A Counterexample to “ Positive Realness Preserving Model Reduction With ${cal H}_{\infty }$ Norm Error Bounds”

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.