论文标题
多面体锥的单调性和收缩
Monotonicity and Contraction on Polyhedral Cones
论文作者
论文摘要
在本说明中，我们研究了单调动力学系统，相对于多面体锥。使用半空间表示和顶点表示，我们提出了三个等效条件，以证明相对于多面体锥体的动态系统的单调性。然后，我们引入了与锥体相关的规范的概念，并为计算与多面体锥相关的计算规范规范提供了封闭式公式。规范的一个关键特征是，单调系统相对于它们的合同性可以通过简单的不平等有效地表征。该结果概括了梅兹勒矩阵的赫维兹度的众所周知的标准，并提供了一种可扩展的方法来搜索单调系统的lyapunov功能相对于多面体锥。最后，我们研究了我们在动态流网络的瞬时稳定性以及具有安全保证的可扩展控制设计中的应用。
In this note, we study monotone dynamical systems with respect to polyhedral cones. Using the half-space representation and the vertex representation, we propose three equivalent conditions to certify monotonicity of a dynamical system with respect to a polyhedral cone. We then introduce the notion of gauge norm associated with a cone and provide closed-from formulas for computing gauge norms associated with polyhedral cones. A key feature of gauge norms is that contractivity of monotone systems with respect to them can be efficiently characterized using simple inequalities. This result generalizes the well-known criteria for Hurwitzness of Metzler matrices and provides a scalable approach to search for Lyapunov functions of monotone systems with respect to polyhedral cones. Finally, we study the applications of our results in transient stability of dynamic flow networks and in scalable control design with safety guarantees.
