  In this paper we propose a solution to the problem of moment matching with
preservation of the port Hamiltonian structure, in the framework of time-domain
moment matching. We characterize several families of parameterized port
Hamiltonian models that match the moments of a given port Hamiltonian system,
at a set of finite interpolation points. We also discuss the problem of Markov
parameters matching for linear systems as a moment matching problem for
descriptor representations associated to the given system, at zero
interpolation points. Solving this problem yields families of parameterized
reduced order models that achieve Markov parameter matching. Finally, we apply
these results to the port Hamiltonian case, resulting in families of
parameterized reduced order port Hamiltonian approximations.
