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Lecture 11
Dr. Ali Karimpour Sep 2015
CONTROL
ENGINEERING
Ali Karimpour
Associate Professor
Ferdowsi University of Mashhad
Lecture 11
Dr. Ali Karimpour Sep 2015
2
Lecture 11
Frequency domain analysis
Topics to be covered include:
Frequency domain specification.
Peak of resonance and resonance frequency.
Bandwidth.
Gain margin.
Phase margin.
Polar plot. Stability analysis with polar plot.
Nichols chart or gain phase plot. Stability analysis with gain phase plot.
Bode plot. Stability analysis with Bode plot.
Effect of adding poles and zeros on loop transfer function.
Frequency domain specification.
Peak of resonance and resonance frequency.
Bandwidth.
Gain margin.
Phase margin.
Lecture 11
Dr. Ali Karimpour Sep 2015
3
One degree-of-freedom configuration
)()()(1
)()()()(
)()(1
1)(
)()(1
)()()( sn
sKsG
sKsGsdsG
sKsGsr
sKsG
sKsGsy d
T(s) T(s)S(s)
S(s) is : Sensitivity Function
T(s) is : Complementary Sensitivity Function Why?
Lecture 11
Dr. Ali Karimpour Sep 2015
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Frequency (rad/s)
T
S
L
Frequency domain specification
PM
P
Peak of resonance ( )PM
Resonance frequency ( )P
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Dr. Ali Karimpour Sep 2015
5
Frequency domain specification
Closed-loop bandwidth ( )B
B
-3
Open - loop bandwidth ( )O
O
Gain crossover frequency ( )c
c
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Stability margins
Stability Is Not A Yes/No Proposition
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Stability margins
)(sG)(sC
)(sGc
+
-
)(sR
ω=ωc
ω=ω180
G (jω) Gc(jω)
Phase Margin
We define phase margin as the phase
(angle) that the frequency response
would have to change to move to the
-1 point.
c : Is the gain crossover frequency
Phase margin (PM) is the most widely
used measure of relative stability.
)()(180cccm
jGjGPM
Physical meaning:
Lecture 11
Dr. Ali Karimpour Sep 2015
8
Example 1: Derive PM and gain crossover frequency of following
system.
)10)(5(
500
sss
)(sC+
-
)(sR
)10)(5(
500)(
ssssG
Stability margins
sec/5.6 radc
10PM......
PM and gain crossover frequency from Nyquist (polar plot)
PM and gain crossover frequency from Bode plot
PM and gain crossover frequency from Nichols (gain phase plot)
Lecture 11
Dr. Ali Karimpour Sep 2015
9
Stability margins
Phase and Gain Margin
-1
Same Phase Margin
Gain Margin ??
Thus we need another measure
of relative stability.
Lecture 11
Dr. Ali Karimpour Sep 2015
10
Stability margins
)(sG)(sC
)(sGc
+
-
)(sR
ω=ωc
ω=ω180
G (jω) Gc(jω)
Gain Margin
We define gain margin as the gain
that the frequency response would
have to increase to move to the -1
point.
180 : Is the phase crossover frequency
Gain margin is another widely used
measure of relative stability.
)()(/1 180180 jGjGGM c
)()(/1log20 180180 jGjGGM c
Physical meaning:
Lecture 11
Dr. Ali Karimpour Sep 2015
11
Example 2: Derive GM and phase crossover frequency of following
system.
)10)(5(
500
sss
)(sC+
-
)(sR
)10)(5(
500)(
ssssG
Stability margins
sec/07.7180
rad 5.1GM......
GM and phase crossover from Nyquist (polar plot)
GM and phase crossover from Bode plot
GM and phase crossover from Nichols (gain phase plot)
dbGM 5.3
Lecture 11
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12
Stability margins
Phase Margin and Gain Margin
-1
Same Phase Margin
Thus we need another measure
of relative stability.
Same Gain Margin
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Dr. Ali Karimpour Sep 2015
13
G (jω) Gc(jω)
Sensitivity peak
)()(1
1)(
jGjGjS
c
)()(1
1)(max
scs
sjGjG
jSM
1
sM)(sG)(sC
)(sGc
+
-
)(sR
The sensitivity peak, Ms is defined as follows:
)()(1 00 jGjG c
0
)( 0
1 jS
Lecture 11
Dr. Ali Karimpour Sep 2015
14
Example 3: Derive the sensitivity peak of following system.
)10)(5(
500
sss
)(sC+
-
)(sR
)10)(5(
500)(
ssssG
Stability margins
sec/?? rads ??
sM......
