USE OF SINGULAR PERTURBATIONS FOR THE REDUCTION OF MANUFACTURING SYSTEM MODELS

Daniel RACOCEANU, Noureddine ZERHOUNI

Laboratoire d’Automatique de Besançon (LAB), UMR CNRS 6596 25, rue Alain Savary, 25000 Besançon (France) email : [email protected]

Abstract: The sense of singular perturbation reduction is close to the meaning of the longand short-run management problem. In this paper, we propose the decomposition of a class of manufacturing systems management using the singular perturbation methods. Controlled Stochastic Petri Nets are used for the modelling. The application of singular perturbation techniques on the Markov Reward associate model, gives a set of marking having different behaviour of its occurrence probability for the original Petri Net. The behaviour of slow part of the management system corresponds to a long-run management policy. Its fast part corresponds to a real time operational control of the manufacturing system. Keywords: Controlled Stochastic Petri Nets, Markov Reward model, Markov Decision Process, Singular Perturbations, Maintenance, Management, Manufacturing System, Safety.

1. THE MULTIPLE SCALE OF THE INDUSTRIAL MANAGEMENT Due to the important number of uncertain parameters, the management of a large class of real manufacturing systems are very difficult to assist using almost existent software. The random parameters of evolution are very hard to analyse and modelling specially when they take into account safety aspects of the manufacturing system. The separation of the short real-time control and the strategic management is a daily challenge of production managers. In the practical industrial management, deciders must use heuristic or empirical experience-based methods for making urgent decisions. The complete model of the system is impracticable for the real-time management. This analysis becomes useful when a long-run decision has to be taken. In this case, the perturbations are not so important, and estimated parameters are well adapted for a detailed long-run analysis of the system behaviour.

Singular perturbation methods of simplification have the same meaning than the long- and short-run management problem. The fast part of the system must be used for the real-time management, and the slow part has to be capable to predict the long run evolution of the system. In this paper, we propose the study of the maintenance management of a safe manufacturing system using singular perturbation decomposition methods developed for the random models (Delebecque, 1978, 1983; Phillips, 1980, 1981a,b; Racoceanu, 1997; Amodeo, 1999). Modelling, analyse and optimisation of real-time control of manufacturing systems with random parameters seem very appropriate to the Controlled Stochastic Petri Nets (COSTPN) models (De Meer, 1997). COSTPNs are mapped on MRMs (Markov Rewards Models) for a numerical analysis. On theses Markov models, we apply the singular perturbations techniques. We obtain models associated to the slow and fast evolution of the Markov process. These entities are characterized by slow and fast evolution of the state probabilities. In the case of the original COSTPN, this corresponds to the detection of a set of marking having a quasi-constant probability of

apparition for the slow part, and fast instable probability values for the fast part. We introduce a correspondence between the communication rate of the controlled stochastic Petri net and the dynamics decoupling of the corresponding Markov reward process. The configurations of the manufacturing system management of the slow part are associated to a long-run management policy and the fast part configurations correspond to a real time operational control.

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