research article hybrid synchronization of general t-s fuzzy complex dynamical...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 384654, 8 pageshttp://dx.doi.org/10.1155/2013/384654

Research ArticleHybrid Synchronization of General T-S Fuzzy ComplexDynamical Networks with Time-Varying Delay

Degang Xu, Weihua Gui, Panlei Zhao, and Chunhua Yang

College of Information Science and Engineering, Central South University, Changsha 410083, China

Correspondence should be addressed to Degang Xu; [email protected]

Received 5 April 2013; Accepted 28 April 2013

Academic Editor: Wenwu Yu

Copyright © 2013 Degang Xu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose and investigate a new general model of fuzzy complex network systems described by Takagi-Sugeno (T-S) fuzzy modelwith time-varying delays. Hybrid synchronization problem is discussed for this general T-S fuzzy complex dynamical networkwith nondelayed and delayed coupling between nodes. Utilizing Lyapunov-Krasovskii functional method, synchronization stabilitycriteria for the networks are established in terms of linear matrix inequalities (LMIs).These criteria reveal the relationship betweencoupling matrices with time-varying delays and synchronization stability of the dynamical network. Numerical simulation isprovided to illustrate the effectiveness and advantage of derived theoretical results.

1. Introduction

In recent years, complex dynamical networked agent systemshave attracted a great deal of attention in various engineeringfields from physics to biology, chemistry, and computerscience [1–3]. The reason can be attributed to their flexibilityand generality for representing virtually any natural andman-made systems. Such systems in the real world usually consistof a large number of highly interconnected dynamical units.Transportation networks, coupled biological and chemicalengineering systems, neural networks in human brains, andthe Internet are only a few of such examples [4].

Synchronization is one of the most significant andinteresting collective behaviors in complex networked agentsystems due to its potential applications in many fieldsincluding secure communization, parallel image processing,and information science [5–7]. On the other hand, timedelays occur commonly in complex networks because ofthe finite speed of signal transmission over the links [8].Since the time delay often causes undesirable performanceand instability of the network, various approaches to syn-chronization analysis for complex dynamical networks withtime delay have been investigated in the literature [9–12].Therefore, synchronization criteria of complex networks withdelays have become a topic of practical importance. The

stability criteria for time delay systems include two categories:delay-dependent ones and delay-independent ones. Sincedelay-dependent stability criteria include the informationon the size of delay, delay-dependent stability criteria aregenerally less conservative than delay-independent ones [10].Moreover, many real-world networks are not static but morelikely to be time-varying evolving, particularly in biologicaland physical networks. Commonly, time-varying delays aregeneral form of time delays. There are a few research works[13–15] considering the time-varying coupling for complexagent systems of dynamical networks.

Furthermore, the uncertainty or vagueness is unavoidablein real modeling problems of agent systems. Fuzzy theory asan efficient tool in approximating a complex nonlinear systemis a feasible method to take vagueness into consideration[16].There are some researchworks investigating the problemof delay-dependent robust controllers and filtering designfor a class of uncertain state-delayed Takagi-Sugeno (T-S)fuzzy systems [17–20]. Recently, the problems of stabilityanalysis, approximation, and stabilization for Takagi-Sugenofuzzy systems with time-varying state delay are investigated[21–24]. So, fuzzy complex networks have advantages overpure complex networks since they incorporate the capabilityof fuzzy reasoning in handling uncertain information. Inthis regard, the fuzzy models to describe complex dynamical

2 Mathematical Problems in Engineering

networks which are subjected to nonlinearity and have time-varying delays are introduced. The synchronization problemfor T-S fuzzy stochastic discrete-time complex networkswith mixed time-varying delays is discussed in a recentwork [25]. By employing the information of probabilitydistribution of time delays, the original system is transformedinto a T-S fuzzy model with stochastic parameter [26, 27].But how to solve the synchronization problem for generalfuzzy complex networks still remains largely unsolved andchallenging. To the best of our knowledge, time-varyingdelay [28, 29] synchronization analysis of general fuzzycomplex dynamical networks has not been reported in theliterature.

Besides, it is noticed that most of the studies on synchro-nization of dynamical networks have been performed undersome implicit assumptions that there exists the informationcommunication of nodes via the edges only at time 𝑡 orat time 𝑡 𝜏 [30]. However, in many circumstances, thissimplification does not match satisfactorily the peculiaritiesof real networks; there exists the information communicationof nodes not only at time 𝑡 but also at time 𝑡 𝜏. Morerecently, coexistence of the hybrid synchronization in chaoticsystems was investigated intensively [31]. However, our gen-eralized dynamical network model has different couplingstrengths for different connections. So it is complex andextended comparing with the simplified network modelin [32]. So our conclusion is more compact and moremeaningful for the generalized network models. To thebest of our knowledge, there are very few studies on thehybrid synchronization of general coupled complex dynam-ical networks with nondelayed and delayed coupling in theliterature.

