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Improved delay-dependent robust stability criteria for uncertain time delay systems
Applied Mathematics and Computation 218 (2011) 2880–2888
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Improved delay-dependent robust stability criteria for uncertain time delay systems Cheng Wang a,b,⇑, Yi Shen a a Department of Control Science and Engineering and the Key Laboratory of Ministry of Education for Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China b College of Mathematics and Computer Science, Huanggang Normal University, Huanggang, Hubei 438000, PR China
a r t i c l e
i n f o
Keywords: Delay-dependent Robust stability Linear matrix inequality (LMI) Uncertain time-delay systems
a b s t r a c t This paper deals with the robust stability analysis for uncertain systems with time-varying delay. New delay-dependent robust stability criteria of uncertain time-delay systems are proposed by exploiting appropriate Lyapunov functional candidate. These developed results have advantages over some previous ones in that they have fewer matrix variables yet less conservatism, due to the introduction of a method to estimate the upper bound of the derivative of Lyapunov functional candidate without ignoring the additional useful terms. Numerical examples are given to demonstrate the effectiveness and the advantage of the proposed method. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Time delays frequently appear in many practical systems such as biological systems, chemical systems, electronic systems and network control systems. The time delays are regarded as the major source of instability and poor performance. Therefore, a great amount of effort has been devoted to the stability analysis of the time-delay systems [1–24]. Depending on whether the existence of time delays or not, stability criteria for time-delay systems can be divided into two types: delay-dependent ones and delay-independent ones. While the delay-independent criteria guarantee the stability of systems irrespective to the size of time delays, delay-dependent criteria can give the maximum delay bounds for making the system stable. Thus, delay-dependent criteria, which make use of information on the length of delay, tend to be less conservative than delay-independent ones [16,17], especially when the time delays are small. Recently, considerable attentions have been focused on delay-dependent stability analysis of time-delay systems. As for systems with time-varying delay, many methods have been taken to deal with delay-dependent stability problems. Fixed model transformation was the main method employed in [2], and inequality methods were used to estimate the upper bound of cross product terms in the derivative of the Lyapunov functional in [4,25]. In order to further improve the performance of delay-dependent stability criteria, free-weighting method was proposed in [6,7,27], where neither system transformation nor bounding techniques on some cross terms was involved, thus avoiding the conservatism induced by model transformation. Note that the free weighting matrix method makes stability criteria complicated, therefore the criteria still leave some room for improvement in accuracy as well as complexity due to the method used. On the other hand, much effort has been devoted to improve the delay-dependent conditions based on the new Lyapunov functionals in [4,5,23]. New classes of Lyapunov functionals and augmented Lyapunov functionals were introduced in [11] to ⇑ Corresponding author. at: Department of Control Science and Engineering and the Key Laboratory of Ministry of Education for Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China. E-mail address: [email protected] (C. Wang). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.031
C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888
2881
study the delay-dependent stability for systems with time-varying delay. Therefore, the results were less conservative than R0 Rt _ some existing ones, but there is room for further investigation. For example, in [2,5,6] the derivative of h tþh x_ T ðsÞZ xðsÞdsdh Rt R tdðtÞ T T T _ _ _ was often estimated as hx_ ðtÞZ xðtÞ tdðtÞ x_ ðsÞZ xðsÞds, and the term th x_ ðsÞZ xðsÞds was ignored, which may lead to considerable conservativeness. Although [11,30] retained these terms and proposed an improved delay-dependent stability criterion for systems with time-varying delay, the stability condition still leaves some room for improvement. For instance, in [30], both terms d(t) and h d(t) were enlarged as h. It is observed that d(t) and h d(t) have an important relationship that there sum is h. So, the above may lead to conservativeness. Thus, a source of conservatism that could be further reduced lies in some other useful terms in the calculation of the upper bound of the derivative of Lyapunov functional, which motivates the present study. In this paper, our purpose is to present some new delay-dependent stability criteria for uncertain systems with timevarying delay. By constructing appropriate Lyapunov–Krasovskii functional and employing an improved inequality, delay-dependent stability criteria are obtained in the linear matrix equality (LMI) format. The main contribution of this paper is that a tighter upper bound of the differential of Lyapunov–Krasovskii functional is obtained by an improved approximation method, without resorting to any model transformation and free weighting matrix technique. The resulting criteria have advantages over some previous ones in that they involve few matrix variables but have less conservatism, which are established theoretically. From a mathematical point of view, they are simple, and improve over some previous ones at the same time. Numerical examples are provided to show the effectiveness and advantage over some existing results. 2. Problem statement Consider the following linear system with time-delay and parameter uncertainties described by
_ xðtÞ ¼ AðtÞxðtÞ þ A1 ðtÞxðt hðtÞÞdt; xðtÞ ¼ uðtÞ;
8t 2 ½h; 0
t>0
;
ð1Þ
where x(t) 2 Rn is the state vector, and u(t) is a continuous and differentiable vector-valued initial function of t 2 [h, 0]. A(t) and A1(t) are matrix functions described as follows
AðtÞ ¼ A þ DAðtÞ;
A1 ðtÞ ¼ A1 þ DA1 ðtÞ;
ð2Þ
where A and A1 are constant matrices of appropriate dimensions while DA(t) and DA1(t) are unknown constant matrices of appropriate dimensions representing time-varying parameter uncertainties, which are assumed to be the following form:
½DAðtÞ DA1 ðtÞ ¼ DFðtÞ½E1 E2 ;
ð3Þ
where D, Ei(i = 1, 2) are appropriately dimensioned constant matrices, and F(t) is an unknown time-varying matrix with Lebesgue measurable elements satisfying
F T ðtÞ FðtÞ 6 I:
ð4Þ
h(t), the time-varying delay of the system, is a differentiable function and is assumed to satisfy
0 6 hðtÞ 6 h;
hðtÞ 6 d:
ð5Þ
where h and d are known constants. The purpose of this paper is to formulate new delay-dependent criteria to check the robust stability of system (1). Let us give the following lemmas, which will play an indispensable role in deriving our criteria. Lemma 1 (Schur complement). Given constant matrices R1, R2 and R3 with appropriate dimensions, where RT1 ¼ R1 and RT2 ¼ R2 , then
