Observability Categorization for Boolean Control Networks

This article studies the observability categorization of Boolean control networks, for which the observability regarding each state pair is classified into four categories: indistinguishable, transient, primitive, and imprimitive ones. The observability categorization is guided by the distinguishable time domain of each state pair, which provides an indication of when to add the observer. First, the notion of the $k$-step distinguishability is presented and fully characterized. Then, two necessary and sufficient conditions are established to determine the observability categorization, respectively, from the graph-theoretic and algebraic perspectives. Finally, the observability categorization for a biological example about the lac operon in Escherichia coli and a constructive example is studied to illustrate the effectiveness of the theoretical methods.