Passivity Analysis of Complex Delayed Dynamical Networks with Output Coupling

Abstract

A new complex dynamical network model with output coupling is proposed. This paper is concerned with input passivity and output passivity of the proposed network model. By constructing new Lyapunov functionals, some sufficient conditions ensuring the input passivity and output passivity are obtained. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed results.

Yan-Fang Kang 1 and Lu-Xian Fang 2 and Yue-Hui Zhao 2 and Shun-Yan Ren 3

Academic Editor:Zhichun Yang

1, School of Economics, Wuhan University of Technology, Wuhan 430070, China
2, The College of Post and Telecommunication, Wuhan Institute of Technology, Wuhan 430070, China
3, School of Computer Science and Software Engineering, Tianjin Polytechnic University, Tianjin 300387, China

Received 12 May 2014; Revised 29 August 2014; Accepted 30 August 2014; 4 June 2015

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.

1. Introduction

Recently, there has been increasing interest in the study of complex dynamical networks. The main reason is that many practical systems can be characterized by various models of complex networks. It is well known that one of the most significant and interesting dynamical phenomena of complex networks is the synchronization of systems. Many interesting results on synchronization have been derived for various complex networks [1-10]. But, it should be noticed that the complex networks with state coupling were considered in these papers.

To our knowledge, Jiang et al. [11] first introduced a complex network model with output coupling without time delays. Some conditions for synchronization were established based on the Lyapunov stability theory. However, time delays always exist in complex networks due to the finite speeds of transmission and/or the traffic congestion, and most of delays are notable. So it is crucial for us to take the delay into the consideration when we study complex networks. Practically, many phenomena in nature can be modeled as complex networks with output coupling. The cooperative control problem of multiple agents has received much attention recently since it has challenging features and many applications, for example, large object moving, formation control, rescue mission, and satellite clustering. It is well known that the state of agent is difficult to be observed or measured because of technology limitations and environmental disturbances. For instance, the measuring of velocity is more difficult than that of position. In some circumstances, the information of velocity is unavailable for agents [12]. Therefore, it is quite necessary to design protocols based on the output variables. In this case, the closed-loop systems can be described by the complex networks with output coupling. Hence, study of complex networks with output coupling is very interesting and important in both theory and application. A complex delayed dynamical network model with output coupling was proposed in [13, 14]. Wang and Wu [13] investigated the output synchronization of the proposed network model, and some criteria on local and global exponential output synchronization were derived.

Passivity [15-33] is an important concept of system theory and provides a nice tool for analyzing the stability of systems and has found applications in diverse areas such as stability, complexity, signal processing, chaos control and synchronization, and fuzzy control. Many researchers have studied the passivity of fuzzy systems [19-22] and neural networks [23-28]. Liang et al. [20] discussed the passivity and passification problems for a class of uncertain stochastic fuzzy systems with time-varying delays. In [26] Song et al. investigated the passivity for a class of discrete-time stochastic neural networks with time-varying delays, and a delay-dependent passivity condition was obtained by constructing proper Lyapunov-Krasovskii functional. However, there are few work on the passivity of complex networks [29, 30, 32, 33]. In [29, 30], Yao et al. obtained some sufficient conditions on passivity properties for linear (or linearized) complex networks with and without coupling delays (constant delay). However, in practical evolutionary processes of the networks, absolute constant delay may be scarce and delays are frequently varied with time. Therefore, it is important to further study the passivity of complex networks with time-varying delays. Wang et al. [32] considered input passivity and output passivity for a generalized complex network with nonlinear, time-varying, nonsymmetric, and delayed coupling. By constructing some suitable Lyapunov functionals, several sufficient conditions ensuring input passivity and output passivity were derived. To the best of our knowledge, the input passivity and output passivity of complex delayed dynamical network model with output coupling have not yet been established. Therefore, it is interesting to study the input passivity and output passivity of complex delayed dynamical network model with output coupling.

Motivated by the above discussions, we propose a new complex delayed dynamical network model with output coupling. The objective of this paper is to study the input and output passivity of the proposed network model. Some sufficient conditions ensuring input passivity and output passivity are obtained by Lyapunov functional method.

