洞見 - System Identification - # Empirical Transfer Function Estimate (ETFE)

Finite-Sample Guarantees for Frequency Domain System Identification

Q: How can the results be extended to handle more general input excitation signals beyond periodic inputs

To extend the results to handle more general input excitation signals beyond periodic inputs, we can consider a broader class of excitation signals that satisfy certain properties. One approach could be to analyze the properties of the excitation signals in terms of their spectral characteristics and design input signals that provide sufficient information for system identification. By studying the properties of different types of excitation signals, such as random signals, multisine signals, or signals with specific spectral properties, we can develop finite-sample guarantees that are applicable to a wider range of input scenarios. Additionally, incorporating the characteristics of the excitation signals into the analysis can help in deriving bounds and rates that are tailored to the specific input signal used in the identification process.

Q: What are the implications of the finite-sample guarantees for the design of optimal experiment design in frequency domain identification

The finite-sample guarantees derived in this work have significant implications for the design of optimal experiment design in frequency domain identification. By providing bounds on the estimation error and convergence rates for non-parametric system identification, these guarantees can guide the selection of input signals and experimental setups to achieve accurate and reliable frequency response estimates. Optimal experiment design involves choosing excitation signals that maximize the information content in the input-output data while minimizing the effects of noise and transient phenomena. The guarantees obtained in this work can inform the selection of excitation signals, the number of samples needed, and the frequency resolution required to achieve a certain level of accuracy in estimating the system's frequency response. By leveraging these guarantees, researchers and practitioners can design experiments that lead to efficient and effective system identification in the frequency domain.

Q: How can the techniques developed in this work be applied to other non-parametric system identification problems, such as time-domain identification or nonlinear system identification

The techniques developed in this work for finite-sample frequency domain system identification can be applied to other non-parametric system identification problems, such as time-domain identification or nonlinear system identification. By adapting the methodology and analysis framework to different domains, researchers can derive similar finite-sample guarantees for system identification in these contexts. For time-domain identification, the same principles of non-parametric estimation and concentration of estimation errors can be applied to derive bounds on the accuracy and convergence rates of system identification algorithms. In the case of nonlinear system identification, the techniques can be extended to handle more complex system dynamics and nonlinearity, providing insights into the finite-sample behavior of non-parametric identification methods in nonlinear systems. Overall, the techniques developed in this work offer a versatile framework for analyzing system identification problems across different domains and can be adapted to various settings to provide valuable insights into the finite-sample performance of system identification algorithms.

核心概念

Under sub-Gaussian colored noise and stability assumptions, the ETFE estimates are concentrated around the true frequency response values, with an error rate of O((du + √dudy)√M/Ntot), where Ntot is the total number of samples, M is the number of desired frequencies, and du, dy are the dimensions of the input and output signals.

摘要

The paper studies non-parametric frequency-domain system identification from a finite-sample perspective. It considers an open-loop scenario where the excitation input is periodic and focuses on the Empirical Transfer Function Estimate (ETFE), where the goal is to estimate the frequency response at certain desired (evenly-spaced) frequencies, given input-output samples.

The key highlights and insights are:

The authors provide finite-sample guarantees for the ETFE under sub-Gaussian colored noise (in time-domain) and stability assumptions. They show that the ETFE estimates are concentrated around the true values, with an error rate of O((du + √dudy)√M/Ntot), where Ntot is the total number of samples, M is the number of desired frequencies, and du, dy are the dimensions of the input and output signals.
The rate remains valid for general irrational transfer functions and does not require a finite order state-space representation, unlike prior non-asymptotic bounds.
By tuning M, the authors obtain a N^(-1/3) finite-sample rate for learning the frequency response over all frequencies in the H∞ norm.
The proof relies on an extension of the Hanson-Wright inequality to semi-infinite matrices, which allows dealing with quadratic forms of a (countably) infinite number of sub-Gaussian variables.
The authors study the finite-sample behavior of ETFE in simulations, validating the theoretical results.

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統計資料

∥ut∥2 ≤ Du, for all t ∈ Z
∥G∥⋆ < ∞
∥R∥⋆ < ∞

引述

"We show that under sub-Gaussian colored noise (in time-domain) and stability assumptions, the ETFE estimates are concentrated around the true values. The error rate is of the order of O((du + √dudy)√M/Ntot), where Ntot is the total number of samples, M is the number of desired frequencies, and du, dy are the dimensions of the input and output signals."
"By tuning M, we obtain a N^(-1/3) finite-sample rate for learning the frequency response over all frequencies in the H∞ norm."

從以下內容提煉的關鍵洞見

Finite Sample Frequency Domain Identification

by Anastasios T... 於 arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.01100.pdf

Finite Sample Frequency Domain Identification

深入探究

How can the results be extended to handle more general input excitation signals beyond periodic inputs

To extend the results to handle more general input excitation signals beyond periodic inputs, we can consider a broader class of excitation signals that satisfy certain properties. One approach could be to analyze the properties of the excitation signals in terms of their spectral characteristics and design input signals that provide sufficient information for system identification. By studying the properties of different types of excitation signals, such as random signals, multisine signals, or signals with specific spectral properties, we can develop finite-sample guarantees that are applicable to a wider range of input scenarios. Additionally, incorporating the characteristics of the excitation signals into the analysis can help in deriving bounds and rates that are tailored to the specific input signal used in the identification process.

What are the implications of the finite-sample guarantees for the design of optimal experiment design in frequency domain identification

The finite-sample guarantees derived in this work have significant implications for the design of optimal experiment design in frequency domain identification. By providing bounds on the estimation error and convergence rates for non-parametric system identification, these guarantees can guide the selection of input signals and experimental setups to achieve accurate and reliable frequency response estimates. Optimal experiment design involves choosing excitation signals that maximize the information content in the input-output data while minimizing the effects of noise and transient phenomena. The guarantees obtained in this work can inform the selection of excitation signals, the number of samples needed, and the frequency resolution required to achieve a certain level of accuracy in estimating the system's frequency response. By leveraging these guarantees, researchers and practitioners can design experiments that lead to efficient and effective system identification in the frequency domain.

How can the techniques developed in this work be applied to other non-parametric system identification problems, such as time-domain identification or nonlinear system identification

The techniques developed in this work for finite-sample frequency domain system identification can be applied to other non-parametric system identification problems, such as time-domain identification or nonlinear system identification. By adapting the methodology and analysis framework to different domains, researchers can derive similar finite-sample guarantees for system identification in these contexts. For time-domain identification, the same principles of non-parametric estimation and concentration of estimation errors can be applied to derive bounds on the accuracy and convergence rates of system identification algorithms. In the case of nonlinear system identification, the techniques can be extended to handle more complex system dynamics and nonlinearity, providing insights into the finite-sample behavior of non-parametric identification methods in nonlinear systems. Overall, the techniques developed in this work offer a versatile framework for analyzing system identification problems across different domains and can be adapted to various settings to provide valuable insights into the finite-sample performance of system identification algorithms.