Abstract

This paper first derives equations of motion of extensible and shearable slender beams with large motions under both deterministic and stochastic external loads. Boundary feedback controllers are then proposed to achieve almost surely globally practically asymptotic stability. The control design, well-posedness, and stability analysis are based on a Lyapunov-type theorem developed for a class of stochastic evolution systems (SESs) in Hilbert space.

References

1.
Fard
,
M. P.
, and
Sagatun
,
S. I.
,
2001
, “
Exponential Stabilization of a Transversely Vibrating Beam by Boundary Control Via Lyapunov's Direct Method
,”
ASME J. Dyn. Syst., Meas., Control
,
123
(
2
), pp.
195
200
.
2.
Liu
,
K.
, and
Liu
,
Z.
,
2000
, “
Boundary Stabilization of a Nonhomogeneous Beam With Rotatory Inertia at the Tip
,”
J. Comput. Appl. Math.
,
114
(1), pp.
1
10
.
3.
Queiroz
,
M. S. D.
,
Dawson
,
M.
,
Nagarkatti
,
S.
, and
Zhang
,
F.
,
2000
,
Lyapunov-Based Control of Mechanical Systems
,
Birkhauser
,
Boston, MA
.
4.
Queiroz
,
M. D.
,
Dawson
,
D. M.
,
Agarwal
,
M.
, and
Zhang
,
F.
,
1999
, “
Adaptive Nonlinear Boundary Control of a Flexible Link Robotic Arm
,”
IEEE Trans. Rob. Autom.
,
15
(
4
), pp.
779
787
.
5.
Do
,
K. D.
, and
Pan
,
J.
,
2008
, “
Boundary Control of Transverse Motion of Marine Risers With Actuator Dynamics
,”
J. Sound Vib.
,
318
(
4–5
), pp.
768
791
.
6.
He
,
W.
,
Ge
,
S. S.
,
Voon
,
B.
,
How
,
E.
, and
Choo
,
Y. S.
,
2014
,
Dynamics and Control of Mechanical Systems in Offshore Engineering
,
Springer
,
London
.
7.
Nguyen
,
T. L.
,
Do
,
K. D.
, and
Pan
,
J.
,
2013
, “
Boundary Control of Two-Dimensional Marine Risers With Bending Couplings
,”
J. Sound Vib.
,
332
(
16
), pp.
3605
3622
.
8.
Luo
,
Z. H.
,
Guo
,
B. Z.
, and
Morgul
,
O.
,
1999
,
Stability and Stabilization of Infinite Dimensional Systems With Applications
,
Springer-Verlag
, London.
9.
Lu
,
L.
,
Chen
,
Z.
,
Yao
,
B.
, and
Wang
,
Q.
,
2013
, “
A Two-Loop Performance Oriented Tip Tracking Control of a Linear Motor Driven Flexible Beam System With Experiments
,”
IEEE Trans. Ind. Electron.
,
60
(
3
), pp.
1011
1022
.
10.
Krstic
,
M.
, and
Smyshlyaev
,
A.
,
2008
, “
Adaptive Control of PDEs
,”
Annu. Rev. Control
,
32
(
2
), pp.
149
160
.
11.
Paranjape
,
A.
,
Chung
,
S.-J.
, and
Krstic
,
M.
,
2013
, “
PDE Boundary Control for Flexible Articulated Wings on a Robotic Aircraft
,”
IEEE Trans. Rob.
,
29
(
3
), pp.
625
640
.
12.
Weiss
,
G.
, and
Curtain
,
R. F.
,
2008
, “
Exponential Stabilization of a Rayleigh Beam Using Collocated Control
,”
IEEE Trans. Autom. Control
,
53
(
3
), pp.
643
654
.
13.
Guo
,
B. Z.
, and
Jin
,
F. F.
,
2013
, “
The Active Disturbance Rejection and Sliding Mode Control Approach to the Stabilization of the Euler-Bernoulli Beam Equation With Boundary Input Disturbance
,”
Automatica
,
49
(
9
), pp.
2911
2918
.
14.
Abd-Elwahab
,
M.
, and
Sherif
,
H. A.
,
2006
, “
Pre-Tensioned Layer Damping as a New Approach for Vibration Control of Elastic Beams
,”
ASME J. Vib. Acoust.
,
128
(
3
), pp.
338
346
.
15.
Tanaka
,
N.
, and
Kikushima
,
Y.
,
1999
, “
Optimal Vibration Feedback Control of an Euler-Bernoulli Beam: Toward Realization of the Active Sink Method
,”
ASME J. Vib. Acoust.
,
121
(
2
), pp.
174
182
.
16.
Omidi
,
E.
, and
Mahmoodi
,
S. N.
,
2015
, “
Multiple Mode Spatial Vibration Reduction in Flexible Beams Using h2 and h∞ Modified Positive Position Feedback
,”
ASME J. Vib. Acoust.
,
137
(1), p.
011016
.
17.
Vatankhah
,
R.
,
Najafi
,
A.
,
Salarieh
,
H.
, and
Alasty
,
A.
,
2015
, “
Lyapunov-Based Boundary Control of Strain Gradient Microscale Beams With Exponential Decay Rate
,”
ASME J. Vib. Acoust.
,
137
(
3
), p.
031003
.
18.
Yang
,
J. S.
, and
Gao
,
R. P.
,
2016
, “
Active Control of a Very Large Floating Beam Structure
,”
ASME J. Vib. Acoust.
,
138
(
2
), p.
021010
.
19.
Eringen
,
A. C.
,
1952
, “
On the Nonlinear Vibration of Elastic Bars
,”
Q. Appl. Math.
,
9
(
4
), pp.
361
369
.
20.
Love
,
A.
,
1920
,
A Treatise on the Mathematical Theory of Elasticity
