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Research Papers

Modeling and Boundary Control of a Hanging Cable Immersed in Water

[+] Author and Article Information
Michael Böhm

Graduate Research Assistant
Institute for System Dynamics,
University of Stuttgart,
Stuttgart 70569, Germany
e-mail: boehm@isys.uni-stuttgart.de

Miroslav Krstic

Professor
Department of Mechanical and
Aerospace Engineering,
University of California, San Diego,
La Jolla, CA 92093-0411
e-mail: krstic@ucsd.edu

Sebastian Küchler

Former Graduate Research Assistant
e-mail: kuechler@isys.uni-stuttgart.de

Oliver Sawodny

Professor
e-mail: sawodny@isys.uni-stuttgart.de
Institute for System Dynamics,
University of Stuttgart,
Stuttgart 70569, Germany

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement and Control. Manuscript received June 20, 2012; final manuscript received May 17, 2013; published online September 23, 2013. Assoc. Editor: Prashant Mehta.

J. Dyn. Sys., Meas., Control 136(1), 011006 (Sep 23, 2013) (15 pages) Paper No: DS-12-1193; doi: 10.1115/1.4024604 History: Received June 20, 2012; Revised May 17, 2013

A nonlinear distributed parameter system model governing the motion of a cable with an attached payload immersed in water is derived. The payload is subject to a drag force due to a constant water stream velocity. Such a system is found, for example, in deep sea oil exploration, where a crane mounted on a ship is used for construction and thus positioning of underwater parts of an offshore drilling platform. The equations of motion are linearized, resulting in two coupled, one-dimensional wave equations with spatially varying coefficients and dynamic boundary conditions of second order in time. The wave equations model the normal and tangential displacements of cable elements, respectively. A two degree of freedom controller is designed for this system with a Dirichlet input at the boundary opposite to the payload. A feedforward controller is designed by inverting the system using a Taylor-series, which is then truncated. The coupling is ignored for the feedback design, allowing for a separate design for each direction of motion. Transformations are introduced, in order to transform the system into a cascade of a partial differential equation (PDE) and an ordinary differential equation (ODE), and PDE backstepping is applied. Closed-loop stability is proven. This is supported by simulation results for different cable lengths and payload masses. These simulations also illustrate the performance of the feedforward controller.

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References

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Figures

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Fig. 1

Offshore crane with cable and attached payload which is subject to a drag force due to the stream velocity V

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Fig. 2

Local displacements and forces acting on an infinitely small cable element

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Fig. 3

Two degree of freedom control strategy diagram

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Fig. 4

Input and desired load trajectories using T = 11 s and w¯*(0,T)=u¯*(0,T)=0.2 and the reference parameters

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Fig. 5

The different coordinate frames for describing the motion of the cable

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Fig. 6

Comparing the actual w(L, t) (dotted) and desired w*(L, t) (solid) payload motion. The input w(0, t) (dashed) is also shown. The transition time is T = 15 s.

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Fig. 7

Spatially varying weighing terms for position and velocity feedback for L = 100 m (solid) and L = 200 m (dashed)

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Fig. 8

Closed-loop simulation results for an orthogonal and tangential displacement of 20 m. The transition time is T = 15 s. The desired trajectory (solid) and the actual trajectory (dotted) are with the left axis, the difference (dash-dotted) is shown on the right axis. Note that two different y-scales have been used to magnify the error.

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Fig. 9

Closed-loop simulation results for an orthogonal and tangential displacement of 20 m. The transition time is T = 15 s. The feedforward control input (solid) is with the left axis, the feedback part (dash-dotted) is shown on the right axis. Note that two different y-scales have been used to magnify the feedback part.

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Fig. 10

Closed-loop simulation results for an orthogonal and tangential displacement of 20 m. The transition time is T = 30 s. The desired trajectory (solid) and the actual trajectory (dotted) are with the left axis, the difference (dash-dotted) is shown on the right axis. Note that two different y-scales have been used to magnify the error.

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Fig. 11

Closed-loop simulation results for a step-shaped disturbance from t = 10 s to t = 35 s. The desired trajectory (solid) and the actual trajectory (dotted) are with the left axis, the difference (dash-dotted) is shown on the right axis. Note that two different y-scales have been used to magnify the error.

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