Letters to the Editor: On Multivariable Pole-Zero Cancellations and the Stability of Feedback Systems

Brian D.O. Anderson; Michel R. Gevers

doi:10.1109/TCS.1981.1085040

Letters to the Editor: On Multivariable Pole-Zero Cancellations and the Stability of Feedback Systems

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Abstract

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.

Original language	English
Pages (from-to)	830-833
Number of pages	4
Journal	IEEE Transactions on Circuits and Systems
Volume	28
Issue number	8
DOIs	https://doi.org/10.1109/TCS.1981.1085040
Publication status	Published - Aug 1981
Externally published	Yes

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title = "Letters to the Editor: On Multivariable Pole-Zero Cancellations and the Stability of Feedback Systems",

abstract = "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.",

author = "Anderson, {Brian D.O.} and Gevers, {Michel R.}",

year = "1981",

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volume = "28",

pages = "830--833",

journal = "IEEE Transactions on Circuits and Systems",

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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.

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DO - 10.1109/TCS.1981.1085040

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JO - IEEE Transactions on Circuits and Systems

JF - IEEE Transactions on Circuits and Systems

IS - 8

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Letters to the Editor: On Multivariable Pole-Zero Cancellations and the Stability of Feedback Systems

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