Synchronization of complex dynamical networks with uncertain inner coupling and successive delays based on passivity theory

This paper is discussed with the problem of passivity based synchronization for a class of complex dynamical networks (CDNs) consisting uncertain inner coupling matrix together with successive time-varying delays via a state feedback delayed controller. Due to occurrence of uncertainties in coupling strengths, the considered CDNs take account of an uncertain inner coupling strength which is more general than the previously existing inner coupling strengths. Specifically, the uncertainties encountering in coupling terms are characterized with the aid of interval matrix approach. Also, by introducing a simple linear transformation, the corresponding error system is formulated. Then, based on the information about control delay term, two cases are considered namely, differentiable and non-differentiable. More precisely, by constructing an appropriate Lyapunov–Krasovskii functional (LKF) containing triple integral terms in respect of Kronecker product, for both the cases, some sufficient criteria are established in terms of linear matrix inequalities (LMIs) to guarantee the robust synchronization of the addressed CDNs based on passivity property. And the established criteria optimistically reduce the L2L2 gain level from the disturbance to the output vector. Subsequently, the desired state feedback gain matrix is designed in terms of the solution to a convex optimization problem. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed theoretical results.