Motivated by the previous discussions, in this paper,we attempt to introduce some more general time-varyingdynamical network models based on the T-S fuzzy modeland investigate the synchronization properties of this model.Based on the Lyapunov-Krasovskii functional method, weuse a linearized method to solve the problem of synchroniza-tion for fuzzy complex networks with time-varying couplingdelay and derive hybrid synchronization conditions for delay-dependent stabilities in terms of LMIs, the time-varyingnetwork model. A numerical example is given to demon-strate the effectiveness and the advantage of the proposedmethod.

The rest of this paper is organized as follows. In Section 2,we present some preliminaries for proving the proof andthe fuzzy time-varying coupled dynamical network model.Hybrid synchronization criterions are derived in Section 3.In Section 4, we provide a numerical simulation to verifythe correctness and effectiveness of the derived results.Conclusions are presented in Section 5.

2. Problem Formulation

Consider the following model of general continuous-timecomplex networks with time-varying coupling delays whichcan be represented by a T-S fuzzy model.

Rule 𝑙: If 𝜃1(𝑡) is 𝐹

𝑙1, 𝜃2(𝑡) is 𝐹

𝑙2,. . ., 𝜃𝑔(𝑡) is 𝐹l𝑔, then

��𝑖(𝑡) = 𝑓 (𝑥

𝑖(𝑡)) + 𝜀

𝑁

∑

𝑗=1

𝑐𝑖𝑗Γ𝑙𝑥𝑗(𝑡)

+ 𝜀

𝑁

∑

𝑗=1

𝑐𝑖𝑗Γ𝑙𝑥𝑗(𝑡 − 𝜏 (𝑡)) ,

𝑡 > 0, 𝑖 = 1, 2, . . . , 𝑁,

𝑥 (𝑡) = 𝜙 (𝑡) , 𝑡 ∈ [−ℎ, 0] ,

(1)

where 𝐹𝑙𝑗(𝑙 = 1, . . . , 𝑟; 𝑗 = 1, . . . , 𝑔) are the fuzzy sets;

𝑟 is the number of rules and 𝜃𝑗(𝑗 = 1, . . . , 𝑔) are the

premise variables; 𝑁 is the number of agent nodes, whereeach agent node is an 𝑛-dimensional dynamical system withnode dynamics �� = 𝑓(𝑥, 𝑡); 𝑥

𝑖= (𝑥

𝑖1, 𝑥𝑖2, . . . , 𝑥

𝑖𝑛)𝑇

∈

𝑅𝑛 are the state variables of node 𝑖; 𝑓 : 𝑅

𝑛→ 𝑅

𝑛 iscontinuously differentiable; Γ

𝑙and Γ𝑙are the constant inner-

coupling matrices of the nodes; the constant 𝜀 > 0 is thecoupling strength; 𝐶 = (𝑐

𝑖𝑗)𝑁×𝑁

and 𝐶 = (𝑐𝑖𝑗)𝑁×𝑁

arethe outer-coupling matrices of the network, in which 𝑐

𝑖𝑗is

defined, as follows: if there is a connection between node 𝑖and node 𝑗 (𝑗 = 𝑖), then 𝑐

𝑖𝑗= 𝑐𝑗𝑖= 1; otherwise, 𝑐

𝑖𝑗= 𝑐𝑗𝑖= 0,

𝑐𝑖𝑗is similar defined and the diagonal elements of matrices 𝐶

and 𝐶 are defined by

𝑐𝑖𝑗= −

𝑁

∑

𝑗=1,𝑖 = 𝑗

𝑐𝑖𝑗= −

𝑁

∑

𝑗=1,𝑖 = 𝑗

𝑐𝑗𝑖,

𝑐𝑖𝑗= −

𝑁

∑

𝑗=1,𝑖 = 𝑗

𝑐𝑖𝑗

= −

𝑁

∑

𝑗=1,𝑖 = 𝑗

𝑐𝑗𝑖, 𝑖 = 1, . . . , 𝑁.

(2)

Suppose that C and 𝐶 are irreducible matrices. Time-varyingdelay 𝜏(𝑡) satisfies

0 ≤ 𝜏 (𝑡) ≤ ℎ, 𝜏 (𝑡) ≤ 𝛾, (3)

in which ℎ and 𝛾 are constants. The initial function 𝜙(𝑡) is acontinuous and differentiable vector-valued function.

