# Flocking of Multi-Agent System with Nonlinear Dynamics via Distributed Event-Triggered Control
## 1 Introduction  
Event-triggered control (ETC) is a kind of control strategy in which the controller is actuated by the occurrence of a specific event [1; 2; 3]. ETC has been widely used for the control of discrete-event systems, Petri Nets, and finite-state machines [4]. The possibility of reducing control costs and saving resources makes ETC appealing in resource-limited control systems. For example, the application of the ETC strategy in multi-agent system (MAS) equipped with microprocessors, such as physically distributed sensor/actuator networks [5], can lower the power consumption and prolong the lifetime of networks. Given these advantages, increasing attention has been paid to event-based control and communication [6; 7; 8; 9; 10; 11; 12; 13]. Within these studies, the practices of ETC have been carried out on the general sampled-data systems [6], sensor systems [7; 8; 9], signal source localization and navigation of multiple robots [10; 11; 12], and human mobility analysis [13]. Several results related to diverse ETC strategies have been reported in recent literature concerning multi-agent cooperative control [14; 15; 16; 17; 18; 19; 20; 21; 22; 23], which involves the consensus, the synchronization, etc. In [16; 17; 18; 19], distributed ETC formulation was established to study first-order consensus. In [20; 21], distributed ETC scheme was presented to study MAS with general linear dynamics. In [22; 23], a synchronization problem was investigated by using distributed ETC strategy. Flocking, which is a conspicuous behavior in life, can be regarded as another form of cooperative control in MAS. The research on flocking problems in MAS has attracted increasing attention (Refs. [24; 25; 26; 27; 28; 29]) due to its wide engineering applications, such as massive distributed sensing using mobile sensor networks, automated parallel delivery of payloads, and cooperative control of unscrewed air vehicles [30; 31]. In recent years, the application of ETC to study flocking problems has been a hot topic for the research community. Distributed ETC, as a new control strategy, is more in line with the biological characteristics of the interaction and decision-making behavior. In addition, it may potentially reduce the control costs compared to continuous-time flocking controlof artificial-intelligence-based MAS. Recently, some distributed ETC algorithms on the flocking problem have been proposed [32; 33; 34; 35]. Among these works, it is assumed that there is a virtual leader to be tracked in the flocking motion. For example, leader-follower flocking based on distributed event-triggered hybrid control has been studied in our previous works [32; 33]. Generally, the flocking motion is easier to achieve when all agents are able to communicate with the leader [34], while for the free (leaderless) flocking problem, limited communication and only local information pose a challenge for the coordination among agents. Motivated by the above consideration, a distributed ETC strategy is proposed to investigate the free flocking problem for MAS in this paper. In the proposed algorithm, we consider only the information of neighbors and no global information is required, which is different from [35]. Both the relative position and the relative velocity information are updated at discrete instants. Distributed triggering conditions are established to execute the update of both the position and velocity information in the controller. The contributions of this paper are as follows. Firstly, in contrast to the abovementioned works, the free flocking motion considering the Lipschitz nonlinear dynamics of each agent is investigated. Secondly, distributed ETC strategy has been introduced to realize flocking motion for MAS. Distributed ETC law with bounded action function is proposed. Thirdly, it is proved that the designed ETC strategy can guarantee a group of agents to achieve stable flocking motion. The rest of the paper is organized as follows. In Section 2, notations, preliminaries on graph theory, and the problem formulation are given.