Karadeniz Technical University ,y Chapter 4-1 -...

Karadeniz Technical UniversityDepartment of Electrical and Electronics Engineering

61080 Trabzon, Turkey

Chapter 4-1

, y

Block Diagram ReductionBlock Diagram Reduction

B d tl d b d i l öğ il i k ll k lBu ders notları sadece bu dersi alan öğrencilerin kullanımına açık olup,üçüncü sahıslara verilmesi, herhangi bir yöntemle çoğaltılıp başkayerlerde kullanılması, yayınlanması Prof. Dr. İsmail H. ALTAŞ’ın yazılıiznine tabidir Aksi durumlarda yasal işlem yapılacaktıriznine tabidir. Aksi durumlarda yasal işlem yapılacaktır.

Chapter 4-2

Chapter 4-3

QUIZ 2- 6 Mayıs 2008 Salı Günü saat 15:30 da HalisDuman Amfisinde yapılacaktır.

6 Mayıs 2008 Salı günü saat 13:00 – 15:00 de arası Ave B gruplarına birlikte toplu ders yapılacaktır.

Ödev 2’nin bugün teslim edilmesi gerektiğiniunutmayınız.

HATIRLATMALAR

Chapter 4-4

BLOCK MANIPULATION RULES

Gain Block

Summing Junction

Pick-off Point

G(s)G(s)EEaa(s)(s) θθm m (s)(s)

+ _+ _

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

Y(s)

Chapter 4-5


G1 G2X Y

G1G2X Y

COMBINING SERIAL BLOCKS

COMBINING PARALLEL BLOCKS

+_G1 G2X YG1

G2

+X Y

m

Chapter 4-6


CLOSING A FEEDBACK LOOP

Y=GE

X=HY

E=U X±

G

H±

+ EU Y

X

YE=

GE=U HY±

Y=U±HY

G

( )Y=G U±HY

Y=GU±GHY

Y GHY=GUm

( )1 GH Y=GUm

GY= U

1 GHm

G+_

1 GH

U Y

Chapter 4-7


CLOSING A FEEDBACK LOOP

G(s)

H(s)−

+ E(s)U(s) Y(s)

X

E(s)=U(s)-H(s)Y(s)

G(s)Y(s)= U(s)

1 G(s)H(s)+

Y(s)=G(s)E(s)

[ ]Y(s)=G(s) U(s)-H(s)Y(s)

Y(s)=G(s)U(s)-G(s)H(s)Y(s)

[ ]Y(s) 1+G(s)H(s) =G(s)U(s)G(s)

1 G(s)H(s)+

U(s) Y(s)

This is a rule that is used extensively.

Chapter 4-8


G1

H±

+ EU YG2

U Y1 2

1 21

G G

G G H±

Chapter 4-9


MOVING A SUMMING JUNCTION AHEAD OF A BLOCK

G

1/G

X

Y

Z

+-

+GX Z

+

+

_Y

MOVING A SUMMING JUNCION PAST A BLOCK

GX

Y

Z+

+

_GX Z

+

+

_

Y G

Chapter 4-10

MOVING A PICKOFF POINT AHEAD OF A BLOCK

GX

Y

Y

G

GXY

Y


MOVING A PICKOFF POINT PAST A BLOCK

GXY

X1/G

XY

X

G

Chapter 4-11

BLOCK DIAGRAM REDUCTION

TTdd(s)(s)

EEaa(s)(s) θθmm(s)(s)++++

++--

EEaa(s)(s) ++++

++--

aa LsR +1

aa LsR +1

GG11(s)(s)

KKiiKKii

GG22(s)(s)

mm JsB +1

mm JsB +1

GG33(s)(s)

s

1

s1

GG44(s)(s)

KKbb

HH11(s)(s)

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

GG11 GG22 GG33(s)(s)

HH11(s)(s)

+ _+ _ GG44(s)(s)EEaa(s)(s) θθmm(s)(s)

Chapter 4-12

Next combine the feedback loop.Next combine the feedback loop.

)(1

)(

1321

321

sHGGG

sGGG

+ GG44(s)(s)θθmm(s)(s)EEaa(s)(s)

The final series combination isThe final series combination is

)(1

)(

1321

4321

sHGGG

sGGGG

+EEaa(s)(s) θθmm(s)(s)

Note:Note: The defined values of the components may be The defined values of the components may be substituted insubstituted in to get the final transfer function in terms to get the final transfer function in terms of system parameters.of system parameters.


Chapter 4-13


Given a control system represented in the block diagram shown. Determine the relationship Y(s)/R(s).

