Research Journal of Applied Sciences, Engineering and Technology 6(11): 1963-1969,... ISSN: 2040-7459; e-ISSN: 2040-7467

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Research Journal of Applied Sciences, Engineering and Technology 6(11): 1963-1969, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: December 14, 2012
Accepted: January 11, 2013
Published: July 25, 2013
Covariance Intersection Kalman Fuser for Two-sensor System with
Time-delayed Measurements
1, 2
Peng Zhang and 1Zi-Li Deng
Department of Automation, Heilongjiang University, Harbin 150080, China
2
Department of Computer and Information Engineering, Harbin Deqiang College of Commerce, Harbin
150025, China
1
Abstract: For a two-sensor linear discrete time-invariant stochastic system with time-delayed measurements, by the
measurement transformation method, an equivalent system without measurement delays is obtained and then using
the Covariance Intersection (CI) fusion method, the covariance intersection steady-state Kalman fuser is presented.
It can handle the estimation fusion problem between local estimation errors for the system with unknown crosscovariances and avoid a large computed burden and computational complexity of cross-covariances. It is proved that
its accuracy is higher than that of each local estimator and is lower than that of optimal Kalman fuser weighted by
matrices with known cross-covariances. A Monte-Carlo simulation example shows the above accuracy relation and
indicates that its actual accuracy is close to that of the Kalman fuser weighted by matrices, hence it has good
performances.
Keywords: Consistency, covariance intersection fusion, information fusion kalman fuser, time-delayed
measurement, unknown cross-covariance
INTRODUCTION
Multisensor information has been applied to many
fields, such as Guidance, defense, robotics, tracking,
signal processing (Hall and Llinas, 1997; Li et al.,
2003; Bar-Shalom and Li, 2001). There are two kinds
of optimal information fusion methods: the centralized
and distributed information fusion, where the
distributed weighted fusion method applies widely
because of its less computational burden. Even if, to
compute the optimal distributed weighted fusion
Kalman estimators (Sun and Deng, 2004; Bar-Shalom
and Campo, 1986), it is required to compute the crosscovariances among local Kalman estimator errors,
which needs a large computational burden, because
these cross-covariances are generally unknown (Deng
et al., 2008) or their computation is very complex (Sun
and Deng, 2009; Sun et al., 2009; Liggins et al., 2009).
In order to overcome this limitation, a Covariance
Intersection (CI) fusion method was presented and
developed in Julier and Uhlmann (1997), Chen et al.
(2002), Julier and Uhlmann (1996), Julier and Uhlmann
(2009) and Arambel et al. (2001), which can solve the
fused filtering problems with unknown crosscovariances and has the consistency and robustness
(Julier and Uhlmann, 1997). It has wide applications
(Julier and Uhlmann, 2007; Guo et al., 2010; Bolzani
et al., 2007). Besides that, the systems with timedelayed measurement exist widely in control and
estimation fields, which results that it is more difficult
to compute the cross-covariances. Now there are
augmented state method (Kailath et al., 2000) and nonaugmented state method (Sun and Deng, 2009; Zhang
and Xie, 2007) to convert the system with time delays
into that without time delays.
In this study, for the two-sensor system with
measurement delays, a measurement transformation
approach is presented, by which the fused estimation
problem is converted into that for systems without
measurement delays (Sun and Deng, 2009; Sun et al.,
2009; Liggins et al., 2009). And, to overcome the
difficulty and complexity of computing crosscovariances, using the CI method (Julier and Uhlmann,
1996, 1997, 2009; Chen et al., 2002; Guo et al., 2010;
Bolzani et al., 2007), a CI fusion steady-state Kalman
fuser is presented, which avoids to find the crosscovariances and can reduce the computational burden.
It can handle the system with unknown crosscovariances and it has higher accuracy comparing with
local estimators. It is rigorously proved that the
accuracy of CI fuser is higher than that of each local
filter and is lower than that of optimal fuser weighted
by matrices. The CI fuser is consistent and the good
performance.
