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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

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,Sun7,

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:



t  1



Φx





 Bu





Γw





, 2.1





 H





 v





,i 1,...,L, 2.2

where xt ∈ R

is the state, ut ∈ R

is the given control, y

t ∈ R

is the measurement

of the sensor i, wt ∈ R

is the input noise, and v

t ∈ R

is the measurement noise. L is

the number of sensors, H

∈ R

×n

is the measurement matrix of the sensor i. Φ, B,andΓ are

constant matrices with compatible dimensions.

Assumption 2.1. wt and v

t are correlated Gaussian white noise with zero means, and































,i 1,...,L, 2.3

where the symbol E denotes the expectation, δ

is Kronecker delta function, that is, δ

 1,

 0 t

 k. The variance matrix of v

t is R

 R

. Combining L measurement equations

of 2.2 yields

I





 H

I





 v

I





, 2.4

where y

I

ty

t,...,y

t

, H

I

H

,...,H



and v

I

tv

t,...,v

t

.Let

the variance of v

I

t be R

I

R

 > 0 and the cross covariance of wt and v

I

t be

S S

,...,S

.

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

t is bounded, that is,









<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



t  1



Φx





 Bu





Γw





 J



I





− H

I





− v

I





, 2.6

where J is a pending matrix. 2.6 can be converted into



t  1





Φx





 u





 w





, 2.7

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