用参数线性规划构建非光滑屏障函数，保障系统状态多面体体积不坍缩至零。
Polytope Volume Monitoring Problem: Formulation and Solution via Parametric Linear Program Based Control Barrier Function
- 基于切比雪夫球思想，设计参数线性规划的非光滑屏障函数。
- 提出确保安全滤波器可行性的二次规划求解方法。
- 适用于需要动态保持状态多面体体积的非线性系统安全控制。
针对多控制屏障函数的可行空间监控与多面体避障问题，本文研究了多面体体积监控（PVM）问题，目标是为非线性系统设计控制律，防止依赖状态的多面体体积减少至零。尽管已有研究尝试将切比雪夫球方法应用于该问题优化，但未解决由非光滑性引发的底层困难。本文首次通过方向导数建立非光滑控制屏障函数与参数优化理论之间的联系，提出基于参数线性规划（PLP）的非光滑屏障函数候选，并给出其成为非光滑CBF的充分条件。基于此，设计了一种保证可行性的二次规划（QP）安全滤波器以解决PVM问题。最后通过数值仿真验证了所提方法的有效性。
原文摘要 · Abstract (English)
Motivated by the latest research on feasible space monitoring of multiple control barrier functions (CBFs) as well as polytopic collision avoidance, this paper studies the Polytope Volume Monitoring (PVM) problem, whose goal is to design a control law for inputs of nonlinear systems to prevent the volume of some state-dependent polytope from decreasing to zero. Recent studies have explored the idea of applying Chebyshev ball method in optimization theory to solve the case study of PVM; however, the underlying difficulties caused by nonsmoothness have not been addressed. This paper continues the study on this topic, where our main contribution is to establish the relationship between nonsmooth CBF and parametric optimization theory through directional derivatives for the first time, to solve PVM problems more conveniently. In detail, inspired by Chebyshev ball approach, a parametric linear program (PLP) based nonsmooth barrier function candidate is established for PVM, and then, sufficient conditions for it to be a nonsmooth CBF are proposed, based on which a quadratic program (QP) based safety filter with guaranteed feasibility is proposed to address PVM problems. Finally, a numerical simulation example is given to show the efficiency of the proposed safety filter.
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