Improved delay-dependent stabilization for a class of networked control systems with nonlinear perturbations and two delay components

Applied Mathematics and Computation 316 (2018) 1–17

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Improved delay-dependent stabilization for a class of networked control systems with nonlinear perturbations and two delay components Guoliang Chen a,b, Jianwei Xia a,∗, Guangming Zhuang a, Junsheng Zhao a a b

School of Mathematics Science, Liaocheng University, Liaocheng 252000, PR China School of Automation, Beijing Institute of Technology, Beijing 100081, PR China

a r t i c l e

i n f o

Keywords: Networked control systems State feedback control Delay-dependent stability Time delay

a b s t r a c t This paper focuses on the problem of delay-dependent state feedback control for a class of networked control systems (NCSs) with nonlinear perturbations and two delay components. Based on the dynamic delay interval (DDI) method and the Wirtinger integral inequality, some improved delay-dependent stability analysis are obtained. Furthermore, the results are extended to the conditions of NCSs with one time delay, and the corresponding stability analysis results and state feedback controller are obtained. Finally, some numerical examples and simulations are given to show the effectiveness of the proposed methods. © 2017 Elsevier Inc. All rights reserved.

1. Introduction In the past years, systems with time delays have received widely consideration due to time delays inevitably exist in various practical systems, such as neural networks, stochastic Markovian jumping system, engineering systems, NCSs [1–5,19]. Specially, in NCSs, sampled data are transmitted to the controller via communication networks, so there often exist time delays in measurements [6–9]. The inevitable encountered time delays usually are the main reason resulting in some complex dynamic behaviors such as oscillation, divergence and instability. Generally, the technique to proof the delay-dependent stability and stabilization criteria can be classified into two types: one is LKF method, for example, Discretized LKF method [10], augmented LKF method [11] , delay-partition-dependent LKF method [12], discontinuous LKF method [13] etc. The other one is estimation of bounding the integral term approach, for example, Jensen inequality approach, free-weighting matrix approach [14], convex optimization approach [7,15], Wirtinger inequality approach [7,16], free-matrix- based integral inequality approach [17], relaxed conditions for stability of time-varying delay systems [18] etc. On the other hand, networked control systems (NCSs) have a relatively new structure where sensors, controllers and plants are often connected over a common networked medium [19] and many results has been reported recently based on this modeling idea. Such as, improved stability and stabilization design for networked control systems using new quadrupleintegral functionals [20], output tracking control of networked control systems via delay compensation controllers [21], event-triggered control for networked control systems using passivity [22]. Due to variable networked transmission conditions, there do exists in the transmission delay and the data packet dropout. The transmission delay generally includes the sensor-to-control delay and the control-to-actuator delay. So the networked control system can take a form with two time delays: x˙ (t ) = Ax(t ) + Bx(t − τ1 (t ) − τ2 (t )). Based on this model, many results have addressed in some articles [23–27]. In ∗

Corresponding author. E-mail addresses: [email protected] (G. Chen), [email protected] (J. Xia), [email protected] (G. Zhuang), [email protected] (J. Zhao).

http://dx.doi.org/10.1016/j.amc.2017.07.072 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

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G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

this articles, the stability of networked control system with additive time-varying delays was discussed and got some delay dependent results. The results of this articles is got by constructing different LKF and the linear inequalities. With the different of [23–26], [27] constructed new LKF by separating the whole delay interval into subintervals. The LKF as a whole to examine its positive definite, rather than restrict each term of it to positive definite as usual. So getting the less conservatism of stability results and new state feedback controller. But there is a problem that we have to addressing, as variable network transmission conditions in different properties, it may be not rational to lump them into one input state delay in some articles. Recently, [28] proposed a new method: dynamic delay interval method (DDI). A novel feature of the DDI is that the new bounds of the time-varying delay are the convex combinations of τ 1 (t) and τ 2 (t), and their fixed bounds. In [28], the stability analysis of neural networks with two delay components based on DDI was discussed. Motivated by our observations, we make a try to apply this method to get the results of the state feedback stabilization controller for NCSs with two delay components. The problem of delay-dependent feedback stabilization for a class of networked control systems (NCSs) with nonlinear perturbations and two delay components is considered in this paper. Firstly, by constructing a new augmented Lyapunov functional based on the DDI method and using the reciprocally convex combination technique and Wirtinger integral inequality, some some improved delay-dependent stability analysis are obtained for for a class of NCSs with two additive input delays and the nonlinearity. Secondly, the state feedback controller is designed by adjusting different parameters with the DDI parameters. Then, the obtained results are extended to the case of NCSs with one time delay. Finally, numerical examples and illustrations are given to show the effectiveness and the significant improvement of the proposed methods. The main contributions of this paper include: 1. A new augmented Lyapunov functional is constructed based on the DDI method. 2. The reciprocally convex combination technique and Wirtinger integral inequality are applied to estimate the derivative of the Lyapunov functional to reduce conservatism of the obtained results. Notation. Throughout this paper, I is the identity matrix with appropriate dimension; MT represents the transpose of the matrix M; Rn denotes the n−dimensional Euclidean space; 0m × n represents a zero matrix with m × n dimensions; The notationsX > 0( ≥ 0) is used to denote a symmetric positive-definite (positive-semidefinite) matrix. In symmetric block matrices or complex matrix expressions, we use an asterisk ∗ to represent a term that is induced by symmetry, and diag{ · } stands for a block-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations . 2. System description and preliminaries Consider the following nonlinear networked control model

x˙ (t ) = Ax(t ) + f (x(t )) + Bu(t ),

(1)

where x(t) ∈ Rn is the state vector; A, B are the known real constant matrices of appropriate dimensions. f(x(t)) is the nonlinear perturbations and satisfy the following:

df (x(t )) = h(x(t )) = DF (x(t ))E. dx(t )

(2)

Here C, D are the known real constant matrices of appropriate dimensions and F(x(t)) is the norm-bounded matrix satisfying FT (x(t))F(x(t)) ≤ I. In the following we focus on the nonlinear networked control model with a controller gain K and two delay components described by

x˙ (t ) = Ax(t ) + f (x(t )) + BKx(t − τ1 (t ) − τ2 (t )),

(3)

τ 1 (t) and τ 2 (t) are two time-varying delays satisfying

τ1 (t ) ≤ τ1 , 0 ≤ τ2 (t ) ≤ τ2 τ˙ 1 (t ) = d1 (t ) ≤ d1 , τ˙ 2 (t ) = d2 (t ) ≤ d2 , 0≤

(4)

where τ 1 , τ 2 , d1 , d2 are constants and τ 1 > 0, τ 2 > 0. We denote

τ (t ) = τ1 (t ) + τ2 (t ), τ = τ1 + τ2 . d (t ) = d1 (t ) + d2 (t ), dm = d1 + d2 .

