Decentralized Information Filter in Federated Form - CiteSeerX

SICE Annual Conference in Fukui, August 4-6, 2003 Fukui University, Japan

Decentralized Information Filter in Federated Form 1

G

Yong-Shik Kim , Keum-Shik Hong2 G 1 Department of Mechanical and Intelligent Systems Engineering, Pusan Natinal University; San 30, Jangjeon-dong, Gumjeong-gu, Busan, 609-735, Korea. Tel: +82-51-510-2454, Email: [email protected] 2 School of Mechanical Engineering, Pusan National University; San 30, Jangjeon-dong, Gumjeong-gu, Busan, 609-735, Korea. Email: [email protected] G Abstract: In this paper, a decentralized information filter in federated structure is developed in multi-sensor environments. Being equivalent to the Kalman filter algebraically, the information filter of Mutambara 11) is extended to N -sensor distributed dynamic systems. In multi-sensor environments, the information-based filter is easier to decentralize, initialize, and fuse than a KFbased filter. The structural features and information sharing principle of the federated information filter is discussed. It is proved that the information state and its information matrix of the information filter, which are weighted in terms of information sharing factor, are equal to those of the centralized information filter under the regular conditions. Keywords: State estimation, Kalman filter, information filter, multi-sensor, track fusion, federated filterG

1. Introduction Data fusion techniques are used in many tracking and surveillance systems as well as in applications where reliability is a main concern 2, 6). One method for design of such systems is to employ a number of sensors (may be different type) and to fuse the information obtained from all these sensors on a central processor. The Kalman filter with the centralized structure in multi-sensor environments is needed in many applications such as military surveillance, air traffic control, and mobile robots and other systems. Target tracking using multiple sensors can provide better performance than using a single sensor. Previous efforts in developing multi-sensor fusion algorithms for a centralized architecture, where measurements from all sensors are sent to the central processor, have shown that significant gains in performance are possible with using multiple sensors. If multi-sensor systems are to be able to process their data in real-time, however, the performance of the centralized Kalman filter will be degraded. In a distributed system a data processing is performed at local sensors and results are transmitted to the data fusion center for track processing to get the final results 16). The algorithms based on parallelization of the Kalman filter equations, as proposed in 8), extend the previous results allowing one to obtain the global estimation using only local estimates without transmission of information between sensors. Rao and Durrant-Whyte 15) considered a fully decentralized system. The method is based on the internodal communications between local processor units without the need of any central processor. In addition, linear sensor fusion algorithms are developed in 2) for configurations: with a feedback from the central processor to local processing units and without such a feedback. Zhu et al. 17) have shown that the track fusion formulas with feedback are, like the track fusion without feedback, exactly equivalent to the

corresponding centralized track fusion formulas. Besides, they have shown that the feedback does reduce the covariance of each local tracking error. Oh et al. 13) derived a suboptimal filtering equation with different types of observations. As other method to improve the track fusion problem, the information filter algebraic-equivalent to the Kalman filter 9) was developed 1, 3, 7, 10-12). The information filter is essentially a Kalman filter expressed in terms of measures of information about state estimates and their associated covariances. This filter has been called the inverse covariance form of the Kalman filter. Despite its potential application, however, it was not widely used and it was thinly covered in literature. Bar-Shalom 3) and Maybeck 10) briefly discussed the idea of information estimation, but did not explicitly derive the algorithm in terms of information as done Mutambara 11), nor did they use it as a principal filtering method. Information filter algorithm by Mutambara 11) is a Kalman filter expressed in terms of informationanalytic variables, which are measures of the amount of information about the parameter (state) of interest. In addition, a decentralized information filter was developed by 5, 11) . Chang et al. 5) provided performance evaluation of the information matrix form of state vector fusion. Closed-form analytical solution of steady state fused covariance has been derived as a measure of performance using this approach. Paik and Oh 14) and Carlson and Berarducci 4) considered the federated structure as another method for data fusion. It is known that the federated Kalman filter has advantages of simplicity and fault-tolerant capability over other decentralized filter techniques. The federated filter method based on rigorous information-sharing principles provides globally optimal or near-optimal estimation accuracy with a high degree of fault tolerance. The federated filter structure employs sensor-dedicated local filters, and a master filter to combine or fuse the local filter outputs.

