Optimal Distributed Estimation Fusion with Transformed Data

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1 Optimal Distribute Estimation Fusion with Transforme Data Zhansheng Duan X. Rong Li Department of Eletrial Engineering University of New Orleans New Orleans LA U.S.A. Abstrat Most of the existing istribute estimation fusion algorithms rely on the existene of the inverses of the orresponing error ovariane matries e.g. istribute estimation fusion algorithms base on the information form of the Kalman filter an the optimal weighte least-square WLS estimator. Theoretially speaing the error ovariane matries are only at least positive semi-efinite an not neessarily invertible. To overome this by taing a linear transformation of the raw measurements reeive by eah loal sensor an optimal istribute estimation fusion sheme is propose in this paper. Compare with the existing istribute estimation fusion shemes the new algorithm is not only optimal in the sense that it is equivalent to the entralize fusion the ommuniation requirements from eah sensor to the fusion enter are equal to or less than most of the existing istribute fusion algorithms. One possible way to relieve the omputational omplexity of the new algorithm is also isusse. Keywors: Estimation fusion istribute fusion entralize fusion reursive estimation linear minimum meansquare error LMMSE. I. INTRODUCTION Estimation fusion or ata fusion for estimation is the problem of how to best utilize useful information ontaine in multiple sets of ata for the purpose of estimating an unnown quantity a parameter or proess at a time 1. There are two basi estimation fusion arhitetures: entralize an eentralize/istribute also referre to as measurement fusion an tra fusion in target traing respetively epening on whether the raw measurements are sent to the fusion enter or not. In entralize fusion all raw measurements are sent to the fusion enter while in istribute fusion eah sensor only sens in the proesse ata. Centralize fusion espite of its heavy omputational buren at the fusion enter an poor survivability an provie globally optimal fuse estimates provie the proessing apability of the proessor at the fusion enter an ommuniation banwith an reliability an satisfy the requirements. With a reue omputational buren at the fusion enter an reue ommuniation emans for the sensor networs istribute fusion usually has faster realtime proessing an stronger fault-tolerane abilities. Usually istribute fusion is more hallenging in terms of performane Researh supporte in part by NSFC grant Projet 863 through grant 2006AA01Z126 an Navy through Planning Systems Contrat # N C The authors are also with the College of Eletroni an Information Engineering Xi an Jiaotong University. hannel apaity reliability survivability information sharing et. an has been a foal point of most fusion researh. The topi of istribute estimation fusion has been researhe for several eaes ue to its nie properties mentione above an there are a lot of results available. Two lasses of optimality riteria were isusse most in the existing istribute estimation fusion algorithms. The first lass tries to reonstrut the entralize fuse estimate from the loally proesse ata e.g. loal estimates. That is the optimality riterion use by the first lass is the equivalene to the entralize estimation fusion. The seon lass is usually optimal onitione on the loally proesse ata. For example 2 an 3 propose a two-sensor tra-to-tra fusion algorithm whih is optimal in the sense of maximum lielihoo ML for the Gaussian ase. For more than two sensors 4 5 propose a tra-to-tra fusion algorithm whih is optimal in the sense of ML for the Gaussian ase an WLS. 6 7 propose a eentralize struture to reonstrut the optimal global estimate when the measurement noises aross sensors are unorrelate. 8 propose a eentralize struture to reonstrut the optimal global estimate when the measurement noises aross sensors are orrelate. 