State estimation for non-linear discrete-time systems with input signals

One of the typical approaches to construct an observer for non-linear 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 non-linear discrete-time system with inputs into a linear system with output injections. This leads to a linearizable error dynamics and thus the estimated state converges to the actual state exponentially in the new coordinates. We also discuss the observability and controllability issues which are related to the proposed change of coordinates.