automatique by meiling Ch en 1 Lesson 11 Bode Diagram
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automatique by meiling Chen 1
Lesson 11
Bode Diagram
automatique by meiling Chen 2
Viewpoints of analyzing control system behavior
• Routh-Hurwitz • Root locus• Bode diagram (plots)• Nyquist plots• Nicols plots• Time domain
)( js
)( js
)( js
)( js
)( js
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L.T.I systemtAtr sin)( )sin()( tBty
Magnitude: Phase:A
B
G(s)
H(s)
＋ －
)(ty)(tr
)()(1
)(
)(
)(
sHsG
sG
sR
sY
jsjs
Magnitude: Phase:
)()(1
)(
jHjG
jG
)]()(1[
)(
jHjG
jG
Steady state response
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1
210log
decDecade :1
22log
octOctave :
1 10 100
Logarithmic coordinate
2 3 4 20
dB
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))()((
))((
)(
)(2
21
21
basspsps
zszsk
sR
sY
Case I : k
Magnitude:
Phase:
)(log20 dBkkdB
0,180
0,0
k
kk
o
o
)(dBGH
GH
1.0 1 10
090
0180
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Case II :
Magnitude:
Phase:
)(log20)(
1dBp
jdB
p
pj
op
)90()(
1
)(dBGH
GH
1.0 1 10
0900180
ps
1
090
1p
1p
2p
2p
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Case III :
Magnitude:
Phase:
)(log20)( dBpjdB
p
pj op )90()(
)(dBGH
GH
1.0 1 10
0900180
ps
0901p
1p
2p
2p0180
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Case IV :1)1
1(
)(
s
aor
as
a
Magnitude:
Phase:
])(1log[10
)(1log20)1(
2
21
a
aaj
dB
aaj
10 tan0)1(
)(dBGH
GH
1.0 1 10
0900180
090
0180
01log100 dBa
a
]log20log20[
log201
adBa
dBaa
ja
oGHa
a 00tan0 1
oGHa
a 90tan 1
01.32log1011 dBja
045a
1a
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Case V :
Magnitude:
Phase:
])(1log[10
)(1log20)1(
2
2
a
aaj
dB
aaj
1tan)1(
)(dBGH
GH
1.0 1 10
0900180
090
0180
)11
()(
sa
ora
as
01log100 dBa
a
adBa
dBaa
ja
log20log20
log201
oGHa
a 00tan0 1
oGHa
a 90tan 1
01.32log1011 dBja
1a
045a
automatique by meiling Chen 10
automatique by meiling Chen 11
Case VI : 22
2
2)(
nn
n
sssT
2
1
2
221
22
2
)(1
2
tan)(2))(1(
1)(
)(
2tan)(
2)()(
n
n
nn
n
n
nn
n
jTj
jT
jTj
jT
1,)log(40
1,)2log(20
1,0
)(
nn
n
n
jT
1,
1,
1,
180
90
0
)( 0
0
n
n
n
o
jT
automatique by meiling Chen 12
n
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Example : )10(
)2(50)(
ss
ssT
)10
10)(
2
2)(
1(10)(
s
s
ssT
Example : page 6-24
Example : page 6-28
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automatique by meiling Chen 15
)1.01)(5.01()(
sss
kskGH
n=[-3 -9]m=[1 –1 –1 –15 0]g2=tf(n,m)bode(g1,g2)
ssss
skskGH
15
)93()(
234
g1=zpk([],[0 –2 -10],[1])bode(g1)
MATLAB Method
g1g2
g1
g2
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Identification
150
150
15
1516)(
sssF
Example 6-39
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0,0,)(
)()(
1
1
iinpz
pss
zsksT
Minimum phase system
Type 0 : (i.e. n=0)
)()(
1
1
ps
pksT p
)(dBGH
11.0 p 1p 110 p
A
AK p log20
0dB/dec
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Type I : (i.e. n=1)
)()(
1
1
pss
pksT v
)(dBGH
11.0 p 1p
110 p
A
-20dB/dec
-40dB/dec
1AKv log20 0
dBj
Kv 0log200
vk0
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Type 2 : (i.e. n=2)
)()(
12
1
pss
pksT a
)(dBGH
11.0 p 1p 110 p
A
-40dB/dec
-60dB/dec
1AKa log200
dBj
Ka 0)(
log202
0
ak20
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A transfer function is called minimum phase when all the poles and zeros are LHP and non-minimum-phase when there are RHP poles or zeros.
Minimum phase system Stable
The gain margin (GM) is the distance on the bode magnitude plot from the amplitude at the phase crossover frequency up to the 0 dB point. GM=-(dB of GH measured at the phase crossover frequency)
The phase margin (PM) is the distance from -180 up to the phase at the gain crossover frequency. PM=180+phase of GH measured at the gain crossover frequency
Relative stability
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Open loop transfer function :
Closed-loop transfer function :
)()( sHsG
)()(1 sHsG
Open loop Stability poles of in LHP)()( sHsG
)0,0()0,1(Re
Im
RHPClosed-loop Stability poles of in left side of (-1,0))()( sHsG
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)(dBGH
GH
0900180
090
0180
0180
0)0,1(
dB
g
p
Gain crossover frequency: g
phase crossover frequency: p
P.M.>0
G.M.>0
Stable system
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)(dBGH
GH
0900180
090
0180
g
pP.M.<0
G.M.<0
Stable system
Unstable system
Unstable system
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