Sequential Fusion Kalman Filter Peng Zhang

Wenjuan Qi, Zili Deng

Department of Automation Heilongjiang University Harbin,China [email protected]

Department of Automation, Heilongjiang University Harbin,China [email protected] The problem is reduced to the inverse operation of several lower-dimensional matrices. It is proved that its accuracy is higher than that of each local Kalman filter, and is lower than that of the batch fusion (BF) Kalman filter weighted by matrices. The geometric interpretation of accuracy relations based on the covariance ellipses is given. Two simulation examples are given to show the above accuracy relations, and verify the correctness.

Abstract—For the multisensor linear discrete time-invariant system, the batch fusion (BF) Kalman filtering algorithm needs the inverse operation of a high-dimensional matrix, which yields a larger computational burden and computational complexity. A sequential fusion (SF) Kalman filter is presented in this paper, which can significantly reduce the computational burden. It is equivalent to several two-sensor Kalman fusers weighting by matrices, and is a recursive two-sensor Kalman fuser. It is proved that its accuracy is higher than that of each local estimator and is lower than that of the batch fusion Kalman filter weighted by matrices. The geometric interpretation of accuracy relations based on the covariance ellipses is given. Two simulation examples for multisensor tracking systems show that its actual accuracy is not very sensitive with respect to the orders of sensors, and is close to the accuracy of the optimal batch fusion Kalman filter.

II.

Consider the multisensor systems

Keywords- multisensor information fusion; sequential fusion; batch fusion; Kalman filter;accuracy; covariance ellipse; sensitivity.

I.

BATCH FUSION KALMAN FILTER WEIGHTING BY MATRICES

x(t + 1) = Φ x(t ) + Γ w(t )

(1)

yi (t ) = H i x(t ) + vi (t ), i = 1, " , L

(2)

where t is the discrete time, x(t ) ∈ R n is the state, yi (t ) ∈ R mi

INTRODUCTION

The multisensor information fusion has received great attentions in the recent years, and has been widely applied to many fields such as guidance, robotics, target tracking, GPS position [1,2]. Its aim is to combine the local estimators obtained from each sensor, to obtain the fused estimator, whose accuracy is higher than that of each local estimator. Basic fusion methods include the centralized and distributed fusion methods, depending on whether raw data are used directly for fusion or not [3]. Compared with the former, the latter method can significantly reduce the computational burden and can facilitate fault detection and isolation more conveniently. The three optimal fusion rules weighted by matrices, diagonal matrices, and scalars were presented in [412], respectively.

and vi (t ) ∈ R mi are the measurement and measurement noise of the ith sensor, respectively. L is the number of sensors, w(t ) ∈ R r and vi (t ) are uncorrelated white noises with zero mean and variances Qw and Qvi , respectively. (Φ , H i ) is a completely observable pair, (Φ , Γ ) is a completely controllable pair. Based on the ith sensor, the local steady-state Kalman filter is given by [14]

It is well known that to compute the fused error variance requires to compute the inverse of a high dimension matrix for batch fusion (BF) Kalman filter, i.e., the optimal fused Kalman filter weighted by matrices, whose dimension depends on the number of sensors, if the number of sensors is large, the computational burden is large. To overcome this disadvantage, by the sequential processing, based on the sequential twosensor fusion Kalman filters with matrix weights, a fast sequential fusion (SF) Kalman filter is presented, which is a fast recursive two-sensor SF Kalman filter. The principle of the proposed sequential processing method is similar to that in [13]. This work is supported by the Natural Science Foundation of China under grant NSFC-60874063, Automatic Control Key Laboratory of Heilongjiang University and Support Program for Young Professional in Regular Higher Education Institutions of Heilongjiang Province with No. 1251G012.