(5)

For each delay components, the dynamic delay interval (DDI) of τ 1 (t) is denoted by [ατ1 (t ), τ1 − ατ1 (t )], and the DDI of τ 2 (t) is denoted by [βτ2 (t ), τ2 − βτ2 (t )], respectively. Therefore, the internal of τ (t) is [a(t), b(t)], where

a(t ) = b(t ) =

ατ1 (t ) + βτ2 (t ), τ − α (τ1 − τ1 (t )) − β (τ2 − τ2 (t )),

(α , β ) ∈

 ℵ  {[0, 1] × [0, 1] − (0, 0 ) ∪ (1, 1 )}, d1 + d2 < 1 {(α , β )|α d1 + β d2 < 1} ∩ ℵ, d1 + d2 ≥ 1

(6)

(7)

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

3

Remark 1. This new stability analysis method(DDI method) mentioned above is first proposed in [28]. A novel feature of the DDI is that the new bound of the time-varying delay are the convex combination of τ 1 (t)) and τ 2 (t), and their fixed bounds. Note that b(t ) − a(t ) = τ − ατ1 − βτ2 , the length of the time-varying delay interval can be adjusted by changing the parameters α and β . It is possible to find optimal dynamic upper and lower bounds of a time-varying delay, which brings the conservatism as little as possible to the LMI criterion for the NCSs with time delay. Lemma 1 ([15]). For given positive integers n, m, a scalar α in the interval (0, 1), a given n × n matrix R > 0, two matrices W1 and W2 in Rn×m . Define for all vector ξ in Rm , the function θ (a, R) given by:

θ (a, R ) =

1

α

ξ T W1T RW1 ξ +

1 ξ T W2T RW2 ξ . 1−α

Then, if there exists a matrix S in Rn×n such that [



W1 ξ min θ (a, R ) ≥ W2 ξ α ∈ ( 0,1 )

T 



R ∗

S R



(8)

R ∗

S ] > 0, then the following inequality holds R

W1 ξ . W2 ξ

(9)

Lemma 2 ([16]). For a given matrix S > 0, the following inequality holds for all continuously differentiable function σ in [a, b] → Rn :



b a

σ˙ T (s )Sσ˙ (s )ds ≥

where  = σ (a ) + σ (b) −

1 b−a



2 b−a



b a

b a

σ˙ (s )ds

T 

b

S a

 σ˙ (s )ds +

3  T S b−a

(10)

σ (s )ds.

Lemma 3 ([28]). Let ε 1 (t) and ε 2 (t) be differential functions, and f(t) is denoted by a continuous function. k(t) is a double

−ε (t ) t integral of f(t), which is denoted by −ε 2(t ) t+θ f (s )dsdθ . Then, we have 1

d k(t ) = (ε1 (t ) − ε2 (t ) ) f (t ) − (1 − ε˙ 1 (t ) ) dt



t −ε2 (t )

t −ε1 (t )

f (s )ds + (ε˙ 1 (t ) − ε˙ 2 (t ) )



t

t −ε2 (t )

f (s )ds.

(11)

3. Main results Based on the Lyapunov–Krasovskii stability theory , the following theorem can be obtained. Theorem 1. For given controller gain K, positive scalars τ 1 , τ 2 > 0, d1 , d2 > 0, and α , β satisfying (7), if there exist a 4n × 4n positive definite symmetric matrix P  Pi j (i, j = 1, . . . , 4 ), a 2n × 2n matrix X{Xij } (i, j = 1, 2 ), n × n positive definite symmetric matrices Q1 , Q2 , Q3 , S1 , S2 , free weighting matrix R, S, T with appropriate dimensions, the system (3) is asymptotically stable for delay functions τ 1 (t) and τ 2 (t) satisfying (4), where

(τ1 (t ), τ2 (t ), d1 (t ), d2 (t )) ⎡

11 12 13 14

22 23 24 ⎢ ∗ ⎢ ∗ ∗

33 34 ⎢ ⎢ ∗ ∗ ∗

44 ⎢ ∗ ∗ ∗ =⎢ ∗ ⎢ ∗ ∗ ∗ ∗ ⎢ ⎢ ∗ ∗ ∗ ∗ ⎣ ∗ ∗

11

12

13

14

15

16

17

18

22

∗ ∗

∗ ∗

∗ ∗

15

25

35

45

55 ∗ ∗ ∗ ∗

16

26

36

46

17

27

37

47

0

0

66 ∗ ∗ ∗

67

77 ∗ ∗

18 0

38 0

58

68

78

88 ∗

⎤

−RD 0 ⎥ −SD ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥<0 0 ⎥ ⎥ 0 ⎥ ⎦ −T D −ε I

T = Q1 + Q3 − 4γ (t )γ S1 − RA − AT RT + P12 + P12 + ε E T E,

= −(1 − a˙ (t ))P12 + (1 − a˙ (t ))P13 − 2γ (t )γ S1 , = −RBK − AT ST − (1 − τ˙ (t ))P13 + (1 − τ˙ (t ))P14 , = −(1 − b˙ (t ))P14 , T = 6γ (t )γ S1 + a(t )P22 ,

= (τ (t ) − a(t ))P23 , = (b(t ) − τ (t ))P24 , = R + P11 − AT T T , = (1 − a˙ (t ))Q2 − (1 − a˙ (t ))Q1 − (1 − τ˙ (t ))Q2 − 4γ (t )γ S1 − 4γ (t )S2 ,

(12)

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G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

23

24

25

26

27

33

34

35

36

= −2γ (t )S2 − γ (t )X11 − γ (t )X12 − γ (t )X21 − γ (t )X22 ,

γ (t )X11 − γ (t )X12 + γ (t )X21 − γ (t )X22 , T T = 6γ (t )γ S1 − (1 − a˙ (t ))a(t )P22 + (1 − a˙ (t ))a(t )P23 , T = 6γ (t )S2 + (1 − a˙ (t ))(τ (t ) − a(t ))P23 + (1 − a˙ (t ))(τ (t ) − a(t ))P33 , = 2γ (t )X12 + 2γ (t )X22 − (1 − a˙ (t ))(b(t ) − τ (t ))P24 + (1 − a˙ (t ))(b(t ) − τ (t ))P34 , T T T T = −8γ (t )S2 +γ (t )X11 +γ (t )X11 +γ (t )X12 − γ (t )X12 − γ (t )X21 + γ (t )X21 − γ (t )X22 − γ (t )X22 − SBK − K T BT ST , = −γ (t )X11 + γ (t )X12 + γ (t )X21 − γ (t )X22 − 2γ (t )S2 , T T = −(1 − τ˙ (t ))a(t )P23 + (1 − τ˙ (t ))a(t )P24 , T T T = 6γ (t )S2 + 2γ (t )X21 + 2γ (t )X22 − (1 − τ˙ (t ))(τ (t ) − a(t ))P23 − (1 − τ˙ (t ))(τ (t ) − a(t ))P33 T +(1 − τ˙ (t ))(τ (t ) − a(t ))P34 , T

37 = −2γ (t )X12 + 2γ (t )X22 + 6γ (t )S2 − (1 − τ˙ (t ))(b(t ) − τ (t ))P34 + (1 − τ˙ (t ))(b(t ) − τ (t ))P44 , T T T

38 = S − K B T − (1 − τ˙ (t ))(b(t ) − τ (t ))P34 ,

44 = −4γ (t )S2 − (1 − b˙ (t ))Q3 , T

45 = −(1 − b˙ (t ))a(t )P24 , T T T

46 = −2γ (t )X12 + 2γ (t )X22 − (1 − b˙ (t ))(τ (t ) − a(t ))P34 , T ˙

47 = 6γ (t )S2 − (1 − b(t ))(b(t ) − τ (t ))P44 ,

55 = −12γ (t )γ S1 , T

58 = a(t )P12 ,

66 = −12γ (t )S2 ,

67 = −4γ (t )X22 , T

68 = (τ (t ) − a(t ))P13 ,

77 = −12γ (t )S2 , T

78 = (b(t ) − τ (t ))P14 ,

88 = T + T T + a(t )S1 + (b(t ) − a(t ))S2 . =

and P12 > P13 > P14 > 0, P22 > P23 > P24 > 0, P23 > P33 > P34 > 0, P24 > P34 > P44 > 0, Q1 > Q2 , > 0. where

1 − a˙ (t ) = 1 − b˙ (t ) = 1 − α d1 − β d2 , 1 − τ˙ (t ) = 1 − d1 (t ) − d2 (t ), 1 − a˙ (t ) ( 1 − α ) τ1 + ( 1 − β ) τ2 γ (t ) = ,γ = . ( 1 − α ) τ1 + ( 1 − β ) τ2 ατ1 + βτ2 Proof. We construct the following LKF

