Algebraic conditions for state equivalence to a discrete-time nonlinear observer canonical form

Algebraic necessary and sufficient conditions for a single output discrete-time system to be state equivalent to a (generalized) nonlinear observer canonical form with (or without) output transformation are derived. Our algebraic conditions are easier to verify for researchers unaccustomed to differential geometry. In addition, our proofs are constructive and contain the state and output transformations for (generalized) nonlinear observer canonical forms.