Extended Elementary Siphons and Their Application to Liveness‐Enforcement of Generalized Petri Nets

Abstract

As a significant structural object, siphons are extensively employed to implement a large number of deadlock prevention and liveness-enforcing methods for flexible manufacturing systems modeled by Petri nets. By linear combinations, a set of elementary siphons is chosen from all strict minimal ones to be controlled and thus the structural complexity of a supervisor is greatly reduced. The concept of elementary siphons is originally proposed for ordinary Petri nets. When applied to generalized Petri nets, their selection and controllability require an additional study. In this work, the concept of augmented siphons is proposed to extend the application of the elementary ones to a class of generalized Petri nets, GLS 3 PR. Based on graph theory, a siphon extraction algorithm is developed to obtain all strict minimal siphons, from which augmented elementary ones are computed. In addition, the controllability conditions of dependent siphons are developed. Through fully investigating the net structure, especially weight information, the set of augmented elementary siphons is more compact and well suits for generalized Petri net models under consideration. Some examples are used to illustrate the proposed method.

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

“…The introduction ofS and l ½S ensures that lS + l ½S = I S always holds in a WS Theorem 12. Let S be a dependent siphon in N. 223 Augmented elementary siphons. By fully investigating the net structure, especially weights information, the set of elementary siphons obtained by the improved method is more compact and well suits for GLS 3 PR.…”

“…Later, the work by Hou and Zhao 225 further updates the method proposed by Hou et al 224 in a more general class of Petri nets, GLS 3 PR. Motivated by finding a set of more proper and compact elementary siphons, the concept of augmented siphons is further proposed by Hou et al 223 within a class of generalized Petri nets, called GLS 3 PR nets. The key contributions by Hou et al 223 are twofold.…”

“…Motivated by finding a set of more proper and compact elementary siphons, the concept of augmented siphons is further proposed by Hou et al 223 within a class of generalized Petri nets, called GLS 3 PR nets. The key contributions by Hou et al 223 are twofold. First, the supremum of the number of augmented elementary siphons is found according to the relationship between siphons and their restrictive induced subgraphs, which is not greater than that of strongly connected restrictive induced digraphs that cannot be composed.…”

“…A dependent siphon can be implicitly controlled by properly arranging the number of tokens that can stay at its elementary siphons. The results mentioned above have widely applied to a variety of elementary siphon-related deadlock control approaches 139,[217][218][219][220][221][222][223][224][225][226][227][228][229][230][231][232][233] in the literature. This article mainly aims to reveal, first, how the elementary siphon theory of Petri nets can be used to troubleshoot deadlock problems, and second, how the refined concept of elementary siphons in a Petri net improves the existing deadlock control policies.…”

“…Based on graph theory, Wang et al 233 propose a polynomial complexity algorithm to find a set of elementary siphons for a linear system of simple sequential processes with resources (LS The related works can be found in the studies by Hou et al 223,224 Li et al 270 propose an iterative algorithm to extract a set of elementary siphons in S 3 PR. By an MIP method, the algorithm finds a maximal unmarked siphon at each iteration, classifies the places in it, extracts an elementary siphon from the classified places, and adds a new constraint in order to extract the next elementary siphon.…”