2140

xˆi (t | t ) = Ψ i xˆi (t − 1 | t − 1) + K i yi (t ), i = 1," , L

(3)

Ψ i = [ I n − K i H i ]Φ

(4)

K i = Σ i H iΤ [ H i Σ i H iΤ + Qvi ]−1

(5)

where I n denotes the n × n unit matrix, Τ denotes the transpose, Σ i satisfies the Riccati equation

Σ i = Φ [Σ i − Σ i H iΤ ( H i Σ i H iΤ + Qvi ) −1 H i Σ i ]Φ Τ + Γ Qw Γ Τ (6) The local filtering error covariance are given

Pi = [ I n − K i H i ]Σ i , i = 1," , L

Pm = P2 − ( P2 − P21 )( P1 + P2 − P12 − P21 )−1 ( P2 − P21 )Τ (16)

(7)

and the cross-covariances Pij of the local filtering errors satisfy the Lyapunov equations [5] Pij = Ψ i PijΨ jΤ + Δ ij , i, j = 1," , L

III.

SEQUENTIAL FUSION KALMAN FILTER WEIGHTING BY MATRICES

In order to reduce the complexity and computational burden, by the sequential processing, based on the ( L − 1) two-sensor Kalman fusers with matrix weights, a recursive two-sensor Kalman fuser with matrix weights is presented. It consists of the ( L − 1) two-sensor Kalman fusers with matrix weights as shown in Fig.1. It is realized by the ( L − 1) steps as follows:

(8)

where Δ ij = [ I n − K i H i ]Γ Qw Γ Τ [ I n − K j H j ]Τ , and we define Pi = Pii .

When Pi and Pij are known, by the batch processing, the batch fusion Kalman filter weighted by matrices is given by

xˆ1 , P1

L

xˆBF (t | t ) = ∑ Ωi xˆi (t | t )

(9)

xˆ1SF , P1SF

xˆ2 , P2

i =1

in the sense that xˆBF (t | t ) is the linear unbiased minimum variance (LUMV) estimate of x(t ) , and the optimal matrix weights are given by [4,5]

xˆ3 , P3

#

⎡ P11 " P1L ⎤ [Ω1 ," , Ω L ] = (eΤ P −1e)−1 eΤ P −1 , P = ⎢⎢ # # ⎥⎥ (10) ⎢⎣ PL1 " PLL ⎥⎦

xˆ2SF , P2SF

#

#

xˆLSF− 2 , PLSF− 2

xˆL , PL

xˆLSF−1 , PLSF−1

where eΤ = [ I n , " , I n ] , the fused error covariance is given by PBF = (eΤ P −1e) −1

xˆSF = xˆLSF−1

(11)

PSF = PLSF−1

From (10) and (11) we see that in order to compute the optimal matrix weights Ω1 ," , Ω L , we need to compute the inverse of the high dimensional matrix P , where P is an nL × nL matrix. Remark 1. Particularly, when L = 2 , the batch fusion Kalman filter weighted by matrices is given by [9]

xˆm (t | t ) = Ω1 xˆ1 (t | t ) + Ω 2 xˆ2 (t | t )

(12)

where Ω1 = ( P2 − P21 )( P1 + P2 − P12 − P21 )

−1

Fig. 1 The sequential fusion Kalman filter

Step 1. Based on the local estimators xˆ1 (t | t ) with covariance P1 and xˆ2 (t | t ) with covariance P2 , and the crosscovariance P12 between the two local estimators, the twosensor matrix weighting Kalman fuser xˆ1SF (t | t ) with the fused error covariance P1SF can be obtained by the two-sensor fused algorithm (12)-(16), i.e.,

or

−1 Ω (1) 2 = ( P1 − P12 )( P1 + P2 − P12 − P21 )

(18)

ˆ xˆ1SF (t | t ) = Ω1(1) xˆ1 (t | t ) + Ω(1) 2 x2 (t | t )

(19)

P1SF = P1 − ( P1 − P12 )( P1 + P2 − P12 − P21 )−1 ( P1 − P12 )Τ

(20)

(14)

The fused error covariance is given by Pm = P1 − ( P1 − P12 )( P1 + P2 − P12 − P21 ) ( P1 − P12 )

(17)

(13)

Ω 2 = ( P1 − P12 )( P1 + P2 − P12 − P21 ) −1

−1

Ω1(1) = ( P2 − P21 )( P1 + P2 − P12 − P21 )−1

Τ

(15) and define the estimation error x1SF (t | t ) = x(t ) − xˆ1SF (t | t ) , xi (t | t ) = x(t ) − xˆi (t | t ) , according to the constrains

