Nonlinear Dynamics

, Volume 93, Issue 3, pp 1219–1230 | Cite as

Cluster oscillatory synchronization of networked Lagrangian systems with the distributed adaptive observers

  • Liyun Zhao
  • Rui Wang
  • Wen Li
  • Quanjun Wu
Original Paper


This paper investigates cluster synchronization problem for uncertain networked Lagrangian systems with nonidentical oscillatory leaders. Firstly, in the case of positive couplings and in the case of positive and negative couplings, we propose two different distributed adaptive observers. Based on these adaptive observers, two adaptive controllers are developed. Then, some cluster synchronization criteria are given to ensure that the desired cluster synchronization scheme can be arrived. Due to introduction of these two adaptive observers, it is no longer necessary for each follower to obtain the frequency information of the corresponding leader system. Finally, the performance and effective of the provided controllers are verified by some numerical examples.


Cluster synchronization Lagrangian systems Positive couplings Positive and negative couplings 



This work is supported by the National Science Foundation of China (Grant No. 61663035), the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 2015MS0122), and Baotou science and technology project (Grant No. 2017S2003-3-8).

Compliance with ethical standards

Conflict of interest

All the authors declare that they have no conflict of interest.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ScienceInner Mongolia University of Science and TechnologyBaotouChina
  2. 2.School of Mathematics and PhysicsShanghai University of Electric PowerShanghaiChina

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