Lead$-$Lag Controller Design via Frequency Analysis

You need to solve the following two problems. MATLAB use is allowed but you must explain all steps and justify all answers.

Consider the system

$G(s)=\frac{10}{(\frac{s}{0\mathrm{.}2}+1)(\frac{s}{0\mathrm{.}5}+1)}$

We want to design a lag controller that provides PM of at least $60\%$ and steady$-$state tracking of constant references within $10\%.$

$($i$)$ For the controller

$KD(s)=0\mathrm{.}4\frac{s+0\mathrm{.}05}{s+0\mathrm{.}02}$

compute the PM and steady$-$state tracking error to verify that the specs are met.

$($ii$)$ Suppose that to get better damping, we increase the PM spec to $70,$ while keeping the same steady$-$state tracking spec as in $($i$).$ Modify the design to achieve the new specs. Verify that your design indeed works.