Article
Transactions of the Institute of
Measurement and Control
2014, Vol. 36(4) 478–486
Ó The Author(s) 2013
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DOI: 10.1177/0142331213507076
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Global stabilization via homogeneous
output feedback for a class of uncertain
nonlinear systems subject to time
delays
Lin Chai
1,2
and Chunjiang Qian
3
Abstract
This paper addresses the problem of global output feedback stabilization for a class of uncertain nonlinear systems subject to time delay. Using the
homogeneous domination approach, we first construct a homogeneous output feedback controller with an adjustable scaling gain. With the aid of a
homogeneous Lyapunov–Krasovskii functional, the scaling gain is adjusted to dominate the time-delay nonlinearities bounded by homogeneous growth
conditions and render the closed-loop system globally asymptotically stable. As a special case, a linear output feedback controller with a tunable scaling
gain is constructed to globally stabilize time-delay systems under a linear growth condition. In addition, we also show the proposed approach is applica-
ble to the time-delay systems under lower-order growth conditions and non-triangular growth conditions.
Keywords
Output feedback control, global stabilization, homogeneous domination approach
Introduction
In this paper, we consider the problem of global output feed-
back stabilization for a class of uncertain time-delay systems
described by
_
x
1
ðtÞ= x
2
ðtÞ+ f
1
ðxðtÞ; xðt tÞÞ
.
.
.
_
x
n1
ðtÞ= x
n
ðtÞ+ f
n1
ðxðtÞ; xðt tÞÞ
_
x
n
ðtÞ= uðtÞ+ f
n
ðxðtÞ; xðt tÞÞ
yðtÞ= x
1
ðtÞ
xðtÞ= CðtÞ; t t 0;
ð1:1Þ
where x ðtÞ= ðx
1
ðtÞ; x
2
ðtÞ; ..., x
n
ðtÞÞ
T
2 R
n
is the system state,
uðtÞ2R is the control input, yðtÞ2R is the system output,
t . 0 is a given time-delay constant, and CðtÞ is the initial
function of the system state vector. The terms
f
i
ðÞ; i = 1; ..., n 1, represent nonlinear perturbations that
are not guaranteed to be precisely known.
It is known that the problem of global output feedback
stabilization for uncertain nonlinear systems is very challen-
ging compared with the state feedback case. In the past
decade, global stabilization by the output feedback domina-
tion method has been achieved for a series of nonlinear sys-
tems. With the help of the output feedback domination
design (Qian and Lin, 2002), some interesting results have
been established under a linear growth condition (Qian and
Lin, 2002) and under a higher-order growth condition (Qian,
2005b). Recently, the homogeneous domination approach
proposed by Qian (2005b) has been used as a universal tool
to solve the problem of global output feedback stabilization
for inherently nonlinear systems (Andrieu et al., 2008; Ding
and Li, 2010; Lei and Lin, 2009; Li et al., 2009; Polendo and
Qian, 2007; Qian, 2005a, b).
However, the aforementioned results have not considered
the time-delay effect which is actually very common in state,
input and output due to the time consumed in sensing, infor-
mation transmitting and controller computing. In the case
when the nonlinearities contain time delay, some interesting
results have been achieved in the literature. Mazenc and
Iggidr (2004) and Mazenc and Bowong (2005) constructed
bounded backstepping state feedback controllers with the
help of a type of variable coordinates and strict Lyapunov
functions. Recently, Bekiaris-Liberis and Krstic (2011a, b),
for a class of nonlinear systems with time-varying input delay,
used a backstepping transformation to design a predictor-
based state feedback controller.Ye (2011) employed an
1
Key Laboratory of Measurement and Control of Complex Systems of
Engineering, Southeast University, Nanjing, Jiangsu, China
2
The School of Automation, Southeast University, Nanjing, Jiangsu, China
3
Department of Electrical and Computer Engineering, University of Texas
at San Antonio, San Antonio, TX, USA
Corresponding author:
Lin Chai, Key Laboratory of Measurement and Control of Complex
Systems of Engineering and the School of Automation, Southeast
University, Nanjing, Jiangsu 210096, China.
Email: chailin1@seu.edu.cn
at Southeast University on May 28, 2014tim.sagepub.comDownloaded from