PROBLEM FORMULATION
Consider the two-sensor linear discrete timeinvariant stochastic system:
Corresponding Author: Zi-Li Deng, Department of Automation, Heilongjiang University, Harbin 150080, China, Tel.:
13804583507
1963
Res. J. App. Sci. Eng. Technol., 6(11): 1963-1969, 2013
x(t +=
1) Φ x(t ) + Γ w(t )
(1)
zi (t=
) H i x(t − τ i ) + ξi (t ), =
i 1, 2
(2)
where, t is the discrete time, τ i is the measurement
delay, 𝑥𝑥(𝑡𝑡)𝜖𝜖𝑅𝑅𝑛𝑛 is the state, 𝑧𝑧𝑖𝑖 (𝑡𝑡)𝜖𝜖𝑅𝑅𝑚𝑚 𝑖𝑖 is the
measurement, w(t) 𝜖𝜖𝑅𝑅𝑟𝑟 and 𝜉𝜉𝑖𝑖 𝜖𝜖(𝑡𝑡)𝑅𝑅𝑚𝑚 𝑖𝑖 are uncorrelated
white noises with zero mean and variances Q w and 𝑄𝑄𝜉𝜉𝜉𝜉 ,
respectively.
The objectives are to find the local steady-state
optimal Kalman estimators 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁), (N<0, N = 0,
N>0), i = 1, 2, the Kalman fusers 𝑥𝑥�𝑚𝑚𝑧𝑧 (𝑡𝑡|𝑡𝑡 +
𝑁𝑁, 𝑥𝑥𝑑𝑑𝑧𝑧𝑡𝑡𝑡𝑡+𝑁𝑁, 𝑥𝑥𝑠𝑠𝑧𝑧𝑡𝑡𝑡𝑡+𝑁𝑁 weighted by matrices, diagonal
matrices and scalars, the CI fusion Kalman fuser
𝑧𝑧 (𝑡𝑡|𝑡𝑡
+ 𝑁𝑁) and compare their accuracy relations.
𝑥𝑥�𝐶𝐶𝐶𝐶
THE STEADY-STATE OPTIMAL
WEIGHTED KALMAN FUSER
Introducing the new measurement y i (t) and
measurement noise v i (t):
(t ) zi (t + τ i )
yi =
vi =
(t ) ξi (t + τ i )
(3)
where, 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁) = 𝑥𝑥(𝑡𝑡) − 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁), 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 +
𝑁𝑁) = 𝑥𝑥(𝑡𝑡) − 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁).
Defining the steady-state estimator error variances
and cross-covariances are 𝑝𝑝𝑖𝑖𝑖𝑖𝑧𝑧 (𝑁𝑁) = 𝐸𝐸[𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 +
and
𝑝𝑝𝑖𝑖𝑖𝑖𝑧𝑧 (𝑁𝑁) = 𝐸𝐸[𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 +
𝑁𝑁)𝑥𝑥�𝑖𝑖𝑧𝑧𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁)]
𝑁𝑁)𝑥𝑥�𝑖𝑖𝑧𝑧𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁)], respectively. So we have obtained
Pijz ( N ) = Pij ( N − τ i , N − τ j ), i = 1, 2
(9)
where, P ij (N-τ i , N-τ j ) = E[𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁 − 𝜏𝜏𝑖𝑖 )𝑥𝑥�𝑖𝑖𝑇𝑇 (𝑡𝑡|𝑡𝑡 +
𝑁𝑁−𝜏𝜏𝑖𝑖
From (7) we can see that the problem of finding the
local steady-state optimal Kalman estimators 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 +
𝑁𝑁 based on the measurement (z i (t+N), z i (t+N-1), …) is
converted t that of finding the optimal steady-state
Kalman estimators 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁 − 𝜏𝜏𝑖𝑖 ) based on the
measurement (y i (t+N-τ i ), y i (t+N-τ i -1), …), For N > τ i ,
N = τ i , N > τ i , the estimators are called the smoothers,
filters and predictors, respectively.
