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TECHNICAL PAPERS

Shaped Time-Optimal Feedback Controllers for Flexible Structures

[+] Author and Article Information
Lucy Y. Pao, Chanat La-orpacharapan

Department of Electrical and Computer Engineering, University of Colorado, Boulder, CO 80309-0425

J. Dyn. Sys., Meas., Control 126(1), 173-186 (Apr 12, 2004) (14 pages) doi:10.1115/1.1637639 History: Received August 29, 2002; Revised July 08, 2003; Online April 12, 2004
Copyright © 2004 by ASME
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References

Barbieri,  E., and Ozguner,  U., 1993, “A New Minimum-Time Control Law for a One-Mode Model of a Flexible Slewing Structure,” IEEE Trans. Autom. Control, 38(1), pp. 142–146.
Book,  W. J., 1993, “Controlled Motion in an Elastic World,” ASME J. Dyn. Syst., Meas., Control, 115(2), pp. 252–261.
Franklin, G. F., Powell, J. D., and Workman, M. L., 1998, Digital Control of Dynamic Systems, Reading, MA: Addison-Wesley, pp. 599–615.
Li,  F., and Bainum,  P. M., 1994, “Analytical Time-Optimal Control Synthesis of Fourth-Order System and Maneuvers of Flexible Structures,” J. Guid. Control Dyn., 17(6), pp. 1171–1178.
Hejmo,  W., 1983, “On the Sensitivity of a Time-Optimal Positional Control,” IEEE Trans. Autom. Control, 28, No. 5, pp. 618–621.
Junkins, J. L., and Kim, Y., 1993, Introduction to Dynamics and Control of Flexible Structures, American Institute of Aeronautics and Astronautics, Washington, D.C.
Newman,  W. S., 1990, “Robust Near Time-Optimal Control,” IEEE Trans. Autom. Control, 35(7), pp. 841–844.
Pao,  L. Y., and Franklin,  G. F., 1993, “Proximate Time-Optimal Control of Third-Order Servomechanisms,” IEEE Trans. Autom. Control, 38(4), pp. 560–580.
Pao,  L. Y., and Franklin,  G. F., 1994, “The Robustness of a Proximate Time-Optimal Controller,” IEEE Trans. Autom. Control, 39, No. 9, pp. 1963–1966.
Ryan,  E. P., 1983, “On the Sensitivity of a Time-Optimal Switching Function,” IEEE Trans. Autom. Control, 25(2), pp. 275–277.
Singer,  N. C., and Seering,  W. P., 1990, “Preshaping Command Inputs to Reduce System Vibration,” ASME J. Dyn. Syst., Meas., Control, 112(1), pp. 76–82.
Workman, M. L., and Franklin, G. F., 1988, “Implementation of Adaptive Proximate Time-Optimal Controllers,” Proc. American Ctrl. Conf., Atlanta, GA, 2 , pp. 1629–1635.
Zinober,  A. S. I., and Fuller,  A. T., 1973, “The Sensitivity of Nominally Time-Optimal Control Systems to Parameter Variation,” Int. J. Control, 17(4), pp. 673–703.

Figures

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Input shaper placed completely outside the closed-loop system. The shaped input is the feedforward input command for the system.
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Closed-loop system with a shaped velocity profile controller.
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Convolution of bang-bang input with impulse sequence yields the shaped bang-bang.
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The (xf,ẋf) phase-plane trajectories resulting from the unshaped bang-bang input (solid) and from the shaped bang-bang input (dashed)
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The solid lines are phase-plane trajectories resulting from the shaped bang-bang when varying the move distance. The dashed lines divide the plane into regions representing different values of control input to be applied.
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In the first through fourth quadrants, the inputs are assigned to be −U,f1(⋅),U, and f2(⋅), respectively, where xe=xr−|L|.
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Typical time response of the closed-loop control law when the acceleration and deceleration rates are different. Here, the deceleration factor α=0.8.  
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Phase-plane trajectory (solid) and switching curves (dashed) for a damped system with different acceleration and deceleration rates. Here, α<a1/a2. If α>a1/a2, then the switching curve r is below q and the rigid body velocity of the state trajectory increases between switching curves g(ẋr) and h(ẋr).  
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Unshaped and shaped input profiles for three move distance ranges.
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Phase-plane trajectory (solid) and switching curves (dashed) for damped systems when the move distance is in Case 2.  
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Phase-plane trajectory (solid) and switching curves (dashed) for a damped system when the move distance is in Case 3.
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Typical time response of the closed-loop control law for Case 1 for a damped system, where the slew rate limit V=60.
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Typical time response of the closed-loop control law for Case 2 for a damped system, where the slew rate limit V=60.
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Typical time response of the closed-loop control law for Case 3 for a damped system, where the slew rate limit V=60.
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Block diagram of a single mode flexible structure driven by a voice coil motor. R,Kt, and J are coil resistance, torque constant, and motor inertia, respectively.
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Typical (xr+xf,ẋr+ẋf) phase-plane trajectories for an undamped system when α=1. No unique control value can be assigned for the region slightly below the switching curve g1(ẋr).
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Current control command (top) and read/write head position time response (bottom, zoomed in on settling phase) due to STOS controller (dashed) and XPTOS controller (solid). The XPTOS controller has a deceleration factor αXPTOS=0.9 (see 312).
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Sensitivity curves relative to normalized rigid body gain b2 for: 1) STOS control law (solid), and 2) shaped feedforward control (dashed). The move distances are 50, 500, and 1000 tracks.
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Sensitivity curves relative to normalized frequency (top) and damping (bottom) for the STOS control law when the move distances |L| are 50 (dashed), 500 (solid), and 1,000 (dashed-dot) tracks.
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Phase-plane trajectories for: normalized frequency, ωactualmodel varying from 0.7 to 2 (top) and normalized damping, ζactualmodel varying from 0 to 10 (bottom). Dashed line is when ωactualmodelactualmodel=1.

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