Due to the proliferation of wind turbines in power networks, participation of doubly-fed induction generator (DFIG)-based wind turbines in the frequency regulation task is attracting more attention during recent decades. It is a challenge to design an effective DFIG's auxiliary frequency controller, since back-to-back converters used in DFIG make the possibility of large deviations in current and speed of rotor during frequency support period. Hence, it is necessary to use exact expression of DFIG's output power in the frequency-related studies. This paper addresses this challenge by developing a nonlinear dynamic model for the DFIG's output power integrated into the dynamic model of power grid. A state feedback controller is proposed by considering whether the DFIGs participate in the frequency regulation task or not. The stability of overall system is studied using a nonlinear control tool, namely, the input-to-state stability (ISS). We provide the sufficient conditions for the controller's parameters to guarantee the power grid with wind speed and load variations to be ISS. The controller is then embedded in the DFIG's detailed model and simulations are performed to evaluate its performance. Negligible recovery period, higher output power, and smoother frequency response are observed by using the proposed controller.