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 $\phi (y_0,y_1,..,y_r)$ satisfying the following conditions: (1) The map $(y_0,..,y_{r+1})\mapsto (\phi (y_0,y_1,..,y_r),\frac{\partial \phi }{\partial y_0}{y}_1+\frac{\partial \phi }{\partial y_1}{y}_2+..+\frac{\partial \phi }{\partial y_r}{y}_{r+1})$ is a submersion on the variety $F(x,p)=0$. (2) The map $y_0\mapsto x_0=\phi (y_0,0,..,0)$ is a diffeomorphism on the equilibrium variety $F(x,0)=0$.