Transcript Slide 1

Chapter 5
Sinusoidal Input
1
Chapter 5
Processes are also subject to periodic, or cyclic, disturbances.
They can be approximated by a sinusoidal disturbance:
0 for t  0
U sin  t   
 A sin t  for t  0
where:
A = amplitude, ω = angular frequency
A
U sin ( s )  2
s  2
Examples:
1. 24 hour variations in cooling water temperature.
2. 60-Hz electrical noise (in USA!)
2
For a sine input to the 1st order process:

K
U ( s)  A 2
and G  s  
2
s 
 s 1
Chapter 5
output is...
0
1s
2
K
A
Y ( s) 
 2

 2
 2
2
2
 s 1 s    s 1 s   s  2
By partial fraction decomposition,
 AK 2
0  2 2
  1
 AK
1  2 2
  1
 AK
2  2 2
  1
3
Inverting,
this term dies out for large t
AK  t 
AK
e

sin(t   )
2 2
2
2
  1
  1
AK
y 
sin(t   )  B sin(t   )
2 2
  1
where
B
K

 amplitude ratio
2
2
A
  1
Chapter 5
y (t ) 
   arctan( )  phase angle
Note that the amplitude ratio and phase angle is not a
function of t but of τ and ω. For large t, y(t) is also
sinusoidal, output sine is attenuated by…
1
2  2  1
4
Figure 13.1 Attenuation and time shift between input and output
sine waves (K= 1). The phase angle φ of the output signal is given
φ    t / P   360 , where t is the (period) shift and P
by
is the period of oscillation.
Frequency Response Characteristics of a
First-Order Process
ˆ sin  ωt  φ  as t   where :
For x(t )  A sin t , y t   A
Aˆ 
KA
ω2 τ 2  1
and φ   tan 1  ωτ 
1. The output signal is a sinusoid that has the same frequency, ,
as the input signal, x t   Asin t
2. The amplitude of the output signal, Aˆ , is a function of the
frequency  and the input amplitude, A:
Aˆ 
KA
ω2 τ 2  1
Dividing both sides by the input signal amplitude A yields the
amplitude ratio (AR)
Aˆ  
K
AR   

A
ω2 τ 2  1
which can, in turn, be divided by the process gain to yield the
normalized amplitude ratio (ARN) (or magnitude ratio):
AR N   
1
ω2 τ 2  1
3. The output has a phase shift, φ, relative to the input. The
amount of phase shift depends on frequency, i.e.,
t
    2   tan 1 
P
Basic Theorem of Frequency
Response
The frequency response of a system can be found
by substituting j for s in the system transfer function,
i.e.
Laplace domain
G s

s  j
ferquency domain
G  j 
8
Properties
G  j   AR    amplitude ratio
G  j       phase angle
Note,
G  j   R    jI    G  j  e jG  j 
G  j   R    I  
2
G  j   tan
1
2
I  
R  
9
Shortcut Method for Finding
the Frequency Response
The shortcut method consists of the following steps:
Step 1. Set s=j in G(s) to obtain G  jω  .
Step 2. Rationalize G(j); We want to express it in the form.
G(j)=R + jI
where R and I are functions of . Simplify G(j) by
multiplying the numerator and denominator by the
complex conjugate of the denominator.
Step 3. The amplitude ratio and phase angle of G(s) are given
by:
AR  R 2  I 2
Memorize 
  tan 1 ( R / I )
Example 13.1
Find the frequency response of a first-order system, with
1
G s 
τs  1
(13-16)
Solution
First, substitute s  jω in the transfer function
1
1
G  jω  

τjω  1 jωτ  1
(13-17)
Then multiply both numerator and denominator by the complex
conjugate of the denominator, that is,  jωτ  1
 jωτ  1
 jωτ  1
G  jω  
 2 2
 jωτ  1  jωτ  1 ω τ  1

1
ω τ 1
2 2
j
 ωτ 
ω τ 1
2 2
 R  jI
(13-18)
R
where:
I
1
(13-19a)
ω τ 1
ωτ
2 2
(13-19b)
ω τ 1
2 2
From Step 3 of the Shortcut Method,
2
1

  ωτ 
AR  R 2  I 2   2 2    2 2 
 ω τ 1   ω τ 1 
or


2
2 2
 ω τ 1
1  ω2 τ 2
AR 
Also,
φ  tan
1 
1
ω2 τ 2  1
2
(13-20a)
I
1
1

tan

ωτ


tan


 ωτ  (13-20b)
 
R
Complex Transfer Functions
Consider a complex transfer G(s),
Ga  s  Gb  s  Gc  s 
G s 
G1  s  G2  s  G3  s 
Substitute s=j,
Ga  jω  Gb  jω  Gc  jω 
G  jω  
G1  jω  G2  jω  G3  jω 
(13-22)
(13-23)
From complex variable theory, we can express the magnitude and
angle of G  jω  as follows:
G  jω  
Ga  jω  Gb  jω  Gc  jω 
(13-24a)
G1  jω  G2  jω  G3  jω 
G  jω   Ga  jω   Gb  jω   Gc  jω  
 [G1  jω   G2  jω   G3  jω  
]
(13-24b)
Transfer Functions in Series
n
Y s
 G  s    Gi  s 
X s
i 1
s  j
n
G  j    Gi  j 
i 1
n
Gi  j 

