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

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.