Sensitivity peak from Nyquist (polar plot)
Sensitivity peak from Bode plot
Sensitivity peak from Nichols (gain phase plot)
Lecture 11
Dr. Ali Karimpour Sep 2015
)(sC)(sL
+
-
)(sR
15
Frequency (rad/s)
T
S
L
Frequency domain specification
1- Peak of resonance ( )PM
3- Resonance frequency ( )P 4- Closed-loop bandwidth ( )B
2- Open - loop bandwidth ( )O
6- Gain crossover frequency ( )c5- Sensitivity Peak ( )s
M
Lecture 11
Dr. Ali Karimpour Sep 2015
16
ω=ωc
ω=ω180
L (jω)
Frequency domain specification
)(sC)(sL
+
-
)(sR5- Sensitivity Peak ( )
sM
6- Gain crossover frequency ( )c
7- Phase crossover frequency ( )180
8- Phase Margin ( )PM
9- Gain Margin ( )GM
Lecture 11
Dr. Ali Karimpour Sep 2015
17
Frequency domain analysis
Topics to be covered include:
Frequency domain specification.
Peak of resonance and resonance frequency.
Bandwidth.
Gain margin.
Phase margin.
Polar plot. Stability analysis with polar plot.
Nichols chart or gain phase plot. Stability analysis with gain phase plot.
Bode plot. Stability analysis with Bode plot.
Effect of adding poles and zeros on loop transfer function.
Lecture 11
Dr. Ali Karimpour Sep 2015
18
-2.5 -2 -1.5 -1 -0.5 0 0.5 1-3
-2.5
-2
-1.5
-1
-0.5
0
Nyquist Diagram
Real Axis
Imagin
ary
Axis
Nyquist chart (polar plot))(sG
)(sC)(sGc
+
-
)(sR
1 10793.2
2 12337.1
3 13882.0
4 15154.0
5 16238.0
6 17127.0
7 17920.0
8 18716.0
9 19312.0
20 22902.0
?c 6.2 ?180 7 ?GM db????PM 45
1
2
)10)(5(
150)()(Let
ssssGsGc
)()( jGjGc
3
7 20
45
10793.2)101)(51(
150)1()1(
jjjjGjGc
Lecture 11
Dr. Ali Karimpour Sep 2015
19
Frequency charts
Nyquist
(polar plot)
Bode plot
Nichols chart
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Dr. Ali Karimpour Sep 2015
20
-270 -225 -180 -135 -90-40
-30
-20
-10
0
10
20Nichols Chart
Open-Loop Phase (deg)
Open-L
oop G
ain
(dB
)
-270 -225 -180 -135 -90-40
-30
-20
-10
0
10
20Nichols Chart
Open-Loop Phase (deg)
Open-L
oop G
ain
(dB
)
)(sG)(sC
)(sGc
+
-
)(sR
)10)(5(
150)()(Let
ssssGsGc
)()(log20 jGjGc
db33.9
db71.2
db71.1
db29.5
db42.8
db23.11
db80.13
db18.16
db39.18
1
3
68
)()( jGjGc
1 10793.2
2 12337.1
3 13882.0
4 15154.0
5 16238.0
6 17127.0
7 17920.0
8 18716.0
9 19312.0
20 22902.0 db77.35
20
2
4
5
?c 6.2 ?180 7 ?GM db8.13?PM 45
Nichols chart (gain phase plot)
Lecture 11
Dr. Ali Karimpour Sep 2015
21
Bode plot )(sG)(sC
)(sGc
+
-
)(sR
)10)(5(
150)()(Let
ssssGsGc
)()( jGjGc )()(log20 jGjGc
1 10793.2 db33.9
2 12337.1 db71.2
3 13882.0 db71.1
4 15154.0 db29.5
5 16238.0 db42.8
6 17127.0 db23.11
7 17920.0 db80.13
8 18716.0 db18.16
9 19312.0 db39.18
20 22902.0 db77.35 ?c 5.2 ?180 7 ?GM db8.13?PM 48
-80
-60
-40
-20
0
20
Magnitu
de (
dB
)
100
101
102
-270
-225
-180
-135
-90P
hase (
deg)
Bode Diagram
Frequency (rad/sec)
-80
-60
-40
-20
0
20
Magnitu
de (
dB
)
100
101
102
-270
-225
-180
-135
-90P
hase (
deg)
Bode Diagram
Frequency (rad/sec)
1003020107654321
Lecture 11
Dr. Ali Karimpour Sep 2015
22
University entrance exam 1393
. اشدفاز کدام یک از توابع زیر دارای کم ترین تغییرات فاز می بدیاگرام-4مثال
3(1
1)(4(
)1(
1)(
1(1
1)(2(
)1(
1)(
22
22
ssg
ssg
ssg
s
ssg
Lecture 11
Dr. Ali Karimpour Sep 2015
University entrance exam 1393
قطبی یک سیستم مرتبه دوم نوعی دیاگرام-5مثال
کدام عبارت صحیح است؟. در دو حالت در زیر ترسیم شده است
