Research Journal of Applied Sciences, Engineering and Technology 6(10): 1872-1878, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: November 20,2012
Accepted: January 11, 2013
Published: July 20, 2013
Covariance Intersection Fusion Kalman Estimators for Multi-Sensor System with
Colored Measurement Noises
Wen-Juan Qi, Peng Zhang and Zi-Li Deng
Institute of Electronic Engineering, Heilongjiang University, Harbin 150080, China
Abstract: For multi-sensor system with colored measurement noises, using the observation transformation, the
system can be converted into an equivalent system with correlated measurement noises. Based on this method, using
the classical Kalman filtering, this study proposed a Covariance Intersection (CI) fusion Kalman estimator, which
can handle the fused filtering, prediction and smoothing problems. The advantage of the proposed method is that it
can avoid the computation of the cross-covariances among the local filtering errors and can reduce the
computational burden significantly, as well as the CI fusion algorithm can be used in the uncertain system with
unknown cross-covariances. Based on classical Kalman filtering theory, the centralized fusion and three weighted
fusion (weighted by matrices, scalars and diagonal) estimators are also presented respectively. Their accuracy
comparisons are given. The geometric interpretations based on covariance ellipses are also given. The experiment
results show that the accuracy of the CI fuser is higher than that of the each local smoothers and is lower that that of
the centralized fusion Kalman smoother or the optimal fuser weighted by matrix. The MSE curves show that the
accuracy of the CI fuser is close to the optimal fuser weighted by matrix in most instances, which means that our
proposed method has higher accuracy and good performance.
Keywords: Covariance intersection fusion, colored measurement noises, the centralized fusion, weighted fusion
INTRODUCTION
Multisensor information fusion filtering has been
widely applied to many fields, including guidance,
navigation, GPS positioning and so on. Now the
commonly used method of information fusion is
centralized and distributed fusion Kalman methods. The
centralized fusion Kalman filters can give the globally
optimal state estimation by directly combing all the
local measurement equations, but the computation
burden is larger. Distributed fusion Kalman filters are
given by three weighted fusion algorithms with the
matrix weights, scalar weights or diagonal matrix
weights (Deng et al., 2005; Sun et al., 2010; Deng
et al., 2012). Compared with the centralized fuser, the
weighted fuser can reduce the calculation burden, but
they are globally suboptimal.
The above weighting fusion filters require to
calculate the cross-covariances of local filtering errors.
However, in many theoretical and application problems,
the cross-covariance is unknown, or the computation of
the cross-covariances is very complicated (Sun et al.,
2010).
In order to overcome the above drawback and
limitation, the covariance intersection fusion Kalman
method is presented in Julier and Uhlman (1997, 2009),
which can avoid computing local cross-covariance and
can solve the fused filtering problems for multi-sensor
systems with unknown cross-covariance. The accuracy
comparison in Deng et al. (2012) is given only for
systems with uncorrelated white measurement noises.
In this study, a Covariance Intersection (CI) fusion
Kalman estimator is presented for multi-sensor system
with colored measurement noises, whose accuracy is
higher than that of each local Kalman estimator and is
lower than that of the centralized fusion estimator or the
optimal Kalman fuser weighted by matrices and the
accuracy comparison is given.
PROBLEM FORMULATION
Consider a multi-sensor tracking system with
colored measurement noises:
x ( t +=
1) Φ x ( t ) + Γ w ( t )
=
zi ( t ) H i x ( t ) + ηi ( t ) , i = 1, 2 , L
(2)
ηi ( t +=
1) Piηi ( t ) + ξi ( t ) , i = 1, 2 , L
(3)
where,
t
x (t) ∈ Rn
z i (t)∈ 𝑅𝑅𝑚𝑚 𝑖𝑖
= The discrete time
= The state
= The measurement of the i th sensor
Corresponding Author: Wen-Juan Qi, Institute of Electronic Engineering, Heilongjiang University, Harbin, 150080, China
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Res. J. App. Sci. Eng. Technol., 6(10): 1872-1878, 2013
w(t) ∈ 𝑅𝑅𝑒𝑒 , ξ i (t) ∈ 𝑅𝑅𝑚𝑚 𝑖𝑖 = White noises with zero mean
and variance Q and Q ξi ,
respectively
= The colored measurement
η i (t) ∈ 𝑅𝑅𝑚𝑚 𝑖𝑖
noise of the i th sensor
�𝑖𝑖 , 𝑃𝑃𝑖𝑖
= The known appropriate
Φ, Γ, 𝐻𝐻
dimension constant matrices
From Eq.
transformation:
(2)
introducing
the
observation
yi ( t )= zi ( t + 1) − Pi zi ( t )
(4)
we have the new measurement equation:
=
yi ( t ) H i x ( t ) + vi ( t )
Σi =
Φ i Σ i − Σ i H iT ( H i Σ i H iT + Ri ) H i Σ i  Φ i T
−1

