A Tunable Universal Formula for Safety-Critical Control

Sontag's universal formula is a widely used technique for stabilizing control through control Lyapunov functions. Recently, it has been extended to address safety-critical control by incorporating control barrier functions (CBFs). However, deriving a universal formula that satisfies requirements on essential properties, including safety, smoothness, and robustness against input disturbances, is still an open problem. To address this challenge, this paper introduces a novel solution - a tunable universal formula - by incorporating a (state-dependent) tunable term into Sontag's formula. This tunable term enables the regulation of safety-critical control performances, allowing the attainment of desired properties through a proper selection of tunable terms. Generally, the tunable universal formula can be seen as a controller that improves the quadratic program (QP)-synthesized controllers in terms of robustness and smoothness, while also reducing the conservatism (corresponding to robustness) in Sontag's formula. Furthermore, we extend the tunable universal formula to address safety-critical control problems with norm-bounded input constraints, showcasing its applicability across diverse control scenarios. Finally, we demonstrate the efficacy of our method through a two-link manipulator safe tracking example, investigating the essential properties including safety, smoothness, and robustness against input disturbances under various tunable terms.