Flatness of nonlinear control systems : a Cartan-Kähler Approach

In this work we consider the concept of differential flatness defined by Fliess, Levine, Martin and Rouchon. The structural property of k-flatness (the case in which flat outputs depends up to the kth derivative of the inputs) is studied based on the properties of the dynamic extension algorithm and the Cartan-Kähler theorems on the existence of integral manifolds of exterior differential systems. In this context, it is shown that there may exist two kind of flat systems — the nonsingular and the singular cases. Since the nonsingular case contains properly the class of state-feedback linearizable systems, this induces a classification of nonlinear flat systems as state-feedback linearizable, nonsingular and singular systems, in an ascending order of complexity. Necessary and sufficient conditions of k-flatness for the nonsingular case are developed. Necessary and sufficient conditions for the singular case are also presented, although they are much more involved and are not in closed form. Examples are presented to illustrate the results.