R C (3)Signals: • Arrow as shown in figure represent signals flowing in or out of the system

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INTRODUCTION

•

Block diagram is graphical representation of the various components of the system.

•

Unlike pure mathematical representation of the system, block diagrams visualize signal flow of the system.

•

Functional Blocks are used to represent the transfer functions of various components.

•

Blocks are connected to each other with arrows indicating the direction of flow of the signals as shown in the figure:

Transfer function G(s)

R C

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Signals:

•

Arrow as shown in figure represent signals flowing in or out of the system.

Summing points:

•

A circle is used to represent a summing point which adds any number of signals. The plus or minus sign indicates if the signal is to be added or subtracted.

•

It is important to note that the quantities being added or subtracted must have the same units.

B A

-

+

^A^-^B

C(s) R(s)

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Transfer function:

𝑪 𝒔 = 𝑹 𝒔 𝑮(𝒔)

Branch points or pick off points:

•

A branch point is a point from which the signal branches to more than one direction, as shown in figure.

𝑪 𝒔 = 𝑹 𝒔 𝑮(𝒔)

R G C

C

Transfer function G(s)

R C

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Blocks in cascade:

𝑪(𝒔) = 𝑹(𝒔)𝑮_𝟏(𝒔)𝑮_𝟐(𝒔)

Blocks in parallel:

𝑪 𝒔 = 𝑹 𝒔 𝑮_𝟏(𝒔) ± 𝑮_𝟐(𝒔)

R C

𝑮

_𝟏

𝑮

_𝟐

+

±

R C

𝑮

_𝟏

± 𝑮

_𝟐

or

R C

𝑮

_𝟏

𝑮

_𝟐

R C

𝑮

_𝟏

𝑮

_𝟐

or

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Moving a pickoff point behind a block Moving a pickoff point ahead of a block

R C

G R

R G C

𝟏 𝑮 R

R G C

C

G G

R C

C

𝑪(𝒔) = 𝑹 𝒔 𝑮 𝒔 𝑪(𝒔) = 𝑹 𝒔 𝑮 𝒔

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𝑹

_𝟏

±

+ C

G

𝑹

_𝟐

𝑹

_𝟏

G 𝟏 𝑮

C

± +

𝑹

_𝟐

Moving a summing point ahead of a block Moving a summing point behind a block

±

+ 𝑮 C

𝑹

_𝟏

𝑹

_𝟐

𝑹

_𝟏

𝑮

+

±

C 𝑹

_𝟐

𝑪 𝒔 = 𝑹_𝟏 𝒔 ± 𝑹_𝟐 𝒔 𝑮 𝒔 𝑪 𝒔 = 𝑹_𝟏 𝒔 ± 𝑹_𝟐 𝒔 𝑮 𝒔

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Manipulation Original Block Diagram Equivalent Block Diagram Equation

1 Combining blocks in cascade 𝑪 = 𝑹(𝑮_𝟏𝑮_𝟐)

2 Combining blocks in parallel

or eliminating a forward loop 𝑪 = 𝑹(𝑮_𝟏 ± 𝑮_𝟐)

3 Moving a pickoff point behind a block

𝑪 = 𝑹𝑮 𝑹 = 𝟏

𝑮𝑪

BLOCK DIAGRAM ALGEBRA (SUMMARY)

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Manipulation Original Block Diagram Equivalent Block Diagram Equation

4 Moving a pickoff

point ahead of a block 𝑪 = 𝑹𝑮

5 Moving a summing

point behind a block 𝑪 = (𝑹_𝟏− 𝑹_𝟐)𝑮

6 Moving a summing

point ahead of a block 𝑪 = (𝑹_𝟏− 𝑹_𝟐)𝑮

BLOCK DIAGRAM ALGEBRA (SUMMARY)

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BLOCK DIAGRAM OF A FEEDBACK SYSTEM

R 𝑮_𝟏

𝟏 ± 𝑮_𝟏𝑯_𝟏

C

𝑮_𝟏

R

+

C

𝑯_𝟏

∓

E

From the above figure:

𝑬 = 𝑹 ∓ 𝑪𝑯_𝟏 𝑪 = 𝑬 𝑮_𝟏 Or:

𝑪 = (𝑹 ∓ 𝑪𝑯_𝟏) 𝑮_𝟏 𝑪(𝟏 ± 𝑮_𝟏 𝑯_𝟏) = 𝑹 𝑮_𝟏

Accordingly:

𝑪

𝑹 = 𝑮_𝟏

(𝟏 ± 𝑮_𝟏 𝑯_𝟏) Or:

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𝑮_𝟏

R C

- +

𝑮_𝟐

R C

-

+

𝑮_𝟏

𝟏 𝑮_𝟐

Moving a block out of the feedback path:

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R(s)

-

+ G(s)

^C(s)

H(s)

E(s)

BLOCK DIAGRAM OF A FEEDBACK SYSTEM

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COMPONENTS OF A FEEDBACK SYSTEM

Control Signal

R(s)

-

+

_Controller _Plant

C(s)

Sensor

Error Signal

System Output Reference

input

Actuator

Block diagram representation of a general feedback control system

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Controller

Control Signal

R(s)

-

+

G_c(s) G_p(s)

C(s)

H(s)

Error Signal

Plant

Sensor

System Output Reference

input

+ +

Disturbance D(s)

FEEDBACK SYSTEM SUBJECT TO A DISTURBANCE

A general feedback control system with disturbance input

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RESPONSE OF A FEEDBACK SYSTEM SUBJECTED TO A DISTURBANCE INPUT

R(s)

-

+

_G

c G_p

C(s)

H

+ +

D(s)

Output 𝑪_𝑹(𝒔) due to input R(s)

i.e. let D(s)=0

𝑪_𝑹(𝒔)

𝑹 𝒔 = 𝑮_𝒄𝑮_𝒑 𝟏 + 𝑮_𝒄𝑮_𝒑𝑯

R(s)

-

+

G_c G_p

𝑪_𝑹(𝒔)

H

Discuss the desired conditions to cancel out the disturbance effect on the output of the system for the following system.