The rest of this paper is organized as follows. A new complex network model is introduced and some useful preliminaries are given in Section 2. Several input and output passivity criteria are established in Section 3. In Section 4, two numerical examples are given to illustrate the effectiveness of the proposed results. Conclusions are finally given in Section 5.

2. Network Model and Preliminaries

Let [figure omitted; refer to PDF] be the [figure omitted; refer to PDF] -dimensional Euclidean space, and let [figure omitted; refer to PDF] be the space of [figure omitted; refer to PDF] real matrices. [figure omitted; refer to PDF] means that matrix [figure omitted; refer to PDF] is real symmetric and semipositive (seminegative) definite. [figure omitted; refer to PDF] means that matrix [figure omitted; refer to PDF] is real symmetric and positive (negative) definite. [figure omitted; refer to PDF] denotes the [figure omitted; refer to PDF] identity matrix. [figure omitted; refer to PDF] denotes the transpose of a square matrix [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is a Banach space of continuous functions mapping the interval [figure omitted; refer to PDF] into [figure omitted; refer to PDF] with the norm [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the Euclidean norm.

In this paper, we consider a complex delayed dynamical network consisting of [figure omitted; refer to PDF] identical nodes with diffusive and output coupling. The mathematical model of the coupled network can be described as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is the time-varying delay with [figure omitted; refer to PDF] .

The function [figure omitted; refer to PDF] , describing the local dynamics of the nodes, is continuously differentiable and capable of producing various rich dynamical behaviors, [figure omitted; refer to PDF] is the state variable of node [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the output of node [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the input vector of node [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are known matrices with appropriate dimensions, [figure omitted; refer to PDF] is inner-coupling matrix, which describes the individual coupling between two connected nodes of the network, [figure omitted; refer to PDF] represents the overall coupling strength, [figure omitted; refer to PDF] is the degree of node [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is a tunable weight parameter, and the real matrix [figure omitted; refer to PDF] is a symmetric matrix with diagonal entries [figure omitted; refer to PDF] and off-diagonal entries [figure omitted; refer to PDF] if node [figure omitted; refer to PDF] and node [figure omitted; refer to PDF] are connected by a link, and [figure omitted; refer to PDF] otherwise.

In this paper, we always assume that complex network (1) is connected. Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The initial condition associated with the complex network (1) is given in the form [figure omitted; refer to PDF]

Next, we give several useful definitions.

Definition 1 (see [32]).

Complex network (1) is called input passive if there exist two constants [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Definition 2 (see [32]).

Complex network (1) is called output passive if there exist two constants [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Definition 3 (see [34]).

Let [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF] . Then the Kronecker product (or tensor product) of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is defined as the matrix [figure omitted; refer to PDF]

The Kronecker product has the following properties: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are matrices with suitable dimensions.

3. Main Results

In this section, we shall investigate the input passivity and output passivity of the complex delayed dynamical networks with output coupling.

In [32, 35], authors make the assumption that the function [figure omitted; refer to PDF] is in the QUAD function class. In this paper, we make similar assumptions.

(A1) There exist a positive definite diagonal matrix [figure omitted; refer to PDF] and a diagonal matrix [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] satisfies the following inequality: [figure omitted; refer to PDF] for some [figure omitted; refer to PDF] and all [figure omitted; refer to PDF] .

For the convenience, we denote [figure omitted; refer to PDF]

In the following, we first give two input passivity criteria.

Theorem 4.

Let (A1) hold, and let [figure omitted; refer to PDF] . The complex network (1) is input passive if there exist matrix [figure omitted; refer to PDF] and a scalar [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

For convenient analysis, we let [figure omitted; refer to PDF] Then, complex network (1) can be rewritten as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is a coupling matrix, accounting for the topology of complex dynamical network. We can rewrite system (12) in a compact form as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Construct Lyapunov functional for system (13) as follows: [figure omitted; refer to PDF] The derivative of [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF] Then, we can get [figure omitted; refer to PDF] According to (A1), we can obtain [figure omitted; refer to PDF] It follows from (9) and (18) that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

By integrating (19) with respect to [figure omitted; refer to PDF] over the time period [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] From the definition of [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The proof is completed.

