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

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 behavior 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.