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MODELING FOR CONTROL

Transfer Matrix Modeling of Systems With Noncollocated Feedback

[+] Author and Article Information
Ryan W. Krauss1

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026rkrauss@siue.edu

Wayne J. Book

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332wayne.book@me.gatech.edu

1

Corresponding author.

J. Dyn. Sys., Meas., Control 132(6), 061301 (Oct 28, 2010) (11 pages) doi:10.1115/1.4002476 History: Received December 28, 2007; Revised December 18, 2009; Published October 28, 2010; Online October 28, 2010

The transfer matrix method (TMM) can be a powerful tool for modeling flexible structures under feedback control. It is particularly well suited to modeling structures composed of serially connected elements. The TMM is capable of modeling continuous elements such as beams or flexible robot links without discretization. The ability to incorporate controller transfer functions into the transfer matrix model of the system makes it a useful approach for control design. A limitation of the traditional formulation of the TMM is that it can only model feedback where the actuators and sensors are strictly collocated. The primary contribution of this paper is an algorithm for modeling noncollocated feedback with the TMM. Two cases of noncollocated sensors are considered (upstream and downstream). The approach is experimentally verified on a flexible robot that has one upstream and one downstream sensor in its feedback loops.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

A two-mass system used to introduce the TMM

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Figure 2

Free body diagram of a mass, showing the sign convention for forces (force in tension is positive)

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Figure 3

Free body diagram of a mass to explain the negative sign in the forcing matrix of Eq. 6

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Figure 4

A two-mass system used to illustrate a downstream noncollocated sensor

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Figure 5

Picture of SAMII showing joint 2 and the two outputs: θ is the angular position of joint 2 and ẅbeam is the acceleration of SAMII’s base (i.e., the end of the cantilever beam)

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Figure 6

Schematic of SAMII showing transfer matrix elements

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Figure 7

Block diagram of the system with position control (θ feedback) and vibration suppression (ẅbeam feedback)

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Figure 8

Open-loop actuator Bode plot θ/v comparing experimental data to the TMM model after system identification

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Figure 9

Open-loop flexible base Bode plot ẅ/v comparing experimental data to the TMM model after system identification

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Figure 10

Block diagram of the open-loop system

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Figure 11

Block diagram of the system with position control (θ feedback)

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Figure 12

Closed-loop actuator Bode plot θ/θd for the system with θ feedback comparing experimental data to the transfer matrix model

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Figure 13

Closed-loop flexible base Bode plot ẅ/θd for the system with θ feedback comparing experimental data to the transfer matrix model

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Figure 14

Closed-loop actuator Bode plot θ/θ̂d for the system with θ and ẅ feedback (vibration suppression) comparing experimental data to the transfer matrix model

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Figure 15

Closed-loop flexible base Bode plot ẅ/θ̂d for the system with θ and ẅ feedback (vibration suppression) comparing experimental data to the transfer matrix model

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