On the Sample Complexity of Set Membership Estimation for Linear Systems with Disturbances Bounded by Convex Sets

TL;DR

This paper analyzes the convergence rates of set membership estimation for linear systems with disturbances bounded by convex sets, relaxing previous assumptions and validating results through numerical experiments.

Contribution

It introduces relaxed assumptions on excitation and disturbance shapes, establishing convergence rates for general convex sets using a block-martingale small-ball condition.

Findings

Convergence rates hold for disturbances bounded by general convex sets.

Validation through numerical experiments confirms theoretical results.

Bridges gap between previous convex set and $ ext{l}_ ext{infinity}$ ball analyses.

Abstract

This paper revisits the set membership identification for linear control systems and establishes its convergence rates under relaxed assumptions on (i) the persistent excitation requirement and (ii) the system disturbances. In particular, instead of assuming persistent excitation exactly, this paper adopts the block-martingale small-ball condition enabled by randomly perturbed control policies to establish the convergence rates of SME with high probability. Further, we relax the assumptions on the shape of the bounded disturbance set and the boundary-visiting condition. Our convergence rates hold for disturbances bounded by general convex sets, which bridges the gap between the previous convergence analysis for general convex sets and the existing convergence rate analysis for $\ell_\infty$ balls. Further, we validate our convergence rates by several numerical experiments. This manuscript contains supplementary content in the Appendix.