Observer construction for nonlinear discrete-time systems with inputs

One of the typical approaches to construct an observer for nonlinear systems is to seek a change of coordinates such that the observer design can be implemented in a relatively easier way. Thus for such an approach the main challenge is how to find such a change of coordinates? In this paper we Will provide an explicit expression of a change of coordinates which can transform a nonlinear discrete-time system with inputs into a linear system with an output injection. We also discuss the observability and controllability issues which are related to the proposed change of coordinates.