  We study $H_\infty$ control design for linear time-invariant port-Hamiltonian
systems. By a modification of the two central algebraic Riccati equations, we
ensure that the resulting controller will be port-Hamiltonian. Using these
modified equations, we proceed to show that a corresponding balanced truncation
approach preserves port-Hamiltonian structure. We illustrate the theoretical
findings using numerical examples and observe that the chosen representation of
the port-Hamiltonian system can have an influence on the approximation
qualities of the reduced order model.
