aaa

Journal of the Franklin Institute 346 (2009) 958–968
Short communication
Robust exponential stability for uncertain
time-varying delay systems with delay dependence
Pin-Lin Liu
Department of Electrical Engineering, Chienkuo Technology University, Changhua 500, Taiwan, ROC
Received 22 October 2008; received in revised form 26 March 2009; accepted 9 April 2009
Abstract
This paper investigates the exponential stability problem for uncertain time-varying delay systems.
Based on the Lyapunov–Krasovskii functional method, delay-dependent stability criteria have been
derived in terms of a matrix inequality (LMI) which can be easily solved using efficient convex
optimization algorithms. These results are shown to be less conservative than those reported in the
literature. Four numerical examples are proposed to illustrate the effectiveness of our results.
r2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Keywords: Convex optimization algorithm; Time-varying delay; Exponential stability
1. Introduction
Time-delay is encountered in various engineering systems, such as the turbojet engine,
microware oscillator, nuclear reactor, rolling mill, chemical process, long transmission
lines and in pneumatic, hydraulic systems. The stability criteria for linear system with
delayed state can be classified in two classes function on their dependence on the size of
delay: delay-independent or delay-dependent [12]. The subject of uncertain time delay
systems, thus, has received a considerable amount of interest from researchers
[2–4,6–9,11,13,15,16]. Recently, the linear matrix inequality (LMI)-based approaches have
been employed to tackle stability and stability problems [1–11,13–16]. An LMI approach
has two advantages. First, it needs no turning of parameters and/or matrix. Second, it can
be efficiently solved numerically by using interior-point algorithm [1]. Such algorithms
ARTICLE IN PRESS
www.elsevier.com/locate/jfranklin
0016-0032/$32.00 r2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.jfranklin.2009.04.005
Tel.: [phone]; fax: [phone].
E-mail address: [email protected]

have recently been developed for solving optimization problems involving LMIs
[1,8,11,15,16]. In order to improve the delay bound. Free weighting matrix was introduced
by Wu et al. [15] for the time delay system. Recently, there are a number of interesting new
ideas on the Lyapunov–Krasovskii methods with improved results on delay-dependent
stability [1–4,6,8–11,13,15,16].
The aim of this paper is to provide a further contribution to the LMI technique of delay-
dependent stability conditions for a class of systems with time-varying delays and
uncertainties. Based on the Lyapunov–Krasovskii functional combining with LMI, delay-
dependent stability criteria are derived. The LMI optimization approaches are used to
obtain a sufficient condition that is very easy to be checked by utilizing the LMI Toolbox
in Matlab. From the illustrated examples, if the delay time lengthens, the decay rate
becomes conservative. We claim that the sharpness of the upper bound delay time hvaries
with the chosen decay rate a. Finally, numerical examples are given to demonstrate the
advantages of the proposed approach.
2. Stability analysis
Consider a nominal time delay system given by
_
xðtÞ¼AxðtÞþBxðthðtÞÞ t40
xðtÞ¼fðtÞ;t2½h;0
((1)
where x(t)AR
n
is the state vector of the system; A,BAR
nn
are constant matrices. f(t)is
the initial condition function that is continuously differentiable on tA[h,0]. Many papers
provide delay-dependent criteria to evaluate the allowable delay magnitude for the
asymptotic stability of time delay systems (1). The time-delay, h(t), is a time-varying
continuous function that satisfies
0hðtÞh;j_ hðtÞj  hdo1 (2)
where hand h
d
denote the upper bound on the time-varying delay and its derivative,
respectively.
When the system contains uncertainty, it can be described by the following linear
differential-difference equation as follows:
_
xðtÞ¼ðAþDAðtÞÞxðtÞþðBþDBðtÞÞxðthðtÞÞ;t40 (3)
The uncertainties are assumed to be the form
½DAðtÞDBðtÞ¼DFðtÞ½EaEb(4)
where D,E
a
, and E
b
are constant matrices with appropriate dimensions, and F(t)is
an unknown, real, and possibly time-varying matrix with Lebesgue-measurable elements
satisfying
FTðtÞFðtÞI8t(5)
For the above system, the main objective is to find the range of hto guarantee stability
for the time delay system (1) or with uncertainties (3). When the time-delay is unknown,
how long time-delay can be tolerated to keep the system stable. To do this, two
fundamental lemmas are reviewed.
ARTICLE IN PRESS
P.-L. Liu / Journal of the Franklin Institute 346 (2009) 958–968 959

