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.