Optimal Distributed Estimation Fusion with Transformed Data

Optimal Distributed Estimation Fusion with Transformed Data Zhansheng Duan

X. Rong Li

Department of Electrical Engineering University of New Orleans New Orleans, LA 70148, U.S.A. Email: {zduan,xli}@uno.edu

Abstract— Most of the existing distributed estimation fusion algorithms rely on the existence of the inverses of the corresponding error covariance matrices, e.g., distributed estimation fusion algorithms based on the information form of the Kalman filter and the optimal weighted least-square (WLS) estimator. Theoretically speaking, the error covariance matrices are only at least positive semi-definite and not necessarily invertible. To overcome this, by taking a linear transformation of the raw measurements received by each local sensor, an optimal distributed estimation fusion scheme is proposed in this paper. Compared with the existing distributed estimation fusion schemes, the new algorithm is not only optimal in the sense that it is equivalent to the centralized fusion, the communication requirements from each sensor to the fusion center are equal to or less than most of the existing distributed fusion algorithms. One possible way to relieve the computational complexity of the new algorithm is also discussed.

Keywords: Estimation fusion, distributed fusion, centralized fusion, recursive estimation, linear minimum meansquared error (LMMSE). I. I NTRODUCTION Estimation fusion, or data fusion for estimation, is the problem of how to best utilize useful information contained in multiple sets of data for the purpose of estimating an unknown quantity—–a parameter or process (at a time) [1]. There are two basic estimation fusion architectures: centralized and decentralized/distributed (also referred to as measurement fusion and track fusion in target tracking, respectively), depending on whether the raw measurements are sent to the fusion center or not. In centralized fusion, all raw measurements are sent to the fusion center, while in distributed fusion, each sensor only sends in the processed data. Centralized fusion, despite of its heavy computational burden at the fusion center and poor survivability, can provide globally optimal fused estimates provided the processing capability of the processor at the fusion center and communication bandwidth and reliability can satisfy the requirements. With a reduced computational burden at the fusion center and reduced communication demands for the sensor networks, distributed fusion usually has faster realtime processing and stronger fault-tolerance abilities. Usually distributed fusion is more challenging in terms of performance, Research supported in part by NSFC grant 60602026, Project 863 through grant 2006AA01Z126 and Navy through Planning Systems Contract # N68335-05-C-0382. The authors are also with the College of Electronic and Information Engineering, Xi’an Jiaotong University.

channel capacity, reliability, survivability, information sharing, etc. and has been a focal point of most fusion research. The topic of distributed estimation fusion has been researched for several decades due to its nice properties mentioned above and there are a lot of results available. Two classes of optimality criteria were discussed most in the existing distributed estimation fusion algorithms. The first class tries to reconstruct the centralized fused estimate from the locally processed data (e.g., local estimates). That is, the optimality criterion used by the first class is the equivalence to the centralized estimation fusion. The second class is usually optimal conditioned on the locally processed data. For example, [2] and [3] proposed a two-sensor track-to-track fusion algorithm which is optimal in the sense of maximum likelihood (ML) for the Gaussian case. For more than two sensors, [4], [5] proposed a track-to-track fusion algorithm which is optimal in the sense of ML (for the Gaussian case) and WLS. [6], [7] proposed a decentralized structure to reconstruct the optimal global estimate when the measurement noises across sensors are uncorrelated. [8] proposed a decentralized structure to reconstruct the optimal global estimate when the measurement noises across sensors are correlated. [1] proposed unified fusion rules in the sense of best linear unbiased estimate (BLUE) and WLS for all fusion architectures with arbitrary correlation of local estimates or observation errors across sensors or across time, which includes the distributed estimation fusion as a special class. [9] proposed a state estimation fusion algorithm which is optimal in the sense of maximum a posteriori (MAP). Most of the existing distributed estimation fusion algorithms, e.g., distributed estimation fusion algorithms based on the information form of the Kalman filter and the optimal WLS estimator, rely on the existence of the corresponding error covariance matrices. The error covariance matrices are at least positive semi-definite, but not necessarily invertible. This means that in these cases we can not always use the existing distributed estimation algorithms directly. One simplest way to solve this problem is to simply replace the traditional inverse by some generalized inverses (e.g., the Moore-Penrose inverse) when the traditional inverse does not exist. But in this way, the optimality of the original distributed estimation fusion algorithm does not necessarily hold. In this paper, by taking a linear transformation of the

