Realization of discrete-time nonlinear input-output equations: polynomial approach

The algebraic approach of differential one-forms has been applied to study the realization problem of nonlinear input-output equations in the classical state space form, both in continuous- and discrete-time cases. Slightly different point of view in the studies of nonlinear control systems is provided by the polynomial approach in which the system is described by two polynomials from the non-commutative ring of skew polynomials that act on input and output differentials. Polynomial approach has been used so far to study the problems of reduction, input-output and transfer equivalence. The aim of the present paper is to apply the polynomial approach also to the realization problem. This allows to simplify the step-by-step algorithm given in terms of the sequence of subspaces of differential one-forms to check realizability and calculate the state coordinates in case the system is realizable. A new formula is presented which allows to compute the subspaces of one-forms directly from the polynomial description of the nonlinear system. The above method is noticeable less time-consuming, more direct and therefore better suited for implementation in computer algebra packages like Mathematica or Maple.