aaa

ARTICLE IN PRESS
Journal of the Franklin Institute 346 (2009) 958–968
www.elsevier.com/locate/jfranklin
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.
r 2009 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
Tel.: +886 7111155; fax: +886 7111129.
E-mail address: [email protected]
0016-0032/$32.00 r 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.jfranklin.2009.04.005
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P.-L. Liu / Journal of the Franklin Institute 346 (2009) 958–968
959
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 delaydependent stability conditions for a class of systems with time-varying delays and
uncertainties. Based on the Lyapunov–Krasovskii functional combining with LMI, delaydependent 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 h varies
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
(
_ ¼ AxðtÞ þ Bxðt hðtÞÞ t40
xðtÞ
xðtÞ ¼ fðtÞ;
(1)
t 2 ½h; 0
where x(t)ARn is the state vector of the system; A,BARn 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;
_
jhðtÞj
hd o1
(2)
where h and hd 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:
_ ¼ ðA þ DAðtÞÞxðtÞ þ ðB þ DBðtÞÞxðt hðtÞÞ;
xðtÞ
t40
(3)
The uncertainties are assumed to be the form
½ DAðtÞ DBðtÞ ¼ DF ðtÞ½ E a
Eb (4)
where D, Ea, and Eb are constant matrices with appropriate dimensions, and F(t) is
an unknown, real, and possibly time-varying matrix with Lebesgue-measurable elements
satisfying
F T ðtÞF ðtÞ I
8t
(5)
For the above system, the main objective is to find the range of h to 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.
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Lemma 1. If there exist symmetric positive-definite matrix X3340 and arbitrary matrices
X11, X12, X13, X22 and X23 such that
2
3
X 11 X 12 X 13
6 T
7
(6a)
X ¼ 4 X 12 X 22 X 23 5 0
X T13 X T23 X 33
Then, we obtain
Z t
_ ds
x_ T ðsÞX 33 xðsÞ
thðtÞ
Z
t
h
xT ðtÞ xT ðt hðtÞÞ
thðtÞ
2
X 11
i6
XT
x_ T ðsÞ 6
4 12
X T13
X 12
X 22
X T23
X 13
32
xðtÞ
3
76
7
X 23 76 xðt hðtÞÞ 7 ds
54
5
0
_
xðsÞ
(6b)
Lemma 2. Given matrices Q ¼ QT, D, E, and R ¼ RT40 of appropriate dimensions,
Q þ DF ðtÞE þ E T F T ðtÞDT o0
(7a)
T
for all F satisfying F (t)F(t)rH, if and only if there exists some e40 such that
Q þ DDT þ 1 E T HEo0
(7b)
Consider the time delay systems (1) and (3) utilize the following transformation:
zðtÞ ¼ eat xðtÞ
(8)
where a40 is stability degree (delay decay rate), to transform (1) and (3) into
z_ðtÞ ¼ ðA þ aIÞzðtÞ þ Beah zðt hðtÞÞ
(9)
and
z_ðtÞ ¼ ðA þ aI þ DAðtÞÞzðtÞ þ ðB þ DBðtÞÞeah zðt hðtÞÞ
(10)
We now present a delay-dependent criterion for exponential stability of systems (9).
