Stability testing of 2-D discrete linear systems by telepolation of an immittance-type tabular test

Abstract

A new procedure for deciding whether a bivariate (two-dimensional, 2-D) polynomial with real or complex coefficients does not vanish in the closed exterior of the unit bi-circle (is stable) is presented. It simplifies a recent immittance-type tabular stability test for 2-D discrete-time systems that creates for a polynomial of degree (n/sub 1/, n/sub 2/) a sequence of n/sub 2/ (or n/sub 1/) centre-symmetric 2-D polynomials (the table) and requires the testing of only one last one dimensional (1-D) symmetric polynomial of degree 2n/sub 1/n/sub 2/ for no zeros on the unit circle. It is shown that it is possible to bring forth (to telescope) the last polynomials by interpolation without the construction of the 2-D table. The new 2-D stability test requires an apparently unprecedentedly low count of arithmetic operations. It also shows that stability of a 2-D polynomial of degree (n/sub 1/, n/sub 2/) is completely determined by n/sub 1/n/sub 2/+1 stability tests (of specific form) of 1-D polynomials of degrees n/sub 1/ or n/sub 2/ for the real case (or 2n/sub 1/n/sub 2/+1 polynomials in the complex cases).