This paper studies the problem of designing a resilient control strategy for cyber-physical systems (CPSs) under denial-of-service (DoS) attacks. By constructing an H∞ observer-based periodic event-triggered control (PETC) framework, the relationship between the event-triggering mechanism and the prediction error is obtained. Then, inspired by the maximum transmission interval, the input-to-state stability of the closed-loop system is proved. Compared with the existing methods, a Zeno-free periodic PETC scheme is designed for a continuous-time CPS with the external disturbance and measurement noise. In particular, the objective of maximizing the frequency and duration of the DoS attacks is achieved without losing robustness. Finally, two examples are given to verify the effectiveness of the proposed approach.

Introduction

In recent years, cyber-physical systems (CPSs) have been widely used in various engineering fields owing to advances in computing and communication technologies. However, the use of networks and heterogeneous digital elements has made these CPSs vulnerable to various cyber attacks, such as deception attacks, replay attacks, bias injection attacks, zero-dynamics attacks, denial-of-service (DoS) attacks and so on. Unlike traditional systems where attacks limit their impact to the cyber level, malicious attacks to CPSs can impact the physical world [1].Thus these is a strong demand for analysis, synthesis and design methods to guarantee the security and reliability of CPSs despite the presence of malicious attacks[2, 3]. Among the various malicious attacks, DoS attacks make the actuator and sensor data to be blocked rather than reach their respective destinations and lead to the absence of data for the related components. Such kind of attack is very common in network communications, and a lot of works have been made for the CPSs under DoS attacks [4–۸]. A basic research field on security problem of CPSs is the stability analysis under DoS attacks. In [9], the authors characterize frequency and duration of the DoS attacks under which input-to-state stability of the closed-loop system can be presented, and the transmission times is scheduled. A resilient control method is presented in [10] to maximize frequency and duration of the DoS attacks under which closed-loop stability is not destroyed. In [11], based on the studies on [10], a control architecture that approximate co-location while enable remote implementation is designed. The input-to-state stable (ISS) control problem for CPSs with multiple transmission channels under DoS attacks is concerned in [12]. In [13], a systematic design framework for output-based dynamic event-triggered control (ETC) systems under DoS attacks is proposed for a class of nonlinear systems using a hybrid model. The traditional control methods are implemented in a time-triggered method where the sampling and the signal transmission are executed periodically, such as sampled-data control [14]. Usually, the wireless communication units of CPSs are power restricted, and the network is often shared with multiple devices. Hence communication resource utilization is needed to be considered. ETC scheme which helps reduce the network utilization has been widely investigated on networked control systems [15–۱۹]. In [20], a state-dependent triggering method is proposed for network-based interconnected systems with delays and packet losses. In [21], periodic event-triggered control (PETC) strategy for linear systems is proposed. By combining time-triggered control and ETC, the event-triggering condition is verified periodically in PETC, and whether or not to compute or to transmit new measurements and control signals is decided at every periodic sampling instant. In [22], a model-based PETC strategy for linear discrete systems is presented, and both sensor-to-controller channels and the controller-to-actuator channels of the systems are communicated through networks. However, the ETC scheme for CPSs under DoS attacks has not been fully investigated. On the other hand, the DoS attacks occurred at the event-triggering intervals are invalid, thus the tolerable DoS attacks can be increased by using the ETC strategy. These are the major motivations of this study.