The objective of this paper is to develop an optimal boundary control strategy for the axially moving material system through a mass-damper-spring (MDS) controller at its right-hand-side (RHS) boundary. The partial differential equation (PDE) describing the axially moving material system is combined with an ordinary differential equation (ODE), which describes the MDS. The combination provides the opportunity to suppress the flexible vibration by a control force acting on the MDS. The optimal boundary control laws are designed using the output feedback method and maximum principle theory. The output feedback method only includes the states of displacement and velocity at the RHS boundary, and does not require any model discretization thereby preventing the spillover associated with discrete parameter models. By utilizing the maximum principle theory, the optimal boundary controller is expressed in terms of an adjoint variable, and the determination of the corresponding displacement and velocity is reduced to solving a set of differential equations involving the state variable, as well as the adjoint variable, subject to boundary, initial and terminal conditions. Finally, a finite difference scheme is used to validate the theoretical results.

1.
Mote
, Jr.,
C. D.
,
1972
, “
Dynamic Stability of Axially Moving Materials
,”
Shock Vib. Dig.
4
, pp.
2
11
.
2.
Wickert
,
J. A.
, and
Mote
, Jr.,
C. D.
,
1989
, “
On the Energetics of Axially Moving Continua
,”
J. Acoust. Soc. Am.
85
, pp.
1365
1368
.
3.
Wickert
,
J. A.
, and
Mote
, Jr.,
C. D.
,
1990
, “
Classical Vibration Analysis of Axially Moving Continua
,”
ASME J. Appl. Mech.
57
, pp.
738
744
.
4.
Lee
,
S. Y.
, and
Mote
, Jr.,
C. D.
,
1997
, “
A Generalized Treatment of the Energetics of Translating Continua, Part I: Strings and Second Order Tensioned Pipes
,”
J. Sound Vib.
204
, No.
5
, pp.
717
734
.
5.
Chen
,
J. S.
,
1997
, “
Nautral Frequencies and Stability of an Axially-Traveling String in Contact with a Stationary Load System
,”
ASME J. Vibr. Acoust.
119
, pp.
152
157
.
6.
Chen
,
G.
,
1979
, “
Energy Decay Estimates and Exact Boundary Value Controllability for the Wave Equation in a Bounded Domain
,”
J. Math. Pures Appl.
58
, pp.
249
273
.
7.
Jai
,
A. E.
, and
Pritchard
,
A. J.
,
1987
, “
Sensors and Actuators in Distributed Systems
,”
Int. J. Control
46
, pp.
1139
1153
.
8.
Yang
,
B.
, and
Mote
, Jr.,
C. D.
,
1991
, “
Active Vibration Control of the Axially Moving String in the S Domain
,”
ASME J. Appl. Mech.
58
, pp.
189
196
.
9.
Chung
,
C. H.
, and
Tan
,
C. A.
,
1995
, “
Active Vibration Control of the Axially Moving String by Wave Cancellation
,”
ASME J. Vibr. Acoust.
117
, p.
49
55
.
10.
Lee
,
S. Y.
and
Mote
, Jr.,
C. D.
,
1996
, “
Vibration Control of an Axially Moving String by Boundary Control
,”
ASME J. Dyn. Syst., Meas., Control
118
, pp.
66
74
.
11.
Lee
,
S. Y.
, and
Mote
, Jr.,
C. D.
,
1999
, “
Wave Characteristics and Vibration Control of Translating Beams by Optimal Boundary Damping
,”
ASME J. Dyn. Syst., Meas., Control
121
, pp.
18
25
.
12.
Lions, J. L., 1971, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York.
13.
Curtain, R. F., and Pritchard, A. J., 1978, Infinite Dimensional Linear Systems Theory, Springer-Verlag, New York.
14.
Balakrishnan, A. V., 1981, Applied Functional Analysis, Springer-Verlag, New York.
15.
Gibson
,
J. S.
,
1983
, “
Linear-quadratic optimal control of hereditary differential systems: infinite dimensional Riccati equations and numerical approximations
,”
SIAM J. Control Optim.
