Root Locus Analysis (1)Root Locus Analysis (1)Hany FerdinandoHany Ferdinando
Dept. of Electrical Eng.Dept. of Electrical Eng.
Petra Christian UniversityPetra Christian University
General OverviewGeneral Overview
This section discusses how to plot the This section discusses how to plot the Root Locus methodRoot Locus method
Step by step procedure is used inline with Step by step procedure is used inline with an examplean example
Finally, some comments are given as the Finally, some comments are given as the complement for this sectioncomplement for this section
Why Root LocusWhy Root Locus
Closed-loop poles’ location determine the Closed-loop poles’ location determine the stability of the systemstability of the system
Closed-loop poles’ location is influenced Closed-loop poles’ location is influenced as the gain is variedas the gain is varied
Root locus plot gives designer information Root locus plot gives designer information how the gain variation influences the how the gain variation influences the stability of the systemstability of the system
Plot ExamplePlot Example
-6 -5 -4 -3 -2 -1 0 1 2-4
-3
-2
-1
0
1
2
3
4Root Locus
Real Axis
Imag
inar
y Ax
is
Important Notes:Important Notes:
Poles are drawn as ‘x’ while zeros are Poles are drawn as ‘x’ while zeros are drawn as ‘o’drawn as ‘o’
Gain at poles is zero, while gain at zeros Gain at poles is zero, while gain at zeros is infinityis infinity
Pole is the starting point and it Pole is the starting point and it mustmust finish finish at zero; therefore, for every pole there at zero; therefore, for every pole there should be corresponding zeroshould be corresponding zero
Root locus is plot on the s planeRoot locus is plot on the s plane
StandardizationStandardization
G(s)
H(s)
+-
R(s) C(s)
)()(1
)(
)(
)(
sHsG
sG
sR
sC
Find Characteristic Equation!!Find Characteristic Equation!!
1 + G(s)H(s) = 01 + G(s)H(s) = 0
How to make it?How to make it?
1.1. Start from the characteristic equationStart from the characteristic equation
2.2. Locate the poles and zeros on the s planeLocate the poles and zeros on the s plane
3.3. Determine the root loci on the real axisDetermine the root loci on the real axis
4.4. Determine the asymptotes of the root lociDetermine the asymptotes of the root loci
5.5. Find the breakaway and break-in pointsFind the breakaway and break-in points
6.6. Determines the angle of departure (angle of Determines the angle of departure (angle of arrival) from complex poles (zeros)arrival) from complex poles (zeros)
7.7. Find the points where the root loci may cross Find the points where the root loci may cross the imaginary axisthe imaginary axis
Example (1)Example (1)
)2)(1()(
sss
KsG 1)( sH
1. Start from the characteristic equation1. Start from the characteristic equation
0)2)(1(
1
sss
K
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Real Part
Imag
inar
y Pa
rt
Example (2)Example (2)
2. Locate the poles and zeros on the s plane2. Locate the poles and zeros on the s plane
Example (3)Example (3)
3. Determine the root loci on the real axis3. Determine the root loci on the real axis
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Real Part
Imag
inar
y Pa
rt
Example (4a)Example (4a)
4. Determine the asymptotes of the root loci4. Determine the asymptotes of the root loci
03
)12(180
ko
103
]0[)]2()1(0[
Example (4b)Example (4b)
4. Determine the asymptotes of the root loci4. Determine the asymptotes of the root loci
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Real Part
Imag
inar
y Pa
rt
Example (5)Example (5)
5. Find the breakaway and break-in point5. Find the breakaway and break-in point
From the characteristic equation, findFrom the characteristic equation, find
)23( 23 sssK
then calculate…then calculate…
0)263( 2 ssds
dK
s = -0.4266 and s = -1.5744s = -0.4266 and s = -1.5744
Example (6)Example (6)
6. Determine the angle from complex pole/zero6. Determine the angle from complex pole/zero
This example has no complex This example has no complex poles and zeros, therefore, poles and zeros, therefore, this step can be skipped!!!this step can be skipped!!!
polepole = 0 – sum from pole + sum from zero= 0 – sum from pole + sum from zero
zerozero = 0 – sum from zero + sum from pole = 0 – sum from zero + sum from pole
Example (7)Example (7)
7. Find the points where the root loci may cross the 7. Find the points where the root loci may cross the imaginary axisimaginary axis
Do this part by substituting jDo this part by substituting j for all s in the for all s in the characteristic equationcharacteristic equation
0)2()3(
0)(2)(3)(32
23
jK
Kjjj
= = ±√2, K = 6 or ±√2, K = 6 or = 0, K = 0 = 0, K = 0
Example (8)Example (8)
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Root Locus
Real Axis
Imag
inar
y Ax
is
CommentsComments
nnthth degree algebraic equation in s degree algebraic equation in s If n-mIf n-m≥2 then a1 is negative sum of the roots of ≥2 then a1 is negative sum of the roots of
the equation and is independent of Kthe equation and is independent of K It means if some roots move on the locus toward the It means if some roots move on the locus toward the
left as K increased then the other roots left as K increased then the other roots mustmust move move towards the right as K is increasedtowards the right as K is increased
n1n
1n
m1m
1m
a...sas
)b...sbK(sH(s)G(s)
Root Locus in MatlabRoot Locus in Matlab
Function rlocus(num,den) draws the Root Locus Function rlocus(num,den) draws the Root Locus of a system. Another version in state space is of a system. Another version in state space is rlocus(A,B,C,D)rlocus(A,B,C,D)
01 den
numThe characteristic equationThe characteristic equation
Those functions are for negative Those functions are for negative feedback (normal transfer function)feedback (normal transfer function)
Next…Next…
Topic for the next meeting is Root Locus Topic for the next meeting is Root Locus in positive feedbackin positive feedback
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