.است( 2)سیستم فراجهشبیشتر از ( 1)سیستم فراجهش1).است( 1)سریع تر از سیستم ( 2)پاسخ سیستم 2).م استسیستمیراینشان دهنده فرکانس نوسانات موهومیفرکانس تالقی با محور 3). هر سه عبارت صحیح است4)
22
2
2nn
n
ss
23
Lecture 11
Dr. Ali Karimpour Sep 2015
24
)(sG)(sC
)(sGc
+
-
)(sR
)10)(5(
150)()(Let
ssssGsGc
-2.5 -2 -1.5 -1 -0.5 0 0.5 1-3
-2.5
-2
-1.5
-1
-0.5
0
Nyquist Diagram
Real Axis
Imagin
ary
Axis
1
2
3
7 20
45
Closed loop values from Nyquist chart
2004.1)1( jT
?)2( jT
?pM
?)1( jT
?)3( jT
?)( jT
Lecture 11
Dr. Ali Karimpour Sep 2015
25
M circles (constant magnitude of T)
Lecture 11
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26
N circles (constant phase of T)
Lecture 11
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27
Constant gain and phase loci in Nichols chart
M circles and N circles
on Nichols chart
GC
cG+
-
R
Lecture 11
Dr. Ali Karimpour Sep 2015
28
Nichols chart specification
How to plot |T| versus
frequency?How to plot <T versus
frequency?
How to derive φm and GM?
How to derive cross over
frequencies?
How to derive open loop
bandwidth?How to derive closed loop
bandwidth?How to derive Mp?
How to derive ωp?How to derive type of
system?How to derive error
coefficient?
GC
cG+
-
R
Lecture 11
Dr. Ali Karimpour Sep 2015
29
Frequency domain analysis
Topics to be covered include:
Frequency domain specification.
Peak of resonance and resonance frequency.
Bandwidth.
Gain margin.
Phase margin.
Polar plot. Stability analysis with polar plot.
Nichols chart or gain phase plot. Stability analysis with gain phase plot.
Bode plot. Stability analysis with Bode plot.
Effect of adding poles and zeros on loop transfer function.
Lecture 11
Dr. Ali Karimpour Sep 2015
30
Effect of adding poles on Bode plot.
-
c2er)(sG
s1
1
Adding poles
-
c2er)(sG
trSystem speedBW
/1
Lecture 11
Dr. Ali Karimpour Sep 2015
31
Adding poles to open loop transfer functions
اضافه کردن قطب به تابع انتقال حلقه باز
-
c2er)(sG
s1
1223
2
22
2)21()(
)()(
nnn
n
ssssR
sCsM
5,2,1,05.01 n
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Step Response
Time (sec)
Am
plit
ude τ=0
τ=1.0
τ=2.0
τ=5.0P.O.
tr
System speed
More problem as
poles go to ??
BW
Lecture 11
Dr. Ali Karimpour Sep 2015
32
Effect of adding zeros on Bode plot.
-
c2er)(sG s1
Adding zeros
-
c2er)(sG
/1
trSystem speedBW
Lecture 11
Dr. Ali Karimpour Sep 2015
33
Adding zeros to open loop transfer functions
اضافه کردن صفر به تابع انتقال حلقه باز
-
c2er)(sG s1
6)62(3
)1(6
)(
)()(
23
22
sss
s
sR
sCsM
10,5,2,5.0,2.0,0
P.O.
tr
System speed
BW
Note: For τ<0 system
is unstable. Why?0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Step Response
Time (sec)
Am
plit
ude
τ=0
τ=0.5
τ=2.0
τ=0.2
τ=5.0
τ=10
Lecture 11
Dr. Ali Karimpour Sep 2015
34
Example 6: Derive the Bode plot of following system.
(rad/sec)Frequency
10101010 3210
0
10
20
30
20
10
90
0
90
0
10
20
30
40
50
Magnitu
de (
dB
)
100
101
102
103
0
45
90P
hase (
deg)
Bode Diagram
Frequency (rad/sec)
Ph
ase
(deg
)
M
agn
itu
de
(db
)
1
1)(
s
sasG
)(log20 jG
1
1log201log20
jja
)( jG
)1
1()1(
jja
1aLet
a/1 /1
alog20
?
)(tan)(tan 11 am
221)tan(
a
am
222
222
)1(
)(2)1)(()tan(
a
aaaam
2222 2)1( aa
a
1
1
1sin
a
am
m
Lecture 11
Dr. Ali Karimpour Sep 2015
35
Example 7: Derive the Bode plot of the following system.
