  With a specific emphasis on control design objectives, achieving accurate
system modeling with limited complexity is crucial in parametric system
identification. The recently introduced deep structured state-space models
(SSM), which feature linear dynamical blocks as key constituent components,
offer high predictive performance. However, the learned representations often
suffer from excessively large model orders, which render them unsuitable for
control design purposes. The current paper addresses this challenge by means of
system-theoretic model order reduction techniques that target the linear
dynamical blocks of SSMs. We introduce two regularization terms which can be
incorporated into the training loss for improved model order reduction. In
particular, we consider modal $\ell_1$ and Hankel nuclear norm regularization
to promote sparsity, allowing one to retain only the relevant states without
sacrificing accuracy. The presented regularizers lead to advantages in terms of
parsimonious representations and faster inference resulting from the reduced
order models. The effectiveness of the proposed methodology is demonstrated
using real-world ground vibration data from an aircraft.