R1 þ RT3 R1 2 R3 < 0: if and only if
"
#
R2 R3 R1 RT3 < 0 or < 0: RT3 R1 R3 R2
Lemma 2 [28]. Given appropriately dimensioned matrices w, H, G with w = wT, Then
w þ HFðtÞG þ GT F T ðtÞHT < 0; holds for all F(t) satisfying FT(t)F(t) 6 I if and only if for some e > 0,
w þ eHHT þ e1 GT G < 0:
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C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888
Lemma 3 [29]. For any symmetric positive definite matrix R > 0 and vector function x(t) : [0, h] ? Rn,such that the integrations concerned are well defined, the following inequality holds
h
Z
h
xT ðsÞRxðsÞds 6
0
Z
h
xT ðsÞds R
0
Z
h
xðsÞds:
0
3. New stability criteria In this section, our objective is to establish new and less conservative stability conditions for system (1) as follows: Theorem 1. Given scalars h > 0 and d, the system (1) with a time-varying delay h(t) subject to (5) is robustly stable if there exist matrices P = PT > 0, Q = QT > 0, Z = ZT > 0, R = RT > 0 and a scalar e > 0, such that the following LMIs hold:
2 6 6 6 6 6 6 /1 ¼ 6 6 6 6 6 6 4 2 6 6 6 6 6 6 /2 ¼ 6 6 6 6 6 6 4
T
PA þ AT P þ Q þ Z R þ eET1 E1
PA1 þ R þ eET1 E2
0
hA R
ð1 dÞQ 3R þ eET2 E2
2R
hA1 R
Z 2R
0
R
eI
T
PD
T
PA þ AT P þ Q þ Z 2R þ eET1 E1
PA1 þ 2R þ eET1 E2
0
hA R
ð1 dÞQ 3R þ eET2 E2
R
hA1 R
Z R
0
R
T
PD
3
7 7 0 7 7 7 7 0 7 7 < 0; 7 7 hRD 7 7 5
ð6Þ
3
7 7 0 7 7 7 7 0 7 7 < 0: 7 7 hRD 7 7 5
ð7Þ
eI
Proof. Define the following Lyapunov functional candidate
Vðxt Þ ¼ xT ðtÞPxðtÞ þ
Z
t
xT ðsÞQxðsÞds þ thðtÞ
Z
t
th
xT ðsÞZxðsÞds þ
Z
0
h
Z
t
_ hx_ T ðsÞRxðsÞdsdh;
ð8Þ
tþh
where P = PT > 0, Q = QT > 0, Z = ZT > 0 and R = RT > 0 are to be determined. Calculating the derivative of V(xt) along the solutions of system (1) yields T _ _ t Þ ¼ 2xT ðtÞPxðtÞ _ þ xT ðtÞZxðtÞ þ xT ðtÞQxðtÞ ð1 hðtÞÞx Vðx ðt hðtÞÞQxðt hðtÞÞ xT ðt hÞZxðt hÞ Z t 2 _ _ hx_ T ðsÞRxðsÞds þ h x_ T ðtÞRxðtÞ th T
6 2x ðtÞP½AðtÞxðtÞ þ A1 ðtÞxðt hðtÞÞ þ xT ðtÞðZ þ Q ÞxðtÞ xT ðt hÞZxðt hÞ ð1 dÞxT ðt hðtÞÞQxðt hðtÞÞ Z thðtÞ Z t 2 _ _ þ h ½AðtÞxðtÞ þ A1 ðtÞxðt hðtÞÞT R½AðtÞxðtÞ þ A1 ðtÞxðt hðtÞÞ hx_ T ðsÞRxðsÞds hx_ T ðsÞRxðsÞds: th
thðtÞ
ð9Þ We know that
R thðtÞ _ _ ðh hðtÞÞx_ T ðsÞRxðsÞds th hðtÞx_ T ðsÞRxðsÞds: Rt Rt Rt _ _ _ thðtÞ hx_ T ðsÞRxðsÞds ¼ thðtÞ hðtÞx_ T ðsÞRxðsÞds thðtÞ ðh hðtÞÞx_ T ðsÞRxðsÞds:
R thðtÞ th
_ hx_ T ðsÞRxðsÞds ¼
Set l = h(t)/h. Then
R thðtÞ th
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C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888
Z
thðtÞ
_ hðtÞx_ T ðsÞRxðsÞds ¼ l
Z
th
thðtÞ
_ hx_ T ðsÞRxðsÞds ¼ l
Z
th
¼ l
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds l th
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds l2 Z
thðtÞ
_ hðtÞx_ T ðsÞRxðsÞds
th
Z
th
l2
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds
th
thðtÞ
_ hðtÞx_ T ðsÞRxðsÞds
th
¼ l
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds l2
Z
th
ln2
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds
th thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds ln2 th
¼ l
Z
_ ðh hðtÞÞx_ T ðsÞRxðsÞds l2
Z
th
Z
Z
_ ðh hðtÞÞx_ T ðsÞRxðsÞds
th thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds ln1 _ ðh hðtÞÞx_ T ðsÞRxðsÞds l2
Z
th
Z
Z
thðtÞ
_ hx_ T ðsÞRxðsÞds
th
thðtÞ
ln2
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
th
6 l
thðtÞ
th
thðtÞ
ln2
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds
th thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds ln1 th
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds:
th
Similarly, we have
Z
t
_ ðh hðtÞÞx_ T ðsÞRxðsÞds ¼ ð1 lÞ
Z
thðtÞ
t
_ hx_ T ðsÞRxðsÞds ¼ ð1 lÞ thðtÞ
ð1 lÞ
Z
Z
t
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
t
_ ðh hðtÞÞx_ T ðsÞRxðsÞds ¼ ð1 lÞ
Z
thðtÞ
ð1 lÞ2
Z
t
t
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
_ hðtÞx_ T ðsÞRxðsÞds ð1 lÞn2
Z
thðtÞ
ð1 lÞn1
Z
Z
t
_ hx_ T ðsÞRxðsÞds 6 ð1 lÞ
t
Z
t
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
_ hðtÞx_ T ðsÞRxðsÞds ð1 lÞn2
Z
thðtÞ
ð1 lÞn1
Z
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
thðtÞ
ð1 lÞ2
t
t
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
t
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
Then, we can get
Z
thðtÞ
_ hx_ T ðsÞRxðsÞds
th
Z
t
_ hx_ T ðsÞRxðsÞds 6 ½1 þ l þ þ ln1
thðtÞ
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds
th
½1 þ ð1 lÞ þ þ ð1 lÞn1
Z
t
_ hðtÞx_ T ðsÞRxðsÞds thðtÞ
Xn1 Z ¼ i¼0 li
thðtÞ
th
_ ðh hðtÞÞx_ T ðsÞRxðsÞds
Xn1
ð1 lÞi i¼0
Z
t
_ hðtÞx_ T ðsÞRxðsÞds:
ð10Þ
thðtÞ
Using Lemma 3 and letting n = 2 in (10) result in
Z
thðtÞ
th
_ hx_ T ðsÞRxðsÞds
Z
t
_ hx_ T ðsÞRxðsÞds 6 ½xðt hðtÞÞ xðt hÞT R½xðt hðtÞÞ xðt hÞ
thðtÞ
½xðtÞ xðt hðtÞÞT R½xðtÞ xðt hðtÞÞ l½xðt hðtÞÞ xðt hÞT R½xðt hðtÞÞ xðt hÞ ð1 lÞ½xðtÞ xðt hðtÞÞT R½xðtÞ xðt hðtÞÞ: Combining (9) and (11), we can furthermore get
ð11Þ
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C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888
_ t Þ 6 xT ðtÞ½PAðtÞ þ AT ðtÞP þ Z þ Q R þ h2 AT ðtÞRAðtÞxðtÞ Vðx 2
þ2xT ðtÞ½PA1 ðtÞ þ h AT ðtÞRA1 ðtÞ þ Rxðt hðtÞÞ þxT ðt hðtÞÞ½ð1 dÞQ 2R þ AT1 ðtÞRA1 ðtÞxðt hðtÞÞ xT ðt hÞðZ þ RÞxðt hÞ þ 2xT ðt hðtÞÞRxðt hÞ l½xðt hðtÞÞ xðt hÞT R½xðt hðtÞÞ xðt hÞ ð1 lÞ½xðtÞ xðt hðtÞÞT R½xðtÞ xðt hðtÞÞ ¼ nT ðtÞ½lu1 þ ð1 lÞu2 nðtÞ; where
2
u1 ¼ 6 4 2
u2 ¼ 6 4
2
2
PAðtÞ þ AT ðtÞP þ Z þ Q R þ h AT ðtÞRAðtÞ
PA1 ðtÞ þ R þ h AT ðtÞRA1 ðtÞ
0
ð1 dÞQ 3R þ AT1 ðtÞRA1 ðtÞ
2R
Z 2R
2
2
PAðtÞ þ AT ðtÞP þ Z þ Q 2R þ h AT ðtÞRAðtÞ
PA1 ðtÞ þ 2R þ h AT ðtÞRA1 ðtÞ
0
ð1 dÞQ 3R þ AT1 ðtÞRA1 ðtÞ
R
Z R
nðtÞ ¼ ½xT ðtÞ;
xT ðt hðtÞÞ;
3 7 5 3 7 5
xT ðt hÞT :
Since P > 0, Q > 0, R > 0, Z > 0 and 0 6 l 6 1. Therefore, if u1 < 0 and u2 < 0, we have lu1 + (1 l)u2 < 0. Then, _ t Þ < ekxðtÞk2 for a sufficiently small constant e > 0, which ensures robust stability of the system. Vðx By Schur Complement Lemma, it is easily seen that u1 < 0 if and only if
2 6 6
PAðtÞ þ AT ðtÞP þ Z þ Q R
PA1 ðtÞ þ R
ð1 dÞQ 3R
R1 ¼ 6 6 4
T
0
hA ðtÞR
7 T hA1 ðtÞR 7 7 < 0; 7 5 Z 2R 0 R 2R
and u2 < 0 if and only if
2 6 6
PAðtÞ þ AT ðtÞP þ Z þ Q 2R
PA1 ðtÞ þ 2R
ð1 dÞQ 3R
R2 ¼ 6 6 4
3
T
0
hA ðtÞR
3
7 T hA1 ðtÞR 7 7 < 0: 7 5 Z R 0 R
ð12Þ
ð13Þ
R
Then by Lemma 2,
R1 ¼ X1 þ PFðtÞC þ CT F T ðtÞPT < 0 and
R2 ¼ X2 þ PFðtÞC þ CT F T ðtÞPT < 0 if and only if there exist a scalar e > 0 such that the following inequalities are satisfied
X1 þ eCT C þ e1 PPT < 0;
ð14Þ
X2 þ eCT C þ e1 PPT < 0;
ð15Þ
and
where
C ¼ ½E1 E2 0 0; 2 6 6 X1 ¼ 6 6 4
P ¼ ½DT P 0 0 hDT RT ;
PA þ AT P þ Z þ Q R
PA1 þ R
ð1 dÞQ 3R
0
T
hA R
3
T 7 hA1 R 7 7 < 0; 7 Z 2R 0 5 R
2R
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C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888
2 6 6
PA þ AT P þ Z þ Q 2R
PA1 þ 2R
ð1 dÞQ 3R
X2 ¼ 6 6 4
0
T
hA R
3
T 7 hA1 R 7 7 < 0: 7 Z R 0 5 R
R
Applying the Schur Complement shows that (14) is equivalent to (6) and (15) is equivalent to (7). This completes the proof. h
