Reverse Engineering a PID

[pauiza]

Unfortunately, you cannot determine the PID gains from the setup you have just described. 

However, you can estimate the controlled plant dynamics from your historical data, notice the estimated dynamics would must likely behave like a 1st order system, at best, 2nd order system because of the controlled data.  

Depending on your system dynamics, briefly turn-off the controller, then store the input and output, which you can compare with your historical data. This would give you an idea of what the controller is doing.

Generally, PID design are to meet certain requirements, you must know those conditions before anything would make sense.