CS 323
CLASSICAL CONTROL SYSTEMS ENGINEERING
Practical 4: Systems Stability – Root Locus
Question 1
!"#
A plant to be controlled is described by a transfer function 𝐺 (𝑠) = !!"$!"%#
Obtain a root locus plot using MATLAB
question1.m
num = [1 5];
den = [1 7 25];
rlocus(num, den);
Output
Question 2
%! ! "#!"&
Plot a root locus for a system with a transfer function 𝐺 (𝑠 ) = ! ! "%!"'
Question2.m
sys = tf([2 5 1],[1 2 3]);
rlocus(sys)
Output
Question 3
Use rlocus method to extract the closed-loop poles and associated feedback gain for
'! ! "&
the transfer function. 𝐺 (𝑠) = (!
" !"$! "#!")
Question3.m
sys = tf([3 0 1],[9 7 5 6]);
[r,k] = rlocus(sys);
disp(r);
disp(k);
Output
r = 3×53 complex
102 ×
-0.0094 + 0.0000i -0.0104 + 0.0000i -0.0105 + 0.0000i -0.0106 +
0.0000i -0.0107 + 0.0000i -0.0108 + 0.0000i -0.0109 + 0.0000i -0.0111
+ 0.0000i -0.0112 + 0.0000i -0.0113 + 0.0000i -0.0115 + 0.0000i -
0.0117 + 0.0000i -0.0119 + 0.0000i -0.0121 + 0.0000i -0.0124 + 0.0000i
-0.0126 + 0.0000i -0.0129 + 0.0000i -0.0132 + 0.0000i -0.0135 + 0.0000i
-0.0139 + 0.0000i -0.0143 + 0.0000i -0.0148 + 0.0000i -0.0152 + 0.0000i
-0.0158 + 0.0000i -0.0163 + 0.0000i -0.0170 + 0.0000i -0.0177 + 0.0000i
-0.0184 + 0.0000i -0.0192 + 0.0000i -0.0201 + 0.0000i -0.0211 + 0.0000i
-0.0222 + 0.0000i -0.0233 + 0.0000i -0.0246 + 0.0000i -0.0259 + 0.0000i
-0.0274 + 0.0000i -0.0290 + 0.0000i -0.0307 + 0.0000i -0.0326 + 0.0000i
-0.0346 + 0.0000i -0.0368 + 0.0000i -0.0392 + 0.0000i -0.0418 + 0.0000i
-0.0446 + 0.0000i -0.0476 + 0.0000i -0.0508 + 0.0000i -0.0543 + 0.0000i
-0.0582 + 0.0000i -0.0623 + 0.0000i -0.0667 + 0.0000i
0.0008 + 0.0084i 0.0006 + 0.0083i 0.0006 + 0.0082i 0.0006 +
0.0082i 0.0006 + 0.0082i 0.0006 + 0.0082i 0.0005 + 0.0082i 0.0005
+ 0.0082i 0.0005 + 0.0082i 0.0005 + 0.0081i 0.0005 + 0.0081i
0.0004 + 0.0081i 0.0004 + 0.0081i 0.0004 + 0.0080i 0.0004 + 0.0080i
0.0003 + 0.0080i 0.0003 + 0.0080i 0.0003 + 0.0079i 0.0002 + 0.0079i
0.0002 + 0.0078i 0.0002 + 0.0078i 0.0002 + 0.0078i 0.0001 + 0.0077i
0.0001 + 0.0077i 0.0001 + 0.0076i 0.0000 + 0.0076i 0.0000 + 0.0075i
-0.0000 + 0.0074i -0.0000 + 0.0074i -0.0000 + 0.0073i -0.0001 + 0.0073i
-0.0001 + 0.0072i -0.0001 + 0.0071i -0.0001 + 0.0071i -0.0001 + 0.0070i
-0.0001 + 0.0070i -0.0001 + 0.0069i -0.0001 + 0.0068i -0.0001 + 0.0068i
-0.0001 + 0.0067i -0.0001 + 0.0067i -0.0001 + 0.0066i -0.0001 + 0.0066i
-0.0001 + 0.0065i -0.0001 + 0.0065i -0.0001 + 0.0064i -0.0001 + 0.0064i
-0.0001 + 0.0064i -0.0001 + 0.0063i -0.0001 + 0.0063i
0.0008 - 0.0084i 0.0006 - 0.0083i 0.0006 - 0.0082i 0.0006 -
0.0082i 0.0006 - 0.0082i 0.0006 - 0.0082i 0.0005 - 0.0082i 0.0005
- 0.0082i 0.0005 - 0.0082i 0.0005 - 0.0081i 0.0005 - 0.0081i
0.0004 - 0.0081i 0.0004 - 0.0081i 0.0004 - 0.0080i 0.0004 - 0.0080i
0.0003 - 0.0080i 0.0003 - 0.0080i 0.0003 - 0.0079i 0.0002 - 0.0079i
0.0002 - 0.0078i 0.0002 - 0.0078i 0.0002 - 0.0078i 0.0001 - 0.0077i
0.0001 - 0.0077i 0.0001 - 0.0076i 0.0000 - 0.0076i 0.0000 - 0.0075i
-0.0000 - 0.0074i -0.0000 - 0.0074i -0.0000 - 0.0073i -0.0001 - 0.0073i
-0.0001 - 0.0072i -0.0001 - 0.0071i -0.0001 - 0.0071i -0.0001 - 0.0070i
-0.0001 - 0.0070i -0.0001 - 0.0069i -0.0001 - 0.0068i -0.0001 - 0.0068i
-0.0001 - 0.0067i -0.0001 - 0.0067i -0.0001 - 0.0066i -0.0001 - 0.0066i
-0.0001 - 0.0065i -0.0001 - 0.0065i -0.0001 - 0.0064i -0.0001 - 0.0064i
-0.0001 - 0.0064i -0.0001 - 0.0063i -0.0001 - 0.0063i
k = 1×53
0 0.4201 0.4542 0.4911 0.5309 0.5740 0.6205
0.6709 0.7253 0.7841 0.8477 0.9165 0.9908 1.0712
1.1581 1.2521 1.3536 1.4634 1.5822 1.7105 1.8493
1.9993 2.1614 2.3368 2.5263 2.7313 2.9529 3.1924
3.4514 3.7313 4.0340 4.3613 4.7151 5.0975 5.5111
5.9581 6.4415 6.9640 7.5289 8.1397 8.8000 9.5138
10.2856 11.1200 12.0220 12.9973 14.0516 15.1915 16.4238
17.7561
Question 4
Consider an open loop system with the following transfer function. 𝐺 (𝑠) =
!"$
!(!"#)(!"&#)(!"%,)
Question4.m
num=[1 7];
den=conv(conv([1 0],[1 5]),conv([1 15],[1 20]));
rlocus(num,den)
axis([-22 3 -15 15])
Output
