I added an integrator to increase the gain at low frequencies (below 5 Hz). In addition, I increased

the band of the differentiator. The schematics for both integrator and differentiator are the following:

The magnetic is stably levitated.

I turned off the light to get rid of 60Hz noise on the photodiode. I tried to measured the

open-loop transfer function of this setup, but somehow the SR560 is always saturate

when I injected the signal from SR785, which produces some weird results at

low-frequencies.

In addition, I found out that when the light is turned on, the levitation

can be stable even when I inverted the sign of the control loop. The control signal

on the osciloscope is the following:

This oscillator is around 120 Hz, which should be  the harmonics of 60 Hz from light pollution.

I am not sure exactly why it is stable when the control-loop sign is flipped. This could

be similar to the Pauli trap in the iron trap, because the coil not only provides a force

but also provides the rigidity. The sign of such rigidity depends on the sign of the control

current. If such oscillating rigidity changes at a frequency much higher than the response

frequency of the magnet, it will stablize  the system simply by significantly increasing

the inertial of the magnet.More investigations are essential to completely understand it.

For information about Pauli trap, one can look at the wikipedia:

http://en.wikipedia.org/wiki/Quadrupole_ion_trap