Suppose you have a control card. We do not have access to the control card's code and we do not know which control algorithm is used (PID, PD, PI, Fuzzy, ...). We can only measure the control inputs we apply and the output signals the card generates. For examples a quadrotor card, inputs are yaw, pitch roller and throttle and the outputs are propeller pwm signals. Can we predict the algorithm and coefficients/properties used? Is there such a method or study? There are many ready to use control cards, we can imitate or improve for robot control. I think it is important to develop a method for extraction/prediction of algorithm on control board. Thank you for your help and your views.
What you are trying to do is to identify a system. Since you kind of know what to expect, your system is not black box, it is grey box or white box.
There is a lot of theory on how to identify these systems. The field itself is called System Identification and you can find a lot of books and publications for all kinds of systems (blackbox, greybox, whitebox etc.)