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Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality
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Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality O.M. Kwona, M.J. Parka, Ju H. Parkb,n, S.M. Leec, E.J. Chad a
School of Electrical Engineering, Chungbuk National University, 52 Naesudong-ro, Heungduk-gu, Cheongju 362-763, Republic of Korea b Nonlinear Dynamics Group, Department of Electrical Engineering, Yeungnam University, 280 Daehak-ro, Kyongsan 712-749, Republic of Korea c School of Electronics Engineering, Daegu University, Gyungsan 712-714, Republic of Korea d Department of Biomedical Engineering, School of Medicine, Chungbuk National University, 52 Naesudong-ro, Heungduk-gu, Cheongju 362-763, Republic of Korea Received 23 January 2014; received in revised form 8 August 2014; accepted 30 September 2014
Abstract In this paper, the problem of stability analysis for linear systems with time-varying delays is considered. By the consideration of new augmented Lyapunov functionals, improved delay-dependent stability criteria for asymptotic stability of the system are proposed for two cases of conditions on time-varying delays with the framework of linear matrix inequalities (LMIs), which can be solved easily by various efficient convex optimization algorithms. The enhancement of the feasible region of the proposed criteria is shown via three numerical examples by the comparison of maximum delay bounds. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author. E-mail addresses: [email protected] (O.M. Kwon), [email protected] (M.J. Park), [email protected] (J.H. Park), [email protected] (S.M. Lee), [email protected] (E.J. Cha). http://dx.doi.org/10.1016/j.jfranklin.2014.09.021 0016-0032/& 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
O.M. Kwon et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
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1. Introduction Stability analysis for dynamic systems is one of the basic research in control society and a prerequisite job before designing controllers. Specially, stability analysis of time-delay systems has been one of the hot issue since physical and industrial systems naturally have time-delays such as transport, communication, or measurement delays [1]. In other words, since the occurrence of time-delays causes undesirable dynamic behaviors such as performance degradation, instability or oscillation, it is strongly needed to investigate the stability problem for time-delay systems before implementing various control strategies. In line with this view, in order to check the conservatism of stability criteria, an important index is to obtain maximum delay bounds for guaranteeing asymptotic stability of time-delay systems as large as possible. Therefore, in the field of stability analysis, how to construct a Lyapunov–Krasovskii functional, augmented vectors and the derivation of a stability condition from the time-derivative of such a functional with some appropriate techniques play key roles in enhancing the feasible regions of stability criteria. For more details, see [2–10] and references therein. Moreover, the field of stability analysis of time-delay systems can be classified into two categories, i.e., delay-dependent stability criteria and delay-independent ones. Also, it is well known that delay-dependent stability criteria, which make the use of the information on the size of time-delays, are less conservative than delay-independent ones. Thus, more attention has been paid to the derivation of delay-dependent stability criteria for time-delay systems. Naturally, in order to improve results for this problem, various methods were introduced. Simply put, the integral inequality lemmas [11–13], Jensen's inequality [14], the reciprocally convex approach [15], the augmented model [16,17], new cross-term bounding technique [18], triple integral terms [16,19–21], parameterized neutral model transformation method [22] and free weighting matrices technique [23,24] had contributed to enhance the feasible regions of stability criteria for systems with time-delays. Specifically, since estimating a lower bound of the quadratic integral Rt term such as th xT ðsÞQxðsÞ dsðQ40Þ is one of the major topics studied in research field on time-delay systems, Jensen's inequality has been used in plenty as a key lemma in obtaining delay-dependent stability criteria during the last decade. Very recently, the Wirtinger-based integral inequality which reduced the conservatism of Jensen's inequality was introduced in [25] and its advantage was shown via the comparison of maximum delay bounds for various systems such as systems with constant and known delay, systems with a time-varying delay, and sampled-data systems. However, the work [25] focused on the application of the Wirtinger-based integral inequality and some new Lyapunov–Krasovskii functionals were not considered as mentioned in [25]. Therefore, there is still room for a further improvement on the reduction of conservatism in stability analysis for a system with time-varying delays, which motivated this research. In this paper, the problem of delay-dependent stability analysis is investigated by utilizing the Wirtinger-based integral inequality [25]. Unlike the method [25], some new Lyapunov– Krasovskii functionals will be introduced and utilized in stability analysis for systems with different conditions on time-varying delays. The obtained stability criteria are derived in terms of LMIs which can be solved efficiently by using standard convex optimization algorithms such as interior-point methods [26]. Finally, three numerical examples are included to illustrate the effectiveness of the proposed methods. Notation: Throughout this paper, the used notations are standard. Rn is the n-dimensional Euclidean vector space, and Rmn denotes the set of all m n real matrices. Cn;h ¼ Cð½ h; 0; Rn Þ denotes the Banach space of continuous functions mapping the interval ½ h; 0 into Rn , with the Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
O.M. Kwon et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
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topology of uniform convergence. For symmetric matrices X and Y, X4Y means that the matrix X Y is positive definite, whereas X Z Y means that the matrix X Y is nonnegative. In, 0n and 0mn denote n n identity matrix, n n and m n zero matrices, respectively. diagf⋯g denotes the block diagonal matrix. For square matrix X, symfXg means the sum of X and its symmetric matrix XT; i.e., symfXg ¼ X þ X T . X ½f ðtÞ A Rmn means that the elements of matrix X ½f ðtÞ include the scalar value of f(t); i.e., X ½f 0 ¼ X ½f ðtÞ ¼ f 0 . 2. Problem statement and preliminaries Consider the following systems with time-varying delays: x_ ðtÞ ¼ AxðtÞ þ Ad xðt hðtÞÞ; xðsÞ ¼ ϕðsÞ; s A ½ hM ; 0;
ð1Þ
where xðtÞA R is the state vector; ϕðsÞA Cn;hM is a given vector-valued initial function; A and Ad are known constant matrices with appropriate dimensions; the delay, h(t), is a time-varying continuous function that satisfies the following two conditions: n
_ (H1) 0r hðtÞr hM and hðtÞr huD , u _ (H2) 0r hðtÞr hM and hD r hðtÞr huD o1,
