A student, Mitch, who is studying process control argues that an open$-$loop unstable process can be $1$

stablised with a feedback controller. Morrie, who is Mitch's research supervisor, instructs Mitch to

analyse the closed$-$loop system to prove that he is right. Mitch considers the transfer function:

$$

$\frac{-kc}{1-r_s-kc},G_p=\frac{K}{1-s}$

where $>0$ and KinR. The closed$-$loop system is shown in Fig. $3$ where $G_c$ is the controller.

$($iv$)$ With $G_c=K_c,$ what is the effect of increasing $K_e$ on the offset of the closed$-$loop system? Will

overshoot and$/$or oscillation ever occur with increasing $K_c?$

$($v$)$ Consider a scenario where a proportional$-$integral controller $G_c=2(1+\frac{1}{{}_{1s}})$ is used. Discuss

qualitatively the effect of ${}_{\text{I}}$ on the closed$-$loop system.

$(6$ marks$)$

$($vi$)$ Consider the scenario where a proportional$-$integral controller $G_e=(1+\frac{1}{{}_{i}s})$ is used. What is the

general trend of the closed$-$loop system? How does varying ${}_{\text{I}}$ affect the trend?

$(10$ marks$)$