Neural-network-based output-feedback adaptive dynamic surface
control for a class of stochastic nonlinear time-delay systems with
unknown control directions
Zhaoxu Yu
a,
n
, Shugang Li
b
a
Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education, East China University of Science and Technology,
Shanghai 200237, PR China
b
Department of Industrial Engineering and Logistics, Shanghai Jiao Tong University, Shanghai 200240, PR China
article info
Article history:
Received 1 April 2012
Received in revised form
21 August 2013
Accepted 6 September 2013
Communicated by Bin He
Available online 23 October 2013
Keywords:
Stochastic time-delay systems
Output-feedback
Neural network
Dynamic surface control
Unknown control direction
abstract
This paper focuses on the problem of output-feedback adaptive stabilization for a class of stochastic
nonlinear time-delay systems with unknown control directions. First, based on a linear state transforma-
tion, the unknown control coefficients are lumped together and the original system is transformed to a
new system for which control design becomes feasible. Then, after the introduction of an observer, an
adaptive neural network (NN) output-feedback control scheme is presented for such systems by using
dynamic surface control (DSC) technique and Lyapunov–Krasovskii method. The designed controller
ensures that all the signals in the closed-loop system are 4-Moment (or 2-Moment) semi-globally
uniformly ultimately bounded. Finally, a numerical example is given to demonstrate the feasibility and
effectiveness of the proposed control design.
& 2013 Elsevier B.V. All rights reserved.
1. Introduction
In recent years, the topic of stability analysis and control design
for stochastic nonlinear systems has been an intensive area
of research [1–8]. Some nonlinear control design methods such
as Lyapunov function approach, backstepping technique, and
approximation-based control method were extended to the case
of stochastic nonlinear systems. Especially, by using neural net-
work or fuzzy system, many adaptive control schemes have been
developed for some classes of stochastic nonlinear systems with
unstructured uncertainties. When only system output can be
measured, the problem of output-feedback stabilization has been
investigated for some classes of stochastic nonlinear systems by
using the systematic backstepping techniques [9–15]. However, an
obvious drawback in the traditional backstepping design is the
problem of ‘explosion of complexity’, which is caused by the
repeated differentiations of some nonlinear functions such as
virtual controllers. To avoid the problem of ‘explosion of complex-
ity’, the DSC method has been proposed by introducing a first-
order filtering of the virtual control law at each step of the
conventional backstepping design procedure for some classes of
deterministic nonlinear systems in [16]. Moreover, the DSC
method was extended to the approximation-based control [17],
decentralized control [18], output-feedback stabilization [19], and
control of time-delay systems [18–20]. Since the DSC has been
extended to the adaptive neural output-feedback stabilization for a
class of stochastic nonlinear systems in [21], it has been used to
solve the output-feedback stabilization problem for stochastic
nonlinear strict-feedback systems [22], stochastic nonminimum
phase nonlinear systems [23], stochastic MIMO (Multiinput and
Multioutput) nonlinear systems [24], and stochastic nonlinear
large-scale systems [25,26]. Unfortunately, these aforementioned
DSC designs for stochastic nonlinear systems did not consider the
time-delay and the unknown control direction.
On the other hand, the unknown control direction may be
encountered in a variety of practical systems, such that the control
design for such systems will be quite difficult. In comparison with
lots of research results in deterministic nonlinear systems [27–33],
there are only a few results on control for stochastic nonlinear
systems. In [3], the problem of adaptive fuzzy control has been
presented for a class of stochastic strict-feedback nonlinear sys-
tems with unknown control direction. In [34], an adaptive neural
controller design was addressed for a general class of stochastic
nonlinear pure-feedback systems with unknown control direc-
tion by combining the decoupled backstepping technique
and the Nussbaum Gain Function (NGF) approach. In addition,
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Neurocomputing
0925-2312/$ - see front matter & 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.neucom.2013.09.005
n
Corresponding author. Tel.: þ86 21 64253396.
E-mail addresses: yu_yyzx@163.com (Z. Yu), maxli@sjtu.edu.cn (S. Li).
Neurocomputing 129 (2014) 540–547