H∞ synchronization of uncertain stochastic time-varying delay systems with exogenous disturbance via intermittent control

Chaos, Solitons and Fractals 127 (2019) 244–256

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

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H∞ synchronization of uncertain stochastic time-varying delay systems with exogenous disturbance via intermittent control Kui Ding a, Quanxin Zhu a,b,∗ a b

School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, PR China MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081 Hunan, PR China

a r t i c l e

i n f o

Article history: Received 25 January 2019 Revised 2 June 2019 Accepted 28 June 2019

Keywords: H∞ synchronization Exogenous disturbance Aperiodically intermittent control Parameter uncertainly Local martingales

a b s t r a c t In this work, the drive-response H∞ synchronization of stochastic time-varying systems with exogenous disturbance and parameter uncertainly via intermittent control is discussed. Based on a novel piecewise time dependent Lyapunov functional (PTDLF) and the strong low of large numbers for local martingales, the sufficient causes presented by linear matrix inequalities are established to ensure the mean square exponential synchronization and almost sure exponential synchronization with L2 -gain performance for the stochastic drive-response system, respectively. In addition, two types of delay independent intermittent controllers are designed in view of the obtained synchronization standard. Finally, two numerical examples are given to illustrate the rationality and effectiveness of the method under investigation.

1. Introduction Stochastic systems have developed rapidly since the middle of last century and been widely used in various fields, such as macroeconomics, chemical reactor control, population dynamics, and others. A host of problems with stochastic systems, such as optimal control [1], observer design [2], stability analysis [3], etc. have been investigated by researchers. However, in recent years, the researchs on the synchronization problem of stochastic systems have also received strong attention from a slice of scholars. Meanwhile, with the development of control theory and engineering, various types of control strategies are applied to such problems, such as event-trigger control [4], adaptive control [5], sampleddata control [6], impulsive control [7–10], intermittent control [11–16], and more. In which, the discontinuous control method includes impulsive control and intermittent control. Compared with continuous controls, researchers are paying more attention to to discontinuous control because it can reduce costs effectively. Furthermore, intermittent control can better control the performance of closed-loop systems compared with impulsive control. Therefore, a large amount of efforts have been spent on intermittent control to solve the synchronization problem of stochastic systems. However, to author’s knowledge, in the existing results of the intermittent control approach [11–16], the stability conclusion is mainly obtained by the ordinary Lyapunov function or the Lya∗

Corresponding author. E-mail address: [email protected] (Q. Zhu).

https://doi.org/10.1016/j.chaos.2019.06.038 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

© 2019 Elsevier Ltd. All rights reserved.

punov functional, namely, the two modes (open-loop and closedloop) share the same Lyapunov Function or Lyapunov functional. The basic idea is to obtain the growth rate in the open-loop mode and the decay rate in the corresponding closed-loop mode according to some traditional inequalities. Then, the closed-loop decay rate is used to suppress the open-loop growth rate. Finally, the stability criterion of the system can be established after some iterations. Moreover, it is well known that the process of dealing with the switching system [17] under the same Lyapunov function or Lyapunov functional is very demanding and the resulting stability result is also very standpat. Intermittent can be considered as a special switch, so these conclusions may also be very conservative. Recently, Chen et al. [18] proposed a strategy for analyzing the stability of intermittently controlled systems by employing a piecewise Lyapunov function and a piecewise Lyapunov functional. This means can make good use of the information inside the two different modalities and the results can also reduce the certain conservativeness compared with the previous methods. Nevertheless, the result given by Chen et al. [18] also has the same precondition as [11–15]: the magnitude of the time delay cannot exceed the control length. Much lately, although some subsequent conclusions [19,20] that can eliminate this condition are presented, the stability conclusions are still largely dependent on the upper and lower bounds of the time delay. This indicates that most of the conclusions obtained by the existing intermittent control method for the processing of the delayed systems only match the delay bounded case. These observations have inspired us to explore the following questions: what conditions should be set for the time-delay system

K. Ding and Q. Zhu / Chaos, Solitons and Fractals 127 (2019) 244–256

to obtain a delay-independent intermittent controller? It needs to point out that the existing works are based on the condition of time-delay dependence, so this problem cannot be solved by the these ways. On the other hand, by all accounts, noise and other exogenous disturbance are inevitable in actual systems, and the disturbance can make the system poor performance, even make the response system unable to synchronize with the drive system. Accordingly, it is very essential to design a delay independent intermittent controller which can effectively suppress the exogenous disturbance and the random noise. Through the study of control theory, H∞ control is recognized as an effective robust control mechanism for solving noisy systems. In addition, so far, there have been very few studies on the H∞ synchronization problem of stochastic systems with time-delay by dint of intermittent control [21–23]. It is also explained here that the main difficulty in solving this problem is that the L2 gain of dynamic behavior in the open-loop mode cannot be assured. Moreover, the existence of random noise and parameter uncertainty in the system greatly increases the complexity of this problem. Therefore, the intermittent H∞ synchronization of uncertain stochastic systems with time-delay and exogenous disturbance is still a meaningful and challenging problem. This is another motivation for this study. Based on the above discussion, this paper focuses on a class of stochastic time delay systems with parameter uncertainty and exogenous disturbance to study the delay independent asymptotically exponential moment synchronization and almost sure exponential synchronization via intermittent control. By applying a novel PTDLF and combining Itoˆ’s formula, the drive-response H∞ synchronization in mean square of the system is studied. An significant characteristic of this PTDLF is that the selected Lyapunov functional does not increase at the moment of switching. Subsequently, by using the law of large numbers for local martingales and the PTDLF, the almost sure exponential H∞ synchronization of the system is also addressed. Based on the results of the previous study, two types of controllers are designed. Finally, two examples and its numerical simulations are given to illustrate the rationality of our research results. This paper mainly emphasizes three major contributions as follows: 1) The control period and control length in the intermittent control strategy planned in this paper are all variable. When the control period and the control length change, it represents the aperiodically intermittent control mode; when the control period and the control length are fixed, it is reduced to the periodically intermittent control mode. Compared with the previous results, either only periodically intermittent control or only aperiodically intermittent control, the intermittent strategy in this paper is more suitable for the actual controller. 2) The constructed delay-independent intermittent controller solves the problem that the time delay cannot exceed the control length in the existing results, and effectively suppresses the exogenous disturbance and the random noise. 3) In the analysis of stochastic synchronization, the paper is based on the updated PTDLF method, which can promote the Lyapunov functional not to increase at the switching time and can well grasp the performance of the closed-loop system [24]. Notation: The notation X > ( ≥ , < , ≤ ) 0 is used to denote a symmetric positive-definite (positive-semidefinite, negative, negative-semi definite) matrix. XT and X −1 denote the transpose and the inverse of any square matrix X, respectively. I is an identity matrix of suitable dimension. Trace ( · ) stands for the trace of the corresponding matrix. N0 refers to the set of non–negative integers, i.e., N0 = {0, 1, 2, . . .}.  .  represents the Euclidean vector norm. E denotes the mathematical expectation. For a scalar d¯ > 0 and for any φ ∈ C ([−d¯, 0]; Rn ), define φd¯ = max−d¯≤s≤0 φ (s ). If

245

e ∈ C ([−d¯, a]; Rn ) with a > 0 and t ∈ [0, a), then et ∈ C ([−d¯, a]; Rn ) is defined by et (θ ) = e(t + θ ), θ ∈ [−d¯, 0]. i ∈ 1, n is said to be i ∈ {1, 2, , n}. 2. Problem Formulation Consider the following uncertain drive syetem

