Multidimensional Schur Coefficients and BIBO Stability

Abstract

In the framework of BIBO stability tests for one-dimensional (1-D) linear systems, the Schur-Cohn stability test has the appealing property of being a recursive algorithm. This is a consequence of the simultaneously algebric and analytic aspect of the Schur coefficients, which can be also regarded as \textit{reflection coefficients. } In the multidimensional setting, this dual aspect gives rise to two extension of the Schur coefficients that are no longer equivalent. This paper presents the two extensions of the Schur-Cohn stability test that derive from these extended Schur coefficients. The reflection-coefficient approach was recently proposed in the 2-D case as a necessary but non sufficient condition of stability. The Schur-type multidimensional approach provides a stronger condition of stability, which is necessary and sufficient condition of stability for multidimensional linear system. This extension is based on so-called slice function associated to $n$-variable analytic functions. Several examples are given to illustrate this approach.