Hybrid event-based asynchronous finite-time control for cyber-physical switched systems under denial-of-service attacks

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Abstract

This paper investigates the event-based asynchronous finite-time control for a class of cyber-physical switched systems under Denial-of-Service (DoS) attacks. Considering the attack’s characteristics, we put forward a novel attack-instant-constrained hybrid event-triggered scheme (HETS), which can not only contribute to reducing the network transmission overload, but also well descibe the network denial service behavior under attack interference. An asynchronous and ZOH-based controller is delicately constructed to mitigate the influence of DoS attacks and network-induced delay. A double-mode dependent Lyapunov–Krasovskii functional (LKF) is developed to set up some sufficient finite-time stability criteria for the concerned systems in view of the asynchronous switching effect. Finally, an application example of the urban railway system is offered to verify the proposed control algorithm.

Introduction

In the fields of theoretical research and engineering applications, it is well known that many equipments have switching characteristics so as to adapt the various working conditions in reality, in which these application processes with complex switching behavior can be modeled as switched systems. The so-called switching system consists of a group of switched subsystems and a switching law for coordinating the switching of subsystems, which is the one of the more important type of the hybrid systems. Because of the powerful capability in modeling and control, switched systems have attracted widespread research interest in recent years [1], [2], [3], [4], [5].

With the wide application of digital computers and communication networks, signal transmission is typically implemented digitally. Then along comes a hot topic in the field of switched systems, i.e., the control problem for the cyber-physical switched systems, in which the data transmission via communication networks. A good deal of momentous results have been acquired for the cyber-physical switched systems, such as sampled-data-based stability analysis [6], [7], event-triggered H filtering [8], [9], finite-time stability analysis [10], [11] and asynchronous switching [12], [13]. And it’s not uncommon to find that sampled-data method has been paid much attention in signal transmission digitally. The sampling sequence of traditional sampling mechanism, as a time-triggered control strategy, is determined by a certain time period whether those sampled data are really beneficial for the dynamic system’s performance. This may lead to the danger of congestion in the condition of limited network resource, cause large consumptions of nodes or even network paralysis. Then, based on the system’s real-time requirements, developing an efficient sampling mechanism to optimize limited network resources is of great significance.

To address these issues so that the sampling mechanism more efficient, event-triggered scheme (ETS) derives from in the late 1990s [14]. Under the impact of the pre-designed ETS, the transmitted signals are sampled on demand instead of fixed time sampling. Correspondingly, these networked continuous dynamic systems with ETS can be classified as hybrid systems with complex hybrid dynamic characteristics, which are widely utilised in various fields [15], [16], [17]. In order to avoid the Zeno behavior and to facilitate handling of transmission delays, Yue et al. [18] developed an event-triggered communication based on periodic sampling data in the continuous-time domain. From hybrid control point of view, the event-triggered stabilization problem is studied in Zhang and Gharesifard [19] by using impulsive control method to help rule out Zeno behavior. In [20], a hybrid time-event-triggered control technique is investigated, i.e., some of the system components are time-triggered while others are event-triggered. For switched system with time-varying delays, Ren et al. [21] mentions an hybrid ETC which transmits both state signal and switching signal. In addition to the focus of saving limited network resources, the issues of network security arouse our research horizons [22], [23] due to the openness of the network. Compared with other types of attacks, DoS attacks can make NCSs out of control and even destroy their stability with minimal cost [24]. Therefore, as the most destructive form of network attacks, it is very valuable to study the network security issues under DoSs attacks, especially how to simultaneously save the network resources [25], [26], [27]. As for switched system, its switching features as well as its complex time series analysis under DoS attacks and event-triggered scheme make the event-triggered control problem under DoS attacks be a big challenge in control field. Therefore, for the cyber-physical switched system under DoS attacks, a study on how to develop a hybrid event-triggered control method that can save network resources and defend network attack is quite meaningful.

Previously, the investigation is centered on Lyapunov asymptotic stability for the most part. Considering practical applications, the following two phenomena have been observed: (a) a system may be asymptotically stable, but it may exhibit terrible transient performance; (b) some practical application systems need to estimate its trajectory bounds over a fixed short period of time, such as missile and satellite, air traffic control systems and power electronic systems. Therefore, the study of transient information of dynamic systems may be more meaningful than the study of its qualitative behavior in infinite interval. As finite-time stability (FTS) is mainly used to describe the transient behavior of the system in finite time, it is more essential to inquiry the finite-time control for practical applications. Actually, FTS was originally called short-time stability, which was put forward by Dorato [28] in 1960’s. Furthermore, the concept of FTS has been extend by Amato et al. [29] to finite-time boundedness (FTB). Since F. Amato discussed the finite-time control for different dynamic systems, many results are addressed in Zhou [30], Hu et al. [31], Naifar et al. [32], Wei et al. [33], Zhang et al. [34], including stability and boundedness. Besides, considering the limited network resource and the hybrid characteristics of the switched systems, there is few result reported on finite-time control for the cyber-physical switched systems subject to DoS attacks, which is another motivation of the work.