Lemma 1. If there exist symmetric positive-definite matrix X
33
40and arbitrary matrices
X
11
,X
12
,X
13
,X
22
and X
23
such that
X¼
X11 X12 X13
XT
12 X22 X23
XT
13 XT 23 X33
2
6
43
7
50 (6a)
Then, we obtain
Zt
thðtÞ
_
xTðsÞX33 _ xðsÞds
Zt
thðtÞ
xTðtÞxTðthðtÞÞ _ xTðsÞ
hi
X11 X12 X13
XT
12 X22 X23
XT
13 XT 23 0
2
6
6
43
7
7
5
xðtÞ
xðthðtÞÞ
_
xðsÞ
2
6
6
43
7
7
5
ds (6b)
Lemma 2. Given matrices Q ¼Q
T
,D,E,and R ¼R
T
40of appropriate dimensions,
QþDFðtÞEþETFTðtÞDTo0 (7a)
for all F satisfying F
T
(t)F(t)rH,if and only if there exists some e40such that
QþDDTþ1ETHEo0 (7b)
Consider the time delay systems (1) and (3) utilize the following transformation:
zðtÞ¼eatxðtÞ(8)
where a40 is stability degree (delay decay rate), to transform (1) and (3) into
_
zðtÞ¼ðAþaIÞzðtÞþBeahzðthðtÞÞ (9)
and
_
zðtÞ¼ðAþaIþDAðtÞÞzðtÞþðBþDBðtÞÞeahzðthðtÞÞ (10)
We now present a delay-dependent criterion for exponential stability of systems (9).
Theorem 1. Give scalars h40and h
d
40, the nominal system (9) is exponential stable with
decay rate aif there exist symmetric positive-definite matrices P40, Q40, R40, and a semi-
positive definite matrix
X¼
X11 X12 X13
XT
12 X22 X23
XT
13 XT 23 X33
2
6
43
7
50
such that the following LMIs hold:
C¼
C11 C12 C13
CT
12 C22 C23
CT
13 CT 23 C33
2
6
43
7
5o0 (11a)
and
RX33 0 (11b)
ARTICLE IN PRESS
P.-L. Liu / Journal of the Franklin Institute 346 (2009) 958–968960

where
C11 ¼ðAþaIÞTPþPðAþaIÞþQþX13 þXT
13 þhX 11
C12 ¼eahPB X13 þXT
23 þhX12
C13 ¼hðAþaIÞTR
C22 ¼ð1hdÞQX23 XT
23 þhX 22
C23 ¼heahBTR
C33 ¼hR
Then, the time delay system (9) is exponential stable within allowable time-delay h.
Proof. Consider the time-delay system (9), using the Lyapunov–Krasovskii functional
candidate in the following form, we can write
VðztÞ¼zTðtÞPzðtÞþZt
thðtÞ
zTðsÞQzðsÞds þZ0
hZt tþy
_
zTðsÞR_
zðsÞds dy&(12)
The time derivative of Eq. (12) along the trajectory of Eq. (9) is given by
_
VðztÞ¼zTðtÞððAþaIÞTPþPðAþaIÞÞzðtÞþzTðtÞPBeahzðthðtÞÞ
þzTðthðtÞÞBTeahPzðtÞþzTðtÞQzðtÞzTðthðtÞÞð1_
hðtÞÞQzðthðtÞÞ
þ_ zTðtÞhR_
zðtÞZt
th
_
zTðsÞR_
zðsÞds
zTðtÞððAþaIÞTPþPðAþaIþQÞzðtÞþzTðtÞPBeahzðthðtÞÞ
þzTðthðtÞÞBTeahPzðtÞzTðthðtÞÞð1hdÞQzðthðtÞÞ þ _
zTðtÞhR_ zðtÞ
Zt
thðtÞ
_
zTðsÞðRX33Þ_
zðsÞds Zt
thðtÞ
_
zTðsÞX33 _ zðsÞds (13)
Applying Lemma 1, if there exist symmetric positive-definite matrix X
33
40 and
arbitrary matrices X
11
,X
12
,X
13
,X
22
,andX
23
such that
X¼
X11 X12 X13
XT
12 X22 X23
XT
13 XT 23 X33
2
6
43
7
50 (14)
Then, we obtain
Zt
thðtÞ
_
zTðsÞX33 _ zðsÞds
Zt
thðtÞ½zTðtÞzTðthðtÞÞ _
zTðsÞ
X11 X12 X13
XT
12 X22 X23
XT
13 XT 23 0
2
6
6
43
7
7
5
zðtÞ
zðthðtÞÞ
_
zðsÞ
2
6
6
43
7
7
5
ds
zTðtÞhX11zðtÞþzTðtÞhX 12xðthðtÞÞ þ zTðthðtÞÞhX T
12zðtÞ
ARTICLE IN PRESS
P.-L. Liu / Journal of the Franklin Institute 346 (2009) 958–968 961

