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相关噪声系统的测量反馈自整定加权测量融合卡尔曼滤波器
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对于具有多个传感器和未知噪声统计信息的线性离散随机系统,通过使用测量反馈,满秩分解和加权最小二乘理论设计了噪声方差和互协方差的在线估计器。 此外,提出了一种自调谐加权测量融合卡尔曼滤波器。 Fadeeva公式用于建立噪声统计未知的ARMA创新模型。 静态和可逆ARMA创新模型的采样相关函数用于识别噪声统计信息。 证明了所提出的自校正加权测量融合卡尔曼滤波器收敛于最优加权测量融合卡尔曼滤波器,这意味着它的渐近全局最优性。 雷达跟踪系统的仿真结果表明了该算法的有效性。
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Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 324296, 16 pages
doi:10.1155/2012/324296
Research Article
Measurement Feedback Self-Tuning
Weighted Measurement Fusion Kalman Filter for
Systems with Correlated Noises
Xin Wang and Shu-Li Sun
Department of Automation, Heilongjiang University, Harbin 150080, China
Correspondence should be addressed to Xin Wang, wangxin@hlju.edu.cn
Received 26 February 2012; Accepted 19 March 2012
Academic Editor: Baocang Ding
Copyright q 2012 X. Wang and S.-L. Sun. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
For the linear discrete stochastic systems with multiple sensors and unknown noise statistics, an
online estimators of the noise variances and cross-covariances are designed by using measure-
ment feedback, full-rank decomposition, and weighted least squares theory. Further, a self-tuning
weighted measurement fusion Kalman filter is presented. The Fadeeva formula is used to establish
ARMA innovation model with unknown noise statistics. The sampling correlated function of the
stationary and reversible ARMA innovation model is used to identify the noise statistics. It is
proved that the presented self-tuning weighted measurement fusion Kalman filter converges
to the optimal weighted measurement fusion Kalman filter, which means its asymptotic global
optimality. The simulation result of radar-tracking system shows the effectiveness of the presented
algorithm.
1. Introduction
With the development of scientific technology, the scale of a control system has become more
and more complex and tremendous, and the accuracy, fault-tolerance, and robustness of a sys-
tem are required much higher, so that single sensor has been unable to satisfy the demands
of high scientific technologies. Thus, the multisensor information fusion technology has been
paid great attention to and become an important research issue.
In early 1980s, Shalom 1, 2 presented the computation formula of cross-covariance
matrix by studying the correlation of two sensor subsystems with independent noises. Carl-
son 3 presented the famous federated Kalman filter by using the upper bound of noise
variance matrix to replace noise variance matrix and supposing that the initial local esti-
mation errors are not correlated. Kim 4 proposed the maximum likelihood fusion estimation
algorithm by requiring the hypothesis that random variables obey normal distributions.
2 Journal of Applied Mathematics
The universal weighted least squares method and the best linear unbiased fusion estimation
algorithm were presented by Li et al. 5 on the basis of a unified linear model for three
estimation fusion architectures of centralized filter, distributed filter, and hybrid filter. Three
weighted fusion algorithms of matrix-weighted, diagonal-matrix-weighted, and scalar-
weighted in the linear minimum variance sense were proposed by Sun and Deng 6,Sun7,
where the matrix-weighted fusion algorithm, maximum likelihood fusion algorithm 4,and
distributed best linear unbiased estimation algorithm 5 have the same result and avoid the
derivation on the basis of hypothesis of normal distribution and linear model. The short-
comings of methods presented in 3–7 are that they have larger calculation b urdens and the
fusion accuracy is global suboptimal.
Based on Kalman filtering, Gan and Chris 8 discussed two kinds of multisensor
measurement fusion method: the centralized measurement fusion CMF and the weighted
measurement fusion WMF. The former is to directly merge the multisensor data through
the augmented measurement vector to calculate the estimation. Its advantage is that it can
obtain globally optimal state estimator. Its shortcoming is that the computational burden is
large since the measurement dimension is high. So it is unsuitable for real-time application.
The latter is to weigh local sensor measurements to obtain a low-dimensional measurement
equation, and then to use a single Kalman filter to obtain the final fused state estimation.