2141

PLSF−1 = PLSF− 2 − ( PLSF− 2 − PLSF− 2, L )( PLSF− 2 + PL − PLSF− 2, L − PLSF− 2,ΤL )−1 ( PLSF− 2 − PLSF− 2, L )Τ

SF Ω1(1) + Ω(1) 2 = I n , the cross-covariance P1,3 between the local

(30)

estimator xˆ3 (t | t ) and the fused estimator xˆ1SF (t | t ) is given as P1,3SF = Ε[ x1SF (t | t ) x3Τ (t | t )] = Ω1(1) P13 + Ω(1) 2 P23

(21)

Step 2. Similarly, based on the first fused estimator xˆ1SF (t | t ) with covariance P1SF and local estimator

The sequential fusion (SF) Kalman filter xˆSF (t | t ) weighted by matrix with the fused error covariance PSF is defined as

SF 1,3

xˆ3 (t | t ) with covariance P3 , and the cross-covariance P

xˆSF (t | t ) = xˆLSF−1 (t | t )

(31)

PSF = PLSF−1

(32)

between xˆ1SF (t | t ) and xˆ3 (t | t ) , the two-sensor matrix weighting Kalman fuser xˆ2SF (t | t ) with the covariance P2SF can be obtained as SF Τ −1 1,3

(22)

Ω(22) = ( P1SF − P1,3SF )( P1SF + P3 − P1,3SF − P1,3SF Τ ) −1

(23)

xˆ2SF (t | t ) = Ω1(2) xˆ1SF (t | t ) + Ω 2(2) xˆ3 (t | t )

(24)

( 2) 1

Ω

SF 2

P

SF 1

=P

SF Τ 1,3

= ( P3 − P

SF 1

SF 1,3

− (P

SF 1

)( P

SF 1

− P )( P

SF 1,3

+ P3 − P

SF 1,3

+ P3 − P

SF 2,4

and the cross-covariance P xˆ4 (t | t ) is written as

−P

)

SF Τ −1 1,3

−P

SF 1

) (P

Theorem 1. For the multisensor system (1) and (2), by the sequential fusion processing, based on the two-sensor fusion Kalman filters with matrix weights, a recursive expression of sequential fusion (SF) Kalman filters are (i ) ˆ xˆiSF (t | t ) = Ω1(i ) xˆiSF −1 (t | t ) + Ω 2 xi +1 (t | t ), i = 1, " , L − 1 (33)

where

SF Τ 1,3

−P ) (25)

SF 2

between xˆ (t | t ) and

Ω1(i ) = ( Pi +1 − Pi −SF1,Τi +1 )( Pi −SF1 + Pi +1 − Pi −SF1,i +1 − Pi −SF1,Τi +1 ) −1

(34)

Ω (2i ) = ( Pi −SF1 − Pi −SF1,i +1 )( Pi −SF1 + Pi +1 − Pi −SF1,i +1 − Pi −SF1,Τi +1 ) −1

(35)

the fused error covariance matrix Pi SF and fused error crosscovariance Pi −SF1,i +1 are given as

SF (2) P2,4 = Ε[ x2SF (t | t ) x4Τ (t | t )] = Ω1( 2) Ω1(1) P14 + Ω1(2) Ω(1) 2 P4 + Ω 2 P34 (26)

Pi SF = Pi −SF1 − ( Pi −SF1 − Pi −SF1, i +1 )( Pi −SF1 + Pi +1 − Pi −SF1,i +1 − Pi −SF1,Τi +1 ) −1 ( Pi −SF1 − Pi −SF1,i +1 ) Τ

(36)

#

Step L − 1 . Based on the L − 2 step fused estimator ˆxLSF− 2 (t | t ) with covariance PLSF− 2 and the local estimator