The local filtering error variances are given as:
 I n − K fi H i  Σ i , i =
1, 2 , L
Pi ( 0 ) =
and the local filtering error cross-covariance satisfies
the Lyapunov equation:
=
Pij ( 0 ) Ψ fi Pij ( 0 )Ψ Tfj + ∆ fij , i, j = 1, , L ,
(6)
=
vi ( t ) H i Γ w ( t ) + ξi ( t )
(7)
From (7) we easily obtain w(t) and v i (t) are
correlated noise with zeros mean and variance:
=
Ri H i Γ QΓ T H iT + Qξ i
vi ( t ) v ( t )  H i Γ QΓ H
Rij E=
=
T
H
The local steady-state kalman predictors: For multisensor system (1) and (5), the local steady-state Kalman
one-step predictor of the i th sensor is given as:
xˆˆi ( t +=
1| t ) Ψ pi x ( t | t − 1) + K pi yi ( t )
+ Κ fi yi ( t + 1)
+ Γ Si ) Qξ−i1
(19)
(20)
(11)
The local one-step predictor error variances are
computed by Eq. (14)
The local one-step predictor error cross-covariance
satisfies the Lyapunov equation:
=
Σ ij Ψ pi Σ ijΨ pjT + ∆pij , i, j = 1, , L , i ≠ j
∆pij Γ
=
Q
−Κ pi   T
 Si
i
i
(23)
The N-step predictor error variances and crosscovariance are given as the following formula:
− N −2
∑Φ
k
k =0
Γ QΓ T (Φ k )
T
， N ≤ −2
(24)
, N≤ −2
(25)
(12)
=
Ρ ij ( N ) Φ − N −1Σ ij (Φ − N −1 ) +
T
(H Σ H
(22)
xˆˆi ( t=
| t + N ) Φ − N −1 xi ( t + N + 1| t + N ) , N ≤ −2
T
T
i
Sj   Γ T 


Rij   −Κ pjT 
(21)
and the N-step predictor of the i th sensor is:
=
Ρ i ( N ) Φ − N −1Σ i (Φ − N −1 ) +
Φ − Ji Η i
 I n − K fi H i  Φ i , Φ=
Ψ=
i
fi
Σi H
T
i
(10)
T
i
xˆˆi ( t + 1|
t + 1) Ψ fi xi ( t | t ) + Ι n − Κ fi Η i  J i yi ( t )
=
J =ΓSR ,
i
=
Qε i Η i Σ i Η iΤ + Ri
The local steady-state kalman filters: Assume that the
system (1) and (5) is completely observable and
completely stable, then the local steady-state Kalman
filter of the i th sensor is given as:
T
i
(ΦΣ H
(18)
(9)
T
j
where, E is the mathematical expectation operator and
the superscript T denotes the transpose.
Based on the observation transformation, the
system with colored noises (1)-(3) is transformed into
the system with correlated measurement noises (1) and
(5). The aim is to find the local, centralized fusion,
three weighted fusion and CI fusion smoothers 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 +
𝑁𝑁) i = 1, 2, …, L, c, m, s, d, CI, N>0. And compare
their accuracies.
−1
K fi
i =
i i
T
j
T
, K pi
Ψ pi= Φ − Κ pi Η=
i
T
( t ) = QΓ
(17)
T
+ J i Rij J Tj  ×  I n − K j H j  + K fi Rij K Tfj
(8)
T
j
T
i
(16)
T
+ Ι n − Κ fi Η i  Γ QΓ − J i S Γ − Γ S j J
=
H i H iΦ − Pi H i
Si = E  w ( t ) v
i≠ j
+ Ι n − Κ fi Η i   J i Rij − Γ S j  K TfjΨ Tfj
T
T
i
(15)
∆ fij= Ψ fi Κ fi  Rij J Tj − SiT Γ T  Ι n − Κ fj Η j 
(5)
(14)

