Impulsive pinning synchronization of discrete-time network
- Zhaoyan Wu^{1}Email author and
- Hengjun Wang^{1}
https://doi.org/10.1186/s13662-016-0766-x
© Wu and Wang 2016
Received: 8 August 2015
Accepted: 20 January 2016
Published: 2 February 2016
Abstract
Combining impulsive and pinning control, we investigate the synchronization problem of discrete-time network. In the proposed pinning control scheme, the controlled nodes are chosen according to the norm of the synchronization errors at different impulsive instants. Based on the Lyapunov function method and mathematical analysis technique, two synchronization criteria with respect to the impulsive gains and intervals are analytically derived. Both undirected and directed discrete-time networks coupled with Chirikov standard maps are performed in numerical examples to verify the effectiveness of the derived results.
Keywords
synchronization discrete-time network impulsive control pinning control1 Introduction
Over the past decade, for better modeling and describing the large-scale physical systems consisting of interactive individuals, many kinds of dynamical networks coupled with continuous- or discrete-time dynamical systems have been introduced [1–26]. For example, discrete-time networks are used to model digitally transmitted signals in a dynamical way [1]. Synchronization, as one of the most important and interesting collective behaviors, has been widely studied, and many valuable results have been obtained. In fact, dynamical networks usually cannot synchronize themselves or synchronize with given orbits without external control. Therefore, many control schemes are proposed to design effective controllers for achieving network synchronization, such as feedback control, intermittent control, impulsive control, pinning control, and so on.
In real world, many complex systems contain large numbers of interactive individuals, that is, the corresponding networks consisting of large numbers of nodes. For this case, applying controllers to all nodes is expensive and even impracticable. Pinning control, as an effective control scheme for reducing the number of controlled nodes, has been extensively used to investigate network synchronization, and many valuable results have been obtained [5–17]. Chen et al. [8] investigated pinning synchronization of complex network using only one controller. Liu et al. [10] studied cluster synchronization in directed networks through combining intermittent with pinning control schemes. Zhang et al. [11] studied pinning control of some typical discrete-time dynamical networks. Mwaffo et al. [12] studied stochastic pinning control of networks of chaotic maps. In [15–17], authors studied synchronization of continuous-time networks through combining impulsive with pinning control schemes. Wherein, the impulsively controlled nodes are chosen according to the norm of the synchronization errors at distinct control instants. As we know, research on synchronization of continuous- and discrete-time dynamical networks has significant differences. Therefore, extension of the results obtained in [15–17] for continuous-time networks to discrete-time networks is an important issue and deserves further study.
Motivated by the above discussions, in this paper, we investigate the impulsive pinning synchronization of discrete-time dynamical network. At different impulsive instants, the pinned nodes are chosen according to the norm of synchronization errors. Since the synchronization errors are time-varying, the pinned nodes become nonidentical at different impulsive instants. Based on the Lyapunov function method and mathematical analysis approach, we analytically derive two synchronization criteria with respect to the impulsive intervals and gains. The obtained results are verified to be effective and correct by two numerical examples.
The rest of this paper is organized as follows. Section 2 introduces the network model and some preliminaries. Section 3 studies the synchronization of discrete-time network via impulsive pinning control. Section 4 provides two numerical examples to verify the effectiveness of the derived results. Section 5 concludes the paper.
2 Model and preliminaries
Assumption 1
3 Main result
In what follows, let \(e(k)=(e_{1}^{T}(k),e_{2}^{T}(k),\ldots,e_{N}^{T}(k))^{T}\), \(F(e(k))=((f(x_{1})-f(s))^{T},\ldots, (f(x_{N})-f(s))^{T})^{T}\), \(\mathcal {T}_{l}=\mathcal{I}_{l}-1-\mathcal{I}_{l-1}\) be the impulsive intervals, \(\lambda^{2}\) be the largest eigenvalue of the matrix \((C^{T}\otimes\Gamma )(C\otimes\Gamma)\) with \(\lambda>0\), \(\beta_{l}=(1+b_{l})^{2}\), and \(\rho _{l}=1-(1-\beta_{l})p/N\).
Theorem 1
Proof
If the impulsive gains \(b_{l}\) and the impulsive intervals \(\mathcal {T}_{l}\) are chosen as a constant \(b_{0}\) and a positive constant \(\mathcal {T}_{0}\), the following corollary can be easily derived.
Corollary 1
Remark 1
In many existing results about pinning control, the outer coupling matrix is assumed to be irreducible, that is, the network is connected. From the proof of Theorem 1 it is clear that the outer coupling matrix C need not be symmetrical or irreducible. That is, the obtained results can be applied to more general networks and even to disconnected networks.
4 Numerical simulations
Example 1
Example 2
5 Conclusions
In this paper, we studied the synchronization problem of discrete-time network via impulsive pinning control. In the proposed control scheme, the impulsive controllers are applied to only a fraction of nodes, and the pinned nodes are chosen according to the norm of the synchronization errors at different control instants. Two sufficient conditions for achieving synchronization are derived based on Lyapunov function method and mathematical analysis technique. From the conditions, for any given networks, we can easily estimate the largest impulsive interval by fixing the impulsive gains and the number of pinned nodes p. Finally, two numerical examples are performed to illustrate the obtained results.
Declarations
Acknowledgements
This work is jointly supported by the National Natural Science Foundation of China under Grant No. 61463022 and the Natural Science Foundation of Jiangxi Educational Committee under Grant No. GJJ14273.
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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