Coordinated motion of Lagrangian systems with auxiliary oscillators under cooperative and cooperative-competitive interactions

The present paper investigates coordinated oscillatory motion of networked Lagrangian systems under diverse interactions. Based on two new auxiliary oscillator systems, we formulate two integral-sliding adaptive controllers to drive the systems to reach disparate coordination behaviors. Specially, in the case of cooperative network, where the correspondence between the agents is cooperating, the systems can achieve complete oscillatory synchronization if interactions network contains a spanning tree. In the case of weighted cooperative-competitive network, cluster oscillatory synchronization issue is solved for Lagrangian systems. Compared with the previous methods, three major advantages for introduction of auxiliary oscillator systems are that, firstly, Lagrangian systems reach the oscillatory synchronization with arbitrary amplitudes and initial phases instead of distributed stabilization or consensus; secondly, synchronization states are explicitly expressed under the integral-sliding controllers; thirdly, multi-partite synchronization is formed on the basis of the newly introduced cooperative-competitive interactions without the balanced couple condition. Finally, two numerical examples are provided to demonstrate effectiveness of the presented algorithms.