Lemma 1: (Sun and Deng, 2009) For the two-sensor
system (1) and (5), the local steady-state Kalman
predictor 𝑥𝑥�𝑖𝑖 (𝑡𝑡 + 1|𝑡𝑡) is given by:
xˆˆ(
=
1| t ) Ψ pi xi (t | t − 1) + K pi y=
1, 2
i t+
i (t ), i
(10)
(4)
where,
From (2), we have the new measurement equation
without measurement delays.
yi (t ) = H i x(t ) + vi (t ), i = 1, 2
= L( yi (t + N − τ i ), yi (t + N − τ i − 1),)
(7)
where 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁 − 𝜏𝜏𝑖𝑖 ) is the linear minimum variance
filters of x(t) based on (y i (t+N-τ i ), y i (t+N-1-τ i ), …)
and 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁) is the linear minimum variance filters
based on L(z i (t+N), z i (t+N-1), …). So the relation
between the estimation errors 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁) and 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 +
𝑁𝑁 is:
xiz (t | t + N=
) xi (t | t + N − τ i ),=
i 1, 2
(12)
The prediction error variance matrix Σ i satisfies the
steady-state 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
and
the
cross-covariance
matrix
(13)
∑𝑖𝑖𝑖𝑖 = 𝐸𝐸[𝑥𝑥�𝑖𝑖 (𝑡𝑡 +
1𝑡𝑡𝑥𝑥𝑖𝑖𝑇𝑇𝑡𝑡+1𝑡𝑡 between local prediction errors satisfies the
Lyapunov equation:
Σ ij =
Ψ pi Σ ijΨ pjΤ + Γ Qw Γ Τ , i, j =
1, 2, i ≠ j
(6)
So we have the relation as:
xˆˆiz (t | t + N=
) xi (t | t + N − τ i )
(11)
(5)=
K pi ΦΣ i H iΤ ( H i Σ i H iΤ + Qvi ) −1
where, the new measurement noise v i (t) is white noise
with zero mean and variance Q vi = Q ζi .
Denoting the linear space spanned by the stochastic
variables (z i (t+N), z i (t+N-1), …) as L (z i (t+N), z i (t+N1), …) and the linear space spanned by the stochastic
variables ( yi (t + N − τ i ), yi (t + N − τ i − 1),) as
L( yi (t + N − τ i ), yi (t + N − τ i − 1),) , we have the
relation between the two linear measurement spaces as:
L( zi (t + N ), zi (t + N − 1),)
Ψ pi= Φ − K pi H i
(14)
The -k i steps steady-state Kalman predictor
𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 ) is given by:
| t + ki ) Φ − ki −1 xi (t + ki + 1| t + ki ), ki ≤ −2
xˆˆ(
i t=
(15)
The -k i steps Kalman prediction error variance
𝑃𝑃𝑖𝑖 (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑖𝑖 ) = 𝐸𝐸[𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )𝑥𝑥�𝑖𝑖𝑇𝑇 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )] and crosscovariance P ij (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑗𝑗 ) = E�𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )𝑥𝑥�𝑖𝑖𝑇𝑇 �𝑡𝑡�𝑡𝑡�𝑡𝑡 + 𝑘𝑘𝑗𝑗 ��
are given by:
(8)
1964
Pi (ki , ki ) = Φ − ki −1Σ iΦ ( − ki −1) Τ
− ki
+ ∑Φ − ki − j Γ Qw Γ ΤΦ ( − ki − j ) Τ , ki ≤ −2
j =2
(16)
Res. J. App. Sci. Eng. Technol., 6(11): 1963-1969, 2013
=
Pij (ki , k j ) Φ − ki −1Σ ijΦ ( − ki −1) Τ +
