Rate-Tunable Control Barrier Functions: Methods and Algorithms for Online Adaptation

Control Barrier Functions offer safety certificates by dictating controllers that enforce safety constraints. However, their response depends on the classK function that is used to restrict the rate of change of the barrier function along the system trajectories. This paper introduces the notion of Rate Tunable Control Barrier Function (RT-CBF), which allows for online tuning of the response of CBF-based controllers. In contrast to the existing CBF approaches that use a fixed (predefined) classK function to ensure safety, we parameterize and adapt the classK function parameters online. Furthermore, we discuss the challenges associated with multiple barrier constraints, namely ensuring that they admit a common control input that satisfies them simultaneously for all time. In practice, RT-CBF enables designing parameter dynamics for (1) a better-performing response, where performance is defined in terms of the cost accumulated over a time horizon, or (2) a less conservative response. We propose a model-predictive framework that computes the sensitivity of the future states with respect to the parameters and uses Sequential Quadratic Programming for deriving an online law to update the parameters in the direction of improving the performance. When prediction is not possible, we also provide point-wise sufficient conditions to be imposed on any user-given parameter dynamics so that multiple CBF constraints continue to admit common control input with time. Finally, we introduce RT-CBFs for decentralized uncooperative multi-agent systems, where a trust factor, computed based on the instantaneous ease of constraint satisfaction, is used to update parameters online for a less conservative response.