Based on the most widely used proportional-integrator controller of the tip-tilt correction loop in an adaptive optical system, the relationship between the optimal closed-loop bandwidth and the atmosphere turbulence as well as the sensor noise is analyzed. Numerical simulation is also performed by using the time serial signals that follow the power spectrum density of the global tilt of Kolmogorov turbulence. Results show that with different atmosphere turbulences and sensor noises, the correction capability of the tip-tilt correction loop can be brought into full play by choosing appropriate closed-loop bandwidth and the residual variance can be reduced.