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PD 제어기의 설계PD 제어기의 설계
( )( ) 1 dD s K T s= +
예제: PD 제어기예제: PD 제어기
1( )( 1)
G ss s
=+( )
K0
lim( 1)v s
KK s Ks s→
= =+
1 1sse
K K= =
vK K
( ) 100(1 0.1 )D s s= +( ) 100(1 0.1 )D s s+
예제: PD 제어기예제: PD 제어기
예제: PD 제어기예제: PD 제어기
진상 제어기(Lead Compensator)의 설계
1( )1
TsD s KTα+
=+1Tsα +
진상 제어기의 설계진상 제어기의 설계
( ) ( )1 11 tan tan1
jT T Tj T
ωφ ω α ωα ω
− −⎛ ⎞+= ∠ = −⎜ ⎟+⎝ ⎠j⎝ ⎠
1 1 1log log logω ⎛ ⎞= +⎜ ⎟ max1ω =
10 max 10 10log log log2 T T
ωα
= +⎜ ⎟⎝ ⎠
max T α
1 1max
1tan tanφ αα
− −= −
1 α− 1 α− 1 sinφ−max
1tan2
αφα
= max1sin1
αφα
=+
max
max
1 sin1 sin
φαφ
=+
진상 제어기의 설계진상 제어기의 설계
10 10120log ( ) ( ) 0 5 20logKG j H jω ω = − ×10 max max 1020 log ( ) ( ) 0.5 20logKG j H jω ωα
×
1T =max
Tω α
=
예제: 진상 제어기예제: 진상 제어기
max 50φ = ° 0.13α =
( )100.5 20log 1/ 9dBα× =( )
1 1 0 1max
1 1 0.1716.7 0.13
Tω α
= = =
1 0.17 1( ) 1001 0.13(0.17 ) 1
Ts sD s KTs sα+ +
= =+ +1 0.13(0.17 ) 1Ts sα + +
예제: 진상 제어기예제: 진상 제어기
예제: 진상 제어기예제: 진상 제어기
MATLAB lead mMATLAB lead.mnum=100;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);[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)[gm,pm,wcg,wcp]=margin(G)figure(2)margin(num,den)
MATLAB lead mMATLAB lead.mhi 50phimax=50;
alpha=(1-sin(pi*phimax/180))/(1+sin(pi*phimax/180))10*log10(1/alpha)g ( / p )
[w' 20*log10(mag) phase ]
wmax=16.5;T=1/(wmax*sqrt(alpha));num1=[T 1];den1=[T*alpha 1];
MATLAB lead mMATLAB lead.mnum conv(num1 num);num=conv(num1,num);den=conv(den1,den);[mag,phase]=bode(num,den,w);[ ] i ( h )[gm,pm,wcg,wcp]=margin(mag,phase,w)figure(3)margin(num,den)
[y]=lsim(feedback(tf(num,den),1),u,t);figure(4)g ( )plot(t,y);grid on
D=tf(num1,den1)Dz=c2d(D,1/2000,'tustin')
MATLAB lead mMATLAB lead.malpha =alpha =
1.3247e-001
ans =
8.7787e+000
ans =
1 0000e+000 3 6990e+001 -1 3500e+0021.0000e+000 3.6990e+001 -1.3500e+002
1.4481e+001 -6.4528e+000 -1.7605e+0021 4993e+001 -7 0545e+000 -1 7618e+0021.4993e+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+0021.7226e+001 9.4618e+000 1.7668e+002
MATLAB lead mMATLAB lead.m
100
Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 5.72 deg (at 9.97 rad/sec)
0
50
agni
tude
(dB)
-50
0
Ma
-90
-135e (d
eg)
2 1 0 1 2-180
Phas
e
10-2
10-1
100
101
102
Frequency (rad/sec)
MATLAB lead mMATLAB lead.m
100
Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 53.4 deg (at 16.6 rad/sec)
0
50gn
itude
(dB)
-100
-50Mag
-90
-135
-90
e (d
eg)
-180
135
Phas
e
10-2
10-1
100
101
102
103
Frequency (rad/sec)
MATLAB lead mMATLAB lead.m
• Digital Implementation2 1 1z −
2 11
2 1 11( ) 1( ) 2 1( ) 1 11s
s
zsT z
zTT zU z TsD z K K zE z Ts TT
α α−=
+
+++
= = =−+ +
1ssT z +
( ) ( ) ( )1( ) 2 ( ) 2 ( 1) 2 ( 1)u k K T T e k K T T e k T T u kα= + + +⎡ ⎤⎣ ⎦( ) ( ) ( )( ) 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
αα
= + + − − + − −⎡ ⎤⎣ ⎦+
MATLAB lead mMATLAB lead.mD=tf(num1,den1)D 2d(D 1/2000 't ti ')Dz=c2d(D,1/2000,'tustin')
Transfer function:0.1665 s + 1-------------0.02206 s + 1
Transfer function:Transfer function:7.475 z - 7.453---------------z 0 9776z - 0.9776
Sampling time: 0.0005
Frequency Response of Digital System
( ) ( ) ( )Y z G z U z=
[ ] ( )( )sin( ) sin
j T j T
z TU z tz e z eω ω
ωω−
= =− −
Z( )( )
( )( )1 2( ) sin( ) ( )gj T j Tj T j T
k z k zG z z TY z Y zz e z ez e z e ω ωω ω
ω−−
= = + +− −− −( )( )
1 2( )ss j T j T
z e z ez e z e
k z k zY z ω ω−= +
1
( )
( )sin ( )
ss j T j T
j T j T
j T j T
z e z eG e T G ek
ω ω
ω ωω
−− −
= =1 2j T j Te e jω ω−−
Frequency Response of Digital System
( ) ( ) , ( )j T j T j j TG e G e e G eω ω θ ωθ= =
1
( )j T jG e ek
ω θ
=1 2
( ) ( )j T j j T j
j
G e e G e eω θ ω θ− −
2
( ) ( )2 2
G e e G e ek
j j= = −
−
1 2( ) ( ) ( ) ( ) sin( )j T k j T k j Tssy kT k e k e G e kTω ω ω ω θ−= + = +
Frequency Response of Digital System
( ) ( ) ( )j T j TG z G e G eω ω→( ) ( ) , ( )j jG z G e G e→
( ) ( ) ( )G s G j G jω ω→( ) ( ) , ( )G s G j G jω ω→
>> G=tf(1 [1 1])>> G=tf(1,[1 1])
Transfer function:1
-----s + 1
>> Gz=c2d(G 0 1)>> Gz=c2d(G,0.1)
Transfer function:0.09516----------z - 0.9048
Sampling time: 0 1Sampling time: 0.1>> figure;bode(G)>> figure;bode(Gz)
0Bode Diagram
-20
-10ag
nitu
de (d
B)
-40
-30Ma
0
-45
Phas
e (d
eg)
0Bode Diagram
10-2
10-1
100
101
102
-90
Frequency (rad/sec)-15
-10
-5
nitu
de (d
B)
-30
-25
-20
Mag
n
0
-90
-45
Phas
e (d
eg)
10-2
10-1
100
101
102
-180
-135P
Frequency (rad/sec)
Bilinear TransformationBilinear Transformation
1 ( / 2)1 ( / 2)
T wzT w
+=
−1 ( / 2)2 1
1
T wzw
T z−
=+
/ 2 / 2
1On the unit circle in z-plane:
2 1 2 1 2
j T
j T j T j T
T zz e
z e e e
ω
ω ω ω−
+=