⎧ 

2  ⎪ ⎪ dx(t ) = Ar (t )x(t − dr (t )) + Br (t ) f (r ) (x(t − dr (t ))) + J (t ) dt ⎪ ⎨ r=1 +g(x(t ))dω (t ), t > 0 ⎪ ⎪ zˆ(t ) = Cx(t ), ⎪ ⎩ x(t ) = ϕ (t ), t ∈ [−d¯, 0], (1) where x(t ) ∈ Rn is the state vector of drive system (1); J (t ) ∈ Rn represents the external input vector; zˆ(t ) ∈ Rq is the controlled output of drive system (1); d1 (t ) = 0, d2 (t ) = d (t ), where d(t) is the time-varying delay satisfying 0 ≤ d (t ) ≤ d¯, d˙ (t ) ≤ dˆ < 1; ϕ (t) is the initial function for t ∈ [−d¯, 0]; ω (t ) ∈ Rm0 stands for the m0 –dimensional Wiener process defined on complete probability pace (, F, {Ft }t≥0 , P ) with a filtration {Ft }t≥0 ; Ar (t ) = Ar + Ar (t ), Br (t ) = Br + Br (t ), r = 1, 2, and here Ar , Br ∈ Rn×n , C ∈ Rq×n are known matrices, whereas Ar (t), Br (t) donate timevarying uncertain matrices, and assume that the following conditions are met:

[A1 (t ) A2 (t ) B1 (t ) B2 (t )] = MF (t )[N1 N2 N3 N4 ],

(2)

where M ∈ N1 , N2 , N3 , N4 ∈ are known matrices, F (t ) is a time-varying unknown matrix, which satisfies the condition for any t, Rn×q ,

Rq×n

F T (t )F (t ) ≤ I.

(3)

f (r ) (x(t )) : Rn → Rn , r = 1, 2, are the nonlinear functions, g(x(t )) :

Rn → Rm0 means the noise intensity function, and satisfy the following assumption.

Assumption 1 [5]. There exist six scalars Li(0 ) , Li(1 ) , Li(2 ) , and Ui(0 ) ,

Ui(1 ) , Ui(2 ) , i = 1, n satisfying

Li(0) ≤ Li(r ) ≤

gi ( a1 ) − gi ( a2 ) ≤ Ui(0) , a1 − a2 fi(r ) (a1 ) − fi(r ) (a2 ) a1 − a2

≤ Ui(r ) , r = 1, 2

for all a1 , a2 ∈ R, a1 = a2 . Let Lg = max {|Li(0 ) |, |Ui(0 ) |}, and L f r = max {|Li(r ) |, |Ui(r ) |}, r = 1≤i≤n

1≤i≤n

1, 2, and then we easily obtain

 g( a 1 ) − g( a 2 )  ≤ L g  a 1 − a 2  ,  f ( a1 ) − f ( r ) ( a2 ) ≤ L f r a1 − a2 . (r )

Assumption 2. There exists a positive matrix D such that for every x, y ∈ Rn , the noise intensity function satisfies



Trace g(x ) − g(y )

T

g( x ) − g( y )



≤ ( x − y )T D ( x − y ).

The corresponding response system is described as follows

⎧  2  ⎪ dy(t ) = Ar (t )y(t − dr (t )) + Br (t ) f (r ) (y(t − dr (t ))) + J (t ) ⎪ ⎪ ⎪ r=1 ⎨

(4) +U (t ) + E1W (t ) dt + g(y(t ))dω (t ), t > 0 ⎪ ⎪ ⎪ ⎪ ⎩z¯ (t ) = Cy(t ) + E2W (t ), y(t ) = φ (t ), t ∈ [−d¯, 0] where y(t ) ∈ Rn is the state vector; z¯ (t ) ∈ Rq is the controlled output of response system (4); W(t) ∈ L2 [0, ∞) is the exogenous disturbance input; φ (t) is the initial function for t ∈ [d¯, 0]; u(t ) ∈ Rn

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K. Ding and Q. Zhu / Chaos, Solitons and Fractals 127 (2019) 244–256

where e(t ) = x(t ) − y(t ), z˜(t ) = zˆ(t ) − z¯ (t ), f˜(r ) (e(t − dr (t )) = f (r ) (x(t − dr (t )) − f (r ) (y(t − dr (t )), g˜(e(t )) = g(x(t )) − g(y(t )), and A˜ i1 (t ) = A˜ i1 + A1 (t ), A˜ i2 (t ) = A˜ i2 + A2 (t ), in which A˜ 11 = A1 − K, A˜ 21 = A1 , A˜ 12 = A˜ 22 = A2 . Now, the stochastic synchronization problem of drive system (1) and response system (4) reduces to find a control gain K such that the stochastic error system (6) is MSES and ASES over S (δ10 , δ11 ; δ 20 , δ 21 ), respectively. Definition 1 [26]. The zero solution of the stochastic error system (6) is mean square exponential stability (MSES) over S (δ10 , δ11 ; δ 20 , δ 21 ), i.e., there two exist positive scalars M and ν such that the following inequality holds:

E e(t )2 ≤ ME  (t )d2¯e−ν t , t ≥ 0. Definition 2 [26]. The zero solution of the stochastic error system (6) is almost surely exponential stability (ASES) over S (δ10 , δ11 ; δ 20 , δ 21 ), if the following inequality holds:

lim sup t→∞

Fig. 1. Drive system and Response system.

is the control input vector to response system (4); E1 ∈ Rn×p , E2 ∈ Rq×p are known constant matrices; In order to synchronize the response system (4) with external disturbance to the drive system (1), the following intermittent controller is given:



U (t ) = K (t ) x(t ) − y(t )

Definition 3 [27]. For the given scalar R and switching time sequences S (δ10 , δ11 ; δ 20 , δ 21 ), the intermittent controller K is designed such that the drive system (1) and response system (4) to be exponentially synchronization with an H∞ performance level R, if the following two conditions are satisfied: (i) The error system (6) with the external disturbance W (t ) = 0 is MSES or ASES. (ii) When the initial condition (t ) = 0, for any bounded external disturbance input, the following inequality holds:

(5)



in which,



K (t ) =

K, 0,

E

t ∈ [T1,k , T2,k ) , k ∈ N0 , t ∈ [T2,k , T1,k+1 )

where K ∈ Rn×n is the intermittent feedback gain to be planned. For the sake of simplification of the notation, we make £1,k = [T1,k , T2,k ), £2,k = [T2,k , T1,k+1 ), where {Ti,k } ∈ S (δ10 , δ11 ; δ20 , δ21 )  {{Ti,k } : δi0 ≤ T3−i,k−1+i − Ti,k ≤ δi1 , k ∈ N0 , i = 1, 2}. Fig. 1 clearly manifests the intermittent control process of the stochastic synchronization scheme described by (1) and (4). Remark 1. Here, we emphasize that the intermittent control mechanism of this paper and sampled-data control mechanism are still very different. First, the intermittent control only exerts an effect on the system in a segmentally continuous interval, i.e., [T1,k , T2,k ), which does not work at the discrete sampled-data instant. Second, the sampled-data control approach can generally be converted to a special form of time delay, such as the work of E. Fridman in [25]:

u(t ) = ud (Ti,k ) = ud (t − (t − Ti,k )) = ud (t − d (t )),

1 lne(t ) < 0, a.s.. t

t 0

z˜(t )2 ds ≤ R2 E



0

t

W (t )2 ds, ∀t ≥ 0.

(7)

Next, in order to obtain the main results, the following lemmas are indispensable. Lemma 1 [28]. Let M = {M (t )}t≥0 be a local martingale with M (0 ) = 0. Then one can obtain lim Mt(t ) = 0, if the following cont→∞

dition is satisfied:



lim

t→∞ 0

t

d M, M s ds < ∞. ( 1 + s )2

Lemma 2 [29]. For given matrices A ∈ Rn×N , B ∈ RN×n and i ∈ RN×N , if there exist several matrices X0 , Pi ∈ Rn×n , i = 1, m, satisfying



i + BX0 A + (BX0 A )T ∗

 ((Xi − X0 )A )T + BPi < 0, i = 1, m, T −Pi − Pi

then the following holds:

i + BXi A + (BXi A )T < 0, i = 1, m.

