Necessary and Sufficient Conditions for State Equivalence to a Nonlinear Discrete-Time Observer Canonical Form

In this technical note, we obtain necessary and sufficient conditions for a multi-input, multi-output, discrete-time nonlinear system to be state equivalent to a nonlinear observer form, and for an uncontrolled multi-output system to be state equivalent to a linear observer form. We adopt a geometric approach, and the proofs are constructive with respect to the required coordinate change.