Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 639580, 21 pages

http://dx.doi.org/10.1155/2013/639580

Research Article

Exponential Synchronization for Neutral Complex Dynamical Networks with Interval Mode-Dependent Delays and Sampled Data

Xinghua Liu and Hongsheng Xi

Department of Auto, School of Information Science and Technology, University of Science and Technology of China, Anhui 230027, China

Received 26 May 2013; Revised 7 August 2013; Accepted 8 August 2013

Academic Editor: Jun-Juh Yan

Copyright © 2013 Xinghua Liu and Hongsheng Xi. 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.

Abstract.

The exponential synchronization and sampled-data controller problem for a class of neutral complex dynamical networks (NCDNs) with Markovian jump parameters, partially unknown transition rates and delays, is investigated in this paper. Both the discrete and neutral delays are considered to be interval mode dependent and time varying, while the sampling period is assumed to be time varying and bounded. Based on a new augmented stochastic Lyapunov functional, the delay-range-dependent and rate-dependent exponential stability conditions for the closed-loop error system are obtained by the Lyapunov-Krasovskii stability theory and reciprocally convex lemma. Then according to the proposed exponential stability conditions, the sampled-data synchronization controllers are designed in terms of the solution to linear matrix inequalities that can be solved effectively by using Matlab. Finally, numerical examples are given to demonstrate the feasibility and effectiveness of the proposed methods.

1. Introduction

It is well known that many practical systems can be described as complex networks such as Internet networks, biological networks, epidemic spreading networks, collaborative networks, social networks and neural networks [1–4]. Thus, during the past decades, the research on the dynamics of complex dynamical networks (CDNs) has attracted extensive attention of scientific and engineering researchers in all fields domestic and overseas since the pioneering work of Watts and Strogatz [5]. As one of the most significant and important collective behaviors in CDNs, synchronization has received more attention; please refer to [6–10] and references therein for more details.

Since time delay inevitably exists and has become an important issue in studying the CDNs, synchronization problems for complex networks with time delays have gained increasing research attention, and considerable progress has been made; see, for example, [6–17] and references therein for more details. However, in some practical applications of communication networks, signal transmission channels often involve network-induced delays, packet dropouts, bit errors, environment disturbances, and so on, which will cause the error of transmission from one system to another. Then past change rate of the state variables affects the dynamics of nodes in the networks. This kind of complex dynamical network is termed as neutral complex dynamical network (NCDN), which contains delays both in its states and the derivatives of its states. There are some results about the synchronization design problem for neutral systems [18–23]. In these works, [19, 20] had studied the synchronization control for a kind of master-response setup and were further extended to the case of neutral-type neural networks with stochastic perturbation. The authors of [18, 22] had researched the synchronization problem for a class of complex networks with neutral-type coupling delays. The authors in [21] had studied the global asymptotic stability of neural networks of neutral type with mixed delays, which include constant delay in the leakage term, time-varying delays, and continuously distributed delays. The authors in [23] had investigated the robust global exponential synchronization problem for an array of neutral-type neural networks. However, much fewer results have been proposed for neutral complex dynamical networks (NCDNs) compared with the rich results for CDNs with only discrete delays.

