Globally Optimal Multisensor Distributed Random Parameter Matrices Kalman Filtering Fusion with Applications

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1 Sensors 2008, 8, ; DOI: /s OPEN ACCESS sensors ISSN Artce Gobay Optma Mutsensor Dstrbuted Random Parameter Matrces Kaman Fterng Fuson wth Appcatons Yngtng Luo, Yunmn Zhu *, Dandan Luo, Je Zhou, Enbn Song and Donghua Wang Department of Mathematcs, Schuan Unversty, Chengdu, Schuan, , P. R. Chna; E-Ma: * Author to whom correspondence shoud be addressed; E-Ma: Receved: 28 August 2008; n revsed form: 26 November 2008 / Accepted: 3 December 2008 / Pubshed: 8 December 2008 Abstract: hs paper proposes a new dstrbuted Kaman fterng fuson wth random state transton and measurement matrces,.e., random parameter matrces Kaman fterng. It s proved that under a md condton the fused state estmate s equvaent to the centrazed Kaman fterng usng a sensor measurements; therefore, t acheves the best performance. More mportanty, ths resut can be apped to Kaman fterng wth uncertan observatons ncudng the measurement wth a fase aarm probabty as a speca case, as we as, randomy varant dynamc systems wth mutpe modes. Numerca exampes are gven whch support our anayss and show sgnfcant performance oss of gnorng the randomness of the parameter matrces. Keywords: Random parameters matrces; Kaman fterng; Centrazed fuson; Dstrbuted fuson. 1. Introducton Lnear dscrete tme system wth random state transton and observaton matrces arse n many areas such as radar contro, msse trac estmaton, satete navgaton, dgta contro of chemca processes, economc systems. Konng [1] gave the Lnear Mnmum Varance recursve estmaton formuae for the near dscrete tme dynamc system wth random state transt and measurement

2 Sensors 2008, matrces wthout detaed rgorous dervaton. Such system can be converted to a near dynamc system wth determnstc parameter matrces and state-dependent process and measurement noses. herefore, the condtons of standard Kaman Fterng are voated and the recursve formuae n [1] can not be derved drecty from the Kaman Fterng heory. In ths paper, a rgorous anayss (many n the appendx) shows that under md condtons, the converted system st satsfes the condtons of standard Kaman Fterng; therefore, the recursve state estmaton of ths system s st of the form of a modfed Kaman fterng. Reference [5] shows that ths resut can be apped to Kaman fterng wth uncertan observatons, as we as randomy varant dynamc systems wth mutpe modes. Many advanced systems now mae use of arge number of sensors n practca appcatons rangng from aerospace and defense, robotcs and automaton systems, to the montorng and contro of a process generaton pants. An mportant practca probem n the above systems s to fnd an optma state estmator gven the observatons. When the processng center can receve a measurements from the oca sensors n tme, centrazed Kaman fterng can be carred out, and the resutng state estmates are optma n the Mean Square Error (MSE) sense. Unfortunatey, due to mted communcaton bandwdth, or to ncrease survvabty of the system n a poor envronment, such as a war stuaton, every oca sensor has to carry on Kaman fterng upon ts own observatons frst for oca requrement, and then transmts the processed data-oca state estmate to a fuson center. herefore, the fuson center now needs to fuse a receved oca estmates to yed a gobay optma state estmate. Under some reguarty condtons, n partcuar, the assumpton of ndependent sensor noses, an optma Kaman fterng fuson was proposed n [11-12], whch was proved to be equvaent to the centrazed Kaman fterng usng a sensor measurements; therefore, such fuson s gobay optma. hen, Song [7] proved that under a md condton the fused state estmate s equvaent to the centrazed Kaman fterng usng a sensor measurements. In the mutsensor random parameter matrces case, sometmes, even f the orgna sensor noses are mutuay ndependent, the sensor noses of the converted system are st cross-correated. Hence, such mutsensor system seems not satsfyng the condtons for the dstrbuted Kaman fterng fuson gven n [11-12]. In ths paper, t was proved that when the sensor noses or the random measurement matrces of the orgna system are correated across sensors, the sensor noses of the converted system are cross-correated. Even f so, smary wth [7], centrazed random parameter matrces Kaman fterng, where the fuson center can receve a sensor measurements, can st be expressed by a near combnaton of the oca estmates. herefore, the performance of the dstrbuted fterng fuson s the same as that of the centrazed fuson under the assumpton that the expectatons of a sensor measurement matrces are of fu row ran. Numerca exampes are gven whch support our anayss and show sgnfcant performance oss of gnorng the randomness of the parameter matrces. he remander of ths paper s organzed as foows. In Secton 2, we present the concept of random parameter matrces Kaman fterng. In Secton 3, we present an optma Kaman fterng fuson wth random parameter matrces and show that under a md condton the fused state estmate s equvaent to the centrazed Kaman fterng wth a sensor measurements. In Secton 4, we show that the resut can be apped to Kaman fterng wth uncertan observatons as we as randomy varant dynamc systems wth mutpe modes. More mportanty, we w see that the Kaman fterng wth fase aarm

