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