Semidefinite approximations of the polynomial abscissa

TL;DR

This paper introduces semidefinite programming-based polynomial approximations for the polynomial abscissa, enabling more tractable control law design by approximating a challenging non-Lipschitz function.

Contribution

It proposes a hierarchy of convex semidefinite programs to approximate the polynomial abscissa with controlled complexity, converging to the true abscissa as degree increases.

Findings

Approximations converge in norm to the true abscissa

Hierarchical semidefinite programming provides controlled complexity

Approximations can be from above or below

Abstract

Given a univariate polynomial, its abscissa is the maximum real part of its roots. The abscissa arises naturally when controlling linear differential equations. As a function of the polynomial coefficients, the abscissa is H{\"o}lder continuous, and not locally Lipschitz in general, which is a source of numerical difficulties for designing and optimizing control laws. In this paper we propose simple approximations of the abscissa given by polynomials of fixed degree, and hence controlled complexity. Our approximations are computed by a hierarchy of finite-dimensional convex semidefinite programming problems. When their degree tends to infinity, the polynomial approximations converge in norm to the abcissa, either from above or from below.