where hM, and huD are positive known constant values. The purpose of this paper is to investigate delay-dependent criteria for asymptotic stability of the systems (1) with Conditions (H1) and (H2). Before deriving our main results, the following lemma will be utilized in main results. Lemma 1 (Seuret and Gouaisbaut [25]). For a given matrix M40, the following inequality holds for all continuously differentiable function x in ½a; b-Rn : Z b 1 T 3 T ξ1 ðt ÞMξ1 ðt Þ þ ξ ðt ÞMξ2 ðt Þ x_ T ðsÞM x_ ðsÞ ds Z b a b a 2 a Rb where ξ1 ðtÞ ¼ xðbÞ xðaÞ and ξ2 ðtÞ ¼ xðbÞ þ xðaÞ ð2=ðb aÞÞ a xðsÞ ds. 3. Main results In this section, new stability criteria for the systems (1) with (H1) and (H2) will be derived in Theorems 1 and 2, respectively. For the sake of simplicity on matrix representation in Theorem 1, the notations of several matrices are defined as ei ¼ ½0nði 1Þn ; I n ; 0nð6 iÞn T ði ¼ 1; 2; …; 6Þ; ζ ðt Þ ¼ col xðtÞ; xðt hðtÞÞ; xðt hM Þ; x_ ðtÞ; Z t Z thðtÞ 1 1 xðsÞ ds; xðsÞ ds ; hðtÞ thðtÞ hM hðtÞ thM Π 1;1 ¼ ½e1 e2 ; e1 þ e2 2e5 ; Π 1;2 ¼ ½e2 e3 ; e2 þ e3 2e6 ; " # diagfR; 3Rg M Ψ¼ ; MT diagfR; 3Rg Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
O.M. Kwon et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
4
2
0n
In
6 Γ ½hðtÞ ¼ 4 0n 0n
3
7 hðtÞI n 5; ðhU hðtÞÞI n
Ξ 1½hðtÞ ¼ symf½e1 ; e5 ; e6 Γ ½hðtÞ P½e4 ; e1 e3 T g þ e1 Q1 eT1 e3 Q1 eT3 þ½e1 ; 06nn Q2 ½e1 ; 06nn T ð1 huD Þ½e2 ; e1 e2 Q2 ½e2 ; e1 e2 T þsym ½e5 ; e1 e5 ðhðtÞQ2 Þ½06nn ; e4 T þ h2M e4 ReT4 ; Ξ 2 ¼ ½Π 1;1 ; Π 1;2 Ψ ½Π 1;1 ; Π 1;2 T ; Ξ½hðtÞ ¼ Ξ 1½hðtÞ Ξ 2 ; Υ ¼ AeT1 þ Ad eT2 eT4 :
ð2Þ
Then, the following theorem is given as a main result. Theorem 1. For given positive scalars hM and huD, the systems (1) are asymptotically stable for _ 0r hðtÞr hM and hðtÞr huD , if there exist positive definite matrices P A R2n2n , Q1 A Rnn , 2n2n Q2 ¼ ½Q2;ij 22 A R , R A Rnn and any matrix M A R2n2n satisfying the following LMIs: ðΥ ? ÞT Ξ½0 Υ ? o0;
ð3Þ
ðΥ ? ÞT Ξ½hM Υ ? o0;
ð4Þ
Ψ Z 0;
ð5Þ
where all the notations in Eqs. (3)–(5) are defined in Eq. (2). Proof. Let us consider the following Lyapunov–Krasovskii functional candidate as Z t Z t T T x ðsÞQ1 xðsÞ ds þ ϖ T2 ðt; sÞQ2 ϖ 2 ðt; sÞ ds V ¼ ϖ 1 ðtÞPϖ 1 ðtÞ þ thM thðtÞ Z t Z t þhM x_ T ðuÞR_x ðuÞ du ds; thM s n o Rt Rt where ϖ 1 ðtÞ ¼ col xðtÞ; thM xðsÞ ds and ϖ 2 ðt; sÞ ¼ col xðsÞ; s x_ ðuÞ du . The time-derivative of V can be calculated as 2 3T xðtÞ R # 6 7 " t 1 x_ ðtÞ 6 7 x ð s Þ ds h ð t Þ hðtÞ thðtÞ 7 V_ r 26 6 7 P xðtÞ xðt hM Þ R thðtÞ 4 5 1 þðhM hðt ÞÞ hM hðtÞ thM xðsÞ ds " þxT ðtÞQ1 xðtÞ xT ðt hM ÞQ1 xðt hM Þ þ "
#T
"
xðtÞ 0n1
#T
" Q2 #
xðtÞ
ð6Þ
#
0n1
xðt hðtÞÞ xðt hðtÞÞ Q2 xðtÞ xðt hðtÞÞ xðtÞ xðt hðtÞÞ Z t " xðsÞ #T " 0 # Z t n1 2 T R Q2 x_ T ðsÞR_x ðsÞ ds: þ2 ds þ hM x_ ðtÞR_x ðtÞ hM t _ ðuÞ du x_ ðtÞ s x thðtÞ thM ð1 hD Þ
ð7Þ
Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
O.M. Kwon et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
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Rt By utilizing Lemma 1, a lower bound of the term hM thM x_ T ðsÞR_x ðsÞ ds can be estimated as Z t Z thðtÞ hM x_ T ðsÞR_x ðsÞ ds þ hM x_ T ðsÞR_x ðsÞ ds thðtÞ
thM
1 1 diagfR; 3RgΠ T1;1 ζ ðt Þ þ ζ T ðt ÞΠ 1;2 diagfR; 3RgΠ T1;2 ζ ðt Þ Z ζ ðt ÞΠ 1;1 αðtÞ 1 αðtÞ T
ð8Þ where αðtÞ ¼ hðtÞ=hM . As a result, by the reciprocally convex approach in [15] with any matrix M, it can be obtained as follows: Z t hM ð9Þ x_ T ðsÞR_x ðsÞ ds r ζ T ðtÞΞ 2 ζðtÞ: thM
Then a new stability condition for the system (1) can be V_ r ζ T ðtÞðΞ 1½hðtÞ Ξ 2 ÞζðtÞo0:
ð10Þ
For any free weighting matrix X with an appropriate dimension, the following zero equality holds: 0 ¼ ζ T ðtÞ X Υ þ Υ T X T ζðtÞ: ð11Þ Therefore, Ξ 1½hðtÞ Ξ 2 þ X Υ þ Υ T X T o0
ð12Þ
hold, then V_ is negative definite. By multiplying ðΥ ? ÞT and ðΥ ? Þ by pre- and post-side of the inequality (12), then it can be obtained as ðΥ ? ÞT ðΞ 1½hðtÞ Ξ 2 ÞðΥ ? Þo0:
ð13Þ
Since the inequality (13) depends on h(t) affinely, if LMIs (3)–(4) hold, then the systems (1) are _ asymptotically stable for 0 r hðtÞ r hU and hðtÞr huD . This completes our proof. □ Rt Remark 1. In RTheorem 1, by calculating the time-derivative of thðtÞ ϖ T2 ðt; sÞQ2 ϖ 2 ðt; sÞ ds t unlike the term t hðtÞ xT ðsÞGxðsÞ dsðG A Rnn 40Þ which has been extensively utilized in delay_ dependent stability criteria when 0r hðtÞr hM and hðtÞr huD , some new augmented terms such as # " #T " xðt hðtÞÞ xðt hðtÞÞ ð1 huD Þ Q2 ð14Þ xðtÞ xðt hðtÞÞ xðtÞ xðt hðtÞÞ and Z
t
2 thðtÞ
"
xðsÞ Rt _ ðuÞ du s x
"
#T Q2
# 0n1 ds x_ ðtÞ
ð15Þ
are incorporated into the stability condition as shown in Eq. (10). Eq. (15) can be represented as Z t " xðsÞ #T " 0 # n1 Rt Q2 2 ds _ x ðuÞ du _ x ðtÞ s thðtÞ Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
O.M. Kwon et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
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Z ¼2
T
t
xðsÞ ds t hðtÞ
Z Q2;12 x_ ðt Þ þ 2 hðtÞxðtÞ
T
t
xðsÞ ds
t hðtÞ
Q2;22 x_ ðt Þ
T
T Z t Z t 1 1 ¼2 xðsÞ ds hðtÞQ2;12 x_ ðt Þ þ 2 xðt Þ xðsÞ ds hðtÞQ2;22 x_ ðt Þ hðtÞ t hðtÞ hðtÞ t hðtÞ ¼ ζT ðtÞ symf½e5 ; e1 e5 ðhðtÞQ2 Þ½06nn ; e4 g ζðtÞ:
ð16Þ
In section, it will be shown that the proposed Lyapunov–Krasovskii functional, R t the next T ϖ ðt; sÞQ 2 ϖ 2 ðt; sÞ ds, plays anRimportant role to reduce the conservatism by comparing the 2 thðtÞ t results with the method employing t hðtÞ xT ðsÞGxðsÞ ds. In the second place, a new stability criterion for the systems (1) with (H2) will be derived in Theorem 2. The notations of several matrices in Theorem 2 are defined for simplicity: ei ¼ ½0nði 1Þn ; I n ; 0nð8 iÞn T
ði ¼ 1; 2; …; 8Þ;
ζ^ ðt Þ ¼ col xðtÞ; xðt hðtÞÞ; xðt hM Þ; x_ ðtÞ; x_ ðt hM Þ; Z t Z thðtÞ 1 1 xðsÞ ds; xðsÞ ds; x_ ðt hðt ÞÞ hðtÞ thðtÞ hM hðtÞ thM Π^ 1;1 ¼ ½e1 ; e2 ; e3 ; e6 ; e7 ; Π^ 1;2 ¼ ½e1 ; e2 ; e3 ; e4 ; e5 ; e8 Π^ 2;1 ¼ ½e1 ; e4 ; 08nn ; Π^ 2;2 ¼ ½e2 ; e8 ; e1 e2 ; Π^ 2;3½hðtÞ ¼ ½hðtÞe6 ; e1 e3 ; hðtÞðe1 e6 Þ; Π^ 2;4 ¼ ½08n2n ; e4 ; Π^ 3;1 ¼ ½e3 ; e5 ; e1 e3 ; Π^ 3;2½hðtÞ ¼ ½ðhM hðtÞÞe7 ; e2 e3 ; ðhM hðtÞÞðe1 e7 Þ; Π^ 4;1 ¼ ½e1 e2 ; e1 þ e2 2e6 ; Π^ 4;2 ¼ ½e2 e3 ; e2 þ e3 2e7 ; 3 2 I nn ; 0n4n 7 6 0 ; I ; 0 nn nn n3n 7 6 7 6 7; 0 ; I ; 0 Γ^ ½hðtÞ ¼ 6 n2n nn n2n 7 6 6 0 ; hðtÞI ; 0 7 nn nn 5 4 n3n 0n4n ; ðhU hðtÞÞI nn 3 2 0n3n ; I nn ; 0n2n 7 6 _ 0n5n ; ð1 hðtÞÞI nn 7 6 7 6 ^Λ _ ¼ 6 7; 0n4n ; I nn ; 0nn ½hðtÞ 7 6 7 6 I ; ð1 hðtÞÞI _ nn nn ; 0n4n 5 4 _ 0nn ; ð1 hðtÞÞI nn ; I nn ; 0n3n ^ T1;2 g þ e1 Q1 eT1 e3 Q1 eT3 þ e4 Q2 eT4 e5 Q2 eT5 Ξ^ 1½hðtÞ;hðtÞ ¼ symfΠ^ 1;1 Γ^ ½hðtÞ P Λ^ ½hðtÞ _ _ Π _ Π^ 2;2 Q3 Π^ þ symfΠ^ 2;3½hðtÞ Q3 Π^ g þΠ^ 2;1 Q3 Π^ 2;1 ð1 hðtÞÞ 2;2 2;4 T
T
T
_ Π^ 2;2 Q4 Π^ Π^ 3;1 Q4 Π^ þ symfΠ^ 3;2½hðtÞ Q4 Π^ g þ h2 e4 ReT ; þð1 hðtÞÞ M 4 2;2 3;1 2;4 T
T
T
Ξ^ 2 ¼ ½Π^ 4;1 ; Π^ 4;2 Ψ ½Π^ 4;1 ; Π^ 4;2 T ; Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
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Ξ^ ½hðtÞ;hðtÞ ¼ Ξ^ 1½hðtÞ;hðtÞ Ξ^ 2 ; _ _ Υ^ ¼ AeT þ Ad eT I n eT : 1
2
7
ð17Þ
4
Theorem 2. For given positive scalars hM and huD, the systems (1) are asymptotically stable for _ 0 r hðtÞ r hM and huD r hðtÞr huD o1, if there exist positive definite matrices P A R5n5n , nn 3n3n Qi A R (i¼ 1,2), Qi A R (i¼ 3,4), R A Rnn and any matrix M A R2n2n satisfying the LMIs (5) and ? ? ðΥ^ ÞT Ξ^ ½0; huD Υ^ o0; ?