R(s) +G1 G2 G3

H2

H1

Y(s)

_

++

_

+

(a)

Chapter 4-14


R(s) +G1 G2 G3

H2

H1

Y(s)

_

++

_

+

(a)

Y(s)R(s)

H2

G1

G1 G2

H1

+ _

+

+G3

+_

Chapter 4-15


Y(s)R(s)

H2

G1

G1 G2

H1

+ _

+

+G3

+_

Y(s)R(s)

H2

G1

G3+

_+

_

G1G2

1- G1G2H1

+

Chapter 4-16


+_R(s) Y(s)G1G2G3

1- G1G2H1+G2G3H2-

R(s) Y(s)G1G2G3

1- G1G2H1+G2G3H2+G1G2G3

Chapter 4-17

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 of rules that allow a systematic approachto finding the overall input/output transfer function.

Basic definitions

Chapter 4-18

Signal Flow Graphs

- It is a graphical tool for control systems analysis and design

- It consists of nodes and branches

- The relationship between the inputs(s) and output(s) are determined by Mason’s gain formula

DEFINITION:

Chapter 4-19

Signal Flow Graphs

• Each branch is unilateral (one direction)

• Each node transmits the sum of all entering signals along each “output” branch

• A forward path is the path travelled by the signal in a forward direction

• A loop is formed when the signal travels and returns to its original source

• Special nodes: Source node - no inputsSink node - no outputs

PROPERTIES OF FLOW GRAPHS:

Chapter 4-20

Signal Flow Graphs

a) construct the signal flow graph either from ablock diagram or from the basic physicalconnection of system components (the transferfunctions of the components must be known).

b) Identify and calculate the various paths andloops in the signal flow graph.

c) With the results from b), apply a formula,Mason’s formula, to determine the overalltransfer function.

The main steps are as follows:

Chapter 4-21

Signal Flow Graphs

node

x1 x2t12

nodebranch

x2 = t12 x1

t12

x2x1

t12 ≡ G12(s)

Construction

Nodes, branches and transmission elements

Chapter 4-22

Signal Flow Graphs Construction

x2

x1

t12

t13

t14

x3

x4

x2 = t12 x1

x3 = t13 x1

x4 = t14 x1

Distribution node

t14

t24

t34

x2

x3

x4

x1

x4 = t14 x1 + t24 x2 + t34 x3

Summation node

Chapter 4-23


V1 V2T12

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)

Chapter 4-24


V3 = T13V1 + T23V2

V1 V3

V2

T13

T23

2. SUM OF TWO BRANCHES

Chapter 4-25


3. PARALLEL BRANCHES

V2 = (T12a + T12b) V1

V2 = T12 V1

V1 = T21 V2

V1 V2

T12b

T12a

T12

V1

T21

V2

Chapter 4-26


V3 = T12 T23 V1

4. CASCADED BRANCHESV1 T12 V2 V3T23

5. NODE ELIMINATION

V3 = T13V1 + T23V2 and V4 = T34V3, then

V4 = (T34T13)V1 + (T34T23)V2

T13 T34V1 V3

T23

V2

V4 V1T34 T13

V4

T34 T23

Chapter 4-27


•• Write down and Write down and label the nodeslabel the nodes from input to from input to output, representing all the important signals.output, representing all the important signals.

•• Draw in all the Draw in all the branchesbranches connecting the nodes connecting the nodes and write down their transmission functions.and write down their transmission functions.

•• Check for any additional nodes and branches Check for any additional nodes and branches required in the required in the feedback pathsfeedback paths..

Chapter 4-28

Signal Flow Graphs Example 1

Ea(s) ++

+-

Ki

Kb

aa LsR +1

mm JsB +1

s

1

Td(s)Td(s)

Ea(s)

+Ki

aa LsR +1

mm JsB +1

s

1

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

H1(s)

x1 x2 x3 x4

x5

θm(s)

xx11 GG11xx22 GG22

xx33 xx44GG33 GG4411EEaa(s)(s) θθmm(s)(s)

--HH11

TTdd(s)(s)11

Servomotor System

Chapter 4-29

Signal Flow Graphs

Mason’s FormulaSource node:Source node: only has outgoing branches.only has outgoing branches.Sink node:Sink node: only has incoming branches.only has incoming branches.Path:Path: continuous unidirectional succession of branchescontinuous unidirectional succession of branches

(passes through no node more than once).(passes through no node more than once).Forward path:Forward path: a path from input to output.a path from input to output.Feedback path or loop:Feedback path or loop: originates and terminates at theoriginates and terminates at the

same node.same node.NonNon--touching paths:touching paths: paths with paths with nono common nodes.common nodes.PathPath gain or loop gain:gain or loop gain: product of branch gains orproduct of branch gains or

transmission functions along the path.transmission functions along the path.

