Stabilization involving condition-based adaptive control of an uncertain wave equation for the vibration of a flexible string

Abstract

This paper investigates the stabilization of an uncertain wave equation which describes the vibration dynamics of a flexible string. All the system parameters are unknown while the disturbance with unknown bound and period is allowed which lead to more serious uncertainties than the related literature, and hence result into the incapability of the traditional schemes. To solve the control problem, a novel condition-based adaptive control framework is developed in the paper. Specifically, as preparation for the adaptive control design, a state-feedback controller is first designed under the assumption that all the system parameters are known. Then, an adaptive controller, together with two continuous adaptive laws and a condition-based updating mechanism for the online tuning of the controller parameters, is proposed for the original uncertain system. Finally, it is proven that the updating of controller parameters under the proposed condition-based updating mechanism stops at a finite time instant, then the proposed controller guarantees the stability of the resulting closed-loop system. The proposed theoretical results are validated by a simulation example.