ð18Þ
?
ðΥ^ ÞT Ξ^ ½0;huD Υ^ o0;
ð19Þ
? ? ðΥ^ ÞT Ξ^ ½hM ; huD Υ^ o0;
ð20Þ
? ? ðΥ^ ÞT Ξ^ ½hM ;huD Υ^ o0;
ð21Þ
where all the notations of Eqs. (18)–(21) are defined in Eq. (17). Proof. Based on a newly Lyapunov–Krasovskii functional given by Z t Z t T T ^ V ¼ϖ ^ 1 ðtÞP ϖ ^ 1 ðtÞ þ x ðsÞQ1 xðsÞ ds þ x_ T ðsÞQ2 x_ ðsÞ ds Z þ
thM
Z
t
thM
thðtÞ
ϖ ^ T2 ðt; sÞQ3 ϖ ^ 2 ðt; sÞ ds þ ϖ ^ T2 ðt; sÞQ4 ϖ ^ 2 ðt; sÞ ds thðtÞ thM Z t Z t þhM ð22Þ x_ T ðuÞR_x ðuÞ du ds; thM s n o Rt R thðtÞ where ϖ ^ 1RðtÞ ¼ col xðtÞ; xðt hðtÞÞ; xðt hM Þ; thðtÞ xðsÞ ds; thM xðsÞ ds and ϖ ^ 2 ðt; sÞ ¼ colf t xðsÞ; x_ ðsÞ; s x_ ðuÞ dug, its time-derivative can be calculated as 3 2 2 3 xðtÞ x_ ðtÞ 7 6 6 7 xðt hðtÞÞ _ x ðt hðtÞÞ 7 6 6 ð1 hðtÞÞ_ 7 7 6 6 7 7 6 6 xðt hM Þ 7 _ x ðt h Þ P V^_ ¼ 26 M 7 R 7 6 t 7 6 6 1 7 hðt Þ hðtÞ x ð s Þ ds _ 7 6 thðtÞ xðtÞ ð1 hðtÞÞxðt hðtÞÞ 5 4 5 4 R thðtÞ 1 _ ð1 hðtÞÞxðt hðtÞÞ xðt hM Þ ðhM hðt ÞÞ h hðtÞ th xðsÞ ds M
M
þx ðtÞQ1 xðtÞ x ðt hM ÞQ1 xðt hM Þ þ x_ T ðtÞQ2 x_ ðtÞ x_ T ðt hM ÞQ2 x_ ðt hM Þ 2 3 2 3 2 3 2 3 xðt hðtÞÞ xðt hðtÞÞ xðtÞ xðtÞ T 6 7 6 7 7 6 7 _ 6 þ4 x_ ðtÞ 5 Q3 4 x_ ðtÞ 5 ð1 hðtÞÞ 4 x_ ðt hðtÞÞ 5Q3 4 x_ ðt hðtÞÞ 5 T
T
xðtÞ xðt hðtÞÞ 0n1 3 " # xðsÞ Z t 02n1 6 x_ ðsÞ 7 þ2 ds 4R 5 Q3 x_ ðtÞ t thðtÞ _ x ðuÞ du s 0n1
2
xðtÞ xðt hðtÞÞ
Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
O.M. Kwon et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
8
2
xðt hðtÞÞ x_ ðt hðtÞÞ
_ 6 þð1 hðtÞÞ 4 2 6 4
2
3T
6 7 5 Q4 4
xðt hðtÞÞ x_ ðt hðtÞÞ
3 7 5
xðtÞ xðt hðtÞÞ xðtÞ xðt hðtÞÞ 3 2 3 xðt hM Þ xðt hM Þ 6 7 x_ ðt hM Þ 7 5Q4 4 x_ ðt hM Þ 5
xðtÞ xðt hM Þ xðtÞ xðt hM Þ 2 3 " # xðsÞ Z thðtÞ 02n1 6 x_ ðsÞ 7 þ2 ds 4R 5Q4 x_ ðtÞ t thM _ ðuÞ du s x Z t x_ T ðsÞR_x ðsÞ ds: þh2M x_ T ðtÞR_x ðtÞ hM
ð23Þ
thM
Then the V^_ has a new upper bound as T ^ V^_ r ζ^ ðtÞðΞ^ 1½hðtÞ Ξ^ 2 ÞζðtÞ:
ð24Þ
From the above inequality, the deriving process of stability condition presented in Eqs. (18)–(21) is very similar to the proof of Theorem 1. Thus, it is omitted. □ Remark 2. In Theorem 2, the state x_ ðt hðtÞÞ was utilized element of the augmenR t as an T ^ ϖ ^ ðt; sÞQ3 ϖ ^ 2 ðt; sÞ ds where ted vector ζðtÞ in Eq. (17). Thus, the term 2 thðtÞ Rt ϖ ^ 2 ðt; sÞ ¼ col xðsÞ; x_ ðsÞ; s x_ ðuÞ du was utilized as one of the Lyapunov–Krasovskii R thðtÞ T functionals. Furthermore, the integral term thM ϖ ^ 2 ðt; sÞQ4 ϖ ^ 2 ðt; sÞ ds which has not been proposed yet was introduced in Theorem 2 and utilized in obtaining the stability criterion of _ Eq. (1) with the condition 0 r hðtÞr hM and huD r hðtÞr huD o1. When huD is larger than one, the stability criterion of Theorem 2 is infeasible because the diagonal term concerning x_ ðt hðtÞÞ cannot be negative definite in all the inequalities (18)–(21). In Theorem 2, the positiveness of V^ is guaranteed such as P40, Q1 40, Q2 40, Qi ði ¼ 3; 4Þ, and R40. Recently, an approach to relaxing the positive-definiteness restriction of the Lyapunov matrices has been proposed in [27,28]. Inspired by these works, a further improved stability condition of Theorem 2 will be introduced as Theorem 3. For the sake of simplicity of vector and matrix representation in Theorem 3, block entry matrices defined in Theorem 1 ei A R6nn will be used. Assume that Qi 40 ði ¼ 1; 2Þ, Qi 40 ði ¼ 3; 4Þ, and R40. Then, the defined Lyapunov– Krasovskii functional V^ in the proof of Theorem 2 has a lower bound as follows: Z t Z t T T ^ V Zϖ ^ 1 ðtÞP ϖ ^ 1 ðtÞ þ x ðsÞQ1 xðsÞ ds þ x_ T ðsÞQ2 x_ ðsÞ ds: ð25Þ thM
thM
Here, by utilizing Corollary 4 in [25] and Lemma 1, it can be obtained as Z t xT ðsÞQ1 xðsÞ ds thM
Z Z ð1=hM Þ
T
t
t hM
xðsÞ ds
Z Q1
t
xðsÞ ds t hM
Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
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9
Z t
T Z t Z s þ ð3=hÞ xðsÞ ds ð2=hM Þ xðuÞ du ds Q1 t hM Z t
Z t Zt shM t hM xðsÞ ds ð2=hM Þ xðuÞ du ds ;
ð26Þ
t hM
Z
t
t hM
t hM
x_ T ðsÞQ2 x_ ðsÞ ds
thM
Z ð1=hM ÞðxðtÞ xðt hM ÞÞT Q2 ðxðtÞ xðt hM ÞÞ
T Z t þ ð3=hM Þ xðtÞ þ xðt hM Þ ð2=hM Þ xðsÞ ds Q2 t hM
Z t xðtÞ þ xðt hM Þ ð2=hM Þ xðsÞ ds :
ð27Þ
t hM
From the above two inequalities, V^ has a new lower bound V^ Z ηT ðtÞΦηðtÞ;
ð28Þ
where Z T T T ηðtÞ ¼ x ðtÞ; x ðt hðtÞÞ; x ðt hM Þ; Z
t
Z
s
t hM
t hðtÞ
T x ðuÞ du ds T
t hM
t
Z x ðsÞ ds;
t hðtÞ
T
xT ðsÞ ds;
t hM
;
ð29Þ
and Φ ¼ ½e1 ; e2 ; e3 ; e4 ; e5 P ½e1 ; e2 ; e3 ; e4 ; e5 þ ð1=hM Þðe4 þ e5 ÞM 1 ðe4 þ e5 ÞT þð3=hM Þðe4 þ e5 ð2=hM Þe6 ÞM 1 ðe4 þ e5 ð2=hM Þe6 ÞT þ ð1=hM Þðe1 e3 ÞM 2 ðe1 e3 ÞT þð3=hM Þðe1 þ e3 ð2=hM Þðe4 þ e5 ÞÞM 2 ðe1 þ e3 ð2=hM Þðe4 þ e5 ÞÞT :
ð30Þ
Therefore, if Φ40 with the condition Qi 40 ði ¼ 1; 2Þ, Qi 40 ði ¼ 3; 4Þ, and R40, the positiveness of V^ in the proof of Theorem 2 can be guaranteed. Therefore, by deleting the positiveness condition of the matrix P and considering Φ40 in Theorem 2, we have the following theorem. Theorem 3. For given positive scalars hM and huD, the systems (1) are asymptotically stable for _ 0 r hðtÞ r hM and huD r hðtÞr huD o1, if there exist positive definite matrices Qi A Rnn 3n3n (i¼ 1,2), Qi A R (i¼ 3,4), R A Rnn and any matrix M A R2n2n satisfying the LMIs (5), (18)–(21) and Φ40. 4. Numerical examples In this section, three numerical examples are introduced to show the improvements of the proposed theorems. In examples, MATLAB, YALMIP [29] and SeDuMi 1.3 [30] are used to solve LMI problems. Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
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Example 1. Consider the system (1) where 2 0 1 0 : A¼ ; Ad ¼ 0 0:9 1 1
ð31Þ
