Differentially observable systems and normal form

By normal form of observability (or canonical form), we mean a controlled dynamical system with a certain triangular structure that respects the input-output map of the system. The problem of transforming a single output controlled dynamical system using a local diffeomorphism (local coordinate change) was solved in the 1980s and 1990s. Under the assumption of uniform observability, it was shown that the system can be locally transformed almost everywhere into the so-called normal form. The global aspect consists of finding an injective transformation that sends the initial system into normal form, while preserving the input-output map. To get around the problem of the singularities involved in constructing this transformation, we have proposed a purely analytic condition which, combined with an observability condition (differential observability), solves this problem.