Optimal Supervisory Control Synthesis

Petri Nets (PN) is an appropriate and useful tool for the study of Discreet-Event Systems (DES) because of their modeling power and their mathematical properties. One of the advantages of PN models in DES control synthesis is the use of a PN marking as a distributed representation of the system state. For control goals represented as state avoidance problems, this often allows a decomposition of the control synthesis problem into a series of smaller problems which are more efficiently solved. In the last decade, the research in the field of controller synthesis of DES became one of the most active domains [1],[2],[3], [4].

In this paper we present an efficient method for synthesizing a controller to solve forbidden state problems for a particular class of DES’s. We consider control specification which can be stated in the PN model. Synchronous composition between model of system and model of specification by existence the uncontrollable transitions, produce the forbidden states.

The forbidden state problem for a DES was introduced in a paper by Ramadge and Wonhom in which they demonstrated several fundamental properties of the solution [?]. The main limitation of such approach is the lack of structure in controlled automata and the large number of states of the related state transition structures.

Li and Wonham [??] have presented an algorithm which calculates the optimal solution for the systems modeled by Petri nets whose uncontrollable sub nets are loop-free. The controller has to solve on-line at each step linear integer programs. The approach will be difficult to apply to the grand systems because of computational complexity. A method for the computation of a maximally permissive controller using theory of region is presented in Ghaffari and Xie []. The main idea is to reduce the reachability graph to the set of admissible markings, i.e., markings that respect the specifications. Control places are added to the original network, so that the closed loop plant respecting the specifications. The theory of region is used on the reachability graph for the computation of the control places. For the computation of control places, a system of linear inequalities must be resolved. The solution of the system must be an integer positive solution. The advantage of this method is that gives a maximally permissive solution of the control problem, but the computation method is difficult and does not guarantee a solution. In this method, for each forbidden state, we must compute a control place. Then there is many control places and it increases the complexity of system. There is many of research for solving the problem of forbidden states [],[],[],[]. Each of them proposed the method for solving this problem. Most relevant to our work are [5], [6]. In [5], author presented a method for construct a optimal controller for the system modeled by Grafcet. The method is based on the equivalence between the set of forbidden states and the set of linear constraints deducted from it. The use of the Yamalidou technique in [5],[6] allows constructing a set of control places which represent the optimal controller. However the number of control places is equal to the number of constraints and can become very important. Intuitively, we realized that the PN solution is not unique from the point of view of the structure, but the reachable space (graph of the markings) is naturally unique. The problem comes from all the linear constraints which can be simplified by holding structural properties of the PN. In this paper is proposed a systematic method of reduction of the number of constraints. The minimal solution is not unique in the general case, but a method to choose the best solution is presented. This approach is based on the hypothesis that the PN models are safe and conservative. This paper is organized as below:

In the second section will be presented the fundamental definitions then how we can construct the linear constraints from forbidden states. Later, in the fourth section, the method to reduce the number and the size of the constraints will be presented. At the end, this method will be explained through an example.

In this paper it is supposed that reader knows the bases of the Petri net paradigm and we present only the notions used in this paper. For more details, see the book in (David and Alla, 2005). A PN is presented by a 4-uplet N = {P, T, Pre, Post} where P is the set of places, T the set of transitions, Pre: P×T →N is the pre-incidence function that defines weighted arcs from places to transitions. Post: T×P →N is the post-incidence function that defines weighted arcs from transitions to places. C= Post -Pre is the incidence matrix.

The reachability graph is an automata = {M, Σ, δ, m 0 } where M is the states set, Σ is the events set, δ : M x Σ → M the evolution function, m 0 is the initial state. This graph corresponds to all the possible evolutions of the PN. The reachability graph consists of nodes whic

…(Full text truncated)…

This content is AI-processed based on ArXiv data.