Chapter 14
Frequency Response
Chapter 14
Force dynamic process with A sin t ,
U (s) 
A
s 
2
2
14.1
1
Chapter 14
Input: A sin  t
Output: Aˆ sin  t   
Aˆ / A is the normalized amplitude ratio (AR)
 is the phase angle, response angle (RA)
AR and  are functions of ω
Assume G(s) known and let
s  j
G  AR 
G
 j  
K1  K 2 j
K1  K 2
2
   G  arctan
2
K2
K1
2
Example 14.1:
Chapter 14
G s 
1
 s 1
G  j  
G  j  
1

1   j
( j   1)
2
1   j 1   j
1
1  
2
K1
2


1  
2
2
j
K2
3
4
Chapter 14
G 
1
1  
2
2
   arctan   

Chapter 14
as    ,    90
0
Use a Bode plot to illustrate frequency response
(plot of log |G| vs. log  and  vs. log )
log coordinates:
G  G1  G 2  G 3
G  G1  G 2  G 3
lo g G  lo g G 1  lo g G 2  lo g G 3
 G   G1   G 2   G 3
G 
G1
G2
lo g G  lo g G 1  lo g G 2
 G   G1   G 2
5
Chapter 14
Figure 14.4 Bode diagram for a time delay, e-qs.
6
Example 14.3
 0.5 s
(20 s  1)(4 s  1)
Chapter 14
G (s) 
5(0.5 s  1) e
7
Chapter 14
F.R. Characteristics of Controllers
Recall that the F.R. is characterized by:
1. Amplitude Ratio (AR)
2. Phase Angle ()
For any T.F., G(s)
A R  G ( j )
   G ( j )
A) Proportional Controller
GC (s)  K C
 AR  K C ,
 0
B) PI Controller

1 
For G C ( s )  K C  1 


s

I

  tan
1
AR  K C
1
 I
2
2
1

1 






I

The Bode plot for a PI controller is shown in next slide.
Note b = 1/I . Asymptotic slope ( 0) is -1 on log-log plot.
8
9
Chapter 14
Ideal PID Controller.
Chapter 14
G c ( s )  K c (1 
1
Is
  D s)
(14  48)
Series PID Controller. The simplest version of the series PID
controller is
 τI s 1 
Gc  s   K c 
  τ D s  1
 τI s 
(14-50)
Series PID Controller with a Derivative Filter. The series
controller with a derivative filter was described in Chapter 8
 τI s 1  τD s 1 
Gc  s   K c 


τ
s
α
τ
s

1
I
D



(14-51)
10
Chapter 14
Figure 14.6 Bode
plots of ideal parallel
PID controller and
series PID controller
with derivative filter
(α = 1).
Ideal 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



11
Chapter 14
Advantages of FR Analysis for Controller Design:
1. Applicable to dynamic model of any order
(including non-polynomials).
2. Designer can specify desired closed-loop response
characteristics.
3. Information on stability and sensitivity/robustness is
provided.
Disadvantage:
The approach tends to be iterative and hence time-consuming
-- interactive computer graphics desirable (MATLAB)
12
Controller Design by Frequency Response
- Stability Margins
Chapter 14
Analyze GOL(s) = GCGVGPGM
(open loop gain)
Three methods in use:
(1) Bode plot |G|,  vs.  (open loop F.R.) - Chapter 14
(2) Nyquist plot - polar plot of G(j) - Appendix J
(3) Nichols chart |G|,  vs. G/(1+G) (closed loop F.R.) - Appendix J
Advantages:
• do not need to compute roots of characteristic equation
• can be applied to time delay systems
• can identify stability margin, i.e., how close you are to instability.
13
14.8
14
Chapter 14
Frequency Response Stability Criteria
Chapter 14
Two principal results:
1. Bode Stability Criterion
2. Nyquist Stability Criterion
I) Bode stability criterion
A closed-loop system is unstable if the FR of the
open-loop T.F. GOL=GCGPGVGM, has an amplitude ratio
greater than one at the critical frequency,  C . Otherwise
the closed-loop system is stable.
• Note:  C  value of  where the open-loop phase angle
is -1800. Thus,
• The Bode Stability Criterion provides info on closed-loop
stability from open-loop FR info.
• Physical Analogy: Pushing a child on a swing or
bouncing a ball.
15
Example 1:
A process has a T.F.,
Chapter 14
G p (s) 
2
(0.5 s  1)
3
And GV = 0.1, GM = 10 . If proportional control is used, determine
closed-loop stability for 3 values of Kc: 1, 4, and 20.
Solution:
The OLTF is GOL=GCGPGVGM or...
GOL ( s ) 
2KC
(0.5 s  1)
3
The Bode plots for the 3 values of Kc shown in Fig. 14.9.
Note: the phase angle curves are identical. From the Bode
diagram:
KC
1
4
20
A R OL
0 .2 5
1 .0
5 .0
S ta b le ?
Yes
C o n d itio n a lly sta b le
No
16
Chapter 14
Figure 14.9 Bode plots for GOL = 2Kc/(0.5s + 1)3.
17
• For proportional-only control, the ultimate gain Kcu is defined to
be the largest value of Kc that results in a stable closed-loop
system.
Chapter 14
• For proportional-only control, GOL= KcG and G = GvGpGm.
AROL(ω)=Kc ARG(ω)
(14-58)
where ARG denotes the amplitude ratio of G.
• At the stability limit, ω = ωc, AROL(ωc) = 1 and Kc= Kcu.
K cu 
1
A R G (ω c )
(14-59)
18
Example 14.7:
Determine the closed-loop stability of the system,
Chapter 14
G p (s) 
4e
s
5s  1
Where GV = 2.0, GM = 0.25 and GC =KC . Find C from the
Bode Diagram. What is the maximum value of Kc for a stable
system?
Solution:
The Bode plot for Kc= 1 is shown in Fig. 14.11.
Note that:
 C  1.69 rad m in
A ROL
 C
 K C m ax =
 0.235
1
A RO L

1
 4.25
0.235
19
14.11
20
Chapter 14
Ultimate Gain and Ultimate Period
Chapter 14
• Ultimate Gain: KCU = maximum value of |KC| that results in a
stable closed-loop system when proportional-only
control is used.
2
• Ultimate Period: PU 
C
• KCU can be determined from the OLFR when
proportional-only control is used with KC =1. Thus
K CU 
1
A RO L
for K C  1
 C
• Note: First and second-order systems (without time delays)
do not have a KCU value if the PID controller action is correct.
21
Gain and Phase Margins
Chapter 14
• The gain margin (GM) and phase margin (PM) provide
measures of how close a system is to a stability limit.
• Gain Margin:
Let AC = AROL at  = C. Then the gain margin is
defined as: GM = 1/AC
According to the Bode Stability Criterion, GM >1  stability
• Phase Margin:
Let g = frequency at which AROL = 1.0 and the
corresponding phase angle is g . The phase margin
is defined as: PM = 180° + g
According to the Bode Stability Criterion, PM >0  stability
See Figure 14.12.
22
23
Chapter 14
Rules of Thumb:
A well-designed FB control system will have:
Chapter 14
1.7  GM  2.0
30  PM  45
Closed-Loop FR Characteristics:
An analysis of CLFR provides useful information about
control system performance and robustness. Typical desired
CLFR for disturbance and setpoint changes and the
corresponding step response are shown in Appendix J.
24
Chapter 14
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