Supervisory Control and High-level Petri nets

The Supervisory Control Theory (SCT) (Ramadge&Wonham, 1989) was developed to provide a formal methodology for the automatic synthesis of controllers for Discrete Event Systems (DES). In this theory, a system, called a plant, is assumed to have uncontrollable behaviours which may violate some desired specifications. Hence, these behaviours have to be controlled by means of a feedback controller, called a controller (or a supervisor), so that the system fulfils the specifications. Primarily, the SCT was studied in the context of automaton based models. More recently, a special interest is given to the Petri net models for the studying of the control problem (Ghaffari et al., 2003; Giua & DiCesare, 1994; Sreenivas & Sreenivas, 1997), since they represent a good trade-off between modelling power and analysis capabilities. For details about the supervisory control problem methods based on Petri nets, one can refer to (Holloway et al., 1997; Su et al., 2005). In addition, high level nets, especially Coloured Petri nets (CP-nets) (Jensen & Rozenberg, 1991), provide a great improvement over the ordinary Petri nets. Notably, the high expressiveness of CP-nets allows to obtain compact models even for large systems, while keeping the same formal analysis capabilities. However, not many works have addressed the supervisory control problem by considering a CP-net as a plant model. In this context, we can cite the method developed in (Makungu et al., 1999). This method addresses the forbidden state problem for a class of CP-nets where the process to be controlled is separated from the control logic. In this chapter, we review our previous works (Abid & Zouari, 2008; Zouari & Ghedira, 2004; Zouari & Zairi, 2005) for the supervisory control problem of DES modelled by CP-nets. The control specifications herein considered are expressed in terms of forbidden states, i.e. states which have to be avoided by the controlled model. In a first approach, we propose to derive a controller for a plant CP-net model by using the theory of regions. According to the control specifications, the desired behaviours are extracted from the rechability graph associated with the plant model. Then, the theory of regions is used in order to design the controller. Thanks to the expressiveness of CP-nets, as a main advantage, the obtained controller is reduced to one single place. Secondly, we propose to optimise the first approach in order to deal efficiently with symmetric systems. Indeed, the reachability graph of a symmetric system can be represented by an optimised version, called symbolic reachability graph (Chiola et al., 1991; 1997), which is quite smaller. Thereby, the use of symbolic graphs allows to alleviate one important drawback of the latter approach which is the well-known problem of the state space explosion. Moreover and consequently, the use of a smaller graph allows to reduce the complexity of the synthesis process. Finally, we propose an approach which considers as plant model a CP-net that is assumed to be structured on a set of generic processes sharing a set