Theorem 5.

Let (A1) hold, and let [figure omitted; refer to PDF] . The complex network (1) is input passive if there exist two matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and a scalar [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

Define the following Lyapunov functional for system (13): [figure omitted; refer to PDF] The derivative of [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF] Then, we can get [figure omitted; refer to PDF] It follows from (18) and (22) that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

By integrating (27) with respect to [figure omitted; refer to PDF] over the time period [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , we can get [figure omitted; refer to PDF] From the definition of [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The proof is completed.

In the above, two sufficient conditions are given to ensure the input passivity of complex network (1). In the following, we shall discuss the output passivity of complex network (1).

Theorem 6.

Let (A1) hold, and let [figure omitted; refer to PDF] . The complex network (1) is output passive if there exist matrix [figure omitted; refer to PDF] and scalar [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

Construct the same Lyapunov functional as (15) for system (13). Then, we can get [figure omitted; refer to PDF] Therefore, we have [figure omitted; refer to PDF] It follows from (18) and (30) that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

By integrating (34) with respect to [figure omitted; refer to PDF] over the time period [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] From the definition of [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The proof is completed.

Theorem 7.

Let (A1) hold, and let [figure omitted; refer to PDF] . The complex network (1) is output passive if there exist two matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and a scalar [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

Construct the same Lyapunov functional as (24) for system (13). Then, we can obtain [figure omitted; refer to PDF] Therefore, we have [figure omitted; refer to PDF] It follows from (18) and (37) that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

By integrating (41) with respect to [figure omitted; refer to PDF] over the time period [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] From the definition of [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The proof is completed.

Remark 8.

In recent years, some researchers have studied the input passivity and output passivity of the complex networks with state coupling, and many interesting results have been derived. To the best of our knowledge, this is the first paper to investigate the input passivity and output passivity of complex delayed dynamical networks with output coupling. By constructing new Lyapunov functionals, some sufficient conditions ensuring the input passivity and output passivity are established in this paper.

4. Examples

In this section, two illustrative examples are provided to verify the effectiveness of the proposed theoretical results.

Example 1.

Consider a three-order dynamical system as the dynamical node of the complex network (1) which is described by [figure omitted; refer to PDF] Clearly, we can take [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Take [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The matrix [figure omitted; refer to PDF] is chosen as follows: [figure omitted; refer to PDF] Obviously, network (1) is connected, and matrix [figure omitted; refer to PDF] is [figure omitted; refer to PDF]

Next, we analyze the input passivity of complex network (1).

Setting [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] .

We can find the following matrix [figure omitted; refer to PDF] satisfying (9) with [figure omitted; refer to PDF] . Consider the following: [figure omitted; refer to PDF] Hence, it follows from Theorem 4 that complex network (1) with above given parameters is input passive.

Example 2.

Consider a three-order dynamical system as the dynamical node of the complex network (1) which is described by [figure omitted; refer to PDF] Clearly, we can take [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Take [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The matrix [figure omitted; refer to PDF] is chosen as follows: [figure omitted; refer to PDF] Obviously, network (1) is connected, and matrix [figure omitted; refer to PDF] is [figure omitted; refer to PDF]

In the following, we analyze the output passivity of complex network (1).

Setting [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] .

We can find the following matrix [figure omitted; refer to PDF] satisfying (30) with [figure omitted; refer to PDF] . Consider the following: [figure omitted; refer to PDF] By Theorem 6, we know that complex network (1) with above given parameters is output passive.

5. Conclusion

A new complex delayed dynamical network model with output coupling has been introduced. We have considered the input passivity and output passivity of the proposed network model. Some input passivity and output passivity criteria have been established by constructing new Lyapunov functionals. Moreover, two illustrative examples have been provided to verify the correctness and effectiveness of the obtained results. In future work, we shall study the input passivity and output passivity of impulsive complex delayed dynamical networks with output coupling.

Acknowledgments

The authors would like to thank the Associate Editor and anonymous reviewers for their valuable comments and suggestions. This work was supported in part by the Fundamental Research Funds for the Central University (2013-YB-017).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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