Ti,k ≤ t < T3−i,k−1+i , d (t ) = t − Ti,k , i ∈ {1, 2}, k ∈ N0 , where ud is discrete sampled-data control, and so ud is related to d(t) and satisfies the requirement d˙ (t ) ≡ 1, but does not require d˙ (t ) < 1. Then, the synchronization error system between drive system (1) and response system (4) can be deduced as

⎧ 2 

 ⎪ ⎪ de(t ) = A˜ ir (t )e(t − dr (t )) + Br (t ) f˜(r ) (e(t − dr (t ))) − E1W (t ) dt ⎪ ⎨ r=1 +g˜(e(t ))dω (t ), t ∈ £ik , i = 1, 2, k ∈ N0 , ⎪ ⎪ z˜(t ) = Ce(t ) − E2W (t ), ⎪ ⎩ e(t ) = (t )  ϕ (t ) − φ (t ), t ∈ [−d¯, 0],

Lemma 3 [30]. For any given matrices Y = Y T ∈ Rn×n , M ∈ Rn×p , N ∈ R p×n , F ∈ R p×p with F T F ≤ I, then the following inequality hold:

Y + MFN + N T F T MT < 0, if and only if there exists a scalar ε > 0 such that

Y + ε −1 MMT + ε N T N < 0. 3. Main results

(6)

In this section, the mean square and almost sure exponential synchronization analysis of system (6) will be developed. For this purpose, the following auxiliary functions are introduced.

K. Ding and Q. Zhu / Chaos, Solitons and Fractals 127 (2019) 244–256

For switching time sequence {Ti,k } ∈ S (δ10 , δ11 ; δ20 , δ21 ) and t ∈ £i,k , i = 1, 2, k ∈ N0 , let us define

i0 (t ) = ρi0k (t ) =

1 , T3−i,k−1+i − Ti,k

(8a)

t − Ti,k , T3−i,k−1+i − Ti,k

 ςi0k (t ) =

i0 (t )−1/δi1 , 1/δi0 −1/δi1

(8b)

1

δi0 ,

system:

⎧ 2   ⎪ ⎨de(t ) = (A˜ ir (t )e(t − dr (t )) + Br f˜(r ) (e(t − dr (t ))) dt r=1

(8c)

Then, from the condition (12), it is known that there is a sufficiently small ν > 0, such that

⎢ 1i jl (t )  ⎣

Then, from the definition of ς i0k (t) we have

1

i0 (t ) = ςi0k (t )

δi0

+ ςi1k (t )

1

δi1

where ςi1k (t ) = 1 − ςi0k (t ), 0, ρ10k (T2−,k ) = ρ20k (T1−,k+1 ) = 1.

˜ 1i jl (t ) + ν Pi j 

,

(9) and ρ10k (T1,k ) = ρ20k (T2,k ) = Besides, setting ρi1k (t ) =

= 0.

Remark 2. It is known from (8c) that for any t ∈ £i,l , when δ i0 = δ i1 , it means aperiodically intermittent, when δ i0 ≡ δ i1 , it will be reduced to periodically intermittent, so the controller will be more flexible when executed. 3.1. Mean-square exponential synchronization Thoerem 1. Consider the error system (6) and assume that Assumptions 1 and 2 hold. If for prescribed scalars μi > 0, i = 1, 2, and n × n matrix K, n × n diagonal matrices Gı , ı = 1, 4, there exist n × n matrices Pij > 0, Q > 0, i = 1, 2, j = 0, 1, n × n diagonal matrices Di j , i = 1, 2, j,  = 0, 1 and scalars ξ i > 0, i = 1, 2, such that the following inequalities hold:

Pi j ≤ ξi I,

(10)

Pi1 ≤ μ3−i P3−i,0 ,

(11)

⎢ ⎢ ⎢ ⎢ ⎣

1i jl (t ) ∗ ∗ ∗ ∗ ∗

Pi j A˜ i2 (t )

i j ∗ ∗ ∗ ∗

Pi j B1 (t ) + Di j0 G2 0 −Di j0 I ∗ ∗ ∗

Pi j B2 (t ) Di j1 G4 0 −Di j1 I ∗ ∗

−Pi j E1 0 0 0 −μ0 R2 I ∗

< 0,

˜ 1i jl (t ) = 

∗ ∗

δil

+

i j = −

Pi j +

1

δil

Q

μi (1 − dˆ) 1

μi

⎤

Pi j B2 (t ) Di j1 G4 ⎥ ⎦ 0 −Di j1 I

(14) ν d¯

ln μi

e Q 1 ˜ ˜T δil Pi j + δil (Pi0 − Pi1 ) + Pi j Ai1 (t ) + Ai1 (t )Pi j + μ (1−dˆ) + i

V (t ) = V0 (t ) +



t

t −d (t )

CT 0 ⎥ 0 ⎥ ⎥ 0 ⎥ T⎦ −E2 −I

(s )eT (s )Qe(s )ds, t ∈ £i,k , i = 1, 2, k ∈ N0 (15)

where V0 (t ) = ψ (t )eT (t )Pik (t )e(t ), Pik (t ) =



 

Q + Di j1 G3 ,

in which μi = min{1, μi }, μi = max{1, μi }, then the stochastic synchronization error of (6) is mean square exponentially stable and the L2 –gain less than level R over S (δ10 , δ11 ; δ 20 , δ 21 ). Proof. First of all, it is proved that when W (· ) = 0, the error system (6) is mean square exponentially stable over under given conditions. Here, we consider (6) as the following stochastic switching

j=0

ρi jk (t )Pσ (t ), j , σ (t ) =

dV (t ) = LV (t )dt + 2ψ (t )eT (t )Pik (t )g˜(e(t ))dω (t ), where

(16)



LV (t ) ≤ −νV (t ) + ψ (t ) eT (t )[(ν + i0 (t )lnμi )Pi j (t ) + i0 (t )(Pi0 − Pi1 ) + Pik (t )A˜ i1 (t ) + A˜ Ti1 (t )Pik (t ) +

eν d

μi (1 − dˆ)

Q]e(t ) −

1

μi

eT (t − d (t )) × Qe(t − d (t ))

+2eT (t )Pik (t )[A˜ i2 (t )e(t − d (t )) + B1 (t ) f˜(1) (e(t ))



+ B2 (t ) f˜(2) (e(t − d (t )))] + Tr[(g˜(e(t ))T Pik (t )g˜(e(t )] . By Assumption 2 and condition (10), we get

Tr[(g˜(e(t )))T Pik (t )g˜(e(t ))] ≤

ξi

1 

ρi jk (t )Tr[(g˜(e(t )))T g˜(e(t ))]

j=0

(Pi0 − Pi1 ) + Pi j A˜ i1 (t ) + A˜ Ti1 (t )Pi, j + Di j0 G1 + ξi D,

1 

−ν (t−s−d¯) 1, t ∈ [−d¯, 0] £1,k ρ (t ) , ψ (t ) = μσi(0tk) , (s ) = e ˙ ,  1−d ( (s )) 2, t ∈ £2,k in which ϖ(s) denotes the inverse of the strictly increasing function s − d (s ). It is easy to demonstrate that the candidate Lyapunov functional V(t) is continuous in each interval £i,k . When t ∈ £i,k , applying the Itoˆ’s formula [26] to V(t) along with error system (6) gives

(12)

ln μi

Pi j B1 (t ) + Di j0 G2 0 −Di j0 I ∗

Di j0 G1 + ξi D. Now, construct the following PTDLF for error system (6):

⎤

where μ0 = μ/μ, μ = max{1, μ1 , μ2 }, μ = min{1, μ1 , μ2 }, and

1i jl (t ) =

i j

∗ ∗ ∗

¯

⎡

Pi j A˜ i2 (t )

< 0,

1 − ρi0k (t ), we can obtain ρ11k (T1,k ) = ρ21k (T2,k ) = 1, ρ11k (T2−,k ) =

ρ21k (T1−,k+1 )

(13)

+g˜(e(t ))dω (t ), t ∈ £ik , e(t )  (t ), t ∈ [−d¯, 0]

⎪ ⎩

⎡

δi0 = δi1 . i f δi0 ≡ δi1 if

247

≤

ξi

1 

ρi jk (t )eT (t )De(t ).

