Vibration Control of an Axially Moving System with Restricted Input

Hindawi Complexity Volume 2019, Article ID 2386435, 10 pages https://doi.org/10.1155/2019/2386435 Research Article Vibration Control of an Axially Moving System with Restricted Input Zhijia Zhao ,1,2,3 Yonghao Ma,1 Guiyun Liu ,1,3 Dachang Zhu ,1 and Guilin Wen1,2,3 1 School of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou 510006, China Advanced Technology Center for Special Equipment, Guangzhou University, Guangzhou 510006, China 3 Center for Intelligent Equipment and Network-Connected System, Guangzhou University, Guangzhou 510006, China 2 Correspondence should be addressed to Guiyun Liu; and Dachang Zhu; Received 11 April 2018; Accepted 6 September 2018; Published 2 January 2019 Guest Editor: Andy Annamalai Copyright © 2019 Zhijia Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, we consider the global stabilization of an axially moving system under the condition of input saturation nonlinearity and external perturbation. Based on Lyapunov redesign method, observer backstepping, and high-gain observers, an output feedback control strategy with an auxiliary system is constructed to eliminate the input saturation constraint effect and suppress the string system vibration, and a boundary disturbance observer is exploited to cope with the external disturbance. The stability of the controlled system is analyzed and proven based on Lyapunov stability without simplifying or discretizing the infinite dimensional dynamics. The presented simulation results show the effectiveness of the derived control. 1. Introduction Axially moving systems are important components in mechanical system and play a significant role in actual production process. However, a few issues exist; nonsmooth input nonlinearities and external perturbations frequently occur and produce severe impacts on system performance. It is worth noting that nonsmooth input nonlinearities containing saturation, backlash, hysteresis, and dead-zone are generally found in industrial control systems, such as mechanical, hydraulic, biomedical, piezoelectric, and physical systems [1– 5]. Such nonlinearities usually arise from inherent physical constraints of the dynamical system and constraints in the controller actuators, which are impossible to be eliminated. If the input nonlinearities are ignored in the system model, it is difficult to make the actual axially moving system stabilized. So far, some results associated with how to achieve the control objective for flexible structure systems with the input saturation have been attained [6–8]. In [6], the vibration control and input saturation problem for a vibrating flexible aerial refueling hose with variable lengths are addressed introducing the hyperbolic tangent function and adopting the backstepping approach. In [7], the authors develop the boundary control for a vibrating riser system with mixed input nonlinearities to suppress the deflection and compensate for the input saturation. In [8], the control schemes are constructed for a flexible beam system to suppress the vibration and eliminate the input saturation and output constraint in the presence of disturbances. However, these results do deal with the input saturation issue for stationary flexible systems, but there is little information on how to handle the input restriction for axially moving systems. In recent decades, many achievements regarding vibration control for axially moving systems have been attained, whose dynamics can be mathematically considered to be distributed parameter systems (DPS) with infinite dimensional feature [9–15]. Effective solutions for controlling the DPS mainly include truncation model-based method, and boundary control. Different from truncation model-based method [16–20], which is employed in different ways to extract a finite dimensional subsystem to be controlled while showing robustness to neglecting the remaining infinite dimensional dynamics in the design, boundary control is the implementation of control design based on infinite dimensional system dynamics, which is generally considered to be physically more realistic due to nonintrusive actuation and sensing [21]. For the past few years, the vibration boundary control scheme design for the axially moving system has made great 2 Complexity achievement [22–28]. In [22], the deflection of the axially moving string is regulated by the proposed adaptive vibration isolation and the practical experiment illustrates the theoretical results. In [23], an adaptive robust control strategy is constructed for controlling the vibrational offset of an axially moving system in the presence of parameter and disturbance uncertainties. In [24], an iterative learning control scheme is exploited for a stretched flexible string to damp out any string oscillation based on continuous and discrete Lyapunov functions. In [25], the vibration of a translating tensioned beam is exponentially stabilized and effectively suppressed via the choice of a proper Lyapunov function candidate. In [26], a stabilizing control law is derived for a translating tensioned strip to suppress the vibration and the closedloop system is proven to be exponentially stable. In [27], simultaneous vibration control scheme design and velocity regulation issue are discussed and good stability is attained in the sense of Lyapunov. In [28], the high-gain observer technique and Lyapunov-based observer backstepping method are integrated into the context of boundary control design to generate a stabilizing robust control law for suppressing the deflection of an accelerative belt system. In this article, the axially moving system with the input saturation is studied under the condition of the external disturbance, which makes the control scheme design more complicated and difficult in comparison with previous research. Moreover, in research achievements [22–28], the control schemes are implemented based on the assumption that all the system state signals consisting of the control law can be directly measured by sensors or obtained by algorithms. However, in practice, the measurement noise derived from sensors is completely unavoidable, and its effect will be further magnified in the procedure to obtain the terms of differentiation to time, which would limit the controller in [22–28] implementation. To resolve this issue, the observer backstepping [29] and high-gain observers [30] can be exploited to estimate the unmeasured system states and then an output feedback boundary control is developed to globally stabilize the considered axially moving system. In this study, our interest lies in how to construct an output feedback control for the global stabilization of the axially moving system and simultaneously for the elimination of input saturation nonlinearity e (...truncated)