Place invariant simplification in optimal supervisor synthesis for FMS

The theory of regions is an important method to derive an optimal and liveness-enforcing supervisor for a flexible manufacturing systems based on Petri nets. It first partitions the reachability graph into a live zone (LZ) and a deadlock zone (DZ). Then, activity places are used to construct place invariants (PIs) to prevent the system from entering DZ and permit all markings in LZ. This work studies the reduction of the number of places to be considered in the optimal PI designs. First, the concepts of critical transitions and critical activity places are defined, and an algorithm is provided to compute the sets of critical and uncritical activity places. Then, the proof of that only critical activity places need to be considered in such optimal PI designs is established.