Sliding mode boundary control of a vibrating string under time-varying distributed disturbance

Abstract

This paper is concerned with sliding-mode boundary control (SMBC) of a vibrating string system under parameter variation. The string is excited by time-varying distributed disturbance. Dynamics of the vibrating string are described by two types of differential equations, namely: (a) non-homogenous hyperbolic partial differential equation (PDE) and (b) ordinary differential equations (ODEs). In the proposed scheme, the perturbations of the vibrating string are damped in the presence of system parameter variation. The boundary control law based on the original infinite dimensional model is applied at the free end of the string. Robustness of this scheme is validated through different lengths of the string system. The suggested control system is proven to be exponentially converging to zero by using Lyapunov direct method. Simulation results show that the proposed design is valid for attenuating the vibrations effectively.