On convexity of the frequency response of a stable polynomial

TL;DR

This paper establishes that the frequency response component of a stable polynomial can be exactly characterized by a linear matrix inequality, linking stability to convexity in the complex plane.

Contribution

It provides a precise algebraic characterization of the frequency response of stable polynomials using linear matrix inequalities.

Findings

The frequency response component including the origin is representable by an LMI if and only if the polynomial is stable.

This characterization connects polynomial stability with convex geometric properties in the complex plane.

The result offers a new perspective on stability analysis through convex algebraic geometry.

Abstract

In the complex plane, the frequency response of a univariate polynomial is the set of values taken by the polynomial when evaluated along the imaginary axis. This is an algebraic curve partitioning the plane into several connected components. In this note it is shown that the component including the origin is exactly representable by a linear matrix inequality if and only if the polynomial is stable, in the sense that all its roots have negative real parts.