(rad/sec)Frequency
10101010 3210
0
10
20
30
20
10
90
0
90
0
10
20
30
40
50
Magnitu
de (
dB
)
100
101
102
103
0
45
90
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Ph
ase
(deg
)
M
agn
itu
de
(db
)
1
1)(
s
sasG
)(log20 jG
1
1log201log20
jja
)( jG
)1
1()1(
jja
1aLet
/1 a/1
alog20
1
1sin
a
am
Lecture 11
Dr. Ali Karimpour Sep 201515025
150:
)25(
150:
2 ssb
ssaanswer
2- The polar plot of an open
loop system with negative
unit feedback is shown.
a) Find the open loop
b) transfer function.
c) Find the closed loop
d) transfer function.
36
Exercises
1- Derive the gain crossover frequency, phase
crossover frequency, GM and PM of following
system by use of Bode plot.
)10(
1
ss
)(sC
200+
-
)(sR
38,,5.12: 180 mc andGMAnswer
Lecture 11
Dr. Ali Karimpour Sep 2015
37
Exercises
3- Bode plot of an open loop system with negative unit feedback is
shown.
a) Find the open loop transfer function.
b) Find the closed loop transfer function.
20020
200:
)20(
200:
2 ssb
ssaanswer
-80
-60
-40
-20
0
20
40
Magnitu
de (
dB
)
100
101
102
103
-180
-135
-90
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Lecture 11
Dr. Ali Karimpour Sep 2015
38
4- The Nichols chart of an open loop system with negative unit
feedback is shown.
a) Find the GM and PM.
b) Find MP.
dbMbPMdbGMaanswer p 8.1:45,14:
-360 -315 -270 -225 -180 -135 -90 -45 0-120
-100
-80
-60
-40
-20
0
20
40
6 dB 3 dB
1 dB 0.5 dB
0.25 dB 0 dB
-1 dB
-3 dB
-6 dB
-12 dB
-20 dB
-40 dB
-60 dB
-80 dB
-100 dB
-120 dB
Nichols Chart
Open-Loop Phase (deg)
Open-L
oop G
ain
(dB
)Exercises
Lecture 11
Dr. Ali Karimpour Sep 2015
39-360 -315 -270 -225 -180 -135 -90 -45 0
-120
-100
-80
-60
-40
-20
0
20
40
6 dB 3 dB
1 dB 0.5 dB
0.25 dB 0 dB
-1 dB
-3 dB
-6 dB
-12 dB
-20 dB
-40 dB
-60 dB
-80 dB
-100 dB
-120 dB
Nichols Chart
Open-Loop Phase (deg)
Open-L
oop G
ain
(dB
)
1.0
5- The Nichols chart of a open loop system with negative unit feedback
is shown.
a) Find the error constants
b) Find the GM and PM and gain crossover frequency and phase
crossover frequency.
c) Find MP , open loop bandwidth and closed loop bandwidth.
sec/3.6sec,/7.4,3.5:
sec/7sec,/75.3,32,10:
0,5,:
180
radBWradBWdbMc
radradPMdbGMb
kkkaanswer
closedlooploopopenp
c
avp
23
57
12
Exercises
Lecture 11
Dr. Ali Karimpour Sep 2015
40
6- Draw Nichols chart of following system (Final exam).
Exercises
7- Draw gain-phase plot of a minimum phase type one system with no
zero and three poles and GM=2 db and PM=45° (Final exam).
8- Bode plot of a minimum phase system is: (Final exam).
a- Derive phase and gain crossover
Frequency, Gm and PM.
b- Determine the nonzero error constant.
c- If 0.01 sec delay added inside the
feedback loop, derive new Bode plot
in the same figure.
d- Derive phase and gain crossover
Frequency, Gm and PM of new system.
Lecture 11
Dr. Ali Karimpour Sep 2015
41
9- Nichols chart of a system is given, determine
a- Gain and phase cross over
frequency.
b- GM and PM.
c- Open loop and closed loop BW.
d- Type of system.
e- Nonzero error constant.
f- …
g- …
Exercises
Lecture 11
Dr. Ali Karimpour Sep 2015
42
Example 8: Derive the GM and PM of following system
by use of Bode plot.
)10)(5(
500
sss
)(sC+
-
)(sR
?c sec/5.6 rad
10PM
?180 sec/2.7 rad
dbGM 3
(rad/sec)Frequency
10101010 3210
40
20
0
20
80
60
90
180
270
0
10
20
30
40
50
Magnitu
de (
dB
)
100
101
102
103
0
45
90
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Ph
ase
(deg
)
M
agn
itu
de
(db
)