A criterion for the differential flatness of a nonlinear control system

TL;DR

This paper presents a new criterion to determine when a nonlinear control system, described implicitly, is differentially flat by using a specific function and submersion conditions.

Contribution

It introduces a novel criterion based on the existence of a function satisfying submersion and diffeomorphism conditions for differential flatness.

Findings

Provides a necessary condition for differential flatness.

Characterizes flatness via a specific function and submersion property.

Offers a practical test for flatness in implicit systems.

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 $F(x,0) = 0$.