Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Synchronization Analysis and Verification for Complex Networked Systems Under Directed Topology

  • Conference paper
  • First Online:
Complex Networks & Their Applications XII (COMPLEX NETWORKS 2023)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 1144))

Included in the following conference series:

  • 1018 Accesses

Abstract

This article investigates synchronization analysis and verification for complex networked systems with nonlinear coupling. Based on general Lyapunov functions beyond the quadratic form, a less conservative synchronization criterion is proposed for the nonlinear networked systems under the directed topology. Then, the synchronization problem for polynomial networks can be converted into a sum-of-squares programming problem, which falls within the convex programming framework, yielding polynomial Lyapunov functions efficiently to realize the automatic synchronization verification in polynomial time. Finally, the effectiveness of the theoretical results is demonstrated by a numerical example, in which our proposed method can guarantee to achieve synchronization by using a smaller lower bound of coupling strength.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 219.99
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

Similar content being viewed by others

References

  1. Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)

    Article  MathSciNet  Google Scholar 

  2. Cornelius, S.P., Kath, W.L., Motter, A.E.: Realistic control of network dynamics. Nat. Commun. 4, 1–11 (2013)

    Article  Google Scholar 

  3. Brask, J.B., Ellis, S., Croft, D.P.: Animal social networks: an introduction for complex systems scientists. J. Complex Netw. 9(2), 1–19 (2021)

    Google Scholar 

  4. Tanner, H.G., Jadbabaie, A., Pappas, G.J.: Flocking in fixed and switching networks. IEEE Trans. Autom. Control 52(5), 863–868 (2007)

    Article  MathSciNet  Google Scholar 

  5. Vicsek, T., Cziók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75(6), 1226–1229 (1995)

    Article  MathSciNet  Google Scholar 

  6. Panteley, E., Loria, A.: Synchronization and dynamic consensus of heterogeneous networked systems. IEEE Trans. Autom. Control 62(8), 3758–3773 (2017)

    Article  MathSciNet  Google Scholar 

  7. Wang, L., Chen, M.Z.Q., Wang, Q.-G.: Bounded synchronization of a heterogeneous complex switched network. Automatica 56(6), 19–24 (2015)

    Article  MathSciNet  Google Scholar 

  8. Stilwell, D., Bollt, E., Roberson, D.: Sufficient conditions for fast switching synchronization in time-varying network topologies. SIAM J. Appl. Dyn. Syst. 5(1), 140–156 (2006)

    Article  MathSciNet  Google Scholar 

  9. Qin, J., Ma, Q., Yu, X., Wang, L.: On synchronization of dynamical systems over directed switching topologies: an algebraic and geometric perspective. IEEE Trans. Autom. Control 65(12), 5083–5098 (2020)

    Article  MathSciNet  Google Scholar 

  10. He, W., et al.: Multiagent systems on multilayer networks: synchronization analysis and network design. IEEE Trans. Syst., Man, Cybern., Syst. 47(7), 1655–1667 (2017)

    Google Scholar 

  11. Belykh, V.N., Belykh, I.V., Hasler, M.: Connection graph stability method for synchronization of coupled chaotic systems. Phys. D 195(1–2), 159–187 (2004)

    Article  MathSciNet  Google Scholar 

  12. DeLellis, P., di Bernardo, M., Russo, G.: On QUAD, Lipschitz, and contracting vector fields for consensus and synchronization of networks. IEEE Trans. Circ. Syst. I, Reg. Papers 58(3), 576–583 (2011)

    Google Scholar 

  13. Wen, G., Duan, Z., Chen, G. Yu, W.: Consensus tracking of multi-agent systems with Lipschitz-type node dynamics and switching topologies. IEEE Trans. Circuits Syst. I, Reg. Papers 61(2), 499–511 (2014)

    Google Scholar 

  14. Zhang, Z., Yan, W., Li, H.: Distributed optimal control for linear multiagent systems on general digraphs. IEEE Trans. Autom. Control 66(1), 322–328 (2021)

    Article  MathSciNet  Google Scholar 

  15. Xiang, J., Chen, G.: On the V-stability of complex dynamical networks. Automatica 43(6), 1049–1057 (2007)

    Article  MathSciNet  Google Scholar 

  16. Zhao, J., Hill, D.J., Liu, T.: Stability of dynamical networks with non-identical nodes: a multiple V-Lyapunov function method. Automatica 47(12), 2615–2625 (2011)

    Article  MathSciNet  Google Scholar 

  17. Papachristodoulou, A., Prajna, S.: On the construction of Lyapunov functions using the sum of squares decomposition. In Proceedings of the IEEE Conference on Decision and Control, Las Vega, Nevada USA (2002)

    Google Scholar 

  18. Papachristodoulou, A., Prajna, S.: A tutorial on sum of squares techniques for systems analysis. In: Proceedings of the American Control Conference, Portland, USA (2005)

