Stabilization of a Wave Equation with a Tip Mass Based on Disturbance Observer of Time-Varying Gain

In this paper, we consider the stabilization problem of a wave equation with a tip mass, which undergoes the external disturbances at the tip mass end. Here, the disturbance may be exponentially increasing. For such a model, the usual sliding mode control method cannot be applied. Therefore, we employ the active disturbance rejection control (ADRC) approach to investigate this problem. At first, by the ADRC method, we design a disturbance observer that has time-varying gain so that the disturbance can be estimated exponentially. We show the disturbance observer is an exponential-type observer. Then, we use the estimate term as negative feedback so as to cancel disturbance. Finally, we prove that the resulted closed-loop system is well-posedness and exponentially stable.