Supervisory control of a class of Petri nets with unobservable and uncontrollable transitions

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This paper uses Petri nets (PNs) as a modeling tool to deal with the forbidden state problem of discrete event systems (DESs) in the presence of both unobservable and uncontrollable events. First of all, it is proved that two state specifications are equivalent if their admissible marking sets coincide. Motivated by this result, we focus on studying how to compute optimal policies with respect to a state specification that is an admissible linear constraint. Thanks to many approaches in the literature that allow one to efficiently transform an arbitrary linear constraint into an admissible one with the admissible marking set unchanged, the proposed result remains useful in the more general case of arbitrary linear constraints. Specifically, focusing on ordinary PNs subject to an admissible linear constraint, we propose an optimal control policy whose computation mainly lies in the computation of the unobservable minimal decrease, a parameter depending on the current observation and the given constraint. A procedure to compute such a parameter with polynomial complexity is proposed provided that a particular subnet, called observation subnet, is acyclic and backward-conflict and backward-concurrent free (BBF). As a result, under such assumptions, the optimal control policy could be computed with polynomial complexity.