Elsevier

Neurocomputing

Volume 400, 4 August 2020, Pages 458-466
Neurocomputing

Distributed event-triggered consensus of multi-agent systems under periodic DoS jamming attacks

https://doi.org/10.1016/j.neucom.2019.03.089Get rights and content

Abstract

This paper is concerned with distributed event-triggered consensus of a generally linear multi-agent system subjected to periodic denial-of-service (DoS) jamming attacks. First, a novel switched time-varying delay system model is established to describe consensus error dynamics. By virtue of the switched system and time delay system approaches, one constraint is obtained as a tradeoff between exponential convergence rate of consensus error and uniform lower bound of sleep intervals of periodic DoS jamming attacks. Second, sufficient conditions are derived to guarantee exponential consensus of the multi-agent system. Third, a co-design method is presented to determine the parameters of both the distributed event-triggered mechanism and consensus protocol. Furthermore, an algorithm is given to find the allowable uniform lower bound of sleep intervals and suboptimal distributed event-triggering parameters. Finally, an illustrative example is provided to verify the effectiveness of the proposed resilient event-triggering consensus design method against periodic DoS jamming attacks.

Introduction

Security of cyber-physical systems (CPSs) is of great theoretical and practical significance due to widespread applications of CPSs in critical infrastructure including power systems, water systems and transportation systems, and thus has attracted increasing research attention [1], [2], [3], [4], [5], [6], [7]. Generally, the physical process of CPSs can be destroyed permeably by cyber attacks through damaging confidentiality, integrity, and availability of control commands and/or sensor measurements [2], [3], [8]. In the literature, the two most commonly studied attacks are denial-of-Service (DoS) attacks and deception attacks, see, e.g., [3], [7], [9], [10], [11], [12], [13], [14]. Denial-of-Service (DoS) attacks aim at obstructing information exchange by launching in communication channels or terminal nodes [11], [15], [16], [17], [18], [19], [20]. In contrast, deception attacks aim at corrupting information integrity by invading information system illegally and tampering datas maliciously [21], [22], [23], [24], [25].

As one significant category of CPSs, multi-agent systems (MASs) has found considerable applications in areas such as satellites, mobile robots, sensor networks, unmanned vehicles and so on [26], [27], [28], [29]. It should be noted that MASs with complex communication topologies face more risks from cyber attacks [24], [30], [31], [32], [33]. Undoubtedly, consensus and formation problems of MASs under DoS attacks have been well studied in some existing literature [34], [35], [36], [37], [38]. It should be noted that the impact of the power limited DoS attacks on MASs is commonly represented by the switching residual communication topologies. For example, the work in [34] classified residual topologies as two types according to whether the residual communication topology has a directed spanning tree; and the consensus of MASs with stochastic linear dynamics was achieved under the continue-time leader-following consensus protocol. Further studies on continue-time resilient consensus protocol can be seen in [37], [38]. From the perspective of sampled-data based consensus, self-triggered coordination of MASs with single-integrator dynamics was considered in [36] under DoS attacks constrained by frequency and duration, in which the adjacency matrix of topology graph was zero over DoS attacks interval. Obviously, zero-topology (ZT) can be viewed as the worst case of DoS attacks and brings serious damage on consensus performance of MASs. Apart from the DoS attacks investigated in [34], [36], periodic DoS jamming attacks represent another common type of DoS attacks with the nature of limited energy, difficult detection and ease of realization [39], [40], [41]. To the best of our knowledge, periodic DoS jamming attacks have not been adequately considered in the MASs consensus researches. On the other hand, it is well-acknowledged that distributed event-triggered mechanism provides an effective transmission strategy for wirelessly distributed and networked systems so as to save possible system device battery power and/or communication resources [42], [43], [44], [45], [46], [47]. For more latest results on event-triggered MASs, we refer to the surveys [48], [49]. Therefore, how to deal with the following two challenging issues, i.e., periodic DoS jamming attacks and constrained communication resources on information exchange among interacting agents for networked MASs, motivates the present study.

In this paper, we study the problem of distributed event-triggered consensus of a general linear multiagent system subjected to periodic DoS jamming attacks. According to Senejohnny et al. [36], Foroush and Martínez [50], DoS jamming attacks are implemented by launching pulse-width modulated (PWM) jamming signals and the attack scenario considered is zero-topology periodic (ZTP) DoS jamming attacks, which means that communication topology is switching between zero-topology (ZT) and initial topology periodically. The main contributions of this paper are summarized as follows. (1) A novel switched time-varying delay system model will be established to describe the consensus error dynamics of the general linear multiagent system under ZTP DoS jamming attacks. Correspondingly, the considered consensus problem of the multiagent system can be converted into an exponential stability problem of a switched time-delay system; (2) a distributed event-triggered mechanism will be developed and a resilient event-triggered coordination protocol will be constructed; (3) a co-design method will be proposed to solve out the parameters of both the distributed event-triggered mechanism and the consensus protocol. An algorithm will be also presented to determine the allowable uniform lower bound of sleep intervals of ZTP DoS jamming attacks and suboptimal parameters of distributed event-triggered mechanisms; and (4) a uniform constraint on each attack cycle will be obtained as one sufficient condition to achieve resilient consensus. It will be shown that such a constraint reflects explicitly the relationship between exponential convergence rate of consensus and ZTP DoS jamming attacks.

The rest of the paper is organized as follows. Section 2 establishes the model of ZTP DoS jamming attacks and consensus error dynamics with distributed event-triggered mechanism. Consensus performance analysis and co-design of distributed event-triggered mechanism and consensus protocol are provided in Section 3. An illustrative example is simulated in Section 4 to illustrate the effectiveness of the proposed method. Section 5 concludes this paper.

