  We study linear quadratic Gaussian (LQG) control design for linear
port-Hamiltonian systems. To this end, we exploit the freedom in choosing the
weighting matrices and propose a specific choice which leads to an LQG
controller which is port-Hamiltonian and, thus, in particular stable and
passive. Furthermore, we construct a reduced-order controller via balancing and
subsequent truncation. This approach is closely related to classical LQG
balanced truncation and shares a similar a priori error bound with respect to
the gap metric. By exploiting the non-uniqueness of the Hamiltonian, we are
able to determine an optimal pH representation of the full-order system in the
sense that the error bound is minimized. In addition, we discuss consequences
for pH-preserving balanced truncation model reduction which results in two
different classical H-infinity-error bounds. Finally, we illustrate the
theoretical findings by means of two numerical examples.
