Delay-dependent robust stabilisation of discrete-time
systems with time-varying delay
X.G. Liu, R.R. Martin, M. Wu and M.L. Tang
Abstract: The stability of discrete systems with time-varying delay is considered. New delay-
dependent stability criteria are devised, which are dependent on the minimum and maximum
delay bounds. An initial analysis leads to a criterion depending on an inequality involving
certain matrices that can be freely chosen. By carefully choosing them to reflect the appropriate
relationship between states at differing times, a stricter criterion is thereby obtained.
Furthermore, new results for delay-dependent robust stabilisation of uncertain systems with
time-varying delay are provided on the basis of a linear matrix inequality (LMI) framework. As
the conditions obtained for the existence of admissible controllers are not expressed using strict
LMI conditions, a cone complementary linearisation procedure is used to find suitable controllers.
Finally, the results obtained, including the stability anal ysis, static output-feedback stabilisation
and dynamic output feedback stabilisation are further extended to discrete time-delay systems
having uncertain but norm-bounded parameters. Numerical examples demonstrate the validity of
the approach proposed.
1 Introduction
Time delays often appear in control systems and are often a
source of instability and oscillations in such systems.
Assessing and controlling the stability of such systems
with delay are of theoretical and practical importanc e.
Increasing attention has been paid to the problem of feed-
back stabilisation of systems with state delay. Most of the
results obtained have been derived using delay-independent
approaches (see, e.g. Li and De Souza [1]). As the time
delay is not taken into consideration using these approaches
to design controllers, the results are generally more conser-
vative than ones using a delay-dependent approach.
However, previous delay-dependent methods for systems
with time-varying delays have mainly considered the con-
tinuous systems [2 –12]. Relatively, few papers have con-
sidered the time-varying delay case for discrete-time
systems [13]. Recently, Gao et al. [14] studied delay-depen-
dent output-feedback stabilisation of discrete-time systems
with time-varying state delay. A new stability condition
was proposed, which is dependent on the delay bounds.
Their results are based on an inequality on the inner
product of two vectors proved by Moon et al. [8], which
we repeat as Lemma 1 here. Given the system state x(k),
where k is discrete time , with time-dependent delay d(k),
this ineq uality is typically used to evaluate the bounds on
a weighted cross-product between x(k) and the difference
x(k) 2 x(k 2 d(k)), needed in the analysis of the delay-
dependent stability problem. The use of this inequality
leads to conservatism in the delay-dependent stability
conditions obtained.
This paper pres ents a new approach to establ ish a stricter
delay-dependent stability criterion for time-varying delay
systems, using relations between all system states x(k),
without requiring any system model transformation. An
initial criterion is found on the basis of an inequality invol-
ving various matrices that can be freely chosen, and an
improved criterion is then found by carefully choosing
these matrices to reflect the correlation between system
states at differing delays. Moon’s inequalities are not
needed in our approach. Our new stability condition is
very simple. Going further, cone complementary linearisa-
tion algorithms as used in El Ghaoui et al. [15] are exploited
to enable us to solve the inequalities needed to provide
static and dynamic output-feedback stabilisation of such
systems. Numerical examples are given to demonstrate
that the asymptotic stability results derived in this paper
are effective and less conservative than those derived by
Gao et al. [14].
2 Problem description
Consider the following discrete-time system with a
time-varying delay in the state
xðk þ 1Þ¼AxðkÞþA
1
xðk dðkÞÞ þ BuðkÞ
yðkÞ¼CxðkÞþC
1
xðk dðkÞÞ
xðkÞ¼
f
ðkÞ for k ¼d
max
; d
max
þ 1; ...; 0
ð1Þ
where k is discrete time, x(k) [ R
n
the state vector,
y(k) [ R
m
the measured output and u(k) [ R
l
the controlled
input. A, A
1
, C and C
1
are system matrices with compatible
dimensions. d(k), appearing in both the dynamic and
measurement equations, is the state delay, as frequently
# The Institution of Engineering and Technology 2006
IEE P roceedings online no. 20050223
doi:10.1049/ip-cta:20050223
Paper first received 9th June and in revised form 9th December 2005
X.G. Liu and M.L. Tang are with the School of Mathematical Science and
Computing Technology, Central South University, Changsha, Hunan 410083,
People’s Republic of China
R.R. Martin and X.G. Liu are with the School of Computer Science, Cardiff
University, Cardiff, UK
M. Wu and X.G. Liu are with the School of Information Science and
Engineering, Central South University, Changsha, Hunan 410083, People’s
Republic of China
E-mail: liuxgliuhua@163.com
IEE P roc.-Control Theory Appl., Vol. 153, No. 6, November 2006
689
Authorized licensed use limited to: Central South University. Downloaded on December 6, 2008 at 08:32 from IEEE Xplore. Restrictions apply.
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