Identification of Linear Systems with Multiplicative Noise from Multiple Trajectory Data

TL;DR

This paper presents a novel least-squares based algorithm for identifying linear systems with multiplicative noise from multiple trajectories, without prior noise knowledge, and analyzes its theoretical properties and practical performance.

Contribution

It introduces a new joint estimation method for system matrices and noise covariance that does not require system stability or noise prior knowledge.

Findings

Algorithm is asymptotically consistent under certain conditions.

High-probability bounds for finite-sample performance are derived.

Numerical simulations validate the effectiveness of the proposed method.

Abstract

The paper studies identification of linear systems with multiplicative noise from multiple-trajectory data. An algorithm based on the least-squares method and multiple-trajectory data is proposed for joint estimation of the nominal system matrices and the covariance matrix of the multiplicative noise. The algorithm does not need prior knowledge of the noise or stability of the system, but requires only independent inputs with pre-designed first and second moments and relatively small trajectory length. The study of identifiability of the noise covariance matrix shows that there exists an equivalent class of matrices that generate the same second-moment dynamic of system states. It is demonstrated how to obtain the equivalent class based on estimates of the noise covariance. Asymptotic consistency of the algorithm is verified under sufficiently exciting inputs and system controllability conditions. Non-asymptotic performance of the algorithm is also analyzed under the assumption that the system is bounded. The analysis provides high-probability bounds vanishing as the number of trajectories grows to infinity. The results are illustrated by numerical simulations.