Synthesis of H₂ optimal static structured controllers: Primal and dual formulations

We consider the design of H₂ optimal static structured feedback gains for large-scale interconnected systems. The design of distributed controllers with access to measurements of a small number of the subsystems imposes particular sparsity constraints on the feedback gains. For this nonconvex constrained optimal control problem, we study both the primal and dual formulations to obtain optimality bounds. We exploit the sparsity structure present in large-scale systems by implementing an efficient quasi-Newton algorithm to solve the primal problem. We employ the subgradient method to solve the dual problem and obtain a lower bound for the optimal value of the performance index. Surprisingly, in many problems of practical interest, the upper bounds from solving primal problems and the lower bounds from solving dual problems are almost identical, suggesting the lack of duality gap in these applications and that the globally optimal structured gains have in fact been attained.