Automatica 49 (2013) 3476–3483
Contents lists available at ScienceDirect
Automatica
journal homepage: www.elsevier.com/locate/automatica
Brief paper
Output feedback control of large-scale nonlinear time-delay systems
in lower triangular form
✩
Xianfu Zhang
a,b,1
, Lu Liu
b
, Gang Feng
b
, Chenghui Zhang
a
a
School of Control Science and Engineering, Shandong University, Jinan, 250061, PR China
b
Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong
a r t i c l e i n f o
Article history:
Received 10 April 2012
Received in revised form
25 May 2013
Accepted 6 August 2013
Available online 17 September 2013
Keywords:
Large-scale systems
Time-delay
Output feedback
Dynamic gain controllers
a b s t r a c t
This paper is concerned with the stabilization problem for a class of large-scale nonlinear time-delay
systems in lower triangular form. The uncertain nonlinearities are assumed to be bounded by continu-
ous functions of the outputs or delayed outputs multiplied by unmeasured states or delayed states. An
observer based output feedback control scheme is proposed using the dynamic gain control design ap-
proach. Based on Lyapunov stability theory, global asymptotic stability of the closed-loop control system
is proved. Contrary to many existing control designs for lower triangular nonlinear systems, the celebrated
backstepping method is not utilized here. An example is finally given to demonstrate the effectiveness of
the proposed design procedure.
© 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Large-scale systems arise in many applications such as power
systems, multi-robot systems, communication networks, trans-
portation networks, and supply chains; they can normally be
viewed as interconnections of multiple subsystems, see Liu, Zhang,
and Jiang (2007) and Ye, Huang, and Unbehaue (2006).
In the last two decades, the constructive control design meth-
ods such as backstepping, forwarding and saturation design, which
were proposed to control single-input nonlinear systems in trian-
gular form, see Krishnamurthy and Khorrami (2007) and Krstic,
Kanellakopoulos, and Kokotovic (1995), have now been widely
used to study large-scale nonlinear systems with multiple inputs,
see Chen and Li (2008), Liu et al. (2007), Mehraeen, Jagannathan,
and Crow (2011), and Ye et al. (2006). Based on the forwarding and
saturation design method, the authors in Ye et al. (2006) studied
asymptotic stabilization for a class of uncertain large-scale nonlin-
ear systems in upper triangular form. Based on the backstepping
method, for large-scale nonlinear systems in lower triangular form,
the authors in Liu et al. (2007) investigated the problem of adap-
tive output feedback stabilization and the authors in Chen and Li
✩
The material in this paper was not presented at any conference. This paper was
recommended for publication in revised form by Associate Editor Emilia Fridman
under the direction of Editor Ian R. Petersen.
E-mail addresses: zhangxianfu@sdu.edu.cn (X. Zhang), lu.liu@cityu.edu.hk
(L. Liu), megfeng@cityu.edu.hk (G. Feng), zchui@sdu.edu.cn (C. Zhang).
1
Tel.: +86 531 88395047; fax: +86 531 88395047.
(2008) and Mehraeen et al. (2011) proposed neural network-based
nonlinear adaptive controllers, respectively.
On the other hand, it is widely known that time delays are
a source of the instability or performance degradation of con-
trol systems (Hale, Lunel, & Verduyn, 1993). Therefore, many re-
searchers have investigated the control problems for large-scale
nonlinear systems with time delays, see Tong, Li, and Zhang (2011),
Ye (2011), and Yoo and Park (2009, 2012). For large-scale time-
delay systems in lower triangular form, the authors in Yoo and Park
(2009) considered the control problem based on full state feed-
back, and the authors in Tong et al. (2011) and Ye (2011) proposed
output feedback stabilization methods. The results of Tong et al.
(2011) and Ye (2011) can only be applied to a limited class of sys-
tems, where time-delayed bounding functions of interaction terms
only depend on measurable output variables. However in prac-
tice, these time-delayed bounding functions are often dependent
on state variables and it presents a significant challenge. Recently,
the authors in Yoo and Park (2012) investigated the adaptive out-
put feedback controller design for large-scale nonlinear time-delay
systems, and proved that all signals of the closed-loop control sys-
tem are semiglobally uniformly bounded.
To the best of our knowledge, up to now, no work has been
reported in literature on the global output feedback stabiliza-
tion problem for large-scale nonlinear time-delay systems, where
bounding functions of interaction terms include both unmeasured
states and unmeasured delayed states.
The dynamic gain control approach has been recently proposed
for stabilization control of single-input nonlinear systems in trian-
gular form, and it has been proved that the dynamic gain control
0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.automatica.2013.08.026