Theorem 1. Give scalars h40 and hd40, the nominal system (9) is exponential stable with
decay rate a if there exist symmetric positive-definite matrices P40, Q40, R40, and a semipositive definite matrix
2
3
X 11 X 12 X 13
6 T
7
X ¼ 4 X 12 X 22 X 23 5 0
T
T
X 13 X 23 X 33
such that the following LMIs hold:
3
2
C11 C12 C13
7
6 T
C ¼ 4 C12 C22 C23 5o0
CT13 CT23 C33
(11a)
and
R X 33 0
(11b)
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where
C11 ¼ ðA þ aIÞT P þ PðA þ aIÞ þ Q þ X 13 þ X T13 þ hX 11
C12 ¼ eah PB X 13 þ X T23 þ hX 12
C13 ¼ hðA þ aIÞT R
C22 ¼ ð1 hd ÞQ X 23 X T23 þ hX 22
C23 ¼ heah BT R
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
Z t
Z 0Z t
zT ðsÞQzðsÞ ds þ
&
(12)
V ðzt Þ ¼ zT ðtÞPzðtÞ þ
z_T ðsÞR_zðsÞ ds dy
thðtÞ
h
tþy
The time derivative of Eq. (12) along the trajectory of Eq. (9) is given by
V_ ðzt Þ ¼ zT ðtÞððA þ aIÞT P þ PðA þ aIÞÞzðtÞ þ zT ðtÞPBeah zðt hðtÞÞ
_
þ zT ðt hðtÞÞBT eah PzðtÞ þ zT ðtÞQzðtÞ zT ðt hðtÞÞð1 hðtÞÞQzðt
hðtÞÞ
Z t
þ z_T ðtÞhR_zðtÞ z_T ðsÞR_zðsÞ ds
th
zT ðtÞððA þ aIÞT P þ PðA þ aI þ QÞzðtÞ þ zT ðtÞPBeah zðt hðtÞÞ
þ zT ðt hðtÞÞBT eah PzðtÞ zT ðt hðtÞÞð1 hd ÞQzðt hðtÞÞ þ z_T ðtÞhR_zðtÞ
Z t
Z t
z_T ðsÞðR X 33 Þ_zðsÞ ds z_T ðsÞX 33 z_ðsÞ ds
(13)
thðtÞ
thðtÞ
Applying Lemma 1, if there exist symmetric positive-definite matrix X3340 and
arbitrary matrices X11, X12, X13, X22, and X23 such that
3
2
X 11 X 12 X 13
7
6 T
(14)
X ¼ 4 X 12 X 22 X 23 5 0
T
T
X 13 X 23 X 33
Then, we obtain
Z t
z_T ðsÞX 33 z_ðsÞ ds
thðtÞ
Z
2
t
thðtÞ
½ zT ðtÞ zT ðt hðtÞÞ
X 11
X 12
6 T
X
z_T ðsÞ 6
4 12
X 22
X T13
X T23
X 13
32
zðtÞ
3
76
7
X 23 76 zðt hðtÞÞ 7 ds
54
5
0
z_ðsÞ
zT ðtÞhX 11 zðtÞ þ zT ðtÞhX 12 xðt hðtÞÞ þ zT ðt hðtÞÞhX T12 zðtÞ
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þ zT ðtÞX 13
Z
t
thðtÞ
þ zT ðt hðtÞÞX 23
Z
z_ðsÞ ds þ xT ðt hðtÞÞhX T12 zðtÞ þ zT ððt hðtÞÞhX 22 zðt hðtÞÞ
Z
Z
t
t
z_ðsÞ ds þ
thðtÞ
thðtÞ
z_T ðsÞ dsX T13 zðtÞ
t
þ
thðtÞ
z_T ðsÞ dsX T23 zðt hðtÞÞ
¼ zT ðtÞhX 11 zðtÞ þ zT ðtÞhX 12 zðt hðtÞÞ þ zT ðtÞX 13 ½zðtÞ zðt hðtÞÞ
þ zT ðt hðtÞÞhX T12 zðtÞ þ zT ðt hðtÞÞhX 22 zðt hðtÞÞ þ zT ðt hðtÞÞX 23 ½zðtÞ zðt hðtÞÞ
þ ½zðtÞ zðt hðtÞÞT X T13 zðtÞ þ ½zðtÞ zðt hðtÞÞT X T23 zðt hðtÞÞ
¼ zT ðtÞhX 11 zðtÞ þ zT ðtÞhX 12 zðt hðtÞÞ þ zT ðtÞX 13 zðtÞ zT ðt hðtÞÞX T13 zðtÞ
þ zT ðt hðtÞÞhX T12 zðtÞ þ zT ðt hðtÞÞhX 22 zðt hðtÞÞ þ zT ðt hðtÞÞX 23 zðtÞ
zT ðt hðtÞÞX 23 zðt hðtÞÞ þ zT ðtÞX T13 zðtÞ zT ðtÞX T13 zðt hðtÞÞ þ zT ðtÞX T23 zðt hðtÞÞ
zT ðt hðtÞÞX T23 zðt hðtÞÞ
¼ zT ðtÞ½hX 11 þ X T13 þ X 13 zðtÞ þ zT ðtÞ½hX 12 X 13 þ X T23 zðt hðtÞÞ
þ zT ðt hðtÞÞ½hX T12 X T13 þ X 23 zðtÞ þ zT ðt hÞ½hX 22 X 23 X T23 zðt hðtÞÞ
(15)
Substituting Eqs. (14) and (15) into Eq. (13), we obtain
Z t
V_ ðzt ÞoxT ðtÞXxðtÞ z_T ðsÞðR X 33 Þ_zðsÞ ds
(16)
thðtÞ
where
T
h
T
T
x ðtÞ ¼ z ðtÞ z ðt hðtÞÞ
i
"
and
X¼
X11
XT12
X12
X22
#
with
X11 ¼ ðA þ aIÞT P þ PðA þ aIÞ þ Q þ X 13 þ X T13 þ hX 11
