Practical Considerations for Discrete-Time Implementations of Continuous-Time Control Barrier Function-Based Safety Filters (2404.12329v1)

Abstract: Safety filters based on control barrier functions (CBFs) have become a popular method to guarantee safety for uncertified control policies, e.g., as resulting from reinforcement learning. Here, safety is defined as staying in a pre-defined set, the safe set, that adheres to the system's state constraints, e.g., as given by lane boundaries for a self-driving vehicle. In this paper, we examine one commonly overlooked problem that arises in practical implementations of continuous-time CBF-based safety filters. In particular, we look at the issues caused by discrete-time implementations of the continuous-time CBF-based safety filter, especially for cases where the magnitude of the Lie derivative of the CBF with respect to the control input is zero or close to zero. When overlooked, this filter can result in undesirable chattering effects or constraint violations. In this work, we propose three mitigation strategies that allow us to use a continuous-time safety filter in a discrete-time implementation with a local relative degree. Using these strategies in augmented CBF-based safety filters, we achieve safety for all states in the safe set by either using an additional penalty term in the safety filtering objective or modifying the CBF such that those undesired states are not encountered during closed-loop operation. We demonstrate the presented issue and validate our three proposed mitigation strategies in simulation and on a real-world quadrotor.

• The paper introduces penalty terms in the CBF objective to enforce timely reversion to backup control policies, mitigating lag-induced safety violations.
• The paper transforms safe sets using rotation, translation, and alternative designs to guarantee persistent safety filter activity despite input delays.
• The paper validates its strategies through simulations and quadrotor experiments, underscoring their effectiveness for safety-critical reinforcement learning.

An Exploration of Discrete-Time Implementations of Continuous-Time CBF-Based Safety Filters

The paper "Practical Considerations for Discrete-Time Implementations of Continuous-Time Control Barrier Function-Based Safety Filters" by Lukas Brunke et al. addresses critical issues arising from the implementation of continuous-time Control Barrier Function (CBF)-based safety filters in a discrete-time setting. As the application of reinforcement learning and learning-based control grows, ensuring the safety of uncertified control policies gains paramount importance. CBFs serve as a proven method to enforce safety constraints, defining conditions under which systems can operate safely within predefined boundaries. However, the transition from continuous to discrete time poses challenges that can undermine these safety guarantees.

Problem Formulation and Challenges

This paper begins by articulating the problem of translating continuous-time CBF-based safety filters into discrete-time frameworks. In continuous systems, CBFs provide safety assurances under the assumption of immediate response times. The shift to discrete-time systems, however, introduces a delay in the application of control inputs, leading to potential issues such as chattering—frequent, undesirable switching between control states—and possible violation of safety constraints due to input lag within sampling intervals. These challenges are amplified when the magnitude of Lie derivatives concerning input dynamics tends to zero, leading to periods of inactivity for the safety filter during which any control input, including potentially unsafe ones, is effectively certified.

Proposed Solutions

To mitigate these issues, the authors propose three key strategies:

1. Penalty Terms in Objective Functions: By augmenting the CBF safety filtering objective with a penalty term, the system prioritizes reverting to a backup control policy when the safety filter potentially becomes inactive. This modification ensures consistent adherence to safety criteria even within undesirable state neighborhoods.
2. Transformation of the Safe Set: The second strategy involves strategically altering the safe set through transformations like rotation or translation. This aims to preemptively circumvent the instances where the system's state tends to render the CBF condition inactive, thereby maintaining an active and effective safety filter operation.
3. Alternative Safe Set Design: This strategy involves crafting an alternative safe set represented by functions chosen to preserve a positive Lie derivative norm concerning the input dynamics. This design restricts the occurrence of states where the safety filter would otherwise be inactive.

Validation and Implications

The solutions are rigorously validated both in simulation and via experiments using a real-world quadrotor system. These empirical tests demonstrate the effectiveness of each mitigation strategy in maintaining safe operations, particularly when standard CBF implementations fail.

The implications of this work are significant for the deployment of learning-based control in real-world scenarios. Learning systems are inherently unpredictable, and safety filters play a critical role in constraining potential failures. The practical solutions from this paper enhance the robustness of such systems, making them more viable for applications that range from autonomous driving to industrial automation.

Future Developments in AI and Control Systems

In the broader context of AI and control systems, this research encourages further investigation into adaptive and resilient system designs that address the latency inherent in digital implementations of continuous systems. The strategies proposed could stimulate new methods in safety-critical reinforcement learning, where temporal dynamics and decision freshness are essential. Future work might further explore adaptive safe set transformations or the automatic tuning of penalty parameters in real-time applications, enhancing both system performance and safety.

In summary, this paper provides a comprehensive analysis and actionable solutions for ensuring the efficacy of CBF-based safety filters within discrete environments. The contributions serve as a foundational reference for researchers and practitioners aiming to enhance safety in complex, real-world control systems.