Exponential-Alpha Safety Criteria of a Class of Dynamic Systems With Barrier Functions

Abstract

A classic kind of researches about the operational safety criterion for dynamic systems with barrier function can be roughly summarized as functional relationship, denoted by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\oplus$</tex> , between the barrier function and its first derivative for time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t$</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\oplus$</tex> can be “=”, “<”, or “>”, etc. This article draws on the form of the stable condition expression for finite time stability to formulate a novel kind of relaxed safety judgement criteria called exponential-alpha safety criteria. Moreover, we initially explore to use the control barrier function under exponential-alpha safety criteria to achieve the control for the dynamic system operational safety. In addition, derived from the actual process systems, we propose multi-hypersphere methods which are used to construct barrier functions and improved them for three types of special spatial relationships between the safe state set and the unsafe state set, where both of them can be spatially divided into multiple subsets. And the effectiveness of the proposed safety criteria are demonstrated by simulation examples.