    Google Scholar 

  19. Zhang, S., Song, S., Wang, L., Xue, B.: Stability verification for heterogeneous complex networks via iterative SOS programming. IEEE Control Syst. Lett. 7, 559–564 (2023)

    Article  MathSciNet  Google Scholar 

  20. Liang, Q., She, Z., Wang, L., Su, H.: General Lyapunov functions for consensus of nonlinear multiagent systems. IEEE Trans. Circuits Syst. II, Exp. Briefs 64(10), 1232–1236 (2017)

    Google Scholar 

  21. Zhang, S., Wang, L., Liang, Q., She, Z., Wang, Q.-G.: Polynomial Lyapunov functions for synchronization of nonlinearly coupled complex networks. IEEE Trans. Cybern. 52(3), 1812–1821 (2022)

    Article  Google Scholar 

  22. Liang, Q., Ong, C., She, Z.: Sum-of-squares-based consensus verification for directed networks with nonlinear protocols. Int. J. Robust Nonlinear Control 30(4), 1719–1732 (2020)

    Article  MathSciNet  Google Scholar 

  23. Godsil, C., Royle, G.: Algebraic Graph Theory. Springer-Verlag, New York (2001)

    Book  Google Scholar 

  24. Yu, W., Chen, G., Cao, M.: Consensus in directed networks of agents with nonlinear dynamics. IEEE Trans. Autom. Control 56(6), 1436–1441 (2011)

    Article  MathSciNet  Google Scholar 

  25. Liu, X., Chen, T.: Synchronization analysis for nonlinearly-coupled complex networks with an asymmetrical coupling matrix. Phy. A. 387(16–17), 4429–4439 (2008)

    Article  Google Scholar 

  26. Papachristodoulou, A., Prajna, S.: Analysis of non-polynomial systems using the sum of squares decomposition, vol. 312, pp. 23–43. Springer (2005). https://doi.org/10.1007/10997703_2

  27. Löfberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: IEEE International Symposium on Computer Aided Control Systems Design, Taipei, Taiwan (2004)

    Google Scholar 

  28. Mosek, A.: The MOSEK optimization toolbox for MATLAB manual, Version 7.1 (2015)

    Google Scholar 

  29. Lu, J., Chen, G., Cheng, D., Celikovsky, S.: Bridge the gap between the Lorenz system and the Chen system. Int. J. Bifurcat. Chaos 12(12), 2917–2926 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

The authors gratefully acknowledge the support from the following foundations: the National Natural Science Foundation of China under Grant 61873017 and Grant 61973017, the Academic Excellence Foundation of BUAA for PhD Students, and the Outstanding Research Project of Shen Yuan Honors College, BUAA under Grant 230121101 and Grant 230122104.

Author information

Authors and Affiliations

  1. School of Automation Science and Electrical Engineering, Beihang University, Beijing, China

    Shuyuan Zhang, Lei Wang & Wei Wang

  2. Shen Yuan Honors College, Beihang University, Beijing, China

    Shuyuan Zhang

Authors
  1. Shuyuan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lei Wang .

Editor information

Editors and Affiliations

  1. University of Burgundy, Dijon Cedex, France

    Hocine Cherifi

  2. Thomas J. Watson College of Engineering and Applied Science, Binghamton University, Binghamton, NY, USA

    Luis M. Rocha

  3. IUT Lumière - Université Lyon 2, University of Lyon, Bron, France

    Chantal Cherifi

  4. Department of Economics, Yildiz Technical University, Istanbul, Türkiye

    Murat Donduran

Appendix. Proof of Theorem 1

Appendix. Proof of Theorem 1

Let \(\bar{x}=\sum _{r=1}^{N}\zeta _{r}x_{r}\) and \(\tilde{V}=\sum _{i=1}^{N}\zeta _{i}V(x_{i}-\bar{x})\), where V is defined in Assumption 1. Then, we have

$$\begin{aligned} \dot{\bar{x}}=\sum _{r=1}^{N}\zeta _{r}f(x_{r})-\sigma \sum _{r=1}^{N}\zeta _{r}\sum _{j=1}^{N}L_{rj}h(x_{j}). \end{aligned}$$
(10)

It follows from Lemma 1 that \(\zeta =[\zeta _{1},...,\zeta _{N}]^{T}\) is the left eigenvector of L corresponding to the eigenvalue 0, then we have

$$\begin{aligned} \zeta ^{T}L=\sum _{r=1}^{N}\zeta _{r}L_{rj}=0, \end{aligned}$$
(11)

i.e.,

$$\begin{aligned} \sum _{r=1}^{N}\zeta _{r}\sum _{j=1}^{N}L_{rj}h(x_{j})=0. \end{aligned}$$
(12)

