Letters to the Editor: 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.