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

2025-08-23

AI Summary

This paper addresses frequency-response identification of low-order linear time-invariant systems from a finite number of noisy measurements. Method: We propose a convex optimization framework that minimizes the ℓ₂-norm mismatch between measured and modeled frequency responses, while explicitly embedding Loewner matrix structure as a nuclear-norm regularizer to enforce low-rank prior knowledge and ensure computational tractability. Contribution/Results: Theoretically, we derive the first finite-sample upper bound on sampling complexity for frequency-response identification and establish an explicit dependence of the full-band identification error on system order, signal-to-noise ratio, and sampling density. Numerical experiments demonstrate that the proposed method significantly outperforms classical approaches—especially under sparse sampling—achieving both high accuracy and strong robustness to noise.

Technology Category

• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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, along with numerical simulations validating the growth rate of the sample complexity bound.

Problem

• Identifying low-order systems from frequency response data
• Minimizing l2 norm error with nuclear norm regularization
• Deriving sample complexity bounds for frequency-domain identification

Innovation

• Frequency-domain system identification with l2 norm minimization
• Nuclear norm regularization of Loewner matrix
• Convex optimization techniques with sample complexity analysis