Question

Consider a negative feedback system whose open-loop transfer function is: G(s)H(s)=K/(s(s+1)) Write a MATLAB program to obtain the root-locus plot of G(s)H(s). [2 marks] What are the locations of poles when K = 0.19. [2 marks] When K = 0.4, what are the locations of poles? [3 marks] Find values of the damping ratio, % overshoot and frequency when K = 0.4. [3 marks] Write a MATLAB program to obtain a bode plot of G(s)H(s) when K = 1. [2 marks] Find the magnitude (dB) and phase angle at a frequency of 10rad/sec.

Answer 1

MATLAB Code

%numerator coefficient of open loop TF
num=[1];
%denominator coefficient of open loop TF
den=[1 1 0];
%initializing the vector of k
k=0:0.01:2;
%open loop transfer function
sys=tf(num,den);
%Plotting the root locus plot
figure(1)
rlocus(sys,k);
%finding locaiton of poles for K=0.19
k=0.19;
r=rlocus(sys,k);
fprintf('The location of the poles for k=%1.2f are at\n',k)
fprintf('s=(%1.4f)+i(%1.4f)\n',real(r(1)),imag(r(1)))
fprintf('s=(%1.4f)+i(%1.4f)\n',real(r(2)),imag(r(2)))
%finding locaiton of poles for K=0.4
k=0.4;
r=rlocus(sys,k);
fprintf('The location of the poles for k=%1.2f are at\n',k)
fprintf('s=(%1.4f)+i(%1.4f)\n',real(r(1)),imag(r(1)))
fprintf('s=(%1.4f)+i(%1.4f)\n',real(r(2)),imag(r(2)))
%bode plot of the open loop TF for k=1
k=1;
figure(2)
bode(k*sys)
%finding magnitude and phase for w=10rad/sec
w=10;
[mag,phase]=bode(k*sys,w);
fprintf('The magnitude(dB) at frequency of 10 rad/sec is %1.4fdB\n',20*log10(mag))
fprintf('The phase angle at te frequency of 10 rad/sec is %1.4f degrees\n',phase)

Output on command window

The location of the poles for k=0.19 are at
s=(-0.7449)+i(0.0000)
s=(-0.2551)+i(0.0000)
The location of the poles for k=0.40 are at
s=(-0.5000)+i(0.3873)
s=(-0.5000)+i(-0.3873)
The magnitude(dB) at frequency of 10 rad/sec is -40.0432dB
The phase angle at te frequency of 10 rad/sec is -174.2894 degrees

The root locus plot is

On the same plot by using the data cursor option, we can find the overshoot, damping ratio and frequency.

The bode plot of the open loop transfer function G(s)H(s) for K=1 is