# Learning the Dynamics of Autonomous Linear Systems From Multiple Trajectories

arXiv: https://arxiv.org/abs/2203.12794 · HTML: https://mlsanity.com/paper/2203.12794/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 19 references that are in the map.
- First posted on arXiv 24 Mar 2022.

## Builds on (most cited first, 9 of 9)

- [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)
- [Finite Sample Analysis of Stochastic System Identification](https://mlsanity.com/paper/1903.09122/) (2019)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [Learning Linear Dynamical Systems with Semi-Parametric Least Squares](https://mlsanity.com/paper/1902.00768/) (2019)
- [Non-asymptotic Identification of Linear Dynamical Systems Using Multiple Trajectories](https://mlsanity.com/paper/2009.00739/) (2020)
- [Linear System Identification Under Multiplicative Noise from Multiple Trajectory Data](https://mlsanity.com/paper/2002.06613/) (2020)

## Built on by (most cited first, 8 of 8)

- [Identifying the Dynamics of a System by Leveraging Data from Similar Systems](https://mlsanity.com/paper/2204.05446/) (2022)
- [Learning Dynamical Systems by Leveraging Data from Similar Systems](https://mlsanity.com/paper/2302.04344/) (2023)
- [Outlier-Robust Linear System Identification Under Heavy-tailed Noise](https://mlsanity.com/paper/2501.00421/) (2024)
- [Learning Linearized Models from Nonlinear Systems with Finite Data](https://mlsanity.com/paper/2309.08805/) (2023)
- [Learning Linearized Models from Nonlinear Systems under Initialization Constraints with Finite Data](https://mlsanity.com/paper/2505.04954/) (2025)
- [Joint Learning of Linear Dynamical Systems under Smoothness Constraints](https://mlsanity.com/paper/2406.01094/) (2024)
- [Learning Dynamic Bayesian Networks from Data: Foundations, First Principles and Numerical Comparisons](https://mlsanity.com/paper/2406.17585/) (2024)
- [Boosting-Enabled Robust System Identification of Partially Observed LTI Systems Under Heavy-Tailed Noise](https://mlsanity.com/paper/2504.18444/) (2025)

## Abstract (arXiv)

We consider the problem of learning the dynamics of autonomous linear systems (i.e., systems that are not affected by external control inputs) from observations of multiple trajectories of those systems, with finite sample guarantees. Existing results on learning rate and consistency of autonomous linear system identification rely on observations of steady state behaviors from a single long trajectory, and are not applicable to unstable systems. In contrast, we consider the scenario of learning system dynamics based on multiple short trajectories, where there are no easily observed steady state behaviors. We provide a finite sample analysis, which shows that the dynamics can be learned at a rate O(frac1√N) for both stable and unstable systems, where N is the number of trajectories, when the initial state of the system has zero mean (which is a common assumption in the existing literature). We further generalize our result to the case where the initial state has non-zero mean. We show that one can adjust the length of the trajectories to achieve a learning rate of O(√fraclogNN) for strictly stable systems and a learning rate of O(frac(logN)d√N) for marginally stable systems, where d is some constant.