Lecture5 Diagram Block Diajarkan

8/12/2019 Lecture5 Diagram Block Diajarkan

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References for reading

1. R.C. Dorf and R.H. Bishop,Modern Control Systems,

11th Edition, Prentice Hall, 2008,Chapter 2.6

2. J.J. DiStefano, A. R. Stubberud, I. J. Williams,Feeedback and Control Systems, Schaum's Outline Series,McGraw-Hill, Inc., 1990Chapters 7

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Outline

Terms and concepts

Canonical form of a feedback control system

Block diagram transformations

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Block diagrams

Block diagrams consist of unidirectional,operational blocks that represent the transferfunction of the variables of interest.

The block diagram representation of a givensystem often can be reduced to a simplified blockdiagram with fewer blocks than original diagram.

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Introduction

A graphical toolcan help us to visualize the modelof a systemand evaluate the mathematical relationships between theirelements, using their transfer functions.

In many control systems, the system of equations can bewritten so that their components do not interact except byhaving the input of one part be the output of another part.

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Oxford University Press 20016


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Component Block Diagram

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Block Diagram

It represents the mathematical relationshipsbetween the elements of the

system.

The transfer functionof each component is placed in box, and the input-

output relationshipsbetween components are indicated by l ines and

arrows.

)()()( 111 sYsGsU

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Block Diagram Algebra

We can solve the equations by graphical simpli f ication, which is ofteneasier and more informative than algebraic manipulation, even thoughthe methods are in every way equivalent.

It is convenient to think of each block as representing an electronicamplifierwith the transfer function printed inside.

The interconnections of blocks include summing points, where anynumber of signals may be added together.

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Diagram Blok sistem tertutup:

Ideal

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

-

+

H(s)

R(s)

B(s)

Titik PenjumlahanTitik Percabangan

R(s)=Referensi sinyal inputE(s)=Sinyal error [E(s)=R(s)-B(s)]

G(s), H(s)=Fungsi Transfer

B(s)= Sinyal feedback

Y(s)=Sinyal output


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)()(

)(sG

sE

sYFFTF

)()()(

)(sHsG

sE

sBOLTF

)()(1

)(

)(

)(

sHsG

sG

sR

sYCLTF

)()()(1

)()( sRsHsG

sGsY

Feed-forward Transfer Function, FFTF

Open-Loop Transfer Function, OLTF

Closed-Loop Transfer Function, CLTF

Hubungan Input Output (Lihat

Diagram Blok):

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

E(s)=R(s)-B(s)

B(s)=H(s)Y(s)

Atau

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

Y(s)+G(s)H(s)Y(s)=G(s)R(s)

(1+G(s)H(s))Y(s)= G(s)R(s)

Atau,


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)()()(1

)(

)(

)(

21

2

sHsGsG

sG

sD

sYD

)()()(1

)()(

)(

)(

21

21

sHsGsG

sGsG

sR

sYR

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

)()()()( 121

2 sDsRsGsHsGsG

sGsYsYsY DR

Response Y(s) terhadap gangguan D(s),

Response Y(s) terhadap referensi input R(s), dengan measumsikan gangguan

sama degan nol

Total Response Y(s),


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Paralel

R(s) Y(s)G2(s)

G(s)

)(...)()()()( 211

sGsGsGsGsG k

k

i

i

Fungsi Transfer hubungan paralel:

G1(s)

Gk(s)

+++


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Feedback

R(s) Y(s)G1(s)

G(s)

)()(1

)()(

21

1

sGsG

sGsG

Fungsi Transfer

G2(s)

+

+-


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Simplikasi Diagram Blok

RG +

+-

B

+

+-

B

G

1/G

Y YR

++-

B

YR

G G

G

+

+-

R

B

Y


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RG

B

Y

G

G

Y YR

R

YR

G G

1/G

R

R

Y


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RG +

+-

B

+

+-H

H

Y YR

++-

YR

G 1/HGH+

+-

R Y

G/H

H


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Contoh1

)()()(

)(

)(sHsGsC

sE

sBOLTF

+-U

C

H

YR

B

EG

Diagram blok dari suatu sistem diberikan seperti gambar berikut, Tentukan:a). Open-Loop Transfer Function, OLTF

b). Closed-Loop Transfer Function, CLTF

Jawab

a). Open-Loop Transfer Function, OLTF

)()()(1

)()(

)(

)(

sHsGsC

sGsC

sR

sYCLTF

b). Closed-Loop Transfer Function, CLTF


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21

Contoh2

+- C2

H3

YRG1

Sederhanakan diagram blok berikut:

C1

+-

H1

G2

H2

+


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Contoh2

+- C2

H3

YRG1

Jawab

C1

+-

H1

G2

H2

+


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Contoh2

+-C1+C2

H2H3

YR G1

1+G1H1

Jawab

G2

+-

H2H3

YR (C1+C2)G1G2

1+G1H1


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Example 2

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Feedback Rule

The gain of a single-loop negative feedback system is given

by the forward gain divided by the sum of 1 plus theloop gain

21

1

1 GG

G

)s(R

)s(Y

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Eliminating a feedback loop

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Closed-loop transfer function

Ea(s) = R(s) - B(s) = R(s) - H(s) Y(s)

Y(s) = G(s) Ea(s)

Y(s) = G(s) [ R(s) - H(s) Y(s) ] Y(s) [ 1 + G(s) H(s) ] = G(s) R(s)

Y(s)/R(s)= G(s) /(1 + G(s) H(s))

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Ea(s) = R(s) - B(s) = R(s) - H(s) Y(s)

Y(s) = G(s) Ea(s)

Ea(s) = R(s) - H(s) G(s) Ea(s) Ea(s) [ 1 + G(s) H(s) ] = R(s)

Ea(s)= R(s)/(1 + G(s) H(s))

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Y(s)= R(s)G(s)/(1 + G(s) H(s))

Ea(s)= R(s)/(1 + G(s) H(s))

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All the transformations can be derived

by simple algebraic manipulation of the

equations representing the blocks.

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Ex. 2.7 Block diagram reduction

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Summary

Using transfer function notations, block

relationships were obtained.

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Moving a pickoff point ahead of

a block

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Moving a pickoff point behind of

a block

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Moving a summing point ahead of a

block

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Lecture5 Diagram Block Diajarkan

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