Normal forms for x-flat two-input control-affine systems in dimension five

TL;DR

This paper derives normal forms for x-flat two-input control-affine systems in five dimensions, relating flatness to feedback linearization and generalizing Brunovsky forms with new structural insights.

Contribution

It introduces normal forms for five-dimensional x-flat control-affine systems, linking flatness to feedback linearization and extending Brunovsky canonical forms.

Findings

x-flat systems can be linearized with up to three prolongations

Normal forms generalize Brunovsky forms with triangular and nontriangular structures

New phenomena arise with three-fold prolongation normal forms

Abstract

In this paper, we give normal forms for flat two-input control-affine systems in dimension five that admit a flat output depending on the state only (we call systems with that property x-flat systems). We discuss relations of x-flatness in dimension five with static and dynamic feedback linearization and show that if a system is x-flat it becomes linearizable via at most three prolongations of a suitably chosen control. Therefore x-flat systems in dimension five can be, in general, brought into normal forms generalizing the Brunovsky canonical form. If a system becomes linear via at most two-fold prolongation, the normal forms are structurally similar to the Brunovsky form: they have a special triangular structure consisting of a linear chain and a nonlinear one with at most two nonlinearities. If a system becomes linear via a three-fold prolongation, we obtain not only triangular structures but also a nontriangular one, and face new interesting phenomena.