Stabilization of a Rotating Disk-Beam System With the Higher Angular Velocity

In this article, we consider the stabilization of a unitized rotating disk-beam system, which consists of a uniform beam attached to the center of a rotating disk. It is known that the torque control exerted on the disk combined with the boundary control can make it rotate with the desired angular velocity, and when the angular velocity is greater than $\sqrt{\mu _{1}}\approx \text{3.516}$, where $\mu _{1}$ is the first eigenvalue of the free beam operator, it is found that there exists at least one positive eigenvalue, so the system can not be stabilized by the collocated static feedback controls only. How to stabilize the system with the higher angular velocity is an interesting problem. In order to improve the admissible high angular velocity, we propose a static feedback composed of collocated and noncollocated boundary measurements on the beam and a nonlinear torque control exerted on the disk. We show that the closed-loop system is well-posed and stable as long as the desired angular velocity of the disk is improved from less than 3.516 to less than $\pi ^{2}$. This relaxes the limitation 3.516 of the admissible angular velocity of the disk in literature. Numerical simulation results demonstrate the effectiveness of the control design.