lecture7.ppt [호환 모드]ccrs.hanyang.ac.kr/webpage_limdj/digitalcontrol/lecture7.pdf ·...

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

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

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)


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)


• 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−−


( ) ( ) , ( )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−= + = +


( ) ( ) ( )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 =


2 tan2wT

Tωω =


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


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


-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)


-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

⎛ ⎞+= =⎜ ⎟−⎝ ⎠

lecture7.ppt [호환 모드]ccrs.hanyang.ac.kr/webpage_limdj/digitalcontrol/lecture7.pdf ·...

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