− − −/ 2 / 2
2 1 2 1 21 1
2j T
j T j T j Tz e
z e e ewT z T e T e e
Tω
ω ω ω
ω
−=
= = =+ + +
2 tan2
2 t
wTj j
TT
ω ω
ω
= =
tan2w T
ω =
Bilinear TransformationBilinear Transformation
2 tan2wT
Tωω =
Bilinear TransformationBilinear Transformation
s
Tω ω
ω<<
sin2 2 22tan2 2w
TT T
TT T T
ωω ωω ωω= = ≈ =2 2cos
2TT T Tω
error is less than 4%2 10Tω π< →
210 10
s
Tωπω < =
10 10T
MATLAB lead digital mMATLAB lead_digital.mN=200;N=200;Ts=1/100Gp=tf(100,[1 1 0])Gz=c2d(Gp,Ts,'zoh')
figure(1)step(feedback(Gz,1),10)
Gw=d2c(Gz,'tustin')figure(2)margin(Gw);
w=logspace(-2,3,N);[mag,phase]=bode(Gw,w);for i=1:Nfor i=1:N
Gw_mag(i)=mag(i);Gw_phase(i)=phase(i);
end
MATLAB lead digital mMATLAB lead_digital.mphimax=50;phimax=50;alpha=(1-sin(pi*phimax/180))/(1+sin(pi*phimax/180))10*log10(1/alpha)
[w' (20*log10(Gw_mag))' Gw_phase']
wmax=16.5;T=1/(wmax*sqrt(alpha));T=1/(wmax*sqrt(alpha));num1=[T 1];den1=[T*alpha 1];
Dw=tf(num1,den1)
figure(3)margin(Dw*Gw)margin(Dw*Gw)
Dz=c2d(Dw,Ts,'tustin')figure(4)step(feedback(Dz*Gz,1),10)
MATLAB lead digital mMATLAB lead_digital.mTransfer function:Transfer function:100
-------s^2 + s
Transfer function:0 004983 z + 0 0049670.004983 z + 0.004967---------------------z^2 - 1.99 z + 0.99
Sampling time: 0.01
Transfer function:-4 167e-006 s^2 - 0 4992 s + 100-4.167e-006 s^2 - 0.4992 s + 100--------------------------------
s^2 + s + 1.11e-013
MATLAB lead digital mMATLAB lead_digital.malpha =alpha =
1.3247e-001
ans =
8 7787e+0008.7787e+000
ans =
1.0000e-002 8.0000e+001 -9.0576e+001
1.4650e+001 -6.6302e+000 -1.8028e+0021.5522e+001 -7.6302e+000 -1.8074e+0021.6447e+001 -8.6300e+000 -1.8121e+0021.7426e+001 -9.6297e+000 -1.8169e+002
MATLAB lead digital mMATLAB lead_digital.mTransfer function:Transfer function:0.1665 s + 1-------------0.02206 s + 1
Transfer function:6 339 z 5 9696.339 z - 5.969---------------
z - 0.6304z 0.6304
Sampling time: 0.01
MATLAB lead digital mMATLAB lead_digital.m2
Step Response
1.4
1.6
1.8
0.8
1
1.2
Ampl
itude
100
Bode DiagramGm = 6.04 dB (at 14.2 rad/sec) , Pm = 2.87 deg (at 9.98 rad/sec)
0 2
0.4
0.6
-50
0
50
nitu
de (d
B)0 1 2 3 4 5 6 7 8 9 10
0
0.2
Time (sec) -150
-100
-50
Mag
n
270
180
225
Phas
e (d
eg)
10-2
100
102
104
106
90
135P
Frequency (rad/sec)
MATLAB lead digital mMATLAB lead_digital.m100
