Stability testing of two-dimensional discrete-time systems by a scattering-type stability table and its telepolation

Stability testing of two-dimensional (2-D) discrete-time systems requires decision on whether a 2-D (bivariate) polynomial does not vanish in the closed exterior of the unit bi-circle. The paper reformulates a tabular test advanced by Jury to solve this problem. The 2-D tabular test builds for a real 2-D polynomial of degree (n(1), n(2)) a sequence of n(2) matrices or 2-D polynomials (the '2-D table'). It then examines its last polynomial - a 1-D polynomial of degree 2n(1)n(2) - for no zeros on the unit circle. A count of arithmetic operations for the tabular test is performed. It shows that the test has O(n(6)) complexity (assuming n(1) = n(2) = n)- a significant improvement compared to previous tabular tests that used to be of exponential complexity. The analysis also reveals that, even though the testing of the condition on the last polynomial requires O(n(4)) operations, the count of operations required for the table's construction makes the overall complexity O(n(6)). Next it is shown that it is possible to telescope the last polynomial of the table by interpolation and circumvent the construction of the 2-D table. The telepolation of the tabular test replaces the table by n(1)n(2) + 1 stability tests of 1-D polynomials of degree n(1) or n(2) of certain form. The resulting new 2-D stability testing procedure requires a very low O(n(4)) count of operations. The paper also brings extension for the tabular test and its simplification by telepolation to testing 2-D polynomials with complex valued coefficients.