Mathematical Problems in Petri Nets Theory and Applications

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Petri net controllers for Generalized Mutual Exclusion Constraints with floor operators

In this paper a special type of nonlinear marking specifications called stair generalized mutual exclusion constraints (stair-GMECs) is defined. A stair-GMEC can be represented by an inequality whose left-hand is a linear combination of floor functions. Stair-GMECs have higher modeling power than classical GMECs and can model legal marking sets that cannot be defined by OR–AND GMECs. We propose two algorithms to enforce a stair-GMEC as a closed-loop net, in which the control structure is composed by a residue counter, remainder counters, and duplicate transitions. We also show that the proposed control structure is maximally permissive since it prevents all and only the illegal trajectories of a plant net. This approach can be applied to both bounded and unbounded nets. Several examples are proposed to illustrate the approach

Deadlock Prevention Policy with Behavioral Optimality or Suboptimality Achieved by the Redundancy Identification of Constraints and the Rearrangement of Monitors

This work develops an iterative deadlock prevention method for a special class of Petri nets that can well model a variety of flexible manufacturing systems. A deadlock detection technique, called mixed integer programming (MIP), is used to find a strict minimal siphon (SMS) in a plant model without a complete enumeration of siphons. The policy consists of two phases. At the first phase, SMSs are obtained by MIP technique iteratively and monitors are added to the complementary sets of the SMSs. For the possible existence of new siphons generated after the first phase, we add monitors with their output arcs first pointed to source transitions at the second phase to avoid new siphons generating and then rearrange the output arcs step by step on condition that liveness is preserved. In addition, an algorithm is proposed to remove the redundant constraints of the MIP problem in this paper. The policy improves the behavioral permissiveness of the resulting net and greatly enhances the structural simplicity of the supervisor. Theoretical analysis and experimental results verify the effectiveness of the proposed method

Basis marking representation of Petri net reachability spaces and its application to the reachability problem

In this paper a compact representation of the reachability graph of a Petri net is proposed. The transition set of a Petri net is partitioned into the subsets of explicit and implicit transitions, in such a way that the subnet induced by implicit transitions does not contain directed cycles. The firing of implicit transitions can be abstracted so that the reachability set of the net can be completely characterized by a subset of reachable markings called basis makings. We show that to determine a max-cardinality-T_I basis partition is an NPhard problem, but a max-set-T_I basis partition can be determined in polynomial time. The generalized version of the marking reachability problem in a Petri net can be solved by a practically efficient algorithm based on the basis reachability graph. Finally this approach is further extended to unbounded nets

Petri Net Controllers for Disjunctive Generalized Mutual Exclusion Constraints

In this paper a type of specifications called OR-GMEC for place/transition nets is defined. Such a specification consists of a set of disjunctive Generalized Mutual Exclusion Constraint, i.e. the requirement is that, at any given time, the controlled system should satisfy at least one of them. We show that a bounded OR-GMEC can be enforced by a special control structure composed by a set of monitor places (one for each constraint) plus a switcher that determines the current active constraint. We also show that such a simple control structure is not maximally permissive, and characterize this problem identifying a special subset of transitions that may be over-restricted. A modified controller that ensures maximal permissiveness is also presented. Finally, we discuss a particular control problem, that consists in preventing the firing of a given set of transitions and show that it can be reduced to an OR-GMEC problem

Confusion Control in Generalized Petri Nets Using Synchronized Events

The loss of conflicting information in a Petri net (PN), usually called confusions, leads to incomplete and faulty system behavior. Confusions, as an unfortunate phenomenon in discrete event systems modeled with Petri nets, are caused by the frequent interlacement of conflicting and concurrent transitions. In this paper, confusions are defined and investigated in bounded generalized PNs. A reasonable control strategy for conflicts and confusions in a PN is formulated by proposing elementary conflict resolution sequences (ECRSs) and a class of local synchronized Petri nets (LSPNs). Two control algorithms are reported to control the appeared confusions by generating a series of external events. Finally, an example of confusion analysis and control in an automated manufacturing system is presented

Design of Optimal Petri Net Controllers for Disjunctive Generalized Mutual Exclusion Constraints

In this paper a type of specifications called OR-AND Generalized Mutual Exclusion Constraints (GMEC) for place/transition nets is defined. Such a specification consists of a disjunction of conjunction of several single GMECs, i.e., the requirement is that, at any given time, the controlled system should satisfy at least one set of conjunctive GMECs. We show that a bounded OR-AND GMEC can be enforced by a special control structure composed by a set of AND-GMEC monitor places plus a switcher that determines the current active ones. We also show that such a simple control structure can be modified to ensure maximal permissiveness. This approach can be used in the framework of supervisory control in Petri netsIn this paper, a type of specifications called ORAND Generalized Mutual Exclusion Constraints (GMEC) for place/transition nets is defined. Such a specification consists of a disjunction of conjunction of several single GMECs, i.e., the requirement is that, at any given time, the controlled system should satisfy at least one set of conjunctive GMECs. We show that a bounded OR-AND GMEC can be enforced by a special control structure composed by a set of AND-GMEC monitor places plus a switcher that determines the current active ones. We also show that such a simple control structure can be modified to ensure maximal permissiveness. This approach can be used in the framework of supervisory control in Petri nets

Design of Optimal Petri Net Controllers for Disjunctive Generalized Mutual Exclusion Constraints

DOI: 10.1109/TAC.2015.2389313International audienceno abstrac