Enforcing Liveness via Supervisory Control in Discrete Event Dynamic Systems Modeled by Completely Controlled Petri Nets

We consider discrete-state plants represented by a class of Controlled Petri nets (cf. 3, 4]) where every transition is individually controllable. We refer to such models as Completely Controlled Petri nets (CCPNs). For these models, a supervisory policy is an implicitly deened table that lists the transitions permitted to re for each reachable marking. A transition in a CCPN is state-enabled at a given marking if every input place to the said transition has a nonzero token-load. Similarly, a transition in a CCPN is control-enabled at a given marking if the supervisory policy permits the ring of the said transition. A transition in a CCPN has to be state-enabled and control-enabled to re. The set of valid ring sequences for a given marking is deened accordingly. A transition in a CCPN is live if for every marking reachable under supervision, there exists a valid ring sequence that results in a marking under which the said transition is control-enabled and state-enabled (cf. level 4 liveness, section 4.1.4, 6]). A supervisory policy enforces liveness if every transition in the CCPN is live under supervision. For the class of CCPNs, we present a necessary and suf-cient condition for the existence of a supervisory policy that enforces liveness.