Control of a Three-Dimensional String System

Abstract

Abstract In this paper, the control design and stability analysis are presented for a three-dimensional string system with the payload dynamics. A set of partial-ordinary differential equations (PDEs-ODEs) are developed by using the Hamilton's principle to describe the motion of the three-dimensional string system. The dynamic model considers the comprehensive effects of environmental loads, which are critical for the analysis 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 disturbances. Uniformly boundedness of the three-dimensional dynamics with the proposed control is achieved. Exponential stability is proved via the Lyapunov's direct method when there is no distributed disturbance. Simulation examples are provided by using the finite difference method, and some useful conclusions are drawn.