V (x(t ), t ) =

ηT (t )Pη (t ) +  +

where



0 −a(t )

 ηT (t ) = xT (t )

t

t

t+s

t −a(t )



t

t −a(t )

xT (s )Q1 x(s )ds +

x˙ T (u )S1 x˙ (u )duds +

t −a(t )

xT (s )ds

t −τ (t )





t −a(t )

t −τ (t )

−a(t ) −b(t )



t

t+s

xT (s )Q2 x(s )ds +

t −b(t )



0 −a(t )



t

x˙ (u )S1 x˙ (u )duds +



T

t+s

= a(t )x˙ T (t )S1 x˙ (t ) − (1 − a˙ (t ))



t

t −a(t )



−a(t )

−b(t )

t

t+s



(13)

 x˙ (u )S2 x˙ (u )duds T

t −a(t )



t

t−0

x˙ T (s )S1 x˙ (s )ds

x˙ T (s )S2 x˙ (s )ds + (a˙ (t ) − b˙ (t ))

t −b(t )

= a(t )x˙ T (t )S1 x˙ (t ) + (b(t ) − a(t ))x˙ T (t )S2 x˙ (t )  t  −(1 − a˙ (t )) x˙ T (s )S1 x˙ (s )ds − (1 − b˙ (t )) t −a(t )

xT (s )Q3 x(s )ds

T

x˙ T (s )S1 x˙ (s )ds + (a˙ (t ) − 0 )

+(b(t ) − a(t ))x˙ T (t )S2 x˙ (t ) − (1 − b˙ (t ))

t −b(t )

xT (s )ds

Firstly, due to b˙ (t ) = a˙ (t ) and by Lemma 1, we have

d dt

t

x˙ T (u )S2 x˙ (u )duds

t −τ (t )

xT (s )ds



t −a(t )

t −b(t )

x˙ T (s )S2 x˙ (s )ds



t

t −b(t )

x˙ T (s )S2 x˙ (s )ds

(14)

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

5

Computing the derivative of V(x(t)) with respect to t along the trajectory of (3) yields

V˙ (x(t ), t ) = 2ηT (t )P η˙ (t ) + xT (t )Q1 x(t ) − (1 − a˙ (t ))xT (t − a(t ))Q1 x(t − a(t )) +(1 − a˙ (t ))xT (t − a(t ))Q2 x(t − a(t )) − (1 − τ˙ (t ))xT (t − a(t ))Q2 x(t − a(t )) +xT (t )Q3 x(t ) − (1 − b˙ (t ))xT (t − b(t ))Q3 x(t − b(t )) + a(t )x˙ T (t )S1 x˙ (t ) + (b(t ) − a(t ))x˙ T (t )S2 x˙ (t )  t  t −a(t ) −(1 − a˙ (t )) x˙ T (s )S1 x˙ (s )ds − (1 − a˙ (t )) x˙ T (s )S2 x˙ (s )ds (15) t −a(t )

t −b(t )

Then, similar in [28], note that if the lower bounds of derivatives of delays are known, the coefficient matrix of d1 (t) and d2 (t) has to be positive. From the term 2ηT (t )P η˙ (t ), we can get P12 > P13 > P14 > 0, P22 > P23 > P24 > 0, P23 > P33 > P34 > 0, P24 > P34 > P44 > 0, Moreover, considering the term (1 − a˙ (t ))xT (t − a(t ))(Q2 − Q1 )x(t − a(t )), we have the conclusion that V˙ (x(t ), t ) is equal or less than 0. So Q1 > Q2 . On the other hand, since ατ1 + βτ2 ≥ a(t ) ≥ 0, 1 − a˙ (t ) = 1 − α d1 − β d2 > 0 and α + β = 0 or 2, which guarantees ατ1 + βτ2 = 0 and (1 − α )τ1 + (1 − β )τ2 = 0. With Lemma 1 and 2, we get the following inequality:

−(1 − a˙ (t ))



t

t −a(t )

x˙ T (s )S1 x˙ (s )ds − (1 − a˙ (t ))

≤−

1 − a˙ (t ) a(t ) ατ1 + βτ2

≤−

1 − a˙ (t ) a(t ) ατ1 + βτ2

 ×



t

t −a(t )



t

t −a(t )



t −a(t )

x˙ T (s )S2 x˙ (s )ds

t −b(t )

x˙ T (s )S1 x˙ (s )ds −

1 − a˙ (t ) (b(t ) − a(t )) ( 1 − α ) τ1 + ( 1 − β ) τ2

x˙ T (s )S1 x˙ (s )ds −

1 − a˙ (t ) ( 1 − α ) τ1 + ( 1 − β ) τ2

b(t ) − a(t ) (τ (t ) − a(t )) τ (t ) − a(t )

  ≤ −ξ (t ) γ (t )T  ξ (t ).



t −a(t )

t −τ (t )

x˙ T (s )S2 x˙ (s )ds +



t −a(t )

t −b(t )

b(t ) − a(t ) (b(t ) − τ (t )) b(t ) − τ (t )



T

where

⎡I ⎢−I ⎢0 ⎢ ⎢0 T = ⎢ ⎢0 ⎢0 ⎣ 0 0

⎡ γ S1 ⎢ ∗ ⎢ ∗ =⎢ ⎢ ∗ ⎣ ∗ ∗

I I 0 0 −2I 0 0 0

0 I −I 0 0 0 0 0

0 3γ S1 ∗ ∗ ∗ ∗

0 I I 0 0 −2I 0 0 0 0 S2 ∗ ∗ ∗

0 0 0 3S2 ∗ ∗

x˙ T (s )S2 x˙ (s )ds

t −τ (t )

t −b(t )

 x˙ T (s )S2 x˙ (s )ds (16)

0 0 I −I 0 0 0 0 0 0 X11 X21 S2 ∗

⎤

0 0 ⎥ I ⎥ ⎥ I ⎥ ⎥, 0 ⎥ 0 ⎥ ⎦ −2I 0

⎤

0 0 ⎥ X12 ⎥ ⎥, X22 ⎥ ⎦ 0 3S2

and

γ (t ) =

1 − a˙ (t ) ( 1 − α ) τ1 + ( 1 − β ) τ2 ,γ = . ( 1 − α ) τ1 + ( 1 − β ) τ2 ατ1 + βτ2

In addition, from the Leibniz-Newton formula, the following equations are true for any matrices R, S, T with appropriate dimensions





2 xT (t )R + xT (t − τ1 (t ) − τ2 (t ))S + x˙ T (t )T    1 × x˙ (t ) − Ax(t ) − h(s )|s=(1−λ)x(t ) x(t )dλ − BKx(t − τ1 (t ) − τ2 (t )) = 0.

