Cluster synchronization for a class of complex dynamical network system with randomly occurring coupling delays via an improved event-triggered pinning control approach

Cluster synchronization for a class of complex dynamical network system with randomly occurring coupling delays via an improved event-triggered pinning control approach

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Cluster synchronization for a class of complex dynamical network system with randomly occurring coupling delays via an improved event-triggered pinning control approach Hongqian Lu, Yue Hu, Chaoqun Guo, Wuneng Zhou PII: DOI: Reference:

S0016-0032(19)30880-4 https://doi.org/10.1016/j.jfranklin.2019.11.076 FI 4311

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

2 May 2019 6 October 2019 27 November 2019

Please cite this article as: Hongqian Lu, Yue Hu, Chaoqun Guo, Wuneng Zhou, Cluster synchronization for a class of complex dynamical network system with randomly occurring coupling delays via an improved event-triggered pinning control approach, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.076

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Cluster synchronization for a class of complex dynamical network system with randomly occurring coupling delays via an improved event-triggered pinning control approach Hongqian Lua,∗, Yue Hua,∗, Chaoqun Guoa , Wuneng Zhoub a School

of Electrical Engineering and Automation ,Qilu University of Technology (Shandong Academy of Sciences),Jinan 250353 b College of Information Science and Technology, Donghua University, Shanghai 201620

Abstract This paper discusses the cluster synchronization problem for a class of complex dynamical networks with coupling time delays. Under the randomly occurring coupling time delays condition, a novel complex network model is established to express the dynamical behavior of the nodes more precisely. An improved event-triggered mechanism is proposed for two purpose containing the reduction of network burden and improvement of cluster synchronization efficiency. The pinning control strategy is adopted for cluster synchronization via controlling a small fraction of the network nodes. By employing Lyapunov-Krasovskii functional method, a stability criterion is obtained to ensure the achievement of cluster synchronization and a event-triggered pinning controller is constructed. Finally, a simulation example is expressed to verify the effectiveness of our main results. Keywords: Cluster synchronization, Event-triggered, Pinning control, Random coupling delays 1. Introduction Complex networks, involved a large number of nodes, various structures, nonlinear terms and stochastic factors [1–3], have received extensive research and attention in ✩ This work is supported by the National Natural Science Foundation of China under Grant No.61573095, Natural Science Foundation of Shandong Province under Grant No.ZR2013FM022 and ZR2014FM019, Shandong Province Higher Educational Science and Technology Program(J15LN04). ∗ Corresponding author

Preprint submitted to Journal of LATEX Templates

December 4, 2019

fields containing physics, mathematics and biology [4–6]. In real world, many systems 5

can be analyzed by means of complex networks, such as aviation networks [7], traffic networks [8], power networks [9]. In particular, with the development of online social network platform, the research with respect to online social complex networks has been a topic [10]. Synchronization is a very ubiquitous phenomenon in complex networks because

10

each node is a dynamic system and there are some couplings among many dynamic systems. Nowadays, synchronization has been extensively researched in various forms [11, 12]. In [13], the global synchronization problem of the complex system was researched. Under the input constraint condition, the synchronization problem in the complex networks was investigated in [14]. In the multi-weighted complex networks, literature [15]

15

analyzed the finite-time synchronization problem. In order to analyze the complete synchronization problem, [16] proposed the approximation method. However, complete synchronization is not always expected in some networks such as brain neural network [17–19]. Different from the complete synchronization, cluster synchronization is the situation that the trajectories of the nodes in same cluster can synchronize

20

to a same value but there is no consensus between different clusters. Recently, a lot of work has been done to research cluster synchronization of complex systems and a series of achievements have been acquired. For discrete uncertain network, cluster synchronization transmission was researched in [20]. Exponential cluster synchronization criterion was derived for directed connection networks [21]. In [22], stochastic factors

25

were considered in cluster synchronization of neural networks. Moreover, time delay as a widely existing phenomenon in various systems which can cause oscillation and instability has been researched for many years [23, 24]. Usually, researchers employ variation rate and variation range of delay to investigate the effect produced by delay [25–27]. But time delay may change suddenly with a small

30

probability in some real objects. For this probabilistic delay, traditional research approaches are not applicable. Based on probabilistic delay, many researchers have applied themselves to analyze the stability problem of systems. For example, the probabilistic random delays problem of the uncertain systems have been considered in [28]. The paper [29] discussed the robust H∞ performance of the fuzzy systems with multi2

35

ple probabilistic delays. Although there have been many results about probabilistic delay, no one has considered probabilistic delay in clustering synchronization of complex networks. In many real complex networks, time delays not always exist in dynamic behavior of nodes because the complex coupling and variable stochastic factors. Unfortunately, in complex networks, the randomly occurring coupling time delay has not