Chapter 4-30

paths.forwardofnumber the;11

ppkMTk

kk L=ΔΔ

= ∑wherewhere 1=Δ

fourofgroupsngnon touchi

ofproductsgain loop⎟⎟⎠

⎞⎜⎜⎝

⎛+∑

threeofgroupsngnon touchi

ofproductsgain loop⎟⎟⎠

⎞⎜⎜⎝

⎛− ∑

pairsngnon touchiof

productsgain loop⎟⎟⎠

⎞⎜⎜⎝

⎛+ ∑

( )gainsloopall−∑

∑T =

FP (1 -∑ ∑ Loops not touching FP)

1 - Loops

∑Δ

Δ= kkM

Signal Flow Graphs Mason’s Formula

Chapter 4-31


paths.forwardofnumber the;11

ppkMTk

kk L=ΔΔ

= ∑

wherewhere

path.forwardthengnot touchiloopsusingonly defined thk kΔ=Δ

gainpath forwardthk kM =

... continued

Chapter 4-32


1. Identify all forward paths and write the path gains Mk.

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 Δ as defined.

6. Identify all loops not touching forward path k, and repeat steps 2 → 5 to calculate Δk.

7. Apply Mason’s formula to calculate the overall transfer function.

Chapter 4-33


Example

Forward paths:Forward paths:

MM11 ⇒⇒ EEaa xx1 1 xx2 2 xx3 3 xx4 4 θθm m Gain = Gain = GG11GG22GG33GG44

Feedback loops:Feedback loops:

LL11 ⇒⇒ xx1 1 xx2 2 xx3 3 xx4 4 xx11 Loop gainLoop gain = = -- GG11GG22GG33HH11

Non touching loop pairs: noneNon touching loop pairs: none

xx11 GG11xx22 GG22

xx33 xx44GG33 GG4411EEaa(s)(s) θθmm(s)(s)

--HH11

TTdd(s)(s))(

)(

sE

s

a

mθServomotor System

Chapter 4-34


Servomotor System ...continued

then, then, ΔΔ = 1 + = 1 + GG11GG22GG33HH11

Loops not touching forward path 1: Loops not touching forward path 1: nonenone

then,then, ΔΔ11 = 1= 1

Apply Mason’s formula.Apply Mason’s formula.

( )1321

432111

a

m

HGGG1

GGGGM

)s(E

sT

+=

Δ

Δ=

θ=

Chapter 4-35


Consider the transfer function from the disturbance input, Td(s) to the output, θm(s) , with (Ea = 0).

The forward path is now

M1 → Td x3 x4 θm Gain = G3G4

The loops are not changed, so Δ and Δ1 are unchanged.

Applying Mason’s formula;

Note: The denominator has not changed.Note: The denominator has not changed.

Servomotor System ...continued

( )1321

4311

1)( HGGG

GGM

sT

sT

d

m

+=

Δ

Δ==

θ

Chapter 4-36


Example

Forward Paths:

M1 ⇒ R’R x3 x4 C Gain = G6G4G5

M2 ⇒ R’R x1 x2 x3 x4 C Gain = G1G2G3G4G5

C(s)C(s)GG11xx11 GG22

xx22 xx33GG33 GG4411RR’’(s)(s)

--HH11

xx44 GG55

R(s)R(s)

--HH22

GG66

Chapter 4-37


Feedback loops:

L1 ⇒ x1 x2 x1 Loop gain = – G2H1

L2 ⇒ x3 x4 x3 Loop gain = – G4H2

Non touching loop pairs:

L1 L2 ⇒ Loop gain = G2G4H1H2

then Δ = 1 – (– G2H1 – G4H2) + (G2G4H1H2)

= 1 + G2H1 + G4H2 + G2G4H1H2

... continued

Chapter 4-38


Loops not touching forward path 1 : Loops not touching forward path 1 : LL11

then, then, ΔΔ11 = 1= 1 –– ((–– GG22HH1 1 ) = 1 + ) = 1 + GG22HH11

Loops not touching forward path 2 : Loops not touching forward path 2 : nonenone

then, then, ΔΔ22 = 1= 1

Now applying Mason’s formulaNow applying Mason’s formula

... continued

21422412

54321125462211

1

)1(

)(

)(

HHGGHGHG

GGGGGHGGGGMM

sR

sCT

+++++

=Δ

Δ+Δ==

Chapter 4-39


Example

CVin A V1 B

F E

V4

V2 V3 DVout

Vout / Vin =ABCD

1 - CEF

Vout / Vin =FP

1 - LP

Chapter 4-40


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 + ABDE1 - ( BG + EF ) + ( BGEF )

Example

Chapter 4-41


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 + ABDEFG1 - DI - FH + DIFH

Example

Chapter 4-42


Vin +1

H

B

CA D E

G

F

+1 Vout.FP1 = (1) (B) (1) = B LOOP #1 = CHFP2 = (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

Example

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