(17)

0

So, from(13) to (17), we obtain

V˙ (xt ) ≤ ξ T (t )ξ (t ). where

 ξ T (t ) = xT (t )

(18)

xT (t − a(t ))

1 τ (t ) − a(t )



t −a(t )

t −τ (t )

xT (t − τ (t ))

xT (s )ds

xT (t − b(t ))

1 b(t ) − τ (t )



t −τ (t )

t −b(t )

1 a(t )

xT (s )ds



t

t −a(t )



x˙ (t )

xT (s )ds

6

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

⎡ 11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ∗ =⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗

12 22

13 23 33

∗ ∗ ∗ ∗ ∗ ∗

∗

14 24 34 44

∗ ∗ ∗ ∗ ∗

15 25 35 45 55

∗ ∗ ∗ ∗

∗ ∗ ∗

16 26 36 46 0

66 ∗ ∗

17 27 37 47 0

67 77 ∗

⎤

R + P11 − Aˆ T T T ⎥ 0 ⎥ S − K T BT T T − (1 − τ˙ (t ))(b(t ) − τ (t ))P34⎥ ⎥ 0 ⎥ T ⎥ a(t )P12 ⎥ T ⎥ (τ (t ) − a(t ))P13 ⎦ T (b(t ) − τ (t ))P14 T + T T + a(t )S1 + (b(t ) − a(t ))S2

T 11 = Q1 + Q3 − 4γ (t )γ S1 − RAˆ − Aˆ T RT + P12 + P12 , 12 = −(1 − a˙ (t ))P12 + (1 − a˙ (t ))P13 − 2γ (t )γ S1 ,

13 14 15 16 17 22 23 24 25 26 27 33 34 35 36

= −RBK − Aˆ T ST − (1 − τ˙ (t ))P13 + (1 − τ˙ (t ))P14 , = −(1 − b˙ (t ))P14 , T = 6γ (t )γ S1 + a(t )P22 ,

= (τ (t ) − a(t ))P23 , = (b(t ) − τ (t ))P24 , = (1 − a˙ (t ))Q2 − (1 − a˙ (t ))Q1 − (1 − τ˙ (t ))Q2 − 4γ (t )γ S1 − 4γ (t )S2 , = −2γ (t )S2 − γ (t )X11 − γ (t )X12 − γ (t )X21 − γ (t )X22 ,

γ (t )X11 − γ (t )X12 + γ (t )X21 − γ (t )X22 ,

=

T T = 6γ (t )γ S1 − (1 − a˙ (t ))a(t )P22 + (1 − a˙ (t ))a(t )P23 , T = 6γ (t )S2 + (1 − a˙ (t ))(τ (t ) − a(t ))P23 + (1 − a˙ (t ))(τ (t ) − a(t ))P33 ,

= 2γ (t )X12 + 2γ (t )X22 − (1 − a˙ (t ))(b(t ) − τ (t ))P24 + (1 − a˙ (t ))(b(t ) − τ (t ))P34 , T T T T = −8γ (t )S2 +γ (t )X11 +γ (t )X11 +γ (t )X12 −γ (t )X12 − γ (t )X21 + γ (t )X21 − γ (t )X22 − γ (t )X22 − SBK − K T BT ST ,

= −γ (t )X11 + γ (t )X12 + γ (t )X21 − γ (t )X22 − 2γ (t )S2 , T T = −(1 − τ˙ (t ))a(t )P23 + (1 − τ˙ (t ))a(t )P24 , T T = 6γ (t )S2 + 2γ (t )X21 + 2γ (t )X22 − (1 − τ˙ (t ))(τ (t ) − a(t ))P23 T T −(1 − τ˙ (t ))(τ (t ) − a(t ))P33 + (1 − τ˙ (t ))(τ (t ) − a(t ))P34 ,

T 37 = −2γ (t )X12 + 2γ (t )X22 + 6γ (t )S2 − (1 − τ˙ (t ))(b(t ) − τ (t ))P34 + (1 − τ˙ (t ))(b(t ) − τ (t ))P44 ,

44 = −4γ (t )S2 − (1 − b˙ (t ))Q3 , T 45 = −(1 − b˙ (t ))a(t )P24 , T T T 46 = −2γ (t )X12 + 2γ (t )X22 − (1 − b˙ (t ))(τ (t ) − a(t ))P34 ,

47 55 66 67 77 Here,

T = 6γ (t )S2 − (1 − b˙ (t ))(b(t ) − τ (t ))P44 ,

= −12γ (t )γ S1 , = −12γ (t )S2 , = −4γ (t )X22 , = −12γ (t )S2 .

⎡ ⎢ ⎢ ⎢ ⎢ ˆ =⎢  ⎢ ⎢ ⎢ ⎣

ˆ 11 

12 22

∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗



+ −DT RT



+ E

ˆ 13  23 33

0

14 24 34 44

∗ ∗ ∗ ∗ ∗

0 ...

∗ ∗ ∗ ∗

−DT ST 0

15 25 35 45 55

T

 0

1

∗ ∗ ∗

0

...

16 26 36 46

R + P11 − AT T T ⎥ (1 − a˙ (t ))(b(t ) − a(t ))P24 ⎥ S − K T BT T T − (1 − τ˙ (t ))(b(t ) − τ (t ))P34⎥ ⎥ 0 ⎥ T ⎥ 0 0 a(t )P12 ⎥ T ⎥ 66 67 (τ (t ) − a(t ))P13 ⎦ T ∗ 77 (b(t ) − τ (t ))P14 T ∗ ∗ T +T  1    T F (s )|s=(1−λ)x(t ) dλ E 0 . . . 0 0 −DT T T

 F (s )|s=(1−λ)x(t ) dλ −DT RT T

⎤

17 27 37 47

0

0

−DT ST

0

...

0



−DT T T .

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

For ∀ε > 0,



−DT RT



+ E ≤

1

...

0 −D R

ε  +ε E

−DT ST

0

T

0

T

0

...

0

T 

1 0

...

0

T 

−DT T T

T 



1

F (s )|s=(1−λ)x(t ) dλ E

0

 F T (s )|s=(1−λ)x(t ) dλ −DT RT

−DT ST

0

0

E

...

0

...

0

0

−DT T T

−DT ST

0

T 

−DT RT

0

7



0

...

...

0

−DT T T

−DT ST

0

...

0

0

0

 −DT T T





0 .

With Schur complements, < 0 is equivalent to  < 0. So V˙ (x(t ), t ) < 0. It is means that the system is asymptotically stable. This completes the proof of the theorem.  Remark 2. In the light of LKF [28], based on the networked system (3), we constructed LKF (13). It is obvious that when α = 0, β = 1 or α = 1, β = 0, the extended LKF (13) is the same as before, which means that our LKF is more general than the existing ones. Furthermore, the value of α , β can choose any value between 0 and 1 except that they are both 0 and 1. So the upper bound of time delay and the result of state feedback stabilization controller K can be changed by adjusting the parameters α , β in Matlab LMI tool box. Remark 3. To getting less conservative criteria, the reciprocally convex combination technique [15] and Wirtinger integral

t

t −a(t ) inequality [16] to estimate −(1 − a˙ (t )) t −a(t ) x˙ T (s )S1 x˙ (s )ds − (1 − a˙ (t )) t −b(t ) x˙ T (s )S2 x˙ (s )ds. It is first used in the stability analysis of neural networks with two delay components [28] to enhance the feasible region of stability criteria. Remark 4. As the LMI cannot handle the time-varying terms in Theorem 1, (τ 1 (t), τ 2 (t), d1 (t), d2 (t)) is convex with respect to τ 1 (t), τ 2 (t), d1 (t), d2 (t). Therefore, (τ 1 (t), τ 2 (t), d1 (t), d2 (t)) < 0 is equivalent to the following inequalities hold: (τ 1 , τ 2 , d1M , d2M ) < 0, (0, 0, d1M , d2M ) < 0, (τ 1 , 0, d1M , d2M ) < 0, (0, τ 2 , d1M , d2M ) < 0. On the basis of Theorem 1, the following Theorem 2 will give a method for designing the state feedback stabilization controller matrixK. Theorem 2. Given positive  scalars τ 1 , τ 2 > 0, d1 , d2 > 0, and α, β satisfying (7), if there exist a 4n × 4n positive definite symmetric matrix Pˆ  Pˆi j (i, j = 1, . . . , 4 ), a 2n × 2n matrix Xˆ  Xˆi j (i, j = 1, 2 ), n × n positive definite symmetric matrices Qˆ1 , Qˆ2 , Qˆ3 , Sˆ1 , Sˆ2 , free weighting matrix R, S, T with appropriate dimensions, the delay functions τ 1 (t) and τ 2 (t) satisfying (4), the system (3) is asymptotically stable and the state-feedback controller gain K = Kˆ H −T with the following LMI holds

