2-D stable polynomials with parameter-dependent coefficients: generalizations and new results

Abstract

Stability of multidimensional systems is a field of intensive research. In this context, different classes of Hurwitz polynomials (in the continuous case) and Schur polynomials (in the discrete case) are in the focus of interest. Although there exist various methods for testing whether a given polynomial belongs to a certain class of the afore mentioned. The type of converse problem, namely the design of stable polynomials is much more tedious. In this paper, a parametric model for the characterization of real or complex two-variable scattering Schur polynomials is given. In other words, the coefficients of the two-dimensional (2-D) polynomial model are functions of real parameters. The following features make it best suited for the design of 2-D systems: no dependencies between the real valued parameters, coverage of the whole class of 2-D scattering Schur polynomials, and the coefficients of the polynomial are rational functions of the parameters. The synthesis of 2-D lossless networks and unitary matrices play a key role in our considerations.