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DYNAMIC MODELING CONTROL AND MANIPULATION AT THE NANOSCALE

Design and Analysis of Discrete-Time Repetitive Control for Scanning Probe Microscopes

[+] Author and Article Information
Ugur Aridogan, Yingfeng Shan

Department of Mechanical Engineering, University of Nevada-Reno, Reno, NV 89557

Kam K. Leang1

Department of Mechanical Engineering, University of Nevada-Reno, Reno, NV 89557kam@unr.edu

1

Corresponding author.

J. Dyn. Sys., Meas., Control 131(6), 061103 (Oct 30, 2009) (12 pages) doi:10.1115/1.4000068 History: Received May 30, 2008; Revised May 22, 2009; Published October 30, 2009

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 nanofabrication is periodic. Dynamic and hysteresis effects in the piezoactuator cause significant tracking error. To minimize the tracking error, commercial SPMs commonly use proportional-integral-derivative (PID) feedback controllers; however, the residual error of PID control can be excessively large, especially at high scan rates. In addition, the error repeats from one operating cycle to the next. To account for the periodic tracking error, a discrete-time RC is designed, analyzed, and implemented on an atomic force microscope (AFM). The advantages of RC include straightforward digital implementation and it can be plugged into an existing feedback control loop, such as PID, to enhance performance. The proposed RC incorporates two phase lead compensators to ensure robustness and minimize the steady-state tracking error. Simulation and experimental results from an AFM system compare the performance among (1) PID, (2) standard RC, and (3) the modified RC with phase lead compensation. The results show that the latter reduces the steady-state tracking error to less than 2% at 25 Hz scan rate, an over 80% improvement compared with PID control.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 3

Magnitude and phase versus frequency for signal generator z−N/(1−z−N), where z=ejωTs

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Figure 4

Techniques to account for hysteresis in RC design: (a) Feedback-linearization approach and (b) feedforward hysteresis compensation (38)

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Figure 5

A block diagram of the experimental AFM system. An external computer running custom C code was used to implement the control algorithm.

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Figure 6

The frequency response of piezoactuator along the x-axis. The solid line is the measured response, the dash-dot line represents the linear continuous-time model G(s), and the dash line is the linear discrete-time model Gp(z) using MATLAB function c2d with zero-order hold and sampling frequency of 10 kHz.

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Figure 7

The measured responses of the PID controller to (a) a step reference and (b) triangle references at 1 Hz, 5 Hz, and 25 Hz. (c) The tracking error for the triangle reference signals associated with plot (b).

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Figure 8

The phase response of the closed-loop feedback system without RC and added phase lead θ2(ω), stability condition Eq. 10. The inset plot shows the cutoff frequency versus the phase lead parameter m2. As m2 increases, the frequency range for stability increases. A maximum is reached when m2=9.

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Figure 9

Simulation results showing the tracking performance and error for scanning at 25 Hz, where (a1) and (b1) belong to RC with krc=0.40 and no phase lead; (a2) and (b2) belong to RC with phase lead m2=7 and krc=1.1; (a3) and (b3) belong to RC with phase leads m1=6, m2=7, and krc=1.1.

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Figure 10

Maximum error versus phase lead parameter m1. For the experiments, m1=6 gave smallest error.

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Figure 11

Digital implementation of repetitive control: (a) Equivalent discrete-time block diagram of the RC loop, (b) linear data vector for implementing the one-period delay and the phase lead compensators, and (c) the flow diagram for implementing the RC loop

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Figure 12

Experimental tracking response and error for PID (dash-dot), RC (dashed line), and RC with phase lead compensation (m1=6 and m2=0) (solid line) for 5 Hz (a1 and b1), 10 Hz (a2 and b2), and 25 Hz (a3 and b3) scanning

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Figure 13

Tracking results for offset triangle scan at 25 Hz

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Figure 14

Atomic force microscope images using measured tracking response along the x-axis at 25 Hz and ±25 μm range. Steady-state tracking error shown below each image. PID control (a1) first pass and (b1) second pass; standard RC (a2) first pass and (b2) second pass; and RC with phase lead compensators (m1=6 and m2=0) (a3) first pass and (b3) second pass. The x-axis is the fast-scanning motion and tip starts at the top and slowly scans down along the y-axis.

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Figure 2

The repetitive control (RC) feedback system: (a) The block diagram of the proposed RC system, (b) positive feedback system for stability analysis, and (c) positive feedback system representing the block diagram in part (a) for stability analysis

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Figure 1

The atomic force microscope (AFM): (a) A schematic of the main components and (b) typical scan paths in the lateral directions during AFM imaging

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