Hybrid Synchronization of General T-S Fuzzy Complex Dynamical Networks with Time-Varying Delay

Abstract

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

1. Introduction

In recent years, complex dynamical networked agent systems have attracted a great deal of attention in various engineering fields from physics to biology, chemistry, and computer science [1–3]. The reason can be attributed to their flexibility and generality for representing virtually any natural and man-made systems. Such systems in the real world usually consist of a large number of highly interconnected dynamical units. Transportation networks, coupled biological and chemical engineering systems, neural networks in human brains, and the Internet are only a few of such examples [4].

Synchronization is one of the most significant and interesting collective behaviors in complex networked agent systems due to its potential applications in many fields including secure communization, parallel image processing, and information science [5–7]. On the other hand, time delays occur commonly in complex networks because of the finite speed of signal transmission over the links [8]. Since the time delay often causes undesirable performance and instability of the network, various approaches to synchronization analysis for complex dynamical networks with time delay have been investigated in the literature [9–12]. Therefore, synchronization criteria of complex networks with delays have become a topic of practical importance. The stability criteria for time delay systems include two categories: delay-dependent ones and delay-independent ones. Since delay-dependent stability criteria include the information on the size of delay, delay-dependent stability criteria are generally less conservative than delay-independent ones [10]. Moreover, many real-world networks are not static but more likely to be time-varying evolving, particularly in biological and physical networks. Commonly, time-varying delays are general form of time delays. There are a few research works [13–15] considering the time-varying coupling for complex agent systems of dynamical networks.

Furthermore, the uncertainty or vagueness is unavoidable in real modeling problems of agent systems. Fuzzy theory as an efficient tool in approximating a complex nonlinear system is a feasible method to take vagueness into consideration [16]. There are some research works investigating the problem of delay-dependent robust controllers and filtering design for a class of uncertain state-delayed Takagi-Sugeno (T-S) fuzzy systems [17–20]. Recently, the problems of stability analysis, approximation, and stabilization for Takagi-Sugeno fuzzy systems with time-varying state delay are investigated [21–24]. So, fuzzy complex networks have advantages over pure complex networks since they incorporate the capability of fuzzy reasoning in handling uncertain information. In this regard, the fuzzy models to describe complex dynamical networks which are subjected to nonlinearity and have time-varying delays are introduced. The synchronization problem for T-S fuzzy stochastic discrete-time complex networks with mixed time-varying delays is discussed in a recent work [25]. By employing the information of probability distribution of time delays, the original system is transformed into a T-S fuzzy model with stochastic parameter [26, 27]. But how to solve the synchronization problem for general fuzzy complex networks still remains largely unsolved and challenging. To the best of our knowledge, time-varying delay [28, 29] synchronization analysis of general fuzzy complex dynamical networks has not been reported in the literature.

Besides, it is noticed that most of the studies on synchronization of dynamical networks have been performed under some implicit assumptions that there exists the information communication of nodes via the edges only at time 𝑡 or at time 𝑡_𝜏 [30]. However, in many circumstances, this simplification does not match satisfactorily the peculiarities of real networks; there exists the information communication of nodes not only at time 𝑡 but also at time 𝑡_𝜏. More recently, coexistence of the hybrid synchronization in chaotic systems was investigated intensively [31]. However, our generalized dynamical network model has different coupling strengths for different connections. So it is complex and extended comparing with the simplified network model in [32]. So our conclusion is more compact and more meaningful for the generalized network models. To the best of our knowledge, there are very few studies on the hybrid synchronization of general coupled complex dynamical networks with nondelayed and delayed coupling in the literature.

