Semi-global cluster synchronization for nonlinear systems under fixed and switching topologies.

In this paper, the cluster synchronization problem for heterogeneous nonlinear systems with input saturation is studied under both fixed and switching topologies. The distributed feedback controllers are first designed by low gain technique to deal with the input saturation. Sufficient conditions for reaching semi-global cluster synchronization are derived by utilizing the algebraic graph theory, Lyapunov method, and low gain technique. It is shown that the semi-global cluster synchronization problem can be solved by using the proposed synchronization protocols on some preconditions for both fixed and switching topologies. Moreover, under the switching topology, the lower bound of the total activation time for the derived topology with a directed spanning tree is explicitly specified. Finally, some examples are presented to demonstrate the effectiveness of the proposed theories.