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A new integral inequality and application to stability of time-delay systems
Applied Mathematics Letters 101 (2020) 106058
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
A new integral inequality and application to stability of time-delay systems Junkang Tian a ,∗, Zerong Ren a , Shouming Zhong b a
School of Mathematics Science, Zunyi Normal College, Zunyi 563002, China School of Mathematics Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China b
article
info
Article history: Received 2 August 2019 Received in revised form 17 September 2019 Accepted 17 September 2019 Available online 28 September 2019 Keywords: Time-delay system Integral inequality Lyapunov–Krasovskii functional (LKF)
abstract This paper is concerned with the delay-dependent stability analysis for linear systems with state and distributed delays. Firstly, based on an integral equality, a new integral inequality is obtained. Then, to show the effectiveness of the new integral inequality, a new delay-dependent stability criterion is derived in terms of linear matrix inequality (LMI). Finally, two numerical examples are given to illustrate the effectiveness of the present result. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Consider the following system with time-delay ∫
t
x(t) ˙ = Ax(t) + Bx(t) + C
x(s)ds
(1)
t−h
x(t) = ϕ(t),
t ∈ [−h, 0]
(2)
where x(t) ∈ Rn is the state vector, A, B, C ∈ Rn×n are constant matrices. h > 0 is a constant time delay, and ϕ(t) is initial condition. During the last two decades, the delay-dependent stability analysis of systems with time-delay via LKF method has received much attention. In order to reduce conservatism of stability criteria, many techniques are presented, such as free-weighting matrix method [1–3], cross term method [4], reciprocally convex approach [5], Jensen inequality method [6,7], Wirtinger-based integral inequality [8–10], various inequality methods [11–14]. It is well known that the triple integral form of LKF can reduce the conservatism of stability ∗ Corresponding author. E-mail addresses: [email protected] (J. Tian), [email protected] (Z. Ren), [email protected] (S. Zhong).
https://doi.org/10.1016/j.aml.2019.106058 0893-9659/© 2019 Elsevier Ltd. All rights reserved.
J. Tian, Z. Ren and S. Zhong / Applied Mathematics Letters 101 (2020) 106058
2
criteria effectively. Recently, without using the Wirtinger integral inequality, a further improved integral inequality presented in [15] is shown more powerful than Jensen inequality. The LKF of [15] includes the )k ∫b( ∫b ∫b∫b triple integral. However, the relationship between a s − a+b x(s)ds and a x(s)ds, a u x(s)dsdu1 , · · · , 2 1 ∫b∫b ∫b · · · x(s)dsdu · · · du was not considered in [15]. This may yield conservative result. k 1 a u1 uk This paper will present a new integral inequality which includes those in [7,10] as special cases by utilizing )k ∫b( ∫b ∫b∫b a new integral equality. The relationship between the a s − a+b x(s)ds and a x(s)ds, a u x(s)dsdu1 , 2 1 ∫b∫b ∫b · · · , a u · · · u x(s)dsduk · · · du1 is considered in our paper. This may yield less conservative result. A 1
k
new stability criterion is obtained by applying the new integral inequality. The advantage of the integral inequality has been illustrated via two numerical examples. 2. Main result Lemma 1. holds
