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8/6/2019 Root Locus2 2
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1
Example #2C(s)+
-
R(s)K
)20(
1
+ss
Closed-Loop Transfer Function is:
K ssK
ssK
ss
K
s RsC ++=
++
+= )20(
)20(
11
)20(
1
)()(
Use Matlab to plot Root Locus
-20 -15 -10 -5 0 5-15
-10
-5
0
5
10
15
Real Axis
I m a g A x i s
8/6/2019 Root Locus2 2
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2
-20 -15 -10 -5 0 5-15
-10
-5
0
5
10
15
Real Axis
I m a g A x i s
Find K for roots with ζ=.707
-p2=-20 -p1=0
s=-10+j10
200
010101
=
++−=+ j ps
200
2010102
=
++−=+ j ps
Use Magnitude Criterion
1*
1
)(
)(
21
=++
= ps ps
K s Den
s NumK
Once a point, s, is known to be on the root locus, then the
magnitude criterion can be used to find the gain K for that
point (i.e., find K at desired root locations)
2001200*200
1=⇒= K K
8/6/2019 Root Locus2 2
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3
-20 -15 -10 -5 0 5-15
-10
-5
0
5
10
15
Real Axis
I m a g A x i s
Find K for a C.E. root at s=-8
-p2=-20 -p1=0
s=-8
8081 =+−=+ ps
122082 =+−=+ ps
96
1
12*8
1
=⇒
=
K
K
Example #3
C(s)+
-
R(s)K
)2)(1(
)3(
+++ss
s
Closed-Loop Transfer Function is:
)3()2)(1(
)3(
)(
)(
++++
+
= sK ss
sK
s R
sC
Use Matlab to find the root locus
8/6/2019 Root Locus2 2
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4
Root Locus for Example #3
-6 -5 -4 -3 -2 -1 0 1 2-1.5
-1
-0.5
0
0.5
1
1.5
Real Axis
I m a g A x i s
Modify Axes>> axis([-5 0 -1 3]) axis([Xmin Xmax Ymin Ymax])
Find K for
minimum
peak time
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-1
-0.5
0
0.5
1
1.5
2
2.5
3
Real Axis
I m a g A x i s
8/6/2019 Root Locus2 2
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5
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-1
-0.5
0
0.5
1
1.5
2
2.5
3
Real Axis
I m a g A x i s
K for minimum peak time
-p1=-1
4.24.12
14.13
22
1
=+=
++−=+ j ps
s=-3+j1.4
-p2=-2
7.14.11
24.13
22
2
=+=
++−=+ j ps
-z1=-34.14.10
34.13
221
=+=
++−=+ j zsXXO
Use magnitude criterion
1*)(
)(
21
1 =++
+=
ps ps
zsK
s Den
s NumK
9.217.1*4.2
4.1≈⇒= K K
0)3(9.2)2)(1( =++++ sssCheck roots of:
8/6/2019 Root Locus2 2
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6
Example #4C(s)+
-
R(s)K
)3)(1(
)5(
+++ss
s
Closed-Loop Transfer Function is:
)5()3)(1(
)5(
)(
)(
++++
+=
sK ss
sK
s R
sC
Is the point s=-4+j2 on the root locus?
Plot the vectors
-5 -4 -3 -2 -1 0
+j1
+j2
-j1
-j2
s=-4+j2
)( 1 zs+∠ )( 1 ps +∠)( 2 ps +∠
8/6/2019 Root Locus2 2
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7
Use Phase Criterion
oo
oo
360180)()()(
360180)(
)(
211 k ps ps zs
k s Den
s Numangle
±=+∠−+∠−+∠
±=
++
=++−∠=+∠ −1
2tan)524()( 1
1 j zs
o4.6312tan)( 1
1 =
++=+∠ − zs
Continue Phase Criterion
−+
=++−∠=+∠ −3
2tan)124()( 1
1 j ps
o3.1463
2tan)( 1
1 =
−+
=+∠ − ps
−+
=++−∠=+∠ −1
2tan)324()( 1
2 j ps
o6.11612tan)( 1
1 =
−+=+∠ − ps
8/6/2019 Root Locus2 2
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8
Add Phase Angles
oooo 5.1996.1163.1464.63
)()()( 211
−=−−
=+∠−+∠−+∠ ps ps zs
s=-4+j2 is not on this
root locus
Rules #1-#3 for Root Locus
• Put polynomial in standard form,
• Plot open-loop poles (X) and zeros (O)on complex plane
• Root locus is on the real axis to the leftof an odd number of open-loop polesand zeros (gives angle of 180°)
0))...()((
))...()((1
)(
)(1)(1
21
21 =++++++
+=+=+n
m
ps ps ps
zs zs zsK
s Den
s NumK sKf
8/6/2019 Root Locus2 2
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9
Rules #4-#5 for Root Locus• Number of linear asymptotes = n-m,
angle to 1st one (equally spaced) is
• If n-m>1, center of linear asymptotesfound by
mna −=
°180
φ
−−−
−= ∑∑
==
m
j j
n
iia z p
mn11
)()(1σ
Rules #6-#8 for Root Locus
• Determine the breakaway points on the real axis (if any).
