School of Science, Jiangnan University, Wuxi 214122, China
† Corresponding author. E-mail:
aihuahu@126.com
Project supported by the National Natural Science Foundation of China (Grant Nos. 61807016 and 61772013) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20181342).
1. IntroductionThe consensus for multi-agent systems has been intensively discussed in the past decade, due to successful applications in various areas, including formation control,[1] vehicle systems,[2] large-scale sensors,[3] obstacle avoidance,[4] etc. It is well-known that the foremost goal of consensus problem is to design an appropriate control protocol,[5] so that all agents can asymptotically attain a consistent quantity based on local interactions.
At present, it should be noted that most of results involve complete consensus.[6–12] However, in many real-world situations, the multi-agent systems have different cooperative tasks, and it is more general to trigger agents in the same subgroup to achieve a consistent state, while in different subgroups they behave non-identically, which can be denoted by group consensus.[13] Moreover, the phenomenon of group consensus is ubiquitous in nature and human society, such as multi-species foraging groups, the pattern formation of bacteria colonies, a school of fish and formation of opinion.[14] Obviously, group consensus is more suitable for addressing complex and flexible problems than complete consensus.
Recently, the group consensus of multi-agent systems has received much attention from researchers, and group consensus on first-order,[15,16] second-order,[13,17] or heterogeneous agents[18,19] have yielded some research results. It is found that the subgroups need to meet an in-degree balanced couple, and the Laplacian matrix
is not similar to that in complete consensus because of the introduction of competition and cooperation mechanism.[13,16] Consequently, it is much more difficult to study group consensus than complete consensus. By now, directed topology, fixed topologies, switching topologies, communication delays, nonlinear input constraints, pinning control and event-triggered control have also been introduced to study the group consensus issue; see, e.g., Refs. [15–23]. However, many existing results of group consensus problem include the virtual leader, which may not ensure that the controllers are distributed; for further information, see Ref. [24]. In Ref. [25], distributed feedback control was used to achieve the group consensus of multi-agent systems without virtual leader. Therefore, in this paper we will further investigate the problem of leaderless group consensus.
In addition, multi-agent systems are often in an indeterminate communication environment, just like networks under cyber-attacks. Some results of the consensus problem with cyber-attack have been published; for example, in Refs. [26–28] Cyber-attacks may destroy the agents or the communication links between agents,[29] thus, cyber-attacks can weaken the stability of the networked systems and undermine the system performance.[30] In fact, with the development of network information technology, cyber security has become a common issue. The synchronization of complex dynamical networks under cyber-attacks has been explored in Ref. [31]. By applying new control strategies, the authors in Ref. [32] studied the stability of the network subjected to cyber-attacks, which would disrupt the communication channels existing in both controllers and observers. Distributed tracking control was proposed for multi-agent systems under two types of attacks, which are connectivity-maintained attacks and connectivity-broken attacks.[33] Inspired by this discussion, in this paper we further focus on the group consensus of multi-agent systems under cyber-attacks. Comparing with the existing papers, the main contributions of this paper include: (i) the multi-agent system subjected to cyber-attacks is considered for group consensus, and an effective algorithm and optimization problem for selecting parameter are designed; (ii) the results of consensus issue for multiple subgroups with cyber-attacks are given. Based on the algebraic graph theory, matrix theory, and Lyapunov stability method, some sufficient conditions are derived to ensure the group consensus with cyber-attacks.
The rest of this paper is organized as follows. In Section 2 the preliminaries about graph theory and the problem formulation are given. In Section 3 the results of consensus problem for two subgroups and multiple subgroups with cyber-attacks are obtained. In Section 4 the numerical examples are presented, and finally in Section 5 some conclusions are drawn from the present study.
The following notations are used throughout this paper: R denotes the set of real numbers,
the set of
real matrices. Notation diag{…} is a diagonal matrix,
has a proper dimension,
is the zero matrix with an appropriate dimension, and
is the Euclidean norm. For any real symmetric matrix
,
indicates that
is a positive matrix.
2. Preliminaries and problem formulationIn this section, some essential concepts on graph theory in Ref. [34] and problem formulation are stated.
2.1. PreliminariesIn general, the topology of a multi-agent system is described as a weighted directed graph
, where
is the set of nodes representing a network containing n + m nodes,
is the set of directed links, and
is the weighted adjacency matrix. Define
if and only if there is a directed link
otherwise,
. Here, we assume
for all
. The subscript set of the agents in the group G is
. Assume
to be the neighboring set of node vi. The Laplacian matrix
is denoted with
when
, and
for i = j. Moreover, the in-degree and out-degree of node vi are denoted respectively by
In graph
G, a directed path is a finite sequence of connected edges. If
G possesses a root node with directed paths to all other nodes, then
G is said to have a directed spanning tree.