21
, pp.
95
139
.
16.
Teo, K. L., and Wu, T. J., 1984, Computational Methods for Optimizing Distributed Systems, Orlando, FL: Academic Press.
17.
Lee. W. H. R., 1986, “On robust control designs for infinite dimensional systems,” Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, Massachusetts.
18.
Pritchard
,
A. J.
, and
Salamon
,
D.
,
1987
, “
The linear quadratic control problem for infinite dimensional systems with unbounded input and output operators
,”
SIAM J. Control Optim.
25
, No.
1
, pp.
121
144
.
19.
Sloss
,
J. M.
,
Sadek
,
I. S.
, and
Bruch
,
J. C.
,
1995
, “
Maximum Principle for the Optimal Control of a Hyperbolic Equation in One Space Dimension, Part 1: Theory
,”
J. Optim. Theory Appl.
87
, pp.
33
45
.
20.
Bruch
,
J. C.
,
Adali
,
S.
,
Sloss
,
J. M.
, and
Sadek
,
I. S.
,
1995
, “
Maximum Principle for the Optimal Control of a Hyperbolic Equation in One Space Dimension, Part 2: Application
,”
J. Optim. Theory Appl.
87
, pp.
287
300
.
21.
Bazezew, A., 1996, “Optimal Control of Structural Vibration Using a Maximum Principle Solved Numerically in the Space-Time Domain,” Ph.D. dissertation, University of California at Santa Barbara.
22.
Wu, S. L., 1997, “Boundary Feedback Control of the Axially Moving String System,” Master’s thesis, Chung Yuan University, Chung-Li, Taiwan.
23.
Fung
,
R. F.
, and
Tseng
,
C. C.
,
1999
, “
Boundary Control of an Axially Moving String Via Lyapunov Method
,”
ASME J. Dyn. Syst., Meas., Control
121
, pp.
105
110
.
24.
Baily
,
T.
, and
Hubbard
, Jr.,
J. E.
,
1985
, “
Distributed Piezoelectric-Polymer Active Vibration Control of a Cantilever Beam
J. Guid. Control
8 pp.
284
291
.
25.
Habib
,
M. S.
, and
Radcliffe
,
C. J.
,
1991
, “
Active Parametric Damping of Distributed Parameter Beam Transverse Vibration
,”
ASME J. Dyn. Syst., Meas., Control
113
,
295
299
.
26.
Tzafestas
,
S. G.
, and
Nightingale
,
J. M.
,
1970
, “
Optimal Distributed Parameter Control Using Classical Variational Calculus
,”
Int. J. Control
12
, No.
4
, pp.
593
608
.
27.
Yang
,
S. M.
, and
Jeng
,
C. A.
,
1998
, “
Structural Control of Distributed Parameter Systems by Output Feedback
,”
ASME J. Dyn. Syst., Meas., Control
120
, pp.
322
327
.
28.
T. S. Tang, 1990, “Optimal Control of a Class of Nonlinear Distributed Parameter Systems,” Ph.D. dissertation, University of A&M at Texas.
29.
P. C. Wang, 1964, “Control of Distributed Parameter Systems,” Advances in Control Systems, 1, Academic, New York, pp. 75–172.
30.
Kuo, Y. L., 1999, “Nonlinear Optimal Boundary Control of Axially Moving Material,” Master’s thesis, Chung Yuan University, Chung-Li, Taiwan.
31.
Sadek, I. S., 1983, “A Maximum Principle for Optimal Control of Systems with Distributed Parameters,” Ph.D. dissertation, University of California at Santa Barbara.
32.
Abhyankar
,
N. S.
,
Hall
,
E. K.
, and
Hanagud
,
S. V.
,
1993
, “
Chaotic Vibrations of Beams: Numerical Solution of partial Differential Equations
,”
ASME J. Appl. Mech.
60
, pp.
167
174
.
You do not currently have access to this content.