Zonotopic fault detection for discrete-time systems with limited communication capability and sensor nonlinearity

Abstract

In this paper, the fault detection filter (FDF) design problem based on a dynamic event-triggered mechanism (DETM) is investigated for discrete-time systems with signal quantization and sensor nonlinearity. In order to conserve the limited network resources, a newly event-triggered mechanism with dynamic threshold is adopted to reduce the number of transmitted data through network more effectively. With the consideration of DETM, signal quantization and sensor nonlinearity, a fault detection filter is constructed to achieve the robustly asymptotic stability of established model with expected fault detection objective. In addition, by influence of DETM, external interference and quantization errors, a zonotopic residual evaluation mechanism is constructed to detect the occurring fault of plant. Finally, a practical example is provided to illustrate the effectiveness of proposed design approach.

Introduction

Over the past few decades, the fault detection problems have been developed into an engaging research subject to network control systems (NCSs) due to the advantage of flexibility, facilitate installation and maintainable in practical systems ranging from chemical processes, industrial production and aerospace, etc. [1], [2], [3], [4]. For instance, the authors designed a local observer to deal with the distributed fault detection problem for homogeneous discrete-time systems [5]. The plug-and-play fault detection and isolation issue was investigated for large-scale nonlinear systems [6]. The author designed a fault detection filter (FDF) to detect the jump fault signals for discrete-time conic-type nonlinear Markov jump systems [7]. It is noteworthy that most above results for fault detection problems were not considered the sensor nonlinearity, which is always existing in practical industrial applications. In [8], the researchers investigated the output feedback control problem for linear systems with sensor nonlinearities. The coordinated motion/force control issue of multi-arm robot with unknown sensor nonlinearity was addressed in Chen et al. [9]. The distributed $H_{\infty }$ filter design problem was considered for switched repeated scalar nonlinear systems with randomly occurred sensor nonlinearities [10]. However, the FDF design problems subject to sensor nonlinearities have not yet been well explored, which is the first principal motivation of this work.

Owing to the limited communication capacity of real NCSs, various network-induced problems tend to emerge inevitably in practical industrial applications, including transmission delays, signal quantization, missing measurements, etc. [11], [12], [13]. These undesirable occurrences may deteriorate the performance of NCSs and even result in the system instability. In order to save the limited network communication resources, many experts designed the event-triggered communication mechanism [14], [15], [16], [17], [18], [19]. The “event” in this communication mechanism is determined by a trigger condition, which depends on the system state and the trigger threshold. The data will be transmitted at the current instants if and only if the event is triggered. It is not difficult to discover that a lower trigger threshold will result in a smaller interval time. Nevertheless, most of related event-triggered works have focused on fixed thresholds, which was generally difficult to reflect the time-varying characteristic of system. Therefore, the adoption of the event-triggered mechanism based on dynamic threshold parameters will be more suitable for dynamic circumstances in physical applications [20]. For instance, the researchers discussed the filter design problems for nonlinear delay systems by designing an adaptive event-triggered strategy mechanism [21]. In [22], the adaptive event-triggered fault detection scheme design was investigated for nonlinear stochastic systems subject to network-induced delays and packet loss. Moreover, the signal quantization problem has received considerable attention for NCSs since it has a substantial impact on system performance [23], [24], [25]. The authors proposed the event-triggered control design for singular systems with quantizations [26]. The quantized feedback control problem was studied by applying the sector bound method for discrete time-varying systems [27]. However, there are few related fault detection results taking dynamic event-triggered mechanism (DETM), signal quantization and sensor nonlinearity into account. This is the second main motivation of this work.

On the other hands, the residual evaluation performs a vital affect in the fault detection model. The crucial factor of residual evaluation is to select an appropriate threshold to quickly and effectively detect the occurring fault of plant on the basis of avoiding false alarms. However, The traditional constant thresholds are usually difficult to obtain good fault detection results in practical applications. To overcome this problem, researchers have found that the zonotopal algebra has enormous potential in control science and control strategy [28]. Recently, the zonotope-based fault detection method has been developed to generate natural thresholds for residual signals [29], [30], [31]. This also attracts the author’s interest.

Based on above discussions, this paper studies the dynamic event-triggered fault detection problem for discrete-time systems subject to signal quantization and sensor nonlinearity. In addition, the zonotopic residual evaluation is applied to evaluate the detection performance. The major difficulties and challenges of this work is that: (i) how to design a dynamic threshold function to save limited network communication resources more effectively. (ii) how to co-design the expected parameters of FDF and DETM for the established fault detection model subject signal quantization and sensor nonlinearity. The essential achievements of this work involve that:

• Based on the limited dynamic communication capability, a FDF design strategy is firstly proposed with the consideration of signal quantization and sensor nonlinearity, simultaneously.

• A novel DETM and signal quantization strategy are applied to conserve the limited network resources.

• A zonotopic-based residual evaluation scheme is constructed to evaluate the detection performance.

Notation: $M^T$ and $M^{-1}$ respectively represent the transpose and inverse of matrix $M$. $M>0$ means that matrix $M$ is a positive definite matrix. $*$ represents the item corresponding to the diagonal matrix. $\text{diag}\{·\}$ defines a diagonal matrix. $\text{sym}\left\{M\right\}$ stands for $M^T+M$. $\oplus$ represses the Minkowski sum. $Z=〈p,H〉=\left\{p+Hz:z\in {\mathbb{B}}^s\right\}$ is a s-order zonotope with ${\mathbb{B}}^s={\left[-1,1\right]}^s$, $p\in {\mathbb{R}}^{n\times 1}$ and $H\in {\mathbb{R}}^{n\times s}$.