40

been introduced to the investigation of cluster synchronization, although it is important and challenging. As a way to get over the shortcoming of period time system, event-triggered mechanism has been extensively studied since it was proposed [30]. Compared with the time-triggered mechanism, event-triggered mechanism can effectively reduce the re-

45

lease times of sampled data via judging whether the system dynamic behavior satisfies the trigger conditions. To further save network transmission resources, a series of improved event-triggered mechanisms have been proposed. By employing a dynamic variable, [31] introduced a dynamic event-triggered scheme. [32] designed an integral-based event-triggering control scheme to nonlinear systems. Network burden

50

in complex networks is especially heavy because of the existence of a large number of nodes. [33] employed a dynamic event-triggered pinning control method to analyze the synchronization of stochastic complex dynamical networks. Recently, a novel eventtriggered scheme was proposed to deal with the filtering problem for the nonlinear systems [34]. When analyzing the clustering synchronization problem of complex net-

55

works, a smart event-triggered scheme should meet the requirement of reaching cluster synchronization faster, which specific performances as high trigger frequency during transient response and low trigger frequency when error state approaches to zero. To the best of authors’ knowledge, in the complex networks, under the randomly occurring coupling delays, no one has ever designed such an event-triggered mechanism for

60

research of cluster synchronization problem. An improved event-triggered mechanism for such a continuous-time complex network will be introduced in this paper. On the other hand, synchronization may not be possible only by relying on structure of the network topology, coupling strength between nodes [36–38]. Thus, imposing control to force the system to achieve synchronization is very necessary. The

65

characteristic of numerous nodes makes the computational cost of traditional control 3

approach higher. It is extremely important for complex networks to reduce control cost and improve control effectiveness. This promotes the development of pinning control strategy which makes the trajectories of nodes achieve synchronization by controlling only a part of nodes. For analyzing the synchronization problem of complex networks, 70

some efforts have been made to the evolution of pinning control. The paper [39] introduced an intermittent pinning control approach to analyze the cluster lag synchronization problem of complex networks. In uncertain network, [40] employed an adaptive pinning control approach to deal with cluster synchronization problem. Under the randomly occurring coupling delays condition, this paper will combine the pinning control

75

strategy with the improved event-triggered mechanism to force the cluster synchronization to be achieved in complex network. Motivated by the above discussions, this paper is devoted to investigating cluster synchronization control for complex networks by event-triggered pinning control strategy. The main contributions are summarized as follows. 1) The randomly occurring

80

coupling delays are introduced into cluster synchronization investigation for complex networks. Compared with the general complex networks model, based on randomly occurring coupling delays condition, the complex networks can describe the actual world more accurately. 2) An improved event-triggered mechanism was proposed for this continuous-time complex network. Under this mechanism, network transmission

85

resources are saved. Simultaneously, the dynamic process to achieve cluster synchronization is shortened by raising the trigger frequency in the start time period. 3) A criterion is derived to guarantee the realization of cluster synchronization by constructing an appropriate Lyapunov-Krasovskii functional and researching the stability of error system. In addition, in this paper, a pinning controller combined with the improved

90

event-triggered scheme is designed for such a complex network. The rest of this article is summarized as follows: In Section 2, based on the randomly occurring coupling delays condition, the model of complex network system is constructed. Simultaneously, the improved event-triggered scheme condition is also introduced. In Section 3, the cluster synchronization problem is analyzed and the sta-

95

bility criteria is derived. In addition, the cluster synchronization controller is designed. In Section 4, we give a numerical example to prove the feasibility of the main results. 4

Finally, conclusions are shown in Section 5. Notations E be the expectation operator. Rn means n dimensional Euclidean space. I denotes appropriate dimensional identity matrix. AT means transpose of 100

matrix A, A−1 presents the inverses of matrix A, A > 0 means that matrix A is positive definite; suppose that He(A) is equal to A+AT . By using an asterisk ∗ to represent a term that is induced by symmetry. Furthermore, suppose that the matrices have appropriate dimensions if their dimensions are not explicitly stated. 2. Problem Formulation

105

Assume there are N nodes in complex network. Define the set γ = {1, 2, ..., N },

and {γ1 , ..., γq } is a subset of γ, such that γl 6= ∅, and ∪ql=1 γl = γ, and γl ∩ γg = ∅, for l 6= g, l, g, = 1, 2, ..., q. Let γl represent the various clusters. Define ¯i denoting

the subscript of the cluster which node i is located and ¯i = 1, ..., q. Define zl is the number of nodes in cluster γl . We number the nodes in each cluster as {zl−1 +1, zl−1 +

110

2, ..., zl }. Let’s consider the complex dynamic network system as follows: x˙ i (t) =fl (xi (t)) + (1 − β(t)) + (β(t))

