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

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 mixed-integer 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.