Group Consensus Using Event-Triggered Control for Second-Order Multi-Agent Systems under Asynchronous DoS Attack

Abstract

This paper explores the group consensus problem of second-order multi-agent systems (MAS) under asynchronous denial-of-service (DoS) attacks. Asynchronous DoS attacks involve the interruption of certain communication links, allowing the MAS to be reimagined as a switching system with a persistent dwell time (PDT). The agents in each group can be divided into three types, which are inter-act agents, intra-act agents with zero in-degree, and other agents. Then, according to the properties of the different agents, suitable agents are pinned. By leveraging the concepts of switching topology and the PDT, a suitable event-triggered control protocol is designed, along with the establishment of conditions to ensure the group consensus of the MAS. Moreover, through the construction of topology-dependent Lyapunov functions, the achievement of group consensus within the MAS under asynchronous DoS attacks is demonstrated. Subsequently, a numerical example is presented to validate the effectiveness of the proposed results.

1. Introduction

In recent years, the consensus problem has gained significant attention within the field of multi-agent systems, owing to its extensive applications across various domains, including formation control [1,2,3], flocking behavior [4,5], and collaborative decision-making processes [6,7]. The primary objective in addressing the consensus problem is to develop an efficient control protocol that ensures all agents converge on a collectively agreed-upon value through iterative local information exchanges with their neighboring agents [8,9,10,11,12]. Generally, there are two approaches for multi-agent systems to achieve consensus: leader-following and leaderless consensus. The leader-following approach allows all followers to converge to the trajectory of a designated leader, thereby utilizing the leader’s state as the target for control. This method facilitates the achievement of prescribed behaviors among all agents and is widely applied in practical scenarios.

The exchange of information between agents occurs through wireless networks, which are inherently vulnerable to cyber-attacks. This vulnerability is particularly concerning in the context of information exchange, as cyber-attacks can manifest in various forms, including false data injection attacks [13,14] and repay attacks [15,16]. Notably, denial-of-service (DoS) attacks are a prevalent type of cyber-attack that can cause significant disruptions in information transmission across networks. Consequently, considerable research efforts have been directed towards establishing consensus in networked systems under the threat of DoS attacks, with the goal of maintaining operational integrity and ensuring communication reliability in these challenging environments [17,18,19].

To achieve consensus in multi-agent systems, various strategies such as containment control [20,21], tracking control [22,23,24], and observer-based control mechanisms [25,26] have been explored and implemented. Each of these approaches plays a distinct role in advancing the overarching goal of attaining systemic consensus within multi-agent environments. However, these control methods are predominantly continuous, which poses practical challenges in real-world applications. Consequently, addressing the constraints imposed by limited resources emerges as a critical issue that necessitates resolution. A promising strategy that offers a resource-efficient alternative is event-triggered control. This methodology predicates the updating and transmission of control signals based on specific events, defined by the violation of pre-established event-triggered thresholds. By employing a selective update mechanism, the frequency of control signal adjustments is inherently reduced, thereby optimizing resource utilization and significantly extending the operational lifespan of network components. In recent years, there has been a notable increase in scholarly investigations focused on the development of event-triggered control protocols, particularly regarding their applicability to systems characterized by both linear [27,28,29,30] and nonlinear [31,32] dynamics. These studies, tailored to various multi-agent systems, utilize system state errors to design the event-triggered function, demonstrating effective resource conservation.

Pinning control is a technique employed in multi-agent systems to tackle the challenges associated with implementing controllers across all nodes, particularly in the context of resource constraints. This method involves applying local feedback to a carefully selected subset of nodes, thereby reducing the number of controlled nodes while still achieving the desired consensus dynamics. Extensive research has been dedicated to refining consensus protocols for multi-agent systems, building upon the foundational concept of pinning control. For example, ref. [33] investigated the consensus of switching multi-agent systems through the application of pinning control, while ref. [34] established several consensus conditions specifically for bipartite consensus in multi-agent systems. Furthermore, to address the complexities of cooperative–competitive networks, ref. [35] derived exponential finite-time couple-group consensus conditions for agents.

In scenarios involving group consensus, agents are organized into subgroups during cooperative tasks, enabling synchronization of states among agents within the same subgroup, while those in different groups operate independently. This notion of group consensus serves not only as a theoretical framework but also reflects natural phenomena, such as the coordinated movement observed in bird flocks and collaborative behaviors in multi-species hunting scenarios. These real-world examples underscore the practical relevance and applicability of group consensus models in understanding and designing complex multi-agent systems. Consequently, there has been a significant research focus on developing group consensus protocols. Notably, ref. [36] introduced a novel group consensus protocol tailored for heterogeneous multi-agent systems, while ref. [37] established effective group consensus conditions for systems subject to additive noise. Additionally, ref. [38] explored finite-time group consensus for second-order multi-agent systems with input saturation, further contributing to the body of knowledge in this area.

However, to the best of the authors’ knowledge, there is little research on the group consensus of second-order multi-agent systems under consecutive asynchronous DoS attacks. In this study, the group consensus issue in second-order multi-agent systems under consecutive asynchronous DoS attacks is examined. The paper outlines several significant contributions to the field, as presented below:

• In this discussion, we focus on asynchronous DoS attack. Different types of DoS attacks may occur during the attack phase, resulting in consecutive changes to the communication topology. Compared with other existing research on DoS attacks, there is little research on asynchronous DoS attacks, which is more practical.
• Some group consensus conditions will be proposed based on switched theory with persistent dwell time. Compared with [39], the system considered in this paper is a second-order multi-agent system, which is more practical. Furthermore, ref. [39] studied normal leader-following consensus; however, we study group consensus in this paper, which is more complex.
• In the DoS phase, MAS can be considered to be a switched system. An event-triggered controller will be developed to promote group consensus in an attack environment, leading to efficient resource utilization.

The rest of this paper is organized as follows. In Section 2, some preliminary theories and model descriptions are given. In Section 3, the main results about the group consensus of multi-agent systems are discussed. In Section 4, a simulation example is given to demonstrate the correctness of results. Finally, a conclusion is given in Section 5.

2. Preliminaries

2.1. Graph Theory and Notations

For the convenience of the following discussion, some notations will be used throughout this paper. $R^n$ denotes the $n-$dimensional Euclidean space. $R^{m\times n}$ denotes an $m\times n$ real matrix space. $I_n$ and $0_n$ denote an $n-$dimensional identity matrix and a zero matrix, respectively. The term $1_n$ denotes an $n-$dimensional column vector with all ones. For any symmetric matrix P, $P<0$ means P is a negative definite matrix, and $P>0$ means P is a positive definite matrix. The superscript ‘T’ means the transpose. The Kronecker product of matrices X and Y is denoted as $X\otimes Y$. In this paper, if not clearly stated, matrices are assumed to have compatible dimensions.

Throughout this paper, the networked system topology is represented by a weighted directed graph $\mathcal{G}=(\mathcal{V},\epsilon ,\mathcal{A})$ with the set of nodes $\mathcal{V}=\{v_1,v_2,\dots ,v_N\}$, the set of directed edges $\epsilon \subseteq \mathcal{V}\times \mathcal{V}$, and a weighted adjacency matrix $\mathcal{A}={\left(a_{ij}\right)}_{N\times N}$, whose entry $a_{ij}$ denotes connection and weight between agents $v_i$ and $v_j$. If node $v_j$ can receive information from node $v_i$, then $a_{ij}>0$; otherwise, $a_{ij}=0$. The in-degree and out-degree of node i are defined as $de{g}_{in}\left(i\right)=\sum _{j=1,j\ne i}^N{a}_{ji}$ and $de{g}_{out}\left(i\right)=\sum _{j=1,j\ne i}^N{a}_{ij}$, respectively, and the Laplacian matrix $L={\left(l_{ij}\right)}_{N\times N}$ is defined as $l_{ij}=-a_{ij},i\ne j,l_{ii}=\sum _{j=1,j\ne i}^N{a}_{ij}$.

In this paper, we will split N agents into r groups $(r\ge 2)$. That means $\mathcal{V}={\cup }_{k=1}^r{\mathcal{V}}_k$ and ${\mathcal{V}}_p\cap {\mathcal{V}}_q=0$ for different p and q $(p\ne q)$. Let the symbol ${\mathcal{V}}_{k_i}$ denote the group to which the ith agent belongs. Further, the number of groups is the same as the number of virtual leaders.

The network topology in this paper we consider to be directed and weakly connected. Firstly, we learn some concepts related to inter-act agents and intra-act agents. Agent i is said to be an inter-act agent if i belongs to ${\tilde{\mathcal{V}}}_{k_i}$, in which agent i can receive information from other groups. An agent that can only exchange information among agents in the group in which the agent belongs is said to be an intra-act agent. The set of intra-act agents is signed as ${\mathcal{V}}_{k_i}\setminus {\tilde{\mathcal{V}}}_{k_i}$.