3 Sensors 2008, probabty s a speca case of Kaman wth random parameter matrces. A smuaton exampe s gven n Secton 5. And fnay, n Secton 6, we present our concusons. 2. Random Parameter Matrces Kaman Fterng Consder a dscrete tme dynamc system: x 1 Fx (1) y Hx, 0,1, 2,... (2) r N r where x R s the system state, y R s the measurement matrx, R s the process nose, N r r and R s the measurement nose. he subscrpt s the tme ndex. F R N r and H R are random matrces. We assume the system has the foowng statstca propertes: { F, H,,, 0,1, 2,...} are a sequences of ndependent random varabes temporay and across sequences as we as ndependent of x 0. Moreover, we assume x and { F, H, 0,1,2,...} are mutuay ndependent. he nta state x 0, the noses, and the parameter matrces F, H have the foowng means and covarance. Where fj and Rewrtng F and , E x E x x P, (3) E E R E E R (4) 0,, 0, EF F, Cov fj, fmn C fj fmn EH H, Covh, h C (5) (6) j mn hh j mn hj are the (, j) th entres of matrces H as: F and H, respectvey. F F F (7) H H H (8) And substtutng (7), (8) nto (1), (2) converts the orgna system to: x 1 Fx (9) y Hx (10) where Fx Hx (11) System (9), (10) has determnstc parameter matrces, but the process nose and observaton nose are dependent on the state. herefore, ths woud not satsfy the we-nown assumptons of the standard Kaman fterng apparenty. In the appendx, we gve a detaed proof that system (9) and (10) satsfy a condtons of the standard Kaman fterng and derve the recursve state estmate of the new system as foows:

4 Sensors 2008, heorem 1. he Lnear Mnmum Varance recursve state estmaton of system (9), (10) s gven by: x x K y H x x K 1ㄧ 1 1ㄧ F x ㄧ 1 1 ㄧ,, 1ㄧ 1 1 K1 1ㄧ ㄧ P H H P H R +1 1ㄧ 1 1 1ㄧ 1 1 ㄧ P F P F R P I K H P R R E F E x x F R R E H E x x H E x x F E x x F E FE xx F R x Ex P Var x E x x Ex Ex P where the superscrpt "+" denotes Moore-Penrose pseudo nverse, x 1 denotes the one-step predcton of x 1, P 1 denotes the covarance of x 1, x 1 1 denotes the update of x 1 and P 1 1 denotes the covarance of x 1 1. Compared wth the standard Kaman fterng and notng the notatons n (5), (6), the random parameter matrces Kaman fterng has one more recurson of E x 1x 1 as foows: E x x FE xx F E F E xx F R where and where X j s the (, ) 1 1 r 1 r 1... (, ) fn f r mn m fnr fm 1 1 E F E x x F C X C X r 1 r 1... (, ) hn h r mn m hnrhm 1 1 X Exx. E H E x x H C X C X j th entres of ( ) 3. Random Parameter Matrces Kaman Fterng wth Mutsensor Fuson In ths secton, a new dstrbuted Kaman fterng fuson wth random parameter matrces s proposed. he framewor of the dstrbuted tracng system s the same as those consdered n [12-15]. he advantages of transmttng sensor estmates other than sensor measurements can be seen n [12-15]. We w show that under a md condton the fused state estmate s equvaent to the centrazed Kaman fterng usng a sensor measurements. herefore, t acheves the best performance. he -sensor dynamc system s gven by: x 1 Fx, 0,1,... (12) y Hx, 1,..., where x r R s the state, y N r R s the measurement matrx n -th sensor, R s the process N nose, and R s the measurement nose. Parameter matrces F and H are random. We assume

