Fixed-time synchronization criteria for complex networks via quantized pinning control

Accepted Manuscript Fixed-time synchronization of complex networks via quantized pinning control Wanli Zhang, Hongfei Li, Chuandong Li, Zunbin Li, Xinsong Yang

PII: DOI: Reference:

S0019-0578(19)30044-8 https://doi.org/10.1016/j.isatra.2019.01.032 ISATRA 3080

To appear in:

ISA Transactions

Received date : 25 April 2018 Revised date : 4 October 2018 Accepted date : 24 January 2019 Please cite this article as: W. Zhang, H. Li, C. Li et al., Fixed-time synchronization of complex networks via quantized pinning control. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.01.032 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Title page showing Author Details

Fixed-time synchronization of complex networks via quantized pinning control Authors: Wanli Zhang, Hongfei Li, Chuandong Li, Zunbin Li, Xinsong Yang Wanli Zhang, Hongfei Li, Chuandong Li and Zunbin Li are with National & Local Joint Engineering Laboratory of Intelligent Transmission and Control Technology (Chongqing); College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China Xinsong Yang is with the School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China Corresponding author author: Chuandong Li (e-mail: [email protected]; [email protected]) The other authors’ e-mail: Wanli Zhang ([email protected]); Hongfei Li ([email protected]); Zunbin Li([email protected]); Xinsong Yang ([email protected]) This work was jointly supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 61374078, 61633011, 61673078, 61703346.

1

*Highlights (for review)

The main contributions are as follows: (a) New control schemes with logarithmic quantization are designed, which not only can reduce control cost but also can save channel resources. (b) The quantized pinning controller with sign function can be used more generally than the quantized pinning controller without sign function, but the quantized pinning controller without sign function can be utilized to overcome the chattering phenomenon in most of existing results. (c) Several sufficient conditions formulated by linear matrix inequalities are obtained to guarantee that the complex networks fixed-timely synchronize with an isolated system. (d) As special cases, several fixed-time synchronization criteria are also presented in corollaries.

*Blinded Manuscript - without Author Details Click here to view linked References

Fixed-time synchronization of complex networks via quantized pinning control I

Abstract In this paper, fixed-time (FDT) synchronization of complex networks (CNs) is considered via quantized pinning controllers (QPCs). New control schemes with logarithmic quantization are designed, which not only can reduce control cost but also can save channel resources. The QPC with sign function can be used more generally than the QPC without sign function, but the QPC without sign function can be utilized to overcome the chattering phenomenon in some existing results. Based on designed Lyapunov function and different control schemes, several FDT synchronization criteria expressed by linear matrix inequalities (LMIs) are presented. Moreover, a numerical example is presented to illustrate the theoretical results. Keywords: FDT synchronization, complex networks, quantized pinning control, non-chattering control.

1. Introduction In recent years, the synchronization of chaotic systems has been extensively considered from different viewpoints. As a result, various types of synchronization are presented, for example, asymptotic synchronization, exponential synchronization, and finite-time (FET) synchronization [1–6]. Synchronization may be not realized until time approaches infinity when asymptotic synchronization or exponential synchronization techniques are utilized. In some cases, there is a desire to synchronize chaotic systems as quickly as possible. FET control methods can make this goal come true within a settling time. Therefore, the rate of convergence can be improved greatly if FET control techniques are utilized. Considering the superiorities of FET control techniques including fast convergence rate and disturbance rejection [7, 8], extensive attention has been attracted to these attractive control techniques [9–11]. Note that the settling time of classical FET synchronization relies heavily on the initial conditions of considered systems. Initial states must be given in advance when the FET control techniques are utilized in considered systems. However, in some cases, initial values of systems may not be available or even impossible to obtain. The shortcoming limits practical applications of FET synchronization. Recently, the authors in [12] proposed a special FET synchronization named FDT synchronization. The settling time of FDT I This work was jointly supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 61374078, 61633011, 61673078, 61703346.

