A counterexample to "positive realness preserving model reduction with H-infinity norm error bounds"

Christopher Guiver; Mark R Opmeer

doi:10.1109/tcsi.2010.2097750

A counterexample to "positive realness preserving model reduction with H-infinity norm error bounds"

Christopher Guiver, Mark R Opmeer

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Abstract

We provide a counterexample to the H-infinity error bound for the difference of a positive real transfer function and its positive real balanced truncation stated in "Positive realness preserving model reduction with H-infinity norm error bounds," IEEE Trans. Circuits Syst, I, Fundam. Theory Appl., vol. 42, no. 1, pp. 23-29 (1995). The proof of the error bound is based on a lemma from an earlier paper, " A tighter relative-error bound for balanced stochastic truncation," Syst. Control Lett., vol. 14, no. 4, 307-317 (1990), which we also demonstrate is false by our counterexample. The main result of this paper was already known in the literature to be false. We state a correct H-infinity error bound for the difference of a proper positive real transfer function and its positive real balanced truncation and also an error bound in the gap metric.

Original language	English
Pages (from-to)	1410-1411
Number of pages	2
Journal	IEEE Transactions on Circuits and Systems. Part I: Regular Papers
Volume	58
Issue number	6
DOIs	https://doi.org/10.1109/tcsi.2010.2097750
Publication status	Published - 2011

Keywords

control theory
reduced order systems

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Cite this

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title = "A counterexample to {"}positive realness preserving model reduction with H-infinity norm error bounds{"}",

abstract = "We provide a counterexample to the H-infinity error bound for the difference of a positive real transfer function and its positive real balanced truncation stated in {"}Positive realness preserving model reduction with H-infinity norm error bounds,{"} IEEE Trans. Circuits Syst, I, Fundam. Theory Appl., vol. 42, no. 1, pp. 23-29 (1995). The proof of the error bound is based on a lemma from an earlier paper, {"} A tighter relative-error bound for balanced stochastic truncation,{"} Syst. Control Lett., vol. 14, no. 4, 307-317 (1990), which we also demonstrate is false by our counterexample. The main result of this paper was already known in the literature to be false. We state a correct H-infinity error bound for the difference of a proper positive real transfer function and its positive real balanced truncation and also an error bound in the gap metric.",

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doi = "10.1109/tcsi.2010.2097750",

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AU - Guiver, Christopher

AU - Opmeer, Mark R

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AB - We provide a counterexample to the H-infinity error bound for the difference of a positive real transfer function and its positive real balanced truncation stated in "Positive realness preserving model reduction with H-infinity norm error bounds," IEEE Trans. Circuits Syst, I, Fundam. Theory Appl., vol. 42, no. 1, pp. 23-29 (1995). The proof of the error bound is based on a lemma from an earlier paper, " A tighter relative-error bound for balanced stochastic truncation," Syst. Control Lett., vol. 14, no. 4, 307-317 (1990), which we also demonstrate is false by our counterexample. The main result of this paper was already known in the literature to be false. We state a correct H-infinity error bound for the difference of a proper positive real transfer function and its positive real balanced truncation and also an error bound in the gap metric.

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KW - reduced order systems

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JO - IEEE Transactions on Circuits and Systems. Part I: Regular Papers

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IS - 6

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A counterexample to "positive realness preserving model reduction with H-infinity norm error bounds"

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