(17)

j=0

From Assumption 1, for any positive scalars dihlk , i = 1, 2, h = 0, 1, l = 1, n, we have

0≤

n 

ι=1

+

di0ιk (Uι(1) eι (t ) − f˜ι(1) (eι (t ))( f˜ι(1) (eι (t )) − Lι(1) eι (t ))

n 

di1ιk (Uι(2) eι (t − d (t )) − f˜ι(2) (eι (t − d (t )))

ι=1 × ( f˜ι(2) (eι (t − d (t ))) − Lι(2) eι (t − d (t )))

248

K. Ding and Q. Zhu / Chaos, Solitons and Fractals 127 (2019) 244–256

⎡

Di0k (t )G1 ⎢ ∗ T = ζ (t )⎣ ∗ ∗

0 Di1k (t )G3 ∗ ∗

Di0k (t )G2 0 −Di0k (t ) ∗

⎤

0 Di1k (t )G4 ⎥ ⎦ζ (t ), 0 −Di1k (t ) (18)

Dihk (t ) = diag(dih1k , dih2k , . . . , dihnk ) =

where

1  j=0

For any t > 0, there exists some k ∈ N0 such that t ∈ £1,k or t ∈ £2,k . Without loss of generality, we assume t ∈ £1,k . From (22) and V (T10 ) = 0, we get

 E

t 0



LV (s )ds = E

ρi jk (t )Di jh , =E

J (t ) = E

ρi jk (t )ςilk (t )ζ (t )1i jl (t )ζ (t )dt



t 0

 ≤ E

(20)

(21)

Next, we consider the V(t) at the switching instant Ti,k , i = 1, 2, k ∈ N0 . From the definition of ψ (t), i = 1, 2, j = 0, 1, k ∈ N0 , we obtain

0

LV (s )ds



ψ (s ) z˜(s )2 − μ0 R2 W (s )2 + LV (s )ds. (27)

eν d

¯

μi (1 − dˆ)

Q + ξi D ] e ( s )

1

eT (s − d (s ))Qe(t − d (s )) + 2eT (t )Pik (s )[A˜ i2 (t )e(s − d (s )) μi + B1 (t ) f˜(1) (e(s )) + B2 (t ) f˜(2) (e(s − d (s )))] + z˜T (s )z˜(s ) − μ0 R2W T (s )W (s ) . (28) −

ψ (Ti,k )eT (Ti,k )Pik (Ti,k )e(Ti,k ) = eT (Ti,k )Pi,1 e(Ti,k ) ≤ μ3−i eT (Ti,k )P3−i,0 e(Ti,k )

Applying (18) to (28) yields



ψ (s ) z˜(s )2 − μ0 R2 W (s )2 + LV (s )

− ρi jk (Ti,k ) T e (Ti,k )P3−i,0 e(Ti,k )

≤ μ3−i

ρ (T − )

= ψ (s )

i jk i,k − − μ3−i eT (Ti,k )P3−i,0 e(Ti,k ) − = V0 (Ti,k ),

1  1 



T ˜ ρi jk (s )ςilk (s )ζ˜ T (s )  1i jl (s ) + H H ζ (s ),

(29)

j=0 l=0

where ζ˜ (s ) = col{ζ (s ), W (s )}, H = [C

which gives − V (Ti,k ) ≤ V (Ti,k ), i = 1, 2, k ∈ N0 .

(22)

Combining (21) with (22), we have

EV (t ) ≤ EV (0 )e−ν t ,

(23)

then

E e(t )2 ≤ ME  2d e−ν t , t ≥ 0

 (s )  1i jl ⎡ ˜ 1i jl (s ) + ν Pi j  ⎢ ∗ ⎢ =⎢ ∗ ⎣ ∗ ∗

(24)

Q νd where M = λμ λμ + λμν (e − 1 ), λ = λmax {Pi j }, λ = λmin {Pi j }, i = 1, 2, j = 0, 1, λQ = λmax {Q }. Thus, the stochastic synchronization error system (6) converges exponentially in the mean square over S (δ10 , δ11 ; δ20 , δ21 ). In what follows, we will prove that the control output satisfies inequality (7) under the initial condition (t ) = 0. To this end, introduce a helper function

0

0

t

+ Pik (s )A˜ i1 (s ) + A˜ Ti1 (s )Pik (s ) +

V0 (Ti,k ) =

J (t ) = E

t





ψ (s ) z˜(s )2 − μ0 R2 W (s )2 + LV (s )  = ψ (s ) eT (s )[(ν + i0 (s )lnμi )Pik (s ) + i0 (t )(Pi0 − Pi1 )

It follows from (11) that



k−1 

V (T2−,m ) − V (T2,m )

For any s ∈ £i,k , the trajectory along the system (6) can be obtained:

ψ (T1,k ) = 1, ψ (T2−,k ) = μ1 , ψ (T2,k ) = 1, ψ (T1−,k ) = μ2 ⇒ ψ (t ) ∈ [μi , μi ].

t





ψ (s ) z˜(s )2 − μ0 R2 W (s )2

+ LV (s )ds − E

(19)

Integrating from Ti,k to t on both sides of (20), then taking the mathematical expectation, we get



V (T2−,m ) − V (T1,m ) + V (T1,m+1 ) − V (T2,m )

By virtue of the well-known Dynkin’s formula [26], the following inequality is satisfied:

Utilizing inequalities (14) and (19), we can deduce

λ

LV (s )ds

(26)

j=0 l=0

=

T1,k



m=0

T

EV (t ) ≤ EV (Ti,k )e−ν (t−Ti,k ) , t ∈ £i,k .

LV (s )ds +

t

≥ 0.

Applying the relations (17)-(18) and convex combination technique to (16) yields

d[eν t V (t )] ≤ 2eν t ψ (t )eT (t )Pik (t )g˜(e(t ))dω (t ), t ∈ £i,k .

T2m



m=0

1 G4 = diag(L1(2) + U1(2) , L2(2) + U2(2) , · · · , Ln(2) + Un(2) ). 2

+2ψ (t )eT (t )Pik (t )g˜(e(t ))dω (t ), t ∈ £i,k .

k−1 

+E V (T1−,m+1 ) − V (T1,m+1 )

G3 = diag(−L1(2)U1(2) , −L2(2)U2(2) , · · · , −Ln(2)Un(2) ),

dV (t ) ≤ −νV (t )dt + ψ (t )

+

T1,m+1

+ V (t ) − V (T1,k ) ≥ E

1 diag(L1(1) + U1(1) , L2(1) + U2(1) , · · · , Ln(1) + Un(1) ), 2

1  1 



 k−1 m=0

G1 = diag(−L1(1)U1(1) , −L2(1)U2(1) , · · · , −Ln(1)Un(1) ),

T2m T1m

m=0

j = 0, 1, k ∈ N0 , and ζ (t ) = [e(t ), e(t − d (t )), f˜(1 ) (e(t )), f˜(2 ) (e(t − d (t )))]T , and

G2 =

k−1  

¯

2

ψ (s ) z˜(s )2 − μ0 R2 W (s ) ds, t ≥ 0.

(25)

Pi j A˜ i2 (s )

i j ∗ ∗ ∗

0

0

Pi j B1 (s ) + Di j0 G2 0 −Di j0 I ∗ ∗

0

− E2 ], and

Pi j B2 (s ) Di j1 G4 0 −Di j1 I ∗

⎤

−Pi j E1 ⎥ 0 ⎥ 0 ⎥. ⎦ 0 −μ0 R2 I

According to inequality (12) and (14), applying the Schur complement to obtain

 ( s ) + H T H < 0.  1i jl

(30)

Thus, we have



ψ (s ) z˜(s )2 − μ0 R2 W (s )2 + LV (s ) < 0. Substituting the above formula into (27) yields J (t ) ≤ 0, namely



E

t 0

ψ (s )z˜(s )2 ds ≤ E



t 0

μ0 R2 ψ (s )W (s )2 ds, t ≥ 0.

