In the light of an increasing share in the electrical grid, wind turbines must be enabled to provide grid stabilizing behavior. This can be achieved by a variation of the turbine's power output depending on the current state of the electrical grid. However, changes of power output excite oscillations in the turbine structure. To reduce the loading caused by the considered frequency droop scheme, in this paper a nonlinear model-based filter design in a Takagi-Sugeno structure is proposed. The design uses Lyapunov function-based linear matrix inequalities for deriving the necessary feedback gains of the filter. The results are obtained for NREL's 5 MW reference turbine. By connecting FAST to an analytic power system model, we study the effects on turbine loading as a result of frequency stabilization in case of a load imbalance. The proposed filter is designed and implemented to reduce the damage equivalent load of the tower fore-aft motion, and its influence on the frequency trajectory is studied.