+ Γ ( Q − Si Ri−1 SiT ) Γ T
+ Ri )
−1
(13)
− N −2
∑Φ
k =0
k
Γ QΓ T (Φ k )
T
where the ∑ i and ∑ ij are the one-step error variances
where, ∑ i is the one-step predicting error variance and
and cross-covariance which are obtained by the Eq.
satisfies the steady-state Riccati equation:
(14) and (21).
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Res. J. App. Sci. Eng. Technol., 6(10): 1872-1878, 2013
The local steady-state kalman smoothers: For multisensor system (1) and (5), the local steady-state Kalman
N-step smoother of the i th sensor is given as:
N
vc ( t ) = v1T ( t ) , , vLT ( t ) 
xˆˆc ( t + 1|
t + 1) Ψ fc xc ( t | t ) + Ι n − Κ fc Η c  J c yc ( t )
=
k =0
(26)
Ψ𝑇𝑇𝑇𝑇
𝑝𝑝𝑝𝑝
+Κ fc yc ( t + 1)
Ι n − Κ fc Η c  Φ c , Φ=
Φ − JcΗ c
Ψ=
c
fc
where, we define that
=
∑ i is the onestep predicting error variance and 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 − 1) is the
local steady-state one-step predictor.
N -step error variance matrix and error crosscovariance of smother are obtained by:
N
,N >0
(28)
k =0
N
r 0=
s 0
=
N
+ ∑∑ Κ i ( r ) Ε ij ( r , s ) Κ
r 0=s 0
=
Τ
j
 R1  R1L  ,
Sc
Rc =     
 RL1  RL 
min ( r , s )
∑
k =1
−Κ pi 

+Γ ( Q − Sc Rc−1 ScT ) Γ T
(42)
The centralized fusion error variance is given by:
Pc ( 0=
)
[Ι n − Kc Η c ] Σ c
(43)
For the system (1) and (34), the centralized fusion
steady-state Kalman N-step predictor is given as:
(30)
xˆˆc ( t | =
t + N ) Φ − N −1 xc ( t + N + 1| t + N ) , N ≤ −2
(44)
Ψ=
Φ c − K pc H c , K pc = Φ c Κ fc
pc
(45)
=
Ρc ( N )
(31)
Ε=
Η i Σ ij Η + Rij
ij ( 0, 0 )
(41)
−1
when min (r, s) = 0:
Τ
j
= [ S1 , , S L ]