5 Sensors 2008, j that { F, H,,, 0,1,2,...},, j 1,..., s a sequence of ndependent varabes. Every snge sensor satsfes the assumpton n the ast secton. Convert system (12) to the foowng one wth determnstc parameter matrces: x 1 Fx, 0,1... (13) y Hx, 1,..., where Fx Hx he staced measurement equaton s wrtten as: y Hx where 1,...,, 1,..., 1,,..., y y y H H H and the covarance of the nose s gven by: Var Consder the covarance of the measurement nose of snge sensor n new system. By the assumpton above, we have: j j j E E H x H x R j j E EH Exx H E x H H x H x x H j j j As shown n the ast part of Secton 2, every entry of the ast matrx term of the above equaton s a near combnaton of E( hh ). Hence, when H and H are correated, n genera, H 0 j. herefore, even f 0 s j E j HE xx E,.e., the orgna sensor noses are mutuay ndependent, the sensor noses of the converted system are st cross-correated,.e., R s nondagona boc matrx. Lucy, when sensor noses are cross-correated, n [7], t was proven that under a md condton the fuse state estmate s equvaent to the centrazed Kaman fterng usng a sensor measurments. Accordng to heorem 1 and the Kaman fterng formuae gven n [8-10], the oca Kaman fterng at the -th sensor s: x ㄧ x ㄧ1Ky Hx ㄧ1 I KH x ㄧ1Ky 1 (14) K P H R ㄧ where R Var wth covarance of fterng error gven by: P ㄧ I KH P ㄧ 1 or P P H R H (15) j

6 Sensors 2008, where x F x ㄧ1 1ㄧ 1 P E x x x x ㄧ ㄧ ㄧ ㄧ1 ㄧ1 ㄧ1 P E x x x x We assume that the system has the foowng propertes: the row dmensons of a sensor measurement matrces H to be ess than or equa to the dmenson of the state, and a H to be of fu row ran. In many practca appcatons, ths assumpton s fufed very often. hus, we now H( H) I. Accordng to [7] and heorem 1, the centrazed Kaman fterng wth a sensor data s gven by: x x K y H x where, R 1 * or where s the -th coumn boc of ㄧ ㄧ1 ㄧ1 I K H x K y ㄧ ㄧ1 ㄧ * I KH x P H R y K 1 1 R ㄧ ㄧ P H * 1 R R H H R H ㄧ 1 P P H R H x (16) P H R ㄧ (17). he covarance of fterng error gven by: 1 P I KH P (18) ㄧㄧ (19) F x ㄧ1 1ㄧ 1 P E x x x x ㄧ ㄧ ㄧ P E x x x x ㄧ1 ㄧ1 ㄧ1 Usng (15) and (18), the estmaton error covarance of the centrazed Kaman fterng s gven by usng the estmaton error covarance of a oca fters: P P H * 1 R R H P P ㄧㄧ ㄧ ㄧ 1 (20) Usng (17): 1 1 ㄧ * PHR * R H H R ㄧ y 1 K y P H R y o express the centrazed fterng 1 1 H R y P K y ㄧ 1 P x I KHx 1 1 P x P x x n terms of the oca fterng, by (14) and (15), we have ㄧ ㄧ ㄧ1 ㄧ ㄧ ㄧ1 ㄧ1 (21) (22)