− max( ki , k j ) − 2
∑
j =0
(17)
Φ j Γ Qw Γ ΤΦ jΤ , ki , k j ≤ −2
The steady-state Kalman filter 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡) is as following:
xˆˆ(
| t ) Ψ fi xi (t − 1| t − 1) + K fi yi (t=
), i 1, 2
i t=
(18)
(19)
=
K fi Σ i H iΤ ( H i Σ i H iΤ + Qvi ) −1
(29)
Pi (ki , ki ) =
Σ i , ( ki =
−1)
(30)
Pij (ki , k j ) =
Σij , (ki =
−1, k j =
−1)
(31)
− ki − 2
∑Φ
j
j =0
Γ Qw Γ ΤΦ jΤ ,
k −k j
Pij (ki , k j ) = Φ − ki −1Ψ pii
−k j −2
∑
+
Σ ijΦ ( − k −1) Τ
i
(33)
Φ − k −1Ψ pik + r +1Γ Qw Γ ΤΦ r Τ +
i
i
r=
− ki −1
− ki − 2
∑Φ
(21)
Γ Qw Γ ΤΦ r Τ , (k j ≤ ki ≤ −2)
r
r =0
( k − ki ) Τ
Pij (ki , k j ) = Φ − ki −1Σ ijΨ pi j
Pij Ψ fi PijΨ fjΤ + ( I n − K fi H i )Γ Qw Γ Τ ( I n − K fj H j )Τ
=
(22)
− ki − 2
∑
+
Φ
( − k j −1) Τ
k j + r +1
( −1− k j ) Τ
k j + r +1
pj
( −1− k j ) Τ
Φ r Γ Qw Γ ΤΨ pj
Φ
(34)
r=
− k j −1
The steady-state Kalman smoother 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 ),
(k i >0) is given by:
ki
xˆˆ(
=
xi (t | t − 1) + ∑ K i (r )ε i (t + r )
i t |t+k
i)
i = 1, 2
(32)
(ki ≤ −2)
The corresponding steady-state Kalman filtering
error variance 𝑃𝑃𝑖𝑖𝑖𝑖 = 𝐸𝐸[𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡)𝑥𝑥�𝑖𝑖𝑇𝑇 (𝑡𝑡|𝑡𝑡)] and crosscovariance 𝑃𝑃𝑖𝑖𝑖𝑖 = 𝐸𝐸[𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡)𝑥𝑥�𝑖𝑖𝑇𝑇 (𝑡𝑡|𝑡𝑡)] are given by:
i, j = 1, 2 , i ≠ j
Pij (ki , k j ) = Pij
=
Pi (ki , ki ) Φ − ki −1Σ iΦ ( − ki −1) Τ +
(20)
P
=
( I n − K fi H i )Σ i
ii
(28)
Case 2: when k i <0, k j <0
where,
Ψ=
( I n − K fi H i )Φ
fi
Pi (ki , ki ) = Pii
− ki − 2
∑
+
Φ Γ Qw Γ Ψ
Τ
r
r=
− k j −1
−k j −2
∑Φ
(23)
r
r =0
Φ
+
Γ Qw Γ ΤΦ rΤ , (ki ≤ k j ≤ −2)
r =0
− k j −1
Pij (ki , k j ) = Ψ pi
where,
+
=
K i (r ) Σ iΨ piΤr H iΤ ( H i Σ i H iΤ + Qvi ) −1
ε i (t ) =
yi (t ) − H i xˆi (t | t − 1)
−k j −2
r =0
(24)
(25)
∑Ψ
ki
Pi (ki , ki ) =
Σ i − ∑ K i (r )( H Σ i H Τ + Qvi ) K iΤ (r ), ki > 0
( − k j −1) Τ
(35)
Γ Qw Γ ΤΦ r Τ , (ki = −1, k j ≤ −2)
=
Pij (ki , k j ) Φ − ki −1Σ ijΨ pi( − ki −1) Τ +
− ki − 2
∑Φ
r
r =0
The Kalman smoothing error variance 𝑃𝑃𝑖𝑖 (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑖𝑖 ) =
𝐸𝐸[𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )𝑥𝑥�𝑖𝑖𝑇𝑇 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )]
and cross-covariance
𝑃𝑃𝑖𝑖𝑖𝑖 �𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑗𝑗 � = 𝐸𝐸�𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )𝑥𝑥�𝑖𝑖𝑇𝑇 �𝑡𝑡�𝑡𝑡 + 𝑘𝑘𝑗𝑗 �� are given by:
r
pi
Σ ijΦ
(36)
Γ Qw Γ ΤΨ pjrΤ , (ki ≤ −2, k j = −1)
Case 3: when k i >0, k j >0
ki
Pii (ki , ki ) =
Σ i − ∑ K i (r )[ H i Σ i H iΤ + Qvi ]K iΤ (r ), (ki > 0)
(37)
r =0
(26)
kj
ki
Pij (ki , k j ) =
∑ [ I nδ r 0 − Ki (r ) H iΨ pir ]Σ ij ∑ [ I nδ s 0 − K j (s) HjΨ pjs ]Τ
r =0
(38)
=
r 0=s 0
