Convex Relaxation of Bilinear Matrix Inequalities Part II: Applications to Optimal Control Synthesis

The first part of this paper proposed a family of penalized convex relaxations for solving optimization problems with bilinear matrix inequality (BMI) constraints. In this part, we generalize our approach to a sequential scheme which starts from an arbitrary initial point (feasible or infeasible) and solves a sequence of penalized convex relaxations in order to find feasible and near-optimal solutions for BMI optimization problems. We evaluate the performance of the proposed method on the $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ optimal controller design problems with both centralized and decentralized structures. The experimental results based on a variety of benchmark control plants demonstrate the promising performance of the proposed approach in comparison with the existing methods.