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PR0001/03/0000-2025 ¥400 © 2003 SICE

The main contributions of this paper are as follows: First, there are no gain or innovation covariance matrices and the maximum dimension of a matrix to be inverted is the state dimension. In multi-sensor systems the state dimension is generally smaller than the observation dimension, hence it is preferable to employ the information filter and invert smaller information matrices than use the Kalman filter and invert larger innovation covariance matrices. Second, initializing the information filter is much easier than for the Kalman filter. This is because information estimates (matrix and state) are easily initialized to zero information. Third, the information filter is easier to distribute and fuse than the Kalman filter. Forth, it is shown that, in terms of information sharing factor, decentralized information filter with federated structure is equal to the centralized information filter. This paper is organized as follows: In Sections 2 we review the centralized and decentralized Kalman filter and information filter algorithms. We then comment on how our formalism can be applied to federated filter in Section 3. In Section 4, we compare the suggested information filter with the centralized information filter by applying to two sensor systems. Section 5 contains conclusions.

2. Problem Formulation In this Section, the centralized/decentralized KF and the centralized/decentralized information filter in multi-sensor environments are briefly discussed. The dynamic system and observation equations are given as x(k  1) F (k ) x(k )  Z (k ), k 0, 1,  , (1) z (k ) H (k ) x(k )  X (k ) , (2) where F (k ) is the system matrix, x(k ) is the state,

Z (k )  ƒ n is the process noise, z (k ) [ z1c (k ),  , z cN (k )]c is the stacked observation vector, H (k ) is the observation matrix, and X (k ) is the observation noise. We assume that (i) {Z (k )}, {X i (k )} , i 0, 1,  , N , are zero-mean white sequences uncorrelated with each other and E[Z (k )Z c( j )] Q (k )G k , j , E[X (k )X c( j ) R (k )G kj ; (ii) The initial state x(0) is a Gaussian random vector, x(0) ~ N ( x (0), P(0)) ; (iii) x(0) is independent of {Z (k )}, {Xi (k )} , i 0, 1, , N .

2.1 The centralized KF equations

We will begin by reviewing the centralized Kalman filtering equations, both as a means of introducing notation and for later comparison with the new decentralized information filter. For the system defined in (1) and (2), the centralized fusion using all observations from multi-sensors is given by: i) Time-update (prediction) xˆ (k | k ) xˆ (k | k  1)  W (k )[ z (k )  H (k ) xˆ (k | k  1)] , P(k | k  1) F (k  1) P(k  1 | k  1) F c(k  1)  Q(k  1) , (3) ii) Measurement-update xˆ (k | k  1) F (k ) xˆ (k  1 | k  1) , xˆ (0 | 0) x (0) ,

P(k | k )

[ I  W (k ) H (k )]P(k | k  1) , P (1 | 0)

P (0) , (4)

where W (k ) P(k | k 1)Hc(k )[H (k)P(k | k  1)Hc(k )  R(k)]1 is the Kalman gain matrix. P(k | k ) is the covariance matrix of the estimate error ~ x (k | k ) xˆ (k | k )  x(k ) . As above, for the system and measurement equation (1) and (2), the Kalman filter provides a recursive solution for the estimate xˆ (k | k ) of the state x(k ) in terms of the estimate xˆ (k | k  1) and the new observation z (k ) . However, The primary limitations of standard (centralized) Kalman filter methods 9) when applied to multi-sensor systems with embedded local filters are (i) heavy computation computational loads, (ii) poor fault-tolerance, and (iii) inability to correctly process prefiltered data in a cascaded filter structure.

2.2 The decentralized KF equations

As an alternative filter implementation method to compensate limitations of the centralized Kalman filter, the decentralized Kalman filter was suggested 16). It is composed of multiple structures involving a master filter at high-level and local filters at low-level. A local filter, related to each observation sensor, estimates the local state variable. The master filter combines the estimates transmitted from local filters and deduces the globally optimal state. We first unstuck the observation vector z (k ) into m subvectors of dimension mi corresponding to the observations made by each individual sensor z (k ) [ z1c (k ),, zcN (k )]c and partition the observation matrix into submatricies corresponding to these observations H (k ) [ H1c (k ),, H cN (k )]c . The noise vector is also partitioned X (k ) [X1c (k ),,X cN (k )]c and it is assumed that these partitions are uncorrelated. Letting Fi (k ) be the sensors local system model, and letting xi (k ) be the states associated with this system, the observation equation can be written as: zi (k ) H i (k ) x(k )  Xi (k ), i 0, 1, , N , (5) where z (k )  ƒm , m

