Event-Based Consensus for General Linear Multiagent Systems under Switching Topologies

In this paper, event-triggered leader-following consensus of general linear multiagent systems under both fixed topology and switching topologies is studied. First, centralised and decentralised event-triggered control strategies based on neighbors’ state estimation are proposed under fixed topology, in which the controller is only updated at the time of triggering. Obviously, compared with the continuous time control algorithms, the event-triggered control strategies can reduce the communication frequency among agents effectively. Meanwhile, event-triggering conditions are derived for systems to achieve consensus by using the Lyapunov stability theory and model transformation method. /en, the theoretical results obtained under the fixed topology are extended to the switching topologies, and the sufficient conditions for the system to achieve leader-following consensus under the switching topologies are given. However, different from fixed topology, the control input of each agent is updated both at event-triggering and topology switching time. Finally, Zeno behaviors can be excluded by proving that the minimum triggering interval of each agent is strictly positive, and the effectiveness of the event-triggered protocol is verified by simulation experiments.


Introduction
In recent years, multi-agent systems have great potential of application in the fields of biology, engineering, physics, society and so on. Also its distributed cooperative control [1,2] has attracted more and more researchers' attention. For example, the controllability [3][4][5][6][7][8][9][10][11] and the consensus problem [12][13][14][15][16][17][18][19] are widely studied in multi-agent systems. Among them, as the basic problem of multi-agent cooperative control, consensus is widely used in formation control [20], cluster control, sensor network and other aspects, which is a research hot issue in the control discipline at present.
In practical applications, the information needed for cooperative control among agents is transmitted through the network. It is necessary to design a reasonable controller to ensure the control performance of the system due to the finite energy of the agent and the limited network bandwidth. It is well known that periodic sampling control [21][22][23] can save resources, but when the system runs in an ideal environment or the system state tends to be consensus gradually, it will cause unnecessary resource waste if the control task is executed periodically. In order to reduce this unnecessary waste of resources, a new simple event-triggered control strategy based on feedback mechanism was proposed in [24]. In short, the event-triggered control strategy means that the control task is executed as required. On the premise of ensuring the closed-loop system has certain performance, only can the task be executed once when a specific event occurs (such as the state error exceeds the preset threshold value). e advantage of event-triggered control strategies is that it can not only guarantee the performance of the system but also save the network and computing resources. At present, event-triggered mechanism has been applied to the research of consensus for multiagent systems effectively. For example, Dimarogonas and Frazoli and Dimarogonas et al. [25,26] studied the consensus of a first-order multiagent system in undirected topology, and Yan et al. [27] investigated the consensus of a second-order multiagent system based on event-triggered mechanism in directed topology. Eventtriggering conditions based on composite measurements were designed in [28] to address the consensus of a multiagent system. Hu et al. and Li et al. [29,30] discussed the leaderfollowing consensus of the second-order multiagent system. e centralized and decentralized event-triggered strategies were proposed in [31], which made the state of all agents converge to the same value gradually. In [32], a consensus control algorithm based on event-triggered mechanism was proposed for the nonlinear multiagent system. In [33], a new average consensus problem under event-triggered control strategy was set up, which limits the measurement error of each agent to a threshold that changes with time. Meng and Chen [34] studied the average consensus of the event-triggered multi-integrator systems under fixed and switching topologies and designed an event-triggered scheme based on quadratic Lyapunov function, which made each agent's state converge to it's initial average eventually. Xiao et al. [35] discussed the average consensus of the network of the integrator system with unidirectional information link. In order to reduce the communication cost, a distributed state consensus sampling data control scheme based on edge-event was proposed. In recent years, more and more attention has been paid to general linear systems. In [36], the consensus of general linear multiagent systems under integral event-triggered strategy has been considered. In [37], the consensus of general linear systems under fixed topology and switching topologies has been investigated. Hu et al. [38] studied the leader-following consensus of general linear multiagent system under fixed topology, and Zhu et al. [39] studied the consensus of general linear multiagent system under mixed event-triggering conditions.
With this background, we consider the event-triggered leader-following consensus of the general linear multiagent system under fixed topology and switching topologies. Event-triggered control mechanism is designed for each agent, respectively. Under the mechanism, multiagent systems can achieve leader-following consensus gradually. Moreover, a continuous event-triggering condition is proposed, which uses the state error between the follower and leader to design the triggering conditions of each agent under the fixed topology and switching topologies, respectively. e consensus problem is transformed into the stability problem by the method of model transformation, and the sufficient conditions for the system to achieve leader-following consensus are obtained by using Lyapunov stability theory. In addition, all the proposed event-triggered mechanism can exclude Zeno behavior. Finally, the accuracy of the conclusion is verified by simulation experiment. e structure of this paper is as follows. Section 2 introduces some concepts of the graph theory and system model. e leader-following consensus of the systems under the fixed/switching topologies is considered in Sections 3 and 4 In addition, the effectiveness of the results is shown through simulation experiment in Section 5. Section 6 summarizes this paper. e following notations are used in this paper. ⊗ denotes the Kronecker product. I N denotes the N-dimensional identity matrix. R n and R m×n indicate the set of n dimensional real vectors and m × n dimensional real matrices, respectively.

