n [7]. Consdeng the smulaton esults n nonmaneuveng cases, the best estmaton s wth ρ =. But n maneuveng condton ρ = wll beadown the algothm. The coespondng smulaton esults show that as the value of ρ deceases the estmaton of the glnt nose paametes would be bette, and the convegence would also be faste. Based on the dscusson n [7] a good choce would be ρ =.9, whee the estmates ae almost not affected by maneuveng. Theefoe, ρ =.9 could be an almost optmum value fo the whole tacng peod ncludng maneuveng and nonmaneuveng cases. Howeve, wth ρ, u s no longe smla toν, and (8) fals. Consequently (9) wll be based. Afte emovng the bas, α wll be estmated as [7]: ˆ α ( ξ + ξ ) ( ξ + ξ ) 4ξ ( ξ + ξ + ξ ) = () ξ 3 whee, 3 ξ3 = ( ρ + ) b ξ = ( ρ + ρ 4) + ( 6ρ ) α 3 b ξ = ( ρ 3ρ + 5ρ + 7) + (8ρ + 8) α b ξ = (ρ + ρ 4) + ( ρ 6) α () α b = () Havng of the glnt whte nose at teaton, the teatve algothm n [8] wll yeld the estmaton of the othe paametes, such as the glnt nose paametes, θ = ( ε, σ, μ). Howeve, n the system unde study the eal nose ae glnt but not whte. We now that the nose n atfcal measued data s whte wth glnt dstbuton and t s the same as η n the AR pocess (8). Thus, the of η can be used to estmate the paametes ε, σ and μ of the glnt nose nstead of the of the coloed glnt nose ν. To pefom the onlne pocess of the above paamete estmaton, we geneate of the glnt whte nose η fom (8) as follows: η u αu () = In the fst teatons of tacng whee few obsevatons ae made, the onllne estmton pocess of the glnt nose paametes s pefomed wth nadequate nput nfomaton whch esults n lage paamete estmtaon eo. Consequently the coespondng tacng pefoamce wll be pooe than the case wth nown nose paamet values. Ths has been shown n the smulaton esutls. As the pocess poceeds, the numbe of obsevatons used n the paamete estmatos nceases and the estmates get bette. Consequently the tacng pefomance conveges to ts nomal eo pefomance n nown nose paametes case. III. SUMMARY OF THE PROPOSED ONLINE ALGORITHM ) At each teaton, say the th teaton, geneate the sgnal u when the th obseved data, s eceved as dscussed n secton.4. [7] and []. ) Estmate the paamete α usng (). ) Geneate the th sample of the glnt whte nose, η fom () usng estmated α fom step (). v) Apply the pepocesso ntoduced n [8] to the of η to estmate the appopate ntal values of θ = ( ε, σ, μ). v) Estmate the paametes θ = ( ε, σ, μ) usng the algothm dscussed n [8] v) Geneate the atfcal data fom (5). v) Apply the IMM algothm n [3]-[4] to the atfcal data fo handlng the poblem of two ndependent dscete uncetantes (modes) of the tacng system: the taget dynamc system, and the glnt nose. The esultant output yelds the estmated state vaables. Rema: In the standad IMM algothm, updatng the flte weghts s set fo the Gaussan nose. In the poposed algothm some modfcatons must be mposed to set t fo the Laplacan nose. To update the Laplacan flte weghts, we use the appoach pesented n [] wth a change n the nput data by usng the atfcal data nstead of the glnt coloed obseved data. Then, we use the geneal Bayes ule wth the Laplacan obsevaton weghts as f ( m Z ) f ( m, Z ) p ˆ = f ( m Z ) = (3) f ( Z )

4 Wold Academy of Scence, Engneeng and Technology 6 7 whee Z epesents the atfcal measued data {,,..., }, and m specfes the th mode. In the poposed algothm thee ae fou modes whee each one coesponds to a Kalman flte n the flte ban. In (3) f ( m, Z ) can be obtaned fom convoluton of f ( H ( m ) m, Z ) and the whte glnt nose densty functon f ( η ). Computng the dect convoluton s dffcult, so we use the appoxmate method ntoduced n [] whch yelds pφ () pˆ c (4) ασ whee p epesents f ( m Z ) and c stands fo the nvese K ( T ) + T of f ( Z ), and α = e, K ( T ) = ln[ M ( T )] and Tx M ( T ) = e f ( x) dx whch s the moment geneatng functon (MGF) of the pobablty densty functon f (.). The vaable T s the value of T evaluated at the th teaton, whch s detemned fom the followng equaton [] K ( T ) = (5) whee s the nown atfcal measued data n the th teaton. Eventually, the pobablty densty functon f ( m, Z ) n (3) at th teaton of the algothm wll be evaluated as: φ () f ( m, Z ) (6) α σ Fo the nomal expanson we obtan φ( ) = π and σ () = K ( T ) [9] and []. In the IMM algothm the pobablty of tanston between dffeent modes ae govened by the Maovan tanston pobablty matx. In the poposed algothm, we wll consde fou dffeent modes