Robust centralized and weighted measurement fusion Kalman estimators for multisensor systems with multiplicative and uncertain-covariance linearly correlated white noises

Highlights

• Multisensor systems with mixed uncertainties and linearly correlated noises.

• A fictitious noise technique to compensate multiplicative noises was presented.

• A unified approach to design the robust Kalman estimators was presented.

• A minimax robust fusion Kalman filtering theory was presented.

• Robust centralized and weighted measurement fusion Kalman estimators.

Abstract

This paper addresses the design of robust centralized and weighted measurement fusion (WMF) Kalman estimators for a class of uncertain multisensor systems with linearly correlated measurement and process white noises. The uncertainties of the systems include the same multiplicative noises in state and measurement matrices, and the uncertain noise variances. By introducing the fictitious noises to compensate the multiplicative noises, the system under consideration is converted into one with only uncertain noise variances. According to the minimax robust estimation principle, based on the worst-case system with the conservative upper bounds of the noise variances, the robust centralized and two WMF time-varying Kalman estimators (predictor, filter, and smoother) are presented in a unified framework. Their robustness is proved by using Lyapunov equation approach, such that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. Using the information filter, their equivalence is proved. Their accuracy relations are proved. The computational complexities are analyzed and compared. Compared with the centralized fusion algorithm, the two WMF algorithms can significantly reduce the computational burden when the number of sensors is larger. The corresponding robust local and fused steady-state Kalman estimators are also presented, and the convergence in a realization between the time-varying and steady-state robust fused Kalman estimators is proved by the dynamic error system analysis (DESA) method. A simulation example with application to signal processing to show the effectiveness and correctness of the proposed results.

Introduction

Multisensor information fusion has received great attention and has been applied to many fields including guidance, target tracking, signal processing and so on [1]. The Kalman filtering [2], [3] is a basic tool of multisensor information fusion. For Kalman filtering-based data fusion, two basic fusion approaches are centralized and distributed fusion approaches, depending on whether raw data are used directly for fusion or not. The distributed fusion approaches can be classified as the measurement fusion and state fusion approaches. Based on the weighted least squares (WLS) method, two weighted measurement fusion algorithms were presented in [4], [5]. Based on the unbiased linear minimum variance(ULMV) weighting fusion rule, three optimal weighted state fusion algorithms weighted by matrices, diagonal matrices, and scalars were presented in [6], [7].

It is worth mentioning that, the traditional Kalman filtering method requires the assumption that the process and measurement noises are white noises, and are uncorrelated with each other. However, in the engineering applications, the correlations often arise. For example, for the target tracking problem, the correlation between the process and measurement noises may arise if both of them are dependent on the system state. Also, the process noise or measurement noise may be autocorrelated across time if it is modeled as the autoregressive moving average (ARMA) model, and there exists cross-correlation among different sensor measurement noises if the various sensors work in a common noisy environment. Recently, the state estimation problems have been widely studied for systems with auto-correlated and/or cross-correlated noises, in [8], [9], [10] for the single sensor systems, and in [7], [11], [12], [13] for the multisensor systems. On the other hand, as is well known, the Kalman filtering method also requires another standard assumption that the model parameters and noise variances of the considered system are precisely known [2]. Such an assumption does not always hold in practical applications due to a number of reasons such as model reduction, stochastic parameters, uncertain perturbations and unmodeled dynamics [14]. When there exist uncertainties in the system model, including the stochastic parameter and/or noise variance uncertainties, the performance of traditional Kalman filters could be largely deteriorated. This has motivated many studies on robust Kalman filter design. The so-called robust Kalman filter is concerned with the design of a fixed filter for a family of system models formed by uncertainties, such that its actual filtering error variances yielded by all admissible uncertainties are guaranteed to have a minimal upper bound [15], and such property is called the robustness.

