A sufficient condition for $k$-contraction in Lurie systems

TL;DR

This paper introduces a new sufficient condition for $k$-contraction in Lurie systems, extending classical stability criteria and ensuring convergence properties in nonlinear feedback systems and networks.

Contribution

It derives a novel sufficient condition for $k$-contraction in Lurie systems, generalizing stability conditions and applying to networked systems.

Findings

For $k=1$, reduces to standard stability condition.

For $k=2$, guarantees convergence to an equilibrium.

Applicable to networked systems for $k$-contractivity.

Abstract

We consider a Lurie system obtained via a connection of a linear time-invariant system and a nonlinear feedback function. Such systems often have more than a single equilibrium and are thus not contractive with respect to any norm. 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. For $k=2$, our condition guarantees well-ordered asymptotic behaviour of the closed-loop system: every bounded solution converges to an equilibrium, which is not necessarily unique. We apply our results to derive a sufficient condition for $k$-contractivity of a networked system.