Preprint submitted to ISA Transactions

October 4, 2018

synchronization is irrelevant to the initial conditions, which is expressed by some parameters which are related to systems and control gains [13]. The remarkable property makes FDT synchronization more preferable than the FET synchronization and promotes its practical application. Therefore, FDT control techniques have attracted wide-scale attention over the past few years, for example, FDT stabilization was investigated in [14]; FDT consensus was considered in [15]; FDT synchronization of delayed memristor neural networks was studied in [16]; FDT stochastic synchronization criteria of CNs were presented in [17]; FDT synchronization of discontinuous systems was investigated in [18]. Via a nonlinear FDT protocol, distributed synchronization of drive-response systems was considered in [19]. As every knows, the controller plays the essential role in realizing synchronization for chaotic systems [20–23]. A simple and effective controller can always be widely utilized in practice. Compared with the other control methods, pinning control is an effective method in realizing synchronization of chaotic systems since only a limited number of network nodes need to be controlled [24–26]. Consider the advantages of pinning control, many results investigated synchronization of CNs via pinning controls [27–29]. Over the past few years, many attempts have been done to avoiding the chattering phenomenon in synchronizing chaotic systems. This is because chattering always brings some undesirable effects to dynamical systems. As we all know, sign function can bring chattering influence to the system state and control signal [30, 31]. Sign function seems to be necessary in previous investigations [9, 10, 16–18, 30, 31]. To avoid the chattering influence brought by sign function, some control schemes without sign function are designed. For example, the authors in [13] designed continuous controllers to realize FDT synchronization of CNs; the authors in [31, 32] proposed 1-norm-based controllers which had a discontinuous point and did not include sign function. However, there are still few synchronization results via non-chattering control up till now. Thus, investigating FDT synchronization of CNs via non-chattering control becomes a motivation. On the other hand, the transmission of signals is usually limited by capacity and bandwidth of communication channels in practice. Quantizing signals before transmission is an effective method to improve the efficiency of communication. The quantization problems have attracted considerable attention in recent years [33–37]. Especially, using quantized control with random packet losses, the stabilization problem of a wireless networked control system was studied in [35]. The authors proposed several synchronization criteria of neural networks with quantization process in [36]. Via quantized intermittent semi-pinning control, the authors considered FET synchronization of coupled networks in [37]. Quantized control techniques can make full use of transmission capacity of the network and reduce channel blocking. Note that there is no paper focus on synchronization of CNs within a fixed time by use of QPC in the open literature. Thus, considering FDT synchronization via QPC becomes another motivation. In this paper, we aim at investigating FDT synchronization of CNs via QPCs. The main contributions are: 1) New pinning non-chattering control scheme is designed, which does not include sign function and can avoid chattering influence; 2) The QPC with sign function can be used generally; 3) These QPCs not 2

only can reduce control cost but also can save channel resources; 4) Several FDT synchronization criteria are established by LMIs which can reduce conservativeness of derived results. Notations R is the space of real number, Rn and Rn×m are the n-dimensional Euclidean space and the set of all n×m real matrices, respectively. I represents the nN -dimensional identity matrix. B = (bij )n×m stands for a matrix of n × m-dimension. B > 0 or B < 0 denotes that B is a symmetric and positive or negative

definite matrix, sym(B) = B + B T and λmin (B) (λmax (B)) represents its minimum (maximum) eigenvalue. For real symmetric matrices X and Y , the notation X > Y means that the matrix X − Y is positive define. ⊗ is the Kronecker product. k · k denotes the standard Euclidean norm of a vector or a matrix. |Ψ| is a vector obtained by taking absolute values of all the elements of the vector Ψ.

2. Preliminaries This section presents the model of CNs and also provides some definitions and necessary assumptions. The model of CNs including N identical nodes is described as follows: x˙ i (t) =Axi (t) + Bg(xi (t)) +

N X j=1

φij Γxj (t), i ∈ N ,

(1)

where N = {1, 2, · · · , N }, xi (t) = (xi1 (t), xi2 (t), · · · , xin (t))T ∈ Rn denotes the state vector of the ith node,

g(·) : Rn → Rn is a continuous function, A, B ∈ Rn×n are constant matrices. Φ = (φij )N ×N is outer-coupling N P φij and Γ = (Γij )n×n is inner-coupling matrix. The matrix, satisfying φij ≥ 0, for i 6= j, φii = − j=1,j6=i

initial value of network (1) is xi (0), i ∈ N .

The target of this article is to fixed-timely synchronize the CNs (1) and the following network y(t) ˙ = Ay(t) + Bg(y(t))

(2)

with initial value y(0) = y0 . Definition 1. [13] The CN (1) is said to be fixed-timely synchronized onto (2), if there is a constat T > 0 which is independent of the initial values x(0) = (xT1 (0), xT2 (0), · · · , xTN (0))T and y0 , such that lim kxi (t)−y(t)k2 = 0 t→T

and kxi (t) − y(t)k2 ≡ 0 for t > T , i ∈ N . This paper utilizes the following assumption. Assumption 1. There exists a constant d ≥ 0 such that kg(x(t)) − g(y(t))k ≤ dkx(t) − y(t)k, ∀x(t), y(t) ∈ Rn . Adding controllers ui (t), the controlled CNs can be presented: x˙ i (t) = Axi (t) + Bg(xi (t)) +

N X j=1

3

φij Γxj (t) + ui (t), i ∈ N .

