This paper extends input shaping control to one-dimensional continua. Unlike discrete systems where the input is shaped only in the temporal domain, temporal and spatial input shaping can produce zero residual vibration in setpoint position control of distributed strings and beams. For collocated and noncollocated boundary control of strings and domain control of strings and pinned beams, the response to step inputs is solved in closed form using delays. For a clamped beam model, a closed form infinite modal series is used. The boundary controlled string can be setpoint regulated using two-pulse zero vibration (ZV) and three-pulse zero vibration and derivative (ZVD) shapers but ZVD is not more robust to parameter variations than ZV, a unique characteristic of second-order partial differential equations systems. Noncollocated ZV and ZVD boundary control enables rigid body translation of a string with zero residual vibration. Domain controlled strings and pinned beams with spatial input distributions that satisfy certain orthogonality conditions (e.g., midspan point load or uniformly distributed load) can be setpoint regulated with shaped inputs. For the cantilevered beam, modal shaping of the input distribution and ZV or ZVD temporal shaping drives the tip to the desired position with zero residual vibration.

1.
Singer
,
N. C.
, and
Seering
,
W. P.
, 1989, “
Design and Comparison of Command Shaping Methods for Controlling Residual Vibration
,”
IEEE International Conference on Robotics Automation
, pp.
888
893
.
2.
Hyde
,
J. M.
, and
Seering
,
W. P.
, 1991, “
Using Input Command Pre-Shaping to Suppress Multiple Mode Vibration
,”
IEEE International Conference on Robotics and Automation
,
Sacramento, CA
, pp.
2604
2609
.
3.
Singh
,
T.
, and
Heppler
,
G. R.
, 1993, “
Shaped Input Control of a System With Multiple Modes
,”
ASME J. Dyn. Syst., Meas., Control
0022-0434,
115
, pp.
341
347
.
4.
Singhose
,
W. E.
,
Seering
,
W. P.
, and
Singer
,
N. C.
, 1990, “
Shaping Inputs to Reduce Vibration: A Vector Diagram Approach
,”
Proceedings of the 1990 IEEE International Conference on Robotics and Automation
, pp.
922
927
.
5.
Singhose
,
W. E.
, 1997, “
Command Generation for Flexible Systems
,” Ph.D. thesis, MIT, Cambridge.
6.
Singhose
,
W. E.
,
Seering
,
W. P.
, and
Singer
,
N. C.
, 1997, “
Time-Optimal Negative Input Shapes
,”
ASME J. Dyn. Syst., Meas., Control
0022-0434,
119
, pp.
198
205
.
7.
Pao
,
L. Y.
, 1999, “
Multi-Input Shaping Design for Vibration Reduction
,”
Automatica
0005-1098,
35
, pp.
81
89
.
8.
Shan
,
J.
,
Liu
,
H. T.
, and
Sun
,
D.
, 2005, “
Modified Input Shaping For a Rotating Single-Link Flexible Manipulator
,”
J. Sound Vib.
0022-460X,
285
, pp.
187
207
.
9.
Gimpel
,
D. J.
, and
Calvert
,
J. F.
, 1952, “
Signal Component Control
,”
AIEE Transactions
,
71
, pp.
339
343
.
10.
Baicu
,
C. F.
,
Rahn
,
C. D.
, and
Nibali
,
B. D.
, 1996, “
Active Boundary Control of Elastic Cables: Theory and Experiment
,”
J. Sound Vib.
0022-460X,
198
, pp.
17
26
.
11.
Dadfarnia
,
M.
,
Jalili
,
N.
,
Xian
,
B.
, and
Dawson
,
D.
, 2004, “
A Lyapunov-Based Piezoelectric Controller for Flexible Cartesian Robot Manipulators
,”
ASME J. Dyn. Syst., Meas., Control
0022-0434,
126
(
2
), pp.
347
358
.
12.
Qu
,
Z.
, 2002, “
An Iterative Learning Algorithm for Boundary Control of a Stretched Moving String
,”
Automatica
0005-1098,
38
, pp.
821
827
.
13.
Zhao
,
H.
, and
Rahn
,
C. D.
, 2005, “
Iterative Learning Velocity and Tension Control for Axially Moving Materials
,”
Proceedings of ASME DETC2005
,
Long Beach, CA
.
14.
Zhao
,
H.
,
Rahn
,
C. D.
, and
Canbolat
,
H.
, 2005, “
Repetitive Control for an Electrostatic Microbridge
,”
Proceedings of ASME IMECE2005
,
Orlando, FL
.
15.
Singh
,
T.
, and
Alli
,
H.
, 1996 “
Exact Time-Optimal Control of the Wave Equation
,”
J. Guid. Control Dyn.
0731-5090,
19
(
1
), pp.
130
134
.
16.
Fortgang
,
J.
, and
Singhose
,
W.
, 2005, “
Input Shaping for Continuum Beams Under Longitudinal Vibration
,”
IEEE/ASME International Conference on Advanced Intelligent Mechatronics
,
Monterey, CA
, pp.
1599
1604
.
17.
Balas
,
M. J.
, 1978, “
Active Control of Flexible Systems
,”
J. Optim. Theory Appl.
0022-3239,
25
(
3
),
415
436
.
18.
Meirovitch
,
L.
, and
Baruh
,
H.
, 1985, “
Implementation of Modal Filters for Control
,”
J. Guid. Control Dyn.
0731-5090,
8
(
6
), pp.
707
716
.
19.
Iwamoto
,
H.
, and
Tanaka
,
N.
, 2005, “
Adaptive Feed-Forward Control of Flexural Waves Propagating in a Beam Using Smart Sensors
,”
Smart Mater. Struct.
0964-1726,
14
, pp.
1369
1376
.
20.
Rahn
,
C. D.
, 2001,
Mechatronic Control of Distributed Vibration and Noise
,
Springer
,
Berlin
.
21.
Meirovitch
,
L.
, 1967,
Analytical Methods in Vibration
,
Macmillan
,
New York
.
22.
Kozak
,
K.
,
Huey
,
J.
, and
Singhose
,
W. E.
, 2003, “
Performance Measures for Input Shaping
,”
Proceedings of IEEE Conference on Control Applications
, Vol.
2
, pp.
1227
1232
.
23.
Lu
,
J.
,
Crocker
,
M. J.
, and
Raju
,
P. K.
, 1989, “
Active Vibration Control Using Wave Control Concepts
,”
J. Sound Vib.
0022-460X,
134
(
2
), pp.
364
368
.
24.
Lee
,
C. K.
, and
Moon
,
F. C.
, 1990, “
Modal Sensors/Actuators
,”
ASME J. Appl. Mech.
0021-8936,
57
, pp.
434
441
.
25.
Pao
,
L. Y.
, and
Singhose
,
W. E.
, 1995, “
On the Equivalence of Minimum Time Input Shaping With Traditional Time-Optimal Control
,”
Proceedings of the Fourth IEEE Conference
,
Albany, NY
, pp.
1120
1125
.
You do not currently have access to this content.