Transforming state equations into the generalized observer form: comparison of algebraic and differential geometric approaches

Two different approaches exist that allow transforming nonlinear discrete-time state equations into the generalized observer form. One of them is based on the algebraic approach of vector fields and the other on the standard differential geometric approach. This paper presents a comprehensive comparison of these two solutions, covering assumptions, two sets of solvability conditions, parametrized state transformation algorithms and domains of validity of the results.