Measurement Feedback Self-Tuning Weighted Measurement Fusion Kalman Filter for Systems with Correlated Noises

Abstract

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 measurement 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 system 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. Carlson [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 estimation errors are not correlated. Kim [4] proposed the maximum likelihood fusion estimation algorithm by requiring the hypothesis that random variables obey normal distributions. 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 shortcomings of methods presented in [3–7] are that they have larger calculation burdens 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 statistics [13].

For self-tuning fusion filters, 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 optimality. 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 matrices.

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 measurements 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 optimality.

2. Problem Formulation

Consider the controlled multisensor time-invariant systems with correlated noises:𝑥()(𝑡)(𝑡)(𝑡),𝑦𝑡+1=Φ𝑥+𝐵𝑢+Γ𝑤_𝑖(𝑡)=𝐻_𝑖𝑥(𝑡)+𝑣_𝑖(𝑡),𝑖=1,…,𝐿, (2.1)(2.2) where 𝑥(𝑡)∈𝑅^𝑛 is the state, 𝑢(𝑡)∈𝑅^𝑝 is the given control, 𝑦_𝑖(𝑡)∈𝑅^𝑚𝑖 is the measurement of the sensor 𝑖, 𝑤(𝑡)∈𝑅^𝑟 is the input noise, and 𝑣_𝑖(𝑡)∈𝑅^𝑚𝑖 is the measurement noise. 𝐿 is the number of sensors, 𝐻_𝑖∈𝑅^{𝑚𝑖×𝑛} is the measurement matrix of the sensor 𝑖. Φ, 𝐵, and Γ are constant matrices with compatible dimensions.

Assumption 2.1. 𝑤(𝑡) and 𝑣_𝑖(𝑡) are correlated Gaussian white noise with zero means, and 𝐸𝑤(𝑡)𝑣_𝑖(𝑡)𝑤^T(𝑘)𝑣^T_𝑗(𝑘)=𝑄_𝑤𝑆_𝑗𝑆^T_𝑖𝑅_𝑖𝑗𝛿_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)(𝑡)=[𝑦^T₁(𝑡),…,𝑦^T_𝐿(𝑡)]^T, 𝐻^(I)=[𝐻^T₁,…,𝐻^T_𝐿]^T and 𝑣^(I)(𝑡)=[𝑣^T₁(𝑡),…,𝑣^T_𝐿(𝑡)]^T. Let the variance of 𝑣^(I)(𝑡) be 𝑅^(I)=(𝑅_𝑖𝑗)>0 and the cross covariance of 𝑤(𝑡) and 𝑣^(I)(𝑡) be 𝑆=[𝑆₁,…,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>0is 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)+𝐽𝑦^(I)(𝑡)−𝐻^(I)𝑥(𝑡)−𝑣^(I)(𝑡), (2.6) where 𝐽 is a pending matrix. (2.6) can be converted into𝑥(𝑡+1)=Φ𝑥(𝑡)+𝑢(𝑡)+𝑤(𝑡), (2.7) where Φ=Φ−𝐽𝐻^(I), 𝑢(𝑡)=𝐵𝑢(𝑡)+𝐽𝑦^(I)(𝑡), 𝑤(𝑡)=Γ𝑤(𝑡)−𝐽𝑣^(I)(𝑡). 𝐽𝑦^(I)(𝑡) as an output feedback becomes a part of the control item. Then, primary system formulae (2.1) and (2.2) are equivalent to the system formed by formulae (2.4) and (2.7). To make 𝐸[𝑤(𝑡)𝑣^(I)T(𝑡)]=0, introduce 𝐽=Γ𝑆𝑅^(I)−1 which ensures that 𝑤(𝑡) and 𝑣^(I)(𝑡) are not correlated. Then, variance matrix of 𝑤(𝑡) is yielded as 𝑄_𝑤=Γ(𝑄_𝑤−𝑆𝑅^(I)𝑆^T)Γ^T. From [18], we know that any nonzero matrix 𝐻^(I) has full-rank decomposition: 𝐻^(I)=𝐹𝐻^(II), (2.8) where 𝐹 is a full column-rank matrix with the rank 𝑟, and 𝐻^(II) is a full row-rank matrix with the rank 𝑟, then measurement model (2.4) can be represented as 𝑦^(I)(𝑡)=𝐹𝐻^(II)𝑥(𝑡)+𝑣^(I)(𝑡). (2.9)

