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Review: Controller Design
in Frequency Domain
Feedback Control System
( ) ( )Closed-loop Transfer Function: ( )
1 ( ) ( )
Open-loop Transfer Function: ( ) ( )
T
D s G sG s
D s G s
D s G s
Root Locus
( ) 1
( ) 11
1
Characteristic Equation: 1 0
Pole: 1
K
Y s KsKR s s K
s
s K
s K
Root Locus
Root Locus
2
2
( ) ( 1)
( )1
( 1)
1 1 40
2
K
Y s Ks s
KR s s s K
s s
Ks s K s
Root Locus
Frequency Response
( )( )
( )
Y sG s
U s ( ) sinu t A t
1 2
( )( )
n
b sG s
s p s p s p
2 2( ) ( ) ( ) ( )
AY s G s U s G s
s
*
0 01 2
1 2
( ) n
n
Y ss p s p s p s j s j
1 2 *
1 2 0 0( ) np tp t p t j t j t
ny t e e e e e
Frequency Response
*
0 0( ) j t j ty t e e
0 2 2
( )
( ) ( )2
( )2
s j
j G j
A AG s s j G j
s j
AG j e
j
*
0 2 2
( )
( ) ( )2
( )2
s j
j G j
A AG s s j G j
s j
AG j e
j
Frequency Response
( ) ( )
( ) ( )
( ) ( ) ( )2 2
( )2 2
( ) sin ( )
j G j j t j G j j t
j t G j j t G j
A Ay t G j e e G j e e
j j
e eA G j
j j
A G j t G j
Frequency Response and Poles
2
22 2
1( )
2 / 2 / 1
n
n n n n
G ss s s s
2
( )
1
/ 2 / 1n n
G s
s s
Pole Locations
( )G s
0.3 0.5
( )G s
0.7 0.9
Bode Plot
10( ) 20log ( )dB
G j G j
Bode Plot
1 2
1 2
( )m
s z s zG s K
s s p s p
1 2
1 2
( ) ( )ms j
j z j zG j G s K
j j p j p
0
1 2
1 1( )
1 1
a b
m
j jG j K
j j j
Bode Plot
10 10 0
1 2
10 0
1 2
10 0 10 10
10 10 1 10 2
1 1( ) 20log ( ) 20log
1 1
1 120log
1 1
20log 20log 1 20log 1
20log 20log 1 20log 1
a b
mdB
a b
m
a b
m
j jG j G j K
j j j
j jK
j j j
K j j
j j j
Bode Plot
0
1 2
0
1 2
1 1( )
1 1
1 1
1 1
a b
m
a b
m
j jG j K
j j j
K j j
j j j
Bode Plot
0
mK s
1 ,1/ 1s s
2 2
/ 2 / 1 ,1/ / 2 / 1n n n ns s s s
0
mK s
10 0 10 0 10
10 0 10
20log 20log 20log
20log 20 log
m mK j K j
K m
위상
0
mK s
0 0
0 0 90
m mK j K j
K m j K m
1s
10 10
10 10
1 20log 1 20log 1 0 1/
1 20log 1 20log 1/dB
dB
j j
j j
2
10 101 20log 1 20log 1dB
j j
1s
1s
1 1 0 1/
1 1 45 1/
1 90 1/
j
j j
j j
1s
1/ 1s
10 10
10 10
120log 1 20log 1 0 1/
1
120log 1 20log 1/
1
dB
dB
jj
jj
1/ 1s
1/ 1s
11 0 1/
1
11 45 1/
1
190 1/
1
j
jj
jj
1/ 1s
Example
1000
( )10
G ss s
1000 100( )
10 0.1 1s j
G js s j j
10
10
10020log
20log 100 20log
40 20log
j
Example
Example
2
1/ / 2 / 1n ns s
102
2
102
120log 1 0
/ 2 / 1
120log /
/ 2 / 1
n
n n dB
n n
n n dB
j j
j j
2
1/ / 2 / 1n ns s
2
1/ / 2 / 1n ns s
2
1 1
2/ 2 / 1n
n nj j
120log 20log 2
2
2
1/ / 2 / 1n ns s
2
2
2 2
2
11 0
/ 2 / 1
12 90
/ 2 / 1
1/ 180
/ 2 / 1
n
n n
n
n n
n n
n n
j j
jj j
j j
2
1/ / 2 / 1n ns s
2
1/ / 2 / 1n ns s
Example
2
10000( )
2 100G s
s s s
2
2
100( )
/100 2 /100 1
100
/10 0.2 /10 1
s j
G js s s
j j j