Inspired by the above discussions, we conduct the investigations on the hybrid event-based asynchronous finite-time control for cyber-physical switched systems under DoS attacks. It is worth noting that DoS attacks may cause data transfer failure, which will seriously affect the continuity of the controller to the system dynamic process control. Hence, the major challenge is to co-design finite-time controller, event-triggered condition, and switching signals in consideration of the characteristics of DoS attacks. The main work of this paper are highlighted as follows:

  • (i)

    In contrast with [35], [36], [37], the designed controller can guarantee the FTB of the cyber-physical switched systems under DoS attacks, rather than the asymptotic stability.

  • (ii)

    Different from Hu et al. [38], Peng and Sun [39], the influences of the DoS attacks are fully considered in this paper, and then an attack-instant-constrained HETS is developed to save limited network resources.

  • (iii)

    Compared with the conventional ETS, the triggered thresholds φ(sr,νh) in the proposed HETS can automatically adjust when the system response varies, which can well balance the system performance and the utility ratio of the communication bandwidth.

  • (iv)

    Taking into account the intermittent nature of DoS attacks, an asynchronous and ZOH-based controller is constructed, which can make full use of the system state information to improve dynamic characteristics of system more effectively when the transmission network is under attack.

Notations In this paper, Rn×n denotes the n-dimensional real matrices, N the set of natural numbers, and λm(P)(λM(P)) the minimum (maximum) eigenvalue of matrix P. In and 0n denote n×n identity matrix and n×n zero matrix, respectively. Notation · denotes the Euclidean norm. PT and P stand the transpose and the inverse of P. For a symmetric matrix P, P>0,0,<0, and 0 stand for positive definite, positive-semidefinite, negative definite, and negative-semidefinite matrix respectively. The block-diagonal matrix of P1 and P2 is represented by diag{P1,P2}. · and · respectively stand for rounding up and rounding down to an integer. col{·} denotes the column vector.

Section snippets

Physical plant

The switched system considered in this paper is described by:x˙(t)=Aσ(t)x(t)+Bσ(t)u(t)+Dσ(t)ω(t),where x(t)Rn×1, u(t)Rm×1, and ω(t)Rl×1 denote the state, the control input, and the disturbance input, respectively. The piecewise constant function σ(t):[0,)S={1,2,,s} is right continuous, which denote the switching signal. {(tk,σ(tk)} is the set of switching sequence, where σ(tk)S,kN, and tk satisfying 0<t1<<tk<. When σ(tk)=iS, denoting ith subsystem (Ai,Bi) be activated. AiRn×n, BiRn×

Main results

Under the action of the HETC, DoS attack, and transmission delay, the FTB analysis of the system Eq. (7) is conducted in this part firstly. Then, a feasible asynchronous controllers shall be designed to guarantee the finite-time control synthesis.

Simulation example

In this second, an application example of the urban railway system is offered to verify the effectiveness of our asynchronous finite-time control design scheme. The mathematical model of railway system can be described bym(t)a(t)=F(t)(a¯0+a¯1(t)v(t)+a¯2v2(t)),where the mass of the train is denoted by m(t), which is a piecewise right continuous system parameter due to the variability of the number of passengers over different time intervals. For given constants a¯0 and a¯2, a¯0+a¯1(t)v(t)+a¯2v2(

Conclusion

Under the action of DoS attacks and HETS, the finite-time control problem of cyber-physical switched systems has been investigated in this paper. By extending the DoS attacks to the event-triggered control of switched system, a novel attack-instant-constrained HETS has been developed. To simplify the finite-time stability analysis, the closed-loop switched system with DoS attacks and HETS has been modeled as a time delay switched system with asynchronous switching characteristics. Moreover,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to thank the associate editor and the reviewers for their very helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China under grant (Nos. 62003311, 62076223, 62103378, 62273254) and the doctoral research fund of Zhengzhou University of Light Industry in 2020 (2020BSJJ005).

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