2.2. Model Description

In this paper, we consider a nonlinear second-order multi-agent system with N agents:

$$\left\{\begin{array}{cc}\hfill & {\dot{x}}_i\left(t\right)=v_i\left(t\right),\hfill \\ \hfill & {\dot{v}}_i\left(t\right)=f(x_i\left(t\right),v_i\left(t\right))+u_i\left(t\right).\hfill & \hfill (i=1,2,\dots ,N)\end{array}\right.$$

(1)

where $x_i\left(t\right),v_i\left(t\right)\in R^n$ denote the position and velocity, respectively, of the ith agent, $u_i\left(t\right)\in R^n$ is the designed control input, and $f(x_i\left(t\right),v_i\left(t\right))$ is a nonlinear vector-valued continuous function describing the dynamics of agent i. The virtual leader of each group for multi-agent system (1) can be described by the following model:

$$\left\{\begin{array}{cc}\hfill & {\dot{\overline{x}}}_{k_i}\left(t\right)={\overline{v}}_{k_i}\left(t\right),\hfill \\ \hfill & {\dot{\overline{v}}}_{k_i}\left(t\right)=f({\overline{x}}_{k_i}\left(t\right),{\overline{v}}_{k_i}\left(t\right)).\hfill & \hfill (i=1,2,\dots ,N)\end{array}\right.$$

(2)

where ${\overline{x}}_i\left(t\right),{\overline{v}}_i\left(t\right)\in R^n$ are the position and velocity, respectively, of the ith virtual leader.

Definition 1. 

The second-order group consensus in multi-agent systems (1) and (2) is said to be achieved if for any initial condition, the solutions of (1) and (2) satisfy

$$\left\{\begin{array}{c}\hfill \underset{t\to \infty }{lim}\left|\right|x_i\left(t\right)-{\overline{x}}_{k_i}\left(t\right)\left|\right|=0,\\ \hfill \underset{t\to \infty }{lim}\left|\right|v_i\left(t\right)-{\overline{v}}_{k_i}\left(t\right)\left|\right|=0.\end{array}\right.$$

Assumption A1. 

For the nonlinear function $f(x_i\left(t\right),v_i\left(t\right))$, there exist positive constants $l_1$, $l_2$ such that for any $x_1,x_2,y_1,y_2\in R^n$, the following inequality holds:

$$\left|\right|f(x_1,v_1)-f(x_2,v_2)\left|\right|\le l_1\left|\right|x_1-x_2\left|\right|+l_2\left|\right|v_1-v_2\left|\right|$$

Lemma 1 

([40]). We have the following linear matrix inequality

$$\left(\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right)>0$$

where $S_{11}^T=S_{11},S_{22}^T=S_{22},S_{12}^T=S_{21}$ is equivalent to one of the following conditions:

(1) 

$S_{11}>0,S_{22}-S_{21}{S}_{11}^{-1}{S}_{12}>0$,

(2) 

$S_{22}>0,S_{11}-S_{12}{S}_{22}^{-1}{S}_{21>}>0$.

3. Main Results

Now we are in a position to present a consensus protocol based on a pinning-event-triggered control approach to ensure that agents in each group can asymptotically follow the corresponding virtual leader.

Suppose the system error in each group is

$${\tilde{x}}_i\left(t\right)=x_i\left(t\right)-{\overline{x}}_{k_i}\left(t\right),{\tilde{v}}_i\left(t\right)=v_i\left(t\right)-{\overline{v}}_{k_i}\left(t\right),i=1,2,\dots ,N$$

DoS attacks can be divided into synchronous attacks or asynchronous attacks according to the attack method. A synchronous attack means that all communication links are interrupted simultaneously. An asynchronous attack means only part of the communication links are interrupted. In this paper, an asynchronous DoS attack is considered, and the sleep phase of an asynchronous DoS attack is regarded as the $\tau$-portion, and the activation phase is regarded as the T-portion.

Consecutive asynchronous attacks will destroy parts of communication links; thus, they may lead to the system switching among different communication topologies during the attack phase. Further, at the end of the attack, the MAS can return to the original topology. It is assumed that there are $\beta$ types of asynchronous DoS attacks. Then, let the piecewise constant function $\delta \left(t\right)$:$[0,\infty ]\to \mathsf{\Theta }=\{0,1,2,\dots ,\beta \}$ denote the switching signal. When $\delta \left(t\right)=0$, there is no DoS attack. When $\delta \left(t\right)=m,m\in \{1,2,\dots ,\beta \}$, it denotes the appearance of the mth DoS attack. Inspired by [41], the following definition of an asynchronous attack is given.

Definition 2. 

A positive constant ${\tau }_D$ is called the persistent dwell time (PDT) if there is an infinite number of disjoint intervals with lengths no smaller than ${\tau }_D$ on which $\delta \left(t\right)$ is constant and consecutive intervals with this property are separated by no more than T, which is called the period of persistence.

Remark 1. 

In practice, an attacker’s energy is limited due to finite resources. The activation time of a DoS attack may have an upper bound. Moreover, there exists a limited number of DoS attacks. Thus, it is rational to introduce the length T and attack frequency f to limit the number of attacks during the attack time.

In this paper, the time instant of consecutive DoS attacks is as shown in Figure 1. As the figure shows, $[t_{s_l},t_{s_{l+1}})$ stands for lth stage, and the DoS attack starts at $t_{s_l+1}$. During the $\tau$-portion $[t_{s_l},t_{s_l+1})$, there is no DoS attack, and the communication topology is represented by $G^0$. During the T-portion $[t_{s_l+1},t_{s_{l+1}})$, the MAS is under a consecutive asynchronous DoS attack, where $t_{s_l+1},t_{s_l+2},\dots ,t_{s_{l+1}}$ mean the DoS instants. The communication topology of this period is represented by $G^m$. Further, the length of the T-portion is represented by $T_l\le T$, and the frequency of the T-portion is represented by $f_l=N(t_{s_l+1},t_{s_{l+1}})/T_l\le f$, where $N(t_{s_l+1},t_{s_{l+1}})$ stands for the switching number during $[t_{s_l+1},t_{s_{l+1}})$. When the DoS attack ends, the communication topology will be restored to the original topology from $t_{s_{l+1}}$.

Let time sequence $\left\{t_k^i\right\}(k=1,2,\dots ;i=1,2,\dots ,N)$ denote the kth event trigger of the ith agent, and inspired by the pinning-event-triggered scheme, the control $u_i\left(t\right)$ for a multi-agent system can be designated as

$$u_i\left(t\right)=\left\{\begin{array}{cc}\hfill & c{\displaystyle \sum _{j=1}^N}{a}_{ij}^{\delta \left(t\right)}[x_j\left(t_k^i\right)-x_i\left(t_k^i\right)+v_j\left(t_k^i\right)-v_i\left(t_k^i\right)]-c{d}_i^{\delta \left(t\right)}[x_i\left(t_k^i\right)-{\overline{x}}_{k_i}\left(t_k^i\right)+v_i\left(t_k^i\right)-{\overline{v}}_{k_i}\left(t_k^i\right)]\hfill \\ & +c{\displaystyle \sum _{j=1}^N}{l}_{ij}^{\delta \left(t\right)}({\overline{x}}_{k_j}\left(t_k^i\right)+{\overline{v}}_{k_j}\left(t_k^i\right)),i\in {\tilde{\mathcal{V}}}_{k_i}\hfill \\ \hfill & c{\displaystyle \sum _{j=1}^N}{a}_{ij}^{\delta \left(t\right)}[x_j\left(t_k^i\right)-x_i\left(t_k^i\right)+v_j\left(t_k^i\right)-v_i\left(t_k^i\right)]-c{d}_i^{\delta \left(t\right)}[x_i\left(t_k^i\right)-{\overline{x}}_{k_i}\left(t_k^i\right)+v_i\left(t_k^i\right)-{\overline{v}}_{k_i}\left(t_k^i\right)],\hfill \\ & i\in {\mathcal{V}}_{k_i}\setminus {\tilde{\mathcal{V}}}_{k_i}and\mathrm{deg}{\left(\mathrm{i}\right)}_{\mathrm{in}}=0\hfill \\ & c{\displaystyle \sum _{j=1}^N}{a}_{ij}^{\delta \left(t\right)}[x_j\left(t_k^i\right)-x_i\left(t_k^i\right)+v_j\left(t_k^i\right)-v_i\left(t_k^i\right)],i\in {\mathcal{V}}_{k_i}\setminus {\tilde{\mathcal{V}}}_{k_i}and\mathrm{deg}{\left(\mathrm{i}\right)}_{\mathrm{in}}\ne 0\hfill \end{array}\right.$$

(3)

where $c>0$ denotes the coupling strength, $a_{ij}^{\delta \left(t\right)}$ denotes the communication link of agent i when the communication topology changes into $G^{\delta \left(t\right)}$, and $d_i^{\delta \left(t\right)}>0$ denotes the pinning control gain of agent i when the communication topology changes into $G^{\delta \left(t\right)}$.

Remark 2. 

It is important to point out that when $\delta \left(t\right)=0$, there is no DoS attack. If agent i is an intra-act agent, one has ${\sum }_{j=1}^N{l}_{ij}^0({\overline{x}}_{k_j}\left(t_k^i\right)+{\overline{v}}_{k_j}\left(t_k^i\right))=0$. When the DoS attack occurs, we do not divide the group again; thus, $l_{ij}^{\delta \left(t\right)}=l^0$ or $l_{ij}^{\delta \left(t\right)}=0$, so we can still obtain ${\sum }_{j=1}^N{l}_{ij}^{\delta \left(t\right)}({\overline{x}}_{k_j}\left(t_k^i\right)+{\overline{v}}_{k_j}\left(t_k^i\right))=0,$ $i\in {\mathcal{V}}_{k_i}\setminus {\tilde{\mathcal{V}}}_{k_i}$. Then $u_i\left(t\right)$ can be simplified into $u_i\left(t\right)=-c({\sum }_{j=1}^N{l}_{ij}^{\delta \left(t\right)}({\tilde{x}}_j\left(t_k^i\right)+{\tilde{v}}_j\left(t_k^i\right))+d_i^{\delta \left(t\right)}({\tilde{x}}_i\left(t_k^i\right)+{\tilde{v}}_i\left(t_k^i\right)))$.

