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间歇通讯下全局最优分布式融合算法
－针对传感器噪声相关的动态系统
罗应婷，朱允民
四川大学数学学院，成都 610064
摘要：由于无线通讯的带宽限制，传感器会间歇地向融合中心发送数据。本文指出过去文献中
间歇通讯时的分布式融合算法可以推广到传感器噪声相关的情况中去。和传统的分布式算法相
比，无论通讯的间歇周期多长，只要是同步间歇通讯，新算法都是全局最优的。
关键词：信号与信息处理，估计融合，分布式融合，丢包，间歇通讯
中图分类号： TN01
A Globally Optimal Distributed Estimation
Fusion Algorithm for Dynamical Systems with
Intermittent Communications and Correlated
Sensor Noises
Yingting Luo, Yunmin Zhu
College of Mathematics, Sichuan University, Chengdu 610064
Abstract: Due to power and bandwidth limitation in wireless sensor network, the sensor
communications to a fusion center are often intermittent. In this article, we point out that
the previous distributed estimation fusion algorithm can be used in the discrete time linear
dynamic system even if the sensor noises are cross-correlated. Compared with existing
distributed fusion algorithms, the biggest advantage of this distributed fusion algorithm can
be globally optimal no matter how long is the intermittent communication period provided
that the intermittent communications are synchronous.
Key words: signal and information process, estimation fusion, distributed fusion, packet
loss, intermittent communications
0 Introduction
Modern surveillance systems often utilize multiple sensors of di�erent types to provide
complementary and overlapping coverage on targets. In order to generate target tracks and
基金项目： SRFDP (20080610001)
作者简介：Yingting Luo(1983-), female, instructor, major research direction: signal and data processing, informa-
tion fusion, and statistics. Correspondence author：Yunmin Zhu(1944-), male, professor, major research direction：
multisensor distributed estimation and decision fusion.
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estimates, the sensor data need to be fused. There are two basic fusion architectures: centralized
and distributed. In the centralized fusion, all raw measurements from the local sensors are sent
to the fusion center in time. While in the distributed fusion, the local sensors send their
processed data.
The distributed fusion is more challenging and has been a focal point of most fusion
research. In the existing literature, many researches on the distributed fusion have been done.
For the case with cross-independent sensor noises, an globally optimal distributed Kalman
�ltering fusion was proposed in [1, 2] respectively, which was proved to be equivalent to the
centralized Kalman �ltering using all sensor measurements. A rigorous performance analysis for
distributed Kalman �ltering fusion with feedback was presented in [3]. When the sensor noises
are cross-correlated, Song et al. [4] proposed a new distributed fusion to reconstruct the optimal
centralized estimate. Duan et al. [5] recently proposed distributed fusion algorithms where local
sensors transmit two di�erent transformations of observations to the fusion center and they have
own advantage and weakness. In practice, the communication environment is sometimes not
perfect. It is meaningful to discuss the distributed fusion under the communication constrains.
When there exists the limitation of communication bandwidth between sensors and fusion
center, researchers [6, 7, 8] discuss how to compress sensor observations or estimates before
the sensors’transmission in order to obtain an optimal estimation at the fusion center. When
there exist intermittent communications or passive packet loss between sensors and fusion
center, researchers proposed some new distributed fusion algorithms in [9, 10]. When the
local estimates are not lost many times, Luo et al. proposed an optimal distributed fusion
which can be equivalent to the corresponding centralized fusion with complete communications
even if the process noise does exist in [10]. However, When the intermittent period is quite
long, only a suboptimal distributed fusion algorithm was proposed there. For the dynamic
system without model noises, Wolfe et al. [9] proposed a simple mechanism for implementing
Luenberger observers that allows the globally optimal estimates at the fusion center to be
constructed by the sum of the current local partial estimates immediately no matter how long
the communications from sensor to the fusion center have paused provided that the fusion center
is synchronously collecting sensor transmissions. However, the condition without model noises
in [9] is too restrictive in practice. In this short article, we point out that this condition can be
removed and the global optimality of the algorithm with any long intermittent communication
period is still true even if the model noises exists and sensor noises are cross-correlated. Pros
and Cons of this algorithm are analyzed in the end of this article.
1 Distributed Fusion Algorithm
Consider the linear dynamic system
xk+1 = �kxk + �k; k = 0; 1; :::; (1)
yik = H
i
kxk + !
i
k; i = 1; : : : ; L; (2)
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where �k 2 <r�r, xk; �k 2 <r, H ik 2 <Ni�r, yik; !ik 2 <Ni . The process noise �k and sensor
noise !ik are both zero-mean random variables independent of each other temporally. The
sensor noise !ik could be cross-correlated. The covariance matrices of the process noise and
sensor noise are known as follows
Cov(�k) = R�k ; Cov(!
i
k) = R!ik :
The stacked measurement equation is written as
yk = Hkxk + !k; (3)
where
yk = (y
10
k ; : : : ; y
L0
k )
0; Hk = (H1
0
k ; : : : ; H
L0
k )
0; !k = (!1
0
k ; : : : ; !
L0
k )
0; (4)
and the covariance of the noise !k is given by
Cov(!k) = R!k : (5)
According to the the centralized Kalman �ltering [11, 12, 13, 14]with all sensor measure-
ments, the optimal estimate at the fusion center is given by
x^k = (I �KkHk)�k�1x^k�1 +Kkyk; (6)
where
Pk=k�1 = �k�1Pk�1=k�1�0k�1 +R�k ; (7)
P�1k=k = P
�1
k=k�1 +H
0
kR
�1
!k
Hk; (8)
Kk = Pk=kH
0
kR
�1
!k
: (9)
Substituting (9) into (10), we have
x^k = (I �KkHk)�k�1x^k�1 + Pk=kH 0kR�1!k yk
= (I �KkHk)�k�1x^k�1 + Pk=kH 0k
PL
i=1R
�1
!k
(�i)yik;
(10)
where R�1!k (�i) denotes the i-th column of R�1!k .
Denote the local partial estimate as
zik = (I �KkHk)�k�1x^k�1 + Pk=kH 0kR�1!k (�i)yik: (11)
If we initialize
LX
i=1
zi0 = x^0; (12)
it is easy to verify that
LX
i=1
zik = x^k 8k: (13)
Thus, if the local sensors send zik to the fusion center, the fusion center only needs to add them
together to get the optimal estimate.
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2 Discussion
The new distributed fusion algorithm that allows the optimal estimates at the fusion center
to be constructed by the sum of the current local partial estimates immediately. It has the
following advantages:
� This algorithm is optimal even if the sensor noises are cross-correlated.
� The fusion criterion of the new algorithm is not recursive, thus it is robust to packet loss
or intermittent communications. No matter how much local data is lost, once the fusion center
receives all the current local data, it could immediately constructed the optimal estimate.
� Because of its robustness to intermittent communications, this algorithm can reduce the
communication frequency between local sensors and fusion center sharply.
There is no doubt that this algorithm has its disadvantages as well as advantages.
� The local partial estimate is not a true estimate of the state.
� The topology of the sensor network must be known a priori.
� There must be some synchronization between the local sensors.
Fortunately, most of these drawbacks can be overcome in practice. The local sensor could
has its own estimate by local �ltering and known the topology of the sensor network with no
hard. Synchronization can be easily accomplished at little energy via GPS.
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