A Suboptimality Approach to Distributed $\mathcal{H}_2$ Optimal Control

This paper deals with the distributed $\mathcal{H}_2$ optimal control problem for linear multi-agent systems. In particular, we consider a suboptimal version of the distributed $\mathcal{H}_2$ optimal control problem. Given a linear multi-agent system with identical agent dynamics and an associated $\mathcal{H}_2$ cost functional, our aim is to design a distributed diffusive static protocol such that the protocol achieves state synchronization for the controlled network and such that the associated cost is smaller than an a priori given upper bound. We first analyze the $\mathcal{H}_2$ performance of linear systems and then apply the results to linear multi-agent systems. Two design methods are provided to compute such a suboptimal distributed protocol. For each method, the expression for the local control gain involves a solution of a single Riccati inequality of dimension equal to the dimension of the individual agent dynamics, and the smallest nonzero and the largest eigenvalue of the graph Laplacian.