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Monotonicity and Contraction on Polyhedral Cones

Saber Jafarpour, Samuel Coogan

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.

Monotonicity and Contraction on Polyhedral Cones

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.
Paper Structure (11 sections, 9 theorems, 51 equations, 2 figures)

This paper contains 11 sections, 9 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and Mathematical Preliminary
  3. Cones, positive operators, and Metzler operators
  4. Gauge and dual gauge norm
  5. Monotone system on polyhedral cones
  6. Applications
  7. Monotone edge flows in dynamic networks
  8. Scalable control design with safety guarantees
  9. Conclusions
  10. Proof of Lemma \ref{['thm:pos-mon']}
  11. Proof of Theorem \ref{['thm:contraction']}

Key Result

Lemma III.1

Let $K\subset \mathbb{R}^n$ be a polyhedral cone with representation $(H,V)$ such that $H\in \mathbb{R}^{m\times n}$ and $V\in \mathbb{R}^{n\times q}$. The following statements are equivalent: Additionally, the following statements are equivalent:

Figures (2)

  • Figure 1: The structure of the graph $G$ in Example \ref{['ex:ave']}
  • Figure 2: The safe set $\mathcal{X}$ (blue) and its under-approximation by the polytope $\mathcal{P}$ (red).

Theorems & Definitions (20)

  • Lemma III.1: $H$-rep and $V$-rep of polyhedral cones
  • Lemma III.2
  • Proposition IV.1: Gauge and dual gauge norms
  • proof
  • Lemma IV.2: Formula for the gauge seminorms
  • proof
  • Theorem IV.3: Characterization of the gauge matrix measures
  • Definition V.1: Monotone systems
  • Theorem V.2: Characterization of monotonicity
  • proof
  • ...and 10 more