Static feedback stabilization of a rotating disk–cable system with measurements from each boundary

Abstract

In this paper, we consider the stabilization of a unitized rotating disk–cable system, which consists of a uniform cable attached to a disk rotating at an angular velocity about an axis through the center of the disk and perpendicular to the disk. When the angular velocity is greater than π2, the system always has at least one positive eigenvalue, which makes the system cannot be stabilized only with the collocated static feedback control. This is why the existing literature usually considers the case where the admissible angular velocity is always less than π2. To overcome this obstacle, we design a static proportional feedback control with both collocated and anti-collocated measurements from each boundary to stabilize the system, where the admissible angular velocity is improved from less than π2 to less than π while ensuring that the system is minimum-phase. Through a detailed spectral analysis, the spectrum of the closed-loop system is proved to be in the left-half complex plane, and the asymptotic expressions of both eigenvalues and eigenfunctions are obtained. By the Riesz basis approach, the stability of the closed-loop system is proved. The effectiveness of the feedback control is verified by numerical simulation.