Hierarchical $\mathcal{H}_{2}$ Control of Large-Scale Network Dynamic Systems

Standard H₂ optimal control of networked dynamic systems tend to become unscalable with network size. Structural constraints can be imposed on the design to counteract this problem albeit at the risk of making the solution nonconvex. In this paper, we present a special class of structural constraints such that the H₂ design satisfies a quadratic invariance condition, and therefore can be reformulated as a convex problem. This special class consists of structured and weighted projections of the input and output spaces. The choice of these projections can be optimized to match the closed-loop performance of the reformulated controller with that of the standard H₂ controller. The advantage is that unlike the latter, the reformulated controller results in a hierarchical implementation which requires significantly lesser number of communication links. We illustrate our design with simulations of a 500-node network.