Preliminaries
2.1. eory of Graph. For a multiagent system composed of a leader and follower agents, its communication topology can be represented by an undirected graph G � (V, E), where V � 0, 1, . . . , N { }. 0 denotes the leader, and 1, . . . , N denote the followers. E ⊆ V × V denotes the edges set. e connection matrix between the follower agent i(i � 1, . . . , N) and leader 0 is D � diag a 10 , . . . , a N0 , where a i0 is the connection weight between the leader 0 and follower i. If a i0 > 0, the follower agent i can receive state information of leader 0; otherwise, a i0 � 0. e communication network among the followers is denoted by G � (V, E, A), where V � 1, 2, . . . , N { } and E ⊆ V × V are obtained from E by removing all edges among the leader 0 and followers in V, and A � (a ij ) ∈ R N×N is the weighted adjacency matrix of graph G, where a ij > 0 for (i, j) ∈ E if agent i obtains information from agent j. We assume that (i, i) ∉ E, and hence a ii � 0. For a given graph G with the adjacency matrix A, the Laplacian matrix used in this paper is L � D − A, where D is a diagonal matrix, its diagonal elements are d ii � j∈N i |a ij |, and define N i as the neighbor set of agent i in V. erefore, the elements of L are A path from the vertex i to vertex k is a sequence of adjacent edges in the form (i, i + 1), (i + 1, i + 2), . . . , (k − 1, k). e undirected graph is said to be connected if there exists a path between any two distinct vertices.

System
Model. Consider a multiagent system composed of the leader 0 and N followers. e dynamics of leader 0 is where x 0 (t) ∈ R n is the state and A ∈ R n×n is constant matrix. Accordingly, each follower has the following linear dynamic equation: where x i (t) ∈ R n and u i (t) ∈ R p are the state and input of the ith follower agent, respectively. A ∈ R n×n and B ∈ R n×p are constant matrices. Denote the initial state of the ith follower as x i (0).

Definition 1.
If there is a control input u i (t), the leader 0 and follower i for any initial state satisfy the following conditions: en, the leader (2) is said to be successfully tracked by follower (3).

Assumption 1.
e communication network topology G among followers is connected.
e pair (A, B) is stabilizable. Based on Assumption 2, there is a symmetric positive definite matrix P that satisfies the following algebraic Riccati and Lyapunov inequality with β > 0: Lemma 1 (see [37]). For an undirected and connected graph G, the eigenvalues of L are real and can be labelled as Lemma 2 (see [40]). For any x, y ∈ R and β > 0, one has the following property: Lemma 3 (see [41]). (Comparison Principle). Consider a differential equation

u) is continuous and satisfies the local Lipschitz condition in t. Let [t 0 , T) be the maximum existence interval of the solution u, where T can be infinite. If, for any
then v(t) ≤ u(t), t ∈ [t 0 , T).

Leader-Following Control of Multiagent Systems under Fixed Topology
In this part, we consider the leader-following control of multiagent systems (2) and (3) under the event-triggered strategy. Based on the general event-triggered control law, we put forward two kinds of piecewise continuous control mechanisms, which are centralized event-triggered mechanism and decentralized event-triggered mechanism with state estimation in order to minimize the frequency of controller updating. e analysis shows that under the two control mechanisms, multiagent system (3) can track the system (2) successfully with appropriate event-triggering function. e minimum interval between any two consecutive event-triggering instants under the two control mechanisms is greater than 0, and Zeno behavior can be excluded.

