Necessary and Sufficient Conditions for Harmonic Control in Continuous Time

TL;DR

This paper establishes precise conditions under which harmonic controls in continuous time have valid time-domain representatives, enabling accurate control design and analysis for linear and bilinear systems.

Contribution

It provides a rigorous mathematical framework linking harmonic controls with their time-domain counterparts, including conditions for existence and equivalence in LTP and bilinear systems.

Findings

One-to-one correspondence between functional spaces and control representatives.

Equivalence between Carathéodory solutions and harmonic differential models.

Design of stabilizing harmonic control laws for practical systems.

Abstract

In this paper, we revisit the concepts and tools of harmonic analysis and control and provide a rigorous mathematical answer to the following question: when does an harmonic control has a representative in the time domain ? By representative we mean a control in the time domain that leads by sliding Fourier decomposition to exactly the same harmonic control. Harmonic controls that do not have such representatives lead to erroneous results in practice. The main results of this paper are: a one-to-one correspondence between ad hoc functional spaces guaranteeing the existence of a representative, a strict equivalence between the Carath{\'e}orody solutions of a differential system and the solutions of the associated harmonic differential model, and as a consequence, a general harmonic framework for Linear Time Periodic (LTP) systems and bilinear affine systems. The proposed framework allows to design globally stabilizing harmonic control laws. We illustrate the proposed approach on a single-phase rectifier bridge. Through this example, we show how one can design stabilizing control laws that guarantee periodic disturbance rejection and low harmonic content.