Pi −SF1,i +1 = Θ1(i −1) P1,i +1 + Θ(2i −1) P2,i +1 + " + Θi(i −1) Pi ,i +1

xˆL (t | t ) with covariance PL and the cross-covariance PLSF− 2, L

where

between xˆLSF− 2 (t | t ) and xˆL (t | t ) , the two-sensor matrix weighting Kalman fuser xˆLSF−1 (t | t ) with the covariance PLSF−1 can be obtained as Ω1( L −1) = ( PL − PLSF− 2,ΤL )( PLSF− 2 + PL − PLSF− 2, L − PLSF− 2,ΤL ) −1

(37)

Θ1(i ) = Ω1( i ) Θ1( i −1) , Θ(2i ) = Ω1(i ) Θ2(i −1) ," , Θi(i ) = Ω1( i ) Θi( i −1) , Θi(+i )1 = Ω(2i ) (38)

(27)

with the initial condition xˆ0SF (t | t ) = xˆ1 (t | t ) , P0SF = P1 , Ω

( L −1) 2

SF L−2

= (P

SF L − 2, L

−P

SF L−2

)( P

SF L − 2, L

+ PL − P

SF Τ −1 L − 2, L

−P

xˆLSF−1 (t | t ) = Ω1( L −1) xˆLSF− 2 (t | t ) + Ω (2L −1) xˆL (t | t )

)

(28)

(29)

P0,SFi+1 = P1,i +1 , Θ1(0) = I n . Proof. Applying two-sensor Kalman fuser with matrix weights (12)-(16), and iterating step 1~step ( L − 1) , the general expressions of SF Kalman filters weighted by matrices (33)(38) can be obtained. The proof is completed.

2142

Theorem 2. For the multisensor system (1) and (2), by the sequential fusion processing, based on the two-sensor fusion Kalman filters with matrix weights, a non-recursive expression of sequential fusion (SF) Kalman filters are SF i

(i ) 1 1

(i ) i i

(i ) i +1 i +1

Proof. Applying the mathematical induction. For L = 2 , from (12)-(15), we have

xˆ (t | t ) = Θ xˆ (t | t ) + " + Θ xˆ (t | t ) + Θ xˆ (t | t ), i = 1, ", L − 1

where Θ

(i ) j

i.e., the accuracy of the SF Kalman filter is higher than that of each local Kalman filter.

(39)

P1SF ≤ P1 , P1SF ≤ P2 , PSF = P1SF

are given by (38).

i.e., (45) holds for the case L = 2 .

Proof. Applying the mathematical induction method, for i = 2 , from Theorem 1, we have

xˆ2SF (t | t ) = Ω1(2) xˆ1SF (t | t ) + Ω 2(2) xˆ3 (t | t )

For L = 3 , from (33), we have

(40)

ˆ xˆ2SF (t | t ) = Ω1(2) xˆ1SF (t | t ) + Ω (2) 2 x3 (t | t )

(47)

P2SF ≤ P1SF , P2SF ≤ P3 , PSF = P2SF

(48)

and

Substituting (19) and (38) into (40), we can obtain SF 2

xˆ (t | t ) (2) ˆ ˆ = Ω1(2) Ω1(1) xˆ1 (t | t ) + Ω1(2) Ω(1) 2 x1 (t | t ) + Ω 2 x3 (t | t )

Substituting (46) into (48), we can obtain

(41)

(2) ˆ ˆ = Θ1(2) xˆ1 (t | t ) + Θ(2) 2 x2 (t | t ) + Θ3 x3 (t | t )

P2SF ≤ P1 , P2SF ≤ P2 , P2SF ≤ P3 , PSF = P2SF

If (39) holds for i = m , i.e.,

If for L = n we have

then when i = m + 1 , from (33),

xˆnSF−1 (t | t ) = Ω1( n −1) xˆnSF− 2 (t | t ) + Ω(2n −1) xˆn (t | t )

(50)

PnSF −1 ≤ Pi , i = 1, " , n

(51)

and

xˆmSF+1 (t | t ) = Ω1( m +1) xˆmSF (t | t ) + Ω(2m +1) xˆm + 2 (t | t )

(43)

