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.
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
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 -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
types of asynchronous DoS attacks. Then, let the piecewise constant function
:
denote the switching signal. When
, there is no DoS attack. When
, 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 is called the persistent dwell time (PDT) if there is an infinite number of disjoint intervals with lengths no smaller than on which 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,
stands for
lth stage, and the DoS attack starts at
. During the
-portion
, there is no DoS attack, and the communication topology is represented by
. During the
T-portion
, the MAS is under a consecutive asynchronous DoS attack, where
mean the DoS instants. The communication topology of this period is represented by
. Further, the length of the
T-portion is represented by
, and the frequency of the
T-portion is represented by
, where
stands for the switching number during
. When the DoS attack ends, the communication topology will be restored to the original topology from
.
Let time sequence
denote the
kth event trigger of the
ith agent, and inspired by the pinning-event-triggered scheme, the control
for a multi-agent system can be designated as
where
denotes the coupling strength,
denotes the communication link of agent
i when the communication topology changes into
, and
denotes the pinning control gain of agent
i when the communication topology changes into
.
Remark 2. It is important to point out that when , there is no DoS attack. If agent i is an intra-act agent, one has . When the DoS attack occurs, we do not divide the group again; thus, or , so we can still obtain . Then can be simplified into .
Suppose
; define
; then choose the event trigger function as
where
is a positive constant. If
, then
will update to
, which can be expressed as follows:
According to (1) and (2), by the use of the Kronecker product, we can obtain the compact form
where
Based on the above analysis, we have the following theorem:
Theorem 1. For given non-zero scalars , prescribed period of persistence , and frequency , if there exist matrices such that the following conditions hold:
- (1).
;
- (2).
;
- (3).
PDT satisfies .
where , , , , and will be given in the proof,
then MAS (1) and (2) can achieve group consensus.
Proof. From condition (1), we can see that
since
; thus,
□
According to Lemma 1, we have
When there is no DoS attack, namely,
, and we assume that there are
n event-triggered time instants, denoted by
, when
, we choose the positive definite Lyapunov function as follows:
where
.
Further, based on Assumption 1 and Young’s inequality, we can obtain
At the same time, one can obtain that
; thus, we have . Similarly, we can obtain
Calculate the derivative of Lyapunov function as
According to the event-triggered condition, we can obtain
where
. Similarly, we can obtain
Because
, we obtain
According to condition (1), we can have
, so
; therefore, we have
For the convenience of the following discussion, we let
then we obtain
, so
.
Let
, then
, and
,
. Then we can obtain
Thus, we can obtain
When the system is under a consecutive asynchronous attack, namely,
, the communication topology is switching; thus, we choose the Lyapunov function as follows:
Similarly, we assume that there are p event-triggered time instants, denoted by
, when
. For convenience, let
. When
, calculate the derivative as
Similar to the analysis when there is no DoS attack, we can obtain
The DoS attack considered here is effective. Therefore,
is semi-positive. So we can obtain
Meanwhile, it is clear that
and
, so we can obtain
. Let
, then
and
,
. We suppose that when
,
. Then we have
When
, we can obtain
According to condition (2), one has
Because
is a right-continuous function, then one has
So we have
. Repeating the above step, we can obtain
Similarly, according to condition (2), we can obtain
When
, there is no DoS attack; we have
Thus, when
, we have
Let
, then we can obtain
Furthermore, it is noted that
and
; then we have
Let
; it is clear that
according to condition (3). Similar to the above analysis, one has
Further, it is clear that
,
Meanwhile, we can obtain
where
,
.
Thus, we can obtain as , 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.
Let
and
; then
According to the proof of theorem 1, we can obtain
.
Then we can obtain
So we have
On the other hand, it is easy to see that
is a sufficient condition for
. At triggering instant
, one can easily obtain that
; then we have
Thus,
Then we can obtain the lower bound of the inter-event time:
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 asThe 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:
Let the nonlinear function be
. From Assumption 1, we can obtain
; thus,
. 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
,
, and
. Choosing the system parameters in
Table 1, we have the pinning matrix
and the coupling strength of the multi-agent system
; then, by some calculations, we can obtain that
, which satisfies condition (1) of Theorem 1. Furthermore, we can obtain
. Moreover, there are three types of asynchronous DoS attacks, which are shown in
Figure 3.
Furthermore, choosing
, based on condition (2) of Theorem 1, we can obtain a persistent dwell time
. 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, 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:
We choose a nonlinear function similar to
Example 1; then, we can obtain
; thus,
. 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
,
, and
. Choosing the system parameters in
Table 3, we have the pinning matrix
and the coupling strength of the multi-agent system
; then, by some calculations, we can obtain that
, which satisfies condition (1) of Theorem 1. Furthermore, we can obtain
. Moreover, there are two types of asynchronous DoS attacks, which are shown in
Figure 13.
In this example, we choose
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
.
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.