TY - JOUR

T1 - Letters to the Editor

T2 - On Multivariable Pole-Zero Cancellations and the Stability of Feedback Systems

AU - Anderson, Brian D.O.

AU - Gevers, Michel R.

PY - 1981/8

Y1 - 1981/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0019606754&partnerID=8YFLogxK

U2 - 10.1109/TCS.1981.1085040

DO - 10.1109/TCS.1981.1085040

M3 - Letter

AN - SCOPUS:0019606754

SN - 0098-4094

VL - 28

SP - 830

EP - 833

JO - IEEE Transactions on Circuits and Systems

JF - IEEE Transactions on Circuits and Systems

IS - 8

ER -