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

Reference:

Guiver, C. and Opmeer, M. R., 2011. A counterexample to "positive realness preserving model reduction with H-infinity norm error bounds". IEEE Transactions on Circuits and Systems. Part I: Regular Papers, 58 (6), pp. 1410-1411.

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Official URL:

http://dx.doi.org/10.1109/tcsi.2010.2097750

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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