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Sampling in Parametric and Nonparametric System Identification: Aliasing, Input Conditions, and Consistency

Rodrigo A. González, Max van Haren, Tom Oomen, Cristian R. Rojas

Abstract

The sampling rate of input and output signals is known to play a critical role in the identification and control of dynamical systems. For slow-sampled continuous-time systems that do not satisfy the Nyquist-Shannon sampling condition for perfect signal reconstructability, careful consideration is required when identifying parametric and nonparametric models. In this letter, a comprehensive statistical analysis of estimators under slow sampling is performed. Necessary and sufficient conditions are obtained for unbiased estimates of the frequency response function beyond the Nyquist frequency, and it is shown that consistency of parametric estimators can be achieved even if input frequencies overlap after aliasing. Monte Carlo simulations confirm the theoretical properties.

Sampling in Parametric and Nonparametric System Identification: Aliasing, Input Conditions, and Consistency

Abstract

The sampling rate of input and output signals is known to play a critical role in the identification and control of dynamical systems. For slow-sampled continuous-time systems that do not satisfy the Nyquist-Shannon sampling condition for perfect signal reconstructability, careful consideration is required when identifying parametric and nonparametric models. In this letter, a comprehensive statistical analysis of estimators under slow sampling is performed. Necessary and sufficient conditions are obtained for unbiased estimates of the frequency response function beyond the Nyquist frequency, and it is shown that consistency of parametric estimators can be achieved even if input frequencies overlap after aliasing. Monte Carlo simulations confirm the theoretical properties.

Paper Structure

This paper contains 10 sections, 5 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. A least-squares estimate for the frequency response function
  4. Statistical properties of the least-squares estimator
  5. Frequency-domain interpretation
  6. Nonparametric to parametric estimators
  7. Simulation studies
  8. Nonparametric system identification
  9. Parametric system identification
  10. Conclusions

Key Result

Theorem III.1

Consider the sampled input and output signals $\{u(kh),y(kh)\}_{k=1}^N$, where $u(t)$ is given by input, $y(kh)$ is measured in a stationary regime, $h\geq \pi/\omega_M$, and $N>2M$. Then, ls is well-defined and is an unbiased estimator of the frequency response vector $\mathbf{G}_0^{\textnormal{f}}

Figures (2)

  • Figure 1: Bode plot of the system (blue), and the mean value of the magnitude and phase of the estimated frequency response function via least-squares, with its $95\%$ confidence interval (red). The least-squares approach provides an unbiased estimate of the system over the Nyquist frequency.
  • Figure 2: Empirical means of the parameter estimates (blue), with corresponding MSEs (log-log subplots, black lines). Parameters $a_i$ and $b_i$ correspond to the denominator and numerator coefficients of $G(p, \bm{\theta})$, respectively. All estimates converge to their true values (green dashed lines), and MSEs decay as $1/N$, indicating the consistency of the estimator minimizing \ref{['vt']}.

Theorems & Definitions (12)

  • Remark II.1
  • Theorem III.1
  • Corollary III.1
  • proof
  • Theorem III.2
  • proof
  • Remark III.1
  • Theorem IV.1
  • proof
  • Remark IV.1
  • ...and 2 more