Research On Linear Constraint Transformation With Uncontrolled Transitions

A discrete event system (DES) is a dynamic system that evolves in accordance with the abrupt occurrence of events which are at. unknown irregular intervals. In essence, DES is a man-made system. The typical DES contains the flexible manufacturing system, large scale computer integrated manufacturing system, traffic management system, database management system, communication network system, etc. In discrete event systems, it is vitally important to avoid forbidden states and satisfy the performance requirements by supervising the DES’s behavior. However, it is quite tough to solve such problem.Compared with mathematical modeling, Petri nets have such advantages as a compact and graphical description of the state space, a variety of analytical methods and solid mathematical foundation. Undoubtedly, Petri nets are widely applied in studies of discrete events systems. In this way, we adopt Petri nets as mathematical tool to study the problem of forbidden state supervision in DES. Based on the design of DES supervisor in Petri nets, the controlling problem can be described with the method of liner constraints. Also, the problem of liner constraints transformation is the focus in the research. However, the existence of uncontrolled transitions makes the problem of constraint transformation much more complex and difficult. Knowing from huge number of research, it is also a big challenge to solve forbidden state problems effectively by getting admissible marking sets from liner constraints.This paper concentrates on the problem of constraint transformation in Petri nets which involve uncontrolled transitions. The detailed work and research results are listed as follows:1. We initially introduce the concept of generalized mutual exclusion constraints which is related to liner constraints. Aiming at the problem of liner constraints with the form of logic or in generalized mutual exclusion constraints, we also propose a new definition of the admitted label sets and get the transformation method which can obtain the maximum admissible constraints.2. We also study such kind of Petri nets whose uncontrollable subsets are α-nets and propose the optimal method for liner constraints transformation. This method firstly obtains the uncontrollable subsets with definitions and then gives the algorithm which can calculate all the sets of forbidden places. Finally, the disjunction of admissible linear constraints which is maximally permissive is constructed according to the sets of forbidden places. This algorithm is base on the concept of the constraints transformation. It simplifies the supervisory problem by transforming the Petri nets with uncontrolled transitions to the ones with controllable transitions. In this way, the systematic complexity is reduced and the control of supervisory problem is more smoothly.3. Aiming at the Petri nets with much more complex uncontrollable subsets, we propose the method of liner constraint transformation based on the maximum increment function. This method transforms the liner constraints by analyzing and solving the maximum increment function, and also put forward the algorithm.Finally, conclusions and future research are provided.