Bode Stability
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Transcript of Bode Stability
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Chapter 14
Frequency Response
Force dynamic process with A sin t ,22
)(
s
AsU
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14.1
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Input:
Output:
is the normalized amplitude ratio (AR) is the phase angle, response angle (RA)
AR and are functions of
Assume G(s) known and let
tsinA
tAsin
1 22 2
1 2
2
1
arctan
s j G j K K j
G AR K K
KG
K
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AA /
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Example 14.1:
21 1
( 1)1 1
j
G j jj j
1
1G s
s
2 2 2 21
1 1G j j
K1
K2
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(plot of log |G| vs. log and vs. log )
2 2
0
1
1
arctan
as , 90
G
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
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Figure 14.4 Bode diagram for a time delay, e-qs
.
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Example 14.3
0.55(0.5 1)( )
(20 1)(4 1)
ss e
G s
s s
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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.
Recall that the F.R. is characterized by:
1. Amplitude Ratio (AR)
2. Phase Angle ()
F.R. Characteristics of Controllers
For any T.F., G(s)
A) Proportional Controller
B) PI Controller
For
( )
( )
AR G j
G j
( ) , 0C C CG s K AR K
2 2
1
1 1( ) 1 1
1tan
C C C
I I
I
G s K AR K
s
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Series PID Controller. The simplest version of the series PID
controller is
Series PID Controller with a Derivative F il ter. The series
controller with a derivative filter was described in Chapter 8
1
1 (14-50)
Ic c D
I
sG s K s
s
1 1
(14-51) 1
I Dc c
I D
s sG s K
s s
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I deal PID Controll er.
1( ) (1 ) (14 48)c c DI
G s K ss
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Figure 14.6 Bode
plots of ideal parallel
PID controller andseries PID controller
with derivative filter
( = 1).
Ideal parallel:
Series withDerivative Filter:
10 1 4 1
210 0.4 1
c
s sG s
s s
1
2 1 410
cG s ss
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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)
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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|, 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.
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Chapter14
14.8
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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.
C
value ofC
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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 ofKc: 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)
COL
KG s
s
3
2( )
(0.5 1)
pG s
s
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Figure 14.9 Bode plots forGOL = 2Kc/(0.5s + 1)3.
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For proportional-only control, the ultimate gainKcu is defined to
be the largest value ofKc that results in a stable closed-loop
system.
For proportional-only control, GOL= KcG and G = GvGpGm.
AROL()=Kc ARG() (14-58)
whereARG denotes the amplitude ratio ofG.
At the stability limit, = c,AROL(c) = 1 andKc= Kcu.
1(14-59)
( )cu
G c
KAR
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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 Cfrom the
Bode Diagram. What is the maximum value ofKc for a stable
system?
Solution:
The Bode plot forKc= 1 is shown in Fig. 14.11.
Note that:
15
4
)(
s
e
sG
s
p
OL
max
1.69rad min
0.235
1 1= 4.25
0.235
C
C
C
OL
AR
KAR
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Chapter14
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Ultimate Gain and Ultimate Period
Ultimate Gain: KCU= maximum value of |KC| that results in astable closed-loop system when proportional-only
control is used.
Ultimate Period:
KCUcan be determined from the OLFR when
proportional-only control is used withKC=1. Thus
Note: First and second-order systems (without time delays)
do not have aKCUvalue if the PID controller action is correct.
2
U
C
P
1for 1
C
CU COL
K KAR
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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 = 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.
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Chapter14
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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 aboutcontrol 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
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