Event-triggered resilient control for cyber-physical systems under asynchronous DoS attacks
Introduction
With the development of sensing, computing and network communication, the study of networked control and cyber-physical systems (CPSs) has become a special research area and receives considerable interests. CPS is integrated with computation, networking and physical processes, the cyber layer and the physical world are conjoined with communication networks. The applications of CPS involve power-grid, autonomous vehicles and water distribution systems [16]. However, the use of communication networks makes these CPSs vulnerable to cyber attacks [4], [26], and the question of cyber-security has become a critical problem in industrial processes.
Cyber attacks can be classified into two main categories: deception attacks and denial-of-service (DoS) attacks. The former can damage the integrality of the measurement and actuator data by manipulating the packets transmitted over the network, such as false injection attacks [2], [5], [19] and replay attacks [21] . On the other hand, DoS attacks aim at jamming the network communication channels to prevent the measurement and actuator data from reaching their destinations. These attacks can be accomplished easily by an adversary [27]. Extensive attention to the security research of the CPSs under DoS attacks has been attracted, and some theoretic methods have been studied in the literature. In [29], optimal DoS attack schedules are investigated from the viewpoint of an energy-constrained attacker. In [3], a probabilistic characterization for the link failures is proposed to study combined effects of malicious and random packet losses, and almost sure stabilization is investigated for the networked control systems. A two-player zero-sum stochastic game framework is formulated to estimate the state for a CPS with multi-channel network under DoS attacks in [8].
A major line of the research on security is the stability analysis of the CPSs under DoS attacks. Persis and Tesi [23] characterizes the frequency and duration of the DoS attacks under which input-to-state stability (ISS) of the closed-loop system can be preserved. Based on the DoS model of frequency and duration, extensions have been concerned in [1], [11], [12], [20], [24]. Nonlinear systems are considered in [24]. In [11], the condition for maximizing frequency and duration of the DoS attacks is presented while simultaneously guaranteeing the closed-loop stability. Control architectures that approximate co-location in [11] while enabling remote implementation are designed in [12]. For nonlinear uncertain interconnected systems under intermittent DoS attacks, the decentralized adaptive fuzzy secure control problem is investigated in [1]. The ISS control problem for CPSs with multiple transmission channels under DoS attacks is investigated in [20]. However, the aforementioned works presume that the adversaries attack only sensor-to-controller (S-C) channel or controller-to-actuator (C-A) channel, or attack the two channels synchronously. Resilient control for CPSs with both S-C and C-A channels under asynchronous DoS attacks has not been fully investigated. It is not a trivial problem because the interconnected channels under DoS attacks can affect each other. This is the major motivation of this paper.
In practice, communications may induce a nontrivial cost in terms of energy in wireless networks, and the networks may be shared and bandwidth-limited in CPSs. In order to save communication resources, event-triggered control (ETC) is widely used in the studies of networked control systems [17], such as [10], [15], [28], [33] for linear systems, [9], [22] for nonlinear systems, [7], [13], [32] for distributed systems and [6], [14] for multi-agent systems. In this paper, an observer-based ETC architecture is constructed for the linear continuous-time plant, and the control system configuration is similar to [10], [28] as shown in Fig. 1. Two different ETC strategies are designed for S-C and C-A channels, respectively. Because of the continuous-time plant, Zeno behaviors can be excluded for the S-C channel by carefully designing the event-triggering coefficients of event-triggering mechanisms (ETMs). For the C-A channel, periodic event-triggered control (PETC) [10], [15] strategy is employed to deal with the packet-based transmission [30], [31], [34]. Then based on the ISS analysis framework, the relationship between the ETMs is obtained, and the stability is proved whether or not the DoS attacks are presented. The packet-based control method with buffer has been briefly analyzed for the systems under different DoS signals in [12]. However, it is a challenge to obtain the coupling relationship of the ETMs, since the event-triggering coefficients of the different ETC strategies depend on each other because of the interconnection of the two channels. Besides, the stability analysis of the closed-loop systems is different.
The main contributions of this paper are characterized as follows: First, two different ETC strategies are designed for CPSs with both S-C and C-A channels under asynchronous DoS attacks, and both disturbance and measurement noise are considered. Second, the relationship between inter-event times of the S-C channel and event-triggering coefficients is proposed to exclude Zeno behaviors. Third, the conditions on attack parameters and the prediction horizon of the C-A channel (buffer size) are proposed to guarantee the closed-loop stability under the worst case of the DoS attacks for both two channels, and the condition for removing the restriction of the buffer size is obtained.
The remainder of this paper is organized as follows. In Section 2, the preliminaries and problem formulation are presented. The stability analysis of ETC strategies for two channels is given in Section 3. In Section 4, the resilient control algorithm is designed for both two channels under asynchronous DoS attacks. Then the simulation is given in Section 5, and Section 6 concludes this paper.
Donate the set of reals by denotes the n-dimensional Euclidean space. Given let be the set of reals greater than or equal to α. Let be the set of natural numbers. Given a vector ‖x‖ indicates its Euclidean norm. Given a matrix A, let AT , ‖A‖ and μA be its transpose, spectral norm and logarithmic norm [25], respectively, where . Given two sets S1 and S2, let S2\S1 be the relative complement of S1 in S2. For an interval its length is denoted by . Given a measurable time function f(t) and a time interval [0, t), the norm of f( · ) on [0, t) is formulated as .
Section snippets
Process dynamics and network
Consider the CPS process shown in Fig. 2. The S-C channel and the C-A channel are all networked, where the communication resources (the batteries for the wireless devices, for instance) are limited. At the same time, the networks of the CPSs may be attacked by asynchronous DoS attacks, that is, an adversary can block both the S-C and C-A communications at different time intervals. The process to be controlled is described by where are system
Stability analysis of the two-channels ETC strategies
In this section, two channels ETC strategies are designed, and an observer-based control framework is proposed to analyze the stability. Besides, the relationship between the lower bound of inter-event time for S-C channel and the ETMs is characterized.
Resilient control under DoS attacks
In 3 Stability analysis of the two-channels ETC strategies, Theorem 3.1 and 3.2 are provided to analyze the stability of the closed-loop system with ETC strategies. Based on these obtained ETMs and ETMc, this section focuses on analyzing the stability of the closed-loop system under asynchronous DoS attacks.
Let i ∈ {s, c} be the first time instants which the ETMi is satisfied after a successful transmission attempt at and that is . Denote by
Simulation example
A numerical example is given in this section. The process to be controlled is open-loop unstable and is characterized by the parameter matrices . The disturbance ω(t) and measurement noise ν(t) are random signals with uniform distribution in [-0.2, 0.2] and [-0.02, 0.02], respectively. The initial conditions are . The H∞ performance index . The observer gain matrix L and the state-feedback gain matrix K are given as
Conclusions
In this paper, an observer-based resilient control strategy for continuous-time CPSs with disturbance and measurement noise is investigated, and the asynchronous DoS attacks to both S-C and C-A channels are considered. Different ETC strategies are designed for the two channels. For the S-C channel, the relationship between inter-event times and event-triggering coefficients is proposed to exclude Zeno behaviors. For the C-A channel, the PETC strategy is employed to deal with the packet-based
Acknowledgments
This work was supported in part by the Funds of the National Natural Science Foundation of China (Grant nos. 61621004 and 61420106016), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant no. 2018ZCX03).
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