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

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

2. Problem Formulation

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

Rule 𝑙: If 𝜃₁(𝑡) is 𝐹_𝑙1, 𝜃₂(𝑡) is 𝐹_𝑙2,…, 𝜃_𝑔(𝑡) is 𝐹_l𝑔, then 𝑥̇_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+𝜀^𝑁_𝑗=1𝑐_𝑖𝑗Γ_𝑙𝑥_𝑗(𝑡)+𝜀^𝑁_𝑗=1𝑐̃_𝑖𝑗Γ_𝑙𝑥_𝑗((𝑡,𝑥(𝑡)(𝑡)[],𝑡−𝜏))𝑡>0,𝑖=1,2,…,𝑁,=𝜙,𝑡∈−ℎ,0 (1) where 𝐹_𝑙𝑗(𝑙=1,…,𝑟;𝑗=1,…,𝑔) are the fuzzy sets; 𝑟 is the number of rules and 𝜃_𝑗(𝑗=1,…,𝑔) are the premise variables; 𝑁 is the number of agent nodes, where each agent node is an 𝑛-dimensional dynamical system with node dynamics 𝑥̇=𝑓(𝑥,𝑡); 𝑥_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛)^𝑇∈𝑅^𝑛 are the state variables of node 𝑖; 𝑓∶𝑅^𝑛→𝑅^𝑛 is continuously differentiable; Γ_𝑙 and Γ_𝑙 are the constant inner-coupling matrices of the nodes; the constant 𝜀>0 is the coupling strength; 𝐶=(𝑐_𝑖𝑗)_𝑁×𝑁 and 𝐶=(𝑐̃_𝑖𝑗)_𝑁×𝑁 are the outer-coupling matrices of the network, in which 𝑐_𝑖𝑗 is defined, as follows: if there is a connection between node 𝑖 and node 𝑗 (𝑗≠𝑖), then 𝑐_𝑖𝑗=𝑐_𝑗𝑖=1; otherwise, 𝑐_𝑖𝑗=𝑐_𝑗𝑖=0, 𝑐̃_𝑖𝑗 is similar defined and the diagonal elements of matrices𝐶 and 𝐶 are defined by 𝑐_𝑖𝑗=−^𝑁_{𝑗=1,𝑖≠𝑗}𝑐_𝑖𝑗=−^𝑁_{𝑗=1,𝑖≠𝑗}𝑐_𝑗𝑖,𝑐̃_𝑖𝑗=−^𝑁_{𝑗=1,𝑖≠𝑗}𝑐̃_𝑖𝑗=−^𝑁_{𝑗=1,𝑖≠𝑗}𝑐̃_𝑗𝑖,𝑖=1,…,𝑁. (2) Suppose that C and 𝐶 are irreducible matrices. Time-varying delay 𝜏(𝑡) satisfies 0≤𝜏(𝑡)≤ℎ,𝜏̇(𝑡)≤𝛾, (3) in which ℎ and 𝛾 are constants. The initial function 𝜙(𝑡) is a continuous and differentiable vector-valued function.

By using the standard fuzzy inference method, the T-S fuzzy network (1) can be expressed by the following model: 𝑥̇_𝑖(𝑡)=^𝑟_𝑙=1𝜇_𝑙(𝜃(𝑡))𝑓𝑥_𝑖(𝑡)+𝜀^𝑁_𝑗=1𝑐_𝑖𝑗Γ_𝑙𝑥_𝑗(𝑡)+𝜀^𝑁_𝑗=1𝑐̃_𝑖𝑗Γ_𝑙𝑥_𝑗(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=1𝜔_𝑙(𝜃(𝑡)), in which 𝐹_𝑙𝑗(𝜃_𝑗(𝑡)) is the grade of membership of 𝜃_𝑗(𝑡) in 𝐹_𝑙𝑗. It is obvious that the fuzzy weighting functions 𝜇_𝑙(𝜃(𝑡)) satisfy 𝜇_𝑙(𝜃(𝑡))≥0,^𝑟_𝑙=1𝜇_𝑙(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) is said to achieve asymptotic synchronization if 𝑥₁(𝑡)=𝑥₂(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=(𝑐_𝑖𝑗)_𝑁×𝑁 satisfies the aforementioned defined conditions, then there exists a unitary matrix, 𝜙=(𝜙₁,…,𝜙_𝑁), such that 𝐶^𝑇𝜙_𝑘=𝜆_𝑘𝜙_𝑘,𝑘=1,2,…,𝑁, where 𝜆_𝑘,𝑘=1,2,…,𝑁, are the eigenvalues of matrix 𝐶.

