Distributed fusion for nonlinear uncertain systems with multiplicative parameters and random delay

Highlights

• The nonlinear system with multiplicative parameters and random delay is considered.

• A new measurement model is reconstructed by statistical linear regression.

• The centralized information filter is proposed via Gaussian mixture realization.

• The distributed implementation is designed via average consensus.

• The state smoothing is obtained to improve estimation precision.

Abstract

This paper proposes an information filtering-based distributed fusion for nonlinear uncertain systems with multiplicative random parameters and randomly delayed measurements in sensor networks. This is a typical nonlinear and non-Gaussian stochastic system with multiple uncertainties including nonlinearity, multiplicative random parameters and random delay, along with additive noises. In the criterion of minimizing the mean square error of state estimate, the centralized information-type filter via Gaussian mixture realization is put forward, by applying statistical linear regression to nonlinear measurement models to reconstruct a new and equivalent one with the linear form. Then, the distributed implementation is designed via average consensus with the corresponding weight of each component being updated distributively, in order to ensure that the Gaussian mixture distribution of the posterior probability density in each processing unit asymptotically approximates the corresponding one in the centralized fusion as closely as possible. Meanwhile, the d-step lag state smoothing is obtained to further improve the estimation precision, in terms of the proposed filtering method. An example with multiplicative random parameters and one-step/two-step random delay in the distributed processing network is simulated to validate the proposed method.

Introduction

Much attention has been paid on the filtering problem for discrete-time nonlinear dynamic systems due to its widespread and practical applications in statistical signal processing [1], [2], nonlinear systems control [3], [4], integrated navigation [5], target tracking and fusion [6] and so on. The recursive Bayesian filtering, as the mainstream, is always intractable to obtain an analytical and optimal solution to minimize the trace of estimate error covariance (or mean square error) for nonlinear stochastic systems. Thus, the major study on the Bayesian filtering is focused on the approximation strategies to achieve cost-effective filters [7]. Up to present, there are two main kinds of approximation strategies to design the corresponding executable nonlinear filters.

The one is the function approximation, which aims at using piece-wise time-varying linear functions or other typical functions to approximate nonlinear dynamical and measurement models, such as the extended Kalman filter via Taylor expansion and divided difference filter based on interpolation polynomial [8], [9]. In general, such filters are computation-effective but sensitive to linearized errors or interpolating points. The other is the posterior probability density approximation, and the existing filters can be primarily summarized into the following two subclasses according to the numerical integral calculation. The first subclass is the Monte Carlo-based method, i.e., the particle filter and its derived filters, by utilizing a large number of weighted particles to approximate the posterior probability density [10]. The particle filter can obtain an accurate estimation precision but owns a huge computational load. The other subclass is the Gaussian/Gaussian mixture filters, where the posterior probability density is approximated as the Gaussian/Gaussian mixture distribution [11], [12], [13]. Generally, the Gaussian/Gaussian mixture filter achieves a tradeoff between the estimation accuracy and computational cost, by rationally selecting several fixed sampling points with the corresponding weights to further calculate the first two order moments of the interested vector via numerical Gaussian integrals. In addition, since the Gaussian mixture distribution can approximate any a probability density as closely as possible with appropriate components [14], the Gaussian posterior probability density assumption is further weakened in Gaussian mixture filters. However, all above nonlinear filters have the same assumption that the system is just perturbed by additive noises and the current measurement should be arrived on time.

Actually, in the large-scale networked multi-sensor sensing system, the dynamical and measurement models not only own inherent nonlinearity, but also exist nonlinear unknown inputs deriving from multiple multiplicative random parameters or noises [15], [16], [17]. For example, in the mobile sensors target tracking, the measurement precision of range or signal intensity often depends on the relative distance between the interested target and sensors in practice [18], which leads to the coexistence of additive and multiplicative noises in the measurement model. Meanwhile, due to bandwidth limitation or geographically remote routing, the random delay often occurs during the measurement data transmitting [7], [19], [20], [21], [22]. In other words, the above characteristic of each sensor and the feature of data transmission between sensors cause the actual system to be complex due to the above-mentioned inevitable multiple nonlinearities and randomness. Although the existing filter designs based on multiplicative noises or random delay can be summarized into the following three categories, they have never considered the coexistence of the above model nonlinearity, multiplicative random parameters/noises and random delay.

