Research Journal of Applied Sciences, Engineering and Technology 6(19): 3602-3609, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: December 31, 2012
Accepted: February 08, 2013
Published: October 20, 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: To handle the estimation fusion problem between local estimation errors for the system with unknown
cross-covariances and to avoid a large computed burden and computational complexity of cross-covariances, 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 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, 2009) and Arambel
et al. (2001), which can solve the fused filtering
problems with unknown cross-covariances 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 time-delayed 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 non-augmented 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, first, 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) for the two-sensor system with
measurement delays. Second, in order to overcome the
difficulty and complexity of computing crosscovariances, by the CI method (Julier and Uhlmann,
1997; Chen et al., 2002; Julier and Uhlmann, 1996,
2009; Guo et al., 2010; Bolzani et al., 2007), a CI
fusion steady-state Kalman fuser is presented, whose
objective is to avoid to find the cross-covariances and
reduce the computational burden. It can handle the
fused estimation problem of the system with unknown
cross-covariances and it has higher accuracy comparing
with local estimators. Third, 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, so it has the
good performance.
PROBLEM FORMULATION
Consider the two-sensor linear discrete timeinvariant stochastic system:
Corresponding Author: Peng Zhang, Department of Automation, Heilongjiang University, Harbin, 150080, China
3602
Res. J. Appl. Sci. Eng. Technol., 6(19): 3602-3609, 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
𝜏𝜏𝑖𝑖
x(t) ∈ 𝑅𝑅𝑛𝑛
𝑧𝑧𝑖𝑖 (t) ∈ 𝑅𝑅 𝑚𝑚 𝑖𝑖
w(t) ∈ 𝑅𝑅𝑟𝑟 , 𝜉𝜉𝑖𝑖 (t)∈ 𝑅𝑅𝑚𝑚 𝑖𝑖
= The discrete time
= The measurement delay
= The state
= The measurement
= Uncorrelated white noises with
zero mean and variances Qw
and Qξ i , respectively
The objectives are to find the local steady-state
optimal Kalman estimators 𝑥𝑥�𝑖𝑖𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁), (N<0, N>), i =
1, 2, the Kalman fusers 𝑥𝑥�𝑚𝑚𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁) 𝑥𝑥�𝑑𝑑𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁),
𝑥𝑥�𝑠𝑠𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁) weighted by matrices, diagonal matrices
𝑧𝑧
(t|𝑡𝑡 + 𝑁𝑁)
and scalars, the CI fusion Kalman fuser 𝑥𝑥�𝐶𝐶𝐶𝐶
and compare their accuracy relations.
THE STEADY-STATE OPTIMAL WEIGHTED
KALMAN FUSER
xiz (t | t + N=
) xi (t | t + N − τ i ),=
i 1, 2
yi =
(t ) zi (t + τ i )
(3)
vi =
(t ) ξi (t + τ i )
(4)
From (2), we have the new measurement equation
without measurement delays:
where,
xiz (t | t + N ) = x(t ) − xˆiz (t | t + N )
xi (t | t + N ) = x(t ) − xˆi (t | t + N )
Defining
steady-state
estimator
error
(N) =
variances and cross-covariances are 𝑃𝑃𝑖𝑖𝑖𝑖𝑧𝑧
E[𝑥𝑥�𝑖𝑖𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁) 𝑥𝑥�𝑖𝑖𝑧𝑧𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁)] and 𝑃𝑃𝑖𝑖𝑖𝑖𝑧𝑧 (N) = E[𝑥𝑥�𝑖𝑖𝑧𝑧 (t|𝑡𝑡 +
𝑁𝑁) 𝑥𝑥𝑖𝑖𝑧𝑧𝑇𝑇(t𝑡𝑡+𝑁𝑁)] , respectively. So we have obtained:
Pijz ( N ) = Pij ( N − τ i , N − τ j ), i = 1, 2
= L( yi (t + N − τ i ), yi (t + N − τ i − 1),)
From (7) we can see that the problem of finding the
local steady-state optimal Kalman estimators 𝑥𝑥�𝑖𝑖𝑧𝑧 (t|𝑡𝑡 +
𝑁𝑁) based on the measurement (𝑧𝑧𝑖𝑖 (t+N), 𝑧𝑧𝑖𝑖(t+N-1),…)
is converted t that of finding the optimal steady-state
Kalman estimators 𝑥𝑥�𝑖𝑖 (t|𝑡𝑡 + 𝑁𝑁−𝜏𝜏𝑖𝑖 ) based on the
measurement (𝑦𝑦𝑖𝑖 (t+N−𝜏𝜏𝑗𝑗 ), 𝑦𝑦𝑖𝑖 (t+N-1),…). For N>𝜏𝜏𝑖𝑖 , N
= 𝜏𝜏𝑖𝑖 , N<𝜏𝜏𝑖𝑖 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 (t+1|𝑡𝑡) is given by:
(5)
(6)
xˆˆ (t | t + N=
) xi (t | t + N − τ i )
(10)
(11)
(12)
The prediction error variance matrix ∑ 𝑖𝑖 satisfies
the steady-state Riccati equation:
So we have the relation as:
z
i
(9)
where,
where, the new measurement noise 𝑣𝑣𝑖𝑖 (t) is white noise
with zero mean and variance 𝑄𝑄𝑣𝑣𝑣𝑣 = 𝑄𝑄𝜉𝜉𝜉𝜉 .
