Stability analysis of systems with two additive time-varying delay components via an improved delay interconnection Lyapunov–Krasovskii functional

Available online at www.sciencedirect.com

Journal of the Franklin Institute 356 (2019) 3457–3473 www.elsevier.com/locate/jfranklin

Stability analysis of systems with two additive time-varying delay components via an improved delay interconnection Lyapunov–Krasovskii functional Meng Liu a,b, Yong He a,b,∗, Min Wu a,b, Jianhua Shen c b Hubei

a School of Automation, China University of Geosciences, Wuhan 430074, China Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems, Wuhan 430074, China c Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

Received 6 September 2018; received in revised form 5 December 2018; accepted 4 February 2019 Available online 25 February 2019

Abstract This paper is concerned with the stability analysis of systems with two additive time-varying delay components in an improved delay interconnection Lyapunov–Krasovskii framework. At first, an augmented vector and some integral terms considering the additive delays information in a new way are introduced to the Lyapunov–Krasovskii functional (LKF), in which the information of the two upper bounds and the relationship between the two upper bounds and the upper bound of the total delay are both fully considered. Then, the obtained stability criterion shows advantage over the existing ones since not only an improved delay interconnection LKF is constructed but also some advanced techniques such as the free-matrix-based integral inequality and extended reciprocally convex matrix inequality are used to estimate the upper bound of the derivative of the proposed LKF. Finally, a numerical example is given to demonstrate the effectiveness and to show the superiority of the proposed method over existing results. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

∗

Corresponding author at: School of Automation, China University of Geosciences, Wuhan 430074, China. E-mail address: [email protected] (Y. He).

https://doi.org/10.1016/j.jfranklin.2019.02.006 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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1. Introduction In practical application, the time-varying delay is a general phenomenon that often occur in the fields of industrial production, network transportation, circuit signal system [1–5]. The existence of time-varying delay may make the system’s performance worse significantly and even instability. Therefore, the stability analysis of systems with time-varying delay has been becoming a great deal of research attention in the past few decades [6,7]. The most important concern of stability analysis is to find a maximum allowable upper bound (MAUB) of the time-delay such that time-delay system remains asymptotically stable for any delay less than the MAUB. Therefore, how to reduce the conservativeness of the stability criterion and obtain the MAUB of time-varying delay is the main direction of the stability study of time-varying delay systems. At present, the most commonly used method is the Lyapunov stability theory [8], which mainly starts from the following two aspects: selecting the appropriate Lyapunov– Krasovskii functional (LKF) and reducing the enlargement when estimating its derivative. It is vital to choose appropriate LKF for obtaining a less conservative criterion. Currently, LKF is mainly constructed in the following ways: discretized LKF [9], delay-partitioning LKF [10], augmented LKF [11], etc. In addition, some triple integral terms [12] are also introduced into the LKF in order to further reduce the conservativeness. However, these LKFs are difficult to balance between the conservativeness of the results and the computational complexity. So it is necessary to select the appropriate functional according to the form of the system for the stability analysis. For the derivative of LKF, the enlargement the quadratic single integral terms appropriately in the derivative plays an important role. At present, various methods are applied for getting a less conservative criterion. The model transformation methods [13] combined with Park or Moon’s inequality were applied to handle the integral terms in the early literature. However, the inequality-baesd cross term bounding needs to be amplified inevitably and leads to great conservatism. The free-weighting-matrix (FWM) approach [14,15] was proposed by He et al., which introduced the free matrix variables and had greater advantages. Later, the improved FWM approach [16] was put forward, which fully considered the relationship between timevarying delay, its upper bound and the difference of both. But the FWM method introduces a large number of matrix variables, which makes the calculation more complicated. In view of the limitations of the FWM approach, in recent years, various integral inequality methods have been proposed, such as Jensen inequality [17], Wirtinger-based inequality [18], auxiliary function-based integral inequality [19], free-matrix-based integral inequalities [20,21], double integral inequality [22], Bessel–Legendre inequalities [23,24] and so on. In addition, during analyzing the systems with a time-varying delay, the general idea is to divide the time delay into two parts, the time-varying delay and the distance from it to the upper bound. Among them, it can be seen that FWM can handle these two parts uniformly. However, some integral inequalities proposed so far cannot treat them together, the simplest treatment is to directly replace the time-varying delay, d(t), and the distance from it to the upper bound, d − d (t ), by its upper bound, d, while the enlargement inevitably leads to the conservatism. Later on, an effective method for this task is to use the reciprocally convex combination lemma (RCCL) [25] directly handling the time-varying delay, which can move d(t) from the denominator to the numerator in the quadratic term, by introducing some slack matrices. And then, relax integral inequality [26], the bounding inequality combined with the RCCL, is becoming the most popular method for estimating the integral terms with time-

