  We investigate two different types of non-Markovian coarse-grained models
extracted from a linear, non-equilibrium microscopic system, featuring a tagged
particle coupled to underdamped oscillators. The first model is obtained by
analytically ``integrating out'' the oscillators and the second is derived
using projection operator techniques. We observe that these two models behave
very differently when the tagged particle is exposed to external harmonic
potentials or pulling forces. Most importantly, we find that the analytic model
has a well defined friction kernel and can be used to extract work, consistent
with the microscopic system, while the projection model corresponds to an
effective equilibrium model, which cannot be used to extract work. We apply the
analysis to two popular non-equilibrium systems, time-delay feedback control
and the active Ornstein-Uhlenbeck process. Finally, we highlight that our study
could have important consequences for dynamic coarse-graining of
non-equilibrium systems going far beyond the linear systems investigated in
this manuscript.