Suppose ${\mathsf{\Phi }}_i^{\delta \left(t\right)}\left(t\right)={\sum }_{j=1}^N{l}_{ij}^{\delta \left(t\right)}({\tilde{x}}_j\left(t\right)+{\tilde{v}}_j\left(t\right))+d_i^{\delta \left(t\right)}({\tilde{x}}_i\left(t\right)+{\tilde{v}}_i\left(t\right))$; define $e_i^{\delta \left(t\right)}\left(t\right)={\mathsf{\Phi }}_i^{\delta \left(t\right)}\left(t_k^i\right)-{\mathsf{\Phi }}_i^{\delta \left(t\right)}\left(t\right)$; then choose the event trigger function as

$$g_i^{\delta \left(t\right)}\left(t\right)=\left|\right|e_i^{\delta \left(t\right)}{\left(t\right)\left|\right|}^2-\alpha \left|\right|{\mathsf{\Phi }}_i^{\delta \left(t\right)}\left(t\right){\left|\right|}^2$$

(4)

where $\alpha >0$ is a positive constant. If $g_i^{\delta \left(t\right)}\left(t\right)\ge 0$, then $x_i\left(t_k^i\right)$ will update to $x_i\left(t_{k+1}^i\right)$, which can be expressed as follows:

$$t_{k+1}^i=inf\{t>t_k^i|g_i^{\delta \left(t\right)}\left(t\right)\ge 0\}$$

According to (1) and (2), by the use of the Kronecker product, we can obtain the compact form

$$\left\{\begin{array}{cc}\hfill \dot{\tilde{x}}\left(t\right)& =\tilde{v}\left(t\right),\hfill \\ \hfill \dot{\tilde{v}}\left(t\right)& =f(x\left(t\right),v\left(t\right))-f(\overline{x}\left(t\right),\overline{v}\left(t\right))-c[((L^{\delta \left(t\right)}+D^{\delta \left(t\right)})\otimes I_n)(\tilde{x}\left(t\right)+\tilde{v}\left(t\right))+e^{\delta \left(t\right)}\left(t\right)].\hfill \end{array}\right.$$

where

$$\begin{array}{cc}\hfill & f(x\left(t\right),v\left(t\right))={[f^T(x_1\left(t\right),v_1\left(t\right)),\dots ,f^T(x_N\left(t\right),v_N\left(t\right))]}^T,\hfill \\ \hfill & f(\overline{x}\left(t\right),\overline{v}\left(t\right))={[f^T({\overline{x}}_{k_1}\left(t\right),{\overline{v}}_{k_1}\left(t\right)),\dots ,f^T({\overline{x}}_{k_N}\left(t\right),{\overline{v}}_{k_N}\left(t\right))]}^T,\hfill \\ \hfill & \tilde{x}\left(t\right)={[{\tilde{x}}_1^T\left(t\right),\dots ,{\tilde{x}}_N^T\left(t\right)]}^T,\tilde{v}\left(t\right)={[{\tilde{v}}_1^T\left(t\right),\dots ,{\tilde{v}}_N^T\left(t\right)]}^T.\hfill \end{array}$$

Based on the above analysis, we have the following theorem:

Theorem 1. 

For given non-zero scalars ${\mu }_1>1,{\mu }_2>1,0<{\mu }_3<1$, prescribed period of persistence $\mathit{T}$, and frequency $\mathit{f}$, if there exist matrices $Q_m>0,Q_n>0,m,n\in \{1,2,3,\dots ,\beta \},m\ne n$ such that the following conditions hold:

(1). 

$(\delta +{\displaystyle \frac{c}{2}})I_N-c{\overline{L}}^0-cD+2c\alpha {\left(H^0\right)}^T{H}^0<0$;

(2). 

${\mathsf{\Gamma }}_m\le {\mu }_1\mathsf{\Omega },Q_m\le {\mu }_2{Q}_n,\mathsf{\Omega }\le {\mu }_3{\mathsf{\Gamma }}_m$;

(3). 

PDT satisfies ${\tau }_D>{\displaystyle \frac{(Tf+1)ln\mu +{\rho }_{2}T}{{\rho }_1}}$.

where ${\overline{L}}^0={\displaystyle \frac{L^0+{\left(L^0\right)}^T}{2}}$, $H^{\delta \left(t\right)}=L^{\delta \left(t\right)}+D^{\delta \left(t\right)}$, $\mu =max\{{\mu }_1,{\mu }_2,{\mu }_3\}$, $\delta =max\{{\displaystyle \frac{3l_1+l_2}{2}},{\displaystyle \frac{3l_2+l_1}{2}}+1\}$, and $\mathsf{\Omega },{\mathsf{\Gamma }}_m,{\rho }_1,{\rho }_2$ will be given in the proof,

then MAS (1) and (2) can achieve group consensus.

Proof. 

From condition (1), we can see that

$$\delta I_N-c{\overline{L}}^0-c{D}^0<0$$

since $\delta >1$; thus,

$$c{\overline{L}}^0+c{D}^0-{\displaystyle \frac{I_N}{2}}>0$$

(5)

 □

According to Lemma 1, we have

$$\left[\begin{array}{cc}c(L^0+{\left(L^0\right)}^T)+2c{D}^0& I_N\\ I_N& I_N\end{array}\right]>0$$

(6)

When there is no DoS attack, namely, $t\in [t_{s_l},t_{s_l+1})$, and we assume that there are n event-triggered time instants, denoted by $\{t_1^i,t_2^i,\cdots ,t_n^i\}$, when $t\in [t_k^i,t_{k+1}^i)$, we choose the positive definite Lyapunov function as follows:

$$V_0\left(t\right)={\displaystyle \frac{1}{2}}{\xi }^T\left(t\right)(\left[\begin{array}{cc}c(L^0+{\left(L^0\right)}^T)+2c{D}^0& I_N\\ I_N& I_N\end{array}\right]\otimes I_n)\xi \left(t\right)$$

where $\xi \left(t\right)={[{\tilde{x}}^T\left(t\right),{\tilde{v}}^T\left(t\right)]}^T$.

Further, based on Assumption 1 and Young’s inequality, we can obtain

$$\begin{array}{cc}\hfill \left|\right|\tilde{x}\left(t\right)\left|\right|·\left|\right|f(x\left(t\right),v\left(t\right))-f(\overline{x}\left(t\right),\overline{v}\left(t\right))\left|\right|& \le l_1\left|\right|\tilde{x}\left(t\right)\left|\right|·\left|\right|\tilde{x}\left(t\right)\left|\right|+l_2\left|\right|\tilde{x}\left(t\right)\left|\right|·\left|\right|\tilde{v}\left(t\right)\left|\right|\hfill \\ \hfill & \le l_1{\tilde{x}}^T\left(t\right)\tilde{x}\left(t\right)+{\displaystyle \frac{l_2}{2}}({\tilde{x}}^T\left(t\right)\tilde{x}\left(t\right)+{\tilde{v}}^T\left(t\right)\tilde{v}\left(t\right))\hfill \\ \hfill & =(l_1+{\displaystyle \frac{l_2}{2}}){\tilde{x}}^T\left(t\right)\tilde{x}\left(t\right)+{\displaystyle \frac{l_2}{2}}{\tilde{v}}^T\left(t\right)\tilde{v}\left(t\right).\hfill \end{array}$$

At the same time, one can obtain that

${\tilde{x}}^T\left(t\right)(f(x\left(t\right),v\left(t\right))-f(\overline{x}\left(t\right),\overline{v}\left(t\right)))\le \left|\right|\tilde{x}\left(t\right)\left|\right|·\left|\right|f(x\left(t\right),v\left(t\right))-f(\overline{x}\left(t\right),\overline{v}\left(t\right))\left|\right|$; thus, we have ${\tilde{x}}^T\left(t\right)(f(x\left(t\right),v\left(t\right))-f(\overline{x}\left(t\right),\overline{v}\left(t\right)))\le (l_1+{\displaystyle \frac{l_2}{2}}){\tilde{x}}^T\left(t\right)\tilde{x}\left(t\right)+{\displaystyle \frac{l_2}{2}}{\tilde{v}}^T\left(t\right)\tilde{v}\left(t\right)$. Similarly, we can obtain ${\tilde{v}}^T\left(t\right)(f(x\left(t\right),v\left(t\right))-f(\overline{x}\left(t\right),\overline{v}\left(t\right)))\le {\displaystyle \frac{l_1}{2}}{\tilde{x}}^T\left(t\right)\tilde{x}\left(t\right)+({\displaystyle \frac{l_1}{2}}+l_2){\tilde{v}}^T\left(t\right)\tilde{v}\left(t\right).$