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original measurement of each sensor, an optimal distributed estimation fusion algorithm with the transformed data is proposed, which is actually equivalent to the centralized estimation fusion. As can be seen later, all operations are taken in the estimatee (quantity to be estimated) space, so the communication requirements from each sensor to the fusion center are equal to or less than most of the existing distributed estimation fusion algorithms. Recursive processing of the transformed data is also discussed, which is intended to relieve the increased computational complexity of the newly proposed distributed estimation fusion algorithm to a certain degree. The paper is organized as follows. Sec. II formulates the multiple sensor distributed estimation fusion problem for dynamic systems. Sec. III describes a new linear transformation of the original sensor measurement. Sec. IV presents the distributed estimation fusion algorithm with the transformed data. Sec. V analyzes the optimality of the proposed distributed estimation fusion algorithm with the transformed data. Sec. VI provides one way to reduce the computational complexity of the proposed distributed estimation fusion algorithm with the transformed data. Sec. VII gives concluding remarks. II. P ROBLEM

For the above described dynamic system which is observed by multiple sensors, suppose that at time instant k−1 the fused (i) (i) estimate and local estimates are x ˆk−1|k−1 , Pk−1|k−1 , i = d 1, 2, · · · , M and xˆdk−1|k−1 , Pk−1|k−1 , respectively, [6] firstly proposed the following distributed estimation fusion algorithm which is equivalent to the centralized estimation if there is no feedback from the fusion center to each local sensor. M   −1  −1 X −1  −1  (i) (i) d d Pk|k = Pk|k−1 + Pk|k − Pk|k−1 i=1

−1  −1  d d x ˆdk|k−1 Pk|k xˆdk|k = Pk|k−1  M  −1  −1 X (i) (i) (i) (i) + Pk|k xˆk|k − Pk|k−1 x ˆk|k−1 i=1

where at the fusion center, x ˆdk|k−1 = Fk−1 xˆdk−1|k−1 d d T Pk|k−1 = Fk−1 Pk−1|k−1 Fk−1 + Gk−1 Qk−1 GTk−1

and at each local sensor i, i = 1, 2, · · · , M

FORMULATION AND BACKGROUND

xk = Fk−1 xk−1 + Gk−1 wk−1

x ˆk|k−1 = Fk−1 xˆk−1|k−1 (1)

(i)

 −1  −1  T  −1 (i) (i) (i) (i) (i) Pk|k = Pk|k−1 + Hk Rk Hk

xk ∈ Rn , E [wk ] = 0nw ×1 cov [wk , wj ] = Qk δkj , Qk ≥ 0

 T  −1  −1  −1 (i) (i) (i) (i) (i) (i) (i) Rk zk x ˆk|k−1 + Hk Pk|k xˆk|k = Pk|k−1

E [x0 ] = x ¯0 , cov [x0 ] = P0 cov [x0 , wk ] = 0n×nw Assume that altogether M sensors are used to observe the system state at the same time (i)

(i)

(i)

(i)

T Pk|k−1 = Fk−1 Pk−1|k−1 Fk−1 + Gk−1 Qk−1 GTk−1

where

zk = Hk xk + vk , i = 1, 2, · · · , M

(i)

(i)

Consider the following generic dynamic system

(2)

where h i (i) (i) zk ∈ Rmi , E vk = 0mi ×1 h i (i) (i) (i) cov vk , vj = Rk δkj h i h i (i) (i) cov wj , vk = 0nw ×mi , cov x0 , vk = 0n×mi Also it is assumed that the measurement noises across sensors (i) are uncorrelated and Rk > 0, i = 1, 2, · · · , M . In distributed estimation fusion, the fusion center tries to get the best estimate of the system state with the processed data received from each local sensor. In this paper, by distributed estimation fusion, we mean only data-processed observations are available at the fusion center, not necessarily the local estimates from each sensor. Systems with only local estimates available at the fusion center, referred to as the standard distributed estimation fusion in [1], are not the focus of this paper.

