Skip to main content
SpringerLink
Log in

Convergence, Consensus and Synchronization of Complex Networks via Contraction Theory

  • Chapter
  • First Online:
Complex Systems and Networks

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

This chapter reviews several approaches to study convergence of networks of nonlinear dynamical systems based on the use of contraction theory. Rather than studying the properties of the collective asymptotic solution of interest, the strategy focuses on finding sufficient conditions for any pair of trajectories of two agents in the network to converge towards each other. The key tool is the study, in an appropriate metric, of the matrix measure of the agents’ or network Jacobian. The effectiveness of the proposed approach is illustrated via a set of representative examples.

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

Access this chapter

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

References

  1. Cao, Y., Yu, W., Ren, W., Chen, G.: An overview of recent progress in the study of distributed multi-agent coordination. IEEE Trans. Ind. Inf. 9(1), 427–438 (2013)

    Article  Google Scholar 

  2. Cornelius, S.P., Kath, W.L., Motter, A.E.: Realistic control of network dynamics. Nat. Commun. 4(1942) (2013)

    Google Scholar 

  3. Liu, Y., Slotine, J., Barabasi, A.L.: Controllability of complex networks. Nature 473(7346), 167–173 (2011)

    Article  Google Scholar 

  4. Pecora, L.M., Carroll, Tl: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80(10), 2019–2112 (1998)

    Article  Google Scholar 

  5. Li, Z., Chen, G.: Global synchronization and asymptotic stability of complex dynamical networks. IEEE Trans. Circuits Syst. II: Express Briefs 53(1), 28–33 (2006)

    Article  Google Scholar 

  6. McMillen, D., Kopell, N., Hasty, J., Collins, J.J.: Synchronizing genetic relaxation oscillators by intercell signaling. Proc. Natl. Acad. Sci. 99(2), 679–684 (2002)

    Google Scholar 

  7. Lohmiller, W., Slotine, J.J.E.: On contraction analysis for non-linear systems. Automatica 34(6), 683–696 (1998)

    Google Scholar 

  8. Russo, G., di Bernardo, M., Slotine, J.J.: A graphical algorithm to prove contraction of nonlinear circuits and systems. IEEE Trans. Circuits Syst. 58(2), 336–348 (2011)

    Google Scholar 

  9. Russo, G., di Bernardo, M., Sontag, E.D.: Global entrainment of transcriptional systems to periodic inputs. PLoS Comput. Biol. 6(4), e1000739 (2010)

    Google Scholar 

  10. Forni, F., Sepulchre, R.: A differential Lyapunov framework for contraction analysis. IEEE Trans. Autom. Control 59(3), 614–628 (2014)

    Google Scholar 

  11. Simpson-Porco, J.W., Bullo, F.: Contraction theory on Riemannian manifolds. Syst. Control Lett. 65, 74–80 (2014)

    Article  MathSciNet  Google Scholar 

  12. Aminzare, Z., Sontag, E.D.: Contraction methods for nonlinear systems: a brief introduction and some open problems. In: Proceedings of IEEE Conference on Decision and Control, pp. 3835–3847 (2014)

    Google Scholar 

  13. Aminzare, Z., Sontag, E.D.: Remarks on diffusive-link synchronization using non-Hilbert logarithmic norms. In: Proceedings of IEEE Conference on Decision and Control, pp. 6086-6091 (2014)

    Google Scholar 

  14. Wang, W., Slotine, J.J.: On partial contraction analysis for coupled nonlinear oscillators. Biol. cybern. 92(1), 38–53 (2005)

    Google Scholar 

  15. Pham, Q.C., Slotine, J.J.: Stable concurrent synchronization in dynamic system networks. Neural Netw. 20(1), 62–77 (2007)

    Article  MATH  Google Scholar 

  16. Russo, G., di Bernardo, M.: How to synchronize biological clocks. J. Comput. Biol. 16(2), 379–393 (2009)

    Google Scholar 

  17. Russo, G., di Bernardo, M.: Solving the rendezvous problem for multi-agent systems using contraction theory. In: Proceedings of the IEEE Conference on Decision and Control, pp. 5821–5826 (2009)

    Google Scholar 

  18. Russo, G., Slotine, J.J.E.: Global convergence of quorum-sensing networks. Phys. Rev. E 82, 041919 (2010)

    Article  MathSciNet  Google Scholar 

  19. Russo, G., di Bernardo, M., Sontag, E.D.: A contraction approach to the hierarchical analysis and design of networked systems. IEEE Trans. Autom. Control 58(5), 1328–1331 (2013)

    Google Scholar 

  20. Hartman, P.: On stability in the large for systems of ordinary differential equations. Can. J. Math. 13(3), 480–492 (1961)

    Google Scholar 

  21. Lewis, D.C.: Metric properties of differential equations. Am. J. Math. 71(2), 294–312 (1949)

    Google Scholar 

  22. Pavlov, A., Pogromsky, A., van de Wouw, N., Nijmeijer, H.: Convergent dynamics, a tribute to Boris Pavlovich Demidovich. Syst. Control Lett. 52(3), 257–261 (2004)

    Article  MATH  Google Scholar 

  23. Angeli, D.: A Lyapunov approach to incremental stability properties. IEEE Trans. Autom. Control 47(3), 410–421 (2002)

