  The efficiency of a feedback mechanism depends on the precision of the
measurement outcomes obtained from the controlled system. Accordingly,
measurement errors affect the entropy production in the system. We explore this
issue in the context of active feedback cooling by modeling a typical cold
damping setup as a harmonic oscillator in contact with a heat reservoir and
submitted to a velocity-dependent feedback force that reduces the random
motion. We consider two models that distinguish whether the sensor continuously
measures the position of the resonator or directly its velocity (in practice,
an electric current). Adopting the standpoint of the controlled system, we
identify the `entropy pumping' contribution that describes the entropy
reduction due to the feedback control and that modifies the second law of
thermodynamics. We also assign a relaxation dynamics to the feedback mechanism
and compare the apparent entropy production in the system and the heat bath to
the total entropy production in the super-system that includes the controller.
In this context, entropy pumping reflects the existence of hidden degrees of
freedom and the apparent entropy production satisfies fluctuation theorems
associated to an effective Langevin dynamics.
