标题： 关于最优衰减控制屏障函数的性质 显示英文标题

标题： On the Properties of Optimal-Decay Control Barrier Functions

摘要： 控制屏障函数提供了一种强大的方法，用于合成安全过滤器，以确保以正向集不变性为框架的安全性。 CBFs 的有效性关键在于系统动态上的简单不等式：$\dot{h} \geq - \alpha(h)$。 然而确定类$\mathcal{K}^e$函数$\alpha$是用户定义的选择，这可能会对结果系统行为产生重大影响。 本文形式化了使用最优衰减控制屏障函数（OD-CBFs）选择$\alpha$的过程。 这些函数修改了传统的 CBF 不等式为：$\dot{h} \geq - \omega \alpha(h)$，其中$\omega \geq 0$由安全过滤器自动确定。 详细阐述了该框架的全面特性，包括 OD-CBF 有效性的可处理条件、状态空间中底层集合的控制不变性、安全集合的正向不变性条件，以及在可行性、Lipschitz 连续性和显式表达式方面对基于优化的安全控制器的讨论。 该框架还扩展了现有的高阶 CBF 技术，解决了相对阶消失的安全约束问题。 所提出的方法在仿真中的卫星控制问题上进行了演示。 显示英文摘要

摘要： Control barrier functions provide a powerful means for synthesizing safety filters that ensure safety framed as forward set invariance. Key to CBFs' effectiveness is the simple inequality on the system dynamics: $\dot{h} \geq - \alpha(h)$. Yet determining the class $\mathcal{K}^e$ function $\alpha$ is a user defined choice that can have a dramatic effect on the resulting system behavior. This paper formalizes the process of choosing $\alpha$ using optimal-decay control barrier functions (OD-CBFs). These modify the traditional CBF inequality to: $\dot{h} \geq - \omega \alpha(h)$, where $\omega \geq 0$ is automatically determined by the safety filter. A comprehensive characterization of this framework is elaborated, including tractable conditions on OD-CBF validity, control invariance of the underlying sets in the state space, forward invariance conditions for safe sets, and discussion on optimization-based safe controllers in terms of their feasibility, Lipschitz continuity, and closed-form expressions. The framework also extends existing higher-order CBF techniques, addressing safety constraints with vanishing relative degrees. The proposed method is demonstrated on a satellite control problem in simulation.