From Time-Invariant to Uniformly Time-Varying Control Barrier Functions: A Constructive Approach

TL;DR

A subclass of (time-invariant) Control Barrier Functions (CBF) that have favorable properties for the construction of uniformly time-varying CBFs and thereby for the satisfaction of uniformly time-varying constraints is defined and analyzed.

Abstract

In this paper, we define and analyze a subclass of (time-invariant) Control Barrier Functions (CBF) that have favorable properties for the construction of uniformly timevarying CBFs and thereby for the satisfaction of uniformly time-varying constraints. We call them Λ-shiftable CBFs where Λ states the extent by which the CBF can be varied by adding a time-varying function. Moreover, we derive sufficient conditions under which a time-varying CBF can be obtained from a time-invariant one, and we propose a systematic construction method. Advantageous about our approach is that a Λ-shiftable CBF, once constructed, can be reused for various control objectives. In the end, we relate the class of Λ-shiftable CBFs to Control Lyapunov Functions (CLF), and we illustrate the application of our results with a relevant simulation example.