Discrete-Time Phase Compensated Repetitive Control for Piezoactuators in Scanning Probe Microscopes

Abstract

This paper studies repetitive control (RC) with linear phase lead compensation to precisely track periodic trajectories in piezo-based scanning probe microscopes (SPMs). Quite often, the lateral scanning motion in SPMs during imaging or fabrication is periodic in time. Because of hysteresis and dynamic effects in the piezoactuator, the tracking error repeats from one scanning period to the next. Commercial SPMs typically employ PID feedback controllers to minimize the tracking error; however, the error repeats from one operating cycle to the next. Furthermore, the residual error can be excessively large, especially at high scan rates. A discrete-time repetitive controller was designed, analyzed, and implemented on an experimental SPM. The design of the RC incorporates two phase lead compensators to provide stability and to minimize the steady-state tracking error. Associated with the lead compensators are two parameters that can be adjusted to control the performance of the repetitive controller. Experimental tracking results are presented that compare the performance of PID, standard RC, and the modified RC with phase lead compensation. The results show that the modified RC reduces the steady-state tracking error to less than 2% at 25 Hz scan rate, an over 80% improvement compared to PID control.

A novel learning control strategy for hysteresis and vibration of piezo-scanners

“…Consider the dynamics part of the nonlinear system (1) satisfying Assumption 1. The proposed controller (15) and learning law (11) Proof: In order to prove Theorem 1, we define the following nonnegative function…”

“…After substituting (15) into (14), the closed-loop filtered tracking error dynamics can be expressed aṡ…”

“…We define u 1 (t) as the control input of the dynamic system in (15). Then, by integrating the above learning and adaptive mechanisms (24) (28), the actual control input v(t) is constructed as…”

“…Considering its good performance, researchers have recently proposed some learning control schemes for specific piezo-actuators [12]- [15]. In [13], Kam et al designed an inversion-based iterative learning controller to obtain better performance for AFM systems.…”

“…Unfortunately, some strong assumptions on the initial states of the systems are strictly required to guarantee the performance. A discrete-time phase compensated repetitive controller was constructed for piezoactuators in scanning probe microscopes [15] to attack the system dynamics without accounting for the effects of hysteresis.…”