Solving of a Minimal Realization Problem in Maple

Abstract

In the computer algebra system Maple, we have created a package MinimalRealization to solve the minimal realization problem for a discrete-time linear time-invariant system. The package enables to construct the minimal realization of a system starting with either a nite sequence of Markov parameters of a system, or a transfer function, or any non-minimal realization. It is designed as a user library and consists of 11 procedures: ApproxEssPoly, ApproxNullSpace, Approxrank, ExactEssPoly, FractionalFactorizationG, FractionalFactorizationMP, MarkovParameters, MinimalityTest, MinimalRealizationG, MinimalRealizationMP, Realization2MinimalRealization. The realization algorithm is based on solving of sequential problems: (1) determination of indices and essential polynimials (procedures ExactEssPoly, ApproxEssPoly), (2) construction of a right fractional factorization of the transfer function (FractionalFactorizationG, FractionalFactorizationMP), (3) construction of the minimal realization by the given fractional factorization (MinimalRealizationG, MinimalRealizationMP, Realization2MinimalRealization). We can solve the problem both in the case of exact calculations (in rational arithmetic) and in the presence of rounding errors, or for input data which are disturbed by noise. In the latter case the problem is ill-posed because it requires nding the rank and the null space of a matrix. We use the singular value decomposition as the most accurate method for calculation of the numerical rank (Approxrank) and the numerical null space (ApproxNullSpace). Numerical experiments with the package MinimalRealization demonstrate good agreement between the exact and approximate solutions of the problem.