Observability Decomposition of Boolean Control Networks Under Several Kinds of Observability

In this article, a uniform method is provided to investigate the observability decomposition of Boolean control networks (BCNs) under different definitions of observability. Since the observability of BCNs is defined via the distinguishability of states, an algorithm is devised to identify and sort distinguishable states. To find the possible control sequences, a truth matrix reflecting the relationship between state pairs and control inputs is constructed. Using the truth matrix, necessary and sufficient conditions for several types of observability are given. Based on criteria for observability and some matrix discriminants, the realizability of observability decomposition is discussed. Notably, the proposed method is applicable to the observability decomposition under different definitions of observability, while the existing result is valid for only one type of observability. Furthermore, an example is introduced to illustrate the feasibility of the presented results.