7 Sensors 2008, hus, substtutng (19), (21) and (22) nto (16) yeds: P x P x H R * R H P x P x ㄧ ㄧ ㄧ1 ㄧ1 ㄧ ㄧ ㄧ1 ㄧ 1 (23) 1 hat s to say that the centrazed fterng (23) and error matrx (20) are expcty expressed n terms of the oca fterng. Hence, the performance of the dstrbuted random parameter matrces Kaman fterng fuson s the same as that of the centrazed random parameter matrces fuson. Remar 1. In ths paper t s assumed that a sensor observatons are synchronous. In practce, ths may be very rarey true. However, n the past 20 years, such assumpton was used very often n the trac fuson communty (for exampe, see [7, 11-13] among others). One of reasons for ths s that t s easy to extend the resuts for synchronous dstrbuted Kaman fterng fuson to the correspondng asynchronous case by ettng x ㄧ = x ㄧ and P 1 ㄧ = P ㄧ when the th sensor does not receve ts 1 observaton y at tme to reduce the asynchronous dstrbuted tracng system to be synchronous. Remar 2.he dstrbuted systems here and n [7, 11-15] have a fuson center and the oca sensor shoud transmt x ㄧ, and to the fuson center. Our purpose n ths paper s to derve a x ㄧ, P 1 ㄧ P ㄧ 1 gobay optma dstrbuted fuson agorthm equvaent to the centrazed Kaman fterng as the fuson center can receve a sensor observatons. In ths framewor, the system has not ony the goba optmaty, but aso the good survvabty n a poor stuaton. Ceary, such systems are dfferent from the system consdered [16]. he system n [16] does not requre any form of centra processng facty, and the agorthm there s hghy resent to oss of one or more sensng nodes, but costs more communcaton and has no rea-tme goba optmaty. hus, both of them have ther own advantages and dsadvantages. he Random Kaman fterng n the framewor of [16] may be another future research drecton. 4. Appcatons of Random Parameter Matrces Kaman Fterng In ths secton, we w see that the resuts n the ast two sectons can be apped to the Kaman fterng wth uncertan observatons as we as randomy varant dynamc systems wth mutpe modes Appcaton to a Genera Uncertan Observaton he Kaman fterng wth uncertan observaton attracted extensve attenton [2-4]. here are two types of uncertan observatons n practce. he frst one s that the estmator can exacty now whether the observaton fuy or partay contans the sgna to be estmated, or just contans nose aone (for exampe, see [2]). By drecty usng the optma estmaton theory, the Kaman fter for the frst type of uncertan observatons can be derved easy. he other uncertan observatons beong to the second type,.e., the estmator cannot now whether the observaton fuy or partay contans the sgna to be estmated or just contans nose aone, but the occurrence probabtes of each case are nown. Ceary, the atter s more practca. By appyng the random measurement matrx Kaman fterng, we

8 Sensors 2008, can derve the Kaman fter wth the second type of uncertan observatons, whch s much more genera than that n [2-4]. Consder a system: x 1 Fx (24) (25) y I H x I 1 1 where a the parameter matrces are non-random and a set of mutpe observaton equatons s seected to represent the possbe observaton case at each tme. he random varabe s ether observabe or unobservabe. If, the measure matrx s H and the observaton nose corresponds to. When the vaue of s observabe at each tme, ths s an uncertan observaton of the frst type and the state estmaton wth measurement equaton (25) s converted to: y Hx (26) whch s obvousy the cassca Kaman fterng,.e., the east mean square estmate usng the varous avaabe observaton of y. o show the appcatons of the random measurement matrx Kaman fterng, we focus on the second type of uncertan observaton,.e., n (25), s unobservabe at each tme, but the probabty of occurrence of every avaabe measurement matrx s nown. Consder that n (25), s unobservabe at each tme, but the probabty of the occurrence of each measurement s nown. Obvousy, (2) s a more genera form of (25) because ony expectaton and covarance of H n (2) are nown other than ts dstrbuton. he expectaton of H can be expressed as: H j j ph j1 (27) H H H, wth probabty p (28) A that remans n order to appy the random measurement matrx Kaman fterng s just to cacuate: R R E H E xx H R p H H E xx H H (29) Substtutng (27}) and (29) nto heorem 1 can mmedatey obtan the random measurement matrx Kaman fterng of mode (1) and (25). In the cassca Kaman fterng probem, the observaton s aways assumed to contan the sgna to be estmated. However, n practce, when the exteror nterference s strong,.e., tota covarance of the measurement nose s arge; the estmator w mstae the nose as the observaton sometmes. In radar termnoogy, ths s caed a fase aarm. Usuay, the estmator cannot now whether ths happens or not, ony the probabty of a fase aarm s nown. In the foowng, we w show that the Kaman Fterng wth a fase aarm probabty s a speca case of the uncertan observatons of the above mode (1), (25) are gven. Consder a dscrete dynamc process: x 1 Fx (30) y hx, 0,1,2,... (31) 1