N X j=1

N X

αij Γxj (t)

j=1

αij Γxj (t − τ (t)) + ui (t),

(1) i ∈ γl , l = 1, ..., q,

where xi (t) = [x1i , ..., xni ] ∈ Rn is the state. fl (·) : Rn → Rn is a continuous nonlinear

vector function. The positive definite diagonal matrix Γ is the inner-coupling matrix between node i and node j. ui (t)∈ Rm represents the control input. α = (αij )N ×N 115

is the coupling configuration matrix of the network. αij 6= 0 if there is a connection PN between node i and j or else αij = 0. In addition, αii = − j=1,j6=i αij . τ (t) denotes

the time varying delay and 0 ≤ τ (t) ≤ τ , τ˙ (t) ≤ µ ≤ 1. β(t) is the Bernoulli random variable describing the randomly occurring coupling delay and satisfying:   1, coupling delay happens, β(t) =  0, coupling delay does not happen, 5

(2)

with P r{β(t) = 1} = β,

(3)

P r{β(t) = 0} = 1 − β. According to the expectation of Bernoulli function, we have: E(β(t) − β) = 0,

(4)

E(β(t) − β)2 = β(1 − β). 120

Let L = (lij )N ×N be the Laplacian matrix of α, lij = −αij and lii =

L is written as:

 L11   L21 L=  ..  .  Lq1

Assumption 1.

X

j∈γ l

L12

...

L22 .. .

... .. .

Lq2

...

L1q



PN

  L2q   ..  . .   Lqq

αij = 0, ∀i ∈ / γ l , l = 1, 2, ..., q.

j=1,j6=i

αij .

(5)

(6)

Remark 1. Assumption 1 is based on in-degree balance conception [41] which extensively exists in real complex networks such as unmanned vehicle systems and vehicle systems. According to Assumption 1, the row sum of connection matrix between two 125

different clusters is zero. That is the row sum of Llj is zero, where l, j = 1, ..., q, l 6= j. PN Due to lii = − j=1,j6=i lij , thus, the row sum of Lll equals to zero. Let sl (t), l = 1, ..., q be the desired solutions of nodes in γl respectively. sl (t)

satisfies that limt→∞ ||sl (t) − sg (t)|| = 6 0, l, g = 1, ..., q, s˙ l (t) = fl (sl (t)). Definition 1. For any initial state, if limt→∞ ||xi (t) − xj (t)|| = 0 when ¯i = ¯j, and 130

limt→∞ ||xi (t)−xj (t)|| = 6 0 when ¯i 6= ¯j, then the cluster synchronization of system (1)

is called to be realized. In other words, for any initial state, limt→∞ ||xi (t) − sl (t)|| = 0, i ∈ γl , l= 1, 2, ...,q.

From Definition 1, cluster synchronization means that the nodes in cluster γl synchronize to the desired solution sl (t) when t → ∞. However, this situation may not 6

135

happen. Thus, we impose a state feedback control to force the states to realize cluster synchronization. Next, we define synchronization error ei (t) as: ei (t) = xi (t) − sl (t),

i ∈ γl , l = 1, ..., q.

(7)

The achievement of cluster synchronization means limt→∞ ei (t) = 0. Thus, the cluster synchronization can be realized by proving the stability of error system. Theorem 1 will give the stability criterion of error system with state feedback control which 140

can guarantee the states to achieve cluster synchronization. Theorem 2 expresses the stabilization criterion of error system to force the states to realize cluster synchronization. In other words, the cluster synchronization problem is converted to stability and stabilization problem of error system. Before giving Theorem 1, we need to deduce the error system model and introduce an improved event-triggered scheme and pinning

145

control strategy to reduce channel transmission burden and save control costs. According to Assumption 1, for i ∈ γ: N X

lij xj (t) =

j=1

=

=

q X X

l=1 j∈γl q X X l=1 j∈γl q X X

lij [xj (t) − sl (t) + sl (t)] q X X lij ej (t) − ( lij )sl (t) l=1 j∈γl

lij ej (t)

l=1 j∈γl

=

N X

lij ej (t).

j=1

Then, e˙ i (t) =fl (ei (t)) − (1 − β(t)) − (β(t))

N X j=1

N X

lij Γej (t)

j=1

lij Γej (t − τ (t)) + ui (t),

(8) i ∈ Nl , l = 1, ..., q.

Next, the pinning state feedback controller is considered as follows:   Ki (ei (t)), i ∈ Nl , l = 1, ..., q, ui (t) =  0, i∈ / Nl , l = 1, ..., q, 7

(9)

4

where Nl denotes the set of selected nodes to be controlled, Nl = {zl−1 + 1, ..., zl−1 +

m} (0 < m < zl , m ∈ Z+ ). This means that we select the first m points in each cluster

150

to control.