1 propose unifie fusion rules in the sense of best linear unbiase estimate BLUE an WLS for all fusion arhitetures with arbitrary orrelation of loal estimates or observation errors aross sensors or aross time whih inlues the istribute estimation fusion as a speial lass. 9 propose a state estimation fusion algorithm whih is optimal in the sense of maximum a posteriori MAP. Most of the existing istribute estimation fusion algorithms e.g. istribute estimation fusion algorithms base on the information form of the Kalman filter an the optimal WLS estimator rely on the existene of the orresponing error ovariane matries. The error ovariane matries are at least positive semi-efinite but not neessarily invertible. This means that in these ases we an not always use the existing istribute estimation algorithms iretly. One simplest way to solve this problem is to simply replae the traitional inverse by some generalize inverses e.g. the Moore-Penrose inverse when the traitional inverse oes not exist. But in this way the optimality of the original istribute estimation fusion algorithm oes not neessarily hol. In this paper by taing a linear transformation of the 1291

2 original measurement of eah sensor an optimal istribute estimation fusion algorithm with the transforme ata is propose whih is atually equivalent to the entralize estimation fusion. As an be seen later all operations are taen in the estimatee quantity to be estimate spae so the ommuniation requirements from eah sensor to the fusion enter are equal to or less than most of the existing istribute estimation fusion algorithms. Reursive proessing of the transforme ata is also isusse whih is intene to relieve the inrease omputational omplexity of the newly propose istribute estimation fusion algorithm to a ertain egree. The paper is organize as follows. Se. II formulates the multiple sensor istribute estimation fusion problem for ynami systems. Se. III esribes a new linear transformation of the original sensor measurement. Se. IV presents the istribute estimation fusion algorithm with the transforme ata. Se. V analyzes the optimality of the propose istribute estimation fusion algorithm with the transforme ata. Se. VI provies one way to reue the omputational omplexity of the propose istribute estimation fusion algorithm with the transforme ata. Se. VII gives onluing remars. II. PROBLEM FORMULATION AND BACKGROUND Consier the following generi ynami system x F 1 x 1 + G 1 w 1 1 x R n E w 0 nw 1 ov w w j Q δ j Q 0 E x 0 x 0 ov x 0 P 0 ov x 0 w 0 n nw Assume that altogether M sensors are use to observe the system state at the same time ov ov z i i x + v i i 1 2 M 2 z i v i vi j w j v i R mi E R i δ j v i 0 nw m i ov 0 mi 1 x 0 v i 0 n mi Also it is assume that the measurement noises aross sensors are unorrelate an R i > 0 i 1 2 M. In istribute estimation fusion the fusion enter tries to get the best estimate of the system state with the proesse ata reeive from eah loal sensor. In this paper by istribute estimation fusion we mean only ata-proesse observations are available at the fusion enter not neessarily the loal estimates from eah sensor. Systems with only loal estimates available at the fusion enter referre to as the stanar istribute estimation fusion in 1 are not the fous of this paper. For the above esribe ynami system whih is observe by multiple sensors suppose that at time instant 1 the fuse estimate an loal estimates are ˆx i 1 1 P i 1 1 i 1 2 M an ˆx 1 1 P 1 1 respetively 6 firstly propose the following istribute estimation fusion algorithm whih is equivalent to the entralize estimation if there is no feeba from the fusion enter to eah loal sensor. 1 P P 1 i1 1 + M i1 1 1 P i P i P ˆx P 1 ˆx 1 M P i i ˆx P i i 1 ˆx 1 at the fusion enter ˆx 1 F 1ˆx 1 1 P 1 F 1P 1 1 F T 1 + G 1Q 1 G T 1 an at eah loal sensor i i 1 2 M P i ˆx i 1 F 1ˆx i 1 1 P i 1 F 1P i 1 1 F T 1 + G 1 Q 1 G T 1 P i 1 1 P i 1 + i 1 1 i ˆx P i i 1 ˆx 1 + T i R i T 1 i R i 1 z i As an be seen from 6 this optimal istribute estimation fusion algorithm ame from an ingenerious equivalent transformation to the original optimal entralize estimation fusioin algorithm in whih all the raw measurements are replae by the orresponing loal preite an upate estimates from eah loal sensor. Also an be seen is that to ensure the optimality of the above istribute estimation fusion algorithm it is require that all the assoiate inverses P 1 1 P 1 P i 1 1 P i 1 i 1 2 M exist. Sine P 1 P P i 1 P i i 1 2 M are all ovariane matries the only nowlege about them in a generi ynami system is that they are all at least positive semi-efinite if there are no further assumption about the system. This property surely an not guarantee that all the assoiate inverses 1 1 P i P 1 P i 1 2 M exist. In this paper we will answer the question about how to obtain the optimal istribute estimation fusion algorithm whih is equivalent to the optimal entralize estimation fusion if the above assoiate inverses o not neessarily exist. P i 1292