Calculate the derivative of Lyapunov function as

$$\begin{array}{cc}\hfill {\dot{V}}_0(t)=& \left[\begin{array}{cc}{\tilde{x}}^T(t)& {\tilde{v}}^T(t)\end{array}\right](\left[\begin{array}{cc}c(L^0+{(L^0)}^T)+2c{D}^0& I_N\\ I_N& I_N\end{array}\right]\otimes I_n)\hfill \\ \hfill & \left[\begin{array}{c}\tilde{v}(t)\\ f(x(t),v(t))-f(\overline{x}(t),\overline{v}(t))-(c(L^0+D^0)\otimes I_n)(\tilde{x}(t)+\tilde{v}(t))-c{e}^0(t)\end{array}\right]\hfill \\ \hfill =& {\tilde{x}}^T(t)((c(L^0+{(L^0)}^T)+2c{D}^0)\otimes I_n)\tilde{v}(t)+{\tilde{x}}^T[f(x(t),v(t))-f(\overline{x}(t),\overline{v}(t))]\hfill \\ \hfill & -{\tilde{x}}^T(t)(c(L^0+D^0)\otimes I_n)(\tilde{x}(t)+\tilde{v}(t))-c{\tilde{x}}^T(t)e^0(t)+{\tilde{v}}^T(t)\tilde{v}(t)\hfill \\ \hfill & +{\tilde{v}}^T(t)[f(x(t),v(t))-f(\overline{x}(t),\overline{v}(t))]-{\tilde{v}}^T(t)(c(L^0+D^0)\otimes I_n)(\tilde{x}(t)+\tilde{v}(t))-c{\tilde{v}}^T(t)e^0(t)\hfill \\ \hfill \le & {\tilde{x}}^T(t)((c(L^0+{(L^0)}^T)+2c{D}^0)\otimes I_n)\tilde{v}(t)+(l_1+{\displaystyle \frac{l_2}{2}}){\tilde{x}}^T(t)\tilde{x}(t)+{\displaystyle \frac{l_2}{2}}{\tilde{v}}^T(t)\tilde{v}(t)\hfill \\ \hfill & -{\tilde{x}}^T(t)(c(L^0+D^0)\otimes I_n)\tilde{x}(t)-{\tilde{x}}^T(t)(c(L^0+D^0)\otimes I_n)\tilde{v}(t)-c{\tilde{x}}^T(t)e^0(t)+{\tilde{v}}^T(t)\tilde{v}(t)\hfill \\ \hfill & +{\displaystyle \frac{l_1}{2}}{\tilde{x}}^T(t)\tilde{x}(t)+(l_2+{\displaystyle \frac{l_1}{2}}){\tilde{v}}^T(t)\tilde{v}(t)-{\tilde{v}}^T(t)(c(L^0+D^0)\otimes I_n)\tilde{x}(t)\hfill \\ \hfill & -{\tilde{v}}^T(t)(c(L^0+D^0)\otimes I_n)\tilde{v}(t)-c{\tilde{v}}^T(t)e^0(t)\hfill \\ \hfill =& {\tilde{x}}^T(t)[(({\displaystyle \frac{3l_1+l_2}{2}})I_N-c{\overline{L}}^0-c{D}^0)\otimes I_n]\tilde{x}(t)-c{\tilde{x}}^T{e}^0(t)-c{\tilde{v}}^T{e}^0(t)\hfill \\ \hfill & +{\tilde{v}}^T(t)[(({\displaystyle \frac{3l_2+l_1}{2}}+1)I_N-c{\overline{L}}^0-c{D}^0)\otimes I_n]\tilde{v}(t)\hfill \end{array}$$

According to the event-triggered condition, we can obtain

$$\begin{array}{cc}\hfill -c{\tilde{x}}^T(t)e^0(t)& =-c{\displaystyle \sum _{i=1}^N}{\tilde{x}}_i^T(t)e_i^0(t)\le c{\displaystyle \sum _{i=1}^N}\left|\right|{\tilde{x}}_i(t)\left|\right|·\left|\right|e_i^0(t)\left|\right|\le {\displaystyle \frac{c}{2}}({\displaystyle \sum _{i=1}^N}{\tilde{x}}_i^T(t){\tilde{x}}_i(t)+{\displaystyle \sum _{i=1}^N}{(e_i^0(t))}^T{e}_i^0(t))\hfill \\ \hfill & ={\displaystyle \frac{c}{2}}({\tilde{x}}^T(t)\tilde{x}(t)+{(e^0(t))}^T{e}^0(t))\le {\displaystyle \frac{c}{2}}({\tilde{x}}^T(t)\tilde{x}(t)+\alpha {({\mathsf{\Phi }}^0)}^T(t){\mathsf{\Phi }}^0(t))\hfill \\ \hfill & ={\displaystyle \frac{c}{2}}({\tilde{x}}^T(t)\tilde{x}(t)+\alpha ({\tilde{x}}^T(t)+{\tilde{v}}^T(t))({(H^0)}^T{H}^0\otimes I_n)(\tilde{x}(t)+\tilde{v}(t)))\hfill \\ \hfill & \le {\displaystyle \frac{c}{2}}{\tilde{x}}^T(t)\tilde{x}(t)+c\alpha {\tilde{x}}^T(t)({(H^0)}^T{H}^0\otimes I_n)\tilde{x}(t)+c\alpha {\tilde{v}}^T(t)({(H^0)}^T{H}^0\otimes I_n)\tilde{v}(t)\hfill \end{array}$$

where ${\mathsf{\Phi }}^0\left(t\right)={[{\left({\mathsf{\Phi }}_1^0\right)}^T\left(t\right),{\left({\mathsf{\Phi }}_2^0\right)}^T\left(t\right),\dots ,{\left({\mathsf{\Phi }}_N^0\right)}^T\left(t\right)]}^T$. Similarly, we can obtain

$$-c{\tilde{v}}^T\left(t\right)e^0\left(t\right)\le {\displaystyle \frac{c}{2}}{\tilde{v}}^T\left(t\right)\tilde{v}\left(t\right)+c\alpha {\tilde{x}}^T\left(t\right)({\left(H^0\right)}^T{H}^0\otimes I_n)\tilde{x}\left(t\right)+c\alpha {\tilde{v}}^T\left(t\right)({\left(H^0\right)}^T{H}^0\otimes I_n)\tilde{v}\left(t\right)$$

Thus, we have

$$\begin{array}{cc}\hfill \dot{V_0}(t)\le & {\tilde{x}}^T(t)[(({\displaystyle \frac{3l_1+l_2}{2}})I_N-c{\overline{L}}^0-c{D}^0)\otimes I_n]\tilde{x}(t)\hfill \\ \hfill & +{\tilde{v}}^T(t)[(({\displaystyle \frac{3l_2+l_1}{2}}+1)I_N-c{\overline{L}}^0-c{D}^0)\otimes I_n]\tilde{v}(t)\hfill \\ \hfill & +{\displaystyle \frac{c}{2}}{\tilde{x}}^T(t)\tilde{x}(t)+{\displaystyle \frac{c}{2}}{\tilde{v}}^T(t)\tilde{v}(t)+2c\alpha {\tilde{x}}^T(t)({(H^0)}^T{H}^0\otimes I_n)\tilde{x}(t)+2c\alpha {\tilde{v}}^T(t)({(H^0)}^T{H}^0\otimes I_n)\tilde{v}(t)\hfill \\ \hfill =& {\tilde{x}}^T(t)[(({\displaystyle \frac{3l_1+l_2}{2}}+{\displaystyle \frac{c}{2}})I_N-c{\overline{L}}^0-cD+2c\alpha {(H^0)}^T{H}^0)\otimes I_n]\tilde{x}(t)\hfill \\ \hfill & +{\tilde{v}}^T(t)[(({\displaystyle \frac{3l_2+l_1}{2}}+1+{\displaystyle \frac{c}{2}})I_N-c{\overline{L}}^0-c{D}^0+2c\alpha {(H^0)}^T{H}^0)\otimes I_n]\tilde{v}(t)\hfill \end{array}$$

Because $\delta =max\{{\displaystyle \frac{3l_1+l_2}{2}},{\displaystyle \frac{3l_2+l_1}{2}}+1\}$, we obtain

$$\begin{array}{cc}\hfill \dot{V_0}(t)\le & {\tilde{x}}^T(t)[((\delta +{\displaystyle \frac{c}{2}})I_N-c{\overline{L}}^0-c{D}^0+2c\alpha {(H^0)}^T{H}^0)\otimes I_n]\tilde{x}(t)\hfill \\ \hfill +& {\tilde{v}}^T(t)[((\delta +{\displaystyle \frac{c}{2}})I_N-c{\overline{L}}^0-c{D}^0+2c\alpha {(H^0)}^T{H}^0)\otimes I_n]\tilde{v}(t)\hfill \end{array}$$

Let

$$\mathsf{\Psi }=\left[\begin{array}{cc}c{\overline{L}}^0+c{D}^0-(\delta +{\displaystyle \frac{c}{2}})I_N-2c\alpha {\left(H^0\right)}^T{H}^0& 0\\ 0& c{\overline{L}}^0+c{D}^0-(\delta +{\displaystyle \frac{c}{2}})I_N-2c\alpha {\left(H^0\right)}^T{H}^0\end{array}\right]$$

According to condition (1), we can have $\mathsf{\Psi }>0$, so ${\lambda }_{min}\left(\mathsf{\Psi }\right)>0$; therefore, we have

$$\dot{V_0}\left(t\right)\le -{\lambda }_{min}\left(\mathsf{\Psi }\right){\xi }^T\left(t\right)\xi \left(t\right)$$

For the convenience of the following discussion, we let

$$\mathsf{\Omega }=\left[\begin{array}{cc}c(L^0+{\left(L^0\right)}^T)+2c{D}^0& I_N\\ I_N& I_N\end{array}\right]$$

then we obtain $V_0\left(t\right)={\displaystyle \frac{1}{2}}{\xi }^T\left(t\right)\mathsf{\Omega }\xi \left(t\right)$, so ${\displaystyle \frac{{\lambda }_{min}\left(\mathsf{\Omega }\right)}{2}}{\xi }^T\left(t\right)\xi \left(t\right)\le V_0\left(t\right)\le {\displaystyle \frac{{\lambda }_{max}\left(\mathsf{\Omega }\right)}{2}}{\xi }^T\left(t\right)\xi \left(t\right)$.