9 Sensors 2008, where { F, h,,, 0,1,2,...} satsfy the assumptons of standard Kaman fterng. F and h are determnstc matrces. he fase aarm probabty of the observaton s1 p. hen,we can rewrte the measurement equatons as foows: y Hx, 0,1,2,... (32) where the observaton matrx H s a bnary-vaued random wth: PrH h p (33) PrH 0 1 p (34) Due to (8): H ph (35) PrH 1 p h p (36) Pr H 1 ph p (37) In the fase aarm case, the state transton matrx s st determnstc, but the measurement matrx s random, by (35), (36) and (37), the covarance of the process and observaton noses can be wrtten as foows: R R (38) R R 1 E HE xx H R p phe xx h (39) hus, the Kaman fterng wth fase aarm probabty n ths case s gven by: x 1ㄧ1 x 1ㄧ K 1 y 1 p 1h1x1ㄧ x F x 1ㄧ ㄧ ㄧ ㄧ 1 1 ㄧ ㄧ 1 1 1,, K p P h p h P h R P F P F R P I p K h P R R p p h E x x h E x x F E x x F R x Ex P Var x E x x Ex Ex P 00 ㄧ In ths secton, we consder the appcaton to a genera uncertan observaton for one sensor case. In a manner anaogous to the dervaton of Secton 4.1, we can aso gve an appcaton to a genera uncertan observaton for mutsensor case usng Secton 3. he procedure s omtted here Appcaton to a Mut-Mode Dynamc Process he mutpe-mode (MM) dynamc process has been consdered by many researchers. Athough the possbe modes consdered n those papers are qute genera and can depend on the state, but no optma agorthm n the mean square error (MSE) sense was proposed n the past a few decades. On the other hand, when some of the MM systems satsfy the assumptons n ths paper, they can be reduced to dynamc modes wth random transton matrx and thus the optma rea-tme fter can be gven drecty accordng to the random transton matrx Kaman fterng proposed n heorem 1.

10 Sensors 2008, Consder a system: x 1 Fx wth probabty p, 1,..., (40) y Hx (41) where F and are ndependent sequence, and H s non-random. We use random matrx F to stand for the state transton matrx. he expectaton of F can be expressed as: F j j pf j1 (42) F F F,wth probabty p (43) A necessary step for mpementng the random Kaman fterng s to cacuate: R R E FE xx F R p F F E xx F F (44) hus, a the recursve formuas of random Kaman fterng can be gven by: x 1ㄧ1 x 1ㄧ K 1y1H1x1ㄧ x F x 1ㄧ ㄧ 1 1ㄧ 1 1 1ㄧ 1 1 ㄧ K P H H P H R P F P F R P I K H P ㄧ ㄧ 0, 0 0 0, R R p F F E x x F F E x x F E x x F R p x Ex P Var x E x x Ex Ex P F F Ex x F F 5. Numerca Exampe In ths secton, three smuatons w be done for a dynamc system wth random parameter matrces modeed as an object movement wth process nose and measurement nose on the pane. he smuatons gve the speca appcatons of resuts n the ast secton and show that fused random parameter matrces Kaman fterng agorthms can trac the object satsfactory. Remember that we have rgorousy proved n Secton 3 that the centrazed agorthm usng a sensor observatons y at the fuson center can be equvaenty converted to be dstrbuted agorthm usng a sensor estmates x. In addton, the computer smuatons we have done show that the smuaton resuts of two agorthms are exacty the same. It turns out that n the foowng numerca exampes, we ony compare the dstrbuted random Kaman fterng and the correspondng standard Kaman fterng that gnores the randomness of the parameter matrces. Wthout oss of generaty our exampes assume the oca sensors send updates each tme when they receve a measurement. Frsty, we consder a three-sensor dstrbuted Kaman fterng fuson probem wth fase aarm probabtes.

