Outline
•
•
•
•
Root Locus Review
M. Sami Fadali
Professor Electrical Engineering
University of Nevada, Reno
What is the root locus?
Rules for sketching root loci.
Examples
MATLAB
1
The Root Locus
What is Required?
• Closed-loop characteristic equation
• Gain
2
• Determine the loci of the closed-loop poles
(root loci) as varies between and .
• Complex equality
gives two
real equalities:
i) Magnitude Condition
ii) Angle Condition
is a design parameter
is the loop gain
= open-loop system zeros
= open-loop system poles
• Use magnitude and angle conditions to
derive rules for sketching the root locus.
3
4
Rules For Sketching Root Loci
Root Loci (Cont.)
1. Number of root locus branches=number of
open-loop poles of
2. Root locus branches start at open-loop poles
and end at open-loop zeros or at .
3. Real axis root loci have an odd number of poles
plus zeros to their right.
4. Branches going to asymptotically approach
the straight lines
2
1 180°
,
0,1,2, … ,
∑
5.
maxima at breakaway points (departure
from real axis), minima at breakin points
(arrival at real axis).
6. Angle of departure from a complex pole
(similarly angle of arrival at complex zeros)
∑
5
6
Solution (i)
Example 5.1
Sketch the root locus plots for the transfer
functions. Comment on the effect of adding
a pole or a zero to the loop gain.
7
• Obtain root locus using MATLAB.
• Rule 1: two root locus branches.
• Rule 2: branches start at (1) and ( 3) and go to
.
• Rule 3: real-axis locus is between ( 1) and ( 3).
• Rule 4 gives the asymptotes
8
Solution (i)
(i) RL of 2nd Order System
Rule 5: Breakaway point
• Characteristic equation
• Breakaway point at a maximum for ,
.
second derivative
• For any system with two real axis poles, the
breakaway point is midway between the two
poles.
9
Solution (ii)
10
(ii) Rule 5: Breakaway Point
• Rule 1: 3 root locus branches.
• Rule 2: each branch starts at an open loop
poles (1,  3,  5).
• Rule 3: real-axis loci are between 1 and
3 and to the left of 5. All branches go to
. One branch remains on the negative
real axis and the other two break away.
breakaway point
(on real axis locus between poles at (1) and (3))
4.155 corresponds to a negative gain (inadmissible).
Gain at the breakaway point (magnitude condition)
11
12
(ii) Intersection With j-axis
(ii) Asymptotes
• Closed-loop characteristic equation:
• Angle:
• Routh table
• Intercept:
13
(ii) RL of 3rd Order System
• Auxiliary Equation:
• Intersection with j-axis:
rad/s.
14
(iii) Solution
• Rule 1: 2 root locus branches.
• Rule 2: each branch starts at an open loop
pole (1,  3). One branch goes to  and
the other goes to the zero ( 5).
15
16
(iii) Rule 5: Breakaway/Breakin
(iii) Breakaway/Breakin
2.172 = breakaway (between open-loop poles)
7.828 = breakin point (to left of zero)
• 2nd derivative: negative for 2.172 and positive
for 7.828
• K maximum at 2.172 and minimum at 7.828
• Rule: root locus is a circle centered at zero with
radius = geometric mean of the distances
between the zero and the two real poles.
17
(iii) RL of 2nd Order System with Zero
18
Root Locus Using MATLAB
» g=tf([1,5],[1,2,10]) % Transfer function
» rlocus(g) % Root locus.
Click mouse at a point on root locus to
obtain the corresponding gain, pole
locations, and time response information.
19
20