As can be seen from [6], this optimal distributed estimation fusion algorithm came from an ingenerious equivalent transformation to the original optimal centralized estimation fusioin algorithm, in which all the raw measurements are replaced by the corresponding local predicted and updated estimates from each local sensor. Also can be seen is that, to ensure the optimality of the above distributed estimation fusion algorithm,    it isrequired  that all  the associated   inverses −1

−1

−1

−1

(i)

−1

(i)

−1

d d Pk|k−1 , Pk|k , Pk|k−1 , Pk|k , i = 1, 2, · · · , M exist. (i) (i) d d Since Pk|k−1 , Pk|k , Pk|k−1 , Pk|k , i = 1, 2, · · · , M are all covariance matrices, the only knowledge about them in a generic dynamic system is that they are all at least positive semi-definite if there are no further assumption about the system. This property surely can not guarantee    that  all the  associated   inverses (i)

−1

(i)

−1

d d Pk|k−1 , Pk|k , Pk|k−1 , Pk|k , i = 1, 2, · · · , M exist. In this paper, we will answer the question about how to obtain the optimal distributed estimation fusion algorithm which is equivalent to the optimal centralized estimation fusion if the above associated inverses do not necessarily exist.

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III. S ENSOR Let

MEASUREMENT TRANSFORMATION

 T  −1 (i) (i) (i) (i) z¯k = Hk Rk zk

(3)

Then from Eq. (2), it follows that  T  −1  T  −1 (i) (i) (i) (i) (i) (i) (i) z¯k = Hk Rk Hk xk + Hk Rk vk Furthermore, let  T  −1 (i) (i) ¯ (i) = H (i) H R Hk k k k

(4)

problem in the measurement space with dimension mi , i = 1, 2, · · · , M . After the linear transformation of Eq. (3), we convert the estimation fusion problem into the estimatee space.  Remark:  For each  local  sensor i, i = 1, 2, · · · , M , if −1

(i)

This is exactly where our idea of using the transformation in Eq. (3) comes from.

 T  −1 (i) (i) (i) (i) v¯k = Hk Rk vk

IV. D ISTRIBUTED

(i)

(i)

(i)

Let (5)

h i  T  −1 h i (i) (i) (i) (i) E v¯k = Hk Rk E vk = 0n×1 h i ¯ (i) = cov v¯(i) R k k  T  −1  −1 (i) (i) (i) (i) (i) = Hk Rk · Rk · Rk Hk

(i)

−1

(i)

 

¯ (1) H k

vkd =

 

(1) v¯k

T

T



(2) z¯k

T

 T ¯ (2) H k



(2) v¯k

T

··· ··· ···

 T T (M) z¯k

 T T ¯ (M) H k

(6)

  T T (M) v¯k

(7)

Then the stacked measurement equation at the fusion center w.r.t the M local sensors can be written as zkd = Hkd xk + vkd

In distributed estimation fusion, each local sensor sends the (i) processed data z¯k (n × 1) and the corresponding measure(i) ¯ (n × n), which is also the covariance matrix ment matrix H k (i) of the measurement noise v¯k , to the fusion center. In total, the communication requirements from each local sensor to the fusion center is n + n2 at any time instant. This is actually equal to or less than most of the existing distributed estimation fusion algorithms. (i) Remark: For a given multi-sensor dynamic system, if Hk (i) and Rk , i = 1, 2, · · · , M are time-varying, then in cen(i) tralized estimation fusion, we also need to transmit Rk to (i) (i) the fusion center besides zk and Hk . But for our new transformed measurement equation (5), although it has a form similar to the original measurement equation (2), there is no ¯ (i) to the fusion center at all even when it is need to transmit R k ¯ (i) . This will certainly time-varying due to its equivalence to H k help save communication requirements from each local sensor to the fusion center and is one of the nice properties of our new distributed estimation fusion algorithm. Remark: We  can clearly  see that the new transformed T

Hkd =

T

  E vkd = 0Mn×1 o n   ¯ (1) , H ¯ (2) , · · · , H ¯ (M) Rkd = cov vkd = diag H k k k

¯ (i) H k

(i)

