Observer design for a class of non-linear systems with linearisable error dynamics

This study aims to study the existence of a change of coordinates which permits to transform a class of non-linear dynamical systems into the so-called non-linear observer canonical form. Moreover, an algorithm used to construct such a change of coordinates is given. Based on this form, the authors can design an observer with a linear error dynamic. This study is motivated by the results obtained by Krener and Respondek in 1985, where they studied the existence of change of coordinates for a class of multi-outputs non-linear systems. In 1989, Xia and Gao improved their work. The objective of this study is to show that the counterexample given by Xia and Gao for one result obtained by Krener and Respondek is not a real counterexample. For this purpose, they consider the case where the difference between any two observability indices is not larger than 1. Moreover, they allow a diffeomorphism on the outputs in the obtained non-linear observer canonical form.