Download Root Locus What is the root locus? System Configurations Closed

What is the root locus?
• Plot of the loci of the closed-loop poles of a
system as its gain is varied from 0 to .
• Plots are rarely obtained for negative gains.
• Plots can be obtained for system parameters
other than the gain (e.g. time constant).
• Useful design tool since closed-loop poles
(and zeros) determine the time response.
Root Locus
M. Sami Fadali
Professor of electrical Engineering
University of Nevada
1
Closed-loop Characteristic Equation
System Configurations

R(s) 
Plant
Compensator


=
Cascade Compensation


2
1+
K
G(s)
C(s)
H(s)
Plant
=
Compensator
• Determined using the loop gain.
1+
Feedback Compensation
• In both cases, use the loop gain for root locus plots.
• Only zeros of the forward path are closed-loop zeros.
3
= open-loop zeros,
= open-loop poles
4
Complex Equality
Complex Numbers
• Forms
Equivalent to two real equalities:
Im{x}
i. Magnitude condition
x
xI
ii.
|x|
Angle Condition
x
xR
Re{x}
5
6
Arithmetic Operations
Transfer Function
• Forms
j
s
s
sa
• Addition/Subtraction

a
a
• Magnitude
• Multiplication
• Division
• Angle
7
8
Example: MATLAB
Second Test Point
Test Point: 2+j/sqrt(2) on root locus?
Test Points: 2+j 3, 2+j/sqrt(2) on root locus?
First test point
[g=zpk([-3,-4],[-1,-2],1);
[angle(evalfr(g, 2+j*3 ) )
-1.2315
o  1.2315*180/ =  70.56 degrees
o 2+j 3 not on root locus
[g=zpk([-3,-4],[-1,-2],1);
[angle(evalfr(g, 2+j/sqrt(2) ) )
-3.1416
2+j/sqrt(2) on root locus
[K=1/abs(evalfr(g,-2+j/sqrt(2)))
9
10
Example
Calculator
• Unity feedback closed-loop system with the
forward gain:
• Closed-loop characteristic equation
 Angle:  180 degrees
,
 On root locus
/
 Gain:
11
12
Root Locus for Example 1
MATLAB
[g=zpk([],[0,0],1);
[rlocus(g)
Root Locus:
Root Locus
1.5
j
s
XX
0.5
Imaginary Axis
Test point s
1

0
-0.5
-1
13
-1.5
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
14
Real Axis
Numerical Computation
Example: Root Locus
• Fix value of K and solve for the roots.
15
16