Liveness enforcing supervisory policies tolerant to controllability failures for discrete-event systems modeled by Petri Nets

A Discrete Event System (DES) modeled by a Petri Net (PN) is live if it is possible to fire any transition, although not necessarily immediately, from any marking that is reachable from the initial marking. A Liveness Enforcing Supervisory Policy (LESP) for a PN enforces liveness by preventing the firing of a subset of transitions called the controllable transitions, which correspond to the preventable events in a DES. In this paper, we consider the existence and synthesis of LESPs for arbitrary PNs in the presence of faults, where a subset of controllable transitions become temporarily uncontrollable, for a finite number of event occurrences. Following the formal specification of the fault model, we present a necessary and sufficient condition for the existence of Fault-Tolerant LESPs (FT-LESPs) for arbitrary PNs. We show that, even when an LESP is given, the existence of an FT-LESP for an arbitrary PN is undecidable. We then identify a class of PNs for which the existence of FT-LESPs is decidable. We conclude with some suggestions for future research.