• Use the Routh-Hurwitz criteria to evaluateimaginary axis crossings (if any)
– look for a row of zeros in the S0 or S1 rows
• Evaluate the angle of departure from complex poles (and angle of arrival at complex zeros) by the angle criteria
8/6/2019 Root Locus2 2
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10
Rules #9-#10 for Root Locus• Sketch in the remainder of the root locus.
– Use the angle criteria to determine suitability of any
questionable points.
• The gain K at any root location can befound by the magnitude criteria,
1...
...
2121 =+++
+++
nm ps ps ps
zs zs zsK
Example #5
C(s)+
-
R(s)K
)4)(3)(1(
)2(
++++
ssss
s
Closed-Loop Transfer Function is:
)2()4)(3)(1(
)2(
)(
)(
+++++
+
= sK ssss
sK
s R
sC
Apply root locus steps 1, 2, 3, 4, 5, and 7 to this problem
8/6/2019 Root Locus2 2
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11
Root Locus #5
-6 -5 -4 -3 -2 -1 0 1 2-8
-6
-4
-2
0
2
4
6
8
Real Axis
I m a g A x i s
Example #6
C(s)+
-
R(s)K
)25)(4(
1
++ sss
Closed-Loop Transfer Function is:
K sss
K
s R
sC
+++= )25)(4()(
)(
Apply root locus steps 1, 2, 3, 4, 5, and 7 to this problem
8/6/2019 Root Locus2 2
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12
Root Locus #6
-30 -25 -20 -15 -10 -5 0 5-30
-20
-10
0
10
20
30
Real Axis
I m a g A x i s
Example #7
C(s)+
-
R(s)K
)5)(1(
)4(
++−ss
ss
Closed-Loop Transfer Function is:
)2()4)(3)(1(
)2(
)(
)(
++++−
+
= sK sss
sK
s R
sC
Apply root locus steps 1, 2, 3, 4, 5, and 7 to this problem
8/6/2019 Root Locus2 2
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8/6/2019 Root Locus2 2
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14
Root Locus #8
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1-4
-3
-2
-1
0
1
2
3
4
Real Axis
I m a g A x i s
Example #9
C(s)+
-
R(s)K
)84(
1
2 ++ sss
Closed-Loop Transfer Function is:
K sss
K
s R
sC
+++=
)84()(
)(
2
Apply root locus steps 1, 2, 3, 4, 5, and 7 to this problem
8/6/2019 Root Locus2 2
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15
Root Locus #9
-8 -6 -4 -2 0 2 4-5
-4
-3
-2
-1
0
1
2
3
4
5
Real Axis
I m a g A x i s
Example #10
C(s)+
-
R(s)K
)172)(5(
)1012(2
2
+++
++
sss
ss
Closed-Loop Transfer Function is:
)1012()172)(5(
)1012(
)(
)(
22
2
++++++
++
= ssK sss
ssK
s R
sC
Apply root locus steps 1, 2, 3, 4, 5, and 7 to this problem
8/6/2019 Root Locus2 2
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16
Root Locus #10
-7 -6 -5 -4 -3 -2 -1 0 1 2-10
-8
-6
-4
-2
0
2
4
6
8
10
Real Axis
I m a g A x i s