(1) z¯k

where

h i (i) (i) ¯ (i) δkj cov v¯k , , v¯j = H k h i (i) (j) cov v¯k , v¯l = 0n×n , i 6= j h i h i (i) (i) cov wj , v¯k = 0nw ×n , cov x0 , v¯k = 0n×n

(i)

=

 

zkd

where

=

ESTIMATION FUSION WITH

TRANSFORMED DATA

Then the above equation can be rewritten as ¯ xk + v¯ , i = 1, 2, · · · , M z¯k = H k k

−1

(i)

Pk|k−1 and Pk|k exist, then it follows from the information form of the Kalman filter that  −1  −1  T  −1 (i) (i) (i) (i) (i) (i) (i) Pk|k xˆk|k − Pk|k−1 xˆk|k−1 = Hk Rk zk

data z¯k = Hk Rk zk can actually be called informational state in the information form of the Kalman filter. That is, originally we are handling the estimation fusion

Assuming that the distributed fused estimate at time instant k − 1 is xˆdk−1|k−1 with the corresponding error covariance d matrix Pk−1|k−1 , then in the sense of LMMSE [1], [10], [11], the optimal distributed fused estimate of the system state at the fusion center at time instant k can be recursively computed as follows (LMMSE Distributed Fusion): x ˆdk|k−1 = Fk−1 xˆdk−1|k−1

(8)

d d T Pk|k−1 = Fk−1 Pk−1|k−1 Fk−1 + Gk−1 Qk−1 GTk−1 (9)   x ˆdk|k = x ˆdk|k−1 + Kkd zkd − Hkd xˆdk|k−1 (10)

d Pk|k


T d
+ d Kkd = Pk|k−1 Hkd Sk
T d
+ d d d d = Pk|k−1 − Pk|k−1 Hkd Sk Hk Pk|k−1
T d Skd = Hkd Pk|k−1 Hkd + Rkd

(11)

where A+ stands for the unique Moore-Penrose pseudoinverse (MP inverse in short) of matrix A. ¯ (i) = H ¯ (i) = Remark: In general, we have R k k  T  −1 (i) (i) (i) d Hk Rk Hk ≥ 0 and Rk = n o (1) ¯ (2) (M) ¯ ¯ diag Hk , Hk , · · · , Hk ≥ 0, and this is the reason why we used MP inverse in the above. But for some special

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¯ (i) > 0, i = 1, 2, · · · , M , e.g., when cases, we do have H k
+ (i) Hk are all full column rank. In this case, Skd in the
d −1 above will be replaced by Sk . V. O PTIMALITY OF

DISTRIBUTED ESTIMATION FUSION WITH TRANSFORMED DATA

Let n o ˘ k = diag H (1) , H (2) , · · · , H (M) H k k k Then from Eqs. (7) and (4), it can be seen that n o ¯ (1) , H ¯ (2) , · · · , H ¯ (M) Rkd = diag H k

Let

=

zkc =

 

zk

Hkc =

 

(1) Hk

vkc =

 

(1)

(1)

vk

T



T

T

(2)

zk

T

 T (2) Hk 

(2)

vk

T

  T T (M) zk

···

 T T (M) ··· Hk   T (M)

···

where E [vkc ] = 0l×1 , l =

(12)

−1

˘ kT (Rkc ) =H

=

cov [vkc ]

= diag

n

Hkd Then

mi

(1) (2) Rk , Rk , · · ·

Hkd (M) , Rk

o

(13)

  xˆck|k = x ˆck|k−1 + Kkc zkc − Hkc x ˆck|k−1 T

c Kkc = Pk|k−1 (Hkc ) (Skc )

Skc =

T

Hkd −1

(16)

c Hkc Pk|k−1

˘k + H ˘ T (Rc )−1 Rc (Rc )−1 H ˘k H k k k k

˘ T (Rc )−1 H c ·H k k k −1

T

= (Hkc ) (Rkc ) ˘ T (Rc ) ·H k k

−1

Hkc

+

 + ˘ kT (Rkc )−1 Skc (Rkc )−1 H ˘k H ˘k H

We further have −1

T

(Hkc ) (Rkc )

