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A harmonic framework for the identification of linear time-periodic systems

Flora Vernerey, Pierre Riedinger, Andrea Iannelli, Jamal Daafouz

TL;DR

This work tackles the identification of continuous-time linear time-periodic systems by exploiting a harmonic modelling approach that maps an LTP system to an infinite-dimensional LTI representation with a block Toeplitz structure. By truncating the harmonic expansion and focusing on a central strip, the authors reduce the problem to a finite-dimensional linear least-squares problem that provably converges to the true infinite-dimensional solution with arbitrarily small error. The method yields several advantages, including avoiding derivative computations and providing robustness to noise and instability, as demonstrated on finite and infinite phasor-order examples and an unstable wind-turbine model. The results offer a computationally tractable framework for identifying Fourier coefficients of the state and input matrices directly in continuous time, expanding applicability beyond steady-state or strongly excited regimes.

Abstract

This paper presents a novel approach for the identification of linear time-periodic (LTP) systems in continuous time. This method is based on harmonic modeling and consists in converting any LTP system into an equivalent LTI system with infinite dimension. Leveraging specific harmonic properties, we demonstrate that solving this infinite-dimensional identification problem can be reduced to solving a finitedimensional linear least-squares problem. The result is an approximation of the original solution with an arbitrarily small error. Our approach offers several significant advantages. The first one is closely tied to the harmonic system's inherent LTI characteristic, along with the Toeplitz structure exhibited by its elements. The second advantage is related to the regularization property achieved through the integral action when computing the phasors from input and state trajectories. Finally, our method avoids the computation of signals' derivative. This sets our approach apart from existing methods that rely on such computations, which can be a notable drawback, especially in continuous-time settings. We provide numerical simulations that convincingly demonstrate the effectiveness of the proposed method, even in scenarios where signals are corrupted by noise.

A harmonic framework for the identification of linear time-periodic systems

TL;DR

This work tackles the identification of continuous-time linear time-periodic systems by exploiting a harmonic modelling approach that maps an LTP system to an infinite-dimensional LTI representation with a block Toeplitz structure. By truncating the harmonic expansion and focusing on a central strip, the authors reduce the problem to a finite-dimensional linear least-squares problem that provably converges to the true infinite-dimensional solution with arbitrarily small error. The method yields several advantages, including avoiding derivative computations and providing robustness to noise and instability, as demonstrated on finite and infinite phasor-order examples and an unstable wind-turbine model. The results offer a computationally tractable framework for identifying Fourier coefficients of the state and input matrices directly in continuous time, expanding applicability beyond steady-state or strongly excited regimes.

Abstract

This paper presents a novel approach for the identification of linear time-periodic (LTP) systems in continuous time. This method is based on harmonic modeling and consists in converting any LTP system into an equivalent LTI system with infinite dimension. Leveraging specific harmonic properties, we demonstrate that solving this infinite-dimensional identification problem can be reduced to solving a finitedimensional linear least-squares problem. The result is an approximation of the original solution with an arbitrarily small error. Our approach offers several significant advantages. The first one is closely tied to the harmonic system's inherent LTI characteristic, along with the Toeplitz structure exhibited by its elements. The second advantage is related to the regularization property achieved through the integral action when computing the phasors from input and state trajectories. Finally, our method avoids the computation of signals' derivative. This sets our approach apart from existing methods that rely on such computations, which can be a notable drawback, especially in continuous-time settings. We provide numerical simulations that convincingly demonstrate the effectiveness of the proposed method, even in scenarios where signals are corrupted by noise.
Paper Structure (12 sections, 8 theorems, 41 equations, 5 figures)

This paper contains 12 sections, 8 theorems, 41 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Harmonic modelling
  3. Problem formulation
  4. Main results
  5. A central strip identification problem
  6. Solving the central strip least squares identification problem
  7. Validation
  8. Illustrative examples
  9. A finite phasor-order example
  10. An infinite phasor-order example
  11. Wind turbine
  12. Conclusion

Key Result

Theorem 1

For a given $X\in L_{loc}^{\infty}(\mathbb{R},\ell^2(\mathbb{C}^n))$, there exists a representative $x\in L^2_{loc}(\mathbb{R},\mathbb{C}^n)$ of $X$, i.e. $X=\mathcal{F}(x)$, if and only if $X \in H$.

Figures (5)

  • Figure 1: Comparison of the true and estimated trajectories of the finite phasor-order example.
  • Figure 2: Comparison of the true and estimated trajectories of the infinite phasor-order example.
  • Figure 3: Moduli of the true and estimated phasors of $A$ with noise on the state measurements.
  • Figure 4: Comparison of the true and estimated trajectories of the wind turbine.
  • Figure 5: Identification error convergence for the wind turbine.

Theorems & Definitions (19)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Definition 2
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 9 more