Remark 1. Theorem 1 provides a novel delay-dependent robust stability criterion for system (1) with h(t) satisfying (5) in terms of LMIs, which can be verified using recently developed standard algorithms. Theorem 1 can be applied to both slow and fast time-varying delays only if dis known. In many cases, the information of the time derivative of delay is unknown. Regarding this circums, the rate-independent criterion can be derived by choosing Q = 0. Rt _ Remark 2. In the previous work such as [6,19], the term th x_ T ðsÞZ xðsÞds in the derivative of Vðx_ t Þ was simply enlarged as Rt R thðtÞ T T _ _ thðtÞ x_ ðsÞZ xðsÞds. It is seen that the term th x_ ðsÞZ xðsÞds in Vðx_ t Þ was ignored, which may lead to considerable conR thðtÞ _ servativeness. In Theorem 1, th x_ T ðsÞRxðsÞds is taken into account when estimating the upper bound of Vðx_ t Þ. On the other hand, we have not introduced any free weighting matrix as references [6,11,21]. Theorem 1 only involves the matrix variables in the Lyapunov functional. Therefore, our method in Theorem 1 is simple and less conservative in point of mathematical view. Next, we provide another delay-dependent robust stability criterion for system (1). Theorem 2. Given scalars h > 0 and d, the system (1) with a time-varying delay h(t) subject to (5) is robustly stable if there exist matrices P = PT > 0, Q = QT > 0, Z = ZT > 0, R = RT > 0 and a scalar e > 0, such that the following LMI holds:
2 6 6 6 6 6 /¼6 6 6 6 4
T
PA þ AT P þ Q þ Z R þ eET1 E1
PA1 þ R þ eET1 E2
0
hA R
ð1 dÞQ 2R þ eET2 E2
R
hA1 R
Z R
0
R
T
PD
3
7 7 0 7 7 7 < 0: 0 7 7 7 hRD 7 5 eI
ð16Þ
Proof. Choose the same Lyapunov functional candidate as in (8) for the system(1). Different from Theorem 1, here, letting n = 1 in (10) and using Lemma 3, we can acquire
Z
thðtÞ
_ hx_ T ðsÞRxðsÞds
th
Z
t
_ hx_ T ðsÞRxðsÞds 6 ½xðt hðtÞÞ xðt hÞT R½xðt hðtÞÞ xðt hÞ ½xðtÞ xðt
thðtÞ
hðtÞÞT R½xðtÞ xðt hðtÞÞ _ t Þ 6 nT ðtÞXnðtÞ, where Similar to the proof of Theorem 1, we obtain Vðx
2 6
2
2
PAðtÞ þ AT ðtÞP þ Z þ Q R þ h AT ðtÞRAðtÞ
PA1 ðtÞ þ R þ h AT ðtÞRA1 ðtÞ
0
ð1 dÞQ 2R þ AT1 ðtÞRA1 ðtÞ
R
Z R
X¼4
3 7 5
nðtÞ ¼ ½xT ðtÞ; xT ðt hðtÞÞ; xT ðt hÞT : Using Lemmas 1–3 to (16) results in X < 0. Therefore, similar to Theorem 1 we can get the result of Theorem 2. This completes the proof. h _ t Þ. The difference between them Remark 3. Theorems 1 and 2 use the same formula (10) to estimate the upper bound of Vðx Rt T _ _ is that in Theorem 1, th hx ðsÞRxðsÞds in formula (10) is enlarged as
Rt _ x_ T ðsÞdsR thðtÞ xðsÞds R thðtÞ T R thðtÞ Rt Rt T _ _ l th x_ ðsÞdsR th xðsÞds ð1 lÞ thðtÞ x_ ðsÞdsR thðtÞ xðsÞds;
R thðtÞ th
x_ T ðsÞdsR
while in Theorem 2,
R thðtÞ th
Rt
th
_ xðsÞds
Rt
thðtÞ
_ hx_ T ðsÞRxðsÞds is enlarged as
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C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888
Z
Z
thðtÞ
x_ T ðsÞdsR th
thðtÞ
_ xðsÞds
th
Z
t
x_ T ðsÞdsR
Z
thðtÞ
t
_ xðsÞds:
thðtÞ
Both the two criteria are closer to the real value of
Rt
th
_ hx_ T ðsÞRxðsÞds than that in [2,3,5,6], and is less conservative.
Remark 4. Theorem 1 is obtained by using n = 2, while Theorem 2 is obtained by using n = 1. When n >= 3, using Lemma 3 and (10), we have
Z
_ t Þ 6 nT ðtÞXnðtÞ l Vðx
x_ T ðsÞds R
th n1
l
Z
!
thðtÞ
!
thðtÞ
x_ T ðsÞds R
th 2
ð1 lÞ
Z
_T
T
6 n ðtÞXnðtÞ l
Z
thðtÞ
thðtÞ
_T
x ðsÞds R
th
Z
Z
thðtÞ
!
t
x_ T ðsÞds R
_ xðsÞds ð1 lÞ
thðtÞ
!
x_ T ðsÞds R
Z
n1
_ xðsÞds ð1 lÞ
th
Z
!
_ xðsÞds
t
!
_T
x ðsÞds R
thðtÞ
!
_ xðsÞds
t
thðtÞ
Z t
!
thðtÞ th
Z
thðtÞ
!
t
!
thðtÞ th
!
_ xðsÞds ð1 lÞ Z
Z
_ xðsÞds l2
th
x ðsÞds R
thðtÞ
!
thðtÞ
th
Z !
t
Z
!
_T
x ðsÞds R
Z
!
t
_ xðsÞds thðtÞ
Z
thðtÞ
t
!
_ xðsÞds thðtÞ
¼ nT ðtÞ½lu1 þ ð1 lÞu2 nðtÞ:
R R R thðtÞ thðtÞ thðtÞ _ In order to get the stability criterion, the terms l2 th x_ T ðsÞds R th xðsÞds ln1 th x_ T ðsÞds R R R R R thðtÞ t t t t _ _ _ ð1 lÞ2 thðtÞ x_ T ðsÞds R thðtÞ xðsÞds ð1 lÞn1 thðtÞ x_ T ðsÞds R thðtÞ xðsÞds are removed R th xðsÞds directly, which will inevitably cause some conservatism in theory. If we retain these terms, we can not get the appropriate stability criterion. Therefore, when n >= 3, the stability criterion is equivalent to that derived from n = 2 (Theorem 1). Remark 5. New technique named ‘‘delay partitioning’’ has been proposed in [32]. The major different between the delay partitioning approach and our method is that the the delay partitioning idea is to represent the time delay h(t) in two parts: the constant part h1 and time-varying part d(t), that is h(t) = d(t) + h1, and partition h1 into m parts. On the one hand, the computational complexity is dependent on the partition number m. For a system, since the total number of decision variables is dependent on the delay partitioning number m and it will increase if m increases, the computational complexity will be increased as the partitioning becomes thinner. On the other hand, the delay partitioning number m becomes larger, the conservatism of the results is further reduced, while the computational cost increases. This is reasonable since m is related to the number of decision variables. Thus we should deal with the relationship between complexity and performance of the stability _ t Þ, we have analysis. Compared with the delay partitioning method, our method is easily understandable. When estimating Vðx not introduced any free weighting matrix and not used any model transformation, thus making Theorems 1 and 2 only involve Rt _ the matrix variables in the Lyapunov functional. In order to reduce the conservatism, th x_ T ðsÞRxðsÞds is not simply enlarged Rt R thðtÞ _ _ as thðtÞ x_ T ðsÞRxðsÞds, but th x_ T ðsÞRxðsÞdsis retained as well. By using an improved approximation method, delay-dependent stability criteria are obtained. From a mathematical point of view, our method is simple.
4. Numerical example In this section, we use two examples and compare our results with the previous to show the effectiveness of ours. Example 1. Consider the uncertain time-delay system (1) with the following parameters:
A¼
0:5 2 ; 1 1
A1 ¼
0:5 1 ; 0 0:6
E¼
0:2 0 ; 0 0:2
E1 ¼
0:2 0 ; 0 0:2
D¼
1 0 : 0 1
The upper bound on the time delay for different d obtained from Theorems 1 and 2 are shown in Table 1. For comparison, the Table 1 also lists the upper bounds obtained from the criteria in [1,2,5,6,10–12], it is easy to see that our proposed stability criteria with fewer matrix variables provide a large stability bound than the other results. Table1 also shows one existing result is the special cases of our results. For example, To our Theorem 2, the maximum allowable upper delay bound is equivalent to that derived from [11, Corollary1]; better than those in [1,2,5,6,10,12]. He’s criterion in [11, Corollary 4] gives the best results. To our Theorem 1, our results outperform all existing ones. Thus, Theorem 1 improves the result in Theorem 2. Example 2. Consider the uncertain time-delay system (1) with the following parameters:
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C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888 Table 1 The Maximum allowable delay bound for Example 1. d
0.5
0.9
any d
Fridman and Shaked [1,2] Jing et al. [5] and Wu et al. [6] Jiang and Han [10] Li and Guo [12] He et al. (Corollary 1) [11] He et al. (Corollary 4) [11] Theorem 2 Theorem 1
0.1820 0.2433 — 0.31 0.3155 0.3420 0.3155 0.3497
— 0.2420 — 0.31 0.3155 0.3378 0.3155 0.3497
— 0.2420 0.2923 0.31 0.3155 0.3356 0.3155 0.3497
Table 2 The Maximum allowable delay bound for Example 2.