(31)

K. Ding and Q. Zhu / Chaos, Solitons and Fractals 127 (2019) 244–256

Noting that μ ≤ ψ (s ) ≤ μ, we obtain

 E

t 0

μz˜(s )2 ds ≤ E

 0

t

μ0 R2 μW (s )2 ds, t ≥ 0,

(32)

which implies the inequality (7) holds. This completes the proof.  Remark 3. The synchronous error system (6) under intermittent control mainly has two modes, one is the closed-loop mode for t ∈ £1,k , and the other is the open-loop mode for t ∈ £2,k . In order to deal with the dynamic behavior of these two modes, we add a model-dependent weighting factor ψ (t) to the constructed PTDLF V(t). Under its action, V(t) can achieve monotonically decreasing in the interval £i,k and will not monotonically increase at the switching instant Ti,k due to the selected parameter ui . What’s more, we also need to explain that because Pij does not need to be equal, it can better demonstrate the degree of freedom, which can reduce the conservative. Remark 4. Although the intermittent control mechanism was investigated in [12,13] to study the exponential synchronization problem in the stochastic systems, the influence of exogenous disturbance and random noise and the uncertainty of parameters on the systems were not considered. Further, Guo et al. [12] only considered periodically intermittent control strategies. Moreover, Liu et al. [21] and Song and Nie [22] used intermittent control to investigate the H∞ performance of error synchronization either do not consider time delay or just consider very small constant time delay. In sharp contrast, this paper considers a class of uncertain stochastic system with time-varying delay, which is more in line with the actual situation.

249

there exist n × n positive matrices Pij , Q, i = 1, 2, j = 0, 1, n × n diagonal matrices Di j , i = 1, 2, j,  = 0, 1 and scalars ξ i > 0, i = 1, 2, such that (10)–(11) and the following inequalities hold:

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

2i jl ∗ ∗ ∗ ∗ ∗

Pi j A˜ i2 (t )

Pi j B1 (t ) + Di j0 G2 0 −Di j0 I ∗ ∗ ∗

i j ∗ ∗ ∗ ∗

Pi j B2 (t ) Di j1 G4 0 −Di j1 I ∗ ∗

⎤

−Pi j E1 0 0 0 −μ0 R2 I ∗

CT 0 ⎥ 0 ⎥ ⎥ < 0, 0 ⎥ ⎦ −E2T −I

(33) where

ln μi

2i jl =

δil

+

+

αi2

! Pi j +

2

Q

1

δil

(Pi0 − Pi1 ) + Pi j A˜ i1 (t ) + A˜ Ti1 (t )Pi j

+ Di j0 G1 + ξi D, i = 1, 2, j, l = 0, 1

μi (1 − dˆ)

then drive system (1) and response system (4) are the almost surely exponentially synchronous with the L2 –gain less than level R over S (δ10 , δ11 ; δ20 , δ21 ). Proof. Similarly, when W (· ) = 0, we will prove that drive system (1) and response system (4) can achieve almost sure exponential synchronization. The condition (33) implies

⎡

˜ 2i jl 

Pi j A˜ i2 (t )

∗ ∗

∗ ∗

⎢ 2i jl (t )  ⎣ ∗

i j

Pi j B1 (t ) + Di j0 G2 0 −Di j0 I ∗

⎤

Pi j B2 (t ) Di j1 G4 ⎥ ⎦ < 0, 0 −Di j1 I (34)

Remark 5. It should also be noted that in [13], a time-invariant Lyapunov function (TILF) was used to study the synchronization problem of a class of stochastic complex networks under intermittent control. A piecewise time-invariant Lyapunov functional was introduced to analyze the synchronism of a class of chaotic systems with multi-delay via intermittent control in [22]. Unlike [13,22], the synchronization criterion of the error system (6) in this paper is based on a novel PTDLF method. For the PTDLF, there are three meaningful features that need to be introduced. The first one, when Pi j = P, ui = 1, the designed PTDLF is equivalent to the general form of TILF V0 (t ) = xT P x. Besides, when Pi j = Pi , ui = max{λmax (Pi )/λmin (P3−i )}, the proposed PTDLF is reduced to the piecewise TILF adopted by Song and Nie [22]. Hence, Zhang et al. [13] and Song and Nie [22] are just a special form of PTDLF designed in this paper. Next one, along the trajectory of the error system (6), the Itoˆ’s differential formula LV (t ) contains a

i0 (t )(Pi0 − Pi1 ), which reveals the upper and lower bounds information of the interval £i,k . However, it is found that the Lyapunov function approach used in [13,22] cannot obtain the information of the upper of £1,k and lower bounds of £2,k . Thus, compared with the Lyapunov strategy of [13,22], the proposed PTDLF is more reasonable and accurate in capturing the switching characteristics of the error system (6). The last one, the introduction of ϱi0 (t) in (8a) in PTDLF accelerates the convergence speed of the system because it combines the convex combination technique, which is particularly obvious in the simulation results. Therefore, these three points can prove that the proposed method can effectively reduce the conservativeness of existing results. 3.2. Almost sure exponential synchronization Thoerem 2. Consider the error system (6) and assume that Assumptions 1 and 2 hold. If for prescribed scalars μi > 0, α i > 0, i = 1, 2, and m × n matrix K, n × n diagonal matrices Gı , ı = 1, 4,

αi2

˜ 2i jl = ( ln μi + where  δil ¯ eν d Q

μi (1−dˆ)

2

)Pi j +

1 ˜ ˜T δil (Pi0 − Pi1 ) + Pi j Ai1 (t ) + Ai1 (t )Pi j +

+ Di j0 G1 + ξi D.

Let V(t) be the Lyapunov functional expressed by (15) of error system (6). For t ∈ £i,k , applying the Itoˆ ’s formula to lnV (t ), we get

dlnV (t ) ≤ LlnV (t )dt + where

LlnV (t ) ≤

2ψ (t ) T e (t )Pik (t )g˜(e(t ))dω (t ) V (t )

(35)

ψ (t )  T  e (t ) i0 (t )lnμi Pik (t ) + i0 (t )(Pi0 − Pi1 ) V (t ) + Pik (t )A˜ i1 (t ) + A˜ Ti1 (t )Pik (t ) + −

1

μi

eν d

¯

μi (1 − dˆ)



Q + ξi D e(t )

eT (t − τ (t ))Qe(t − τ (t ))



+2eT (t )Pik (t ) A˜ i2 (t )e(t − τ (t )) + B1 (t ) f˜(1) (e(t )) + B2 (t ) f˜(2) (e(t − τ (t )))



.

(36)

Integrating both sides of the above inequality from Ti,k to t yields

lnV (t ) ≤ lnV (Ti,k ) +



t Tik

1  1 ψ (s )  ρi jk (s )ςilk (s )ζ T (s ) V (s ) j=0 l=0

× 2i jl (t )ζ (s )ds  t ψ (s ) T +2 e (s )Pi j (s )g˜(e(s ))dω (s ), Ti,k V (s )

(37)

Resorting to the Eq. (22), we get

V (T1k ) ≤ V (T1−k ), V (T2k ) ≤ V (T2−k ), k ∈ N0 .

(38)

250

K. Ding and Q. Zhu / Chaos, Solitons and Fractals 127 (2019) 244–256

Accordingly, we have

lnV (t ) < lnV (0 ) + M1 (t ) + M2 (t ), where

(39)

1  1 " t ψ (s )  ρi jk (s )ςilk (s )ζ T (s )2i jl (t )ζ (s )ds, 0 V (s )

M1 (t ) =

j=0 l=0

" t (s ) T M2 (t ) = 2 0 ψ e (s )Pik (s )g˜(e(s )) dω(s), i = 1, 2, j, l = 0, 1. Since V (s ) the right-hand side of (34) is negative definite, we derive that M1 (t) < 0. Besides, it follows from Assumption 1 that

M2 (t ), M2 (t ) ≤ 4



t

0



ψ (s ) T e (s )Pi j (t )g˜(e(s ))2 ds V0 (s )

≤ 4tLg LTg .

