Frequency Response Identification of Low-Order Systems: Finite-Sample Analysis

TL;DR

The paper addresses finite-sample frequency-domain identification of low-order LTI systems by casting the problem as a convex Loewner-nuclear-norm regularized regression. It derives a non-asymptotic error bound showing the identification error at sampled frequencies scales as $O\big(\sqrt{ \frac{M \ln M}{N} }\big)$, with constants depending on system stability, the Loewner-operator conditioning, and noise level. The approach promotes low-order models without explicit stability constraints by using the nuclear norm of the Loewner matrix as a regularizer and provides a detailed operator-theoretic analysis via the Loewner operator, its adjoint, and concentration inequalities. Numerical experiments demonstrate that the method yields low-order, accurate frequency responses and validate the predicted scaling of the sample complexity, highlighting its potential for practical frequency-domain system identification and model-order reduction.

Abstract

This paper proposes a frequency-domain system identification method for learning low-order systems. The identification problem is formulated as the minimization of the l2 norm between the identified and measured frequency responses, with the nuclear norm of the Loewner matrix serving as a regularization term. This formulation results in an optimization problem that can be efficiently solved using standard convex optimization techniques. We derive an upper bound on the sampled-frequency complexity of the identification process and subsequently extend this bound to characterize the identification error over all frequencies. A detailed analysis of the sample complexity is provided, along with a thorough interpretation of its terms and dependencies. Finally, the efficacy of the proposed method is demonstrated through an example, and numerical simulations validating the growth rate of the sample complexity bound.

Figures (7)

• Figure 1: Frequency Points $\bm{z}$: Equally spaced points on the upper half of the unit circle with margin $\delta_m$ from $0$ and $\pi$.
• Figure 2: Bode Diagrams: The solid blue line represents the frequency response of the ground truth system. The frequency responses of the systems identified by Averaging and LNNM are represented by red and green dashed lines, respectively. It is observed that at low frequencies, LNNM exhibits less fluctuation, resulting in a response that is closer to the baseline.
• Figure 3: $\mathcal{H}_\infty$ Identification Error: The $\mathcal{H}_\infty$ identification error of the proposed LNNM method is compared with that of the Averaging approach, in which the $N$ frequency response measurements at each frequency are averaged. A Monte Carlo simulation was performed by repeating the experiment 20 times for a fair comparison.
• Figure 4: Singular Values of Loewner Matrix: The decay of the singular values indicates that the system identified using Averaging has a higher order, whereas the system identified using LNNM is of lower order and closer to the low-order ground truth system.
• Figure 5: $\mathcal{H}_\infty$ Identification Error as a Function of $N$: The results are presented in a logarithmic scale, where the red dotted line represents a reference slope of $0.5$, confirming the $\mathcal{O}(N^{-\frac{1}{2}})$ scaling in sample complexity.

Theorems & Definitions (32)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Lemma 1
• proof
• Lemma 2
• Remark 7