(3)

Subtracting (2) from (3) yields that z˙i (t) = Azi (t) + Bf (zi (t)) +

N X j=1

φij Γzj (t) + ui (t), i ∈ N ,

(4)

where zi (t) = xi (t) − y(t), f (zi (t)) = g(xi (t)) − g(y(t)). Therefore, fixed-timely synchronizing the dynamical networks (3) and (2) corresponds to stabilizing the error system (4) at the origin within a fixed time. Lemma 1. [15] Let η1 , η2 , · · · , ηN ≥ 0, 0 < α ≤ 1, β > 1. Then, one can derive that: N X

ηiα

i=1

≥

X N

ηi

i=1

α

,

N X i=1

ηiβ

≥N

1−β

X N i=1

ηi

β

.

Lemma 2. [38] For any dimension-compatible matrices M , H and E with H T H ≤ I and a scalar ε > 0, then the following inequality holds: M HE + E T H T M T ≤ εM M T + ε−1 E T E. Lemma 3. [39] (Schur Complement) The linear matrix inequality   D11 D12 <0 D= T D22 D12 is equivalent to any one of the following two conditions: −1 T (L1 ) D11 < 0, D22 − D12 D11 D12 < 0,

−1 T (L1 ) D22 < 0, D11 − D12 D22 D12 < 0,

T T . , D22 = D22 where D11 = D11

Lemma 4. [40] Let a nonnegative function V(t) satisfy ˙ V(t) ≤ −ηV α (t) − ξV β (t), where ξ > 0, η > 0, 1 > α > 0, β > 1. Then, V(t) ≡ 0, if t≥

1 1 + . η(1 − α) ξ(β − 1)

3. FDT synchronization via QPCs In this section, QPCs are designed. Via the new pinning controllers, several FDT synchronization criteria which formulated by LMIs are established. Rearrange the order of the nodes and control the first m nodes. Designing the following QPC:    −li ϕ(zi (t)) − ξ[ϕ(zi (t))] γθ − η[ϕ(zi (t))] υs , i = 1, 2, · · · , m, ui (t) =   −ξ[ϕ(zi (t))] γθ − η[ϕ(zi (t))] υs , i = m + 1, m + 2, · · · , N, 4

(5)

where li > 0 to be determined, ξ > 0, η > 0 are tunable constants, and γ, θ, s, and υ are positive odd integers satisfying γ > θ, s < υ. [ϕ(zi (t))]σ = ([ϕ(zi1 (t))]σ , [ϕ(zi2 (t))]σ , · · · , [ϕ(zin (t))]σ )T , σ =

γ θ

or σ =

s υ.

ϕ(·) : R → Ω is a quantizer, where Ω = {±ωi : ωi = ρi ω0 , i = 0, ±1, ±2, · · · } ∪ {0} with ω0 > 0. For ∀τ ∈ R, the quantizer ϕ(τ ) is constructed as follows:  1   ωi < τ ≤ ωi , if 1+δ    ϕ(τ ) = 0, if τ = 0,     −ϕ(−τ ), if τ < 0.

where δ =

1−ρ 1+ρ ,

1 1−δ ωi ,

0 < ρ < 1. According to the analysis in [37], there exists a Filippov solution ∆ ∈ [−δ, δ) such

that ϕ(τ ) = (1 + ∆)τ . Remark 1. The QPC (5) is more practical than the following continuous controller which can also be regarded as a special case of controller (5)    ui (t) =  

γ

s

−li zi (t) − ξ[zi (t)] θ − η[zi (t)] υ , i = 1, 2, · · · , m, γ

s

−ξ[zi (t)] θ − η[zi (t)] υ , i = m + 1, m + 2, · · · , N.

Considering logarithmic quantization is necessary due to the limited transmission of signals and finite

bandwidth of communication channels. Remark 2. Compared with other controllers, only some parts of nodes need to be controlled for pinning control. Thus, it always be used to decrease the control gains of chaotic systems and further reduces control cost. Moreover, controller (5) does not include sign function. So, the chattering influence can be avoided by utilizing controller (5). T Let f (z(t)) = (f T (z1 (t)), f T (z2 (t)), · · · , f T (zN (t)))T , z(t) = (z1T (t), z2T (t), · · · , zN (t))T , [ϕ(z(t))]σ =

(([ϕ(z1 (t))]σ )T , ([ϕ(z2 (t))]σ )T , · · · , ([ϕ(zN (t))]σ )T )T , A = IN ⊗ A, B = IN ⊗ B, Φ = Φ ⊗ Γ, L = diag(l1 , l2 , · · · , lm , 0, 0, · · · , 0) ⊗ In . By use of the Kronecker product, the network (4) with controller (5) is rewritten as: γ

s

z(t) ˙ = (A + Φ − L − LΛ(t))z(t) + Bf (z(t)) − ξ[ϕ(z(t))] θ − η[ϕ(z(t))] υ ,

(6)

where Λ(t) = diag(Λ1 (t), Λ2 (t), · · · , ΛN (t)), Λi (t) = diag(Λi1 (t), Λi2 (t), · · · , Λin (t)), Λij (t) are Filippov solutions with Λij ∈ [−δ, δ), i = 1, 2, · · · , N , j = 1, 2, · · · , n. Theorem 1. Let the Assumption 1 hold and ξ > 0, η > 0. Then, for given control gain L, the network (3) can be fixed-timely synchronized with (2) via the QPC (5) if there exist positive definite diagonal matrix P , and scalars ǫ1 > 0, ε1 > 0 such that



Π1

  T  B P  LP δ

PB

P Lδ

−ǫ1 I

0

0

−ε1 I 5



   < 0. 

