Invariance causale contrôlée pour les systèmes max-plus linéaires

In this paper, the concept of causal controlled invariance, in the framework of max-plus linear systems, is explored in depth. This concept is very useful and applicable to various control problems for such systems, and although it has already been presented in the literature, it is currently represented, and defined only through sufficient conditions that allow to verify if a module has this property. In order to design appropriate control laws for these systems, in which specifications and constraints are given in terms of vector space, it is indeed necessary to consider the concepts of controlled invariance and causal controlled invariance. However, for the latter no algorithm has so far been designed to test the sufficient conditions already defined or other equivalent conditions. It is for this reason that, in addition to various considerations on this topic, a preliminary version of an algorithm is presented within this work. In this way, it makes possible to check more easily whether a module is causally controlled invariant or not. We introduce the concept of causal projection relative to a matrix, which plays a central role in the development of the concepts that are presented. The described algorithm has been implemented on ScicosLab software package. It consists in a procedure that is applied recursively to each row of the matrix under consideration, until convergence is achieved, which is ensured within a defined number of steps. Some examples are finally provided, to illustrate how the algorithm permits to confirm whether or not a given module is causally controlled invariant.