# Learning the Uncertainty Sets for Control Dynamics via Set Membership: A Non-Asymptotic Analysis

arXiv: https://arxiv.org/abs/2309.14648 · HTML: https://mlsanity.com/paper/2309.14648/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 85 references that are in the map.
- First posted on arXiv 26 Sep 2023.

## Builds on (most cited first, 30 of 40)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [A General Safety Framework for Learning-Based Control in Uncertain Robotic Systems](https://mlsanity.com/paper/1705.01292/) (2017)
- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [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)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [Non-asymptotic Identification of LTI Systems from a Single Trajectory](https://mlsanity.com/paper/1806.05722/) (2018)
- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [Learning nonlinear dynamical systems from a single trajectory](https://mlsanity.com/paper/2004.14681/) (2020)
- [Black-Box Control for Linear Dynamical Systems](https://mlsanity.com/paper/2007.06650/) (2020)
- [Beyond Confidence Regions: Tight Bayesian Ambiguity Sets for Robust MDPs](https://mlsanity.com/paper/1902.07605/) (2019)
- [Distributed Reinforcement Learning for Decentralized Linear Quadratic Control: A Derivative-Free Policy Optimization Approach](https://mlsanity.com/paper/1912.09135/) (2019)
- [Non-asymptotic Identification of Linear Dynamical Systems Using Multiple Trajectories](https://mlsanity.com/paper/2009.00739/) (2020)
- [Safely Learning to Control the Constrained Linear Quadratic Regulator](https://mlsanity.com/paper/1809.10121/) (2018)
- [Active Learning for Identification of Linear Dynamical Systems](https://mlsanity.com/paper/2002.00495/) (2020)
- [On the Regret Analysis of Online LQR Control with Predictions](https://mlsanity.com/paper/2102.01309/) (2021)
- [Reinforcement Learning with Fast Stabilization in Linear Dynamical Systems](https://mlsanity.com/paper/2007.12291/) (2020)
- [Online Optimal Control with Affine Constraints](https://mlsanity.com/paper/2010.04891/) (2020)
- [Thompson Sampling for Linear-Quadratic Control Problems](https://mlsanity.com/paper/1703.08972/) (2017)
- [Finite Sample Identification of Bilinear Dynamical Systems](https://mlsanity.com/paper/2208.13915/) (2022)
- [Safe Adaptive Learning-based Control for Constrained Linear Quadratic Regulators with Regret Guarantees](https://mlsanity.com/paper/2111.00411/) (2021)
- [Online switching control with stability and regret guarantees](https://mlsanity.com/paper/2301.08445/) (2023)
- [Linear robust adaptive model predictive control: Computational complexity and conservatism -- extended version](https://mlsanity.com/paper/1909.01813/) (2019)
- [Thompson Sampling Achieves tilde O(√T) Regret in Linear Quadratic Control](https://mlsanity.com/paper/2206.08520/) (2022)
- [Robust adaptive MPC using control contraction metrics](https://mlsanity.com/paper/2209.11713/) (2022)
- [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)
- [Regret Analysis of Distributed Online LQR Control for Unknown LTI Systems](https://mlsanity.com/paper/2105.07310/) (2021)

## Built on by (most cited first, 6 of 6)

- [On the Sample Complexity of Set Membership Estimation for Linear Systems with Disturbances Bounded by Convex Sets](https://mlsanity.com/paper/2406.00574/) (2024)
- [Interpolation Conditions for Data Consistency and Prediction in Noisy Linear Systems](https://mlsanity.com/paper/2504.08484/) (2025)
- [Frequency Response Identification of Low-Order Systems: Finite-Sample Analysis](https://mlsanity.com/paper/2508.17142/) (2025)
- [Online Learning for Supervisory Switching Control](https://mlsanity.com/paper/2603.14762/) (2026)
- [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 studies uncertainty set estimation for unknown linear systems. Uncertainty sets are crucial for the quality of robust control since they directly influence the conservativeness of the control design. Departing from the confidence region analysis of least squares estimation, this paper focuses on set membership estimation (SME). Though good numerical performances have attracted applications of SME in the control literature, the non-asymptotic convergence rate of SME for linear systems remains an open question. This paper provides the first convergence rate bounds for SME and discusses variations of SME under relaxed assumptions. We also provide numerical results demonstrating SME's practical promise.