Achieving $\widetilde{O}(1/\epsilon)$ Sample Complexity for Bilinear Systems Identification under Bounded Noises

TL;DR

This paper establishes that for bilinear systems with bounded noise, the feasible parameter set's diameter decreases at a rate of approximately 1/epsilon with finite samples, improving understanding of system identification.

Contribution

It provides the first finite-sample analysis showing $ ilde{O}(1/ ext{epsilon})$ sample complexity for bilinear systems under bounded disturbances, extending results beyond linear systems.

Findings

Feasible set diameter shrinks at $ ilde{O}(1/ ext{epsilon})$ with sample size.

Simulation confirms theoretical bounds and estimator effectiveness.

Method handles trajectory-dependent regressors and marginally stable dynamics.

Abstract

This paper studies finite-sample set-membership identification for discrete-time bilinear systems under bounded symmetric log-concave disturbances. Compared with existing finite-sample results for linear systems and related analyses under stronger noise assumptions, we consider the more challenging bilinear setting with trajectory-dependent regressors and allow marginally stable dynamics with polynomial mean-square state growth. Under these conditions, we prove that the diameter of the feasible parameter set shrinks with sample complexity $\widetilde{O}(1/\epsilon)$. Simulation supports the theory and illustrates the advantage of the proposed estimator for uncertainty quantification.