Controllability and Observability of Boolean Control Networks via Sampled-Data Control

Abstract

In this paper, the controllability and observability of sampled-data Boolean control networks (SDBCNs) are investigated. New phenomena are observed in the study of the controllability and observability of SDBCNs. We routinely convert SDBCNs into linear discrete-time systems by the semitensor product of matrices. Necessary and sufficient conditions are derived for the controllability of SDBCNs. After that, we combine two SDBCNs with the same transition matrix into a new SDBCN to study the observability. Using an iterative algorithm, a stable row vector ${\mathcal {U}}^*$ , called the observability row vector, in finite iterations, is obtained. It is proved that an SDBCN is observable, if and only if $\Vert {\mathcal {U}}^*\Vert _1=N^2-N$ with $N:=2^n$ , where $n$ is the number of state-variables of BNs. Moreover, based on graph theory, a more effective algorithm is given to determine the observability of SDBCNs. Its complexity is not related to the length of the sampling period. In addition, some equivalent necessary and sufficient conditions are put forward for the observability of SDBCNs. Numerical examples are given to demonstrate the effectiveness of the obtained results.