Characterizing safety: minimal barrier functions from scalar comparison systems
R. Konda, A. Ames, S. Coogan
IEEE Control Systems Letters, April 2021
Abstract
Verifying set invariance, considered a fundamental problem in dynamical systems theory and practically motivated by problems in safety assurance, has classical solutions stemming from the seminal work by Nagumo. Defining sets via a smooth barrier function constraint inequality results in computable flow conditions for guaranteeing set invariance. While a majority of these historic results on set invariance consider flow conditions on the boundary, recent results on control barrier functions extended these conditions to the entire set - although they still reduced to the classic Nagumo conditions on the boundary and thus require regularity conditions on the barrier function. This paper fully characterizes set invariance through minimal barrier functions by directly appealing to a comparison result to define a novel flow condition over the entire domain of the system. A considerable benefit of this approach is the removal of regularity assumptions of the barrier function. This paper also outlines necessary and sufficient conditions for a valid differential inequality condition, giving the minimum conditions for this type of approach and allowing for the verification of the largest class of invariant sets. We also show when minimal barrier functions are necessary and sufficient for set invariance. This paper further discusses extensions into time varying and control formulations, and outlines the connections between the proposed minimal barrier function and the historic boundary-based conditions for set invariance.