 
i 1
G  j  e
  Gi  j  e
  Gi  j   e
i 1
 i 1

n
 n

ln G  j   jG  j   ln  Gi  j    j  Gi  j 
i 1
 i 1


G  j 
n
Gi  j 
n
n
ln G  j    ln Gi  j 
i 1
n
G  j    Gi  j 
i 1
14
Transfer Functions in Series
Gi  s  
Kp
e  s

zi

s 1
1
 pi s  1
1
2 2
 i s  2 i i s  1
 0   i  1
15
Bode Diagrams
• A special graph, called the Bode diagram or
Bode plot, provides a convenient display of the
frequency response characteristics of a transfer
function model. It consists of plots of AR and
phase angle as a function of frequency.
• Ordinarily, frequency is expressed in units of
radians/time.
16
Bode Plot of A First-order System
Recall:
AR N 
1
ω2 τ 2  1
and φ   tan 1  ωτ 
 At low frequencies ( ω  0 and ω
AR N  1 and   
 At high frequencies ( ω  0 and ω
1) :
1) :
AR N  1/ ωτ and   
17
Figure 13.2 Bode diagram for a first-order process.
• Note that the asymptotes intersect at ω  ωb  1/ τ, known as the
break frequency or corner frequency. Here the value of ARN
from (13-21) is:
AR N  ω  ωb  
1
 0.707
11
(13-30)
• Some books and software defined AR differently, in terms of
decibels. The amplitude ratio in decibels ARd is defined as
AR d  20 log AR
(13-33)
Negative Zeros
Consider a process zero term,
G  s   K ( sτ  1)
Substituting s=j gives
G  jω   K ( jωτ  1)
Thus:
AR  G  jω   K ω2 τ 2  1
φ  G  jω    tan 1  ωτ 
Note: In general, a multiplicative constant (e.g., K) changes
the AR by a factor of K without affecting φ .
20
Negative Zeros
1
AR N    ω τ  1 and φ  tan
 ωτ 
 At low frequencies ( ω  0 and ω
1) :
2 2
AR N  1 and   
 At high frequencies ( ω  0 and ω
1) :
AR N  ωτ and   
21
Positive Zeros
G  s   1  s
AR N    ω2 τ 2  1 and φ   tan 1  ωτ 
 At low frequencies ( ω  0 and ω
AR N  1 and   
1) :
 At high frequencies ( ω  0 and ω
1) :
AR N  ωτ and   
22
Examples
1
 2
1
1s  1 2 s  1
 z1s  1

p1
s  1 p 2 s  1
23
n-th Power of s
G  s   sn
n  1, 2,
G  j   j 
n
AR    
n
n
Im G  j 
    tan
Re G  j 
1
24
Integrating Elements
The transfer function for an integrating element
was given in Chapter 5:
Y s
1

G s 
U s s
1
1

AR  G  j  
j 
φ  G  jω   K       90
25
Differentiating Element
G s  s
AR  G  j   
φ  G  jω   tan
1 
0
 90
26
Second-Order System
G s 
K
τ 2 s 2  2ζτs  1
G  j  

K 1   2 2

1  
2 2

2
 4  
j
2 2 2

2 K 
1  
2 2

2
 4 2 2 2
1
ARN   

1  
   tan

2 2

2
  2 
2
2 
1   2 2 


1 
27
Second-Order System
Low-frequency asymptote (slope=0):
  0  ARN  1 and   0
High-frequency asymptote (slope=-2):
    ARN 
1

2
2
and   
Beaking frequency:
1

1
b  1  ARN b  
and    tan   
2
2
28
Resonant Peak
d  ARN 
1
0 
d  
2

 2 2  1   2 2   0

3
2
 2  2  1   2 2   8 2 


  1  2 2
1

2

*

1

2





1
 AR* 

2 1  2 2
Notice that, since  and  must both be real,

1
( 0.707)  
2
29
Figure 13.3 Bode diagrams for second-order processes.
Time Delay
Its frequency response characteristics can be obtained
according to
G  jω   e jωθ
which can be written in rational form by substitution of the
Euler identity,
G  jω   e jωθ  cosωθ  j sin ωθ
Thus
AR  G  jω   cos 2 ωθ  sin 2 ωθ  1
 sin ωθ 
φ  G  jω   tan 1  

cos
ωθ


or
φ  ωθ
Figure 13.6 Bode diagram for a time delay, e θs.
Figure 13.7 Phase angle plots for e θs and for the 1/1 and 2/2
Padé approximations (G1 is 1/1; G2 is 2/2).
Example
e s
G s 
s 1
Let
1
G1  s   e and G2  s  
s 1
AR1    G1  j   1
s
AR2   
1
1 2
AR    AR1    AR2  
    1    2      tan 1 
34
Nyquist Diagrams
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)
35
Figure 13.12 The Nyquist diagram for G(s) = 1/(2s + 1)
plotting Re  G  jω   and Im  G  jω   .
36
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)
37
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 . Therefore,
AR c  K c
(13-58)
and
φc  0
(13-59)
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=j:
(13-60)

 jτ I  1 

1 
1
Gc  j  Kc 1 
  Kc 
  Kc 1 
 τ I j 
 jτ I 
 τI 

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)
 10 s  1 
G
s

2
Figure 13.9 Bode plot of a PI controller, c  


10
s


Ideal Proportional-Derivative Controller. For the ideal
proportional-derivative (PD) controller (cf. Eq. 8-11)
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)
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 
PID Controller Forms
• 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 
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


