  Model reduction by projection-based approaches is often associated with
losing some of the important features that contribute towards the dynamics of
the retained scales. As a result, a mismatch occurs between the predicted
trajectories of the original system and the truncated one. We put forth a
framework to apply a continuous time control signal in the latent space of the
reduced order model (ROM) to account for the effect of truncation. We set the
control input using parameterized models by following energy transfer
principles. Our methodology relies on observing the system behavior in the
physical space and using the projection operator to restrict the feedback
signal into the latent space. Then, we leverage the forward sensitivity method
(FSM) to derive relationships between the feedback and the desired
mode-dependent control. We test the performance of the proposed approach using
two test cases, corresponding to viscous Burgers and vortex merger problems at
high Reynolds number. Results show that the ROM trajectory with the applied FSM
control closely matches its target values in both the data-dense and
data-sparse regimes.
