Stability Testing of 2-D Discrete Linear Systems by Telepolation of an Immittance-Type Tabular Test

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