Question $2$

$[20$ Marks$]$

$($a$)$

Given the system of Figure $2\mathrm{.}1$

$($i$)$ Design the value of $K_1,$ as well as $a$ in the feedback path of the minor loop, to yield a settling time of $4$ seconds with $5\%$ overshoot for the step response.

$[7$ Marks$]$

$($ii$)$ Design the value of K to yield a major$-$loop response with $10\%$ overshoot and a peak time of $2\mathrm{.}3$ seconds for a step input.

$[5$ Marks$]$

$($b$)$

$($i$)$ How can you tell from the root locus if the settling time does not change over a region of gain?

$[2$ Marks$]$

$($ii$)$ If $KG(s)H(s)=\frac{25}{{?}_{180}},$ for what value of gain is s a point on the root locus?

$[2$ Marks$]$

$($iii$)$ How can you tell from the root locus that the natural frequency does not change over a region of gain?

$[2$ Marks$]$

$($iv$)$ Why is there more improvement in steady$-$state error if a PI controller is used instead of a lag network?

$[2$ Marks$]$

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