LMI sufficient conditions for the consensus of linear agents with nearly-periodic resets

The paper focuses on the problem of consensus in the framework of hybrid systems. We consider networks of scalar agents interconnected via directed graphs. Precisely, the network consists of a large number of agents belonging to several clusters. Each cluster is represented by a fixed directed strongly connected graph and almost all the time there is no link between different clusters. In each cluster there exists a specific agent called leader. At specific instants, the leaders interact via a fixed directed strongly connected graph. In this model the agents have continuous dynamics but the states of the leaders are reseted at the instants when they interact between them. In the paper, we first characterize the consensus value of this model and show it depends only on the initial condition and the interaction topologies. Next, we provide sufficient condition in Linear Matrix Inequality (LMI) form for the global uniform exponential stability of the consensus in presence of an almost periodic reset rule. The numerical implementation of this LMI condition requires a polytopic embedding which is provided before the illustrative example.