Notation: The sign ⊗ is matrix Kronecker product. diag{w1,w2,,wn} denotes the diagonal matrix with diagonal elements wi. exp { · } represents exponent function. And we introduce symbol [*] as the symmetric term of matrix. 1 and 0 represent column vector of which all elements take value 1 and 0 respectively. And define norm ϕmsupt[h,0]{ϕ(t),ϕ˙(t)}.

Section snippets

Graph theory

A directed graph is represented by G={P,E,W} where P={p1,p2,,pN} denotes set of nodes, EP × P is set of edges and element {eij=(pi,pj)}E means information flow from node pi to node pj, W={wij} is weighted adjacency matrix where wij > 0 if eji ∈ E, otherwise wij=0. The Laplacian matrix of graph G is W¯=W˜W where W˜=diag{W˜1,,W˜N} and W˜i=wij. A directed path (pi1,pis) of G consists of the following ordered pairs of edges (pi1,pi2), (pi2,pi3), , (pis1,pis). G has a directed spanning tree

Multiagent system consensus analysis

According to the switched time delay system model (8), we have the subsystem Σi(i=1,2):y˙(t)=(IN1Ai)y(t)(E1W¯E2BiK)[y(tξ(t))η(t)],where Ai=A, B1=B and B2=0.

Choose a Lyapunov–Krasovskii functional candidate for Σi asVi(t)=yT(t)Piy(t)+thtyT(θ)Rie(1)iαi(tθ)y(θ)dθ+thtθty˙T(φ)(Mi+Ni)e(1)iαi(tφ)y˙(φ)dφdθ.

In this section, we estimate the decay law of Lyapunov–Krasovskii functional candidate (11) along the trajectory of subsystem (10). With the framework of switched system method,

An illustrative example

In this section, a simulation example is illustrated to validate the effectiveness of the proposed consensus coordination method. In simulation, the multi-agent system consists of six agents each of which is described by F-18 aircraft model. According to Adams and Buffington [55] and Ge et al. [56], the longitudinal linear model of F-18 aircraft is represented by[x˙1(t)x˙2(t)]=Along[x1(t)x2(t)]+Blong[u1(t)u2(t)],where x1(t), x2(t) represent angle of attack and pitch rate, respectively; x˙1(t), x

Conclusions

Consensus of MASs subjected to ZTP DoS jamming attacks has been explored. The switched time-delay system model has been established to describe the consensus error dynamics of the multiagent system with distributed event-triggered mechanism. Sufficient conditions in form of LMIs have been derived by using time-delay system approach and switched system theory, under which the multiagent system can reach exponential consensus under ZTP DoS jamming attacks. An algorithm has also been given to

Declaration of interests

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

This work was supported by the National Natural Science Foundation of China under Grant (No. 61533010, 61633016), the National Natural Science Foundation of China (Grant no. 61673223), the “Six Talent Peaks Project” of Jiangsu Province of China (RLD201810), the QingLan Project of Jiangsu Province of China (Grant no. QL 04317006), the NUPTSF (Grant no. XJKY15001).

Zihao Cheng received the M.S. degree in School of Electrical Engineering and Automation from the Henan Polytechnic University in 2016, Jiaozhuo, China. He is currently pursing his Ph.D. degree in School of Computer Science, Nanjing University of Posts and Telecommunications, Nanjing, China. His research interests include analysis and synthesis of networked control system and its security control, multi-agent systems, fuzzy control system, power system.

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    Zihao Cheng received the M.S. degree in School of Electrical Engineering and Automation from the Henan Polytechnic University in 2016, Jiaozhuo, China. He is currently pursing his Ph.D. degree in School of Computer Science, Nanjing University of Posts and Telecommunications, Nanjing, China. His research interests include analysis and synthesis of networked control system and its security control, multi-agent systems, fuzzy control system, power system.

    Dong Yue (SM’08) received the Ph.D. degree in control science and engineering from the South China University of Technology, Guangzhou, China, in 1995. He is currently a Professor and the Dean of the Institute of Advanced Technology, Nanjing University of Posts and Telecommunications, Nanjing, China, and a Changjiang Professor with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China. He has published over 100 papers in international journals, domestic journals, and international conferences. His current research interests include analysis and synthesis of networked control systems, multiagent systems, optimal control of power systems, and Internet of Things. Dr. Yue is currently an Associate Editor of the IEEE Control Systems Society Conference Editorial Board and an Associate Editor of the IEEE Transactions on Neural Networks and Learning Systems, the Journal of the Franklin Institute, and the International Journal of Systems Science.

    Songlin Hu received the Ph.D. degree in control science and engineering from the Huazhong University of Science and Technology, Wuhan, China, in 2012. Since 2013, he has been with the College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, China, where he is currently an Associate Professor with the Institute of Advanced Technology. His current research interests include networked/event-triggered control, T--S fuzzy systems, and time delay systems.

    Hui Ge (M’17) received the B.S. and M.S. degree in Mechatronics Engineering and Automation in 2006 and Theory and New Technology of Electrical Engineering in 2009 from Nanjing Normal University, Nanjing, China, respectively. He received the Ph.D. degree in Information Acquisition and Control at Nanjing University of Posts and Telecommunications in 2018, Nanjing, China. He is now a lecture with the department of Nanjing University of Posts and Telecommunications. His current research interests include analysis and synthesis of networked control systems, fault diagnosis and CPS security control.

    Lei Chen received the B.Eng. degree in electrical engineering from Tongda College of Nanjing University of Posts and Telecommunications, Yangzhou, China, in 2016. He is currently working towards the Ph.D. degree in College of Automation and College of Artificial Intelligence at Nanjing University of Posts and Telecommunications, Nanjing, China. His research interests include systems modeling and vulnerability analysis in cyber-physical power systems.

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