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varying delays. According to the RCCL, Zhang et al. proposed an extended reciprocally convex matrix inequality in 2017 [27]. It can not only contain relax integral inequality but also reduce estimation gap of the RCCL-based matrix inequality and the number of decision variables. Soon after, the extended reciprocally convex matrix inequality has been further improved [28], which can be used to handle more cases because the proposed matrix inequality does not require the same integrand of two integral terms. In addition, in the previous paper, the quadratic integral terms, which derive from the double integral term and the triple integral term, were estimated separately. However, it can be seen that there are some common features between these quadratic integral terms. In accordance with the proposed matrix inequality, these terms can be processed together. Thus, a stability criterion of a linear system with a time-delay is obtained in [28]. This greatly shows the advantage of this method over other methods. A large number of stability criteria for systems with time delay are usually considered in a single delay in the state. Furthermore, for the stability problems of systems with two constant time delays, in 2006, He et al. constructed the LKF from a new perspective in [29], which contains an integral terms of the state with respect to the relationship between two constant delays. However, this paper only considers the stability of systems with constant time delays. So it leads to a certain degree of conservatism, which leaves room for further investigation. In many practical applications, signals transmitted from one point to another may experience a few different segments of networks, which may result in successive delays of different properties due to different network transmission conditions. One simple example of such situations can be found in the networked control system. It can be seen that there are two delays: d1 (t) is used to represent the delay from sensor to controller and d2 (t) is used to represent the delay from controller to actuator. It is unreasonable to lump them into one state delay [30]. There are two main reasons: (i) The properties of these two delays may be different in nature due to the network transmission conditions. (ii) When d1 (t ) + d2 (t ) reaches its maximum, we do not necessarily have both d1 (t) and d2 (t) reach their maxima at the same time. So, it is of vital importance to investigate the systems with two additive time-varying delays. Recently, many approaches have been proposed to analyze the stability of systems with two delay components. For example, [30] presented a stability criterion for a system with two additive delay components by using the free-weighting-matrix (FWM) approach. In [31], improved results on stability are derived by exploiting a new Lyapunov–Krasovskii functional, which makes full use of the information about time-varying delays, d1 (t) and d2 (t). In [32,33], by considering the relationship between the time-varying delay and its upper bound when estimating the upper bound of the derivative of Lyapunov functional, some less conservative stability criteria are proposed to guarantee the systems with two successive delay components to be robustly asymptotically stable for all admissible parameter uncertainties. In [34], a new stability result was presented for system with additive time-varying delay, where the LKF as a whole was positive definite, rather than each term of it to positive definite as usual. In [35], by decomposing one delay interval into multiple subintervals which may be unequal, an appropriate LKF on the subintervals is constructed to analyse the stability of system with additive time-varying delay. In addition, many scholars have studied the stability of neural networks with two additive time-varying delays through simple LKF [36–39]. Later on, with regard to mixed-delay-dependent robust stability problem for uncertain linear neutral systems with discrete and distributed delays. Chen et al. proposed new augmented

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LKFs in [40,41], which contain some interconnected terms reflecting the relationships between discrete delay, neutral delay and distributed delay. Recently, based on these ideas, in view of the relationships between the upper bounds of the two time-varying delays, a new augmented LKF is constructed by using the information of the two upper bounds, and the free-matrixbased integral inequality is used to estimate the derivative of the constructed LKF. Then, a less conservative criterion is derived to assess stability in [42]. However, the LKF used in existing results only contained information of the relationships between the upper bounds of the two time-varying delays, d1 and d2 , which does not make full use of information of the system with two successive delay components. Besides, in [42], two single integral terms with augmented vectors are added into the LKF. That is, when the augmented vector takes into account one of the time-varying delays, the lower limit of the integration considers the other, which is used to ensure that the relationship between the upper bounds of the two time-varying delays and the upper bound of the total time delay. However, we can see that this method introduces augmented vectors in single integral terms, which greatly increases the computational complexity. On the other hand, most of the existing results used the FWM approach, Jensen inequality or free-matrix-based inequalities to estimate the derivative of the LKF, which are generally conservative to some extent. Recently, several stability criteria with less conservatism have been established by using the extended reciprocally convex matrix inequality [43,44]. Therefore, there still exists room for further improvement. This motivates the current research. Motivated by the above ideas, the objective of this paper is to further study the stability of the system with two additive time-varying delay components by taking into account both the choice of the LKF and the estimation of its derivative. An improved delay interconnection LKF with more general form, not only considering the information of the two upper bounds and the relationship between the two upper bounds and the upper bound of the total delay in some singer and double integral terms but also containing an augmented vector and a triple integral term, is constructed to improve the results. More information of system with additive time-varying delays is utilized, which in turn reduces the conservatism. Then, some advanced techniques, such as the free-matrix-based integral inequality and extended reciprocally convex matrix inequality, are applied to estimate the derivative of the proposed LKF such that a less conservative stability criterion is derived. Among them, the free-matrix-based integral inequality is used to estimate the quadratic integral terms, which derive from some double integral terms with additive time-varying delays. In addition, inspired by the previous work [28], other the quadratic integral terms which containing the R1 -dependent term and the Z-dependent term are treated together by using the extended reciprocally convex matrix inequality, which can enhance the feasibility region of stability criterion. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method. The reminder of the paper is organized as follows. Section 2 gives problem formulation and preliminary. A stability criterion is developed in Section 3. In Section 4, A numerical example is used to demonstrate the effectiveness and superiority of the proposed criterion. Conclusions are given in Section 5. Notations: Throughout this paper, the superscripts T and −1 mean the transpose and the inverse of a matrix, respectively; Rn denotes the n-dimensional Euclidean space; Rn×m is the set of all n × m real matrices; P > 0( ≥ 0) means that P is a real symmetric and positive-definite (semi-positive-definite) matrix; diag{} denotes a block-diagonal matrix; I (0) represent an appropriately dimensioned identity (zero) matrix; symmetric term in a symmetric matrix is denoted by ∗ ; and Sym{X } = X + X T .

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2. Problem formulation and preliminary Consider the following linear system with two additive time-varying delay components:  x˙(t ) = Ax(t ) + Ad x(t − d1 (t ) − d2 (t )) (1) x(t ) = φ(t ), ∀t ∈ [−d, 0] where x(t ) = [x1 (t ) x2 (t ) · · · xn (t )]T ∈ Rn is the state vector; A ∈ Rn×n , Ad ∈ Rn×n are known constant matrices with appropriate dimensions; the initial condition φ(t) is a continuous vector-valued function in t ∈ [−d, 0]; and the time delay, d1 (t) and d2 (t), represent two additive delay components in the state. They are continuous differentiable functions of time that satisfying d˙1 (t ) ≤ μ1 . d˙2 (t ) ≤ μ2 .

