Root Locus - David Trumpertrumper.mit.edu/sites/default/files/Root_Locus.pdf · Root Locus Editor...
Transcript of Root Locus - David Trumpertrumper.mit.edu/sites/default/files/Root_Locus.pdf · Root Locus Editor...
Root Locus
• Root locus definition
• Root locus sketching rules
• Examples
• Root locus is no longer used much as a computational tool
• But, the perspective of how poles move as influenced by loop singularities is still extremely useful as a guide for design
Configuration for studying root locus
100
101
102
103
104
105
!270
!225
!180
!135
!90P.M.: 54.8 degFreq: 1.03e+03 rad/sec
Frequency (rad/sec)
Phase (
deg)
!100
!50
0
50
100
150
G.M.: !26.2 dBFreq: 112 rad/secStable loop
Open!Loop Bode Editor for Open Loop 1 (OL1)
Magnitude (
dB
)
!2500 !2000 !1500 !1000 !500 0 500!1000
!800
!600
!400
!200
0
200
400
600
800
1000Root Locus Editor for Open Loop 1 (OL1)
Real Axis
Imag A
xis
Use sisotool.m to get root locus and loop shape plots; see Matlab routine pidloop.m
!1200 !1000 !800 !600 !400 !200 0!1000
!800
!600
!400
!200
0
200
400
600
800
1000
Pole!Zero Map
Real Axis
Imagin
ary
Axis
Fine settling dominated by doublet near -50
This doublet gives overshoot
‘Dominant’ pole pair
Closed-loop poles and zeros
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
0.2
0.4
0.6
0.8
1
1.2
1.4
Step Response
Time (sec)
Am
plit
ude
0.025 0.03 0.035 0.04 0.045 0.05
0.97
0.98
0.99
1
1.01
1.02
Step Response
Time (sec)
Am
plit
ud
e
20 msec time constant in fine settling due to pole at about -50, which was attracted in root locus sense to the zero in that vicinity
For Ti = Td = 4 msec:
!2500 !2000 !1500 !1000 !500 0!250
!200
!150
!100
!50
0
50
100
150
200
250
Pole!Zero Map
Real Axis
Ima
gin
ary
Axis
125
130
135
140
145
150
155
160
Ma
gn
itu
de
(d
B)
101
102
103
104
105
!90
!45
0
45
90
Ph
ase
(d
eg
)
Bode Diagram
Frequency (rad/sec)
!3000 !2500 !2000 !1500 !1000 !500 0 500!300
!200
!100
0
100
200
300
Root Locus
Real Axis
Ima
gin
ary
Axis
!1200 !1000 !800 !600 !400 !200 0!1000
!800
!600
!400
!200
0
200
400
600
800
1000
Pole!Zero Map
Real Axis
Ima
gin
ary
Axis
Closed-loop poles and zeros
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050
0.2
0.4
0.6
0.8
1
1.2
1.4
Step Response
Time (sec)
Am
plit
ud
e
100
101
102
103
104
105
!270
!225
!180
!135
!90P.M.: 53.4 degFreq: 997 rad/sec
Frequency (rad/sec)
Ph
ase
(d
eg
)
!80
!60
!40
!20
0
20
40
60
80
100
G.M.: !12.8 dBFreq: 251 rad/secStable loop
Open!Loop Bode Editor for Open Loop 1 (OL1)
Ma
gn
itu
de
(d
B)
!2500 !2000 !1500 !1000 !500 0 500!1000
!800
!600
!400
!200
0
200
400
600
800
1000Root Locus Editor for Open Loop 1 (OL1)
Real Axis
Ima
g A
xis