2.2. Problem formulationSuppose that there are n + m nodes in the multi-agent system, then each agent will have the dynamics as follows:
where
is the position of the agent
vi at time
t, and
is the control protocol to be designed later.
First, consider that n + m agents are all split into two subgroups. Suppose that n agents (
) belong to the subgroup
and the last m agents (
) are in the subgroup
. Let
and
be the subscript sets of the agents in each group, respectively.
Remark 1 It is clear that
and
, and the neighboring sets of node vi are correspondingly defined as
and
. Thus, one has
. Furthermore, the concept of in-group links and out-group links are presented in this paper, where in-group refers to that the links exist between nodes in the same subgroup, and out-group implies that the links exist between nodes in the different subgroups.
The following group consensus protocol is presented for each agent:
where
for all
otherwise,
for all
.
Assumption 1
.
Remark 2 Obviously, both cooperation interaction and competition interaction are considered for the agents, namely,
indicates that the agents vi and vj are cooperative, while
can be regarded as the agents vi and vj being competitive, which is close to a practical system. Assumption 1 implies that an interaction in-degree balance exists between two subgroups.
Definition 1[16] (Group consensus) Under the group consensus protocol (2), system (1) is said to realize group consensus asymptotically, if for any initial condition, the states of agents satisfy:
(i)
,
(ii)
,
. 2pt
Define the group consensus error of system (1) as follows:
For the sake of obtaining a compact form of the dynamics of the agents, one can denote
Then, the multi-agent system (
1) under protocol (2) can be represented as
where
is the Laplacian matrix of the graph representing communication topology.
Before putting forward the main results, a necessary lemma is stated.
Lemma 1[25] Under the proposed group consensus error (3), system (1) is transformed into the following reduced-order system:
where
and
. Additionally, it follows from Ref. [
25] that
with
,
, and
with
,
.
Remark 3 Through this analysis, if the reduced-order system (5) satisfies the condition
, then one can obtain that system (1) under protocol (2) will asymptotically achieve the group consensus.
2.3. Multi-agent systems subjected to cyber-attacksSince the mass data transmit through the channel of communication topology, which is vulnerable to attack, cyber-attacks will be introduced into the network environment.
First, the definition of cyber-attacks is given.
Definition 2[29 A cyber-attack is called “a successful but recoverable attack” if the attack changes the directed network G containing a spanning tree into
that does not contain a spanning tree. However, after a short period of time, the original directed path of G can be reverted. In other words,
can be reconstructed into G.
Figure 1 shows the communication topology graphs under the cyber-attacks. Obviously,
denotes the time when the cyber-attacks occur, and t2 refers to the time when the recovered system works. Specifically, from t1 to t2, the network G becomes paralyzed. Since the influence of the cyber-attacks vanishes at t2, the topology of network G will be reverted to its original setting during t2 to t3. Then next attack occurs at time t3. Without loss of generality, suppose that
is the moment of attack and the network works well during
, therefore, the network is paralyzed during
, and
is denoted as the duration of the attack.
It can be seen from Fig. 1 that the cyber-attacks will affect the communication channels; therefore, an attack will destroy original links. To achieve group consensus, it is necessary to satisfy the in-degree balanced condition for subgroups. Thus, in this paper we consider that the cyber-attacks just affect the links existing between nodes in the same subgroup, in other words, cyber-attacks will attack in-group links of
and
.
Let
represent the states of the agents, with state
and
, respectively. The
has a block form
where
and
, and
and
are the Laplacian matrices of two subgroups, respectively. Thus, the multi-agent system (
1) can be represented as
Apparently, according to Lemma 1, the decomposition of
is
where
,
,
, and
.
Then, group consensus error (5) can be rewritten equivalently as
In the time intervals
, suppose that in-group links of
and
are subjected to the
types of cyber-attacks simultaneously. Denote the corresponding Laplacian matrix of the communication topology
as
and
as
,
as a piecewise constant function. Then equation (7) can be rewritten as
and the dynamics of the group consensus error (9) can also be expressed as follows:
Remark 4 Since the network is subjected to different types of cyber-attacks, the values of
may be non-identical.
Assumption 2 Assume that there exist two positive constants Tm and TM, so that
,
.