On the other hand, network mode switching is also a universal phenomenon in CDNs of the actual systems, and sometimes the network has finite modes that switch from one to another with certain transition rate; then such switching can be governed by a Markovian chain. The stability and synchronization problems of complex networks and neural networks with Markovian jump parameters and delays are investigated in [16, 24–30], and references therein. The authors in [24] have established sufficient global exponential stability conditions on Markovian jump neural networks with impulse control and time varying delays. The authors in [27] have studied synchronization in an array of coupled neural networks with Markovian jumping and random coupling strength. Particularly, Ma et al. [25] have considered the stability and synchronization problems for Markovian jump delayed neural networks with partly unknown transition probabilities, which have not been fully investigated and need to propose more good results. Besides, sampled-data systems have attracted great attention because the digital signal processing methods require better reliability, accuracy, and stable performance with the rapid development in digital measurement and intelligent instrument. There are some important and essential results which have been reported in the literature [31–35]. What is worth mentioning is that the sampled-data synchronization control problem has been investigated for a class of general complex networks with time-varying coupling delays in [34, 35], where conditions have been presented to ensure the exponential stability of the closed-loop error system, and the desired sampled data feedback controllers have been designed. However, few results are available for neutral complex dynamical networks (NCDNs) with sampled data. To the best of the authors' knowledge, the NCDNs are difficult to treat and they are very challenging, especially in the presence of Markovian jump parameters, mode-dependent time-varying delays, and sampled data. Motivated by the above analysis, the exponential synchronization and sampled-data controller problem for a class of NCDNs with Markovian jump parameters and mode-dependent time-varying delay is investigated in this paper. The addressed NCDNs consist of modes, and the networks switch from one mode to another according to a Markovian chain with partially known transition rate.

In this paper, the synchronization and sampled-data controller problem is studied for NCDNs with Markovian jump parameters and partially known transition rates. The sampling period considered here is assumed to be time varying and bounded, while the neutral and discrete delays are interval mode-dependent and time varying. Firstly, by constructing a new augmented stochastic Lyapunov functional, exponential stability conditions are derived based on the Lyapunov stability theory and reciprocally convex lemma. Then the design method of the desired sampled-data controllers is solved on the basis of the obtained conditions. Moreover, all the derived results are in terms of LMIs that can be solved numerically, which are proved to be less conservative than the existing results.

The remainder of the paper is organized as follows. Section 2 presents the problem and preliminaries. Section 3 gives the main results, which are then verified by numerical examples in Section 4. Section 5 concludes the paper.

Notations. The following notations are used throughout the paper. denotes the dimensional Euclidean space and is the set of all matrices. (), where and are both symmetric matrices, which means that is negative (positive) definite. is the identity matrix with proper dimensions. For a symmetric block matrix, we use to denote the terms introduced by symmetry. stands for the mathematical expectation; is the Euclidean norm of vector , , while is spectral norm of matrix , . is the eigenvalue of matrix with maximum (minimum) real part. The Kronecker product of matrices and is a matrix in which is denoted as . Let and denote the family of continuous function , from to with the norm . Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

2. Problem Statement and Preliminaries

Given a complete probability space with a natural filtration satisfying the usual conditions, where is the sample space, is the algebra of events and is the probability measure defined on . Let be a right-continuous Markov chain taking values in a finite state space with a generator , , which is given by

where , , and is the transition rate from mode to , and for any state or mode , it satisfies

It is assumed that is irreducible and available at time , but the transition rates of the Markov chain are partially known in this paper, which means that some elements in matrix are inaccessible. For instance, in the system with six operation modes, the jump rates matrix may be viewed as

where represents the unknown element. Furthermore, let , be lower and upper bound for the diagonal element or , for all . For notation clarity, we denote that , for all , and

If , than it is further described as

where represent the th known element of the set in the th row of the transition rate matrix . It should be noted that if , which means that any information between the th mode and the other modes is not accessible, then MJSs with modes can be regarded as ones with modes.

The following neutral complex dynamical network (NCDN) consisting of identical nodes with Markovian jump parameters and interval time-varying delays over the space is investigated in this paper:

where , are state variable and the control input of the node , respectively. describes the evolution of the mode. , , and represent the inner-coupling matrices linking between the subsystems in mode . , , and are the coupling configuration matrices of the networks representing the coupling strength and the topological structure of the NCDNs in mode , in which is defined as follows: if there exists a connection between th and th nodes, then ; otherwise , and

and denote the mode-dependent time-varying neutral delay and retarded delay, respectively. They are assumed to satisfy

where , , , and , , are real constant scalars.

is a parametric matrix with real values in mode . is a continuous vector-valued nonlinear function. It is assumed to satisfy the following sector-bounded condition [36]:

where , are two constant matrices. Such a description of nonlinear functions has been exploited in [37–39] and is more general than the commonly used Lipschitz conditions. The sector-bounded condition would be possible to reduce the conservatism of the main results caused by quantifying the nonlinear functions via a matrix inequality technique.