Then for L = n +1 , we xˆnSF (t | t ) = Ω1( n ) xˆnSF−1 (t | t ) + Ω(2n ) xˆn +1 (t | t ) holds and

Substituting (38) and (42) into (43), we have xˆmSF+1 (t | t ) =Ω

(m) 1 1

(Θ

xˆ (t | t ) + " + Θ

(49)

i.e., (45) holds for the case L = 3 .

xˆmSF (t | t ) = Θ1( m ) xˆ1 (t | t ) + " + Θ(mm ) xˆm (t | t ) + Θ(mm+)1 xˆm +1 (t | t ) (42)

( m +1) 1

(46)

(m) m +1 m +1

xˆ

(t | t )) + Ω

( m +1) 2 m+2

xˆ

(t | t )

= Θ1( m +1) xˆ1 (t | t ) + " + Θ(mm++21) xˆm + 2 (t | t )

have

PnSF ≤ PnSF −1 ≤ Pi , i = 1, " , n

(52)

PnSF ≤ Pn +1

(53)

PSF = PnSF ≤ Pi , i = 1," , L

(54)

so we can obtain (44) IV.

THE ACCURACY ANALYSIS OF SF KALMAN FUSER WEIGHTED BY MATRICES

Theorem 3. For the multisensor system (1) and (2), the local filters and SF Kalman fusers have the accuracy relations

PSF ≤ Pi , i = 1," , L

The proof is completed. Theorem 4. For the multisensor system (1) and (2), the BF and SF Kalman filter have the accuracy relations PBF ≤ PSF

(45)

(55)

where PBF is the covariance of the BF Kalman filter xˆBF (t | t ) , which is given by (9). i.e., the inequality (55) means that the

2143

accuracy of the SF Kalman filter is lower than that of the BF Kalman filter. Proof. The batch fusion Kalman filter with matrix weights xˆBF (t | t ) given by (9) is the linear unbiased minimum variance (LUMV) estimate of x(t ) based the linear space L = L( xˆ1 (t | t )," , xˆ L (t | t )) spanned by the unbiased estimates xˆ1 (t | t )," , xˆL (t | t ) . From theorem 2,

xˆSF (t | t ) = xˆLSF−1 (t | t ) ∈ L( xˆ1 (t | t )," , xˆL (t | t )) , so the inequality of (55) holds.

is not very sensitive with respect to the orders of sensors, and its actual accuracy is close or equal to that of the BF Kalman filter, so it has good performance. Remark 2. From (9)-(11), it is showed that to obtain the BF Kalman filter, we need to compute the inverse of a nL × nL matrix, which has the computational load O(n3 L3 ) , however, by the sequential fusion processing, from theorem 1, the computational load of SF Kalman filter is O(( L − 1) × n3 ) , when L is large, the SF Kalman filter can significant reduce the computational burden compared with BF filter. VI.

V.

THE SF KALMAN FUSER WITH DIFFERENT ORDERS OF SENSORS

Example 1. Consider the tracking system with 3 sensors

For the SF Kalman filter weighted by matrices, the fused schemes can be different with respect to different orders of sensors (i.e. the order of the sequential processing). For example, In the case with L = 4 , there exist the 12 SF fused orders as shown Fig.2, where the notation “ SFijkr ” denotes the fused order that the fuser of the local estimators i and j is fused with the local estimator k , and the obtained fuser is fused with the local estimator r . xˆ1 , P1

SIMULATION EXAMPLES

⎡ 0.5T02 ⎤ ⎡1 T0 ⎤ x(t + 1) = ⎢ x ( t ) + ⎢ ⎥ w(t ) ⎥ ⎣0 1 ⎦ ⎣ T0 ⎦

(56)

yi (t ) = H i x(t ) + vi (t ), i = 1, 2,3

(57)

H1 = [1 0] , H 2 = I 2 , H 3 = [1 0]

(58)

where T0 is the sample period, x(t ) ∈ R 2 is the state, x(t ) = [ x1 (t ), x2 (t )]Τ , x1 (t ) and x2 (t ) are the position and

xˆ1SF , P1SF

velocity of target at time tT0 , yi (t ) is the measurement for

xˆ2 , P2

sensor i , w(t ) and vi (t ) are white noise with zero mean and

xˆ3 , P3

variances Qw and Qvi , respectively. In simulation we take

xˆ2SF , P2SF

that T0 = 0.4 , Qw = 2 , Qv1 = 1 , Qv 2 = diag(49, 0.25) , Qv 3 = diag(4.2,1.5) .