In order to further enhance the system dynamics and decrease the burden of channel in the control loop, we introduce an improved event-triggered mechanism. Under the event-triggered mechanism, ui (t) becomes ui (t) = Ki ei (tik h), k = 1, 2, ..., where tik h, denotes the release instant of node i. Next, based on literature [30], for node i, suppose that the delay from sensor to actuator is zero, and define ρk = min{n|tik h + nh ≥ tik+1 h, n = 0, 1, 2, ...}, where h is sampling period. The release interval [tik h, tik+1 h) can be rewritten as [tik h, tik+1 h) =

ρk [

Πin ,

n=1 155

where Πin = [tik h + (n − 1)h, tik h + nh), n = 1, 2, ..., ρk − 1, Πiρk = [tik h + (ρk − 1)h, tik+1 h).

   tik h, t ∈ Πi1       ti h + h, t ∈ Πi2 k t= .. ..    . .      ti h + (ρ − 1)h, t ∈ Πi , k ρk k

   0, t ∈ Πi1       ei (ti h) − ei (ti h + h), t ∈ Πi2 k k i δk (t) = .. ..    . .      e (ti h) − e (ti h + (ρ − 1)h), t ∈ Πi . i k i k k ρk

(10)

(11)

(12)

For t ∈ [tik h, tik+1 h), the event-triggered condition is:

tik+1 h = inf {t ∈ Z+ : t > tik h, and |δki (t)|2 − σki (t)|ei (t)|2 ≥ 0}, 8

(13)

where σki (t) is a nonnegative time-varying parameter and i σki (t) = σkn ,

t ∈ Πin , n = 1, 2, ..., ρk ,

160

(14)

T

i =σ ¯+ σk(n+1)

i i δkn δkn T

i i λ + δkn δkn

i (σkn −σ ¯ ),

(15)

i δkn = ei (tik h) − ei (tik h + (n − 1)h), n = 1, 2, ..., ρk ,

(16)

i and known constant λ > 0, σ ¯ ∈ [0, 1] is the supremum of σki (t), 0 < σkn <σ ¯ , and i i σk1 is the initial value of σkn . Obviously, σki (t) is monotonically increasing.

Remark 2. When the inequality in (13) is satisfied, the sampled error state will be 165

transmitted. In the improved event-triggered mechanism, σki (t) is time-varying which can increase the release times at the beginning times to shorten the system dynamics. Then, ui (t) produced by zero-order holder can be written as: ui (t) = Ki ei (tik h) = Ki (ei (t) + δki (t)),

t ∈ [tik h, tik+1 h),

i ∈ Nl .

(17)

Thus,  PN   e˙ i (t) = fl (ei (t)) − (1 − β(t)) j=1 lij Γej (t)      PN   −(β(t)) j=1 lij Γej (t − τ (t)) + Ki (ei (t) + δki (t)), i ∈ Nl , l = 1, ..., q, PN    e˙ i (t) = fl (ei (t)) − (1 − β(t)) j=1 lij Γej (t)     PN   −(β(t)) j=1 lij Γej (t − τ (t)), i∈ / Nl , l = 1, ..., q, (18)

where fl (ei (t)) = fl (xi (t)) − fl (sl (t)). 170

Based on the above mentioned preparations and deal with model (18), then the error

system (18) can be rewritten as: e˜˙ l (t) = F (˜ el (t)) − (1 − β(t)) − β(t)

q X j=1

q X (Llj ⊗ Γ)˜ ej (t) j=1

˜ l (˜ (Llj ⊗ Γ)˜ ej (t − τ (t)) + K el (t) + δ˜kl (t)), 9

(19)

where ˜ l =diag{Kz +1 , Kz +2 , ..., Kz +m , 01×(z −m) }, K l−1 l−1 l−1 l e˜l (t) =[eTzl−1 +1 (t), eTzl−1 +2 (t), ..., eTzl (t)]T , z +1 T z +2 T z +m T δ˜kl (t) =[δkl−1 (t), δkl−1 (t), ..., δkl−1 (t), 01×(zl −m) ]T , z

σ ˜kl (t) =diag{σkl−1

+1

z

(t), σkl−1

+2

z

(t), ..., σkl−1

+m

(t), 01×(zl −m) },

F (˜ el (t)) =[flT (ezl−1 +1 (t)), flT (ezl−1 +2 (t)), ..., flT (ezl (t))]T , e˜l (t − τ (t)) =[eTzl−1 +1 (t − τ (t)), eTzl−1 +2 (t − τ (t)), ..., eTzl (t − τ (t))]T . Assumption 2. For any x, y ∈ Rn , t ≥ 0, there exist matrices Pl , ∆l such that (x − y)T Pl (fl (x) − fl (y) − ∆l (x − y)) ≤ 0

(20)

holds, where Pl = diag{p1l , p2l , ..., pnl }, ∆l = diag{δl1 , δl2 , ..., δln } are positive definite appropriate dimensional diagonal matrices.

