Analysis on Fault Detectability of Boolean Control Networks: A Labeled Graph Approach

In this article, fault detectability of Boolean control networks (BCNs) is analyzed via a labeled graph approach. First, matrix-based representations of nonfault BCNs and fault BCNs are constructed by using the semi-tensor product (STP) of matrices. Based on these matrix representations, labeled graphs are further developed for nonfault BCNs and fault BCNs, respectively. Then, the passive fault detectability (PFD) is solved by labeled graph of the nonfault BCN. Meanwhile, based on the labeled graphs of nonfault BCNs and fault BCNs, the active fault detectability (AFD) is further studied. By leveraging labeled graphs, two sufficient criteria for strong AFD and AFD can be derived without the need for iterative matrix calculations, thereby significantly reducing computational complexity. Furthermore, the corresponding necessary and sufficient criteria are further derived when these two sufficient criteria are invalid. Finally, a biological system for the lac operon in $Escherichia~coli$ is elaborated to verify the effectiveness of obtained results.