˘ k = (Hkc )T (Skc )−1 Skc (Rkc )−1 H ˘k H

and from matrix inversion lemma [10], it follows that d (Skc )−1 = (Rkc )−1 − (Rkc )−1 Hkc Uk−1 Pk|k−1 (Hkc )T (Rkc )−1

(17)

+ Rkc

where T

d Uk = Pk|k−1 (Hkc ) (Rkc )

−1

Hkc + I

Thus

For the given dynamic system with multiple sensors, the following theorems hold. d c Theorem 1 If Pk−1|k−1 = Pk−1|k−1 , then for the above LMMSE distributed estimation fusion, we have
T d
+ d T −1 Sk Hk = (Hkc ) (Skc ) Hkc Hkd Proof: Since

−1

T

(15)

−1

−1

c c c Pk|k = Pk|k−1 − Pk|k−1 (Hkc ) (Skc ) c Hkc Pk|k−1 (Hkc )T


+

· (Hkc ) (Rkc )

(14)

Gk−1 Qk−1 GTk−1

T

Skd

˘k = (Hkc )T (Rkc )−1 H

 ˘ T (Rc )−1 H c P d ˘k H = (Hkc )T (Rkc )−1 H k k k k|k−1

x ˆck|k−1 = Fk−1 x ˆck−1|k−1 +


T


T

˘ T (Rc )−1 H c ·H k k k

Assuming that the centralized fused system state estimate at time instant k − 1 is xˆck−1|k−1 with the corresponding error c , then in the sense of LMMSE [1], covariance matrix Pk−1|k−1 [10], [11], the optimal centralized fused estimate of the system state at the fusion center at time instant k can be recursively computed as follows:

=

Hkc

 ˘k H ˘ T (Rc )−1 H c P d H k k|k−1 k k + −1 ˘ T ˘k ˘ T c −1 H · (Hkc ) (Rkc ) H k + Hk (Rk )

zkc = Hkc xk + vkc

c T Fk−1 Pk−1|k−1 Fk−1

˘k H

= (Hkc ) (Rkc )

Then the stacked measurement equation at the fusion center w.r.t the M local sensors can be written as

c Pk|k−1

k

T

vk

M X

k

−1 (Rkc )

Furthermore, it follows from Eqs. (6) and (4) that h iT ¯ (1) H ¯ (2) · · · H ¯ (M) Hkd = H k k k

i=1

Rkc

˘ kT H

(Hkc )T (Skc )−1 −1

T

= (Hkc ) (Rkc )

T

−1

− (Hkc ) (Rkc )

d Hkc Uk−1 Pk|k−1

−1

T

· (Hkc ) (Rkc )   d = I − (Hkc )T (Rkc )−1 Hkc Uk−1 Pk|k−1 (Hkc )T (Rkc )−1 Then −1

˘k (Hkc ) (Rkc ) H   T −1 T −1 d = I − (Hkc ) (Rkc ) Hkc Uk−1 Pk|k−1 (Hkc ) (Rkc ) T

d c Pk−1|k−1 = Pk−1|k−1

it follows from Eqs. (9) and (15) that

· Skc (Rkc )

d c Pk|k−1 = Pk|k−1

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−1

˘k H

Note that  

T

(Hkc ) =

(1)

Hk



T

 T (2) Hk

··· 

fusion is globally optimal in the sense that it is equivalent to the centralized estimation fusion, that is,

 T  (M) Hk

x ˆdk|k = x ˆck|k

In×n In×n · · · In×n  T  T  T  (1) (2) (M) · diag Hk , Hk , · · · , Hk =

d c Pk|k = Pk|k

Proof: Since xˆdk−1|k−1 = x ˆck−1|k−1 , it follows from Eqs. (8) and (14) that x ˆdk|k−1 = x ˆck|k−1

Let IM = Then



In×n

In×n

···

In×n



˘ kT (Hkc ) = IM H

d c Also, since Pk−1|k−1 = Pk−1|k−1 , it follows from Eqs. (9) and (15) that d c Pk|k−1 = Pk|k−1

T −1 ˘ (Hkc ) (Rkc ) H k   T −1 d ˘ T (Rc )−1 = I − (Hkc ) (Rkc ) Hkc Uk−1 Pk|k−1 IM H k k

d c Since Pk−1|k−1 = Pk−1|k−1 , it follows from Theorem 1, Eqs. (11) and (17) that

