  This paper studies a structure-preserving model reduction problem for
large-scale second-order dynamical systems via the framework of time-domain
moment matching. The moments of a second-order system are interpreted as the
solutions of second-order Sylvester equations, which leads to families of
parameterized second-order reduced models that match the moments of an original
second-order system at selected interpolation points. Based on this, a
two-sided moment matching problem is addressed, providing a unique second-order
reduced system that match two distinct sets interpolation points. Furthermore,
we also construct the reduced second-order systems that matches the moments of
both zero and first order derivative of the original second-order system.
Finally, the Loewner framework is extended to the second-order systems, where
two parameterized families of models are presented that retain the second-order
structure and interpolate sets of tangential data.
