We have shown how to use the closed-loop characteristic equation to determine the closed-loop stability of processes. Except for P-only control, we have not discussed the effect of controller tuning parameters on the feedback system performance (closed-loop response).
Our discussion here is based on open-loop stable processes. To better understand the effect of controller tuning parameters, you should experiment with a number of transfer functions. You can also determine the closed-loop poles as a function of the tuning parameters, for a particular process.