Remarks on the realization of time-varying systems

The realization problem of nonlinear time-varying input-output equations is considered. Differentials of the state coordinates, necessary for realization, are determined by the vector space of differential one-forms, spanned over the field of meromorphic functions. Formulas for computing the basis one-forms are given, based on the Euclidean division of non-commutative polynomials. Moreover, it is shown that in the case of a reducible system, the subspace admits a basis with certain structure, explicitly related to reduced input-output equations.