Thus, we have

$$\dot{V_0}\left(t\right)\le -{\displaystyle \frac{2{\lambda }_{min}\left(\mathsf{\Psi }\right)}{{\lambda }_{max}\left(\mathsf{\Omega }\right)}}{V}_0\left(t\right)$$

Let ${\rho }_1={\displaystyle \frac{2{\lambda }_{min}\left(\mathsf{\Psi }\right)}{{\lambda }_{max}\left(\mathsf{\Omega }\right)}}$, then ${\rho }_1>0$, and ${\dot{V}}_0\left(t\right)\le -{\rho }_1{V}_0\left(t\right)$,$t\in [t_k^i,t_{k+1}^i)$. Then we can obtain

$$\begin{array}{cc}\hfill & V_0\left(t\right)\le e^{-{\rho }_1(t-t_n^i)}{V}_0\left(t_n^i\right),t\in [t_n^i,t_{s_l+1})\hfill \\ \hfill & V_0\left(t_n^i\right)\le e^{-{\rho }_1(t_n^i-t_{n-1}^i)}{V}_0\left(t_{n-1}^i\right)\hfill \end{array}$$

Thus, we can obtain

$$\begin{array}{cc}\hfill V_0\left(t\right)& \le e^{-{\rho }_1(t-t_n^i)}{V}_0\left(t_n^i\right)\hfill \\ \hfill & \le e^{-{\rho }_1(t-t_{n-1}^i)}{V}_0\left(t_{n-1}^i\right)\hfill \\ \hfill & \le \cdots \hfill \\ \hfill & \le e^{-{\rho }_1(t-t_{s_l})}{V}_0\left(t_{s_l}\right)\hfill \end{array}$$

When the system is under a consecutive asynchronous attack, namely, $t\in [t_{s_l+1},t_{s_{l+1}})$, the communication topology is switching; thus, we choose the Lyapunov function as follows:

$$\begin{array}{c}\hfill V_{\delta \left(t\right)}\left(t\right)={\displaystyle \frac{1}{2}}{\xi }^T\left(t\right)(\left[\begin{array}{cc}{Q}_{\delta \left(t\right)}& 0\\ 0& Q_{\delta \left(t\right)}\end{array}\right]\otimes I_n){\xi }^T\left(t\right)\end{array}$$

Similarly, we assume that there are p event-triggered time instants, denoted by

$\{t_1^i,t_2^i,\cdots ,t_p^i\}$, when $\delta \left(t\right)=m$. For convenience, let ${\mathsf{\Gamma }}_m=\left[\begin{array}{cc}{Q}_m& 0\\ 0& Q_m\end{array}\right]$. When $t\in [t_k^i,t_{k+1}^i)$, calculate the derivative as

$$\begin{array}{cc}\hfill {\dot{V}}_m=& \left[\begin{array}{cc}{\tilde{x}}^T\left(t\right)& {\tilde{v}}^T\left(t\right)\end{array}\right](\left[\begin{array}{cc}{Q}_m& 0\\ 0& Q_m\end{array}\right]\otimes I_n)\hfill \\ \hfill & \left[\begin{array}{c}\tilde{v}\left(t\right)\\ f(x\left(t\right),v\left(t\right))-f(\overline{x}\left(t\right),\overline{v}\left(t\right))-c[((L^m+D^m)\otimes I_n)(\tilde{x}\left(t\right)+\tilde{v}\left(t\right))+e^m\left(t\right)]\end{array}\right]\hfill \\ \hfill =& {\tilde{x}}^T\left(t\right)(Q_m\otimes I_n)\tilde{v}\left(t\right)+{\tilde{v}}^T\left(t\right)(Q_m\otimes I_n)(f(x\left(t\right),v\left(t\right))-f(\overline{x}\left(t\right),\overline{v}\left(t\right)))\hfill \\ \hfill & -c\tilde{v}\left(t\right)(Q_m\otimes I_n)e^m\left(t\right)-c\tilde{v}\left(t\right)(Q_m\otimes I_n)((L^m+D^m)\otimes I_n)(\tilde{x}\left(t\right)+\tilde{v}\left(t\right))\hfill \end{array}$$

Similar to the analysis when there is no DoS attack, we can obtain

$$\begin{array}{cc}\hfill {\dot{V}}_m\le & {\displaystyle \frac{1}{2}}({\tilde{x}}^T\left(t\right)(Q_m\otimes I_n)\tilde{x}\left(t\right)+{\tilde{v}}^T\left(t\right)(Q_m\otimes I_n)\tilde{v}\left(t\right))+{\displaystyle \frac{l_1}{2}}{\tilde{x}}^T\left(t\right)(Q_m\otimes I_n)\tilde{x}\left(t\right)\hfill \\ \hfill & +({\displaystyle \frac{l_1}{2}}+l_2){\tilde{v}}^T\left(t\right)(Q_m\otimes I_n)\tilde{v}\left(t\right)+c\alpha {\lambda }_{max}\left(Q_m\right){\tilde{x}}^T\left(t\right){\left(H^{\delta \left(t\right)}\right)}^T{H}^m\tilde{x}\left(t\right)\hfill \\ \hfill & +c\alpha {\lambda }_{max}\left(Q_m\right){\tilde{v}}^T\left(t\right){\left(H^m\right)}^T{H}^m\tilde{v}\left(t\right)+{\displaystyle \frac{c}{2}}{\tilde{v}}^T\left(t\right)(Q_m\otimes I_n)\tilde{v}\left(t\right)\hfill \\ \hfill & +{\displaystyle \frac{c}{2}}{\tilde{x}}^T\left(t\right)(Q_m{H}^m\otimes I_n)\tilde{x}\left(t\right)-{\displaystyle \frac{c}{2}}{\tilde{v}}^T\left(t\right)(Q_m{H}^m\otimes I_n)\tilde{v}\left(t\right)\hfill \end{array}$$

The DoS attack considered here is effective. Therefore, $Q_m{H}^m$ is semi-positive. So we can obtain

$$\begin{array}{cc}\hfill {\dot{V}}_m\left(t\right)\le & {\displaystyle \frac{1}{2}}({\tilde{x}}^T\left(t\right)(Q_m\otimes I_n)\tilde{x}\left(t\right)+{\tilde{v}}^T\left(t\right)(Q_m\otimes I_n)\tilde{v}\left(t\right))+{\displaystyle \frac{l_1}{2}}{\tilde{x}}^T\left(t\right)(Q_m\otimes I_n)\tilde{x}\left(t\right)\hfill \\ \hfill & +({\displaystyle \frac{l_1}{2}}+l_2){\tilde{v}}^T\left(t\right)(Q_m\otimes I_n)\tilde{v}\left(t\right)+c\alpha {\lambda }_{max}\left(Q_m\right){\tilde{x}}^T\left(t\right)({\left(H^m\right)}^T{H}^m\otimes I_n)\tilde{x}\left(t\right)\hfill \\ \hfill & +c\alpha {\lambda }_{max}\left(Q_m\right){\tilde{v}}^T\left(t\right)({\left(H^m\right)}^T{H}^m\otimes I_n)\tilde{v}\left(t\right)+{\displaystyle \frac{c}{2}}{\tilde{v}}^T\left(t\right)(Q_m\otimes I_n)\tilde{v}\left(t\right)\hfill \\ \hfill & +{\displaystyle \frac{c}{2}}{\tilde{x}}^T\left(t\right)(Q_m{H}^m\otimes I_n)\tilde{x}\left(t\right)\hfill \end{array}$$

Let

$$K=\left[\begin{array}{cc}{\displaystyle \frac{1+l_1}{2}}{Q}_m+c\alpha {\lambda }_{max}\left(Q_m\right){\left(H^m\right)}^T{H}^m+{\displaystyle \frac{c}{2}}{Q}_m{H}^m& 0\\ 0& {\displaystyle \frac{1+l_1+2l_2+c}{2}}{Q}_m+c\alpha {\lambda }_{max}\left(Q_m\right){\left(H^m\right)}^T{H}^m\end{array}\right]$$

Then we can obtain

$$\begin{array}{c}\hfill {\dot{V}}_m\left(t\right)\le {\xi }^T\left(t\right)(K\otimes I_n)\xi \left(t\right)\end{array}$$