A¼
2
0
0
1
d
0.5
0.9
Yue and Won [13] Han [3] Wu et al. [6] Parlakci [18] Peng et al. (Corollary 1) [15] Peng et al. (Theorem 1, s1 = 0) [15] Zhao et al. (Corollary 2) [20] Yan et al. (Corollary 2) [31] Theorem 2 Theorem 1
0.2195 0.5 0.9247 0.9264 0.9264 0.9322 0.9247 0.9428 0.9428 0.9561
0.1561 0.08 0.6954 0.6954 0.6954 0.7590 0.6954 0.8189 0.8189 0.8919
;
A1 ¼
1
0
1 1
;
E¼
1:6
0
0
0:05
;
E1 ¼
0:1
0
0
0:3
;
D¼
1 0 0 1
:
Under different levels of the upper bounds of the time-delay (d = 0.5 and d = 0.9 respectively), Table 2 lists the results of the maximum allowable delay bounds derived from various methods including the one proposed in this paper, [3,6,13,15,18,20,31]. It can be seen from Table 2 that the maximum allowable delay h decreases as d increases. Table 2 also shows that from our Theorems 1 and 2, To our Theorem 2, our results in this paper and Yan’s results in [31] are equivalent, and are better than all other results from[3,6,13,15,18,20]. We obtain the same results with less variables than those in [31]. Theorem 1 gives much better results than those obtained by the methods in [3,6,13,15,18,20], and better results than [31]. For d = 0.1, by Theorem 1, we can obtain the maximum upper bound on the allowable size to be h = 1.1075. However, applying criteria in [20,26, Theorem 3 (c = 0)], the maximum value of h for the above system is 0.92 and 1.1072. Hence, it is obvious that the results obtained from our simple method with less variables are less conservative than those obtained from the existing methods. 5. Conclusion In this note, new delay-dependent stability criteria have been developed for uncertain time delay systems. New method to estimate the upper bound of the derivative of Lyapunov functional candidate has been proposed without introducing any slack variables. The resulting criteria, though involved with fewer matrix variables have been show simple and less conservative than the existing ones. Numerical examples have been given to verify its low level of conservativeness. Acknowledgment This work was supported by the Key National Natural Science Foundation of China Grant (No. 61134012), the 2011 Excellent Youth Project (No. Q20112907) of Hubei Provincial Department of Education, and the Research Project (No. 10CB146) of Huanggang Normal University, China. References [1] [2] [3] [4]
E. Fridman, U. Shake, An improved stabilization method for linear time-delay systems, IEEE Transactions Automatic Control 47 (2002) 1931–1937. E. Fridman, U. Shake, Delay-dependent stability and H1 control: Constant and time-varying delays, International Journal of Control 76 (2003) 48–60. Q.L. Han, On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty, Automatica 40 (2004) 1087–1092. Y.S. Lee, Y.S. Moon, W.H. Kwon, P.G. Park, Delay-dependent robust H1 control for uncertain systems with a state-delay, Automatica 40 (2004) 65–72.
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[5] J.X. Jing, D.L. Tan, Y.C. Wang, An LMI approach to stability of systems with severe time-delay, IEEE Transactions Automatic Control 49 (2004) 1192– 1195. [6] M. Wu, Y. He, J.H. She, G.P. Liu, Delay-dependent criteria for robust stability of time-varying delay systems, Automatica 40 (2004) 1435–1439. [7] Y. He, M. Wu, J.H. She, G.P. Liu, Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Systems and Control Letters 51 (2004) 57–65. [8] C. Lin, Q.G. Wang, T.H. Lee, A less conservative robust stability test for linear uncertain time-delay systems, IEEE Transactions Automatic Control 51 (2006) 87–91. [9] X.M. Zhang, M. Wu, J.H. She, Y. He, Delay-dependent stabilization of linear systems with time-varying state and input delays, Automatica 41 (2005) 1405–1412. [10] X. Jiang, Q.L. Han, On H1 control for linear systems with interval time-varying delay, Automatica 41 (2005) 2099–2106. [11] Y. He, Q.G. Wang, L.H. Xie, L. Chong, Further improvement of free-weighting matrices technique for systems with time-varying delay, IEEE Transactions Automatic Control 52 (2007) 293–299. [12] T. Li, L. Guo, Y. Zhang, Delay-range-dependent robust stability and stabilization for uncertain systems with time-varying delay, International Journal of Robust and Nonlinear Control 18 (2008) 1372–1387. [13] O.M. Kown, J.H. Park, S.M. Lee, On robust stability criterion for dynamic systems with time-varying delays and nonlinear perturbations, Applied Mathematics and Computation 203 (2008) 937–942. [14] D. Yue, S. Won, An improvement on delay and its time-derivative dependent robust stability of time-delay linear systems with uncertainty, IEEE Transactions on Automatic Control 47 (2002) 407–408. [15] C. Peng, Y.C. Tian, Dleay-dependent robust stability criteria for uncertain systems with interval time-varying delay, Journal of Computational and Applied Mathematics 214 (2008) 480–494. [16] S.Y. Xu, T. Chen, Robust H1 control for uncertain stochastic systems with state delay, IEEE Transactions on Automatic Control 47 (2002) 2089–2094. [17] S.Y. Xu, J. Lam, C. Yang, Robust H1 control for uncertain linear neutral delay systems, Optimal Control Applications and Methods 23 (2002) 113–123. [18] M.N.A. Parlakci, Robust stability of uncertain time-varying state-delayed systems, Control Theory Application 153 (2006) 469–477. [19] S.Y. Xu, J. Lam, T.W. Chen, Robust H1 control for uncertain discrete stochastic time-delay systems, Systems and Control Letters 51 (2004) 03–215. [20] Z.R. Zhao, W. Wang, B. Yang, Delay-dependent robust stability of neutral control system, Applied Mathematics and Computation 187 (2007) 1326– 1332. [21] P. Park, J.W. Ko, Stability and robust stability for systems with a time-varying delay, Automatica 43 (2007) 1855–1858. [22] J.H. Park, O. Kwon, Novel stability criterion of time-delay systems with nonlinear uncertainties, Applied Mathematics Letters 18 (2005) 683–688. [23] S. Xu, J. Lam, Y. Zou, New results on delay-dependent robust H1 control for systems with time-varying delays, Automatica 42 (2006) 343–348. [24] O.M. Kwon, J.H. Park, Robust stabilization of uncertain systems with delays in control input: a matrix inequality approach, Applied Mathematics and Computation 172 (2006) 1067–1077. [25] Y.S. Moon, P.G. Park, W.H. Kwon, Y.S. Lee, Delay-dependent robust stabilization of uncertain state-delay systems, International Journal of Control 74 (2001) 1447–1455. [26] F. Qiu, B.T. Cui, Y. Ji, Further results on robust stability of neutral system with mixed time-varying delays and nonlinear perturbtions, Nonlinear Analysis: Real World Applications 11 (2010) 895–906. [27] Y. He, M. Wu, J.H. She, G.P. Liu, Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties, IEEE Transactions Automatic Control 49 (2004) 828–832. [28] S. He, F. Liu, Observer-based passive control for nonlinear uncertain time-delay jump systems, Acta Mathematica Scientia 29A (2009) 334–343. [29] Y. Chen, A.K. Xue, J.H. Wang, Delay-dependent passive control of stochastic delay systems, Acta Automatica Sinica 35 (2009) 324–327. [30] Y. He, Q.G. Wang, C. Lin, M. Wu, Delay-range-dependent stability for systems with time-varying delay, Automatica 43 (2007) 371–376. [31] H.C. Yan, H. Zhang, M.Q. Meng, Delay-range-dependent robust H1 control for uncertain systems with interval time-varying delays, Neurocomputing 73 (2010) 1235–1243. [32] X.Y. Meng, J. Lam, B.Z. Du, H.J. Gao, A delay-partitioning approach to the stability analysis of discrete-time systems, Automatica 46 (2010) 610–614.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Improved delay-dependent robust stability criteria for uncertain time delay systems Cheng Wang a,b,⇑, Yi Shen a a Department of Control Science and Engineering and the Key Laboratory of Ministry of Education for Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China b College of Mathematics and Computer Science, Huanggang Normal University, Huanggang, Hubei 438000, PR China
