  Quantum control techniques applied at macroscopic scales provide us with
opportunities in fundamental physics and practical applications. Among them,
measurement-based feedback allows efficient control of optomechanical systems
and quantum-enhanced sensing. In this paper, we propose a near-resonant
narrow-band force sensor with extremely low optically added noise in an
optomechanical system subject to a feedback-controlled in-loop light. The
membrane's intrinsic motion consisting of zero-point motion and thermal motion
is affected by the added noise of measurement due to the backaction noise and
imprecision noise. We show that, in the optimal low-noise regime, the system is
analogous to an optomechanical system containing a near quantum-limited optical
parametric amplifier coupled to an engineered reservoir interacting with the
cavity. Therefore, the feedback loop enhances the mechanical response of the
system to the input while keeping the optically added noise of measurement
below the standard quantum limit. Moreover, the system based on feedback offers
a much larger amplification bandwidth than the same system with no feedback.
