How to infer equivalent PID controller coefficients from an existing black box ccontroller?
If response of any linear time invariant system to an input is the convolution of the input with the impulse response of the system.

So if you have measured the impulse response then you can find the output using convolution. 

If you cant create the conditions to accurately measure an impulse response you can measure the step response.  The impulse response is then the derivative of the step response.

http://lpsa.swarthmore.edu/Transient/TransInputs/TransImpulseTime.html

Obviously as time progresses a brute force convolution requires too much computation because you would be numerically integrating over ever longer times.

The solution is typically to either limit the scope of the convolution in time by modeling the system as a Finite Impulse Response system (FIR).

The other way is to use some sort of mathematical shortcut (such as recursion) to determine the next output based on the input signal and the prior output.  This is Infinite Impulse Response (IIR).