On multivariable pole- zero cancellations and the stability of feedback systems

We study conditions for pole-zero cancellation including unstable pole-zero cancellation in the product of two transfer function matrices G and H. Pole-zero cancellation is defined using McMillan degree ideas, and conditions for cancellation are phrased in terms of the coprimeness of matrices associated with matrix fraction descriptions of G and H. Using the condition for unstable pole-zero cancellation, we obtain a new set of conditions for the stability of linear MIMO feedback systems. We show that such a feedback system is stable if and only if there is no unstable pole-zero cancellation in GH and if
(I+GH)^{-1}
is stable. On the other hand, if there is no unstable pole-zero cancellation in GH and any or all of
(I+ HG)^{-1}, G(I+ HG)^{-1}
, and
H(I+ GH)^{-1}
are stable, the closed-loop may be unstable- but only if there is an unstable pole-zero cancellation in HG.