Given that 𝐹 is a full column-rank matrix, it follows that 𝐹^T𝑅^(I)𝐹 is nonsingular. Then, the weighted least squares (WLS) [19] method is used and the Gauss-Markov estimate of 𝐻^(II)𝑥(𝑡) is yielded as 𝑦^(II)(𝑡)=𝐹^T𝑅^(I)−1𝐹⁻¹𝐹^T𝑅^(I)−1𝑦^(I)(𝑡), (2.10) substituting (2.9) into (2.10) yields𝑦^(II)(𝑡)=𝐻^(II)𝑥(𝑡)+𝑣^(II)(𝑡),𝑣^(II)(𝑡)=𝐹^T𝑅^(I)−1𝐹⁻¹𝐹^T𝑅^(I)−1𝑣^(I)(𝑡). (2.11)(2.12)

The variance matrix 𝑅^(II)=E[𝑣^(II)(𝑡)𝑣^(II)T(𝑡)] of 𝑣^(II)(𝑡) is given by𝑅^(II)=𝐹^T𝑅^(I)−1𝐹⁻¹. (2.13)

For systems (2.4) and (2.7), and (2.7) and (2.11), respectively, using standard Kalman filtering algorithm [20], we can obtain CMF and WMF Kalman estimators 𝑥^(𝑖)(𝑡∣𝑡+𝑗), 𝑖=I, II, 𝑗=0, 𝑗<0, and 𝑗>0, and their error variance matrices 𝑃^(𝑖)(𝑡∣𝑡+𝑗). It is proved in [11] that the weighted measurement fusion steady-state Kalman filter 𝑥^(II)(𝑡∣𝑡) for the weighted measurement fusion system (2.7) and (2.11) has the global optimality, that is, it is numerically identical to the CMF steady-state Kalman filter 𝑥^(I)(𝑡∣𝑡) if they have the same initial values.

The above WMF method can obviously reduce the computational burden since the dimension of the measurement vector for the centralized measurement fusion is 𝑚×1, 𝑚=𝑚₁+𝑚₂+⋯+𝑚_𝐿, while that for the weighted measurement fusion is 𝑟×1, and 𝑚 is much larger than 𝑟 generally.

2.2. Optimal Measurement Fusion Steady-State Kalman Filter

By the above WMF methods, the corresponding optimal steady-state Kalman filter is given as [21]𝑥^(II)(𝑡+1∣𝑡+1)=𝑥^(II)(𝑡+1∣𝑡)+𝐾^(II)_𝑓𝜀^(II)(𝑡+1),𝑥^(II)(𝑡+1∣𝑡)=Ψ^(II)_𝑝𝑥^(II)(𝑡∣𝑡−1)+𝐾^(II)_𝑝𝑦^(II)(𝑡)+𝑢(𝑡),𝑥^(II)(𝑡∣𝑡+𝑁)=𝑥^(II)(𝑡∣𝑡)+^𝑁_𝑗=1𝐾^(II)𝑗𝜀^(II)𝑡+𝑗,𝑁>0,𝑢(𝑡)=𝐵𝑢(𝑡)+𝐽𝑦^(I)(𝑡),𝐽=Γ𝑆𝑅^(I)−1,𝜀^(II)(𝑡)=𝑦^(II)(𝑡)−𝐻^(II)𝑥^(II)(),𝐾𝑡∣𝑡−1_(II)𝑓=Σ^(II)𝐻^(II)T𝐻^(II)Σ^(II)𝐻^(II)T+𝑅^(II)−1,𝐾^(II)_𝑝=ΦΣ^(II)𝐻^(II)T𝐻^(II)Σ^(II)𝐻^(II)T+𝑅^(II)−1,Ψ^(II)_𝑝=Φ−𝐾^(II)_𝑝𝐻^(II),𝐾^(II)𝑗=Σ^(II)𝐼_𝑛−𝐾^(II)_𝑓𝐻^(II)TΦ^T𝑗𝐻^(II)T𝐻^(II)Σ^(II)𝐻^(II)T+𝑅^(II)−1,𝑃^(II)=𝐼_𝑛−𝐾^(II)_𝑓𝐻^(II)Σ^(II), (2.14) where Ψ^(II)_𝑝 is a stable matrix [19] and Σ^(II) satisfies the following Riccati equation:Σ^(II)=ΦΣ^(II)−Σ^(II)𝐻^(II)T𝐻^(II)Σ^(II)𝐻^(II)T+𝑅^(II)−1𝐻^(II)Σ^(II)Φ^T+𝑄_𝑤. (2.15)

