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

Ahmed Eshag Ahmed Eshag,Zhang Zhang,Sun Sun,Ma Ma

doi:10.3390/app9235122

Ahmed Eshag Ahmed Eshag, Zhang Zhang + Show 2 more

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https://doi.org/10.3390/app9235122

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Abstract

By integrating H∞ 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 H∞ 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 H∞ 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.

Highlights

With the evolution of manufacturing, flexible structure-type systems are commonly used in a variety of engineering areas such as moving strips, adjustable marine risers, and drill cable strings [1,2,3].The string system is a distributed parameter system (DPS) which is composed of one partial differential equation (PDE) and two ordinary differential equations (ODEs)
In this study a H∞ iterative learning boundary control (ILBC)-based method of lines (MOL) scheme was proposed for decreasing the perturbations of a certain and uncertain vibrating string under iteration-varying distributed disturbance and boundary disturbance
The PDE and ODEs described the dynamic characteristic of the vibrating string under iteration-varying external disturbances

Summary

Introduction

With the evolution of manufacturing, flexible structure-type systems are commonly used in a variety of engineering areas such as moving strips, adjustable marine risers, and drill cable strings [1,2,3].The string system is a distributed parameter system (DPS) which is composed of one partial differential equation (PDE) and two ordinary differential equations (ODEs). The presence of external disturbances and system parameters uncertainty in the string system makes the dynamics solution and control design indispensable. It has become more challenging and received increasing attention [4]. ILC is a practical approach to control such systems that execute similar tasks in a repetitive manner [14], such as a robotic arm in industry [15], a chemical reactor production [16], a nonlinear process [17], and position control of nano systems [18]. ILC plays a vital role in DPS, including a Timoshenko beam [21], a class of mixed

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