- Research
- Open Access
Almost sure synchronization control for stochastic delayed complex networks based on pinning adaptive method
- Tianbo Wang^{1}Email author,
- Shouwei Zhao^{1},
- Wuneng Zhou^{2} and
- Weiqin Yu^{1}
https://doi.org/10.1186/s13662-016-1013-1
© Wang et al. 2016
- Received: 21 May 2016
- Accepted: 25 October 2016
- Published: 29 November 2016
Abstract
This paper investigates the almost sure synchronization control problem for a class of stochastic delayed complex networks by using the stochastic differential equation theory and the Kronecker product technique. Different from the existing works, the considered problem is that all the nodes in the complex networks can synchronize with each other although the target node is unknown. Some sufficient conditions which guarantee the complex networks to have almost sure synchronization are derived and two kinds of controllers are designed, respectively. Finally, a numerical example is given to illustrate the effectiveness of the main results.
Keywords
- complex network
- almost sure synchronization
- stochastic disturbance
- pinning adaptive control
1 Introduction
Complex dynamical networks are composed of a family of interconnected nodes, in which each node denotes an individual element in the network and adjusts its behavior by the information received from its neighbor nodes. They can be used to model some complex nonlinear dynamical systems in science and engineering. Thus, in recent years, complex networks have attracted increasing attention in various fields such as biology [1, 2], sociology [3], and physics [4, 5].
In the dynamical behaviors of complex networks, synchronization motion is one of the important elements. Synchronization means that all the nodes’ action in complex networks will attain the same dynamic behavior along with the time evolution. For example, both a group of fish swarming together and a flock of birds synchronously flying belong to the synchronization phenomena. Up to now, there exists much literature such as [6–27] studying the synchronization control problem of complex networks by using different methods. For instance, [6, 7] study the synchronization control problem of discrete complex dynamical networks with a time varying delay by using the method of partitioning time delay and chief stability function, respectively. For the continuous complex networks with different characters, such as time delayed complex networks [8–10], stochastic complex networks [11, 12], complex networks with switching topology [13–15], there have existed a great deal of papers to study the synchronization control problem. The control methods mainly include pinning control [16–18], impulsive control [19, 20], adaptive control [21–27], etc. The adopted theories mainly include Lyapunov stability theory, the chief stability function method, and M-matrix theory.
It should be noted that most of these works required all the nodes in complex networks to synchronize with the target node or the isolated node given beforehand. If the target node is unknown, then these results and methods could fail to achieve the synchronization because the designed controllers are usually based on the state information of the target node. In fact, when the target node is known, each node in the complex networks could adjust its behavior according to the error with the target node. However, if the target node is unknown, each node can only adjust its behavior according to the information from its adjacent nodes, at the same time, the adjacent nodes are varying. Practically, the phenomena of the unknown target node also exist in the real world. For example, for a complex network consisting of some multi-agents without leader, all the agents achieve consensus by adjusting the information received from its adjacent agents. In addition, sometimes it is difficult to precisely describe the state equation of the target node when systems are disturbed by stochastic noise and transmission time delay. Hence, it is necessary to analyze the synchronization control problem of complex networks with unknown target node.
On the other hand, the stochastic complex networks models are very common and there also exists much literature such as [11, 12] and [24–27] studying synchronization problems of stochastic complex network. However, these papers mainly focus on the synchronization in mean square. For the almost sure synchronization of complex networks, there exist few results. Especially, the complex network which is the almost sure synchronization could not have synchronization in mean square, the relative counter-example can be found in [28–30].
Motivated by the above discussion, in this paper, we will consider the almost sure synchronization control problem for a class of stochastic delayed complex networks. The contributions of our paper are as follows. (i) The almost sure synchronization control other than synchronization in mean square is investigated. (ii) The provided results can suit for the synchronization of complex networks with the target node unknown. (iii) The obtained results only depend on the complex network’s parameters.
The rest of this paper is organized as follows. In Section 2, we introduce the stochastic delayed complex dynamical network model and some useful lemmas. In Section 3, some criteria which ensure that the complex network synchronizes well are derived and some synchronization controllers are given. In Section 4, a numerical example is provided to illustrate the effectiveness of our proposed results. Finally, this paper ends with conclusions in Section 5.
Notation
\(R^{n}\) and \(R^{n\times m}\) denote the n-dimensional Euclidean space and the set of all \(n\times m\) dimensional real matrices, respectively. For a vector \(v=(v_{1},v_{2},\ldots,v_{n})^{T} \in R^{n}\), whose 2-norm is denoted by \(\|v\|_{2}= \sqrt{\sum^{n}_{i=1}v^{2}_{i} }\). \(A^{T}\), \(\operatorname{tr}(A)\), and \(\operatorname{det}(A)\) represent the transpose, trace, and determinant of the matrix A, respectively. \(\lambda_{\mathrm{min}}(A)\) and \(\lambda_{\mathrm{max}}(A)\) represent the minimum and maximum eigenvalues of the matrix A, respectively. \(X\geq Y\) (respectively, \(X>Y\)) means that \(X-Y\) is a symmetric positive semi-definite matrix (respectively, positive definite matrix), where X, Y are symmetric matrices. \(I_{n}\) is the \(n\times n\) identity matrix, ⊗ is the Kronecker product. The set \((\Omega, {\mathcal{F}}, \{{\mathcal{F}}_{t}\}_{t\geq0},{\mathcal{P}})\) denotes the complete probability space with a filtration \(\{ {\mathcal{F}}_{t}\}_{t\geq0}\) satisfying right continuity and \({\mathcal{F}}_{0}\) containing all \({\mathcal{P}}\)-null sets. \(C^{b}_{\mathcal{F}_{0}}([-\tau,0];R^{n})\) denotes the family of all bounded \({\mathcal{F}}_{0}\)-measurable \(C([-\tau,0];R^{n})\) valued random variables. Throughout this paper, all matrices have the appropriate dimensions.
