Controlling Identical Linear Multi-Agent Systems Over Directed Graphs

We consider the problem of synchronizing a multiagent system (MAS) composed of several identical linear systems connected through a directed graph. To design a suitable controller, we construct conditions based on Bilinear Matrix Inequalities (BMIs) that ensure state synchronization. Since these conditions are non-convex, we propose an iterative algorithm based on a suitable relaxation that allows us to formulate Linear Matrix Inequality (LMI) conditions. As a result, the algorithm yields a common static state-feedback matrix for the controller that satisfies general linear performance constraints. Our results are achieved under the mild assumption that the graph is time-invariant and connected.