# Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functions

arXiv: https://arxiv.org/abs/2208.10733 · HTML: https://mlsanity.com/paper/2208.10733/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 66 references that are in the map.
- First posted on arXiv 23 Aug 2022.

## Builds on (most cited first, 30 of 37)

- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances](https://mlsanity.com/paper/1709.07523/) (2017)
- [A General Safety Framework for Learning-Based Control in Uncertain Robotic Systems](https://mlsanity.com/paper/1705.01292/) (2017)
- [A predictive safety filter for learning-based control of constrained nonlinear dynamical systems](https://mlsanity.com/paper/1812.05506/) (2018)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [Learning for Safety-Critical Control with Control Barrier Functions](https://mlsanity.com/paper/1912.10099/) (2019)
- [Reinforcement Learning for Safety-Critical Control under Model Uncertainty, using Control Lyapunov Functions and Control Barrier Functions](https://mlsanity.com/paper/2004.07584/) (2020)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [Robust Control Barrier-Value Functions for Safety-Critical Control](https://mlsanity.com/paper/2104.02808/) (2021)
- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Safe Learning of Regions of Attraction for Uncertain, Nonlinear Systems with Gaussian Processes](https://mlsanity.com/paper/1603.04915/) (2016)
- [Neural Certificates for Safe Control Policies](https://mlsanity.com/paper/2006.08465/) (2020)
- [Guaranteeing Safety of Learned Perception Modules via Measurement-Robust Control Barrier Functions](https://mlsanity.com/paper/2010.16001/) (2020)
- [Robustness of Control Barrier Functions for Safety Critical Control](https://mlsanity.com/paper/1612.01554/) (2016)
- [Adaptive Safety with Control Barrier Functions](https://mlsanity.com/paper/1910.00555/) (2019)
- [Feedback Linearization based on Gaussian Processes with event-triggered Online Learning](https://mlsanity.com/paper/1911.06565/) (2019)
- [Safe Multi-Agent Interaction through Robust Control Barrier Functions with Learned Uncertainties](https://mlsanity.com/paper/2004.05273/) (2020)
- [Control Barrier Functions for Unknown Nonlinear Systems using Gaussian Processes](https://mlsanity.com/paper/2010.05818/) (2020)
- [Learning Robust Output Control Barrier Functions from Safe Expert Demonstrations](https://mlsanity.com/paper/2111.09971/) (2021)
- [Episodic Learning with Control Lyapunov Functions for Uncertain Robotic Systems](https://mlsanity.com/paper/1903.01577/) (2019)
- [Bayesian Learning-Based Adaptive Control for Safety Critical Systems](https://mlsanity.com/paper/1910.02325/) (2019)
- [Control Barriers in Bayesian Learning of System Dynamics](https://mlsanity.com/paper/2012.14964/) (2020)
- [SABLAS: Learning Safe Control for Black-box Dynamical Systems](https://mlsanity.com/paper/2201.01918/) (2022)
- [Input-to-State Safety With Control Barrier Functions](https://mlsanity.com/paper/1803.03035/) (2018)
- [Control Barrier Functions for Stochastic Systems](https://mlsanity.com/paper/2003.03498/) (2020)
- [Towards Robust Data-Driven Control Synthesis for Nonlinear Systems with Actuation Uncertainty](https://mlsanity.com/paper/2011.10730/) (2020)
- [Pointwise Feasibility of Gaussian Process-based Safety-Critical Control under Model Uncertainty](https://mlsanity.com/paper/2106.07108/) (2021)
- [Control Barrier Functions and Input-to-State Safety with Application to Automated Vehicles](https://mlsanity.com/paper/2206.03568/) (2022)

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

- [Can Learning Deteriorate Control? Analyzing Computational Delays in Gaussian Process-Based Event-Triggered Online Learning](https://mlsanity.com/paper/2305.08169/) (2023)
- [Model-Assisted Probabilistic Safe Adaptive Control With Meta-Bayesian Learning](https://mlsanity.com/paper/2307.00828/) (2023)

## Abstract (arXiv)

Learning-based control has recently shown great efficacy in performing complex tasks for various applications. However, to deploy it in real systems, it is of vital importance to guarantee the system will stay safe. Control Barrier Functions (CBFs) offer mathematical tools for designing safety-preserving controllers for systems with known dynamics. In this article, we first introduce a model-uncertainty-aware reformulation of CBF-based safety-critical controllers using Gaussian Process (GP) regression to close the gap between an approximate mathematical model and the real system, which results in a second-order cone program (SOCP)-based control design. We then present the pointwise feasibility conditions of the resulting safety controller, highlighting the level of richness that the available system information must meet to ensure safety. We use these conditions to devise an event-triggered online data collection strategy that ensures the recursive feasibility of the learned safety controller. Our method works by constantly reasoning about whether the current information is sufficient to ensure safety or if new measurements under active safe exploration are required to reduce the uncertainty. As a result, our proposed framework can guarantee the forward invariance of the safe set defined by the CBF with high probability, even if it contains a priori unexplored regions. We validate the proposed framework in two numerical simulation experiments.