Chapter 14 Frequency Response -...
Transcript of Chapter 14 Frequency Response -...
Chapter 14
Frequency Response
Force dynamic process with A sin ωt , 22
)(ωω+
=s
AsU
Chapter 14
14.1
1
Input:
Output:
is the normalized amplitude ratio (AR)
φ is the phase angle, response angle (RA)
AR and φ are functions of ω
Assume G(s) known and let
tsinA ω
( )φω +tAsinˆ
( ) 1 2
2 2
1 2
2
1
arctan
s j G j K K j
G AR K K
KG
K
ω ω
φ
= = +
= = +
= ∠ =
Chapter 14
AA /ˆ
2
Example 14.1:
( ) 21 1( 1)
1 1
jG j j
j j
τ ωω
τ ω τ ω−
= ⋅ = −+ −
( ) 1
1G s
sτ=
+
( ) 2 2 2 2
1
1 1G j j
ωτω
ω τ ω τ= −
+ +
K1 K2
Chapter 14
3
Chapter 14
4
(plot of log |G| vs. log ω and φ vs. log ω)
( )
2 2
0
1
1
arctan
as , 90
Gω τ
φ ωτ
ω φ
=+
= −
→∞ → −
Use a Bode plot to illustrate frequency response
log coordinates:
1 2 3
1 2 3
1 2 3
1 2 3
1
2
1 2
1 2
log log log log
log log log
G G G G
G G G G
G G G G
G G G G
GG
G
G G G
G G G
= ⋅ ⋅
= ⋅ ⋅
= + +
∠ = ∠ + ∠ + ∠
=
= −
∠ = ∠ −∠
Chapter 14
5
Figure 14.4 Bode diagram for a time delay, e-θs.
Chapter 14
6
Chapter 14
Example 14.3
0.55(0.5 1)( )
(20 1)(4 1)
ss eG s
s s
−+=
+ +
7
The Bode plot for a PI controller is shown in next slide.
Note ωb = 1/τI . Asymptotic slope (ω→ 0) is -1 on log-log plot.
Recall that the F.R. is characterized by:
1. Amplitude Ratio (AR)
2. Phase Angle (φ)
F.R. Characteristics of Controllers
For any T.F., G(s)
A) Proportional Controller
B) PI Controller
For
( )
( )
AR G j
G j
ω
φ ω
=
=∠
( ) , 0C C CG s K AR K φ= ∴ = =
2 2
1
1 1( ) 1 1
1tan
C C C
I I
I
G s K AR Ksτ ω τ
φτ ω
−
= + = +
= −
Chapter 14
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Chapter 14
9
Series PID Controller. The simplest version of the series PID
controller is
Series PID Controller with a Derivative Filter. The series
controller with a derivative filter was described in Chapter 8
( ) ( )τ 1τ 1 (14-50)
τ
Ic c D
I
sG s K s
s
+= +
( ) τ 1 τ 1(14-51)
τ ατ 1
I Dc c
I D
s sG s K
s s
+ += +
Chapter 14
Ideal PID Controller.
1( ) (1 ) (14 48)
c c D
I
G s K ss
ττ
= + + −
10
Figure 14.6 Bode
plots of ideal parallel
PID controller and
series PID controller
with derivative filter
(α = 1).
Ideal parallel:
Series with
Derivative Filter:
( ) 10 1 4 12
10 0.4 1c
s sG s
s s
+ + = +
( ) 12 1 4
10cG s s
s
= + + C
hapter 14
11
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)
Chapter 14
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Controller Design by Frequency Response
- Stability Margins
Analyze GOL(s) = GCGVGPGM (open loop gain)
Three methods in use:
(1) Bode plot |G|, φ vs. ω (open loop F.R.) - Chapter 14
(2) Nyquist plot - polar plot of G(jω) - Appendix J
(3) Nichols chart |G|, φ 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.
Chapter 14
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Chapter 14
14.8
14
Frequency Response Stability Criteria 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, . Otherwise
the closed-loop system is stable.
• Note: 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.
Cω
value of C
ω ω≡
Chapter 14
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Example 1:
A process has a T.F.,
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...
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 AROL Stable?
1 0.25 Yes
4 1.0 Conditionally stable
20 5.0 No
3
2( )
(0.5 1)
C
OL
KG s
s=
+
3
2( )
(0.5 1)p
G ss
=+
Chapter 14
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Figure 14.9 Bode plots for GOL = 2Kc/(0.5s + 1)3.
Chapter 14
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• 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.
• 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.
1(14-59)
(ω )cu
G c
KAR
=Chapter 14
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Example 14.7:
Determine the closed-loop stability of the system,
Where GV = 2.0, GM = 0.25 and GC =KC . Find ωC 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:
15
4)(
+=
−
s
esG
s
p
OL
max
1.69rad min
0.235
1 1= 4.25
0.235
C
C
C
OL
AR
KAR
ω ω
ω
=
=
=
∴ = =
Chapter 14
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Chapter 14
14.11 20
Ultimate Gain and Ultimate Period
• Ultimate Gain: KCU = maximum value of |KC| that results in a
stable closed-loop system when proportional-only
control is used.
• Ultimate Period:
• KCU can be determined from the OLFR when
proportional-only control is used with KC =1. Thus
• Note: First and second-order systems (without time delays)
do not have a KCU value if the PID controller action is correct.
2U
C
Pπω
≡
1for 1
C
CU C
OL
K KAR
ω ω=
= =
Chapter 14
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Gain and Phase Margins
• 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 ω = ωC. Then the gain margin is
defined as: GM = 1/AC
According to the Bode Stability Criterion, GM >1 ⇔ stability
• Phase Margin:
Let ωg = frequency at which AROL = 1.0 and the
corresponding phase angle is φg . The phase margin
is defined as: PM = 180° + φg
According to the Bode Stability Criterion, PM >0 ⇔ stability
See Figure 14.12.
Chapter 14
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Chapter 14
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Rules of Thumb:
A well-designed FB control system will have:
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.
1.7 2.0 30 45GM PM≤ ≤ ≤ ≤o o
Chapter 14
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25