(7)

Moreover, the settling time is estimated as T =

θ υ , + ˆ − θ) ηˆ(υ − s) ξ(γ

(8)



b 2 = ǫ2 d2 I + sym(P A + Φ) , ξˆ = ξ(N n) θ−γ 2θ (1 − where Π1 = (ǫ1 d2 + ε1 )I + sym(P A + Φ − L) , Π γ

δ) θ λmin (P )(λmax (P ))−

γ+θ 2θ

s

and ηˆ = η(1 − δ) υ λmin (P )(λmax (P ))−

s+υ 2υ

.

Proof. Consider Lyapunov function V(t) = z T (t)P z(t),

(9)

It follows from (6) that  γ s  ˙ V(t) =2z T (t)P (A + Φ − L − LΛ(t))z(t) + Bf (z(t)) − ξ[ϕ(z(t))] θ − η[ϕ(z(t))] υ .

By Assumption 1, one can derive that

dkzi (t)k = dkxi (t) − y(t)k ≥ kg(xi )(t) − g(y(t))k = kf (zi (t))k. Furthermore d2 ziT (t)zi (t) = d2 kzi (t)k2 ≥ kf (zi (t))k2 = f (zi (t))f T (zi (t)). It follows that d2 z T (t)z(t) − f T (z(t))f (z(t)) ≥ 0. From ǫ1 > 0, it yields ǫ1 d2 z T (t)z(t) − ǫ1 f T (z(t))f (z(t)) ≥ 0. It is derived that ˙ V(t) ≤z T (t)P (A + Φ − L + δL)z(t) + z T (t)(A + Φ − L + δL)T P z(t) γ

s

+ z T (t)P Bf (z(t)) + f T (z(t))BT P z(t) − 2ξz T (t)P [ϕ(z(t))] θ − 2ηz T (t)P [ϕ(z(t))] υ + ǫ1 d2 z T (t)z(t) − ǫ1 f T (z(t))f (z(t)) γ

s

=ΞT (t)Π(t)Ξ(t) − 2ξz T (t)P [ϕ(z(t))] θ − 2ηz T (t)P [ϕ(z(t))] υ , b 1 (t) = ǫ1 d2 I + P (A + Φ − L + δL) + (A + Φ − L + δL)T P and where Ξ(t) = (z T (t), f T (z(t)))T , Π   b 1 (t) P B Π . Π(t) =  BT P −ǫ1 I 6

(10)

Then, one derives    T 2 b Π1 (t) =ǫ1 d I + P A + Φ − L + δL + A + Φ − L + δL P e 1 + δP L + δLP, =Π


e 1 = ǫ1 d2 I + sym(P A + Φ − L) . where Π
T Let M = (δP L)T , 0 , E = (I, 0)T , H = I, where I represents the 2nN -dimensional identity matrix.

Then HH T = I and

One can derive that

M HE T + EHM T =((δP L)T , 0)T I(I, 0) + (I, 0)T I((δP L)T , 0)     δLP 0 δLP 0 + . = 0 0 0 0 

Π(t) = 



=

b 1 (t) Π

PB

BT P

−ǫ1 I

e1 Π

PB

T

B P

−ǫ1 I

 



 + M HE T + EHM T .

By Lemma 3 and conditions (7), one derives   Π1 PB   + ε1−1 M M T < 0. BT P −ǫ1 I

(11)

(12)

On the other hand, by Lemma 2

T T M HE T + EH T M T ≤ ε−1 1 M M + ε1 EE .

It can be derived from (11)–(13) that  Π(t) =  

≤ 

=

e1 Π

PB

T

B P

−ǫ1 I

e1 Π

PB

T

B P

−ǫ1 I

Π1

PB

T

−ǫ1 I

B P

From inequalities (10) and (14), one has

(13)



 + M HE T + EHM T



T T  + ε−1 1 M M + ε1 EE



T  + ε−1 1 M M < 0.

γ s ˙ V(t) ≤ −2ξz T (t)P [ϕ(z(t))] θ − 2ηz T (t)P [ϕ(z(t))] υ .