(40)

Remark 7. It is worth noting that the literature [19] is the same as the time delay discussed in this paper, but like [11–16], its stability conclusion is established by applying the common Lyapunov functional method. The resulting stability conditions involve the transfer of the roots of the equation, which undoubtedly increases the complexity of the calculation. Different from [19], the stability analysis in Theorems 1 and 2 uses a piecewise Lyapunov functional analysis method, which gives a looser stability condition. Furthermore, this stability condition consists of linear matrix inequalities that can be easily calculated by numerical software.

4. Controller design

Thereby, taking advantage of Lemma 1, the following holds:

lim

t→∞

M2 (t ) =0 t

a.s.

(41)

Combing inequality (39) with relations (40) and (41) gives

lim sup t→∞

lnV (t ) <0 t

a.s.,

(42)

which implies

lim sup t→∞

lne(t ) <0 t

a.s.

(43)

Therefore, we have that the zero solution of (6) is almost surely exponentially stable. Afterwards, under the initial value (t ) = 0 and based on the previous similar processing H∞ synchronization skills, from (25) to (31), we can get



E

0

t

z˜(s )2 ds ≤ E



0

t

μ0 R2 W (s )2 ds, t ≥ 0.

1i jl

In this section, we will give sufficient conditions for the existence of intermittent control gain matrices based on the previously obtained mean square and almost sure exponential synchronization results. The finite dimensional gain matrix K can be gained by solving the following two theorems.

⎡ #  1i jl ⎢ ∗ ⎢ ⎢ ∗ ⎢ ∗ =⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

(44)

Thoerem 3. Consider the error system (6) and assume that assumptions 1 and 2 hold. If for prescribed scalars μi > 0, i = 1, 2, there exist n × n matrices K¯ , X1 > 0, Pij > 0, Q > 0, i = 1, 2, j = 0, 1, diagonal matrices Di j , i = 1, 2, j,  = 0, 1, and scalars ξ i > 0, ε ij > 0, β j > 0, i = 1, 2, j = 0, 1, such that (10)–(11) and the following LMIs hold:



11 jl ∗



I T P1 j − X1 − β j K¯ T −β j (X1 + X1T )

i j ∗ ∗ ∗ ∗ ∗ ∗

Pi j B1 + Di j0 G2 0 −Di j0 I ∗ ∗ ∗ ∗ ∗

Pi j B2 Di j1 G4 0 −Di j1 I ∗ ∗ ∗ ∗

Remark 6. Different from Theorem 1, the synchronization criterion obtained by Theorem 2 fully considers the influence of random noise on the stabilization in the system (6). By observing the conditions in Theorem 2, the known Theorem 2 includes Theorem 1, which was explained in Corollary 4.3 of [26]. Additionally, a conclusion similar to Theorem 2 can also be obtained by applying Theorem 3 in [26] and using the similar processing technique of [31]. Nonetheless, we have to say that this surface-like conclusion is relatively conservative. Moreover, because of the existence of a very important factor in the system, namely, the diffusion term g(x(t)) in the drive system (1) and the diffusion term g(y(t)) in the response system (4) must be controllable, the required intermittent controller is difficult to be designed. Whereas, the Theorem 2 based on the strong law of large numbers of local martingale does not indicate this requirement.

(45)

(46)

−Pi j E1 0 0 0 −μ0 R2 I ∗ ∗ ∗

#  11 jl =

ln μ1

δ1l

0

P1 j +

0

0

1

δ1l

0

0

0

0], and



T (P10 − P11 ) + P1 j A1 − K¯ + P1 j A1 − K¯

Q

+ D1 j0 G1 + ξ1 D, μ1 (1 − dˆ) ln μ2 1 = P + (P − P ) + P2 j A1 + AT1 P2 j δ2l 2 j δ2l 20 21 +

#  12 jl

⎤ εi j N1T εi j N2T ⎥ ⎥ εi j N3T ⎥ εi j N4T ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎦ −εi j I

Pi j M 0 0 0 0 0 −εi j I ∗

CT 0 0 0 −E2T −I ∗ ∗

in which I = [I Hence, from Definition 3, we see that the drive system (1) and the response system (4) are almost sure exponential synchronization over S (δ10 , δ11 ; δ 20 , δ 21 ) and the L2 –gains level is less than R. The proof is completed. 

< 0,

12 jl < 0, where

Pi j A2



+

Q

μ2 (1 − dˆ)

+ D2 j0 G1 + ξ2 D,

then the stochastic synchronization error system (6) under controller (5) with gain K = X1−1 K¯ is mean square exponentially stable and the L2 –gain less than level R over S (δ10 , δ11 ; δ20 , δ21 ). Proof. Set K = X1−1 K¯ . On the basis of Lemma 2 and condition (45), we obtain



11 jl + BX1 A + (BX1 A )T ∗

< 0, j, l = 0, 1,





T (P1 j − X1 )A + BX j −X j − X jT

K. Ding and Q. Zhu / Chaos, Solitons and Fractals 127 (2019) 244–256

where [I 0 0

11 jl

0

X j = β j X1 , B = col{−K T , 0, 0, 0, 0, 0, 0, 0}, 0 0 0 0]T , and

⎡# 11 jl ⎢ ∗ ⎢ ⎢ ∗ ⎢ ∗ =⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣

P1 j B1 + D1 j0 G2 0 −D1 j0 ∗ ∗ ∗ ∗ ∗

P1 j A2

1 j

∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

ln μ1

1

A=

−P1 j E1 0 0 0 −μ0 R2 I ∗ ∗ ∗

P1 j B2 D1 j1 G4 0 −D1 j1 ∗ ∗ ∗ ∗

CT 0 0 0 −E2T −I ∗ ∗

in which I = [I

0

0

11 jl =

δ1l

+

P1 j +

δ1l

Q

μ1 (1 − dˆ)

Then, it yields

(P10 − P11 ) + P1 j A1 +

# 

AT1 P1 j

21 jl

11 jl + BP1 j A + BP1 j A

T

(48)

⎡# ⎤ 11 jl P1 j A2 P1 j B1 + D1 j0 G2 P1 j B2 −P1 j E1 C T ⎢ ∗ 1 j 0 D1 j1 G4 0 0 ⎥ ⎢ ⎥ ∗ −D1 j0 I 0 0 0 ⎥ ⎢ ∗ ⎢ =⎢ ∗ ∗ ∗ −D1 j1 I 0 0 ⎥ ⎥, 2 ⎢ ∗ ∗ ∗ ∗ −μ0 R I −E2T ⎥ ⎣ ⎦ ∗

M = [ MT P1 j N = [ N1

0

N2

∗

0 N3

0

0

N4

0

∗

∗

−I

0 ]T , 0 ].

It is easy to verify that (48) is equivalent to (12). In other words, (12) can be procured from (45) with i = 1. In the same way, it can be proved that (12) can be introduced by (46) with i = 2. For the intermittent controller in the case of almost sure exponential synchronization, the control gain matrix can be obtained by using the proof idea of Theorem 3. The process is as follows.  Thoerem 4. Consider the error system (6) and assume that Assumptions 1 and 2 hold. If for prescribed scalars μi > 0 and α i > 0, i = 1, 2, there exist n × n matrices K¯ , X2 > 0, Pij > 0, Qi > 0, i = 1, 2, j = 0, 1, diagonal matrices Di j , i = 1, 2, j,  = 0, 1, and scalars ξ i , ε ij > 0, β j > 0, i = 1, 2, j = 0, 1, such that (10)–(11) and the following LMIs hold:



21 jl ∗



I T P1 j − X2 − β j K¯ −β j (X2 + X2T )

 T

< 0,

(49)

+

2i jl

∗

i j ∗ ∗ ∗ ∗ ∗ ∗

Pi j B2 Di j1 G4 0 −Di j1 I ∗ ∗ ∗ ∗

−Pi j E1 0 0 0 −μ0 R2 I ∗ ∗ ∗

CT 0 0 0 −E2T −I ∗ ∗

0], and

+

α12

+

! P1 j +

2

T

α22 2

Q

μ2 (1 − dˆ)

+

!