m1    mN , H i (k )  ƒmi un is

X (k ) ~ N (0, R (k )) , and m u n matrix, Xi (k )  ƒmi R (k ) diag[ R1 (k ),, RN (k )] . Applying the standard Kalman filtering equations to the N -sensor distributed dynamic system, the decentralized structure is given as follows: i) Time update (prediction) xˆi (k | k  1) Fi (k ) xˆi (k  1 | k  1) , Pi (k | k  1) Fi (k  1) Pi (k  1 | k  1) Fi (k  1)  Qi (k  1) . (6) ii) Measurement update  xi (k | k ) xˆi (k | k  1)  Wi (k )[ zi (k )  H i (k ) xˆi (k | k  1)] ,  Pi (k | k) Pi (k | k  1)  Wic(k)[Hi (k)P(k | k 1)Hic(k)  Ri (k)]Wic(k) , (7)  1 where the gain matrix is Wi (k ) Pi (k | k ) H ic (k ) Ri (k ) and   the x and P denote the partial estimate and its covariance based only on the i th node’s own observation. Using the above formulas, track fusions without feedback have been done by Hashemipour et al. 8) and Oh et

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al. 13). The global and local forms of assimilation equation results in: i) Assimilation of state xˆ (k | k ) P(k | k )[P1(k | k  1) xˆ (k | k  1)

Pi (k | k )[Pi1(k | k  1) xˆi (k | k  1) N    ¦{Pj1(k | k ) x j (k | k )  Pj1(k | k  1) xˆ j (k | k  1)}].

where the information prediction coefficient L j (k | k  1) is given by the expression (15)

ii) Information matrix

(8)

Yi (k | k )

ii) Covariance matrix

N  Yi (k | k  1)  ¦{Y j (k | k )  Yi (k | k  1)} . (17) j 1

N

 [ P (k | k  1)  ¦{Pj1 (k | k )  Pj1 (k | k  1)}]1 ,

In

j 1

Pi (k | k )

Y j (k | k  1) F (k )Y j1 (k  1 | k  1) .

j 1

j 1

1

(14)

Then, the assimilation equation is as follows: i) Information state N  yˆi (k | k ) yˆi (k | k  1)  ¦{ y j (k | k )  yˆi (k | k  1)} ,(16)

j 1

P(k | k )

Y j (k | k  1)  H cj (k ) R j 1 (k ) H j (k ) ,

L j (k | k  1)

N    ¦{Pj1(k | k ) x j (k | k )  Pj1(k | k  1) xˆ j (k | k  1)}],

xˆi (k | k )

 Y j (k | k )

N  [ Pi1 (k | k  1)  ¦{Pj1 (k | k )  Pj1 (k | k  1)}]1 . j 1

(9) Remark 1: As above, the decentralized filter has the same form as the centralized Kalman filter in real-time implementation, since the master model includes N estimates. In general, however, in case that the system model is the same and the observation model is decomposed to each local filter, the filter structure is not optimal. The estimate of local filter is affected by the system model due to the system model’s overlapping use. Therefore, as one method to settle this problem, an information-type filter is required.

13)

, the new sub-optimal estimate xˆk |k of the state

vector x k based on the all observations was constructed from the estimates xˆ1k |k , , xˆkN|k by the following formula xˆk |k

N

N

¦ cki xˆki |k , ¦ cki

i 1 c1k , , ckN

I,

i 1

where are weighting coefficients determined by the MMSE criterion. Method pursuing in this paper, which is similar to 13), is that a globally optimal fusion is obtained by applying a weighting factor to the information state and its information matrix at local filter. The federated filter satisfies this requirement. It weights the decentralized information filter with information sharing factor.

2.3 Centralized information filter '

Denote the information matrix as Y (k | k ) P 1 (k | k ) , information state as

'

yˆ (k | k ) P 1 (k | k ) xˆ (k | k ) . The

centralized information filter equations are given as

11)

:

i) Time update (prediction) yˆ (k | k  1) L(k | k  1) yˆ (k  1 | k  1) , Y (k | k  1) [ F (k )Y 1 (k  1 | k  1) F c(k )  Q(k )]1 . (10) ii) Measurement update yˆ (k | k ) yˆ (k | k  1)  H c(k ) R 1 (k ) z (k ) , Y (k | k ) Y (k | k  1)  H c(k ) R 1 (k ) H (k ) , (11) where the information prediction coefficient L(k | k  1) is given by the expression L(k | k  1)

Y (k | k  1) F (k )Y 1 (k  1 | k  1) .