min( ki , k j )
Pij (ki , k j ) =
Σ ij − ∑ K i (r ) H Σ ij H Τ K Τj (r ), ki , k j > 0
+
(27)
min( ki , k j ) r −1
∑ ∑ K (r ) H Ψ
=
r 0=
u 0
i
i
uΤ
pi
Γ Qw Γ ΤΨ pjuΤ H Τj K Τj (r ), (i ≠ j )
r =0
Case 4: when k i <0, k j <0≥
Lemma 2: (Sun and Deng, 2009) For the two-sensor
system (1) and (5), the steady-state Kalman filtering
error variance P ij (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑖𝑖 ) = E [𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )𝑥𝑥�𝑖𝑖𝑇𝑇 (𝑡𝑡|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )]
and cross-covariance P ij (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑗𝑗 ) = E�𝑥𝑥�𝑖𝑖 �𝑡𝑡�𝑡𝑡 +
𝑘𝑘𝑗𝑗𝑥𝑥𝑖𝑖𝑇𝑇𝑡𝑡𝑡𝑡+𝑘𝑘𝑖𝑖, (k i = N-τ i , k j = N- τ i ) are given by:
=
Pij (ki , k j ) Φ − ki −1Σ ijΨ pj( −1− ki ) Τ −
Case 1: when k i = 0, k j = 0
1965
kj
∑Φ
− ki −1
s =0
− ki − 2
∑Φ
r
r =0
Σ ijΨ pj( s − k −1) Τ H Τj K Τj ( s ) +
i
Γ Qw Γ ΤΨ pjrΤ +
k j − ki − 2
∑ ∑Φ
r 0
=s 0=
r
Γ Qw Γ ΤΨ pj( s + r ) Τ H Τj K Τj ( s )
(39)
Res. J. App. Sci. Eng. Technol., 6(11): 1963-1969, 2013
Case 5: when k i ≥0, k j <0
− k j −1
=
Pij (ki , k j ) Ψ pj
ki
Σ ijΦ ( − k −1) Τ −
∑ Ki (s) H iΨ pi
( s − k j −1)
s =0
+
−k j −2
∑Ψ
=r 0
where the optimal weighted diagonal matrices A j is
given as:
i
Σ ijΦ
(40)
( − k j −1) Τ
ki − k j − 2
r
pi
Γ Qw Γ ΤΦ rΤ + ∑
∑ K ( s) H Ψ
i
=s 0=r 0
j
s+r
pi
 a j1

Aj = 



Γ Qw Γ ΤΦ rΤ
Lemma 3: (Sun and Deng, 2004) For the two-sensor
system (1) and (5), the optimal Kalman fuser weighted
by matrices is given by:
z
xˆˆˆ(
N ) Ω1 x1z (t | t + N ) + Ω2 x2z (t | t + N )
m t | t +=
= Ω1 xˆˆ(
1 t | t + N − τ 1 ) + Ω2 x2 (t | t + N − τ 2 )
 xˆˆzj1 
  z
z
ˆˆ
x j =
  , xd
=
 xˆˆzjn 
 
−1 Τ
−1
=
P(ki , k j ) (=
Pij (ki , k j )) NL× NL , (i, j 1, 2)
(42)
(43)
where, e = [I n , I n ] and the fused error variance matrix
P m is given as:
−1
−1
Pm = [e P (ki , k j )e]
The optimal Kalman fuser
by scalars is given by:
(44)
𝑥𝑥�𝑠𝑠𝑧𝑧 (𝑡𝑡|𝑡𝑡
+ 𝑁𝑁) weighted
z
xˆˆˆ(
N ) a1 x1z (t | t + N ) + a2 x2z (t | t + N )
s t | t +=
= a1 xˆˆ(
1 t | t + N − τ 1 ) + a2 x2 (t | t + N − τ 2 )
 xdz1 
 
  
 xdnz 
 
(52)
(45)
2
∑a
j =1
jl
x zjl (t | t=
+ N ), l 1, , n
esΤ Ptr−1
esΤ Ptr−1es
=
al
a1l a2l ]
[=
edΤ ( P ll ) −1
edΤ ( P ll ) −1 ed
where 𝑒𝑒𝑑𝑑𝑇𝑇 = [1 … 1] and defining:
 P (ll )
P ll =  1(ll )
 P21
P12(ll ) 

P2(ll ) 
 trP (k , k ) trP12 (k1 , k2 ) 
Ptr =  1 1 1

 trP21 (k2 , k1 ) trP2 (k2 , k2 ) 
(48)
The corresponding optimal fused estimation error
variance P s is given by:
2
From the proposed interpretation, it is clear that the
computation of getting the cross-covariances between
local filtering is very complex and the computed burden
is very large. So a CI fusion method was presented
(Julier and Uhlmann, 1997; Chen et al., 2002).