Lemma 3 (see [12]). Let 𝑋 and 𝑌 be arbitrary 𝑛-dimensional real vectors, and let 𝐾 be an 𝑛×𝑛 positive definite matrix. 𝑃∈𝑅^𝑛×𝑛 is an arbitrary real matrix. Then, the following matrix inequality holds: 2𝑋^𝑇𝑃𝑌≤𝑋^𝑇𝑃𝐾⁻¹𝑃^𝑇𝑋+𝑌^𝑇𝐾𝑌. (7)

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

3. Main Results

In this section, we focus on investigating the hybrid synchronization problem of fuzzy complex dynamical networks with nondelayed and delayed coupling. Before deriving our main results, the following lemma will be utilized.

Lemma 4. Consider the T-S fuzzy dynamical network (4). Let 0=𝜆₁>𝜆₂≥⋯≥𝜆_𝑁 and 0=𝜆₁>𝜆₂≥⋯≥𝜆_𝑁, respectively, be the eigenvalues of outer coupling matrix C and 𝐶. If the following 𝑁−1 time-varying delayed differential equations are asymptotically stable about their zero solution: 𝑤̇_𝑘(𝑡)=^𝑟_𝑙=1𝜇_𝑙(𝜃(𝑡))𝐽(𝑡)+𝜀𝜆_𝑘Γ_𝑙𝑤_𝑘(𝑡)+𝜀𝜆_𝑘Γ_𝑙𝑤_𝑘(𝑡−𝜏(𝑡)),𝑘=2,3,…,𝑁, (8) where 𝐽(𝑡) is the Jacobin of 𝑓(𝑥(𝑡)) at 𝑠(𝑡), then synchronized states in fuzzy complex networks (1) are asymptotically stable.