(τ1 (t ), τ2 (t ), d1 (t ), d2 (t )) = ⎡ 11 12 13 14 15 22 23 24 25 ⎢ ∗ ⎢ ∗ ∗ 33 34 35 ⎢ ⎢ ∗ ∗ ∗ 44 45 ⎢ ∗ ∗ ∗ 55 ⎢ ∗ ⎢ ∗ ∗ ∗ ∗ ∗ ⎢ ⎢ ∗ ∗ ∗ ∗ ∗ ⎣ ∗ ∗

∗ ∗

∗ ∗

∗ ∗

∗ ∗

16 26 36 46

17 27 37 47

0

0

66 ∗ ∗ ∗

67 77 ∗ ∗

18 0

38 0

58 68 78 88 ∗

⎤

HE T 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ < 0. 0 ⎥ ⎥ 0 ⎥ ⎦ 0 −δ I

T 11 = Qˆ1 + Qˆ3 − 4γ (t )γ Sˆ1 − AH T − HAT + Pˆ12 + Pˆ12 + δ DDT ,

12 13 14 15

= −(1 − a˙ (t ))Pˆ12 + (1 − a˙ (t ))Pˆ13 − 2γ (t )γ Sˆ1 ,

16 17 18 22 23 24 25

= (τ (t ) − a(t ))Pˆ23 , = (b(t ) − τ (t ))Pˆ24 ,

= −BKˆ − λ1 HAT − (1 − τ˙ (t ))Pˆ13 + (1 − τ˙ (t ))Pˆ14 + λ1 δ DDT , = −(1 − b˙ (t ))Pˆ14 , T = 6γ (t )γ S1 + a(t )Pˆ22 ,

= H T + Pˆ11 − λ2 HAT + λ2 δ DDT , = (1 − a˙ (t ))Qˆ2 − (1 − a˙ (t ))Qˆ1 − (1 − τ˙ (t ))Qˆ2 − 4γ (t )γ Sˆ1 − 4γ (t )Sˆ2 , = −2γ (t )Sˆ2 − γ (t )Xˆ11 − γ (t )Xˆ12 − γ (t )Xˆ21 − γ (t )Xˆ22 ,

γ (t )Xˆ11 − γ (t )Xˆ12 + γ (t )Xˆ21 − γ (t )Xˆ22 , T T = 6γ (t )γ S1 − (1 − a˙ (t ))a(t )Pˆ22 + (1 − a˙ (t ))a(t )Pˆ23 , =

(19)

8

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17 T 26 = 6γ (t )Sˆ2 + (1 − a˙ (t ))(τ (t ) − a(t ))Pˆ23 + (1 − a˙ (t ))(τ (t ) − a(t ))Pˆ33 ,

27 = 2γ (t )Xˆ12 + 2γ (t )Xˆ22 − (1 − a˙ (t ))(b(t ) − τ (t ))Pˆ24 + (1 − a˙ (t ))(b(t ) − τ (t ))Pˆ34 , T T T 33 = −8γ (t )Sˆ2 + γ (t )Xˆ11 + γ (t )Xˆ11 + γ (t )Xˆ12 − γ (t )Xˆ12 − γ (t )Xˆ21 + γ (t )Xˆ21 T −γ (t )Xˆ22 γ (t )Xˆ22 − λ1 BKˆ − λ1 Kˆ T BT + λ21 δ DDT ,

34 = −γ (t )Xˆ11 + γ (t )Xˆ12 + γ (t )Xˆ21 − γ (t )Xˆ22 − 2γ (t )Sˆ2 , T T 35 = −(1 − τ˙ (t ))a(t )Pˆ23 + (1 − τ˙ (t ))a(t )Pˆ24 , T T 36 = 6γ (t )Sˆ2 + 2γ (t )Xˆ12 + 2γ (t )Xˆ22 − (1 − τ˙ (t ))(τ (t ) − a(t ))Pˆ23 T T −(1 − τ˙ (t ))(τ (t ) − a(t ))Pˆ33 + (1 − τ˙ (t ))(τ (t ) − a(t ))Pˆ34 , T 37 = −2γ (t )Xˆ12 + 2γ (t )Xˆ22 + 6γ (t )Sˆ2 − (1 − τ˙ (t ))(b(t ) − τ (t ))Pˆ34 + (1 − τ˙ (t ))(b(t ) − τ (t ))Pˆ44 ,

38 = λ1 H T − λ2 Kˆ T BT − (1 − τ˙ (t ))(b(t ) − τ (t ))Pˆ34 + λ1 λ2 δ DDT , 44 = −4γ (t )Sˆ2 − (1 − b˙ (t ))Qˆ3 , T 45 = −(1 − b˙ (t ))a(t )Pˆ24 , T T T 46 = −2γ (t )Xˆ21 + 2γ (t )Xˆ22 − (1 − b˙ (t ))(τ (t ) − a(t ))Pˆ34 , T 47 = 6γ (t )Sˆ2 − (1 − b˙ (t ))(b(t ) − τ (t ))Pˆ44 ,

55 = −12γ (t )γ Sˆ1 , T 58 = a(t )Pˆ12 , 66 = −12γ (t )Sˆ2 , 67 = −4γ (t )Xˆ22 , T 68 = (τ (t ) − a(t ))Pˆ13 , 77 = −12γ (t )Sˆ2 , T 78 = (b(t ) − τ (t ))Pˆ14 , 88 = λ2 H T + λ2 H + λ22 δ DDT + a(t )Sˆ1 + (b(t ) − a(t ))Sˆ2 . ˆ > 0. and Pˆ12 > Pˆ13 > Pˆ14 > 0, Pˆ22 > Pˆ23 > Pˆ24 > 0, Pˆ23 > Pˆ33 > Pˆ34 > 0, Pˆ24 > Pˆ34 > Pˆ44 > 0, Qˆ1 > Qˆ2 , where

1 − a˙ (t ) = 1 − b˙ (t ) = 1 − α d1 − β d2 , 1 − τ˙ (t ) = 1 − d1 (t ) − d2 (t ), 1 − a˙ (t ) ( 1 − α ) τ1 + ( 1 − β ) τ2 γ (t ) = ,γ = . ( 1 − α ) τ1 + ( 1 − β ) τ2 ατ1 + βτ2 Proof. Set S = λ1 R, T = λ2 R. Based on the feasibility of (12), we know that R is reversible. So setting H = R−1 . On the other hand,pre-multiplying diag{H, H, H, H, H, H, H, H, I}. And post-multiplying diag{HT , HT , HT , HT , HT , HT , HT , HT , IT }. We introduce some new variables Qˆ1 = H Q1 H T , Qˆ2 = H Q2 H T , Qˆ3 = H Q3 H T , Sˆ1 = H S1 H T , Pˆ11 = H P11 H T , Pˆ12 = H P12 H T , Pˆ13 = H P13 H T , Pˆ14 = H P14 H T , Pˆ22 = H P22 H T , Pˆ23 = H P23 H T , Pˆ24 = H P24 H T , Pˆ33 = H P33 H T , Pˆ34 = H P34 H T , Pˆ44 = H P44 H T .Xˆ11 = H X11 H T , Xˆ12 = H X12 H T , Xˆ21 = H X21 H T , Xˆ22 = H X22 H T , Kˆ = KH T , δ = 1 . ε

Based on the Schur complements, we can get LMI (19). This completes the proof.



