Classification of solutions of matrix equation related to parallel structure of a Petri net

Han

2003

Int. J. Intell. Syst.

This article presents a new way of generating a generalized state equation that is useful for analyzing the token flow of the Petri Net (PN) with priority. The transition values in the firing vector as used in the conventional state equation are replaced with transition variables, which are generated by multiplying a series of firing condition functions taking the weighted inhibitor arc into account. The actual value of a transition variable is determined by taking priority and the present marking into account. The proposed state equation generalizes the conventional one by using the transition variable form and by containing the formulation of priority. Given the initial marking, the subsequent marking evolution can be determined successively from the generalized state equation as the simultaneous firings occur. A PN with deadlock is analyzed as an example to establish the validity of the generalized state equation.

Section: Petri Netmentioning

confidence: 99%

Generalized state equation of Petri Nets with priority

Han

2003

Int. J. Intell. Syst.

“…When the solution of the state equation exists, its solution may not be unique. Miyazawa et al [35] classify the solutions only related to the parallel structure of PNs and point out that the number of these solutions must be finite. For a given NIS, we need to determine whether there is a legal firing sequence (LFS) corresponding to it.…”

Section: Introductionmentioning

confidence: 99%

A State-Equation-Based Method to Non-Reachability Analysis of Ordinary Petri Nets With Token-Free Circuit-Based Subnets

2022

IEEE Access

Petri nets (PNs) can model various event-driven systems well and detect a fault via reachability analysis. Some state-equation-based approaches are designed for only subclasses of PNs with specific structures. A key issue for PNs' reachability analysis approach based on state equation is that a non-negative integer solution (NIS) to a state equation is necessary but not sufficient for a marking's reachability. In this paper, we propose a circuit-based subnet. For ordinary Petri nets with token-free circuit-based subnets, we present an algorithm to determine that the state equation has NISs but the marking is non-reachable. We adopt a divide-and-conquer strategy to consider each circuit-based subnet. The proposed method is tested on a PN-based flexible manufacturing system with unreliable resources. When unreliable resources are lost, the non-reachable markings can be determined effectively. Besides, simulations are done, and the results verify the efficiency of the proposed algorithm compared with the reachability-tree-based method.

Advances in Mechanical Engineering

“…The studies on the state equation of a variety of Petri nets are essential to their applications, leading to a lot of results, [32][33][34][35][36][37][38][39][40][41][42][43][44][45] since they can provide an algebraic approach to the analysis of a Petri net. Burns and Bidanda30 use the concept of transition variables to translate a safe Petri net into sequential Boolean equations but not to formulate the state equation.…”

Section: Introductionmentioning

confidence: 99%

A class of generalized Petri nets and its state equation

Zhu

Zhang

Yang

2017

By analyzing the evolution nature of a Petri net, this article reports a generalized state equation for Petri nets, including Petri nets with inhibitor and enabling arcs. By proposing the generalized state equations, all enabled transitions meeting the firing condition can fire concurrently. Conflicts can be found when any component of a resulting marking vector by firing the enabled transitions at some marking becomes negative. We first formulate a novel state equation for regular Petri nets. Then, it is extended to the nets with inhibitor and enabling arcs. A classical problem with conflicts and concurrency, that is, the dining philosophers problem, is taken as an example to validate the proposed state equations of Petri nets.