Question $1$

$[20$ Marks$]$

$($a$)$

Consider the unity feedback system shown in Figure $1\mathrm{.}1$

$G(s)=\frac{K}{(s+5{)}^3}$

$($i$)$ Find the location of the dominant poles to yield a $1\mathrm{.}2$ second settling time and an overshoot of $15\%.$

$[2$ Marks$]$

$($ii$)$ If a compensator with a zero at $-1$ is used to achieve the conditions of part $(i),$ what must the angular contribution of the compensator pole be$?$

$[2$ Marks$]$

$($iii$)$ Find the location of the compensator pole.

$[3$ Marks$]$

$($iv$)$ Find the gain required to meet the requirements stated in Part $($i$)$

$[3$ Marks$]$

$($b$)$

$($i$)$ Cascade compensation to improve the steady$-$state error is based upon what polezero placement of the compensator? Also, state the reasons for this placement.

$[3$ Marks$]$

$($ii$)$ Cascade compensation to improve the transient response is based upon what polezero placement of the compensator? Also, state the reasons for this placement

$[3$ Marks$]$

$($iii$)$ A lag compensator with the zero $25$ times as far from the imaginary axis as the compensator pole will yield approximately how much improvement in steady$-$state error?

$[3$ Marks$]$

$($iv$)$ How would you determine whether a root locus plot crossed the real axis?

$[1$ Marks$]$

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