How to Derive the PID controller Formula

Hi all

I want to know how the formula here (Real-time Interactive Tuning Using Moku:Go’s PID Controller the transfer function for the controller given at 6:53 ) is derived, from the block diagram in Open Loop Tuning lab using Moku:Go’s PID Controller The Figure 4: Block diagram with PID controller

Thank you very much

While the video and the application note you linked are related, they are using different plants. You can see the picture in the app note the tube is much longer while the video using a 3D printed setup which is much smaller and has different air flow mechanics.

You can determine the transfer function given at 6:53 from the bode plot given in the video. The poles and zeros of the PID controller are given as corner and crossover frequencies so you can derive the equation this way.

Relating this to the app note you linked, this equation would be C(s) which is the controller. G(s), the plant’s transfer function, is not as easy to determine but there are various methods to do so. In the app note, an approximation is used where we assume the plant acts as a translating mass system. In the video, the plant’s transfer function is a bit harder to define mathematically, but we could use the Moku:Go Frequency Response Analyzer to determine this.