Abstract : This PHD thesis aims to study the differential flatness property on non-linear bond graph models (BGs) and to contribute to the resolution of the two principal problems encountered in practice, mainly the problems of flat outputs identification and differential parameterisation.
In order to reach this objective, new concepts and graphical tools are introduced. Particularly, thanks to the use of the Kähler differentials, the notion of tangent or variationnel BG (VBG) model is defined. A BG method based on the use of the BGV model enables identifying the bases of the differential module associated with the VBG model, which become the flat outputs of the original non-linear system after integration.
Besides, by defining the notion of non-commutative ring BGs, a new gain rule known as Riegle's gain formula is extended to BGs. Then, by considering a VBG model as a particular case of non-commutative ring BGs, the problem of differential parameterisation is then solved using Riegle's gain formula and the concept of bicausality.
Finally, in order to introduce further concepts of differential algebra and modules theory to the BG methodology, the case of non-linear BG models governed by polynomial differential equations is approached. In this context, the BG allows to conduct a direct analysis of the main properties of the system from its associated BG model, such as the choices of inputs, the dynamics corresponding to these choices, the calculation of differential (non-differential) transcendence degrees, etc... It is also shown that Riegle's gain rule can be extended to this class of BGs models.