Lemma 1. [25] For any appropriate dimensional matrices Ω1 > 0, Ω2 , Ω3 > 0, the following inequality

175

 Ω  1 ∗

Ω2 Ω3

T is equivalent to Ω1 − Ω2 Ω−1 3 Ω2 > 0.



>0

3. Main results The cluster synchronization problem will be analyzed in Theorem 1. The stability criterion of system (19) will be obtained to ensure cluster synchronization achieving. The main results will be given as follows. 180

Theorem 1. For given Bernulli probability parameters β, time-varying appropriate dimensional diagonal matrix σk (t) = diag{˜ σk1 (t), σ ˜k2 (t), ..., σ ˜kq (t)}, scalar 0 < µ < 1, ˜ 1, K ˜ 2 , ..., K ˜ q } be and 0 ≤ τ (t) ≤ τ . Let the controller gain matrix K = diag{K

given, if there exist appropriate dimensional matrices P = diag{P˜1 , P˜2 , ..., P˜q } > ˜ 1, ∆ ˜ 2 , ..., ∆ ˜ q } > 0, Q1 = diag{Q ˜ 11 , Q ˜ 12 , ..., Q ˜ 1q } > 0, Q2 = 0, ∆ = diag{∆ 10

185

˜ 21 , Q ˜ 22 , ..., Q ˜ 2q } > 0 such that the following LMI holds: diag{Q   Ξ11 Ξ12 0 PK      ∗ Ξ22 0 0   < 0, Ξ=    ∗ ∗ −Q1 0    ∗ ∗ ∗ −I

(21)

then the cluster synchronization can be guaranteed, where

˜ l =diag{∆z +1 , ∆z +2 , ..., ∆z }, ∆ l−1 l−1 l P˜l =diag{Pzl−1 +1 , Pzl−1 +2 , ..., Pzl }, ˜ 1l =diag{Q1z +1 , Q1z +2 , ..., Q1z }, Q l−1 l−1 l ˜ 2l =diag{Q2z +1 , Q2z +2 , ..., Q2z }, Q l−1 l−1 l z

σ ˜kl (t) =diag{σkl−1

+1

z

(t), σkl−1

+2

z

(t), ..., σkl−1

+m

(22) (t), 01×(zl −m) },

Ξ11 =He(P ∆ + P K) − (1 − β)P L + Q1 + Q2 + δk (t)I, Ξ12 = − βP L, Ξ22 = − (1 − µ)Q2 . Proof. Choosing the Lyapunov-Krasovkii functional as follows: V (t) = V1 (t) + V2 (t) + V3 (t),

V1 (t) =

V2 (t) =

V3 (t) = 190

(23)

Pq

˜Tl (t)P˜l e˜l (t), l=1 e Rt l=1 t−τ

Pq

˜ 1l e˜l (s)ds, e˜Tl (s)Q

Rt ˜ 2l e˜l (s)ds, ˜Tl (s)Q l=1 t−τ (t) e

Pq

(24)

˜ 1l = diag{Q1z +1 , Q1z +2 , ..., where P˜l = diag{Pzl−1 +1 , Pzl−1 +2 , ..., Pzl } > 0, Q l−1 l−1

˜ 2l = diag{Q2z +1 , Q2z +2 , ..., Q2z } > 0. Q1zl } > 0, Q l−1 l−1 l

11

Consider that the randomly occurring coupling delays obey Bernoulli distribution and by the derivative of V (t) E{V˙ (t)} = E{V˙ 1 (t) + V˙ 2 (t) + V˙ 3 (t)}.

(25)

then, E{V˙ 1 (t)} = E{2

q X

e˜Tl (t)P˜l e˜˙ l (t)},

l=1

q X
˜ 1l e˜l (t) − e˜Tl (t − τ )Q ˜ 1l e˜l (t − τ ) }, E{V˙ 2 (t)} = E{ e˜Tl (t)Q l=1

q X
˜ 2l e˜l (t) − (1 − τ˙ (t))˜ ˜ 2l e˜l (t − τ (t)) }. E{V˙ 3 (t)} = E{ e˜Tl (t)Q eTl (t − τ (t))Q l=1

195

Among them,

E{V˙ 1 (t)} =E{2

q X l=1

e˜Tl (t)P˜l [F (˜ el (t)) − (1 − β(t))

q X j=1

(Llj ⊗ Γ)˜ ej (t)

q X ˜ l (˜ − β(t) (Llj ⊗ Γ)˜ ej (t − τ (t)) + K el (t) + δ˜kl (t))]}.