þ hðA þ aIÞT RðA þ aIÞ
X12 ¼ PBeah X 13 þ X T23 þ hX 12 þ heah ðA þ aIÞT RB
X22 ¼ ð1 hd ÞQ X 23 X T23 þ hX 22 þ he2ah BT RB
Finally, using the Schur complements, with some effort we can show that Eq. (16)
guarantees of V_ ðxt Þo0 if Xo0. In condition (11a) and (11b) of the present Theorem 1 is
satisfied, if V_ ðxt Þo0, then Oo0 and RX33Z0, 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 h40 and hd40, the uncertain system (10) is exponential stable
with decay rate a if there exist symmetric positive-definite matrices P40, Q40, R40, e40,
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and a semi-positive definite matrix
2
3
X 11 X 12 X 13
6 T
7
X ¼ 4 X 12 X 22 X 23 5 0
X T13 X T23 X 33
such that the following LMIs
2
C11 þ E Ta E a
6 CT þ eah E T E
6
b a
C̄ ¼ 6 12
4 CT13
DT P
are true:
3
C12 þ eah E Ta E b
C13
PD
C22 þ e2ah E Tb E b
CT23
C23
C33
7
0
7
7o0
hRD 5
0
hDT R
I
(17a)
and
R X 33 0
(17b)
where C11, C12, C13, C22, C23, and C33 are defined in Eq. (11a) and (11b).
Then, the time delay uncertain system (10) is exponential stable with decay rate a and
allowable time-delay h. Then the time delay system (10) is exponential stable, that is, the
uncertain parts of the nominal system can be tolerated.
Proof. Replacing A+aI and Beah in Eq. (9) with A+aI+DF(t)Ea and (B+DF(t)Eb)eah,
respectively, we find Eq. (11a) and (11b) for system (10) is equivalent to the following
condition:
3
2 T 3
2
PD
Ea
h
i
7
6 ah T 7 T T
6
ah
E
e
E
0
F
ðtÞ
O þ 40
(18)
þ
e
5
4 E b 5F ðtÞ D P 0 hDT R o0
a
b
hRD
0
By Lemma 2, a sufficient condition guaranteeing (11a) and (11b) for uncertainties (4) is
that there exists a positive number e40 such that
2 T 3
2
3
PD
Ea
h
i
6 ah T 7 E eah E 0
7 T
1 6
T
o0
(19)
O þ 40
5 D P 0 hD R þ 4 e E b 5 a
b
hRD
0
Applying the Schur complement shows that Eq. (19) is equivalent to Eq. (17a) and
(17b)1. This completes the proof. &
Remark 1. As in the stability problem, the upper bound h which ensure that timedelays (9) (or (10)) is robust stabilizable for any h can be determined by solving
the following quasi-convex optimization problem when the other bound of time-delay h
is known:
(
Maximize
h
(20)
Subject to ð11Þ ðor ð17ÞÞ P40; Q40; R40 and X 0 ðor 40Þ
Inequality (20) is a quasi-convex optimization problem and can be obtained efficiently
using MATLAB LMI Toolbox.
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3. Illustrative examples
Example 1. Consider the uncertain time-varying delay systems as follows:
_ ¼ ðA þ DAðtÞÞxðtÞ þ ðB þ DBðtÞÞxðt hðtÞÞ
xðtÞ
(21)
where
"
1:2
0:1
0:1
1
"
#
A¼
; B¼
0:6
0:7
1
0:8
#
D ¼ I; E a ¼ diagfb; bg; E b ¼ diagfg; gg
Now, our problem is to estimate the bound of delay time h to keep the stability of
system. Concurrently, the uncertain parts of the nominal system can be tolerated for with
allowable time-delay h.
