Controller synthesis with very simplified linear constraints in PN model

A unifying framework for the control of discrete event systems is provided by Ramadge andWonham (1987, 1989), and so far their supervisory control theory is the most general and comprehensive theory presented for logical DES. In Ramadge & Wonham (1987, 1989), it is shown that we can model the systems by using of formal languages and finite automata. Methods exist for designing controllers based on automata system models, however these methods often involve exhaustive searches or simulations of the system behaviour, making them impractical for systems with large numbers of states and transitions causing event (Moody, Antsaklis 2000). For solving these problems, we can model systems with Petri nets (PNs). Petri nets are a very appropriate and useful tool for the study of discrete event systems because of their modelling power and their mathematical properties. In the last decade, the research in the field of controller synthesis of discrete event systems has become one of the most active domains (Achour et al. 2004;Giua A & Xie X., 2005). In Giua et al., (1992) it is shown that it is possible to represent the forbidden states with linear constraints. With the idea presented by Yamalidou et al. (1996) we can design a controller by adding a control place to the PN mode. Each control place corresponds to a constraint. However, when the number of forbidden states is large, then the number of constraints has the same size. In Giua (1992b) it is shown that we can simplify some constraints. But there is no systematic method for simplifying these constraints in general case. This simplification is done by using the PNs structural properties. In Dideban and Alla (2005) a systematic method for reducing the size and the number of constraints was introduced. It was applicable for safe and conservative PNs. In Dideban and Alla (2008), an efficient method for constructing a controller based on safe PN is presented. A set of linear constraints allow forbidding the reachability of specific states. The number of these so-called forbidden states, and consequently the number of constraints are large and lead to a large number of control places. A systematic method for reducing the size and the number of constraints for safe PN was introduced there. With applying the idea in Dideban & Alla (2008), it is possible that sometimes the final number of constraints may remian large. In this paper our objective is to develop the idea in Dideban & Alla (2008) and to present a systematic method for safe PN which allows to do more simplification than in the previous approaches. In this paper, we introduce the concept of Partial Place invariant and then, we present the simplification method. We apply this method to a concrete example to highlights its advantages. We use constraints which are equivalent to forbidden states. These constraints can be calculated in two different ways. They can be given directly as specifications or they can be deduced thanks to the Kumar approach (Kumar & Holloway, 1996). By applying the idea in Dideban & Alla (2008) and the new approach, it is possible to make a powerful simplification. The advantage of the new idea is to reduce more the number of forbidden states. This paper is oraganized as follows: in section 2, some important definitions are introduced. In section 3, the idea of passage from forbidden states to linear constraints will be introduced. The idea in Dideban & Alla (2008) and a practical example are presented in section 4. The basic original idea of simplification and the calculation of control places are presented in section 5. Finally, the conclusion is given in the last section.

In this paper, it is supposed that the reader is familiar with the PNs basis (David & Alla, 2005) and the theory of supervisory control (Ramadge & Wonham, 1987, 1989). We briefly introduce the idea that was presented in (Dideban & Alla, 2008). For more detail, the reader can refer to this reference. . A PN is represented by a quadruplet R={P, T, W, M 0 } where P is the set of places, T is the set of transition, W the incidence matrix and M 0 is the initial marking. Here we show the number of tokens in a place P i with m i . The PN is assumed to be safe; then the marking of each place is Boolean.

M R denotes the set of PN reachable markings. In M R , two subsets can be distinguished; the set of authorized states M A and the set of forbidden states M F . The set of forbidden states correspond to two groups: 1) The set of reachable states (M F ′ ) which either don’t respect the specifications or are deadlock states.

2. The set of states for which the occurrence of uncontrollable events leads to states in M F ′ .

The set of authorized states are the reachable states without the set of forbidden states:

Among the forbidden states, an important subset is constituted by the border forbidden states denoted as M B (Kumar & Holloway, 1996).

Definition 1. let M B be the set of border forbidden states:

Where ∑ c is the set

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