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 (x) over dot = Ax + Bu, y = Cx + Du is via a transfer function matrix H(s) obtained by measurements or complicated computations for frequencies s = j omega. Thus H(s) is nonrational or rational of high order. Its state equation approximation means obtaining A, B, C, D in the rational transfer matrix C(sI - A)(-1) B + D approximate to H(s). This approximation problem is difficult because it is nonlinear and often ill conditioned. This paper describes a methodology for fitting the columns h(sf of H(s) by two linear procedures. First theta (s)h(s) is fitted with a set of prescribed poles, where B(s) is an unknown rational function with the same poles as theta (s)h(s). Then the poles for h(s) are calculated as the zeros of theta (s). With the poles known, the unknown residues and constant terms are calculated for h(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. Illustrative examples for the application of vector fitting are given for a power transformer, a transmission line, and a network of transmission lines.