Synchronization forMulti-network with Two Types of Inter-network Coupling Faults: Pinning Control Effects

Investigation of simultaneous occurrence of permanent and/or recoverable faults of coupled nodes within different networks in a larger network structure is an appealing timely research topic. For the modelling of such a multi-network structure, the coupling terms in the same network and the coupling terms among different networks are described separately in order to explain clearly the multi-network feature here. The dimension-transformation matrix is used to deal with the mismatched dimensions of nodes in the different networks. Then a synchronisation controller is designed from the underlying theory of pinning control schemes. Furthermore, following the ideas of Lyapunov stability theory, a sufficient stability condition under the pinning control is derived which guarantees nodes in the same network synchronisation. Computer simulation results for a numerical case study of three coupled networks with nodes that possess rather complex non-linear dynamics demonstrate that the proposed controllers can enforce synchronisation despite different types of faults being occurred in the multi-network. It is also shown that the specific pinning control scheme is more effective than the random pinning.