State Feedback Controller Design for the Synchronization of Master-Slave Probabilistic Boolean Networks

The state feedback design for the synchronization of master-slave probabilistic Boolean networks is investigated in this paper, where the master system is a probabilistic Boolean network (PBN) and the slave system is a probabilistic Boolean control network (PBCN). Firstly, by means of the semi-tensor product of matrices, an auxiliary system in the form of a PBCN is constructed in terms of the algebraic representation. It is proved that the synchronization problem of master-slave PBNs is equivalent to a set stabilization problem of the corresponding auxiliary system. Secondly, through the analysis of the auxiliary system, a necessary and sufficient existence condition of a synchronizing state feedback controller is derived for master-slave PBNs. Finally, a constructive design approach to a synchronizing state feedback controller is provided. All controllers designed by our approach guarantee that master-slave PBNs can achieve synchronization in finite steps with probability one.