Optimal Robust Stabilization and Dissipativity Synthesis by Behavioral Interconnection

Given a nominal plant, together with a fixed neighborhood of this plant, the problem of robust stabilization is to find a controller that stabilizes all plants in that neighborhood (in an appropriate sense). If a controller achieves this design objective, we say that it robustly stabilizes the nominal plant. In this paper we formulate the robust stabilization problem in a behavioral framework, with control as interconnection. We also formulate a relevant behavioral $\mathcal{H}_{\infty}$ synthesis problem, which will be instrumental in solving the robust stabilization problem. We use both rational and polynomial representations for the behaviors under consideration. Necessary and sufficient conditions for the existence of robustly stabilizing controllers are obtained using the theory of dissipative systems. We will also find the optimal stability radius, i.e., the smallest upper bound on the radii of the neighborhoods for which there exists a robustly stabilizing controller. This smallest upper bound is expressed in terms of certain storage functions associated with nominal control system.