When noise variance matrices 𝑄_𝑤, 𝑆_𝑖, and 𝑅_𝑖𝑗(𝑖,𝑗=1,…,𝐿) are unknown, the problem is to find a self-tuning WMF Kalman filter 𝑥^(II)𝑠(𝑡∣𝑡) for the fused system (2.7) and (2.11). Then, the key to the problem is how to find the consistent estimates of the noise variance and cross-covariance matrices 𝑄_𝑤, 𝑆_𝑖, and 𝑅_𝑖𝑗.

3. Online Estimators of Variances and Cross-Covariances

Lemma 3.1 (matrix inverse Fadeeva formula [13]). The matrix inverse formula is given by 𝐼_𝑛−𝑞⁻¹Φ⁻¹=adj𝐼_𝑛−𝑞⁻¹Φdet𝐼_𝑛−𝑞⁻¹Φ=𝐹𝑞⁻¹𝐴𝑞⁻¹, (3.1) where Φ is an 𝑛×𝑛 matrix, 𝑞⁻¹ is a backward shift operator, 𝐼_𝑛 is an 𝑛×𝑛 unit matrix, and 𝐴𝑞⁻¹=1+𝑎₁𝑞⁻¹+⋯+𝑎_𝑛𝑞^−𝑛,𝐹𝑞⁻¹=𝐼_𝑛+𝐹₁𝑞⁻¹+⋯+𝐹_𝑛−1𝑞^{−(𝑛−1)} (3.2) then the coefficients 𝑎_𝑖 and 𝐹_𝑖 can be computed recursively as 𝑎_𝑖1=−𝑖traceΦ𝐹_𝑖−1,𝑖=1,…,𝑛,𝐹_𝑖=Φ𝐹_𝑖−1+𝑎_𝑖𝐼_𝑛,𝑖=1,…,𝑛−1,𝐹₀=𝐼_𝑛,𝑎₀=1. (3.3) Suppose the greatest common factor of 𝐴(𝑞⁻¹) and 𝐹(𝑞⁻¹) as scalar polynomial of 𝜇(𝑞⁻¹), that is, 𝐹𝑞⁻¹=𝜇𝑞⁻¹𝐹𝑞⁻¹,𝐴𝑞⁻¹=𝜇𝑞⁻¹𝐴𝑞⁻¹. (3.4) Eliminate the greatest common factor 𝜇(𝑞⁻¹) of numerator and denominator in (3.1), we have the irreducible Fadeeva formula: 𝐼_𝑛−𝑞⁻¹Φ⁻¹=𝐹𝑞⁻¹𝐴𝑞⁻¹,𝐹𝑞⁻¹=𝐼_𝑛+𝐹₁𝑞⁻¹+⋯+𝐹_𝑛𝑓𝑞^−𝑛𝑓,𝐴𝑞⁻¹=1+𝑎₁𝑞⁻¹+⋯+𝑎_𝑛𝑎𝑞^−𝑛𝑎,𝑎_𝑛𝑎≠0,𝑛_𝑎≤𝑛−𝑛_𝜇,𝐹_𝑖=Φ𝐹_𝑖−1+𝑎_𝑖𝐼_𝑛,𝑖=1,…,𝑛_𝑓,𝐹₀=𝐼_𝑛,𝑛_𝑓=𝑛−𝑛_𝜇−1. (3.5)

