Distributed Event-Triggered Consensus Control of Multi-Agent Systems under Denial-of-Service Jamming Attacks

This work focuses on the consensus problem of multi-agent systems (MASs) under event-triggered control (ETC) subject to denial-of-service (DoS) jamming attacks. To reduce the cost of communication networks, a novel event-triggering mechanism (ETM) is applied to the sleeping interval to determine whether the sampled signal should be transmitted or not. Unlike periodic DoS attacks model, the DoS attacks occurrence are irregular , where attack attributes such as attack frequency and attack duration are taken into account. Moreover, compared with the ﬁxed topological graph, the communication topologies may change due to DoS jamming attacks in this work. In view of this, based on the piecewise Lyapunov functional, suﬃcien-t conditions are derived to guarantee that consensus problem of the MASs can be solved. Finally, the eﬀec-tiveness and correctness of the theoretical results are veriﬁed by a numerical example.

Generally speaking, consensus control aims to design a valid distributed control protocol that uses the information of adjacent agents in the system to make the state of all agents reach an agreement. Until now, various distributed control schemes have been developed for leader-following consensus problems [7,8] and leaderless consensus problems [9][10][11][12].
However, when discussing the consensus problems of MASs, some existing results are based on ideal conditions of infinite network resources [9][10][11][12]. To minimize the waste of communication and computing resources, a time-triggering mechanism was first proposed for MASs in [13], which transmits measurements after fixed time intervals. Unfortunately, this strategy may still cause unnecessary communication and consumption of computing resources. To further address this issue, various ETC strategies were proposed [14][15][16][17][18][19], where the data will be broadcasted depends whether the pre-designed event-triggering condition is met or not. For example, the authors in [14] proposed a centralized output feedback ETC to achieve consensus of linear MASs. By applying the centralized ETC protocol, leader-following consensus issue of MASs with different structure of topologies was investigated in [15]. Wang et al. [16] introduced a centralized ETC scheme with internal dynamic variables to investigate the consensus issue of MASs. It is noteworthy that centralized ETC approaches require information from all agents of MASs, which may cause unnecessary waste of resources. To improve the limited network resources, distributed ETC strategies were proposed for MASs in [17][18][19][20]. A new iterative event-triggered analysis method was proposed in [17] to avoid continuous sampling transmission between adjacent states in MASs. In [18], the authors proposed a new event-triggered distributed predictive control method, which not only realizes the asynchronous exchange of information, but also achieves the balance between efficient resource utilization and control performance. To solve the issue of robust cooperative output regulation for uncertain linear MASs with and without extra disturbances, two adaptive ETC protocols based on internal model theories were proposed in [19], where the frequency of information exchange between agents is greatly reduced and Zeno behavior is excluded.
Furthermore, with the development of spatially distributed technology, MASs are usually considered to be a class of networked control systems that are brittle to cyber-attacks [21]. The communication channel between agents is easily interrupted or broken due to cyber-attacks, which will lead to the unavailable of information exchange between agents and system instability [22]. Therefore, it is essential to discuss the impacts of cyber-attacks in the investigation of various MASs [23][24][25][26]. In general, cyber-attacks can be categorized into deception attacks [27][28][29], replay attacks [30][31][32] and DoS jamming attacks [33][34][35]. Among them, DoS jamming attack is the most important and difficult to address. The authors considered asynchronous DoS jamming attacks in [36], where MASs can achieve consensus by restricting the frequency and duration of valid DoS jamming attacks. In [37], the authors established a hybrid dynamic model for the formation control of nonlinear MASs under DoS jamming attacks, which gives the calculation method of transmission interval with DoS attacks. Based on a novel security controller, the frequency and duration of DoS jamming attacks were obtained in [7] to guarantee that the tracking error system converges to zero for MASs under the DoS jamming attacks. The authors in [8] developed a distributed security consensus control method for leader-following MASs with DoS jamming attacks, which successfully overcome the issue of inaccurate control input calculation within the attacking periods. By taking the DoS jamming attacks into account, leader-following robust H ∞ consensus of heterogeneous MASs was presented in [33]. However, how to design a distributed eventtriggered controller to achieve consensus of MASs in the presence of DoS jamming attacks, which still an open challenge issue and motivates this work.
In this article, we investigate the consensus issue of MASs under DoS jamming attacks by a distributed event-triggered controller. The main contributions of this paper are summarized as follows.
1) Compared with [22,38,39] where the periodic DoS jamming attacks are considered, the aperiodic DoS jamming attacks are proposed by a time-sequence way in this work, which are more general in practice.
2) Different from the centralized ETC approaches in [14][15][16], a distributed discrete ETC method is pro-posed in this work to avoid continuous information exchange among adjacent agents, which not only improves the network communication but also achieves consensus control for MASs under DoS jamming attacks.
3) Different from [21,40] where the DoS jamming attacks active/sleeping periods need to known, this work does not need such restrictions as the attack frequency and duration of DoS jamming attacks are analyzed. Notations: Kronecker product λ max (X) the maximum eigenvalue of X λ min (X) the minimum eigenvalue of X N the set of non-negative integers P > 0 real symmetric matrix P is positive definite

