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Defense of Mr. Monsieur Guilherme ESPINDOLA WINCK's thesis9:30 am | POLYTECH ANGERS | AMPHI E | 62, avenue Notre-Dame du Lac 49000 ANGERS

Subject : On the stochastic filtering of max-plus linear systems.

Director of thesis : Mr. Laurent HARDOUIN

Abstract

In a number of man-made systems, such as - telecommunication networks; - production systems; - computer systems; in which the evolution is governed by one-off events, typically the arrival of a signal or the completion of a task, are called since the early 1970s "Discrete Event Systems" (DES). Among the DES, a particular class of systems involving synchronisation and delay phenomena can be modelled by linear equations in algebras of type (max,+). This property has motivated the development of what is commonly called the theory of Max-Plus Linear (MPL) systems. This theory has many analogies with the conventional theory of continuous linear systems and allows in particular to address control problems. The control problem studied in this thesis concerns the state estimation of dynamic MPL systems. Based on recent work on stochastic filtering of MPL systems and inspired by Bayesian filtering, we propose a new approach that is less costly, but just as efficient. As for the classical Bayesian filtering, our approach is composed of two steps: a prediction step and a correction step. The prediction stage developed in this thesis is based on Max-Plus polyhedra. This new approach allows, in the prediction stage, to compute the support of the a priori probability density (PDF) with polynomial complexity. We also show that, under certain conditions related to the observability matrix, the correction step, which corresponds to taking into account the information provided by the measurement in order to refine the support of the a priori PDF, is computable with quadratic complexity. We then look for a solution to improve the performance of the filter. The objective is to evaluate the quality of the prediction with respect to the errors of the measurements induced by the noise.

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