A frequency-independent and parallel algorithm for computing the zeros of strictly proper rational transfer functions

Abstract

We develop an algorithm for computation of the zeros of a strictly proper rational transfer function in partial fraction form, by transforming the problem of finding the roots of the determinant of a frequency-dependent matrix into one of finding the eigenvalues of a companion matrix comprised of the determinants of a binomial-based set of frequency-independent matrices. The proposed algorithm offers a fundamentally new approach that avoids solving severely ill-conditioned system of linear equations, where condition numbers increase rapidly with frequency. The developed algorithm is straightforward, and enables parallel computation of the characteristic polynomial coefficients an′ that comprise the companion matrix to the characteristic polynomial ∑nansn of the frequency-dependent matrix. Additionally, the algorithm allows for relatively inexpensive computation of asymptotically accurate approximants of the transfer function, such that an′ need be computed only for selected powers of s=jω, where the number of required determinant operations are shown to be relatively small. Additionally, limitations of the developed algorithm are highlighted, where the computational cost is shown to be on the order O(2Np) determinant operations on matrices of dimensions (Np+1)×(Np+1), and Np is the number of poles. Illustrative numerical examples are selected and discussed to provide further insight about pros/cons of the proposed method, and to identify potential areas for further research.