Multivariable Nyquist theory

Abstract

It is shown that a multivariable digital filter is stable if and only if A(z₁, z₂…,z_n)≠0 when | z₁ 1| = | z₂|=… = |z_n| = 1 and the Nyquist plot for the single-variable function A(z, z, … z) does not encircle zero. Here A(z₁, z₂… z_n) is the denominator polynomial of the relatively prime rational transfer function of the digital filter.