xˆˆ(
=
1| t ) Ψ pi xi (t | t − 1) + K pi y=
1, 2
i t +
i (t ), i
Denoting the linear space spanned by the stochastic
variables (𝑧𝑧𝑖𝑖 (t+N), 𝑧𝑧𝑖𝑖 (t+N-1),…) as L(𝑧𝑧𝑖𝑖 (t+N), 𝑧𝑧𝑖𝑖 (t+Nwhere,
1),…) and the linear space spanned by the stochastic
variables (𝑦𝑦𝑖𝑖 (t+N-𝜏𝜏𝑖𝑖 ), 𝑦𝑦𝑖𝑖 (t+N-𝜏𝜏𝑖𝑖 −1),…) as L(𝑦𝑦𝑖𝑖 (t+NΨ pi= Φ − K pi H i
𝜏𝜏𝑖𝑖 ), 𝑦𝑦𝑖𝑖 (t+N-𝜏𝜏𝑖𝑖 -1),…), we have the relation between the
two linear measurement spaces as:
=
K pi ΦΣ i H iΤ ( H i Σ i H iΤ + Qvi ) −1
L( zi (t + N ), zi (t + N − 1),)
(8)
Pij ( N − τ i , N − τ j ) =
Ε[ xi (t | t + N − τ i ) xiΤ (t | t + N − τ j )]
Introducing the new measurement 𝑦𝑦𝑖𝑖 (t) and
measurement noise 𝑣𝑣𝑖𝑖 (t):
yi (t ) = H i x(t ) + vi (t ), i = 1, 2
𝜏𝜏𝑖𝑖 −1),…) and 𝑥𝑥�𝑖𝑖𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁) is the linear minimum
variance filters based on L(𝑧𝑧𝑖𝑖 (t+N), 𝑧𝑧𝑖𝑖 (t+N-1),…). So
the relation between the estimation errors 𝑥𝑥�𝑖𝑖𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁)
and 𝑥𝑥�𝑖𝑖 (t|𝑡𝑡 + 𝑁𝑁) is:
(7)
where, 𝑥𝑥�𝑖𝑖 (𝑡𝑡|𝑡𝑡 + 𝑁𝑁 − 𝜏𝜏𝑖𝑖 ), is the linear minimum
variance filters of x(t) based on (𝑦𝑦𝑖𝑖 (t+N-𝜏𝜏𝑖𝑖 ), 𝑦𝑦𝑖𝑖 (t+N-
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
(13)
and the cross-covariance matrix ∑ 𝑖𝑖𝑖𝑖 = E[𝑥𝑥�𝑖𝑖
(t+1|𝑡𝑡) 𝑥𝑥�𝑖𝑖𝑇𝑇 (t+1|𝑡𝑡)] between local prediction errors
satisfies the Lyapunov equation:
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Res. J. Appl. Sci. Eng. Technol., 6(19): 3602-3609, 2013
Σ ij =
Ψ pi Σ ijΨ pjΤ + Γ Qw Γ Τ , i, j =
1, 2, i ≠ j
(14)
K i (r ) Σ iΨ piΤr H iΤ ( H i Σ i H iΤ + Qvi ) −1
=
The -k i steps steady-state Kalman predictor 𝑥𝑥�𝑖𝑖
(t|𝑡𝑡 + 𝑘𝑘𝑖𝑖 ) is given by:
xˆˆ(
| t + ki ) Φ
i t=
− ki −1
xi (t + ki + 1| t + ki ), ki ≤ −2 (15)
The -k i steps Kalman prediction error variance
𝑃𝑃𝑖𝑖 (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑖𝑖 ) = E[𝑥𝑥�𝑖𝑖 (t|𝑡𝑡 + 𝑘𝑘𝑖𝑖 ) 𝑥𝑥�𝑖𝑖𝑇𝑇 (t|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )] and crosscovariance 𝑃𝑃𝑖𝑖𝑖𝑖 (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑗𝑗 ) = E[𝑥𝑥�𝑖𝑖 (t|𝑡𝑡 + 𝑘𝑘𝑖𝑖 ) 𝑥𝑥�𝑗𝑗𝑇𝑇 (t|𝑡𝑡 + 𝑘𝑘𝑗𝑗 )] are