For any continuous function f (x) : [a, b] −→ Rn , and any x ∈ (a, b), the following equality ∫
b
∫
b
dx1 x
Proof . Let In (x) =
1 n!
∫b x
b
∫ dx2 . . .
1 n!
f (xn+1 )dxn+1 =
x1
xn
∫
b
(y − x)n f (y)dy
(3)
x
(y − x)n f (y)dy, then ∫ b 1 dIn (x) =− (y − x)n−1 f (y)dy = −In−1 (x) dx (n − 1)! x
(4)
For In (b) = 0, we obtain ∫
b
In (x) =
b
∫ In−1 (x1 )dx1 =
x ∫ b
b
∫
x
∫
x b
x1
∫
(5)
I0 (xn )dxn
b
∫
b
f (y)dy = xn
x
In−2 (x2 )dx2 = · · ·
xn−1
I0 (xn ) = Then, based on (5) and (6), we obtain ∫ b ∫ In (x) = dx1
b
x1
dx2 · · ·
dx1
=
∫ dx1
b
x1
f (xn+1 )dxn+1
(6)
xn
∫
b
dx2 · · ·
∫
b
dxn xn−1
f (xn+1 )dxn+1
(7)
xn
This completes the proof. Lemma 2 ([7]). For a positive definite matrix Q > 0, and any continuously differentiable function x : [a, b] −→ Rn , the following inequality holds ∫ b 3 5 1 (8) Ω T QΩ1 + Ω T QΩ2 + Ω T QΩ3 x˙ T (s)Qx(s)ds ˙ ≥ b−a 1 b−a 2 b−a 3 a where Ω1 = x(b) − x(a) ∫ b 2 Ω2 = x(b) + x(a) − x(s)ds b−a a ∫ b ∫ b∫ b 6 12 Ω3 = x(b) − x(a) + x(s)ds − x(s)dsdu b−a a (b − a)2 a u
J. Tian, Z. Ren and S. Zhong / Applied Mathematics Letters 101 (2020) 106058
3
To reduce the conservatism of inequality (8), a new integral inequality is proposed as follows. Lemma 3. For a positive definite matrix Q > 0, and any continuously differentiable function x : [a, b] −→ Rn , the following inequality holds ∫
b
x˙ T (s)Qx(s)ds ˙ ≥
a
3 5 7 1 Ω T QΩ1 + Ω T QΩ2 + Ω T QΩ3 + Ω T QΩ4 b−a 1 b−a 2 b−a 3 b−a 4
(9)
where Ω1 , Ω2 , and Ω3 are the same as in (8), and Ω4 = x(b) + x(a) −
12 b−a
∫
b
x(s)ds + a
60 (b − a)2
∫
b
∫
b
x(s)dsdu − a
u
120 (b − a)3
∫
b
∫
b
∫
b
x(s)dsdvdu a
u
v
Proof . Based on Remark 1 in [15], the following inequality holds: ∫
b
x˙ T (s)Qx(s)ds ˙ ≥
a
1 ¯T 12 ¯ T ¯ 1 (x) ¯ 2 (x) Ω (x)Q ˙ Ω ˙ + Ω (x)Q ˙ Ω ˙ b−a 1 (b − a)3 2 180 ¯ T 2800 ¯ T ¯ 3 (x) ¯ 4 (x) + Ω3 (x)Q ˙ Ω ˙ + Ω (x)Q ˙ Ω ˙ 5 (b − a) (b − a)7 4
(10)
where ¯ i (x) Ω ˙ =
∫
b
φi (s)x(s)ds, ˙ i = 1, 2, 3, 4, φ1 (s) = 1, φ2 (s) = s − a ( )2 a+b (b − a)2 , φ3 (s) = s − − 2 12 ( )3 a+b 3(b − a)2 a+b φ4 (s) = s − − (s − ), 2 20 2
a+b , 2
Then, based on Lemma 1, we obtain ¯ 1 (x) Ω ˙ =
∫
b
φ1 (s)x(s)ds ˙ = x(b) − x(a)
(11)
a
( ) ) ∫ b( a+b b−a φ2 (s)x(s)ds ˙ = s− x(s)ds ˙ = (s − a) − x(s)ds ˙ 2 2 a a a ∫ b∫ b ∫ b ∫ b b−a b−a = x(s)dsdu ˙ − x(s)ds ˙ = (x(b) + x(a)) − x(s)ds 2 2 a u a a } )2 ∫ b ∫ b {( 2 a + b (b − a) ¯ 3 (x) Ω ˙ = φ3 (s)x(s)ds ˙ = s− − x(s)ds ˙ 2 12 a a } ∫ b{ (b − a)2 x(s)ds ˙ = (s − a)2 − (s − a)(b − a) + 6 a ∫ b ∫ b ∫ (b − a)2 b = (s − a)2 x(s)ds ˙ − (b − a) (s − a)x(s)ds ˙ + x(s)ds ˙ 6 a a a ∫ b∫ b∫ b ∫ b∫ b ∫ (b − a)2 b =2 x(s)dsdvdu ˙ − (b − a) x(s)dsdv ˙ + x(s)ds ˙ 6 a u v a v a ∫ b ∫ b∫ b (b − a)2 = (x(b) − x(a)) + (b − a) x(s)ds − 2 x(s)dsdu 6 a a u
¯ 2 (x) Ω ˙ =
∫
b
∫
b
(12)
(13)
J. Tian, Z. Ren and S. Zhong / Applied Mathematics Letters 101 (2020) 106058
4
( )} a + b ¯ 4 (x) x(s)ds ˙ Ω ˙ = φ4 (s)x(s)ds ˙ = s− 2 a a } ∫ b{ 3 3 (b − a)3 (s − a)3 − (s − a)2 (b − a) + (s − a)(b − a)2 − x(s)ds ˙ = 2 5 20 a ∫ b ∫ b ∫ b 3 3 = (s − a)3 x(s)ds (s − a)x(s)ds ˙ ˙ − (b − a) (s − a)2 x(s)ds ˙ + (b − a)2 2 5 a a a ∫ b 3 − x(s)ds ˙ (b − a)3 20 a ∫ b∫ b∫ b ∫ b∫ b∫ b∫ b x(s)dsdudv ˙ x(s)dsdβdvdu ˙ − 3(b − a) =6 ∫
b
a
∫
u
v
b
{(
a+b s− 2
)3
3(b − a)2 − 20
β
a
v
(14)
u
∫ ∫ ∫ 3(b − a)2 b b (b − a)3 b + x(s)dsdu ˙ − x(s)ds ˙ 5 20 a u a ∫ ∫ b∫ b (b − a)3 3(b − a)2 b = (x(b) + x(a)) − x(s)ds + 3(b − a) x(s)dsdu 20 5 a a u ∫ b∫ b∫ b +6 x(s)dsdudv a
v
u
