Vibration Control of an Axially Moving String by Boundary Control

1996 ◽  
Vol 118 (1) ◽  
pp. 66-74 ◽  
Author(s):  
Seung-Yop Lee ◽  
C. D. Mote
Keyword(s):  
Boundary Control ◽  
Mechanical Energy ◽  
Semigroup Theory ◽  
External Boundary ◽  
Vibration Energy ◽  
Transverse Motion ◽  
Feedback Gain ◽  
Axially Moving String ◽  
Axially Moving ◽  
Time Required

The stabilization of the transverse vibration of an axially moving string is implemented using time-varying control of either the boundary transverse motion or the external boundary forces. The total mechanical energy of the translating string is a Lyapunaov functional and boundary control laws are designed to dissipate the total vibration energy of the string at the left and/or right boundary. An optimal feedback gain determined by minimizing the energy reflected from the boundaries, is the radio of tension to the propagation velocity of an incident wave to the boundary control. Also the maximum time required to stabilize all vibration energy of the system for any initial disturbance is the time required for a wave to propagate the span of the string before hitting boundary control. Asymptotic and exponential stability of the axially moving string under boundary control are verified analytically through the decay rate of the energy norm and the use of semigroup theory. Simulations are used to verify the theoretically predicted, optimal boundary control for the stabilization of the translating string.

Boundary Control of an Axially Moving String Via Lyapunov Method

1999 ◽  
Vol 121 (1) ◽  
pp. 105-110 ◽  
Author(s):  
Rong-Fong Fung ◽  
Chun-Chang Tseng
Keyword(s):  
Boundary Control ◽  
Lyapunov Method ◽  
Sliding Mode ◽  
Semigroup Theory ◽  
Active Vibration Control ◽  
Nonlinear Partial Differential Equation ◽  
Distributed Parameter ◽  
Axially Moving String ◽  
Active Vibration ◽  
Axially Moving

This paper presents the active vibration control of an axially moving string system through a mass-damper-spring (MDS) controller at its right-hand side (RHS) boundary. A nonlinear partial differential equation (PDE) describes a distributed parameter system (DPS) and directly selected as the object to be controlled. A new boundary control law is designed by sliding mode associated with Lyapunov method. It is shown that the boundary feedback states only include the displacement, velocity, and slope of the string at RHS boundary. Asymptotical stability of the control system is proved by the semigroup theory. Finally, finite difference scheme is used to validate the theoretical results.

Stabilization of an Axially Moving String by Nonlinear Boundary Feedback

1999 ◽  
Vol 121 (1) ◽  
pp. 117-121 ◽  
Author(s):  
Rong-Fong Fung ◽  
Jinn-Wen Wu ◽  
Sheng-Luong Wu
Keyword(s):  
Nonlinear Boundary ◽  
Mechanical Energy ◽  
Control Laws ◽  
Axially Moving String ◽  
Boundary Feedback ◽  
Boundary Velocity ◽  
Total Mechanical Energy ◽  
Axially Moving ◽  
The Right ◽  
Nonlinear Boundary Feedback

In this paper, we consider the system modeled by an axially moving string and a mass-damper-spring (MDS) controller, applied at the right-hand side (RHS) boundary of the string. We are concerned with the nonlinear string and the effect of the control mechanism. We stabilize the system through a proposed boundary velocity feedback control law. Linear and nonlinear control laws through this controller are proposed. In this paper, we find that a linear boundary feedback caused the total mechanical energy of the system to decay an asymptotically, but it fails for an exponential decay. However, a nonlinear boundary feedback controller can stabilize the system exponentially. The asymptotic and exponential stability are verified.

Suppression of Vibration in a Nonlinear Axially Moving String by the Boundary Control

1997 ◽  
Author(s):  
Shahram M. Shahruz
Keyword(s):  
Boundary Control ◽  
Negative Feedback ◽  
Linear Boundary ◽  
Axially Moving String ◽  
Axially Moving ◽  
Nonlinear String

Abstract In this note, a nonlinear axially moving string is considered. It is proved that the nonlinear string can be stabilized by the linear boundary control, which is the negative feedback of the transversal velocity of the string at one end.

Adaptive Boundary Control of an Axially Moving String System

2002 ◽  
Vol 124 (3) ◽  
pp. 435-440 ◽  
Author(s):  
Rong-Fong Fung ◽  
Jinn-Wen Wu ◽  
Pai-Yat Lu
Keyword(s):  
Boundary Control ◽  
Tracking Error ◽  
Torque Control ◽  
Control Force ◽  
Unknown Parameters ◽  
System Equation ◽  
Axially Moving String ◽  
Coupling System ◽  
Axially Moving ◽  
Adaptive Boundary Control

This paper proposes an adaptive boundary control to an axially moving string system, which couples with a mass-damper-spring (MDS) controller at its right-hand-side (RHS) boundary. Unknown parameters appearing in the system equation are assumed constant and estimated on-line by using adaptation laws. The adaptive computed-torque control algorithm applied to robot manipulators of lumped systems is extended to design the adaptive boundary controller for the coupling system. It is found that the control force and update laws depend only on the displacement, velocity and slope of the string at the RHS boundary. Lyapunov stability guarantees the convergence of the tracking error to zero. Finally, the performance of the proposed controller is demonstrated by numerical simulations.

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