(12)

2.4 Decentralized information filter 11) In local filter, the information filtering equation is given by i) Time update yˆ j (k | k  1) L j (k | k  1) yˆ j (k  1 | k  1) , Y j (k | k  1)

[ F (k )Y j1 (k  1 | k  1) F c(k )  Q(k )]1 . (13)

ii) Measurement update  y j (k | k ) yˆ j (k | k  1)  H cj (k ) R j 1 (k ) z j (k ) ,

3. Information Filter Fusion Using Federated Structure The federated Kalman filter, suggested by Carlson 4), can be considered a special form of the decentralized Kalman filter. The federated Kalman filter can obtain the globally optimal estimate by applying the “principle of information sharing” to each local filter and then fusing the estimates of local filters. For the system with local filter structure like (13) and (14), optimal information state and its information matrix equations are as follows: Ymaster Y1    YN , (18) yˆ master

N

¦ yˆi .

(19)

i 1

Theorem 1. For system (1) and (5), and the local filter structure (13) and (14), the solution of the federated information filter, (18) and (19), is equal to the solution of the centralized information filter, (10) and (11), in the case Conditions a)-c) are satisfied. a) The initial value of the information matrix, the initial information state and process noise covariance are distributed to local filters as follows: 1 1 1 Pi1 (0) Yi (0) P (0) Y (0) , i 1,, N , (20)

Ji

Ji

( Pi1xˆi )(0) yˆi (0) ( P 1xˆ )(0) yˆ (0) , i 1,, N , (21) Qi J i Q , i 1,, N . (22) b) The information state and its information matrix, which

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are calculated using (18) and (19), are distributed to local filters as follows: 1 Yi Ymaster , i 1,, N , (23)

Ji yˆi yˆ master , i 1,, N . (24) c) An information-sharing factor is defined as follows: N 1 1 1, 0 d d 1. (25) ¦ Ji i 1J i Proof: we shall prove using a mathematical induction. First, assuming that at k  1 time epoch information state and its information matrix of master filter is optimal as follows: Ymaster (k  1 | k  1) Y (k  1 | k  1) , i 1,, N , (26) yˆ (k  1 | k  1) , i

yˆ master (k  1 | k  1)

1,, N , (27)

where yˆ and Y are the optimal information state and its information matrix, respectively. The fused information state and its information matrix are sent to local filter as follows: 1 (28) Yi (k  1 | k  1) Ymaster ,

Ji (29) yˆi (k  1 | k  1) yˆ master (k  1 | k  1) . Prediction procedure at each local filter using equation of the centralized information filter (10) and (11) is rewritten as follows: Yi (k | k  1) [ F (k ){Yi (k  1 | k  1)}1 F c(k )  Qi (k )]1 1 [ F (k ){ Ymaster (k  1 | k  1)}1 F c(k )  J i Q(k )]1 Ji 1 1 [ F (k )Ymaster (k  1 | k  1) F c(k )  Q(k )]1 (30) Ji 1 [ F (k )Y 1 (k  1 | k  1) F c(k )  Q(k )]1 Ji 1 Y (k | k  1), i 1,, N , Ji yˆi (k | k  1) Li (k | k  1) yˆi (k  1 | k  1) Li (k | k  1) yˆ master (k  1 | k  1) (31) Li (k | k  1) yˆ (k  1 | k  1)

yˆ (k | k  1). Measurement update of the information matrix at each local filter can be obtain as follows: Yi (k | k ) Yi (k | k  1)  H ic (k ) Ri1 (k ) H i (k ) (32) 1 Ymaster (k | k  1)  H ic (k ) Ri1 (k ) H i (k ).

Y (k | k ).

(33)

dy

Reference Sensor #1

y

LF#1

z 1

Sensor #2 z

R

LF #2 2

ym , Ȗ 1Ym y, Y 1 1 y , Ȗ Y m 2 m y , Y 2 2 y , Ȗ Y m N m

Sensor #N

Master Filter (Fusion)

LF#N

y , Y m m

y , Y N N N Fig. 1 The decentralized information filter proposed in federated form. z

The measurement update of the information state at local filters can be written as yˆi (k | k ) yˆi (k | k  1)  H ic (k ) Ri1(k ) zi (k ) . (34) Therefore, the assimilation equation in the master filter is given by yˆ master

yˆ1    yˆ N

N

¦ yˆi (k | k )

i 1

N

¦ [ yˆi (k | k  1)  H ic (k ) Ri1(k ) zi (k )]

i 1

(35)

N

yˆ (k | k  1)  ¦ H ic (k ) Ri1 (k ) zi (k ) i 1

ˆ

y (k | k ). ̣ Remark 2: The difference between the federated Kalman filter and other decentralized KF is that the former uses an information-sharing factor and contains a fusionreset procedure of the state variable. Under regular conditions, the federated filter can obtain the same optimal solution as that of the centralized information filter. Also, contrary to other decentralized filter, the master filter combines the only filtered information state and its information matrix of a local filter. Then it diminished the number of variable transmitted from local filters to the master filter. The decentralized information filter in federated structure is shown in Fig. 1.

