Consider the dynamic system

$G(s)=\frac{1}{(s+1{)}^3}$

We want to control $G(s)$ with a controller $K(s)$ negative feedback with it$.$ We want the closed loop system to track a unit step with an error $e(t)$ such that $|e(t)|<mn>0</mn><mn>.</mn><mn>1</mn>$ with the additional constraint the $K(s)$ has not poles in $s=0.$

a$)$ Notice that by using a simple proportional controller $K(s)=k$ with $k>0,$ we cannot meet the step tracking requirement and at the same time keep the closed loop stable. Why? $($Hint: simulate the system $)$

b$)$ Design a controller $K(s)$ that stabilizes the system and meets the tracking requirement. $($Hint: Take a look at the phase margin when using a proportional controller $k$ with $k$ just large enough to meet the tracking requirements, then choose a desired crossover frequency $($suggestion $w_{gc}^{**}=2)$ and use a controller to gain as much phase as possible at $w_{gc}^{**}=2))$

c$)$ Verify that the controller designed at b$)$ makes the system stable with a satisfying error bound. $($Barely stable if you followed the hints at point b$),$ but stable nonetheless$)$