Contraction and $k$-contraction in Lurie systems with applications to networked systems

TL;DR

This paper introduces new conditions for k-contraction in Lurie systems, enabling analysis of systems with multiple equilibria and applications to neural networks, opinion dynamics, and power systems.

Contribution

It derives a novel sufficient condition for k-contraction in Lurie systems, extending stability analysis beyond traditional single-equilibrium frameworks.

Findings

Established a sufficient condition for k=1 contraction matching the small gain theorem.

Extended the condition to k=2, ensuring convergence to equilibria in systems with multiple solutions.

Applied the theory to neural networks, opinion models, and power systems, demonstrating practical relevance.

Abstract

A Lurie system is the interconnection of a linear time-invariant system and a nonlinear feedback function. We derive a new sufficient condition for $k$-contraction of a Lurie system. For $k=1$, our sufficient condition reduces to the standard stability condition based on the bounded real lemma and a small gain condition. However, Lurie systems often have more than a single equilibrium and are thus not contractive with respect to any norm. For $k=2$, our condition guarantees a well-ordered asymptotic behaviour of the closed-loop system: every bounded solution converges to an equilibrium, which is not necessarily unique. We demonstrate our results by deriving a sufficient condition for $k$-contraction of a general networked system, and then applying it to guarantee $k$-contraction in a Hopfield neural network, a nonlinear opinion dynamics model, and a 2-bus power system.