# Theory of Regions for Control Synthesis without Computing Reachability Graph
## 1 Introduction  
Petri nets (PNs) present an effective tool to model and analyze Discrete Event Systems [1]. They have compact structures and can be represented in the form of matrices. Indeed, PNs can be analyzed by linear algebras. They play an important role in addressing the deadlock problems and analyzing the behavior of flexible manufacturing system [2; 3; 4; 5]; a flexible manufacturing system can be defined as a computer controlled production system capable of processing a variety of part types. The flexibility of the system gives manufacturing firms an advantage in a quickly changing manufacturing environment [6]. In the framework of PNs, several approaches, such as the theory of regions, have been conducted for synthesizing PN controllers. The objective is to determine a convenient set of places and arcs connecting them to the transitions that, once added to a given PN plant model, will prevent the whole system from reaching forbidden states This PN controller is a designed subpart of the PN added to the controlled model. The theory of regions (TR) was initially proposed by [PERSON] and [PERSON] [7] to design a bounded PN from a given reachability graph. Indeed, the set of all possible markings reachable from the initial marking with their all possible firing transitions is denoted a reachability graph (RG) [1] A PN is said bounded if it is k-bounded for k \(>\) 0, i.e., the number of tokens in all places of the PN does not exceed k. Nevertheless, TR has been newly interpreted for control synthesis problem by [PERSON] et al. to calculate an optimal PN supervisor. This PN controller is capable ofsatisfying a given control specification [8; 9]. Indeed, this theory can be presented by a linear system composed of three classes of conditions: the reachability conditions, the Marking/transition separation instances (MTSI) conditions and basic cycle equations. The resolution of this linear system leads to the design of a PN controller which is a set of control places. However, the steps of generating and analyzing the reachability graph present difficult tasks that weigh down heavily on the process of calculating PN controllers. Moreover, the combinatorial explosion of states in the RG complicates the calculation of PN controllers. Since computing time costs money, the objective of this work is to apply the TR for supervisory control without computing the RG in order to ease the supervisor calculation and eventually reduce the production cost. In an industrial context, different products can be performed at the same time which can increase the control constraints and incite the monitor to make the necessary adjustments in order to control the system and maximize the production with less time. In fact, many studies have been conducted on reachability graph analysis which represents a significant technique for synthesizing optimal/near-optimal PN controllers [10; 11; 12; 13]. In [13], a RG based method is developed to calculate an optimal PN supervisor with the fewest control places for FMS modeled by PN. The RG is divided into a live zone and deadlock zone. The states in live zone are legal ones for a Flexible manufacturing system (FMS), and those in deadlock zone are deadlock or will lead to deadlocks. In [10], a vector covering method is used to reduce the number of markings to consider in the synthesis of PN controllers. An optimal supervisor with the fewest control places can be designed by solving a Minimal number of control places problem (MCCP). The PN controller obtained is optimal and structurally minimal in terms of the number of control places. However, it has a limitation: the computational burden is extremely hard, mainly for large models. In addition, in [PERSON] et al. [14], a new method for supervisory control is developed using a divide-and-conquer strategy. Compared with most of traditional global-conquer methods of [PERSON] and [PERSON] [15; 16], the computational efficiency of PN supervisors is improved.