Modelling and vibration control for a flexible string system in three-dimensional space

In this study, the control design and stability analysis are presented for a three-dimensional (3D) string system with the payload dynamics. A set of partial differential equations and ordinary differential equations (PDEs-ODEs) are developed to describe the motion of the 3D string system. The dynamic model considers the comprehensive effects of environmental loads, which are critical for the design of a string system. Based on the Lyapunov´s direct method and the properties of the string system dynamics, three boundary control inputs are applied at the boundary to suppress the vibrations of the system under the external loads. Uniform boundedness of the 3D dynamics with the proposed control is achieved. Exponential stability is proved via the Lyapunov´s direct method when there is no distributed load. Simulation examples are provided by using the finite difference method, and some useful conclusions are drawn.