Estimation of Markov Jump Systems with Mode Observation One-Step Lagged to State Measurement

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1 Estmaton of Marov Jump Systems wth Mode Observaton One-Step Lagged to State Measurement Yan Lang, Zengfu Wang, Ll We, Yongme Cheng, Quan Pan College of Automaton Northwestern Polytechncal Unversty X an, Chna {langyan@nwpu.edu.cn, wangzengfu@gmal.com, quanpan@nwpu.edu.cn} Abstract The estmaton of Marov Jump systems (MJS s wdely used n target tracng, fault detecton, sgnal processng and dgtal communcatons. However, the above researches all assume that state measurement and addtonal mode observaton are synchronous whch means both state measurement and mode observaton at each samplng tme arrve at the fuson centre at the same tme. The problem of estmaton of MJS that mode observaton s one-step lagged to ts correspondng state measurement s consdered. Along state-augmentaton approach and the dervaton of mage-enhanced nteractng multple model (IE-IMM, a new generc estmaton algorthm s proposed. It s shown by smulaton result that the proposed algorthm s effectve. Keywords: Marov Jump system, IE-IMM, Asynchronous fuson, Target tracng. Introducton Marov Jump Systems (MJS are lnear systems whose parameters evolve wth tme accordng to a fnte state Marov chan [4]. These models are wdely used n target tracng, fault detecton, sgnal processng and dgtal communcatons [4]. To the MJS estmaton, the underlyng state space s hybrd, consstng of the contnuous-valued state and dscrete-valued mode. It s well nown that the optmal estmaton or smoothng of JMS s the NP-hard problem because ts mode sequence ncreases exponentally wth tme. The suboptmal estmators nclude generalzed pseudo-bayes (GPB and nteractng multple model (IMM and etc [5].Obvously, accurate state estmaton of MJS severely depends on effectve mode dentfcaton. Through ntroducng more nformaton, such estmaton can be mproved n the followng three categores. The frst, called smoothng, s to obtan the current estmaton usng the future state measurements. The smooth algorthms based on IMM nclude one step fxed lagged smooth [0], arbtrary step fxed lagged smooth [2],[3], and fxed nterval smooth[9]. The second, called varable structure, adaptvely adds lely models whle deletes unlely models n order to obtan better adaptaton to tme-varant outsde world. Here such onlne decson about model addton and elmnaton allows usng new nformaton. The last, called heterogeneous fuson, s based on state measurement and addtonal mode observaton. Up to now, some sensors, e.g. mage sensors, can supply addtonal mode observaton nformaton. Research result shows that t can mprove performance of system n the applcaton of maneuverng target tracng [6],[8]. Sworder [6] consders that the observaton from mage contans more mode nformaton than that of state measurement, so that only mage nformaton s used n the update of model probablty. Evans [] uses both state measurement and mode observaton to updatng model probablty. The man dea of these mageenhanced target tracng algorthms s to use the relatonshp between target s azmuth and acceleraton [7]. However, above results are all based on the case that state measurement and addtonal mode observaton are synchronously obtaned. No matter that mode observaton come from complcated mage-based target recognton or uncertanty reasonng, huge computaton s nevtable. Thus t s the usual case that model observaton s lagged to state measurement. Base on above consderaton, our paper s the frst attempt to deal wth estmaton of MJS wth mode observaton lagged to ts correspondng state measurement. 2 Problem Formulaton MJS wth an addtonal mode observaton s descrbed by system equaton: X F X + G W ( X s state vector of target at tme. F and G are proper matrxs. W s a random sequence wth zero-mean whte Gaussan wth covarance Q. Superscrpt ( denotes quanttes pertnent to mode M. The ump th th from mode at tme to mode at tme s assumed to obey a homogeneous Marov Chan wth transton probabltes []. { M } π p M (2, {,..., m} (. The state measurement equaton s