a r t i c l e
i n f o
Keywords: Delay-dependent Robust stability Linear matrix inequality (LMI) Uncertain time-delay systems
a b s t r a c t This paper deals with the robust stability analysis for uncertain systems with time-varying delay. New delay-dependent robust stability criteria of uncertain time-delay systems are proposed by exploiting appropriate Lyapunov functional candidate. These developed results have advantages over some previous ones in that they have fewer matrix variables yet less conservatism, due to the introduction of a method to estimate the upper bound of the derivative of Lyapunov functional candidate without ignoring the additional useful terms. Numerical examples are given to demonstrate the effectiveness and the advantage of the proposed method. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Time delays frequently appear in many practical systems such as biological systems, chemical systems, electronic systems and network control systems. The time delays are regarded as the major source of instability and poor performance. Therefore, a great amount of effort has been devoted to the stability analysis of the time-delay systems [1–24]. Depending on whether the existence of time delays or not, stability criteria for time-delay systems can be divided into two types: delay-dependent ones and delay-independent ones. While the delay-independent criteria guarantee the stability of systems irrespective to the size of time delays, delay-dependent criteria can give the maximum delay bounds for making the system stable. Thus, delay-dependent criteria, which make use of information on the length of delay, tend to be less conservative than delay-independent ones [16,17], especially when the time delays are small. Recently, considerable attentions have been focused on delay-dependent stability analysis of time-delay systems. As for systems with time-varying delay, many methods have been taken to deal with delay-dependent stability problems. Fixed model transformation was the main method employed in [2], and inequality methods were used to estimate the upper bound of cross product terms in the derivative of the Lyapunov functional in [4,25]. In order to further improve the performance of delay-dependent stability criteria, free-weighting method was proposed in [6,7,27], where neither system transformation nor bounding techniques on some cross terms was involved, thus avoiding the conservatism induced by model transformation. Note that the free weighting matrix method makes stability criteria complicated, therefore the criteria still leave some room for improvement in accuracy as well as complexity due to the method used. On the other hand, much effort has been devoted to improve the delay-dependent conditions based on the new Lyapunov functionals in [4,5,23]. New classes of Lyapunov functionals and augmented Lyapunov functionals were introduced in [11] to ⇑ Corresponding author. at: Department of Control Science and Engineering and the Key Laboratory of Ministry of Education for Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China. E-mail address: [email protected] (C. Wang). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.031
C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888
2881
study the delay-dependent stability for systems with time-varying delay. Therefore, the results were less conservative than R0 Rt _ some existing ones, but there is room for further investigation. For example, in [2,5,6] the derivative of h tþh x_ T ðsÞZ xðsÞdsdh Rt R tdðtÞ T T T _ _ _ was often estimated as hx_ ðtÞZ xðtÞ tdðtÞ x_ ðsÞZ xðsÞds, and the term th x_ ðsÞZ xðsÞds was ignored, which may lead to considerable conservativeness. Although [11,30] retained these terms and proposed an improved delay-dependent stability criterion for systems with time-varying delay, the stability condition still leaves some room for improvement. For instance, in [30], both terms d(t) and h d(t) were enlarged as h. It is observed that d(t) and h d(t) have an important relationship that there sum is h. So, the above may lead to conservativeness. Thus, a source of conservatism that could be further reduced lies in some other useful terms in the calculation of the upper bound of the derivative of Lyapunov functional, which motivates the present study. In this paper, our purpose is to present some new delay-dependent stability criteria for uncertain systems with timevarying delay. By constructing appropriate Lyapunov–Krasovskii functional and employing an improved inequality, delay-dependent stability criteria are obtained in the linear matrix equality (LMI) format. The main contribution of this paper is that a tighter upper bound of the differential of Lyapunov–Krasovskii functional is obtained by an improved approximation method, without resorting to any model transformation and free weighting matrix technique. The resulting criteria have advantages over some previous ones in that they involve few matrix variables but have less conservatism, which are established theoretically. From a mathematical point of view, they are simple, and improve over some previous ones at the same time. Numerical examples are provided to show the effectiveness and advantage over some existing results. 2. Problem statement Consider the following linear system with time-delay and parameter uncertainties described by
_ xðtÞ ¼ AðtÞxðtÞ þ A1 ðtÞxðt hðtÞÞdt; xðtÞ ¼ uðtÞ;
8t 2 ½h; 0
t>0
;
ð1Þ
where x(t) 2 Rn is the state vector, and u(t) is a continuous and differentiable vector-valued initial function of t 2 [h, 0]. A(t) and A1(t) are matrix functions described as follows
AðtÞ ¼ A þ DAðtÞ;
A1 ðtÞ ¼ A1 þ DA1 ðtÞ;
ð2Þ
where A and A1 are constant matrices of appropriate dimensions while DA(t) and DA1(t) are unknown constant matrices of appropriate dimensions representing time-varying parameter uncertainties, which are assumed to be the following form:
½DAðtÞ DA1 ðtÞ ¼ DFðtÞ½E1 E2 ;
ð3Þ
where D, Ei(i = 1, 2) are appropriately dimensioned constant matrices, and F(t) is an unknown time-varying matrix with Lebesgue measurable elements satisfying
F T ðtÞ FðtÞ 6 I:
ð4Þ
h(t), the time-varying delay of the system, is a differentiable function and is assumed to satisfy
0 6 hðtÞ 6 h;
hðtÞ 6 d:
ð5Þ
where h and d are known constants. The purpose of this paper is to formulate new delay-dependent criteria to check the robust stability of system (1). Let us give the following lemmas, which will play an indispensable role in deriving our criteria. Lemma 1 (Schur complement). Given constant matrices R1, R2 and R3 with appropriate dimensions, where RT1 ¼ R1 and RT2 ¼ R2 , then