Then, we have

$$\begin{aligned} \dot{x_{i}}-\dot{\bar{x}}=f(x_{i})-\sum _{r=1}^{N}\zeta _{r}f(x_{r})-\sigma \sum _{j=1}^{N}L_{ij}h(x_{j}). \end{aligned}$$
(13)

We now show that \(\tilde{V}\) is a GLF for network (1). Taking the derivative of \(\tilde{V}\), yields that

$$\begin{aligned} \dot{\tilde{V}} &= \sum _{i=1}^{N}\zeta _{i} \nabla V(x_{i}-\bar{x})(\dot{x_{i}}-\dot{\bar{x}}) \nonumber \\ &= \sum _{i=1}^{N}\zeta _{i} \nabla V(x_{i}-\bar{x})(f(x_{i})-\sum _{r=1}^{N}\zeta _{r}f(x_{r})) \nonumber \\ &\ \ \ -\sigma \sum _{i=1}^{N}\zeta _{i} \nabla V(x_{i}-\bar{x})\sum _{j=1}^{N}L_{ij} h(x_{j}) \nonumber \\ &= \sum _{i=1}^{N}\sum _{r=1}^{N} \nabla V(x_{i}-\bar{x}) U_{ir} f(x_{r}) \nonumber \\ &\ \ \ -\sigma \sum _{i=1}^{N}\zeta _{i} \nabla V(x_{i}-\bar{x})\sum _{j=1}^{N}L_{ij} (h(x_{j})-h(\bar{x})) \nonumber \\ &= -\sum _{i=1}^{N-1}\sum _{r=i+1}^{N} G_{\nabla V}(x_{i}-\bar{x},x_{r}-\bar{x}) U_{ir} G_{f}(x_{i},x_{r}) \nonumber \\ &\ \ \ -\sigma \sum _{i=1}^{N}\zeta _{i} \nabla V(x_{i}-\bar{x})\sum _{j=1}^{N}L_{ij} G_{h}(x_{j}, \bar{x}), \end{aligned}$$
(14)

where \(U=(U_{ir}) \in \mathbb {R}^{N \times N}=\varXi -\zeta \zeta ^{T}\), which is a symmetric matrix with zero-sum rows and negative off-diagonal entries satisfying Lemma 3. According to (4) and (5), we have

$$\begin{aligned} \dot{\tilde{V}} &\le -\vartheta \sum _{i=1}^{N-1}\sum _{r=i+1}^{N} U_{ir} G_{q,W} (x_{i}-\bar{x}, x_{r}-\bar{x}) \nonumber \\ &\ \ \ -\sigma \sum _{i=1}^{N}\sum _{j=1}^{N} \zeta _{i} L_{ij} q^{T}(x_{i}-\bar{x})W q(x_{j}-\bar{x}) \nonumber \\ &= \vartheta \sum _{i=1}^{N}\sum _{r=1}^{N} U_{ir} q^{T}(x_{i}-\bar{x})W q(x_{r}-\bar{x}) \nonumber \\ &\ \ \ -\sigma \sum _{i=1}^{N}\sum _{j=1}^{N} \tilde{L}_{ij} q^{T}(x_{i}-\bar{x})W q(x_{j}-\bar{x}) \nonumber \\ &= \vartheta Q^{T}(x-\bar{x})(U \otimes W)Q(x-\bar{x}) \nonumber \\ &\ \ \ -\sigma Q^{T}(x-\bar{x})(\tilde{L}\otimes W)Q(x-\bar{x}) \nonumber \\ &\le Q^{T}(x-\bar{x})((\vartheta \varXi -\sigma \tilde{L})\otimes W)Q(x-\bar{x}), \end{aligned}$$
(15)

where \(Q(x-\bar{x})=[q^{T}(x_{1}-\bar{x}),\ldots ,q^{T}(x_{N}-\bar{x})]^{T}\). Based on Eq. (2), one sees that \(\dot{\tilde{V}} \le (\vartheta -\sigma a_{\zeta }(L))Q^{T}(x-\bar{x})(\varXi \otimes W)Q(x-\bar{x})\). Thus, if \(\vartheta -\sigma a_{\zeta }(L)<0\) holds, we have \(\dot{\tilde{V}}<0\), which indicates that the synchronization of network (1) under the directed topology is achieved.    \(\blacksquare \)

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Zhang, S., Wang, L., Wang, W. (2024). Synchronization Analysis and Verification for Complex Networked Systems Under Directed Topology. In: Cherifi, H., Rocha, L.M., Cherifi, C., Donduran, M. (eds) Complex Networks & Their Applications XII. COMPLEX NETWORKS 2023. Studies in Computational Intelligence, vol 1144. Springer, Cham. https://doi.org/10.1007/978-3-031-53503-1_36

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-53503-1_36

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-53502-4

  • Online ISBN: 978-3-031-53503-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation