  It has been shown recently that the Jarzynski equality is generalized under
nonequilibrium feedback control [T. Sagawa and M. Ueda, Phys. Rev. Lett. {\bf
104}, 090602 (2010)]. The presence of feedback control in physical systems
should modify both Jarzynski equality and detailed fluctuation theorem [K. H.
Kim and H. Qian, Phys. Rev. E {\bf 75}, 022102 (2007)]. However, the
generalized Jarzynski equality under forward feedback control has been proved
by consider that the physical systems under feedback control should locally
satisfies the detailed fluctuation theorem. We use the same formalism and
derive the generalized detailed fluctuation theorem under nonequilibrium
feedback control. It is well known that the exponential average in one
direction limits the calculation of precise free energy differences. The
knowledge of measurements from both directions usually gives improved results.
In this aspect, the generalized detailed fluctuation theorem can be very useful
in free energy calculations for system driven under nonequilibrium feedback
control.
