Q$2.$ Figure Q$2$ shows the amplitude observations from an experimental frequency

response test.

a$)$ Suggest a transfer function model for the system;

b$)$ Explain why the ultimate gain and ultimate period cannot be determined for

the system;

c$)$ Suggest a FOPTD model base don Skogestad method;

d$)$ Find the ultimate gain and ultimate period for the model in part c;

e$)$ Suggest the closed $-$loop $ZN$ tuning parameters based on the parameters in

part d;

f$)$ Suggest the open$-$loop ZN tuning method based on the model in part c;

g$)$ Compare the performance of the two controllers in e and $f($via Matlab$)$;

h$)$ Suggest PID tuning parameters based on ISA, IAE, and ITAE for the system

based on model in part $c$

Hint: For part d; approximate the time delay by $e^{-s}$~~$\frac{1-\frac{}{2}}{1+\frac{}{2}}$