chapter 2 - automatic control

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!"#$%&'( (Dr.-Ing. Wei Wang,Dept. of Automation, Shanghai Jiao TongUniv.

")#*$+ (Automatic Control System) 1

1

Chapter 2 Model of control systems

1. Differential equation of control systems(from physical and chemical principle)

2. Transfer function (definition and how to get)

3. Block diagram

4. Signal flow chart

2

Transfer function:

)()(

)( s R sC

sG =

)]([)( t c L sC =

)]([)( t r L s R =

where

C(t) is system output

r(t) is system input





3

Different forms of transfer function:

011

1

011

1)(a sa saa

b sb sb sb sG n

nn

mm

mm

++++++++= −

−

−−

L

L

]1)(2)[()1(

]1)(2)[()1()(

2

11

2

11

21

21

+++

+++=

∏∏

∏∏

==

==

l l

l

n

l j

n

j

k k

k

m

k i

m

i

s s sT s

s s s K

sG

ωξ

ω

ωξ

ωτ

ν

∏∏

+=

=

+

+=++++++=

q

v j j

m

ii

q

mr

p s s

z s

p s p s p s s z s z s z s K

sG

1

1

21

21

)(

)(

)())(()())((

)( ν

ν L

L

4

Example:

∫

∫ =

=+

dt t iC

t u

t udt t iC

t Ri

o

i

)(1

)(

)()(1

)(

ui

i

uo

R





5

Eliminating i(t), get

)()()(

t ut udt

t du RC io

o =+ui

i

uo

make Laplace transform:

)()()1( sU sU RCs io =+

11

)()(

+=

RCs sU sU

i

o

R

6

1. Amplifying factor:

K sG =)(

Typical factors in transfer functions:

2. Inertial factor :

( )1Ts

1sG +

=





7

3. differential factor

( ) Ks=sG

4. First order numerator term

( ) 1sG += sτ

5. Second order numerator term

( ) 12sG 2 ++= s s ξττ

8

6. integral factor

( )s1

sG =

7. oscillatory factor

( )nn

n

s s sG

ωζω

ω

++=

22

8. Time delay term

se τ− =G(s)





9

Block diagrams

Basic Elements in block diagrams:

( )G s( ) R s+

-

( )C s

( ) ( ) R s C s−( )r t ( )c t

( )C s

( )C s

( )a ( )b ( )c ( )d

( ) R s

)( s R ( )C s

, a- signal line .

, b- pickoff point ., c- summing junction ., d- block .

10

Ø Serial configuration

Ø General feedback configuration





11

Serial configuration:

G1( s)G1( s) G2( s) G 3( s)U(s) V(s) W(s) X(s)

G1(s)G 2(s)G 3(s)

12

General feedback configuration:

G(s)

H(s)

R(s) E(s) C(s)

E(s) = R(s) - H(s)C(s)C(s) = G(s) E (s)hence

C (s)= G(s)E(s) = G(s) [ R(s)- C(s)H(s) ]= G(s)R(s) G(s)C(s)H(s)

C(s)[1+G(s)H(s)] = G(s)R(s)

-

)()(1)(

)()(

s H sG sG

s R sC

+=





13

Block manipulation rules

in the process of

simplifying a block diagram

Simplification of block diagram

14

A

GH

G

1

Bl o

c k m

a ni p ul a t i onr ul e s





A

B

C

A B C − + A

B

C

A B C − +

A

B

AG B−

A

B

AG B−

1

G

B

AG BG−

B

A AG BG−

G

AG

AG

AG

AG

GG

GG

G

B C

A B C − +

C B

A A B C − +

AG

G

AG

AG A A

AG

1G

AG

A

A B−

A B− B

B

A A B−

A B−

A

1G

2G

1 2 AG AG+ A

1G2G

2

1G

1 2 AG AG+

A

A

A A B

B B1G

2G

2

1G 1G2G

1G1G2G

2G

1 AG 1 AG 1 AG 1 AG





17

Example:

From the principles of the circuits:

(t)iR

(t)ur(t)1

1

1 =−

∫ −= (t)]dti(t)[iC1

(t)u 211

1

(t)iR

c(t)(t)u2

2

1 =−

∫ = (t)dtiC1

c(t) 22

18





19

Transfer functions when multiple inputs

G1(s) G 2(s)

H(s)

R(s)E(s) C(s)

B(s)-

N(s)

20





21

G1(s) G 2(s)

H(s)

R(s) E(s)

N(s)C(s)

B(s)-

)()()()(

)(21 sG sG sG

s E

sC =⋅=

)()()(

s H sC s B =

22

)()()()()()(

)(21 s H sG s H sG sG

s E

s B =⋅⋅=

G1(s) G 2(s)

H(s)

R(s) E(s) N(s)

C(s)

B(s)-

)()(1)(

)()()(1)()(

)()(

21

21

s H sG sG

s H sG sG sG sG

s R sC

+=

+=

)()(1)(

)()()(1)(

)()( 2

21

2

s H sG sG

s H sG sG sG

s N sC

+=

+=





23

)()(11

)()(

s H sG s R s E

+=

G1(s) G 2(s)

H(s)

R(s) E(s) N(s) C(s)

B(s)-

)()(1)()(

)()( 2

s H sG s H sG

s N s E

+−=

24

)()()(1

)()(

)()(1)(

)( 2 s N s H sG

sG s R

s H sG sG

sC ++

+=

)()()(1

)()()(

)()(11

)( 2 s N s H sG

s H sG s R

s H sG s E +

−+

=

G1(s) G 2(s)

H(s)

R(s) E(s)

N(s)

C(s)

B(s)-





25

2 x

3 x

4 x

5 x

a b c

d e

f

1 x

The signal flow chart :

Signal Flow Chart

Concepts in signal flow chart /

Input node :

2 x

3 x

4 x

5 x

a b c

d e

f

1 x

x1, x5

Output node : x4





27

2 x

3 x

4 x

5 x

a b c

d e

f 1 x

Mixed nodes :

x2, x3

A path:

x1! x2! x3,

x2! x3

! x2

28

2 x

3 x

4 x

5 x

a b c

d e

f

1 xA forward path:

x1! x2

! x3! x4

A loop:x2! x3! x2

Path gain:

Path gain of the path x 1! x2

! x3 is ab





29

Forward-path gain: 2 x

3 x

4 x

5 x

a b c

d e

f 1 x

The forward path ( x 1! x2

! x3! x4 ) gain in the example is abc.

Loop gain:

The loop gain of the loop x 2! x3

! x2 is be

Nontouching loops:

Concepts in signal flow chart /

30

The manipulation of signal flow chart:

5514413312211 ya ya ya ya y +++=

1166 ya y =

1177 ya y =

1188 ya y =





31

32

y1 y2 y3 y4 y5 y6

a12 a23 a34 a45 a56

y1 y6

a12a23a34a45a56





33

ii

L

1i" p#

"

1/G

=== inout y y

whereyin = input node variable, y out = output node variableG = gain between y in and y out , L = total number of forward pathsP i = gain of the ith forward path

L+−+−=∆ ∑∑∑m

mm

mm

m P P P 3211

i = the for that part of the signal flow chart which isnontouching with the ith forward path.

Pmr = gain product of the mth possible combination

of r nontouching loops

The Mason Gain Formula

34

Example:

( ) R s ( )C s1G 2G 3G 4G 5G

6G

7G

1 H

2 H

+

-

+ +

-

++

Signal flow chart:

( ) R s ( )C s1G

2G 3G

4G 5G

6G7G

1 H −

2 H −

1





35

( ) R s ( )C s1G

2G 3G

4G 5G

6G7G

1 H −

2 H −

1

individual loops:

L1 = -G 4H1 L2 = -G 2G7H2

L4 = -G 2G3G4G5H2L3 = -G 6G4G5H2

The two nontouching loop is L 1 L2

36

( ) R s ( )C s1G

2G 3G

4G 5G

6G7G

1 H −

2 H −

1

forward paths:

P1= G 1G2G3G4G5 "

1=1

P3= G 1G2G7"

3=1-L 1

P2= G 1G6G4G5 "

2=1





37

214321

321

LL)LLL(L-1

1

++++=

+−+−=∆ ∑∑∑ Lm

mm

mm

m P P P P1= G 1G2G3G4G5

"1=1

P2= G 1L6G4G5 "

2=1

P3= G 1G2G7 "

3=1-L 1

2721425432254627214

14721546154321

332211

HGGHGHGGGGHGGGHGGHG1)HG(1GGGGGGGGGGGG

)" p" p" p("1G(s)

R(s)C(s)

++++++++=

++==

L1 = -G 4H1 L2 = -G 2G7H2

L3 = -G 6G4G5H2

L4 = -G 2G3G4G5H2

38

B E

1

1C s 2

1 R 2

1C s1

1 R

( ) R s ( )C s D

( ) R s ( )C s A

B

C D

E 1

1 R 1

1C s 11 2

1 R 2

1C s 1

1−

1− 1−

Signal flow chart:

Example2:




39

( ) R s ( )C s A

B

C D

E 1

1 R 1

1C s 11 2

1 R 2

1C s 1

1−

1− 1−

Example2:

individual loops:

sCR 1

L11

1

−=sCR

1L

222

−=sCR

1L

123

−=

22211

21 sCsR CR 1

LL = Nontouching loop:

22211122211

21321

321

sCR CR 1

sCR 1

sCR 1

sCR 1

1

LL)LL(L1

1

++++=

+++−=

+−+−=∆ ∑∑∑ Lm

mm

mm

m P P P

40

( ) R s ( )C s A

B

C D

E

1

1 R 1

1C s 11 2

1 R 2

1C s 1

1−

1− 1−

forward path:

1sCCR R

1P 12

21211

=∆=

sCR CR 1

sCR 1

sCR 1

sCR 1

1

LL)LL(L1

1

2211122211

21321

321

++++=

+++−=

+−+−=∆ ∑∑∑ Lm

mm

mm

m P P P

1sCR sCR sCR sCCR R 1

"

"PG

R(s)C(s)

2111222

2121

11

++++===

chapter 2 - automatic control

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