S j   Γ T  Τ( s −k ) Τ
Η j + Rij δ rs

Ψ
Rij   −Κ pjT  pj
Q
× T
 Si
(40)
Σc =
Φ c Σ c − Σ c H cT ( H c Σ c H cT + Rc ) H c Σ c  Φ c T
(s), N > 0
Η iΨ pir − k Γ
−1
where, ∑ c satisfies the steady-state Riccati equation:
(29)
where, ∑ ij is the one-step predicting error crosscovariance and E ij (r, s) = E[𝜀𝜀1 (𝑡𝑡 + 𝑟𝑟)𝜀𝜀2𝑇𝑇 (𝑡𝑡 + 𝑠𝑠)] .
when min (r, s)>0, we have:
Ε ij ( r , s ) =
Η iΨ pir Σ ijΨ pjΤΤs Η j +
(39)
J c=
= Γ Sc Rc−1 , Κ fc Σ c Η cT Η c Σ c Η cT + Rc 
N
s
Ρ ij ( N ) =
Σ ij − ∑ Κ i ( r )Η iΨ pir Σ ij − ∑ Σ ijΨ pjTΤΤ
Η j Κ j (s)
N
(38)
(27)
�Ψ𝑇𝑇𝑝𝑝𝑝𝑝 �𝑘𝑘,
Ρ i ( N=
) Σ i − ∑ Κ i ( k ) Qε i Κ iT ( k )
(37)
For the system (1) and (34), the centralized fusion
steady-state Kalman filter is given as:
xˆˆi ( t | t + N
=
) xi ( t | t − 1) + ∑ Κ i ( k ) ε i ( t + k ) , N > 0 , i = 1, 2 , L
−
k
ΤΤ1
=
Κ i ( k ) Σ=
0, , N
iΨ pi Η i Qε i , k
T
r−


Ε ij ( r , 0 ) =Η iΨ pir Σ ij Η Τ1
j + Η iΨ pi  Γ S j − Κ pi Rij 
(32)
s
Ε ij ( 0,
s ) Η i Σ ijΨ pjΤΤT
Η j T+  Si Γ T − Rij Κ pj ΨT pjT ( s −1) Η j
=
(33)
Φ − N −1Σ c (Φ − N −1 ) +
T
− N −2
∑Φ
k =0
k
Γ QΓ T (Φ k )
T
, N ≤ −2
(46)
For the system (1) and (34), the centralized fusion
steady-state Kalman smoother is given as:
N
xˆˆc ( t | t + =
N ) xc ( t | t − 1) + ∑ Κ c ( k ) ε c ( t + k ) , N > 0
(47)
k =0
ΤΤ1
k
−
The centralized fusion steady-state kalman =
Κ c ( k ) Σ=
0, , N
cΨ pc Η c Qε c , k
estimators: Introducing the augmented measurement
equation:
Ψ pc= Φ − Κ pc Η c
=
yc ( t ) H c x ( t ) + vc ( t )
(34)
=
K pc
with the definitions:
yc ( t ) =  y
T
1
( t ) , , y ( t )
T
L
T
c
H cT + Γ Sc ) Qξ−c1
=
Qε c Η c Σ c Η cΤ + Rc
T
L
H c =  H , , H 
T
1
(ΦΣ
T
(48)
(49)
(50)
(51)
(35)
 R1  R1L 
, S = [ S1 , , S L ]
Rc =      c
 RL1  RL 
(36)
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Res. J. App. Sci. Eng. Technol., 6(10): 1872-1878, 2013
Ai = diag ( ai1 , , ain ) ， ∑ a=
1,=
j 1, , n
ij
where, 𝑥𝑥�𝑐𝑐 (𝑡𝑡|𝑡𝑡 − 1) is the certralized fusion steady-state
one step predictor:
xˆˆc ( t | t=
− 1) Ψ pc xc ( t − 1| t − 2 ) + K pc yc ( t − 1)
L
(53)
The optimal diagonal matrix weights are given by:
−1
−1
−1
 a1 j , , aLj  = eT ( P ii ) e  eT ( P ii )


