Boundary position feedback control of Kirchhoff's non‐linear strings

Kobayashi, Toshihiro

doi:10.1002/mma.440

Cited by 7 publications

(10 citation statements)

References 21 publications

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“…which are possible, since z 0 , z 1 ∈ W m for m 2. It should be noted that (13) and (14) are a system of ordinary differential equations in t, which is known to have a local solution z m (t), m (t) in an interval [0, t m ). After the estimate below the approximate solution z m (t), m (t) will be extended to the interval [0, T ] for any T >0.…”

Section: Proofmentioning

confidence: 99%

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Adaptive stabilization of Kirchhoff's non‐linear strings by boundary displacement feedback

Kobayashi

Sakamoto

2007

Math Methods in App Sciences

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SUMMARYThis paper is concerned with adaptive global stabilization of an undamped non-linear string in the case where any velocity feedback is not available. The linearized system may have an infinite number of poles and zeros on the imaginary axis. In the case where any velocity feedback is not available, a parallel compensator is effective. The adaptive stabilizer is constructed for the augmented system which consists of the controlled system and a parallel compensator. It is proved that the string can be stabilized by adaptive boundary control.

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Section: Proofmentioning

confidence: 99%

“…The boundary stabilization of non-linear strings was investigated in [11,12], where the boundary velocity feedback is used. In the case where any velocity feedback is not available, the stabilization of non-linear strings by using position feedback was also considered in [13].…”

Section: Introductionmentioning

confidence: 99%

Adaptive stabilization of Kirchhoff's non‐linear strings by boundary displacement feedback

Kobayashi

Sakamoto

2007

Math Methods in App Sciences

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“…Furthermore, a similar approach is used in [15] and [7] to prove the stability for various nonlinear axially moving strings controlled by the linear boundary feedback. In the case where any velocity feedback is not available, the stabilization of the Kirchhoff strings by using position feedback was also considered in [11]. This paper deal with the finite element approximation problem for the nonlinear Kirchhoff strings.…”

Section: Introductionmentioning

confidence: 99%

Finite element approximation of the nonlinear Kirchhoff string with boundary control

Shen

2014

Proceedings of the 33rd Chinese Control Conference

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This paper addresses the finite element approximation problem of a nonlinear Kirchhoff string with the nonlinear boundary control. The nonlinear boundary control is the negative feedback of the transversal velocity of the string at one end, which satisfies a sector constrain. Based on the equivalent variational formulation of the nonlinear string equations, the finite element approximation of the Kirchhoff string with boundary control input is derived in the Lagrange polynomial space of degree 1. Simulation examples are presented to show the effectiveness of our main result.

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“…It should be mentioned that a string here refers a one-dimensional continuum without bending stiffness and with a straight line equilibrium configuration. The modeling of a loose string (a cable) with sag [10] will not be treated here.…”

Section: Introductionmentioning

confidence: 99%

Two nonlinear models of a transversely vibrating string

Chen

Ding²

2007

Arch Appl Mech

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Modeling transverse vibration of nonlinear strings is investigated via numerical solutions of partialdifferential equations and an integro-partial-differential equation. By averaging the tension along the deflected string, the classic nonlinear model of a transversely vibrating string, Kirchhoff's equation, is derived from another nonlinear model, a partial-differential equation. The partial-differential equation is obtained via neglecting longitudinal terms in a governing equation for coupled planar vibration. The finite difference schemes are developed to solve numerically those equations. An index is proposed to compare the transverse responses calculated from the two models with the transverse component calculated from the coupled equation. A steel string and a rubber string are treated as examples to demonstrate the differences between the two models of transverse vibration and their deviation from the full model of coupled vibration. The numerical results indicate that the differences increase with the amplitude of vibration. Both models yield satisfactory results of almost the same precision for vibration of small amplitudes. For large amplitudes, the Kirchhoff equation gives better results.

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Boundary position feedback control of Kirchhoff's non‐linear strings

Cited by 7 publications

References 21 publications

Adaptive stabilization of Kirchhoff's non‐linear strings by boundary displacement feedback

Adaptive stabilization of Kirchhoff's non‐linear strings by boundary displacement feedback

Finite element approximation of the nonlinear Kirchhoff string with boundary control

Two nonlinear models of a transversely vibrating string

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