1

δ1l

(P10 − P11 ) + P1 j A1 − K¯

Q

μ1 (1 − dˆ)

P2 j +

1

δ2l

+ L1 D1 j0 + ξ1 D,

(P20 − P21 ) + P2 j A1 + AT1 P1 j

+ L1 D2 j0 + ξ2 D.

Remark 8. The intermittent control gain matrix in (5) is due to the existence of the KPij term in Theorems 1 and 2, so that it cannot be easily combined. However, with the aid of Lemma 2, and the control variable X1 is introduced in Theorem 3 and the control variable X2 is introduced in Theorem 4, this situation can be effectively avoided. Then, by giving a few necessary parameters and matrices, the desired gain matrix K can be obtained by the MATLAB’ LMI toolbox. Remark 9. In order to reduce the noise generated by the control matrix, the following two convex optimization procedures are designed for the intermittent control gain matrix obtained by Theorems 3 and 4:

min s.t

γ1 

−X1

K¯



< 0, −η1 γ12 I X1 ≥ η1 I, (10 )−(11 ), (45 )−(46 ).

(51)

If a feasible solution can be obtained from (51), then the gain matrix K satisfies K ≤ γ 1min .

min

where

Pi j B1 + Di j0 G2 0 −Di j0 I ∗ ∗ ∗ ∗ ∗

0

Proof. The proof is omitted here because this treatment is the similar to Theorem 3. 

(50)

Pi j A2

0

Then, the drive system (1) and response system (4) under controller (5) with gain K = X2−1 K¯ are the almost surely exponentially synchronous with the L2 –gain less than level R over S (δ10 , δ11 ; δ20 , δ 21 ).

s.t

22 jl < 0,

⎡ #  2i jl ⎢ ∗ ⎢ ⎢ ∗ ⎢ ∗ =⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗

δ2l

(47)

where

∗

δ1l

ln μ2

ˆ 22 jl = 



T T T   + MF N + N F T M < 0 , 11 jl + BP1 j A + BP1 j A

11 jl

=



Applying Schur Complement and Lemma 3 to (47), we can get

 

ln μ1

+ P1 j A1 − K¯

< 0, j, l = 0, 1.

0

0 −ε1 j I

+ D1 j0 G1 + ξ1 D.



0

⎤ ε1 j N1T ε1 j N2T ⎥ ⎥ ε1 j N3T ⎥ T⎥ ε1 j N4 ⎥, 0 ⎥ ⎥ 0 ⎥ ⎦

P1 j M 0 0 0 0 0 −ε1 j I ∗

in which

#

251

Pi j M 0 0 0 0 0 −εi j I ∗

γ2 

−X2

K¯



< 0, −η2 γ22 I X2 ≥ η2 I, (10 )−(11 ), (49 )−(50 ).

⎤ εi j N1T εi j N2T ⎥ ⎥ εi j N3T ⎥ T⎥ εi j N4 ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎦ −εi j I

(52)

252

K. Ding and Q. Zhu / Chaos, Solitons and Fractals 127 (2019) 244–256

If a feasible solution can be obtained from (52), then the gain matrix K satisfies K ≤ γ 2min . Algorithm 1: Controller design algorithm. Step 1: Set a larger initial upper bound γi , i = 1, 2, and an adjustable step size h. Step 2: Solving the LMIs by dint of the convex optimization process (51) (or (52)) to compute K¯ . Step 3: In step 2, if (51) (or (52)) is solvable, then let γi = γi − h, go back to step 2 and continue the operation until we get the γi min , which means that the minimum gain matrix K is obtained at this time.

Remark 10. The following algorithm is given to solve the convex optimization process procedures (51) (or (52)). 5. Illustrated example

Fig. 2. The trajectory of Brownian motion.

In this section, in order to verify the effectiveness of this method, we will provide the following two examples. Example 1. Consider the exponential synchronization for the drive system (1) and response system (4) with the following parameters:

$

A1 =

−am1 1 0

$ B1 =

a ( m1 − m0 ) 0 0

$ B2 =

0 0 0

$ M=

0 0 0

0.01 0.02 0.03

%

a −1 −15.87

%

%

0 1 , A2 = 0 0 0 , 0

$

1 0 0

$

=

−0.1 −0.1 0.2

%

0 0 0

0 0 , CT = m2

, NıT

$

0.03 0 0.05

%

$

, E1 =

%

0 0 0

−0.3 −0.3 −0.5

%

0 0 , −0.1

% ,

, ı = 1, 4

E2 = −0.1, J (t ) = 03×1 , F (t ) = sin(3π t ),

Fig. 3. The chaotic behavior of drive system.

τ (t ) = 1.5 + 0.26 sin(t ), where a = 9, m0 = − 71 , m1 =

2 7,

m2 = (1 )

1 9.

The stochastic function is fi (x ) = fi(2 ) (x ) = 12 (|x + 1| − |x − 1| ), i = 1, 2, 3. Let the noise intensity function be g(x(t )) = Gx(t ) with G = diag(−0.3, 0.2, 0.2 ) and g( · ) satisfy Assumption 2 with D = diag(0.1, 0.2, 0.3 ). Fig. 2 reveals the stochastic perturbation. One can verify that Assumption 1 is satisfied with Li(1 ) = Li(2 ) = 0,

Ui(1 ) = Ui(2 ) = 1, i = 1, 2, 3. Then, G1 = G3 = 0, G2 = G4 = 12 I3 , L fr = 1, r = 1, 2, and Lg = 1. The simulation results show that there exhibit a chaotic attraction behavior in the drive system when the delay τ (t) ≡ 1 with the initial value ϕ (t ) = (0.2, 0.1, −0.5 )T , t ∈ [−1, 0], as rewritten in Fig. 3. 5.1. Mean-square exponential synchronization First, we study the relationship between the control rate and the L2 -gain R under the condition of aperiodically intermittent control mechanism by the result of Theorem 1. We define Rd as the control rate and set {Tik } ∈ S (0.8, 1; 0.1, 0.169) (the switching signal process is shown in Fig. 4). By solving the inequalities (45)– (49) with adjusting the parameters β 0 , β 1 , μ1 , μ2 can obtained the maximum control rate Rd , which is listed in Table 1. From Table 1,

Fig. 4. The switching signal σ (t).

K. Ding and Q. Zhu / Chaos, Solitons and Fractals 127 (2019) 244–256

253

Table 1 Relationship between R and Rd under MSES. R

Rd

(β 0 , β 1 , μ1 , μ2 )

0.5 1 3 5 10

88.02% 78.79% 69.54% 67.38% 66.26%

(0.07941, (0.12789, (0.11476, (0.10569, (0.09608,

0.11203, 0.11315, 0.10286, 0.09351, 0.08501,

1.10712, 1.10934, 1.22027, 1.34230, 1.47653,

0.31843) 0.31995) 0.35195) 0.38714) 0.42585)

Table 2 Relationship between R and Rd under ASES. R

Rd

(β 0 , β 1 , μ1 , μ2 )

0.5 1 3 5 10

83.41% 75.23% 67.39% 62.42% 53.00%

(0.07764, (0.07151, (0.07037, (0.03932, (0.00523,

0.07281, 0.06933, 0.06888, 0.03888, 0.00649,

1.01025, 1.00924, 1.00813, 1.00674, 1.00169,

0.37035) 0.36813) 0.35750) 0.33242) 0.19387)

Fig. 6. The time evolution of the E (e(t ) ).