3 III. SENSOR MEASUREMENT TRANSFORMATION z i i T R i from Eq. 2 it follows that T 1 z i i R i i x + Furthermore let i v i i i T T R i R i 1 z i 3 i the above equation an be rewritten as E R i z i v i i T R i 1 v i 1 i 4 1 i v x + v i i 1 2 M 5 i ov v i T ov i i ov T R i R i v i vi j 1 E 1 R i v i R i i δ j 0 n 1 1 i ov v i vj l 0 n n i j 0 nw n ov x 0 v i 0 n n w j v i In istribute estimation fusion eah loal sensor sens the proesse ata z i n 1 an the orresponing measurement matrix i n n whih is also the ovariane matrix of the measurement noise v i to the fusion enter. In total the ommuniation requirements from eah loal sensor to the fusion enter is n + n 2 at any time instant. This is atually equal to or less than most of the existing istribute estimation fusion algorithms. an R i Remar: For a given multi-sensor ynami system if i i 1 2 M are time-varying then in entralize estimation fusion we also nee to transmit R i to the fusion enter besies z i an i. But for our new transforme measurement equation 5 although it has a form similar to the original measurement equation 2 there is no i nee to transmit R to the fusion enter at all even when it is i time-varying ue to its equivalene to. This will ertainly help save ommuniation requirements from eah loal sensor to the fusion enter an is one of the nie properties of our new istribute estimation fusion algorithm. Remar: We an learly see that the new transforme T 1 z i ata z i i R i an atually be alle informational state in the information form of the Kalman filter. That is originally we are hanling the estimation fusion problem in the measurement spae with imension m i i 1 2 M. After the linear transformation of Eq. 3 we onvert the estimation fusion problem into the estimatee spae. Remar: For eah loal sensor i i 1 2 M if 1 1 P i 1 an P i exist then it follows from the information form of the Kalman filter that 1 1 T 1 P i i ˆx P i i 1 ˆx 1 i R i i z This is exatly our iea of using the transformation in Eq. 3 omes from. z IV. DISTRIBUTED ESTIMATION FUSION WIT TRANSFORMED DATA T T z 1 z 2 1 v T v 1 T T 2 v 2 T T T z M T T M 6 v M T T E v 0Mn 1 R ov v { iag 1 2 M the stae measurement equation at the fusion enter w.r.t the M loal sensors an be written as z x + v Assuming that the istribute fuse estimate at time instant 1 is ˆx 1 1 with the orresponing error ovariane matrix P 1 1 then in the sense of LMMSE the optimal istribute fuse estimate of the system state at the fusion enter at time instant an be reursively ompute as follows LMMSE Distribute Fusion: 7 ˆx 1 F 1ˆx P 1 F 1P 1 1 F 1 T + G 1 Q 1 G T 1 9 ˆx ˆx 1 + K z ˆx 1 10 K P 1 T S + P P 1 P 1 T S + P 1 11 S P 1 T + R A + stans for the unique Moore-Penrose pseuoinverse MP inverse in short of matrix A. Remar: In general we have Ri T 1 i R i i 0 an R { iag 1 2 M 0 an this is the reason why we use MP inverse in the above. But for some speial i 1293

4 i ases we o have > 0 i 1 2 M e.g. when i are all full olumn ran. In this ase S + in the above will be replae by 1. S V. OPTIMALITY OF DISTRIBUTED ESTIMATION FUSION WIT TRANSFORMED DATA z T z 1 T 1 v T v 1 z 2 2 v 2 T T T E v 0 l 1 l M i1 z M M v M m i T T T T 12 T T { R ov v iag R 1 R2 RM 13 the stae measurement equation at the fusion enter w.r.t the M loal sensors an be written as z x + v Assuming that the entralize fuse system state estimate at time instant 1 is ˆx 1 1 with the orresponing error ovariane matrix P 1 1 then in the sense of LMMSE the optimal entralize fuse estimate of the system state at the fusion enter at time instant an be reursively ompute as follows: ˆx 1 F 1ˆx P 1 F 1P 1 1 F 1 T + G 1Q 1 G T 1 15 ˆx ˆx 1 + K z ˆx 1 16 K P 1 T S 1 P P 1 P 1 T S 1 P 1 17 S P 1 T + R For the given ynami