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 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.