(26)

j=1

Define that e(t) = [˜ eT1 (t), e˜T2 (t), ..., e˜Tq (t)]T , P = diag{P˜1 , P˜2 , ..., P˜q }, ∆ = ˜ 1, ∆ ˜ 2 , ..., ∆ ˜ q }. According to Assumption 2, we have diag{∆ E{2

q X l=1

e˜Tl (t)P˜l F (˜ el (t))} ≤ E{2

q X

˜ l e˜l (t)} e˜Tl (t)P˜l ∆

l=1

(27)

T

= E{2e (t)P ∆e(t)}. In addition, in E{V˙ 1 (t)}, E{2(1 − β(t))

q X

= E{2(1 − β)

e˜Tl (t)P˜l

l=1 q X

e˜Tl (t)P˜l

q X (Llj ⊗ Γ)˜ ej (t)} j=1 q X j=1

l=1 T

= E{2(1 − β)e (t)P Le(t)}, 12

(Llj ⊗ Γ)˜ ej (t)}

(28)

where

  L11 ⊗ Γ L12 ⊗ Γ . . . L1q ⊗ Γ     L21 ⊗ Γ L22 ⊗ Γ . . . L2q ⊗ Γ .  L= .. .. ..  ..   . . . .   Lq1 ⊗ Γ

similarly,

q X

E{2β(t)

= E{2β

e˜Tl (t)P˜l

l=1 q X l=1

Lq2 ⊗ Γ

e˜Tl (t)P˜l

q X

...

Lqq ⊗ Γ

(Llj ⊗ Γ)˜ ej (t − τ (t))}

j=1 q X j=1

(Llj ⊗ Γ)˜ ej (t − τ (t))}

(29)

= E{2βeT (t)P Le(t − τ (t))}, 200

and E{2

q X

˜ l (˜ e˜Tl (t)P˜l [K el (t) + δ˜kl (t))]} = E{2eT (t)P K[e(t) + δk (t)]},

(30)

l=1

˜ 1, K ˜ 2 , ..., K ˜ q }, δk (t) = diag{δ˜1 , δ˜2 , ..., δ˜q }. where K = diag{K k k k Thus,

E{V˙ 1 (t)} ≤E{2eT (t)P [∆e(t) − (1 − β)Le(t) − βLe(t − τ (t)) + Ke(t) + Kδk (t)]}. (31) q X
˜ 1l e˜l (t) − e˜Tl (t − τ )Q ˜ 1l e˜l (t − τ ) } E{V˙ 2 (t)} = E{ e˜Tl (t)Q l=1

(32)

= E{eT (t)Q1 e(t) − eT (t − τ )Q1 e(t − τ )},

E{V˙ 3 (t)} = E{eT (t)Q2 e(t) − (1 − τ˙ (t))eT (t)Q2 e(t)} ≤ E{eT (t)Q2 e(t) − (1 − µ)eT (t − τ (t))Q2 e(t − τ (t))}, where eT (t) = {˜ eT1 (t), e˜T2 (t), ..., e˜Tq (t)}T , ˜ 11 , Q ˜ 12 , ..., Q ˜ 1q }, Q1 = diag{Q ˜ 21 , Q ˜ 22 , ..., Q ˜ 2q }. Q2 = diag{Q 13

(33)

205

Combining (31)-(33), we obtain that E{V˙ (t)} ≤ E{V˙ (t) + σk (t)eT (t)e(t) − δk (t)T δk (t)}

(34)

≤ E{η T (t)Ξη(t)}, where σk (t) = diag{˜ σk1 (t), σ ˜k2 (t), ..., σ ˜kq (t)}, η T (t) = [eT (t),

eT (t − τ (t)),

eT (t − τ ),

δkT (t)]T ,

Ξ is defined in Theorem 1. If η T (t)Ξη(t) < 0,

(35)

then system (19) is asymptotically stable, which means the complex network in this paper can achieve cluster synchronization. The proof is complete. Theorem 2. Given the scalars 0 < µ < 1 and β, appropriate dimensional matrix 210

σk (t), the complex dynamic network system can be synchronized if there exist appropriate dimensional matrices P > 0, Q1 > 0, Q2 > 0, ∆ > 0 and controller gain K = P −1 YK , such that  0 Ξ  11   ∗ Ξ=   ∗  ∗

holds, where

Ξ12

0

Ξ22

0

∗

−Q1

∗

∗

YK



  0  <0  0   −I

(36)

0

Ξ11 =He(P ∆ + Yk ) − (1 − β)P L + Q1 + Q2 + δk (t)I, (37)

Ξ12 = − βP L, Ξ22 = − (1 − µ)Q2 , P , Q1 , Q2 , ∆, K and σk (t) are defined in Theorem 1. 215

Proof. According to Theorem 1, the synchronization of complex network system is

14

guaranteed if the following inequality holds,  Ξ Ξ12 0  11   ∗ Ξ22 0 Ξ=   ∗ ∗ −Q1  ∗ ∗ ∗

where

PK



  0   < 0,  0   −I

(38)