of opeatons as follows: constant velocty matched to Gaussan obsevaton nose, constant acceleaton matched to Gaussan obseavaton nose, constant velocty matched to Laplacan obsevaton nose, and constant acceleaton matched to Laplacan obsevaton nose. Thus, the Maovan tanston matx ( IMM tanston 4 4 matx ) A R wll be: q( ( q)( A = q( ( q)( ( q)( q( ( q)( q( ( q) ε ( q) ε ( q) ε ( q) ε (7) q shows the pobablty of usng the same dynamc system n two consecutve tme, o the pobablty of no change n the dynamc of the system, and s vey close to unty n pactce. IV. SIMULATION RESULTS In ths secton, we have pesented some smulaton esults fo the poposed algothm. Fo smplcty as t was stated befoe we pefom the esults on one sngle channel o axs. Due to ndependency of the dmensons, the smulaton pocess on the othe channels wll end up to smla esults. The taget s taced fo, and s assumed to move wth constant velocty of m / s between the fst and the th. Then the velocty nceases by a constant acceleaton of 4 m / s fom the th sample up to the th sample. It contnues movng wth constant velocty between the th and th. The acceleaton used hee s = 399 and = 4 The samplng tme s consdeed to be T =. 5 s. The vaance of the pocess nose fo constant velocty s assumed to be. and fo the constant acceleaton (maneuveng) would be.. We assume that the dynamcs of the taget ae nown and the only unnowns ae the paametes of the glnt nose that would be estmated though the algothm pocesses. Thus, choosng small pocess nose vaance s consstent wth havng nowledge about the dynamcs of the system. To analye the pefomance of the poposed algothm esults an deal system wth followng nown paametes s taen nto consdeaton: The coelaton coeffcent of the glnt coloed nose s assumed to beα =. 8, and the paametes of the whte glnt nose η ae consdeed to be ε =., σ = 5m, and μ = m. The onlne estmaton of the paametes s teatvely pefomed as the tacng pocess advances. The constants ρ =. 9, and β =. 99 have been chosen n the estmaton pocess of the paameteα, and q s set to unty n the pobablty tanston matx. Monte Calo uns ae caed out and the aveage s epesented by the oot mean squae eo (RMSE) cteon as a measue of the pefomance n ths smulaton: m RMSE ( ) = ( ˆ ) =,,...,, m = m = whee ˆ denotes the state estmate of the th Monte Calo un fo the th sample. We fst compae the poposed method eo pefomance (RMSE) esults wth smla esults obtaned fom the deal system whee all the paametes α, ε, σ, and μ ae assumed to be nown. The coespondng RMSE esults fo poston, velocty and acceleaton ae shown n Fgs., 3, and 4. It s seen that the esults of the poposed algothm wth unnown paametes ae n vey good ageement wth the esults obtaned fom the deal system wth nown paametes.

5 Wold Academy of Scence, Engneeng and Technology 6 7 ( unnown nose paametes ) Tacng wth nown nose paametes To study the effect of dffeent types of obseved data noses on pefomance of the poposed algothm the followng tests have been caed out: RMS eo of poston (m) Fg. Poston estmaton RMSE ) Dsablng the decoelatng pocess. ) Dsablng the glnt nose pocess. In the fst test we teat the obseved data nose as a whte nose pocess, and gnoe ts coloed chaactestc. Thus, thee s no need to decoelate the nose and consequently, we dsable the decoelatng pocess of the algothm and leave the sgnal undecoelated. The coespondng esults fo the taget poston, velocty and acceleaton ae llustated n Fgs. 5, 6, and 7. ( unnown nose paametes ) Tacng wth nown nose paametes Wthout coloed nose decoelato RMS eo of velocty (m/s) RMS eo of poston (m) Fg. 3 Velocty estmaton RMSE The slght changes between the two esults at the fst teatons, s the effect of the mpefect paamete estmatons due to nsuffcent eceved data. As the tacng pocess advances, the estmaton esults get bette and bette and eventually each the steady state values, whee the dffeences between the two coespondng esults, ae vey small. 7 ( unnown nose paametes ) Tacng wth nown nose paametes RMS eo of velocty (m/s) Fg. 5 Poston estmaton RMSE wthout Coloed nose decoelato Wthout coloed nose decoelato RMS eo of acceleaton (m/s ) 5 3 Fg. 4 Acceleaton estmaton RMSE Fg. 6 Velocty estmaton RMSE wthout Coloed nose decoelato It s qute obvous that the eo pefomance has nceased compaed to the poposed algothm esults. Theefoe, ths test epesents the capablty of the poposed algothm n cancelng the effect of the coloed nose.