By using the standard fuzzy inference method, the T-Sfuzzy network (1) can be expressed by the following model:

��𝑖(𝑡) =

𝑟

∑

𝑙=1

𝜇𝑙(𝜃 (𝑡)) [

[

𝑓 (𝑥𝑖(𝑡)) + 𝜀

𝑁

∑

𝑗=1


+𝜀

𝑁

∑

𝑗=1

𝑐𝑖𝑗Γ𝑙𝑥𝑗(𝑡 − 𝜏 (𝑡))]

]

,

(4)

where 𝜃(𝑡) = [𝜃1(𝑡), 𝜃2(𝑡), . . . , 𝜃

𝑔(𝑡)], and 𝜇

𝑙(𝜃(𝑡)) = 𝜔

𝑙(𝜃(𝑡))/

∑𝑟

𝑙=1𝜔𝑙(𝜃(𝑡)), in which 𝐹

𝑙𝑗(𝜃𝑗(𝑡)) is the grade of membership

Mathematical Problems in Engineering 3

of 𝜃𝑗(𝑡) in 𝐹

𝑙𝑗. It is obvious that the fuzzy weighting functions

𝜇𝑙(𝜃(𝑡)) satisfy

𝜇𝑙(𝜃 (𝑡)) ≥ 0,

𝑟

∑

𝑙=1

𝜇𝑙(𝜃 (𝑡)) = 1. (5)

In the following, some elementary situations are introduced,which play an important role in proving the main result.

Definition 1. The dynamical networked agent system (1) issaid to achieve asymptotic synchronization if

𝑥1(𝑡) = 𝑥

2(𝑡) = ⋅ ⋅ ⋅ = 𝑥

𝑁(𝑡) = 𝑠 (𝑡) , 𝑡 → ∞, (6)

where 𝑠(𝑡) is a solution of an isolate node, satisfying 𝑠(𝑡) =

𝑓(𝑠(𝑡)).

Lemma 2 (see [7]). If 𝐶 = (𝑐𝑖𝑗)𝑁×𝑁

satisfies the aforemen-tioned defined conditions, then there exists a unitary matrix,𝜙 = (𝜙

1, . . . , 𝜙

𝑁), such that 𝐶𝑇𝜙

𝑘= 𝜆𝑘𝜙𝑘, 𝑘 = 1, 2, . . . , 𝑁,

where 𝜆𝑘, 𝑘 = 1, 2, . . . , 𝑁, are the eigenvalues of matrix C.

Lemma 3 (see [12]). Let 𝑋 and 𝑌 be arbitrary n-dimensionalreal vectors, and let 𝐾 be an n × n positive definite matrix.𝑃 ∈ 𝑅

𝑛×𝑛 is an arbitrary realmatrix.Then, the followingmatrixinequality holds:

2𝑋𝑇𝑃𝑌 ≤ 𝑋

𝑇𝑃𝐾−1𝑃𝑇𝑋 + 𝑌

𝑇𝐾𝑌. (7)

The aim of this paper is to investigate synchronizationproblem of the fuzzy complex dynamical network with time-varying delay (4).

3. Main Results

In this section, we focus on investigating the hybrid synchro-nization problem of fuzzy complex dynamical networks withnondelayed and delayed coupling. Before deriving our mainresults, the following lemma will be utilized.

Lemma 4. Consider the T-S fuzzy dynamical network (4). Let0 = 𝜆

1> 𝜆2≥ ⋅ ⋅ ⋅ ≥ 𝜆

𝑁and 0 = ��

1> ��2≥ ⋅ ⋅ ⋅ ≥ ��

𝑁,

respectively, be the eigenvalues of outer coupling matrix C and𝐶. If the following 𝑁 − 1 time-varying delayed differentialequations are asymptotically stable about their zero solution:

��𝑘(𝑡) =

𝑟

∑

𝑙=1

𝜇𝑙(𝜃 (𝑡)) [ (𝐽 (𝑡) + 𝜀𝜆

𝑘Γ𝑙) 𝑤𝑘(𝑡)

+𝜀��𝑘Γ𝑙𝑤𝑘(𝑡 − 𝜏 (𝑡))] ,

𝑘 = 2, 3, . . . , 𝑁,

(8)

where 𝐽(𝑡) is the Jacobin of 𝑓(𝑥(𝑡)) at 𝑠(𝑡), then synchronizedstates in fuzzy complex networks (1) are asymptotically stable.

