Analysis Of Structural Behavior And Structural Identification Of Petri Nets

The theory and application study of Petri nets have been developed rapidly since Petri nets were introduced in1962. Petri nets have been used to model and analyze various kinds of dynamic event-driven systems. Petri net theory already has been one of the systemic and unaided subjects. Up to the present, the analysis of structural behavior of Petri nets is still an important investigative topic. Based on the analysis of structural behavior of Petri nets, some significant structural properties of Petri nets are discussed in this dissertation. Necessary and/or sufficient conditions for some structural properties of Petri nets are obtained, and then the properties and controllability of the marking trajectories of timed continuous Petri nets are investigated. In succession the structural identification problems of Petri nets are researched, which is an inverse problem about the structural behavior analysis of Petri nets.Structural behavior properties and structural identification of systems are studied by combining linear algebra and Petri net theory. The theoretic and applicable aspects of Petri nets allow precise modeling and analysis of system behavior, while the graphical representation of Petri nets enable visualization of the modeled system state changes. This combination is the main reason for the great success of Petri nets. At present, Petri nets are an optimum tool to study and model discrete event dynamic systems with parallelism, concurrency, asynchronous, nondeterministic and stochastic. At the same time, more and more researchers have given their attention to the issues on the analysis of the structural behavior properties and structural identification of Petri nets. The structural properties of Petri nets are independent of the initial marking and the firing policy, and the behavior properties depend on the initial marking. The identification of net structure consists of designing a Petri net model which can describe discrete event system from the observation of the structural characteristic evolution of the net systems’external behaviors and the unlabelled graph, which is generated by the coverability graph or modified coverability graph.The dissertation focuses on the investigation of static characteristic (structural conservativeness and consistency, etc.) and dynamic characteristic (weak boundedness and weak conservativeness, etc.) for several Petri net systems by making use of linear inequalities and matrix theory. Moreover, a Petri net model is identified from a given unlabelled graph and other knowledge about the system’s partially structural characteristic. The main work of this dissertation is as follows:The definition of the product incidence matrix of a generalized pure Petri net is given. Based on the rank and eigenvalues of the product incidence matrix, a new approach for the study of the important structural properties of net systems is presented. In terms of Cramer Rule, an efficient algorithm is proposed to compute some sub-determinants of the product incidence matrix, and these results are made up of a set of linear combination inequalities. The (partially) boundedness,(partially) conservativeness,(partially) repetitiveness, and (partially) consistency, etc. of a generalized pure Petri net are determined by the signs of the inequalities. Moreover, a linear representation of P-and T-invariant is generated. The relations are further improved among Petri nets, linear inequalities and matrix analysis.The weak boundeness and weak conservativenss of a timed continuous Petri net are redefined. Based on the equivalent transform of the state trajectories of a net, under infinite server semantics a timed continuous Petri net with joins can be transformed to a join-free type with the equivalent dynamics properties. Since the system matrix of timed continuous Petri nets without joins is a Metzler matrix. The weak boundeness and weak conservativenss of a timed continuous Petri net are expressed in terms of the eigenvalues of the system matrix or the correlative linear inequalities. The necessary and/or sufficient conditions for the weak properties of a timed continuous Petri net are obtained.Under the infinite service semantics, if the weak boundedness and/or weak conservation of a timed continuous Petri net were not satisfied, a state feedback matrix will been designed. Based on the state feedback control technique of the positive systems and the equivalent transform of the inequalities, a linear programming problem is constructed. An optimal state feedback matrix is found such that the dynamic trajectories of the timed continuous Petri nets satisfying the desired weak properties.The problems of identifying free-labeled and λ-free labeled unbounded Petri nets are presented. Given an (modified) unlabelled graph that represents the (modified) coverability graph of a net and partially structural characteristic, an integer linear programming problem is obtained based on the partition of the nodes in the unlabelled graph. The unknowns to be determined are the elements of the pre-and post-incidence matrices and the initial marking of the net. The isomorphism is validated between the (modified) coverability graph of the acquired Petri net and the known (modified) unlabelled graph. At the same time, the complexity of the algorithm is discussed. Finally, the concluding remarks are summarized, and the future works which may be further studied are presented. Overall, the dissertation has explored the structural behavior properties and structural identification problem of several kinds of Petri nets. The computing complexity is discussed. The research methods of the system theory and identification of Petri nets are extended and enriched. Some numerical examples illustrate the rationality and validity of the proposed methods and conclusions.