On a Minimally Restrictive Supervisory Policy that Enforces Liveness in Partially Controlled Free Choice Petri Nets

On supervisory policies that enforce liveness in completely controlled petri nets with directed cut-places and cut-transitions. On supervisory policies that enforce liveness in in a class of completely controlled petri nets obtained via reenement. Starting from the observation that the set of initial markings (N) for which there is a supervisory policy that enforces liveness in a partially controlled Free Choice Petri net (FCPN) N(m 0) is right-closed, following a series of observations, we (i) developed a test for the existence of a supervisory policy that enforces liveness in N(m 0), and (ii) synthesized a minimally restrictive supervisory policy that enforces liveness in N(m 0). This synthesis procedure supposes the ready availability of the nite set of minimal-elements of (N). The computation of this set is suggested as a future research topic. References 1] M.H.T. Hack. Extended state machine allocatable nets (esma): An extension of free-choice petri net results. A survey of Petri net methods for controlled discrete event systems. On the existence of supervisory policies that enforce liveness in discrete event dynamic systems modeled by controlled Petri nets. 8 if t i 2 T u , then b P(m) i = 1, if t i 2 T c , then b P(m) i = 1 only if either (i) t i = 2 T e (m; N), or (ii) m ! t i ! m in N(m), and 9f m 2 ?(((N)), such that m f m (i.e. m 2 (N)). Since the set ?(((N)) is nite, we know that the test that is required for the case when t i 2 T c is computable. In theorem 4.2 we show that b P is a minimally restrictive policy that enforces liveness in N(m 0). Before we state and prove theorem 4.2, we rst make an observation about the markings that are reachable under the supervision of b P in N(m 0). Observation 4.6 Given a partially controlled FCPN structure N = ((; T;), an initial marking m 0 2 (N), and the supervisory policy b P as deened above, then if m 0 ! ! m 1 under the supervision of b P in N(m 0), then m 1 2 (N). This observation can be established by an induction argument over the length of. The base-case is trivially established when is the empty string. The induction hypothesis supposes the observation is true for all of length k 2 …