Solution: Let b ¼ g ¼ 0.1 and a ¼ hd ¼ 0.9, by using the LMI Toolbox in MATLAB
(with accuracy 0.01), then, the solutions of the LMI given in Eq. (21) are found to be
"
#
"
#
0:0156
0:0025
0:5038
0:0513
P¼
; Q¼
,
0:0025 0:0195
0:0513 0:5957
"
#
26:5419 0:6711
R¼
0:6711 27:7434
"
#
"
#
1:0823
0:0422
0:9926 0:1420
X 11 ¼
; X 12 ¼
,
0:0422 1:1329
0:2048
1:0447
"
#
"
#
3:9772 0:3798
1:0124
0:0367
; X 22 ¼
,
X 13 ¼
0:6769
4:1906
0:0367 1:0856
"
#
"
#
3:9349 0:3742
21:0164 0:5507
X 23 ¼
; X 33 ¼
0:1383 4:2504
0:5507 22:2364
e ¼ 10.1710 and ho5.3921. Applying the criteria in [2,5,9], the maximum value of h
for the stability of system under consideration is listed in Table 1. It is easy to see that
the stability criterion in this paper gives a much less conservative result than those
in [2,5,9].
Example 2. Consider the uncertain time-varying delay systems are described as follows:
_ ¼ ðA þ DAðtÞÞxðtÞ þ ðB þ DBðtÞÞxðt hðtÞÞ
xðtÞ
(22)
where
"
A¼
2
0
0
1
#
"
1
0
1
1
; B¼
#
E a ¼ diagf1:6; 0:05g; E b ¼ diagf0:1; 0:3g; D ¼ I
Now, our problem is to estimate the bound of delay time h to keep the stability of
system.
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Table 1
Comparison of upper bounds of delays for example 2(a ¼ 0).
Uncertainties
b ¼ 0, g ¼ 0.1
b ¼ 0.1, g ¼ 0.1
hd
hd ¼ 0
hd ¼ 0.5
hd ¼ 2
hd ¼ 0
hd ¼ 0.5
hd ¼ 2
Cao and Lem [2]
Han [5]
Li and Guan [9]
Theorem 2
0.6811
1.3279
1.7423
22.1218
0.5467
0.6743
1.1424
17.4573
–
–
0.7315
4.9747
0.6129
1.2503
1.8753
18.7519
0.4950
1.0097
1.0097
17.1050
–
–
0.7147
4.6585
Solution: Let a ¼ hd ¼ 0.9, by using the LMI Toolbox in MATLAB (with accuracy
0.01), then, the solutions of the LMI given in Eq. (22) are found to be
"
#
"
#
"
#
0:1974 0
0:1974 0
8:2497 0
P¼
; Q¼
; R¼
0
0:3182
0
0:3182
0
7:3389
"
#
"
#
0:2433 0
0:2543 0
; X 12 ¼
,
X 11 ¼
0
0:2309
0
0:2257
"
#
"
#
1:1667 0
0:2899 0
X 13 ¼
; X 22 ¼
,
0
0:9956
0
0:2524
"
#
"
#
1:1692 0
6:7726 0
; X 33 ¼
X 23 ¼
0
0:9945
0
5:7996
e ¼ 0.1965 and ho5.7699.
For comparison, Table 2 also lists the upper bounds from the criteria in [3,4,6,8,
11,13,15,16]. It is clear that Theorem 2 gives much better results than those obtained
[3,4,6,8,11,13,15,16].
Example 3. Consider the uncertain time-varying delay systems are described as follows:
_ ¼ ðA þ DAðtÞÞxðtÞ þ ðB þ DBðtÞÞxðt hðtÞÞ
xðtÞ
(23)
where
A¼
0:5
1
0:5
;B ¼
0
1
2
1
0:6
; E a ¼ E b ¼ diagf0:2; 0:2g; D ¼ I
Now, our problem is to estimate the bound of delay time h to keep the stability of
system.
Solution: Let a ¼ 0.1, hd ¼ 0.9, by using the LMI Toolbox in MATLAB (with accuracy
0.01), then, the solutions of the LMI given in Eq. (23) are found to be
"
#
"
#
0:8259
0:0988
0:0388
0:0017
P¼
; Q¼
,
0:0988 0:2709
0:0017 0:0125
"
#
11:9071 0:0713
R¼
0:0713 7:5941
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Table 2
Comparison between the result in this paper and previous results (a ¼ 0).