system with multiple sensors the following theorems hol. Theorem 1 If P 1 1 P 1 1 then for the above LMMSE istribute estimation fusion we have T S + T S 1 Proof: Sine P 1 1 P 1 1 it follows from Eqs. 9 an 15 that P 1 P 1 { iag 1 2 M from Eqs. 7 an 4 it an be seen that R 2 M iag { 1 T R 1 Furthermore it follows from Eqs. 6 an 4 that 1 T S + T R 1 2 M T T R 1 T R 1 T R 1 P 1 T R 1 + T R 1 + T R 1 T R 1 T R 1 P 1 T R 1 + T R 1 R R 1 + T R 1 T R 1 T R 1 S R 1 + T R 1 We further have T R 1 T S 1 S R 1 an from matrix inversion lemma 10 it follows that S 1 R 1 R 1 U 1 P 1 T R 1 Thus T S 1 U P 1 T R 1 + I T R 1 T R 1 U 1 P 1 T R 1 I T R 1 U 1 T R 1 I T R 1 U 1 S R 1 P 1 P 1 T T R 1 T R

5 Note that T T T 1 2 I n n I n n I n n T T iag Thus { 1 2 I M I n n I n n I n n T I M T T R 1 I T R 1 U 1 S R 1 P 1 Taing transpose on both sies we have T R 1 M M T T I M T R 1 T R 1 S R 1 I T M I P 1 V 1 T R 1 V T R 1 P 1 + I T S + I T R 1 U 1 I M T R 1 S R 1 + T R 1 S R 1 P 1 T R 1 S R 1 I T M I P 1 V 1 T R 1 I T R 1 U 1 P 1 I M T R 1 S R 1 I T M I P 1 V 1 T R 1 I T R 1 U 1 P 1 T R 1 S R 1 I P 1 V 1 T R 1 T S 1 S S 1 T S 1 This ompletes the proof. Theorem 2 If ˆx 1 1 ˆx 1 1 an P 1 1 P 1 1 then the above bath LMMSE istribute estimation fusion is globally optimal in the sense that it is equivalent to the entralize estimation fusion that is ˆx ˆx P P Proof: Sine ˆx 1 1 ˆx 1 1 it follows from Eqs. 8 an 14 that ˆx 1 ˆx 1 Also sine P 1 1 P 1 1 it follows from Eqs. 9 an 15 that P 1 P 1 Sine P 1 1 P 1 1 it follows from Theorem 1 Eqs. 11 an 17 that P P From the almost sure uniqueness of the LMMSE estimators that two LMMSE estimators of the same estimatee using the same set of ata are almost surely iential if an only if their MSE matries are equal it follows that ˆx ˆx This ompletes the proof. Remar: From the above theorems we an see that the propose istribute estimation fusion algorithm is globally optimal in the sense that it is equivalent to the entralize fusion algorithm. Another nie property of it is that the ommuniation requirements from eah loal sensor to the fusion enter is just n + n 2 whih is equal to or less than most of the existing istribute estimation fusion algorithms. A thir nie property is that the inverses of the orresponing error ovariane matries were never use an our initial goal is ahieve. Remar: A omparison with other existing istribute estimation fusion algorithms e.g. WLS metho 4 an MAP metho 9 has also been one whih shows that the propose istribute estimation fusion algorithm is better either from the aspet of optimality equivalene to entralize estimation fusion or from the aspet of ommuniation transmission or from both aspets. But ue to spae limitation it is not provie in this paper. VI. REDUCTION OF COMPUTATIONAL COMPLEXITY From the above we an see that when the inverses of the orresponing error ovariane matries o not exist most of the existing istribute estimation fusion algorithms whih highly epen on the existene of these inverses o not wor any more. While our propose algorithm still wors in this ase but there is really no free lunh an the prie we nee to pay is the highly inrease omputational omplexity ue to the involvement of the MP inverse S + of a imension of Mn Mn. In the following we show that even when MP 1295

6 inverse is involve we an still reursively proess the transforme ata. In this way the heavy omputational omplexity ue to the MP inverse an be alleviate to a ertain egree. Theorem 3 Reursive LMMSE Distribute Fusion For the above ynami system with multiple sensors the istribute estimation fusion an also be ompute reursively as follows. Time upate: ˆx 0 ˆx 1 F 1ˆx 1 1 P 0 P 1 F 1P 1 1 F T 1 + G 1 Q 1 G T 1 Measurement upate by sensor i i 1 2 M: ˆx i ˆxi 1 + K i z i i ˆxi 1 Finally P i P i 1 K i S i P i 1 K i i P i 1 i S i K i T Si i ˆx ˆxM P P M T + T + i Proof: In the sense of LMMSE it is very easy to get the time upate of x at the fusion enter as follows P 0 ˆx 0 ˆx 1 E x z1 z2 z 1 F 1ˆx 1 1 P 1 ˆx MSE 1 T E x ˆx 1 x ˆx 1 F 1 P 1 1 F T 1 + G 1Q 1 G T 1 Given z 1 the LMMSE estimator of x is given as follows. ˆx 1 E x z1 z2 z 1 z 1 ˆx 1 + P 1 z 1 1 ˆx 0 + K1 P 1 MSE ˆx 1 P 1 P 1 P 0 K1 S 1 ˆx 1 1 z 1 1 K 1 T S1 1 ˆx0 T S1 T P 1 { { z