given by:
Pi (ki , ki ) = Φ
− ki −1
Σ iΦ
r =0
min( ki , k j )
Pij (ki , k j ) =
Σ ij − ∑ Ki (r ) H Σ ij H Τ K Τj (r ), ki , k j > 0
(27)
(16)
Lemma 2: Sun and Deng (2009) For the two-sensor
system (1) and (5), the steady-state Kalman filtering
error variance 𝑃𝑃𝑖𝑖 (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑖𝑖 ) = E[𝑥𝑥�𝑖𝑖 (t|𝑡𝑡 + 𝑘𝑘𝑖𝑖 ) 𝑥𝑥�𝑖𝑖𝑇𝑇 (t|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )]
and cross- covariance 𝑃𝑃𝑖𝑖𝑖𝑖 (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑖𝑖 )= E[𝑥𝑥�𝑖𝑖 (t|𝑡𝑡 +
𝑘𝑘𝑖𝑖) 𝑥𝑥𝑗𝑗𝑇𝑇(t𝑡𝑡+𝑘𝑘𝑗𝑗)], (𝑘𝑘𝑖𝑖 = N-𝜏𝜏𝑖𝑖, 𝑘𝑘𝑗𝑗= N−𝜏𝜏𝑗𝑗) are given by
j =2
=
Pij (ki , k j ) Φ − ki −1Σ ijΦ ( − ki −1) Τ +
j =0
ki
Pi (ki , ki ) =
Σ i − ∑ K i (r )( H Σ i H Τ + Qvi ) K iΤ (r ), ki > 0 (26)
r =0
− ki
∑
(25)
The Kalman smoothing error variance 𝑃𝑃𝑖𝑖 (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑖𝑖 ) =
E[𝑥𝑥�𝑖𝑖 (t|𝑡𝑡 + 𝑘𝑘𝑖𝑖 ) 𝑥𝑥�𝑖𝑖𝑇𝑇 (t|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )] and cross-covariance
𝑃𝑃𝑖𝑖𝑖𝑖 (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑗𝑗 ) = E[𝑥𝑥�𝑖𝑖 (t|𝑡𝑡 + 𝑘𝑘𝑖𝑖 ) 𝑥𝑥�𝑗𝑗𝑇𝑇 (t|𝑡𝑡 + 𝑘𝑘𝑗𝑗 )] are given by:
( − ki −1) Τ
+ ∑Φ − ki − j Γ Qw Γ ΤΦ ( − ki − j ) Τ , ki ≤ −2
− max( ki , k j ) − 2
ε i (t ) =
yi (t ) − H i xˆi (t | t − 1)
(24)
Φ j Γ Qw Γ ΤΦ jΤ , ki , k j ≤ −2
(17)
The steady-state Kalman filter 𝑥𝑥�𝑖𝑖 (t|𝑡𝑡)
following:
is as
xˆˆ(
| t ) Ψ fi xi (t − 1| t − 1) + K fi yi (t=
), i 1, 2
i t=
(18)
where,
Ψ=
( I n − K fi H i )Φ
fi
(19)
=
K fi Σ i H iΤ ( H i Σ i H iΤ + Qvi ) −1
(20)
The corresponding steady-state Kalman filtering
error variance 𝑃𝑃𝑖𝑖𝑖𝑖 = E[𝑥𝑥�𝑖𝑖 (t|𝑡𝑡) 𝑥𝑥�𝑖𝑖𝑇𝑇 (t|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )] and crosscovariance 𝑃𝑃𝑖𝑖𝑖𝑖 = E[𝑥𝑥�𝑖𝑖 (t|𝑡𝑡) 𝑥𝑥�𝑗𝑗𝑇𝑇 (t|𝑡𝑡 + 𝑘𝑘𝑖𝑖 )] are given by:
P
( I n − K fi H i )Σ i
=
ii
Case 1: When 𝑘𝑘𝑖𝑖 = 0，𝑘𝑘𝑗𝑗 = 0:
Pi (ki , ki ) = Pii
(28)
