Fault detection filtering for MNNs with dynamic quantization and improved protocol

Highlights

• An improved dynamic event-triggering protocol, whose multiple threshold functions are dynamically adjustable, is presented.

• Two mutually independent Bernoulli variables are given to depicting the randomly occurring cyber-attacks.

• A dynamic quantizer is established to account for restricted bandwidth efficiently.

Abstract

This paper concerns the fault detection filtering problem for discrete-time memristive neural networks with mixed time delays. An improved dynamic event-triggering protocol, whose multiple threshold functions are dynamically adjustable, is presented to decrease the utilization of limited resources and achieve desired performance. Two mutually independent Bernoulli variables are given to depicting the randomly occurring cyber-attacks. Meanwhile, a dynamic quantizer is established to account for restricted bandwidth efficiently. Based on the Lyapunov theory, sufficient conditions are derived to ensure the filtering error system is exponential mean square stable and desired performance. In the end, a numerical example is provided to verify the effectiveness of the proposed methodology.

Introduction

Memristive neural networks (MNNs) have attracted extensive attention owing to the memory resistor and other merits (i.e. lower energy consumption and larger storage capacity), which can be seen as remarkable superiorities to the traditional neural networks. In recent years, reiterative research interest has been devoted to the dynamical analysis of MNNs, such as stability, state estimation, filtering, etc [1], [2], [3]. To date, MNNs s have been found to be a successful application in various areas, such as pattern recognition, disease diagnosis, and power system. In view of their huge potential application, MNNs have attracted ever-increasing attention and many results have been reported [4]. On the other hand, the time delay can be seen as one source of the oscillation, divergence, or instability [5]. Therefore, it is of great significance to exploit the dynamic behavior of discrete-time MNNs with time-varying delays and probabilistic distributed delays. To date, quite a little effort has been devoted to the FD filtering of discrete-time MNNs, which partly motivates the present study.

The filtering problem has been observed as one of the fundamental research topics in the control community over the past decades, which is an important measure to suppress and prevent interference [6]. The purpose of filtering is to reconstruct the state of the system through a designed filter. Meanwhile, with respect to the security and reliability of control systems, fault detection (FD)-based on the filter has been widely concerned in the control community [7], [8]. The main principle of the FD filter is to estimate the dynamic output based on the analytic formula of the system and obtain a residual signal by comparing it with the output value of the actual measurement.

In some networked systems with limited communication bandwidth, compared with time-triggering protocol [9], [10], [11], [12], [13], event-triggering protocol (ETP) [14], [15] can transmit signals only when the event triggering condition is met, which effectively degrades the system resource consumption while ensuring eligible system performance. Different from static ETP (SETP) [16], dynamic ETP (DETP) [17] achieves more flexibility and better capacity in saving network resources. In this regard, DETP has attracted increasing attention. As such, many valuable ETPs have been forwarded, such as switching event-triggering case [18], [19], adaptive event-triggering case [20], [21], memory-based time-triggering case [15], [22], and improved event triggering case [23]. On the other hand, quantification schemes are also seen to be an effective way of saving information communication burden [24]. Therefore, past decades have observed the achievement of quantitative control systems. In [25], the quantized measurement of interconnected systems has been studied via a logic quantizer. In [26], the variable quantization density of the logarithm quantizer has been proposed from a practical viewpoint, which is more efficient than the conventional constant case. In [27], the dynamic quantizer that consists of a static quantizer and dynamic parameters has been introduced. To our knowledge, how to make a trade-off between the desired performance and rational resource utilization needs further investigation.

Meanwhile, the network control systems are vulnerable to malicious cyber-attacks, which result in performance degradation or instability. In consideration of the network security, denial-of-service (DoS) attack [28], [29], deception attacks [30], [31], replay attack [32] and false data injection (FDI) attack [33] are well-known four typical cyber-attacks. Among them, the DoS attacks are regarded as one of the most harmful cyber-attacks, which mainly destroy data availability by preventing information exchange. FDI attacks are capable of misleading the system into an insecure situation via injecting false information.

Inspired by the previous observation, we devote ourselves to exploiting the FD problem for MNNs with dynamic quantization and IDETM. Compared with existing studies, the main contributions are concluded as following aspects: 1) a new framework of time-delay-dependent discrete-time MNNs is formulated, and the FD filter problem is quite comprehensive that involves mixed delays, quantization effects, communication protocol, and cyber-attacks; 2) Apart from the conventional ETPs, a generalized IDETP with variable thresholds is developed to seek a rational trade-off mechanism between reducing network resources and desired performance; 3) Different from the static quantization strategy, the dynamic quantizer with a feasible adjustable parameter is introduced on the basis of randomly occurring cyber-attacks.

Notations: ${\mathbb{R}}^n$ designates the $n$-dimensional Euclidean space. $\parallel ·\parallel$ means the Euclidean norm of vectors/matrices. $\mathcal{E}\{·\}$ represents the mathematical expectation. $\text{diag}\{·\}$ means diagonal matrix and the symmetric term in a symmetric matrix is denoted by $*$. $\mathbf{He}\left\{\mathcal{L}\right\}$ refers to $\mathcal{L}+{\mathcal{L}}^{\top }$. $I_{n^2}$ signifies an $n^2$-dimensional identity matrix.