11 Sensors 2008, Exampe 1. he object dynamcs and measurement equatons are modeed as foows: x 1 Fx y Hx, 1,2,3 where { F, H,,, 0,1,2,...} satsfy the assumptons of standard Kaman fterng. he state transton matrx F cos2 / 300 sn 2 / 300 F sn 2 / 300 cos2 / 300 s a constant. he measurement matrx s gven by: H, 1,2, H he fase aarm probabty of the -th sensor s gven by: p 0.01,1 p 0.02,1 p he nta state x 0 (50,0), P0 0 I. he covarance of the noses are dagona, gven by R 1, R 2, 1,2,3.. Usng a Monte-Caro method of 50 runs, we can evauate tracng performance of an agorthm by estmatng the second moment of the tracng error, gven by: E x x 50 ㄧ 1 Fgure 1 shows that the second moments of tracng error for three sensors Kaman fterng fuson wthout consderng the fase aarm (.e. standard Kaman fterng) and three sensors random Kaman fterng fuson consderng the fase aarm (.e. random Kaman fterng), respectvey. It can be shown that even f the fase aarm probabty s very sma, the dstrbuted Random Kaman fterng fuson performs much better than the standard Kaman fterng. Fgure 1. Comparson of standard Kaman fterng fuson and random Kaman fterng fuson.

12 Sensors 2008, In Exampe 1, both the sensor noses and the random measure matrces of the orgna system are mutuay ndependent, so the sensor nose of the converted system are mutuay ndependent. Now, we consder another exampe that both the noses and the random measure matrces of the orgna system are cross-correated. Exampe 2. he object dynamcs and measurement equatons are modeed as foows: x 1 Fx y Hx, wth probabty of p, wth probabty of 1 p where 1, 2,3, { F, H,,,, 0,1,2,...} satsfy the assumptons of standard Kaman fterng. he state transton matrx F and the measurement matrces H are the same as Exampe 1 and s a arge transton nose. When t happens, the sensors w mstae the transton nose as the observaton. he fase aarm probabty of the transton nose s gven by1 p hough the sensor noses, 1,2,3 are mutuay ndependent and ndependent of, but the tota measurement noses are cross-correated here. he covarance of the noses are dagona, gven by R 1, R 2 R 0.5, 1, 2, 3. he nta state x0 (50,0), P0 0 I. In ths exampe, both the measurement noses and the random measure matrces of the orgna system are cross-correated. Hence, the sensor noses of the converted system are cross-correated. Fgure 2 shows that the random Kaman fterng fuson gven n Secton 3 st wors better than the standard Kaman fterng wthout consderng the fase aarm. hs mpes that the standard Kaman fterng ncorrecty assumes that sensor noses are ndependent. Fgure 2. Comparson of standard Kaman fterng fuson and random Kaman fterng fuson. Exampe 3. In ths smuaton, there are three dynamc modes wth the correspondng probabtes of occurrence avaabe. he object dynamcs and measurement matrx n (40) are gven by:

13 Sensors 2008, cos 2 / 300 sn 2 / F wth probabty 0.1 sn 2 / 300 cos 2 / 300 cos 2 / 250 sn 2 / F wth probabty 0.2 sn 2 / 250 cos 2 / 250 cos 2 /100 sn 2 / 50 3 F wth probabty 0.7 sn 2 / 50 cos 2 / H 1 1 he covarance of the noses are dagona, gven by R 2, R 1. In the foowng, we compare our numerca resuts wth the IMM. Snce n ths exampe, the occurrence probabty of each mode at every tme s nown and mutuay ndependent, t s aso the transton probabty n the IMM. herefore, the transton probabty matrx at each tme n the IMM s fxed and gven by: (, j) here means the transton probabty of mode to mode j. hs assumpton aso mpes that the mode probabty n the IMM s fxed as foows: , 0.2, 0.7, 1, 2... Fgure 3 shows that the random Kaman fterng gven n secton 4.2 st wors better than the IMM wth the fxed transton probabty and mode probabty. hs maes sense snce the former s optma n the MSE sense but the atter s not. However,n practce, the occurrence probabty of each mode s very often dependent on state or observaton, and therefore not ndependent of each other n tme. In ths case, our new method offers no advantage. Fgure 3. Comparson of IMM and random Kaman fterng.

14 Sensors 2008, Concusons In the mutsensor random parameter matrces case, t was proven n ths paper that when the sensor noses, or the measurement matrces of the orgna system are correated across sensors, the sensor noses of the converted system are cross-correated. Hence, such mutsensor system seems not to satsfy the condtons for the standard dstrbuted Kaman fterng fuson. hs paper propose a new dstrbuted Kaman fterng fuson wth random parameter matrces Kaman fterng and proves that under a md condton the fused state estmate s equvaent to the centrazed Kaman fterng usng a sensor measurements, therefore, t acheves the best performance. More mportanty, ths resut can be apped to Kaman fterng wth uncertan observatons as we as randomy varant dynamc systems wth mutpe modes. he Kaman fterng wth fase aarm s a speca case of Kaman fterng wth uncertan observatons. Numerca exampes are gven whch support our anayss and show sgnfcant performance oss of gnorng the randomness of the parameter matrces. Appendx In the Appendx, we w gve the proof of heorem 1. Frsty, we gve a detaed proof that system (9) and (10) satsfy a condtons of the standard Kaman fterng as foows: Lemma 1. Suppose random matrx F and random vector x are ndependent, then: E Fxx F E FE xx F Proof. By the propertes of condtona expectaton, we have that: E EFxx F F EFExx F F E FExx F E Fxx F Lemma 2. ae 0, E 0; b1ex0v 0, b2ex 0 0 c1e 0, c2e 0, c3e 0, de Rv, E, R where R R, E F E xx F R R EH. E xx H Proof. (a): By the assumptons on the mode (1) and (2), and notatons n (11), t s obvous. (b1): Snce F,, 0,1, 2,... s ndependent of x 0, Q.E.D.

15 Sensors 2008, Ex0xF 0. E x. (b2): Smary, E x E x F x E x F x F E x x F F E x F x F F E x x F F F E x x F F F F F E x x E F E F E F E F E F 0 0 (c1): Wthout oss of generaty, we consder the case of ony. E E x F Fx Fxx F For x s neary dependent on F,, 0,1,2,... s ndependent of x 0,... F F... FF x,, F F... F, 2,3,...,, and E x F EF F1x1 EF F1F2x2 2 EF F1F2x2 EF F12 E F x E F F x E F F Fx E F F F E F E F E F E 0. Notng that x and F, 0,1, 2,... are ndependent, by Lemma 1 we have: E F x x F E F F x x F EF F 1x 1x F E F F F x x F F F x F F 1F 2... F 1Fx x F E F F F F Fx x F E F F F F x F E F F F F Fx x F E F 0. E E F F F F FE x x F

16 Sensors 2008, Hence, E 0 (c2): Aso consder the case of ony. Snce x s neary dependent on F 1F 2... FF 1 0x0,, 1 F 1F 2... F 1, 2,3,...,, F, H,,, 0,1, 2,... s ndependent of x 0, and F, H, 0,1, 2,... s ndependent of x.moreover: E E xh Hx HxxH EH xx H EH F1x1 1x H EH F1x1x H E H F F x x H H F x H (c3): When when,, EH F 1F 2... F 1Fx x H E H F1F2... F 1 x H 0. E E x H F x F x x H. E EF xx H EF F1x1 1x H EF F 1x 1x H EF F 1F 2x 2x H EF F1 2x H... E F F F... F Fx x H F F F... F x H E x 1F 1H Ex 2F 2H E2F1H E x H E F x H 0, E x F F F H E F F H

