I have gone through the entirety of K. Ogata's Modern Control Engineering, and I do not understand how can I translate all the transfer function models into differential equations? For example, how would a closed loop control system with PID compensation look like as differntial equations??
You assign variables for each node that branches off or where signals join (eg the feedback adder). Then you write down the transfer functions between those variables. And as a last step you convert the functions back from Laplace domain into the time domain. Et voilà, you have differential equations.
You actually don't need that many variables. Only input, output and one per integrator and per differentiator is required. But starting off with more variables makes the functions simpler and thus the back transformation. At the end you can always get rid of the excess variables.