Ξ11 =He(P ∆ + P K) − (1 − β)P L + Q1 + Q2 + δk (t)I, Ξ12 = − βP L, Ξ22 = − (1 − µ)Q2 . Clearly, if (38) holds, then (36) holds. However, in inequality (38), there exists nonlinear term P K. We need to design the controller gain K to eliminate the term P K. Define K = P −1 YK , then P K becomes Yk . Consequently, we replace Ξ11 with 0

Ξ11 , where 0

Ξ11 = He(P ∆ + Yk ) − (1 − β)P L + Q1 + Q2 + δk (t)I. Thus, inequality (36) in Theorem 2 holding can force the complex network to synchronize. The proof is complete. In order to prove the feasibility of this new method, we will give a numerical ex220

ample in next section.

4. Numerical Example Example 1. Consider a complex network (1) with randomly occurring coupling delays which is shown in Fig. 1. Suppose  that thereare two clusters, and the inner connected L11 L21 , coupling Laplacian matrix L =  ∗ L22

15

Cluster 1 Cluster 2

2

1

4

-0.5

1 6

0.5

1

1

1

1

1

5

3 1

S1 s2

1

Figure 1: The considered complex network. where L11

L22

    −1 1 0 0 0 0.5  , L12 =  , = 0 0 0 0 0 −0.5   −1 0 1 0      1 −2 0 1  . =  0 1 −2 1   0 0 0 0

Moreover, the inner coupling matrix Γ = 0.6I. Define xi (t) = (x1i (t), x2i (t))T , i = 1, ..., 6, sl (t) = (s1l (t), s2l (t))T , l = 1, 2, ei (t) = xi (t) − sl (t), i ∈ γl . Thus, ei (t) = (ei1 (t), ei2 (t))T , i ∈ γl . Suppose the nonlinear function satisfying h iT fl (xi (t)) = (−1.2sin(x1i (t)) + 0.3tan(x2i (t)), 0.6sin(x2i (t)) − tanh(x1i (t))) . According to Assumption 2, suppose that the nonlinear term parameter ∆=diag

˜ 1, ∆ ˜ 2 }, where ∆ ˜ 1 =diag{1.5, 1.6, 2.5, 1.4}, ∆ ˜ 2 =diag{2.1, 1.4, 2.6, 1.5, 2.5, 2.0, 1.6, {∆ 225

1.5, }. Furthermore, in the complex network, set the probabilistic parameter of ran-

domly occurring coupling time delay as β = 0.5 and the event-triggered parameters 16

σk1 =0.01, σ ¯ =0.6, h = 0.16, the initial values of s1 (t), s2 (t) are selected as [−20, 5], ˜ 1, K ˜ 2 }, [15, 5] respectively. Applying the matlab toolbox, controller gain K = diag{K ˜ 1 =diag{K1 , K2 }, K ˜ 2 =diag{K3 , K4 , K5 , K6 } can be obtained as follows: where K Table 1: The controller gains K1 , K2 , K3 , K4 , K5 , K6 t K1

K2

,

K3

K4

K5

K6

230

 5.5725  0.1773  9.2112  3.6141  1.2606  1.0619  25.9347  30.4083  32.5951  31.7114  6.4164  0.6443

0

T 0.1773  6.2654 T 3.6141  2.6890  1.7675  1.4461  15.3230  6.8392  7.8930  13.8733  2.4798  2.2125

 7.7317  0.2312  4.4035  1.2438  5.5706  0.0646  2.3648  0.1610  16.9291  5.6226  8.8605  0.1613

1

T 0.2312  3.5491 T 1.2438  3.8896  0.6626  0.4345  0.1610  3.3713  29.9787  22.5106  0.1613  6.3334

... ...

...

...

...

...

...

In this note, cluster synchronization problem is converted to the stability problem of error states ei (t) i ∈ γl . An improved event-triggered pinning controller is applied on the complex network to force the trajectories of nodes in cluster γl to synchronize

to the desired solution sl (t), which means that limt→∞ ei (t) = 0. The responses of error states are shown in Fig. 2–Fig. 9. Among them, Fig. 2–Fig. 5 are the responses 235

of error states under the case that without control. From Fig. 2, Fig. 3, we can see that xi (t) can’t synchronize to s1 (t) where i ∈ γ1 . Similarly, in Fig. 4 and Fig. 5, xi (t) also can’t synchronize to s2 (t) where i ∈ γ2 . However, if we added the improved

event-triggered pinning controller, the consequences are shown in Fig. 6–Fig. 9. In Fig. 6, we obtain that x1i (t), i ∈ γ1 can synchronize to s11 (t) at 3 seconds. In Fig. 7, 240

we obtain that x2i (t), i ∈ γ1 can synchronize to s21 (t) at 2 seconds. From 8 and Fig. 9,

x1i (t), i ∈ γ2 can synchronize to s12 (t) and x2i (t), i ∈ γ2 can synchronize to s22 (t) at 8 17

seconds. Furthermore, in order to demonstrate the effectiveness of the proposed event-triggered scheme, we made further analysis and discussion. 245

