Minimal observability of Boolean networks

In this study, the minimum observability of Boolean networks (BNs) is investigated by using the semi-tensor product (STP) of matrices. First, a new system based on the considered BN is obtained to analyze states pair dynamic trajectories, from which a necessary and sufficient condition for the observability of BNs is determined. Second, adding a new observer improves the observability without affecting the observable states. Thus, an algorithm is presented to design an observer for an unobservable BN. In addition, a necessary condition is obtained to determine the minimum number of nodes required to be directly measurable. Further, an algorithm to address the minimal observability is presented. Finally, examples are provided to demonstrate the effectiveness of the obtained results.