A result on the relationship between Petri net and directed graph-real time fault diagnosis based on Petri net model

The Petri net (PN) is a well-known bipartite graph theory used to model and analyze discrete event systems. The properties of PNs can be classified into two types, i.e., behavioral properties and structural properties. Many behavioral properties are investigated in association with the markings of PNs. On the other hand, the structural properties are just considered based on the PN structure without markings. In this meaning, a PN has been classified to normal, cycle and parallel structures according to its homogenous state matrix equation. As a PN is a bipartite graph, its structure can be transformed into a directed graph and Mason's theorem can be applied to obtain the properties of the original net. In this paper, the authors discuss the relationship between PN structure and directed graphs, and describe a result for the cycle structure of PNs. This result is applied to analyze the structure of a sequential function chart (SFC) and to carry out fault diagnosis within real-times. SFC is a kind of representation form defined in the international standard IEC 1131-3 as a common element of languages of programmable controllers (PCs). Because SFC aims at cyclic processing like sequential control, there are many cycle loops in a program of SFC. The cycle structure of an SFC is analyzed offline, the real-time fault diagnosis is carried out online according to the results of analysis.