Proof. For the synchronized states of complex networks (4),we have

𝑥𝑖(𝑡) = 𝑠 (𝑡) + 𝑒

𝑖(𝑡) , 𝑖 = 1, 2, . . . , 𝑁. (9)

Substituting (9) into (4), we obtain

𝑒𝑖(𝑡) =

𝑟

∑

𝑙=1

𝜇𝑙(𝜃 (𝑡)) [

[

𝑓 (𝑠 (𝑡) + 𝑒𝑖(𝑡)) − 𝑓 (𝑠 (𝑡))

+ 𝜀

𝑁

∑

𝑗=1

𝑐𝑖𝑗Γ𝑙𝑒𝑗(𝑡)

+𝜀

𝑁

∑

𝑗=1

𝑐𝑖𝑗Γ𝑙𝑒𝑗(𝑡 − 𝜏 (𝑡))]

]

.

(10)

Considering that 𝑓(𝑠(𝑡)) is continuous differentiable, it iseasy to know that the origin of the complex networks (8) isan asymptotically stable equilibrium point for the followinglinear time delay systems:

𝑒𝑖(𝑡) =

𝑟

∑

𝑙=1

𝜇𝑙(𝜃 (𝑡)) [

[

𝐽 (𝑡) 𝑒𝑖(𝑡) + 𝜀

𝑁

∑

𝑗=1

𝑐𝑖𝑗Γ𝑙𝑒𝑗(𝑡)

+𝜀

𝑁

∑

𝑗=1

𝑐𝑖𝑗Γ𝑙𝑒𝑗(𝑡 − 𝜏 (𝑡))]

]

=

𝑟

∑

𝑙=1

𝜇𝑙(𝜃 (𝑡))

× [𝐽 (𝑡) 𝑒𝑖(𝑡)+𝜀Γ

𝑙(𝑒1(𝑡) , . . . , 𝑒

𝑁(𝑡) (𝑐𝑖1, . . . , 𝑐

𝑖𝑁)𝑇

+ 𝜀Γ𝑙[𝑒1(𝑡 − 𝜏 (𝑡)) , . . . , 𝑒

𝑁(𝑡 − 𝜏 (𝑡) ]

× (𝑐𝑖1, . . . , 𝑐

𝑖𝑁)𝑇

] .

(11)

Let 𝑒(𝑡) = (𝑒1(𝑡), . . . , 𝑒

𝑁(𝑡)) ∈ 𝑅

𝑁×𝑁; we can obtain

𝑒𝑖(𝑡)

=

𝑟

∑

𝑙=1

𝜇𝑙(𝜃 (𝑡)) [𝐽 (𝑡) 𝑒

𝑖(𝑡) + 𝜀Γ

𝑙𝑒 (𝑡) 𝐶

𝑇+ 𝜀Γ𝑙𝑒 (𝑡 − 𝜏 (𝑡)) 𝐶

𝑇] .

(12)

According to Lemma 2, there exist nonsingular matrices,Φ1= (𝜙11, . . . , 𝜙

1𝑁), Φ2= (𝜙21, . . . , 𝜙

2𝑁), such that 𝐶𝑇Φ

1=

Φ1Λ1and 𝐶𝑇Φ

2= Φ2Λ2, with Λ

1= diag(𝜆

1, . . . , 𝜆

𝑁), Λ2=

diag(��1, . . . , ��

𝑁). Using the nonsingular transform 𝑒(𝑡)Φ =

𝑤(𝑡), 𝑒(𝑡 − 𝜏)Φ = 𝑤(𝑡 − 𝜏), then we obtain

��𝑘(𝑡) =

𝑟

∑

𝑙=1

𝜇𝑙(𝜃 (𝑡)) [ (𝐽 (𝑡) + 𝜀𝜆


+𝜀��𝑘Γ𝑙𝑤𝑘(𝑡 − 𝜏 (𝑡))] ,

𝑘 = 1, . . . , 𝑁.

(13)

Note that 𝜆1= 0, ��

1= 0 correspond to the synchroniza-

tion of the network states (4), where the state 𝑠(𝑡) is an orbital


stable solution of the isolate node as assumed before in (4). Ifthe following 𝑁 − 1 pieces of 𝑛-dimensional linear multipletime delay differential equations

��𝑘(𝑡) =

𝑟

∑

𝑙=1

𝜇𝑙(𝜃 (𝑡)) [ (𝐽 (𝑡) + 𝜀𝜆


+𝜀��𝑘Γ𝑙𝑤𝑘(𝑡 − 𝜏 (𝑡))] ,

𝑘 = 1, . . . , 𝑁

(14)

are asymptotically stable, then 𝑒(𝑡) will tend to the originasymptotically, which implies that synchronized states ofcomplex networks (8) are asymptotically stable.

The proof is thus completed.

