KONTROL SİSTEMİ

7/30/2019 KONTROL SSTEM

1/20

26 September 2004 21:30 2-1 i.h Alta - 2003

Department of Electrical and Electronics Engineering - Electrical and Control Area

ELKE 405 Automatic Control Systems

Eyll 2004

www.altas.org

Blok

Diyagramlar

ve

aret AkGrafikleri




Block Diagram

BLOCK MANIPULATION RULES

Gain Block

Summation Junction

Pick-off Point

G(s)Ea(s) m (s)

+ _

R(s) E(s)

Y(s)


2/20




Block DiagramBLOCK MANIPULATION RULES

RULE

#

PROCESS ORIGINAL BLOCK

DIAGRAM

EQUIVALENT BLOCK

DIAGRAM

1

2

3

COMBINING

SERIAL

BLOCK

COMBINING

PARALLEL

BLOCK

CLOSING A

FEEDBACK

LOOP

G1

G1

G2

G2

+_G1 G2

G1G2

G G

+_

1 GHH

+ +_

-+

+

X

X

X

Y

Y

Y X Y

X

X

Y

Y




4

5

6

7

MOVING

A SUMMING

JUNCTION

AHEAD OF

A BLOCK

MOVING

A SUMMING

JUNCION

PAST A BLOCK

MOVING A

PICKOFF POINT

AHEAD OF

A BLOCK

MOVING A

PICKOFF POINT

PAST A BLOCK

G

GG

G

G

G

G

G

1/G

1/G

X

X

X

X

X

X

X X

Y

Y

Y

Y

Y

ZZ

Z

Z

XX

Y

Y

+-

+

+

+

++

++

_

_

_

Y

Y

Y

G

G

Block Diagram

BLOCK MANIPULATION RULES


3/20




Block Diagram Reduction

An overall input/output transfer function can be

obtained from the block diagram by applying someblock diagram reduction rules.

Series rule:

Feedback rule:

G1(s) G2(s) G1(s) G2(s)

G(s)

H(s)

)(1

)(

sGH

sG

++ _




Feedback rule derivation:

G(s)

H(s)

+ _R(s) E(s) C(s)

)()(1

)(

)(

)(

)]()()()[()(

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

sHsG

sG

sR

sC

sCsHsRsGsC

sEsGsCsCsHsRsE

+=

=

==

This is a rule

that is usedextensively.

then

and



4/20




Servomotor Block Diagram Reduction

Td(s)

Ea(s) m(s)+

+

+

--

Ki

Kb

aa LsR +

1

mm JsB +

1

s

1

Td(s)

Ea(s) ++

+

--

Kiaa LsR +

1

mm JsB +

1

s

1G1(s) G2(s) G3(s) G4(s)

H1(s)

With Td(s)=0 , first combine the inner forward path.

G1 G2 G3(s)

H1(s)

+ _G4(s)

Ea(s) m(s)




Next combine the feedback loop.

)(1

)(

1321

321

sHGGG

sGGG

+ G4(s)

m(s)Ea(s)

The final series combination is

)(1

)(

1321

4321

sHGGG

sGGGG

+

Ea(s)m(s)

Note: The defined values of the components may be substituted in

to get the final transfer function in terms of system parameters.

Servomotor Block Diagram Reduction


5/20




Block Diagram


An ExampleGiven a control system represented inthe block diagram shown. Determine

the relationship Y(s)/R(s).

R(s) +G1 G2 G3

H2

H1

Y(s)

_

+

+

_

+

(a)




Block Diagram


An Example

Y(s)R(s)

H2

G1

G1 G2

H1

+ _

+

+G3

+

_

(b)


6/20




Block Diagram


An Example

Y(s)R(s)

H2

G1

G3+

_+

_

G1G2

1- G1G2H1




Block Diagram ReductionAn Example

+_R(s) Y(s)

(d)

G1G2G3

1- G1G2H1+G2G3H2-

R(s) Y(s)

(e)

G1G2G3

1- G1G2H1+G2G3H2+G1G2G3


7/20




Signal Flow Graphs

The block diagram reduction method works well for

relatively simple block diagrams, but it gets very

confusing for more complicated models.

A signal flow graph represents the same information

as the block diagram, however it leads to a set ofrules that allow a systematic approach to finding the

overall input/output transfer function.

Basic definitions




Signal Flow Graphs

DEFINITION: - It is agraphicaltool for control systems

analysis and design

- It consists ofnodes and branches

- The relationship between the inputs(s) andoutput(s) are determined byMasons gain

formula

PROPERTIES OF FLOW GRAPHS:

Each branch is unilateral (one direction)

Each node trasnsmits the sum of all entering signals along each output branch

Aforward path is the path travelled by the signal in a forward directionA loop is formed when the signal travels and returns to its original source

Special nodes: Source node - no inputs

Sink node - no outputs

Basic definitions


8/20




The main steps are as follows:

a) construct the signal flow graph either from a

block diagram or from the basic physical

connection of system components (the transfer

functions of the components must be known).

b) Identify and calculate the various paths and

loops in the signal flow graph.

c) With the results from b), apply a formula,

Masons formula, to determine the overall

transfer function.

Signal Flow Graphs Basic definitions




Nodes, branches and transmission elements

Summation node Distribution node

node

x1 x2t12

nodebranch

x2 = t12x1

t12 x2x1

t12 G12(s)

x2

x1

t12t13

t14 x3

x4

t14t24

t34x2

x3

x4

x1

x4 = t14x1 + t24x2 + t34x3

x2 = t12x1

x3 = t13x1x4 = t14x1

Signal Flow Graphs Construction


9/20




ConstructionSignal Flow Graphs

1. A SINGLE BRANCH

V1,V2 are called nodes and T12 is called a branch

This single branch represents the equation V2 = T12 V1

Note: V1 = V2/T12 (each branch is unilateral)

2. SUM OF TWO BRANCHES

V3 = T13V1 + T23V2

V1 V2

T12

V1 V3

V2

T13

T23




Construction

3. PARALLEL BRANCHES

V2 = (T12a + T12b) V1

V2 = T12 V1

V1 = T21 V2

V1 V2

T12b

T12a

T12

V1

Signal Flow Graphs


10/20




Construction

4. CASCADED BRANCHES

V3 = T12 + T23 V1

5. NODE ELIMINATION

V3 = T13V1 + T23V2 and V4 = T34V3 , then

V4 = (T34T13)V1 + (T34T23)V2

V1 T12 V2 V3T23

T13 T34V1 V3

T23

V2

V4 V1T34 T13

V4

T34 T23

Signal Flow Graphs




Write down and label the nodes from input to

output, representing all the important signals.

Draw in all the branches connecting the nodes and

write down their transmission functions.

Check for any additional nodes and branches

required in the feedback paths.

ConstructionSignal Flow Graphs


11/20




Td(s)

Ea(s) ++

+

--

Ki

Kb

aa LsR +

1

mm JsB +

1

s

1

Td(s)

Ea(s)

+

--

Kiaa LsR +

1

mm JsB +

1

s

1

G1(s) G2(s) G3(s) G4(s)

H1(s)

x1 x2 x3 x4

x5

m(s)

x1 G1x2 G2

x3 x4G3 G41Ea(s)m(s)

-H1

Td(s)1

Example 1: Servomotor System

Signal Flow Graphs




Source node: only has outgoing branches.

Sink node: only has incoming branches. Path: continuous unidirectional succession of branches

(passes through no node more than once).

Forward path: a path from input to output.

Feedback path or loop: originates and terminates at thesame node.

Non-touching paths: paths with no common nodes.

Path gain or loop gain: product of branch gains ortransmission functions along the path.

Definitions

Masons Formula

Signal Flow Graphs


12/20


13/20




1. Identify all forward paths and write the path gainsMk.

2. Identify all loops and write the loop gains.3. Identify all non touching loop pairs and write down

the loop gain products.

4. Do the same for groups of 3, 4, non touching loops.

5. Calculate as defined.

6. Identify all loops not touching forward path k, and

repeat steps 2 -> 5 to calculate k .

7. Apply Masons formula to calculate the overall

transfer function.

A Systematic ApproachMasons Formula

Signal Flow Graphs




Forward paths:

M1 Eax1x2x3x4 m Gain = G1G2G3G4 Feedback loops:

L1

x1

x2

x3

x4

x1

Loop gain = - G1

G2

G3

H1

Non touching loop pairs: none

x1 G1 x2 G2 x3 x4G3 G41Ea(s) m(s)

-H1

Td(s)

)(

)(

sE

s

a

m

Masons FormulaExample 1

Servomotor System

Signal Flow Graphs


14/20




then, = 1 + G1G2G3H1Loops not touching forward path 1: none

then, 1 = 1

Apply Masons formula.

( )1321

432111

a

m

HGGG1GGGGM

)s(EsT

+=

==


Servomotor System ...continued

Signal Flow Graphs




Consider the transfer function from the disturbance

input, Td(s) to the output, m(s) , with (Ea = 0). The forward path is now

M1 Tdx3x4 m Gain = G3G4 The loops are not changed, so and 1 are unchanged.

Applying Masons formula

( )

1321

4311

1)( HGGG

GGM

sT

sT

d

m

+=

==

Note: The denominator has not changed.


Servomotor System ...continued

Signal Flow Graphs


15/20




Forward Paths:

M1 RR x

3x

4C Gain = G

6G

4G

5

M2 RR x1x2x3x4 C Gain = G1G2G3G4G5

C(s)G1x1 G2

x2 x3G3 G41R(s)

-H1

x4 G5

R(s)

-H2

G6





Feedback loops:

L1 x1x2x1 Loop gain =G2H1L2 x3x4x3 Loop gain =G4H2

Non touching loop pairs:

L1L2 Loop gain = G2G4H1H2

then = 1 (G2H1G4H2) + (G2G4H1H2)= 1 + G2H1 + G4H2 + G2G4H1H2


... continued

Signal Flow Graphs


16/20


17/20




Masons Formula Example 3Low frequency ac model of a single transistor circuit ... continued

CIRCUIT EQUATIONS:

SIGNAL FLOW GRAPH:

Vs +1 Vb +1 Vbè Vout

-Rs -R b

Is

1/R1

-GmRL

(1) Vb = Vs - Rs Is

( 2) Vbè = Vb -R b Is

(3) Is = Vbè / Ri

(4) Vout = -Gm Vbè RL

Signal Flow Graphs




Masons Formula Example 3

Low frequency ac model of a single transistor circuit ... continued

FP1 = Vs + Vb + Vbè + Vout

= (1) (1) (-Gm RL)But ,

FP2 Vs + Vb + Vbè + Is + Vbè +Vout (This contains a loop)

LOOP #1 = Vb + Vbè + Is + Vb

= (1) (1/R1) (-Rs) = - Rs/Ri

LOOP #2 = Vbè + Is + Vbè

= (1/Ri) (-Rb`) = -Rb`/Ri

LOOP #3 = (1) (-Rb`) (-Rs)

Signal Flow Graphs


18/20





Low frequency ac model of a single transistor circuit ... continued

T1 = Vout/Vs =(1) (1) (-Gm RL) [ 1 - 0]

1 - (-Rs/Ri -Rb`/Ri)

FP1

=-GmRLRi

Ri + Rs + Rb`

T2 = Vb`e/Vs =(1) (1) [1 - 0 ]

1 + Rs/Ri + Rb`/Ri

}

FP2

=Ri

Ri + Rs + Rb`

Signal Flow Graphs





CVin A V1 B

F E

V4

V2 V3 DVout

Vout / Vin =

ABCD

1 - CEF

Vout / Vin =FP

1 - LP

Signal Flow Graphs


19/20





Vin A V1 B

G

C

D

V2 E

F

Vout

FP1 = ACE , FP2 = ABDE

LOOP #1 = BG , LOOP #2 = EF

T = Vout/Vin = ACE [ 1 - 0 ] + ABDE [ 1 - 0 ]1 - ( BG + EF ) + ( BGEF )

=ACE + ABDE

1 - ( BG + EF ) + ( BGEF )

Signal Flow Graphs





Vin A B

C

D

I

E F

H

G Vout

.F.P.1 = ACFG , LOOP #1 = DI

F.P.2 = ABDEFG , LOOP #2 = FH

T = Vout/Vin =ACFG [ 1 - DI ] + ABDEFG [ 1 - 0 ]

1 - ( DI + FH ) + ( DIFH )

= ACFG - ACFGDI + ABDEFG

1 - DI - FH + DIFH

Signal Flow Graphs


20/20





Vin +1

H

B

CA D E

G

F

+1 Vout.

FP1 = (1) (B) (1) = B LOOP #1 = CH

FP2 = (1) (A) (C) (D) (E) (F) (1) = ACDEF LOOP #2 = EG

T = Vout/Vin =B [ 1 - ( CH + EG ) + ( CHEG ) ] + ACDEF [ 1 - 0 ]

1 - ( CH + EG ) + ( CHEG )

T = Vout/Vin =B - BCH + BEG + BCHEG + ACDEF

1 - CH - EG + CHEG

Signal Flow Graphs

Documents

KONTROL SİSTEMİ