Meanwhile, it is clear that ${\dot{V}}_m\left(t\right)\le {\lambda }_{max}\left(K\right){\xi }^T\left(t\right)\xi \left(t\right)$ and ${\displaystyle \frac{1}{2}}{\lambda }_{min}\left(Q_m\right){\xi }^T\left(t\right)\xi \left(t\right)\le V_m\left(t\right)\le {\displaystyle \frac{1}{2}}{\lambda }_{max}\left(Q_m\right){\xi }^T\left(t\right)\xi \left(t\right)$, so we can obtain ${\dot{V}}_m\left(t\right)\le {\displaystyle \frac{2{\lambda }_{max}\left(K\right)}{{\lambda }_{min}\left(Q_m\right)}}{V}_m\left(t\right)$. Let ${\rho }_2={\displaystyle \frac{2{\lambda }_{max}\left(K\right)}{{\lambda }_{min}\left(Q_m\right)}}$, then ${\rho }_2>0$ and ${\dot{V}}_m\left(t\right)\le {\rho }_2{V}_m\left(t\right)$, $t\in [t_k^i,t_{k+1}^i)$. We suppose that when $\delta \left(t\right)=m$, $t\in [t_{s_l+m},t_{s_l+m+1})$. Then we have

$$\begin{array}{cc}\hfill & V_m\left(t\right)\le e^{{\rho }_2(t-t_p^i)}{V}_m\left(t_p^i\right),t\in [t_p^i,t_{s_l+m+1})\hfill \\ \hfill & V_m\left(t_p^i\right)\le e^{{\rho }_2(t_p^i-t_{p-1}^i)}{V}_m\left(t_{p-1}^i\right)\hfill \end{array}$$

Thus, we can obtain

$$\begin{array}{cc}\hfill V_m\left(t\right)& \le e^{{\rho }_2(t-t_p^i)}{V}_m\left(t_p^i\right)\hfill \\ \hfill & \le e^{{\rho }_2(t-t_{p-1}^i)}{V}_m\left(t_{p-1}^i\right)\hfill \\ \hfill & \le \cdots \hfill \\ \hfill & \le e^{{\rho }_2(t-t_{s_l+m})}{V}_m\left(t_{s_l+m}\right)\hfill \end{array}$$

When $t\in [t_{s_{l+1}-1},t_{s_{l+1}})$, we can obtain

$$V_{\delta \left(t\right)}\left(t\right)\le e^{{\rho }_2(t-t_{s_{l+1}-1})}{V}_{\delta \left(t_{s_{l+1}-1}\right)}\left(t_{s_{l+1}-1}\right)$$

According to condition (2), one has

$$V_{\delta \left(t_{s_{l+1}-1}^{+}\right)}\left(t_{s_{l+1}-1}^{+}\right)\le {\mu }_2{V}_{\delta \left(t_{s_{l+1}-1}^{-}\right)}\left(t_{s_{l+1}-1}^{-}\right)$$

Because $V_{\delta \left(t\right)}\left(t\right)$ is a right-continuous function, then one has

$$V_{\delta \left(t_{s_{l+1}-1}\right)}\left(t_{s_{l+1}-1}\right)\le {\mu }_2{V}_{\delta \left(t_{s_{l+1}-1}^{-}\right)}\left(t_{s_{l+1}-1}^{-}\right)\le {\mu }_2{e}^{{\rho }_2(t_{s_{l+1}-1}-t_{s_{l+1}-2})}{V}_{\delta \left(t_{s_{l+1}-2}\right)}\left(t_{s_{l+1}-2}\right)$$

So we have $V_{\delta \left(t\right)}\left(t\right)\le {\mu }_2{e}^{{\rho }_2(t-t_{s_{l+1}-2})}{V}_{\delta \left(t_{s_{l+1}-2}\right)}\left(t_{s_{l+1}-2}\right)$. Repeating the above step, we can obtain

$$V_{\delta \left(t\right)}\left(t\right)\le {\mu }_2^{N(t_{s_l}+1,t_{s_{l+1}})-1}{e}^{{\rho }_2(t-t_{s_l+1})}{V}_{\delta \left(t_{s_l+1}\right)}\left(t_{s_l+1}\right)$$

Similarly, according to condition (2), we can obtain

$$\begin{array}{cc}\hfill & V_{\delta \left(t_{s_{l+1}}^{+}\right)}\left(t_{s_{l+1}}^{+}\right)\le {\mu }_3{V}_{\delta \left(t_{s_{l+1}}^{-}\right)}\left(t_{s_{l+1}}^{-}\right)\hfill \\ \hfill & V_{\delta \left(t_{s_l+1}^{+}\right)}\left(t_{s_l+1}^{+}\right)\le {\mu }_1{V}_{\delta \left(t_{s_l+1}^{-}\right)}\left(t_{s_l+1}^{-}\right)\hfill \end{array}$$

When $t\in [t_{s_l},t_{s_l+1})$, there is no DoS attack; we have

$$\begin{array}{cc}\hfill V_{\delta \left(t_{s_l+1}\right)}\left(t_{s_l+1}\right)& \le {\mu }_1{V}_{\delta \left(t_{s_l+1}^{-}\right)}\left(t_{s_l+1}^{-}\right)\hfill \\ \hfill & \le {\mu }_1{e}^{-{\rho }_1(t_{s_l+1}-t_{s_l})}{V}_{\delta \left(t_{s_l}\right)}\left(t_{s_l}\right)\hfill \end{array}$$

Thus, when $t\in [t_{s_{l+1}-1},t_{s_{l+1}})$, we have

$$V_{\delta \left(t\right)}\left(t\right)\le {\mu }_1{\mu }_2^{N(t_{s_l+1},t_{s_{l+1}})-1}{e}^{{\rho }_2(t-t_{s_l+1})}{e}^{-{\rho }_1(t_{s_l+1}-t_{s_l})}{V}_{\delta \left(t_{s_l}\right)}\left(t_{s_l}\right)$$

Moreover, we can obtain

$$\begin{array}{cc}\hfill V_{\delta \left(t_{s_{l+1}}\right)}\left(t_{s_{l+1}}\right)& \le {\mu }_3{V}_{\delta \left(t_{s_{l+1}}^{-}\right)}\left(t_{s_{l+1}}^{-}\right)\hfill \\ \hfill & \le {\mu }_3{e}^{{\rho }_2(t_{s_{l+1}}-t_{s_{l+1}-1})}{V}_{\delta \left(t_{s_{l+1}-1}\right)}\left(t_{s_{l+1}-1}\right)\hfill \\ \hfill & \le {\mu }_3{\mu }_1{\mu }_2^{N(t_{s_l+1},t_{s_{l+1}})-1}{e}^{{\rho }_2(t_{s_{l+1}}-t_{s_l+1})}{e}^{-{\rho }_1(t_{s_l+1}-t_{s_l})}{V}_{\delta \left(t_{s_l}\right)}\left(t_{s_l}\right)\hfill \end{array}$$

Let $\mu =max\{{\mu }_1,{\mu }_2,{\mu }_3\}$, then we can obtain

$$V_{\delta \left(t_{s_{l+1}}\right)}\left(t_{s_{l+1}}\right)\le {\mu }^{N(t_{s_l+1},t_{s_{l+1}})+1}{e}^{{\rho }_2(t_{s_{l+1}}-t_{s_l+1})}{e}^{-{\rho }_1(t_{s_l+1}-t_{s_l})}{V}_{\delta \left(t_{s_l}\right)}\left(t_{s_l}\right)$$

Furthermore, it is noted that $T_l\le T,f_l\le f$ and $N(t_{s_l+1},t_{s_{l+1}})=T_l{f}_l\le Tf,{\tau }_l\ge {\tau }_D$; then we have

$$V_{\delta \left(t_{s_{l+1}}\right)}\left(t_{s_{l+1}}\right)\le {\mu }^{Tf+1}{e}^{{\rho }_{2}T-{\rho }_1{\tau }_D}{V}_{\delta \left(t_{s_l}\right)}\left(t_{s_l}\right)$$

Let $\omega ={\mu }^{Tf+1}{e}^{{\rho }_{2}T-{\rho }_1{\tau }_D}$; it is clear that $\omega <1$ according to condition (3). Similar to the above analysis, one has

$$V_{\delta \left(t_{s_l}\right)}\left(t_{s_l}\right)\le {\omega }^{l-1}{V}_{\delta \left(t_{s_1}\right)}\left(t_{s_1}\right)$$

Further, it is clear that $\forall t\in [t_{s_l},t_{s_{l+1}})$,

$$V_{\delta \left(t\right)}\left(t\right)\le {\mu }^{Tf}{e}^{{\rho }_{2}T}{V}_{\delta \left(t_{s_l}\right)}\left(t_{s_l}\right)\le {\mu }^{Tf}{e}^{{\rho }_{2}T}{\omega }^{l-1}{V}_{\delta \left(t_{s_1}\right)}\left(t_{s_1}\right)$$

Meanwhile, we can obtain

$$\begin{array}{c}\hfill V_{\delta \left(t\right)}\left(t\right)\ge {\nu }_1{\left|\right|\xi \left(t\right)\left|\right|}^2,V_{\delta \left(t_{s_1}\right)}\left(t_{s_1}\right)\le {\nu }_2\left|\right|\xi \left(t_{s_1}\right){\left|\right|}^2\end{array}$$

where ${\nu }_1=min\{{\lambda }_{min}\left(Q_m\right),{\lambda }_{min}\left(\mathsf{\Omega }\right)\}$,${\nu }_2=min\{{\lambda }_{max}\left(Q_m\right),{\lambda }_{max}\left(\mathsf{\Omega }\right)\}$.