Ji

Hence, the assimilation equation in the master filter is expressed as follows: Ymaster (k | k ) N

¦

N

¦ Yi (k | k )

i 1

1

i 1J i

N

Ymaster (k | k  1)  ¦ H ic (k ) Ri1 (k ) H i (k ) i 1

N

Y (k | k  1)  ¦ H ic (k ) Ri1 (k ) H i (k ) i 1

4. Example To compare the centralized information filter and decentralized information filter in federated structure, the system model and two types of observations model corrupted by additive white noises are given by x(k ) x(k  1) , (36) z1 (k ) x(k )  X1 (k ) , (37) z2 ( k ) x( k )  X 2 (k ) , (38)

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where x(k ), z1 (k ), z2 (k )  ƒ , and X1 (k ) ~ N (0, r1 ) , X 2 (k ) ~ N (0, r2 ) are independent Gaussian noises. Let x(0) ~ N ( x (0), P (0)) . The optimal information state yˆ (k | k ) based on observations Z1 (k ) {z1 (1),, z1 (k )} , Z 2 (k ) {z2 (1),, z2 (k )} (39) is determined by the Centralized information filter. Substituting F (k ) 1 , G (k ) Q (k ) 0 , H (k ) [1 1]c , R(k ) diag[r1, r2 ] , and L(k ) [ L1 (k ) L2 (k )] into (10)(12), we have yˆ (k | k  1) yˆ (k  1 | k  1) , yˆ (k | k )

yˆ (k  1 | k  1) 

yˆ (0 | 0)

Y (0 | 0) xˆ (0 | 0)

where r12

Y (0) x (0) ,

Y (k  1 | k  1) , Y (0 | 0) I,

Y (k | k  1) L(k | k  1)

Y (k | k )

r2 r z1(k )  1 z2 (k ) , r1r2 r1r2 P 1 (0 | 0) Y (0) ,

Y (0)  kr12 '

(40)

yˆ1 (k  1 | k  1) 

1  z1(k ) , y1 (0 | 0) r1

Y1(k | k  1) Y1(k  1 | k  1) , Y1(0 | 0) L1(k | k  1) 1 ,

Y (0) x (0) ,

Table 1. Comparison of the information matrix of two filter structures. k

Y1 (k )

Y2 (k )

1 J1

1 J2

Ymaster (k )

Y (k )

H (k )

0

2

2

.

.

4

2

1

1

3

5/2

0.7

0.3

11/2

7/2

1/4

2

4

3

0.7

0.3

14/2

10/2

1/7







0.7

0.3







10

12

7

0.7

0.3

38/2

34/2

1/31







0.7

0.3







100

102

52

0.7

0.3

308/2

304/2

1/301

Acknowledgement

Y1 (k  1 | k  1) 

k . (42) r2 Then, for the decentralized information filter in federated structure, the information matrix of the master filter, from (18) and (19) at N 2 , is as follows: Ymaster 2Y (0)  kr12 , (43) Y2 (k | k )

This work was supported by Korea Research Foundation, Grant No. KRF-2001-041-E00075.

References

Y ( 0) 

r1  r2 and Ymaster is the actual information r1r2 matrix for the solution of the federated information filter. The results of comparison for two structures are performed at the valves of parameters: r1 1, r2 2, Y (0) 2 . Here, H (k ) | {(Y (k | k )  Ymaster (k | k )} / Y (k | k ) | (%) is relative error of the suggested filter. As is seen from Table 1, the information matrix for the where r12

In this paper, a decentralized information filter in federated structure is provided. The suggested filter structure reduces the amount of communications and avoids the need to calculate inverse covariance matrices. Also, it includes fault-tolerant capability over other decentralized filter techniques. It is mathematical shown that, in terms of information sharing factor, the decentralized information filter with federated structure is equal to the centralized information filter. To quantitative comparison of two structures, for two sensors, an example is provided.

