Introduction

System is differentially flat if the state and the input can be reconstructed from current and previous outputs.

Formal Definition

System is differentially flat if there exists an integer $k\in \mathbb{N}$ , and functions $\alpha$ and $\beta$ , such that the state and the input can be reconstructed from the outputs $y^{(t)}$ as follows:

In such cases, the output $y(t)$ is called a flat output.

s-Sparse Flat Systems

Nonlinear control system ${\mathrm{\Sigma }}_a$ :

is said to be s-sparse flat if for every set $\mathrm{\Gamma }\subseteq \{1,\dots ,p\}$ with $|\mathrm{\Gamma }|=s$ , the system ${\mathrm{\Sigma }}_{\overline{\mathrm{\Gamma }}}$ :