Its advantages are that the computational burden can be obviously reduced and the globally
optimal state estimation can be obtained 8–12.
It is known that the existing information fusion Kalman filtering is only effective
when the model parameters and noise statistics are exactly known. But this restricts its
applications in practice. In real applications, the model parameters and noise statistics are
completely or partially unknown in general. The filtering problem for systems with unknown
model parameters and/or noise statistics yields the self-tuning filtering. Its basic principle
is the optimal filter plus a recursive identifier of model parameters and/or noise statis-
tics 13.
For self-tuning fusion fi lters, there are two methods of self-tuning weighted state
fusion and self-tuning measurement fusion. Weighted state fusion method is used by Sun 14
and Deng et al. 15, respectively, but the used distributed state fusion algorithm is globally
suboptimal and the acquired self-tuning estimator cannot reach globally asymptotic opti-
mality. For the self-tuning measurement fuser, 9, 16 considered the uncorrelated input
noise and measurement noise. Ran and Deng 17 presented a self-tuning measurement
fusion Kalman filter under the assumption that all sensors have the same measurement mat-
rices.
This paper is concerned with the self-tuning filtering problem for a multisensor system
with unknown noise variances, different measurement matrices, and correlated noises.
Firstly, transform the system with correlated input noise and measurement noise into one
with uncorrelated input noise and measurement noise by using the measurement feedback
and taking measurement data as a part of system control item. Then, weigh all the meas-
urements by using full-rank decomposition and weighted least squares theory. The Fadeeva
formula is used to establish ARMA innovation model with unknown noise covariance
matrices and the sampling correlated function of a stationary and reversible ARMA
innovation model is used to identify the noise covariance matrices. It is rigorously proved
that the presented self-tuning weighted measurement fusion Kalman filter converges to the
optimal weighted measurement fusion Kalman filter, that is, it has asymptotic global optima-
lity.
Journal of Applied Mathematics 3
2. Problem Formulation
Consider the controlled multisensor time-invariant systems with correlated noises:
x
t 1
Φx
t
Bu
t
Γw
t
, 2.1
y
i
t
H
i
x
t
v
i
t
,i 1,...,L, 2.2
where xt ∈ R
n
is the state, ut ∈ R
p
is the given control, y
i
t ∈ R
m
i
is the measurement
of the sensor i, wt ∈ R
r
is the input noise, and v
i
t ∈ R
m
i
is the measurement noise. L is
the number of sensors, H
i
∈ R
m
i
×n
is the measurement matrix of the sensor i. Φ, B,andΓ are
constant matrices with compatible dimensions.
Assumption 2.1. wt and v
i
t are correlated Gaussian white noise with zero means, and
E
w
t
v
i
t
w
T
k
v
T
j
k
Q
w
S
j
S
T
i
R
ij
δ
tk
,i 1,...,L, 2.3
where the symbol E denotes the expectation, δ
tk
is Kronecker delta function, that is, δ
tt
1,
δ
tk
0 t
/
k. The variance matrix of v
i
t is R
ii
R
i
. Combining L measurement equations
of 2.2 yields
y
I
t
H
I
x
t
v
I
t
, 2.4
where y
I
ty
T
1
t,...,y
T
L
t
T
, H
I
H
T
1
,...,H
T
L
T
and v
I
tv
T
1
t,...,v
T
L
t
T
.Let
the variance of v
I
t be R
I
R
ij
> 0 and the cross covariance of wt and v
I
t be
S S
1
,...,S
L
.
Assumption 2.2. Φ,H
I
is a detectable pair and Φ, Γ is a controllable pair, or Φ are stable.
Assumption 2.3. Measurement data y
i
t is bounded, that is,
y
i
t
<c, i 1,...,L, 2.5
where ·is the norm of a vector and c>0 is a positive real number.
2.1. CMF and WMF Kalman Filter
To convert the systems 2.1 and 2.4 into the uncorrelated system, 2.1 is equivalent to
x
t 1
Φx
t
Bu
t
Γw
t
J
y
I
t
− H
I
x
t
− v
I
t
, 2.6
where J is a pending matrix. 2.6 can be converted into
x
t 1
Φx
t
u
t
w
t
, 2.7
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