Pij (ki , k j ) = Pij
(29)
Case 2: When 𝑘𝑘𝑖𝑖 < 0，𝑘𝑘𝑗𝑗 < 0:
Pi (ki , ki ) =
Σ i , ( ki =
−1)
(30)
Pij (ki , k j ) =
Σij , (ki =
−1, k j =
−1)
(31)
=
Pi (ki , ki ) Φ − ki −1Σ iΦ ( − ki −1) Τ +
(21)
j =0
j
Γ Qw Γ ΤΦ jΤ ,
(32)
k −k j
Pij ( ki , k j ) = Φ − ki −1Ψ pii
−k j −2
∑
Σ ijΦ ( − k −1) Τ
i
Φ − k −1Ψ pik + r +1Γ Qw Γ ΤΦ r Τ +
i
(33)
i
− ki −1
r=
=
Pij Ψ fi PijΨ fjΤ + ( I n − K fi H i )Γ Qw Γ Τ ( I n − K fj H j )Τ
− ki − 2
∑Φ
(22)
r
r =0
( k j − ki ) Τ
+
(23)
+
− ki − 2
∑
− ki − 2
∑
Φ Γ Qw Γ Ψ
r
−k j −2
∑Φ
r =0
3604
( − 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
r=
− k j −1
where,
Φ
Φ
(34)
r=
− k j −1
ki
r =0
Γ Qw Γ ΤΦ r Τ , ( k j ≤ ki ≤ −2)
Pij (ki , k j ) = Φ − ki −1Σ ijΨ pi
The steady-state Kalman smoother 𝑥𝑥�𝑖𝑖 (t|𝑡𝑡 + 𝑘𝑘𝑖𝑖 ),
( 𝑘𝑘𝑖𝑖 > 0) is given by:
xˆˆ(
xi (t | t − 1) + ∑ K i (r )ε i (t + r ) i = 1, 2
i t | t + k=
i)
∑Φ
(ki ≤ −2)
+
i, j = 1, 2 , i ≠ j
− ki − 2
r
Τ
Φ
Γ Qw Γ ΤΦ r Τ , (ki ≤ k j ≤ −2)
+
Res. J. Appl. Sci. Eng. Technol., 6(19): 3602-3609, 2013
− k j −1
Pij (ki , k j ) = Ψ pi
+
−k j −2
∑Ψ
r =0
r
pi
Σ ijΦ
[Ω1 , Ω2 ] = (eΤ P −1 (k1 , k2 )e) −1 eΤ P −1 (k1 , k2 )
( − k j −1) Τ
Γ Qw Γ ΤΦ r Τ , (ki = −1, k j ≤ −2)
(35)
=
P (ki , k j ) (=
Pij (ki , k j )) NL× NL , (i, j 1, 2)
=
Pij (ki , k j ) Φ − ki −1Σ ijΨ pi( − ki −1) Τ +
− ki − 2
∑Φ
r
r =0
Γ Qw Γ ΤΨ pjr Τ , (ki ≤ −2, k j = −1)
Case 3: When 𝑘𝑘𝑖𝑖 > 0，𝑘𝑘𝑗𝑗 > 0:
ki
Pii (ki , ki ) =
Σ i − ∑ K i (r )[ H i Σ i H iΤ + Qvi ]K iΤ (r ),(ki > 0)
(36)
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 )
kj
ki
=r 0=s 0
+
min( ki , k j ) r −1
∑ ∑ K (r ) H Ψ
i
=r 0=
u 0
i
uΤ
pi
(38)
uΤ
pj
Τ
j
Τ
j
a=
Case 4: When 𝑘𝑘𝑖𝑖 < 0，𝑘𝑘𝑗𝑗 ≥ 0:
=
Pij (ki , k j ) Φ − ki −1Σ ijΨ pj( −1− ki ) Τ −
kj
∑Φ
− ki −1
s =0
− ki − 2
∑Φ
r
r =0
i
∑ ∑Φ
r
Γ Qw Γ ΤΨ pj( s + r ) Τ H Τj K Τj ( s )
− k j −1
ki
∑ K (s) H Ψ
s =0
+
i
i
−k j −2
∑Ψ
=
r 0
(48)
2
(49)
=i 1 =j 1
( − k j −1) Τ
ki − k j − 2
r
pi
 trP (k , k ) trP12 (k1 , k2 ) 
Ptr =  1 1 1

