A criterion for the differential flatness of a nonlinear control system

Abstract

Let's consider a control system described by the implicit equation $F(x,\dot x) = 0$. If this system is differentially flat, then the following criterion is satisfied : For some integer $r$, there exists a function $\varphi(y_0, y_1, ..,y_r)$ satisfying the following conditions: (1) The map $(y_0,..,y_{r+1}) \mapsto ( \varphi( y_0, y_1, ..,y_r), \frac {\partial \varphi}{\partial y_0}y_1 +\frac {\partial \varphi}{\partial y_1}y_2+ ..+ \frac {\partial \varphi}{\partial y_r} y_{r+1})$ is a submersion on the variety $F(x,p) = 0$. (2) The map $y_0 \mapsto x_0 = \varphi(y_0,0,..,0)$ is a diffeomorphism on the equilibrium variety $F(x,0) = 0$.