When there is no Lipschitz nonlinear perturbation in system (3), the physical plant is a linear system and the controller is a linear state-feedback one with a controller gain K, the networked control system takes the following form:

x˙ (t ) = Ax(t ) + BKx(t − τ1 (t ) − τ2 (t )).

(20)

Based on the proof of Theorems 1 and 2, we can get the Corollary 1 and 2 for system (20). Corollary 1. For given controller gain K, positive scalars τ 1 , τ 2 > 0, d1 , d2 > 0, and α , β satisfying (7), if there exist a 4n × 4n positive definite symmetric matrix P  Pi j (i, j = 1, . . . , 4 ), a 2n × 2n matrix X{Xij } (i, j = 1, 2 ), n × n positive definite symmetric matrices Q1 , Q2 , Q3 , S1 , S2 , free weighting matrix R, S, T with appropriate dimensions, the system (20) is asymptotically

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

9

stable for delay functions τ 1 (t) and τ 2 (t) satisfying (4), where

(τ1 (t ), τ2 (t ), d1 (t ), d2 (t )) ⎡ 12 13 14 11 22 23 24 ⎢ ∗ ⎢ ∗ ∗ 33 34 ⎢ ∗ ∗ 44 ⎢ ∗ =⎢ ∗ ∗ ∗ ⎢ ∗ ⎢ ∗ ∗ ∗ ∗ ⎣ ∗ ∗

11 12 13 14 15 16 17 18 22 23 24 25 26 27 33 34 35 36 37 38 44

∗ ∗

∗ ∗

∗ ∗

15 25 35 45 55 ∗ ∗ ∗

16 26 36 46

17 27 37 47

0

0

66 ∗ ∗

67 77 ∗

18 ⎤ 0 ⎥ 38 ⎥ ⎥ 0 ⎥ < 0. 58 ⎥ ⎥ ⎥ 68 ⎦ 78 88

(21)

T = Q1 + Q3 − 4γ (t )γ S1 − RA − AT RT + P12 + P12 ,

= −(1 − a˙ (t ))P12 + (1 − a˙ (t ))P13 − 2γ (t )γ S1 , = −RBK − AT ST − (1 − τ˙ (t ))P13 + (1 − τ˙ (t ))P14 , = −(1 − b˙ (t ))P14 , T = 6γ (t )γ S1 + a(t )P22 ,

= (τ (t ) − a(t ))P23 , = (b(t ) − τ (t ))P24 , = R + P11 − AT T T , = (1 − a˙ (t ))Q2 − (1 − a˙ (t ))Q1 − (1 − τ˙ (t ))Q2 − 4γ (t )γ S1 − 4γ (t )S2 , = −2γ (t )S2 − γ (t )X11 − γ (t )X12 − γ (t )X21 − γ (t )X22 ,

γ (t )X11 − γ (t )X12 + γ (t )X21 − γ (t )X22 , T T = 6γ (t )γ S1 − (1 − a˙ (t ))a(t )P22 + (1 − a˙ (t ))a(t )P23 , T = 6γ (t )S2 + (1 − a˙ (t ))(τ (t ) − a(t ))P23 + (1 − a˙ (t ))(τ (t ) − a(t ))P33 , = 2γ (t )X12 + 2γ (t )X22 − (1 − a˙ (t ))(b(t ) − τ (t ))P24 + (1 − a˙ (t ))(b(t ) − τ (t ))P34 , T T T T = −8γ (t )S2 +γ (t )X11 +γ (t )X11 +γ (t )X12 −γ (t )X12 − γ (t )X21 + γ (t )X21 − γ (t )X22 − γ (t )X22 − SBK − K T BT ST , = −γ (t )X11 + γ (t )X12 + γ (t )X21 − γ (t )X22 − 2γ (t )S2 , T T = −(1 − τ˙ (t ))a(t )P23 + (1 − τ˙ (t ))a(t )P24 , T T T T = 6γ (t )S2 + 2γ (t )X21 + 2γ (t )X22 − (1 − τ˙ (t ))(τ (t ) − a(t ))P33 + (1 − τ˙ (t ))(τ (t ) − a(t ))P34 , T T = −2γ (t )X12 + 2γ (t )X22 + 6γ (t )S2 − (1 − τ˙ (t ))(b(t ) − τ (t ))P34 + (1 − τ˙ (t ))(b(t ) − τ (t ))P44 , = S − KBT − (1 − τ˙ (t ))(b(t ) − τ (t ))P34 , = −4γ (t )S2 − (1 − b˙ (t ))Q3 , T 45 = −(1 − b˙ (t ))a(t )P24 , T T T 46 = −2γ (t )X21 + 2γ (t )X22 − (1 − b˙ (t ))(τ (t ) − a(t ))P34 , T 47 = 6γ (t )S2 − (1 − b˙ (t ))(b(t ) − τ (t ))P44 , 55 = −12γ (t )γ S1 , T 58 = a(t )P12 , 66 = −12γ (t )S2 , 67 = −4γ (t )X22 , T 68 = (τ (t ) − a(t ))P13 , 77 = −12γ (t )S2 , T 78 = (b(t ) − τ (t ))P14 , T 88 = T + T + a(t )S1 + (b(t ) − a(t ))S2 . =

and P12 > P13 > P14 > 0, P22 > P23 > P24 > 0, P23 > P33 > P34 > 0, P24 > P34 > P44 > 0, Q1 > Q2 , > 0. where

1 − a˙ (t ) = 1 − b˙ (t ) = 1 − α d1 − β d2 , 1 − τ˙ (t ) = 1 − d1 (t ) − d2 (t ), 1 − a˙ (t ) ( 1 − α ) τ1 + ( 1 − β ) τ2 γ (t ) = ,γ = . ( 1 − α ) τ1 + ( 1 − β ) τ2 ατ1 + βτ2

10

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

Corollary 2. Given positive  scalars τ 1 , τ 2 > 0, d1 , d2 > 0, and α, β satisfying (7), if there exist a 4n × 4n positive definite symmetric matrix Pˆ  Pˆi j (i, j = 1, . . . , 4 ), a 2n × 2n matrix Xˆ  Xˆi j (i, j = 1, 2 ), n × n positive definite symmetric matrices Qˆ1 , Qˆ2 , Qˆ3 , Sˆ1 , Sˆ2 , free weighting matrix R, S, T with appropriate dimensions, the delay functions τ 1 (t) and τ 2 (t) satisfying (4), the system (20) is asymptotically stable and the state-feedback controller gain K = Kˆ H −T with the following LMI holds

ϒ (τ1 (t ), τ2 (t ), d1 (t ), d2 (t )) ⎡ϒ ϒ12 ϒ13 ϒ14 11 ϒ22 ϒ23 ϒ24 ⎢ ∗ ⎢ ∗ ∗ ϒ33 ϒ34 ⎢ ∗ ∗ ϒ44 ⎢ ∗ =⎢ ∗ ∗ ∗ ⎢ ∗ ⎢ ∗ ∗ ∗ ∗ ⎣ ∗ ∗

∗ ∗

∗ ∗

∗ ∗

ϒ15 ϒ25 ϒ35 ϒ45 ϒ55 ∗ ∗ ∗

ϒ16 ϒ26 ϒ36 ϒ46

ϒ17 ϒ27 ϒ37 ϒ47

0

0

ϒ66 ∗ ∗

ϒ67 ϒ77 ∗

ϒ18 ⎤ 0 ⎥ ϒ38 ⎥ ⎥ 0 ⎥ < 0. ϒ58 ⎥ ⎥ ⎥ ϒ68 ⎦ ϒ78 ϒ88

(22)