7

(14)

(15)

From γ > θ and s < υ, one obtains

γ+θ 2θ

> 1 and 0 <

s+υ 2υ

< 1. By Lemma 1, it can be derive that

γ

γ

−z T (t)P [ϕ(z(t))] θ ≤ −λmin (P )z T (t)[ϕ(z(t))] θ = −λmin (P )

N X n X

θ−γ 2θ

γ+θ 2θ

i=1 j=1

≤ −λmin (P )(N n) ≤ −(N n)

γ

2 (1 + Λij ) θ (zij (t))

θ−γ 2θ

γ

(1 − δ) θ (z T (t)z(t))

γ+θ 2θ

γ

(1 − δ) θ λmin (P )(λmax (P ))−

γ+θ 2θ

V

γ+θ 2θ

(t),

(16)

and s

s

−eT (t)P [ϕ(z(t))] υ (t) ≤ −(1 − δ) υ λmin (P )(λmax (P ))−

s+υ 2υ

V

s+υ 2υ

(t).

(17)

It follows from (15)–(17) that s+υ ˆ γ+θ ˙ 2θ (t) − 2ˆ η V 2υ (t). V(t) ≤ −2ξV

Based on Lemma 4, it follows that, V(t) ≡ 0 for t ≥ T . Moreover, the system error z(t) will converge to zero within T . Consequently, the CN (3) is fixed-timely synchronized onto (2) within T , which can be described by (8). This completes the proof. Remark 3.

The settling time of Theorem 1 does not depend on the system initial states x(0) and y0

but only related to the parameters of controller, the group order N and the dimension of node n. The FDT synchronization are widely considered in [12–18]. However, few papers consider this issue via quantized control. Via quantized intermittent pinning control, FET synchronization for dynamical networks was investigated in [37]. In [37], the settling time of synchronization is related to initial values of systems, and the controller includes sign function which can bring chattering influence to the system state and control signals. Theorem 1 realizes FDT synchronization by using of QPCs in this paper. Compared with some FDT synchronization such as the results in [15–18], the synchronization criteria of this paper formulated by LMIs which can reduce the conservativeness of obtained results. When l = N , the controller (5) turns out to be quantized non-chattering control scheme γ

s

ui (t) = −li ϕ(zi (t)) − ξ[ϕ(zi (t))] θ − η[ϕ(zi (t))] υ ,

(18)

Next, an important corollary will be derived according to (18). Corollary 1. Let the Assumption 1 hold and ξ > 0, η > 0. Then for given control gain L, the network (3) with controller (18) can be fixed-timely synchronized with (2) if there exist positive definite diagonal matrix P , and scalars ǫ1 > 0, ε1 > 0 such that (7) hold. Moreover, the settling time is given by (8).

8

Remark 4. Corollary 1 is also a very useful result. Controller (18) can be seen the special case of controller (5). Via controller (18), many important results have been obtained. Especially, by means of a special case of controller (18), FDT synchronization of CNs with impulsive effects was obtained in [13]. A general QPC is designed as follows    −li ϕ(zi (t)) − ξsign(ϕ(zi (t)))|(ϕ(zi (t)))|β − ηsign(ϕ(zi (t)))|ϕ(zi (t))|α , i = 1, 2, · · · , m, ui (t) =   −ξsign(ϕ(zi (t)))|(ϕ(zi (t)))|β − ηsign(ϕ(zi (t)))|ϕ(zi (t))|α , i = m + 1, m + 2, · · · , N,

(19)

where sign(·) is the standard sign function, and sign(ϕ(zi (t))) = diag(sign(ϕ(zi1 (t))), sign(ϕ(zi2 (t))), · · · , sign(ϕ(zin (t)))), 0 < α < 1, β > 1, ξ and η are defined in controller (5). Remark 5. The QPC (19) is more practical than the following controller    −li zi (t) − ξsign(zi (t))|zi (t)|β − ηsign(zi (t))|zi (t)|α , i = 1, 2, · · · , m, ui (t) =   −ξsign(zi (t))|zi (t)|β − ηsign(zi (t))|zi (t)|α , i = m + 1, m + 2, · · · , N.

(20)

Theorem 2. Let the Assumption 1 hold and ξ > 0, η > 0. Then, for given control gain L, the network (3) can be fixed-timely synchronized with (2) via the QPC (19) if there exist positive definite diagonal matrix P , and scalars ǫ1 > 0, ε1 > 0 such that (7) hold. Moreover, the settling time is estimated as 1 1 T = ¯ , + ξ(β − 1) η¯(1 − α)

(21)

1+β 1−β 1+α where ξ¯ = ξ(N n) 2 (1 − δ)β λmin (P )(λmax (P ))− 2 and η¯ = η(1 − δ)α λmin (P )(λmax (P ))− 2 .