Re-rejecting (11)–(14) into (10) yields (9). This completes the proof. Lemma 4 ([10]). For a positive definite matrix Q > 0, and any continuously differentiable function x : [a, b] −→ Rn , the following inequality holds ∫ b∫ b ¯ T QΩ ¯ 5 + 4Ω ¯ T QΩ ¯ 6 + 6Ω ¯ T QΩ ¯7 (15) x˙ T (s)Qx(s)dsdu ˙ ≥ 2Ω 5 6 7 a
u
where ¯ 5 = x(b) − Ω
1 b−a
∫
b
x(s)ds a
∫ b∫ b 6 x(s)ds − x(s)dsdu (b − a)2 a u a ∫ b ∫ b∫ b ∫ b∫ b∫ b 24 60 ¯ 7 = x(b) − 3 Ω x(s)ds + x(s)dsdu − x(s)dsdvdu b−a a (b − a)2 a u (b − a)3 a u v ¯ 6 = x(b) + Ω
2 b−a
∫
b
Based on the above lemmas, an improved less conservative stability criterion for time-delay systems is established. Theorem 1. For given scalar h > 0, then the system (1) is asymptotically stable if there exist symmetric positive matrices P ∈ R4n×4n , Q1 , Q2 , Q3 ∈ Rn×n , such that the following LMI holds h2 T ε Q3 ε0 − Σ3T Q2 Σ3 − 3Σ4T Q2 Σ4 2 0 − 5Σ5T Q2 Σ5 − 7Σ6T Q2 Σ6 − 2Σ7T Q3 Σ7 − 4Σ8T Q3 Σ8 − 6Σ9T Q3 Σ9 < 0
Φ =sym(Σ1T P Σ2 ) + εT1 Q1 ε1 − εT2 Q1 ε2 + h2 εT0 Q2 ε0 +
(16)
where [ ]T 2 , Σ2 = εT0 εT1 − εT2 hεT1 − εT3 h2 εT1 − εT4 , 2 6 12 12 60 120 Σ3 = ε1 − ε2 , Σ4 = ε1 + ε2 − ε3 , Σ5 = ε1 − ε2 + ε3 − 2 ε4 , Σ6 = ε1 + ε2 − ε3 + 2 ε4 − 3 ε5 , h h h h h h 1 2 4 3 24 60 Σ7 = ε1 − ε3 , Σ8 = ε1 + ε3 − 2 ε4 , Σ9 = ε1 − ε3 + 2 ε4 − 3 ε5 , ε0 = Aε1 + Bε2 + Cε3 h h h h h h [ ] and εi ∈ Rn×5n is defined as εi = 0n×(i−1)n In 0n×(5−i)n for i = 1, 2, . . . , 5. [ Σ1 = εT1
εT3
εT4
εT5
]T
J. Tian, Z. Ren and S. Zhong / Applied Mathematics Letters 101 (2020) 106058
Proof . Consider a LKF candidate the same as in [10] ∫ t ∫ t ∫ t x˙ T (s)Q2 x(s)dsdu V (xt ) =η T (t)P η(t) + xT (s)Q1 x(s)ds + h ˙ t−h t−h u ∫ t ∫ t∫ t x˙ T (r)Q3 x(r)drdsdu ˙ + t−h
u
5
(17)
s
where [ ]T ∫t ∫t ∫t ∫t ∫t∫t η(t) = xT (t) t−h xT (s)ds t−h u xT (s)dsdu t−h u s xT (r)drdsdu The derivative of V (xt ) along the solution of system (1) as follows h2 ˙ V˙ (xt ) =2η T (t)P η(t) ˙ + xT (t)Q1 x(t) − xT (t − h)Q1 x(t − h) + h2 x˙ T (t)Q2 x(t) ˙ + x˙ T (t)Q3 x(t) 2 ∫ t ∫ t ∫ t −h x˙ T (s)Q2 x(s)ds ˙ − x˙ T (s)Q3 x(s)dsdu ˙ t−h
t−h
u
Then, we can obtain } { h2 V˙ (xt ) =ξ T (t) sym(Σ1T P Σ2 ) + εT1 Q1 ε1 − εT2 Q1 ε2 + h2 εT0 Q2 ε0 + εT0 Q3 ε0 ξ(t) 2 ∫ t ∫ t ∫ t −h x˙ T (s)Q2 x(s)ds ˙ − x˙ T (s)Q3 x(s)dsdu ˙ t−h
[
ξ(t) = xT (t) xT (t − h)
t−h
∫t
xT (s)ds t−h
(18)
u
∫t
∫t
xT (s)dsdu u
t−h
∫t
∫t∫t
t−h u
s
]T xT (r)drdsdu
Applying Lemmas 3 and 4 yields ∫ t −h x˙ T (s)Q2 x(s)ds ˙ ≤ ξ T (t)(−Σ3T Q2 Σ3 − 3Σ4T Q2 Σ4 − 5Σ5T Q2 Σ5 − 7Σ6T Q2 Σ6 )ξ(t)
(19)
t−h
∫
t
∫
t
− t−h
x˙ T (s)Q3 x(s)dsdu ˙ ≤ ξ T (t)(−2Σ7T Q3 Σ7 − 4Σ8T Q3 Σ8 − 6Σ9T Q3 Σ9 )ξ(t)
(20)
u
Thus, we have V˙ (xt ) ≤ ξ T (t)Φξ(t). This completes the proof. 3. Numerical examples In this section, two numerical examples are given to illustrate the merits of the proposed stability criterion. Example 1. Consider system (1) with: [ ] [ 0.2 0 0 A= ,B= 0.2 0.1 0
] [ 0 −1 ,C= 0 −1
] 0 . −1
The upper bounds of h which keep the system stability are derived by our result and those results in [2,3,7– 10]. The obtained results are listed in Table 1. Table 1 shows that our method is more effective than those in [2,3,7–10]. Example 2. Consider system (1) with: [ 0 A= −100
] [ 1 0 ,B= −1 0.1
] [ 0.1 0 ,C= 0.2 0
] 0 . 0
The upper bounds of h which keep the system stability are derived by our result and those results in [3,7– 10]. The obtained results are listed in Table 2. Table 2 shows that our method is more effective than those in [3,7–10].