Let be the state trajectory of the unforced node ; then the synchronization error is defined to be . So the error dynamics of NCDN (6) can be derived as follows:

where , .

The control signal is assumed to be generated by using a zero-order-hold (ZOH) function with a sequence of hold times . Thus the state feedback controller takes the following form:

where is sampled-data feedback controller gain matrix to be determined. is discrete measurement of at the sampling instant satisfying . It is assumed that for any integer , where represents the largest sampling interval.

With (10) and (11), the error dynamics of NCDN (6) can be replaced as

where and .

For simplicity of notations, we denote , , , , , by , , , , for . By utilizing the Kronecker product of the matrices, (12) can be written in a more compact form as

where , , , , and

The purpose of this paper is to design a serial of sampled-data controllers (11) to ensure the exponential synchronization of NCDN (6). Before proceeding with the main results, we present the following definitions and lemmas.

Definition 1 (see [40]). Define operator as . is said to be stable if the homogeneous difference equation

is uniformly asymptotically stable. In this paper, that is, .

Definition 2 (see [41]). Define the stochastic Lyapunov-Krasovskii function of the error system (13) as , where its infinitesimal generator is defined as

Definition 3. The NCDN (6) is said to be exponentially synchronized if the error system (13) is exponentially stable; that is, [42] there exist two constants and such that for all, ,

where is the exponential decay rate; .

Lemma 4 (see [43]). Given matrices , , , and with appropriate dimensions and scalar , by the definition of the Kronecker product, the following properties hold:

Lemma 5. For any constant matrix , continuous functions , constant scalars , and constant such that the following integrations are well defined:

Proof. (a) is directly obtained from [44]. In addition, from and , it is held that . Then,

(b) is thus true by [45].

Lemma 6 (see [46]). For functions , , , and with and with and matrices , , then there exists matrix such that

and the following inequality holds:

According to Definition 3, the aim will be achieved if we obtain the gain matrices such that the error system (13) is exponentially stable. So we give the main results as follows.

3. Main Results

In this section, sufficient conditions are presented to ensure that the error system (13) is exponentially stable. Then, we propose a design method of the sampled-data controllers for NCDN (6).

3.1. Exponential Stability for the Error System

Theorem 7. For given scalars , , , , , , , , and and constant scalar satisfying , the error system (13) with partially known transition rates and sector-bounded condition (9) is exponentially stable with decay rate and if , and there exist symmetric positive matrices , , , , , , , , , , , , , , and matrices , , , for any scalar , and symmetric matrices , , , , such that the following matrix inequalities hold.

When ; then

When , then

and

where

where

where are block entry matrices; that is,

Proof. Construct the following stochastic Lyapunov functional:

where

Taking as its infinitesimal generator along the trajectory of error system (13), we obtain the following from Definition 3 and (34)-(35):

Define

Due to , the following zero equations hold for arbitrary matrices:

that is,

Based on (9), we can obtain the following inequality for any :

From (42) and (53), we have

From (8) we easily obtain that

Notice (a) of Lemma 5; then,

Notice (b) of Lemma 5; then,

For , the following is held from (a) of Lemma 5:

where

By Lemma 6, there exists matrix with appropriate dimensions such that

Similarly, considering and following the same procedure, there exists matrix with appropriate dimensions such that

For , with the same matrix inequalities technique, we obtain the following:

For , we obtain the following:

Consider and , which are directly estimated by (a) of Lemma 5; that is,

In addition, with the error system (53), we obtain and

where and have been defined in Theorem 7.

Substituting (43)–(52) and (55)–(65) into (54), we obtain

where

On the other hand, for , the integral terms

are disposed and estimated by Lemma 6.

are directly estimated by (a) of Lemma 5. Therefore,

With (66) and (70), the following inequality is held for if (23)–(28) are satisfied:

From the stochastic Lyapunov functional (34) and (71), it is held that

Moreover, we have

Then from (72) and (73), it is readily seen that

where .

