The balanced truncation bound is tight for SISO systems when the truncated system is state-space symmetric

Balanced truncation model reduction for linear dynamical systems yields a reduced-order model that satisfies a well-known error bound involving the system's Hankel singular values. We show that this bound holds with equality for a certain class of single-input, single-output systems exhibiting a particular state-space symmetry. This result extends to singular perturbation balancing. We illustrate the result with an example from power-system modeling.