Plane geometry and convexity of polynomial stability regions

TL;DR

This paper explores the convexity of stabilizing controller sets in two-parameter control problems, using algebraic geometry techniques to explain observed convexity and deriving an LMI formulation for design.

Contribution

It introduces a real algebraic geometry approach to explain convexity in stabilizing controller sets and provides an LMI formulation for two-parameter fixed-order controller design.

Findings

Stabilizing controller sets often appear convex in benchmark problems.

Elementary algebraic geometry techniques can explain this convexity.

A convex LMI formulation for two-parameter control design is derived.

Abstract

The set of controllers stabilizing a linear system is generally non-convex in the parameter space. In the case of two-parameter controller design (e.g. PI control or static output feedback with one input and two outputs), we observe however that quite often for benchmark problem instances, the set of stabilizing controllers seems to be convex. In this note we use elementary techniques from real algebraic geometry (resultants and Bezoutian matrices) to explain this phenomenon. As a byproduct, we derive a convex linear matrix inequality (LMI) formulation of two-parameter fixed-order controller design problem, when possible.