This article aims at giving a new answer for the challenging problem of the parametrisation of multi input multi-output matrix fraction descriptions. In order to reach this goal, new parametrisations of matrix fraction descriptions, called fully-parametrised left matrix fraction descriptions (F-LMFD), are first introduced. Their structural properties as well as their suitability for multi-input multi-output model description are more precisely analysed. As any over-parametrised model description, the F-LMFD cannot describe a transfer function uniquely. The structure of the space of equivalent F-LMFD is then investigated through the determination of its basis. The study carried out in this article is the prelude to a computational improvement of the identification of matrix fraction descriptions with gradient-based optimisation methods. (C) 2016 Elsevier Ltd. All rights reserved.