The fused error variance is given by:
N
(54)
Ρ c ( N=
) Σ c − ∑ Κ c ( k ) Qε c Κ cT ( k )
k =0
The three weighted fusion steady-state kalman
estimators: When the local smoothing error variance
Pi(N) (i = 1, …, L) and cross-covariances P ij (N) (i≠j)
are exactly known, the three weighted fusion steadystate Kalman estimators are given as follows:
The optimal fusion estimator weighted by matrices
is:
L
∑ Ω x (t | t + N ) ,
xˆˆm ( t =
|t + N)
i
i =1
i
N = 0, N < 0, N > 0
(55)
−1
where eT = [1, …, 1], Pii = P ii = ( Pskii )
(63)
L× L
, s, k = 1, , L ,
Pskii is the ( i, i ) diagonal element of Psk and the fused
error variance is given by:
L
L
Pd ( N ) = ∑∑ Ai Pij ( N ) ATj
(64)
=i 1 =j 1
The Covariance Intersection (CI) fusion steady-state
kalman smoother: When the local smoothing error
variance Pi ( i = 1, , L ) are exactly known, but the
cross-covariances Pij , ( i ≠ j ) are unknown, using the CI
with the constraints ∑𝐿𝐿𝑖𝑖=1 Ω𝑖𝑖 = 𝐼𝐼𝑛𝑛 .
The optimal matrix weights are given by:
Ω1 , , Ω L  = ( eT P −1e ) eT P −1
(62)
i =1
fusion algorithm ,the CI fusion Kalman smoother
without cross-covariances is presented as follows:
(56)
L
=
xˆˆCI ( t | t + N ) PCI ( N ) ∑ ωi Pi −1 ( N ) xi ( t | t + N )
(65)
i =1
where, eT = [I n , …, I n ],P = (P ij (N)) nL×Ln the optimal
fused error variance matrix is:
Pm ( N ) = ( eT P −1e )
−1
(57)
The optimal fusion smoother weighted by scalars is:
L
∑ ω x (t | t + N )
xˆˆs ( t | =
t + N)
i i
i =1
(58)
−1
eT Ptr−1
ωi
(59)
where, the symbol tr denotes the trace of matrix and eT
= [1, …, 1], P tr = (trP ij (N)) L×L .
The optimal fused error variance is given as:
L
L
Ps ( N ) = ∑∑ ωiω j Pij ( N )
xˆˆd ( t | =
t + N)
∑ A x (t | t + N )
i =1
with the constraints:
i i
(61)
−1

 
 L

tr   ∑ ωi Pi −1 ( N )  
ωi ∈[ 0,1]
i
=
1




1
ω1 ++ ωL =


min
T
PCI ( N ) = E  xCI ( t | t + N ) xCI
( t | t + N )
(60)
The optimal fusion smoother weighted by diagonal
matrix is:
(66)
(67)
This is a nonlinear optimization problem with
constraints in Euclidean space R L , it can be solved by
“fmincon” function in MATLAB toolbox.
The cross-covariances can be obtained by the local
steady-state Kalman smoothing formula (29), so the
actual fused error variance PCI ( N ) is given by:
=i 1 =j 1
L
−1
with the constraints ∑𝐿𝐿𝑖𝑖=1 𝜔𝜔𝑖𝑖 = 1, 𝜔𝜔𝑖𝑖 ≥ 0, where, the
optimal weighted coefficients 𝜔𝜔𝑖𝑖 (1, … , 𝐿𝐿) are
determined by minimizing the performance index such
that:
min tr PCI ( N ) =
with the constraints ∑𝐿𝐿𝑖𝑖=1 𝜔𝜔𝑖𝑖 = 1.
The optimal scalar weights are given by:
[ω1 ,ωL ] = ( eT Ptr−1e )
 L

