Average Sampled-Data Consensus Driven by Edge Events Feng Xiao 1 , Xiangyu Meng 2 , Tongwen Chen 2 1. School of Automation, Beijing Institute of Technology, Beijing 100081, China 2. Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta T6G 2V4, Canada Abstract: This paper considers the average consensus problem in networks of multiple integrators with unidirectional informa- tion links. To reduce the communication cost, we set up a scheme of sampled-data control driven by edge events for distributed state consensus. These edge events are defined independently for each information link, and their occurrence activates the mu- tually state sampling and controller update of the corresponding two neighboring agents. A set of event-triggering rules are first proposed for the asynchronous data sampling. They are implemented in a complete distributed fashion and no more information exchange is needed between event times. Then this result is further revised to incorporate periodically time-driven event detec- tion. This treatment eliminates the possibility of infinitesimal inter-event time periods and also makes the presented protocol valid in the traditional sampled-data control framework. Key Words: Multi-agent systems, sampled-data consensus, event-driven control, edge events. 1 Introduction Considering communication constraints is an important is- sue in design of distributed consensus protocols of multi- agent systems, especially in cases with unreliable informa- tion channels, and limited data sensing and transmitting. In- tuitively, the sampled-data control technique offers an effec- tive tool to deal with such concerns [1]. It relaxes the re- quirement of continuous signal transmission to the intermit- tent interaction at some discrete time instants, and thus re- duces the communication cost tremendously. Furthermore, sampled-data control protocols also improve the robustness of systems against, for example, information link failure and transmission time-delays. In [2], by using periodic sampling and zero-order hold devices, Xie et al. proposed a sampled-data control proto- col, derived from the continuous-time linear consensus pro- tocol studied in [3], and presented algebraic-type necessary and sufficient conditions for networks with fixed topologies and with or without sampling delays. Xie et al. also pro- posed a sufficient condition for average consensus with sam- pling delays and switching topologies with the help of a common Lyapunov function [4]. In [5], Cao et al. inves- tigated an asynchronous sampled-data version of the Vic- sek model, where each agent sampled the headings of its neighbors at some discrete event times and changed its head- ing from one way-point to the other in a monotonic and piecewise-continuous manner. By the concept of “analytic synchronization”, sufficient conditions were presented for the consensus convergence. Another version of sampled- data single-integrator network models was investigated in [6], where the control input applied to each agent was a lo- cal state feedback, decoupled from other agents, and data exchange and parameter adjustment only occurred at sam- pling times; the authors presented sufficient conditions in terms of jointly connected topologies and bounded sampling This work was supported by the Natural Sciences and Engineering Re- search Council of Canada, a University of Alberta Killam Postdoctoral Fel- lowship, NSFC (60904062, 60925011), and SRFDP (20091101120019). E-mail addresses: fxiao@ualberta.ca (F. Xiao), xmeng2@ece.ualberta.ca (X. Meng), tchen@ualberta.ca (T. Chen). Feng Xiao is currently a Postdoctoral Fellow in the Department of Elec- trical and Computer Engineering, University of Alberta. time periods. It can be observed that all the data sampling schemes under the above protocols were scheduled at some specific time instants; in other words, they were time-driven. The same idea of data-sampling schedule was also employed in many results on double-integrator networks [7–10]. Un- surprisingly, the event-driven control of multi-agent systems is also attracting the interest of researchers due to its many favorable advantages over the pure time-driven control in many real applications with regard to communication cost. In [11], Dimarogonas et al. designed several event-driven controllers for the first-order consensus problem, whose up- date depended on the ratio of a certain measurement er- ror with respect to the norm of a function of the state; to avoid continuous monitoring of the measurement error, these controller laws were further revised by a self-triggering ap- proach. In this paper, we set up a scheme of sampled-data con- trol driven by edge events for distributed state consensus. We model the multi-agent system by an undirected graph, in which each edge connecting two agents represents the communication link between them. With this understand- ing, we define edge events independently for the commu- nication links, and their occurrence will activate the mutual state sampling and controller update of the corresponding two neighboring agents. This feature constitutes the first as- pect of novelty of this paper. A similar idea can be seen in [12], where an asynchronous distributed optimization prob- lem on a complete graph was addressed under the assump- tion that the each agent had the knowledge of its estimated states by other agents. Note that in [11], the communication- triggering events were defined with respect to agents, and each event triggered the communication of the agent with all its neighbors. We first propose a set of event-triggering rules for the asynchronous data sampling and controller up- date. But these rules cannot guarantee a lower bound on the inter-event time periods. Then we remove this limitation and present a novel distributed event-triggering principle with periodically time-driven event detection, which is applica- ble in the traditional sampled-data control framework. With the above two kinds of principles, the event-triggering condi- tions are only checked at specified discrete time instants, and 3URFHHGLQJV RI WKH VW &KLQHVH &RQWURO &RQIHUHQFH -XO\   +HIHL &KLQD 6239