During the last two decades, this example has been numerously utilized in checking the conservatism _ r hu , the obtained of delay-dependent stability criteria. In Table 1, when 0 r hðtÞr hM and hðtÞ D maximum delay bounds by applying Theorem 1 to the above system are listed and compared with the other literature [18,20,24]. From Table 1, it is clear that all the results of Theorem 1 are larger than the other literature, which means that the proposed Theorem 1 reduces the conservatism of stability criterion. To explain Rthe contribution of Theorem 1, let us refer Corollary R1 when Lyapunov– t t Krasovskii functional thðtÞ ϖ T2 ðt; sÞQ2 ϖ 2 ðt; sÞ ds in Theorem 1 is replaced by t hðtÞ xT ðsÞGxðsÞ ds where G A Rnn 40. One can confirm that the results of Corollary 1 are still larger than the other literature listed in Table 1 and smaller than those of Theorem 1. This means that the proposed Rt Lyapunov–Krasovskii functional thðtÞ ϖ T2 ðt; sÞQ2 ϖ 2 ðt; sÞ ds plays an important role to enhance the _ feasible region of stability criterion. When 0 r hðtÞ r hM and huD r hðtÞr huD , the obtained results by applying Theorem 2 are listed and compared with the results published in recent literature [13,16,17,21,25]. From Table 1, it should be noted that all the results listed in Table 1 are more conservative than those of Theorem 2. This implies that the proposed Theorem 2 effectively reduces _ is considered unlike Theorem 1. the conservatism of stability criteria when lower bound of hðtÞ While the number of decision variables of Theorem 7 in [25] is 10n2 þ 3n, the number of decision variables of Theorem 1 is 9n2 þ 3n. It should be noted that the obtained results of Theorem 1 is equal to or slightly larger than those of Theorem 7 in [25] when huD is less than 0.5 in spite of not _ and utilizing fewer number of decision variable. considering lower bound of hðtÞ Example 2. Consider the system (1) with the parameters: 0 0 0 1 A¼ ; Ad ¼ : 0 1 1 1
ð32Þ
Table 2 shows the comparison of maximum delay bounds obtained by the method in other _ literature [11,18,23] and Theorem 1 when 0r hðtÞr hM and hðtÞr huD for different values of huD. Table 1 _ (Example 1). Upper bounds of time-varying delays with different conditions of hðtÞ huD
0
0.1
0.2
0.5
0.8
1
Condition
[24] [20] [18] Theorem 1 Corollary 1
4.472 4.472 4.97 6.0593 5.2305
3.605 3.611 3.86 4.7046 4.1324
3.039 3.047 3.20 3.8347 3.4383
2.043 2.072 2.33 2.4203 2.3845
1.492 1.590 1.93 2.1130 2.1130
1.345 1.529 1.86 2.1130 2.1130
_ rhu hðtÞ D _ rhu hðtÞ D _ rhu hðtÞ D _ rhu hðtÞ D _ rhu hðtÞ
1.499 – 1.529 1.991 2.128 –
_ r hu huD r hðtÞ D u _ r hu hD r hðtÞ D _ r hu huD r hðtÞ D _ r hu huD r hðtÞ D _ r hu huD r hðtÞ D _ r hu huD r hðtÞ D
[13] [21] [16] [17] [25] Theorem 2
4.472 5.3005 4.664 5.120 6.059 6.0593
3.622 3.7304 3.768 4.801 4.703 4.8117
3.087 3.2336 3.257 3.448 3.834 4.1013
2.529 2.5021 2.529 2.528 2.420 3.0615
1.725 2.1728 2.209 2.152 2.137 2.6125
D
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From the results in Table 2, one can see that all the results obtained by Theorem 1 are larger than those of other literature listed in Table 2, which shows that Theorem 1 reduces the conservatism of stability criteria. Example 3. Consider the system (1) where 0 1 0 0 A¼ ; Ad ¼ : 1 2 1 1
ð33Þ
_ huD , the comparison of our results obtained by Theorem 2 When 0 r hðtÞ r hM and huD r hðtÞr with the results in [13,16,17,25] is conducted in Table 3 which shows the less conservatism of Theorem 2 clearly. Furthermore, the results of Theorem 3 show the proposed condition of Theorem 3 which gives slightly larger delay bounds comparing with those of Theorem 2. 5. Conclusions In this paper, the three delay-dependent stability criteria for systems with time-varying delays have been proposed by the use of Lyapunov method and LMI framework. In Theorem 1, by constructing the augmented Lyapunov–Krasovskii functional and utilizing Wirtinger-based inequality, the sufficient condition for asymptotic stability of the system with time-varying delays was derived. Based on the result of Theorem 1, the delay-dependent stability criterion for _ was proposed in Theorem 2 by introducing the systems having the different constraints in hðtÞ newly augmented Lyapunov–Krasovskii functional. With the property of the positiveness of Lyapunov–Krasovskii functional, the further improved stability condition is presented in Theorem 3. Via three numerical examples dealt with in previous works, the improvements of the proposed stability criteria have been successfully verified. Table 2 _ rhu (Example 2). Upper bounds of time-varying delays with conditions of hðtÞ D huD
0
0.05
0.1
0.5
3
[23] [11] [18] Theorem 1
1.82 1.99 2.52 3.0340
1.76 1.81 2.17 2.5502
1.71 1.75 2.02 2.3670
1.38 1.61 1.62 1.6962
– 1.60 1.60 1.6480
Table 3 u _ Upper bounds of time-varying delays with conditions of jhðtÞjrh D (Example 3). huD
0.1
0.2
0.5
0.8
[13] [16] [17] (Theorem 3.1) [17] (Theorem 3.3) [25] Theorem 2 Theorem 3
5.537 5.901 5.823 6.320 6.5906 7.1250 7.1250
3.367 3.839 3.824 3.949 3.6728 4.4131 4.4133
1.183 2.023 2.0083 1.995 1.4118 2.2420 2.2430
0.947 1.404 1.356 1.332 1.2759 1.6599 1.6626
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Acknowledgements The authors would like to thank the editor and anonymous reviewers for their valuable comments and suggestions. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2008-0062611), and by a grant of the Korea Healthcare Technology R & D Project, Ministry of Health & Welfare, Republic of Korea (A100054)
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[25] A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: application to time-delay systems, Automatica 49 (2013) 2860–2866. [26] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [27] S. Xu, J. Lam, B. Zhang, Y. Zou, New insight into delay-dependent stability of time-delay systems, Int. J. Robust Nonlinear Control, http://dx.doi.org/10.1002/rnc.3120. [28] S. Xu, J. Lam, B. Zhang, Y. Zou, A new result on the delay-dependent stability of discrete systems with timevarying delays, Int. J. Robust Nonlinear Control, http://dx.doi.org/10.1002/rnc.3006. [29] J. Löfberg, YAPLMI: a toolbox for modeling and optimization in MATLAB, in: Proceedings of the CACSD Conference, 2004. [30] J.F. Sturm, Using SeDuMi 10.2, a Matlab toolbox for optimization over symmetric cones, Optim. Methods Softw. 11 (1999) 625–653.