Theorem 3.2. For the 𝑖th subsystem of systems (2.1) and (2.2) under the Assumptions of 2.1, 2.2, and 2.3, the innovation model of CARMA 𝐴_𝑖𝑞⁻¹𝑦_𝑖(𝑡)=𝐶_𝑖𝑞⁻¹𝑢(𝑡)+𝐷_𝑖𝑞⁻¹𝜀_𝑖(𝑡) (3.6) is stable. The innovation 𝜀_𝑖(𝑡)∈𝑅^𝑚𝑖 is a white noise with zero mean and variance matrix 𝑄_𝜀𝑖, 𝐶_𝑖𝑞⁻¹=𝐻_𝑖𝐹𝑞⁻¹𝐵𝑞⁻¹,𝐵_𝑖𝑞⁻¹=𝐻_𝑖𝐹𝑞⁻¹Γ𝑞⁻¹, (3.7) where the polynomial matrices of 𝐷_𝑖(𝑞⁻¹), 𝐴_𝑖(𝑞⁻¹), 𝐶_𝑖(𝑞⁻¹), and 𝐵_𝑖(𝑞⁻¹) have the form as 𝑋_𝑖𝑞⁻¹=𝑋_𝑖0+𝑋_𝑖1𝑞⁻¹+⋯+𝑋_{𝑖𝑛𝑥𝑖}𝑞^{−𝑛𝑥𝑖},𝑖=1,2,…,𝐿; (3.8) with 𝐴_𝑖0=𝐼_𝑚𝑖,𝐷_𝑖0=𝐼_𝑚𝑖,𝐵_𝑖0=0, (3.9) we have 𝐷_𝑖𝑞⁻¹𝜀_𝑖(𝑡)=𝐵_𝑖𝑞⁻¹𝑤(𝑡)+𝐴_𝑖𝑞⁻¹𝑣_𝑖(𝑡). (3.10)

Proof. From (2.1) and (2.2), we have 𝑦_𝑖(𝑡)=𝐻_𝑖𝐼_𝑛−𝑞⁻¹Φ⁻¹𝑞⁻¹[𝐵𝑢(𝑡)+Γ𝑤(𝑡)]+𝑣_𝑖(𝑡). (3.11) Applying the extended Fadeeva formula of (3.5), we have 𝐴_𝑖𝑞⁻¹𝑦_𝑖(𝑡)=𝐶_𝑖𝑞⁻¹𝑢(𝑡)+𝐵_𝑖𝑞⁻¹𝑤(𝑡)+𝐴_𝑖𝑞⁻¹𝑣_𝑖(𝑡). (3.12) Suppose (𝐴_𝑖(𝑞⁻¹)𝐼_𝑚𝑖, 𝐶_𝑖(𝑞⁻¹), 𝐵_𝑖(𝑞⁻¹)) left-coprime, and (𝐵_𝑖(𝑞⁻¹), 𝐴_𝑖(𝑞⁻¹)𝐼_𝑚𝑖) or their greatest left factor’s determinant has no zero point on the unit circles. Note that it needs to be left-coprime factorization if it is not left-coprime. Then, there is an MA process of 𝐷_𝑖(𝑞⁻¹)𝜀_𝑖(𝑡), which makes (3.10) hold and guarantees 𝐷_𝑖(𝑞⁻¹) stable. We have (3.6) from (3.10) and (3.12). The proof is completed.

Define a new measurement process:𝑧_𝑖(𝑡)=𝐴_𝑖𝑞⁻¹𝑦_𝑖(𝑡)−𝐶_𝑖𝑞⁻¹𝑢(𝑡). (3.13)

From (3.6), we have𝑧_𝑖(𝑡)=𝐷_𝑖𝑞⁻¹𝜀_𝑖(𝑡). (3.14)

From (3.10), we have𝑧_𝑖(𝑡)=𝐵_𝑖𝑞⁻¹𝑤(𝑡)+𝐴_𝑖𝑞⁻¹𝑣_𝑖(𝑡). (3.15)