Remark 5. We use Lemma 4 to linearize the complicatedsystem to deal with the complex networks by using fuzzytheory. It is an available way to investigate the interconnecteddynamical agents of complex networks. Then some newsubsystems can be obtained, which are easy to be analyzed.

Now, the main result is stated in the following theorem.

Theorem 6. If there exist positive definite symmetric matrices𝑃𝑘= 𝑃𝑇

𝑘> 0, 𝑄

𝑘= 𝑄𝑇

𝑘> 0, 𝑅

𝑘= 𝑅𝑇

𝑘> 0, and 𝑍

𝑘= 𝑍𝑇

𝑘> 0,

such that the following LMI holds for all 𝑙 = 1, 2, . . . , 𝑟:

[[[[[

[

Θ𝑘

ℎΩ1𝑘

ℎΩ2𝑘

ℎΩ3𝑘

ℎΠ𝑇

𝑘𝑍𝑘

∗ −ℎ𝑍𝑘

0 0 0

∗ ∗ −ℎ𝑍𝑘

0 0

∗ ∗ ∗ −ℎ𝑍𝑘

0

∗ ∗ ∗ ∗ −ℎ𝑍𝑘

]]]]]

]

< 0, (15)

where

Ω1𝑘

=

[[[

[

𝑁1

1𝑘

𝑁2

1𝑘

𝑁2

1𝑘

]]]

]

, Ω2𝑘

=

[[[

[

𝑁1

2𝑘

𝑁2

2𝑘

𝑁2

2𝑘

]]]

]

, Ω3𝑘

=

[[[

[

𝑁1

3𝑘

𝑁2

3𝑘

𝑁2

3𝑘

]]]

]

,

for 𝑘 = 2, 3, . . . , 𝑁,

Θ𝑘= Θ𝑘1+ Θ𝑘2+ Θ𝑇

𝑘2,

Θ𝑘1

=[

[

𝑃𝑘(𝐽 (𝑡) + 𝜀𝜆

𝑘Γ𝑙) + (𝐽 (𝑡) + 𝜀𝜆

𝑘Γ𝑙)𝑇

𝑃𝑘

𝜀��𝑘Γ𝑙𝑃𝑘

0

∗ − (1 − 𝛾) 0

∗ ∗ −𝑅𝑘

]

]

,

Θ𝑘2

= [Ω𝑘1 + Ω𝑘3

−Ω𝑘1+ Ω𝑘2

−Ω𝑘2− Ω𝑘3] ,

Π𝑘= [𝐽 (𝑡) + 𝜀𝜆

𝑘Γ𝑙𝜀��𝑘Γ𝑙

0] ,

(16)

where ∗ denotes the symmetric terms in a symmetric matrix,then the asymptotic synchronization of complex network sys-tem (8) can be achieved.

Proof. Construct a Lyapunov-Krasovskii function as

𝑉𝑘(𝑡) = 𝑤

𝑇

𝑘(𝑡) 𝑃𝑘𝑤𝑘(𝑡) + ∫

𝑡

𝑡−𝜏(𝑡)

𝑤𝑇

𝑘(𝑠) 𝑄𝑘𝑤𝑘(𝑠) 𝑑𝑠

+ ∫

𝑡

𝑡−ℎ

𝑤𝑇

𝑘(𝑠) 𝑅𝑘𝑤𝑘(𝑠) 𝑑𝑠

+ ∫

0

−ℎ

∫

𝑡

𝑡+𝜃

��𝑇

𝑘(𝑠) 𝑍𝑘��𝑘(𝑠) 𝑑𝑠 𝑑𝜃.

(17)

According to Lemma 4, calculating the time derivative of𝑉𝑘(𝑡) along the trajectories of complex networks (8), we

obtain

��𝑘(𝑡) = 2𝑤

𝑇

𝑘(𝑡) 𝑃𝑘��𝑘(𝑡) + 𝑤

𝑇

𝑘(𝑡) 𝑄𝑘𝑤𝑘(𝑡)

− (1 − 𝜏 (𝑡)) 𝑤𝑇

𝑘(𝑡 − 𝜏 (𝑡)) 𝑄

𝑘𝑤𝑘(𝑡 − 𝜏 (𝑡))

+ 𝑤𝑇

𝑘(𝑡) 𝑅𝑘𝑤𝑘(𝑡) − 𝑤

𝑇

𝑘(𝑡 − ℎ) 𝑅

𝑘��𝑘(𝑡 − ℎ)

+ ℎ��𝑇

𝑘(𝑠) 𝑍𝑘��𝑘(𝑠) − ∫

𝑡

𝑡−ℎ

��𝑇

𝑘(𝑠) 𝑍𝑘��𝑘(𝑠)

