Observer design for non-linear systems that are not uniformly observable

The observer design problem is studied for a class of single-output non-linear systems that are not necessarily uniformly observable. In the first part of this paper, the non-linear observer canonical form is generalized to the partial non-linear observer canonical form which is quite similar to the decomposition of a linear system into its observable and unobservable states. A necessary and sufficient condition is given for a non-linear system to be transformable into this canonical form by the change of coordinates. Based on this canonical form, a sufficient condition is given for the existence of observers. In the second part, a high gain observer for a class of non-linear systems that are not uniformly observable will be developed in order to relax the system structure of the partial non-linear observer canonical form. To this end, a new canonical form is proposed that generalizes the canonical form of uniform observability and a sufficient condition is provided based on the proposed canonical form.