Event-triggered consensus control for general second-order multi-agent systems

The consensus problem of multi-agent systems with general second-order dynamics is studied. A distributed event-triggering strategy is proposed to reduce the communication frequency of agents and the update frequency of event-triggering controller. Under the fixed topology, a consensus protocol and event-triggering function are designed for each agent. The sufficient conditions of consensus are obtained by the stability theory, and the theoretical proof of excluding Zeno behaviour is presented. Finally, a simulation example is given to illustrate the effectiveness of the obtained results.


Introduction
In recent years, the cooperative control of multi-agent systems has become the research focus of many scholars Ji & Yu, 2017;Lou et al., 2020;Qin & Yu, 2013;Qu et al., 2020;Zou et al., 2019Zou et al., , 2020. As an important basis of cooperative control, consensus is widely used in robot formation control, UAV task assignment, smart grid and so on. Therefore, great achievements have been obtained in the study of consensus problem Richert & Cortés, 2013;Sakurama & Ahn, 2019;Sun et al., 2020Sun et al., , 2019Zhang et al., 2019).
In the early research, in order to achieve consensus of multi-agent systems, it is usually assumed that agents exchange information continuously with their neighbours through the local network. However, in reality, the reserve energy of agents and the bandwidth of the network are limited. In order to ensure the stability of system performance and reduce the consumption of resources, a new control method needs to be designed. After leading into the event-triggered control strategy, the agent transmits information after completing specific tasks for the controller to update the received information, which effectively avoids continuous information exchange. Based on the event-triggered control, a centralized control strategy was proposed after the firstorder multi-agent system was studied in Dimarogonas and Johansson (2009), and all agents share a triggering function. In Dimarogonas et al. (2012), a distributed control strategy was proposed to design a triggering function for each agent. Compared with Dimarogonas and Johansson (2009), the distributed control can reduce the communication frequency more effectively. In Xie et al. (2015), CONTACT Zhijian Ji jizhijian@pku.org.cn the consensus problem of the second-order multi-agent system under the directed graph is studied. The centralized and distributed control strategies were given respectively, and a new control protocol was designed.
In Li et al. (2015), the consensus of second-order tracking control was studied under directed fixed topology and switching topology, and a new distributed eventtriggered sampling scheme was proposed. The consensus problem of tracking control for two order multi-agent systems with nonlinear dynamics was studied in Zhao et al. (2018). In Cao et al. (2015), the consensus problem of second-order multi-agent systems was discussed by using sampling control and edge event driven control strategies. The mixed control strategy of periodic sampling and event driven was given in  and Cao et al. (2016). A fast convergence method for achieving consensus was studied in Qu et al. (2018). In Liang et al. (2019), the consensus problem of general linear systems based on the information of eventtriggered state and event-triggered observer was studied. Hou et al. (2017) discussed the consensus conditions of general second-order multi-agent systems with communication delay. According to the above work, at present, the research on the consensus problems of second-order systems is mostly focused on second-order integrators, while the results about general second-order systems are relatively few. And in the study of event-triggered control results, continuous communication still exists in eventtriggered detection. Therefore, we further study the event-triggered consensus problems of general secondorder systems, and propose a new event-triggered control scheme. The triggering conditions in the scheme only use the information received at the event-triggering instant, and Zeno behaviour is excluded. The sufficient conditions for the system state to achieve consensus are obtained.
The paper is organized as follows: Section 2 gives the definitions that will be used in this paper. Section 3 discusses the consensus conditions based on eventtriggered control. The simulation example in Section 4 verifies the validity of the theoretical derivation. Finally, Section 5 presents the conclusion of this paper.
Notations: In this paper, the symbol R n represents the n dimensional real vectors space. The symbol R m×n represents the sets of m × n dimensional real matrix. I N denotes the N dimensional identity matrix. 1 indicates an appropriate dimensional column vector composed of 1.

Preliminaries
The information exchange topology network among n agents can be depicted by an undirected graph G = {V, E} with N nodes. V = {1, 2, . . . , N} and E ⊆ V × V is the set of nodes and the set of edges, respectively. ε i,j ∈ E means that nodes i and j are neighbours of each other, and node i can receive information from node j. If there is a path between a point and all other points, then the undirected graph is connected. The associated adja- In this paper, we consider a multi-agent system composed of N agents, each agent with general second-order linear dynamics is described as the position, velocity and control input of the ith agent, respectively. a and b denote any real numbers. This paper will discuss the consensus problem of system (1) . . , N, so that the system (1) can achieve consensus under eventtriggered control mechanism and avoid Zeno behaviour.
Assumption 2.1: The undirected topology G is connected.