, 3rd ed.,
Cambridge University Press
, Cambridge, UK.
21.
Do
,
K. D.
, and
Pan
,
J.
,
2009
, “
Boundary Control of Three-Dimensional Inextensible Marine Risers
,”
J. Sound Vib.
,
327
(
3–5
), pp.
299
321
.
22.
Do
,
K. D.
,
2011
, “
Global Stabilization of Three-Dimensional Flexible Marine Risers by Boundary Control
,”
Ocean Syst. Eng.
,
1
(
2
), pp.
171
194
.
23.
Kokarakis
,
J. E.
, and
Bernitsas
,
M. M.
,
1987
, “
Nonlinear Three-Dimensional Dynamic Analysis of Marine Risers
,”
ASME J. Energy Resour. Technol.
,
109
(
3
), pp.
105
111
.
24.
Athisakul
,
C.
,
Monprapussorn
,
T.
, and
Chucheepsakul
,
S.
,
2011
, “
A Variational Formulation for Three-Dimensional Analysis of Extensible Marine Riser Transporting Fluid
,”
Ocean Eng.
,
38
(
4
), pp.
609
620
.
25.
Gawarecki
,
L.
, and
Mandrekar
,
V.
,
2011
,
Stochastic Differential Equations in Infinite Dimensions With Applications to Stochastic Partial Differential Equations
,
Springer
,
Berlin
.
26.
Do
,
K. D.
,
2016
, “
Stability of Nonlinear Stochastic Distributed Parameter Systems and Its Applications
,”
ASME J. Dyn. Syst., Meas., Control
,
138
(
10
), p.
101010
.
27.
Khalil
,
H.
,
2002
,
Nonlinear Systems
,
Prentice Hall
, Upper Saddle River, NJ.
28.
Krstic
,
M.
,
Kanellakopoulos
,
I.
, and
Kokotovic
,
P.
,
1995
,
Nonlinear and Adaptive Control Design
,
Wiley
,
New York
.
29.
Banks
,
H. T.
,
Smith
,
R. C.
, and
Wang
,
Y.
,
1996
,
Smart Material Structures
,
Wiley
,
Paris, France
.
30.
Evans
,
L.
,
2000
,
Partial Differential Equations
,
American Mathematical Society
,
Providence, RI
.
31.
Berrimi
,
S.
, and
Messaoudi
,
S. A.
,
2004
, “
Exponential Decay of Solutions to a Viscoelastic Equation With Nonlinear Localized Damping
,”
Electron. J. Differ. Equations
,
88
, pp.
1
10
.
32.
Cavalcanti
,
M. M.
,
Cavalcanti
,
V. N. D.
, and
Soriano
,
J. A.
,
2002
, “
Exponential Decay for the Solution of Semilinear Viscoelastic Wave Equations With Localized Damping
,”
Electron. J. Differ. Equations
,
44
, pp.
1
14
.
33.
Do
,
K. D.
,
2016
, “
Boundary Control of Elastic Systems
,”
J. Appl. Math. Comput.
,
51
(
1
), pp.
315
339
.
34.
Lacarbonara
,
W.
,
2013
,
Nonlinear Structural Mechanics
,
Springer
,
New York
.
35.
Orthwein
,
W. C.
,
1968
, “
A Nonlinear Stress-Strain Relation
,”
Int. J. Solids Struct.
,
4
(
3
), pp.
371
382
.
36.
Do
,
K. D.
, and
Pan
,
J.
,
2009
,
Control of Ships and Underwater Vehicles
,
Springer
, London.
37.
Adams
,
R. A.
, and
Fournier
,
J. J. F.
,
2003
,
Sobolev Spaces
, 2nd ed.,
Academic Press
,
Oxford, UK
.
38.
Prevot
,
C.
, and
Rockner
,
M.
,
2007
,
A Concise Course on Stochastic Partial Differential Equations
,
Springer
,
Berlin
.
39.
Prato
,
G. D.
, and
Zabczyk
,
J.
,
1992
,
Stochastic Equations in Infinite Dimensions
,
Cambridge University Press
,
Cambridge, UK
.
40.
Deng
,
H.
,
Krstic
,
M.
, and
Williams
,
R.
,
2001
, “
Stabilization of Stochastic Nonlinear Systems Driven by Noise of Unknown Covariance
,”
IEEE Trans. Autom. Control
,
46
(
8
), pp.
1237
1253
.
41.
Lions
,
J.
,
1978
, “
On Some Questions in Boundary Value Problems of Mathematical Physics
,”
Contemporary Developments in Continuum Mechanics and Partial Differential Equations
, Vol.
30
,
G. M.
de La Penha
and
L. A. J.
Medeiros
, eds., Elsevier, North Holland, The Netherlands, pp.
284
346
.
42.
Liu
,
K.
,
2006
,
Stability of Infinite Dimensional Stochastic Differential Equations With Applications
,
Chapman and Hall/CRC
,
Boca Raton, FL
.
43.
Bernitsas
,
M. M.
,
Kokarakis
,
J. E.
, and
Imron
,
A.
,
1985
, “
Large Deformation Three-Dimensional Static Analysis of Deep Water Marine Risers
,”
Appl. Ocean Res.
,
7
(
4
), pp.
178
187
.
44.
Niedzwecki
,
J. M.
, and
Liagre
,
P. Y. F.
,
2003
, “
System Identification of Distributed-Parameter Marine Riser Models
,”
Ocean Eng.
,
30
(
11
), pp.
1387
1415
.
45.
Sarpkaya
,
T.
, and
Isaacso
,
M.
,
1981
,
Mechanics of Wave Forces on Offshore Structures
,
Van Nostrand Reinhold
,
New York
.
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