Technical Briefs

Optimizing Learning Convergence Speed and Converged Error for Precision Motion Control

[+] Author and Article Information
Douglas A. Bristow

Department of Mechanical and Aerospace Engineering, Missouri University of Science and Technology, Rolla, MO 65409dbristow@mst.edu

Andrew G. Alleyne1

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801alleyne@uiuc.edu

Marina Tharayil

 Xerox Corporation, Webster, NY 14580marina.tharayil@xeroxlabs.com


Corresponding author.

J. Dyn. Sys., Meas., Control 130(5), 054501 (Jul 29, 2008) (8 pages) doi:10.1115/1.2936844 History: Received October 30, 2006; Revised February 13, 2008; Published July 29, 2008

This brief paper considers iterative learning control (ILC) for precision motion control (PMC) applications. This work develops a methodology to design a low pass filter, called the Q-filter, that is used to limit the bandwidth of the ILC to prevent the propagation of high frequencies in the learning. A time-varying bandwidth Q-filter is considered because PMC reference trajectories can exhibit rapid changes in acceleration that may require high bandwidth for short periods of time. Time-frequency analysis of the initial error signal is used to generate a shape function for the bandwidth profile. Key parameters of the bandwidth profile are numerically optimized to obtain the best tradeoff in converged error and convergence speed. Simulation and experimental results for a permanent-magnet linear motor are included. Results show that the optimal time-varying Q-filter bandwidth provides faster convergence to lower error than the optimal time-invariant bandwidth.

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

μ-RD manufacturing machine (1)

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Figure 2

Reference trajectory

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Figure 3

Learning algorithm design procedure

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Figure 4

PMC architecture

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Figure 5

Q-filter design limitations for a typical L design (see Refs. 12-13,22) showing (a) bandwidth limitations and (b) corresponding convergence speed and precision limitations

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Figure 6

Initial tracking error without the use of ILC

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Figure 7

Time-frequency decomposition of e0(k) and bounding function

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Figure 8

Parametrized Ω profile

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Figure 9

Gaussian filter with S=1000Hz shown in (a) time domain and (b) frequency domain

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Figure 10

LTI Q-filters with (a) bandwidths in the very high frequency region result in (b) large learning transients, but better asymptotic tracking (see Refs. 12-13,22)

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Figure 11

X axis frequency response and model

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Figure 12

Magnitude plots using a LTI Q(z) with a 60Hz bandwidth. ∣Q(eiω)(1−L(eiω)P(eiω))∣ is close to 1, demonstrating that ωH=60Hz.

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Figure 13

Numerical optimization results showing the tradeoff* between converged error and learning time. A log-log plot is used to show results over a large range of asymptotic error and learning duration.

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Figure 14

Numerical optimization results for η1 and η2. rms convergence profiles are shown in plots (a) and (b), bandwidth profiles are shown in (c) and (d), and asymptotic errors are shown in (e) and (f). Note that horizontal scales are not identical in (a) and (b).

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Figure 15

Experimental results and comparison of the LTV and LTI Q-filters




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