Stability tests of N-dimensional discrete time systems using polynomial arrays

Tabular method has been applied to the difficult problem of stability test for multidimensional (N-D) systems. This paper further investigates a general test procedure using an (N-1)-level polynomial array structure. Each entry of the structure at the same level (say level p) is an p-D polynomial, or a set of numbers in an p-D space. A systematic procedure is proposed to compute those polynomial entries which involves only calculations of a set of second order determinants with numerical numbers. A simple formula is also given to form the auxiliary polynomial required in the test procedure. Another key issue is the countability of the zeros of an N-D polynomial. It is clarified that the properly defined zero clusters of an N-D polynomial with respect to some specified regions are countable. To illustrate the proposed method a step-by-step algorithm of the stability test for 3-D systems is described