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Harmonic Modeling and Control under Variable-Frequency

Maxime Grosso, Pierre Riedinger, Jamal Daafouz, Serge Pierfederici, Hicham Janati Idrissi, Blaise Lapôtre

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.

Harmonic Modeling and Control under Variable-Frequency

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.
Paper Structure (18 sections, 2 theorems, 106 equations, 7 figures, 1 table)

This paper contains 18 sections, 2 theorems, 106 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Fixed-Frequency Harmonic Modeling
  3. The variable-frequency SFD
  4. Exact Variable-Frequency Harmonic Model
  5. Parameter-Varying (PV) Approximation
  6. An approximated PV harmonic model
  7. Analysis and comparison with classical heuristics
  8. Affine Linear phase-periodic Systems
  9. Exact Lyapunov stability condition
  10. Harmonic control design
  11. Classical harmonic controllers as special cases
  12. Application to Variable-Speed Motor Drives
  13. PMSM as a phase-periodic system
  14. Control objectives and synthesis
  15. Simulation results
  16. ...and 3 more sections

Key Result

Corollary 1

Let $\mathcal{P}\xspace$ and $P(\theta)$ satisfy the conditions of Theorem main. Then the Lyapunov functions $\mathcal{V}\xspace(X(t)) = {X(t)}^* \mathcal{P}\xspace X(t)$ and $v(t,x) = {x(t)}^\top P(\theta(t)) x(t)$ guarantee asymptotic stability of the systems eq:harmo and eq:auto, respectively, fo

Figures (7)

  • Figure 1: $\epsilon(t)$ provides an accurate measure of the modeling error, whereas the heuristic $\vert \dot{\omega}/\omega\vert$ and $\dot\omega$ are poor indicators.
  • Figure 2: Phase currents $i_\mathrm{abc}$, current $i_{dq}$, rotor speed $\omega_m$ (blue), $\omega_m^{\mathrm{ref}}$ (yellow) and $\omega_{m,0}^{\mathrm{ref}}$ (red), load torque $\Gamma_L$ and the applied control $u=v_\mathrm{abc}$.
  • Figure 3: Magnitude of harmonics illustrating targeted suppression in $i_a$ and $i_d$. Top to bottom: $I_{a,k}$, $I_{d,k}$, $\Omega_{m,k}$$\Gamma_k$ and $V_{a,k}$, $k=0,1,\ldots,8$.
  • Figure 4: Simulation results without harmonic mitigation: the phase currents $i_{\mathrm{abc}}$, the $dq$ currents $i_{dq}$, the rotor speed $\omega_m$ and the applied control $v_{abc}$, for the same load torque and reference speed profile as in the previous case.
  • Figure 5: Magnitude of harmonics without harmonic mitigation objective. Top to bottom: $I_{a,k}$, $I_{d,k}$, $\Omega_{m,k}$ and $V_{a,k}$, $k=0,1,\ldots,8$.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 1
  • Definition 5
  • Definition 6
  • Definition 7
  • Corollary 1
  • Remark 2
  • ...and 2 more