On Nonexistence of a Maximally Permissive Liveness-Enforcing Pure Net Supervisor

Behavioral permissiveness is one of the well-accepted criteria for evaluating the performance of a liveness-enforcing Petri net supervisor. However, a maximally permissive liveness-enforcing pure net supervisor does not always exist for any Petri net. This paper focuses on the nonexistence of maximally permissive liveness-enforcing pure net supervisors for a subclass of Petri nets. It shows that if a net contains a siphon that cannot be optimally controlled by a P-invariant, the net does not have a maximally permissive liveness-enforcing pure net supervisor. For a class of Petri nets that can model flexible manufacturing systems, sufficient conditions are proposed to decide whether there exists a siphon that cannot be optimally controlled. This paper also shows that a Petri net system does not have a maximally permissive liveness-enforcing pure net supervisor if it contains some special reachable markings.