R1 þ RT3 R1 2 R3 < 0: if and only if
"
#
R2 R3 R1 RT3 < 0 or < 0: RT3 R1 R3 R2
Lemma 2 [28]. Given appropriately dimensioned matrices w, H, G with w = wT, Then
w þ HFðtÞG þ GT F T ðtÞHT < 0; holds for all F(t) satisfying FT(t)F(t) 6 I if and only if for some e > 0,
w þ eHHT þ e1 GT G < 0:
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C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888
Lemma 3 [29]. For any symmetric positive definite matrix R > 0 and vector function x(t) : [0, h] ? Rn,such that the integrations concerned are well defined, the following inequality holds
h
Z
h
xT ðsÞRxðsÞds 6
0
Z
h
xT ðsÞds R
0
Z
h
xðsÞds:
0
3. New stability criteria In this section, our objective is to establish new and less conservative stability conditions for system (1) as follows: Theorem 1. Given scalars h > 0 and d, the system (1) with a time-varying delay h(t) subject to (5) is robustly stable if there exist matrices P = PT > 0, Q = QT > 0, Z = ZT > 0, R = RT > 0 and a scalar e > 0, such that the following LMIs hold:
2 6 6 6 6 6 6 /1 ¼ 6 6 6 6 6 6 4 2 6 6 6 6 6 6 /2 ¼ 6 6 6 6 6 6 4
T
PA þ AT P þ Q þ Z R þ eET1 E1
PA1 þ R þ eET1 E2
0
hA R
ð1 dÞQ 3R þ eET2 E2
2R
hA1 R
Z 2R
0
R
eI
T
PD
T
PA þ AT P þ Q þ Z 2R þ eET1 E1
PA1 þ 2R þ eET1 E2
0
hA R
ð1 dÞQ 3R þ eET2 E2
R
hA1 R
Z R
0
R
T
PD
3
7 7 0 7 7 7 7 0 7 7 < 0; 7 7 hRD 7 7 5
ð6Þ
3
7 7 0 7 7 7 7 0 7 7 < 0: 7 7 hRD 7 7 5
ð7Þ
eI
Proof. Define the following Lyapunov functional candidate
Vðxt Þ ¼ xT ðtÞPxðtÞ þ
Z
t
xT ðsÞQxðsÞds þ thðtÞ
Z
t
th
xT ðsÞZxðsÞds þ
Z
0
h
Z
t
_ hx_ T ðsÞRxðsÞdsdh;
ð8Þ
tþh
where P = PT > 0, Q = QT > 0, Z = ZT > 0 and R = RT > 0 are to be determined. Calculating the derivative of V(xt) along the solutions of system (1) yields T _ _ t Þ ¼ 2xT ðtÞPxðtÞ _ þ xT ðtÞZxðtÞ þ xT ðtÞQxðtÞ ð1 hðtÞÞx Vðx ðt hðtÞÞQxðt hðtÞÞ xT ðt hÞZxðt hÞ Z t 2 _ _ hx_ T ðsÞRxðsÞds þ h x_ T ðtÞRxðtÞ th T
6 2x ðtÞP½AðtÞxðtÞ þ A1 ðtÞxðt hðtÞÞ þ xT ðtÞðZ þ Q ÞxðtÞ xT ðt hÞZxðt hÞ ð1 dÞxT ðt hðtÞÞQxðt hðtÞÞ Z thðtÞ Z t 2 _ _ þ h ½AðtÞxðtÞ þ A1 ðtÞxðt hðtÞÞT R½AðtÞxðtÞ þ A1 ðtÞxðt hðtÞÞ hx_ T ðsÞRxðsÞds hx_ T ðsÞRxðsÞds: th
thðtÞ
ð9Þ We know that
R thðtÞ _ _ ðh hðtÞÞx_ T ðsÞRxðsÞds th hðtÞx_ T ðsÞRxðsÞds: Rt Rt Rt _ _ _ thðtÞ hx_ T ðsÞRxðsÞds ¼ thðtÞ hðtÞx_ T ðsÞRxðsÞds thðtÞ ðh hðtÞÞx_ T ðsÞRxðsÞds:
R thðtÞ th
_ hx_ T ðsÞRxðsÞds ¼
Set l = h(t)/h. Then
R thðtÞ th
2883
C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888
Z
thðtÞ
_ hðtÞx_ T ðsÞRxðsÞds ¼ l
Z
th
thðtÞ
_ hx_ T ðsÞRxðsÞds ¼ l
Z
th
¼ l
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds l th
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds l2 Z
thðtÞ
_ hðtÞx_ T ðsÞRxðsÞds
th
Z
th
l2
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds
th
thðtÞ
_ hðtÞx_ T ðsÞRxðsÞds
th
¼ l
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds l2
Z
th
ln2
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds
th thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds ln2 th
¼ l
Z
_ ðh hðtÞÞx_ T ðsÞRxðsÞds l2
Z
th
Z
Z
_ ðh hðtÞÞx_ T ðsÞRxðsÞds
th thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds ln1 _ ðh hðtÞÞx_ T ðsÞRxðsÞds l2
Z
th
Z
Z
thðtÞ
_ hx_ T ðsÞRxðsÞds
th
thðtÞ
ln2
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
th
6 l
thðtÞ
th
thðtÞ
ln2
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds
th thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds ln1 th
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds:
th
Similarly, we have
Z
t
_ ðh hðtÞÞx_ T ðsÞRxðsÞds ¼ ð1 lÞ
Z
thðtÞ
t
_ hx_ T ðsÞRxðsÞds ¼ ð1 lÞ thðtÞ
ð1 lÞ
Z
Z
t
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
t
_ ðh hðtÞÞx_ T ðsÞRxðsÞds ¼ ð1 lÞ
Z
thðtÞ
ð1 lÞ2
Z
t
t
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
_ hðtÞx_ T ðsÞRxðsÞds ð1 lÞn2
Z
thðtÞ
ð1 lÞn1
Z
Z
t
_ hx_ T ðsÞRxðsÞds 6 ð1 lÞ
t
Z
t
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
_ hðtÞx_ T ðsÞRxðsÞds ð1 lÞn2
Z
thðtÞ
ð1 lÞn1
Z
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
thðtÞ
ð1 lÞ2
t
t
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
t
_ hðtÞx_ T ðsÞRxðsÞds
thðtÞ
Then, we can get
Z
thðtÞ
_ hx_ T ðsÞRxðsÞds
th
Z
t
_ hx_ T ðsÞRxðsÞds 6 ½1 þ l þ þ ln1
thðtÞ
Z
thðtÞ
_ ðh hðtÞÞx_ T ðsÞRxðsÞds
th
½1 þ ð1 lÞ þ þ ð1 lÞn1
Z
t
_ hðtÞx_ T ðsÞRxðsÞds thðtÞ
Xn1 Z ¼ i¼0 li
thðtÞ
th
_ ðh hðtÞÞx_ T ðsÞRxðsÞds
Xn1
ð1 lÞi i¼0
Z
t
_ hðtÞx_ T ðsÞRxðsÞds:
ð10Þ
thðtÞ
Using Lemma 3 and letting n = 2 in (10) result in
Z
thðtÞ
th
_ hx_ T ðsÞRxðsÞds
Z
t
_ hx_ T ðsÞRxðsÞds 6 ½xðt hðtÞÞ xðt hÞT R½xðt hðtÞÞ xðt hÞ
thðtÞ
½xðtÞ xðt hðtÞÞT R½xðtÞ xðt hðtÞÞ l½xðt hðtÞÞ xðt hÞT R½xðt hðtÞÞ xðt hÞ ð1 lÞ½xðtÞ xðt hðtÞÞT R½xðtÞ xðt hðtÞÞ: Combining (9) and (11), we can furthermore get
ð11Þ
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C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888
_ t Þ 6 xT ðtÞ½PAðtÞ þ AT ðtÞP þ Z þ Q R þ h2 AT ðtÞRAðtÞxðtÞ Vðx 2
þ2xT ðtÞ½PA1 ðtÞ þ h AT ðtÞRA1 ðtÞ þ Rxðt hðtÞÞ þxT ðt hðtÞÞ½ð1 dÞQ 2R þ AT1 ðtÞRA1 ðtÞxðt hðtÞÞ xT ðt hÞðZ þ RÞxðt hÞ þ 2xT ðt hðtÞÞRxðt hÞ l½xðt hðtÞÞ xðt hÞT R½xðt hðtÞÞ xðt hÞ ð1 lÞ½xðtÞ xðt hðtÞÞT R½xðtÞ xðt hðtÞÞ ¼ nT ðtÞ½lu1 þ ð1 lÞu2 nðtÞ; where
2
u1 ¼ 6 4 2
u2 ¼ 6 4
2
2
PAðtÞ þ AT ðtÞP þ Z þ Q R þ h AT ðtÞRAðtÞ
PA1 ðtÞ þ R þ h AT ðtÞRA1 ðtÞ
0
ð1 dÞQ 3R þ AT1 ðtÞRA1 ðtÞ
2R
Z 2R
2
2
PAðtÞ þ AT ðtÞP þ Z þ Q 2R þ h AT ðtÞRAðtÞ
PA1 ðtÞ þ 2R þ h AT ðtÞRA1 ðtÞ
0
ð1 dÞQ 3R þ AT1 ðtÞRA1 ðtÞ
R
Z R
nðtÞ ¼ ½xT ðtÞ;
xT ðt hðtÞÞ;
3 7 5 3 7 5
xT ðt hÞT :
Since P > 0, Q > 0, R > 0, Z > 0 and 0 6 l 6 1. Therefore, if u1 < 0 and u2 < 0, we have lu1 + (1 l)u2 < 0. Then, _ t Þ < ekxðtÞk2 for a sufficiently small constant e > 0, which ensures robust stability of the system. Vðx By Schur Complement Lemma, it is easily seen that u1 < 0 if and only if
2 6 6
PAðtÞ þ AT ðtÞP þ Z þ Q R
PA1 ðtÞ þ R
ð1 dÞQ 3R
R1 ¼ 6 6 4
T
0
hA ðtÞR
7 T hA1 ðtÞR 7 7 < 0; 7 5 Z 2R 0 R 2R
and u2 < 0 if and only if
2 6 6
PAðtÞ þ AT ðtÞP þ Z þ Q 2R
PA1 ðtÞ þ 2R
ð1 dÞQ 3R
R2 ¼ 6 6 4
3
T
0
hA ðtÞR
3
7 T hA1 ðtÞR 7 7 < 0: 7 5 Z R 0 R
ð12Þ
ð13Þ
R
Then by Lemma 2,
R1 ¼ X1 þ PFðtÞC þ CT F T ðtÞPT < 0 and
R2 ¼ X2 þ PFðtÞC þ CT F T ðtÞPT < 0 if and only if there exist a scalar e > 0 such that the following inequalities are satisfied
X1 þ eCT C þ e1 PPT < 0;
ð14Þ
X2 þ eCT C þ e1 PPT < 0;
ð15Þ
and
where
C ¼ ½E1 E2 0 0; 2 6 6 X1 ¼ 6 6 4
P ¼ ½DT P 0 0 hDT RT ;
PA þ AT P þ Z þ Q R
PA1 þ R
ð1 dÞQ 3R
0
T
hA R
3
T 7 hA1 R 7 7 < 0; 7 Z 2R 0 5 R
2R
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C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888
2 6 6
PA þ AT P þ Z þ Q 2R
PA1 þ 2R
ð1 dÞQ 3R
X2 ¼ 6 6 4
0
T
hA R
3
T 7 hA1 R 7 7 < 0: 7 Z R 0 5 R
R
Applying the Schur Complement shows that (14) is equivalent to (6) and (15) is equivalent to (7). This completes the proof. h