 trP21 (k2 , k1 ) trP2 (k2 , k2 ) 
2
i
Σ ijΦ
(47)
Ps = ∑∑ ai a j Pij (ki , k j )
Σ ijΦ ( − k −1) Τ −
( s − k j −1)
pi
esΤ = [1 1]
The corresponding optimal fused estimation error
variance 𝑃𝑃𝑠𝑠 is given by:
Case 5: When 𝑘𝑘𝑖𝑖 ≥0，𝑘𝑘𝑗𝑗 < 0:
=
Pij (ki , k j ) Ψ pj
(46)
(39)
Γ Qw Γ ΤΨ pjrΤ +
=s 0=
r 0
esΤ Ptr−1
esΤ Ptr−1es
where,
Σ ijΨ pj( s − k −1) Τ H Τj K Τj ( s ) +
k j − ki − 2
(45)
where the optimal weighted coefficients a = [𝑎𝑎1 𝑎𝑎1 ] is
given as:
Γ Qw Γ Ψ H K (r ),(i ≠ j )
Τ
(44)
The optimal Kalman fuser 𝑥𝑥�𝑠𝑠𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁) weighted
by scalars is given by:
r =0
Pij (ki , k j ) =
∑ [ I nδ r 0 − Ki (r ) H iΨ pir ]Σ ij ∑ [ I nδ s0 − K j (s) HjΨ pjs ]Τ
(43)
where, 𝑒𝑒 𝑇𝑇 = [𝐼𝐼𝑛𝑛 𝐼𝐼𝑛𝑛 ] and the fused error variance matrix
𝑃𝑃𝑚𝑚 is given as:
Pm = [eΤ P −1 (ki , k j )e]−1
(37)
(42)
Γ Qw Γ ΤΦ rΤ + ∑
∑
=s 0=
r 0
The optimal Kalman fuser 𝑥𝑥�𝑑𝑑𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁) weighted by
diagonal matrices is given by:
K i ( s ) H jΨ pis + r Γ Qw Γ ΤΦ r Τ
(40)
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 +=
(41)
= Ω1 xˆˆ(
1 t | t + N − τ 1 ) + Ω2 x2 (t | t + N − τ 2 )
z
xˆˆˆ(
N ) A1 x1z (t | t + N ) + A2 x2z (t | t + N )
d t | t +=
= A1 xˆˆ(
1 t | t + N − τ 1 ) + A2 x2 (t | t + N − τ 2 )
(50)
where, the optimal weighted diagonal matrices 𝐴𝐴𝑗𝑗 is
given as:
where, the weighted matrix is given by:
3605
 a j1

Aj = 



a j2

 , j = 1, 2




a jn 
(51)
Res. J. Appl. Sci. Eng. Technol., 6(19): 3602-3609, 2013
by:
The component expression of 𝑥𝑥�𝑗𝑗𝑧𝑧 and 𝑥𝑥�𝑑𝑑𝑧𝑧 are given
 xˆˆzj1 
  z
=
xˆˆ =
  , xd
 xˆˆzjn 
 
 xdz1 
 
  
z 
 xdn
 
z
j
(52)
2
∑a
j =1
jl
x zjl (t | t=
+ N ), l 1, , n
J = min trPCI = min tr{[ω P1−1 + (1 − ω ) P2−1 ]−1} (59)
ω
ω∈[0,1]
where, the notation tr denotes the trace of matrix. The
optimal weighting coefficient 𝜔𝜔 can be solved by the
gold section method or Fibonacci method (Julier and
Uhlmann, 1996).
