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

arXiv: https://arxiv.org/abs/2508.17142 · HTML: https://mlsanity.com/paper/2508.17142/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 84 references that are in the map.
- First posted on arXiv 23 Aug 2025.

## Builds on (most cited first, 30 of 32)

- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [A Tour of Reinforcement Learning: The View from Continuous Control](https://mlsanity.com/paper/1806.09460/) (2018)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Finite Sample Analysis of Stochastic System Identification](https://mlsanity.com/paper/1903.09122/) (2019)
- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [Formulas for Data-driven Control: Stabilization, Optimality and Robustness](https://mlsanity.com/paper/1903.06842/) (2019)
- [Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers](https://mlsanity.com/paper/2006.07579/) (2020)
- [Data-Enabled Predictive Control: In the Shallows of the DeePC](https://mlsanity.com/paper/1811.05890/) (2018)
- [Non-asymptotic Identification of Linear Dynamical Systems Using Multiple Trajectories](https://mlsanity.com/paper/2009.00739/) (2020)
- [Single Trajectory Nonparametric Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2202.08311/) (2022)
- [A Tutorial on the Non-Asymptotic Theory of System Identification](https://mlsanity.com/paper/2309.03873/) (2023)
- [Learning from many trajectories](https://mlsanity.com/paper/2203.17193/) (2022)
- [Sample Complexity Lower Bounds for Linear System Identification](https://mlsanity.com/paper/1903.10343/) (2019)
- [Improved rates for prediction and identification of partially observed linear dynamical systems](https://mlsanity.com/paper/2011.10006/) (2020)
- [Regularized linear system identification using atomic, nuclear and kernel-based norms: the role of the stability constraint](https://mlsanity.com/paper/1507.00564/) (2015)
- [Learning low-dimensional dynamical-system models from noisy frequency-response data with Loewner rational interpolation](https://mlsanity.com/paper/1910.00110/) (2019)
- [Learning the Uncertainty Sets for Control Dynamics via Set Membership: A Non-Asymptotic Analysis](https://mlsanity.com/paper/2309.14648/) (2023)
- [Robust recovery of complex exponential signals from random Gaussian projections via low rank Hankel matrix reconstruction](https://mlsanity.com/paper/1503.02893/) (2015)
- [Finite Sample Identification of Partially Observed Bilinear Dynamical Systems](https://mlsanity.com/paper/2501.07652/) (2025)
- [Generalization Error Analysis for Selective State-Space Models Through the Lens of Attention](https://mlsanity.com/paper/2502.01473/) (2025)
- [Computationally Efficient and Statistically Optimal Robust High-Dimensional Linear Regression](https://mlsanity.com/paper/2305.06199/) (2023)
- [Online Optimization Perspective on First-Order and Zero-Order Decentralized Nonsmooth Nonconvex Stochastic Optimization](https://mlsanity.com/paper/2406.01484/) (2024)
- [Ensuring Both Positivity and Stability Using Sector-Bounded Nonlinearity for Systems with Neural Network Controllers](https://mlsanity.com/paper/2406.12744/) (2024)
- [Outlier-Robust Linear System Identification Under Heavy-tailed Noise](https://mlsanity.com/paper/2501.00421/) (2024)
- [Sample Complexity Bounds for Linear System Identification from a Finite Set](https://mlsanity.com/paper/2409.11141/) (2024)
- [Convex Data-Driven Contraction With Riemannian Metrics](https://mlsanity.com/paper/2412.20283/) (2024)
- [Frequency Domain Gaussian Process Models for H^∞ Uncertainties](https://mlsanity.com/paper/2211.15923/) (2022)
- [Safe Control for Pursuit-Evasion with Density Functions](https://mlsanity.com/paper/2505.15718/) (2025)
- [Local Stability and Region of Attraction Analysis for Neural Network Feedback Systems under Positivity Constraints](https://mlsanity.com/paper/2505.22889/) (2025)
- [Frequency-Domain Identification of Discrete-Time Systems using Sum-of-Rational Optimization](https://mlsanity.com/paper/2312.15722/) (2023)

## Abstract (arXiv)

This paper proposes a frequency-domain estimator for low-order systems from repeated noisy measurements. The estimator minimizes a quadratic data-fitting term regularized by the nuclear norm of a Loewner matrix, subject to a convex stability constraint enforced via a semidefinite program. We prove a finite-sample error bound at the sampled frequencies and extend it to all frequencies through rational interpolation. The bound characterizes the dependence on the number of repeated experiments, number of frequency points, system order, and noise level. Numerical experiments on SISO and MIMO systems demonstrate the low-order-promoting effect of the method and validate the predicted scaling laws.