≤ 2𝑤𝑇


𝑇

𝑘(𝑡) 𝑄𝑘𝑤𝑘(𝑡)

− (1 − 𝛾)𝑤𝑇

𝑘(𝑡 − 𝜏 (𝑡)) 𝑄


+ 𝑤𝑇

𝑘(𝑡) 𝑅𝑘𝑤𝑘(𝑡) − 𝑤

𝑇


𝑘��𝑘(𝑡 − ℎ)

+ ℎ��𝑇

𝑘(𝑠) 𝑍𝑘��𝑘(𝑠) − ∫

𝑡

𝑡−ℎ

��𝑇

𝑘(𝑠) 𝑍𝑘��𝑘(𝑠) .

(18)

Based on the Newton-Leibniz formula, the followingequations can be obtained for arbitrary matrices Ξ

𝑘with

appropriate dimensions:

2𝜉𝑇

𝑘(𝑡) Ξ𝑘[𝑤𝑘(𝑡) − 𝑤

𝑘(𝑡 − 𝜏) − ∫

𝑡

𝑡−𝜏

𝑤𝑘(𝑠) 𝑑𝑠] = 0, (19)

where 𝜍𝑇𝑘(𝑡) = (𝑤

𝑇

𝑘(𝑡), 𝑤𝑇

𝑘(𝑡 − 𝜏)) and Ξ

𝑇

𝑘= (𝑋𝑇

1, 𝑋𝑇

2).

Therefore, using Lemma 3, we have

��𝑘(𝑡) = 2𝑤

𝑇


𝑇

𝑘(𝑡) (𝑄

𝑘+ 𝑅𝑘) 𝑤𝑘(𝑡)

− (1 − 𝛾)𝑤𝑇

𝑘(𝑡 − 𝜏 (𝑡)) 𝑄


− 𝑤𝑇


𝑘��𝑘(𝑡 − ℎ) + ℎ��

𝑇

𝑘(𝑡) 𝑍𝑘1��𝑘(𝑡)

− ∫

𝑡

𝑡−𝜏(𝑡)

��𝑇

𝑘(𝑠) 𝑍𝑘1(𝑠) 𝑑𝑠

− ∫

𝑡−𝜏(𝑡)

𝑡−ℎ

��𝑇

𝑘(𝑠) 𝑍𝑘1��𝑘(𝑠) 𝑑𝑠

− ∫

𝑡

𝑡−ℎ

��𝑇

𝑘(𝑠) 𝑍𝑘1��𝑘(𝑠) 𝑑𝑠

+ 2Δ𝑇

𝑘(𝑡) Ω1𝑘[𝑤𝑘(𝑡) − 𝑤

𝑘(𝑡 − 𝜏 (𝑡))

−∫

𝑡

𝑡−𝜏(𝑡)

��𝑇

𝑘(𝑠) 𝑑𝑠]


+ 2Δ𝑇

𝑘(𝑡) Ω2𝑘[𝑤𝑘(𝑡 − 𝜏 (𝑡)) − 𝑤

𝑘(𝑡 − ℎ)

− ∫

𝑡−𝜏(𝑡)

𝑡−ℎ

��𝑇


+ 2Δ𝑇

𝑘(𝑡) Ω3𝑘[𝑤𝑘(𝑡) − 𝑤

𝑘(𝑡 − ℎ)

−∫

𝑡

𝑡−ℎ

��𝑇


<

𝑟

∑

𝑖=1

𝜇𝑙(𝜃 (𝑡)) {Φ

𝑇

𝑘(𝑡) [Θ

𝑘+ ℎΓ𝑇

𝑘𝑍𝑘Γ𝑘

+ ℎΩ1𝑘𝑍−1

𝑘Ω𝑇

1𝑘


𝑘Ω𝑇

2𝑘

+ℎΩ3𝑘𝑍−1

𝑘Ω𝑇

3𝑘]Φ𝑘(𝑡)

− ∫

𝑡

𝑡−𝜏(𝑡)

[Φ𝑇

𝑘(𝑡) Ω1𝑘+ ��𝑇

𝑘(𝑠) 𝑍𝑘] 𝑍−1

𝑘

× [Ω𝑇

1𝑘Φ𝑘(𝑡) + 𝑍

𝑘��𝑘(𝑠)] 𝑑𝑠

− ∫

𝑡−𝜏(𝑡)