SF 3

xˆSF = xˆ ,

xˆ4 , P4

As shown in Fig.2, for L = 3 , the 3 fused orders are as follows:

PSF = P3SF

SF123, SF132, SF 231

(a) The order 1: SF1234

# xˆ3 , P3

SF 1

In the simulation, the local filtering error covariance Pi (i = 1, 2,3) , the batch fused error covariance PBF and sequential fused filtering error covariances with different orders of sensor PSFijk (i, j , k = 1, 2,3, i ≠ j ≠ k ) are as follows:

SF 1

xˆ , P

xˆ4 , P4 xˆ2 , P2

xˆ2SF , P2SF

xˆ1 , P1

xˆSF = xˆ3SF ,

(59)

⎡ 0.4481 0.4047 ⎤ ⎡1.3673 0.0642 ⎤ P1 = ⎢ ⎥ , P2 = ⎢ 0.0642 0.1649 ⎥ , 0.4047 0.8047 ⎣ ⎦ ⎣ ⎦ ⎡0.8085 0.29 ⎤ ⎡0.1932 0.0346 ⎤ P3 = ⎢ ⎥ , PBF = ⎢0.0346 0.1521⎥ , 0.29 0.5092 ⎣ ⎦ ⎣ ⎦ (60) 0.1932 0.035 0.1936 0.0357 ⎡ ⎤ ⎡ ⎤ PSF 123 = ⎢ ⎥ , PSF 132 = ⎢0.0357 0.1549 ⎥ , ⎣ 0.035 0.1548 ⎦ ⎣ ⎦

PSF = P3SF

( l ) The order : SF 3421 Fig. 2 The fused orders of the SCI fusers in the L = 4 case

It is obviously that the accuracy of the SF Kalman filter is related to the fused order. The problem is whether the accuracy of the SF Kalman filter is very sensitive with respect to the fused orders of sensors or not. The following simulation examples will show that the accuracy of the SF Kalman filter

2144

⎡0.1949 0.0352⎤ PSF 231 = ⎢ ⎥ ⎣0.0352 0.1523⎦

TABLE I.

In order to give a powerful geometric interpretation with respect to the accuracy relation among the local and fused filters, the concept on the covariance ellipse was introduced [3]. The covariance ellipse for a covariance matrix P is defined as the locus of points {x : x Τ P −1 x = c} where c is a constant. In the sequel, c = 1 will be assumed without loss of generality. The following facts were proved [13]: P1 ≤ P2 is equivalent to that the covariance ellipse of P1 is enclosed in the covariance ellipse of P2 . For the 3 orders as an example, the contained relations among the covariance ellipses for the local and fused filters are shown in Fig.3, combining with (60), we see that (i) the ellipse of PBF is enclosed in the ellipses of

THE SENSITIVITY OF ACCURACY OF SF FUSER WITH RESPECT TO THE ORDERS OF SENSORS FOR 3 SENSORS

trP1

1.2928

trP2

1.5322

trP3

1.3177

trPBF

0.34526

trPSF 123

0.34803

trPSF 132

0.34851

trPSF 231

0.34715

Example 2. Consider the tracking system with 4 sensors

⎡ 0.5T02 ⎤ ⎡1 T0 ⎤ x(t + 1) = ⎢ x(t ) + ⎢ ⎥ w(t ) ⎥ ⎣0 1 ⎦ ⎣ T0 ⎦

(61)

yi (t ) = H i x(t ) + vi (t ), i = 1, 2,3, 4

(62)

batch fused filtering error covariance PBF , and the sequential fused filtering errors variances PSFijk (i, j , k = 1, 2,3, i ≠ j ≠ k ) is