So the Eq. (50) is equivalent with:
xˆˆdlz (t =
| t + N)
where, 𝑃𝑃𝑖𝑖 ≜ 𝑃𝑃𝑖𝑖 (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑖𝑖 ), I = 1, 2, 𝑃𝑃12 ≜ 𝑃𝑃12 (𝑘𝑘𝑖𝑖 , 𝑘𝑘2 ), 𝑃𝑃21
≜ 𝑃𝑃21 (𝑘𝑘2 , 𝑘𝑘1 ) and the weighted coefficient 𝜔𝜔 ∈[0,1]
and minimizes the performance index:
(53)
THE CONSISTENCY AND ACCURACY
OF CI KALMAN FUSER
where, the optimal weighted coefficient vector is:
=
al
a1l a2l ]
[=
Τ
d
ll −1
e (P )
edΤ ( P ll ) −1 ed
(54)
=
PCI PCI [ω 2 P1−1 + ω (1 − ω ) P1−1 P12 P2−1 +
ω (1 − ω ) P2−1 P21 P1−1 + (1 − ω ) 2 P2−1 ]PCI
where, 𝑒𝑒𝑑𝑑𝑇𝑇 = [1 … 1] and defining:
 P (ll )
P =  1(ll )
 P21
ll
P12(ll ) 

P2(ll ) 
Theorem 1: The actual CI fusion error variance matrix
𝑃𝑃�𝐶𝐶𝐶𝐶 is given by:
(55)
Proof: From (58), we can obtain 𝑃𝑃�𝐶𝐶𝐶𝐶 [𝜔𝜔𝑃𝑃1−1 + (1 −
𝜔𝜔)𝑃𝑃1−1 ] = I, which yields:
=
x(t ) PCI [ω P1−1 x(t ) + (1 − ω ) P2−1 x(t )]
𝑃𝑃𝑖𝑖𝑖𝑖𝑙𝑙𝑙𝑙
where,
is the (l,l) th diagonal component of
𝑃𝑃𝑖𝑖𝑖𝑖 (𝑘𝑘𝑖𝑖 , 𝑘𝑘𝑗𝑗 ) and the optimal component fused error
variance 𝑃𝑃𝑑𝑑𝑑𝑑 and the optimal fused error variance 𝑃𝑃𝑑𝑑
are given by:
Τ
d
ll −1
−1
=
Pdl [e=
( P ) ed ] , l 1, , n
2
2
Pd = ∑∑ Ai Pij AΤj
THE STEADY-STATE CI KALMAN FUSER
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
𝑃𝑃1 and 𝑃𝑃2 are known, but the cross-covariance 𝑃𝑃12 is
unknown, the CI Kalman fuser without 𝑃𝑃12 is given by:
xˆˆCIz=
(t | t + N ) PCI [ω P1−1 x1z (t | t + N ) +
(1 − ω ) P2−1 xˆ2z (t | t + N )]
(57)
where, the CI fusion error variance matrix 𝑃𝑃𝐶𝐶𝐶𝐶 is given
by:
P=
[ω P1−1 + (1 − ω ) P2−1 ]−1
CI
(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 ) +
(1 − ω ) P2−1 x2 (t | t + N )]
(56)
=i 1 =j 1
(60)
(62)
The actual CI fusion error variance 𝑃𝑃�𝐶𝐶𝐶𝐶 =
𝑇𝑇
(t|𝑡𝑡 + 𝑁𝑁)], where E denotes the
E[𝑥𝑥�𝐶𝐶𝐶𝐶 (t|𝑡𝑡 + 𝑁𝑁) 𝑥𝑥�𝐶𝐶𝐶𝐶
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 (t|𝑡𝑡 + 𝑁𝑁) is the Unbiased Linear
Minimum Variance (ULMV) estimate of x(t) based on
(58)
the linear space 𝐿𝐿𝑚𝑚 = 𝐿𝐿𝑚𝑚 (𝑥𝑥�1𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁), 𝑥𝑥�2𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁))