Centralized Event-Triggered Control Strategy.
Under the centralized event-triggered strategy, all agents i in system (3) are triggered synchronously at the time t k (k � 0, 1, . . . , ). At the triggering instants, all agents send their states information to neighbours and update the control law with the received state information. Compared with the control protocol in continuous time, each agent i only updates the control input at the event instants under the event-triggered mechanism. So, u i is a piecewise continuous function, and the updating frequency can be reduced. We consider the following control input for the ith follower: where t ∈ [t k , t k+1 ), K ∈ R p×n is the control gain matrix to be designed, and x i (t k ) is the sampling state of agent i at the kth triggering instant. Since there does not exist control input for leader 0, we take e event-triggering time sequence t k is determined by the following triggering functions: e state error between the follower and leader is defined as Combining (2), (3), and (12), we get _ rough the model transformation, the leaderfollowing control problem between systems (2) and (3) can be interpreted by the stability problem of system (13).
Next, we will give the following consensus conditions under the centralized event-triggering protocol (10).

Complexity
Proof. We consider a candidate Lyapunov function, as follows: Along with the trajectories of the state as described in (8), the time derivative of Lyapunov function is where where W � PBK.

Decentralized Event-Triggered Control Strategy.
e centralized event-triggered mechanism given in the previous section sets a global state error threshold for all agents. Once the system error reaches the threshold, all agents in the system perform control tasks at the same time. In this section, an error threshold based on the state of its neighbor node is set for each agent. When the state error of the agent reaches the set threshold, the agent triggers the event independently and executes the control task. e triggering time of the kth event of the i agent is defined as t i k (k � 0, 1, . . .). In the design of this section, it should be noted that the agent triggers asynchronously, that is, each agent has its own event-triggering sequence. e measurement error of agent i is defined as e i (t) � For a multiagent system composed of (2) and (3), we consider the following decentralized event-triggered control protocol: where t ∈ [t i k , t i k+1 ), t j k ′ � arg min l∈N,t≥t j l t − t j l represents the latest event-triggering time before t for agent j. According to (35), agent i will update control input u i at both its triggering instants (t i 0 , t i 1 , . . .) and neighbor agent j event instants (t j 0 , t j 1 , . . .). e event-triggering instant sequence t i k for agent i is determined by the following decentralized event-triggering function: where 0 < κ < 1, W � PBB T P. According to the definition of measurement error and x i (t) � x i (t) − x 0 (t), (35) can be rewritten as e(t)) . (38) Theorem 3. Under Assumption 1, the multiagent systems (3) with protocol (35) can track system (2) successfully under the event-triggering condition (36), where K � B T P and W � PBB T P.

Complexity 7
Proof. Define the Lyapunov function Following the same proof as that of eorem 1, the time derivation of V 2 along the trajectory of system (38) is obtained: According to (6) and event-triggering condition (36), we can find that It can be seen from the abovementioned formula that V 2 is not increasing; therefore, (42) at is to say, ||x(t)|| ≤ ��������������� 2(V 2 (0)/λ min (P)) � 0. According to LaSalle's invariance principle, we can obtain that system (38) can achieve consensus, that is, e proof is completed.

Theorem 4. Under the conditions of eorem 3, system (38) does not exhibit Zeno behavior. e interval between any two consecutive event-triggering instants of the system is not less than
Proof. It is similar to the proof of eorem 2. e event interval between t i k and t i k+1 is (‖e i (t)‖/‖x i (t)‖) which grows from 0 to ((1 − κ)(a i0 λ min (W)/(2d ii + a i0 )λ max (W))) 1/2 . e time derivative of (||e where L i is the row i of the Laplace matrix L. Consider that a nonnegative function ψ(t, ψ 0 ) satisfies _ ψ � (||A|| + ‖BK((L i + a i0 ) ⊗ I n )‖)(1 + ψ) 2 and ψ 0 � 0, according to Lemma 3, z i ≤ ψ(t, 0). It can be seen from (36), Hence, Similar to eorem 2, that Zeno behavior that does not occur can be proved by contradiction, which is omitted here.

