Automatique by meiling Chen1 Lesson 11 Bode Diagram.

automatique by meiling Chen 1

Lesson 11

Bode Diagram


Viewpoints of analyzing control system behavior

• Routh-Hurwitz • Root locus• Bode diagram (plots)• Nyquist plots• Nicols plots• Time domain

)( js

)( js

)( js

)( js

)( js


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


1

210log

decDecade :1

22log

octOctave :

1 10 100

Logarithmic coordinate

2 3 4 20

dB


))()((

))((

)(

)(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


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


Case III :

Magnitude:

Phase:

)(log20)( dBpjdB

p

pj op )90()(

)(dBGH

GH

1.0 1 10

0900180

ps

0901p

1p

2p

2p0180


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


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


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


n


Example : )10(

)2(50)(

ss

ssT

)10

10)(

2

2)(

1(10)(

s

s

ssT

Example : page 6-24

Example : page 6-28


)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


Identification

150

150

15

1516)(

sssF

Example 6-39


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


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


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


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


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


)(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


)(dBGH

GH

0900180

090

0180

g

pP.M.<0

G.M.<0

Stable system

Unstable system

Unstable system

Automatique by meiling Chen1 Lesson 11 Bode Diagram.

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