Fig. 5. The state trajectory of error e(t).

it can be found that the L2 -gain has an inverse proportional relationship with the control rate Rd , scilicet, when the control gain becomes larger, the control rate becomes smaller. Second, to reflect the effect the aperiodically intermittent H∞ synchronization, given R = 1, applying Theorem 3 to solve (51) with the choice of β0 = 0.00806, β1 = 0.07728, μ1 = 1.10934, μ2 = 0.1806, γ 1min is 12.84136, and the corresponding intermittent H∞ synchronization controller gain is

$

K=

12.1005 0.2673 −0.7558

0.6959 11.8995 0.6332

%

−0.7584 −0.2245 . 11.4211

Fig. 7. The state trajectory of error e(t).

0.36813, α1 = 0.36, α2 = 0.21, γ2 min = 10.007, and the corresponding intermittent H∞ synchronization controller gain is

$ K=

(53)

In the numerical results, the initial state of response system (4) is selected as ψ (t ) = (−0.4, 2.3, 1.5)T , the external disturbance (t ) input is assumed as W (t ) = sin . The time evolution of the state 1+et trajectory of error system (6) is displayed in Fig. 5. Meanwhile, Fig. 6 illustrates the time evolution of the L2 -norm of e(t). 5.2. Almost sure exponential synchronization In the same way, we first give the relationship between the L2 gain R and the control rate Rd based on the almost sure exponential stability, and is shown in Table 2. Then, for given R = 1, applying Theorem 4 to solve (52) with the choice of β0 = 0.20136, β1 = 0.11093, μ1 = 1.00924, μ2 =

8.4958 0.4432 −1.7850

2.2570 7.3714 1.0161

%

−1.3258 −0.3016 7.3102

(54)

The simulation results under the gains expressed in (54) are given in Figs. 7 and 8, which indicates that the response system (4) is almost surely exponentially synchronized with the drive system (1). Remark 11. It can be seen from the following Fig. 9 that under the same L2 -gain R, the control rate under the almost sure exponential stability condition is lower than the exponential stability condition in mean square. Besides, the gain is larger, and the result is more obvious. Example 2. In order to highlight the advantages of the proposed method and have certain practical significance, we consider the standard Chua’s circuit system (see Fig. 10), which can be repre-

254

K. Ding and Q. Zhu / Chaos, Solitons and Fractals 127 (2019) 244–256

Fig. 10. Standard framework of Chua’s system. Fig. 8. The time evolution of the E (e(t ) ).

Fig. 11. u − I characteristics diagram of Chua’s diode. Table 3 Comparisons of minimum control length γ .

Fig. 9. The time evolution of the control rate.

sented as follows [32]:

⎧ duD 1 1 ⎪ ⎨D1 dt = R (uD2 − uD1 ) − hNR (uD1 ), duD D2 dt 2 = R1 (uD1 − uD2 ) − IL , ⎪ ⎩ duD2 L

dt

(55)

uD

uD

L ables x1 = E1 , x2 = E2 , x3 = EG ,τ = system (56) can be converted to

dτ

(2, 1.5)

(4, 3.5)

(8, 7)

(10, 8.5)

14.3725 1.7651

11.5134 1.3224

15.3906 1.1585

12.9337 1.1132

18.1835 2.1040

can be rewritten as system (1) with the following parameters:

$

I

Gt D2

with E = 1V and G =

1 R,

⎧ dx1 ⎪ ⎨ dτ = p(x2 − f (x1 )), dx2

(1, 0.6)

= −uD2 ,

where IL means that the current through inductor L. uD1 and uD2 are voltages that pass through capacitors D1 and D2 , respectively. hNR (uD1 ) indicates the u − I characteristic of the Chua’s diode, expressed as a piecewise function is hNR (uD1 ) = Gb uD1 + 12 (Ga − Gb )(|uD1 + E | − |uD1 − E | ) (see Fig. 11). Then, by defining the vari-

dτ ⎪ ⎩ dx3

(W, δ ) Theorem 1 of [16] Theorem 1

= x1 − x2 + x3 ,

(56)

= −qx2 ,

where p =

D2 D1

and q =

D2 G2 L

are bifurcation positive parameters. The

nonlinear function is h(x1 ) = w1 x1 + 12 (w0 − w1 )(|x1 + v| − |x1 −

v| ) with w0 =

G1 G

G

< 0, w1 = Gb < 0, v = 1, and set parameters w0 = −0.1428, w1 = 0.2856, p = 9, q = 14.28. In this case, the system

A1 =

−pw1 − 1 1 0

f (1) (x1 (t )) =

p −2 −q

%

0 1 , B1 = 0

$

%

−p(w0 − w1 ) 0 , 0

1 (|x1 (t ) + 1| − |x1 (t ) − 1| ) 2

In order to compare the results of the Theorem 1 of this paper with [16], we let δi0 = δi1 , i = 1, 2 (that is the periodic intermittent  control case indicated by the Remark 2: W = 2i=1 δi0 , δ = δ10 ), and the control gain matrix is K = γ I. For five different groups of (W, δ ) = (1, 0.6 ), (2, 1.5), (4, 3.5), (8, 7), and (10, 8.5), the corresponding minimum control length γ is calculated, and the obtained results are expanded in Table 3. It can be clearly seen that we have obtained less conservativeness than [16]. On the other hand, we also found that the stability conditions of [18–20] have a large dependence on the delay upper bound d¯max . For dˆ = 0 and the given (W, δ ) of Table 3, the maximum allowable upper bounds d¯max obtained by these methods and our Theorems 1–2 are shown in Table 4. The results show that when

K. Ding and Q. Zhu / Chaos, Solitons and Fractals 127 (2019) 244–256 Table 4 The upper bound d¯max for dˆ = 0.

(W, δ ) Theorem Theorem Theorem Theorem

1 of [18] 1 of [19] 1 of [20] 1–2

255

Declaration of Competing Interest

(1, 0.6)

(2, 1.5)

(4, 3.5)

(8,7)

(10,8.5)

0.5791 0.4217 0.2153 Any

1.4362 0.9436 0.2782 Any

3.1279 1.4621 0.8731 Any

4.6820 3.9776 1.8347 Any

6.3409 4.7138 2.4310 Any

The authors declare that they have no potential conflict of interest. Acknowledgements This work was jointly supported by the National Natural Science Foundation of China (61773217, 61374080), the Natural Science Foundation of Jiangsu Province (BK20161552), the Scientific Research Fund of Hunan Provincial Education Department (18A013) and the Construct Program of the Key Discipline in Hunan Province.

References

Fig. 12. The time evolution of the E (e(t ) ) of Chua’s system.

the control width is smaller, those results can only be established with a small time delay, and our time delay is not affected by the control width, which can be larger than the control width. Thus, it can be said that we have made a lot of improvements to the existing results. Further, applying Theorem 3 to design a aperiodically intermittent controller to achieve (1) and (4) synchronization. Let δ10 = 0.8, δ11 = 1, δ20 = 0.2, δ21 = 0.3, and using MATLAB’s LMI toolbox to process (45)–(49) with adjustable parameters β0 = 0.0129, β1 = 0.4726, μ1 = 1.008, μ2 = 0.0297, γ1 min = 10, the gain matrix is derived as

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6. Conclusions This paper has addressed the H∞ synchronization for uncertain time-varying delay system with disturbance and stochastic perturbation via intermittent control. With the help of a PTDLF and the strong low of the large numbers for local martingales, we have obtained some sufficient conditions to make the synchronization error system reach the mean square stable and almost sure stable with weighted H∞ performance, respectively. Finally, in the numerical simulation section, we demonstrate the effectiveness of our design approach. It should be pointed out that the methods mentioned in this paper may be applied to stochastic reaction-diffusion system [33,34], fuzzy system [35,36], memristor-based neural networks [37–39], which will be studied in our future works.

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