# Safely Learning Dynamical Systems

arXiv: https://arxiv.org/abs/2305.12284 · HTML: https://mlsanity.com/paper/2305.12284/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 58 references that are in the map.
- First posted on arXiv 20 May 2023.

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

- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [End-to-End Safe Reinforcement Learning through Barrier Functions for Safety-Critical Continuous Control Tasks](https://mlsanity.com/paper/1903.08792/) (2019)
- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [A predictive safety filter for learning-based control of constrained nonlinear dynamical systems](https://mlsanity.com/paper/1812.05506/) (2018)
- [Conservative Safety Critics for Exploration](https://mlsanity.com/paper/2010.14497/) (2020)
- [Learning for Safety-Critical Control with Control Barrier Functions](https://mlsanity.com/paper/1912.10099/) (2019)
- [Cautious Adaptation For Reinforcement Learning in Safety-Critical Settings](https://mlsanity.com/paper/2008.06622/) (2020)
- [Safely Learning to Control the Constrained Linear Quadratic Regulator](https://mlsanity.com/paper/1809.10121/) (2018)
- [A New Approach to Learning Linear Dynamical Systems](https://mlsanity.com/paper/2301.09519/) (2023)
- [Learning Dynamical Systems with Side Information](https://mlsanity.com/paper/2008.10135/) (2020)
- [Learning-based Model Predictive Control for Safe Exploration and Reinforcement Learning](https://mlsanity.com/paper/1906.12189/) (2019)
- [On the sample complexity of stabilizing linear dynamical systems from data](https://mlsanity.com/paper/2203.00474/) (2022)
- [Robust Model Predictive Control with Polytopic Model Uncertainty through System Level Synthesis](https://mlsanity.com/paper/2203.11375/) (2022)
- [Convex computation of the maximum controlled invariant set for polynomial control systems](https://mlsanity.com/paper/1303.6469/) (2013)
- [Safely Learning Dynamical Systems from Short Trajectories](https://mlsanity.com/paper/2011.12257/) (2020)
- [Data-driven stabilization of nonlinear polynomial systems with noisy data](https://mlsanity.com/paper/2011.07833/) (2020)
- [Data-driven control via Petersen's lemma](https://mlsanity.com/paper/2109.12175/) (2021)
- [Safe Exploration for Identifying Linear Systems via Robust Optimization](https://mlsanity.com/paper/1711.11165/) (2017)
- [Robust-to-Dynamics Optimization](https://mlsanity.com/paper/1805.03682/) (2018)
- [System Level Synthesis-based Robust Model Predictive Control through Convex Inner Approximation](https://mlsanity.com/paper/2111.05509/) (2021)
- [Data-driven design of safe control for polynomial systems](https://mlsanity.com/paper/2112.12664/) (2021)

## Abstract (arXiv)

A fundamental challenge in learning an unknown dynamical system is to reduce model uncertainty by making measurements while maintaining safety. We formulate a mathematical definition of what it means to safely learn a dynamical system by sequentially deciding where to initialize trajectories. The state of the system must stay within a safety region for a horizon of T time steps under the action of all dynamical systems that (i) belong to a given initial uncertainty set, and (ii) are consistent with information gathered so far. First, we consider safely learning a linear dynamical system involving n states. For the case T=1, we present an LP-based algorithm that either safely recovers the true dynamics from at most n trajectories, or certifies that safe learning is impossible. For T=2, we give an SDP representation of the set of safe initial conditions and show that lceil n/2 rceil trajectories generically suffice for safe learning. For T = ∞, we provide SDP-representable inner approximations of the set of safe initial conditions and show that one trajectory generically suffices for safe learning. We extend a number of our results to the cases where the initial uncertainty set contains sparse, low-rank, or permutation matrices, or when the system has a control input. Second, we consider safely learning a general class of nonlinear dynamical systems. For the case T=1, we give an SOCP-based representation of the set of safe initial conditions. For T=∞, we provide semidefinite representable inner approximations to the set of safe initial conditions. We show how one can safely collect trajectories and fit a polynomial model of the nonlinear dynamics that is consistent with the initial uncertainty set and best agrees with the observations. We also present some extensions to cases where the measurements are noisy or the dynamical system involves disturbances.