Sequential Inverse Covariance Intersection Fusion
Steady-state Kalman Filter for Multi-Sensor Systems
with Multiple Time-delayed Measurements
Tianmeng Shang ,Qi Liu,Yuan Gao
*
,Chenjian Ran,Yinfeng Dou
Department of Automation, Heilongjiang University, Harbin 150080, China
E-mail:gaoyuan_hlju@163.com
Abstract—In order to settle the fusion problem of multi-
sensor systems with multiple time-delayed measurements, a
steady-state suboptimal Kalman filter is derived, and the Inverse
Covariance Intersection (ICI) fusion method is adopted which
can be utilized in the system with unknown common information
shared between sensors, and whose major advantage is that it is
proved to be more accurate and represents an optimal-consistent
and tight-solution to the problem of fusing estimates which share
unknown common information. Then the Sequential ICI fusion
steady-state Kalman filter is obtained. A simulation example
shows the accuracy, the consistence and the tightness of the ICI
fusing method, and the effectiveness of the presented fusion
Kalman filter.
Keywords—Multiple time-delayed measurements; ICI fusion;
Steady-stste Kalman filter
I. INTRODUCTION
The problem of fusing state estimation with time delay
system has received particularly strong attention. Even though
optimal fusion method for such systems have been rapidly
developed, much less reported on multiple time delay system
at present
[1]
. In order to solve the problem of multiple time-
delayed system, the augmented optimal Kalman filter method
has been derived, which transforms the original system into a
new system without time delay and then applies the standard
Kalman filtering algorithm to obtain the final estimation
[2]
.
The disadvantage is that it requires a mass of complicated
compute, particularly when the delays are large
[3]
. In addition
to this method another is the re-organized innovation approach,
which involves the computation of multiple filters with the
same dimension as the original system in series
[4]
.
In multiple time-delayed system, storing and keeping
track of the cross-covariance information is cumbersome and
sometimes impossible if sensor nodes, which are equipped
with own estimate system, are supposed to operate
dependently. In order to improve the quality of fusing
estimation, the calculation of the cross-covariance matrix has
been a challenging problem. Although some optimal weighted
by matrices
[5]
fusion method, such as diagonal matrices and
scalars, have been derived, those weighted methods still have
the disadvantage and drawback on the complicated
computation of the cross-covariance matrices among local
estimators
[6]
. Besides, the local Kalman estimator is difficult
to be obtained because it needs several other estimators to
provide different information, that is, at one local sensor node,
many multi-step smoothers, predictors and filters are
preparing to supply data for one-step estimation. This
calculation process is very complicated. Therefore, in order to
avoid compute cross-covariance which is unknown in some
practical application systems, a novel method, Inverse
Covariance Intersection (ICI), is adopted. Similar to the SCI
algorithm, a Sequential Inverse Covariance Intersection (SICI)
fuser, which can be applied to multi-sensor system, is
presented. Compared with the SCI algorithm
[7]
, the Sequential
ICI algorithm provides less conservative and well consistent
fusion result
[8]
.
In this paper, based on the Sequential Covariance
Intersection (SCI) fusion algorithm, a steady-state Kalman
filter is presented and a Sequential Inverse Covariance
Intersection algorithm is applied to deal with the multiple
time-delayed systems with unknown cross-covariance. It is
proved that the SICI fuser has higher accuracy than ICI and
local estimators, and the computational burden can be reduced
significantly.
Ⅱ. P
ROBLEM FORMULATION
Consider the multi-sensor discrete time-invariant system
with multiple time-delayed measurements
1()()
txtwt
(1)
() () ()
0
(), 1,2, ,
K
i
d
ii i
k
k
yt Hxtkvti l
(2)
where
t
is the discrete time,
n
tR
is the state,
i
i
m
yt R
is the measurement of the
i
th sensor,
()
r
wt R
,
()
i
m
i
vt R
are the state and the measurement noises,
respectively, and
()
,,
i
k
are known time-invariant matrices
with compatible dimensions,
i
d
is the measurement delay of
the
i
th sensor, with 0
i
dd, and
d
is a fixed positive
integer.
978-1-5386-5373-9/18/$31.00 ©2018 IEEE
2018 IEEE 4th Information Technology and Mechatronics Engineering Conference (ITOEC 2018)