As an important part of a hybrid AC/DC microgrid, the interlinking converter not only needs to allocate the power between the AC microgrid and the DC microgrid, but also needs good dynamic performance. In order to solve the problem that the interlinking converter based on the traditional droop control has small inertia and poor transient performance in the islanded mode, the interlinking converter control strategy based on adaptive droop control is proposed. By introducing the difference between the frequency of the AC microgrid and the voltage of the DC microgrid in the droop coefficient, the system inertia is dynamically increased and the system transient performance is improved. In order to avoid frequent switching of the interlinking converter rectification and the inverter mode, a hysteretic control method is proposed to improve system stability. Finally, the simulation results of Matlab/Simulink show that the adaptive droop control method in the islanded mode can effectively increase the inertia of the system and improve the stability of the dynamic process. The hysteresis control method can effectively reduce the power difference between the AC and DC sides caused by the dead zone addition. This work is supported by National Key Research and Development Program of China (No. 2017YFB0903100) and Science and Technology Project of State Grid Corporation of China (No. 521104170043).