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Journal of the Franklin Institute ] (]]]]) ]]]–]]] www.elsevier.com/locate/jfranklin
Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality O.M. Kwona, M.J. Parka, Ju H. Parkb,n, S.M. Leec, E.J. Chad a
School of Electrical Engineering, Chungbuk National University, 52 Naesudong-ro, Heungduk-gu, Cheongju 362-763, Republic of Korea b Nonlinear Dynamics Group, Department of Electrical Engineering, Yeungnam University, 280 Daehak-ro, Kyongsan 712-749, Republic of Korea c School of Electronics Engineering, Daegu University, Gyungsan 712-714, Republic of Korea d Department of Biomedical Engineering, School of Medicine, Chungbuk National University, 52 Naesudong-ro, Heungduk-gu, Cheongju 362-763, Republic of Korea Received 23 January 2014; received in revised form 8 August 2014; accepted 30 September 2014
Abstract In this paper, the problem of stability analysis for linear systems with time-varying delays is considered. By the consideration of new augmented Lyapunov functionals, improved delay-dependent stability criteria for asymptotic stability of the system are proposed for two cases of conditions on time-varying delays with the framework of linear matrix inequalities (LMIs), which can be solved easily by various efficient convex optimization algorithms. The enhancement of the feasible region of the proposed criteria is shown via three numerical examples by the comparison of maximum delay bounds. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author. E-mail addresses: [email protected] (O.M. Kwon), [email protected] (M.J. Park), [email protected] (J.H. Park), [email protected] (S.M. Lee), [email protected] (E.J. Cha). http://dx.doi.org/10.1016/j.jfranklin.2014.09.021 0016-0032/& 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
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1. Introduction Stability analysis for dynamic systems is one of the basic research in control society and a prerequisite job before designing controllers. Specially, stability analysis of time-delay systems has been one of the hot issue since physical and industrial systems naturally have time-delays such as transport, communication, or measurement delays [1]. In other words, since the occurrence of time-delays causes undesirable dynamic behaviors such as performance degradation, instability or oscillation, it is strongly needed to investigate the stability problem for time-delay systems before implementing various control strategies. In line with this view, in order to check the conservatism of stability criteria, an important index is to obtain maximum delay bounds for guaranteeing asymptotic stability of time-delay systems as large as possible. Therefore, in the field of stability analysis, how to construct a Lyapunov–Krasovskii functional, augmented vectors and the derivation of a stability condition from the time-derivative of such a functional with some appropriate techniques play key roles in enhancing the feasible regions of stability criteria. For more details, see [2–10] and references therein. Moreover, the field of stability analysis of time-delay systems can be classified into two categories, i.e., delay-dependent stability criteria and delay-independent ones. Also, it is well known that delay-dependent stability criteria, which make the use of the information on the size of time-delays, are less conservative than delay-independent ones. Thus, more attention has been paid to the derivation of delay-dependent stability criteria for time-delay systems. Naturally, in order to improve results for this problem, various methods were introduced. Simply put, the integral inequality lemmas [11–13], Jensen's inequality [14], the reciprocally convex approach [15], the augmented model [16,17], new cross-term bounding technique [18], triple integral terms [16,19–21], parameterized neutral model transformation method [22] and free weighting matrices technique [23,24] had contributed to enhance the feasible regions of stability criteria for systems with time-delays. Specifically, since estimating a lower bound of the quadratic integral Rt term such as th xT ðsÞQxðsÞ dsðQ40Þ is one of the major topics studied in research field on time-delay systems, Jensen's inequality has been used in plenty as a key lemma in obtaining delay-dependent stability criteria during the last decade. Very recently, the Wirtinger-based integral inequality which reduced the conservatism of Jensen's inequality was introduced in [25] and its advantage was shown via the comparison of maximum delay bounds for various systems such as systems with constant and known delay, systems with a time-varying delay, and sampled-data systems. However, the work [25] focused on the application of the Wirtinger-based integral inequality and some new Lyapunov–Krasovskii functionals were not considered as mentioned in [25]. Therefore, there is still room for a further improvement on the reduction of conservatism in stability analysis for a system with time-varying delays, which motivated this research. In this paper, the problem of delay-dependent stability analysis is investigated by utilizing the Wirtinger-based integral inequality [25]. Unlike the method [25], some new Lyapunov– Krasovskii functionals will be introduced and utilized in stability analysis for systems with different conditions on time-varying delays. The obtained stability criteria are derived in terms of LMIs which can be solved efficiently by using standard convex optimization algorithms such as interior-point methods [26]. Finally, three numerical examples are included to illustrate the effectiveness of the proposed methods. Notation: Throughout this paper, the used notations are standard. Rn is the n-dimensional Euclidean vector space, and Rmn denotes the set of all m n real matrices. Cn;h ¼ Cð½ h; 0; Rn Þ denotes the Banach space of continuous functions mapping the interval ½ h; 0 into Rn , with the Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
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3
topology of uniform convergence. For symmetric matrices X and Y, X4Y means that the matrix X Y is positive definite, whereas X Z Y means that the matrix X Y is nonnegative. In, 0n and 0mn denote n n identity matrix, n n and m n zero matrices, respectively. diagf⋯g denotes the block diagonal matrix. For square matrix X, symfXg means the sum of X and its symmetric matrix XT; i.e., symfXg ¼ X þ X T . X ½f ðtÞ A Rmn means that the elements of matrix X ½f ðtÞ include the scalar value of f(t); i.e., X ½f 0 ¼ X ½f ðtÞ ¼ f 0 . 2. Problem statement and preliminaries Consider the following systems with time-varying delays: x_ ðtÞ ¼ AxðtÞ þ Ad xðt hðtÞÞ; xðsÞ ¼ ϕðsÞ; s A ½ hM ; 0;
ð1Þ
where xðtÞA R is the state vector; ϕðsÞA Cn;hM is a given vector-valued initial function; A and Ad are known constant matrices with appropriate dimensions; the delay, h(t), is a time-varying continuous function that satisfies the following two conditions: n
_ (H1) 0r hðtÞr hM and hðtÞr huD , u _ (H2) 0r hðtÞr hM and hD r hðtÞr huD o1,