Graph theory
A directed graph is represented by G= (P, B, W) where P = {1, 2, . . . , N } denotes a set of nodes, B ⊆ {(i, j), i, j ∈ P} denotes a set of edges, and W= [w ij ] N ×N denotes a weighted adjacency matrix with non-negative element w ij , i, j = 1, 2, . . . , N . The ordered pair of nodes (p i , p j ) represents an edge b ij in the graph G , and b ij ∈ B if and only if w ij > 0. And an element {b ij = (p i , p j )} ∈ B denotes that node i can obtain information from node j.
If there exists one node that can reach any other node through a directed path, we can say G has a directed spanning tree.

System Model
Consider the MASs consisting of N agents, a model of the ith agent shown in Fig. 1 can be represented as: where x i (t) ∈ R n is the state variable, u i (t) ∈ R m is the control input of the ith agent. A, B, and C are real matrices with appropriate dimensions. f (x i (t)) is a nonlinear vector-valued function satisfying the following assumption.

Assumption 1:
The nonlinear function f (x i (t)) satisfies the following condition: where H is a known constant matrix representing the upper bound of the nonlinearity.

Denial-of-Service Attack Model
As a result of the openness of networks, the communication channels are easily destroyed by malicious attacks, which is one of the main factors threatening the security of the system. Then, the sampled data transmitted over the communication networks may be lost and the whole network may collapse in serious cases. In addition, the cyber-attacks may also destroy the communication topology of MASs.
This paper considers the case that the communication networks are disrupted by DoS jamming attacks when transmitting the measurement signals. It should be noted that DoS jamming attacks require a certain amount of energy. Assume that malicious attackers will consume a certain amount of energy when sending DoS jamming signals. Therefore, the attackers need to enter a sleep state to save energy for the next attacks. Fig. 2 shows an example of signal transmission via an eventtriggering mechanism (ETM) under the DoS jamming attacks model.
Let {l n } n∈N denote the time sequence of the DoS jamming attacks, and a DoS jamming attack is launched at time instants l n . L 1,n = [l n , l n + △ n ) denotes the interval of the (n+1)-th DoS jamming attacks, where △ n indicates the length of the (n + 1)-th DoS jamming attacks satisfying l n+1 > l n + △ n . L 2,n = [l n + △ n , l n+1 ) represent the sleeping period of the DoS jamming attacks. Fig. 2: Example of transmitted signals via an ETM under DoS jamming attacks.
Similar to [41], for given t and ι a (ι, t) and c (ι, t) denote sets of time intervals for the active and sleeping intervals of DoS jamming attacks, respectively.
The following assumptions are introduced to describe the attack duration and the attack frequency.

A Distributed Event-Triggered Consensus Protocol
In order to alleviate the communication load of the networks, inspired by [40], a distributed ETM is designed. By using this mechanism, sample data of each agent will not be transmitted through the network unless the event-triggering condition is met. Under the mechanism, the triggering instant is described by: Remark 1: From (4), we can see that only the sampled data satisfying the triggering condition will be transmitted through the network, which greatly reduces the network resources. Note that h denotes the discrete sampling period of the sensors, so-called Zeno behavior will not happen here. If σ = 0, then the proposed ETM will turn into a time-triggering mechanism.
Remark 2: Combined with the more common scenario in practice, assuming that the triggering condition is not met in L 2,n , there will be a situation that t km+1,n h does not exist. Compared with [22] and [43] where the above case is not considered, we define t km+1,n h = l n+1 when the triggering condition is invalid in this paper, which makes the article more organized.
According to the above analysis, the ETC is described by: where α > 0 denotes the coupling strength of the MASs, K is the controller gain, σ(t) is a piecewise function, In what follows, we can define τ L,k (t) ∈ [0, h) as time varying delay variable Further, the event-triggered sampled states x i (t k,n h) and x j (t k,n h) can be given as follows: Then, the controller (5) can be rewritten as Substituting (7) into (1), we havė where The aim of this article is to devise an event-triggering controller to converge the states of all agents to a consistent level. In view of this, we need the following lemma. Lemma 1: [44] For any matrices F ∈ R n×n and Λ ∈ R n×n that satisfy   , and * is utilized to depict the entries implied by symmetry.

