# Learning to Stabilize Unknown LTI Systems on a Single Trajectory under Stochastic Noise

arXiv: https://arxiv.org/abs/2406.00234 · HTML: https://mlsanity.com/paper/2406.00234/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 44 references that are in the map.
- First posted on arXiv 31 May 2024.

## Builds on (most cited first, 22 of 22)

- [Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator](https://mlsanity.com/paper/1801.05039/) (2018)
- [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)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Non-asymptotic Identification of LTI Systems from a Single Trajectory](https://mlsanity.com/paper/1806.05722/) (2018)
- [Learning Linear-Quadratic Regulators Efficiently with only √T Regret](https://mlsanity.com/paper/1902.06223/) (2019)
- [Formulas for Data-driven Control: Stabilization, Optimality and Robustness](https://mlsanity.com/paper/1903.06842/) (2019)
- [Black-Box Control for Linear Dynamical Systems](https://mlsanity.com/paper/2007.06650/) (2020)
- [The Gap Between Model-Based and Model-Free Methods on the Linear Quadratic Regulator: An Asymptotic Viewpoint](https://mlsanity.com/paper/1812.03565/) (2018)
- [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)
- [Data-Driven Model Predictive Control with Stability and Robustness Guarantees](https://mlsanity.com/paper/1906.04679/) (2019)
- [Stabilizing Dynamical Systems via Policy Gradient Methods](https://mlsanity.com/paper/2110.06418/) (2021)
- [Finite-time Analysis of Approximate Policy Iteration for the Linear Quadratic Regulator](https://mlsanity.com/paper/1905.12842/) (2019)
- [Linear Systems can be Hard to Learn](https://mlsanity.com/paper/2104.01120/) (2021)
- [Efficient Reinforcement Learning for High Dimensional Linear Quadratic Systems](https://mlsanity.com/paper/1303.5984/) (2013)
- [Geometric Exploration for Online Control](https://mlsanity.com/paper/2010.13178/) (2020)
- [On the Sample Complexity of Stabilizing LTI Systems on a Single Trajectory](https://mlsanity.com/paper/2202.07187/) (2022)
- [Policy Learning of MDPs with Mixed Continuous/Discrete Variables: A Case Study on Model-Free Control of Markovian Jump Systems](https://mlsanity.com/paper/2006.03116/) (2020)
- [Nonasymptotic Regret Analysis of Adaptive Linear Quadratic Control with Model Misspecification](https://mlsanity.com/paper/2401.00073/) (2023)
- [Identification of Linear Systems with Multiplicative Noise from Multiple Trajectory Data](https://mlsanity.com/paper/2106.16078/) (2021)
- [Robust Control of Unknown Switched Linear Systems from Noisy Data](https://mlsanity.com/paper/2311.11300/) (2023)

## Built on by (most cited first, 2 of 2)

- [Learning Stabilizing Policies via an Unstable Subspace Representation](https://mlsanity.com/paper/2505.01348/) (2025)
- [Learning Fractional-Order Dynamics from a Single Trajectory](https://mlsanity.com/paper/2609.18127/) (2026)

## Abstract (arXiv)

We study the problem of learning to stabilize unknown noisy Linear Time-Invariant (LTI) systems on a single trajectory. It is well known in the literature that the learn-to-stabilize problem suffers from exponential blow-up in which the state norm blows up in the order of Theta(2n) where n is the state space dimension. This blow-up is due to the open-loop instability when exploring the n-dimensional state space. To address this issue, we develop a novel algorithm that decouples the unstable subspace of the LTI system from the stable subspace, based on which the algorithm only explores and stabilizes the unstable subspace, the dimension of which can be much smaller than n. With a new singular-value-decomposition(SVD)-based analytical framework, we prove that the system is stabilized before the state norm reaches 2O(k log n), where k is the dimension of the unstable subspace. Critically, this bound avoids exponential blow-up in state dimension in the order of Theta(2n) as in the previous works, and to the best of our knowledge, this is the first paper to avoid exponential blow-up in dimension for stabilizing LTI systems with noise.