where hM, and huD are positive known constant values. The purpose of this paper is to investigate delay-dependent criteria for asymptotic stability of the systems (1) with Conditions (H1) and (H2). Before deriving our main results, the following lemma will be utilized in main results. Lemma 1 (Seuret and Gouaisbaut [25]). For a given matrix M40, the following inequality holds for all continuously differentiable function x in ½a; b-Rn : Z b 1 T 3 T ξ1 ðt ÞMξ1 ðt Þ þ ξ ðt ÞMξ2 ðt Þ x_ T ðsÞM x_ ðsÞ ds Z b a b a 2 a Rb where ξ1 ðtÞ ¼ xðbÞ xðaÞ and ξ2 ðtÞ ¼ xðbÞ þ xðaÞ ð2=ðb aÞÞ a xðsÞ ds. 3. Main results In this section, new stability criteria for the systems (1) with (H1) and (H2) will be derived in Theorems 1 and 2, respectively. For the sake of simplicity on matrix representation in Theorem 1, the notations of several matrices are defined as ei ¼ ½0nði 1Þn ; I n ; 0nð6 iÞn T ði ¼ 1; 2; …; 6Þ; ζ ðt Þ ¼ col xðtÞ; xðt hðtÞÞ; xðt hM Þ; x_ ðtÞ; Z t Z thðtÞ 1 1 xðsÞ ds; xðsÞ ds ; hðtÞ thðtÞ hM hðtÞ thM Π 1;1 ¼ ½e1 e2 ; e1 þ e2 2e5 ; Π 1;2 ¼ ½e2 e3 ; e2 þ e3 2e6 ; " # diagfR; 3Rg M Ψ¼ ; MT diagfR; 3Rg Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
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4
2
0n
In
6 Γ ½hðtÞ ¼ 4 0n 0n
3
7 hðtÞI n 5; ðhU hðtÞÞI n
Ξ 1½hðtÞ ¼ symf½e1 ; e5 ; e6 Γ ½hðtÞ P½e4 ; e1 e3 T g þ e1 Q1 eT1 e3 Q1 eT3 þ½e1 ; 06nn Q2 ½e1 ; 06nn T ð1 huD Þ½e2 ; e1 e2 Q2 ½e2 ; e1 e2 T þsym ½e5 ; e1 e5 ðhðtÞQ2 Þ½06nn ; e4 T þ h2M e4 ReT4 ; Ξ 2 ¼ ½Π 1;1 ; Π 1;2 Ψ ½Π 1;1 ; Π 1;2 T ; Ξ½hðtÞ ¼ Ξ 1½hðtÞ Ξ 2 ; Υ ¼ AeT1 þ Ad eT2 eT4 :
ð2Þ
Then, the following theorem is given as a main result. Theorem 1. For given positive scalars hM and huD, the systems (1) are asymptotically stable for _ 0r hðtÞr hM and hðtÞr huD , if there exist positive definite matrices P A R2n2n , Q1 A Rnn , 2n2n Q2 ¼ ½Q2;ij 22 A R , R A Rnn and any matrix M A R2n2n satisfying the following LMIs: ðΥ ? ÞT Ξ½0 Υ ? o0;
ð3Þ
ðΥ ? ÞT Ξ½hM Υ ? o0;
ð4Þ
Ψ Z 0;
ð5Þ
where all the notations in Eqs. (3)–(5) are defined in Eq. (2). Proof. Let us consider the following Lyapunov–Krasovskii functional candidate as Z t Z t T T x ðsÞQ1 xðsÞ ds þ ϖ T2 ðt; sÞQ2 ϖ 2 ðt; sÞ ds V ¼ ϖ 1 ðtÞPϖ 1 ðtÞ þ thM thðtÞ Z t Z t þhM x_ T ðuÞR_x ðuÞ du ds; thM s n o Rt Rt where ϖ 1 ðtÞ ¼ col xðtÞ; thM xðsÞ ds and ϖ 2 ðt; sÞ ¼ col xðsÞ; s x_ ðuÞ du . The time-derivative of V can be calculated as 2 3T xðtÞ R # 6 7 " t 1 x_ ðtÞ 6 7 x ð s Þ ds h ð t Þ hðtÞ thðtÞ 7 V_ r 26 6 7 P xðtÞ xðt hM Þ R thðtÞ 4 5 1 þðhM hðt ÞÞ hM hðtÞ thM xðsÞ ds " þxT ðtÞQ1 xðtÞ xT ðt hM ÞQ1 xðt hM Þ þ "
#T
"
xðtÞ 0n1
#T
" Q2 #
xðtÞ
ð6Þ
#
0n1
xðt hðtÞÞ xðt hðtÞÞ Q2 xðtÞ xðt hðtÞÞ xðtÞ xðt hðtÞÞ Z t " xðsÞ #T " 0 # Z t n1 2 T R Q2 x_ T ðsÞR_x ðsÞ ds: þ2 ds þ hM x_ ðtÞR_x ðtÞ hM t _ ðuÞ du x_ ðtÞ s x thðtÞ thM ð1 hD Þ
ð7Þ
Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
O.M. Kwon et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
5
Rt By utilizing Lemma 1, a lower bound of the term hM thM x_ T ðsÞR_x ðsÞ ds can be estimated as Z t Z thðtÞ hM x_ T ðsÞR_x ðsÞ ds þ hM x_ T ðsÞR_x ðsÞ ds thðtÞ
thM
1 1 diagfR; 3RgΠ T1;1 ζ ðt Þ þ ζ T ðt ÞΠ 1;2 diagfR; 3RgΠ T1;2 ζ ðt Þ Z ζ ðt ÞΠ 1;1 αðtÞ 1 αðtÞ T
ð8Þ where αðtÞ ¼ hðtÞ=hM . As a result, by the reciprocally convex approach in [15] with any matrix M, it can be obtained as follows: Z t hM ð9Þ x_ T ðsÞR_x ðsÞ ds r ζ T ðtÞΞ 2 ζðtÞ: thM
Then a new stability condition for the system (1) can be V_ r ζ T ðtÞðΞ 1½hðtÞ Ξ 2 ÞζðtÞo0:
ð10Þ
For any free weighting matrix X with an appropriate dimension, the following zero equality holds: 0 ¼ ζ T ðtÞ X Υ þ Υ T X T ζðtÞ: ð11Þ Therefore, Ξ 1½hðtÞ Ξ 2 þ X Υ þ Υ T X T o0
ð12Þ
hold, then V_ is negative definite. By multiplying ðΥ ? ÞT and ðΥ ? Þ by pre- and post-side of the inequality (12), then it can be obtained as ðΥ ? ÞT ðΞ 1½hðtÞ Ξ 2 ÞðΥ ? Þo0:
ð13Þ
Since the inequality (13) depends on h(t) affinely, if LMIs (3)–(4) hold, then the systems (1) are _ asymptotically stable for 0 r hðtÞ r hU and hðtÞr huD . This completes our proof. □ Rt Remark 1. In RTheorem 1, by calculating the time-derivative of thðtÞ ϖ T2 ðt; sÞQ2 ϖ 2 ðt; sÞ ds t unlike the term t hðtÞ xT ðsÞGxðsÞ dsðG A Rnn 40Þ which has been extensively utilized in delay_ dependent stability criteria when 0r hðtÞr hM and hðtÞr huD , some new augmented terms such as # " #T " xðt hðtÞÞ xðt hðtÞÞ ð1 huD Þ Q2 ð14Þ xðtÞ xðt hðtÞÞ xðtÞ xðt hðtÞÞ and Z
t
2 thðtÞ
"
xðsÞ Rt _ ðuÞ du s x
"
#T Q2
# 0n1 ds x_ ðtÞ
ð15Þ
are incorporated into the stability condition as shown in Eq. (10). Eq. (15) can be represented as Z t " xðsÞ #T " 0 # n1 Rt Q2 2 ds _ x ðuÞ du _ x ðtÞ s thðtÞ Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
O.M. Kwon et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
6
Z ¼2
T
t
xðsÞ ds t hðtÞ
Z Q2;12 x_ ðt Þ þ 2 hðtÞxðtÞ
T
t
xðsÞ ds
t hðtÞ
Q2;22 x_ ðt Þ
T
T Z t Z t 1 1 ¼2 xðsÞ ds hðtÞQ2;12 x_ ðt Þ þ 2 xðt Þ xðsÞ ds hðtÞQ2;22 x_ ðt Þ hðtÞ t hðtÞ hðtÞ t hðtÞ ¼ ζT ðtÞ symf½e5 ; e1 e5 ðhðtÞQ2 Þ½06nn ; e4 g ζðtÞ:
ð16Þ
In section, it will be shown that the proposed Lyapunov–Krasovskii functional, R t the next T ϖ ðt; sÞQ 2 ϖ 2 ðt; sÞ ds, plays anRimportant role to reduce the conservatism by comparing the 2 thðtÞ t results with the method employing t hðtÞ xT ðsÞGxðsÞ ds. In the second place, a new stability criterion for the systems (1) with (H2) will be derived in Theorem 2. The notations of several matrices in Theorem 2 are defined for simplicity: ei ¼ ½0nði 1Þn ; I n ; 0nð8 iÞn T
ði ¼ 1; 2; …; 8Þ;
ζ^ ðt Þ ¼ col xðtÞ; xðt hðtÞÞ; xðt hM Þ; x_ ðtÞ; x_ ðt hM Þ; Z t Z thðtÞ 1 1 xðsÞ ds; xðsÞ ds; x_ ðt hðt ÞÞ hðtÞ thðtÞ hM hðtÞ thM Π^ 1;1 ¼ ½e1 ; e2 ; e3 ; e6 ; e7 ; Π^ 1;2 ¼ ½e1 ; e2 ; e3 ; e4 ; e5 ; e8 Π^ 2;1 ¼ ½e1 ; e4 ; 08nn ; Π^ 2;2 ¼ ½e2 ; e8 ; e1 e2 ; Π^ 2;3½hðtÞ ¼ ½hðtÞe6 ; e1 e3 ; hðtÞðe1 e6 Þ; Π^ 2;4 ¼ ½08n2n ; e4 ; Π^ 3;1 ¼ ½e3 ; e5 ; e1 e3 ; Π^ 3;2½hðtÞ ¼ ½ðhM hðtÞÞe7 ; e2 e3 ; ðhM hðtÞÞðe1 e7 Þ; Π^ 4;1 ¼ ½e1 e2 ; e1 þ e2 2e6 ; Π^ 4;2 ¼ ½e2 e3 ; e2 þ e3 2e7 ; 3 2 I nn ; 0n4n 7 6 0 ; I ; 0 nn nn n3n 7 6 7 6 7; 0 ; I ; 0 Γ^ ½hðtÞ ¼ 6 n2n nn n2n 7 6 6 0 ; hðtÞI ; 0 7 nn nn 5 4 n3n 0n4n ; ðhU hðtÞÞI nn 3 2 0n3n ; I nn ; 0n2n 7 6 _ 0n5n ; ð1 hðtÞÞI nn 7 6 7 6 ^Λ _ ¼ 6 7; 0n4n ; I nn ; 0nn ½hðtÞ 7 6 7 6 I ; ð1 hðtÞÞI _ nn nn ; 0n4n 5 4 _ 0nn ; ð1 hðtÞÞI nn ; I nn ; 0n3n ^ T1;2 g þ e1 Q1 eT1 e3 Q1 eT3 þ e4 Q2 eT4 e5 Q2 eT5 Ξ^ 1½hðtÞ;hðtÞ ¼ symfΠ^ 1;1 Γ^ ½hðtÞ P Λ^ ½hðtÞ _ _ Π _ Π^ 2;2 Q3 Π^ þ symfΠ^ 2;3½hðtÞ Q3 Π^ g þΠ^ 2;1 Q3 Π^ 2;1 ð1 hðtÞÞ 2;2 2;4 T
T
T
_ Π^ 2;2 Q4 Π^ Π^ 3;1 Q4 Π^ þ symfΠ^ 3;2½hðtÞ Q4 Π^ g þ h2 e4 ReT ; þð1 hðtÞÞ M 4 2;2 3;1 2;4 T
T
T
Ξ^ 2 ¼ ½Π^ 4;1 ; Π^ 4;2 Ψ ½Π^ 4;1 ; Π^ 4;2 T ; Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
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Ξ^ ½hðtÞ;hðtÞ ¼ Ξ^ 1½hðtÞ;hðtÞ Ξ^ 2 ; _ _ Υ^ ¼ AeT þ Ad eT I n eT : 1
2
7
ð17Þ
4
Theorem 2. For given positive scalars hM and huD, the systems (1) are asymptotically stable for _ 0 r hðtÞ r hM and huD r hðtÞr huD o1, if there exist positive definite matrices P A R5n5n , nn 3n3n Qi A R (i¼ 1,2), Qi A R (i¼ 3,4), R A Rnn and any matrix M A R2n2n satisfying the LMIs (5) and ? ? ðΥ^ ÞT Ξ^ ½0; huD Υ^ o0; ?
