Fast projection methods for minimal design problems in linear system theory

Abstract

The so-called minimal design problem (or MDP) of linear system theory is to find a proper minimal degree rational matrix solution of the equation H( z) D( z)= N( z), where { N( z), D( z)} are given p× r and m× r polynomial matrices with D( z) of full rank r≦ m. We describe some solution algorithms that appear to be more efficient (in terms of number of computations and of potential numerical stability) than those presently known. The algorithms are based on the structure of a polynomial echelon form of the left minimal basis of the so-called generalized Sylvester resultant matrix of { N( z), D( z)}. Orthogonal projection algorithms that exploit the Toeplitz structure of this resultant matrix are used to reduce the number of computations needed for the solution.