A counterexample to "positive realness preserving model reduction with ℋ∞ norm error bounds"
Opmeer, Mark R.
IEEE Transactions on Circuits and Systems I: Regular Papers
Institute of Electrical and Electronics Engineers (IEEE)
We provide a counterexample to the H∞ 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∞ 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∞ 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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Vol. 58 (6), pp. 1410 - 1411