Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design

TL;DR

This paper presents an elementary algorithm for constructing convex inner approximations of nonconvex semialgebraic sets, useful for fixed-order controller design, by leveraging polynomial optimization and semidefinite programming.

Contribution

The paper introduces a simple, practical algorithm to generate convex inner approximations of nonconvex sets using semidefinite programming, applicable to control system design.

Findings

Algorithm preserves convex boundaries effectively.

Applicable to fixed-order controller design.

Uses Gloptipoly 3 for polynomial optimization.

Abstract

We describe an elementary algorithm to build convex inner approximations of nonconvex sets. Both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials. Even though no optimality guarantees can be given (e.g. in terms of volume maximization for bounded sets), the algorithm is designed to preserve convex boundaries as much as possible, while removing regions with concave boundaries. In particular, the algorithm leaves invariant a given convex set. The algorithm is based on Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial optimization problems with the help of convex semidefinite programming (optimization over linear matrix inequalities, or LMIs). We illustrate how the algorithm can be used to design fixed-order controllers for linear systems, following a polynomial approach.