Fixed/Prescribed-Time Synchronization and Energy Consumption for Kuramoto-Oscillator Networks

To evaluate the energy-saving effect of the controller, obtaining upper bounds on energy consumption and control time has become a worthwhile and meaningful issue to study. This article mainly discusses three contents about the Kuramoto oscillator network, including fixed-time synchronization (FxTS), prescribed-time synchronization (PTS), and energy consumption estimation. First, to reach FxTS, two sufficient conditions are proposed to guarantee that the Kuramoto oscillator network can reach fixed-time phase agreement and frequency synchronization. Unlike finite/fixed-time controllers, the prescribed-time controller in this article includes a time-varying function term, which is essential to ensure that the system achieves the prescribed-time phase agreement and frequency synchronization. At the same time, the setting-time for PTS is independent of the system initial values or controller parameters, which expands the application prospects of the system. Then, with limited setting-time as a premise, the energy consumed during the fixed/prescribed-time control process is obtained, which helps to evaluate the working time of the system. Finally, an example of a 5-node network is used to illustrate the effectiveness of FxTS and PTS in Kuramoto-oscillator networks.