Suppose h = 0.16, λ = 0.01, then σk (t) and release instants can be shown in Fig. 10. In Fig. 10, there are only 33 measurement signals which takes 17.6% of the whole measurement signals being transmitted during t ∈ [0, 30]. Correspondingly, due to each agent must satisfy the improved event-triggered scheme in the complex networks, thus, the triggering time intervals for each agent are almost the same. In addition, if set h = 0.26, λ = 0.01, consider σk (t) varying from 0.01 to 0.6. From Fig. 11 and Fig. 12, it can be seen that the improved static event-triggered release times in initial times is more than general event-triggered scheme.

0

ei1(t), i=1,2

250

e(11) e(21)

-1

-2

0

2

4

6

8 10 12 14 16 18 20 time(second)

Figure 2: Synchronization error ei1 (t), i ∈ γ1 of complex network without control.

18

e i2 (t), i=1,2

1 0 -1 -1.5 -2 -2.5 0 2

e(12) e(22) 4

6

8 10 12 14 16 18 20 time(second)

e i1 (t), i=3,4,5,6

Figure 3: Synchronization error ei2 (t), i ∈ γ1 of complex network without control.

-1

-0.5 0

0.5 1

e(31) e(41) e(51) e(61)

1.5 2 2.5 0

2

4

6

8

10

time(second)

12

14

16

18

20

Figure 4: Synchronization error ei1 (t), i ∈ γ2 of complex network without control.

19

e i2 (t), i=3,4,5,6

-0.5 0 0.5 1

e(12) e(22) e(32) e(42)

1.5 2

0

2

4

6

8

10

time(second)

12

14

16

18

20

Figure 5: Synchronization error ei2 (t), i ∈ γ2 of complex network without control.

e i1 (t), i=3,4,5,6

0.5

0 e(11) e(21) e(31) e(41)

-0.5

-1

0

2

4

6

8

10

12

14

16

18

20

time(second) Figure 6: Synchronization error ei1 (t), i ∈ γ2 of complex network with control.

20

ei1(t), i=1,2

0.2 e(11)

0.1

e(21)

0

-0.1 -0.2

0

5

10

15

time(second)

20

e i2 (t),i=1,2

Figure 7: Synchronization error ei1 (t), i ∈ γ1 of complex network with control.

0.2 e(12)

0.1

e(22)

0 -0.1 -0.2

0

2

4

6

8

10

time(second)

12

14

16

18

20

Figure 8: Synchronization error ei2 (t), i ∈ γ1 of complex network with control.

21

e i2(t), i=3,4,5,6

1 e(12) e(22) e(32) e(42)

0.5 0

-0.5 -1

0

2

4

6

8

10

time(second)

12

14

16

18

20

The release instants The variation of the σ (t) k and release interval

Figure 9: Synchronization error ei2 (t), i ∈ γ2 of complex network with control.

0.6 0.4 0.2 0 2

0

5

10

t(s) 15

20

25

30

0

5

10

t(s)15

20

25

30

1

0

Figure 10: The variation of σk (t), release instants and release intervals, h=0.16.

22

The release instants and the release interval

4 3 2 1 0

0

5

t(s) 10

15

20

Figure 11: The release instants and release interval under general event-triggered scheme, h=0.26. 5. Conclusions In the complex dynamic network, the cluster synchronization problem has been 255

investigated in this paper. A novel system model of complex network with randomly occurring coupling time delays has been constructed, which reflects the real circumstances more accurately. The dynamic behavior of nodes can switch between different situation about coupling time delays. In order to reduce the burden of information transmission and improve the efficiency of continuous-time clustering synchronization,

260

this paper has introduced an improved event-triggered mechanism which can adjust the trigger rate in different times. The improved event-triggered has been combined with pinning control strategy. The cluster synchronization problem was converted to stability problem of error system. Then, stability criterion is derived which can ensure that the trajectories of nodes synchronize to the corresponding expected value by using

265

Lyapunov-Krasovskii stability analysis method. Moreover, an effective event-triggered pinning controller is designed. Finally, a simulation example has been presented to illustrate the effectiveness of theoretical results in this paper. In the future, the research 23

The release instants and the release interval

2

1.5 1

0.5 0

0

5

t(s) 10

15

20

Figure 12: The release instants and release interval under the improved event-triggered scheme, h=0.26. topics can be analyzed the fault estimation of the complex networks. Declaration of Interest Statement : The authors declare that there is no conflict of 270

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28