Observer design for a class of nonlinear systems with linearizable error dynamics

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