Along with de-carbonisation, the penetration of power electronic converters has increased, and the power system has become a power electronic-based power system. In such a situation, the stability of the power system faces great challenges. In the event of a large disturbance, the power grid will lack the ability to maintain a stable voltage and frequency. In order to improve the stability of the power grid, the traditional grid-following (GFL) control is needed to be converted to the grid-forming (GFM) control. This paper analyses the control schemes of the GFL and GFM converters by investigating their state-space models, and the eigenvalue trajectories of both control schemes are shown to analyse the stability of the systems. Moreover, a case study is exemplified to compare the performance of the two control strategies while responding to frequency disturbances. Finally, a time-domain simulation model of a 15 kW grid-connected converter is built in Matlab/Simulink to benchmark the performance of the GFL and GFM converters under different working conditions. The result reveals that the GFL converter may encounter some instabilities when applied in power electronic-based systems, while the GFM converter is more suitable for the weak power grid.