𝑡−ℎ

[Φ𝑇

𝑘(𝑡) Ω2𝑘+ ��𝑇


𝑘

× [Ω𝑇


𝑘��𝑘(𝑠)] 𝑑𝑠

− ∫

𝑡

𝑡−ℎ

[Φ𝑇

𝑘(𝑡) Ω3𝑘+ ��𝑇


𝑘

× [Ω𝑇


𝑘��𝑘(𝑠)] 𝑑𝑠} ,

(20)

where Δ𝑇𝑘(𝑡) = [𝑤

𝑇

𝑘(𝑡), 𝑤𝑇

𝑘(𝑡 −𝜏), 𝑤

𝑇

𝑘(𝑡 −ℎ)]

𝑇. Because the lastthree terms are all less than zero, if

Θ𝑘+ ℎΓ𝑇

𝑘𝑍𝑘Γ𝑘+ ℎΩ1𝑘𝑍−1

𝑘Ω𝑇

1𝑘+ ℎΩ2𝑘𝑍−1

𝑘Ω𝑇

2𝑘


𝑘Ω𝑇

3𝑘< 0,

(21)

then ��𝑘(𝑡) < −𝜅‖𝑤(𝑡)‖

2 for sufficiently small 𝜅 > 0. It iseasy to see that system (8) is globally synchronized. By Schurcomplements, we know that the function (21) is equivalent tothe function (15).

The proof is thus completed.

Remark 7. For the delay-dependent synchronization prob-lems of complex networks, the Lyapunov-Krasovskii condi-tion has been attracted owing to the structural advantage.The key point is the introduction of the integral inequalitytechnique, a fundamental trick to derive delay-dependentstability criteria containing the size or the bounds of delaysand their derivatives. However, almost results in this fieldhave employed partial information on the relationship amongdelay-related terms. Being differently from the commonconstruction, we use not only the time-varying-delayed

0 1 2 3 4 5 6−2

−1.5

−1

−0.5

00.51

1.52

Time

Error𝑒

𝑖1

Figure 1: Synchronization error 𝑒𝑖1for the coupling delayed net-

works with ℎ = 𝛾 = 0.5.

0 1 2 3

3

4 5 6−1.5

−1

−0.5

00.51

1.5

2.52

Time

Error𝑒

𝑖2



state but also the delay-upper-bounded state to exploitall possible information when constructing the Lyapunov-Krasovskii functional. Our approach in Theorem 6 reducesthe conservatism of the existing methods to a certain extent.

The aforementioned results can be extended to the case ofbound constant time delay. If the time-varying delay 𝜏(𝑡) is aconstant time delay in system (1), then fuzzy system (4) canbe expressed in the following model:

��𝑖(𝑡) =

𝑟

∑

𝑙=1

𝜇𝑙(𝜃 (𝑡)) [

[

𝑓 (𝑥𝑖(𝑡)) + 𝜀

𝑁

∑

𝑗=1


+𝜀

𝑁

∑

𝑗=1

𝑐𝑖𝑗Γ𝑙𝑥𝑗(𝑡 − 𝜏)]

]

.

(22)

We obtain the following corollary.

Corollary 8. If there exist positive definite symmetric matrices𝑃𝑘, 𝑄𝑘, and 𝑅

𝑘(2 ≤ 𝑘 ≤ 𝑁) and two arbitrary matrices 𝑁

1,


Table 1: Upper bounds of ℎ for different 𝜀 and 𝛾.

Coupling strength 𝜀 0.3 0.4 0.5 0.6𝛾 = 0 1.832 1.136 0.813 0.637𝛾 = 0.2 1.461 1.053 0.716 0.562𝛾 = 0.5 1.263 0.851 0.641 0.514𝛾 = 0.9 0.953 0.710 0.560 0.473

Error𝑒

𝑖3

0 10 20 30 40 50−1.5

−1

−0.5

0

0.5

1

1.5

2

Time



𝑁2with appropriate dimensions, such that the following LMIs

hold for all 𝑙 = 1, 2, . . . , 𝑟:

Ω𝑘=

[[[

[

Θ ��𝑘𝜀𝑃𝑘Γ𝑙− 𝑁1+ 𝑁𝑇

2𝜏𝐽𝑇(𝑡) 𝑅𝑘

𝜏𝑁1

∗ −𝑄𝑘− 𝑁2− 𝑁𝑇

2𝜏𝑐��𝑘Γ𝑇

𝑙𝑅𝑘

𝜏𝑁2

∗ ∗ −𝜏𝑅𝑘

0

∗ ∗ ∗ −𝜏𝑅𝑘

]]]

]

< 0,

(23)

where

Θ = 𝑃𝑘(𝐽 (𝑡) + 𝜀𝜆

𝑘Γ𝑙) + (𝐽 (𝑡) + 𝜀𝜆

𝑘Γ𝑙)𝑇

𝑃𝑘+ 𝑄𝑘+ 𝑁𝑇

1+ 𝑁1,

(24)

then the asymptotic synchronization of system (22) inDefinition 1 can be achieved.

