# On Singularities of Flat Affine Systems With $n$ States and $n-1$ Controls

## TL;DR

This paper investigates the intrinsic singularities of flat affine systems with n states and n-1 controls using Lie-Bäcklund atlas, providing computable conditions for flat outputs and analyzing singularity sets.

## Contribution

It introduces two new sufficient conditions for constructing flat outputs in generic and degenerate cases, advancing the understanding of singularities in flat affine systems.

## Key findings

- Intrinsic singularities include points where the strong accessibility rank condition fails.
- Singularities are contained within the set where the associated distribution is singular.
- Examples illustrate apparent singularities in both generic and degenerate cases.

## Abstract

We study the set of intrinsic singularities of flat affine systems with $n-1$ controls and $n$ states using the notion of Lie-B\"acklund atlas, previously introduced by the authors. For this purpose, we prove two easily computable sufficient conditions to construct flat outputs as a set of independent first integrals of distributions of vector fields, the first one in a generic case, namely in a neighborhood of a point where the $n-1$ control vector fields are independent, and the second one at a degenerate point where $p-1$ control vector fields are dependent of the $n-p$ others, with $p>1$. We show that the set of intrinsic singularities includes the set of points where the system does not satisfy the strong accessibility rank condition and is included in the set where the distribution of vector fields, introduced in the generic case, is singular. We conclude this analysis by three examples of apparent singularites of flat systems in generic and non generic degenerate cases.