Repetitive Control Design for Piezoelectric Actuators

Piezoactuators exhibit hysteresis and dynamic effects which often cause significant positioning error in a wide variety of motion control applications, especially in applications where the reference trajectory is periodic in time, such as the raster motion in scanning probe microscopy. A feedback-based approach known as repetitive control (RC) is well-suited to track periodic reference trajectories and/or to reject periodic disturbances. However, when an RC is designed with a linear dynamics model and subsequently applied to a system with hysteresis, stability and good tracking performance may not be guaranteed. In this work, the effect of hysteresis on the closed-loop stability of RC is analyzed. In the analysis, the hysteresis effect is represented by the Prandtl-Ishlinskii hysteresis model. Using this model, stability conditions are provided for an RC designed for piezoelectric actuators which commonly exhibit hysteresis. The approach is applied to a custom-designed piezo-driven nanopositioner for tracking periodic trajectories.