Then we can obtain

$${\left|\right|\xi \left(t\right)\left|\right|}^2\le {\displaystyle \frac{{\nu }_2}{{\nu }_1}}{\mu }^{Tf}{e}^{{\rho }_{2}T}{\omega }^{l-1}\left|\right|\xi \left(t_{s_1}\right){\left|\right|}^2$$

Thus, we can obtain $\left|\right|\xi \left(t\right)\left|\right|\to 0$ as $t\to \infty$, which means the error system is asymptotically stable. So the group consensus of MAS (1) and (2) with control protocol (3) can be achieved under a consecutive DoS attack.

Theorem 2. 

Under control protocol (3) and event-triggered function (4), the Zeno behavior can be excluded.

Proof. 

$$\begin{array}{cc}\hfill D^{+}\left|\right|e^{\delta \left(t\right)}\left(t\right)\left|\right|& \le \left|\right|{\dot{e}}^{\delta \left(t\right)}\left(t\right)\left|\right|=\left|\right|{\dot{\mathsf{\Phi }}}^{\delta \left(t\right)}\left(t\right)\left|\right|\hfill \\ & =\left|\right|((L^{\delta \left(t\right)}+D^{\delta \left(t\right)})\otimes I_n)(\tilde{v}\left(t\right)+f(x\left(t\right),v\left(t\right))-f(\overline{x}\left(t\right),\overline{v}\left(t\right))-c{\mathsf{\Phi }}^{\delta \left(t\right)}\left(t_k^i\right))\left|\right|\hfill \\ \hfill & \le \left|\right|L^{\delta \left(t\right)}+D^{\delta \left(t\right)}\left|\right|\left(\right||\tilde{v}\left(t\right)\left|\right|+l_1\left|\right|\tilde{x}\left(t\right)\left|\right|+l_2\left|\right|\tilde{v}\left(t\right)\left|\right|+\left|\right|c{\mathsf{\Phi }}^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|)\hfill \\ \hfill & =\left|\right|L^{\delta \left(t\right)}+D^{\delta \left(t\right)}\left|\right|(l_1\left|\right|\tilde{x}\left(t\right)\left|\right|+(l_2+1)\left|\right|\tilde{v}\left(t\right)\left|\right|+\left|\right|c{\mathsf{\Phi }}^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|)\hfill \end{array}$$

Let $\overline{\delta }=max\{l_1\left|\right|L^{\delta \left(t\right)}+D^{\delta \left(t\right)}\left|\right|,(l_2+1)\left|\right|L^{\delta \left(t\right)}+D^{\delta \left(t\right)}\left|\right|\}$ and $\left|\right|{\mathsf{\Lambda }}^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|=\left|\right|L^{\delta \left(t\right)}+D^{\delta \left(t\right)}\left|\right|·\left|\right|c{\mathsf{\Phi }}^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|$; then

$$\begin{array}{cc}\hfill \left|\right|\dot{e}\left(t\right)\left|\right|& \le \overline{\delta }\left(\right||\tilde{x}\left(t\right)\left|\right|+\left|\right|\tilde{v}\left(t\right)\left|\right|)+\left|\right|{\mathsf{\Lambda }}^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|\hfill \\ \hfill & \le 2\overline{\delta }\left|\right|\xi \left(t\right)\left|\right|+\left|\right|{\mathsf{\Lambda }}^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|\hfill \end{array}$$

According to the proof of theorem 1, we can obtain $\left|\right|\xi \left(t\right)\left|\right|\le \sqrt{{\displaystyle \frac{{\nu }_2}{{\nu }_1}}}{\mu }^{{\displaystyle \frac{Tf}{2}}}{e}^{{\displaystyle \frac{{\rho }_{2}T}{2}}}{\omega }^{{\displaystyle \frac{p-1}{2}}}\left|\right|\xi \left(t_{s_1}\right)\left|\right|$.

Then we can obtain

$$\begin{array}{cc}\hfill D^{+}\left|\right|e_i^{\delta \left(t\right)}\left(t\right)\left|\right|\le \left|\right|{\dot{e}}_i^{\delta \left(t\right)}\left(t\right)\left|\right|\le \left|\right|{\dot{e}}^{\delta \left(t\right)}\left(t\right)\left|\right|& \le 2\overline{\delta }\sqrt{{\displaystyle \frac{{\nu }_2}{{\nu }_1}}}{\mu }^{{\displaystyle \frac{Tf}{2}}}{e}^{{\displaystyle \frac{{\rho }_{2}T}{2}}}{\omega }^{{\displaystyle \frac{p-1}{2}}}\left|\right|\xi \left(t_{s_1}\right)\left|\right|+\left|\right|{\mathsf{\Lambda }}^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|\hfill \end{array}$$

So we have

$$\left|\right|e_i^{\delta \left(t\right)}\left(t\right)\left|\right|\le {\int }_{t_k^i}^t\left|\right|{\dot{e}}_i^{\delta \left(t\right)}\left(s\right)\left|\right|ds\le (2\overline{\delta }\sqrt{{\displaystyle \frac{{\nu }_2}{{\nu }_1}}}{\mu }^{{\displaystyle \frac{Tf}{2}}}{e}^{{\displaystyle \frac{{\rho }_{2}T}{2}}}{\omega }^{{\displaystyle \frac{p-1}{2}}}\left|\right|\xi \left(t_{s_1}\right)\left|\right|+\left|\right|{\mathsf{\Lambda }}^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|)(t-t_k^i)$$

On the other hand, it is easy to see that $\left|\right|e_i^{\delta \left(t\right)}\left(t\right)\left|\right|\le {\displaystyle \frac{\sqrt{\alpha }}{1+\sqrt{\alpha }}}\left|\right|{\mathsf{\Phi }}^{\delta {\left(t\right)}_i}\left(t_k^i\right)\left|\right|$ is a sufficient condition for $\left|\right|e_i^{\delta \left(t\right)}\left(t\right)\left|\right|\le \sqrt{\alpha }\left|\right|{\mathsf{\Phi }}_i^{\delta \left(t\right)}\left(t\right)\left|\right|$. At triggering instant $t_{k+1}^i$, one can easily obtain that

$\left|\right|e_i^{\delta \left(t\right)}\left(t_{k+1}^i\right)\ge {\displaystyle \frac{\sqrt{\alpha }}{1+\sqrt{\alpha }}}\left|\right|{\mathsf{\Phi }}_i^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|$; then we have

$$\left|\right|e_i^{\delta \left(t\right)}\left(t_{k+1}^i\right)\left|\right|\le (2\overline{\delta }\sqrt{{\displaystyle \frac{{\nu }_2}{{\nu }_1}}}{\mu }^{{\displaystyle \frac{Tf}{2}}}{e}^{{\displaystyle \frac{{\rho }_{2}T}{2}}}{\omega }^{{\displaystyle \frac{p-1}{2}}}\left|\right|\xi \left(t_{s_1}\right)\left|\right|+\left|\right|{\mathsf{\Lambda }}^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|)(t_{k+1}^i-t_k^i)$$

Thus,

$${\displaystyle \frac{\sqrt{\alpha }}{1+\sqrt{\alpha }}}\left|\right|{\mathsf{\Phi }}_i^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|\le (2\overline{\delta }\sqrt{{\displaystyle \frac{{\nu }_2}{{\nu }_1}}}{\mu }^{{\displaystyle \frac{Tf}{2}}}{e}^{{\displaystyle \frac{{\rho }_{2}T}{2}}}{\omega }^{{\displaystyle \frac{p-1}{2}}}\left|\right|\xi \left(t_{s_1}\right)\left|\right|+\left|\right|{\mathsf{\Lambda }}^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|)(t_{k+1}^i-t_k^i)$$

Then we can obtain the lower bound of the inter-event time:

$$t_{k+1}^i-t_k^i\ge {\displaystyle \frac{\frac{\sqrt{\alpha }}{1+\sqrt{\alpha }}\left|\right|{\mathsf{\Phi }}_i^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|}{2\overline{\delta }\sqrt{\frac{{\nu }_2}{{\nu }_1}}{\mu }^{\frac{Tf}{2}}{e}^{\frac{{\rho }_{2}T}{2}}{\omega }^{\frac{p-1}{2}}\left|\right|\xi \left(t_{s_1}\right)\left|\right|+\left|\right|{\mathsf{\Lambda }}^{\delta \left(t\right)}\left(t_k^i\right)\left|\right|}}>0$$

which means the Zeno behavior has been excluded.  □

Remark 3. 

A multi-agent system with an undirected communication topology is a special directed case. In an undirected topology, each agent can be affected by other nodes. Therefore, we just need to choose the inter-act agents to be pinned. Then the control protocol can be designated as

$$u_i\left(t\right)=\left\{\begin{array}{cc}\hfill & c{\displaystyle \sum _{j=1}^N}{a}_{ij}[x_j\left(t_k^i\right)-x_i\left(t_k^i\right)+v_j\left(t_k^i\right)-v_i\left(t_k^i\right)]-c{d}_i[x_i\left(t_k^i\right)-x_{k_i}\left(t_k^i\right)+v_i\left(t_k^i\right)-v_{k_i}\left(t_k^i\right)]\hfill \\ & +c{\displaystyle \sum _{j=1}^N}{l}_{ij}(x_{k_j}\left(t_k^i\right)+v_{k_j}\left(t_k^i\right)),i\in {\tilde{\mathcal{V}}}_{k_i}\hfill \\ \hfill & c{\displaystyle \sum _{j=1}^N}{a}_{ij}[x_j\left(t_k^i\right)-x_i\left(t_k^i\right)+v_j\left(t_k^i\right)-v_i\left(t_k^i\right)],i\in {\mathcal{V}}_{k_i}\setminus {\tilde{\mathcal{V}}}_{k_i}\hfill \end{array}\right.$$

The conditions for group consensus of multi-agents with an undirected communication topology are similar to those of Theorems 1 and 2. We omit them here.