6
J. Tian, Z. Ren and S. Zhong / Applied Mathematics Letters 101 (2020) 106058
Table 1 Upper bound of h for Example 1. Methods
Maximum h allowed
NoDv
[2] Theorem Theorem Theorem Theorem Theorem Theorem
1.6339 1.877 1.9504 2.0395 2.0395 2.0402 2.0439
85 16 59 75 27 45 45
Maximum h allowed
NoDv
0.126 0.126 0.577 0.577 0.675 0.728
16 59 75 27 45 45
6 1 3 1 1 1
[8] [9] [3] [7] [10]
Table 2 Upper bound of h for Example 2. Methods Theorem Theorem Theorem Theorem Theorem Theorem
6 1 3 1 1 1
[8] [9] [3] [7] [10]
Acknowledgments This work is supported by Guizhou Provincial Science and Technology Foundation (Qian ke he J zi [2015]2147, Qian ke he LH zi [2015]7006); Guizhou province natural science foundation in China (Qian Jiao He KY [2015]451); High-level Innovative Talents in Guizhou Province (Zun shi ke he rencai[2016]13); Youth Science and Technology Talents Development Project of Education Department of Guizhou Province (Qian ke he KY zi [2017]255); Major Research Projects of Innovative Groups of Education Department of Guizhou Province (Qian jiao he KY [2016]046); New Academic Talents and Innovation Exploration Project of Zunyi Normal College (Qian ke he pingtai rencai[2017]5727-19); The doctoral scientific research Zunyi Normal College (BS[2014]18); Innovation and Entrepreneurship Training Program for College Students of Guizhou Province (2018520891). References [1] Y. He, L. Xie, C. Lin, Further improvement of free-weighting technique for systems with time-varying delay, IEEE Trans. Automat. Control 52 (2) (2007) 293–299. [2] W.H. Chen, W.X. Zheng, Delay-dependent robust stabilization for uncertain neutral systems with distributed delays, Automatica 43 (1) (2007) 95–104. [3] H.B. Zeng, Y. He, M. Mu, J. She, New results on stability ananlysis for systems with discrete distributed delay, Automatica 60 (2015) 189–192. [4] J.H. Kim, Note on stability of linear systems with time-varying delay, Automatica 47 (2011) 2118–2121. [5] P. Park, J.W. Ko, C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica 47 (1) (2011) 235–238. [6] K. Gu, V.L. Kharitonov, J. Chen, Stability of Time-Delay Systems, Birkhauser, Boston, 2003, Control engineering. [7] L.V. Hien, H.M. Trinh, Refined Jensen-based ineuqality approach to stability analysis of time-delay systems, IET Control Theory Appl. 9 (14) (2015) 218–219. [8] A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: Application to time-delay systems, Automatica 49 (9) (2013) 2860–2866. [9] M.J. Park, O.M. Kwon, J.H. Park, S.M. Lee, E.J. Cha, Stability of time-delay systems via Wirtinger-based double integral inequality, Automatica 55 (2015) 204–209. [10] N. Zhao, C. Lin, B. Chen, Q.G. Wang, A new double in tegral inequlity and application to stability test for time-delay systems, Appl. Math. Lett. 65 (2017) 26–31. [11] X.M. Zhang, Q.L. Han, New stability criterion using a matrix-based quadratic convex approach and some novel integral inequalities, IET Control Theory Appl. 8 (12) (2014) 1054–1061. [12] C.K. Zhang, Y. He, L. Jiang, M. Wu, H.B. Zeng, Stability analysis of systems with time-varying delay via relaxed integral inequalities, Systems Control Lett. 92 (2016) 52–61. [13] A. Seuret, F. Gouaisbaut, Hierarchy of LMI conditions for the stability analysis of time-delay systems, Systems Control Lett. 81 (2015) 1–7.