Remark 3.3. When the noise variances 𝑄_𝑤 and 𝑅_𝑖 are known, the Gevers-Wouters [20] algorithm can be used to construct ARMA innovation model and obtain 𝐷_𝑖(𝑞⁻¹) and 𝑄_𝜀𝑖.
In (3.14), 𝑧_𝑖(𝑡) is a stationary stochastic process, whose correlated function 𝑅_𝑧𝑖𝑗(𝑘) has cut-off property, and suppose it be cut-off at 𝑛_𝑧𝑖𝑗, that is, 𝑅_𝑧𝑖𝑗(𝑘)=E𝑧_𝑖(𝑡)𝑧^T_𝑗(𝑡−𝑘),𝑘=0,1,…,𝑛_𝑧𝑖𝑗,𝑅_𝑧𝑖𝑗(𝑘)=0,𝑘>𝑛_𝑧𝑖𝑗,𝑖,𝑗=1,2,…,𝐿. (3.16)
At the end of time 𝑡, the sampling estimation of the correlated function 𝑅_𝑧𝑖𝑗(𝑘)(𝑘=0,1,…,𝑛_𝑧𝑖𝑗) based on measurements (𝑧_𝑖(𝑡),𝑧_𝑖(𝑡−1),𝑧_𝑖(𝑡−2),…) can be defined as 𝑅^𝑡_𝑧𝑖𝑗(𝑘)=1𝑡^𝑡_𝛼=1𝑧_𝑖(𝛼)𝑧^T_𝑗(𝛼−𝑘), (3.17) then we have its recursive form: 𝑅^𝑡_𝑧𝑖𝑗(𝑘)=𝑅_𝑧^𝑡−1_𝑖𝑗(𝑘)+1𝑡𝑧_𝑖(𝑡)𝑧^T_𝑗(𝑡−𝑘)−𝑅_𝑧^𝑡−1_𝑖𝑗(𝑘),𝑡=2,3,…, (3.18) with the initial value 𝑅¹_𝑧𝑖𝑗(𝑘)=𝑧_𝑖(1)𝑧^T_𝑗(1−𝑘).
Computing the correlated function on both sides of (3.14), we have 𝑅_𝑧𝑖𝑗(𝑘)=^𝑛0_𝑢=𝑘𝐵_𝑖𝑢𝑄_𝑤𝐵^T_{𝑗,𝑢−𝑘}+^𝑛0_𝑢=𝑘𝐴_𝑖𝑢𝑆^T_𝑖𝐵^T_{𝑗,𝑢−𝑘}+^𝑛0_𝑢=𝑘𝐵_𝑖𝑢𝑆_𝑗𝐴^T_{𝑗,𝑢−𝑘}+^𝑛0_𝑢=𝑘𝐴_𝑖𝑢𝑅_𝑖𝑗𝐴^T_{𝑗,𝑢−𝑘},𝑘=0,1,…,𝑛_𝑧𝑖𝑗,𝑖,𝑗=1,…,𝐿, (3.19) where 𝑛₀=max(𝑛_𝑎𝑖,𝑛_𝑏𝑖,𝑛_𝑎𝑗,𝑛_𝑏𝑗). 𝐴_𝑖𝑗 and 𝐵_𝑖𝑗 are known, 𝐵_𝑖𝑗=0(𝑗>𝑛_𝑏𝑖), 𝐴_𝑖𝑗=0(𝑗>𝑛_𝑎𝑖).
However (3.19) is a matrix equations set. Substituting the sampling estimates at the time 𝑡𝑅^𝑡_𝑧𝑖𝑗(𝑘)(𝑘=0,…,𝑛_𝑧𝑖𝑗) into (3.19), and solving the matrix equations set, then we have the estimates 𝑆_𝑖(𝑡), 𝑄_𝑤(𝑡), and 𝑅_𝑖𝑗(𝑡) at the time 𝑡.
From the ergodicity of the stationary stochastic process (3.14) and the Assumption 2.3, when 𝑡→∞, we have 𝑆_𝑖(𝑡)→𝑆_𝑖, 𝑄_𝑤(𝑡)→𝑄_𝑤 and 𝑅_𝑖𝑗(𝑡)→𝑅_𝑖𝑗.

4. Self-Tuning WMF Kalman Filter

When the statistical features of the noise are unknown, the self-tuning weighted measurement fusion Kalman estimator can be obtained through the following three steps.

Step 1. For different sensor systems, (3.16)– (3.19) are used to identify online the estimates 𝑄_𝑤(𝑡), 𝑅_𝑖𝑗(𝑡), and 𝑆_𝑖(𝑡),𝑖,𝑗=1,…,𝐿, of noise variances of 𝑄_𝑤, 𝑅_𝑖𝑗, and 𝑆_𝑖 at the time 𝑡, which will yield the following available estimates at the time 𝑡: 𝑆=𝑆₁,…,𝑆_𝐿,𝑅^(I)=𝑅₁₁⋯𝑅_1𝐿𝑅⋮⋮_𝐿1⋯𝑅_𝐿𝐿,𝐽̂=Γ𝑆𝑅^(I)−1,𝑢(𝑡)=𝐵𝑢(𝑡)+𝐽𝑦̂^(I)(𝑡),𝑄=𝑄_𝑤−𝑆𝑅^(I)−1𝑆^T,𝑄_𝑤=Γ𝑄Γ^T,𝑦^(II)(𝑡)=Ω𝑦^(I)(𝑡),Ω=𝐹^T𝑅^(I)−1𝐹⁻¹𝐹^T𝑅^(I)−1,𝑅^(II)=(𝐹^T𝑅^(I)−1𝐹)⁻¹,Φ=Φ−𝐽𝐻̂^(I). (4.1)

