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

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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.
LanguageEnglish
Pages1410-1411
Number of pages2
JournalIEEE Transactions on Circuits and Systems. Part I: Regular Papers
Volume58
Issue number6
DOIs
StatusPublished - 2011

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Transfer functions
Networks (circuits)

Keywords

  • control theory
  • reduced order systems

Cite this

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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.",
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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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