Skip to main content
SpringerLink
Log in
Book cover

International Conference on Intelligent Robotics and Applications

ICIRA 2018: Intelligent Robotics and Applications pp 59–70Cite as

Optimizing Pinning Control of Directed Networks Using Spectral Graph Theory

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10984))

Abstract

Pinning control of a complex network aims at aligning the states of all the nodes to an external forcing signal by controlling a small number of nodes in the network. An algebraic graph-theoretic condition has been proposed to optimize pinning control for both undirected and directed network. The problem we are trying to solve in this paper is the optimization of pinning control in directed networks, where the effectiveness of pinning control can be measured by the smallest eigenvalue of the submatrix obtained by deleting the rows and columns corresponding to the pinned nodes from the symmetrized Laplacian matrix of the network when coupling strength and individual node dynamics are given. By analysing the spectral properties using the topology information of the directed network, a necessary condition that ensure the smallest eigenvalue greater than 0 is obtained. Upper bounds and lower bounds of the smallest eigenvalue are proved, and then an algorithm that optimize pinning scheme to improve the smallest eigenvalue is proposed. Illustrative examples are shown to demonstrate our theoretical results in the paper.

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61773175 61403154, and the Natural Science Foundation of Hubei Province under Grant No. 2017CFB426.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Strogatz, S.H.: Exploring complex networks. Nature 410(7), 268–276 (2001)

    Article  Google Scholar 

  2. Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47–97 (2002)

    Article  MathSciNet  Google Scholar 

  3. Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of networks. Adv. Phys. 51(4), 1079–1187 (2002)

    Article  Google Scholar 

  4. Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 988–1001 (2003)

    Article  MathSciNet  Google Scholar 

  5. Dörfler, F., Chertkov, M., Bullo, F.: Synchronization in complex oscillator networks and smart grids. Proc. Natl. Acad. Sci. USA 110(6), 2005–2010 (2013)

    Article  MathSciNet  Google Scholar 

  6. Zanette, D.H.: Dynamics of rumor propagation on small-world networks. Phys. Rev. E 65(4), 041908 (2002)

    Article  Google Scholar 

  7. Wang, X., Chen, G.: Pinning control of scale-free dynamical networks. Phys. A. Stat. Mech. Its Appl. 310(3), 521–531 (2002)

    Article  MathSciNet  Google Scholar 

  8. Li, X., Wang, X., Chen, G.: Pinning a complex dynamical network to its equilibrium. IEEE Trans. Circ. Syst. I Reg. Pap. 51(10), 2074–2087 (2004)

    Article  MathSciNet  Google Scholar 

  9. Zhou, J., Lu, J.-A., Lü, J.: Pinning adaptive synchronization of a general complex dynamical network. Automatica 44(4), 996–1003 (2008)

    Article  MathSciNet  Google Scholar 

  10. Song, Q., Cao, J.: On pinning synchronization of directed and undirected complex dynamical networks. IEEE Trans. Circ. Syst. I Reg. Pap. 57(3), 672–680 (2010)

    Article  MathSciNet  Google Scholar 

  11. Chen, T., Liu, X., Lu, W.: Pinning complex networks by a single controller. IEEE Trans. Circ. Syst. I Reg. Pap. 54(6), 1317–1326 (2007)

    Article  MathSciNet  Google Scholar 

  12. Chen, G.: Pinning control and synchronization on complex dynamical networks. Int. J. Control Autom. Syst. 12(2), 221–230 (2014)

    Article  Google Scholar 

  13. Yu, W., Chen, G., Lü, J., Kurths, J.: Synchronization via pinning control on general complex networks. SIAM J. Control Optim. 51(2), 1395–1416 (2013)

    Article  MathSciNet  Google Scholar 

  14. Wang, J.-L., Wu, H.-N., Huang, T., Ren, S.-Y., Wu, J.: Pinning control for synchronization of coupled reaction-diffusion neural networks with directed topologies. IEEE Trans. Syst. Man Cybern. Syst. 46(8), 1109–1120 (2016)

    Article  Google Scholar 

  15. Chen, L., Yu, X., Sun, C.: Characteristic modeling approach for complex network systems. IEEE Trans. Syst. Man Cybern. Syst. (2017). https://doi.org/10.1109/TSMC.2017.2737438

  16. Wu, C.W.: Localization of effective pinning control in complex networks of dynamical systems. In: Proceedings of the IEEE International Symposium on Circuits and Systems, pp. 2530–2533 (2008)