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RESPONSE OF A FEEDBACK SYSTEM SUBJECTED TO A DISTURBANCE INPUT

Output 𝑪_𝑫(𝒔) due to input R(s) i.e. let R(s)=0

It can easily be shown from the resulting diagram that:

𝑪_𝑫(𝒔)

𝑫 𝒔 = 𝑮_𝒑

𝟏 + 𝑮_𝒄𝑮_𝒑𝑯

G_c G_p

-H

+ +

D(s)

𝑪_𝑫(𝒔)

G_c

G_p

H

+ -

D(s) 𝑪_𝑫(𝒔)

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RESPONSE OF A FEEDBACK SYSTEM SUBJECTED TO A DISTURBANCE INPUT

 Meanwhile, ensuring the rejection of the disturbance input 𝑫 𝒔 , requires that:

𝑮

_𝒄

≫ 𝑮

_𝒑

Or:

𝑮

_𝒄

𝑮

_𝒑

≫ 𝟏

The resulting output response 𝑪 𝒔 is given as:

𝑪 𝒔 = 𝑪_𝑹 𝒔 + 𝑪_𝑫(𝒔)

Or:

𝑪 𝒔 = 𝑮_𝒄𝑮_𝒑

𝟏 + 𝑮_𝒄𝑮_𝒑 𝑯 𝑹 𝒔 + 𝑮_𝒑

𝟏 + 𝑮_𝒄𝑮_𝒑 𝑯 𝑫 𝒔

At steady state conditions:

Assuming unity feedback system i.e. 𝑯 = 𝟏

 Perfect tracking of the input 𝑹 𝒔 , requires that:

𝑮

_𝒄

𝑮

_𝒑

≫ 𝟏

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CRUISE CONTROL SYSTEM

engine

speedo- meter

desired speed

actual speed

cruise

control vehicle

wind, hills

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THERMOSTAT EXAMPLE

desired temp.

thermo- stat

furnace or AC

thermo- stat

actual temp.

room air

external air

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TOILET FLUSH EXAMPLE

valve

float

desired level

actual level

float water

tank

flush

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EXAMPLE #1

-

G₁

R C

+

G₂

H₁

G₃ H₂

-

Move here

+ +

Simplify the following block diagram to obtain the overall relation between C(s) & R(s) for the following block diagram.

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SOLUTION

-

G₁

R C

+

G₂

H₁

G₃ 𝑯_𝟐

𝑮_𝟏

-

Reduce this loop

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SOLUTION

-

R

+

_G C

3

𝑯_𝟐 𝑮_𝟏

-

𝑮_𝟏𝑮_𝟐 𝟏 − 𝑮_𝟏𝑮_𝟐𝑯_𝟏

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SOLUTION

R _𝑮_𝟏_𝑮_𝟐_𝑮_𝟑

𝟏 − 𝑮_𝟏𝑮_𝟐𝑯_𝟏 + 𝑮_𝟐𝑮_𝟑𝑯_𝟐 + 𝑮_𝟏𝑮_𝟐𝑮_𝟑

C

Or:

𝑪

𝑹 = 𝑮_𝟏𝑮_𝟐𝑮_𝟑

𝟏 − 𝑮_𝟏𝑮_𝟐𝑯_𝟏 + 𝑮_𝟐𝑮_𝟑𝑯_𝟐 + 𝑮_𝟏𝑮_𝟐𝑮_𝟑

-

R C

+

^𝑮^𝟏^𝑮^𝟐^𝑮^𝟑

𝟏 − 𝑮_𝟏𝑮_𝟐𝑯_𝟏+ 𝑮_𝟐𝑮_𝟑𝑯_𝟐

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EXAMPLE #2

𝑮

_𝟑

𝑮

_𝟒

R(s) 𝑮

_𝟏

+ C(s)

𝑮

_𝟐

⁺ +

−

− +

Reduce these parallel blocks

Simplify the following block diagram to obtain the overall relation between C(s) & R(s) for the following block diagram.

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𝑮

_𝟏

+𝑮

_𝟐

SOLUTION

𝑮

_𝟑

𝑮

_𝟒

R(s) C(s)

+

−

− +

Reduce these parallel blocks

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R(s) 𝑮

_𝟏

+𝑮

_𝟐

C(s) 𝟏 + 𝑮

_𝟏

+𝑮

_𝟐

𝑮

_𝟑

−𝑮

_𝟒

SOLUTION

𝑮

_𝟑

−𝑮

_𝟒

R(s) C(s)

𝑮

_𝟏

+𝑮

_𝟐

− +

Reduce the feedback loop