Synchronization Analysis of Multirate Kuramoto Networks Based on an Energy Function Method

The Kuramoto oscillator network is able to model the synchronization dynamics for a variety of practical systems. This thesis focuses on the synchronization problems of a generalized Kuramoto model consisting of first- and second-order Kuramoto oscillators, i.e., a multirate Kuramoto network. Based on an energy function method, we propose analytic conditions for synchronizing multirate Kuramoto systems with various configurations. These conditions explicitly relate to the network characteristics. The contributions of this thesis are as follows. Firstly, we present synchronization conditions in estimating a positively invariant set (PIS) and calculating lower bounds of the coupling strengths for an undirected multirate Kuramoto network. We show that the phase differences of oscillators with uniform natural frequencies converge into zero when its initial state values are in the estimated PIS. When the natural frequencies are non-uniform, lower bounds on the coupling strengths between the oscillators are derived to ensure the existence of a PIS and the ultimate boundedness on the relative phases. For an arbitrary topology, two different conditions on the coupling strengths are obtained by using graph spectral properties, or by analysing the graph path. These methods provide alternatives in checking the coupling strength conditions. Secondly, three types of controllers are applied to synchronize an undirected multirate Kuramoto network with weak coupling strengths. These approaches include distributed, decentralized and pinning controls. Distributed control requires the global system information by using a communication network, whereas decentralized control only needs local information. These two types of controllers are located in each oscillator. In contrast, pinning control is applied to some selected oscillators. Proposed methods increase the flexibility in implementing controllers. We design proportional-like control laws using the phase information of oscillators. Lower bounds are derived on the control gains to regulate the phase differences of oscillators within and into desired regions. Finally, stability conditions are given for a directed multirate Kuramoto network to achieve boundedness and ultimate boundedness on the system trajectories, which can be viewed as a relaxed-type. Besides, we show that a directed network achieves complete synchronization if the oscillators are with zero natural frequencies