Proof. For the synchronized states of complex networks (4), we have 𝑥_𝑖(𝑡)=𝑠(𝑡)+𝑒_𝑖(𝑡),𝑖=1,2,…,𝑁. (9) Substituting (9) into (4), we obtain 𝑒_𝑖̇(𝑡)=^𝑟_𝑙=1𝜇_𝑙(𝜃(𝑡))𝑓𝑠(𝑡)+𝑒_𝑖(𝑡)−𝑓(𝑠(𝑡))+𝜀^𝑁_𝑗=1𝑐_𝑖𝑗Γ_𝑙𝑒_𝑗(𝑡)+𝜀^𝑁_𝑗=1𝑐̃_𝑖𝑗Γ_𝑙𝑒_𝑗(𝑡−𝜏(𝑡)). (10)
Considering that 𝑓(𝑠(𝑡)) is continuous differentiable, it is easy to know that the origin of the complex networks (8) is an asymptotically stable equilibrium point for the following linear time delay systems: 𝑒_𝑖̇(𝑡)=^𝑟_𝑙=1𝜇_𝑙(𝜃(𝑡))𝐽(𝑡)𝑒_𝑖(𝑡)+𝜀^𝑁_𝑗=1𝑐_𝑖𝑗Γ_𝑙𝑒_𝑗(𝑡)+𝜀^𝑁_𝑗=1𝑐̃_𝑖𝑗Γ_𝑙𝑒_𝑗((𝑡=𝑡−𝜏))_𝑟𝑙=1𝜇_𝑙(𝜃(𝑡×𝐽(𝑡)𝑒))_𝑖(𝑡)+𝜀Γ_𝑙𝑒₁(𝑡),…,𝑒_𝑁(𝑡)𝑐_𝑖1,…,𝑐_𝑖𝑁^𝑇+𝜀Γ_𝑙𝑒₁(𝑡−𝜏(𝑡)),…,𝑒_𝑁((𝑡×𝑐̃𝑡−𝜏)_𝑖1,…,𝑐̃_𝑖𝑁^𝑇. (11)
Let 𝑒(𝑡)=(𝑒₁(𝑡),…,𝑒_𝑁(𝑡))∈𝑅^𝑁×𝑁; we can obtain 𝑒_𝑖̇(𝑡)=^𝑟_𝑙=1𝜇_𝑙(𝜃(𝑡))𝐽(𝑡)𝑒_𝑖(𝑡)+𝜀Γ_𝑙𝑒(𝑡)𝐶^𝑇+𝜀Γ_𝑙𝑒(𝑡−𝜏(𝑡))𝐶^𝑇. (12)
According to Lemma 2, there exist nonsingular matrices, Φ₁=(𝜙₁₁,…,𝜙_1𝑁), Φ₂=(𝜙₂₁,…,𝜙_2𝑁), such that 𝐶^𝑇Φ₁=Φ₁Λ₁ and 𝐶^𝑇Φ₂=Φ₂Λ₂, with Λ₁=diag(𝜆₁,…,𝜆_𝑁), Λ₂=diag(𝜆₁,…,𝜆_𝑁). Using the nonsingular transform 𝑒(𝑡)Φ=𝑤(𝑡), 𝑒(𝑡−𝜏)Φ=𝑤(𝑡−𝜏), then we obtain 𝑤̇_𝑘(𝑡)=^𝑟_𝑙=1𝜇_𝑙(𝜃(𝑡))𝐽(𝑡)+𝜀𝜆_𝑘Γ_𝑙𝑤_𝑘(𝑡)+𝜀𝜆_𝑘Γ_𝑙𝑤_𝑘(𝑡−𝜏(𝑡)),𝑘=1,…,𝑁. (13)
Note that 𝜆₁=0, 𝜆₁=0 correspond to the synchronization of the network states (4), where the state 𝑠(𝑡) is an orbital stable solution of the isolate node as assumed before in (4). If the following 𝑁−1 pieces of 𝑛-dimensional linear multiple time delay differential equations 𝑤̇_𝑘(𝑡)=^𝑟_𝑙=1𝜇_𝑙(𝜃(𝑡))𝐽(𝑡)+𝜀𝜆_𝑘Γ_𝑙𝑤_𝑘(𝑡)+𝜀𝜆_𝑘Γ_𝑙𝑤_𝑘(𝑡−𝜏(𝑡)),𝑘=1,…,𝑁 (14) are asymptotically stable, then 𝑒(𝑡) will tend to the origin asymptotically, which implies that synchronized states of complex networks (8) are asymptotically stable.
The proof is thus completed.

Remark 5. We use Lemma 4 to linearize the complicated system to deal with the complex networks by using fuzzy theory. It is an available way to investigate the interconnected dynamical agents of complex networks. Then some new subsystems 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 matrices 𝑃_𝑘=𝑃^𝑇_𝑘>0, 𝑄_𝑘=𝑄^𝑇_𝑘>0,𝑅_𝑘=𝑅^𝑇_𝑘>0, and 𝑍_𝑘=𝑍^𝑇_𝑘>0, such that the following LMI holds for all 𝑙=1,2,…,𝑟: Θ_𝑘ℎΩ_1𝑘ℎΩ_2𝑘ℎΩ_3𝑘ℎΠ^𝑇_𝑘𝑍_𝑘∗−ℎ𝑍_𝑘000∗∗−ℎ𝑍_𝑘00∗∗∗−ℎ𝑍_𝑘0∗∗∗∗−ℎ𝑍_𝑘<0, (15) where Ω_1𝑘=𝑁¹_1𝑘𝑁²_1𝑘𝑁²_1𝑘,Ω_2𝑘=𝑁¹_2𝑘𝑁²_2𝑘𝑁²_2𝑘,Ω_3𝑘=𝑁¹_3𝑘𝑁²_3𝑘𝑁²_3𝑘,Θ𝑓𝑜𝑟𝑘=2,3,…,𝑁,_𝑘=Θ_𝑘1+Θ_𝑘2+Θ^𝑇_𝑘2,Θ_𝑘1=𝑃_𝑘𝐽(𝑡)+𝜀𝜆_𝑘Γ_𝑙+𝐽(𝑡)+𝜀𝜆_𝑘Γ_𝑙^𝑇𝑃_𝑘𝜀𝜆_𝑘Γ_𝑙𝑃_𝑘0∗−1−𝛾0∗∗−𝑅_𝑘,Θ_𝑘2=Ω_𝑘1+Ω_𝑘3−Ω_𝑘1+Ω_𝑘2−Ω_𝑘2−Ω_𝑘3,Π_𝑘=𝐽(𝑡)+𝜀𝜆_𝑘Γ_𝑙𝜀𝜆_𝑘Γ_𝑙0, (16) where ∗ denotes the symmetric terms in a symmetric matrix, then the asymptotic synchronization of complex network system (8) can be achieved.

