A comparison of tracking step inputs with a piezo stage using model predictive control and saturated linear quadratic Gaussian control

Abstract

Compressed Sensing for Atomic Force Microscopy is a newer imaging mode that requires the piezo stage be driven rapidly between measurement locations. In contrast to raster scanning applications, this translates to a setpoint tracking problem. This paper considers the setpoint tracking performance of a piezo nano-positioning stage subject to rate-of-change limitations on the control signal, which is derived from the current limit of the power amplifier. To compensate the vibrational dynamics of the stage, a model predictive control scheme (MPC) and a linear quadratic Gaussian (LQG) controller which saturates the control increment are considered. In both cases, hysteresis and drift are compensated via dynamic inversion. To design the weighting matrices required by the MPC and linear feedback designs, an extension to classic reciprocal root locus ideas is proposed. The robustness of both schemes using classical methods like gain margin, phase margin, and gain of the sensitivity function at low frequencies is analyzed. The overall settle times achieved by both controllers (in both simulation and experiment) across a range of control weights where the reference input is a sequence of step inputs of varying amplitudes are compared. The results show that the best simulation settle time is achieved by MPC using the smallest control weight. However under experimental conditions, the best settle time is achieved by a much larger control weight and the performance of MPC becomes comparable with that of saturated linear feedback. This result is explained by showing that robustness increases with larger control weights.