Transcript Slide 1

Chapter 13
Frequency Response Characteristics of
Feedback Controllers
Proportional Controller. Consider a proportional controller with
positive gain
Gc  s   Kc
(13-57)
In this case Gc  jω   K c , which is independent of w.
Therefore,
AR c  K c
(13-58)
and
φc  0
(13-59)
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Chapter 13
Proportional-Integral Controller. A proportional-integral (PI)
controller has the transfer function (cf. Eq. 8-9),

 τI s 1 
1 
Gc  s   Kc 1 
  Kc 

τ
s
τ
s
I 

 I 
Substitute s=jw:
(13-60)

 jwτ I  1 

1 
1
Gc  jw  Kc 1 
  Kc 
  Kc 1 
 τ I jw 
 jwτ I 
 τI w

j

Thus, the amplitude ratio and phase angle are:
AR c  Gc  jω   K c 1 
1
 ωτ I 
2
 Kc
 ωτ I 
2
1
ωτ I
φc  Gc  jω   tan 1  1/ ωτ I   tan 1  ωτ I   90
(13-62)
(13-63)
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Chapter 13
 10 s  1 
G
s

2
Figure 13.9 Bode plot of a PI controller, c  


10
s


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Ideal Proportional-Derivative Controller. For the ideal
proportional-derivative (PD) controller (cf. Eq. 8-11)
Chapter 13
Gc  s   Kc 1  τ D s 
(13-64)
The frequency response characteristics are similar to those of a
LHP zero:
AR c  K c
 ωτ D 
2
1
φ  tan 1  ωτ D 
(13-65)
(13-66)
Proportional-Derivative Controller with Filter. The PD controller
is most often realized by the transfer function
 τDs 1 
Gc  s   Kc 

ατ
s

1
 D

(13-67)
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Chapter 13
Figure 13.10 Bode
plots of an ideal PD
controller and a PD
controller with
derivative filter.
Idea: Gc  s   2  4s  1
With Derivative
Filter:
 4s  1 
Gc  s   2 

 0.4 s  1 
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PID Controller Forms
Chapter 13
Parallel PID Controller. The simplest form in Ch. 8 is


1
Gc  s   Kc 1 
 τDs 
 τ1s

Series PID Controller. The simplest version of the series PID
controller is
 τ1s  1 
Gc  s   Kc 
(13-73)
  τ D s  1
 τ1s 
Series PID Controller with a Derivative Filter.
 τ1s  1  τ D s  1 
Gc  s   Kc 


 τ1s   τ D s  1 
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Chapter 13
Figure 13.11 Bode
plots of ideal parallel
PID controller and
series PID controller
with derivative filter
(α = 1).
Idea parallel:
1


Gc  s   2 1 
 4s 
 10 s

Series with
Derivative Filter:
 10 s  1  4 s  1 
Gc  s   2 


10
s
0.4
s

1



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Nyquist Diagrams
Chapter 13
Consider the transfer function
1
G s 
2s  1
(13-76)
with
AR  G  jω  
1
 2ω  1
2
(13-77a)
and
φ  G  jω    tan 1  2ω 
(13-77b)
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Chapter 13
Figure 13.12 The Nyquist diagram for G(s) = 1/(2s + 1)
plotting Re  G  jω   and Im  G  jω   .
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Chapter 13
Figure 13.13 The Nyquist diagram for the transfer
function in Example 13.5:
5(8s  1)e6s
G( s) 
(20s  1)(4s  1)
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