Main results
The following section mainly consists of two theorems, before given Theorem 1, we first present the Lemma 2 . Then, sufficient conditions are derived to guarantee the exponential stability of the closed-loop system (8) under DoS jamming attacks in Theorem 1. In Theorem 2, we solve the problem of system parameters.
Consider following piecewise Lyapunov functional: Lemma 1 For given scalars h > 0, µ 1 > 0, µ 2 > 0, σ > 0, and matrices K, H, the functions defined in (9) satisfy if there exist matrices P f > 0, R f > 0, Z f > 0, and Ω > 0, U f with appropriate dimensions satisfying : where the elements of ω f are described in detail in Appendix A.
Proof See Appendix B.
Combining (19) and (20), it is easy to get: Besides, according to the property of the Assumption 3, we can know the number of DoS jamming attacks in the interval [0, t) is N f (0, t) = n + 1. Therefore, one obtains Note that Substituting (23) and (24) into (22) yields Based on (16) and (17), (25) can be written as where Combining (26) and (27), we can obtain which implies that the system (8) is exponentially stable under DoS jamming attacks. This proof is completed.
In Theorem 1, we assume that the controller gain K is known. In Theorem 2, the unknown controller gain K of MASs can be derived on the basis of the foregoing research. Theorem 2. For given scalars h > 0, µ 1 > 0, µ 2 > 0, λ 1 > 0, λ 2 > 0, a sr , s = 1, 2, 3, r = 1, 2, system (8) is exponential stable, if there exist matricesR f > 0, Z f > 0, f = 1, 2 andΩ > 0,Ū f with appropriate dimensions and the following conditions hold: where the element ofω f is given in Appendix C. Moreover, the consensus protocol and ETC parameters can be calculated as follows: Proof For any positive scalars a sr , due to (S−a −1 , then pre-multiply and post-multiply both side of ω 1 < 0 by △ 1 and both side of (12) by △ 2 .
Similarly, employing the schur complement lemma,ω 2 < 0 is equivalent to ω 2 < 0 and (30) is equivalent to (12). Thus, we can derive that (29) and (30) can guarantee (11) and (12) hold. In addition, premultiply and post-multiply the first inequality in (13) and the second inequality in (13) byX 2 andX 1 , respectively, and applying the Schur complement lemma, (31) is equivalent to the first inequality in (13) and (32) is equivalent to the second inequality in (13), respectively. By using similar derivations, we can derive that the LMIs (33) and (34) ensure the (14) and (15) hold, respectively. According to the results of Theorem 1 and the above analysis, the system (8) can achieve exponential stable. Due to Y = KX 1 , the consensus controller gain can be computed as K = Y X −1 1 . This completes the proof.

Numerical example
In this section, we will provide a numerical simulation to illustrate the feasibility of the obtained results. Similar to [38], consider a nonlinear MAS described by ] .
According to the topology graph shown in Fig. 3, the L can be obtained as follows: The initial conditions of the agents are described as follows In this example, suppose that the DoS jamming attacks occur in [0, 2), [10,11), [29,31), [37,39) and [43,47), respectively. According to Theorem 1, when choose µ * 1 = 0.0311 and T D = 0.1, the attack frequency and attack duration of DoS jamming attacks can be obtained as  Fig. 6, which shows that MASs can achieve consensus under the DoS jamming attacks.

Conclusion
In this paper, the event-triggered consensus problem of MASs under DoS jamming attacks has been investigated. The communication topologies may change due to DoS jamming attacks. To mitigate the cost of communication networks, a novel ETC protocol is proposed to guarantee that consensus performance of the MASs can be realized under the DoS jamming attacks. The attack frequency and attack duration of DoS jamming signals are discussed. Then, by using piecewise Lyapunov functional, sufficient conditions are derived to guarantee that consensus problem of MASs can be solved. In addition, controller gains have been obtained in terms of linear matrix inequalities. Finally, a numerical example is given to verify the effectiveness of the proposed method. Future work will be directed to heterogeneous MASs with multiple cyber-attacks by ETC scheme.

Appendix A
Elements of ω f in Lemma 2.
The proof of Lemma 2.
The following cases with f = 1 and f = 2 will be given, respectively.
This proof is completed.
Appendix C Element ofω f in Theorem 2.