- Research
- Open Access
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 control
1 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
References
- Liang, J, Wang, Z, Liu, Y, Liu, X: Global synchronization control of general delayed discrete-time networks with stochastic coupling and disturbances. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 38, 1073-1083 (2008) View ArticleGoogle Scholar
- Cao, J, Wang, Y: Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems. Nonlinear Anal., Real World Appl. 14, 842-851 (2013) View ArticleMathSciNetMATHGoogle Scholar
- Cao, J, Alofi, A, Al-Mazrooei, A, Elaiw, A: Synchronization of switched interval networks and applications to chaotic neural networks. Abstr. Appl. Anal. 2013, 940573 (2013) MathSciNetGoogle Scholar
- Pan, L, Cao, J, Hu, J: Synchronization for complex networks with Markov switching via matrix measure approach. Appl. Math. Model. 39, 5636-5649 (2015) View ArticleMathSciNetGoogle Scholar
- Wang, X, Chen, G: Pinning control of scale-free dynamical networks. Physica A 310, 521-531 (2002) View ArticleMathSciNetMATHGoogle Scholar
- Li, L, Cao, J: Cluster synchronization in an array of coupled stochastic delayed neural networks via pinning control. Neurocomputing 74, 846-856 (2011) View ArticleGoogle Scholar
- Wang, T, Li, T, Yang, X, Fei, S: Cluster synchronization for delayed Lur’e dynamical networks based on pinning control. Neurocomputing 83, 72-82 (2012) View ArticleGoogle Scholar
- Chen, T, Liu, X, Lu, W: Pinning complex networks by a single controller. IEEE Trans. Circuits Syst. I 54, 1317-1326 (2007) View ArticleMathSciNetGoogle Scholar
- Ma, Q, Lu, J: Cluster synchronization for directed complex dynamical networks via pinning control. Neurocomputing 101, 354-360 (2013) View ArticleGoogle Scholar
- Liu, X, Chen, T: Cluster synchronization in directed networks via intermittent pinning control. IEEE Trans. Neural Netw. 22, 1009-1020 (2011) View ArticleGoogle Scholar
- Zhang, H, Li, K, Fu, X: On pinning control of some typical discrete-time dynamical networks. Commun. Nonlinear Sci. Numer. Simul. 15, 182-188 (2010) View ArticleGoogle Scholar
- Mwaffo, V, DeLellis, P, Porfiri, M: Criteria for stochastic pinning control of networks of chaotic maps. Chaos 24, 013101 (2014) View ArticleMathSciNetGoogle Scholar
- Tang, Y, Leung, SYS, Wong, WK, Fang, J: Impulsive pinning synchronization of stochastic discrete-time networks. Neurocomputing 73, 2132-2139 (2010) View ArticleGoogle Scholar
- Deng, L, Wu, Z, Wu, Q: Pinning synchronization of complex network with non-derivative and derivative coupling. Nonlinear Dyn. 73, 775-782 (2013) View ArticleMathSciNetMATHGoogle Scholar
- Lu, J, Kurths, J, Cao, J, Mahdavi, N, Huang, C: Synchronization control for nonlinear stochastic dynamical networks: pinning impulsive strategy. IEEE Trans. Neural Netw. Learn. Syst. 23, 285-292 (2012) View ArticleGoogle Scholar
- Mahdavi, N, Menhaj, MB, Kurths, J, Lu, J, Afshar, A: Pinning impulsive synchronization of complex dynamical networks. Int. J. Bifurc. Chaos 22, 1250239 (2012) View ArticleMathSciNetGoogle Scholar
- Lu, J, Wang, Z, Cao, J, Ho, DWC, Kurths, J: Pinning impulsive stabilization of nonlinear dynamical networks with time-varying delay. Int. J. Bifurc. Chaos 22, 1250176 (2012) View ArticleGoogle Scholar
- Cao, J, Wang, Y, Alofi, A, Al-Mazrooei, A, Elaiw, A: Global stability of an epidemic model with carrier state in heterogeneous networks. IMA J. Appl. Math. 80, 1025-1048 (2015) View ArticleMathSciNetGoogle Scholar
- Zhou, J, Chen, T, Xiang, L: Chaotic lag synchronization of coupled delayed neural networks and its applications in secure communication. Circuits Syst. Signal Process. 27, 833-845 (2005) MathSciNetGoogle Scholar
- Luan, X, Liu, F, Shi, P: Robust finite-time \(H_{\infty}\) control for nonlinear jump systems via neural networks. Circuits Syst. Signal Process. 29, 481-498 (2010) View ArticleMathSciNetMATHGoogle Scholar
- Liu, B, Xia, Y, Mahmoud, MS, Wu, H, Cui, S: New predictive control scheme for networked control systems. Circuits Syst. Signal Process. 31, 945-960 (2012) View ArticleMathSciNetMATHGoogle Scholar
- Li, C, Xu, C, Sun, W, Xu, J, Kurths, J: Outer synchronization of coupled discrete-time networks. Chaos 19, 013106 (2009) View ArticleMathSciNetGoogle Scholar
- Zhang, Q, Lu, J, Zhao, J: Impulsive synchronization of general continuous and discrete-time complex dynamical networks. Commun. Nonlinear Sci. Numer. Simul. 15, 1063-1070 (2010) View ArticleMathSciNetMATHGoogle Scholar
- Yang, X, Cao, J, Ho, DWC: Exponential synchronization of discontinuous neural networks with time-varying mixed delays via state feedback and impulsive control. Cogn. Neurodyn. 9, 113-128 (2015) View ArticleGoogle Scholar
- Rao, P, Wu, Z, Liu, M: Adaptive projective synchronization of dynamical networks with distributed time delays. Nonlinear Dyn. 67, 1729-1736 (2012) View ArticleMathSciNetMATHGoogle Scholar
- Cao, J, Sivasamy, R, Rakkaiyappan, R: Sampled-data \(H_{\infty}\) synchronization of chaotic Lur’e systems with time delay. Circuits Syst. Signal Process. (2015). doi:10.1007/s00034-015-0105-6 View ArticleGoogle Scholar
- Chirikov, BV: A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263-379 (1979) View ArticleMathSciNetGoogle Scholar