Efficiency of two feedback controls for maintenance process using cellular automata approach

The traditional approach of systems theory includes modelling , model analysis and control. These three components have allowed the theoretical study of many concepts ( controllability , observation, stability) of continuous systems such as partial differential equations. The realization of these concepts to practical situations, however, presents difficulties. To this end, discrete systems such as cellular automata (CA) are presented as an alternative \cite { Chopard }. More precisely, a CA is a discrete lattice $\mathcal{L}\subset \mathbb{Z}^n$ which consists of cells. Each cell $c$ has a neighborhood $N(c)$ and a state $s_t(c)\in \mathcal{S}$ which evolves over time depending on its neighborhood state $s_t(N(c))$ according to a transition function $f:\mathcal{S}^m\to \mathcal{S}, s_t(N(c))\mapsto s_{t+1}(c)$ . In this work, we consider the controllability of the CA. It refers to the search for a function $u$ defined on the system state in order to achieve a desired state $s_d$ of the system to a final time starting from its initial state $s_0$ \cite { El_yacoubi }. The feedback control is adopted to reduce the cost or the frequency of the control application seen that the system state does not require action all the time. More practically, we consider an industrial maintenance problem "maintain all equipments in good working order during a time horizon". The problem approach was a feedback control on a CA \cite { Ouardouz }. The formulated control through Voronoi diagram (principle of nearest neighbor) consists of available technicians allocation to failure equipments . We formulate this time the control over the CA through an integer programming. We integrate the solving algorithm in the CA simulator that we developed in Java object-oriented. Through simulations, we compare the effectiveness of the both control formulation with respect to the considered problem.