Question

# A unity feedback system with the forward transfer function G)2)(s +5) is operating with a closed-... A unity feedback system with the forward transfer function G)2)(s +5) is operating with a closed-loop step response that has 15% overshoot. Do the following: a) Evaluate the settling time for a unit step input; b) Design a PD control to yield a 15% overshoot but with a threefold reduction in settling time; c) Evaluate the settling time, overshoot, and steady-state error with the PD control.

I have used MATLAB to find out the gain K where the overshoot is 15%.

MATLAB code to plot the rlocus is given below.

clc;
close all;
clear all;

s = tf('s');
k = 0:0.1:20;
G = 1/(s*(s+2)*(s+5));
figure;rlocus(G,k); From the above figure, it is observed that at gain 10.9, the overshoot is 15%.

Therefore k = 10.9.

settling time for unit step is given by Ts = 4 time constants = 4/ |real part of dominant pole = 4/ 0.724 = 5.52 sec

Now it is required to design a PD controller to meet 15% overshoot but settling time = 5.52/3 = 1.84 sec.

I have used root locus method to determine the PD controller. Therefore the controller is given by C(s) = 19.0245(s+2.3961)

I have used MATLAB to plot he response with the controller.

code:

C = 19.0245*(s+2.3961);
figure;step(feedback(C*G,1));

Response is plotted below. From the above response it is observed that the settling time is 1.84 sec. steady state error is zero and peak overshoot is 17%

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