4. Numerical Simulation

In this section, we will give two simulation examples to demonstrate the efficacy of the results.

Example 1: Suppose that there are six nodes in the communication topology, which is shown in Figure 2. The adjacency matrix of this topology is given as follows:

$$A^0=\left(\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0\end{array}\right)$$

Let the nonlinear function be $f(x_i\left(t\right),v_i\left(t\right))=6cos\left(4t\right)+0.01cos\left(2x_i\left(t\right)\right)+0.01sin\left(4v_i\left(t\right)\right)$. From Assumption 1, we can obtain $l_1=l_2=0.001$; thus, $\delta =1.002$. From the original communication topology, we can see that agents 1 and 4 are inter-act agents, and agent 6 is an intra-act agent with zero degree. So we should choose agents 1, 4, and 6 to be pinned. Furthermore, we divide the six agents into three group, where ${\mathcal{V}}_1=\{1,2\}$, ${\mathcal{V}}_2=\{3,4\}$, and ${\mathcal{V}}_3=\{5,6\}$. Choosing the system parameters in

Table 1. System parameters.

System Parameters
c = 15   $\alpha$ = 0.05   $\mu$ = 1.1    $T=2$   $f=5.5$   $d_i=4.2,i=1,4,6$

, we have the pinning matrix $D^0=diag\{4.2,0,0,4.2,0,4.2\}$ and the coupling strength of the multi-agent system $c=15$; then, by some calculations, we can obtain that ${\lambda }_{max}((\delta +{\displaystyle \frac{c}{2}})I_N-c\overline{L^0}-c{D}^0+2c\alpha {\left(H^0\right)}^T{H}^0)=-0.6348<0$, which satisfies condition (1) of Theorem 1. Furthermore, we can obtain ${\rho }_1=0.7557,{\rho }_2=1.56$. Moreover, there are three types of asynchronous DoS attacks, which are shown in Figure 3.

Furthermore, choosing $T=2,f=5.5$, based on condition (2) of Theorem 1, we can obtain a persistent dwell time $\tau =5.8$. The switching signal of the changing communication topology can be seen in Figure 4.

Here, we depict two types of event-triggered functions. The first one is the event-triggered function designed using the combinational measurements in this paper. The second one is the event-triggered function implemented by utilizing a sampled-event detector, which mainly depends on the state error, such as in [42]. Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 demonstrate the comparison of the two different event-triggered functions. From Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, we can see that the multi-agent system achieves consensus at around 3 s with the event-triggered function designed using the combinational measurements; however, the multi-agent system achieves consensus at around 6 s with the event-triggered function designed using the state error. Further, Figure 11 demonstrates the event-triggered time instants of two different event-triggered functions. Moreover, in Table 1, we can see that the multi-agent system will trigger less frequently with the event-triggered function designed using combinational measurement, which can greatly save resources. (For a brief review, in

Table 2. The comparison of two event-triggered functions.

	Agent	Agent 1	Agent 2	Agent 3	Agent 4	Agent 5	Agent 6
Performance
trigger number (method 1)		397	193	136	443	121	422
trigger number (method 2)		918	387	257	912	184	940

, method 1 stands for the event-triggered function designed using combinational measurement, and method 2 stands for the event-triggered function designed using the state error.) Thus, regardless of the time the agent’s position and velocity states achieve consensus or the event-triggered time instants for a multi-agent system, it is clear that the method presented in this paper is more effective.

Furthermore, compared with [43], which also studies group consensus of a multi-agent system under DoS attack, there are two improved aspects in this paper. First, we consider asynchronous DoS attacks in this paper, but [43] studies synchronous DoS attacks. An asynchronous DoS attack means part of the communication links will be interpreted, but a synchronous DoS attack means all communication links are interpreted. Thus, an asynchronous DoS attack is more general, and a synchronous DoS attack can be seen as a special case of asynchronous DoS attack. Second, [43] makes use of feedback control to achieve group consensus, which is a continuous control method. However, we use event-triggered control in this paper, which is a discontinuous control method. Considering the limited resources in practical engineering, event-triggered control can avoid unnecessary control signal processing, which can greatly save resources.

Example 2: In this example, we increase the number of agents and increase the DoS attack intensity. Suppose that there are nine nodes in the communication topology, which is shown in Figure 12. The adjacency matrix of this topology is given as follows:

$$A^0=\left(\begin{array}{ccccccccc}0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0\end{array}\right)$$

We choose a nonlinear function similar to Example 1; then, we can obtain $l_1=l_2=0.001$; thus, $\delta =1.002$. From the original communication topology, we can see that agents 3 and 4 are inter-act agents, and agent 8 is an intra-act agent with zero degree. So we should choose agents 3, 4 and 8 to be pinned. Furthermore, we divide the nine agents into three group, where ${\mathcal{V}}_1=\{1,2,3\}$, ${\mathcal{V}}_2=\{4,5,6\}$, and ${\mathcal{V}}_3=\{7,8,9\}$. Choosing the system parameters in

Table 3. System parameters.

System Parameters
c = 15   $\alpha$ = 0.05   $\mu$ = 1.1    $T=2.2$   $f=5.5$   $d_i=4.2,i=1,4,6$

, we have the pinning matrix $D^0=diag\{0,0,4.2,4.2,0,0,0,4.2,0\}$ and the coupling strength of the multi-agent system $c=15$; then, by some calculations, we can obtain that ${\lambda }_{max}((\delta +{\displaystyle \frac{c}{2}})I_N-c\overline{L^0}-c{D}^0+2c\alpha {\left(H^0\right)}^T{H}^0)=-0.4686<0$, which satisfies condition (1) of Theorem 1. Furthermore, we can obtain ${\rho }_1=0.9325,{\rho }_2=1.24$. Moreover, there are two types of asynchronous DoS attacks, which are shown in Figure 13.

In this example, we choose $T=2.2,f=5.5$ to describe the DoS attack. Compared with example 1, we lift the attack intensity by extending the active time of the DoS attack. Then, based on condition (2) of Theorem 1, we can obtain the persistent dwell time $\tau =3.6$. Figure 14, Figure 15 and Figure 16 show 344 the position states of follower agents and leader agents, Figure 17, Figure 18 and Figure 19 show the velocity 345 states of follower agents and leader agents. Thus, from Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19, we can see that the 346 group consensus of the multi-agent system can still be achieved by our method even in more complex conditions. The switching 343 signal of the changing communication topology can be seen in Figure 20 and the event-triggered time instants for agents can be seen in Figure 21.

5. Conclusions

In this paper, we mainly solve the group consensus of a second-order multi-agent system under an asynchronous DoS attack. An asynchronous DoS attack leads to continuous changes in the communication topologies; thus, we can change the multi-agent system into a switching system; then, we can use switching theory to solve the group consensus problem. Moreover, we use the persistent dwell time (PDT) to describe a DoS attack in this paper, which is greatly different from the average dwell time (ADT). For an ADT, there exists the chatter bound to constrain the number of switching intervals that are less than the average time interval length. Furthermore, an ADT requires that the length between two time intervals cannot be too short. But for the PDT, we use the T-portion to describe these short time intervals, which means that all of the time intervals will be considered equally no matter the number and length of short time intervals, greatly liberalizing the limit of the ADT. Thus, for general consensus of a multi-agent system under a DoS attack, we can try to combine the switching theory and the concept of the PDT to solve the consensus for multi-agents.

In the real world, group consensus of multi-agent systems is general, such as express deliveries with unmanned aerial vehicles. By the method in this paper, the unmanned aerial vehicles can be divided into different groups to deliver different freight to different destinations. Further, because of the vulnerability of the wireless communication, the formation of unmanned aerial vehicles may suffer cyber-attacks easily. Thus, by the method in this paper, we can embed the event-triggered controller in each agent and set the event-triggered threshold of our methods; then, if the intensity of the cyber-attack is strong enough to violate the event-triggered threshold, the controller will process the control signal to the agent. Then, the formation of unmanned aerial vehicles will be maintained.

However, there are also some limitations existing in our study. Firstly, there are limits to the multi-agent system model. While a second-order multi-agent system model effectively captures many complex scenarios, it cannot encompass all real-world complexities. For instance, helicopter systems may involve numerous variables, necessitating a high-order multi-agent system model. In this situation, it is hard to construct a Lyapunov function for an error system like in this paper. Thus, our method may be less effective, but one can try to combine a backstepping method and the event-triggered control of this paper to explore the group consensus of a high-order multi-agent system, which will be our future work. Secondly, our analysis is limited to DoS attacks. However, various other types of cyber-attacks exist in real-world applications. For example, deception attacks can also exist in the wireless communication of agents. But the PDT is not suitable to describe a deception attack. Thus, how to describe a deception attack is a problem that we will solve in the future.