Proof. Construct a Lyapunov-Krasovskii function as 𝑉_𝑘(𝑡)=𝑤^𝑇_𝑘(𝑡)𝑃_𝑘𝑤_𝑘(𝑡)+^𝑡_{𝑡−𝜏(𝑡)}𝑤^𝑇_𝑘(𝑠)𝑄_𝑘𝑤_𝑘(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ}𝑤^𝑇_𝑘(𝑠)𝑅_𝑘𝑤_𝑘(𝑠)+𝑑𝑠_{0−ℎ𝑡𝑡+𝜃}𝑤̇^𝑇_𝑘(𝑠)𝑍_𝑘𝑤̇_𝑘(𝑠)𝑑𝑠𝑑𝜃. (17) According to Lemma 4, calculating the time derivative of 𝑉_𝑘(𝑡) along the trajectories of complex networks (8), we obtain 𝑉̇_𝑘(𝑡)=2𝑤^𝑇_𝑘(𝑡)𝑃_𝑘𝑤̇_𝑘(𝑡)+𝑤^𝑇_𝑘(𝑡)𝑄_𝑘𝑤_𝑘(𝑡)−(1−𝜏̇(𝑡))𝑤^𝑇_𝑘(𝑡−𝜏(𝑡))𝑄_𝑘𝑤_𝑘(𝑡−𝜏(𝑡))+𝑤^𝑇_𝑘(𝑡)𝑅_𝑘𝑤_𝑘(𝑡)−𝑤^𝑇_𝑘(𝑡−ℎ)𝑅_𝑘𝑤̇_𝑘()𝑤̇𝑡−ℎ+ℎ^𝑇_𝑘(𝑠)𝑍_𝑘𝑤̇_𝑘(𝑠)−^𝑡_𝑡−ℎ𝑤̇^𝑇_𝑘(𝑠)𝑍_𝑘𝑤̇_𝑘(𝑠)≤2𝑤^𝑇_𝑘(𝑡)𝑃_𝑘𝑤̇_𝑘(𝑡)+𝑤^𝑇_𝑘(𝑡)𝑄_𝑘𝑤_𝑘(𝑡)−1−𝛾𝑤^𝑇_𝑘(𝑡−𝜏(𝑡))𝑄_𝑘𝑤_𝑘(𝑡−𝜏(𝑡))+𝑤^𝑇_𝑘(𝑡)𝑅_𝑘𝑤_𝑘(𝑡)−𝑤^𝑇_𝑘(𝑡−ℎ)𝑅_𝑘𝑤̇_𝑘()𝑤̇𝑡−ℎ+ℎ^𝑇_𝑘(𝑠)𝑍_𝑘𝑤̇_𝑘(𝑠)−^𝑡_𝑡−ℎ𝑤̇^𝑇_𝑘(𝑠)𝑍_𝑘𝑤̇_𝑘(𝑠). (18)
Based on the Newton-Leibniz formula, the following equations can be obtained for arbitrary matrices Ξ_𝑘 with appropriate dimensions: 2𝜉^𝑇_𝑘(𝑡)Ξ_𝑘𝑤_𝑘(𝑡)−𝑤_𝑘(𝑡−𝜏)−^𝑡_𝑡−𝜏𝑤_𝑘̇(𝑠)𝑑𝑠=0, (19) where 𝜍^𝑇_𝑘(𝑡)=(𝑤^𝑇_𝑘(𝑡),𝑤^𝑇_𝑘(𝑡−𝜏)) and Ξ^𝑇_𝑘=(𝑋^𝑇₁,𝑋^𝑇₂).
