Stability testing of two-dimensional discrete linear system polynomials by a two-dimensional tabular form

A new test for determining whether a bivariate polynomial does not vanish in the closed exterior of the unit bicircle (is stable) is developed. A stable bivariate polynomial is the key for stability of two-dimensional (2-D) recursive linear discrete systems. The 2-D stability test stems from a modified stability test for one-dimensional (1-D) systems that has been developed by the author. It consists of a 2-D table, a sequence of centro-symmetric matrices, and a set of accompanying necessary and sufficient conditions for 2-D stability imposed on it. The 2-D table is constructed by a three-term recursion of these matrices or corresponding bivariate polynomials. The minimal set of necessary and sufficient conditions for stability consists of testing two univariate polynomial, one before and one after completing the table, for no zeros outside and no zeros on the unit circle, respectively. A larger set of useful conditions that are necessary for 2D stability, and may indicate earlier instability, is also shown.