T ϒ11 = Qˆ1 + Qˆ3 − 4γ (t )γ Sˆ1 − AH T − HAT + Pˆ12 + Pˆ12 ,

ϒ12 ϒ13 ϒ14 ϒ15

= −(1 − a˙ (t ))Pˆ12 + (1 − a˙ (t ))Pˆ13 − 2γ (t )γ Sˆ1 ,

ϒ16 ϒ17 ϒ18 ϒ22 ϒ23 ϒ24 ϒ25

= (τ (t ) − a(t ))Pˆ23 , = (b(t ) − τ (t ))Pˆ24 ,

= −BKˆ − λ1 HAT − (1 − τ˙ (t ))Pˆ13 + (1 − τ˙ (t ))Pˆ14 , = −(1 − b˙ (t ))Pˆ14 , T = 6γ (t )γ Sˆ1 + a(t )Pˆ22 ,

= H T + Pˆ11 − λ2 HAT , = (1 − a˙ (t ))Qˆ2 − (1 − a˙ (t ))Qˆ1 − (1 − τ˙ (t ))Qˆ2 − 4γ (t )γ Sˆ1 − 4γ (t )Sˆ2 , = −2γ (t )Sˆ2 − γ (t )Xˆ11 − γ (t )Xˆ12 − γ (t )Xˆ21 − γ (t )Xˆ22 ,

γ (t )Xˆ11 − γ (t )Xˆ12 + γ (t )Xˆ21 − γ (t )Xˆ22 , T T = 6γ (t )γ Sˆ1 − (1 − a˙ (t ))a(t )Pˆ22 + (1 − a˙ (t ))a(t )Pˆ23 , T ϒ26 = 6γ (t )Sˆ2 + (1 − a˙ (t ))(τ (t ) − a(t ))Pˆ23 + (1 − a˙ (t ))(τ (t ) − a(t ))Pˆ33 , ϒ27 = 2γ (t )Xˆ12 + 2γ (t )Xˆ22 − (1 − a˙ (t ))(b(t ) − τ (t ))Pˆ24 + (1 − a˙ (t ))(b(t ) − τ (t ))Pˆ34 , T T T T ϒ33 = −8γ (t )Sˆ2 +γ (t )Xˆ11 +γ (t )Xˆ11 +γ (t )Xˆ12 − γ (t )Xˆ12 − γ (t )Xˆ21 + γ (t )Xˆ21 − γ (t )Xˆ22 γ (t )Xˆ22 − λ1 BKˆ − λ1 Kˆ T BT , ϒ34 = −γ (t )Xˆ11 + γ (t )Xˆ12 + γ (t )Xˆ21 − γ (t )Xˆ22 − 2γ (t )Sˆ2 , T T ϒ35 = −(1 − τ˙ (t ))a(t )Pˆ23 + (1 − τ˙ (t ))a(t )Pˆ24 , T T T T ˆ ˆ ˆ ϒ36 = 6γ (t )S2 + 2γ (t )X12 + 2γ (t )X22 − (1 − τ˙ (t ))(τ (t ) − a(t ))Pˆ23 − (1 − τ˙ (t ))(τ (t ) − a(t ))Pˆ33 T +(1 − τ˙ (t ))(τ (t ) − a(t ))Pˆ34 , T T ϒ37 = −2γ (t )Xˆ12 + 2γ (t )Xˆ22 + 6γ (t )Sˆ2 − (1 − τ˙ (t ))(b(t ) − τ (t ))Pˆ34 + (1 − τ˙ (t ))(b(t ) − τ (t ))Pˆ44 , T T T ϒ38 = λ1 H − λ2 Kˆ B − (1 − τ˙ (t ))(b(t ) − τ (t ))Pˆ34 , ϒ44 = −4γ (t )Sˆ2 − (1 − b˙ (t ))Qˆ3 , T ϒ45 = −(1 − b˙ (t ))a(t )Pˆ24 , T T T ˆ ϒ46 = −2γ (t )X21 + 2γ (t )Xˆ22 − (1 − b˙ (t ))(τ (t ) − a(t ))Pˆ34 , T ˙ ˆ ˆ ϒ47 = 6γ (t )S2 − (1 − b(t ))(b(t ) − τ (t ))P44 , ϒ55 = −12γ (t )γ Sˆ1 , T ϒ58 = a(t )Pˆ12 , ϒ66 = −12γ (t )Sˆ2 , ϒ67 = −4γ (t )Xˆ22 , T ϒ68 = (τ (t ) − a(t ))Pˆ13 , ˆ ϒ77 = −12γ (t )S2 , T ϒ78 = (b(t ) − τ (t ))Pˆ14 , T ϒ88 = λ2 H + λ2 H + a(t )Sˆ1 + (b(t ) − a(t ))Sˆ2 . =

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

11

Table 1 The results of K with different α , β .

α, β

α = 0.4, β = 0.4

α = 0.5, β = 0.5

α = 0.6, β = 0.6

α = 0.7, β = 0.7

K

[−1.1582 − 1.4334]

[−1.0827 − 1.4387]

[−0.9706 − 1.4319]

[−0.8163 − 1.3361]

Fig. 1. Simulation results of Example 1 with the state feedback controller K.

ˆ > 0. and Pˆ12 > Pˆ13 > Pˆ14 > 0, Pˆ22 > Pˆ23 > Pˆ24 > 0, Pˆ23 > Pˆ33 > Pˆ34 > 0, Pˆ24 > Pˆ34 > Pˆ44 > 0, Qˆ1 > Qˆ2 , where

1 − a˙ (t ) = 1 − b˙ (t ) = 1 − α d1 − β d2 , 1 − τ˙ (t ) = 1 − d1 (t ) − d2 (t ), 1 − a˙ (t ) ( 1 − α ) τ1 + ( 1 − β ) τ2 γ (t ) = ,γ = . ( 1 − α ) τ1 + ( 1 − β ) τ2 ατ1 + βτ2 Remark 5. The proof of Corollary 1 and 2 is similar by constructing the same LKF. So the detailed proof process is omitted. Based on the results of Corollary 1, we can give a comparative with [23–27] in conservatism. 4. Simulation examples Example 1. Consider the system proposed in (3) with parameters as follows:



A=

0.5 0





2 0.2 sin(x2 (t )) , f (x(t )) = 0.1 0.5 sin(x1 (t ))

It is easy to obtain that



df (x(t )) cos(x1 (t )) h(x(t )) = =D 0 dx(t ) where

 D=

√0 0.5

√ √  0.2 0.5 ,E = 0 0



 

,B =

1 , 1



0 E cos(x2 (t ))



√0 . 0.2

Based on this example, setting d1 = 0, 2, d2 = 0.7, λ1 = 0.05, λ2 = 0.6, τ1 = 0.1, τ2 = 0.2, we solved the LMIs in Theorem 2 and got the state feedback controller is listed in Table 1 with different α , β . The simulation results are shown in Figs. 1–4 for the different results of K. We find that all the state feedback controller K can keep the state trajectory of the system (3) converge to zero.

12

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

Fig. 2. Simulation results of Example 1 with the state feedback controller K.

Fig. 3. Simulation results of Example 1 with the state feedback controller K.

In this here, we need to point Fig. 1 that the value of α , β is not only in Table 1. For this example, setting α = 0.1, β = 0.9, the state Fig. 2 feedback controller K = [−0.7760 − 1.2571]. It is also can can keep the state trajectory of the system (3) converge to zero Fig. 3 and illustrating the system(3) is asymptotically stable. Example 2. Consider the system proposed in (3) with parameters as follows:

⎡

0 ⎢−48.6 A=⎣ 0 19.5

1 −1.25 0 0

0 48.6 0 −19.5

⎡

⎤

⎤ ⎡ ⎤ 0 0 0 21.6⎥ ⎢ 0⎥ 0 ⎢ ⎥ ⎥ , f (x(t )) = ⎣ ⎦, B = ⎢ ⎣ 0 ⎦, 1⎦ 0 0 0 −3.33 sin(x3 (t ))

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

13

Fig. 4. Simulation results of Example 1 with the state feedback controller K.