Proof. Select a Lyapunov-Krasovskii function as

𝑉𝑘(𝑡) = 𝑤

𝑇

𝑘(𝑡) 𝑃𝑘𝑤𝑘(𝑡) + ∫

𝑡

𝑡−𝜏

𝑤𝑇

𝑘(𝑠) 𝑄𝑘𝑤𝑘(𝑠) 𝑑𝑠

+ ∫

𝑡

𝑡−𝜏

∫

𝑡

𝜃

��𝑇

𝑘(𝑠) 𝑅𝑘��𝑘(𝑠) 𝑑𝑠 𝑑𝜃.

(25)

The rest of the proof is similar to that of Theorem 6; thus onecan easily obtain the result. Thus it is omitted here. The proofis completed.

Remark 9. Corollary 8 presents new hybrid synchronizationconditions for fuzzy complex networks with constant delays.

We consider the delay which involve general form of delays.Based on Lyapunov-Krasovskii function method, stabilitycriteria are obtained in the form of LMIs which can be easilysolved. And the results are usually less conservative thandelay-independent ones especially when the size of the timedelay is small.

4. Numerical Example

To illustrate the previous results we obtain, we consider thefollowing complex dynamical networks with time-varyingdelays consisting of three nodes:

[

[

��1(𝑡)

��2(𝑡)

��3(𝑡)

]

]

= [

[

−𝑥1

−2𝑥2

−3𝑥3

]

]

(26)

which is asymptotically stable at 𝑠(𝑡) = 0, with Jacobin givenby 𝐽 = diag{−1, −2, −3}. Assume that inner-couplingmatricesare Γ = Γ = diag{1, 1, 1}, and the coupling configurationmatrices are

𝐶 = 𝐶 = [

[

−1 0.5 0.5

0.5 −1 0.5

0.5 0.5 −1

]

]

,

𝜇1(𝜃 (𝑡)) = sin2 (𝑥

1(𝑡)) ,

𝜇2(𝜃 (𝑡)) = cos2 (𝑥

1(𝑡)) .

(27)

The eigenvalues of 𝐶 and 𝐶 are 𝜆(𝐶) = ��(𝐶) = {0, −1.5,

−1.5}.Using MATLAB LMI toolbox, the upper bounds on the

time delay for different values of the coupling strength 𝜀 canbe obtained fromTheorem 6. A detailed comparison is givenin Table 1, where the achieved upper bounds of time delayin the previous system are listed for their respective lowerbounds.

Assume that the coupling strength is 𝜀 = 0.4; it is foundthat the maximum delay bound is ℎ = 0.73 in [27], for whichthe synchronized states of the network are asymptoticallystable. By Theorem 6 in this paper, however, it is found thatthe maximum delay bound for the synchronized states to beasymptotically stable is ℎ = 0.851. It can be seen that themethod proposed in this paper is better.

For example, when the delay is 𝛾 = 0.5, by employing theLMI Toolbox in Matlab, we simulate system (26) for givenchosen initial parameter 𝜀 = 0.3. We define the errors ofsynchronization as follows:

𝑒𝑖𝑗(𝑡) ≜ 𝑥

𝑖𝑗(𝑡) − 𝑥

(𝑖+1)𝑗(𝑡) , 𝑖 = 1, 2, 𝑗 = 1, 2, 3. (28)


For random initial conditions, Figures 1, 2, and 3 show thesynchronization errors between the states of node 𝑖 and node𝑖 + 1 with ℎ = 0.5. We find that the synchronization errorsconverge to zero under the previous conditions.

5. Conclusions

In this paper, we have proposed a class of general complexdynamical network models based on T-S fuzzy theory andinvestigated hybrid synchronization of the proposed complexdynamical networks. Time-varying delays in network cou-plings of dynamical nodes have been considered. New delay-dependent synchronization criteria in terms of LMIs havebeen derived based on an appropriate Lyapunov functional.Synchronization criteria are obtained in this paper whichcan be applicable to networks with different topologies anddifferent sizes. Numerical simulation is also provided toillustrate the usefulness and advantage of the synchronizationcriteria.

Acknowledgments

This research was partially supported by the National NaturalScience Foundation of PR China (Grant nos. 61104135,61025015, 61174132, and 61134006), China Postdoctoral Sci-ence Fund (Grant no. 2012T50705), and Postdoctoral Fundof the Central South University. The authors also gratefullythank the Editor and the reviewers for their helpful commentsand suggestions, which have improved the quality of thispaper.

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