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Generalized rational Prony and Bernoulli methods

Tamás Dózsa, Matthias Voigt, Zoltán Szabó, József Bokor, Péter Kovács

TL;DR

The paper addresses recovering the linear and nonlinear parameters of Prony signals $f=\sum_{k=1}^M c_k v_{\lambda_k}$ by linking generalized Prony (GOP) with Bernoulli schemes through a rational formulation. It introduces a rational variant of GOP (GROP) that maps Prony problems to pole-finding in a finite-dimensional $H_2(\mathbb{D})$ subspace, enabling direct recovery of nonlinear parameters from pole structures and subsequent linear coefficients via a stable triangular system. It further shows that generalized Bernoulli schemes can solve any Prony problem, even when the model order $M$ is unknown, by iteratively extracting $\\\lambda_k$ and projecting out recovered components. The authors advocate using Takenaka-Malmquist bases to form upper-triangular linear systems for the linear parameters, improving conditioning compared with Vandermonde-based approaches, especially when poles lie near the unit circle. Numerical experiments on time-delayed LTI system identification and RKHS kernel parameter recovery illustrate the methods’ robustness and potential for reduced-order modeling and practical system identification.

Abstract

The generalized operator-based Prony method is an important tool for describing signals which can be written as finite linear combinations of eigenfunctions of certain linear operators. On the other hand, Bernoulli's algorithm and its generalizations can be used to recover the parameters of rational functions belonging to finite-dimensional subspaces of $H_2$ Hardy-Hilbert spaces. In this work, we discuss several results related to these methods. We discuss a rational variant of the generalized operator-based Prony method and show that in fact, any Prony problem can be treated this way. This realization establishes the connection between Prony and Bernoulli methods and allows us to address some well-known numerical pitfalls. Several numerical experiments are provided to showcase the usefulness of the introduced methods. These include problems related to the identification of time-delayed linear systems and parameter recovery problems in reproducing kernel Hilbert spaces.

Generalized rational Prony and Bernoulli methods

TL;DR

The paper addresses recovering the linear and nonlinear parameters of Prony signals by linking generalized Prony (GOP) with Bernoulli schemes through a rational formulation. It introduces a rational variant of GOP (GROP) that maps Prony problems to pole-finding in a finite-dimensional subspace, enabling direct recovery of nonlinear parameters from pole structures and subsequent linear coefficients via a stable triangular system. It further shows that generalized Bernoulli schemes can solve any Prony problem, even when the model order is unknown, by iteratively extracting and projecting out recovered components. The authors advocate using Takenaka-Malmquist bases to form upper-triangular linear systems for the linear parameters, improving conditioning compared with Vandermonde-based approaches, especially when poles lie near the unit circle. Numerical experiments on time-delayed LTI system identification and RKHS kernel parameter recovery illustrate the methods’ robustness and potential for reduced-order modeling and practical system identification.

Abstract

The generalized operator-based Prony method is an important tool for describing signals which can be written as finite linear combinations of eigenfunctions of certain linear operators. On the other hand, Bernoulli's algorithm and its generalizations can be used to recover the parameters of rational functions belonging to finite-dimensional subspaces of Hardy-Hilbert spaces. In this work, we discuss several results related to these methods. We discuss a rational variant of the generalized operator-based Prony method and show that in fact, any Prony problem can be treated this way. This realization establishes the connection between Prony and Bernoulli methods and allows us to address some well-known numerical pitfalls. Several numerical experiments are provided to showcase the usefulness of the introduced methods. These include problems related to the identification of time-delayed linear systems and parameter recovery problems in reproducing kernel Hilbert spaces.

Paper Structure

This paper contains 11 sections, 6 theorems, 79 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The generalized operator-based Prony method
  3. Preliminaries
  4. The GOP method for rational problems
  5. Generalized Bernoulli schemes
  6. Rational Prony and Bernoulli methods
  7. Examples
  8. Linear parameter recovery with Vandermonde and Takenaka-Malmquist matrices
  9. Pole identification of time-delayed continuous-time LTI systems
  10. Recovering the parameters of reproducing kernels
  11. Conclusions

Key Result

Theorem 2.1

Let ${(S_{m, j})}_{m,j \ge 0} := {(\mathcal{F}_m \circ \Psi_{\varphi}^{j})}_{m,j \ge 0}$ be an admissible sampling scheme over the $M$-dimensional signal space $\mathcal{M}$ as in eq:ma. Then every $f \in \mathcal{M}$ is uniquely determined by the sampling matrix where $N \geq M-1$.

Figures (4)

  • Figure 1: Nonlinear parameters (mirror image poles) describing the considered systems.
  • Figure 2: Nonlinear parameters of the transfer function of a time-delayed SISO LTI system recovered by the GROP method (top row), the GB method (middle row) and the classical Prony algorithm (bottom row).
  • Figure 3: Nonlinear parameters $\lambda_k$ recovered by the GOP method. The proximity of the parameters to be recovered can cause the Prony sampling matrix to become ill-conditioned which results in poor nonlinear parameter recovery.
  • Figure 4: The GB method can find the actual nonlinear parameters in an iterative fashion. When attempting to reduce the size of the Prony sampling matrix (i.e., by reducing $M$), there are no guarantees that the GOP method will find the actual nonlinear parameters.

Theorems & Definitions (19)

  • Definition 2.1: Iteration operator
  • Definition 2.2: (Admissible) evaluation scheme
  • Definition 2.3: (Realizable/admissible) sampling scheme
  • Theorem 2.1: Generalized operator-based Prony (GOP) method
  • Remark 2.1
  • Theorem 3.1: Generalized Bernoulli (GB) schemes
  • Remark 3.1
  • Theorem 4.1: Generalized operator-based Prony methods and pole finding problems
  • proof
  • Remark 4.1
  • ...and 9 more