On Driftless Systems with m controls and 2m or 2m-1 states that are Flat by Pure Prolongation

TL;DR

This paper introduces new, simpler conditions for identifying flatness by pure prolongation in driftless systems with m controls and 2m or 2m-1 states, expanding the class of systems recognized as differentially flat.

Contribution

It provides novel sufficient conditions for flatness by pure prolongation in specific driftless systems, with a simple algorithm for flat output computation.

Findings

Conditions are more restrictive than general flatness but easier to verify.

Algorithm for flat output computation is remarkably simple.

Broader class of systems recognized as differentially flat.

Abstract

It is widely recognized that no tractable necessary and sufficient conditions exist for determining whether a system is, in general, differentially flat. However, specific cases do provide such conditions. For instance, driftless systems with two inputs have known necessary and sufficient conditions. For driftless systems with three or more inputs, the available conditions are only sufficient. This paper presents new findings on determining whether a system with m inputs and $2m$ or $2m-1$ states is flat by pure prolongation, a specific subclass of differential flatness. While this condition is more restrictive than general differential flatness, the algorithm for computing flat outputs remains remarkably simple, and the verification requirements are relatively lenient. Moreover, the conditions proposed in this work broaden the class of systems recognized as differentially flat, as our sufficient condition differs from existing criteria.