Example
Example
10 10
120log 20log 5 14
2dB
Example
Nyquist Plot
Nyquist path
Nyquist Plot
Nyquist path
Nyquist Plot
Nyquist Plot
Stability Margin
Stability Margin
Stability Margin
Stability Margin
Gain Margin
Phase Margin
Stability Margin
Gain Margin
Phase Margin
180 180
10 10 180 180
10
180 180
0 ( ) ( )
20log 1 20log ( ) ( )
120log
( ) ( )
dBGM dB G j H j
G j H j
G j H j
180 ( ) ( )c cPM G j H j
Stability Margin
Gain Margin
Phase Margin
Control System
Continuous-time controller
PD Controller
( ) 1 dD s K T s
Example
1( )
( 1)G s
s s
0lim
( 1)v
s
KK s K
s s
1 1ss
v
eK K
( ) 100(1 0.1 )D s s
Example
Example
Lead Compensator
1( )
1
TsD s K
Ts
Lead Compensator
1 11tan tan
1
jTT T
j T
10 max 10 10
1 1 1log log log
2 T T
max
1
T
1 1
max
1tan tan
max
1tan
2
max
1sin
1
max
max
1 sin
1 sin
Lead Compensator
10 max max 10
120log ( ) ( ) 0.5 20logKG j H j
max
1T
Example
100.5 20log 1/ 9dB
max
1 10.17
16.7 0.13T
1 0.17 1( ) 100
1 0.13(0.17 ) 1
Ts sD s K
Ts s
max 50 0.13
Example
Example
MATLAB lead.m
num=100;den=[1 1 0];G=tf(num,den)
u=linspace(1,1,200);t=linspace(0,10,200);
[y]=lsim(feedback(G,1),u,t);figure(1)plot(t,y);grid on
w=logspace(0,3,200);[mag,phase]=bode(num,den,w);[gm,pm,wcg,wcp]=margin(G)figure(2)margin(num,den)
MATLAB lead.m
phimax=50;
alpha=(1-sin(pi*phimax/180))/(1+sin(pi*phimax/180))
10*log10(1/alpha)
[w' 20*log10(mag) phase ]
wmax=16.5;
T=1/(wmax*sqrt(alpha));
num1=[T 1];
den1=[T*alpha 1];
MATLAB lead.m
num=conv(num1,num);den=conv(den1,den);[mag,phase]=bode(num,den,w);[gm,pm,wcg,wcp]=margin(mag,phase,w)figure(3)margin(num,den)
[y]=lsim(feedback(tf(num,den),1),u,t);figure(4)plot(t,y);grid on
D=tf(num1,den1)Dz=c2d(D,1/2000,'tustin')
MATLAB lead.m
alpha =
1.3247e-001
ans =
8.7787e+000
ans =
1.0000e+000 3.6990e+001 -1.3500e+002
1.4481e+001 -6.4528e+000 -1.7605e+0021.4993e+001 -7.0545e+000 -1.7618e+0021.5522e+001 -7.6562e+000 -1.7631e+0021.6071e+001 -8.2580e+000 -1.7644e+0021.6638e+001 -8.8599e+000 -1.7656e+0021.7226e+001 -9.4618e+000 -1.7668e+002
MATLAB lead.m
-50
0
50
100M
agnitu
de (
dB
)
10-2
10-1
100
101
102
-180
-135
-90
Phase (
deg)
Bode Diagram
Gm = Inf dB (at Inf rad/sec) , Pm = 5.72 deg (at 9.97 rad/sec)
Frequency (rad/sec)
MATLAB lead.m
-100
-50
0
50
100
Magnitu
de (
dB
)
10-2
10-1
100
101
102
103
-180
-135
-90
Phase (
deg)
Bode Diagram
Gm = Inf dB (at Inf rad/sec) , Pm = 53.4 deg (at 16.6 rad/sec)
Frequency (rad/sec)
Digital Implementation
Tustin’s Method
2 1
1
2 11
1( ) 1( )
2 1( ) 11
1s
s
zs
T z
s
zT
T zU z TsD z K K
zE z TsT
T z
1
( ) 2 ( ) 2 ( 1) 2 ( 1)2
s s s
s
u k K T T e k K T T e k T T u kT T