    Article  MathSciNet  Google Scholar 

  24. Lohmiller, W., Slotine, J.J.E.: Contraction analysis of non-linear distributed systems. Int. J. Control 78(9), 678–688 (2005)

    Google Scholar 

  25. Slotine, J.J., Li, W.: Applied Nonlinear Control. Prentice Hall, Englewood Cliffs (1990)

    Google Scholar 

  26. Bertsekas, D., Tsitsiklis, J.: Parallel and Distributed Computation: Numerical Methods. Prentice-Hall, Englewood Cliffs (1989)

    MATH  Google Scholar 

  27. Jouffroy, J.: Some ancestors of contraction analysis. In: Proceedings of the IEEE Conference on Decision and Control, pp. 5450–5455 (2005)

    Google Scholar 

  28. Vidyasagar, M.: Nonlinear Systems Analysis. SIAM, Philadelphia (2002)

    Google Scholar 

  29. Khalil, H.K.: Nonlinear Systems. Prentice Hall, Upper Saddle River (2002)

    MATH  Google Scholar 

  30. DeLellis, P., di Bernardo, M., Gorochowski, T.E., Russo, G.: Synchronization and control of complex networks via contraction, adaptation and evolution. Circuits Syst. Mag. IEEE. 10(3), 64–82 (2010)

    Google Scholar 

  31. Dorfler, F., Bullo, F.: Synchronization in complex networks of phase oscillators: a survey. Automatica 50(6), 1539–1564 (2014)

    Google Scholar 

  32. Olfati-Saber, R., Fax, J.A., Murray, R.M.: Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95(1), 215–233 (2007)

    Article  Google Scholar 

  33. Ren, W., Cao, Y.: Distributed Coordination of Multi-agent Networks. Springer, London (2011)

    Book  MATH  Google Scholar 

  34. Elowitz, M.B., Leibler, S.: A synthetic oscillatory network of transcriptional regulators. Nature 403(6767), 335–338 (2000)

    Article  Google Scholar 

  35. Garcia-Ojalvo, J., Elowitz, M.B., Strogatz, S.H.: Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing. Proc. Natl. Acad. Sci. USA 101(30), 10955–10960 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Russo, G., Di Bernardo, M.: Contraction theory and master stability function: linking two approaches to study synchronization of complex networks. IEEE Trans. Circuits Syst. 56(2), 177–181 (2009)

    Article  MATH  Google Scholar 

  37. Russo, G., Di Bernardo, M.: Design of coupling protocols for consensus and cluster synchronization using contraction theory (In preparation)

    Google Scholar 

  38. FitzHugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophys. 17(4), 257–278 (1955)

    Article  Google Scholar 

  39. Golubitsky, M., Stewart, I., Török, A.: Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J. Appl. Dyn. Syst. 4(1), 78–100 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Russo, G., di Bernardo, M., Sontag, E.D.: Stability of networked systems: a multi-scale approach using contraction. In: Proceedings of IEEE Conference on Decision and Control, pp 6559–6564 (2010)

    Google Scholar 

  41. Karimi-Ghartemani, M., Iravani, M.R.: A method for synchronization of power electronic converters in polluted and variable-frequency environments. IEEE Trans. Power Syst. 19(3), 1263–1270 (2004)

    Article  Google Scholar 

  42. Chopra, A.K.: Dynamics of Structures. Prentice Hall, Upper Saddle River (2011)

    Google Scholar 

  43. di Bernardo, M., Budd, C., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications, vol. 163. Springer Science & Business Media, London (2008)

    Google Scholar 

  44. di Bernardo, M., Liuzza, D., Russo, G.: Contraction analysis for a class of nondifferentiable systems with applications to stability and network synchronization. SIAM J. Control Optim. 52(5), 3203–3227 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lu, W., di Bernardo, M.: Contraction and incremental stability of switched Carathéodory systems using multiple norms (Submitted to Automatica)

    Google Scholar 

  46. di Bernardo, M., Liuzza, D.: Incremental stability of planar Filippov systems. In: Proceedings of the IEEE European Control Conference, pp 3706–3711 (2013)

    Google Scholar 

  47. di Bernardo, M., Fiore, D.: Incremental stability of bimodal Filippov systems in \({\mathbb{R}}^{n}\). In: Proceedings of IEEE Conference on Decision and Control, pp 4679–4684 (2014)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering and Information Technology, University of Naples Federico II, Via Claudio 21, 80125, Naples, Italy

    Mario di Bernardo, Davide Fiore, Giovanni Russo & Francesco Scafuti

Authors
  1. Mario di Bernardo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Davide Fiore
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Giovanni Russo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Francesco Scafuti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mario di Bernardo .

Editor information

Editors and Affiliations

  1. Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Jinhu Lü

  2. School of Electrical and Computer Engineering, RMIT University, Melbourne, Australia

    Xinghuo Yu

  3. Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China

    Guanrong Chen

  4. Department of Mathematics, Southeast University, Nanjing, China

    Wenwu Yu

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

di Bernardo, M., Fiore, D., Russo, G., Scafuti, F. (2016). Convergence, Consensus and Synchronization of Complex Networks via Contraction Theory. In: Lü, J., Yu, X., Chen, G., Yu, W. (eds) Complex Systems and Networks. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47824-0_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-47824-0_12

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-47823-3

  • Online ISBN: 978-3-662-47824-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation