Continuity and Boundedness of Minimum-Norm CBF-Safe Controllers

TL;DR

This paper analyzes the regularity and boundedness properties of minimum-norm control barrier function (CBF) safe controllers, providing conditions for their continuity and boundedness to improve safe control design.

Contribution

It characterizes the discontinuity set of the minimum-norm safe controller and identifies conditions for its boundedness or unboundedness, independent of the specific CBF used.

Findings

Discontinuity set depends only on the safe set, not the CBF.

Conditions for the controller to be bounded or grow unbounded are identified.

Examples illustrate the impact of these conditions on controller behavior.

Abstract

The existence of a Control Barrier Function (CBF) for a control-affine system provides a powerful design tool to ensure safety. Any controller that satisfies the CBF condition and ensures that the trajectories of the closed-loop system are well defined makes the zero superlevel set forward invariant. Such a controller is referred to as safe. This paper studies the regularity properties of the minimum-norm safe controller as a stepping stone towards the design of general continuous safe feedback controllers. We characterize the set of points where the minimum-norm safe controller is discontinuous and show that it depends solely on the safe set and not on the particular CBF that describes it. Our analysis of the controller behavior as we approach a point of discontinuity allows us to identify sufficient conditions to ensure it grows unbounded or it remains bounded. Examples illustrate our results, providing insight into the conditions that lead to (un)bounded discontinuous minimum-norm controllers.