Harmonic Modeling and Control under Variable-Frequency

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.