Block Diagram Algebra

8/7/2019 Block Diagram Algebra

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ME2142/TM3142 Feedback Control Systems1

Block Diagram Algebra

Block Diagram Algebra

ME2142/ ME2142E Feedback Control SystemsME2142/ ME2142E Feedback Control Systems


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The Transfer Function Block

Block Diagram RepresentationBlock Diagram Representation

A block diagramis agraphicaltoolcan help us to visualize themodelof a system and evaluate the mathematical relationshipsbetween their elements, using their transfer functions.

A block diagramis agraphicaltoolcan help us to visualize themodelof a system and evaluate the mathematical relationshipsbetween their elements, using their transfer functions.

G(s)

System

Input OutputR(s) C(s)

)(

)()(

sR

sCsG =

The transfer function G(s) is

defined only for a linear time-invariant system and notfor nonlinear systems.

Is a property of the system and is independent of the

input to the system.

Commutative1221

GGGG = Associative

1221GGGG +=+


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The Summing Point

Block Diagram ElementsBlock Diagram Elements

Any number of inputs. Only one output

Signed inputs

X

-

Z

Y

++ +X+ Y-Z

output


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Blocks in series or cascaded blocks

Block Diagram AlgebraBlock Diagram Algebra

When manipulating block diagrams, theoriginal relationships, or

equations, relating the various variablesmust remain the same.

When manipulating block diagrams, theoriginal relationships, or

equations, relating the various variablesmust remain the same.

When blocks are connected in series, there must be noloading effect.

G1 G2XY Z

G1G

2ZX


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Blocks in parallel


G1

G2

++ YX

G1 + G2YX

Y=G1X+G2X Y=G1X+G2X



G+

+ ZX

Y

G+

+ ZX

Y

G

X Y

X

G

X Y

X

XG

Y

Z

+

+XG

Y

Z

+

+


G

G

+

+ ZX

Y

GX Y

1/GX

+

+ Z

Y

X

G

1/G

Z=G(X+Y)

Z=GX+GY

Z=GX+Y

Z=G(X+Y/G)



Closed-Loop Feedback SystemClosed-Loop Feedback System

is called the open-loop transfer functionGHE

B =

is called the feedforward transfer functionGE

C=

R is called the reference inputC is the output or controlled variableB is the feedbackE = (R B) is the error

RG

C

-

+ E

HB


8/16ME2142/TM3142 Feedback Control Systems

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Closed-Loop Feedback SystemClosed-Loop Feedback System

C = GE

= G(R B)= G(R HC)

C(1 + GH) = GR

GH

G

R

C

+=

1

is the closed-loop transfer functionR

C

Also andG

CE=

GHR

C

GR

E

+==

1

11

is called the error transfer functionR

E

R G C

-

+ E

HB



9

Closed-Loop Control Feedback SystemClosed-Loop Control Feedback System

is the open-loop transfer functionHGGE

Bpc

=

is the feedforward transfer functionpcGGE

C=

Gc is the controller transfer function

Gp is the plant transfer functionM is the manipulated variableD is the external disturbance

R Gp

C

-

+ E

HB

Gc

+ +

D

M



10

HGG

GG

GH

G

R

C

pc

pc

+=

+=

11

Assuming R = 0, we can re-draw

HGGG

GHG

DC

cp

p

+=+= 11

Closed-Loop Control Feedback SystemClosed-Loop Control Feedback System

RG

p

C

-

+ E

HB

Gc

+ +

D

M

DG

p

C

-

+

GcH


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Block Diagram ManipulationBlock Diagram Manipulation

Example: Determine C(s)/R(s)Example: Determine C(s)/R(s)

R G-

+

H

F+

+

-

+

I

++

ED

C

a Da

When manipulating blocks,must ensure C(s) does not

change, so that C(s)/R(s)

remains same.


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b = Fa-Fc+Da

Assume names of

signals as shown

We wish to move this

signal to before block F


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b = Fa-Fc+Da

Assume names of

signals as shown

b = Fa-Fc+(D/F)aF

To move signal b to

after Block G


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R G-

+

H

F+

+

-

+

I

++

ED

C

aDa

bEb

R G-

+

H

F-

+

I

++

ED/F

C+

a(D/F)a

Gb

R

-

+

H

FG-

+

I

++

E/G

C1+D/F

Gb

Eb

R

-

+

H

FG-

+

I

1+E/GC

1+D/F

C = Gb + Eb

C = Gb + (E/G)Gb


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ExampleExample

R

-

+

H

FG-

+

I

1+E/GC

1+D/F R

-

+

I

1+E/GC

1+D/FFGH1

FG

+

R

-

+

I

C

+

+

+

G

E

FGH

FG

F

D1

11

CR

IG

E

FGH

FG

F

D

G

E

FGH

FG

F

D

+

+

++

+

+

+

1111

11

1

GH

G

+1


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End

Block Diagram Algebra

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