Step 2. From (2.15), solving the following Riccati equation, we get the estimation value Σ^(II) of Σ^(II) at the time 𝑡: Σ^(II)=ΦΣ^(II)−Σ^(II)𝐻^(II)T𝐻^(II)Σ^(II)𝐻^(II)T+𝑅^(II)−1𝐻^(II)Σ^(II)Φ^T+𝑄_𝑤. (4.2)

Step 3. Equations (2.14) are applied, and the self-tuning weighted measurement fusion state estimator is given by 𝑥^(II)𝑠(𝑡+1∣𝑡+1)=𝑥^(II)𝑠(𝑡+1∣𝑡)+𝐾^(II)_𝑓𝜀̂^(II)(𝑡+1),𝑥^(II)𝑠(𝑡+1∣𝑡)=Ψ^(II)_𝑝𝑥^(II)𝑠(𝑡∣𝑡−1)+𝐾^(II)_𝑝𝑦^(II)(𝑡)+𝑢(𝑡),𝑥^(II)𝑠(𝑡∣𝑡+𝑁)=𝑥^(II)𝑠(𝑡∣𝑡)+^𝑁_𝑗=1𝐾^(II)𝑗𝜀̂^(II)𝑡+𝑗,𝑁>0,𝜀̂^(II)(𝑡)=𝑦^(II)(𝑡)−𝐻^(II)𝑥^(II)𝑠(),𝐾𝑡∣𝑡−1_(II)𝑓=Σ^(II)𝐻^(II)T𝐻^(II)Σ^(II)𝐻^(II)T+𝑅^(II)−1,𝐾^(II)_𝑝=ΦΣ^(II)𝐻^(II)T𝐻^(II)Σ^(II)𝐻^(II)T+𝑅^(II)−1,Ψ^(II)_𝑝=Φ−𝐾^(II)_𝑝𝐻^(II),𝐾^(II)𝑗=Σ^II𝐼_𝑛−𝐾^(II)_𝑓𝐻^(II)TΦ^T𝑗𝐻^(II)T𝐻^(II)Σ^(II)𝐻^(II)T+𝑅^(II)−1,𝑃^(II)=𝐼_𝑛−𝐾^(II)_𝑓𝐻^(II)Σ^(II). (4.3)

Remark 4.1. At every moment, the iteration method to solve Riccati equation (4.2) causes comparatively large computation burden, which is not convenient for the real applications. To reduce the computational burden, a computing period (dead band) 𝑇_𝑑 of (4.2) is set and the estimation value Σ^(II) keeps invariant in the period 𝑇_𝑑. So, Σ^(II) is only computed at the moments of 𝑡=𝑇_𝑑,2𝑇_𝑑,3𝑇_𝑑,…, which can reduce computation burden, and can be called Riccati equation with a dead band [9, 17].

5. Simulation Example

The self-tuning 𝛼−𝛽 radar track system with colored measurement noise: 𝑥(𝑡+1)=1𝑇₀𝑥(𝑡)+010.5𝑇²₀𝑇₀𝑤(𝑡),𝑦_𝑖(𝑡)=𝐻_0𝑖𝑥(𝑡)+𝑣_𝑖(𝑡)𝑣,𝑖=1,2,_𝑖(𝑡+1)=𝜌_𝑖𝑣(𝑡)+𝜉_𝑖(𝑡), (5.1) where 𝑤(𝑡) and 𝜉_𝑖(𝑡), 𝑖=1,2, are independent Gaussian white noises with zero means and variances of 𝑄_𝑤=1, 𝑄_𝜉1=0.04, and 𝑄_𝜉2=0.7, 𝑣_𝑖(𝑡) is colored measurement noise, 𝐻₀₁=[10] and 𝐻₀₂=[01]. 𝑇₀=1 is the sampling period, 𝑥(𝑡)=[𝑥₁(𝑡)𝑥₂(𝑡)]^T, 𝑥₁(𝑡) and 𝑥₂(𝑡), are, respectively, the position and speed of the moving object at 𝑡𝑇₀, 𝜌₁=−0.4, 𝜌₂=−0.5.