H1 = [1 0] , H 2 = I 2 , H 3 = [1 0] , H 4 = I 2

(63)

very close to each other, the corresponding covariance ellipses for PBF and PSFijk are almost coincident and indistinguishable.

where T0 , x(t ), yi (t ), w(t ) and vi (t ) are defined as in Example 1. In simulation we take that T0 = 0.5 , Qw = 3 , Qv1 = 1.5 ,

PSF , which means PBF ≤ PSF . (ii) the ellipse of PSF is enclosed in the ellipses of Pi , which means PSF ≤ Pi , i = 1, 2,3 . (iii) since the sequential fused filtering errors covariances PSFijk (i, j , k = 1, 2,3, i ≠ j ≠ k ) are very close to the

Qv 2 = diag(25, 0.36) , Qv 3 = diag(4, 2) , Qv 4 = diag(0.49,16) .

1

The sensitivity of the accuracy for the SF fuser with respect to the orders of sensors is shown as Table 2.

P1

0.8 0.6

P3

0.4

TABLE II.

PSF 123 P

0.2

SF 132

0

P2

PSF 231

THE SENSITIVITY OF ACCURACY OF SF FUSER WITH RESPECT TO THE ORDERS OF SENSORS FOR 4 SENSORS

trP1

2.2969

-0.2

trP2

1.6951

-0.4

trP3

1.8789

-0.6

trP4

1.2657

-0.8

trPBF

0.38006

trPSF 1234

0.3874

trPSF 1243

0.38959*

trPSF 1324

0.38744

trPSF 1342

0.38822

-1 -1.5

PBF

-1

-0.5

0

0.5

1

1.5

Fig. 3 The accuracy comparison of Pi , i = 1, 2,3, SF123, SF132, SF 231, BF

The sensitivity of the accuracy for the SF fuser with respect to the orders of sensors is shown as Table 1, where the notation tr means the trace operation of matrix. From Table 1, we can see that the accuracy of SF Kalman filter is higher than that of each local Kalman filter and the accuracies of SF Kalman filters with different orders of sensors are close to each other, which shows that the SF Kalman filter is not very sensitive with respect to the orders of the sensors. Compared with trPBF and trPSFijk , the accuracies of SF Kalman filters with different orders of sensors are close to the BF Kalman filter.

trPSF 1423

0.38759

trPSF 1432

0.38725

trPSF 2314

0.38201

trPSF 2341

0.38248

trPSF 2413

0.38469

trPSF 2431

0.38135**

trPSF 3412

0.38521

trPSF 3421

0.38269

From Table 2, we can also obtain the conclusions: (i) the accuracy of SF Kalman filter is higher than that of each local Kalman filter, where the lowest and highest accuracies are

2145

[1]

denoted by asterisks * and **. (ii) the accuracies of SF Kalman filters with different orders of sensors are close to each other, which shows that the SF Kalman filter is not very sensitive with respect to the orders of the sensors. (iii) compared with trPBF and trPSFijkr , the accuracies of SF

[3]

Kalman filters with different orders of sensors are all close to the BF Kalman filter, so it has good performances.

[4]

VII. CONCLUSION For the multisensor systems with L sensors, applying sequence processing, based on the ( L − 1) two-sensor batch fusion Kalman filters, a fast recursive two-sensor weighting Kalman fuser has been presented, which is called the sequential fusion (SF) Kalman filter. Compared with the batch fusion (BF) Kalman filter, when L is large, it can obviously reduce the computational burden and decrease complexity. It was rigorously proved that its accuracy of the SF Kalman filter is higher than that of each local estimator, and is lower than that of the BF Kalman filter with matrix weights. Two simulation examples show that the accuracies of the SF Kalman filters with different orders of sensors are close to that of the BF Kalman filter, and are obviously higher than that of each local estimator. Therefore, the SF Kalman filter has good performances.

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[5] [6]

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ACKNOWLEDGMENT This work is supported by the Natural Science Foundation of China under grant NSFC-60874063, Automatic Control Key Laboratory of Heilongjiang University and Support Program for Young Professional in Regular Higher Education Institutions of Heilongjiang Province with No. 1251G012.

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