$(5$ pts$)($Pole$-$zero cancellation$)$ Consider the standard feedback control system with a PI controller, where the plant has an unstable pole,

$P(s)=\frac{1}{s-1},C(s)=k_p+\frac{k_i}{s}.$

Looking at the opportunity, a greedy engineer chooses $k_p=1,k_i=-1,$ so that the unstable plant pole and controller zero may be canceled, i$.$e$.$

$P(s)C(s)=\frac{1}{(s-1)}*\frac{(s-1)}{s}=\frac{1}{s}.$

$($a$)(2$ pts$)$ Obtain the closed loop transfer function for this $($ideal$)$ system and plot its step response using Matlab.

$($b$)(3$ pts$)$ However, this ideal scenario often fails in practice as the engineer will soon find out. Suppose that due to small environmental perturbation, the system's pole is now shifted to $1\mathrm{.}0001,$ i$.$e$.$

$P(s)=\frac{1}{s-1\mathrm{.}0001}$

Keeping the same controller as before, repeat part $($a$).$ What do you observe?