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[14] Q. Song, Q. Yu, Z. Zhao, Y. Liu, F.E. Alsaadi, Boundedness and global robust stability analysis of delayed complex-valued neural networks with interval parameter uncertainties, Neural Netw. 103 (2018) 55–62. [15] J.H. Kim, Further improvement of Jensen inequality and application to stability of time-delayed systems, Automatica 64 (2016) 121–125.
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
A new integral inequality and application to stability of time-delay systems Junkang Tian a ,∗, Zerong Ren a , Shouming Zhong b a
School of Mathematics Science, Zunyi Normal College, Zunyi 563002, China School of Mathematics Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China b
article
info
Article history: Received 2 August 2019 Received in revised form 17 September 2019 Accepted 17 September 2019 Available online 28 September 2019 Keywords: Time-delay system Integral inequality Lyapunov–Krasovskii functional (LKF)
abstract This paper is concerned with the delay-dependent stability analysis for linear systems with state and distributed delays. Firstly, based on an integral equality, a new integral inequality is obtained. Then, to show the effectiveness of the new integral inequality, a new delay-dependent stability criterion is derived in terms of linear matrix inequality (LMI). Finally, two numerical examples are given to illustrate the effectiveness of the present result. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Consider the following system with time-delay ∫
t
x(t) ˙ = Ax(t) + Bx(t) + C
x(s)ds
(1)
t−h
x(t) = ϕ(t),
t ∈ [−h, 0]
(2)
where x(t) ∈ Rn is the state vector, A, B, C ∈ Rn×n are constant matrices. h > 0 is a constant time delay, and ϕ(t) is initial condition. During the last two decades, the delay-dependent stability analysis of systems with time-delay via LKF method has received much attention. In order to reduce conservatism of stability criteria, many techniques are presented, such as free-weighting matrix method [1–3], cross term method [4], reciprocally convex approach [5], Jensen inequality method [6,7], Wirtinger-based integral inequality [8–10], various inequality methods [11–14]. It is well known that the triple integral form of LKF can reduce the conservatism of stability ∗ Corresponding author. E-mail addresses: [email protected] (J. Tian), [email protected] (Z. Ren), [email protected] (S. Zhong).
https://doi.org/10.1016/j.aml.2019.106058 0893-9659/© 2019 Elsevier Ltd. All rights reserved.
J. Tian, Z. Ren and S. Zhong / Applied Mathematics Letters 101 (2020) 106058
2
criteria effectively. Recently, without using the Wirtinger integral inequality, a further improved integral inequality presented in [15] is shown more powerful than Jensen inequality. The LKF of [15] includes the )k ∫b( ∫b ∫b∫b triple integral. However, the relationship between a s − a+b x(s)ds and a x(s)ds, a u x(s)dsdu1 , · · · , 2 1 ∫b∫b ∫b · · · x(s)dsdu · · · du was not considered in [15]. This may yield conservative result. k 1 a u1 uk This paper will present a new integral inequality which includes those in [7,10] as special cases by utilizing )k ∫b( ∫b ∫b∫b a new integral equality. The relationship between the a s − a+b x(s)ds and a x(s)ds, a u x(s)dsdu1 , 2 1 ∫b∫b ∫b · · · , a u · · · u x(s)dsduk · · · du1 is considered in our paper. This may yield less conservative result. A 1
k
new stability criterion is obtained by applying the new integral inequality. The advantage of the integral inequality has been illustrated via two numerical examples. 2. Main result Lemma 1. holds
For any continuous function f (x) : [a, b] −→ Rn , and any x ∈ (a, b), the following equality ∫
b
∫
b
dx1 x
Proof . Let In (x) =
1 n!
∫b x
b
∫ dx2 . . .
1 n!