0 ≤ d1 (t ) ≤ d1 , 0 ≤ d2 (t ) ≤ d2 ,

(2)

where di , i = 1, 2 and μi , i = 1, 2 are constants. Let d = d1 + d2 , μ = μ1 + μ2 , d (t ) = d1 (t ) + d2 (t ). The following Jensen inequality is used to estimate the double integral terms. In addition, for the estimation of single integral terms, the popular methods in current research are the free-matrix-based integral inequality and the extended reciprocally convex matrix inequality, respectively shown as following lemmas. Lemma 1 (Jensen’s inequality [17]). Let x be a differentiable signal in [α, β] → Rn , for a positive definite matrix R ∈ Rn×n , the following inequality holds:   β   β  β T T (3) (β − α) x (s)Rx(s)ds ≥ x(s)ds R x(s)ds α

(β − α)2 2



β α

α

β

θ



x T (s)Rx(s)dsdθ ≥

α

β

β

x(s)dsdθ α

θ



 T

β





β

R

x(s)dsdθ α

(4)

θ

Lemma 2 (Wirtinger-based inequality [18]). Let x be a continuously differentiable signal in [α, β] → Rn , for a symmetric matrix R ∈ Rn×n > 0, the following inequality holds:  β 3 1 χ1T Rχ1 + χ T Rχ2 x˙T (s)Rx˙(s)ds ≥ (5) β −α β −α 2 α where χ1 = x(β ) −x(α),

χ2 = x(β ) + x(α) −

2 β −α



β

x(s)ds α

Lemma 3 (Free-matrix-based integral inequality [20,21]). Let x be a differentiable signal in [α, β] → Rn , for symmetric matrices R ∈ Rn×n , X, Z ∈ R3n×3n , and any matrices Y ∈ R3n×3n , N1 , N2 ∈ R3n×n satisfying ⎡ ⎤ X Y N1 ⎣ ∗ Z N2 ⎦ ≥ 0 ∗ ∗ R the following inequality holds:  β ˆ − x˙T (s)Rx˙(s)ds ≤  T  α

(6)

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where

  β 1  = x T (β ) x T (α) x T (s)ds T β −α α   ˆ = (β − α) X + 1 Z + Sym{N1 1 + N1 2 }  3 1 = e1 − e2 ,

2 = 2e3 − e1 − e2

e1 = [I 0 0], e2 = [0 I 0], e3 = [0 0 I ]. Lemma 4 (Extended reciprocally convex matrix inequality [28]). For a real scalar α ∈ (0, 1), symmetric matrices X1 > 0 and X2 > 0, and any matrices S1 and S2 , the following matrix inequality holds: ⎡ ⎤ 1 

X 0 (1 − α)S1 + αS2 X1 + (1 − α)T1 ⎢α 1 ⎥ (7) ⎣ ⎦≥ 1 ∗ X2 + αT2 X2 0 1−α where T1 = X1 − S2 X2−1 S2T and T2 = X2 − S1T X1−1 S1 . Lemma 5 (Schur complement [45]). Given symmetric matrix Rr×r . The following three conditions are equivalent: (i) S < 0 (ii ) S11 < 0, (iii ) S22 < 0,

⎡

S11 S=⎣ ∗

⎤

S12 ⎦ , where S11 ∈ S22

T −1 S22 − S12 S11 S12 < 0

−1 T S11 − S12 S22 S12 < 0.

3. Main results In this section, an improved delay interconnection LKF is constructed at first. Then, the free-matrix-based integral inequality and the extended reciprocally convex matrix inequality are introduced to handle the derivative of the novel LKF. Finally, a further improved stability criterion is derived by using proposed inequalities and an improved delay interconnection LKF with more information of the two time-varying delays. Firstly, for simplicity of vector and matrix representations, the following notations are denoted:

  t  t−d1  t−d1  t−d2 ξ1 (t ) = x T (t ) x T (s)ds x T (s)ds x T (s)ds x T (s)ds T t−d t−d2 t−d t−d  T  T T T T ξ2 (t ) = x (t ) x (t − d1 ) x (t − d2 ) x (t − d ) x (t − d1 (t )) x T (t − d2 (t )) T

  t  t −d1 (t ) 1 1 ξ3 (t ) = x T (t − d (t )) x T (s)ds x T (s)ds T d1 (t ) t −d1 (t ) d1 − d1 (t ) t−d1

  t  t −d2 (t )  t 1 1 1 T T T ξ4 (t ) = x (s)ds x (s)d s x (s)d s T d2 (t ) t −d2 (t ) d2 − d2 (t ) t−d2 d (t ) t −d (t )

 t−d1  t −d (t )  t−d1 1 1 1 ξ5 (t ) = x T (s)d s x T (s)d s x T (s)d s d2 − d1 t−d2 d − d (t ) t−d d − d1 t−d

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1 d − d2



t−d2

3463

 x (s)ds T

T

t−d

ξ (t ) = [ξ2 (t ) ξ3 (t ) ξ4 (t ) ξ5 (t )]T ei = [0n×(i−1)n In 0n×(16−i)n ]T , i = 1, 2, . . . , 16.

(8)

Theorem 1. For given scalars d1 > 0, d2 > 0 and μ1 , μ2 , the system (1) with two additive time-varying delay components satisfying condition (2) is asymptotically stable if there exist positive definite matrices P ∈ R5n×5n , Qi ∈ Rn×n , (i = 1, 2, 3, 4, 5, 6, 7), Ri ∈ Rn×n , (i = 1, 2, 3, 4, 5, 6), Z ∈ Rn×n , symmetric matrices Xj , Yj , Tj , Uj , Wj , Aj , B j ∈ R3n×3n , (j = 1, 3), and any matrices X2 , Y2 , T2 , U2 , W2 , A2 , B2 ∈ R3n×3n , Xk , Yk , Tk , Uk , Wk , Ak , Bk ∈ R3n×n , (k = 4, 5), S1 , S2 ∈ R2n×2n , such that the following LMIs are satisfied: ⎡ d1 T d2 T T ⎤ (0, d ) − S S 2 2 2   ⎢ d 4 d 5 1 ⎥ ⎢ ⎥ (0, 0) − 1 T4 S2 d 1 ⎢ ⎥ < 0, < 0, (9) ∗ − R˜ 1 0 ⎢ ⎥ ˜ ∗ −R1 d ⎣ ⎦ d2 ∗ 0 − (R˜ 1 + Z˜ ) d ⎡

X2 X3 ∗

d1 T T ⎤ S  d 5 1 ⎥ ⎥ (d1 , d2 ) − 4 ⎥< 0, 0 ⎥ ∗ ⎦ d1 ˜ ˜ − ( R1 + Z ) d ⎤ ⎡ ⎤ X4 Y1 Y2 Y4 ⎥ ⎢ ⎥ X5 ⎦ ≥ 0,

2 = ⎣ ∗ Y3 Y5 ⎦ ≥ 0, R2 ∗ ∗ R2

T1 ⎢

3 = ⎣ ∗ ∗

T2 T3 ∗

⎤ T4 ⎥ T5 ⎦ ≥ 0, R3

⎡ W1 ⎢

5 = ⎣ ∗ ∗

W2 W3 ∗

⎤ W4 ⎥ W5 ⎦ ≥ 0, R4

(13)