Remark 1. Theorem 1 provides a novel delay-dependent robust stability criterion for system (1) with h(t) satisfying (5) in terms of LMIs, which can be verified using recently developed standard algorithms. Theorem 1 can be applied to both slow and fast time-varying delays only if dis known. In many cases, the information of the time derivative of delay is unknown. Regarding this circums, the rate-independent criterion can be derived by choosing Q = 0. Rt _ Remark 2. In the previous work such as [6,19], the term th x_ T ðsÞZ xðsÞds in the derivative of Vðx_ t Þ was simply enlarged as Rt R thðtÞ T T _ _ thðtÞ x_ ðsÞZ xðsÞds. It is seen that the term th x_ ðsÞZ xðsÞds in Vðx_ t Þ was ignored, which may lead to considerable conR thðtÞ _ servativeness. In Theorem 1, th x_ T ðsÞRxðsÞds is taken into account when estimating the upper bound of Vðx_ t Þ. On the other hand, we have not introduced any free weighting matrix as references [6,11,21]. Theorem 1 only involves the matrix variables in the Lyapunov functional. Therefore, our method in Theorem 1 is simple and less conservative in point of mathematical view. Next, we provide another delay-dependent robust stability criterion for system (1). Theorem 2. Given scalars h > 0 and d, the system (1) with a time-varying delay h(t) subject to (5) is robustly stable if there exist matrices P = PT > 0, Q = QT > 0, Z = ZT > 0, R = RT > 0 and a scalar e > 0, such that the following LMI holds:
2 6 6 6 6 6 /¼6 6 6 6 4
T
PA þ AT P þ Q þ Z R þ eET1 E1
PA1 þ R þ eET1 E2
0
hA R
ð1 dÞQ 2R þ eET2 E2
R
hA1 R
Z R
0
R
T
PD
3
7 7 0 7 7 7 < 0: 0 7 7 7 hRD 7 5 eI
ð16Þ
Proof. Choose the same Lyapunov functional candidate as in (8) for the system(1). Different from Theorem 1, here, letting n = 1 in (10) and using Lemma 3, we can acquire
Z
thðtÞ
_ hx_ T ðsÞRxðsÞds
th
Z
t
_ hx_ T ðsÞRxðsÞds 6 ½xðt hðtÞÞ xðt hÞT R½xðt hðtÞÞ xðt hÞ ½xðtÞ xðt
thðtÞ
hðtÞÞT R½xðtÞ xðt hðtÞÞ _ t Þ 6 nT ðtÞXnðtÞ, where Similar to the proof of Theorem 1, we obtain Vðx
2 6
2
2
PAðtÞ þ AT ðtÞP þ Z þ Q R þ h AT ðtÞRAðtÞ
PA1 ðtÞ þ R þ h AT ðtÞRA1 ðtÞ
0
ð1 dÞQ 2R þ AT1 ðtÞRA1 ðtÞ
R
Z R
X¼4
3 7 5
nðtÞ ¼ ½xT ðtÞ; xT ðt hðtÞÞ; xT ðt hÞT : Using Lemmas 1–3 to (16) results in X < 0. Therefore, similar to Theorem 1 we can get the result of Theorem 2. This completes the proof. h _ t Þ. The difference between them Remark 3. Theorems 1 and 2 use the same formula (10) to estimate the upper bound of Vðx Rt T _ _ is that in Theorem 1, th hx ðsÞRxðsÞds in formula (10) is enlarged as
Rt _ x_ T ðsÞdsR thðtÞ xðsÞds R thðtÞ T R thðtÞ Rt Rt T _ _ l th x_ ðsÞdsR th xðsÞds ð1 lÞ thðtÞ x_ ðsÞdsR thðtÞ xðsÞds;
R thðtÞ th
x_ T ðsÞdsR
while in Theorem 2,
R thðtÞ th
Rt
th
_ xðsÞds
Rt
thðtÞ
_ hx_ T ðsÞRxðsÞds is enlarged as
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C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888
Z
Z
thðtÞ
x_ T ðsÞdsR th
thðtÞ
_ xðsÞds
th
Z
t
x_ T ðsÞdsR
Z
thðtÞ
t
_ xðsÞds:
thðtÞ
Both the two criteria are closer to the real value of
Rt
th
_ hx_ T ðsÞRxðsÞds than that in [2,3,5,6], and is less conservative.
Remark 4. Theorem 1 is obtained by using n = 2, while Theorem 2 is obtained by using n = 1. When n >= 3, using Lemma 3 and (10), we have
Z
_ t Þ 6 nT ðtÞXnðtÞ l Vðx
x_ T ðsÞds R
th n1
l
Z
!
thðtÞ
!
thðtÞ
x_ T ðsÞds R
th 2
ð1 lÞ
Z
_T
T
6 n ðtÞXnðtÞ l
Z
thðtÞ
thðtÞ
_T
x ðsÞds R
th
Z
Z
thðtÞ
!
t
x_ T ðsÞds R
_ xðsÞds ð1 lÞ
thðtÞ
!
x_ T ðsÞds R
Z
n1
_ xðsÞds ð1 lÞ
th
Z
!
_ xðsÞds
t
!
_T
x ðsÞds R
thðtÞ
!
_ xðsÞds
t
thðtÞ
Z t
!
thðtÞ th
Z
thðtÞ
!
t
!
thðtÞ th
!
_ xðsÞds ð1 lÞ Z
Z
_ xðsÞds l2
th
x ðsÞds R
thðtÞ
!
thðtÞ
th
Z !
t
Z
!
_T
x ðsÞds R
Z
!
t
_ xðsÞds thðtÞ
Z
thðtÞ
t
!
_ xðsÞds thðtÞ
¼ nT ðtÞ½lu1 þ ð1 lÞu2 nðtÞ:
R R R thðtÞ thðtÞ thðtÞ _ In order to get the stability criterion, the terms l2 th x_ T ðsÞds R th xðsÞds ln1 th x_ T ðsÞds R R R R R thðtÞ t t t t _ _ _ ð1 lÞ2 thðtÞ x_ T ðsÞds R thðtÞ xðsÞds ð1 lÞn1 thðtÞ x_ T ðsÞds R thðtÞ xðsÞds are removed R th xðsÞds directly, which will inevitably cause some conservatism in theory. If we retain these terms, we can not get the appropriate stability criterion. Therefore, when n >= 3, the stability criterion is equivalent to that derived from n = 2 (Theorem 1). Remark 5. New technique named ‘‘delay partitioning’’ has been proposed in [32]. The major different between the delay partitioning approach and our method is that the the delay partitioning idea is to represent the time delay h(t) in two parts: the constant part h1 and time-varying part d(t), that is h(t) = d(t) + h1, and partition h1 into m parts. On the one hand, the computational complexity is dependent on the partition number m. For a system, since the total number of decision variables is dependent on the delay partitioning number m and it will increase if m increases, the computational complexity will be increased as the partitioning becomes thinner. On the other hand, the delay partitioning number m becomes larger, the conservatism of the results is further reduced, while the computational cost increases. This is reasonable since m is related to the number of decision variables. Thus we should deal with the relationship between complexity and performance of the stability _ t Þ, we have analysis. Compared with the delay partitioning method, our method is easily understandable. When estimating Vðx not introduced any free weighting matrix and not used any model transformation, thus making Theorems 1 and 2 only involve Rt _ the matrix variables in the Lyapunov functional. In order to reduce the conservatism, th x_ T ðsÞRxðsÞds is not simply enlarged Rt R thðtÞ _ _ as thðtÞ x_ T ðsÞRxðsÞds, but th x_ T ðsÞRxðsÞdsis retained as well. By using an improved approximation method, delay-dependent stability criteria are obtained. From a mathematical point of view, our method is simple.
4. Numerical example In this section, we use two examples and compare our results with the previous to show the effectiveness of ours. Example 1. Consider the uncertain time-delay system (1) with the following parameters:
A¼
0:5 2 ; 1 1
A1 ¼
0:5 1 ; 0 0:6
E¼
0:2 0 ; 0 0:2
E1 ¼
0:2 0 ; 0 0:2
D¼
1 0 : 0 1
The upper bound on the time delay for different d obtained from Theorems 1 and 2 are shown in Table 1. For comparison, the Table 1 also lists the upper bounds obtained from the criteria in [1,2,5,6,10–12], it is easy to see that our proposed stability criteria with fewer matrix variables provide a large stability bound than the other results. Table1 also shows one existing result is the special cases of our results. For example, To our Theorem 2, the maximum allowable upper delay bound is equivalent to that derived from [11, Corollary1]; better than those in [1,2,5,6,10,12]. He’s criterion in [11, Corollary 4] gives the best results. To our Theorem 1, our results outperform all existing ones. Thus, Theorem 1 improves the result in Theorem 2. Example 2. Consider the uncertain time-delay system (1) with the following parameters:
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C. Wang, Y. Shen / Applied Mathematics and Computation 218 (2011) 2880–2888 Table 1 The Maximum allowable delay bound for Example 1. d
0.5
0.9
any d
Fridman and Shaked [1,2] Jing et al. [5] and Wu et al. [6] Jiang and Han [10] Li and Guo [12] He et al. (Corollary 1) [11] He et al. (Corollary 4) [11] Theorem 2 Theorem 1
0.1820 0.2433 — 0.31 0.3155 0.3420 0.3155 0.3497
— 0.2420 — 0.31 0.3155 0.3378 0.3155 0.3497
— 0.2420 0.2923 0.31 0.3155 0.3356 0.3155 0.3497
Table 2 The Maximum allowable delay bound for Example 2.