T

Thus

−1

· Skc (Rkc )

d c Pk|k = Pk|k

˘k H

From the almost sure uniqueness of the LMMSE estimators that two LMMSE estimators of the same estimatee using the same set of data are almost surely identical if and only if their MSE matrices are equal [10], [12], it follows that

Taking transpose on both sides, we have ˘ T (Rc )−1 H c H k k k ˘ T (Rc )−1 S c (Rc )−1 H ˘ k IT =H k k k M  k  −1 d c T c −1 · I − Pk|k−1 Vk (Hk ) (Rk ) Hkc

x ˆdk|k = x ˆck|k This completes the proof.

where Vk =

(Rkc )−1

(Hkc )T

d Hkc Pk|k−1

 Remark: From the above theorems, we can see that the proposed distributed estimation fusion algorithm is globally optimal in the sense that it is equivalent to the centralized fusion algorithm. Another nice property of it is that the communication requirements from each local sensor to the fusion center is just n + n2 , which is equal to or less than most of the existing distributed estimation fusion algorithms. A third nice property is that the inverses of the corresponding error covariance matrices were never used and our initial goal is achieved. Remark: A comparison with other existing distributed estimation fusion algorithms, e.g., WLS method [4] and MAP method [9], has also been done, which shows that the proposed distributed estimation fusion algorithm is better either from the aspect of optimality (equivalence to centralized estimation fusion) or from the aspect of communication transmission or from both aspects. But due to space limitation, it is not provided in this paper.

+I

Then
T d
+ d Hkd S k Hk   d = I − (Hkc )T (Rkc )−1 Hkc Uk−1 Pk|k−1

˘ kT (Rkc )−1 Skc (Rkc )−1 H ˘k · IM H  + ˘ T (Rc )−1 S c (Rc )−1 H ˘k · H k k k k

˘ T (Rc )−1 S c (Rc )−1 H ˘ k IT ·H k k k M k  −1 −1 d c T · I − Pk|k−1 Vk (Hk ) (Rkc ) Hkc   T −1 d = I − (Hkc ) (Rkc ) Hkc Uk−1 Pk|k−1

˘ T (Rc )−1 S c (Rc )−1 H ˘ k IT · IM H k k k k M   −1 d c T c −1 · I − Pk|k−1 Vk (Hk ) (Rk ) Hkc   T −1 d = I − (Hkc ) (Rkc ) Hkc Uk−1 Pk|k−1 −1

T

−1

· (Hkc ) (Rkc ) Skc (Rkc ) Hkc   d · I − Pk|k−1 Vk−1 (Hkc )T (Rkc )−1 Hkc T

−1

Skc (Skc )

T

−1

Hkc

= (Hkc ) (Skc )

= (Hkc ) (Skc )

−1

VI. R EDUCTION

Hkc

This completes the proof.  d Theorem 2 If x ˆdk−1|k−1 = xˆck−1|k−1 and Pk−1|k−1 = c Pk−1|k−1 , then the above batch LMMSE distributed estimation

OF COMPUTATIONAL COMPLEXITY

From the above, we can see that when the inverses of the corresponding error covariance matrices do not exist, most of the existing distributed estimation fusion algorithms, which highly depend on the existence of these inverses, do not work any more. While our proposed algorithm still works in this case, but there is really no free lunch and the price we need to pay is the highly increased computational
+ complexity due to the involvement of the MP inverse Skd of a dimension of M n × M n. In the following, we show that even when MP

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inverse is involved, we can still recursively process the transformed data. In this way, the heavy computational complexity due to the MP inverse can be alleviated to a certain degree. Theorem 3 (Recursive LMMSE Distributed Fusion) For the above dynamic system with multiple sensors, the distributed estimation fusion can also be computed recursively as follows. Time update: (0)

xˆk|k = x ˆdk|k−1 = Fk−1 xˆdk−1|k−1

Let n o n o (1) (2) (i−1) (i) (i) z¯ki = z¯k , z¯k , · · · , z¯k , z¯k = z¯ki−1 , z¯k

for  i = 2, 3, · · · , M .  Since the LMMSE estimator d E ∗ xk |z1d , z2d , · · · , zk−1 , z¯ki always has the quasi-recursive form [11], we have   (i) d x ˆk|k = E ∗ xk |z1d , z2d, · · · , zk−1 , z¯ki h i (i) d = E ∗ xk |z1d , z2d , · · · , zk−1 , z¯ki−1 , z¯k (i−1)

