Article
Transactions of the Institute of
Measurement and Control
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DOI: 10.1177/0142331217714524
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Vibration control for the payload
at the end of a nonlinear
three-dimensional Euler–Bernoulli
beam with input constraints
Ning Ji and Jinkun Liu
Abstract
In this paper, the vibration control problem for the payload at the end of a three-dimensional Euler–Bernoulli beam in the presence of input constraints
and input disturbances is addressed. Disturbance observers are designed to estimate the disturbances on the tip payload. Based on the disturbance
observers, a boundary control scheme is designed to suppress elastic vibration for the payload at the end of the beam. The smooth hyperbolic function
is applied for the proposed control scheme, which can satisfy physical conditions and input constraints. It is proved that the proposed control scheme
can be guaranteed in handling input constraints and disturbances. Finally, numerical simulations illustrate the effectiveness of the results.
Keywords
Three-dimensional Euler–Bernoulli beam, disturbance observer, smooth hyperbolic function, input constraint, boundary control
Introduction
Euler-Bernoulli beam system has the advantages of high
speed, low energy consumption, contact impact, and so forth,
so it has more and more application in aerospace and industrial
fields. For example, inspired by birds and bats, the control
problem for flexible wings described by hybrid partial differen-
tial equations (PDEs) and ordinary differential equations
(ODEs) of a robotic aircraft is addressed by using boundary
control schemes.
The vibration control for the payload at the end of a flexible
Euler–Bernoulli beam plays an important role in the practical
engineering application. Given that vibrations of the drogue at
the end of the hose will make it difficult to couple with the recei-
ver, a boundary control scheme in Liu et al. (2016) and Liu
et al. (2017a) is proposed based on the original PDEs to regulate
the hose’s vibration. In He et al. (2011), boundary control is
proposed at the top boundary of the riser based on Lyapunov’s
direct method to regulate the riser’s vibration.
There has been a lot of interest shown in the control design
and stability analysis of the flexible mechanical system in
recent years (Guo et al., 2017; He and Ge, 2016; He et al.,
2016; Liu et al., 2013, 2016; Nguyen et al., 2013; Zhao et al.,
2016, 2017). In Wu et al. (2012), fuzzy boundary control and
distributed H ‘ fuzzy controllers with constraint are designed
for nonlinear parabolic and hyperbolic distributed parameter
systems, respectively. In Ge et al. (2010), boundary control
for a coupled nonlinear flexible marine riser with two actua-
tors in transverse and longitudinal directions is developed to
reduce the riser’s vibrations. Boundary control is proposed
based on the original PDEs to regulate the hose’s vibration
and a disturbance observer is designed to estimate the input
disturbance in Liu et al. (2016). In He and Zhang (2017),
boundary control schemes are used to control the flexible
wings of a robotic aircraft. In He et al. (2015), a boundary
controller is presented for a flexible marine riser to suppress
the riser’s vibration with a top tension constraint.
In practice, the constraint characteristic may cause huge
instability and influence the performance of the system.
Therefore, the effect of the input constraint should be taken
into consideration for control design. The problem for the
saturation constraints is a topic of great importance, and
many methods to solve this problem are proposed, such as
neural networks (NN) control (Chen et al., 2010), model pre-
dictive control (MPC) (Adetola et al., 2009), anti-windup
methods (Ailon, 2010), and so forth. In Liu et al. (2017b), a
boundary control is developed to regulate angular position
and suppress elastic vibration simultaneously, and applying
smooth hyperbolic functions to satisfy physical conditions
and input restrictions. In He et al. (2016), boundary control
scheme is developed to suppress transverse vibration of a flex-
ible marine riser with input saturation in the ocean environ-
ment and an auxiliary system is proposed in order to
School of Automation Science and Electrical Engineering, Beihang
University, Beijing, PR China
Corresponding author:
Jinkun Liu, School of Automation Science and Electrical Engineering,
Beihang University, Xueyuan Road No. 37, HaiDian District, Beijing
100191, PR China.
Email: ljk@buaa.edu.cn