Item $1$:

Consider a simplified process represented by the block diagram below:

For a process with:

$G=G_{IP}{G}_v{G}_p=\frac{2\mathrm{.}5}{(2s+1)(2s+1)}$;$,$

$K_M=0\mathrm{.}5$;$,$

$G_d=\frac{0\mathrm{.}5}{8s+1}$

$G_c=K_c(1+\frac{1}{{}_{\text{I}}s})=K_c\frac{{}_{\text{I}}s+1}{{}_{\text{I}}s}$

$($a$)$ Write the open$-$loop transfer function, $G_{OL},$ in the gain$-$and$-$time constants form.

$($b$)$ Write the servo and regulator transfer functions.

$($c$)$ Write the characteristic equation. $($Note that the characteristic equation must have an equal sign, or it is not an equation!$)$

$($d$)$ Find the range of $K_c$ for which the feedback control loop will be stable for ${}_{\text{I}}=0\mathrm{.}5.$ Note: you should find both a lower bound and an upper bound for $K_c.$

Item $2$:

$($a$)$ Repeat Item $1,$ parts $($a$),($c$)$ and $($d$)$ for the case where the controller is a PI controller with ${}_{\text{I}}=2.$ Note that this will cause a pole$-$zero cancelation in $G_{OL},$ which simplifies the transfer function.

$($b$)$ Discuss: What happens to the characteristic equation when you made ${}_{\text{I}}$ equal to one of the time constants in $G(s)?$ Can the process be made unstable by choosing a $K_c$ that is too large?