Excercise 4: PI control for a first-order plant. Suppose you are to design a feedback controller for a first-order plant depicted in the figure below: Controller Plant К kp TS 1 S This configuration is referred to as a proportional-integral (PI) controller. You are to design the controller to 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 the form: 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. This controller 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. wo This 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 one plot, and include a legend (h) Analyze the ability of the system to reject sensor noise n by plotting the step-response of Gun What 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'?