Algebraic calculation for safety synthesis of networked logical finite state machines

This paper investigates the safety synthesis problem of networked logical finite state machines with bounded communication delays in observation and control channels using the semi-tensor product (STP) of matrices. First, algebraic formulations for networked state estimation and state prediction are constructed. By defining a set of critical states that may transition to illegal states through observable events under communication delays, a more refined algorithm for state evolution is proposed. Then, based on the concept of minimal constraints, necessary and sufficient algebraic conditions are derived for the existence of a networked supervisor that satisfies the given safety specifications. The proposed method exhibits polynomial time complexity. Finally, the theoretical results are validated through an illustrative example.