f (xn+1 )dxn+1 =
x1
xn
∫
b
(y − x)n f (y)dy
(3)
x
(y − x)n f (y)dy, then ∫ b 1 dIn (x) =− (y − x)n−1 f (y)dy = −In−1 (x) dx (n − 1)! x
(4)
For In (b) = 0, we obtain ∫
b
In (x) =
b
∫ In−1 (x1 )dx1 =
x ∫ b
b
∫
x
∫
x b
x1
∫
(5)
I0 (xn )dxn
b
∫
b
f (y)dy = xn
x
In−2 (x2 )dx2 = · · ·
xn−1
I0 (xn ) = Then, based on (5) and (6), we obtain ∫ b ∫ In (x) = dx1
b
x1
dx2 · · ·
dx1
=
∫ dx1
b
x1
f (xn+1 )dxn+1
(6)
xn
∫
b
dx2 · · ·
∫
b
dxn xn−1
f (xn+1 )dxn+1
(7)
xn
This completes the proof. Lemma 2 ([7]). For a positive definite matrix Q > 0, and any continuously differentiable function x : [a, b] −→ Rn , the following inequality holds ∫ b 3 5 1 (8) Ω T QΩ1 + Ω T QΩ2 + Ω T QΩ3 x˙ T (s)Qx(s)ds ˙ ≥ b−a 1 b−a 2 b−a 3 a where Ω1 = x(b) − x(a) ∫ b 2 Ω2 = x(b) + x(a) − x(s)ds b−a a ∫ b ∫ b∫ b 6 12 Ω3 = x(b) − x(a) + x(s)ds − x(s)dsdu b−a a (b − a)2 a u
J. Tian, Z. Ren and S. Zhong / Applied Mathematics Letters 101 (2020) 106058
3
To reduce the conservatism of inequality (8), a new integral inequality is proposed as follows. Lemma 3. For a positive definite matrix Q > 0, and any continuously differentiable function x : [a, b] −→ Rn , the following inequality holds ∫
b
x˙ T (s)Qx(s)ds ˙ ≥
a
3 5 7 1 Ω T QΩ1 + Ω T QΩ2 + Ω T QΩ3 + Ω T QΩ4 b−a 1 b−a 2 b−a 3 b−a 4
(9)
where Ω1 , Ω2 , and Ω3 are the same as in (8), and Ω4 = x(b) + x(a) −
12 b−a
∫
b
x(s)ds + a
60 (b − a)2
∫
b
∫
b
x(s)dsdu − a
u
120 (b − a)3
∫
b
∫
b
∫
b
x(s)dsdvdu a
u
v
Proof . Based on Remark 1 in [15], the following inequality holds: ∫
b
x˙ T (s)Qx(s)ds ˙ ≥
a
1 ¯T 12 ¯ T ¯ 1 (x) ¯ 2 (x) Ω (x)Q ˙ Ω ˙ + Ω (x)Q ˙ Ω ˙ b−a 1 (b − a)3 2 180 ¯ T 2800 ¯ T ¯ 3 (x) ¯ 4 (x) + Ω3 (x)Q ˙ Ω ˙ + Ω (x)Q ˙ Ω ˙ 5 (b − a) (b − a)7 4
(10)
where ¯ i (x) Ω ˙ =
∫
b
φi (s)x(s)ds, ˙ i = 1, 2, 3, 4, φ1 (s) = 1, φ2 (s) = s − a ( )2 a+b (b − a)2 , φ3 (s) = s − − 2 12 ( )3 a+b 3(b − a)2 a+b φ4 (s) = s − − (s − ), 2 20 2
a+b , 2
Then, based on Lemma 1, we obtain ¯ 1 (x) Ω ˙ =
∫
b
φ1 (s)x(s)ds ˙ = x(b) − x(a)
(11)
a
( ) ) ∫ b( a+b b−a φ2 (s)x(s)ds ˙ = s− x(s)ds ˙ = (s − a) − x(s)ds ˙ 2 2 a a a ∫ b∫ b ∫ b ∫ b b−a b−a = x(s)dsdu ˙ − x(s)ds ˙ = (x(b) + x(a)) − x(s)ds 2 2 a u a a } )2 ∫ b ∫ b {( 2 a + b (b − a) ¯ 3 (x) Ω ˙ = φ3 (s)x(s)ds ˙ = s− − x(s)ds ˙ 2 12 a a } ∫ b{ (b − a)2 x(s)ds ˙ = (s − a)2 − (s − a)(b − a) + 6 a ∫ b ∫ b ∫ (b − a)2 b = (s − a)2 x(s)ds ˙ − (b − a) (s − a)x(s)ds ˙ + x(s)ds ˙ 6 a a a ∫ b∫ b∫ b ∫ b∫ b ∫ (b − a)2 b =2 x(s)dsdvdu ˙ − (b − a) x(s)dsdv ˙ + x(s)ds ˙ 6 a u v a v a ∫ b ∫ b∫ b (b − a)2 = (x(b) − x(a)) + (b − a) x(s)ds − 2 x(s)dsdu 6 a a u
¯ 2 (x) Ω ˙ =
∫
b
∫
b
(12)
(13)
J. Tian, Z. Ren and S. Zhong / Applied Mathematics Letters 101 (2020) 106058
4
( )} a + b ¯ 4 (x) x(s)ds ˙ Ω ˙ = φ4 (s)x(s)ds ˙ = s− 2 a a } ∫ b{ 3 3 (b − a)3 (s − a)3 − (s − a)2 (b − a) + (s − a)(b − a)2 − x(s)ds ˙ = 2 5 20 a ∫ b ∫ b ∫ b 3 3 = (s − a)3 x(s)ds (s − a)x(s)ds ˙ ˙ − (b − a) (s − a)2 x(s)ds ˙ + (b − a)2 2 5 a a a ∫ b 3 − x(s)ds ˙ (b − a)3 20 a ∫ b∫ b∫ b ∫ b∫ b∫ b∫ b x(s)dsdudv ˙ x(s)dsdβdvdu ˙ − 3(b − a) =6 ∫
b
a
∫
u
v
b
{(
a+b s− 2
)3
3(b − a)2 − 20
β
a
v
(14)
u
∫ ∫ ∫ 3(b − a)2 b b (b − a)3 b + x(s)dsdu ˙ − x(s)ds ˙ 5 20 a u a ∫ ∫ b∫ b (b − a)3 3(b − a)2 b = (x(b) + x(a)) − x(s)ds + 3(b − a) x(s)dsdu 20 5 a a u ∫ b∫ b∫ b +6 x(s)dsdudv a
v
u