Consider a two-sensor system (1) (2) and (5), when
P 1 and P 2 are known, but the cross-covariance P 12 is
unknown, the CI Kalman fuser without P 12 is given by:
xˆˆCIz=
(t | t + N ) PCI [ω P1−1 x1z (t | t + N ) +
(49)
(1 − ω ) P2−1 xˆ2z (t | t + N )]
=i 1 =j 1
The optimal Kalman fuser 𝑥𝑥�𝑑𝑑𝑧𝑧 (𝑡𝑡|𝑡𝑡|𝑡𝑡 + 𝑁𝑁) weighted
by diagonal matrices is given by:
z
xˆˆˆ(
N ) A1 x1z (t | t + N ) + A2 x2z (t | t + N )
d t | t +=
ˆˆ(
= A1 x1 t | t + N − τ 1 ) + A2 x2 (t | t + N − τ 2 )
(56)
THE STEADY-STATE CI KALMAN FUSER
(47)
2
(55)
=i 1 =j 1
(46)
e = [1 1]
Ps = ∑∑ ai a j Pij (ki , k j )
(54)
where 𝑃𝑃𝑖𝑖𝑖𝑖𝑙𝑙𝑙𝑙 is the (l, l)th diagonal component of P ij (k i ,
k j ) and the optimal component fused error variance P dl
and the optimal fused error variance P d are given by:
Pd = ∑∑ Ai Pij AΤj
where,
Τ
s
(53)
where the optimal weighted coefficient vector is:
Τ
ll −1
−1
where the optimal weighted coefficients a = [a 1 , a 2 ] is=
Pdl [e=
1, , n
d ( P ) ed ] , l
given as:
2
2
a=
(51)
So the equation (50) is equivalent with:
T
Τ
j = 1, 2
,
(41)
xˆˆdlz (t =
| t + N)
[Ω1 , Ω2 ] = (e P (k1 , k2 )e) e P (k1 , k2 )
−1






a jn 
The component expression of 𝑥𝑥�𝑗𝑗𝑧𝑧 and 𝑥𝑥�𝑑𝑑𝑧𝑧 are given by:
where the weighted matrix is given by:
Τ
a j2
(57)
where the CI fusion error variance matrix P CI is given
by:
(50)
1966
P=
[ω P1−1 + (1 − ω ) P2−1 ]−1
CI
(58)
Res. J. App. Sci. Eng. Technol., 6(11): 1963-1969, 2013
where, Pi  Pi (ki , ki ), i = 1, 2 , P12  P12 (k1 , k2 ) , P21  P21 (k2 , k1 ) and
the weighted coefficient 𝜔𝜔𝜔𝜔[0, 1] and minimizes the
performance index:
J = min trPCI = min tr{[ω P1−1 + (1 − ω ) P2−1 ]−1}
ω
(59)
ω∈[0,1]
where the notation tr denotes the trace of matrix. The
optimal weighting coefficient ω can be solved by the
gold section method or Fibonacci method (Julier and
Uhlmann, 1996).
� 𝐶𝐶𝐶𝐶 ≤ 𝑃𝑃
� 𝐶𝐶𝐶𝐶 and
which yields (63). It has been proven 𝑃𝑃
trP m ≤trP d ≤trP s ≤trP i , = 1, 2 in the reference (Sun and
Deng, 2004; Julier and Uhlmann, 1997; Chen et al.,
2002), so we have (64) and (65) hold. In the
performance index (59), taking ω = 0, we have J = trP 2
and taking ω = 1, we have J = trP 1 . Hence the
optimal weighting coefficient 𝜔𝜔𝜔𝜔[0, 1] yields
trP CI ≤trP 1 ≤trP CI ≤trP 2 , i.e. trP CI ≤ trP i , i = 1, 2. Taking
trace operation for (63) and (64), we have (66) holds.
