automatic control by meiling CHEN 1
Lesson 9
Root locus
Automatic control2. Analysis
automatic control by meiling CHEN 2
Poles and zeros
)())((
)())(()(
21
21
n
m
pspsps
zszszsksF
n
m
ppp
zzz
,,
,,
21
21 zerospoles
axisRe
axisIm
pole
zero
automatic control by meiling CHEN 3
2
2
2)(
2nn
n
sssT
Closed-loop transfer function :
axisRe
axisIm
21 nj
n
n
%100..
4
1
cos
21
2
eom
T
T
ns
n
p
j
)cos1sin(1
1)( 12
2
te
ty n
tn
automatic control by meiling CHEN 4
j
1p
2p
3p
321
123
321
321
321
321
......
sss
nnn
ppp
TTT
sososo
TTT
n
21 nj
automatic control by meiling CHEN 5
j
1p2p3p
321
123
321
321
321
321
......
sss
nnn
ppp
TTT
TTT
sososo
n
..)(
1)(
)(
2
soiii
Tii
Ti
pn
sn
automatic control by meiling CHEN 6
j
0
01
1
01
0
1
10
10
11 1
1
1
1
10
0
0
Overdamped
Critically damped
Underdamped
Undamped
Negative damped
automatic control by meiling CHEN 7
Root locus
k G(s)
H(s)
+-
)(ty)(tr
)()(1
)(
)(
)()(
sHskG
skG
sR
sysT
0)()(1 sHskG poles
automatic control by meiling CHEN 8
)12()()(
1)()(
1)()(
0)()(1
nsHskG
sHskG
sHskG
sHskGOpen loop transfer function
Using open loop transfer function + system parameters to analyze the closed-loop system response
0k
Draw the s-plan root locus
automatic control by meiling CHEN 9
Root locus properties:(i) The locus segments are symmetrical about the real axis.(ii)
(iii)
0,)()(
1k
sHsGk
zerossHsGk
polessHsGk
)()(,
)()(,0
j0s
1
2
4
3)()()( 432100 sHsG
automatic control by meiling CHEN 10
Root locus construction
(i) Loci Branches each locus from poles to zeros 0k k
mn if for excess zeros or poles, locus segments extend from infinity.
(1) 0mn
(2) 0mn
branchesmn
zerosbranchesnm
automatic control by meiling CHEN 11
(ii) Real axis segments
Poles + zeros = odd
Poles + zeros = even
0180
01801)()( sHskG
automatic control by meiling CHEN 12
(iii) Asymptotic angles ,2,1,0,)12(
kmn
kk
0454
1802,6 mnif
045
045
automatic control by meiling CHEN 13
(iv) Centroid of the asymptotes
mn
zerospoles
)186)(2(
3)()(
2
sss
ssHsG
Zero : 0Poles: -2, -3+j3, -3-j3 4
13
0)33332(
jj
09013
180
example
automatic control by meiling CHEN 14
(v) Breakaway and entry points 0ds
dk
example)2)(1(
sss
kkGH
01 kGH The characteristic function of closed loop system
0)2)(1(
231
23
sss
kssskGH
577.1,423.0
0263
)23(
2
23
s
ssds
dk
sssk
automatic control by meiling CHEN 15
(vi) Angle of departure and approach
)()(180
)()(1800
0
sHsG
sHsG
A
D
example)1)(1(
)2(
jsjs
skkGH
Angle of departure from the pole: js 1
0
0
0
0
225
)11()21(180
180)11()21(
180)1()2(
180)1()1()2(
D
D
D
D
jjj
jjj
jss
jsjss
automatic control by meiling CHEN 16
Angle of approach to the zero:
example )1(
))((
ss
jsjskkGH
js
0
0
0
0
0
135
)1()(180
180)1()(
180)1()(
180)1()()(
A
A
A
A
jjjj
jjjj
ssjs
ssjsjs
automatic control by meiling CHEN 17
(vii) The cross point of root locus and Im-axis
example)22)(3( 2
ssss
kkGH
The characteristic function of closed loop system:
0685
0)22)(3(234
2
kssss
kssss
ks
ks
ks
s
ks
0
1
2
3
4
34
252045
3465
81
16.8
034
25204
k
k
095.1
05
34 2
js
ks
automatic control by meiling CHEN 18
)1.01)(5.01( sss
k
+ -
)(sC)(sR
)1.01)(5.01()(
sss
kskGH
,,
10,2,0
zeros
poles
6003
180
403
0)10()2(0
k
05.7,945.0
0)6.005.0(
06.005.0
0)1.01)(5.01(
21
23
23
ss
sssds
d
ds
dk
ksss
ksss(i)
(ii)
(iii)
automatic control by meiling CHEN 19
ks
ks
ks
s
ksss
6.0
05.06.06.0
105.0
06.005.0
1
2
3
23
5.4
0126.0
122
js
s
k
)0(
0
k)0(
2
k)0(
10
k
)12(5.4 kj
)12(5.4 kj
945.0s4
060
automatic control by meiling CHEN 20
)1.01)(5.01()(
sss
kskGH
MATLAB method
n=[-3 -9]m=[1 –1 –1 –15 0]gh=tf(n,m)rltool(gh)
ssss
skskGH
15
)93()(
234
gh=zpk([],[0 –2 -10],[1])rltool(gh)