  It is well known that any port-Hamiltonian (pH) system is passive, and
conversely, any minimal and stable passive system has a pH representation.
Nevertheless, this equivalence is only concerned with the input-output mapping
but not with the Hamiltonian itself. Thus, we propose to view a pH system
either as an enlarged dynamical system with the Hamiltonian as additional
output or as two dynamical systems with the input-output and the Hamiltonian
dynamic. Our first main result is a structure-preserving Kalman-like
decomposition of the enlarged pH system that separates the controllable and
zero-state observable parts. Moreover, for further approximations in the
context of structure-preserving model-order reduction (MOR), we propose to
search for a Hamiltonian in the reduced pH system that minimizes the
$\mathcal{H}_2$-distance to the full-order Hamiltonian without altering the
input-output dynamic, thus discussing a particular aspect of the corresponding
multi-objective minimization problem corresponding to $\mathcal{H}_2$-optimal
MOR for pH systems. We show that this optimization problem is uniquely
solvable, can be recast as a standard semidefinite program, and present two
numerical approaches for solving it. The results are illustrated with three
academic examples.