ð18Þ
?
ðΥ^ ÞT Ξ^ ½0;huD Υ^ o0;
ð19Þ
? ? ðΥ^ ÞT Ξ^ ½hM ; huD Υ^ o0;
ð20Þ
? ? ðΥ^ ÞT Ξ^ ½hM ;huD Υ^ o0;
ð21Þ
where all the notations of Eqs. (18)–(21) are defined in Eq. (17). Proof. Based on a newly Lyapunov–Krasovskii functional given by Z t Z t T T ^ V ¼ϖ ^ 1 ðtÞP ϖ ^ 1 ðtÞ þ x ðsÞQ1 xðsÞ ds þ x_ T ðsÞQ2 x_ ðsÞ ds Z þ
thM
Z
t
thM
thðtÞ
ϖ ^ T2 ðt; sÞQ3 ϖ ^ 2 ðt; sÞ ds þ ϖ ^ T2 ðt; sÞQ4 ϖ ^ 2 ðt; sÞ ds thðtÞ thM Z t Z t þhM ð22Þ x_ T ðuÞR_x ðuÞ du ds; thM s n o Rt R thðtÞ where ϖ ^ 1RðtÞ ¼ col xðtÞ; xðt hðtÞÞ; xðt hM Þ; thðtÞ xðsÞ ds; thM xðsÞ ds and ϖ ^ 2 ðt; sÞ ¼ colf t xðsÞ; x_ ðsÞ; s x_ ðuÞ dug, its time-derivative can be calculated as 3 2 2 3 xðtÞ x_ ðtÞ 7 6 6 7 xðt hðtÞÞ _ x ðt hðtÞÞ 7 6 6 ð1 hðtÞÞ_ 7 7 6 6 7 7 6 6 xðt hM Þ 7 _ x ðt h Þ P V^_ ¼ 26 M 7 R 7 6 t 7 6 6 1 7 hðt Þ hðtÞ x ð s Þ ds _ 7 6 thðtÞ xðtÞ ð1 hðtÞÞxðt hðtÞÞ 5 4 5 4 R thðtÞ 1 _ ð1 hðtÞÞxðt hðtÞÞ xðt hM Þ ðhM hðt ÞÞ h hðtÞ th xðsÞ ds M
M
þx ðtÞQ1 xðtÞ x ðt hM ÞQ1 xðt hM Þ þ x_ T ðtÞQ2 x_ ðtÞ x_ T ðt hM ÞQ2 x_ ðt hM Þ 2 3 2 3 2 3 2 3 xðt hðtÞÞ xðt hðtÞÞ xðtÞ xðtÞ T 6 7 6 7 7 6 7 _ 6 þ4 x_ ðtÞ 5 Q3 4 x_ ðtÞ 5 ð1 hðtÞÞ 4 x_ ðt hðtÞÞ 5Q3 4 x_ ðt hðtÞÞ 5 T
T
xðtÞ xðt hðtÞÞ 0n1 3 " # xðsÞ Z t 02n1 6 x_ ðsÞ 7 þ2 ds 4R 5 Q3 x_ ðtÞ t thðtÞ _ x ðuÞ du s 0n1
2
xðtÞ xðt hðtÞÞ
Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
O.M. Kwon et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
8
2
xðt hðtÞÞ x_ ðt hðtÞÞ
_ 6 þð1 hðtÞÞ 4 2 6 4
2
3T
6 7 5 Q4 4
xðt hðtÞÞ x_ ðt hðtÞÞ
3 7 5
xðtÞ xðt hðtÞÞ xðtÞ xðt hðtÞÞ 3 2 3 xðt hM Þ xðt hM Þ 6 7 x_ ðt hM Þ 7 5Q4 4 x_ ðt hM Þ 5
xðtÞ xðt hM Þ xðtÞ xðt hM Þ 2 3 " # xðsÞ Z thðtÞ 02n1 6 x_ ðsÞ 7 þ2 ds 4R 5Q4 x_ ðtÞ t thM _ ðuÞ du s x Z t x_ T ðsÞR_x ðsÞ ds: þh2M x_ T ðtÞR_x ðtÞ hM
ð23Þ
thM
Then the V^_ has a new upper bound as T ^ V^_ r ζ^ ðtÞðΞ^ 1½hðtÞ Ξ^ 2 ÞζðtÞ:
ð24Þ
From the above inequality, the deriving process of stability condition presented in Eqs. (18)–(21) is very similar to the proof of Theorem 1. Thus, it is omitted. □ Remark 2. In Theorem 2, the state x_ ðt hðtÞÞ was utilized element of the augmenR t as an T ^ ϖ ^ ðt; sÞQ3 ϖ ^ 2 ðt; sÞ ds where ted vector ζðtÞ in Eq. (17). Thus, the term 2 thðtÞ Rt ϖ ^ 2 ðt; sÞ ¼ col xðsÞ; x_ ðsÞ; s x_ ðuÞ du was utilized as one of the Lyapunov–Krasovskii R thðtÞ T functionals. Furthermore, the integral term thM ϖ ^ 2 ðt; sÞQ4 ϖ ^ 2 ðt; sÞ ds which has not been proposed yet was introduced in Theorem 2 and utilized in obtaining the stability criterion of _ Eq. (1) with the condition 0 r hðtÞr hM and huD r hðtÞr huD o1. When huD is larger than one, the stability criterion of Theorem 2 is infeasible because the diagonal term concerning x_ ðt hðtÞÞ cannot be negative definite in all the inequalities (18)–(21). In Theorem 2, the positiveness of V^ is guaranteed such as P40, Q1 40, Q2 40, Qi ði ¼ 3; 4Þ, and R40. Recently, an approach to relaxing the positive-definiteness restriction of the Lyapunov matrices has been proposed in [27,28]. Inspired by these works, a further improved stability condition of Theorem 2 will be introduced as Theorem 3. For the sake of simplicity of vector and matrix representation in Theorem 3, block entry matrices defined in Theorem 1 ei A R6nn will be used. Assume that Qi 40 ði ¼ 1; 2Þ, Qi 40 ði ¼ 3; 4Þ, and R40. Then, the defined Lyapunov– Krasovskii functional V^ in the proof of Theorem 2 has a lower bound as follows: Z t Z t T T ^ V Zϖ ^ 1 ðtÞP ϖ ^ 1 ðtÞ þ x ðsÞQ1 xðsÞ ds þ x_ T ðsÞQ2 x_ ðsÞ ds: ð25Þ thM
thM
Here, by utilizing Corollary 4 in [25] and Lemma 1, it can be obtained as Z t xT ðsÞQ1 xðsÞ ds thM
Z Z ð1=hM Þ
T
t
t hM
xðsÞ ds
Z Q1
t
xðsÞ ds t hM
Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
O.M. Kwon et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
9
Z t
T Z t Z s þ ð3=hÞ xðsÞ ds ð2=hM Þ xðuÞ du ds Q1 t hM Z t
Z t Zt shM t hM xðsÞ ds ð2=hM Þ xðuÞ du ds ;
ð26Þ
t hM
Z
t
t hM
t hM
x_ T ðsÞQ2 x_ ðsÞ ds
thM
Z ð1=hM ÞðxðtÞ xðt hM ÞÞT Q2 ðxðtÞ xðt hM ÞÞ
T Z t þ ð3=hM Þ xðtÞ þ xðt hM Þ ð2=hM Þ xðsÞ ds Q2 t hM
Z t xðtÞ þ xðt hM Þ ð2=hM Þ xðsÞ ds :
ð27Þ
t hM
From the above two inequalities, V^ has a new lower bound V^ Z ηT ðtÞΦηðtÞ;
ð28Þ
where Z T T T ηðtÞ ¼ x ðtÞ; x ðt hðtÞÞ; x ðt hM Þ; Z
t
Z
s
t hM
t hðtÞ
T x ðuÞ du ds T
t hM
t
Z x ðsÞ ds;
t hðtÞ
T
xT ðsÞ ds;
t hM
;
ð29Þ
and Φ ¼ ½e1 ; e2 ; e3 ; e4 ; e5 P ½e1 ; e2 ; e3 ; e4 ; e5 þ ð1=hM Þðe4 þ e5 ÞM 1 ðe4 þ e5 ÞT þð3=hM Þðe4 þ e5 ð2=hM Þe6 ÞM 1 ðe4 þ e5 ð2=hM Þe6 ÞT þ ð1=hM Þðe1 e3 ÞM 2 ðe1 e3 ÞT þð3=hM Þðe1 þ e3 ð2=hM Þðe4 þ e5 ÞÞM 2 ðe1 þ e3 ð2=hM Þðe4 þ e5 ÞÞT :
ð30Þ
Therefore, if Φ40 with the condition Qi 40 ði ¼ 1; 2Þ, Qi 40 ði ¼ 3; 4Þ, and R40, the positiveness of V^ in the proof of Theorem 2 can be guaranteed. Therefore, by deleting the positiveness condition of the matrix P and considering Φ40 in Theorem 2, we have the following theorem. Theorem 3. For given positive scalars hM and huD, the systems (1) are asymptotically stable for _ 0 r hðtÞ r hM and huD r hðtÞr huD o1, if there exist positive definite matrices Qi A Rnn 3n3n (i¼ 1,2), Qi A R (i¼ 3,4), R A Rnn and any matrix M A R2n2n satisfying the LMIs (5), (18)–(21) and Φ40. 4. Numerical examples In this section, three numerical examples are introduced to show the improvements of the proposed theorems. In examples, MATLAB, YALMIP [29] and SeDuMi 1.3 [30] are used to solve LMI problems. Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
O.M. Kwon et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
10
Example 1. Consider the system (1) where 2 0 1 0 : A¼ ; Ad ¼ 0 0:9 1 1
ð31Þ
