Solving siphons with the minimal cardinality in Petri nets and its applications to deadlock control

Abstract

Siphons can be used to characterise deadlock states and solve deadlock problems in Petri nets that model flexible manufacturing systems. By modifying the objective function and adding new constraints to the mixedinteger programming (MIP) method proposed by Park and Reveliotis, this paper presents a revised MIP (RMIP) to directly solve siphons, called smart siphons, with the minimal cardinality as well as the minimal number of resource places. Accordingly, an iterative siphon-based control (ISC) method adds a proper control place (CP) to make each smart siphon max-controlled until the controlled system is live. Since any siphon that has more resource places can be composed of those containing less resource places, a small number of CPs are required during the iterative control process due to the solved smart siphons containing the minimal number of resource places. Compared with the existing methods in the literature, the proposed ISC method using the RMIP avoids computing a maximal deadly marked siphon from which a minimal siphon is then derived, adds a small number of proper CPs, and leads to a liveness-enforcing supervisor with a simple structure. Finally, a case study demonstrates the effectiveness of the proposed ISC method.

Policy to cope with deadlocks and livelocks for flexible manufacturing systems using the max′‐controlled new smart siphons

“…By analysing the relationship between the states of deadlocks or livelocks and the presence of smart siphons, and then adding new constraints to an MIP method in [31], the RMIP method is described below…”

“…That is to say, it is need for p to be sufficiently marked and then ν p = 1 can lead to z t = 1. To distinguish the solution of the MIP method in [31], the solution of the above RMIP method is called NSS.…”

“…By means of the structural analysis of Petri nets and the concept of smart siphons [31], new constrains [(8) and (9) in Section 3] are generated and then added to the MIP method in [31], called revised MIP (RMIP). This RMIP can detect smart siphons that cause deadlocks or livelocks in Petri net models directly.…”

“…Previous research indicates that siphons are closely related to the deadlocks in Petri net models of FMS. Many scholars focus on solving and controlling siphons and have developed a large number of siphon-based controlled (SC) policies [5,8,9,12,13,15,22,[27][28][29][30][31][32][33][34][35][36]. It is well known that the number of strict minimal siphons (SMSs) in a Petri net grows quickly or and may grow exponentially with respect to its size [1] such that it is difficult and timeconsuming to find out the SMSs.…”

Construct the closed-form solution of A-net of Petri nets by case study

“…[18][19][20][21][22][23] The structural analysis based deadlock prevention has been extensively studied by researchers, in which siphon computation and control play an essential role. [24][25][26][27][28][29][30] However, current advanced approaches such as those in Chen et al 31 produce maximally permissive supervisors while not being able to synthesize large controllers since reachability analysis of the PN must be employed;…”