Re-rejecting (11)–(14) into (10) yields (9). This completes the proof. Lemma 4 ([10]). For a positive definite matrix Q > 0, and any continuously differentiable function x : [a, b] −→ Rn , the following inequality holds ∫ b∫ b ¯ T QΩ ¯ 5 + 4Ω ¯ T QΩ ¯ 6 + 6Ω ¯ T QΩ ¯7 (15) x˙ T (s)Qx(s)dsdu ˙ ≥ 2Ω 5 6 7 a
u
where ¯ 5 = x(b) − Ω
1 b−a
∫
b
x(s)ds a
∫ b∫ b 6 x(s)ds − x(s)dsdu (b − a)2 a u a ∫ b ∫ b∫ b ∫ b∫ b∫ b 24 60 ¯ 7 = x(b) − 3 Ω x(s)ds + x(s)dsdu − x(s)dsdvdu b−a a (b − a)2 a u (b − a)3 a u v ¯ 6 = x(b) + Ω
2 b−a
∫
b
Based on the above lemmas, an improved less conservative stability criterion for time-delay systems is established. Theorem 1. For given scalar h > 0, then the system (1) is asymptotically stable if there exist symmetric positive matrices P ∈ R4n×4n , Q1 , Q2 , Q3 ∈ Rn×n , such that the following LMI holds h2 T ε Q3 ε0 − Σ3T Q2 Σ3 − 3Σ4T Q2 Σ4 2 0 − 5Σ5T Q2 Σ5 − 7Σ6T Q2 Σ6 − 2Σ7T Q3 Σ7 − 4Σ8T Q3 Σ8 − 6Σ9T Q3 Σ9 < 0
Φ =sym(Σ1T P Σ2 ) + εT1 Q1 ε1 − εT2 Q1 ε2 + h2 εT0 Q2 ε0 +
(16)
where [ ]T 2 , Σ2 = εT0 εT1 − εT2 hεT1 − εT3 h2 εT1 − εT4 , 2 6 12 12 60 120 Σ3 = ε1 − ε2 , Σ4 = ε1 + ε2 − ε3 , Σ5 = ε1 − ε2 + ε3 − 2 ε4 , Σ6 = ε1 + ε2 − ε3 + 2 ε4 − 3 ε5 , h h h h h h 1 2 4 3 24 60 Σ7 = ε1 − ε3 , Σ8 = ε1 + ε3 − 2 ε4 , Σ9 = ε1 − ε3 + 2 ε4 − 3 ε5 , ε0 = Aε1 + Bε2 + Cε3 h h h h h h [ ] and εi ∈ Rn×5n is defined as εi = 0n×(i−1)n In 0n×(5−i)n for i = 1, 2, . . . , 5. [ Σ1 = εT1
εT3
εT4
εT5
]T
J. Tian, Z. Ren and S. Zhong / Applied Mathematics Letters 101 (2020) 106058
Proof . Consider a LKF candidate the same as in [10] ∫ t ∫ t ∫ t x˙ T (s)Q2 x(s)dsdu V (xt ) =η T (t)P η(t) + xT (s)Q1 x(s)ds + h ˙ t−h t−h u ∫ t ∫ t∫ t x˙ T (r)Q3 x(r)drdsdu ˙ + t−h
u
5
(17)
s
where [ ]T ∫t ∫t ∫t ∫t ∫t∫t η(t) = xT (t) t−h xT (s)ds t−h u xT (s)dsdu t−h u s xT (r)drdsdu The derivative of V (xt ) along the solution of system (1) as follows h2 ˙ V˙ (xt ) =2η T (t)P η(t) ˙ + xT (t)Q1 x(t) − xT (t − h)Q1 x(t − h) + h2 x˙ T (t)Q2 x(t) ˙ + x˙ T (t)Q3 x(t) 2 ∫ t ∫ t ∫ t −h x˙ T (s)Q2 x(s)ds ˙ − x˙ T (s)Q3 x(s)dsdu ˙ t−h
t−h
u
Then, we can obtain } { h2 V˙ (xt ) =ξ T (t) sym(Σ1T P Σ2 ) + εT1 Q1 ε1 − εT2 Q1 ε2 + h2 εT0 Q2 ε0 + εT0 Q3 ε0 ξ(t) 2 ∫ t ∫ t ∫ t −h x˙ T (s)Q2 x(s)ds ˙ − x˙ T (s)Q3 x(s)dsdu ˙ t−h
[
ξ(t) = xT (t) xT (t − h)
t−h
∫t
xT (s)ds t−h
(18)
u
∫t
∫t
xT (s)dsdu u
t−h
∫t
∫t∫t
t−h u
s
]T xT (r)drdsdu
Applying Lemmas 3 and 4 yields ∫ t −h x˙ T (s)Q2 x(s)ds ˙ ≤ ξ T (t)(−Σ3T Q2 Σ3 − 3Σ4T Q2 Σ4 − 5Σ5T Q2 Σ5 − 7Σ6T Q2 Σ6 )ξ(t)
(19)
t−h
∫
t
∫
t
− t−h
x˙ T (s)Q3 x(s)dsdu ˙ ≤ ξ T (t)(−2Σ7T Q3 Σ7 − 4Σ8T Q3 Σ8 − 6Σ9T Q3 Σ9 )ξ(t)
(20)
u
Thus, we have V˙ (xt ) ≤ ξ T (t)Φξ(t). This completes the proof. 3. Numerical examples In this section, two numerical examples are given to illustrate the merits of the proposed stability criterion. Example 1. Consider system (1) with: [ ] [ 0.2 0 0 A= ,B= 0.2 0.1 0
] [ 0 −1 ,C= 0 −1
] 0 . −1
The upper bounds of h which keep the system stability are derived by our result and those results in [2,3,7– 10]. The obtained results are listed in Table 1. Table 1 shows that our method is more effective than those in [2,3,7–10]. Example 2. Consider system (1) with: [ 0 A= −100
] [ 1 0 ,B= −1 0.1
] [ 0.1 0 ,C= 0.2 0
] 0 . 0
The upper bounds of h which keep the system stability are derived by our result and those results in [3,7– 10]. The obtained results are listed in Table 2. Table 2 shows that our method is more effective than those in [3,7–10].
