Differentially flat systems

Differential flatness has become an important concept in the development of steering methods. It was introduced by Fliess, Lévine, Martin, and Rouchon in [344]; see also [726]. Intuitively, a system is said to be differentially flat if a set of variables called flat outputs can be found for which all states and actions can be determined from them without integration. This specifically means that for a system ${\dot x}= f(x,u)$ with $X = {\mathbb{R}}^n$ and $U = {\mathbb{R}}^m$, there exist flat outputs of the form

$$y = h(x,u,{\dot u},\ldots,u^{(k)})$$ (15.163)

such that there exist functions $g$ and $g'$ for which

$$x = g(y,{\dot y},\ldots,y^{(j)})$$ (15.164)

and

$$u = g'(y,{\dot y},\ldots,y^{(j)}) .$$ (15.165)

One example is the simple car pulling trailers, expressed in (13.19); the flat outputs are the position in ${\cal W}= {\mathbb{R}}^2$ of the last trailer. This property was used for motion planning in [578]. Recent works on the steering of differentially flat systems include [578,813,833].