Inner approximations for polynomial matrix inequalities and robust stability regions

TL;DR

This paper introduces a hierarchy of LMI-based inner approximations for nonconvex polynomial matrix inequality sets, enabling better robust control design with convergence guarantees and flexibility in convexity constraints.

Contribution

It develops a hierarchy of LMI problems for inner approximations of PMI feasible sets that converge strongly to the original set, with options for nesting and convexity.

Findings

Inner approximations converge to the original feasible set

Hierarchy can be made nested or convex

Exploiting stability region geometry improves convergence

Abstract

Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modelled by parametrized polynomial matrix inequalities (PMI). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner approximations modelled by a single polynomial sublevel set. Those inner approximations converge in a strong analytic sense to the nonconvex original feasible set, with asymptotically vanishing conservatism. One may also impose the hierarchy of inner approximations to be nested or convex. In the latter case they do not converge any more to the feasible set, but they can be used in a convex optimization framework at the price of some conservatism. Finally, we show that the specific geometry of nonconvex polynomial stability regions can be exploited to improve convergence of the hierarchy of inner approximations.