제어시스템 설계

Chaper 2 ~ chaper 3 허승현

1

Contents

I. Chapter 2 동적 시스템 모델링

II. Chapter 3 제어시스템의 성능 및 안정도

1) Laplace

2) 부분분수 정리

5) 동적 시스템 모델링

3) Solution of linear Ordinary Equation

4) SFG

1) Routh - Huwitz

2

I.Chapter 2 동적 시스템 모델링

3


1) Laplace Definition of Laplace transform

Given the real function f(t) that satisfied the condition

for some finite real , the Laplace transform of f(t) is defined as

=> One-sided Laplace transform

※ The response of a causal system does not precede the input.

causal = physically realizable

0( ) tf t e dt

0

( ) ( ) ( )stF s f t e dt L f t

operatorLaplacejs : 0 : ( ) 0 :

0 : ( ) :

t f t ignore

t f t defined

4


1) Laplace

( ) tf t e Ex1)

Multiplication by a constant

Sum and difference

( ) ( )L kf t kF s

1 2 1 2( ) ( ) ( ) ( )L f t f t F s F s

Important theorems of Laplace transform

0

( )

0

( )

1

t st

s t

F s e e dt

efor

s s

5


1) Laplace

Important theorems of Laplace transform(cont.) Differentiation

Integration

0

11 2

10

1 2 ( 1)

( )( ) lim ( ) ( ) (0)

( ) ( ) ( )( ) lim ( )

( ) (0) '(0) (0)

t

n nn n

n nt

n n n n

df tL sF s f t sF s f

dt

d f t df t d f tL sF s s f t s

dt dt dt

s F s s f s f f

1 2

0 0 00

1 2 10 0 0

( )( ) ( ) ( )

( )( )

n

st stt t

t t t

n n

F s e eL f d f d f t dt

s s sF s

L f d dt dt dts

6


1) Laplace Important theorems of Laplace transform(cont.)

Shift in time

Initial-value theorem

Final-value theorem: If sF(s) is analytic on the imaginary axis and in the right half of the s-plane, then

Ex)

( ) ( ) ( )

( ) :

Tss

s

L f t T u t T e F s

where u t Unit step function

0lim ( ) lim ( )t s

f t sF s

0lim ( ) lim ( )t s

f t sF s

2

0

5( )

( 2)

5lim ( ) lim ( )

2t s

F ss s s

f t sF s

7


1) Laplace

Important theorems of Laplace transform(cont.) Complex shifting

Real convolution(Complex multiplication)

In General

Complex convolution(Real multiplication)

( ) ( )tL e f t F s

1 2

1 2 1 20

1 20

1 2

( ) 0, ( ) 0, 0

( )* ( ) ( ) ( )

( ) ( )

( ) ( )

t

t

f t f t for t

L f t f t L f f t d

f t f t d

F s F s

11 2 1 2( ) ( ) ( ) ( )L F s F s f t f t

1 2 1 2( ) ( ) ( )* ( )L f t f t F s F s

Dual

8


2) 부분분수 정리 (Partial Fraction expansion)

G(s) has simple poles(cont.) Example

Sol)

5 3( )

( 1)( 2)( 3)

sG s

s s s

31 2

1 11

2 2

3 3

( )1 2 3

5 3( 1) ( ) 1

( 2)( 3)

( 2) ( ) 7

( 3) ( ) 6

1 7 6( )

1 2 3

ss

s

s

KK KG s

s s s

sK s G s

s s

K s G s

K s G s

G ss s s

9



G(s) has simple complex-conjugate poles

( )( )

( )( )

( ) ( )

( ) ( )

j j

j s j

P sG s

s j s j

K K

s j s j

K s j G s

10


2) 부분분수 정리 (Partial Fraction expansion) G(s) has simple complex-conjugate poles(cont.)

Example

2 2( ) ( )

2

2

1( )

( ) ( )

( )2 2

sin 1 01

n

Ls

Lt

j t j t t j t j tn n

tnn

u ts

e f t F s

g t e e e e ej j

e t t

2

2 2

2

2

2

( )2

( ) ( ) 1

2

1 1( )

2 ( ) ( )

n

n n

nj j

n

nj

n

G ss s

K Kwhere

s j s j

Kj

G sj s j s j

11



G(s) has multiple-order poles

1 2

1 2

1 2

1 22

( )

( )

( ) ( )( ) ( 1, 2, , )

( ) ( )( ) ( )( )

( ) ( )

where ( ) ( ) ( 1, 2, , )

1( ) ( )

( )!

n r

j j

rn r i

ss s

n r

rr

i i i

s i s s

r kr

k ir k s

Q s Q sG s where i n r

P s s s s s s s s s

KK K

s s s s s s

A A A

s s s s s s

K s s G s j n r

dA s s G s

r k ds

( 1, 2, , )is

k r

12



G(s) has multiple-order poles(cont.) Example

Sol)

3

1( )

( 1) ( 2)G s

s s s

0 32 1 22 3

0

2

03

3 0 11

2

1

3

( )2 1 ( 1) ( 1)

1

21

2

1 1( 1) ( ) 1

(3 3)! ( 2)

0

1

1 1 1 1( )

2 2( 2) 1 ( 1)

ss

K AK A AG s

s s s s s

K

K

dA s G s

ds s s

A

A

G ss s s s

13



2

2

(0) 1( ) ( )3 2 ( ) 5 ( ),

'(0) 2s

yd y t dy ty t u t

ydt dt

Example

Sol)

2 5( ) (0) '(0) 3 ( ) 3 (0) 2 ( )s Y s sy y sY s y Y s

s

22

2 2

2

2

5 5( 3 2) ( ) 1

5 5( )

( 3 2) ( 1)( 2)

5 5 3

2 1 ( 2)

5 3( ) 5 0

2 2t t

s ss s Y s s

s s

s s s sY s

s s s s s s

s s s s

y t e e t

steady-state solutionor particular integral

transient solution or homogeneous solution0

lim ( ) lim ( )t s

y t sY s

14



Example –cont. Transient

response

Steady-state response

Steady-state error( )sse 0

( ) ( )lim lim1 ( ) 1 ( )x x

R s sR sG s G s

Type of Control system2

1 2 1 22

1 2

(1 )(1 )......(1 )( )

(1 )(1 )....(1 )m m

ja b n n

K T s T s T s T sG s

s T s T s T s T s

15



Example –cont.

16

Type of System

I

Err Const

Step RampParaboli

c

0 K 0 0

1 K 0 0

2 K 0 0

3 0 0 0

sse

Kp Kv Ka

1

R

K

R

K

R

K


4) SFG(Signal flow graph)

( )R s ( )E s ( )Y s)(sG

+_

)(sH

)(sR )(sE

( )G s

)(sY

1

)(sY

1

( )H s

17



Can only be applied between an input node and an output node

A SFG : N forward paths

L loops

- the gain between the input node and output node

where

1

Nout k k

kin

y MM

y

1 2 3i

1k in out

i j kj k

M k y y

L L L

gain of the th forward path between and

( , , , )

(1 )mrL m m i j k

r r L

gain product of the th possible combination of

nontouching loops

k

k

the for that part of the SFG that is nontouching

with the th forward path

18


4) SFG(Signal flow graph) Ex)

is the same regardless of which input and output nodes are chosen.

2y 3y 4y 5y

12a23a

32a 43a

34a

44a

45a

24a

25a

5

1

1 12 23 34 45

2 12 24 45

3 12 25

?y

y

M a a a a

M a a a

M a a

11 23 32

21 34 43

31 24 43 32

41 44

12 23 32 44

L a a

L a a

L a a a

L a

L a a a

11 21 31 41 12

23 32 34 43 24 43 32 44 23 32 44

1 2 3 34 43 44

5 1 1 2 2 3 3 12 23 34 45 12 25 34 43 44 12 24 45

1 23 32 34 43 24 43 32 44

1 ( )

1 ( )

1, 1, 1 ( )

(1 )

1 ( )

L L L L L

a a a a a a a a a a a

a a a

y M M M a a a a a a a a a a a a

y a a a a a a a a

23 32 44a a a

1y

19

3 2 4 3 2 42

1

4 1 2 4

1

6 7 1 2 3 4 1 5 3 2

1 1

1 1 3 2 4 1 2 3 3

1 1 3 2 1 1 4 3 2 4 1 2 3 3 4 1 3 1 2 4

1

(1 )

(1 )

1

G H H G H Hy

y

y G G H

y

y y G G G G G G G H

y y

G H G H H G G G H

G H G H G H H G H H G G G H H G G H H H



Ex)

2y 3y4y 5y

1 1G 4G

5G4H

1

3H

1y

2G 3G

6y 7y1H 2H

20


3) 동적 시스템 모델링 전기 시스템

Kirchhoff 의 법칙 적용

- - 0di

V L Ridt

diL Ri V

dt

( / )

( ) ( )

1 1( )

( ) ( / )

( ) 1 R L t

ELs R I s

s

E EI s

s Ls R R s s R L

Ei t e

R

21

문제 2.4

디스크 헤드의 위치 제어와 같은 기전 시스템에서 흔히 볼 수 있는 모터구동시스템 모델링

Ke : 기전력 상수 , Kt : 토크상수 , L : 전기자 인덕턴스R : 저항 , J1 : 회전자의 회전 관성

모터

구동부분회전축의 관성무시 . Ct : 등가 비틀림 점성 마찰계수 . t : 비틀림 스프링상수 , J2 : 부하의 회전 관성

22

II.Chapter 3 제어시스템의 성능 및 안정도

23

ll. Chapter 3 제어시스템의 성능 및 안정도

1) Routh - Huwitz

4 3 2

4

3

2

1

0

) 2 3 5 10 0

2 3 10

1 5 0

3 107 10 0

135 10

6.43 0 07

10 0 0

ex s s s s

s

s

s

s

s

2 roots in the right-half s-plane

roots : 1.0055 0.9331

0.7555 1.444

s j

s j

25

Special Cases When Routh’s Tabulation Terminates Prematurely① The 1st element in any one row of Routh’s tabulation is

zero, but the others are not.

2 roots in the right-half

s-plane

roots :

4 3 2

4

3

2

1

0

) 23 2 3 0

1 2 3

1 2 0

2 20 3 0

12 3 3

0 0

3 0 0

ex s s s

s

s

s

s

s

0.09057 0.902

0.4057 1.2928

s j

s j

0

0

1) Routh - Huwitz


26

2

roots :

( ) 4 4 0

( )8

( )coefficients of

marginally stable

s j

A s s

dA ss

ds

dA s

ds

5 4 3 2

5

4

3

2

1

1

0

) 4 8 8 7 4 0

1 8 7

4 8 4

32 8 28 46 6 0

4 448 24 4 4

624 24

( 0 0)4

8 0

4 0

ex s s s s s

s

s

s

s

s

s

s

1) Routh - Huwitz


Ex)

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감사합니다… .28

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