Therefore, using Lemma 3, we have 𝑉̇_𝑘(𝑡)=2𝑤^𝑇_𝑘(𝑡)𝑃_𝑘𝑤̇_𝑘(𝑡)+𝑤^𝑇_𝑘(𝑡)𝑄_𝑘+𝑅_𝑘𝑤_𝑘(𝑡)−1−𝛾𝑤^𝑇_𝑘(𝑡−𝜏(𝑡))𝑄_𝑘𝑤_𝑘(𝑡−𝜏(𝑡))−𝑤^𝑇_𝑘(𝑡−ℎ)𝑅_𝑘𝑤̇_𝑘(𝑡−ℎ)+ℎ𝑤̇^𝑇_𝑘(𝑡)𝑍_𝑘1𝑤̇_𝑘(𝑡)−^𝑡_{𝑡−𝜏(𝑡)}𝑤̇^𝑇_𝑘(𝑠)𝑍_𝑘1(𝑠)𝑑𝑠−^{𝑡−𝜏(𝑡)}_𝑡−ℎ𝑤̇^𝑇_𝑘(𝑠)𝑍_𝑘1𝑤̇_𝑘(𝑠)−𝑑𝑠_{𝑡𝑡−ℎ}𝑤̇^𝑇_𝑘(𝑠)𝑍_𝑘1𝑤̇_𝑘(𝑠)𝑑𝑠+2Δ^𝑇_𝑘(𝑡)Ω_1𝑘𝑤_𝑘(𝑡)−𝑤_𝑘((𝑡−𝑡−𝜏))_{𝑡𝑡−𝜏(𝑡)}𝑤̇^𝑇_𝑘(𝑠)𝑑𝑠+2Δ^𝑇_𝑘(𝑡)Ω_2𝑘𝑤_𝑘(𝑡−𝜏(𝑡))−𝑤_𝑘()−𝑡−ℎ_{𝑡−𝜏(𝑡)𝑡−ℎ}𝑤̇^𝑇_𝑘(𝑠)𝑑𝑠+2Δ^𝑇_𝑘(𝑡)Ω_3𝑘𝑤_𝑘(𝑡)−𝑤_𝑘()−𝑡−ℎ_{𝑡𝑡−ℎ}𝑤̇^𝑇_𝑘(𝑠)<𝑑𝑠_𝑟𝑖=1𝜇_𝑙(𝜃(𝑡))Φ^𝑇_𝑘(𝑡)Θ_𝑘+ℎΓ^𝑇_𝑘𝑍_𝑘Γ_𝑘+ℎΩ_1𝑘𝑍_𝑘⁻¹Ω^𝑇_1𝑘+ℎΩ_2𝑘𝑍_𝑘⁻¹Ω^𝑇_2𝑘+ℎΩ_3𝑘𝑍_𝑘⁻¹Ω^𝑇_3𝑘Φ_𝑘(𝑡)−^𝑡_{𝑡−𝜏(𝑡)}Φ^𝑇_𝑘(𝑡)Ω_1𝑘+𝑤̇^𝑇_𝑘(𝑠)𝑍_𝑘𝑍_𝑘⁻¹×Ω^𝑇_1𝑘Φ_𝑘(𝑡)+𝑍_𝑘𝑤̇_𝑘(𝑠−)𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−ℎ}Φ^𝑇_𝑘(𝑡)Ω_2𝑘+𝑤̇^𝑇_𝑘(𝑠)𝑍_𝑘𝑍_𝑘⁻¹×Ω^𝑇_2𝑘Φ_𝑘(𝑡)+𝑍_𝑘𝑤̇_𝑘(𝑠−)𝑑𝑠_{𝑡𝑡−ℎ}Φ^𝑇_𝑘(𝑡)Ω_3𝑘+𝑤̇^𝑇_𝑘(𝑠)𝑍_𝑘𝑍_𝑘⁻¹×Ω^𝑇_3𝑘Φ_𝑘(𝑡)+𝑍_𝑘𝑤̇_𝑘(𝑠)𝑑𝑠, (20) where Δ^𝑇_𝑘(𝑡)=[𝑤^𝑇_𝑘(𝑡),𝑤^𝑇_𝑘(𝑡−𝜏),𝑤^𝑇_𝑘(𝑡−ℎ)]^𝑇. Because the last three terms are all less than zero, if Θ_𝑘+ℎΓ^𝑇_𝑘𝑍_𝑘Γ_𝑘+ℎΩ_1𝑘𝑍_𝑘⁻¹Ω^𝑇_1𝑘+ℎΩ_2𝑘𝑍_𝑘⁻¹Ω^𝑇_2𝑘+ℎΩ_3𝑘𝑍_𝑘⁻¹Ω^𝑇_3𝑘<0, (21) then 𝑉̇_𝑘(𝑡)<−𝜅‖𝑤(𝑡)‖² for sufficiently small 𝜅>0. It is easy to see that system (8) is globally synchronized. By Schur complements, we know that the function (21) is equivalent to the function (15).
The proof is thus completed.

