Sequential convex relaxation for robust static output feedback structured control

We analyse the very general class of uncertain systems that have Linear Fractional Representations (LFRs), and uncertainty blocks in a convex set with a finite number of vertices. For these systems we design static output feedback controllers. In the general case, computing a robust static output feedback controller with optimal performance gives rise to a bilinear matrix inequality (BMI). In this article we show how this BMI problem can be efficiently rewritten to fit in the framework of sequential convex relaxation, a method that searches simultaneously for a feasible controller and one with good performance. As such, our approach does not rely on being supplied with a feasible initial solution to the BMI. This sets it apart from methods that depend on a good initial, feasible starting point to progress from there using an alternating optimization scheme. In addition to using the proposed method, the controller matrices can be of a predetermined fixed structure. Alternatively, an L₁ constraint can be easily added to the optimization problem as a convex variant of a cardinality constraint, in order to induce sparsity on the controller matrices.