Firstly, for the filter design with multiplicative parameters, it always needs to consider the coupling effect between multiplicative parameters and the interested state in the second-order statistics estimation (i.e., the recursive calculation of the estimate error covariance or the second-order moment matrix). In [23], the linear minimum variance recursive estimator for discrete-time linear systems with random state transition and measurement matrices was proposed. Based on this method, the random coefficient matrices Kalman filter was further applied to multiple targets and sensors tracking association problem [24], which was suboptimal in the mean square error sense since it violated some independence conditions with respect to the optimality. Moreover, a robust finite horizon Kalman filter was designed for discrete time-varying uncertain systems with both additive and multiplicative noises to guarantee an upper bound on the state estimate error covariance for admissible uncertainties [25].

Secondly, for the filter design with randomly delayed measurements, since the actual measurement is coming from the ideal measurement at the current or previous sampling epoch, it leads to a new form of measurement prediction and innovation calculation, which further affects state update. A stochastic extended Kalman filter was proposed for interconnected power systems with partially or totally dimensional delayed measurements [26]. Then, the extended and unscented Kalman filter were presented for nonlinear stochastic systems with one-step random measurement delay [19], and further a general Gaussian filter was derived for such systems based on Gaussian assumptions approximating the posterior probability densities [27]. Moreover, a novel unscented filter was extended to deal with multi-step random measurement delay through the state and measurement noises augmentation [20]. By considering that the measurement delay may depend on its previous delay, the Gaussian-consensus filter was proposed for nonlinear systems with multi-step random measurement delay obeying the first-order Markov chain in sensor networks [7].

Thirdly, for the filter design with random delay coupling with multiplicative parameters, the existing studies just consider the first two order statistical moments recursion of the interested state for linear systems, which is not suitable and also impossible to extend to nonlinear systems. Meanwhile, they are not implemented in Bayesian filtering framework. In [15], the finite-horizon filter was derived for linear systems with multiplicative noises and random measurement delay in terms of two Riccati difference equations and one Lyapunov difference equation, by applying the reorganized innovation approach. In [28], an optimal estimation for linear systems with multiplicative noises, random delays and multiple packet dropouts was derived by state augmentation.

As discussed above, all these three types of filters are not suitable to the considered system with multiple nonlinearities and randomness, i.e., the nonlinear uncertain system with multiplicative parameters and random delay. Meanwhile, for the multi-sensor fusion estimation, except for the Gaussian-consensus filter [7], all other remained filters are in a centralized fusion structure. Actually, in the large-scale networked sensing systems or sensor networks, the distributed fusion estimation is always necessary, which is also a hot and interesting research topic [7], [16], [22], [29], [30]. In addition, for such a system with multiple nonlinearities and randomness in sensor networks, the conditional posterior probability density of the state is definitely non-Gaussian in the Bayesian filtering perspective. However, the existing studies not only do not refer to the coexistence of these nonlinearities and randomness, but also mainly follow with the Gaussian filtering framework. Since the Gaussian mixture can approximate any a probability density as closely as possible, the distributed filter design based on the Gaussian mixture realization is an important but still open problem.

Motivated by the above facts, a novel information filtering-based distributed fusion estimation via Gaussian mixture realization is proposed in this paper for nonlinear uncertain systems with multiplicative parameters and random delay in sensor networks with the following technical contributions. Firstly, a new measurement model is reconstructed to provide a cornerstone for the subsequent information filter design, through applying the statistical linear regression to the original one. Secondly, a novel information-type filter based on Gaussian mixture approximation is derived to realize the centralized fusion, abbreviated as CGMIF. Thirdly, the distributed implementation of the proposed CGMIF (termed as DGMIF) is put forward via average consensus to pursue an asymptotically consistent estimate, along with the corresponding Gaussian component weight being updated distributively. It is necessary to emphasize that the proposed DGMIF not only obtains an asymptotically consistent estimate, but also guarantees that the posterior probability density Gaussian mixture approximation is asymptotically consistent in each processing unit. Meanwhile, we need to point out that the state smoothing with d-step lag is also developed based on the proposed filtering method to further improve the estimation precision, where the time-varying d is the maximum step of random delay.

The rest of this paper is organized as follows. The problem formulation is presented in Section 2. The information filtering-based centralized fusion via Gaussian mixture realization is derived in Section 3. Then, the corresponding distributed implementation is designed in Section 4. An example in the multi-sensor processing network is simulated to validate the proposed method in Section 5. Finally, the conclusion is supplied in Section 6. All proofs are shown in the Appendix.

Throughout this paper, the symbol “ ≔ ” means definition. ‖ · ‖ represents the 2-norm of a vector or matrix. G_{q, k|l}(ϑ_{n, k}) denotes the qth Gaussian component $\mathcal{N}({\vartheta }_{n,k};{\widehat{\vartheta }}_{q,n,k|l},P_{q,n,k|l}^{\vartheta \vartheta })$ in the nth processing unit. Besides, a variable without subscript n represents the related term in the case of centralized fusion.