Vector fitting by pole relocation for the state equation approximation of nonrational transfer matrices

Often the information available for a state equation description in the form\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\dot x = Ax + Bu$$ \end{document},y=Cx+Du is via a transfer function matrixH(s) obtained by measurements or complicated computations for frequenciess=jω. ThusH(s) is nonrational or rational of high order. Its state equation approximation means obtainingA, B, C, D in the rational transfer matrixC(sI-A)−1B+D≈H(s). This approximation problem is difficult because it is nonlinear and often ill conditioned. This paper describes a methodology for fitting the columnsh(s) ofH(s) by two linear procedures. First θ(s)h(s) is fitted with a set of prescribed poles, where θ(s) is an unknown rational function with the same poles as θ(s)h(s). Then the poles forh(s) are calculated as the zeros of θ(s). With the poles known, the unknown residues and constant terms are calculated forh(s). If necessary, the procedure is repeated with the new poles taken as prescribed poles. The procedure is accurate and robust, and uses only standard numerical linear algebra computations.