The proof is completed.
SIMULATION EXAMPLE
THE CONSISTENCY AND ACCURACY
OF CI KALMAN FUSER
Consider the tracking system with two-sensor and
with time-delayed measurements:
Theorem1: The actual CI fusion error variance matrix
𝑃𝑃�𝐶𝐶𝐶𝐶 is given by:
=
PCI PCI [ω 2 P1−1 + ω (1 − ω ) P1−1 P12 P2−1 +
0.5T02 
1 T0 
=
x(t + 1) 

 x(t ) + 
0
1


 T0 
) H i x(t − τ i ) + ξi (t ), =
zi (t=
i 1, 2
(60)
ω (1 − ω ) P2−1 P21 P1−1 + (1 − ω ) 2 P2−1 ]PCI
Proof: From (58), we can obtain
which yields
𝑃𝑃𝐶𝐶𝐶𝐶 [𝜔𝜔𝑃𝑃1−1
=
x(t ) PCI [ω P1−1 x(t ) + (1 − ω ) P2−1 x(t )]
+
𝑃𝑃2−1 ]
= 𝐼𝐼,
(61)
Subtracting (57) from (61), we get the actual CI
fuser error:
xCI=
(t | t + N ) PCI [ω P1−1 x1 (t | t + N ) +
H1
=
The actual CI fusion error variance 𝑃𝑃�𝐶𝐶𝐶𝐶 =
𝑇𝑇 (𝑡𝑡|𝑡𝑡
𝐸𝐸[𝑥𝑥�𝐶𝐶𝐶𝐶 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁)𝑥𝑥�𝐶𝐶𝐶𝐶
+ 𝑁𝑁)], where Ε denotes the
mathematical expectation, Τ denotes the transpose.
Substituting (62) into 𝑃𝑃�𝐶𝐶𝐶𝐶 yields that (60) holds. The
proof is completed.
Theorem 2: For local and fused Kalman filters, we have
the accuracy relations:
Pm ≤ Pd , Pm ≤ Ps , Pm ≤ PCI , Pm ≤ Pi , i =
1, 2
(63)
PCI ≤ PCI
(64)
trPm ≤ trPd ≤ trPs ≤ trPi , i =
1, 2
(65)
trPm ≤ trPCI ≤ trPCI ≤ trPi , i =
1, 2
(66)
Proof: from (41), 𝑥𝑥�𝑚𝑚𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁) is the Unbiased Linear
Minimum Variance (ULMV) estimate of x(t ) based on
the linear space 𝐿𝐿𝑚𝑚 = 𝐿𝐿𝑚𝑚 �𝑥𝑥�1𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁), 𝑥𝑥�2𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁)�
spanned by �𝑥𝑥�1𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁), 𝑥𝑥�2𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁)�. From (45),
(50), (57), we have 𝑥𝑥�𝜃𝜃𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁), 𝜖𝜖𝐿𝐿𝑚𝑚 , 𝜃𝜃 = 𝑠𝑠, 𝑑𝑑, 𝐶𝐶𝐶𝐶, 𝑖𝑖,
1 0 
0] , =
H2 
,τ 1 1,=
τ2 2
=
0 1 
where, 𝑥𝑥(𝑡𝑡)𝜖𝜖𝑅𝑅 2 is the state, z i (t) is the measurement
with delays for sensor i, w(t) and ζ i (t) are white noise
with zero mean and variances Q w and Q ζi , respectively.
According to the corresponding transformation, we
obtain the measurement y i (t) without time delays:
yi (t ) = zi (t + τ i ) = H i x(t ) + vi (t )
(62)
(1 − ω ) P2−1 x2 (t | t + N )]
[1
where v i (t) = ζ i (t+τ i ) is the transformed measurement
white noise with zero mean and variance Q vi = Q ζi .
