root locus techniques
TRANSCRIPT
Root Locus Techniques
EE-371 / EE-502 Control SystemsMilwaukee School of Engineering
Fall Term 2005Dr. Glenn Wrate, P.E.
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Why Root Locus?
• What happens when the gain of the controller changes?– Will the system
be stable?– Will the response
change?• The root locus
tells us!-4 -3 -2 -1 0 1 2 3 4
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Root Locus
Real Axis
Imag
inar
y Ax
is
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Closed Loop System
K G
H
Forward Transfer Function
Feedback
OutputInput
-
+
R(s) C(s)E(s)
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Transfer Function
• Overall transfer function
• Poles when
( ) ( )( ) ( )1
KG sT sKG s H s
=+
( ) ( ) ( )1 1 2 1 180KG s H s k= − = ∠ +
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Two Parts to Consider
• The magnitude of the characteristic equation
• The angle of the characteristic equation
( ) ( )( ) ( )
11KG s H s KG s H s
= =
( ) ( ) ( )2 1 180KG s H s k∠ = ∠ +
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Rules for Root Locus
• Number of branches = closed loop poles
• The root locus is symmetric about the real axis
• The root locus segments lie on the real axis to the left of an odd number of open loop poles and zeros
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Rules Continued
• The root locus begins at the poles and ends at the zeros (finite and infinite) of G(s)H(s)
• Asymptotes
( )# #
2 1# #
a
a
finite poles finite zerosfinite poles finite zeros
kfinite poles finite zeros
σ
πθ
−=
−
+=
−
∑ ∑
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Rules Continued
• Break-out and Break-in points
( ) ( )[ ]
( ) ( ) ( ) ( )
0
0
d G s H sds
d dN s D s N s D sds ds
=
− =
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Low Order Loci
• Use only the first few rules– Use the rules in order
• Practice sketching loci to gain proficiency
• The following are eight examples of low order loci
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One Pole
1
12
2sp+= −
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Two Poles
2
1
2
16 824
s spp
+ += −
= −
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One Zero, Two Poles
2
1
1
2
36 8324
ss szpp
++ += −
= −
= −
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Zero Outside Two Poles
2
1
1
2
56 8524
ss szpp
++ += −
= −
= −
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Three Poles
3 2
1
2
3
112 44 48246
s s sppp
+ + += −
= −
= −
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No Breakaway Points
3 2
1
2,3
18 37 5023 4
s s spp j
+ + += −
= − ±
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With Breakaway Points
+ + += −
= − ±
3 2
1
2,3
125 193 169112 5
s s spp j
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One Zero, Three Poles
3 2
1
1
2
3
211 34 242146
ss s szppp
++ + += −
= −
= −
= −
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Comments on Last Slide
• Asymptotes intersect the real axis at:
• Breakaway point at:
( ) ( )1 4 6 24.5
3 1aσ− − − − −
= = −−
root 2 s3⋅ 17 s2
⋅+ 44 s⋅+ 44+ s,( ) 4.957−=