3606
Res. J. Appl. Sci. Eng. Technol., 6(19): 3602-3609, 2013
spanned by (𝑥𝑥�1𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁), 𝑥𝑥�2𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁)). From (45), (50),
(57), we have (𝑥𝑥�0𝑧𝑧 (t|𝑡𝑡 + 𝑁𝑁) ∈ 𝐿𝐿𝑚𝑚 , θs. D, CI, i, which
yields (63). It has been proven 𝑃𝑃�𝐶𝐶𝐶𝐶 ≤ 𝑃𝑃𝐶𝐶𝐶𝐶 and tr𝑃𝑃𝑚𝑚 ≤
tr𝑃𝑃𝑑𝑑 ≤ tr𝑃𝑃𝑠𝑠 ≤ tr𝑃𝑃𝑖𝑖 , 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 = tr𝑃𝑃2 and taking ω = 1,
we have J = tr𝑃𝑃1 . Hence the optimal weighting
coefficient 𝜔𝜔 ∈ [0, 1] yields tr𝑃𝑃𝐶𝐶𝐶𝐶 ≤ tr𝑃𝑃1 , tr𝑃𝑃𝐶𝐶𝐶𝐶 tr𝑃𝑃2 , i.e.
tr𝑃𝑃𝐶𝐶𝐶𝐶 ≤ tr𝑃𝑃𝑖𝑖 , I = 1, 2. Taking trace operation for (63) and
(64), we have (66) holds. The proof is completed.
SIMULATION EXAMPLE
Consider the tracking system with two-sensor and
with time-delayed measurements:
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
H1
=
[1
1 0 
0] , =
H2 
,τ 1 1,=
τ2 2
=
0 1 
where, x(t)∈ 𝑅𝑅2 is the state, 𝑧𝑧𝑖𝑖 (t) is the measurement
with delays for sensor i, w(t) and 𝜉𝜉𝑖𝑖 (t) are white noise
with zero mean and variances 𝑄𝑄𝑤𝑤 and 𝑄𝑄𝜉𝜉𝜉𝜉 , respectively.
According to the corresponding transformation, we
obtain the measurement 𝑦𝑦𝑖𝑖 (t) without time delays:
yi (t ) = zi (t + τ i ) = H i x(t ) + vi (t )
where, 𝑣𝑣𝑖𝑖 (t) = 𝜉𝜉𝑖𝑖 (t+𝜏𝜏𝑖𝑖 ) is the transformed measurement
white noise with zero mean and variance 𝑄𝑄𝑣𝑣𝑣𝑣 = 𝑄𝑄𝜉𝜉𝜉𝜉 .
Thus the problem of finding the local steady-state
optimal Kalman smoother 𝑥𝑥�𝑖𝑖𝑧𝑧 (t|𝑡𝑡 + 1), (i = 1, 2) based
on the measurement 𝑧𝑧𝑖𝑖 (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)
MSE1
MSEd
MSE2
MSECI
Fig. 1: The accuracy comparison of 𝑃𝑃1 , 𝑃𝑃2 , 𝑃𝑃0 , 𝑃𝑃𝑠𝑠 , 𝑃𝑃𝑑𝑑 , 𝑃𝑃𝐶𝐶𝐶𝐶 ,
and 𝑃𝑃�𝐶𝐶𝐶𝐶
based on the new measurement 𝑦𝑦𝑖𝑖 (t), respectively.
Further, applying the Kalman filtering method yields the
steady-state optimal fusion Kalman smoothers 𝑥𝑥�𝑚𝑚𝑧𝑧 (t|𝑡𝑡 +
1), 𝑥𝑥�𝑑𝑑𝑧𝑧 (t|𝑡𝑡 + 1), 𝑥𝑥�𝑠𝑠𝑧𝑧 (t|𝑡𝑡 + 1) weighted by matrices,
diagonal matrices and scalars and CI fused Kalman
𝑧𝑧
(t|𝑡𝑡 + 1). In simulation, we take 𝑇𝑇0 = 0.4,
smoother 𝑥𝑥�𝐶𝐶𝐶𝐶
4
0
�.