During the last two decades, this example has been numerously utilized in checking the conservatism _ r hu , the obtained of delay-dependent stability criteria. In Table 1, when 0 r hðtÞr hM and hðtÞ D maximum delay bounds by applying Theorem 1 to the above system are listed and compared with the other literature [18,20,24]. From Table 1, it is clear that all the results of Theorem 1 are larger than the other literature, which means that the proposed Theorem 1 reduces the conservatism of stability criterion. To explain Rthe contribution of Theorem 1, let us refer Corollary R1 when Lyapunov– t t Krasovskii functional thðtÞ ϖ T2 ðt; sÞQ2 ϖ 2 ðt; sÞ ds in Theorem 1 is replaced by t hðtÞ xT ðsÞGxðsÞ ds where G A Rnn 40. One can confirm that the results of Corollary 1 are still larger than the other literature listed in Table 1 and smaller than those of Theorem 1. This means that the proposed Rt Lyapunov–Krasovskii functional thðtÞ ϖ T2 ðt; sÞQ2 ϖ 2 ðt; sÞ ds plays an important role to enhance the _ feasible region of stability criterion. When 0 r hðtÞ r hM and huD r hðtÞr huD , the obtained results by applying Theorem 2 are listed and compared with the results published in recent literature [13,16,17,21,25]. From Table 1, it should be noted that all the results listed in Table 1 are more conservative than those of Theorem 2. This implies that the proposed Theorem 2 effectively reduces _ is considered unlike Theorem 1. the conservatism of stability criteria when lower bound of hðtÞ While the number of decision variables of Theorem 7 in [25] is 10n2 þ 3n, the number of decision variables of Theorem 1 is 9n2 þ 3n. It should be noted that the obtained results of Theorem 1 is equal to or slightly larger than those of Theorem 7 in [25] when huD is less than 0.5 in spite of not _ and utilizing fewer number of decision variable. considering lower bound of hðtÞ Example 2. Consider the system (1) with the parameters: 0 0 0 1 A¼ ; Ad ¼ : 0 1 1 1
ð32Þ
Table 2 shows the comparison of maximum delay bounds obtained by the method in other _ literature [11,18,23] and Theorem 1 when 0r hðtÞr hM and hðtÞr huD for different values of huD. Table 1 _ (Example 1). Upper bounds of time-varying delays with different conditions of hðtÞ huD
0
0.1
0.2
0.5
0.8
1
Condition
[24] [20] [18] Theorem 1 Corollary 1
4.472 4.472 4.97 6.0593 5.2305
3.605 3.611 3.86 4.7046 4.1324
3.039 3.047 3.20 3.8347 3.4383
2.043 2.072 2.33 2.4203 2.3845
1.492 1.590 1.93 2.1130 2.1130
1.345 1.529 1.86 2.1130 2.1130
_ rhu hðtÞ D _ rhu hðtÞ D _ rhu hðtÞ D _ rhu hðtÞ D _ rhu hðtÞ
1.499 – 1.529 1.991 2.128 –
_ r hu huD r hðtÞ D u _ r hu hD r hðtÞ D _ r hu huD r hðtÞ D _ r hu huD r hðtÞ D _ r hu huD r hðtÞ D _ r hu huD r hðtÞ D
[13] [21] [16] [17] [25] Theorem 2
4.472 5.3005 4.664 5.120 6.059 6.0593
3.622 3.7304 3.768 4.801 4.703 4.8117
3.087 3.2336 3.257 3.448 3.834 4.1013
2.529 2.5021 2.529 2.528 2.420 3.0615
1.725 2.1728 2.209 2.152 2.137 2.6125
D
Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
O.M. Kwon et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
11
From the results in Table 2, one can see that all the results obtained by Theorem 1 are larger than those of other literature listed in Table 2, which shows that Theorem 1 reduces the conservatism of stability criteria. Example 3. Consider the system (1) where 0 1 0 0 A¼ ; Ad ¼ : 1 2 1 1
ð33Þ
_ huD , the comparison of our results obtained by Theorem 2 When 0 r hðtÞ r hM and huD r hðtÞr with the results in [13,16,17,25] is conducted in Table 3 which shows the less conservatism of Theorem 2 clearly. Furthermore, the results of Theorem 3 show the proposed condition of Theorem 3 which gives slightly larger delay bounds comparing with those of Theorem 2. 5. Conclusions In this paper, the three delay-dependent stability criteria for systems with time-varying delays have been proposed by the use of Lyapunov method and LMI framework. In Theorem 1, by constructing the augmented Lyapunov–Krasovskii functional and utilizing Wirtinger-based inequality, the sufficient condition for asymptotic stability of the system with time-varying delays was derived. Based on the result of Theorem 1, the delay-dependent stability criterion for _ was proposed in Theorem 2 by introducing the systems having the different constraints in hðtÞ newly augmented Lyapunov–Krasovskii functional. With the property of the positiveness of Lyapunov–Krasovskii functional, the further improved stability condition is presented in Theorem 3. Via three numerical examples dealt with in previous works, the improvements of the proposed stability criteria have been successfully verified. Table 2 _ rhu (Example 2). Upper bounds of time-varying delays with conditions of hðtÞ D huD
0
0.05
0.1
0.5
3
[23] [11] [18] Theorem 1
1.82 1.99 2.52 3.0340
1.76 1.81 2.17 2.5502
1.71 1.75 2.02 2.3670
1.38 1.61 1.62 1.6962
– 1.60 1.60 1.6480
Table 3 u _ Upper bounds of time-varying delays with conditions of jhðtÞjrh D (Example 3). huD
0.1
0.2
0.5
0.8
[13] [16] [17] (Theorem 3.1) [17] (Theorem 3.3) [25] Theorem 2 Theorem 3
5.537 5.901 5.823 6.320 6.5906 7.1250 7.1250
3.367 3.839 3.824 3.949 3.6728 4.4131 4.4133
1.183 2.023 2.0083 1.995 1.4118 2.2420 2.2430
0.947 1.404 1.356 1.332 1.2759 1.6599 1.6626
Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021
12
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Acknowledgements The authors would like to thank the editor and anonymous reviewers for their valuable comments and suggestions. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2008-0062611), and by a grant of the Korea Healthcare Technology R & D Project, Ministry of Health & Welfare, Republic of Korea (A100054)
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Please cite this article as: O.M. Kwon, et al., Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j. jfranklin.2014.09.021