Synchronization Of High-dimensional Kuramoto Model

A multi-agent dynamical network is a large scale system composed of many subsystems connected by a network, which is also called a complex dynamical network. Kuramoto model is a typical multi-agent dynamical network, which has extensive applications in brain science and power systems and has been paid great attention by physicists, biologists and control theory scientists.The classical Kuramoto model can be regarded as a dynamical equation limited on the unit circle of the plane. In recent years, the classical Kuramoto model has been generalized to high-dimensional spaces. But generally speaking the research results on the high-dimensional Kuramoto model are not much. Many results on the traditional Kuramoto model have not been generalized to the case of high-dimensional spaces. For the traditional Kuramoto model, there have been lots of applications. But for the high-dimensional Kuramoto model, application results are not much. In the thesis, some theoretical results on the traditional Kuramoto model are generalized to the high-dimensional model and are applied to design consensus protocols for multi-agent systems. In details, the contributions of this thesis are list as follows: 1) As the agents have the same dynamical equation, for complete ?-partite graphs, the synchronization of the high-dimensional Kuramoto model is proved.2) If the agents have different dynamical equations,a positive invariant set is obtained. A new concept called sub-synchronization is given and a new lower bound of the coupling gains is obtained to guarantee the sub-synchronization.3) With the high-dimensional Kuramoto model,a new nonlinear consensus protocol is designed to realize the consensus on ellipsoids in high-dimensional spaces.