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

arXiv: https://arxiv.org/abs/2406.00574 · HTML: https://mlsanity.com/paper/2406.00574/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 49 references that are in the map.
- First posted on arXiv 1 Jun 2024.

## Builds on (most cited first, 11 of 11)

- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Naive Exploration is Optimal for Online LQR](https://mlsanity.com/paper/2001.09576/) (2020)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [Robust Adaptive Model Predictive Control: Performance and Parameter Estimation](https://mlsanity.com/paper/1911.00865/) (2019)
- [Set Membership identification of linear systems with guaranteed simulation accuracy](https://mlsanity.com/paper/2001.07628/) (2020)
- [Learning the Uncertainty Sets for Control Dynamics via Set Membership: A Non-Asymptotic Analysis](https://mlsanity.com/paper/2309.14648/) (2023)
- [Non-asymptotic System Identification for Linear Systems with Nonlinear Policies](https://mlsanity.com/paper/2306.10369/) (2023)
- [Uncertainty Quantification of Set-Membership Estimation in Control and Perception: Revisiting the Minimum Enclosing Ellipsoid](https://mlsanity.com/paper/2311.15962/) (2023)
- [Robust adaptive model predictive control with persistent excitation conditions](https://mlsanity.com/paper/2211.09275/) (2022)
- [Online learning for robust voltage control under uncertain grid topology](https://mlsanity.com/paper/2306.16674/) (2023)
- [Adaptive robust predictive control with sample-based persistent excitation](https://mlsanity.com/paper/2211.12478/) (2022)

## Built on by (most cited first, 3 of 3)

- [Interpolation Conditions for Data Consistency and Prediction in Noisy Linear Systems](https://mlsanity.com/paper/2504.08484/) (2025)
- [Achieving widetildeO(1/ε) Sample Complexity for Bilinear Systems Identification under Bounded Noises](https://mlsanity.com/paper/2603.20819/) (2026)
- [Set Membership Estimation for Stochastic Linear Contextual Bandits](https://mlsanity.com/paper/2606.20022/) (2026)

## Abstract (arXiv)

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 ℓ_∞ balls. Further, we validate our convergence rates by several numerical experiments. This manuscript contains supplementary content in the Appendix.