# Pointwise Feasibility of Gaussian Process-based Safety-Critical Control under Model Uncertainty

arXiv: https://arxiv.org/abs/2106.07108 · HTML: https://mlsanity.com/paper/2106.07108/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 25 references that are in the map.
- First posted on arXiv 13 Jun 2021.

## Builds on (most cited first, 13 of 13)

- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [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)
- [Safe Learning of Regions of Attraction for Uncertain, Nonlinear Systems with Gaussian Processes](https://mlsanity.com/paper/1603.04915/) (2016)
- [Data-Enabled Predictive Control: In the Shallows of the DeePC](https://mlsanity.com/paper/1811.05890/) (2018)
- [Adaptive Safety with Control Barrier Functions](https://mlsanity.com/paper/1910.00555/) (2019)
- [Safe Multi-Agent Interaction through Robust Control Barrier Functions with Learned Uncertainties](https://mlsanity.com/paper/2004.05273/) (2020)
- [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)
- [Towards Robust Data-Driven Control Synthesis for Nonlinear Systems with Actuation Uncertainty](https://mlsanity.com/paper/2011.10730/) (2020)
- [Gaussian Process-based Min-norm Stabilizing Controller for Control-Affine Systems with Uncertain Input Effects and Dynamics](https://mlsanity.com/paper/2011.07183/) (2020)
- [The Impact of Data on the Stability of Learning-Based Control- Extended Version](https://mlsanity.com/paper/2011.10596/) (2020)
- [Learning Min-norm Stabilizing Control Laws for Systems with Unknown Dynamics](https://mlsanity.com/paper/2004.10331/) (2020)

## Built on by (most cited first, 13 of 13)

- [SABLAS: Learning Safe Control for Black-box Dynamical Systems](https://mlsanity.com/paper/2201.01918/) (2022)
- [In-Distribution Barrier Functions: Self-Supervised Policy Filters that Avoid Out-of-Distribution States](https://mlsanity.com/paper/2301.12012/) (2023)
- [Distributionally Robust Statistical Verification with Imprecise Neural Networks](https://mlsanity.com/paper/2308.14815/) (2023)
- [Safety Filters for Black-Box Dynamical Systems by Learning Discriminating Hyperplanes](https://mlsanity.com/paper/2402.05279/) (2024)
- [End-to-End Imitation Learning with Safety Guarantees using Control Barrier Functions](https://mlsanity.com/paper/2212.11365/) (2022)
- [Random Features Approximation for Control-Affine Systems](https://mlsanity.com/paper/2406.06514/) (2024)
- [DADEE: Well-calibrated uncertainty quantification in neural networks for barriers-based robot safety](https://mlsanity.com/paper/2407.00616/) (2024)
- [Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functions](https://mlsanity.com/paper/2208.10733/) (2022)
- [Safety-Critical Learning of Robot Control with Temporal Logic Specifications](https://mlsanity.com/paper/2109.02791/) (2021)
- [Model-Assisted Probabilistic Safe Adaptive Control With Meta-Bayesian Learning](https://mlsanity.com/paper/2307.00828/) (2023)
- [Real-Time Adaptive Safety-Critical Control with Gaussian Processes in High-Order Uncertain Models](https://mlsanity.com/paper/2402.18946/) (2024)
- [ProBF: Learning Probabilistic Safety Certificates with Barrier Functions](https://mlsanity.com/paper/2112.12210/) (2021)
- [Safe Online Dynamics Learning with Initially Unknown Models and Infeasible Safety Certificates](https://mlsanity.com/paper/2311.02133/) (2023)

## Abstract (arXiv)

Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) are popular tools for enforcing safety and stability of a controlled system, respectively. They are commonly utilized to build constraints that can be incorporated in a min-norm quadratic program (CBF-CLF-QP) which solves for a safety-critical control input. However, since these constraints rely on a model of the system, when this model is inaccurate the guarantees of safety and stability can be easily lost. In this paper, we present a Gaussian Process (GP)-based approach to tackle the problem of model uncertainty in safety-critical controllers that use CBFs and CLFs. The considered model uncertainty is affected by both state and control input. We derive probabilistic bounds on the effects that such model uncertainty has on the dynamics of the CBF and CLF. We then use these bounds to build safety and stability chance constraints that can be incorporated in a min-norm convex optimization-based controller, called GP-CBF-CLF-SOCP. As the main theoretical result of the paper, we present necessary and sufficient conditions for pointwise feasibility of the proposed optimization problem. We believe that these conditions could serve as a starting point towards understanding what are the minimal requirements on the distribution of data collected from the real system in order to guarantee safety. Finally, we validate the proposed framework with numerical simulations of an adaptive cruise controller for an automotive system.