6 Wold Academy of Scence, Engneeng and Technology Wthout coloed nose decoelato 7 Wthout Laplacan fltes RMS eo of acceleaton ( m/s ) 5 3 Samples Fg. 7 Acceleaton estmaton wthout Coloed nose decoelato In the second test we neglect the glnt chaactestc of the obseved data nose. In othe wods we teat the nose as coloed Gaussan nose wth unown paametes. In ths case thee s no need to use the Laplacan fltes n place of the Gaussan fltes to model the glnt nose n the IMM algothm. So, we un the algothm all wth Gaussan fltes. The coespondng esults fo poston, velocty, and acceleaton ae shown n Fgs. 8, 9, and. RMS eo of acceleaton (m/s ) 5 3 Fg. Acceleaton estmaton RMSE wth no Laplacan flte Agan, the eo pefomance has nceased compaed to the poposed algothm esults.. Theefoe, the second test epesents the capablty of the poposed algothm n elmnatng the effect of the glnt nose data. The oveall esults show that the poposed algothm s able to smultaneously deal wth the poblem of glnt and coloed nose, and also estmate the unnown glnt nose paametes n manueveng taget tacng. RMS eo of poston (m) Wthout Laplacan fltes V. CONCLUSION The smulaton esults show the emaable stength of the poposed algothm n smultaneous emoval of all the undesed degadng factos n optmal opeaton of the Kalman flte fo taget tacng. The tests caed out n ths pape llustate that dsablng the effect of any of the degadng factos n the algothm wll esult n a sgnfcant ncease n the coespondng RMSE values. RMS eo of velocty (m/s) Fg. 8 Poston estmaton RMSE wth no Laplacan flte Wthout Laplacan fltes Fg. 9 Velocty estmaton RMSE wth no Laplacan flte REFERENCES [] Daepou, E. and Ba-Shalom, Y. An Inteactng Multple Model Appoach fo Taget Tacng wth Glnt Nose. IEEE Tansactons on Aeospace and Electonc Systems, vol. 3, no., (995). [] Daepou, E. and Ba-Shalom, Y. IMM Tacng of Maneuveng Tagets n the Pesence of Glnt. IEEE Tansactons on Aeospace and Electonc Systems, vol. 34, no. 3, 996- (Jul-998). [3] Heyda, K. Maneuveng Taget Tacng n Glnt Nose. M. S. Thess, Sha Unvesty, Ian, (998). [4] Masnad-Sha, M. A., Heyda, K., and Sabbagh-Jafa, S. M. Maneuveng Taget Tacng n glnt Nose. The 8 th Ianan Confeence on Electcal Engneeng Poceedngs (May-). [5] Wu, W. R., and Kundu, A. Kalman Flteng n Non-Gaussan Envonment Usng Effcent Scoe Functon Appoxmaton. In Poceedngs of 989 IEEE Intenatonal Symposum on Ccuts and Systems, (989). [6] Gholson, N. H., and Moose, R. L. Maneuveng Taget Tacng Usng Adaptve State Estmaton. IEEE Tansacton on Aeospace and Electonc Systems, vol. AES-3, no. 3, 3-37 (May-977). [7] Wu, W. R., and Chang, D. C. Maneuveng Taget Tacng wth Coloed Nose. IEEE Tansactons on Aeospace and Electonc Systems, vol. 3, no. 4, 3-39 (Oct- 996). [8] Wu, W. R Maxmum Lelhood Identfcaton of Glnt Nose. IEEE Tansactons on Aeospace and Electonc Systems, vol. 3, no., 4-5 (Jan-996). 3

7 Wold Academy of Scence, Engneeng and Technology 6 7 [9] Wu, W. R. Taget Tacng wth Glnt Nose. IEEE Tansactons on Aeospace and Electonc Systems, vol. 9, no., (JAN-993). [] Wu, W. R., and Chang, P. P. A Nonlnea IMM Algothm fo Maneuveng Taget Tacng. IEEE Tansactons on Aeospace and Electonc Systems, vol. 3, no. 3, , (Jul -994). [] Guu, J. A., and We, C. H. Tacng Technque fo Maneuveng Taget wth Coelated Measuement Nose and Unnown Paametes. IEE Poceedng, PT. F, 38, 3, (Jun -99). 4