[Paper Review] Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functions

Fernando Castañeda, Jason J. Choi+4 more|arXiv (Cornell University)|Aug 23, 2022

Gaussian Processes and Bayesian Inference4 citations

TL;DR

This paper proposes a probabilistic online learning framework that ensures recursive feasibility of safety-critical control using Gaussian Process-embedded Control Barrier Functions (GP-CBF). By formulating a robust, uncertainty-aware CBF optimization with event-triggered data collection, the method guarantees forward invariance of a safe set with high probability, even in unexplored regions, while dynamically determining when new data is needed to maintain safety.

ABSTRACT

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.

Motivation & Objective

• Address the safety dilemma in online learning for control systems, where insufficient data prevents safe deployment but deployment is required to collect data.
• Bridge the gap between approximate system models and real dynamics by incorporating Gaussian Process (GP) regression to model uncertainty in control barrier functions.
• Ensure that the safety-critical controller remains feasible at all times by deriving richness conditions on available data and system dynamics.
• Develop an event-triggered data collection strategy that determines when new measurements are required to maintain recursive feasibility and safety.
• Provide formal high-probability guarantees of forward invariance of the safe set, even in a priori unexplored regions of the state space.

Proposed method

• Reformulate Control Barrier Functions (CBFs) using Gaussian Process regression to model uncertainty in system dynamics, resulting in a GP-CBF-SOCP (Second-Order Cone Program) formulation.
• Introduce a feasibility condition based on the sign of a scalar quantity $ p $, which determines whether a safe control input exists given current data and model uncertainty.
• Define an event-triggered data collection mechanism that activates when $ p \leq 0 $, indicating insufficient information to guarantee safety, prompting new measurements.
• Use a posterior variance update rule to ensure that new data reduces uncertainty along the critical direction $ e_{\dagger} $, improving feasibility over time.
• Leverage strong convexity and constraint qualifications to prove local Lipschitz continuity of the safe control law, enabling local existence and uniqueness of closed-loop trajectories.
• Apply theoretical results from parametric optimization to ensure that the GP-CBF-SOCP remains feasible and that the safe control input can be computed in real time.

Experimental results

Research questions

• RQ1 Under what conditions is the GP-CBF-based safety controller recursively feasible given model uncertainty and limited data?
• RQ2 How can we formally determine when new data collection is required to maintain safety during online learning?
• RQ3 Can we guarantee forward invariance of a safe set with high probability even in unexplored regions of the state space?
• RQ4 What data collection strategy ensures that the system remains safe while improving model confidence over time?
• RQ5 How does the integration of GP regression with CBFs enable safe, adaptive learning without requiring offline data or perfect models?

Key findings

• The proposed GP-CBF-SOCP is feasible if and only if the scalar $ p > 0 $, which geometrically corresponds to the hyperplane of the CBF constraint lying above the asymptote of the second-order cone.
• When $ p \leq 0 $, the CBF constraint becomes infeasible, signaling that new data is required to restore the ability to compute a safe control input.
• Event-triggered data collection based on the $ p $-condition ensures that the system only collects data when necessary, maintaining recursive feasibility and safety.
• After adding new data points, the posterior variance along the critical direction $ e_{\dagger} $ decreases, ensuring that $ \lambda_{\dagger} < 0 $ and feasibility is restored.
• The safe control input $ u_{\text{safe}} $ is locally Lipschitz continuous due to strong convexity and constraint qualifications, ensuring well-posedness of the closed-loop system.
• Formal guarantees of forward invariance of the safe set are provided with high probability, even in regions not previously explored, due to the probabilistic uncertainty model and adaptive data collection.