6
J. Tian, Z. Ren and S. Zhong / Applied Mathematics Letters 101 (2020) 106058
Table 1 Upper bound of h for Example 1. Methods
Maximum h allowed
NoDv
[2] Theorem Theorem Theorem Theorem Theorem Theorem
1.6339 1.877 1.9504 2.0395 2.0395 2.0402 2.0439
85 16 59 75 27 45 45
Maximum h allowed
NoDv
0.126 0.126 0.577 0.577 0.675 0.728
16 59 75 27 45 45
6 1 3 1 1 1
[8] [9] [3] [7] [10]
Table 2 Upper bound of h for Example 2. Methods Theorem Theorem Theorem Theorem Theorem Theorem
6 1 3 1 1 1
[8] [9] [3] [7] [10]
Acknowledgments This work is supported by Guizhou Provincial Science and Technology Foundation (Qian ke he J zi [2015]2147, Qian ke he LH zi [2015]7006); Guizhou province natural science foundation in China (Qian Jiao He KY [2015]451); High-level Innovative Talents in Guizhou Province (Zun shi ke he rencai[2016]13); Youth Science and Technology Talents Development Project of Education Department of Guizhou Province (Qian ke he KY zi [2017]255); Major Research Projects of Innovative Groups of Education Department of Guizhou Province (Qian jiao he KY [2016]046); New Academic Talents and Innovation Exploration Project of Zunyi Normal College (Qian ke he pingtai rencai[2017]5727-19); The doctoral scientific research Zunyi Normal College (BS[2014]18); Innovation and Entrepreneurship Training Program for College Students of Guizhou Province (2018520891). References [1] Y. He, L. Xie, C. Lin, Further improvement of free-weighting technique for systems with time-varying delay, IEEE Trans. Automat. Control 52 (2) (2007) 293–299. [2] W.H. Chen, W.X. Zheng, Delay-dependent robust stabilization for uncertain neutral systems with distributed delays, Automatica 43 (1) (2007) 95–104. [3] H.B. Zeng, Y. He, M. Mu, J. She, New results on stability ananlysis for systems with discrete distributed delay, Automatica 60 (2015) 189–192. [4] J.H. Kim, Note on stability of linear systems with time-varying delay, Automatica 47 (2011) 2118–2121. [5] P. Park, J.W. Ko, C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica 47 (1) (2011) 235–238. [6] K. Gu, V.L. Kharitonov, J. Chen, Stability of Time-Delay Systems, Birkhauser, Boston, 2003, Control engineering. [7] L.V. Hien, H.M. Trinh, Refined Jensen-based ineuqality approach to stability analysis of time-delay systems, IET Control Theory Appl. 9 (14) (2015) 218–219. [8] A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: Application to time-delay systems, Automatica 49 (9) (2013) 2860–2866. [9] M.J. Park, O.M. Kwon, J.H. Park, S.M. Lee, E.J. Cha, Stability of time-delay systems via Wirtinger-based double integral inequality, Automatica 55 (2015) 204–209. [10] N. Zhao, C. Lin, B. Chen, Q.G. Wang, A new double in tegral inequlity and application to stability test for time-delay systems, Appl. Math. Lett. 65 (2017) 26–31. [11] X.M. Zhang, Q.L. Han, New stability criterion using a matrix-based quadratic convex approach and some novel integral inequalities, IET Control Theory Appl. 8 (12) (2014) 1054–1061. [12] C.K. Zhang, Y. He, L. Jiang, M. Wu, H.B. Zeng, Stability analysis of systems with time-varying delay via relaxed integral inequalities, Systems Control Lett. 92 (2016) 52–61. [13] A. Seuret, F. Gouaisbaut, Hierarchy of LMI conditions for the stability analysis of time-delay systems, Systems Control Lett. 81 (2015) 1–7.
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