hd
Method
Li and de Souza [8]
Kim [6]
Yue and Won [16]
Moon et al. [11]
Fridman and Shaked [4]
Wu et al. [15])
Parlakic [13]
Chen et al. [3]
Theorem 2
"
X 11 ¼
"
X 13 ¼
"
X 23 ¼
1:6575
0:0324
0:0324
1:6892
1:4226
0:2264
1:2398
0:0896
0
0.5
0.9
0.2013
0.2412
0.2412
0.7059
1.1490
1.1490
1.1623
6.6889 101
6.8051 101
–
o0.2
0.2195
–
0.9247
0.9247
0.9264
2.2359
7.8056
–
o0.1
0.1561
–
0.6710
0.6954
0.6954
1.2017
7.2699
#
"
; X 12 ¼
0:0515
#
; X 22
0:9871
#
0:0097
; X 33 ¼
1:2280
1:6170
0:0141
#
0:0814 1:6695
"
#
1:6409 0:0590
¼
0:0590 1:6822
"
#
6:9053
0:3626
0:3626 5:0731
e ¼ 1.1983 and ho0.6549. As shown in Fig. 1, the simulation of system (23) for h ¼ 0.65. As the
diagram indicates, system (23) would be asymptotically stable if the delay time h is less than 0.65.
For comparison, Table 3 also lists the upper bounds obtained from the criteria
[4,6,8,10,13,15].
It is clear that Theorem 2 gives much better results than those obtained by
[4,6,8,10,13,15]. It is illustrated that the proposed robust stability criteria are effective in
comparison to earlier and newly published results existing in the literature.
Example 4. Consider the uncertain time-varying delay systems [8,14]
_ ¼ ðA þ DAðtÞÞxðtÞ þ ðB þ DBðtÞÞxðt hðtÞÞ
xðtÞ
(24)
where
A¼
2 0
0
2:5
; B¼
1
1
" pffiffiffiffiffiffiffi
#
0:5 0
pffiffiffiffiffiffiffi
; Ea ¼ Eb ¼ D ¼
1
0
0:6
0
Solution: Let a ¼ hd ¼ 0, by using the LMI Toolbox in MATLAB (with accuracy 0.01),
then, the solutions of the LMI given in Eq. (24) are found to be
"
#
"
#
0:0928
0:0011
0:0447
0:0060
; Q¼
P¼
0:0011 0:0963
0:0060 0:0425
"
#
1:6958
0:0096
R¼
0:0096 1:8014
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Time delay 0.65sec
1
0.8
0.6
Output x1,x2
0.4
x2
0.2
0
-0.2
-0.4
x1
-0.6
-0.8
-1
0
1
2
3
4
5
6
7
8
9
10
Time (sec)
Fig. 1. The simulation of Example 3 for h ¼ 0.65 s.
Table 3
Comparison of the maximal allowable delays h for different approaches (a ¼ 0).
hd
Method
Li and Souza [8]
Kim [6]
Moon et al. [10]
Fridman and Shaked [4]
Wu et al. [15]
Parlakic [13]
Theorem 2
"
X 11 ¼
"
X 13 ¼
X 23
0:0123
0:0002
0:0002
0:0132
0:0972
0:0015
0
0.5
0.9
0.3010
0.3513
0.5799
0.6812
0.8435
1.8542
4.3236
–
0.2587
–
0.1820
0.2433
0.3067
3.8866
–
0.0825
–
–
0.2420
0.2512
2.2715
#
"
; X 12 ¼
#
"
0:0109 0:0000
0:0006
0:0119
0:0139
0:0002
#
#
; X 22 ¼
0:1013
0:0002 0:0150
"
#
"
#
0:0972 0:0018
1:2685 0:0088
; X 33 ¼
¼
0:0032 0:1011
0:0088 1:3206
0:0000
e ¼ 0.0286 and ho13.0029. The delay bound for guaranteeing asymptotic stability of the
system given in [8] is 0rhr0.1035 and [14] is 0rhr11.0136, which is less conservative
than the results in the literature [8,14].
ARTICLE IN PRESS
968
P.-L. Liu / Journal of the Franklin Institute 346 (2009) 958–968
4. Conclusion
The objective of this paper is to investigate the robust exponential stability condition
that provide better insights into the effects of delay terms on the system behavior, and to
use these conditions and insights in control problems. Based on the Lyapunov–Krasovskii
functional combined with LMI technique, simple and improved delay-dependent
exponential stability criteria have been derived. By comparing our results with others
through numerical examples, it has been shown that the derived criterion is less
conservative than those in the literature.
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