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Observability Synthesis of Boolean Control Networks

Observability is a fundamental property in finite-state systems, which describes that whether the initial state of a system can be detected based on the values of input and output observations issued by the system. In this paper, using semi-tensor product technique, we investigate the synthesis problem of observability for Boolean control networks (BCNs) based on the graph theory. First, we give a sufficient condition to verify the observability of BCNs, which is different from the existing ones. It decides whether the sum of the number of different paths from non-diagonal vertices to diagonal vertices is equal to the number of all input in weighted pair graph of the BCN. Then, we study the synthesis problem whether the observability of a BCN can be influenced by state feedback with exogenous input. We show that state feedback with exogenous input sometimes can make an unobservable BCN observable, sometimes cannot, and can also make an observable BCN unobservable. Finally, we obtain a result that describes the difference between state-feedback controller with exogenous input and state-feedback controller (without exogenous input) for influencing observability of BCNs. Specifically, we show that if a BCN can be made observable by state-feedback controller, then it can also be made observable by state-feedback controller with exogenous input, but the conversion does not hold.