Root locus example
Basis
• Plot of the closed-loop system roots, as k
varies from 0 to ∞
• Or plot of 1+kG(s)=0
• Sketch by looking at open-loop transfer
function, G(s)
1
s( s  1)( s  5)( s  10)
a=conv([1 0], [1 1])
b=conv([1 5], [1 10])
D=conv(a,b)
N=1
sys=tf(N, D)
rlocus(sys)
roots([4 16*3 65*2 50])
Break-in Break-out points
1
1
D( s)
k


G( s) N ( s)
N (s)
D( s)
dk
0
ds
Matlab output
Transfer function:
1
---------------------------s^4 + 16 s^3 + 65 s^2 + 50 s
Solve for breakin/out points
-8.2399e+000
-3.3005e+000
-4.5963e-001
Root locus
• Open Loop Response
– Poles: n = 4
– Zeros: m =0
• Asymptote angles 45,135,225,315 degrees
• Asymptote centroid s=-4
Root Locus
25
20
15
Imaginary Axis
10
5
0
-5
-10
-15
-20
-25
-30
-25
-20
-15
-10
-5
Real Axis
0
5
10
15
20
Shape of Asymptotes
• Asymptotes lines are symmetric about the
real axis
• Number of asymptotes = n-m
n-m = # poles - # zeros
Can’t have more zeros than poles for a proper
transfer function for a real system.
• Shape of asymptote lines is often called a
Butterworth pattern.
Root Locus
25
20
15
Imaginary Axis
10
5
0
-5
-10
-15
-20
-25
-30
-25
-20
-15
-10
-5
Real Axis
0
5
10
15
20