On Minimum Realization of Boolean Control Networks

This paper investigates the relationship between realization and observability of Boolean control networks (BCNs) and gives some invariants under minimum realization (MR), such as, decoupling, invertibility, and so on. It is proved that the MR of a BCN exists uniquely up to coordinate transformations and a BCN is an MR of itself if and only if it is weakly observable. Based on these, the MR of given BCN can be constructed. Moreover, the observability decomposition of BCNs is discussed and an equivalent condition is obtained to determine the decomposability, which guarantees the existence and uniqueness of the regular MR (RMR) that is a special type of MR. Then the RMR of given BCN can be obtained. Finally, an example is given to show the effectiveness of obtained results.