Problem $1$

Consider the following system

$G(s)=\frac{s+0\mathrm{.}1}{s(s+2)}$

which we want to control with a proportional controller $F(s)=K,$ where $K>0.$

a$)$ Sketch the asymptotic Bode diagram of $G(s).$ What is the minimum $K$ for which ${}_c>6ra\frac{d}{s}?[0\mathrm{.}75$p$]$

b$)$ Pick now $K=10$ and sketch the asymptotic Bode diagram of $G_o(s)=KG(s).$ Then, use it to compute the gain and phase margins. $[0\mathrm{.}75$pt$]$

Pick now $K=10.$ The output measurement is affected by a noise $n(t)$ at ${}_n=100ra\frac{d}{s}.$

c$)$ Propose a strategy to attenuate this noise by a factor of at least $20$ while keeping ${}_c>6$ and ${}_m>60.$ If necessary, you can change the structure of $F(s),$ adding poles and zeros. Help yourself with MATLAB. $[3$pt$]$