Direct equivalence between geometric conditions and algebraic conditions for discrete-time nonlinear observer

Unlike the continuous-time case, algebraic necessary and sufficient conditions for a single output discrete-time system to be state equivalent to a nonlinear observer canonical form have been found and are easier to verify for those who are not accustomed to differential geometry. The geometric conditions look very different from the algebraic conditions. In this paper, we show direct equivalence of the geometric conditions and the algebraic conditions in order to enhance the understanding of the geometric conditions.