Proof. Consider the Lyapunov function (9). Utilizing similar procedure as that presented in Theorem 1, one derives the following inequality ˙ V(t) ≤ −2ξz T (t)sign(ϕ(z(t)))P |ϕ(z(t))|β − 2ηeT (t)sign(ϕ(z(t)))P |ϕ(z(t))|α , where sign(ϕ(z(t))) = diag(sign(ϕ(z1 (t))), sign(ϕ(z2 (t))), · · · , sign(ϕ(zN (t)))). δ ∈ (0, 1) implies that sign(ϕ(e(t))) = sign(e(t)). Thus ˙ V(t) ≤ −2ξ|z T (t)|P |ϕ(z(t))|β − 2η|z T (t)|P |ϕ(z(t))|α ¯ ≤ −2ξV

1+β 2

(t) − 2¯ ηV

1+α 2

(t).

The rest part is same as Theorem 1. Thus, the FDT synchronization of network (2) and (3) is realized. This completes the proof. Corollary 2. Let the Assumption 1 hold and ξ > 0, η > 0. Then for given control gain L, the network (3) with controller (20) can be fixed-timely synchronized with (2) if there exist positive definite diagonal matrix P , and scalars ǫ1 > 0, ε1 > 0 such that (7) hold. Moreover, the settling time is given by (21). 9

Remark 6. From Theorem 2, one can see that sign(ϕ(z(t))) = sign(z(t)) plays an important role. Combined with LMIs, quantized techniques and pinning control can play important effects in realizing synchronization which can be seen from the later simulations. Via controller (20), many important results can be obtained. Several FET synchronization criteria of CNs are established in [17, 18] via some controllers which are similar with the special cases of controller (20). However, the logarithmic quantization is not considered in [17, 18].

4. Numerical Example In this section, some numerical simulations are given to illustrate that the derived synchronization criteria are viable. Here, the quantizer density is taken as ρ = 0.7. Consider a CNs model, in which each subsystem is a Chua’s circuit described as [41] y(t) ˙ = Ay(t) + Bg(y(t)),

(22)

where y(t) = (y1 (t), y2 (t), y3 (t))T , g(y(t)) = (|y1 (t) + 1| − |y1 (t) − 1|, 0, 0)T , B = diag(27/7, 0, 0),   9 0 − 19 7     A= 1 −1 1 .   1 −14.28 0

Chaotic trajectory of the Chua’s circuit with initial condition y(0) = (0.65, 0.2, 0.8)T is shown in Fig. 1.

10

y3

5

0

−5

−10 1 0 −1 y2

−6

−2

−4

0

2

4

6

y1

Fig. 1. Chaotic trajectory of Chua’s circuit with initial value y(0) = (0.65, 0.2, 0.8)T

Consider the CN as follows: x˙ i (t) =Axi (t) + Bg(xi (t)) +

5 X j=1

10

φij Γxj (t), i ∈ 1, 2, · · · , 5,

(23)

8

7

7

6

6 kzi (t)k, i = 1, 2, . . . , 5

kϕ(zi (t))k, i = 1, 2, . . . , 5

8

5 4 3

5 4 3

2

2

1

1

0

0 0

Fig. 2.

0.05

0.1

0.15 t

0.2

0.25

0.3

0

0.05

0.1

0.15 t

0.2

0.25

0.3

Trajectories kϕ(zi (t)k and kzi (t)k (i = 1, 2, · · · , 5) via controller (5) with γ = 5, θ = 3, s = 3, υ = 5.

where xi (t) = (xi1 (t), xi2 (t), xi3 (t))T , Γ = diag(1, 1, 1),  −3 1    1 −2   Φ = 7 2 2    1 0  1 2

1

0

1



  0 0 1    −7 0 3 .   1 −3 1   0 2 −5

By computation, the Assumption 1 is satisfied with d = 2. Take the control gains in (5) are l1 = 33, l2 = 30,

l3 = 15, l4 = l5 = 0, ξ = 6, η = 6. Solving the LMIs (7) in Matlab derives that P = diag(0.0174, 0.0352, 0.0152, 0.0193, 0.0403, 0.0175, 0.0082, 0.0178, 0.0077, 0.0175, 0.0738, 0.0176, 0.0156, 0.0520, 0.0153), ǫ1 = 0.0319, ε1 = 0.0751. The initial values of CN (23) are chosen from (−5, 5). According to Theorem 1, the CN (23) with quantized controller (5) is synchronized with (21) within settling time T = 32.2875, where γ = 5, θ = 3, s = 3, υ = 5. We have the trajectories ϕ(zi (t)) and zi (t), i = 1, 2, · · · , 5 presented in Fig. 2, which show that synchronization is realized within T = 32.2875. Similarly, the CN (23) with quantized controller (19) is synchronized on (21) within T = 26.9700 according to Theorem 2, which is illustrated by Fig. 3. Here, we take α = 1/3, β = 5/3. Remark 7. From Figs. 2 and 3 one can find that if the control signals are quantized, time response of kϕ(zi (t))k has some fold lines which implies that the control signals have some changes. These changes can be seen from the describes of the quantizer.

