H∞ Iterative Learning Boundary Vibration Control of Uncertain Vibrating String with Iteration-Varying Distributed Disturbance and Boundary Disturbance

Abstract

By integrating <i>H</i><sub>∞</sub> control into iterative learning boundary control (ILBC) with the method of lines (MOL), this paper suggests a novel scheme to reduce the vibrations of the uncertain vibrating string system in the presence of iteration-varying distributed/boundary disturbances. The dynamics of the string system are defined by two kinds of differential equations, namely: (a) non-homogenous hyperbolic partial differential equation (PDE) and (b) ordinary differential equations (ODEs). Firstly, MOL is employed to attain the string dynamics in the form of a state-space system instead of a PDE system. Secondly, ILBC is developed in a super-vector framework and integrated with the <i>H</i><sub>∞</sub> control for decreasing the perturbations of the uncertain string system in the presence of iteration-varying distributed/boundary disturbances. Along the time, position, and iteration coordinates: (a) the boundary deflections of the string system are controlled; (b) the vibrations along the string are attenuated to zero; and (c) the external disturbances are excluded. Based on the <i>H</i><sub>∞</sub> algebraic approach, performance/stability conditions and global convergence of the closed-loop string system are assured. Conducted simulations illustrate that the suggested scheme is efficient for diminishing the vibrations of certain and uncertain vibrating string system.