Observer canonical form for a class of multi-outputs nonlinear systems

This paper aims to study the existence of the change of coordinates which permits to transform a class of nonlinear dynamical systems into the so-called nonlinear observer canonical form. Moreover, an algorithm 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 result obtained by Krener & Respondek in 1985, where the authors studied the existence of change of coordinates for a class of multi-outputs nonlinear systems. In 1989, Xia & Gao improved the result of Krener & Respondek by introducing an additional condition for the considered systems, which guarantees that any output and its derivatives do not affect the other outputs in the nonlinear observer canonical form. In particular, they showed that if the observability indices of a considered system are all equal, then the additional condition is not necessary. Here, we extend this result to the case where the difference between any two observability indices is not larger than 1. For this purpose, inspired by Krener & Respondek, instead of using the additional condition given by Xia & Gao, we allow a diffeomorphism of the outputs in the obtained nonlinear observer canonical form.