Remark 5 On the one hand, in order to achieve group consensus with cyber-attacks, the stability of the attacked system should be available. Therefore, the Zenoʼs behavior must be excluded, which means that the dwell time
should be positive, accordingly, suppose that the value of lower bound Tm exists. On the other hand, it is impossible that the network is always paralyzed or working, then assumption of the existence of an upper bound for
is consistent with the realistic situation.
Assumption 3 Subgraph of each group in system (1) is assumed to contain a directed spanning tree.
Remark 6 For system (1), assume that a directed spanning tree exists in each group. Thus, information about the root node can be diffused in each group, which guarantees the feasibility of the group consensus.
Remark 7 Based on Definition 2, the cyber-attacks considered in this paper can be “successful” due to Assumption 3.
3. Main resultsIn this section, the main theoretical results are given and discussed.
3.1. Group consensus in multi-agent systems with two subgroupsTo achieve group consensus in multi-agent system under cyber-attacks, we introduce the following algorithm to obtain the corresponding parameters.
Algorithm 1
i) Solve the following linear matrix inequality (LMI)
then a matrix
will be obtained.
ii) Search for a positive constant α to satisfy the following LMI:
Next, the following main theorem is established.
Theorem 1 Consider that in-group links of
and
are subjected to cyber-attacks at the same time, and the cyber-attacks can be recovered. Suppose that Assumptions 1–3 hold, and LMIs (12) and (13) have feasible solutions. Furthermore, if the following condition holds:
where
, and
γ is determined by converting the optimization problem into minimizing
γ subject to
then the group consensus of multi-agent system (
1) with cyber-attacks can be achieved.
Proof Construct the Lyapunov function candidate as follows:
where
is a solution for inequality (
12) in Algorithm 1.
For
, with no cyber-attack occurring, according to Eq. (13), we have
When
, cyber-attacks affect in-group links, therefore, the corresponding topology graph will change. Take the derivative of each
V(
t) with respect to
t on the trajectory of error system (
11) and select
γ in view of optimization problem (
15), then we will have
By combining Eqs. (17) and (18), one can obtain that
Substituting condition (14) into the above inequality (19) leads to
According to the fact that
and time intervals
are bounded, we can easily conclude that
. This proof demonstrates that the group consensus of multi-agent system (1) with cyber-attacks is achieved.
Remark 8 Actually, the optimization problem (15) is a convex optimization issue, which can be directly solved by the “mincx” LMI toolbox in MATLAB. The optimization method is also applied to the control of complex dynamic networks in Ref. [32]. In this paper, we mainly use the optimization technique to solve the problem of group consensus.
Remark 9 Under condition (14), i.e.,
, the minimum γ can extend the range of feasible solutions for this condition.
In view of Remark 5, one can derive the following corollary.
Corollary 1 Consider that in-group links of
and
are subjected to cyber-attacks at the same time, and the cyber-attacks can be recovered. Suppose that Assumptions 1–3 hold, LMIs (12) and (13) have feasible solutions, and obtain a parameter γ by optimization problem (15). Furthermore, if the following condition holds:
where
, then group consensus of multi-agent system (
1) with cyber-attacks can be achieved.
3.2. Group consensus in multi-agent systems with multiple subgroupsIn this subsection, we mainly extend the above results to the consensus issue of multiple subgroups with cyber-attacks.
Suppose that the agents in system (1) can be partitioned into q groups, i.e., there are q proper subgroups
,
. The weighted directed graph
is considered to have a set of nodes
. Let
, then, the r-th subgroup Gr has the node set
and the state
. Besides, the indexes of agents in Gr are described as a finite set
.
For multi-agent system (1), we give the following consensus protocol:
where
,
, and
for all
, if nodes
vi and
vj are in different subgroups.
Next, the assumption, definition and lemma are given.
Assumption 4
,
.
Remark 10 Assumption 4 means that the in-degree balanced condition exists between any two subgroups.
Definition 3 Multi-agent system (1) with multiple subgroups is said to achieve group consensus asymptotically under protocol (22), if for any initial condition, the states of agents satisfy
Denote the group consensus error vector as
According to Lemma 1, one can obtain a lemma below.
Lemma 2 Under the proposed consensus error (23), system (1) is transformed into the following reduced-order system
where
Furthermore,
with
, and
with
.
Obviously, the error vector
can describe the error between the agents in the same subgroup.
Remark 11 Group consensus of system (1) with multiple subgroups under protocol (22) is equivalent to the asymptotical stability of error system (24), that is,
.
