Excercise 4: PI control for a first-order plant. Suppose you are to design a feedback controller for afirst-order plant depicted in the figure below:ControllerPlantКkpTS 1SThis configuration is referred to as a proportional-integral (PI) controller. You are to design the controllerto satisfy some given time-domain specifications.(a) Find the (closed-loop) transfer function Gyr from r to y (see hw02).(b) Determine the steady-state error for a unit step input (Hint: e r --y)(c) Find the transfer function Gun from n to u.(d) Determine k, and ki such that the feedback controlled system has damping ratio Ç = 0.5 and fre-quency wo. (Hint: the desired denominator polynomial for a closed-loop transfer function is of theform: d(s) s2+ 2Çwos +w2.)From now on, let K = 1, t = 1.(e) Find the values for kp and ki so that the frequency of the closed-loop system is 1, i.e. wo = 1. Thiscontroller will be referred to as controller 1.(f) Also, find the values for k, and k so that the frequency of the closed-loop system is 0.1, i.e. woThis controller will be referred to as controller 2.= 0.1(g) Analyze the tracking ability of both controller 1 and 2 by simulation in MATLAB. For instance,consider the step response of Gur. What is the input and what the output? Plot the results on oneplot, and include a legend(h) Analyze the ability of the system to reject sensor noise n by plotting the step-response of GunWhat is the input and what the output? Plot the responses of both controller 1 and 2 on one plot.(i) Based on your results, which controller do you think is 'best'?

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Step 1

Hi! Thank you for the question. I am solving the Question for first 3 sub-parts as you have posted the question with multiple sub-parts. Kindly post the remaining sub-parts as separate Question. Thank you.

Part (a):

The closed-loop transfer function from r to y is obtained as follows:

Step 2

From Figure 2, obtain the required closed-loop transfer function from r to y as follows:

Step 3

Part (b):

Write the expression for steady-state erro...

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