i z 1 z2 zi 1 z i z i 1 z i for i 2 3 M. Sine the LMMSE estimator E x z1 z 2 z 1 zi always has the quasi-reursive form 11 we have that is Also ˆx i E x z 1 z 2 z 1 zi z i i 1 zi E x z 1 z 2 z 1 zi 1 ˆx i 1 + C i 1i C + z i i 1 z i i 1 z i z i i z i i E z z 1 z 2 z 1 zi 1 E i x z1 z2 z 1 z i 1 i ˆxi 1 x ˆx i 1 + v i i C z S i i 1 C i 1i i 1 C x z i i 1 ˆx i P i 1 T i E x z1 z2 z 1 z i ˆx i 1 + K i z i i ˆxi 1 P i MSE ˆx i MSE P i 1 ˆx i 1 K i C i 1i C + z i i 1C T i 1i S i K i This is atually also the reursive LMMSE estimator with transforme ata z i 1 an z i sine ˆx i epens on zi 1 only through ˆx i 1. Repeating the same proeure until the transforme ata from sensor M is also use we have ˆx ˆxM ˆx M 1 P P M P M 1 + K M z M K M T S M M K M ˆx M 1 This ompletes the proof. Remar: From the above theorem we an see that the original MP inverse S + with imension Mn Mn is now Si + replae by M MP inverses eah having imension n n for i 1 2 M. The omputational omplexity T 1296

7 is inee reue greatly by the reursive proessing of the transforme ata from eah sensor. Remar: In general the reursive LMMSE estimation epens on the orer in whih the ata are use an it may iffer from the bath LMMSE estimation. As was shown in the above our reursive LMMSE istribute estimation fusion is equivalent to the bath LMMSE istribute estimation fusion. Remar: Following a similar reursive proessing iea the optimal asynhronous istribute estimation algorithm whih uses the transforme ata in Eq. 5 has also been obtaine. But ue to spae limitation it is not presente in this paper. 12 X. R. Li an K. S. Zhang Optimal linear estimation fusion - part iv: Optimality an effiieny of istribute fusion in Proeeings of the 4th International Conferene on Information Fusion Montreal QC Canaa August 2001 pp. WeB1 19 WeB1 26. VII. CONCLUSIONS Most of the existing istribute estimation fusion algorithms rely on the existene of the inverses of the orresponing error ovariane matries. For a general ynami system with multiple sensors the existene of these inverses an not neessarily be ensure whih limits the appliability of the existing istribute estimation fusion algorithms. By taing a linear transformation of the original measurement of eah sensor an optimal istribute estimation fusion algorithm has been presente in this paper. The ommuniation requirements from eah sensor to the fusion enter is equal to or less than most of the existing istribute estimation fusion algorithms. Reursive proessing of the transforme ata to relieve the inrease omputational omplexity of the propose istribute estimation fusion algorithm is also isusse. REFERENCES 1 X. R. Li Y. M. Zhu J. Wang an C. Z. an Optimal linear estimation fusion - part i: Unifie fusion rules IEEE Transations on Information Theory vol. 49 no. 9 pp September Y. Bar-Shalom an L. Campo The effet of the ommon proess noise on the two-sensor fuse-tra ovariane IEEE Transations on Aerospae an Eletroni Systems vol. 22 no. 6 pp November K. C. Chang R. K. Saha an Y. Bar-Shalom On optimal tra-totra fusion IEEE Transations on Aerospae an Eletroni Systems vol. 33 no. 4 pp Otober M. Chen T. Kirubarajan an Y. Bar-Shalom Performane limits of tra-to-tra fusion versus entralize estimation: theory an appliation IEEE Transations on Aerospae an Eletroni Systems vol. 39 no. 2 pp April K.. Kim Development of tra to tra fusion algorithms in Proeeings of the 1994 Amerian Control Conferene Baltimore MD June 1994 pp C. Y. Chong ierarhial estimation in Proeeings of the MIT/ONR Worshop on C3 Monterey CA R. ashemipour S. Roy an A. J. Laub Deentralize strutures for parallel Kalman filtering IEEE Transations on Automati Control vol. 33 no. 1 pp January E. B. Song Y. M. Zhu J. Zhou an Z. S. You Optimal Kalman filtering fusion with ross-orrelate sensor noises Automatia vol. 43 no. 8 pp August K. C. Chang Z. Tian an S. Mori Performane evaluation for map state estimate fusion IEEE Transations on Aerospae an Eletroni Systems vol. 40 no. 2 pp April X. R. Li Applie Estimation an Filtering. Course Notes University of New Orleans February Reursibility an optimal linear estimation an filtering in Proeeings of the 43r IEEE Conferene on Deision an Control Atlantis Paraise Islan Bahamas Deember pp