Resposta em Freqüência de Malha Fechada a partir da Resposta em Freqüência de Malha Aberta

 

      Para um sistema em malha fechada estável, sua resposta em freqüência pode ser obtida diretamente a partir de sua resposta em freqüência de malha aberta.      Considere um sistema com realimentação unitária cuja função de transferência de malha fechada é dada por:

T j

1

j G jT j M e

G j

A relação entre e pode ser obtida em termos de variáveis complexas no plano-. As coordenadas real e imaginária do plano- são e , respectivamente. Assim:Lugares de magnitude constante (círculos-M): A magnitude da resposta em malha fechada de um sistema é dada por:

T j G j

G j

G ju

v

G j u jv M

1 22 2

1 22 21 1 1

u vG j u jvM

G j u jv u v

Elevando ao quadrado ambos os membros da equação acima e rearranjando:Observe que, quando , a equação anterior descreve uma linha reta no plano complexo.Dividindo esta equação por e adicionando o termo a ambos os lados:ou

2 2 2 2 2 21 1 2M u M v M u M 1M

1 2u

21 M

22 21M M

2 22 2 2 22 2

2 2 2 2

2

1 1 1 1

M u M M Mu v

M M M M

2 222

2 21 1

M Mu v

M M

Curves of constant module and phase of the closed loop

0-1

|F| = 1

|F| = 0.5

|F| = 2Re (G)

Im (G)

1+G G

arg(G)arg(1+G)

222 1·1

GGkkG

G

222222 ·1· yxykxk

22

22

2

2

22

11

k

ky

k

kx

1;

1 22

2

k

kr

k

kc

Circles of constant gain of F

0-1 -0.5

r

1+G G

c

Re(G)

Im(G)

GG

αψ1

arg

222 5.0cr

cr

ψ

5.02/tg

25.0;cotg5.0 2 crψc

Circles of constant phase of F

Gα 1arg

Garg

(centre = -0.5+jc, radius = r)

ψψψ

ψψψψ

ψψ

ψψ

ψψ

c

cψc

ψc

cψ

rcr

ψcr

cotg5.0

2sincos

25.02/·sin2/cos

2/cos2/sin·25.0

2/sin2/cos

2/cos2/sin

·25.02/tg·2/tan

1125.0

2/tg2/tg

5.05.0

2/tg5.05.0

2/tg;5.0

22

2

22

222

222

Derivation of r and c

0-1

|F| = 1

|F| = 0.5

|F| = 0.707|F| = 1.414

|F| = 2

Re (G)

Im (G)

3π/1 ·e414.1ω jjF

3π/22 ·e414.1ω jjF

8π/73 ·e5.0ω jjF

Nichols Chart

Ope

n-Lo

op P

hase

(de

g)

Open-Loop Gain (dB)

-18

0-1

35

-90

-10 -5 0 5 10 15 20 25

6 dB

3 dB

1 dB

0.5 dB

0.25

dB

phase margin 45o

ssss

jG

234 33

3,0ω

Nichols Chart: ln(G) =ln|G|+j·arg(G)

Ope

n-Lo

op P

hase

(de

g)

Open-Loop Gain (dB)

-180

-135

-90

-10 -5 0 5 10 15 20 25

6 d

B

3 d

B

1 d

B

0.5

dB

0.2

5 dB

phase margin 45o

ssss

jG

234 33

3.0ω

Limitations of Transfer Functions Developed from Pulse Tests

• They require an open loop time constant to complete.

• Disturbances can corrupt the results.

• Bode plots developed from pulse tests tend to be noisy near the crossover frequency which affects GM and PM calculations.

Closed Loop Frequency Response

Y sp (s) Y(s)G p (s)

Y s(s)

D(s)

++

G c(s) G a(s)-

+

G s(s)

G d (s)

Example of a Closed Loop Bode Plot

0

0.4

0.8

1.2

0.01 0.1 1 10 100

Ar

pf

Analysis of Closed Loop Bode Plot

• At low frequencies, the controller has time to reject the disturbances, i.e., Ar is small.

• At high frequencies, the process filters (averages) out the variations and Ar is small.

• At intermediate frequencies, the controlled system is most sensitive to disturbances.

Peak Frequency of a Controller

• The peak frequency indicates the frequency for which a controller is most sensitive.

Diagrama de Nichols

Nichols Chart

Open-Loop Phase (deg)

Open-L

oop G

ain

(dB

)

-180 -135 -90 -45 0-50

-40

-30

-20

-10

0

10

20

jwG

dB

jwG

10.9 Relation between Closed and Open-Loop Freq. Response

의 Bode diagram ∵

Fig.10.44(p.643)의 Nyquist diagram에서 에 대한 gain M(폐루프시스템의 이득 ) 과

phase 를 구할 수 있다 . 그러나 이 그림에서 open-loop gain이 변하면 Nyquist diagram이

변하게 되어 복잡해짐 Fig.10.47(p.645)의 Nichols chart

(Re, Im 축을 Mag, Phase 축으로 변환 )

)(1

)()(

sG

sGsT

)( jT

( 가 크면 high freq도 통과시키므로 속응성이 증가하기 때문에 )

, ,

10.9 Relation between Closed and Open-Loop Freq. Response

의 Bode diagram ∵

Fig.10.44(p.643)의 Nyquist diagram에서 에 대한 gain M(폐루프시스템의 이득 ) 과

phase 를 구할 수 있다 . 그러나 이 그림에서 open-loop gain이 변하면 Nyquist diagram이

변하게 되어 복잡해짐 Fig.10.47(p.645)의 Nichols chart

(Re, Im 축을 Mag, Phase 축으로 변환 )

sT pT rTBWω

BWω

)(1

)()(

sG

sGsT

)( jT

Figure 10.44

Nyquist diagram for Example 10.11 and constant M and N circles

Figure 10.47

Nichols chart with frequency response for superimposed. Values for and are shown

)]2)(1(/[)( sssKsG1K 16.3K

* MATLAB을 이용하면 의 Bode 선도를 정확히 그릴 수 있으므로 Nichols chart의

의미가 반감됨 .

* Closed-loop transfer function이 standard 2nd order system일 때에 , open-loop

system의 phase margin과 closed-loop system의 가 Fig.10.48과 같다 .(p.648)

10.11 Steady-state Error Specs from the Open-loop Frequency

Fig.10.51(p.651)

)( jT

100

PM 단 , 일 때에70PM

Figure 10.48

Phase margin vs. damping ration

Figure 14.15 A Nichols chart. [The closed-loop amplitude ratio ARCL ( ) and phase angle are shown in families of curves.] φCL