Combining Convex-Concave Decompositions and Linearization Approaches for solving BMIs, with application to Static Output Feedback

TL;DR

This paper introduces a new optimization approach for solving bilinear matrix inequalities by decomposing them into convex parts and iteratively linearizing the concave component, with applications to static output feedback control.

Contribution

It presents a novel method combining convex-concave decomposition and linearization for BMIs, enabling convex subproblems with LMI constraints in control synthesis.

Findings

Effective on benchmark datasets from COMPleib library.

Applicable to various output feedback controller synthesis problems.

Outperforms some existing methods in convergence and solution quality.

Abstract

A novel optimization method is proposed to minimize a convex function subject to bilinear matrix inequality (BMI) constraints. The key idea is to decompose the bilinear mapping as a difference between two positive semidefinite convex mappings. At each iteration of the algorithm the concave part is linearized, leading to a convex subproblem.Applications to various output feedback controller synthesis problems are presented. In these applications the subproblem in each iteration step can be turned into a convex optimization problem with linear matrix inequality (LMI) constraints. The performance of the algorithm has been benchmarked on the data from COMPleib library.