Therefore, the error system (13) with partially known transition rates and sector-bounded condition (9) is exponentially stable with a decay rate . This completes the proof.

In some situations, due to the complexity of NCDN (6), the information on the transition rates may be completely unknown, which can be viewed as switched complex dynamical networks with arbitrary switching. The following corollary is therefore given to guarantee the exponential stability for this case.

Corollary 8. For given scalars , , , , , , , , and and constant scalar satisfying , the error system (13) with entirely unknown transition rates and sector-bounded condition (9) is exponentially stable with decay rate and if , and there exist symmetric positive matrices , , , , , , , , , , and matrices , , , for any scalar , and symmetric matrices , , such that (28) and the following matrix inequalities hold:

where

Other notations are the same as those in Theorem 7.

Proof. Since the elements of transition rates are entirely unknown, we choose the Lyapunov functional as follows:

where

and , , are the same as (39), (40), (41).

Then we follow a similar line as in proof of Theorem 7 and obtain the result.

3.2. The Sampled-Data Controllers for NCDN

In this subsection, we give the design method of the desired sampled-data controllers to ensure the NCDN (6) exponentially synchronized on the basis of Theorem 7.

Theorem 9. The NCDN (6) with partially known transition rates and sector-bounded condition (9) is exponentially synchronized by controllers of the form (11) if there exist symmetric positive definite matrices , , , , , , , , , , , , , , , matrices and matrices , , , for any scalar , and symmetric matrices , , , , such that (23)–(28) and the following linear matrix inequalities hold:

In addition, the desired sampled-data controllers gain matrices are given by

where

Other notations are the same as those in Theorem 7.

Proof. Considering (27) and (28), by applying the Schur Complement Lemma, we have

Define the block diagonal matrix , where . Noting that , with (ii) of (86) and (iv) of (87), we define and obtain the following:

Due to and , we know that . Thus (88) are held if (ii) of (82) and (iv) of (83) are satisfied, which implies that (ii) of (27) and (iv) of (28) are held.

According to Definition 3, by the sampled-data controllers gain matrices (84), the NCDN (6) with partially known transition rates and sector-bounded condition (9) is exponentially synchronized. This completes the proof.

Remark 10. It should be mentioned that the sampled-data synchronization problem has been solved for NCDN (6) in Theorem 9, and the desired controllers can be obtained when LMIs (23)–(26), (82), and (83) are feasible. In this paper, we construct the novel stochastic Lyapunov functional (34) containing some triple-integral terms; which is very effective in the reduction of conservatism [45]. Besides, we apply Lemma 5 to the corresponding terms, the method by using reciprocally convex lemma [46] can achieve less conservative results. Thus the obtained delay-range-dependent and decay rate-dependent stability condition for NCDN (6) in Theorem 7 is less conservative than the previous ones, which can achieve larger maximum value of sampling period and will be verified in Section 4.

Remark 11. It should be pointed out that the given results can be extended to more general neutral delay in the form of NCDN (6). For instance, ; we can choose satisfying and construct the stochastic Lyapunov functional on , , , and . Following the same thought, the time delay in the control input can also be extended to be more general.

Corollary 12. The NCDN (6) with completely unknown transition rates and sector-bounded condition (9) is exponentially synchronized by controllers of the form (11) if there exist symmetric positive definite matrices , , , , , , , , , , , matrices and matrices , , , for any scalar , and symmetric matrices , , such that (28), (75), (76), and the following linear matrix inequalities hold:

In addition, the desired sampled-data controllers gain matrices are given by

where

Other notations are the same as those in Corollary 8.

4. Numerical Examples

In this section, numerical examples are used to illustrate the effectiveness of the results derived above.

