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.
Modeling and Boundary Control of a Hanging Cable Immersed in Water
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.
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Böhm, M., Krstic, M., Küchler, S., and Sawodny, O. (September 23, 2013). "Modeling and Boundary Control of a Hanging Cable Immersed in Water." ASME. J. Dyn. Sys., Meas., Control. January 2014; 136(1): 011006. https://doi.org/10.1115/1.4024604
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