PI controller response frequency
You might want to model the rain as a unidirectional disturbance with "popcorn noise" characteristics. Since you have no D (derivative) control characteristic the controller responds only to the error and the integral of error. 

Depending on the system characteristics you will likely see the error increase during a sudden downpour, followed by a correction to zero error as the integral term takes effect. There will be some (likely small) perturbations on the error as individual raindrops impact. You can estimate the maximum from a single drop from basic physics, assuming the controller does nothing to correct immediately. If the downpour stops suddenly, the ball would be expected to overshoot a bit upwards, then come back down to zero average error. 

The controller bandwidth will typically be a lot higher than the closed loop system bandwidth. The proportional term on an analog controller can easily respond far faster than macroscopic normal mechanical systems (hundreds of kHz or MHz). The closed loop system is limited in practice by how much control authority you have. The math will tell you that you can make a battleship turn on a dime with the right controller, but there is not enough power in the engines to make that happen in reality.