2 where Z H X + V ( (3 Z s measurement vector of state and H s measurement matrx. V s zero-mean a random sequence wth whte Gaussan wth covarances R. W and V are mutually ndependent. The mode observaton model s 0 λ f p{ N M} (4 λ d f 0 N {0,,..., m} s the observaton of mode M. N 0 represents that there s no useful estmaton of mode at tme whle N for 0 ndcates that t th s beleved that the target movement at tme obeys mode. Smlar to [], we assume that when N the mode observer outputs a nonzero mode estmate wth probablty λ and that ths estmate s mode wth probablty d. It s assumed that N s condtonally ndependent of all other random varables gven M so that the rates λ and dscernbltes d fully specfy the mode observaton process. More detals about (4 please refer to []. Our am s to obtan the followng MMSE estmaton X, E{ X Z, N } (5 { T P, E ( X X, ( Z, N } (6 2 where ( denotes the same content as that n prevous parenthess. In the followng formulas, I ( denotes the ndcator functon and φ ( x; yp, represents Gaussan densty functon: /2 T φ( x; yp, : 2πP exp( ( x y P ( x y 2 3 Estmaton Algorthm Dervaton Augment the state vector. (0 T ( T T X [ X, X ] (7 ( l where X Xl ( l 0,. Then ( and (3 can be transformed as (0 (0 X F 0 X G ( ( + W (8 X I 0 X 0 X Z H V (0 0 + ( X For smplcty, (8 and (9 are wrtten as: X F X + G W (0 (9 Z H X + V ( The correspondng estmaton and ts covarances are: X : E X Z, N (2, { }, T { } : E ( X X ( Z, N,, (0,0 (0,,, (,0 (, P,, s a postve sem-defnte matrx. So that (0,, { } (3 ( l ( l X, : E X Z, N ( l 0, (4 (0 (0 ( ( T { },, : E ( X X ( X X Z, N { } (, ll ( l ( l T,, (5 : E ( X X ( Z, N (6 (5 and (6can be got through lettng l n (4 and (5. To descrbe algorthm convenently, we ntroduce the followng mars: u : p( M Z, N (7 2, 2 X : E( X M, Z, N (8,, 2 2,, 2 T 2 { }, : E ( X X ( Z, N, (9 u : p( M Z, N (20 X : E( X M, Z, N (2,,,, : E(( X X ( Z, N T,, (22

3 0 u, : p( M Z, N (23 X : E( X M, Z, N (24 0,, 0,, : E(( X X ( Z, N u,, 0 T,, : pm ( Z, N pm ( M / pm ( Z, N u π / u (, {,..., m} 0,,,, (25 (26 X : E( X M, Z, N (27 : E(( X X ( Z, N (28 0 T,,,, Theorem Consder the MJS (-(4, we have (a model probablty (b u u (( λ I( N 0,, 2 + λ din ( / K,,,, 2 (29 X X (30 (3,,,, 2 where K s a normalzed constant. Proof. (a Gven M, N s condtonally ndependent to all other random varables. From Bayes rule, we can obtan u u p( N M, Z, N / 2,, 2 pn ( Z, N 2 u p( N M K, 2 By ntal condton (7 and (20, (29 can be obtaned. (b From Bayes rule, we can obtan (32 px ( M, Z, N px ( M, Z, N pn ( X, M, Z, N / pn ( M, Z, N px ( M, Z, N pn ( M / pn px ( M, Z, N ( M 2 (33 Then by (8, (9, (2 and (22, (30 and (3 are obtaned. The basc dea of IMM s appled for ths augmented system(34-(35, and a Gaussan dstrbuton s used to approxmate { p X Z, N } [2]. And based on IE- IMM, ths algorthm ncludes the followng sx steps. Intalzatng u, 2, X,,, 2 P,, 2 for,, m.. For,, m, update model probablty by (29 usng Mode observaton. For,, m, 2. Input nteracton 0. u, u, π (36 X u X (37 0,,,,,, 2 u ( + 0,,,,, 3. State predcton ( X X ( 0 T,,,, 0,,,, (38 X F X (39 F ( F + G Q ( G (40 0 T T,,,, 4. State update usng state measurement r Z H X Z H X (4 (0,,,, S H H + R (42 (0,0 ( T,, K ( P ( H ( S (43 l ( (0, T T,,

4 X X + K r (44 ( l ( l ( l,,,, K S K (45 (. l l (, l l ( l ( l T,,,, ( K S K l (46 (0. l (0, l (0 ( l T,,,, ( ( 0, 5. Update the probablty of model usng state measurement u u φ( r ; 0, S K (47 0,, 3 6. Output ntegraton X, u, X,, (48, u ( + ( X X ( T,,,,,, (49 Fnally, we note that the state estmaton (5 and covarance (6 are: ( X X (50,, P P (5 (,,, Addtonally, we can get the estmaton of model probablty through the followng theorem. Assumpton Assume that { Z, N } can be approxmated by X (,, whch can be approxmated by X. And condtonal probablty densty 2 px ( M, Z, N can be approxmated by a Gaussan dstrbuton. Theorem 2 Consder systems (-(4, Under Assumpton, we have model probablty pm Z N (, u, 2 φ( X ; X, / K ( (0 (0,,, 2,, 2 4 Where K 4 s a normlazed const. Proof. Under assumpton, we have where pm Z N (, 2 ( pm ( Z, N, X, u p( X M, Z, N K ( 2, 2, 4 (52 (53 ( 2 px (, M, Z, N 2 px ( M, Z, N (0 (0 φ( X ; X,, 2,,, 2 φ( X ; X, ( (0 (0,,, 2,, 2 Put (54 nto (53, (52 s obtaned. 4 Smulaton (54 To test our algorthm, we consder the maneuverng target tracng example []. Observaton perod s Ts s. The state vector of target s defned by X [ η η ξ ξ] T, where η and ξ present two coordnates n Cartesan coordnate respectvely, and let η : dη / dt, ξ : dξ / dt. The system we consder has three modes of operaton[], Mode represents nearly constant velocty moton, Mode 2 and 3 correspond to nearly constant speed turns wth angular veloctes of ω 2 and ω 3, respectvely. Here the smulaton s based on followng Marov chan samplng channel smlar to []., 0 < 24, 2, 24 < 36, M :, 36 < 60, 3, 60 < 72,, 72 < 96 Two man mage-based observaton scenaros (scenaro A and scenaro B are consdered: (A (B λa 0.9 D a λb 0.7 D b The frst set of parameters (wth subscrpt A represent a stuaton wth farly good mage-based observatons whle the second set (wth subscrpt B reflect somewhat lower qualty modal observatons []. We wll focus on the relatve performance of our algorthm (we denote t IE-IMM-FLO aganst the IE- IMM n paper [], fxed lagged smoothng algorthm wth one-step n [0] (we denote t IMM-FLO and IMM n paper [3]. The model probablty estmaton of IMM, IE IMM, IMM-FLO, IE-IMM-FLO are p( M Z,

