Skip to main content
SpringerLink
Log in

On PID control for synchronization of complex dynamical network with delayed nodes

  • Article
  • Published:
Science China Technological Sciences Aims and scope Submit manuscript

Abstract

Over the past two decades, synchronization, as an interesting collective behavior of complex dynamical networks, has been attracting much attention. To reveal and analyze the inherent mechanism of synchronization in complex dynamical networks with time delays in nodes, this paper attempts to use PD and PI control protocols to achieve synchronization. Based on a classical network model, we investigate the PD and PI control for synchronization of complex dynamical networks with delayed nodes and obtain some sufficient conditions. By using Lyapunov functions and appropriate state transformations, we prove that global synchronization can be achieved via the above control protocols. Finally, some simulation examples are illustrated to validate the effectiveness of the proposed theoretical results.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Boccaletti S, Latora V, Moreno Y, et al. Complex networks: Structure and dynamics. Phys Rep, 2006, 424: 175–308

    Article  MathSciNet  MATH  Google Scholar 

  3. Liu K, Zhu H, Lu J. Cooperative stabilization of a class of LTI plants with distributed observers. IEEE Trans Circuits Syst I, 2017, 64: 1891–1902

    Article  MathSciNet  Google Scholar 

  4. Rubenstein M, Cornejo A, Nagpal R. Programmable self-assembly in a thousand-robot swarm. Science, 2014, 345: 795–799

    Article  Google Scholar 

  5. Czirok A, Vicsek T. Collective behavior of interacting self-propelled particles. Physica A, 2000, 281: 17–29

    Article  Google Scholar 

  6. Barabási A L, Gulbahce N, Loscalzo J. Network medicine: A network-based approach to human disease. Nat Rev Genet, 2011, 12: 56–68

    Article  Google Scholar 

  7. Wei X, Wu X, Chen S, et al. Cooperative epidemic spreading on a two-layered interconnected network. SIAM J Appl Dyn Syst, 2018, 17: 1503–1520

    Article  MathSciNet  MATH  Google Scholar 

  8. Wang Y F, Wu X Q, Feng H, et al. Topology inference of uncertain complex dynamical networks and its applications in hidden nodes detection. Sci China Tech Sci, 2016, 59: 1232–1243

    Article  Google Scholar 

  9. Zhang S, Wu X, Lu J A, et al. Recovering structures of complex dynamical networks based on generalized outer synchronization. IEEE Trans Circuits Syst I, 2014, 61: 3216–3224

    Article  Google Scholar 

  10. Watts D J, Strogatz S H. Collective dynamics of ‘small-world’ networks. Nature, 1998, 393: 440–442

    Article  MATH  Google Scholar 

  11. Tan S, Lü J, Lin Z. Emerging behavioral consensus of evolutionary dynamics on complex networks. SIAM J Control Optim, 2016, 54: 3258–3272

    Article  MathSciNet  MATH  Google Scholar 

  12. Arenas A, Díaz-Guilera A, Kurths J, et al. Synchronization in complex networks. Phys Rep, 2008, 469: 93–153

    Article  MathSciNet  Google Scholar 

  13. Ning D, Wu X, Lu J, et al. Driving-based generalized synchronization in two-layer networks via pinning control. Chaos, 2015, 25: 113104

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  15. Liu K X, Wu L L, Lu J H, et al. Finite-time adaptive consensus of a class of multi-agent systems. Sci China Tech Sci, 2016, 59: 22–32

    Article  Google Scholar 

  16. Wang X F, Chen G. Pinning control of scale-free dynamical networks. Physica A, 2002, 310: 521–531

    Article  MathSciNet  MATH  Google Scholar 

  17. Mei G, Wu X, Ning D, et al. Finite-time stabilization of complex dynamical networks via optimal control. Complexity, 2016, 21: 417–425

    Article  MathSciNet  Google Scholar 

  18. Li Y, Wu X, Lu J, et al. Synchronizability of duplex networks. IEEE Trans Circuits Syst II, 2016, 63: 206–210

    Article  Google Scholar 

  19. Bidram A, Lewis F L, Davoudi A. Distributed control systems for small-scale power networks: Using multiagent cooperative control theory. IEEE Control Syst, 2014, 34: 56–77

    MathSciNet  Google Scholar 

  20. Chen G R. Problems and challenges in control theory under complex dynamical network environments. Acta Automat Sin, 2013, 39: 312–321

    Article  Google Scholar 

  21. Zhou J, Lu J, Lu J. Adaptive synchronization of an uncertain complex dynamical network. IEEE Trans Automat Contr, 2006, 51: 652–656

    Article  MathSciNet  MATH  Google Scholar 

  22. Lü J, Chen G, Bernardo M di. On some recent advances in synchronization and control of complex networks. In: Proceedings of 2010 IEEE International Symposium on Circuits and Systems. Paris, 2010. 3773–3776

    Google Scholar 

  23. Yu W W, DeLellis P, Chen G R, et al. Distributed adaptive control of synchronization in complex networks. IEEE Trans Automat Contr, 2012, 57: 2153–2158

    Article  MathSciNet  MATH  Google Scholar 

  24. Yu W W, Chen G R, Cao M, et al. Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics. IEEE Trans Syst Man Cybern B, 2010, 40: 881–891

    Article  Google Scholar 

  25. Li H, Liao X, Dong T, et al. Second-order consensus seeking in directed networks of multi-agent dynamical systems via generalized linear local interaction protocols. Nonlinear Dyn, 2012, 70: 2213–2226

    Article  MathSciNet  MATH  Google Scholar 

  26. Chen Y, Lü J, Yu X, et al. Consensus of discrete-time second-order multiagent systems based on infinite products of general stochastic matrices. SIAM J Control Optim, 2013, 51: 3274–3301

