Distributed non-fragile $l_2-l_{\infty }$ filtering over sensor networks with random gain variations and fading measurements

Abstract

This paper investigates the distributed non-fragile $l_2-l_{\infty }$ filter design for a class of discrete-time nonlinear systems with random gain variations and fading measurements. Two mutually independent random sequences with known distributions are utilized to describe the probabilistic properties of the random gain variations phenomenon and fading measurements, respectively. Based on stochastic analysis and Lyapunov function approach, the sufficient condition is presented to guarantee the mean-square exponential stability and $l_2-l_{\infty }$ disturbance attenuation performance of the augmented filtering error system. The solutions of the desired distributed non-fragile filter gains are characterized by solving linear matrices inequalities to keep the sparsity of the weighted adjacency matrix of sensor networks. Finally, an illustrative example is provided to show the effectiveness of the proposed design approach.

Introduction

Sensor networks are generally composed of a large number of sensors that are capable of sensing, computation, and signal transmission. The sensor nodes are spatially distributed to construct a network with certain topology structure, such that they work in coordination way. Owning to the low cost and power consumption, and customizable and distributed structure, sensor networks have been successfully applied during the past decades in lots of areas, including military, industrial, agricultural and environmental fields [1], [2], [3], [4]. Among the wide-scope domains of applications, distributed estimation and filtering have gain considerable research interest in the past years. Distributed Kalman filtering theory and algorithm have been proposed for target plants with apriori information of exogenous Gaussian noises [5], [6], [7]. On the other hand, for many practical systems whose noises statistical values are unavailable, it is usually assumed that the exterior noises are square-integrable or square-additive, respectively, for continuous-time [8] or discrete-time systems [9]. Then, by minimizing the L₂-induced norm and energy-to-peak norm, the H_∞ filtering [8], [10], [11] and $L_2-L_{\infty }$ filtering [11], [12], [13], [14] strategies have been presented, respectively. The energy-to-peak norm in discrete-time systems is generally called $l_2-l_{\infty }$ one, and the corresponding discrete-time $l_2-l_{\infty }$ filtering issue has been considered [15]. The concept of H_∞ filtering has been extended to the distributed H_∞ filtering over sensor networks [16], [17], [18].

When considering the ubiquitous unideal measurement phenomenon in networked systems [19], the problem of distributed H_∞ filtering over sensor networks with missing measurements, expressed by 0-1 Bernoulli distribution, has been dealt with in recent years [20], [21]. With regard to distributed filtering over sensor networks with missing measurements, the redundant-channels-based method is useful to improve the performance [22], [23]. In fact, many imperfect measurements cannot be simply described as Bernoulli sequences [24]. By contract, fading measurements/channels models are presented to the stochastic measurement decays phenomena characterized by random sequences with known mathematical expectations and variances [25], [26]. The distributed filtering issues for continuous- and discrete-time systems over sensor networks with measurements fading have been thoroughly investigated in [27], [28] and [29], [30], respectively. However, the problem of distributed $l_2-l_{\infty }$ filtering for nonlinear discrete-time systems over sensor networks with fading measurements still remains open, and it is the first motivation of the current paper.

It deserves to note that, in real implementations of controllers/filters, the gains uncertainties resulting from the numerical roundoff errors, A-D conversion errors, limited word length, and some other reasons, will degrade the systems performance, and even lead to instability. So, the influences of gain variations in controllers/filters should be taken into account, and the designed controllers/filters being able to tolerant theses gain perturbations are said to be resilient or non-fragile [31], [32], [33], [34]. To cope with norm-bounded multiplicative gain uncertainties, a non-fragile state estimator has been constructed for discrete-time Markovian jumping neural networks with mode-dependent time delays [35]. The resulting estimation error system in [35] is mean-square exponentially stable and it is not sensitive to the gain variations. More recently, [36] has studied the non-fragile finite-time state estimation issue for a class of periodic time-delay neural networks with multiplicative gain fluctuations over multiple-packet transmission, in terms of linear matrix inequality (LMI) technique. A non-fragile $l_2-l_{\infty }$ state estimator for discrete-time semi-Markovian neural networks with randomly occurring sensor delay and lossy measurements is constructed in [37] to counteract the effects from additive norm-bounded gain perturbations.