Multiplicative noises (also called state-dependent noises) constitute an important class of stochastic parameter uncertainties. The treatments of the multiplicative noises play an important role in several engineering applications. For example, in airborne synthetic aperture radar (SAR) imaging systems, the speckle noise is modeled as multiplicative noise [16]. Another practical example with respect to robust equalizer on communication channel with multiplicative noise is given in [17]. For systems with multiplicative noises and correlated noises, some results regarding optimal or robust state estimation have been reported in the past few decades [13]. For linear time-varying systems with the same multiplicative noises in the state and measurement matrices, by using the linear matrix inequality (LMI) method, a robust recursive Kalman filtering algorithm was presented in [13], [17], [18], [19], [20], [21], [22], [23]. An optimal recursive state estimator for discrete-time linear systems with multiplicative measurement noises and time-correlated additive measurement noise has been proposed in [18], based on linear minimum mean square error rule, however, the multiplicative noises in the system state matrix are not considered. For a class of uncertain systems with deterministic norm-bounded uncertainties in the measurement matrix and multiplicative noises in the state matrix, and with finite-step correlated process noise and missing measurements, based on the min-max game theory, a recursive robust filter is designed in [19]. However, in [19] the multiplicative noises in the measurement matrix are not considered, and the process noise is assumed to be uncorrelated with measurement noise. For the linear discrete-time systems with stochastic parameters, an optimal linear estimator was proposed by transforming the original system to one with deterministic parameters and multiplicative noises, and by introducing the fictitious noises to compensate multiplicative noises [20]. However, in [20] the considered multiplicative noises in the state and measurement matrices are different, which has little complexity. For a class of uncertain systems with the same multiplicative noises in the state and measurement matrices, and with stochastic nonlinearities, autocorrelated missing measurements, and correlated noises, a robust recursive filtering algorithm is presented in [21], where the process noise and measurement noise are arbitrary finite-step autocorrelated and cross-correlated. However, [17], [18], [19], [20], [21] are all limited to the single-sensor systems. For multisensor systems with multiplicative noises, randomly delayed measurements and sensor failures, based on the optimal fusion algorithm weighted by matrices, a robust distributed state fusion Kalman filter is derived in [22], where the measurement noises are assumed to be cross-correlated, but the process noises and measurement noises are uncorrelated. For a class of uncertain systems with multiplicative noises and with autocorrelated and cross-correlated noises, the suboptimal Kalman-type local filter and distributed fusion filter are designed [13] by using the matrix-weighted fusion estimation algorithm. However, [13], [22] only considered the multiplicative noises in the state matrix, while the multiplicative noises in the measurement matrix are not taken into account. For the multisensor systems with multiplicative noises in the state and measurement matrices, and with missing measurements and unknown measurement disturbances, based on the ULMV rule, the Kalman-like centralized and distributed fusion state one-step predictors are presented in [23]. However, the common limitation of references [13], [17], [18], [19], [20], [21], [22], [23] is that only the stochastic parameter uncertainties are considered, but the noise variance uncertainties are not taken into account.

Recently, for the linear discrete time-varying multisensor stochastic systems with uncertain noise variances, according to the minimax robust estimation principle [24], based on the worst-case conservative systems with the conservative upper bounds of noise variances, the robust local and fused Kalman filters, predictors, smoothers, and white noise deconvolution smoothers have been presented in [25], [26], [27], [28], respectively. The actual estimation error variances or their traces of each estimator are guaranteed to have a corresponding minimal upper bound for all the admissible uncertainties of noise variances. However, the common limitation of references [25], [26], [27], [28] is that only the noise variance uncertainties are considered, while the model parameters of the considered systems are all assumed to be known exactly, and in [25], [26], [27], [28] the multiplicative noises and correlated noises were not considered.

Up to now, to the best of the authors knowledge, the robust information fusion Kalman filtering problems for multisensor systems with multiplicative noises, uncertain noise variances and linearly correlated white noises are not reported. This situation motivates our current investigation. In the present paper, the problem of design robust centralized and weighted measurement fusion Kalman estimators is considered for multisensor systems with the same multiplicative noises in the state and measurement matrices, and with uncertain noise variances and linearly correlated measurement and process white noises.