17 Sensors 2008, hus, E 0. (d): EF xx1f1h E Fxx H E Fx F x H E F x F x F H E F x F x F F H E F x x F F 1... F 1H E F E x x F E F F H E E xf Fx Fxx F R EF xxf R EF E xx F Smary, E R E H Exx H. Q.E.D. By Lemma 2, system (9), (10) satsfes a condtons of the standard Kaman Fterng. Hence, we derve the recursve state estmate of the new system,.e., heorem 1 mmedatey. Acnowedgements Supported n part by NSF of Chna (# , ) and Project 863 through grant 2006AA12A104. References and Notes 1. DeKonng, W.L. Optma Estmaton of Lnear Dscrete-tme Systems wth Stochastc Parameters. Automatca 1984, 20, Hah, N.E. Optma Recursve Estmaton Wth Uncertan Observaton. IEEE rans. Inf. heory 1969, 15, Snopo, B.; Schenato, L.; Franceschett, M.; Pooa, K.; Jordan, M.I.; Sastry, S.S. Kaman Fterng wth Intermttent Observatons. IEEE rans. Autom. Contro 2004, 49, Lu, X.H.; Godsmth, A. Kaman Fterng wth Parta Observaton Losses. In Proceedngs of 43rd IEEE Conference on Decson and Contro, Atants, Paradse Isand, Bahamas, Dec.14-17, 2004; pp Luo, D.D; Zhu, Y.M. Appcatons of Random Parameter Matrces Kaman Fterng n Uncertan Observaton and Mut-Mode Systems. In Proceedngs of the Internatona Federaton of the Automatc Contro, Seou, Korea, Juy 6-11, 2008.

18 Sensors 2008, Zhu, Y.M. Mutsensor Decson and Estmaton Fuson; Kuwer Academc Pubshers: Boston, USA, Song, E.B.; Zhu, Y.M.; Zhou, J.; You, Z.S. Optmaty Kaman Fterng fuson wth crosscorreated sensor noses. Automatca 2007, 43, Goodwn, G.C.; Payne, R. Dynamc System Identfcaton: Expermenta Desgn and Data Anayss; Academc Press: New Yor, USA, Hayn, S. Adaptve Fter heory; Prentce-Ha: Engewood Cffs, NJ, Ljung, L. System Identfcaton: heory for the User; Prentce-Ha: Engewood Cffs, NJ, Chong, C.Y.; Chang, K.C.; Mor, S. Dstrbuted racng n Dstrbuted Sensor Networs. In Proceedngs of 1986 Amercan Contro Conf., Seatte, WA, June Hashmpour, H.R.; Roy, S.; Laub, A.J. Decentrazed Structures for Parae Kaman Fterng, IEEE rans. Autom. Contro 1988, 33, Bar-Shaom, Y. Muttarget-mutsensor tracng: Advanced appcatons; Artech House: Norwood, MA, 1990; Voume Bar-Shaom, Y.; L, X. R. Muttarget-Mutsensor racng: Prncpes and echnques; YBS Pubshng: Storrs, C, Lggns, M.E.; Chong, C.-Y.; Kadar, I.; Aford, M.G.; Vanncoa, V.; homopouos, S. Dstrbuted Fuson Archtectures and Agorthms for arget racng. Proc. IEEE 1997, Rao.B.S.; Durrant-Whyte H.F. Fuy decentrazed agorthm for mutsensor Kaman fterng, In IEE Proceedngs on Contro heory and Appcatons, Sep. 1991; Part D, vo. 138, pp by the authors; censee Moecuar Dversty Preservaton Internatona, Base, Swtzerand. hs artce s an open-access artce dstrbuted under the terms and condtons of the Creatve Commons Attrbuton cense (