On Uniformly Time-Varying Control Barrier Functions

This paper investigates the design of a subclass of time-varying Control Barrier Functions (CBFs), specifically that of uniformly time-varying CBFs. Leveraging the fact that CBFs encode a system's dynamic capabilities relative to a state constraint, we decouple the design of uniformly time-varying CBFs into a time-invariant and a time-varying component. We characterize the subclass of time-invariant CBFs that yield a uniformly time-varying CBF when combined with a specific type of time-varying function. A detailed analysis of those conditions under which the time-varying function preserves the CBF property of the time-invariant component is provided. These conditions allow for selecting the time-varying function such that diverse variations in the state constraints can be captured while avoiding the redesign of the time-invariant component. From a technical point of view, the analysis requires the derivation of novel relations for comparison functions, not previously reported in the literature. We further relax the requirements on the time-varying function, showing that forward invariance can still be ensured even when the uniformly time-varying value function does not strictly constitute a CBF. Finally, we discuss how existing CBF construction methods can be applied to design suitable time-invariant CBFs, and demonstrate the effectiveness of the approach through detailed numerical examples.