Fuzzy output system on the basis of the modified fuzzy Petri nets

The development of information technologies requires improvement of simulation methods and mathematical apparatus. The mathematical apparatus of Petri nets is used for simulation of parallel asynchronous systems and has a broad scope. The modified fuzzy Petri nets expand the modeling capabilities of Petri nets by combining the properties of different extensions. Modified extension can serve for construction of models with a complex structure and logic of operation with the use of fuzzy logic apparatus for control on the basis of the system of production rules.


Introduction
Modeling of information processes and systems becomes an urgent task for business and industry. Information support of production processes is a guarantee of effective performance and determines the success of the enterprise to a large extent. Stable operation is provided by the application of simulation on every stage of the life cycle of information and technical systems. The urgency of developing new approaches to modeling and development of the existing mathematical apparatus increases, considering the complexity of modern software products and technical means.
The article introduces the mathematical apparatus of the modified fuzzy Petri nets (MFPN). A description of the structure of the MFPN apparatus is provided. The possibility of constructing a system of fuzzy output on the basis of MFPN, which can enable fuzzy control implementation in Petri net models, is described.

Mathematical apparatus
The mathematical apparatus of the modified fuzzy Petri nets (PN) is developed on the basis of various extensions of Petri nets. The basis of the MFPN are coloured [1,2] and fuzzy PN. The modeling process includes the properties of priority, hierarchical and timed PN in order to simplify the process and to increase visibility and modeling capacity of the modified extension (Fig. 1). MFPN is the modified PN extension, CPN is the coloured PN, FPN is the fuzzy PN, PPN is the priority PN, TPN is the timed PN.

4.
A is the finite set of arcs; 5.
N is the incidence function C is the function of position type definition  → c P C : G is the function of transitions triggering conditions such as ] ))) ( p is position incidental to the given arc; 9.
The inclusion of new types of positions and transitions does not allow to use the existing rules of CPN or FPN functioning. The MFPN rules take into account the presence of coloured and fuzzy positions and transitions in one model.

MFPN application
The possibility of using the marks of a complex format is achieved by combining the properties of various extensions in MFPN in order to enable modeling of information exchange. Setting the time stamps to transitions and labels can help to adequately simulate objects functioning in time. The application of complex triggering conditions and the setting of expressions on arcs allow simulation of the complex logic of the simulated objects. The use of fuzzy PN as a basic extension allows simulation of fuzzy production rules and control on the basis of fuzzy output.
Coloured Petri nets are used for modeling of telecommunication equipment, data transmission protocols. Based on this we can define the area of MFPN application.
Models based on apparatus can be built using CPN Tools, or any other tool that has the possibility of constructing models on the basis of Petri nets with assignment of different types of marks, expressions on the arcs, the trigger conditions of transitions and time transitions stamps. Integrated CPN Tools programming language CPN-ML allows implementation of fuzzy transitions triggering rules.

MFPN-based representation of fuzzy output system
The work [3] presents the models based on fuzzy Petri nets for modeling stages of aggregation of sub-condition and activation of sub-conclusions. The MFPN allows simulating of all stages of fuzzy output by extending the capabilities of fuzzy PN.

Conclusions
The PN apparatus is widely used for solving problems connected with the simulation of information, production and business processes. Such popularity is explained by the presence of a large number of extensions, designed to address specific applied tasks, by the developed analytical apparatus and the availability of software modeling.
The MFPN continues the trend for the extension of the modeling capabilities of PN. By combining the properties of basic expansions, it allows to simulate various technical and informational processes with the use of the apparatus of fuzzy logic for control.