线性系统的安全镇定全阶仿射控制障碍函数（扩展版）

中文摘要 AI 辅助

控制障碍函数安全滤波器通过修改标称输入来强制执行约束，但即使标称模式和滤波模式各自稳定，由此产生的切换也可能破坏闭环的稳定性。本文针对具有单个全相对阶仿射约束的线性系统，提出了一种安全且全局指数镇定的控制器设计框架。我们证明滤波模式的谱由障碍函数调谐决定，且与对象、标称控制器以及二次规划权重无关。这一结构产生了一个显式的标称控制器，使得安全滤波器在全局范围内保持非激活状态。对于给定的标称控制器，我们证明标称模式和滤波模式存在强公共二次李雅普诺夫函数，当且仅当标称特征多项式与障碍多项式之比是强严格正实函数。这一等价性刻画了公共二次李雅普诺夫函数的存在性，并提供了一个标量频域测试以及显式的容许增益区间。基于这些结果，扩展分析推导出显式的公共存储函数，并将现有的LMI综合条件简化为单一矩阵变量的可行性测试。一个灵活的两质量示例解释了已知的不稳定机制，并展示了所提设计如何恢复安全性和全局指数稳定性。

英文摘要

Control barrier function safety filters enforce constraints by modifying a nominal input, but the resulting switching can destabilise the closed loop even when the nominal and filtered modes are individually stable. This paper presents a design framework for safe and globally exponentially stabilising controllers for linear systems with a single full-relative-degree affine constraint. We show that the filtered-mode spectrum is fixed by the barrier tuning and is independent of the plant, nominal controller, and quadratic-program weighting. This structure yields an explicit nominal controller for which the safety filter remains inactive everywhere. For a prescribed nominal controller, we prove that the nominal and filtered modes admit a strong common quadratic Lyapunov function if and only if the ratio of the nominal characteristic polynomial to the barrier polynomial is strongly strictly positive real. This equivalence characterises the existence of a common quadratic Lyapunov and provides a scalar frequency-domain test, along with an explicit interval of admissible gains. Building on these results, the extended analysis derives an explicit common storage function and reduces an existing LMI synthesis condition to a feasibility test in a single matrix variable. A flexible two-mass example explains a known instability mechanism and demonstrates how the proposed design restores safety and global exponential stability.

发表机构

• The University of Manchester（曼彻斯特大学）

机构由 AI 辅助整理，请以论文原文为准。

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9 pages, 3 figures