Thus the problem of finding the local steady-state
optimal Kalman smoother 𝑥𝑥�𝑖𝑖𝑧𝑧 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁), (𝑖𝑖 = 1, 2)
based on the measurement z i (t) is converted into the
problem of finding the local steady-state optimal
Kalman filter 𝑥𝑥�1 (𝑡𝑡|𝑡𝑡) and Kalman predictor 𝑥𝑥�2 (𝑡𝑡|𝑡𝑡 − 1)
based on the new measurement y i (t), respectively.
Further, applying the Kalman filtering method yields the
steady-state optimal fusion Kalman smoothers 𝑥𝑥�𝑚𝑚𝑧𝑧 (𝑡𝑡 +
1, 𝑥𝑥𝑑𝑑𝑧𝑧𝑡𝑡𝑡𝑡+1, 𝑥𝑥𝑠𝑠𝑧𝑧𝑡𝑡𝑡𝑡+1 weighted by matrices,
Fig. 1: The accuracy comparison of P 1 , P 2 , P 0 , P s , P CI and
𝑃𝑃�𝐶𝐶𝐶𝐶
1967
Res. J. App. Sci. Eng. Technol., 6(11): 1963-1969, 2013
0.78
0.68
0.58
0.48
0.38
0.28
20
40
60
80 100 120 140 160 180 200 220 240 260 280 300
MSE1
MSEd
MSE2
MSECI
MSE0
trP1
MSEs
trP2
Fig. 2: The curves of MSEi(t)
diagonal matrices and scalars and CI fused Kalman
𝑧𝑧 (𝑡𝑡|𝑡𝑡
+ 1). In simulation, we take
smoother 𝑥𝑥�𝐶𝐶𝐶𝐶
0 
4
=
T0 0.4,=
Qw 1,=
Qξ 1 0.36,=
Qξ 2 

 0 0.16 
In order to give a powerful geometric interpretation
with respect to accuracy relations of local and fusers,
the covariance ellipse for a covariance matrix P is
defined as the locus of points {x: xTP-1x = c} where c is
a constant. In the sequel, c = 1 will be assumed without
loss of generality. It was proved (Julier and Uhlmann,
1996) that P a ≤ P b is equivalent to that the ellipse for P a
is enclosed in the ellipse for P b . The simulation results
are shown in Fig. 1.
From Fig. 1, it is clearly shown that the covariance
ellipse for P m lies within the intersection of ellipses for
P 1 and P 2 , the ellipse for CI fused theoretical covariance
P CI encloses the intersection region of ellipses for P 1
and P 2 (Chen et al., 2002) and passes through the four
points of intersection of ellipses for P 1 and P 2 . The
ellipse for CI fused actual covariance 𝑃𝑃�𝐶𝐶𝐶𝐶 lies within the
ellipse for theoretical covariance 𝑃𝑃�𝐶𝐶𝐶𝐶 .
In order to verify the above theoretical results for
accuracy relation, 200 Monte-Carlo runs are performed
for t = 1, …, 300 the results are shown in Fig. 2.
where the straight lines denote trP i , i = 0, 1, 2, CI and
trPCI , the curves denote the corresponding mean square
errors MSE i (t). We see the CI fuser has good
performance, whose accuracy is approximates to the
accuracy of the optimal smoother, because MSE CI (t) is
approximates to MSE m (t).
CONCLUSION
matrices, a CI fusion steady-state Kalman fuser has
been presented, which has the advantage that the
computation of cross-covariances can be avoided. It is
proved that its accuracy is higher than that of each local
estimator and is lower than that of optimal fuser
weighted by matrices. A two-sensor system MonteCarlo simulation example shows that its accuracy is
approximates to the accuracy of the optimal fuser and is
higher than those of the fusers weighted by diagonal
matrices and scalars.
ACKNOWLEDGMENT
This study is supported by the Natural Science
Foundation of China under grant NSFC-60874063, the
2012 Innovation and Scientific Research Foundation of
graduate student of Heilongjiang Province under grant
YJSCX2012-263HLJ and the Support Program for
Young Professionals in Regular Higher Education
Institutions of Heilongjiang Province under grant
1251G012.
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1969
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