Necessary and sufficient liveness condition of GS³PR Petri nets

Abstract

Structural analysis is one of the most important and efficient methods to investigate the behaviour of Petri nets. Liveness is a significant behavioural property of Petri nets. Siphons, as structural objects of a Petri net, are closely related to its liveness. Many deadlock control policies for flexible manufacturing systems (FMS) modelled by Petri nets are implemented via siphon control. Most of the existing methods design liveness-enforcing supervisors by adding control places for siphons based on their controllability conditions. To compute a liveness-enforcing supervisor with as much as permissive behaviour, it is both theoretically and practically significant to find an exact controllability condition for siphons. However, the existing conditions, max, max , and max -controllability of siphons are all overly restrictive and generally sufficient only. This paper develops a new condition called max * -controllability of the siphons in generalised systems of simple sequential processes with resources (GS 3 PR), which are a net subclass that can model many real-world automated manufacturing systems. We show that a GS 3 PR is live if all its strict minimal siphons (SMS) are max * -controlled. Compared with the existing conditions, i.e., max-, max -, and max -controllability of siphons, max * -controllability of the SMS is not only sufficient but also necessary. An example is used to illustrate the proposed method.

Deciding the liveness for a subclass of weighted Petri nets based on structurally circular wait

“…Resource sharing and allocation is an important topic for lots of real systems such as computer operating system (Moharrami & Karimov, 2012), grid computing (Shojafar, Barzegar, & Meibody, 2010), multi-processor systemon-chip (Xiao & Lee, 2008), and manufacturing system (Ezpeleta, Colom, & Martinez, 1995;Liu & Barkaoui, 2013;Liu, Li, & Zhou, 2013a, 2013bPiroddi, Cordone, & Fumagalli, 2008). One of the main features of these systems is that finite shared resources are used among multiple concurrent processes.…”

Design of Optimal Petri Net Supervisors for Flexible Manufacturing Systems via Weighted Inhibitor Arcs

“…It is robust to the local change of a modeled system. Thus, there are a number of PN‐based approaches to solve deadlock problems . Meanwhile, PNs have been extensively applied to supervisory control problems , scheduling and planning , and performance evaluation .…”

Deadlock analysis and control based on Petri nets: A siphon approach review

“…Let S be a strict siphon in a well-marked GS 3 PR net (N, M 0 ). 241 S is said to be max H -marked (non-max H -marked) at M 2 R(N , M 0 ) if at least one (none) of the following conditions hold:…”

“…Let (N , M 0 ) be a well-marked GS 3 PR net and P 6 ¼ [ be the set of SMS. 241 It is live if 8S 2 P, S is max H -controlled.…”