As the randomly occurring missing measurements [20], [21], [22], [23] and sensor delay [37], the filter gain variations may also take place in a probabilistic way subject to random factors in a networked situation. Thus, non-fragile H_∞ filtering with randomly occurring gain variations (ROGVs) characterized by Bernoulli process, has been considered by [38]. Meanwhile, [39] has been concerned with H_∞ and $l_2-l_{\infty }$ filtering in finite horizon case for time-varying stochastic nonlinear systems with quantization effects and randomly occurring gain uncertainties. The finite horizon filters of [39] can be computed online by solving a set of recursive LMIs. Furthermore, for filter gain perturbations described by any a random sequence with known mathematical expectation and variance, but not be restricted to Bernoulli distribution, [40] has developed the non-fragile H_∞ filter design method for a class of nonlinear systems with fading channels. In [41], the non-fragile distributed H_∞ filter has been established for T–S fuzzy systems over sensor networks to resist against additive gain perturbations in the form of norm-bounded type. As is known, the massive number of sensor nodes in sensor networks more easily induce some sudden inner components failures, stochastic environmental changes and network-induced random events. In other words, during the implementation of a distributed filter over sensor networks, it’s more prone to randomly varying gain variations. Thus, from both the theoretical and applied points of view, the issue of distributed non-fragile filtering over sensor networks with random gain variations is meaningful.

In consideration of the above discussions, our attention of this paper is focused on the distributed non-fragile $l_2-l_{\infty }$ filtering problem for discrete-time nonlinear systems with randomly varying filter gain uncertainties and fading measurements over sensor networks, which has not been investigated in the existing literature. By making use of stochastic analysis method and Lyapunov stability theory, the $l_2-l_{\infty }$ performance criterion is established. And the non-fragile distributed $l_2-l_{\infty }$ filter is designed to effectively resist against the randomly varying gain fluctuations and fading measurements. The validity of the design method is verified by a numerical example. The main contributions of this paper can be summarized as follows:

(1) The filter gains uncertainties and fading measurements are considered, and their stochastic properties are described by two mutually independent random sequences.

(2) The analysis result is established to ensure that the augmented filtering error system is mean-square exponentially stable with the prescribed $l_2-l_{\infty }$ performance level.

(3) The designed distributed $l_2-l_{\infty }$ filter is non-fragile to the random filter gain variations and the sparsity of the weighted adjacency matrix is retained.

The rest of this paper is organized as follows. In Section 2, the problem is formulated and preliminaries are outlined briefly. Section 3 presents the main theorems. An illustrative example is given in Section 4 to demonstrate the correctness of the results. Conclusions are drawn in Section 5.

Notations: The notations used in this paper are fairly standard except where otherwise noted. I and 0 represent, respectively, the identity matrix and the zero matrix of appropriate dimensions. A real, symmetric and positive definite matrix X is denoted as X > 0. A ⊗ B is the Kronecker product of matrices A and B. $\text{diag}\{A_1,A_2,\dots ,A_n\}$ indicates the block diagonal matrix with diagonal blocks being the matrices A₁, A₂, $\dots ,$ A_n. λ_min(A) and λ_max(A) represent the minimal and maximal eigenvalues of matrix A, respectively. $\mathbf{E}\left\{\alpha \right\}$ and $\mathbf{E}\left\{\alpha \right|\beta \}$ are, respectively, the mathematical expectation of the stochastic variable α and the expectation of α conditional on β. $l_2[0,+\infty )$ stands for the space of square additive vector function over [0, ∞). ‖ · ‖ represents the Euclidean vector norm and the asterisk “*” in a symmetric matrix is used as a symbol for those terms induced by symmetry. Matrices, if their dimensions are not stated, are assumed to be compatible for algebraic operations.