þzTðtÞX13 Zt
thðtÞ
_
zðsÞds þxTðthðtÞÞhXT
12zðtÞþzTððthðtÞÞhX 22zðthðtÞÞ
þzTðthðtÞÞX23 Zt
thðtÞ
_
zðsÞds þZt
thðtÞ
_
zTðsÞdsX T
13zðtÞ
þZt
thðtÞ
_
zTðsÞdsX T
23zðthðtÞÞ
¼zTðtÞhX 11zðtÞþzTðtÞhX 12zðthðtÞÞþzTðtÞX13½zðtÞzðthðtÞÞ
þzTðthðtÞÞhX T
12zðtÞþzTðthðtÞÞhX 22zðthðtÞÞ þ zTðthðtÞÞX23½zðtÞzðthðtÞÞ
þ½zðtÞzðthðtÞÞTXT
13zðtÞþ½zðtÞzðthðtÞÞTXT 23zðthðtÞÞ
¼zTðtÞhX 11zðtÞþzTðtÞhX 12zðthðtÞÞþzTðtÞX13zðtÞzTðthðtÞÞXT
13zðtÞ
þzTðthðtÞÞhX T
12zðtÞþzTðthðtÞÞhX 22zðthðtÞÞ þ zTðthðtÞÞX23zðtÞ
zTðthðtÞÞX23zðthðtÞÞþzTðtÞXT
13zðtÞzTðtÞXT 13zðthðtÞÞ þ zTðtÞXT 23zðthðtÞÞ
zTðthðtÞÞXT
23zðthðtÞÞ
¼zTðtÞ½hX11 þXT
13 þX13zðtÞþzTðtÞ½hX 12 X13 þXT 23zðthðtÞÞ
þzTðthðtÞÞ½hX T
12 XT 13 þX23zðtÞþzTðthÞ½hX 22 X23 XT 23zðthðtÞÞ (15)
Substituting Eqs. (14) and (15) into Eq. (13), we obtain
_
VðztÞoxTðtÞXxðtÞZt
thðtÞ
_
zTðsÞðRX33Þ_
zðsÞds (16)
where
xTðtÞ¼ zTðtÞzTðthðtÞÞ
hi
and X¼X11 X12
XT
12 X22
"#
with
X11 ¼ðAþaIÞTPþPðAþaIÞþQþX13 þXT
13 þhX11
þhðAþaIÞTRðAþaIÞ
X12 ¼PBeahX13 þXT
23 þhX12 þheahðAþaIÞTRB
X22 ¼ð1hdÞQX23 XT
23 þhX22 þhe2ahBTRB
Finally, using the Schur complements, with some effort we can show that Eq. (16)
guarantees of _ VðxtÞo0ifXo0. In condition (11a) and (11b) of the present Theorem 1 is
satisfied, if _ VðxtÞo0, then Oo0 and RX
33
Z0, if and only if Eq. (11a) and (11b) holds.
Therefore, the time delay system (9) is exponential stable with decay rate a.
Now, extending Theorem 1 to system (10) with time-varying structured uncertainties
yields the following theorem.
Theorem 2. Give scalars h40and h
d
40, the uncertain system (10) is exponential stable
with decay rate aif there exist symmetric positive-definite matrices P40, Q40, R40, e40,
ARTICLE IN PRESS
P.-L. Liu / Journal of the Franklin Institute 346 (2009) 958–968962