Monotonicity and Contraction on Polyhedral Cones

TL;DR

This paper develops new criteria for certifying monotonicity and contractivity of dynamical systems on polyhedral cones, enabling scalable stability analysis and control design.

Contribution

It introduces equivalent conditions for monotonicity, defines gauge norms for polyhedral cones, and links contractivity to Lyapunov functions, generalizing existing stability criteria.

Findings

Provides closed-form formulas for gauge norms of polyhedral cones.

Characterizes contractivity of monotone systems using simple inequalities.

Applies results to transient stability and safety-guaranteed control in networks.

Abstract

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.