A sufficient condition for k-contraction in Lurie systems

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.