我们在整个所需轨迹上开发了迭代学习控制的收敛标准，以获得滞后补偿滞后系统中的前馈输入。 在分析中，利用Prandtl-Ishli...
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Research Article
Iterative Learning Control of Hysteresis in
Piezoelectric Actuators
Guilin Zhang,
1
Chengjin Zhang,
2
and Chaoyang Wang
3
1
College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China
2
School of Mechanical, Electrical and Information Engineering, Shandong University at Weihai, Weihai 264209, China
3
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
Correspondence should be addressed to Chengjin Zhang; cjzhang@sdu.edu.cn
Received April ; Accepted May ; Published May
Academic Editor: Qingsong Xu
Copyright © Guilin Zhang et al. is 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.
We develop convergence criteria of an iterative learning control on the whole desired trajectory to obtain the hysteresis-
compensating feedforward input in hysteretic systems. In the analysis, the Prandtl-Ishlinskii model is utilized to capture the
nonlinear behavior in piezoelectric actuators. Finally, we apply the control algorithm to an experimental piezoelectric actuator
and conclude that the tracking error is reduced to .% of the total displacement, which is approximately the noise level of the
sensor measurement.
1. Introduction
Piezoelectric actuators (PEAs) have been widely used in nan-
opositioning systems due to their fast response and nanome-
ter scale resolution [–]. However, the hysteresis existing in
PEAs can greatly limit system performance [, ]. Control
of hysteretic system is an important area of control system
research and a challenging problem [–]. Research on feed-
back and model-based feedforward control has been stud-
ied to achieve relatively high-precision positioning [–].
Iterativemethodscanbeusedtoimprovethepositioning
performance if the positioning application is repetitive.
erefore, many researchers study the iterative and adaptive
control methods to minimize the adverse eect of hysteresis
[–].
e main challenge in iterative approaches for hysteretic
systems is to assure convergence of the iterative algorithm.
Leang and Devasia divide a general desired trajectory into
some monotonicity partition [, ]. Aerwards, they prove
the convergence of iterative learning control (ILC) algorithm
oneachsinglebranch.Inthispaper,westudythedesignof
(ILC) algorithm to compensate for hysteresis-caused error in
PEAs. e main contribution of our work is proving conver-
gence of ILC algorithm on whole tracking trajectory.
e remainder of this paper is organized as follows. First,
we state the problem in the next section. Aerwards, we
briey review the Prandtl-Ishlinskii model in the context of
thisworkandproveconvergenceoftheILCalgorithmwe
designed. Finally, we implement the ILC algorithm on exper-
imental stage and show our experimental results and conclu-
sions.
2. Problem Statement
Consider a hysteretic system of the following form:
(
)
=
[
V
(
)
]
,
()
where V() ∈ R is the input, () ∈ R is the output, and
denotes the hysteresis function R → R. For a given desired
trajectory
𝑑
()dened on the nite time interval ∈[0,],
theobjectiveistondaninputV
𝑑
()by way of the following
iterative learning control (ILC) algorithm:
V
𝑘+1
(
)
=V
𝑘
(
)
+
𝑘
(
)
,
()
where
𝑘
()=
𝑑
()−
𝑘
(), is a constant (to be determined),
and V
𝑘
()and
𝑘
()are the input and output at the th itera-
tion, respectively. Figure depicts the block diagram of the
ILC algorithm. e goal of the ILC algorithm is to generate
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2014, Article ID 856706, 6 pages
http://dx.doi.org/10.1155/2014/856706
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