Remark 7. For the delay-dependent synchronization problems of complex networks, the Lyapunov-Krasovskii condition has been attracted owing to the structural advantage. The key point is the introduction of the integral inequality technique, a fundamental trick to derive delay-dependent stability criteria containing the size or the bounds of delays and their derivatives. However, almost results in this field have employed partial information on the relationship among delay-related terms. Being differently from the common construction, we use not only the time-varying-delayed state but also the delay-upper-bounded state to exploit all possible information when constructing the Lyapunov-Krasovskii functional. Our approach in Theorem 6 reduces the conservatism of the existing methods to a certain extent.

The aforementioned results can be extended to the case of bound constant time delay. If the time-varying delay 𝜏(𝑡) is a constant time delay in system (1), then fuzzy system (4) can be expressed in the following model: 𝑥̇_𝑖(𝑡)=^𝑟_𝑙=1𝜇_𝑙(𝜃(𝑡))𝑓𝑥_𝑖(𝑡)+𝜀^𝑁_𝑗=1𝑐_𝑖𝑗Γ_𝑙𝑥_𝑗(𝑡)+𝜀^𝑁_𝑗=1𝑐̃_𝑖𝑗Γ_𝑙𝑥_𝑗(𝑡−𝜏). (22) We obtain the following corollary.

Corollary 8. If there exist positive definite symmetric matrices 𝑃_𝑘,𝑄_𝑘, and 𝑅_𝑘(2≤𝑘≤𝑁) and two arbitrary matrices 𝑁₁, 𝑁₂ with appropriate dimensions, such that the following LMIs hold for all 𝑙=1,2,…,𝑟: Ω_𝑘=Θ𝜆_𝑘𝜀𝑃_𝑘Γ_𝑙−𝑁₁+𝑁^𝑇₂𝜏𝐽^𝑇(𝑡)𝑅_𝑘𝜏𝑁₁∗−𝑄_𝑘−𝑁₂−𝑁^𝑇₂𝜏𝑐𝜆_𝑘Γ^𝑇_𝑙𝑅_𝑘𝜏𝑁₂∗∗−𝜏𝑅_𝑘0∗∗∗−𝜏𝑅_𝑘<0, (23) where Θ=𝑃_𝑘𝐽(𝑡)+𝜀𝜆_𝑘Γ_𝑙+𝐽(𝑡)+𝜀𝜆_𝑘Γ_𝑙^𝑇𝑃_𝑘+𝑄_𝑘+𝑁^𝑇₁+𝑁₁, (24) then the asymptotic synchronization of system (22) in Definition 1 can be achieved.