A2 A3 ∗

⎤ A4 ⎥ A5 ⎦ ≥ 0, R5

(14)

B2 B3 ∗

⎤ B4 ⎥ B5 ⎦ ≥ 0. R6

(15)

⎢ ⎢ ⎢ ⎢ ⎣

(d1 , 0) − 3 ∗ ∗ ⎡

X1 ⎢

1 = ⎣ ∗ ∗ ⎡

⎡

A1 ⎢

6 = ⎣ ∗ ∗ ⎡

B1 ⎢

7 = ⎣ ∗ ∗

d2 T S2 d 4 d2 − R˜ 1 d 0

⎡ U1 ⎢

4 = ⎣ ∗ ∗

U2 U3 ∗

⎤ U4 ⎥ U5 ⎦ ≥ 0, R3

 T5 S1T < 0, (10) −R˜ 1 − Z˜

(11)

(12)

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where (d1 (t ), d2 (t )) = 1 (d1 (t ), d2 (t )) + 2 + 3 (d1 (t ), d2 (t )) + 4   1 (d1 (t ), d2 (t )) = Sym T1 P 2 2 = eT1 (Q1 + Q2 + Q3 + Q4 )e1 − (1 − μ1 − μ2 )eT7 Q1 e7 − (1 − μ1 )eT5 Q2 e5 − (1 − μ2 )eT6 Q3 e6 − eT4 (Q4 + Q5 + Q6 )e4 + eT2 (Q5 + (d2 − d1 )Q7 )e2 + eT3 (Q6 − (d2 − d1 )Q7 )e3

3 (d1 (t ), d2 (t )) = T3 (d 2 R1 + d12 R2 + d22 R3 + (d2 − d1 )2 R4 + (d − d1 )2 R5 + (d − d2 )2 R6 ) 3     1 + d1 d1 (t ) T6 X1 + X3 6 + d1 Sym T6 X4 7 + T6 X5 8 3     1 + d1 (d1 − d1 (t )) T9 Y1 + Y3 9 + d1 Sym T9 Y4 10 + T9 Y5 11 3     1 + d2 d2 (t ) T12 T1 + T3 12 + d2 Sym T12 T4 13 + T12 T5 14 3     1 + d2 (d2 − d2 (t )) T15 U1 + U3 15 + d2 Sym T15U4 16 + T15U5 17 3     1 + (d2 − d1 )2 T18 W1 + W3 18 +(d2 − d1 )Sym T18W4 19 + T18W5 20 3     1 + (d − d1 )2 T21 A1 + A3 21 + (d − d1 )Sym T21 A4 22 + T21 A5 23 3     1 + (d − d2 )2 T24 B1 + B3 24 + (d − d2 )Sym T24 B4 25 + T24 B5 26 3  2  d Z 3 − 2 T27 Z 27 − 2 T28 Z 28 4 = T3 2 ⎡ ⎤  

 d1 + d R˜ + d1 Z˜ d1 S + d2 S  1 1 2 S1 4 4 T ⎢ d 4 d d d ⎥ , 2 = ⎦ d2 + d 5 ⎣ R˜ 1 5 5 ∗ R˜ 1 d ⎡ ⎤

 d2 + d R˜ + d2 Z˜ d2 S + d1 S 



  1 1 2 S2 4 4 T ⎢ d 4 4 T R˜ 1 d d d ⎥ 3 = , = ⎣ ⎦ 4 d1 + d 5 5 5 ∗ 2R˜ 1 5 ∗ R˜ 1



 d 0 0 R Z 1 , R˜ 1 = Z˜ = 0 3R1 0 3Z



4 T 2R˜ 1 + Z˜ 1 = 5 ∗

  1 = eT1 AT + eT7 ATd eT1 − eT4 eT2 − eT3 eT2 − eT4 eT3 − eT4 T ,  2 = eT1 (d1 (t ) + d2 (t ))eT12 + (d − d1 (t ) − d2 (t ))eT14 (d2 − d1 )eT13 (d − d1 )eT15    (d − d2 )eT16 T , 3 = eT1 AT + eT7 ATd T ,

 4 = eT1  6 = eT1  9 = eT5  12 = eT1  15 = eT6  18 = eT2  21 = eT2  24 = eT3  27 = eT1

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   − eT7 eT1 + eT7 − 2eT12 T , 5 = eT7 − eT4 eT7 + eT4 − 2eT14 T ,      eT5 eT8 T , 7 = eT1 − eT5 T , 8 = 2eT8 − eT1 − eT5 T ,      eT2 eT9 T , 10 = eT5 − eT2 T , 11 = 2eT9 − eT5 − eT2 T ,      eT6 eT10 T , 13 = eT1 − eT6 T , 14 = 2eT10 − eT1 − eT6 T ,      eT3 eT11 T , 16 = eT6 − eT3 T , 17 = 2eT11 − eT6 − eT3 T ,      eT3 eT13 T , 19 = eT2 − eT3 T , 20 = 2eT13 − eT2 − eT3 T ,      eT4 eT15 T , 22 = eT2 − eT4 T , 23 = 2eT15 − eT2 − eT4 T ,      eT4 eT16 T , 25 = eT3 − eT4 T , 26 = 2eT16 − eT3 − eT4 T ,    − eT12 T , 28 = eT7 − eT14 T .

