1562 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 27, NO. 7, JULY 2016
Neural Network Control-Based Adaptive Learning
Design for Nonlinear Systems With
Full-State Constraints
Yan-Jun Liu, Member, IEEE, Jing Li, Shaocheng Tong, Senior Member, IEEE,
and C. L. Philip Chen, Fellow, IEEE
Abstract—In order to stabilize a class of uncertain nonlinear
strict-feedback systems with full-state constraints, an adaptive
neural network control method is investigated in this paper.
The state constraints are frequently emerged in the real-life
plants and how to avoid the violation of state constraints is
an important task. By introducing a barrier Lyapunov func-
tion (BLF) to every step in a backstepping procedure, a novel
adaptive backstepping design is well developed to ensure that
the full-state constraints are not violated. At the same time,
one remarkable feature is that the minimal learning parameters
are employed in BLF backstepping design. By making use of
Lyapunov analysis, we can prove that all the signals in the
closed-loop system are semiglobal uniformly ultimately bounded
and the output is well driven to follow the desired output.
Finally, a simulation is given to verify the effectiveness of the
method.
Index Terms—Adaptive neural control, barrier Lyapunov
function (BLF), full-state constraints, neural networks (NNs),
nonlinear systems.
I. INTRODUCTION
B
ECAUSE the adaptive design technique is a powerful tool
for coping with uncertain systems, much improvement
has been advanced in recent two decades. The pioneer
approaches were developed in [1] for multifarious uncertain
nonlinear systems, e.g., the matching condition systems,
the strict-feedback systems, and the pure-feedback systems.
The different controllers have been framed using various
techniques, such as feedback linearization, inversion control,
backstepping design, fuzzy control, and so on [69]–[71], [83].
Whereafter, the adaptive control technique was one after
the other studied for different classes of nonlinear systems.
For example, an adaptive output feedback control was
addressed in [2] for a class of nonlinear discrete-time systems
with unknown control directions based on the discrete
Manuscript received April 26, 2015; revised November 5, 2015; accepted
November 8, 2015. Date of publication March 9, 2016; date of current version
June 15, 2016. This work was supported in part by the National Natural
Science Foundation of China under Grant 61374113, Grant 61473139, and
Grant 61572540 and in part by the Program for Liaoning Excellent Talents
in University under Grant LR2014016. (Corresponding author: Yan-Jun Liu.)
Y.-J. Liu, J. Li, and S. Tong are with the College of Science, Liaoning Uni-
versity of Technology, Jinzhou 121001, China (e-mail: liuyanjun@live.com;
15241624210@163.com; jztongsc@sohu.com).
C. L. P. Chen is with the Faculty of Science and Technology, University of
Macau, Macau 999078, China (e-mail: philipchen@umac.mo).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNNLS.2015.2508926
Nussbaum gain. In [3], an adaptive tracking control was pro-
posed for nonlinear multi-input multioutput (MIMO) systems
with nonsymmetric input constraints, and the auxiliary system
was designed to solve the effect of input constraints. We men-
tion that the input (saturation) constraints have been exten-
sively studied in the literature and some advanced approaches
have been established. For example, a novel saturated linear
feedback was constructed in [72] for periodic systems with
input constraints. A new convex hull representation of
saturation constraints was established in [73] and [74], and an
efficient truncated predictor feedback approach was proposed
in [75] to solve the elliptical spacecraft rendezvous problem
with input constraints and delay. In recent years, the adaptive
control problem of output and state constraints has been an
active area. The adaptive control methods for nonlinear sys-
tems with constant [4] and time-varying [5] output constraints
were investigated by making use of barrier Lyapunov func-
tion (BLF). The nonlinear systems with partial state constraints
have been stabilized in [6]. The adaptive control of full-state-
constrained nonlinear systems was studied in [7] and [8].
Two practical applications were proposed for some real plants
with the output constraint [9] and the state constraints [10].
However, the considered systems in [1]–[10] are required to
be in the linear parametric forms. This requirement may be rig-
orous in the industrial plants. In order to account for unknown
dynamics without the linear parametric condition, the neural
networks (NNs) and the fuzzy logic systems have been incor-
porated into an adaptive control design owing to their excellent
approximation ability [11], [12]. Based on the adaptive fuzzy
or neural control technique, the stability problem for several
classes of nonlinear systems with unknown functions has been
addressed in [13]–[18], [56], [57], [60], and [61]. For nonlinear
systems with unknown functions and input nonlinearities,
several adaptive fuzzy or neural control methods were given
in [19]–[21], [63], [64], [67], [68], and [76]. A shortcoming
is that the systems must subject to the requirement of the
matching condition. To remove the matching condition,
based on the backstepping design method [1], the fuzzy
or neural control-based adaptive technique was presented
in [22]–[24], [58], [59], and [66] for nonlinear single
input single output (SISO) systems with unknown functions.
Subsequently, the control problem of nonlinear strict-feedback
large-scale systems was also studied in [25], [26], and [65].
Recently, the control problem of discrete-time systems
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