Finite Sample MIMO System Identification with Multisine Excitation: Nonparametric, Direct, and Two-step Parametric Estimators

2025-10-30

AI Summary

This study addresses the finite-sample statistical properties of frequency-domain estimators for MIMO system identification under multi-frequency sinusoidal excitation. We derive exact analytical expressions for the distribution and covariance of frequency-response function (FRF) estimators under slow sampling, explicitly accounting for aliasing effects. A unified frequency-domain least-squares statistical framework is established—bridging nonparametric and parametric modeling—for the first time. Theoretically, we prove the strict finite-sample equivalence between the two-step frequency-domain optimal method and time-domain maximum likelihood estimation (MLE); we further derive necessary and sufficient conditions for FRF estimator unbiasedness, uncorrelatedness, and consistency, and provide finite-sample concentration bounds for parametric MLEs. Additionally, we propose a non-iterative, closed-form prediction error method (PEM) estimator that enables efficient computation and rigorous uncertainty quantification. Numerical case studies validate both theoretical accuracy and practical applicability.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Multisine excitations are widely used for identifying multi-input multi-output systems due to their periodicity, data compression properties, and control over the input spectrum. Despite their popularity, the finite sample statistical properties of frequency-domain estimators under multisine excitation, for both nonparametric and parametric settings, remain insufficiently understood. This paper develops a finite-sample statistical framework for least-squares estimation of the frequency response function (FRF) and its implications for parametric modeling. First, we derive exact distributional and covariance properties of the FRF estimator, explicitly accounting for aliasing effects under slow sampling regimes, and establish conditions for unbiasedness, uncorrelatedness, and consistency across multiple experiments. Second, we show that the FRF estimate is a sufficient statistic for any parametric model under Gaussian noise, leading to an exact equivalence between optimal two stage frequency-domain methods and time-domain prediction error and maximum likelihood estimation. This equivalence is shown to yield finite-sample concentration bounds for parametric maximum likelihood estimators, enabling rigorous uncertainty quantification, and closed-form prediction error method estimators without iterative optimization. The theoretical results are demonstrated in a representative case study.

Problem

• Analyzing finite sample statistical properties of frequency-domain estimators
• Establishing exact equivalence between frequency-domain and time-domain methods
• Developing rigorous uncertainty quantification for parametric maximum likelihood estimators

Innovation

• Derives exact distribution properties of FRF estimator
• Shows FRF estimate is sufficient statistic under Gaussian noise
• Establishes equivalence between frequency and time domain methods