When 𝜌_𝑖, 𝑄_𝑤 and 𝑄_𝜉𝑖(𝑖=1,2) are unknown, the problem is to find the self-tuning WMF 𝛼−𝛽 tracking filter 𝑥^(II)𝑠(𝑡∣𝑡).

The parameter convergence results of subsystem 1 are shown in Figures 1, 2, 3, 4, and 5, and the parameter convergence results of subsystem 2 are shown in Figures 6, 7, 8, 9, and 10, where the curved lines denote the estimates and the straight lines denote true values. The self-tuning WMF Kalman filter 𝑥^(II)𝑠(𝑡∣𝑡) is shown in Figures 11, 12, 13, 14, and 15, where Figure 11 is the convergence of Σ^(II). Using the iteration algorithm with dead band 𝑇_𝑑=200, we can see that the parameter estimates in Σ^(II) shown in ladder-shape curves converge to the corresponding real values.

Figure 1 

𝜌1 and the convergence of estimation value 𝜌1(𝑡).

Figure 2 

𝑑11 and the convergence of estimation value 𝑑11(𝑡).

Figure 3 

𝑑12 and the convergence of estimation value 𝑑12(𝑡).

Figure 4 

𝑄𝜀1and the convergence of estimation value 𝑄𝜀1(𝑡).

Figure 5 

𝑄𝜉1 and the convergence of estimation value 𝑄𝜉1(𝑡).

Figure 6 

𝜌2 and the convergence of estimation value 𝜌2(𝑡).

Figure 7 

𝑑21 and the convergence of estimation value 𝑑21(𝑡).

Figure 8 

𝑄𝜀2 and the convergence of estimation value 𝑄𝜀2(𝑡).

Figure 9 

𝑄𝑤 and the convergence of estimation value 𝑄𝑤(𝑡).

Figure 10 

𝑄𝜉2 and the convergence of estimation value 𝑄𝜉2(𝑡).

Figure 11 

∑(II) and the convergence of estimation value ∑(II)(𝑡).

Figure 12 

Positon 𝑥1 and self-tuning WMF Kalman filter 𝑥(II1)𝑠(𝑡∣𝑡).

Figure 13 

Speed 𝑥2 and self-tuning WMF Kalman filter 𝑥(II2)𝑠(𝑡∣𝑡).

Figure 14 

The curve of position errors 𝑒1(𝑡)=𝑥(II)1(𝑡∣𝑡)−𝑥(II1)𝑠(𝑡∣𝑡) for optimal and self-tuning WMF Kalman filter.

Figure 15 

The curve of velocity errors 𝑒2(𝑡)=𝑥(II)2(𝑡∣𝑡)−𝑥(II2)𝑠(𝑡∣𝑡) for optimal and self-tuning WMF Kalman filter.

The object-tracking curves of self-tuning WMF Kalman filter is shown in Figures 12, 13, 14, and 15. We can see that the system can gradually track the position and speed of the objects as the time increases, where the real lines are real values and the dashed lines are estimates. The error curves between the optimal and self-tuning WMF Kalman filter is shown in Figures 14 and 15. We see that the error curves take on the funnel shape, which demonstrates that the self-tuning filter has better convergence.

6. Conclusion

For multisensor linear discrete time-invariant stochastic control system with different measurement matrices and correlated noise, an online identification method is designed when the input noise and measurement noise variance are unknown. It firstly uses Fadeeva formula to construct ARMA innovation model with unknown noise covariance matrices, and then uses the ergodicity of sampling-correlated function in the stationary and inverse ARMA innovation model to identify the noise covariance matrices. Further, a self-tuning WMF Kalman filter has been presented from a steady-state global optimal measurement fusion Kalman filter by matrix full-rank decompostion, weighted least squares method, and measurement feedback. It has asymptotic global optimality. Compared to the centralized fusion algorithm, it can reduce the computational burden.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant NSFC 61174139, Key Project of Chinese Ministry of Education under Grant 209038, Key Laboratory of Electronics Engineering, College of Heilongjiang Province under Grant DZZD 20100038, Science and Technology Research Foundation of Heilongjiang Province Education Department under Grant 12521420, Open Laboratory Project of Heilongjiang University in China 2012, and High Level Talents Support Project of Heilongjiang University in China under Grant Hdtd2010-03.