5 (, p M Z N, p( M Z, p( M, Z N, respectvely. The state estmaton of IMM, IE-IMM, IMM-FLO, IE-IMM-FLO are E( X Z, E( X Z, N, E( X Z + +, E( X Z, N, respectvely. We gve the comparson of four algorthms under 00 Monte Carlo smulatons under these two scenaros. The smulatons are performed on a PC wth P4 2.4G CPU and 52M RAM usng the software pacage MATLAB (verson 6.5. In the followng fgures, the sold red lne s the ndcator of Mode. Fgure : Model estmaton and state estmaton under scenaro A Fgure 2: Model estmaton and state estmaton under scenaro B In the case that the mage-based observaton s of good qualty as shown n fgure, IE-IMM-FLO s better than IE-IMM n poston estmaton. Ths s derved from the mprovement of estmaton va model-condtonal flters. Because the addtonal mode observaton provdes enough mode nformaton, there s a lttle mprovement n model probablty estmaton. In the case that the mage-based observaton s of somewhat lower qualty, there are obvous mprovements n stable model probablty and state estmaton as shown n fgure 2. It s due to that the addtonal mode observaton provdes comparatvely less nformaton. Thereby the usage of smoothng brngs much. Comparng wth IMM-FLO, n the case that the magebased observaton s of good qualty, IE-IMM-FLO s much better n model probablty estmaton and state estmaton as shown n fgure. The reason s that IE- IMM-FLO has an addtonal mode observaton channel whch provdes mode nformaton. IE-IMM-FLO can detect the swtch of target s operatonal mode much more qucly through usng addtonal mode observaton so that the models used match the operatonal modes more tmely. As a result, the state estmaton of each model-condtonal flter s better. Consequently the pea error of state s reduced and the state estmaton s mproved smultaneously. In the case that the mage-based observaton s of somewhat lower qualty, there s a lttle mprovement as shown n fgure 2. The reason s that not much nformaton s provded. In the terms of computaton cost, the average tme IE- IMM-FLO, IMM-FLO, IE-IMM and IMM need s.0373s,.037s, s and s, respectvely. IE- IMM-FLO has an addtonal mode observaton channel and completes smoothng through augmentng state le IMM-FLO. However, only a lttle computaton cost s ncreased. 5 Concluson In ths paper we have consdered the estmaton problem for MJS when mode-based observaton s obtaned onestep fx-lagged to state measurement. Along stateaugmentaton approach and the dervaton of mageenhanced nteractng multple model (IE-IMM, a new generc estmaton algorthm s proposed. The performance of new algorthm whch uses mage sensor as the mode observer (IE-IMM-FLO was compared numercally wth other approaches ncludng IMM, IMM- FLO and IE-IMM. The numercal results showed that proposed algorthm always gave good performance mprovement over other three algorthms wth only a lttle ncremental computaton cost. The mprovement s obvous especally when the mode observatons are of good qualty. Ths s sgnfcant, whch decdes the proposed algorthm wll have a wde applcaton area n realty. Acnowledgments Ths paper s supported by NSFC (No References [] X.R. L, V.P. Jlov, Survey of maneuverng target tracng. Part V: Multple-model methods, IEEE Transactons on Aerospace and Electronc Systems, Vol. 4, No. 4, pp , October [2] B. Chen, J.K.Tugnat, Interactng multple model fxed-lag smoothng algorthm for Marovan swtchng systems, IEEE Trans on Aerospace and Electronc systems, Vol. 36, No., , 2000.

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