With the renewable energy continuously access to the microgrid, the microgrid itself does not have the disadvantages of the inertia of the synchronous generator in the traditional grid is increasingly apparent, for which the generalized droop control (GDC) strategy is proposed. In the normal operation of the system, GDC has the advantages of large inertia and small active power overshoot, but when subjected to external frequency perturbations, the grid-supporting inverter with large virtual inertia is prone to large transient active power overshoot and oscillation. To address this problem, the article establishes small-signal models under different perturbations based on the GDC compared with the virtual synchronous generator (VSG) and further proposes two adaptive inertia control strategies: GDC adaptive inertia (GDCAI) and adaptive inertia for operation mode switching (AIOMS). By adjusting the virtual inertia appropriately, the rate of change of frequency (RoCoF) is maintained in the set range while improving the transient performance of the system, the methods of reduce-order of the system model and control parameter design is given, to reduce the frequency fluctuation while greatly reducing the active power overshoot and improving the response speed of the system. The effectiveness of the two control strategies is verified by MATLAB/Simulink simulation and StarSim hardware-in-the-loop (HIL) experiment.