(16)

Proof. Construct the following LKF: V (xt ) = V1 (xt ) + V2 (xt ) + V3 (xt ) + V4 (xt )

(17)

where V1 (xt ) = ξ1T (t )Pξ1 (t )  t  T V2 (xt ) = x (s)Q1 x(s)ds + t −d (t )  t

+

x T (s)Q4 x(s)ds +

t−d



+ (d2 − d1 )  V3 (xt ) = d



0

−d

V4 (xt ) =

t

−d2

θ

x (s)Q2 x(s)ds +

t −d1 (t )  t−d1

x T (s)Q5 x(s)ds +

t+θ



−d

x T (s)Q3 x(s)ds

x T (s)Q6 x(s)ds

t−d

x T (s)Q7 x(s)ds 

0 −d1



t

t+θ

x˙T (s)R2 x˙(s)d sd θ

x˙T (s)R3 x˙(s)d sd θ + (d2 − d1 ) −d1

t

t −d2 (t )  t−d2

t−d t−d1

x˙T (s)R1 x˙(s)d sd θ + d1

+ (d − d1 )  t  t t t−d

 T

t−d2

t+θ 0  t



+ d2

t



t



−d1 −d2

x˙ (s)R5 x˙(s)dsdθ + (d − d2 ) T

t+θ



t

x˙T (s)R4 x˙(s)dsdθ

t+θ −d2



−d



t

t+θ

x˙T (s)R6 x˙(s)dsdθ

x˙T (s)Z x˙(s)d sd ud θ

u

Calculating the derivative of V(xt ) along the solution of system (1) yields: V˙ (xt ) = V˙1 (xt ) + V˙2 (xt ) + V˙3 (xt ) + V˙4 (xt )

(18)

Calculating the derivative of V1 (xt ), we get V˙1 (xt ) = 2ξ1T (t )Pξ˙1 (t ) = ξ T (t )( T1 P 2 + T2 P 1 )ξ (t )   = ξ T (t )Sym T1 P 2 ξ (t ) = ξ T (t )1 (d1 (t ), d2 (t ))ξ (t )

(19)

By the time derivative of V2 (xt ), it can be given as: V˙2 (xt ) = ξ T (t )[eT1 (Q1 + Q2 + Q3 + Q4 )e1 − (1 − d˙1 (t ) − d˙2 (t ))eT7 Q1 e7 − (1 − d˙1 (t ))eT5 Q2 e5 −(1 − d˙2 (t ))eT6 Q3 e6 − eT4 Q4 e4 + eT2 Q5 e2 − eT4 Q5 e4 + eT3 Q6 e3 − eT4 Q6 e4

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+ (d2 − d1 )eT2 Q7 e2 − (d2 − d1 )eT3 Q7 e3 ]ξ (t ) ≤ ξ T (t )[eT1 (Q1 + Q2 + Q3 + Q4 )e1 − (1 − μ1 − μ2 )eT7 Q1 e7 − (1 − μ1 )eT5 Q2 e5 −(1 − μ2 )eT6 Q3 e6 − eT4 (Q4 + Q5 + Q6 )e4 + eT2 (Q5 + (d2 − d1 )Q7 )e2 + eT3 (Q6 − (d2 − d1 )Q7 )e3 ]ξ (t ) = ξ T (t )2 ξ (t )

(20)

Taking the derivative of V3 (xt ), we have: V˙3 (xt ) = x˙T (t )(d 2 R1 + d12 R2 + d22 R3 + (d2 − d1 )2 R4 + (d − d1 )2 R5 + (d − d2 )2 R6 )x˙(t )  t  t −d x˙T (s)R1 x˙(s)ds − d1 x˙T (s)R2 x˙(s)ds t−d t

t−d1

 − d2



x˙ (s)R3 x˙(s)ds − (d2 − d1 ) T

t−d2



− (d − d1 )

t−d1

t−d2 t−d1

x˙T (s)R4 x˙(s)ds 

x˙T (s)R5 x˙(s)ds − (d − d2 )

t−d

t−d2

x˙T (s)R6 x˙(s)ds

t−d

= ξ T (t ) T3 (d 2 R1 + d12 R2 + d22 R3 + (d2 − d1 )2 R4 + (d − d1 )2 R5 + (d − d2 )2 R6 ) 3 ξ (t )  t  t T −d x˙ (s)R1 x˙(s)ds − d1 x˙T (s)R2 x˙(s)ds t−d t

t−d1

 − d2



x˙T (s)R3 x˙(s)ds − (d2 − d1 )

t−d2



− (d − d1 )

t−d1

t−d2 t−d1

x˙ (s)R5 x˙(s)ds − (d − d2 ) T

t−d

x˙T (s)R4 x˙(s)ds 

t−d2

x˙T (s)R6 x˙(s)ds

(21)

t−d

If inequality (11) holds, applying Lemma 3 to estimate the R2 -dependent term in Eq. (21) yields  t − d1 x˙T (s)R2 x˙(s)ds t−d1

= −d1



t

 x˙ (s)R2 x˙(s)ds − d1 T

t −d1 (t )

t −d1 (t )

x˙T (s)R2 x˙(s)ds

t−d1

    1 ≤ ξ T (t ) d1 d1 (t ) T6 (X1 + X3 ) 6 + d1 Sym T6 X4 7 + T6 X5 8 ξ (t ) 3      T  1 T T T + ξ (t ) d1 (d1 − d1 (t )) 9 Y1 + Y3 9 + d1 Sym 9 Y4 10 + 9 Y5 11 ξ (t ) (22) 3 Similarly, if inequality (12) holds, using Lemma 3 to estimate the R3 -dependent term in Eq. (21) yields  t −d2 x˙T (s)R3 x˙(s)ds t−d2

 t −d2 (t ) x˙T (s)R3 x˙(s)ds − d2 x˙T (s)R3 x˙(s)ds t −d2 (t ) t−d2       1 ≤ ξ T (t ) d2 d2 (t ) T12 T1 + T3 12 + d2 Sym T12 T4 13 + T12 T5 14 ξ (t ) 3 = −d2



t

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    1 +ξ T (t ) d2 (d2 − d2 (t )) T15 (U1 + U3 ) 15 +d2 Sym T15U4 16 + T15U5 17 ξ (t ) (23) 3 Assume inequality (13) holds, then using Lemma 3 to estimate the R4 -dependent term in Eq. (21) yields  t−d1 −(d2 − d1 ) x˙T (s)R4 x˙(s)ds t−d2    T  1 T 2 T T ≤ ξ (t ) (d2 − d1 ) 18 (W1 + W3 ) 18 + (d2 − d1 )Sym 18W4 19 + 18W5 20 ξ (t ) 3 (24) By using Lemma 3, it is clear that if inequality (14) holds, the estimation of the R5 dependent term in Eq. (21) can be get as follows  t−d1 −(d − d1 ) x˙T (s)R5 x˙(s)ds t−d     1 ≤ ξ T (t ) (d − d1 )2 T21 (A1 + A3 ) 21 + (d − d1 )Sym T21 A4 22 + T21 A5 23 ξ (t ) (25) 3 Applying Lemma 3, if inequality (15) holds, we have  t−d2 −(d − d2 ) x˙T (s)R6 x˙(s)ds t−d    T  1 T 2 T T ≤ ξ (t ) (d − d2 ) 24 (B1 + B3 ) 24 + (d − d2 )Sym 24 B4 25 + 24 B5 26 ξ (t ) (26) 3 Thus, combining Eq. (21) and inequalities (22)–(26), we can derive the estimation of V˙3 (xt ) as follows  t V˙3 (xt ) ≤ ξ T (t )3 (d1 (t ), d2 (t ))ξ (t ) − d x˙T (s)R1 x˙(s)ds (27) t−d

