Harmonic Modeling and Control under Variable-Frequency

TL;DR

This paper introduces a harmonic-domain framework for analyzing and controlling systems with variable fundamental frequency, enabling exact modeling, stability analysis, and control synthesis without conservative assumptions, exemplified on a motor system.

Contribution

It develops a novel harmonic-domain approach for variable-frequency systems, including a sliding Fourier decomposition, exact nonlinear models, and convex LMI conditions for linear systems.

Findings

Exact harmonic model for nonlinear systems under variable frequency.

Convex LMI conditions for stability and control of linear systems.

Successful application to a variable-speed motor example.

Abstract

This paper develops a harmonic-domain framework for systems with variable fundamental frequency. A variable-frequency sliding Fourier decomposition is introduced in the phase domain, together with necessary and sufficient conditions for time- domain realizability. An exact harmonic-domain differential model is derived for general nonlinear systems under variable frequency, without assumptions on the frequency variation. An explicit parameter-varying approximation is then obtained, along with a tight error bound expressed in terms of local relative frequency variation, providing a non-conservative validity criterion and clarifying the limitations of classical heuristics. A main result shows that, for linear phase-periodic systems with affine frequency dependence, stability analysis and control synthesis can be carried out without approximation and without assumptions on the frequency variation, provided the frequency evolves within a prescribed interval. As a consequence, both problems reduce to harmonic Lyapunov inequalities evaluated at the two extreme frequency values, yielding a convex LMI characterization. The framework is illustrated on a variable-speed permanent magnet synchronous motor.