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 conditions, the solutions of the interconnected systems are ultimately bounded. Simulation results using FitzHugh–Nagumo neural oscillators illustrate the performance of the control scheme.