控制Lyapunov-屏障函数从CLF-CBF对的构造

中文摘要 AI 辅助

本文研究从给定的控制Lyapunov函数（CLF）$V$和控制屏障函数（CBF）$h$构造控制Lyapunov-屏障函数（CLBF）的问题。我们考虑形如$W=F(V,h)$的函数，其随CLF值增加而增大，且不随屏障值增加而增大，并证明CLBF的下降条件完全由一个非负标量权重刻画，该权重控制CLF和CBF的相对贡献。由于该权重仅依赖于$(V,h)$，在具有相同CLF和CBF值的所有状态处，该权重必须满足相同的下降条件，这导致一个联合水平集容许性条件。然后，我们通过一阶偏微分方程（PDE）从容许权重构造CLBF，并得到显式的乘法和幂型族作为特例。我们进一步识别出一个可积性障碍，表明仅靠容许性并不能保证函数的正常性。两个非线性例子说明了这些构造，并在基于原始CLF的反馈违反安全约束的情况下展示了安全镇定。

英文摘要

This paper studies the construction of control Lyapunov-barrier functions (CLBFs) from a given control Lyapunov function (CLF) $V$ and control barrier function (CBF) $h$. We consider functions of the form $W=F(V,h)$ that increase with the CLF value and do not increase with the barrier value, and show that the CLBF decrease condition is completely characterized by a nonnegative scalar weight that governs the relative contributions of the CLF and CBF. Because this weight depends only on $(V,h)$, the same value must satisfy the decrease condition at all states sharing the same CLF and CBF values, which leads to a joint-level-set admissibility condition. We then construct CLBFs from admissible weights through a first-order partial differential equation (PDE) and obtain explicit multiplicative and power-type families as special cases. We further identify an integrability obstruction showing that admissibility alone does not guarantee properness. Two nonlinear examples illustrate the constructions and demonstrate safe stabilization in cases where feedback based on the original CLF violates the safety constraint.

发表机构

• The City College of New York, The City University of New York（纽约市立学院，纽约市立大学）
• University of California San Diego（加州大学圣地亚哥分校）

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