Finally, calculating the derivative of V4 (xt ), it can be easily obtained as  t  t d2 T ˙ V4 (xt ) = x˙ (t )Z x˙(t ) − x˙T (s)Z x˙(s)d sd θ 2 t−d θ  t  t d2 x˙T (s)Z x˙(s)d sd θ ≤ ξ T (t ) T3 ( Z ) 3 ξ (t ) − 2 t −d (t ) θ  t  t −d (t )  t −d (t ) x˙T (s)Z x˙(s)d sd θ − (d − d (t )) x˙T (s)Z x˙(s)ds − t−d

θ

(28)

t −d (t )

For the double integral terms of V˙4 (xt ), we can do the following treatment by using Lemma 1  t  t −d (t )  t −d (t )  t − x˙T (s)Z x˙(s)d sd θ − x˙T (s)Z x˙(s)d sd θ t −d (t ) θ t−d θ  t   t   t  t 2 ≤− 2 x˙(s)dsdθ T Z x˙(s)dsdθ d (t ) t −d (t ) θ t −d (t ) θ

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 t −d (t )  t −d (t )   t −d (t )  t −d (t )  2 T − x˙(s)dsdθ Z x˙(s)dsdθ (d − d (t ))2 t−d θ t−d θ      t  t 1 1 T ≤ −2 x(t ) − x(s)ds Z x(t ) − x(s)ds d (t ) t −d (t ) d (t ) t −d (t )      t −d (t )  t −d (t ) 1 1 −2 x(t − d (t )) − x(s)ds T Z x(t − d (t )) − x(s)ds d − d (t ) t−d d − d (t ) t−d ≤ ξ T (t )(−2 T27 Z 27 − 2 T28 Z 28 )ξ (t )

(29)

From inequality (29), it can be seen that V˙4 (xt ) in inequality (28) can be written as  t T ˙ V4 (xt ) ≤ ξ (t )4 ξ (t ) − (d − d (t )) x˙T (s)Z x˙(s)ds (30) t −d (t )

Next, by using Lemma 2 to estimate the integral terms of V˙3 (xt ), V˙4 (xt ) in inequalities (27) and (30), we have  t  t −d x˙T (s)R1 x˙(s)ds − (d − d (t )) x˙T (s)Z x˙(s)ds t−d





t −d (t ) t

 t −d (t ) x˙T (s)Z x˙(s)ds − d x˙T (s)(R1 + Z )x˙(s)ds − d x˙T (s)R1 x˙(s)ds t −d (t ) t −d (t ) t−d

 d d T T ˜ T T ˜ T ˜ ˜ ≤ ξ (t ) 4 Z 4 ξ (t ) − ξ (t ) (R1 + Z ) 4 + R1 5 ξ (t ) (31) d (t ) 4 d − d (t ) 5 = d (t )

t

For matrices S1 , S2 , it follows Lemma 4 that

 d d T T ˜ T ˜ ˜ −ξ (t ) (R1 + Z ) 4 + R1 5 ξ (t ) d (t ) 4 d − d (t ) 5



d − d (t ) 4 T R˜ 1 + Z˜ − S2 R˜ 1−1 S2T T T ˜ T ≤ −ξ (t ) 4 (R1 + Z˜ ) 4 + 5 R˜ 1 5 + 5 ∗ d



 

 d (t ) S2 4 T 0 4 4 + ξ (t ) 5 ∗ R˜ 1 − S1T (R˜ 1 + Z˜ )−1 S1 5 d 5

S1 0



(32)

Combining inequalities (31) and (32) can be written as  t  t T −d x˙ (s)R1 x˙(s)ds − (d − d (t )) x˙T (s)Z x˙(s)ds t −d (t )

t−d

˜ 1 (t ), d2 (t ))ξ (t ) ≤ −ξ (t ) (d T

where ˜ 1 (t ), d2 (t )) = T4 (R˜ 1 + Z˜ ) 4 + T5 R˜ 1 5 − T4 Z˜ 4 (d



  d − d1 (t ) − d2 (t ) 4 T R˜ 1 + Z˜ − S2 R˜ 1−1 S2T S1 4 + 5 ∗ 0 5 d



  d1 (t ) + d2 (t ) 4 T 0 S2 4 + T ˜ ˜ 5 ∗ R1 − S1 (R1 + Z˜ )−1 S1 5 d = T4 R˜ 1 4 + T5 R˜ 1 5

(33)

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  d − d1 (t ) − d2 (t ) 4 T R˜ 1 + Z˜ S1 4 5 ∗ 0 5 d



  d1 (t ) + d2 (t ) 4 T 0 S2 4 + 5 ∗ R˜ 1 5 d d − d1 (t ) − d2 (t ) T d1 (t ) + d2 (t ) T T − 4 S2 R˜ 1−1 S2T 4 − 5 S1 (R˜ 1 + Z˜ )−1 S1 5 d d

+

So, from inequalities (19), (20), (27), (30) and (33), an upper bound of V˙ (xt ) is derived as ˜ 1 (t ), d2 (t ))]ξ (t ) V˙ (xt ) ≤ ξ T (t )[1 (d1 (t ), d2 (t )) + 2 + 3 (d1 (t ), d2 (t )) + 4 − (d T ˜ = ξ (t )[(d1 (t ), d2 (t )) − (d1 (t ), d2 (t ))]ξ (t )

(34)

