Controllability Conditions of Resultant Siphons in a Class of Petri Nets

Strict minimal siphons (SMSs) play a key role in the development of deadlock prevention policies by using Petri nets for flexible manufacturing systems (FMSs). In this paper, given two SMSs and their resultant siphon, the concept of key resource subsets is proposed which is shown to be the critical factor in deciding the controllability conditions of the latter. A necessary and sufficient condition is then proposed under which the resultant siphon can be always marked if its two SMSs are M-controlled, i.e., invariant controlled via the method proposed by Moody and Antsaklis. As for a resultant siphon that is composed by more than two composable SMSs, a sufficient condition is proposed under which it can be always marked if its SMSs are M-controlled. They are established by analyzing the structural characteristics and markings of the resource subnets in a class of Petri nets called L-S³PR. When they are applied to some classes of S³PR nets, i.e., those whose controlled ones are maximally permissive and live once their original SMSs are M-controlled, the number of monitors may be reduced. An FMS example is used to illustrate the application of the results.