  Balanced truncation and singular perturbation approximation for linear
dynamical systems yield reduced-order models that satisfy a well-known error
bound involving the Hankel singular values. We show that this bound holds with
equality for single-input, single-output systems, if the sign parameters
corresponding to the truncated Hankel singular values are all equal. These
signs are determined by a generalized state-space symmetry property of the
corresponding linear model. For a special class of systems having arrowhead
realizations, the signs can be determined directly from the off-diagonal
entries of the corresponding arrowhead matrix. We describe how such arrowhead
systems arise naturally in certain applications of network modeling, and
illustrate these results with a power system model that motivated this study.