Y (0)

1 k Y ( 0)  (41) r1 r1 and, by the same method as that used in (41), in case of j 2 the following equation can be obtained: Y1(k | k )

5. Conclusions

r1  r2 and Y (k | k ) is the actual information r1r2

matrix. Now, for N 2 the centralized information filtering equations (40) is extended to the decentralized information filter (13)-(15). We denote the information states of the unknown parameter x(k ) based on observations Z1(k ) and Z 2 (k ) by yˆ1 (k | k ) and yˆ 2 (k | k ) , respectively. Using the decentralized information filter equations (13)(15) for j 1, 2 , we obtain the equations for yˆ1 (k | k ) and yˆ 2 (k | k ) as follows: yˆ1(k | k  1) yˆ1(k  1 | k  1) ,  y1(k | k )

decentralized information filter with federated structure is practically the same as the information matrix of the centralized information filter.

[1] B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice Hall, Englewood Cliffs, NJ, 1979, pp. 138-141. [2] Y. Bar-Shalom and X. Li, Multitarget-Multisensor Tracking: Principles and Techniques, YBS, Storrs, CT, 1995, Chapter 8, pp. 429-527. [3] Y. Bar-Shalom, X. Li, and T. Kirubarajan, Estimation with Applications to Tracking and Navigation, John Wiley & Sons, INC, New York, 2001, Chapter 7, pp. 303-308. [4] N. A. Carlson and M. P. Berarducci, “Federated Kalman Filter Simulation Results,” Journal of The Institute of Navigation, vol. 41, no. 3, pp. 297-321, 1994. [5] K. C. Chang, T. Zhi, and R. K. Saha, “Performance Evaluation of Track Fusion with Information Matrix

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Filter,” IEEE Transaction on Aerospace and Electronic Systems, vol. 38, no. 2, pp. 455-466, 2002. [6] C. Y. Chong, S. Mori, and K. C. Chang, “Distributed Multitarget MultiSensor Tracking,” in Y. Bar-Shalom (Ed.), Multitarget-Multisensor Tracking: Advanced Applications, Artech House, Norwood, MA, 1990, Chapter 8, pp. 247-295. [7] M. Farooq and S. Bruder, “Information Type Filters for Tracking a Maneuvering Target,” IEEE Transaction on Aerospace and Electronic Systems, vol. 26, no. 3, pp. 441-454, 1990. [8] H. R. Hashemipour, S. Roy, and A. J. Laub, “Decentralized Structures for Parallel Kalman Filtering,” IEEE Transaction on Automatic Control, vol. 33, no. 1, pp. 88-94, 1988. [9] R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problem,” ASME Journal of Basic Engineering, vol. 82, no. 1, pp. 35-45, March 1960. [10] P. S. Maybeck, Stochastic Models, Estimation and Control, Vol. I., Academic Press, New York, 1979, Chapter 5, pp. 236-242. [11] A. G. O. Mutambara, Decentralized Estimation and Control for Multisensor Systems, CRC Press, Boca Raton, 1998, Chapters 2-3, pp. 19-79. [12] A. G. O. Mutambara and M. S. Al-Haik, “State and Information Estimation for Linear and Nonlinear systems,” Transactions of the ASME Journal of Dynamic Systems, Measurement, and Control, vol. 121, no. 2, pp. 318-320, 1999. [13] M. Oh, V. I. Shin, Y. Lee, and U. J. Choi, “Suboptimal Discrete Filters for Stochastic Systems with Different Types of Observations,” Computers & Mathematics with Applications, vol. 35, no. 3, pp. 17-27, 1998. [14] B. S. Paik and J. H. Oh, “Gain Fusion Algorithm for Decentralized Parallel Kalman Filters,” IEE Proceedings-Control Theory and Applications, vol. 147, no. 1, pp. 97-103, 2000. [15] B. S. Rao and H. F. Durrant-Whyte, “Fully decentralised Algorithm for Multisensor Kalman Filtering,” IEE Proceedings-D, vol. 138, no. 5, pp. 413420, 1991. [16] D. Willner, C. B. Chang, and K. P. Dunn, “Kalman Filter Algorithms for a Multisensor system,” Proceedings of 15th IEEE Conference on Decision and Control, Clearwater, FL, USA, December 1-3, pp. 570574, 1976. [17] Y. Zhu, Z. You, J. Zhao, K. Zhang, and X. Li, “The Optimality for the Distributed Kalman Filtering Fusion,” Automatica, vol. 37, no. 9, pp. 1489-1493, 2001.

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