˜ 1 (t ), d2 (t )) < 0, then V˙ (xt ) < 0. However, it is dependent on two If (d1 (t ), d2 (t )) − (d additive time-varying delays d1 (t) and d2 (t), which cannot be solved directly by using an LMI tool. Based on convex combination technique, it is easy to get that (d1 (t ), d2 (t )) − ˜ 1 (t ), d2 (t )) < 0 is satisfied for all d1 (t) ∈ [0, d1 ] and d2 (t) ∈ [0, d2 ], that is to say, V˙ (xt ) < 0 (d holds if the following four inequalities hold (0, 0) − 1 − T4 S2 R˜ 1−1 S2T 4 < 0

(35)

(0, d2 ) − 2 −

d1 T d2 S2 R˜ 1−1 S2T 4 − T5 S1T (R˜ 1 + Z˜ )−1 S1 5 < 0 d 4 d

(36)

(d1 , 0) − 3 −

d2 T d1 S2 R˜ 1−1 S2T 4 − T5 S1T (R˜ 1 + Z˜ )−1 S1 5 < 0 d 4 d

(37)

(d1 , d2 ) − 4 − T5 S1T (R˜ 1 + Z˜ )−1 S1 5 < 0

(38)

which are respectively guaranteed by LMIs (9) and (10) based on Schur complement. Therefore, if LMIs (9)–(15) are feasible, there exists a sufficiently small ε > 0, such that V˙ (xt ) < −εx(t )2 , which ensures that system (1) with additive time-varying delays satisfying condition (2) is asymptotically stable. This completes the proof.  Remark 1. In Theorem 1, an improved delay interconnection LKF is constructed and it helps to reduce the conservatism of stability criterion due to the following three aspects. (1) The augmented vector ξ 1 (t) in LKF (17) takes into account the state vector xT (t) and t  t−d  t−d  t−d the integral terms t−d x T (s)ds, t−d21 x T (s)ds, t−d 1 x T (s)ds, and t−d 2 x T (s)ds, some of which are not included in the LKF of [42]. Among them, compared with [42], the  t−d proposed augmented treatment not only contains the integral term t−d21 x T (s)ds, but  t−d  t−d also introduces the delay interconnection integral terms t−d 1 x T (s)ds and t−d 2 x T (s)ds, which establishes more information between the cross terms by considering the relationship between the upper bound of the total time delay and the upper bound of each time delay. That is to say, more information of system with additive time-varying delays is utilized via the proposed augmented treatment, which in turn reduces the conservatism.

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(2) Different from the LKF in [42], two delay interconnection single integral terms, such  t−d  t−d as t−d 1 x T (s)Q5 x(s)ds and t−d 2 x T (s)Q6 x(s)ds, are introduced to the V2 (xt ) of LKF in this paper in order to take into account the relationship between the upper bounds of the two time-varying delays and the upper It is observed  t bound of the total timedelay. t that these terms have the same effect as t−d1 ς1T (s)Q4 ς1 (s)ds and t−d2 ς2T (s)Q5 ς2 (s)ds proposed in [42]. However, it can be seen that the proposed single integral terms can more clearly show the relationship between d1 and d, d2 and d, and the form is more simple and visualized. Thus, the augmented vectors don’t need to be introduced and the decision variables are reduced without affecting the conservatism. (3) In addition, similarly, the relationship between d1 and d, d2 and d is also considered in the delay interconnection double integral terms. Obviously, it establishes the cross relationship between the vectors of x(t), x(t − d1 ), x(t − d2 ) and x(t − d ), which may theoretically reduce the conservativeness of our criterion. Remark 2. During the deriving of the stability criterion, some new techniques are used to estimate the derivative of V(xt ) and contribute to reducing the conservatism of stability criterion. (1) This paper mainly employs the free-matrix-based integral inequality t  t to estimate the quadratic integral terms, such as −d1 t−d1 x˙T (s)R2 x˙(s)ds, −d2 t−d2 x˙T (s)R3 x˙(s)ds,  t−d  t−d  t−d −(d2 − d1 ) t−d21 x˙T (s)R4 x˙(s)ds, −(d − d1 ) t−d 1 x˙T (s)R5 x˙(s)ds and −(d − d2 ) t−d 2 x˙T (s)R6 x˙(s)ds, which derived from the double integral terms with additive time-varying delays. It can directly deal with the quadratic integral terms containing the time-delay term without introducing the reciprocally convex combination lemma. (2) In addition, according to the quadratic integral terms in inequalities (27) and (30) it can be seen that some quadratic integral terms derived from the double integral term and triple integral term have the same upper and lower limits. Therefore, we consider combining these integral terms with the same structure to estimate, as shown in inequality (31). Firstly, the three terms are estimated using Lemma 2, respectively. Then, we consider the latter two as a whole, and the extended reciprocally convex matrix inequality is used to estimate it, as shown in inequality (32). Obviously, it can be seen that this method not only reduces conservatism but also greatly reduces the introduction of decision variables. 4. A numerical example This section provides a numerical example to verify the effectiveness and the superiority of the presented criterion. Example: Consider the following two additive time-varying delay components system (1) with:



 0 0 −2 −1 , Ad = A= 0 −0.9 −1 −1 Supposing d˙1 (t ) ≤ 0.1 and d˙2 (t ) ≤ 0.8. This example often appears in the literature. For this example, the maximum allowable upper bounds (MAUBs) d1 , d2 of two additive time-varying delays d1 (t) and d2 (t) are calculated when other information is known. This example has been studied in [30–35,42] and

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Table 1 The maximal allowable upper bounds of d2 for various d1 . Methods

Theorem 1 [30] Theorem 1 [31] Theorem 1 [32] Proposition 2 [33] Theorem 1 [34] Theorem 1 [35] Theorem 1 [42] Theorem 1

d1

NoVs

1

1.1

1.2

1.5

0.415 0.512 0.872 0.873 0.989 1.198 0.9999 1.2136

0.376 0.457 0.772 0.773 0.914 1.027 1.0770 1.1136

0.340 0.406 0.672 0.673 0.836 0.980 0.9725 1.0137

0.248 0.283 0.371 0.373 0.564 0.610 0.6807 0.7137

12.5n2 + 4.5n 19.5n2 + 3.5n 32.5n2 + 8.5n 7n 2 + 5n 8.5n 2 + 3.5n 13.5n2 + 4.5n 189n2 + 30n 195.5n2 + 30.5n

