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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
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