Adaptive boundary feedback control of a rigid–flexible system under unknown time-varying disturbance

This paper studies the boundary feedback control of a rotating disk-cable-mass system (DCMS). The system consists of a flexible cable with a rigid disk connected at its upper end and a tip mass at the lower boundary. We assume that the disk and the tip mass are affected by unknown time-varying disturbances, while the cable is influenced by distributed disturbance. We design two boundary controls to suppress the vibration of the flexible cable and simultaneously adjust the system’s rotating speed to a desired value. The first controller is designed assuming the system’s parameters are accurately known. The second controller is conceived under circumstances where some system parameters are unknown, for which an adaptive technique is employed to regulate the DCMS. For these two controls, we choose two Lyapunov functions to prove the stability of the closed-loop system. Theoretical results demonstrate that the states of the closed-loop system are ultimately uniformly bounded. At the end of the paper, we validate two boundary controls through numerical tests.