Example 1. As shown in the example, a three-node NCDN (6) with Markovian switching between two modes is taken into consideration; that is, and . The parametric matrices of the NCDN are given as follows:

The partially known transition rate matrix is considered as in the following two cases:

Furthermore, the nonlinear function is given by

Then, it is easy to verify that

The interval mode-dependent time-varying neutral delays and discrete delays are, respectively, assumed to be

They are governed by the Markov process and shown in Figures 1 and 2. It can be readily obtained that

Given decay rate and the maximum value of sampling period , we choose and and seek to solve the sampled-data controllers.

For the case of , with the above parameters, by Theorem 9, we can obtain the sampled-data controllers as follows:

Furthermore, the state trajectories of the error system (13) are given in Figure 3, where , , , and .

For another case of , the sampled-data controllers can be readily obtained by Corollary 12.

(a)

(b)

Figure 1: Time-varying neutral delay at mode 1 and mode 2.

(a)

(b)

Figure 2: Time-varying discrete delay at mode 1 and mode 2.

Figure 3: State trajectories of the error system in Example 1.

Example 2. Consider NCDN (6) with three nodes but only one mode; that is, and . So the subscript of the mode is omitted here. The outer-coupling matrices are assumed to be

In fact, the NCDN in consideration here has become a common CDN.

The nonlinear function is taken as

(101) can be easily verified that

The time-varying delay is chosen as ; then we know that , , and . Besides, we choose ,  satisfying and consider the following two cases.

Case  1. The inner-coupling matrices are given as and . According to the above parameters, the maximum value of sampling period , which satisfies LMIs (23)–(26), (82), and (83) in Theorem 9, can be calculated by solving a quasiconvex optimization problem. The degraded NCDN has also been considered in [34, 35]. The results on the maximum value of sampling period are compared in Table 1, where it can be seen that our method is less conservative than [34, 35].

Case  2. The inner-coupling matrices are given as and . The maximum value of sampling period also can be solved by Theorem 9. The results are listed in Table 2. In addition, the proposed method in [34] is not applicable here. From Table 2, it also can be seen that our method is less conservative than existing ones.

Table 1: Maximum value of sampling period with and .

Methods

Sampled-data controllers

Li et al. [34]

, ,

Wu et al. [35]

, ,

Theorem 9

, ,

Table 2: Maximum value of sampling period with and .

Methods

Sampled-data controllers

Li et al. [34]
—
—

Wu et al. [35]

, ,

Theorem 9

, ,

5. Conclusions

In this paper, the sampled-data synchronization problem has been solved for a class of NCDNs with Markovian jump parameters and partially known transition rates. The discrete and neutral delays are considered to be interval mode-dependent and time varying, while the sampling period here is bounded and time-varying. With a novel stochastic Lyapunov functional, the delay-range-dependent and rate-dependent exponential synchronization conditions have been proposed by Lyapunov theory and better technique of matrix inequalities. Then the sampled-data controllers have been designed on the basis of the obtained conditions. These theoretical results are successfully verified through numerical examples, whose simulation results are less conservative than the previous results. Finally, the main contributions of this paper can be summarized as follows. (i) To achieve exponential synchronization for NCDN (6), the desired sampled-data feedback controllers have been designed in terms of the solution to certain LMIs. The proposed results are expressed in a new representation, which are theoretically and numerically proved to be less conservative than some existing ones. (ii) The constructed stochastic Lyapunov functional contains some triple-integral terms, which are very effective in the reduction of conservativeness and have not appeared in the context of NCDNs. (iii) The bound of the delay is fully utilized in this paper; that is, improved bounding technique is used to reduce the conservativeness. (iv) The reciprocally convex lemma is used to derive the delay-range-dependent and rate-dependent stability conditions, which can well reduce the conservativeness of the investigated systems.

Acknowledgments

This work was supported in part by the National Key Scientific Research Project (61233003), the National Natural Science Foundation of China (60935001, 61174061, and 61074033), the Doctoral Fund of the Ministry of Education of China (20093402110019), the Anhui Provincial Natural Science Foundation (11040606M143), the Fundamental Research Funds for the Central Universities, and the Program for New Century Excellent Talents in University.

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