Network synchronization using invariant-manifold-based diffusive dynamic couplings with time-delay

We address the problem of controlled synchronization in networks of nonlinear systems interconnected through diffusive time-delayed dynamic couplings. These couplings can be realized as dynamic output feedback controllers constructed by combining nonlinear observers and time-delayed feedback interconnection terms. Using Immersion and Invariance techniques, we present a general tool for constructing the dynamics of the couplings. Sufficient conditions on the systems to be interconnected, the network topology, the couplings, and the time-delay that guarantee (global) state synchronization are derived. The asymptotic stability of the synchronization manifold is proved using Lyapunov-Razumikhin methods. Moreover, using Lyapunov-Krasovskii functionals and the notion of semipassivity, we prove that under some mild assumptions, the solutions of the interconnected systems are ultimately bounded. Simulation results using FitzHugh-Nagumo neural oscillators illustrate the performance of the control scheme.