Extended observer forms for discrete-time nonlinear systems

The paper addresses the problem of transforming observable, time-reversible and multi-input single-output discrete-time state equations into the extended observer form which comprises a linear observable component and a nonlinear injection term depending on the inputs, output, and a finite number of their past values. The intrinsic necessary and sufficient conditions for existence of the extended observer form are provided in terms of a certain vector field, defined by the system output and its past values. The algorithm is presented to find a parametrized state transformation that takes the state equations into the considered extended observer form. Two examples, one of them academic, illustrate the theory.