=x ˆk|k

(0)

d d T Pk|k = Pk|k−1 = Fk−1 Pk−1|k−1 Fk−1 + Gk−1 Qk−1 GTk−1

(i)

(i−1)

(i)

(i−1)

Kk = Pk|k

(i) (i−1) ¯ (i) x = z¯k − H k ˆk|k   (i−1) (i) ¯ (i) xk − x =H ˆ + v¯k k k|k

T  + (i) ¯ (i) H S¯k k

 T (i) ¯ (i) P (i−1) H ¯ (i) ¯ (i) S¯k = H +H k k k k|k

(i) ∗ Cz˜i|i−1 = S¯k

Finally, (M)

k|k

(M)

Proof: In the sense of LMMSE, it is very easy to get the time update of xk at the fusion center as follows =

=E

∗



= Fk−1 x ˆdk−1|k−1

xk |z1d , z2d , · · ·

d , zk−1

i|i−1



¯ (i) H k

T

  (i) d x ˆk|k = E ∗ xk |z1d , z2d , · · · , zk−1 , z¯ki   (i) (i) (i−1) ¯ (i) xˆ(i−1) =x ˆk|k + Kk z¯k − H k k|k Also,   (i) (i) Pk|k = M SE xˆk|k   (i−1) T = M SE xˆk|k − Ci−1,i Cz˜+∗ Ci−1,i i|i−1  T (i) (i) (i) (i−1) = Pk|k − Kk S¯k Kk



  (0) d = M SE xˆdk|k−1 Pk|k = Pk|k−1   T  d d = E xk − xˆk|k−1 xk − x ˆk|k−1

This is actually also the recursive LMMSE estimator with (i) (i) transformed data z¯ki−1 and z¯k since x ˆk|k depends on z¯ki−1 (i−1)

d T = Fk−1 Pk−1|k−1 Fk−1 + Gk−1 Qk−1 GTk−1 (1)

Given z¯k , the LMMSE estimator of xk is given as follows. h i (1) (1) d xˆk|k = E ∗ xk |z1d , z2d , · · · , zk−1 , z¯k  T  + d ¯ (1) ¯(1) =x ˆdk|k−1 + Pk|k−1 H S k k   (1) (1) d ¯ · z¯k − Hk xˆk|k−1   (0) (1) (1) (0) ¯ (1) x =x ˆk|k + Kk z¯k − H k ˆk|k   (1) (1) Pk|k = M SE x ˆk|k

= Pk|k

that is,

d Pk|k = Pk|k

xˆdk|k−1

(i−1)

Ci−1,i = Cx˜(i−1) z˜∗

x ˆdk|k = xˆk|k

(0) x ˆk|k

∗ z˜i|i−1

h i (i) (i) ∗ d , z¯ki−1 z˜i|i−1 = z¯k − E ∗ z¯k |z1d , z2d , · · · , zk−1 h i (i) i−1 d ¯ (i) xk |z1d , z2d , · · · , zk−1 = z¯k − E ∗ H , z ¯ k k

 T (i) (i) (i) − Kk S¯k Kk 

i|i−1

where

Measurement update by sensor i, i = 1, 2, · · · , M :   (i) (i−1) (i) (i) (i−1) ¯ (i) x x ˆk|k = xˆk|k + Kk z¯k − H ˆ k k|k Pk|k = Pk|k

+ Ci−1,i Cz˜+∗

 T  + d d ¯ (1) ¯(1) H ¯ (1) P d = Pk|k−1 − Pk|k−1 H S k|k−1 k k k  T (0) (1) ¯(1) (1) = Pk|k − Kk Sk Kk

only through x ˆk|k . Repeating the same procedure until the transformed data from sensor M is also used, we have (M)

x ˆdk|k = x ˆk|k

(M−1)

=x ˆk|k

(M)

