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An Interpolation-based Scheme for Rapid Frequency-Domain System Identification

Jared Jonas, Bassam Bamieh

TL;DR

This work addresses rapid frequency-domain system identification for high-order, lightly damped LTI systems where individual sinusoidal experiments are expensive. It proposes an interpolation-based scheme using barycentric interpolation with weights chosen by convex optimization, leveraging transient responses and an adaptive frequency grid to minimize a data-driven error proxy related to $H^2$ and $H^\infty$ norms. Stability guarantees are achieved via a convex LMI relaxation, yielding a stable identified model with provable bounds. Numerical results on a 270-state ISS model demonstrate high accuracy with far fewer experiments, with the adaptive frequency selection providing particularly robust performance.

Abstract

We present a frequency-domain system identification scheme based on barycentric interpolation and weight optimization. The scheme is related to the Adaptive Antoulas-Anderson (AAA) algorithm for model reduction, but uses an adaptive algorithm for selection of frequency points for interrogating the system response, as would be required in identification versus model reduction. The scheme is particularly suited for systems in which any one sinusoidal response run is long or expensive, and thus there is an incentive to reduce the total number of such runs. Two key features of our algorithm are the use of transient data in sinusoidal runs to both optimize the barycentric weights, and automated next-frequency selection on an adaptive grid. Both are done with error criteria that are proxies for a system's $H^2$ and $H^\infty$ norms respectively. Furthermore, the optimization problem we formulate is convex, and can optionally guarantee stability of the identified system. Computational results on a high-order, lightly damped structural system highlights the efficacy of this scheme.

An Interpolation-based Scheme for Rapid Frequency-Domain System Identification

TL;DR

This work addresses rapid frequency-domain system identification for high-order, lightly damped LTI systems where individual sinusoidal experiments are expensive. It proposes an interpolation-based scheme using barycentric interpolation with weights chosen by convex optimization, leveraging transient responses and an adaptive frequency grid to minimize a data-driven error proxy related to and norms. Stability guarantees are achieved via a convex LMI relaxation, yielding a stable identified model with provable bounds. Numerical results on a 270-state ISS model demonstrate high accuracy with far fewer experiments, with the adaptive frequency selection providing particularly robust performance.

Abstract

We present a frequency-domain system identification scheme based on barycentric interpolation and weight optimization. The scheme is related to the Adaptive Antoulas-Anderson (AAA) algorithm for model reduction, but uses an adaptive algorithm for selection of frequency points for interrogating the system response, as would be required in identification versus model reduction. The scheme is particularly suited for systems in which any one sinusoidal response run is long or expensive, and thus there is an incentive to reduce the total number of such runs. Two key features of our algorithm are the use of transient data in sinusoidal runs to both optimize the barycentric weights, and automated next-frequency selection on an adaptive grid. Both are done with error criteria that are proxies for a system's and norms respectively. Furthermore, the optimization problem we formulate is convex, and can optionally guarantee stability of the identified system. Computational results on a high-order, lightly damped structural system highlights the efficacy of this scheme.

Paper Structure

This paper contains 11 sections, 2 theorems, 37 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Problem Formulation
  3. Problem formulation
  4. Algorithms and Main Results
  5. System ID experiments
  6. Weight optimization
  7. Enforcing stability
  8. Frequency selection
  9. Computational results
  10. Conclusion
  11. Detecting the Onset of Steady State

Key Result

Theorem III.1

Consider the set of $m$ input-output pairs $\left\{(u^{(k)}, \, y^{(k)})\right\}_{k=1}^m$ where $u^{(k)}, \, y^{(k)}\in\mathbb{R}^{n_k}$ and form the empirical covariance matrix where $X_k$ is the covariance matrix of the signal $x^{(k)} := \mathcal{N} u^{(k)} - \mathcal{M}y^{(k)}$, i.e. $X_k := \mathrm{cov}\left(x^{(k)}, \, x^{(k)}\right)$ where $\mathrm{cov}\left(x, \, z\right) := \frac{1}{n} \

Figures (5)

  • Figure 1: A Bode plot of the input ISS system (in black) and resulting 43 order system constructed using the gridded frequency selection approach (in red). The circles indicate the frequency at which a system ID experiment was ran as well as the measured response data.
  • Figure 2: A Bode plot of the input ISS system (in black) and resulting 43 order system constructed using the adaptive frequency selection approach (in red). The circles indicate the frequency at which a system ID experiment was ran as well as the measured response data. The circle color indicates the order the experiments were ran, with black being oldest and white being newest.
  • Figure 3: A plot showing the $\sf H^2$ norm of the error system, i.e. $R-G$, against the number of poles of each system $R$ generated from the two frequency selection strategies with the input ISS system $G$. The blue line and red line indicate the error norm of systems generated with the gridded frequency selection strategy and the adaptive frequency selection strategy respectively.
  • Figure 4: A plot showing the $L_\infty$ norm of the error system, i.e. $R-G$, against the number of poles of each system $R$ generated using two different optimization approaches with the input ISS system $G$. The blue line and red line indicate the error norm of systems generated with the stability-enforced optimization problem and the explicit/unconstrained optimization problem respectively.
  • Figure 5: A Bode plot of the ISS system in black and two 11 order systems generated using the adaptive approach with the stability constrained optimization in red and the unconstrained optimization in yellow.

Theorems & Definitions (4)

  • Theorem III.1
  • proof
  • Theorem III.2
  • proof