A Novel Attempt of Barrier Function to the Safety of Caputo's fractional-order systems

The applications of barrier functions to safety analysis, diagnosis, and control have become a popular re-search direction. Barrier functions have been widely applied to solve the safety problems for different types of integer-order systems. This paper starts to discuss the relationship between barrier function and system state-safety for fractional-order nonlinear dynamical systems. First, by using Caputos fractional derivative, this paper revisits the safety theories via barrier functions (BFs). And based on less-zero barrier function, reciprocal barrier function and zeroing barrier function, three extended BFs under the Caputo fractional-order descriptions are designed. Then it establishes a framework by proposing the definitions and criteria of safety, safety-and-stability for a class of fractional-order nonlinear dynamical systems.

I. Introduction

The researches on barrier functions applied to the safety of integer-order systems has becoming a hot topic. There are several common barrier functions, such as less-zero barrier function [1], exponential condition of barrier function [2], exponential-alpha barrier function [3], general barrier function [4], reciprocal barrier function [5], [6], and zeroing barrier function [7]. In addition, there are many other inter-esting works in the field of barrier functions, such as vector barrier function [8], barrier function considering extreme-point distribution [9], [10], barrier functions with converse theorems [11], [12], barrier functions that satisfy safety and asymptotic stability synchronously [13], and barrier functions with input-to-state safety (ISSf) [14], [15]. The directions of synchronously satisfying safety and stability (or called stabilization with guaranteed safety) and ISSf considering the robustness of safety are vital studies for dynamic systems and their control applications.