Adaptive neural control for a class of pure-feedback nonlinear
time-delay systems with asymmetric saturation actuators
Zhaoxu Yu
a,
n
, Shugang Li
b
, Zhaosheng Yu
c
a
Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education, East China University of Science and Technology,
Shanghai 200 237, PR China
b
School of Management, Shanghai University, Shanghai 200444, PR China
c
School of Electric Power, South China University of Technology, Guangzhou 510640, PR China
article info
Article history:
Received 3 December 2014
Received in revised form
21 May 2015
Accepted 8 September 2015
Communicated by Long Cheng
Available online 25 September 2015
Keywords:
Nonlinear system
Neural network
Time-varying delay
Asymmetric saturation actuator
Razumikhin lemma
abstract
This paper addresses the problem of adaptive tracking control for a class of uncertain pure-feedback
nonlinear time-delay systems with unknown asymmetric saturation actuators. The considered problem
is challenging due to the existence of unknown distributed time-varying delays and asymmetric
saturation actuator. In particular, the difficulties from distributed time-varying delays and unknown
asymmetric saturation nonlinearity are processed by using the mean value theorem for integrals and a
Gaussian error function-based continuous differentiable model, respectively. Then, based on a novel
combination of mean value theorem, Razumikhin functional method, variable separation technique and
Neural Network (NN) parameterization, an adaptive neural controller which involves only one parameter
to be updated is presented for such systems via Dynamic Surface Control (DSC) technique. Moreover, the
DSC technique can overcome the problem of ‘explosion of complexity’ in the traditional backstepping
design. All signals in the closed-loop system remain semi-globally uniformly ultimately bounded
(SGUUB), and the tacking error converges to a small neighborhood of the origin. Finally, simulation
results are given to verify the effectiveness of the proposed design.
& 2015 Elsevier B.V. All rights reserved.
1. Introduction
Delays frequently occur in a variety of practical systems such
as communication networks, chemical processes, biosystem,
teleoperation systems and underwater vehicles. The presence of
delays always degrades the performance of system, even results in
the instability of system. The topic of stability analysis and control
design for such time-delay systems thus has received much more
attentions [1,2]. Specifically, for some classes of nonlinear time-
delay systems, many adaptive control schemes have been pre-
sented by using the well-known backstepping technique [3–7].In
order to overcome the drawback of “explosion of complexity” in
the traditional of backstepping design, the DSC technique was
employed to design the controller for some classes of nonlinear
time-delay systems [8–10]. Unfortunately, the aforementioned
works did not consider the distributed delays case. In fact, systems
with distributed delays often arise when the number of summands
in a system equation is increased and the differences between
neighboring argument values are decreased. In the modeling of
combustion control in rocket motor chambers, one typical appli-
cation of distributed delay systems can be encountered [11,12].In
comparison with many research results on linear systems with
distributed delays [11–13], a few results are available for nonli-
near systems with distributed delays. For some classes of strict-
feedback nonlinear systems with distributed time-varying delays,
several adaptive control schemes were developed in [14–17].
In addition, pure-feedback nonlinear system which has a more
representative form than the strict-feedback systems has no affine
appearance of state variables to be used as the virtual controls and
the actual control. This makes the control of pure-feedback non-
linear time-delay systems, especially such systems with dis-
tributed time-varying delays, more difficult and more challenging.
As a consequence, it is of importance to address the control design
for such a more general class of nonlinear systems with distributed
time-varying delays.
Usually, the Lyapunov–Krasovskii method [3–17] and the Lya-
punov–Razumikhin method [18–22] are used for the stability
analysis and control design for time-delay systems. Compared
with the classical Krasovskii technique which often requires the
time-varying delay
τðtÞ to satisfy some conservative conditions,
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journal homepage: www.elsevier.com/locate/neucom
Neurocomputing
http://dx.doi.org/10.1016/j.neucom.2015.09.020
0925-2312/& 2015 Elsevier B.V. All rights reserved.
n
Corresponding author.
E-mail addresses: yu_yyzx@163.com (Z. Yu), westside_li@163.com (S. Li),
zsyu@scut.edu.cn (Z. Yu).
Neurocomputing 173 (2016) 1461–1470