Scanning probe microscopes, used in a diverse range of fields, are commonly actuated by piezoelectric nanopositioners. Because of coupling between hysteresis and fast dynamics, precise control of piezo-actuated systems is a difficult task. To solve this problem, we propose the use of an adaptive servocompensator. First, we present an adaptive law augmented with parameter projection, and establish its robustness to matched disturbances. Next, we show that with well designed hysteresis inversion, there exists an asymptoticly stable periodic solution to the closed-loop system. This facilitates the treatment of the error in hysteresis inversion as an exogenous periodic disturbance. Using adaptation, we can modify the parameters of the servocompensator to track the reference signal and reduce the effect of hysteresis on the tracking error. We confirm our theoretical results through experimentation on a commercial nanopositioner, in which we test tracking and adaptation performance for different input signals. Tracking performance of the proposed adaptive servocompensator is comparable to that of an iterative control algorithm, yet without requiring any knowledge about the spectrum or period of the reference. In addition, we demonstrate robustness of the controller to varying loading conditions.