Complexity

Volume 2019, Article ID 8194606, 14 pages

https://doi.org/10.1155/2019/8194606

## Nonfragile Integral-Based Event-Triggered Control of Uncertain Cyber-Physical Systems under Cyber‐Attacks

^{1}College of Mechanical & Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China^{2}Department of Electrical, Computer and Software Engineering, The University of Auckland, Auckland 1142, New Zealand

Correspondence should be addressed to Shen Yan; moc.liamg@hdznehsnay

Received 29 June 2019; Revised 23 August 2019; Accepted 14 September 2019; Published 3 November 2019

Academic Editor: Xinggang Yan

Copyright © 2019 Shen Yan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This article studies the problem of nonfragile integral-based event-triggered control for uncertain cyber-physical systems under cyber-attacks. An integral-based event-triggered scheme is proposed to reduce the data transmissions and save the limited network resources. The triggering condition is related to the mean of system state over a finite time interval instead of instant system state. Random cyber-attacks in a communication channel are taken into account and described by a stochastic variable subject to Bernoulli distribution. A novel Lyapunov–Krasovskii functional based on Legendre polynomials is constructed, and the Bessel–Legendre inequality technique is employed to handle the integral term induced by the integral-based event-triggered scheme. Resorting to these treatments, sufficient conditions are established via a set of linear matrix inequalities to guarantee the asymptotic mean-square stability of the closed-loop system. Finally, a numerical example shows that the presented method is effective.

#### 1. Introduction

Cyber-physical systems (CPSs) are systems where computational and physical resources are tightly integrated via a shared network. They have been applied in industrial fields, such as medical devices, chemical process industries, and transportation systems [1–3]. Security is a critical issue in CPSs, as a great deal of communication signals are exchanged over the network channel that is open to potential cyber-attacks. Cyber-attacks, introduced by external attackers, could cause negative impact on the communication channels and deteriorate system stability performance. Thus, it has attracted much attention to study the control and filtering problems for CPSs under the threats of network attacks [4–7]. The two representative kinds of cyber-attacks are, namely, deception attacks and denial-of-service (DoS) attacks. The deception attacks aim at injecting deceitful information into the transmitted signal, and the DoS attacks could block or break the communication link among system nodes. Some recent results on the study of cyber-attacks are reported in [8, 9]. The distributed recursive filtering problem of stochastic systems with time delays and deception attacks is investigated in [8]. The finite-horizon tracking control issue of a stochastic system subject to hybrid attacks is developed in [9], where both the stochastic deception attacks and stochastic DoS attacks are taken into consideration.

Note that, in the above results, communication signals are transmitted through the network without considering some practical constraints, such as limited network bandwidth and computation burden. Some effective manners such as event-triggered scheme (ETS) [10], signal quantization [11], and real-time scheduling [12, 13] have been proposed to save communication resources and reduce computation burden. Recently, ETS has gained growing attention and interests in existing research studies [14–23] for its simplicity of execution and effectiveness of reducing data transmissions. Due to this fact, the idea of ETS is introduced into security control for CPSs and has produced many interesting results [24–29]. To be specific, in [24], a resilient load frequency controller is designed to stabilize the multiarea power systems with cyber-attacks. The issue of event-triggered controller design for networked systems under the resilient event-triggered mechanism and periodic DoS jamming attacks is developed in [25]. For multiagent systems with deception attacks, [26] is concerned with the event-triggered output consensus control issue. Zha et al. [27] investigate the design of decentralized event-triggered controller of delayed neural networks under deception attacks. Liu et al. [28] address the distributed event-triggered control issue for networked control systems with stochastic cyber-attacks, and a decentralized triggering strategy is utilized to release the network communication burden. In [29], the resilient load frequency control problem for multiarea power systems with cyber-attacks is investigated by using an event-triggered communication mechanism. In these studies, only the instant system information, such as current system state and the last triggered signal, is applied to design the ETS conditions. In practical systems, the system state could have abrupt fluctuations over some time interval incurred by external disturbances, system uncertainties, or environment noises. If such state with abrupt fluctuations is triggered to controller, it is negative for stabilizing the system performance. In [30], the idea of introducing the accumulation error information or system dynamics over a finite time interval into the design of triggering rule is proposed. Considering systems with stochastic measurement noises, an integral-based ETS utilizing the average state over a finite time interval is proposed to reduce the effect of measurement noises in [31], where an approximation approach is used to deal with the integral triggering condition. This method to treat the integral term will result in approximation error, which needs further investigation to exclude such approximation error and obtain less conservative results. In addition, the controllers in these works are assumed to be implemented perfectly. However, uncertainties caused by execution errors or unknown noises are usually observed in engineerings, which may lead to system performance deterioration. To conquer this problem, Sakthivel et al. [32] handle the uncertainties by norm-bounded variations to design the dissipative analysis-based nonfragile controller for network-based singular systems with ETS. In [33], the issue of nonfragile event-triggered control of linear systems under unreliable communication links is addressed. For linear systems against actuator saturation and disturbances, Liu and Yang [34] study the design of nonfragile dynamic output feedback controller via an event-triggered communication scheme. It is noted that the cyber-attacks are not considered in communication channels, and the mean of system state is not taken into account for ETS design in [32–34]. To our best knowledge, few results have been investigated for integral-based event-triggered control problem of uncertain systems with cyber-attacks and gain uncertainties, which inspire the current study.

This paper focuses on the co-design of integral-based ETS and nonfragile controller for uncertain CPSs with cyber-attacks and gain uncertainties. The main contributions of this paper are summarized as follows:(1)An integral-based ETS that utilizes the mean of system state over a fixed time interval is proposed to reduce the wastage of limited network resources. It covers the conventional ETS [10] as a special case as the fixed time interval approaches to zero and can further reduce unnecessary data transmissions.(2)The closed-loop system with the integral-based ETS, cyber-attacks, and gain variations is established as a distributed delay system, in which a Bernoulli variable is adopted to describe the stochastic cyber-attacks. A novel augmented Lyapunov–Krasovskii functional (LKF) based on Legendre polynomials is constructed. With the help of the novel LKF and Bessel–Legendre inequality, the integral term induced by integral-based ETS is handled without the approximation error in [31].

The organization of this paper is given as follows. Section 2 presents the problem statement and some preliminaries. The main results of stability analysis and control synthesis are derived in Section 3. To show the effectiveness of the proposed approach, a numerical example is given in Section 4. Some conclusions and future research topics are shown in Section 5.

##### 1.1. Notation

In this paper, and denote the *n*-dimensional Euclidean space and the set of all real matrices, respectively. is the Euclidean norm in . means the mathematical expectation. The notation represents that *X* is symmetric and positive (negative) definite. The superscript “” stands for the transpose of a vector or matrix. denotes the kernel of *X*. equals to . The notation means the binomial coefficients given by . ⊗ refers to the Kronecker product. is the set of nonnegative integers.

#### 2. Preliminaries

Consider the following uncertain continuous-time system aswhere is the system state, is system input, *A* and *B* are known system matrices, and , where is a stochastic variable with expectation and variance and means the deviation from the nominal model. Different from the commonly used uncertainty model with exactly known bounds, we focus on the statistic of the uncertainty deviating from the nominal part.

Since the network bandwidth is limited, unnecessary data transmission will lead to the wastage of limited network resources. In order to decrease communication frequency, an integral-based event-triggered scheme, drawn in Figure 1, is presented aswhere ; , is the mean of system state; and , the mean of the current and next triggered control signal, respectively; *τ* is the length of the mean of system state over ; and is the positive weighting matrix.