Abstract models for systems identification

In this paper we try to give some contributions, in order to solve the problem of system identification (or minimal realization), for various classes of systems. In order to give some more insight into this problem, we shall define some abstract versions of the minimal realization process. The models we discuss are used for the study of subsets (or subcategories) in which there exists the minimal realization and it is unique. In section 1 we recall the minimal realization problem and some of its properties. Section 2 develops the first model, which is based on equivalence relations. The sets which support minimal realization are in connection with the systems of ropresentants of an equivalence relation. This model coresponds to the external behaviour point of view. In section 3 we give a model based on ordering relations. This versioncorresponds to the qualificative "minimal". 6oth models in sections 2 and 3 include systems identification. This is not however, the only example ; one can find other, as the "integer part" function and the ''congruence modulo n".The most powerful model seems to be the categorical one5 which is introduced in section 4. The admissible subcategories are those one for which the inclusion functor aumits an adjoint. Some conclusions and further developments of the subject are discussed in the last section.