11

8

7

7

6

6 kzi (t)k, i = 1, 2, . . . , 5

kϕ(zi (t))k, i = 1, 2, . . . , 5

8

5 4 3

5 4 3

2

2

1

1

0

0 0

0.05

Fig. 3.

0.1

0.15 t

0.2

0.25

0.3

0

0.05

0.1

0.15 t

0.2

0.25

0.3

Trajectories kϕ(zi (t))k and kzi (t)k (i = 1, 2, · · · , 5) via controller (19) with α = 1/3, β = 5/3.

5. Conclusions This paper investigates FDT synchronization of CNs. Two types of pinning controllers with logarithmic quantization are designed. One control scheme is continuous and does not include sign function, hence the chattering influence can be avoided. The other control scheme including sign function can be generally applied than the controller without sign function if the chattering influence is ignored. Several FDT synchronization criteria presented by LMIs are derived. These synchronization criteria are very general and can be easily deduced to many special cases, which includes some existing results. Numerical simulations demonstrated the correctness and effectiveness of the theoretical results. Note that time-delays are unavoidable in CNs due to the finite speed of signal transmission and traffic congestion. Time-delay may change the dynamics of CNs and make it difficult to achieve synchronization. Thus, investigating time-delayed dynamical systems is practical [42, 43]. Moreover, there exist some uncertain and inevitable factors in real world. Therefore, chaotic systems with uncertain disturbance is worth considering [44]. Fixed-time synchronization of CNs with time delays and uncertain disturbance is interesting. This is our next research topic, which is challenging.

References References [1] Huang T, Li C, Duan S, Starzyk JA. Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects. IEEE Trans Neural Netw Learn Syst 2012; 23; 866–875. [2] Wang J, Zeng C. Synchronization of fractional-order linear complex networks. ISA Trans 2015; 55; 129– 134. 12

[3] Chen Z, Shi K, Zhong S. New synchronization criteria for complex delayed dynamical networks with sampled-data feedback control. ISA Trans 2016; 63; 154–169. [4] Ahmed H, Salgado I, R´ıos H. Robust synchronization of master-slave chaotic systems using approximate model: An experimental study. ISA Trans 2018; 73; 141–146. [5] Zhang H, Ma T, Huang G, Wang Z. Robust global exponential synchronizaiton of uncertain chaotic delayed neural networks via dual-stage impulsive control. IEEE Trans Syst, Man, Cybern B, Cybern 2010; 40; 831–844. [6] Vincent U E, Guo R. Finite-time synchronization for a class of chaotic and hyperchaotic systems via adaptive feedback controller. Phys Lett A 2011; 375; 2322–2326. [7] Haimo V T. Finite-time controllers. SIAM J Control Optim 1986; 24; 760–770. [8] Bowong S, Kakmeni F. Chaos control and duration time of a class of uncertain chaotic systems. Phys Lett A 2003; 316; 206–217. [9] Yang X, Cao J. Finite-time stochastic synchronization of complex networks. Appl Math Model 2010; 34; 3631–3641. [10] Aghababa MP, Aghababa HP. Synchronization of mechanical horizontal platform systems in finite time. Appl Math Model 2012; 36; 4579–4591. [11] Hou H, Zhang Q. Finite-time synchronization for second-order nonlinear multi-agent system via pinning exponent sliding mode control. ISA Trans 2016; 65; 96–108. [12] Polyakov A. Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans Autom Control 2012; 57; 2106–2110. [13] Yang X, Lam J, Ho DWC, Feng Z. Fixed-Time synchronization of complex networks with impulsive effects via nonchattering control. IEEE Trans Autom Control 2017; 62; 5511–5521. [14] Polyakov A, Efimov D, Perruquetti W. Finite-time and fixed-time stabilization: Implicit Lyapunov function approach. Automatica 2015; 51; 332–340. [15] Zuo Z, Tie L, Distributed robust finite-time nonlinear consensus protocols for multi-agent systems. Int J Syst Sci 2016; 47; 1366–1375. [16] Cao J, Li R. Fixed-time synchronization of delayed memristor-based recurrent neural networks. Sci China Inf Sci 2017; 60; 032201. [17] Zhang W, Li C, Huang T, Huang J. Fixed-time synchronization of complex networks with nonidentical nodes and stochastic noise perturbations. Physica A 2018; 492; 1531–1542.

