On the linear static output feedback problem: the annihilating polynomial approach

TL;DR

This paper explores the problem of stabilizing linear systems via static output feedback by analyzing the annihilating polynomials of the resulting system matrix, offering solutions for special cases and proposing computational methods for the general case.

Contribution

It introduces a novel approach using annihilating polynomials to address the static output feedback stabilization problem, providing solutions for single input/output systems and rank-one feedback matrices.

Findings

Simple solutions for single input or output systems.

Characterization of annihilating polynomials for rank-one feedback matrices.

Numerical evidence supporting the approach's plausibility.

Abstract

One of the fundamental open problems in control theory is that of the stabilization of a linear time invariant dynamical system through static output feedback. We are given a linear dynamical system defined through \begin{align*} \mydot{w} &= Aw + Bu y &= Cw . \end{align*} The problem is to find, if it exists, a feedback $u=Ky$ such that the matrix $A+BKC$ has all its eigenvalues in the complex left half plane and, if such a feedback does not exist, to prove that it does not. Substantial progress has not been made on the computational aspect of the solution to this problem. In this paper we consider instead `which annihilating polynomials can a matrix of the form $A+BKC$ possess?'. We give a simple solution to this problem when the system has either a single input or a single output. For the multi input - multi output case, we use these ideas to characterize the annihilating polynomials when $K$ has rank one, and suggest possible computational solutions for general $K.$ We also present some numerical evidence for the plausibility of this approach for the general case as well as for the problem of shifting the eigenvalues of the system.