Probabilistic Discrete Time Robust H2 Controller Design

Optimal {{\mathcal{H}}_2} control theory is appealing, since it allows for optimizing a performance index frequently arising in practical situations. Moreover, in the state feedback case, the resulting closed loop system has an infinite gain margin and a phase margin of at least 60o. However, these properties no longer hold in the output feedback case, where it is well known that there exist cases where the system is arbitrarily fragile. Motivated by this observation, since the early 1980's a large research effort has been devoted to the problem of designing robust {{\mathcal{H}}_2} controllers. To this effect several relaxations of the original problem have been introduced, but all of these lead to conservative solutions. Surprisingly, the original problem remains, to date, still open. To address this issue, in this paper we present a randomization based algorithm that seeks to solve a relaxation of the original problem. Contrary to existing approaches, the performance of the resulting controller can be made - in a sense precisely defined in the paper - arbitrarily close to the optimal one. These results are illustrated with an academic example.