PCI ( N ) =  ∑ ωi Pi −1 ( N ) 
 i =1

 L L

= PCI ( N )  ∑∑ ωi Pi −1 ( N ) Pij ( N ) Pj−1 ( N ) ω j  PCI ( N )
i
j
=
=
1
1


(68)
Form Eq. (66) the P CI (N) only depend on the local
error variance P i (i = 1, …, L), but is independent on
cross-covariances P ij , (i ≠ j). Form (68) the actual error
variance 𝑃𝑃�𝐶𝐶𝐶𝐶 (𝑁𝑁) not only depend on the local error
variance P i (i = 1, …, L) but also dependent on cross-
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Res. J. App. Sci. Eng. Technol., 6(10): 1872-1878, 2013
covariances P ij , (i ≠ j). So P CI (N) is the common upper
bound of the actual fused error variance.
The accuracy comparison of local and fused kalman
smoothers: For the multi-sensor system (1)and (5)with
exactly known local error variances P i (i = 1, …, L) and
cross-covariance P ij , (i ≠ j), the accuracy relations
based on error covariance matrix and their trace are
given as:
tr Pc ( N ) ≤ tr Pm ( N ) ≤ tr Pd ( N ) ≤ tr Ps ( N ) ≤ tr Pi ( N )
(69)
tr Pc ( N ) ≤ tr Pm ( N ) ≤ tr PCI ( N ) ≤ tr PCI ( N ) ≤ tr Pi ( N )
(70)
Pc ( N ) ≤ Pm ( N ) ≤ PCI ( N ) ≤ PCI ( N )
(71)
Pm ( N ) ≤ Pd ( N ) , Pm ( N ) ≤ Ps ( N ) , Pm ( N ) ≤ Pi ( N )
(72)
Table 1: The accuracy comparison of the local and fused Kalman
smoothers
trP 1 (−2)
trP 2 (−2)
trP 3 (−2)
trP c (−2)
trP m (−2)
0.83743
0.75414
0.64807
0.2902
0.31242
trP d (−2)
trP s (−2)
trP CI (−2)
𝑡𝑡𝑡𝑡𝑃𝑃�𝐶𝐶𝐶𝐶 (−2)
0.35446
0.39103
0.41673
0.60245
trP 1 (0)
trP 2 (0)
trP 3 (0)
trP c (0)
trP m (−2)
0.57428
0.61503
0.43132
0.18285
0.20153
trP d (0)
trP s (0)
trP CI (−2)
𝑡𝑡𝑡𝑡𝑃𝑃�𝐶𝐶𝐶𝐶 (0)
0.21842
0.25805
0.2703
0.4048
trP 1 (2)
trP 2 (2)
trP 3 (2)
trP c (−2)
trP m (2)
0.40457
0.54129
0.29834
0.13706
0.152
trP d (0)
trP s (0)
trP CI (0)
𝑡𝑡𝑡𝑡𝑃𝑃�𝐶𝐶𝐶𝐶 (0)
0.1564
0.18525
0.20226
0.27976
where, the matrix inequality A≤B means that B−A≥0 is
positive semi-definite.
Remark1: the accuracy of the local and fused
smoothers is defined as the trace of their error
variances, the smaller trace means higher accuracy and
the larger trace means lower accuracy.
Fig. 1: The variance ellipses of filters
SIMULATION RESULTS
Consider the 3-sensor tracking system with colored
measurement noises:
x ( t +=
1) Φ x ( t ) + Γ w ( t )
=
zi ( t ) H i x ( t ) + ηi ( t ) , i = 1, 2,3
ηi ( t +=
1) Piηi ( t ) + ξi ( t ) , i = 1, 2, 3
(73)
(74)
(75)
In the simulation, we take:
2
0
�1 = [1 0],
T0 = 0.2, Φ = �1 T0 � , Γ = �0.5𝑇𝑇
�,
𝐻𝐻
𝑇𝑇0
0 1
�2 �1 0�, 𝐻𝐻
�3 = [1 0], N = 0, −2, 2, P 1 = 0.3, P 2 =
𝐻𝐻
0 1
diag (0.16, 0.3), P3 = 0.4, t = 1, …, 300, Q = 1, Q ζ1 = 1,
Q ζ2 = diag (2, 0.15), Q ζ1 = 0.49
The accuracy comparison of the local, the
centralized fuser, three weighted fusers and the CI fuser
are compared in Table 1.
Form Table 1, we see that the accuracy of the
centralized fuser trP c (N) is the highest, the actual
accuracy 𝑡𝑡𝑡𝑡𝑃𝑃�𝐶𝐶𝐶𝐶 (𝑁𝑁) is higher than the local smoothers,
but lower than the optimal fusion smoother weighted by
matrix trP m (N). The accuracy weighted by scalars
trP s (N) is close to that weighted by diagonal matrix
trP d (N) and both of them are lower than the
accuracy weighted by matrix. Table 1 verifies the
accuracy relations (69) and (70).
Fig. 2: The variance ellipses of predictors