The novelty of the topic for this paper can be highlighted as follows:

• The system model considered is comprehensive, mixed, or hybrid, where the same multiplicative noises in the state and measurement matrices, the uncertain noise variances and linearly correlated noises are considered simultaneously for the first time, which better reflect the reality. Specially, the same multiplicative noises in the state and measurement matrices have important application backgrounds. For example, for the network system with multiple packet dropouts [29] or the network system with random sensor delays, multiple packet dropouts and uncertain observations [30], by the augmented state approach, the considered system can be transformed to one with stochastic parameter matrices, further, replacing the stochastic parameter matrices with their mean matrices, and introducing the fictitious noises to compensate the deviations of stochastic parameters with respect to their means, it can be converted into one with constant parameter matrices and same multiplicative noises in the state and measurement matrices.

• The considered linearly correlated noises in this paper are different from the correlated noises in [7-13,17-23]. The so-called linearly correlated noises mean that the measurement noise is a linear function of the process noise, which are encountered frequently in practical applications. For example, using the singular value decomposition, the descriptor system can be transformed into two reduced-order non-singular coupled subsystems, where in the first subsystem, the measurement noise is linearly correlated with the process noise [31], [32]. For the system with colored measurement noise, using the measurement differencing transformation, the system can be converted into an equivalent system with linearly correlated noises [33], [34]. By the state-space approach, the ARMA signal filtering problem can be converted to one of system with linearly correlated noises [35].

The main contributions of this paper can be highlighted as follows:

• A fictitious noise technique to compensate multiplicative noises is presented, by which the system with multiplicative noises and uncertain noise variances can be converted into one only with uncertain noise variances, whose robust fusion Kalman estimators can be designed by the Lyapunov equation approach and the robust fusion Kalman filtering theory as shown in [25], [26], [27], [28]. This fictitious noise technique constitutes an important methodology to solve the problems of designing the robust fusion Kalman estimators for multisensor systems with mixed uncertainties including multiplicative noises, missing measurements, random delays, packet dropouts, and uncertain noise variances, and so on. It is called the extended fictitious noise technique. The existing fictitious noise technique [17], [18], [20] for solving the optimal filtering problems for systems with multiplicative noises but known noise variances has been extended and developed.

• A unified approach to design the robust fusion Kalman estimators (predictor, filter, smoother) is presented, whose principle is to design robust Kalman filter and smoother based on the robust Kalman predictor, which overcomes the disadvantage of references [25], [26], [27], where the robust Kalman filter, predictor, and smoother are separately designed based on different Lyapunov equations, and the references [25], [26], [27] do not handle the multiplicative noises and linearly correlated noises, and the robust centralized and WMF Kalman smoothers by the augmented state approach [27] require a larger computational burden.

• A new unified robust fusion time-varying and steady-state Kalman filtering theory based on the minimax robust estimation principle is proposed for multisensor systems with mixed uncertainties, which including the robust centralized and two WMF Kalman estimators, and their robustness, equivalence, accuracy relations, complexity and convergence. Specially, the robustness of the proposed robust fused estimators is proved by the Lyapunov equation approach, which is different from the game theory approach [32], the linear matrix inequalities approach [36] and Riccati equation approach [37].

The remaining part of the paper is organized as follows: The problem formulation is given in Section 2, and preliminary knowledge is given in Section 3. The centralized and weighted fusion measurement equations are given in Section 4. The robust centralized and WMF time-varying Kalman estimators are presented in Section 5, and their equivalence is proved in Section 6. The complexity analysis is presented in Section 7. The convergence analysis is presented in Section 8. An application background to robust fused filtering of AR signal is presented in Section 9, and a simulation example is given in Section 10. The conclusions are presented in Section 11.

Notations: Rⁿ denotes the n-dimensional Euclidean space, $R^{n\times n}$ is the set of $n\times n$ real matrices, $\mathrm{tr}(·)$ denotes the trace of a matrix, the superscript T denotes the transpose, $\mathrm{E}\left[·\right]$ denotes the expectation of a random variable, $\mathit{diag}(·)$ denotes the block-diagonal matrix, and the matrix inequality $A\le B$ means that $B-A\ge 0$ is positive semi-definite.