A¼
2
0
0
1
d
0.5
0.9
Yue and Won [13] Han [3] Wu et al. [6] Parlakci [18] Peng et al. (Corollary 1) [15] Peng et al. (Theorem 1, s1 = 0) [15] Zhao et al. (Corollary 2) [20] Yan et al. (Corollary 2) [31] Theorem 2 Theorem 1
0.2195 0.5 0.9247 0.9264 0.9264 0.9322 0.9247 0.9428 0.9428 0.9561
0.1561 0.08 0.6954 0.6954 0.6954 0.7590 0.6954 0.8189 0.8189 0.8919
;
A1 ¼
1
0
1 1
;
E¼
1:6
0
0
0:05
;
E1 ¼
0:1
0
0
0:3
;
D¼
1 0 0 1
:
Under different levels of the upper bounds of the time-delay (d = 0.5 and d = 0.9 respectively), Table 2 lists the results of the maximum allowable delay bounds derived from various methods including the one proposed in this paper, [3,6,13,15,18,20,31]. It can be seen from Table 2 that the maximum allowable delay h decreases as d increases. Table 2 also shows that from our Theorems 1 and 2, To our Theorem 2, our results in this paper and Yan’s results in [31] are equivalent, and are better than all other results from[3,6,13,15,18,20]. We obtain the same results with less variables than those in [31]. Theorem 1 gives much better results than those obtained by the methods in [3,6,13,15,18,20], and better results than [31]. For d = 0.1, by Theorem 1, we can obtain the maximum upper bound on the allowable size to be h = 1.1075. However, applying criteria in [20,26, Theorem 3 (c = 0)], the maximum value of h for the above system is 0.92 and 1.1072. Hence, it is obvious that the results obtained from our simple method with less variables are less conservative than those obtained from the existing methods. 5. Conclusion In this note, new delay-dependent stability criteria have been developed for uncertain time delay systems. New method to estimate the upper bound of the derivative of Lyapunov functional candidate has been proposed without introducing any slack variables. The resulting criteria, though involved with fewer matrix variables have been show simple and less conservative than the existing ones. Numerical examples have been given to verify its low level of conservativeness. Acknowledgment This work was supported by the Key National Natural Science Foundation of China Grant (No. 61134012), the 2011 Excellent Youth Project (No. Q20112907) of Hubei Provincial Department of Education, and the Research Project (No. 10CB146) of Huanggang Normal University, China. References [1] [2] [3] [4]
E. Fridman, U. Shake, An improved stabilization method for linear time-delay systems, IEEE Transactions Automatic Control 47 (2002) 1931–1937. E. Fridman, U. Shake, Delay-dependent stability and H1 control: Constant and time-varying delays, International Journal of Control 76 (2003) 48–60. Q.L. Han, On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty, Automatica 40 (2004) 1087–1092. Y.S. Lee, Y.S. Moon, W.H. Kwon, P.G. Park, Delay-dependent robust H1 control for uncertain systems with a state-delay, Automatica 40 (2004) 65–72.
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[5] J.X. Jing, D.L. Tan, Y.C. Wang, An LMI approach to stability of systems with severe time-delay, IEEE Transactions Automatic Control 49 (2004) 1192– 1195. [6] M. Wu, Y. He, J.H. She, G.P. Liu, Delay-dependent criteria for robust stability of time-varying delay systems, Automatica 40 (2004) 1435–1439. [7] Y. He, M. Wu, J.H. She, G.P. Liu, Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Systems and Control Letters 51 (2004) 57–65. [8] C. Lin, Q.G. Wang, T.H. Lee, A less conservative robust stability test for linear uncertain time-delay systems, IEEE Transactions Automatic Control 51 (2006) 87–91. [9] X.M. Zhang, M. Wu, J.H. She, Y. He, Delay-dependent stabilization of linear systems with time-varying state and input delays, Automatica 41 (2005) 1405–1412. [10] X. Jiang, Q.L. Han, On H1 control for linear systems with interval time-varying delay, Automatica 41 (2005) 2099–2106. [11] Y. He, Q.G. Wang, L.H. Xie, L. Chong, Further improvement of free-weighting matrices technique for systems with time-varying delay, IEEE Transactions Automatic Control 52 (2007) 293–299. [12] T. Li, L. Guo, Y. Zhang, Delay-range-dependent robust stability and stabilization for uncertain systems with time-varying delay, International Journal of Robust and Nonlinear Control 18 (2008) 1372–1387. [13] O.M. Kown, J.H. Park, S.M. Lee, On robust stability criterion for dynamic systems with time-varying delays and nonlinear perturbations, Applied Mathematics and Computation 203 (2008) 937–942. [14] D. Yue, S. Won, An improvement on delay and its time-derivative dependent robust stability of time-delay linear systems with uncertainty, IEEE Transactions on Automatic Control 47 (2002) 407–408. [15] C. Peng, Y.C. Tian, Dleay-dependent robust stability criteria for uncertain systems with interval time-varying delay, Journal of Computational and Applied Mathematics 214 (2008) 480–494. [16] S.Y. Xu, T. Chen, Robust H1 control for uncertain stochastic systems with state delay, IEEE Transactions on Automatic Control 47 (2002) 2089–2094. [17] S.Y. Xu, J. Lam, C. Yang, Robust H1 control for uncertain linear neutral delay systems, Optimal Control Applications and Methods 23 (2002) 113–123. [18] M.N.A. Parlakci, Robust stability of uncertain time-varying state-delayed systems, Control Theory Application 153 (2006) 469–477. [19] S.Y. Xu, J. Lam, T.W. Chen, Robust H1 control for uncertain discrete stochastic time-delay systems, Systems and Control Letters 51 (2004) 03–215. [20] Z.R. Zhao, W. Wang, B. Yang, Delay-dependent robust stability of neutral control system, Applied Mathematics and Computation 187 (2007) 1326– 1332. [21] P. Park, J.W. Ko, Stability and robust stability for systems with a time-varying delay, Automatica 43 (2007) 1855–1858. [22] J.H. Park, O. Kwon, Novel stability criterion of time-delay systems with nonlinear uncertainties, Applied Mathematics Letters 18 (2005) 683–688. [23] S. Xu, J. Lam, Y. Zou, New results on delay-dependent robust H1 control for systems with time-varying delays, Automatica 42 (2006) 343–348. [24] O.M. Kwon, J.H. Park, Robust stabilization of uncertain systems with delays in control input: a matrix inequality approach, Applied Mathematics and Computation 172 (2006) 1067–1077. [25] Y.S. Moon, P.G. Park, W.H. Kwon, Y.S. Lee, Delay-dependent robust stabilization of uncertain state-delay systems, International Journal of Control 74 (2001) 1447–1455. [26] F. Qiu, B.T. Cui, Y. Ji, Further results on robust stability of neutral system with mixed time-varying delays and nonlinear perturbtions, Nonlinear Analysis: Real World Applications 11 (2010) 895–906. [27] Y. He, M. Wu, J.H. She, G.P. Liu, Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties, IEEE Transactions Automatic Control 49 (2004) 828–832. [28] S. He, F. Liu, Observer-based passive control for nonlinear uncertain time-delay jump systems, Acta Mathematica Scientia 29A (2009) 334–343. [29] Y. Chen, A.K. Xue, J.H. Wang, Delay-dependent passive control of stochastic delay systems, Acta Automatica Sinica 35 (2009) 324–327. [30] Y. He, Q.G. Wang, C. Lin, M. Wu, Delay-range-dependent stability for systems with time-varying delay, Automatica 43 (2007) 371–376. [31] H.C. Yan, H. Zhang, M.Q. Meng, Delay-range-dependent robust H1 control for uncertain systems with interval time-varying delays, Neurocomputing 73 (2010) 1235–1243. [32] X.Y. Meng, J. Lam, B.Z. Du, H.J. Gao, A delay-partitioning approach to the stability analysis of discrete-time systems, Automatica 46 (2010) 610–614.