Proof. Select a Lyapunov-Krasovskii function as 𝑉_𝑘(𝑡)=𝑤^𝑇_𝑘(𝑡)𝑃_𝑘𝑤_𝑘(𝑡)+^𝑡_𝑡−𝜏𝑤^𝑇_𝑘(𝑠)𝑄_𝑘𝑤_𝑘(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏𝑡𝜃}𝑤̇^𝑇_𝑘(𝑠)𝑅_𝑘𝑤̇_𝑘(𝑠)𝑑𝑠𝑑𝜃. (25) The rest of the proof is similar to that of Theorem 6; thus one can easily obtain the result. Thus it is omitted here. The proof is completed.

Remark 9. Corollary 8 presents new hybrid synchronization conditions for fuzzy complex networks with constant delays. We consider the delay which involve general form of delays. Based on Lyapunov-Krasovskii function method, stability criteria are obtained in the form of LMIs which can be easily solved. And the results are usually less conservative than delay-independent ones especially when the size of the time delay is small.

4. Numerical Example

To illustrate the previous results we obtain, we consider the following complex dynamical networks with time-varying delays consisting of three nodes: 𝑥̇₁(𝑡)𝑥̇₂(𝑡)𝑥̇₃(𝑡)=−𝑥₁−2𝑥₂−3𝑥₃ (26) which is asymptotically stable at 𝑠(𝑡)=0, with Jacobin given by 𝐽=diag{−1,−2,−3}. Assume that inner-coupling matrices areΓ=Γ=diag{1,1,1}, and the coupling configuration matrices are,𝜇𝐶=𝐶=−10.50.50.5−10.50.50.5−1₁(𝜃(𝑡))=sin²𝑥₁(𝑡),𝜇₂(𝜃(𝑡))=cos²𝑥₁(𝑡). (27)

The eigenvalues of 𝐶 and 𝐶 are 𝜆(𝐶)=𝜆(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 can be obtained from Theorem 6. A detailed comparison is given in Table 1, where the achieved upper bounds of time delay in the previous system are listed for their respective lower bounds.

Table 1 

Upper bounds of ℎ for different 𝜀 and 𝛾.

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

For example, when the delay is𝛾=0.5, by employing the LMI Toolbox in Matlab, we simulate system (26) for given chosen initial parameter 𝜀=0.3. We define the errors of synchronization as follows: 𝑒_𝑖𝑗(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 the synchronization errors between the states of node 𝑖 and node 𝑖+1 with ℎ=0.5. We find that the synchronization errors converge to zero under the previous conditions.

Figure 1 

Synchronization error 𝑒𝑖1 for the coupling delayed networks with ℎ=𝛾=0.5.

Figure 2 

Synchronization error 𝑒𝑖1 for the coupling delayed networks with ℎ=𝛾=0.5.

Figure 3 

Synchronization error 𝑒𝑖3 for the coupling delayed networks with ℎ=𝛾=0.5.

5. Conclusions

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

Acknowledgments

This research was partially supported by the National Natural Science Foundation of PR China (Grant nos. 61104135, 61025015, 61174132, and 61134006), China Postdoctoral Science Fund (Grant no. 2012T50705), and Postdoctoral Fund of the Central South University. The authors also gratefully thank the Editor and the reviewers for their helpful comments and suggestions, which have improved the quality of this paper.