Please include all calculations and graphs and Matlab Code. Do not just leave a formula or write an answer without calculations. If you are not going to do these, please do not answer this question. This question is related to the field of Control engineering

Problem$-3.$ Considering a similar control system as of the previous problem, where the plant is now

$G_p(s)=\frac{1}{s(s+1)(s^2+10s+26)}$

You are expected to

$($a$)$ Find the settling time for the system when it is operating with $15\%$ overshoot.

$($b$)$Find the zero of a compensator and the gain, $K,$ so that the settling time is $7$ seconds. Assume

that the pole of the compensator is located at $-15.$

$($c$)$ Use MATLAB or any other computer program to simulate the system's step response to test the

compensator.

$($d$)$ For the same plant, design a PID controller by means of an appropriate Ziegler$-$Nichols method.

Plot the response of the system using in MATLAB or Simulink.

$($e$)$ Now, propose another PID controller by halving the proportionality constant and doubling the

derivative time of the previous PID controller. Plot the response of the system on the same plane

as the previouse.