Probability-One Homotopy Algorithms for Robust Controller Analysis and Synthesis with Fixed-Structure Multipliers

Abstract

To enable the development of M-K (i.e., multiplier-controller) iteration schemes that do not require (suboptimal) curve fitting, mixed structured singular value analysis tests that allow the structure of the multipliers to a priori be specified, have been developed. These tests have recently been formulated as linear matrix inequality (LMI) feasibility problems. The least conservative of these tests always results in unstable multipliers and hence requires a stable coprime factorization of the multiplier before the control synthesis phase of the M-K iteration. This paper first reviews the LMI formulations of robustness analysis. It then develops alternative formulations that directly synthesize the stable factorizations and are based on the existence of positive definite solutions to certain Riccati equations. These problems, unlike the LMI problems, are not convex. The feasibility problem is approached by posing an associated optimization problem that cannot be solved using standard descent methods. Hence, we develop probability-one homotopy algorithms to find a solution. These results easily extend to provide computationally tractable algorithms for fixed-architecture, robust control design, which appear to have some advantages over the bilinear matrix inequality (BMI) approaches resulting from extensions of the LMI framework for robustness analysis.