Leader-Following Control of Multiagent Systems under Switching Topologies
In this part, we consider the extended case that the interconnection network switches according to signal σ(t) and is not connected all the time. It is worth noting that, unlike the fixed topology, the controller updates only when the event is triggered. In the switching topologies, the controller updates in the following two cases: (1) event-triggering instant. (2) Communication topology switching instant. e control input of the ith agent is defined as follows: where t ∈ [t i k , t i k+1 ). Different from control protocols (10) and (35), N i (t) and a ij (t) in (48) are changed under the switching topologies. Matrices L σ(t) and D σ(t) in G σ(t) represent Laplacian matrix and connection matrix between leader and agent, respectively. Switching signal σ(t): [0, ∞) ⟶ P is a piecewise continuous constant function, which is used to describe the switching law of communication topology. Also, G σ(t) : p ∈ P is a set of graphs that are switched within a finite set P � 1, 2, . . . { } in any finite time interval. Consider a nonempty and continuous infinite sequence [t s , t s+1 ), where k � 0, 1, . . . and t 0 � 0. Suppose that G σ(t) is switched only at and remains unchanged in t ∈ [t s , t s+1 ).

Remark 2.
It should be noted that graph G σ(t) may be connected or unconnected in interval [t s , t s+1 ).

Simulation
In this part, we consider the trajectories of the state errors between the follower and leader under the fixed topology and the switching topology, respectively, where the dynamic equations of the leader and the follower are given by (2) and (3), respectively, and the communication network topology among agents is shown in Figures 1 and 2. Assume that x i � [x i1 , x i2 ] T , and A and B are chosen as follows: , it is easy to prove that the

Example 1.
Under the centralized event-triggering protocol (10), the leader-following consensus of the multiagent system composed of (2) and (3) is considered. e communication network among agents is shown in Figure 1, and the corresponding weights are all 1. It can be seen from Figures 3 and 4 that followers can successfully follow the leader. Figure 5 shows the event instants of each follower with the centralized event-triggering protocol (10). It can be seen that protocol (10) can effectively reduce the number of communications among agents, thus reducing the waste of resources. Also, there is no Zeno behavior.
Example 2. In this example, we illustrate the leader-following consensus of the multiagent system under the distributed event-triggering protocol (35). e communication network among agents is shown in Figure 1. It can be seen from Figures 6 and 7 that followers can successfully follow the leader. Figure 8 shows the event triggering time of each follower under the decentralized event triggering protocol (35), and Zeno behavior is excluded. e simulation results verify eorems 3 and 4.
Example 3. Finally, the leader-following consensus of the multiagent system under the control protocol (48) is considered. e communication network among agents will randomly switch between G 1 and G 2 , as shown in Figure 2, where G 1 is a connected graph and G 2 is an unconnected graph. e state errors between the follower agent i and leader 0 are shown in Figures 9 and 10, respectively. It indicates that all followers can successfully follow the leader. Figure 11 shows the event-triggering instants of each follower under (48), and there is no Zeno behavior.

Conclusions and Future Work
In this paper, the leader-following control of general linear multiagent systems based on event-triggering mechanism under both fixed topology and switching topologies have been studied. Under the fixed topology, two different control protocols are designed in order to reduce waste of resources. Based on these two control protocols, we propose two different triggering functions, i.e., centralized event-triggering function and decentralized event-triggering function with state error between the follower and leader. When the triggering function exceeds 0, the agent will update the control input at the triggering instants. rough theoretical analysis, the sufficient conditions are derived for the system to achieve leader-following consensus under two control protocols and event-triggering conditions. e conditions obtained under fixed topology are extended to switching topologies (different from the fixed topology, the controller update at the triggering time, and also the switching time).
e results show that the conditions to achieve leader-following are also valid under switching topologies. Finally, it is proved that the system can realize leader-following control without Zeno behavior. e simulation results verify the effectiveness of the theoretical analysis. In the future, we will further study the leader-following control of the linear multiagent system with interference, delay, and other factors.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.