+ Kk



(M)

z¯k

(M−1) ¯ (M) x −H ˆk|k k



(M)

d Pk|k = Pk|k

(M−1)

= Pk|k

(M)

− Kk

 T (M) (M) S¯k Kk

This completes the proof.  Remark: From the above theorem, we can see that the
+ original MP inverse Skd with dimension M n × M n is now  + (i) replace by M MP inverses S¯ , each having dimension k

n × n for i = 1, 2, · · · , M . The computational complexity

1296

is indeed reduced greatly by the recursive processing of the transformed data from each sensor. Remark: In general, the recursive LMMSE estimation depends on the order in which the data are used and it may differ from the batch LMMSE estimation. As was shown in the above, our recursive LMMSE distributed estimation fusion is equivalent to the batch LMMSE distributed estimation fusion. Remark: Following a similar recursive processing idea, the optimal asynchronous distributed estimation algorithm which uses the transformed data in Eq. (5) has also been obtained. But due to space limitation, it is not presented in this paper.

[12] X. R. Li and K. S. Zhang, “Optimal linear estimation fusion - part iv: Optimality and efficiency of distributed fusion,” in Proceedings of the 4th International Conference on Information Fusion, Montreal, QC, Canada, August 2001, pp. WeB1–19–WeB1–26.

VII. C ONCLUSIONS Most of the existing distributed estimation fusion algorithms rely on the existence of the inverses of the corresponding error covariance matrices. For a general dynamic system with multiple sensors, the existence of these inverses can not necessarily be ensured which limits the applicability of the existing distributed estimation fusion algorithms. By taking a linear transformation of the original measurement of each sensor, an optimal distributed estimation fusion algorithm has been presented in this paper. The communication requirements from each sensor to the fusion center is equal to or less than most of the existing distributed estimation fusion algorithms. Recursive processing of the transformed data to relieve the increased computational complexity of the proposed distributed estimation fusion algorithm is also discussed. R EFERENCES [1] X. R. Li, Y. M. Zhu, J. Wang, and C. Z. Han, “Optimal linear estimation fusion - part i: Unified fusion rules,” IEEE Transactions on Information Theory, vol. 49, no. 9, pp. 2192–2208, September 2003. [2] Y. Bar-Shalom and L. Campo, “The effect of the common process noise on the two-sensor fused-track covariance,” IEEE Transactions on Aerospace and Electronic Systems, vol. 22, no. 6, pp. 803–805, November 1986. [3] K. C. Chang, R. K. Saha, and Y. Bar-Shalom, “On optimal track-totrack fusion,” IEEE Transactions on Aerospace and Electronic Systems, vol. 33, no. 4, pp. 1271–1276, October 1997. [4] H. M. Chen, T. Kirubarajan, and Y. Bar-Shalom, “Performance limits of track-to-track fusion versus centralized estimation: theory and application,” IEEE Transactions on Aerospace and Electronic Systems, vol. 39, no. 2, pp. 386–400, April 2003. [5] K. H. Kim, “Development of track to track fusion algorithms,” in Proceedings of the 1994 American Control Conference, Baltimore, MD, June 1994, pp. 1037–1041. [6] C. Y. Chong, “Hierarchical estimation,” in Proceedings of the MIT/ONR Workshop on C3, Monterey, CA, 1979. [7] H. R. Hashemipour, S. Roy, and A. J. Laub, “Decentralized structures for parallel Kalman filtering,” IEEE Transactions on Automatic Control, vol. 33, no. 1, pp. 88–94, January 1988. [8] E. B. Song, Y. M. Zhu, J. Zhou, and Z. S. You, “Optimal Kalman filtering fusion with cross-correlated sensor noises,” Automatica, vol. 43, no. 8, pp. 1450–1456, August 2007. [9] K. C. Chang, Z. Tian, and S. Mori, “Performance evaluation for map state estimate fusion,” IEEE Transactions on Aerospace and Electronic Systems, vol. 40, no. 2, pp. 706–714, April 2004. [10] X. R. Li, Applied Estimation and Filtering. Course Notes, University of New Orleans, February 2006. [11] ——, “Recursibility and optimal linear estimation and filtering,” in Proceedings of the 43rd IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, December 14-17 2004, pp. 1761– 1766.

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