# Safe Learning of Regions of Attraction for Uncertain, Nonlinear Systems with Gaussian Processes

arXiv: https://arxiv.org/abs/1603.04915 · HTML: https://mlsanity.com/paper/1603.04915/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Mar 2016.

## Built on by (most cited first, 30 of 39)

- [EnsembleDAgger: A Bayesian Approach to Safe Imitation Learning](https://mlsanity.com/paper/1807.08364/) (2018)
- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Neural Lyapunov Control of Unknown Nonlinear Systems with Stability Guarantees](https://mlsanity.com/paper/2206.01913/) (2022)
- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [GoSafe: Globally Optimal Safe Robot Learning](https://mlsanity.com/paper/2105.13281/) (2021)
- [There Is No Turning Back: A Self-Supervised Approach for Reversibility-Aware Reinforcement Learning](https://mlsanity.com/paper/2106.04480/) (2021)
- [Safe Model-Based Reinforcement Learning with an Uncertainty-Aware Reachability Certificate](https://mlsanity.com/paper/2210.07553/) (2022)
- [A Human-Centered Safe Robot Reinforcement Learning Framework with Interactive Behaviors](https://mlsanity.com/paper/2302.13137/) (2023)
- [Model Error Propagation via Learned Contraction Metrics for Safe Feedback Motion Planning of Unknown Systems](https://mlsanity.com/paper/2104.08695/) (2021)
- [Pointwise Feasibility of Gaussian Process-based Safety-Critical Control under Model Uncertainty](https://mlsanity.com/paper/2106.07108/) (2021)
- [Statistical Safety and Robustness Guarantees for Feedback Motion Planning of Unknown Underactuated Stochastic Systems](https://mlsanity.com/paper/2212.06874/) (2022)
- [Learning the Koopman Eigendecomposition: A Diffeomorphic Approach](https://mlsanity.com/paper/2110.07786/) (2021)
- [Stability-informed Bayesian Optimization for MPC Cost Function Learning](https://mlsanity.com/paper/2404.12187/) (2024)
- [Learning Control Barrier Functions and their application in Reinforcement Learning: A Survey](https://mlsanity.com/paper/2404.16879/) (2024)
- [Uniform Error and Posterior Variance Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/2101.05328/) (2021)
- [Safe and Stable Closed-Loop Learning for Neural-Network-Supported Model Predictive Control](https://mlsanity.com/paper/2409.10171/) (2024)
- [Safety Beyond the Training Data: Robust Out-of-Distribution MPC via Conformalized System Level Synthesis](https://mlsanity.com/paper/2602.12047/) (2026)
- [SAAC: Safe Reinforcement Learning as an Adversarial Game of Actor-Critics](https://mlsanity.com/paper/2204.09424/) (2022)
- [Effects of Safety State Augmentation on Safe Exploration](https://mlsanity.com/paper/2206.02675/) (2022)
- [Inverse Reinforcement Learning: A Control Lyapunov Approach](https://mlsanity.com/paper/2104.04483/) (2021)
- [Formal Verification of Unknown Dynamical Systems via Gaussian Process Regression](https://mlsanity.com/paper/2201.00655/) (2021)
- [Physically Consistent Learning of Conservative Lagrangian Systems with Gaussian Processes](https://mlsanity.com/paper/2206.12272/) (2022)
- [System-level Safety Guard: Safe Tracking Control through Uncertain Neural Network Dynamics Models](https://mlsanity.com/paper/2312.06810/) (2023)
- [Lyapunov-based uncertainty-aware safe reinforcement learning](https://mlsanity.com/paper/2107.13944/) (2021)
- [Gaussian Control Barrier Functions : A Non-Parametric Paradigm to Safety](https://mlsanity.com/paper/2203.15474/) (2022)
- [Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functions](https://mlsanity.com/paper/2208.10733/) (2022)
- [Stochastic Reinforcement Learning with Stability Guarantees for Control of Unknown Nonlinear Systems](https://mlsanity.com/paper/2409.08382/) (2024)
- [OpReg-Boost: Learning to Accelerate Online Algorithms with Operator Regression](https://mlsanity.com/paper/2105.13271/) (2021)
- [Learning To Estimate Regions Of Attraction Of Autonomous Dynamical Systems Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2111.09930/) (2021)
- [Model-free Learning of Regions of Attraction via Recurrent Sets](https://mlsanity.com/paper/2204.10372/) (2022)

## Abstract (arXiv)

Control theory can provide useful insights into the properties of controlled, dynamic systems. One important property of nonlinear systems is the region of attraction (ROA), a safe subset of the state space in which a given controller renders an equilibrium point asymptotically stable. The ROA is typically estimated based on a model of the system. However, since models are only an approximation of the real world, the resulting estimated safe region can contain states outside the ROA of the real system. This is not acceptable in safety-critical applications. In this paper, we consider an approach that learns the ROA from experiments on a real system, without ever leaving the true ROA and, thus, without risking safety-critical failures. Based on regularity assumptions on the model errors in terms of a Gaussian process prior, we use an underlying Lyapunov function in order to determine a region in which an equilibrium point is asymptotically stable with high probability. Moreover, we provide an algorithm to actively and safely explore the state space in order to expand the ROA estimate. We demonstrate the effectiveness of this method in simulation.