    Article  MathSciNet  MATH  Google Scholar 

  27. Guo W, Lü J, Chen S, et al. Second-order tracking control for leader-follower multi-agent flocking in directed graphs with switching topology. Syst Control Lett, 2011, 60: 1051–1058

    Article  MathSciNet  MATH  Google Scholar 

  28. Li C, Xu H, Liao X, et al. Synchronization in small-world oscillator networks with coupling delays. Physica A, 2004, 335: 359–364

    Article  MathSciNet  Google Scholar 

  29. Lu W, Chen T, Chen G. Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay. Physica D, 2006, 221: 118–134

    Article  MathSciNet  MATH  Google Scholar 

  30. Qin J, Gao H, Zheng W X. Second-order consensus for multi-agent systems with switching topology and communication delay. Syst Control Lett, 2011, 60: 390–397

    Article  MathSciNet  MATH  Google Scholar 

  31. Qian Y, Wu X, Lü J, et al. Consensus of second-order multi-agent systems with nonlinear dynamics and time delay. Nonlinear Dyn, 2014, 78: 495–503

    Article  MathSciNet  MATH  Google Scholar 

  32. Mensour B, Longtin A. Synchronization of delay-differential equations with application to private communication. Phys Lett A, 1998, 244: 59–70

    Article  MATH  Google Scholar 

  33. Zhang Q J, Lu J A, Lu J H, et al. Adaptive feedback synchronization of a general complex dynamical network with delayed nodes. IEEE Trans Circuits Syst II, 2008, 55: 183–187

    Article  Google Scholar 

  34. Wei X, Wu X, Lu J, et al. Counterpart synchronization of duplex networks with delayed nodes and noise perturbation. J Stat Mech, 2015, 2015: P11021

    Article  MathSciNet  Google Scholar 

  35. Li Z. Global synchronization of general delayed dynamical networks. Chin Phys Lett, 2007, 24: 1869–1872

    Article  Google Scholar 

  36. Jiang M, Shen Y, Jian J, et al. Stability, bifurcation and a new chaos in the logistic differential equation with delay. Phys Lett A, 2006, 350: 221–227

    Article  MathSciNet  MATH  Google Scholar 

  37. Astrom K J, Hagglund T. PID Controllers: Theory, Design and Tuning. 2nd ed. Research Triangle Park: Instrument Society of America, 1995

    Google Scholar 

  38. Zhao C, Guo L. PID controller design for second order nonlinear uncertain systems. Sci China Inf Sci, 2017, 60: 022201

    Article  MathSciNet  Google Scholar 

  39. Cheng L, Wang Y, Ren W, et al. Containment control of multiagent systems with dynamic leaders based on a PIn-type approach. IEEE Trans Cybern, 2016, 46: 3004–3017

    Article  Google Scholar 

  40. Burbano D, Bernardo M di. Consensus and synchronization of complex networks via proportional-integral coupling. In: 2014 IEEE International Symposium on Circuits and Systems. Melbourne VIC, 2014. 1796–1799

    Google Scholar 

  41. Burbano D A, Bernardo M di. Distributed PID control for consensus of homogeneous and heterogeneous networks. IEEE Trans Control Netw Syst, 2015, 2: 154–163

    Article  MathSciNet  MATH  Google Scholar 

  42. Burbano D. Distributed PID control for consensus and synchronization of multi-agent networks. Dissertation for Doctoral Degree. Naples: University of Naples Federico II, 2015

    MATH  Google Scholar 

  43. Gu H, Wang X, Lü J, et al. Synchronization of complex network with delayed nodes via proportional-derivative control. In: IECON 2017—43rd Annual Conference of the IEEE Industrial Electronics Society. Beijing, 2017. 5610–5615

    Google Scholar 

  44. Silva G J, Datta A, Bhattacharyya S P. PID Controllers for Time-Delay Systems. Boston: Birkhauser, 2005

    Book  MATH  Google Scholar 

  45. Lu W, Chen T. New approach to synchronization analysis of linearly coupled ordinary differential systems. Physica D, 2006, 213: 214–230

    Article  MathSciNet  MATH  Google Scholar 

  46. Horn R A, Johnson C R. Topics in Matrix Analysis. Cambridge: Cambridge University Press, 1991

    Book  MATH  Google Scholar 

  47. Ren W, Bear R W, Atkins E M. Information consensus in multivehicle cooperative control. IEEE Control Syst, 2007, 27: 71–82

    Article  Google Scholar 

  48. Boyd S, Ghaoui L E, Feron E, et al. Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994

    Book  MATH  Google Scholar 

  49. Bernstein D S. Matrix Mathematics: Theory, Facts, and Formulas. 2nd ed. Princeton: Princeton University Press, 2008

    Google Scholar 

  50. Zhang L, Zhu Y, Zheng W X. Synchronization and state estimation of a class of hierarchical hybrid neural networks with time-varying delays. IEEE Trans Neural Netw Learning Syst, 2016, 27: 459–470

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    HaiBo Gu & JinHu Lü

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    HaiBo Gu & JinHu Lü

  3. School of Automation Science and Electrical Engineering, State Key Laboratory of Software Development Environment, and Beijing Advanced Innovation Center for Big Data and Brain Machine Intelligence, Beihang University, Beijing, 100083, China

    JinHu Lü

  4. Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA, 22904-4743, USA

    ZongLi Lin

Authors
  1. HaiBo Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. JinHu Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. ZongLi Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JinHu Lü.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gu, H., Lü, J. & Lin, Z. On PID control for synchronization of complex dynamical network with delayed nodes. Sci. China Technol. Sci. 62, 1412–1422 (2019). https://doi.org/10.1007/s11431-018-9379-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11431-018-9379-8

Keywords

Search

Navigation