Bode DiagramGm = 20.6 dB (at 88.6 rad/sec) , Pm = 48.7 deg (at 16.6 rad/sec)
0
50
100
itude
(dB)
-100
-50Mag
n
270 1.4Step Response
180
225
hase
(deg
)
1
1.2
10-2
100
102
104
106
90
135P
Frequency (rad/sec) 0.6
0.8
Ampl
itude
0.2
0.4
0 1 2 3 4 5 6 7 8 90
0.2
Time (sec)
Gz 와 Gw 의 GM 과 PMGz 와 Gw 의 GM 과 PMN=200;N=200;Ts=1/100Gp=tf(40,[0.1 1 0])Gz=c2d(Gp,Ts,'zoh')
figure(1)margin(Gz);
Gw=d2c(Gz,'tustin')figure(2)margin(Gw);
w=logspace(-2,2,N);[mag,phase]=bode(Gw,w);for i=1:Nfor i=1:N
Gw_mag(i)=mag(i);Gw_phase(i)=phase(i);
end
Gz 와 Gw 의 GM 과 PMphimax=50;
Gz 와 Gw 의 GM 과 PMphimax=50;alpha=(1-sin(pi*phimax/180))/(1+sin(pi*phimax/180))10*log10(1/alpha)
[w' (20*log10(Gw mag))' Gw phase'][w (20*log10(Gw_mag)) Gw_phase ]
wmax=33;T=1/(wmax*sqrt(alpha));num1=[T 1];num1=[T 1];den1=[T*alpha 1];
Dw=tf(num1,den1)
figure(3)margin(Dw*Gw)
Dz=c2d(Dw Ts 'tustin')Dz=c2d(Dw,Ts, tustin )figure(4)margin(Dz*Gz)
Gz 와 Gw 의 GM 과 PMGz 와 Gw 의 GM 과 PM100
Bode DiagramGm = 14.1 dB (at 44.4 rad/sec) , Pm = 22.7 deg (at 18.8 rad/sec)
-50
0
50
Mag
nitu
de (d
B)
-100
-180
-135
-90
se (d
eg)
10-1
100
101
102
103
-270
-225Phas
Frequency (rad/sec) 50
100
dB)
Bode DiagramGm = 14.1 dB (at 45.1 rad/sec) , Pm = 22.7 deg (at 18.8 rad/sec)
-100
-50
0
Mag
nitu
de (d
135
180
225
270Ph
ase
(deg
)
10-1
100
101
102
103
104
105
106
90
135
Frequency (rad/sec)
Gz 와 Gw 의 GM 과 PMGz 와 Gw 의 GM 과 PM20
40
Bode DiagramGm = 15.5 dB (at 134 rad/sec) , Pm = 58.1 deg (at 32.4 rad/sec)
-40
-20
0
20
Mag
nitu
de (d
B)
-60
180
225
270
se (d
eg)
100
101
102
103
104
105
106
90
135Phas
Frequency (rad/sec)0
20
40
(dB)
Bode DiagramGm = 15.5 dB (at 118 rad/sec) , Pm = 58.1 deg (at 32.1 rad/sec)
-60
-40
-20
Mag
nitu
de
-90
-225
-180
-135
90
Phas
e (d
eg)
100
101
102
103
-270
Frequency (rad/sec)
1 ( / 2)T1 ( / 2)1 ( / 2)2 1
T wzT w
z
+=
−−
/2 /2
1On the unit circle in z-plane:
2 1 2 1 2
j T
j T j T j T
wT z
z e ω
ω ω ω−
=+
=/2 /2
/2 /2
2 1 2 1 21 1
2 t
j T
j T j T j T
j T j T j Tz e
z e e ewT z T e T e e
Tj j
ω
ω ω ω
ω ω ω
ω
−
−=
− − −= = =
+ + +
tan2
2 tan2
w
w
j jT
TT
ω
ωω
= =
=2
1 ( / 2)1 ( / 2)
j T w
w
TT jeT j
ω ωω
+=
−
1 ( /2)1 (
( ) ( ) T ww zG w G z +
=−
=/2)
1 ( / 2)1 ( / 2)
1 ( / 2)
T w
T wGT w
T j
⎛ ⎞+= ⎜ ⎟−⎝ ⎠
⎛ ⎞1 ( / 2)( ) ( )1 ( / 2)
j Tww w
w
T jG j G G eT j
ωωωω
⎛ ⎞+= =⎜ ⎟−⎝ ⎠