It is easy to obtain that

⎡

0 df (x(t )) ⎢0 h(x(t )) = = D⎣ 0 dx(t ) 0 where

⎡

0 ⎢0 D=⎣ 0 0

0 0 0 0

0 0 0 0

⎤

0 0 √0 3.33

0 0 cos(x3 (t )) 0

⎡

0 0 0⎥ ⎢0 ,E = ⎣ 0⎦ 0 0 0

0 0 0 0

⎤

0 0⎥ E. 0⎦ 0

0 √0 − 3.33 0

⎤

0 0⎥ ⎦. 0 0

This example is a single-link flexible Fig. 4 joint robot system [29,30]. For d1 = 0.02, d2 = 0.1, λ1 = 0.05, λ2 = 0.01, τ1 = 0.05, τ2 = 0.06, α = 0.8, β = 0.9, we solved the LMIs in Theorem 2 and got Fig. 5 the state feedback controller K = [−0.2818 − 0.1151 − 0.3063 − 0.2855] such that the system (3) is asymptotically stable. The Fig. 6 simulation results of this example is shown in Fig. 5 with state feedback controller K. Example 3. Consider the system proposed in (20) with parameters as follows:



A=

−2 0





0 1 ,B = −0.9 0



0 , 1

The state negative feedback gain matrix is



K =

−1 −1



0 . −1

As in [23–27], it is assumed that τ˙ 1 (t ) ≤ 0.1, τ˙ 2 (t ) ≤ 0.8. With the help of Corollary 1, setting α = 0.1, β = 0.1, we get the admissible Fig. 7 upper bounds of τ 2 under different value of τ 1 . From Table 2, it is clear that the stability criteria of Corollary 1 is less conservative than Fig. 8 those in [23–27]. Example 4. Consider the Figure 9 system proposed in (20) with parameters as follows:



A=

0.12 1







0 0.2 ,B = . −0.9 −0.5

Based on this Table 3 example, setting d1 = 0, 1, d2 = 0.2, λ1 = 0.1, λ2 = 0.6, τ1 = 1, τ2 = 2, we solved the LMIs in Corollary 2 and got the state feedback controller is listed in Table 1 with different α , β . The simulation results are shown

14

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

Fig. 5. Simulation results of Example 2 with the state feedback controller K.

Fig. 6. Simulation results of Example 4 with the state feedback controller K.

Table 2 Admissible upper bounds of τ 2 under different value of τ1. Method

d1

1

1.2

1.5

[23] [24] [25] [26] [27] Corollary 1 (α , β = 0.1)

d2 d2 d2 d2 d2

0.415 0.512 0.583 0.873 0.988

0.376 0.406 0.519 0.673 0.836

0.248 0.283 0.421 0.373 0.563

d2

3.372

3.172

2.872

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

15

0.3 x1(t) x2(t)

0.25 0.2

x(t)

0.15 0.1 0.05 0 −0.05 −0.1

0

5

10

15

20

25 30 time(sec)

35

40

45

50

Fig. 7. Simulation results of Example 4 with the state feedback controller K.

Fig. 8. Simulation results of Example 4 with the state feedback controller K.

in Figs. 6–9 with the different results of K. We find that all the state feedback controller K can keep the state trajectory of the system (20) converge to zero.

5. Conclusion In this paper, we focuses on the problem of delay-dependent feedback control for a class of NCSs with two delay components. Based on the DDI method, the delay-dependent stability analysis problem has been investigated by constructing a new augmented Lyapunov functional for a class of NCSs. The reciprocally convex combination technique and Wirtinger integral inequality was applied to estimate the derivative of the Lyapunov functional. Then, based on the stability results, state feedback stabilization was studied. The new designed controller has been obtained and the closed-loop system was

16

G. Chen et al. / Applied Mathematics and Computation 316 (2018) 1–17

Fig. 9. Simulation results of Example 4 with the state feedback controller K. Table 3 The results of K with different α , β .

α, β

α = 0.4, β = 0.4

α = 0.5, β = 0.5

α = 0.6, β = 0.6

α = 0.7, β = 0.7

K

[−1.4103 0.1615]

[−1.4090 0.1748]

[−1.3825 0.1762]

[−1.3570 0.1696]

asymptotically stable by the delay-dependent conditions. Numerical examples were given to show the effectiveness and the significant improvement of the proposed methods. Acknowledgement This work was supported by Natural Science Foundation of China (No. 61573177 and 61403178), the Natural Science Foundation of Shandong Province for Outstanding Young Talents in Provincial Universities under Grant ZR2016JL025. References [1] L.G. Chen, J.W. Xia, G.M. Zhuang, Delay-dependent stability and dissipativity analysis of generalized neural networks with Markovian jump parameters and two delay components, J. Frankl. Inst. 353 (2016) 2137–2158. [2] J.W. Xia, J.H. Park, T.H. Lee, B.Y. Zhang, h∞ tracking of uncertain stochastic time-delay systems: memory state-feedback controller design, Appl.Math.Comput. 249 (2014) 356–370. [3] H. Shen, J.H. Park, L.X. Zhang, Z.G. Wu, Robust extended dissipative control for sampled data Markov jump systems, Int. J. Control 87 (2014) 1549–1564. [4] H. Shen, Z.G. Wu, J.H. Park, Reliable mixed passive and filtering for semi-Markov jump systems with randomly occurring uncertainties and sensor failures, Int. J. Robust Nonlinear Control 25 (2015) 3231–3251. [5] D. Yue, E.G. Tian, Q.L. Han, A delay system method for designing event-triggered controllers of networked control systems, IEEE Trans. Autom. Control, 58 (2) (2013) 475–481. [6] J. Sun, G.P. Liu, State feedback and output feedback control of a class of nonlinear systems with delayed measurements, Nonlinear Anal. 67 (2007) 1623–1636. [7] E. Gyurkovics, D. Eszes, Sufficient conditions for stability and stabilization of networked control systems with uncertainties and nonlinearities, Int. J. Robust Nonlinear Control, 25 (2015) 3004–3022. [8] T.H. Lee, J.H. Park, Stability analysis of sampled-data systems via free-matrix-based time-dependent discontinuous Lyapunov approach, IEEE Trans. Autom. Control 62 (2017) 3653–3657. [9] T.H. Lee, J.H. Park, Improved criteria for sampled-data synchronization of chaotic Lure systems using two new approaches, Nonlinear Anal. Hybrid Syst., 24 (2017) 132–145. [10] K. Gu, Discretized LMI set in the stability problem for linear uncertain time-delay systems, Int.J. Control 68 (4) (1997) 923–934. [11] J. Sun, G.P. Liu, J. Chen, Improved delay-range-dependent stability criteria for linear systems with time-varying delays, Automatica, 46 (2) (2010) 466–470. [12] H.B. Zeng, Y. He, M. Wu, Improved conditions for passivity of neural networks with a time-varying delay, IEEE Trans. Cybern., 44 (2014) 785–792. [13] K. Liu, E. Fridman, Networked-based stabilization via discontinuous Lyapunov functionals, Int. J. Robust Nonlinear Control, 22 (2012) 420–436. [14] Y. He, M. Wu, J.H. She, G.P. Liu, Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic type uncertainties, IEEE Trans. Automat. Control, 49 (2004) 828–832. [15] P. Park, J.W. Ko, C.K. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47 (2011) 235–238. [16] A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: application to time-delay systems, Automatica, 49 (9) (2013) 2860–2866.

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