Table 2 The maximal allowable upper bounds of d1 for various d2 . Methods

Theorem 1 [30] Theorem 1 [31] Theorem 1 [32] Proposition 2 [33] Theorem 1 [34] Theorem 1 [35] Theorem 1 [42] Theorem 1

d2

NoVs

0.3

0.4

0.5

1.324 1.453 1.572 1.573 1.707 1.708 1.8804 1.9137

1.039 1.214 1.472 1.473 1.637 1.645 1.7798 1.8137

0.806 1.021 1.372 1.373 1.557 1.574 1.6759 1.7136

12.5n2 + 4.5n 19.5n2 + 3.5n 32.5n2 + 8.5n 7n 2 + 5n 8.5n 2 + 3.5n 13.5n2 + 4.5n 189n2 + 30n 195.5n2 + 30.5n

we take this example for comparison with the criteria proposed in these literatures. Given d1 = 1, 1.1, 1.2 and 1.5, the maximum value of upper bound d2 yielded by Theorem 1 that guarantees the stability of system (1) is listed in Table 1. From Table 1, it is clearly shown that the upper bounds obtained in this paper are markedly better than the existing ones since in which the relationship between the upper bounds of the two time-varying delays is fully considered and the extended reciprocally convex matrix inequality is utilized, which reflects that the new criterion has less conservatism. Given d2 = 0.3, 0.4 and 0.5, the allowable upper bounds of d1 obtained by different methods are shown in Table 2. From Table 2, it can be seen clearly that Theorem 1 in this paper can indeed provide much larger admissible upper bounds than the stability criteria in [30–35,42], which shows the advantage of the stability result in this paper over others. For the analysis of the number of decision variables, many comparisons about the number of decision variables between the presented criterion and the existing ones have been added in Tables 1 and 2. The number of decision variables is represented as NoVs in Tables 1 and 2. As we can see, although the number of decision variables (NoVs) of our result are a little larger than the ones reported in [42], it is less conservative than [42] by the verification of numerical examples. 5. Conclusions This paper has investigated the stability for systems with additive time-varying delays. Firstly, an improved delay interconnection LKF is constructed, which not only introduces the

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information between the upper bounds of the two time-varying delays, but also considers the relationship between the upper bounds of the two time-varying delays and the upper bound of the total delay. According to this new LKF, the more effectively interactions between those cross term of the vectors x(t), x(t − d1 ), x(t − d2 ), x(t − d ) were established. Then, several advanced techniques, such as the free-matrix-based integral inequality and extended reciprocally convex matrix inequality, are employed to estimate the derivative of the constructed LKF. These all contribute to the less conservative delay-dependent stability criterion. Finally, a numerical example has been given to demonstrate the effectiveness and advantage of the criterion. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grants 61573325 and 11571088, the Hubei Provincial Natural Science Foundation of China under Grant 2015CFA010, and the 111 project under Grant B17040. References [1] K. Gu, V.L. Kharitonov, J. Chen, Stability of Time-delay Systems, Birkhauser, 2003. [2] E. Fridman, Introduction to Time-delay Systems: Analysis and Control, Birkhauser, 2014. [3] J. Chen, C. Lin, B. Chen, Q.G. Wang, Improved stability criterion and output feedback control for discrete time-delay systems, Appl. Math. Model. 52 (2017) 82–93. [4] P.G. Park, J.W. Ko, Stability and robust stability for systems with a time-varying delay, Automatica 43 (10) (2007) 1855–1858. [5] E. Fridman, Introduction to time-delay systems, Sys. Control: Found. Appl. 72 (4) (2014) 591–597. [6] X. Zhao, C. Lin, B. Chen, Q.G. Wang, A novel Lyapunov–Krasovskii functional approach to stability and stabilization for TCS fuzzy systems with time delay, Neurocomputing 313 (2018) 288–294. [7] X.M. Zhang, Q.L. Han, X. Ge, D. Ding, An overview of recent developments in Lyapunov–Krasovskii functionals and stability criteria for recurrent neural networks with time-varying delays, Neurocomputing 313 (2018) 392–401. [8] J.H. Kim, Note on stability of linear systems with time-varying delay, Automatica 47 (9) (2011) 2118–2121. [9] K. Gu, Discretized LMI set in the stability problem for linear uncertain time-delay systems, Int. J. Control 68 (4) (1997) 923–934. [10] H.B. Zeng, Y. He, M. Wu, S.P. Xiao, Less conservative results on stability for linear systems with a time-varying delay, Opt. Control Appl. Methods 34 (6) (2013) 670–679. [11] Y. He, Q.G. Wang, M. Wu, Augmented Lyapunov functional and delay-dependent stability criteria for neural systems, Int. J. Robust Nonlinear Control 15 (8) (2005) 923–933. [12] J. Sun, G.P. Liu, J. Chen, D. Rees, Improved delay-range-dependent stability criteria for linear systems with time-varying delays, Automatica 46 (2) (2010) 466–470. [13] E. Fridman, U. Shaked, Delay-dependent stability and h∞ control: constant and time-varying delays, Int. J. Control 76 (1) (2003) 48–60. [14] M. Wu, Y. He, J.H. She, Delay-dependent criteria for robust stability of time-varying delay systems, Automatica 40 (8) (2004) 1435–1439. [15] Y. He, M. Wu, J.H. She, Parameter-dependent Lyapunov functional for stability of time-delay systems with poly topic-type uncertainties, IEEE Trans. Autom. Control 49 (5) (2004) 828–832. [16] Y. He, Q.G. Wang, L. Xie, C. Lin, Further improvement of free-weighting matrices technique for systems with time-varying delay, IEEE Trans. Autom. Control 52 (2) (2007) 293–299. [17] Q.L. Han, Absolute stability of time-delay systems with sector-bounded nonlinearity, Automatica 41 (12) (2005) 2171–2176. [18] A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: application to time-delay systems, Automatica 49 (9) (2013) 2860–2866. [19] P.G. Park, W. Lee, S.Y. Lee, Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems, J. Frankl. Inst. 352 (4) (2015) 1378–1396.

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