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