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Research Papers

Controller Design Procedure for Two-Mass Systems With Single Flexible Mode

[+] Author and Article Information
Young Joo Shin

FAB Equipment Development Team, Samsung Electronics Co., Ltd., Hwasung-City, Gyeonggi-Do, 445-701 Korea

Peter H. Meckl

School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907-2088

J. Dyn. Sys., Meas., Control 130(3), 031002 (Apr 09, 2008) (13 pages) doi:10.1115/1.2837311 History: Received June 29, 2006; Revised June 06, 2007; Published April 09, 2008

Many manufacturing machines must execute motions as quickly as possible to achieve profitable high-volume production. Most of them exhibit some flexibility, which makes the settling time longer and controller design difficult. This paper develops a control strategy that combines feedforward and feedback control with command shaping for systems with collocated actuator and sensor. First, a feedback controller is designed to increase damping and eliminate steady-state error. Next, an appropriate reference profile is generated using command-shaping techniques to ensure fast point-to-point motions with minimum residual vibration. Finally, a feedforward controller is designed to speed up the transient response. The proposed proportional-integral-derivative (PID) controller design ensures that two important resonant frequencies nearly match, making the design of the input commands much simpler. The resulting control strategy is successfully demonstrated for a generic dimensionless system that incorporates some modeling errors to assess robustness.

Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

A simple model with two linked masses

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Figure 2

Bode plots of rigid-body model and two plant models with different mass ratios Mr

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Figure 3

Block diagram of proposed control system

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Figure 4

Root locus of GL* when Mr=0.25 and qw=10

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Figure 5

Root locus of GL* when Mr=4.0 and qw=10

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Figure 6

Locations of plant zero depending on Mr and controller zero depending on qw

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Figure 7

Error bound of the plant with ±10% modeling errors in natural frequency and gain

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Figure 8

Root loci with ±10% frequency model errors when Mr=4.0 and qw=1000

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Figure 9

Root loci with ±10% gain model errors when Mr=4.0 and qw=1000

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Figure 10

Derivative gain KD of optimal PID controller and minimum derivative gain KDmins depending on Mr and qw for stability of closed-loop system with the full-order plant having ±10% model errors in natural frequency and gain when ζ=0.05 and α*=0.02

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Figure 11

Block diagram of closed-loop system showing two frequencies ω12* and ωr*

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Figure 12

Two regions in which controller zero can exist with respect to plant zero in the upper half plane

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Figure 13

Root locus in which the most lightly damped closed-loop pole approaches plant zero ZP when Mr=0.25

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Figure 14

Root locus in which the most lightly damped closed-loop pole approaches controller zero ZC when Mr=0.25

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Figure 15

Root locus with closed-loop poles satisfying frequency difference and angle conditions

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Figure 16

Root locus having no crossing with the limit boundaries for the frequency difference condition when Mr=10 and qw=10,000

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Figure 17

Root locus having closed-loop poles with relatively high damping ratio at KLminc when Mr=0.25 and qw=100

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Figure 18

Root locus having four closed-loop poles on the crossing depending on KL when Mr=0.1 and qw=1000

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Figure 19

Root locus having two sets of closed-loop poles, depending on KL when Mr=0.1 and qw=1000

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Figure 20

Derivative gain KD of optimal PID controller and minimum derivative gain KDminc depending on Mr and qw for the closeness of ω12* and ωr* when ζ=0.05 and α*=0.02

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Figure 21

Process to generate shaped reference force and position input

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Figure 22

Variation of frequency ω12* as function of Mr

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Figure 23

Frequency difference between ω12* and ωr* as function of Mr and qw

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Figure 24

Force responses of nominal plant and actual plant with −20% model error in gain when a shaped input with ωo*ωnTr∕(2πPFR)=5.656 is used and Mr=0.25

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Figure 25

Force responses of nominal plant and actual plant with −20% model error in gain when a shaped input with ωo*ωnTr∕(2πPFR)=6.559 is used and Mr=0.25

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Figure 26

PFR versus percent model error when Mr=0.25 and 4.0, respectively

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Figure 27

Shaped force input uc* for the closed-loop systems with Mr=0.25 and 4.0

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Figure 28

Frequency spectra corresponding to shaped force inputs uc* for the closed-loop systems with Mr=0.25 and 4.0

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Figure 29

Position response when the plant with Mr=0.25 has +10% model error in natural frequency

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Figure 30

Control force response when the plant with Mr=0.25 has +10% model error in natural frequency

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Figure 31

Position response when the plant with Mr=4.0 has −10% model error in gain

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Figure 32

Control force response when the plant with Mr=4.0 has −10% model error in gain

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