Robust global synchronization of two complex dynamical networks

Abstract

We investigate the synchronization of two coupled complex dynamical networks, a problem that has been termed outer synchronization in the literature. Our approach relies on (a) a basic lemma on the eigendecomposition of matrices resulting from Kronecker products and (b) a suitable choice of Lyapunov function related to the synchronization error dynamics. Starting from these two ingredients, a theorem that provides a sufficient condition for outer synchronization of the networks is proved. The condition in the theorem is expressed as a linear matrix inequality. When satisfied, synchronization is guaranteed to occur globally, i.e., independently of the initial conditions of the networks. The argument of the proof includes the design of the gain of the synchronizer, which is a constant square matrix with dimension dependent on the number of dynamic variables in a single network node, but independent of the size of the overall network, which can be much larger. This basic result is subsequently elaborated to simplify the design of the synchronizer, to avoid unnecessarily restrictive assumptions (e.g., diffusivity) on the coupling matrix that defines the topology of the networks and, finally, to obtain synchronizers that are robust to model errors in the parameters of the coupled networks. An illustrative numerical example for the outer synchronization of two networks of classical Lorenz nodes with perturbed parameters is presented.