In the time intervals
, suppose that the in-group links of
,
are subjected to the
types of cyber-attacks simultaneously. Denote the Laplacian matrix of the communication topology
as
, then the Laplacian matrix of graph G will be expressed as
Based on the above analysis, error system (24) can be represented as
where
and
are the same as those in Lemma 2.
Then, the main theorem is established as follows.
Theorem 2 Consider that in-group links of
are subjected to cyber-attacks at the same time, and the cyber-attacks can be recovered. Suppose that Assumptions 2–4 hold, the LMIs (12) and (13) have feasible solutions. Moreover, if the following condition holds:
where
, and
is determined by the following optimization problem:
Minimize
subject to
then the group consensus of multi-agent system (
1) with multiple subgroups subjected to cyber-attacks can be achieved.
The proof of Theorem 2 is similar to that of Theorem 1, therefore, we omit it here.
4. Numerical simulationsIn this section, three simulation examples are provided to demonstrate the main results.
Example 1 (Two subgroups with one type of cyber-attack). Consider a multi-agent system (1) consisting of N = 14 agents. The communication graph is described in Fig. 2, where the blue agents belong to subgroup 1, and the green agents belong to subgroup 2. This topology is commonly existent in reality. Apparently, G satisfies Assumptions 1 and 3. The
is taken as the initial state.
For the network under cyber-attacks, the in-group links
,
,
,
,
are assumed to be destroyed. In the simulation, let
By using Algorithm 1 and Theorem 1, we can obtain
α = 0.1229 and
γ = 1.4931; the condition (
14) is satisfied. The state trajectories are shown in Fig.
3 and group consensus errors are shown in Fig.
4. It shows that the group consensus is reached on the proposed protocol (2).
Example 2 (Two subgroups with two types of cyber-attacks). Suppose that a multi-agent system contains 7 nodes with two subgroups G1 and G2, which is shown in Fig. 5(a), then Agents 1–3 will be in G1, and agents 4–7 in the other subgroup G2. Note that subgroups G1 and G2 satisfy in-degree balanced condition, and G satisfies Assumptions 1 and 3, then the initial state is set to be
.
Case 1 Without loss of generality, suppose that there are two kinds of cyber-attacks in subgroups G1 and G2, which are shown in Figs. 5(b) and 5(c) specifically, when the attack happens to the in-group links of G1 and G2. Let
. According to Algorithm 1 and optimization problem in Theorem 1, we can obtain α = 0.3492 and γ = 1.8631. Obviously, condition (14) is satisfied. Under the group consensus protocol (2), one can conclude that the group consensus under cyber-attacks is achieved. Figure 6 and 7 show the simulation results for the state variables and group consensus errors.
Case 2 If we add the edges of
,
,
, their weights are set to be −1, and the edges of
and
are deleted, the subgroups G1 and G2 are not in-degree balanced, Assumption 1 is not satisfied. In addition, we still choose the two types of cyber-attacks mentioned above. Even if the calculated α = 0.3428, γ = 1.3504 satisfy condition (14), the ultimate errors will not close to zero, which means that system (1) cannot achieve group consensus. The trajectories of the error vectors are plotted in Fig. 8.
Example 3 (Multiple subgroups). Suppose that a multi-agent system consists of 10 nodes with three subgroups, where agents 1–3 are in subgroup 1, agents 4–7 are in subgroup 2, and agents 8–10 are in subgroup 3. The original communication topology is shown in Fig. 9(a). Note that G1, G2, and G3 are in-degree balanced, and G satisfies Assumptions 3 and 4. If the cyber-attack happens to the in-group links of G1, G2 and G3, without loss of generality, we suppose that there is one type of cyber-attack given in Fig. 9(b). When an attack occurs, according to Algorithm 1 and optimization problem (15), we can obtain α = 0.8062 and γ2 = 4.9247. Finally, Figure 10 shows the state trajectory of the agents with initial value
, and the trajectories of group consensus error vectors are depicted in Fig. 11.
5. ConclusionsIn this paper, the group consensus problem for the multi-agent systems in the presence of cyber-attack has been investigated. When attacks happen, the network topology containing a directed spanning tree can be damaged. By using algebraic graph theory, linear matrix inequality, and Lyapunov stability theory, the group consensus conditions with cyber-attacks are established. The effectiveness of the theoretical results is demonstrated by numerical simulations. However, the network needs to satisfy the condition of in-degree balanced couple. Our future work will focus on studying the group consensus for more general topology and second-order multi-agent systems with cyber-attacks.
