Root Locus Techniques
-
Upload
vishwajeet-singh -
Category
Documents
-
view
42 -
download
2
Transcript of Root Locus Techniques
![Page 1: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/1.jpg)
Root Locus Techniques
EE-371 / EE-502 Control SystemsMilwaukee School of Engineering
Fall Term 2005Dr. Glenn Wrate, P.E.
![Page 2: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/2.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 2
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
![Page 3: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/3.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 3
Closed Loop System
K G
H
Forward Transfer Function
Feedback
OutputInput
-
+
R(s) C(s)E(s)
![Page 4: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/4.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 4
Transfer Function
• Overall transfer function
• Poles when
( ) ( )( ) ( )1
KG sT sKG s H s
=+
( ) ( ) ( )1 1 2 1 180KG s H s k= − = ∠ +
![Page 5: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/5.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 5
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∠ = ∠ +
![Page 6: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/6.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 6
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
![Page 7: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/7.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 7
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
−=
−
+=
−
∑ ∑
![Page 8: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/8.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 8
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
=
− =
![Page 9: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/9.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 9
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
![Page 10: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/10.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 10
One Pole
1
12
2sp+= −
![Page 11: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/11.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 11
Two Poles
2
1
2
16 824
s spp
+ += −
= −
![Page 12: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/12.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 12
One Zero, Two Poles
2
1
1
2
36 8324
ss szpp
++ += −
= −
= −
![Page 13: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/13.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 13
Zero Outside Two Poles
2
1
1
2
56 8524
ss szpp
++ += −
= −
= −
![Page 14: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/14.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 14
Three Poles
3 2
1
2
3
112 44 48246
s s sppp
+ + += −
= −
= −
![Page 15: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/15.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 15
No Breakaway Points
3 2
1
2,3
18 37 5023 4
s s spp j
+ + += −
= − ±
![Page 16: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/16.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 16
With Breakaway Points
+ + += −
= − ±
3 2
1
2,3
125 193 169112 5
s s spp j
![Page 17: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/17.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 17
One Zero, Three Poles
3 2
1
1
2
3
211 34 242146
ss s szppp
++ + += −
= −
= −
= −
![Page 18: Root Locus Techniques](https://reader036.fdocuments.us/reader036/viewer/2022081907/54f4cc934a795997318b4b20/html5/thumbnails/18.jpg)
10/18/2005 © 2005, Milwaukee School of Engineering 18
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−=