    Google Scholar 

  17. Xu, C., Zheng, Y., Su, H., Chen, M.Z.Q., Zhang, C.: Cluster consensus for second-order mobile multi-agent systems via distributed adaptive pinning control under directed topology. Nonlinear Dyn. 83(4), 1975–1985 (2016)

    Article  MathSciNet  Google Scholar 

  18. Liu, X., Li, P., Chen, T.: Cluster synchronization for delayed complex networks via periodically intermittent pinning control. Neurocomputing 162, 191–200 (2015)

    Article  Google Scholar 

  19. Cai, S., Jia, Q., Liu, Z.: Cluster synchronization for directed heterogeneous dynamical networks via decentralized adaptive intermittent pinning control. Nonlinear Dyn. 82(1–2), 689–702 (2015)

    Article  MathSciNet  Google Scholar 

  20. Li, H., Chen, G., Xiao, L.: Event-triggered sampling scheme for pinning control in multi-agent networks with general nonlinear dynamics. Neural Comput. Appl. 27(8), 2587–2599 (2016)

    Article  Google Scholar 

  21. Delellis, P., Garofalo, F., Iudice, F.L.: The partial pinning control strategy for large complex networks. Autom. A J. IFAC Int. Fed. Autom. Control 89, 111–116 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Liu, H., Xu, X., Chen, G., Lu, J.-A.: Optimizing pinning control of complex dynamical networks based on spectral properties of grounded Laplacian matrices. arxiv.org/abs/1804.10818v1 [math.OC] (2018)

  23. Pirani, M., Sundaram, S.: On the smallest eigenvalue of grounded Laplacian matrices. IEEE Trans. Autom. Control 61(2), 509–514 (2016)

    MathSciNet  MATH  Google Scholar 

  24. Pirani, M., Sundaram, S.: Spectral properties of the grounded Laplacian matrix with applications to consensus in the presence of stubborn agents. In: Proceedings of the 2014 American Control Conference, pp. 2160–2165 (2014)

    Google Scholar 

  25. Pirani, M., Shahrivar, E. M., Sundaram, S.: Coherence and convergence rate in networked dynamical systems. In: Proceedings of the 54th IEEE Conference on Decision and Control, pp. 968–973 (2015)

    Google Scholar 

  26. Agaev, R., Chebotarev, P.: On the spectra of nonsymmetric Laplacian matrices. Linear Algebra Its Appl. 399(1), 157–168 (2005)

    Article  MathSciNet  Google Scholar 

  27. Hao, H., Barooah, P.: Asymmetric control achieves size-independentstability margin in 1-D flocks. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, pp. 3458–3463 (2011)

    Google Scholar 

  28. Xia, W., Cao, M.: Analysis and applications of spectral properties of grounded Laplacian matrices for directed networks. Automatica 80, 10–16 (2017)

    Article  MathSciNet  Google Scholar 

  29. Bapat, R.B.: Graphs and Matrices. Springer, New York (2010). https://doi.org/10.1007/978-1-84882-981-7

    Book  Google Scholar 

  30. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, New York (1994)

    MATH  Google Scholar 

  31. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, 2nd edn. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-53622-3

    Book  MATH  Google Scholar 

  32. Physicians network dataset - KONECT, April 2017. http://konect.uni-koblenz.de/networks/moreno_innovation

  33. Coleman, J., Katz, E., Menzel, H.: The diffusion of an innovation among physicians. Sociometry 20, 253–270 (1957)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Automation, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Xuanhong Xu, Hui Liu & Jiangqiao Xu

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Jun-An Lu

Authors
  1. Xuanhong Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hui Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jun-An Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jiangqiao Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hui Liu .

Editor information

Editors and Affiliations

  1. University of Newcastle, Callaghan, New South Wales, Australia

    Zhiyong Chen

  2. University of Newcastle, Callaghan, New South Wales, Australia

    Alexandre Mendes

  3. University of Newcastle, Callaghan, New South Wales, Australia

    Yamin Yan

  4. Shenzhen Institutes of Advanced Technology, Shenzhen, China

    Shifeng Chen

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Xu, X., Liu, H., Lu, JA., Xu, J. (2018). Optimizing Pinning Control of Directed Networks Using Spectral Graph Theory. In: Chen, Z., Mendes, A., Yan, Y., Chen, S. (eds) Intelligent Robotics and Applications. ICIRA 2018. Lecture Notes in Computer Science(), vol 10984. Springer, Cham. https://doi.org/10.1007/978-3-319-97586-3_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-97586-3_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-97585-6

  • Online ISBN: 978-3-319-97586-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation