Adaptive Stabilization of High-order Nonlinear Parameterized
Systems with Time-varying Control Coefficient
Huifang Min
1
, Shengyuan Xu
1
, Na Duan
2
1. School of Automation, Nanjing University of Science and Technology, Nanjing 210094, P. R. China
2. School of Electrical Engineering & Automation, Jiangsu Normal University, Xuzhou 221116, P. R. China
E-mail: syxu@njust.edu.cn
Abstract: This notes focuses on the state-feedback stabilization problem for a class of high-order nonlinear parameterized sys-
tems in presence of time-varying control coefficient. The conditions on parameterized nonlinear functions and time-varying
control coefficient are further relaxed by adopting Nussbaum function approach and backsetpping technique. In addition, the u-
niform ultimate boundedness of the closed-loop system is guaranteed by designing an adaptive state-feedback controller. Finally,
a simulation example is given to illustrate the effectiveness of the proposed scheme.
Key Words: High-order Nonlinear Parameterized Systems, State-Feedback Stabilization, Time-Varying Control Coefficient,
Nussbaum Function
1 Introduction
The design and analysis of nonlinear systems has under-
gone rapid and important progress during the past decades.
With many innovative approaches being proposed, the semi-
global or global control for nonlinear systems have obtained
remarkable results; e.g. [1]-[3] and the references therein.
When it comes to high-order nonlinear systems, since Ja-
cobian linearizations are neither controllable nor feedback
linearizable, the design of such systems becomes more chal-
lenging. [4] and [5] proposed a new technique of adding a
power integrator, which overcomes the above difficulty. S-
ince then, adding a power integrator approach and homoge-
nous domination approach proposed by [5]-[6] have been
proven to be effective design methods for high-order non-
linear systems in [7]-[12].
On the other hand, due to nonlinear parametrization exist-
s in various practical control problems such as biochemical
processes and machines with frictions, much attention has
been paid on these kinds of systems. Based on adding a
power order technique, recent years have witnessed fruitful
results of high-order nonlinear parameterized systems [13]-
[15]. In [13]-[14], the authors showed that the nonlinear
parameters can be separated from the nonlinear functions,
which resulted in the linear parameterization conditions be-
ing removed. Then, the latest result in [15] relaxed powers
in the growth rate to take values continuously on an inter-
val. However, [15] still needed somewhat restrictive nonlin-
ear growth conditions from theoretical and practical points
of view. In addition, [13]-[15] didn’t consider time-varying
control coefficient, which may present the motion of system-
s under any control and is referred to as the high-frequency
gain.
To the authors’ knowledge, Numbaum-type gains tech-
nique proposed in [16]-[17] is an effective design tool to
handle unknown control coefficient and has been frequently
used for nonlinear systems. Therefore, interesting problems
are put forward spontaneously:
Can we further weaken the conditions of nonlinear func-
tions? Under the weaker conditions, can an adaptive state-
This work is supported by National Natural Science Foundation of Chi-
na (Nos. 61374087, 61573172).
feedback controller be obtained for high-order nonlinear pa-
rameterized systems with time-varying control coefficient by
using Nussbaum function?
This paper will concentrate on addressing the above prob-
lems. The main contributions are listed as follows:
(i). The growth conditions on nonlinear parameterized
functions are further relaxed in this paper. In addition, the
time-varying control coefficient is considered for high-order
nonlinear parameterized systems under much weaker condi-
tions by introducing Nussbaum function.
(ii). An adaptive state-feedback controller is successful-
ly designed to guarantee the closed-loop system to be uni-
formly ultimately bounded (UUB). An example is given to
illustrate the effectiveness of the designed scheme.
The remainder of this paper is organized as follows. Sec-
tion 2 begins with mathematical preliminaries. The design
and analysis of the adaptive state-feedback controller are
given in Section 3 and Section 4, respectively. In Section
5, a simulation example is given. Section 6 concludes the
paper.
2 Mathematical Preliminaries
Notations: R
+
denotes the set of all non-negative real
numbers, R
n
is the n-dimensional Euclidean space, C
2
de-
notes the family of all the functions with continuous sec-
ond partial derivations. For simplicity, we sometimes denote
X(t) by X for any variable X(t).
We first introduce Nussbaum function approach to handle
the unknown time-varying control coefficient g(t).
Definition 1[16]: A function N(ζ) is said to be a
Nussbaum-type function if the following properties hold:
lim sup
s→∞
s
s
0
N(ζ)dζ =+∞,
lim inf
s→∞
s
s
0
N(ζ)dζ = −∞.
In terms of Definition 1, it is easy to prove that for a s-
mooth Nussbaum function N(ζ), N
q
(ζ) is also a smooth
Nussbaum function with q>1 being an odd number.
Lemma 1[13]: For any real-valued continuous function
f(x, y), where x ∈ R
m
, y ∈ R
n
, there are smooth scalar
Proceedings of the 35th Chinese Control Conference
Jul
y
27-29, 2016, Chen
g
du, China
414
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