In order to give a geometric interpretation of the
accuracy relations, the variance ellipse is defined as the
locus of points {x: xTP−1x = c}, where P is the variance
matrix and c is a constant. Generally, we select c = 1. It
has been proved in Deng et al. (2012) that P 1 ≤P 2 is
equivalent to that the variance ellipse of P1 is enclosed
in that of P2 .the accuracy comparison of the variance
ellipses is shown in Fig. 1-3.
Form Fig. 1-3, we see that the covariance ellipse of
P m (N) is enclosed in the ellipse of 𝑃𝑃�𝐶𝐶𝐶𝐶 (𝑁𝑁) and the
ellipse of PCI ( N ) is enclosed in that of P CI (N). The
ellipse of P s (N) and P d (N) encloses the ellipse of
P m (N). the ellipse of P c (N) is enclosed in all ellipses for
P i (N), i
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Res. J. App. Sci. Eng. Technol., 6(10): 1872-1878, 2013
Fig. 3: The variance ellipses of smoothers
Fig. 6: The MSE curves of local and fused smoothers
In order to verify the above theoretical accuracy
relations, the Mean Square Error (MSE) value at time t ,
ρ = 200 for local and fused Kalman smoothers is
defined as:
1
MSE i =
(t )
ρ
(
× x(
ρ
∑ ( x( ) ( t ) − xˆ ( ) ( t | t + N ) )
j)
j
j =1
j
Τ
i
(76)
0, −2, 2
( t ) − xˆi( j ) ( t | t + N ) ) , N =
(𝑗𝑗 )
𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁) or x(j)(t) denotes the jth realization of
(𝑗𝑗 )
Fig. 4: The MSE curves of the local and fused filters
𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁)or x(t).
The MSE curves of the local and fused estimators
are shown in Fig. 4-6.
Form Fig. 4-6, we see that the MSE i (t) values of
the local and fused Kalman estimators are close to the
corresponding theoretical trace values, when t is large
enough, according to the ergodicity of the sample
function, we have:
MSEi ( t ) → trPi ( N ) , ρ → ∞, t → ∞ i = 1, 2,3, m, s, d
MSE CI ( t ) → trPCI ( N ) , ρ → ∞, t → ∞
(77)
(78)
Form Fig4-6, we see that the accuracy relations
(69) and (70) hold.
CONCLUSION
For the multi-sensor system with colored
measurement noises, it converted into an equivalent
system with correlated noises by the observation
transformation. Based on the classical Kalman filtering,
the CI fuser without cross-covariance has been
presented. The centralized fusion Kalman filter and
Fig. 5: The MSE curves of the local and fused predictors
three weighted fusers have been also presented and the
accuracy comparisons of these fusers were given by a
= 1, 2, 3, P θ (N), m, s, d, 𝑃𝑃�𝐶𝐶𝐶𝐶 (𝑁𝑁) and P CI (N), which
Mote-Carlo simulation example. In this study, the CI
fusion results of the Deng et al. (2012) have been
verifies the correctness of matrix relations (71) and
extended to the case with colored measurement noises.
(72).
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Res. J. App. Sci. Eng. Technol., 6(10): 1872-1878, 2013
ACKNOWLEDGMENT
This study was supported by the National Natural
Science Foundation of China under Grant NSFC60874063, the Graduate innovation fund supported by
Educational commission of Heilongjiang Province
(YJSCX2012-263HLJ), Young Professionals in
Regular Higher Education Institutions of Heilongjiang
Province（1251G012).
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