A hierarchy of LMI inner approximations of the set of stable polynomials

TL;DR

This paper introduces a hierarchy of LMI-based inner approximations for the set of stable polynomials, leveraging spectral properties of Toeplitz matrices, with convergence to a known lifted LMI set as the hierarchy level increases.

Contribution

It proposes a new hierarchy of convex LMI inner approximations for stable polynomials, providing a systematic approach with proven convergence to existing lifted LMI sets.

Findings

Hierarchy converges to the lifted LMI approximation as m increases

Simple sufficient conditions for positivity of trigonometric polynomials via LMI

Provides a practical method for approximating stable polynomial sets

Abstract

Exploiting spectral properties of symmetric banded Toeplitz matrices, we describe simple sufficient conditions for positivity of a trigonometric polynomial formulated as linear matrix inequalities (LMI) in the coefficients. As an application of these results, we derive a hierarchy of convex LMI inner approximations (affine sections of the cone of positive definite matrices of size $m$) of the nonconvex set of Schur stable polynomials of given degree $n < m$. It is shown that when $m$ tends to infinity the hierarchy converges to a lifted LMI approximation (projection of an LMI set defined in a lifted space of dimension quadratic in $n$) already studied in the technical literature.