13

[18] Zhu X, Yang X, Alsaadi F E, Hayat T. Fixed-time synchronization of coupled discontinuous neural networks with nonidentical perturbations. Neural Process Lett 2017, doi: 10.1007/s11063-017-9770-8. [19] Zhao W, Liu G, M X, He B, Dong Y. Distributed synchronization of networked drive-response systems: A nonlinear fixed-time protocol. ISA Trans 2017; 71; 178–184. [20] Xu Y, Ma S, Zhang H. Hopf bifurcation control for stochastic dynamical system with nonlinear random feedback method. Nonlinear Dyn 2011; 65; 77–84. [21] Xu Y, Gu R, Zhang H, Li D. Chaos in diffusionless lorenz system with a fractional order and its control. Int J Bifurcation and Chaos 2012; 22; 1250088. [22] Liu Q, Xu Y, Kurths J. Active vibration suppression of a novel airfoil model with fractional order viscoelastic constitutive relationship. J Sound Vib 2018; 432; 50–64. [23] Liu Q, Xu Y, Xu C, Kurths J. The sliding mode control for an airfoil system driven by harmonic and colored Gaussian noise excitations. Appl Math Model 2018; 64; 249–264. [24] Song Q, Cao J. On pinning synchronization of directed and undirected complex dynamical networks. IEEE Trans Circ Syst I 2010; 57; 672–680. [25] Wang J, Ma Q, Chen A, Liang Z. Pinning synchronizaiton of fractional-order complex networks with lipschitz-type nonlinear dynamics. ISA Trans 2015; 57; 111–116. [26] Lu J, Zhong J, Huang C, Cao J. On pinning controllability of boolean control networks. IEEE Trans Autom Control 2016; 61; 1658–1663. [27] Rakkiyappan R, Kaviarasan B, Rihan FA, Lakshmanan S. Synchronization of singular Markovian jumping complex networks with additive time-varying delays via pinning control. J Frank Inst 2015; 352; 3178– 3195. [28] Liu X, Chen T. Finite-time and fixed-time cluster synchronization with or without pinning control. IEEE Trans Cybern 2018; 48; 240–252. [29] Dharani S, Rakkiyappan R, Park JH. Pinning sampled-data synchronization of coupled inertial neural networks with reaction-diffusion terms and time-varying delays. Neurocomputing 2017; 227; 101–107. [30] Yang X, Ho DWC, Lu J, Song Q. Finite-time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays. IEEE Trans Fuzzy Syst 2015; 23; 2302–2316. [31] Zhang W, Yang X, Xu C, Feng J, Li C. Finite-time synchronization of discontinuous neural networks with delays and mismatched parameters. IEEE Trans Neural Netw Learn Syst 2018; 29; 3761–3771. [32] Yang X, Lu J. Finite-time synchronization of coupled networks with Markovian topology and impulsive effects. IEEE Trans Autom Control 2016; 61; 2256–2261. 14

[33] Brockett RW, Liberzon D. Quantized feedback stabilization of linear systems. IEEE Trans Autom Control 2000; 45; 1279–1289. [34] Tian E, Yue D, Peng C. Quantized output feedback control for networked control systems. Inf Sci 2008; 178; 2734–2749. [35] Qu F, Hu B, Guan Z, Wu Y, He D, Zheng D. Quantized stabilization of wireless networked control systems with packet losses. ISA Trans 2016; 64; 92–97. [36] Wan Y, Cao J, Wen G. Quantized synchronization of chaotic neural networks with scheduled output feedback control. IEEE Trans Neural Netw Learn Syst 2017; 28; 2638–2647. [37] Xu C, Yang X, Lu J, Feng J, Alsaadi FE, Hayat T. Finte-time synchronization of networks via quantized intermittent pinning control. IEEE Trans Cyb 2017, doi: 10.1109/TCYB.2017.2749248. [38] Wang Y, Xie L, de Souza CE. Robust control of a class of uncertain nonlinear systems. Syst Control Lett 1992; 19; 139–149. [39] Boyd S, Ghaoui L El, Feron E, Balakrishnan V. Linear matrix inequalities in system and control theory. SIAM Philadelphia PA 1994. [40] Lu W, Liu X, Chen T. A note on finite-time and fixed-time stability. Neural Netw 2016; 81; 11–15. [41] Matsumoto T, Chua L, Komuro M. The double scroll. IEEE Trans Circuits Syst 1985; 32; 797–818. [42] Zhang H, Wang Z, Liu D. Robust exponential stability of recurrent neural networks with multiple timevarying delays. IEEE Trans Circ Syst II 2007; 54; 730–734. [43] Zhang H, Wang Z, Liu D. A comprehensive revies of stability analysis of continuous-time recurrent neural networks. IEEE Trans Neural Netw Learn Syst 2014; 25; 1229–1262. [44] Xu Y, Liu Q, Guo G, Xu C, Liu D. Dynamical response of airfoil models with harmonic excitation under uncertain disturbance. Nonlinear Dyn 2017; 89: 1579–1590.

15