Harmonic Pole Placement

TL;DR

This paper introduces a novel method for designing harmonic state feedback controllers that assign desired pole locations in linear harmonic systems, using an infinite-dimensional Sylvester equation approach.

Contribution

It develops a new control design technique based on solving an infinite-dimensional Sylvester equation with invertibility conditions, extending classical finite-dimensional pole placement methods.

Findings

Method successfully applied to unstable linear periodic systems.

Provides conditions for invertibility of the Sylvester equation.

Includes a counterexample highlighting differences from finite-dimensional cases.

Abstract

In this paper, we propose a method to design state feedback harmonic control laws that assign the closed loop poles of a linear harmonic model to some desired locations. The procedure is based on the solution of an infinite-dimensional harmonic Sylvester equation under an invertibility constraint. We provide a sufficient condition to ensure this invertibility and show how this infinite-dimensional Sylvester equation can be solved up to an arbitrary small error. The results are illustrated on an unstable linear periodic system. We also provide a counterexample to illustrate the fact that, unlike the classical finite dimensional case, the solution of the Sylvester equation may not be invertible in the infinite dimensional case even if an observability condition is satisfied.