Input shaping techniques reduce the residual vibration in flexible structures by convolving the command input with a sequence of impulses. The exact cancellation of the residual structural vibration via input shaping is dependent on the amplitudes and instances of impulse application. A majority of the current input shaping schemes are inherently open-loop where impulse application at inaccurate instances can lead to system performance degradation. In this paper, we develop a closed-loop control design framework for input shaped systems. This framework is based on the realization that the dynamics of input shaped systems give rise to time delays in the input. Thus, we exploit the feedback control theory of time delay systems for the closed-loop control of input shaped flexible structures. A Riccati equation-based and a linear matrix inequality-based frameworks are developed for the stabilization of systems with uncertain, multiple input delays. Next, the aforementioned framework is applied to two input shaped flexible structure systems. This framework guarantees closed-loop system stability and performance when the impulse train is applied at inaccurate instances. Two illustrative numerical examples demonstrate the efficacy of the proposed closed-loop input shaping controller. [S0022-0434(00)00103-9]

Issue Section:
Technical Papers
Keywords:
flexible structures, delay systems, feedback, Riccati equations, matrix algebra, vibration control, closed loop systems, uncertain systems, stability, Input Shaping, Flexible Structures, Time-Delay Control, Riccati Equation, Linear Matrix Inequality
Topics:
Control equipment, Delays, Flexible structures, Impulse (Physics), Closed loop systems, Pole placement, Feedback, Design, Dynamics (Mechanics), Stability
1.
Tzes
,
A.
, and
Yurkovich
,
S.
,
1993
, “
An Adaptive Input Shaping Control Scheme for Vibration Suppression in Slewing Flexible Structure
,”
IEEE Trans. Control Syst. Technol.
,
1
, pp.
114
–
121
.
2.
Bodson
,
M.
,
1998
, “
An Adaptive Algorithm for the Tuning of Two Input Shaping Methods
,”
Automatica
,
34
, pp.
771
–
776
.
3.
Magee, D. P., Cannon, D. W., and Book, W. J., 1997, “Combined Command Shaping and Inertial Damping for Flexure Control,” Proc. Am. Control. Conf., pp. 1330–1334, Albuquerque, NM.
4.
Pao
,
L. Y.
, and
Lau
,
M. A.
,
1999
, “
Expected Residual Vibration of Traditional and Hybrid Input Shaping Designs
,”
J. Guid. Control Dyn.
,
22
, pp.
162
–
165
.
5.
Singer
,
N. C.
, and
Seering
,
W. P.
,
1990
, “
Preshaping Command Inputs to Reduce System Vibration
,”
ASME J. Dyn. Syst., Meas., Control
,
112
, pp.
76
–
82
.
6.
Singhose
,
W. E.
,
Derezinski
,
S.
, and
Singer
,
N. C.
,
1996
, “
Extra-Insensitive Input Shaper for Controlling Flexible Spacecraft
,”
J. Guid. Control Dyn.
,
19
, pp.
385
–
391
.
7.
Singhose
,
W. E.
,
Porter
,
L. J.
,
Tuttle
,
T. D.
, and
Singer
,
N. C.
,
1997
, “
Vibration Reduction using Multi-Hump Input Shapers
,”
ASME J. Dyn. Syst., Meas., Control
,
119
, pp.
320
–
326
.
8.
Swigert
,
C. J.
,
1980
, “
Shaped Torque Techniques
,”
J. Guid. Control
,
3
, pp.
460
–
467
.
9.
Tallman
,
G. H.
, and
Smith
,
G. H.
,
1958
, “
Analog study of Dead-Beat Posicast Control
,”
IEEE Trans. Autom. Control
,
3
, pp.
14
–
21
.
10.
Singh
,
T.
, and
Vadali
,
S. R.
,
1993
, “
Robust Time Delay Control
,”
ASME J. Dyn. Syst., Meas., Control
,
115
, pp.
303
–
306
.
11.
Tzes, A., and Yurkovich, S., 1989, “Adaptive Precompensators for Flexible Link Manipulator Control,” Proc. IEEE Conf. Dec. Control., pp. 2083–2088, Tampa, FL.
12.
Dugard, L., and Verriest, E. I., 1997, Stability and Control of Time-Delay Systems, Springer-Verlag, New York, NY.
1.
Haddad, W. M., Kapila, V., and Abdallah, C. T., 1997, “Stabilization of Linear and Nonlinear Systems with Time Delay,” Proc. Amer. Contr. Conf., pp. 3220–3225, Albuquerque, NM;
2.
see also, Stability and Control of Time-Delay Systems, Dugard L., and Verriest, E., eds., pp. 205–217.
1.
Kapila
,
V.
,
Haddad
,
W. M.
, and
Grivas
,
A. D.
,
1999
, “
Stabilization of Linear Systems with Simultaneous State, Actuation, and Measurement Delays
,”
Int. J. Control
,
72
, pp.
1619
–
1629
.
2.
Kim
,
J. H.
,
Jeung
,
E. T.
, and
Park
,
H. B.
,
1996
, “
Robust Control for Parameter Uncertain Delay Systems in State and Control Input
,”
Automatica
,
32
, pp.
1337
–
1339
.
3.
Moheimani, S. O. R., and Petersen, I. R., 1995, “Optimal Quadratic Guaranteed Cost Control of a Class of Uncertain Time-Delay Systems,” Proc. IEEE Conf. Dec. Control., pp. 1513–1518, New Orleans, LA.
4.
Mori
,
T.
,
Noldus
,
E.
, and
Kuwahara
,
M.
,
1983
, “
A Way to Stabilize Linear Systems with Delayed State
,”
Automatica
,
19
, pp.
571
–
573
.
5.
Niculescu, S.-I., de Souza, C. E., Dion, J. M., and Dugard, L., 1994, “Robust Stability and Stabilization of Uncertain Linear Systems with State Delay: Single Delay Case (I),” Proc. IFAC Workshop Robust Control. Design, pp. 469–474, Rio de Janeiro, Brazil.
6.
Niculescu, S.-I., Fu, M., and Li, H., 1997, “Stability of Linear Systems with Input Delay: An LMI Approach,” Proc. IEEE Conf. Dec. Control., pp. 1623–1628, San Diego, CA.
7.
Shen
,
J. C.
,
Chen
,
B. S.
, and
Kung
,
F. C.
,
1991
, “
Memoryless Stabilization of Uncertain Dynamic Delay Systems: Riccati Equation Approach
,”
IEEE Trans. Autom. Control
,
36
, pp.
638
–
640
.
8.
Verriest
,
E. I.
, and
Ivanov
,
A. F.
,
1994
, “
Robust Stability of Systems with Delayed Feedback
,”
Circuits Syst. Signal Process.
,
13
, pp.
213
–
222
.
9.
Meirovitch, L., 1980, Computational Methods in Structural Dynamics, Sijthoff and Noordhoff Int. Publ., Rockville, MD.
10.
Bhat
,
K. P. M.
, and
Koivo
,
H. N.
,
1976
, “
Modal Characterizations of Controllability and Observability in Time Delay Systems
,”
IEEE Trans. Automat. Control.
,
21
, pp.
292
–
293
.
11.
Boyd, S., El-Ghaoui, L., Feron, E., and Balakrishnan, V., 1994, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA.
12.
Dorato, P., Abdallah, C., and Cerone, V., 1995, Linear-Quadratic Control: An Introduction, Prentice-Hall, Englewood Cliffs, NJ.
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