Event-triggered control
In this section, we propose a distributed event-triggered control strategy. t i k denotes the kth event-triggering instant of agent i. The measurement error of agent i is defined as The consensus protocol designed for each agent is (2) where k 1 and k 2 are nonnegative real numbers. Eventtriggering instant sequence {t i k } of agent i is defined as: That is, f i (t) is the trigger function, where η = min{ k 1 56 , k 2 56 }, μ = max{k 1 , k 2 } and ν are positive real number.
Remark 3.1: The event-triggering condition (4) in this article utilizes the latest event-triggered state values. The introduction of exponential term can exclude Zeno behaviour. However, the event-triggering conditions in the existing references generally utilized the real-time state values of each agent and its neighbours. Therefore, the event-triggering condition (3) effectively reduces the communication frequency. (1) and (2), we can get the error vector z satisfying the following dynamics: The above inequality satisfies the following equivalent conditions: Theorem 3.1: Consensus protocol (2) and event-triggering condition (3) enable multi-agent system (1) to achieve consensus if Assumption 2.1 holds and parameters a, b, k 1 and Proof: Consider a Lyapunov functions constructed as follows: where δ = k 1 + k 2 , and because of I N > 0, δL + I N − I N ≥ 0, according to Lemma 3.3 , V(t) ≥ 0. The derivative of V 1 along the trajectory of (4) iṡ From a ij = a ji , z xi − z xj = x i − x j =x i −x j − e xi + e xj and Lemma 3.2, −z T x Lx and z T x Lṽ in (6) can be amplified and transformed respectively.
Similarly, according to a ij = a ji ,z vi − z vj = v i − v j =ṽ i − v j − e vi + e xi and Lemma 3.2, we can obtain Substitute (10)-(13) into (9) to geṫ Choose 1 2 ≤ k 1 k 2 ≤ 2. On the basis of Lemma 3.1, we can get thaṫ According to Lemma 3.3, if the above conditions are established, the conditions should be satisfied as follows: Obviously, under the conditions of (5)- (7),V(t) ≤ 0. The error system (4) is asymptotically stable, and the consensus of multi-agent system (1) can be achieved.

Event interval analysis
Next, the theorem of excluding Zeno behaviour is proposed.

Theorem 3.2:
The multi-agent system (1) does not exhibit Zeno behaviour under the consensus protocol (2) and the event-triggering condition (3).
For agent i, the event-triggering function (3) shows that the event interval is the time of e 2 xi + e 2 vi from 0 to the threshold. From the definition of e xi , e vi , (1) and (3), we can get that the derivative of e 2 xi + e 2 vi with respect to t ∈ [t i k , t i k+1 ) is as follows: It is known that c, w i are bounded positive real numbers. According to the above formula, we can get Consider a nonnegative function ϕ : [0, ∞) → R ≥0 , which satisfies the following relatioṅ The solution of the above differential equation is ϕ(t) = ω i c (e ct − 1). According to Lemma 3.4, we can obtain e 2 xi (t) + e 2 vi (t) ≤ ϕ(t − t i k ). It can be seen from the trigger function that if Then f i (t) ≤ 0. Therefore, the lower bound of the interval between event-triggering times t i k and t i k+1 of agent i can be determined by the time when ϕ(t − t i k ) increases from 0 to e −νt d i . The lower bound τ i k = t i k+1 − t i k can be obtained by the following equation: The solution of the above equation is Assuming that Zeno behaviour occurs, there is a positive constant t * satisfying lim k→∞ t i k = t * . Let δ = 1 c ln(1 + c w i d i e −νt * ), then τ i k ≥ δ. According to lim k→∞ t i k = t * , there exist a positive integer N * for ∀k ≥ N * to make t * − δ < t i k ≤ t * . Thus, t * < t i k + τ i k ≤ t i k+1 when k ≥ N 0 . This is contradictory to t i k+1 ≤ t * , k ≥ N 0 . Therefore, Zeno behaviour is strictly excluded.

Simulation
In this section, the results are verified by simulation experiments. A multi-agent system consisting of 5 agents is considered. Its topology G is shown in Figure 1. The Laplacian matrix is A multi-agent system (1), (2) based on event triggered control (3) is considered. Choose the parameters k 1 = 7, k 2 = 8, a = 0.15, b = 0.1. The initial state of the system is Figures 2 and 3 show the simulation results based on event-triggering condition (3) and event-triggering conditions utilizing real-time state, respectively. In contrast, the communication frequency of Figure 2 is lower.
It can be seen from Figure 2 that under the eventtriggered control, the position and velocity of each agent achieve consensus. And event-triggering time instants of each agent under the event-triggering condition (3) show that the proposed control strategy effectively reduces the communication frequency between agents and reduces the communication consumption of the system.

Conclusions
In this paper, the consensus problem of general secondorder multi-agent systems is considered. In order to reduce the communication consumption, the event trigger control protocol is designed so that each agent does not need to update the control input at any time, and its trigger condition update does not need to communicate with each neighbour at any time. Through theoretical analysis, it is proved that the system can achieve consistency under this control protocol. And the Zeno behaviour is strictly excluded. The correctness of theoretical analysis is verified by simulation examples. In the future work, we will focus on extending the results to unknown inputs problems and time-delay systems.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported by the National Natural Science Foundation of China [grant number 61873136].