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

摘要

We provide a counterexample to the ${cal H}_{infty}$ 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 ${cal H}_{infty}$ 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 ${cal H}^{infty}$ 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.