Synchronization of Lur’e-type nonlinear systems in linear dynamical networks having fast convergence rate and large DC gain

Abstract

This paper studies the problem of synchronization in a dynamical network, where we identify the nodes of the undirected graph as identical Lur’e-type nonlinear systems and the edges as heterogeneous and asymptotically stable linear systems, i.e., both of the nodes and edges are dynamical systems. It is shown that sufficiently fast convergence rate of each edge guarantees the dynamic edges degenerate into their associated static maps, equivalent to assigning the constant weights to the edges (as in the usual synchronization problems). Thus, in effect, the dynamical network becomes the static one characterized by the Laplacian matrix of the graph with the weights being the gains of the static maps. Then, the large DC gains of the edges, i.e., the large weights, are shown to yield the synchronization of the state variables of the Lur’e-type systems in the static network. The synchronization in the dynamical network is at last guaranteed by the combination of these two results. The results in this paper are applicable to the synchronization of the terminal voltages of the Lur’e-type circuits that are interconnected with each other via the lines having their associated line admittances.