2 Problem formulation and preliminaries
Definition 1
Remark 1
Remark 2
- (H1)Assume that there exist two constants \(M_{1}\geq0\) and \(M_{2}\geq0\) such thatfor any \(\xi_{1}(t), \xi_{2}(t)\in R^{n}\) and \(t>0\).$$\begin{aligned}& \bigl\Vert f\bigl(t,\xi_{1}(t),\xi_{1}(t- \tau)\bigr)-f\bigl(t,\xi_{2}(t),\xi_{2}(t-\tau)\bigr)\bigr\Vert ^{2} \\& \quad \leq M_{1}\bigl\Vert \xi_{1}(t)-\xi_{2}(t) \bigr\Vert ^{2}+M_{2}\bigl\Vert \xi_{1}(t-\tau)- \xi_{2}(t-\tau)\bigr\Vert ^{2} \end{aligned}$$(5)
- (H2)Assume that there exists a constant \(L\geq0\) such thatfor any \(\xi_{1}(t), \xi_{2}(t)\in R^{n}\).$$\bigl\Vert g\bigl(\xi_{1}(t)\bigr)-g\bigl(\xi_{2}(t)\bigr) \bigr\Vert ^{2}\leq L\bigl\Vert \xi_{1}(t)- \xi_{2}(t)\bigr\Vert ^{2} $$
Remark 3
Without loss of generality, in this paper, we assume that the eigenvalues of the matrix C are \(0=\lambda_{1} \geq\lambda_{2} \geq\lambda_{3} \geq\cdots\geq\lambda_{N} \).
Lemma 2
The eigenvalues of the matrix C are composed of the eigenvalues of matrix C̃ and 0.
Proof
From Lemma 2, we know that the eigenvalues of the matrix C̃ are \(\lambda_{2}, \lambda_{3}, \ldots, \lambda_{N}\), respectively.
Lemma 3
[33]
Lemma 4
[34]
- (1)
\((A+B) \otimes C=A \otimes C+B\otimes C\), \(C \otimes(A+B)=C \otimes A+C \otimes B\);
- (2)
\((A\otimes B)^{T}=A^{T} \otimes B^{T}\);
- (3)
\((A\otimes C)(B\otimes D)=AB\otimes CD\);
- (4)
\(\lambda(A\otimes B)=\{\gamma_{i}\theta_{j}, i=1,2,\ldots,n, j=1,2,\ldots,m\}\),
3 Synchronization analysis and control
3.1 Synchronization analysis
Theorem 1
Proof
Remark 4
Inequalities (9) and (10) are linear matrix inequalities and can be easily solved by the LMI’s toolbox in Matlab. Moreover, these inequalities only depend on the networks’ parameters and the time delay. In addition, different from [12, 21], the inner-coupling matrix Γ need not to be a diagonal matrix in this paper.
3.2 Synchronization controller design
In general, a complex network is not able to achieve synchronization without a control input. In this section, we will design appropriate controllers such that the closed-loop complex network has almost sure synchronization. Next, we introduce two methods, respectively.
From Theorem 1, the following result is obtained.
Theorem 2
From Theorem 2, the following corollary is obtained.
Corollary 1
Similar to Theorem 1, we obtain the following result.
Theorem 3
Proof
Remark 5
It is worth mentioning that the pinning control is one of important control methods for complex network and has been studied in some literature such as [16–18]. Obviously, this method also suits for the case of controlling every node in complex network. Specially, while controlling \(N-1\) nodes, inequalities (30) and (31) must exist feasible solutions because \(k^{\ast}\) can be chosen sufficiently large, which shows that complex network (1) could achieve synchronization under the action of adaptive controller (28). Furthermore, the synchronization speed can be adjusted by tuning δ.
4 A numerical example
In this section, we provide a numerical example to illustrate the effectiveness of our proposed methods.
Example 1
5 Conclusions
This paper has investigated the almost sure synchronization control problem for a class of stochastic delayed complex networks based on the stochastic differential equation theory. Some synchronization criteria and two kinds of pinning controllers have been proposed. These results reflect the relation of synchronization to the parameters of complex networks. A numerical example has shown that our method is effective.
This paper investigated the almost sure synchronization control other than synchronization in mean square of complex networks, and the obtained results may be appropriate for the synchronization of complex networks with the target node unknown. Specially, the results obtained only depend on the complex network’s parameters.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (11602134), the Shanghai Natural Science Foundation (16ZR1413700), and the Natural Science Foundation of Shanghai University of Engineering Science (nhrc-2015-06).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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