Distributed Fusion Filter for Multi-rate Multi-sensor Systems with
Multiplicative Noises
JIN Hao
1
, MA Jing
2
1. School of Computer and Information Engineering, Harbin University of Commerce, Harbin 150080, China
E-mail: jinhao0205@163.com
2. School of Mathematics Science, Heilongjiang University, Harbin 150080, China
E-mail: jingma427@163.com
Abstract: This paper is concerned with the distributed fusion filtering problem for a class of multi-rate multi-sensor systems
with white multiplicative noises. The stochastic dynamic system sampled uniformly. Sampling period of each sensor is uniform
and the integer multiple of the state update period. Moreover, different sensors have the different sampling rates. Firstly, the local
filters (LFs) at state update points are obtained by smoothing of LFs at measurement sampling points. Then, the corresponding
error variance matrices (EVMs) are derived to compute the fusion weights. Finally, the distributed fusion filter is given by the
well-known covariance intersection fusion (CIF) algorithm. Simulation research supports the correctness of the proposed results.
Key Words: Multi-rate, multi-sensor, covariance intersection fusion, multiplicative noise
1 Introduction
In many practical systems, there often exist various
uncertainties due to the unknown or partially unknown
parameters and environmental disturbances. The
uncertainties can be approximated mathematically by an
additive noise or a multiplicative noise [1-6]. These systems
are widely used in target tracking, detection, signal
processing and other areas. Thus the research on systems
with additive and multiplicative noises has the important
practical significance. For single sensor system, the results
include the nonlinear polynomial state estimators [1-3], the
least mean square linear filter [4-5]. For multiple sensor
systems, the results include the distributed fusion filter [6]
and the centralized fusion filter [7]. However, the above
references are only suitable for single rate systems.
For multi-rate systems, the first important work goes back
to the switch decomposition technique proposed by Kranc
[8]. Generally, there are two basic methods for the state
estimation problem for multi-rate systems. One is based on
multiscale system theory and the other is based on Kalman
filtering theory. On the basis of multiscale system theory,
many famous fusion strategies are proposed for systems
with the sampling rate ratio being one or positive integer
power to two. However, the state estimators are very
complex and high computational burden. On the basis of
Kalman filtering theory, many useful filtering strategies are
also proposed such as optimal signal reconstruction method
[9], asynchronous centralized fusion algorithm [10],
sequential filtering algorithm [11], left synchronously lifting
technology [12], and measurement augmented approach
[13]. But, the computational cost of the above filtering
strategies is high since they are given by state/measurement
augmentation. In order to avoid augmentation, the multi-rate
fusion problem is transformed into an equivalent single rate
fusion problem. For non-uniform sampling systems, a
distributed fusion filter is given [14], for uniform sampling
*
This work is supported by National Natural Science Foundation (NNSF)
of China under Grant 61403131.
systems, the corresponding distributed fusion filters are also
given in [15]. However, the multiplicative noises are not
taken into account. In the recent study [16], a distributed
fusion filter is studied for multi-rate systems with
multiplicative noise. However, the cross-covariance
matrices are needed to obtain the fusion weights.
The new LFs in this paper are obtained by smoothing the
LFs at the measurement sampling points. Then, the
well-known CIF algorithm is used to fuse all the LFs. The
proposed new LFs can reduce the computational cost since
state augmentation is avoided. Moreover, differently from
the LFs in [16] where the LFs are obtained by predictions of
LFs at the observation sampling points. Here, the LFs are
obtained by smoothing of LFs at the observation sampling
points. Hence, the accuracy of the proposed LFs is higher
than that in [16]. Further, the proposed covariance
intersection fusion filter (CIFF) can reduce the
computational burden since the cross-covariance matrices
are avoided.
2 Problem Formulation
Assume the S-sensor linear discrete-time dynamic
multi-rate stochastic system is given by
() () ()
tb b x tb w tb
)*
(1)
()( ())() ()
ll l ll l l ll
z tb H tb H x tb v tb
D
,
ll
bcb , {1, , }lS " (2)
where
()
n
xtb \
is the state vector at the tb time moment,
()
l
n
ll
ztb \
, {1, , }lS " are the measured outputs at
the
l
tb time moment. ()
q
wtb \ and ()
l
n
ll
vtb\ ,
{1, , }lS " are white noises. The state ()
tb is updated at
the highest rate with a period
b and the l th sensor
measurement
()
l
n
ll
ztb \ is sampled at a lower rate with a
period
ll
bcb where
l
c is a positive integer. The matrices
)
,
*
,
l
and
l
are constant, known and of suitable
dimensions. Multiplicative noises
()
ll
tb
are scalar white
Proceedings of the 34th Chinese Control Conference
Jul
28-30, 2015, Han
zhou, China
4679