This paper deals with the power acquisition control of variable-speed wind energy conversion systems under inaccurate wind speed measurements. The control goal is to optimize the power capture from wind by tracking the maximum power curve. Firstly, the controller is designed for the case with known aerodynamic torque, which is a common assumption in many literatures. In this controller, the need for the exact knowledge of the system model is waived by using adaptive technologies. The chattering phenomenon in the generator torque, which can result in high mechanical stress, is avoided by adopting a modified robust term. Then, by utilizing an online approximator to learn an auxiliary term induced by the uncertain aerodynamics, the need for the exact knowledge of the aerodynamic torque is waived. Both of the proposed controllers are capable of providing good performance under inaccurate wind speed measurements. The control objective is obtained in the sense that the tracking error is guaranteed to converge to an arbitrarily small set. It is theoretically proved that all the signals in the closed-loop system are bounded via Lyapunov synthesis. Finally, the performance of our proposed controller is shown by simulating on a 1.5 MW three-blade wind turbine using the FAST (Fatigue, Aerodynamics, Structures, and Turbulence) code developed by the National Renewable Energy Laboratory.