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

摘要

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 “2-D 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 1, n2) a sequence of n2 (or n1 ) centre-symmetric 2-D polynomials (the “2-D table”) and requires the testing of only one last one dimensional (1-D) symmetric polynomial of degree 2n1n2 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 (n1, n2) is completely determined by n1n2+1 stability tests (of specific form) of 1-D polynomials of degrees n1 or n2 for the real case (or 2n1n2+1 polynomials in the complex cases)