Generalized observer forms for discrete-time nonlinear systems

The paper addresses the problem of transforming single-output discrete-time state equations into the generalized observer form, which comprises a linear observable component and a nonlinear injection term, depending on the inputs, output, and a finite number of their known past values. The intrinsic necessary and sufficient transformability conditions are provided, under two mild assumptions, in terms of a certain vector field, defined by the system output and its past values. The first assumption requires the state transition map to be invertible with respect to the state variable, and the second requires the constructibility rank condition to be satisfied. The algorithm is presented to find the required parametrized state transformation. The generalized observer form can be applied, under mild conditions, to also jointly estimate the states and disturbances. Two examples illustrate the theory.