Q w = 1, Q 𝜉𝜉1 = 0.36, Q 𝜉𝜉2 = �
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:𝑥𝑥 𝑇𝑇 𝑃𝑃−1 x = 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 𝑃𝑃𝑎𝑎 ≤𝑃𝑃𝑏𝑏 is equivalent to that the
ellipse for 𝑃𝑃𝑎𝑎 is enclosed in the ellipse for 𝑃𝑃𝑏𝑏 .The
simulation results are shown in Fig. 1.
From Fig. 1, it is clearly shown that the covariance
ellipse for 𝑃𝑃𝑚𝑚 lies within the intersection of ellipses for
𝑃𝑃1 and 𝑃𝑃2 , the ellipse for CI fused theoretical covariance
𝑃𝑃𝐶𝐶𝐶𝐶 encloses the intersection region of ellipses for 𝑃𝑃1 an
MSE0
trP1
MSEs
trP2
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
Fig. 2: The curves of MSE i (t)
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Res. J. Appl. Sci. Eng. Technol., 6(19): 3602-3609, 2013
𝑃𝑃2 (Chen et al., 2002) and passes through the four
points of intersection of ellipses for 𝑃𝑃1 and 𝑃𝑃2 . The
ellipse for CI fused actual covariance 𝑃𝑃�𝐶𝐶𝐶𝐶 lies within the
ellipse for theoretical covariance 𝑃𝑃𝐶𝐶𝐶𝐶 .
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
� 𝐶𝐶𝐶𝐶 , the curves denote the corresponding mean square
tr𝑃𝑃
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).
Bolzani, J.C., C. Ferreira and J. Waldmann, 2007.
Covariance intersection-based sensor fusion for
sounding rocket tracking and impact area
prediction. Cont. Eng. Pract., 15: 389-409.
Chen, L.J., P.O. Arambel and R.K. Mehra, 2002.
Estimation under unknown correlation: Covariance
intersection revisited. IEEE T. Autom. Cont., 47:
1879-1882.
Deng, Z.L., Y. Gao, C.B. Li and G. Hao, 2008. Selftuning decoupled information fusion wiener state
component filters and their convergence.
Automatica, 44: 685-695.
Guo, Q., S.Y. Chen, H.R. Leung and S.T. Liu, 2010.
Covariance intersection based image fusion
CONCLUSION
technique with application to pansharpening in
remote sensing. Inform. Sci., 180: 3434-3443.
For the two-sensor system with time-delayed
Hall, D.L. and J. Llinas, 1997. An introduction to
measurements and with unknown cross-covariance
multisensor data fusion. IEEE Proc., 85: 6-23.
matrices, a CI fusion steady-state Kalman fuser has
Julier, S.J. and J.K. Uhlmann, 1996. General
been presented, which has the advantage that the
decentralized data fusion with unknown cross
computation of cross-covariances can be avoided. It is
covariances. Proceeding of the SPIE Aerosence
proved that its accuracy is higher than that of each local
Conference, SPIE., 2775: 536-547.
estimator and is lower than that of optimal fuser
Julier, S.J. and J.K. Uhlmann, 1997. Non-divergent
weighted by matrices. A two-sensor system Monteestimation algorithm in the presence of unknown
Carlo simulation example shows that its accuracy is
correlations. Proceeding of the IEEE American
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ACKNOWLEDGMENT
3-20.
Julier, S.J. and J.K. Uhlmann, 2009. General
This study is supported by the Natural Science
Decentralized Data Fusion with Covariance
Foundation of China under grant NSFC-60874063, the
Intersection, In: Liggins, M.E.D.L. and J.L. Hall
2012 Innovation and Scientific Research Foundation of
(Eds.), Handbook of Multisensor Data Fusion,
graduate student of Heilongjiang Province under grant
Theory and Practice. 2nd Edn., CRC Press, Boca
YJSCX2012-263HLJ and the Support Program for
Raton, pp: 872, ISBN: 1420053086.
Young Professionals in Regular Higher Education
Kailath, T., A.H. Sayed and B. Hassibi, 2000. Linear
Institutions of Heilongjiang Province under grant
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1251G012.
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Li, X.R., Y.M. Zhu, J. Wang and C.J. Han, 2003.
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