Control de Sistemas Aeroespaciales y Mecánicos: Diagramas de Bloques, Diapositivas de Sistemas de Control

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MAE

-^

Control

of

Aerospace

&^

Mechanical

Systems

SECTION

2:

BLOCK

DIAGRAMS

&

SIGNAL

FLOW

GRAPHS

K.^ Webb

MAE

Block

Diagram

Manipulation

K.^ Webb

MAE

Standard

Block

Diagram

Forms

^

The

basic

input/output

relationship

for

a^

single

block

is:

ܻ

ݏ^

ܷൌ

ݏ^

ݏ ܩ ⋅

^

Block

diagram

blocks

can

be

connected

in

three

basic

forms:

^

Cascade

^

Parallel

^

Feedback

^

We’ll

next

look

at

each

of

these

forms

and

derive

a^

single

‐

block

equivalent

for

each

K.^ Webb

MAE

Cascade

Form

^

Blocks

connected

in

cascade

ݏ^

ଵ

ݏ^

ܺൌ^

ଵ

ݏ^

ݏ^

ܷൌ^

ݏ^

ݏ^

௘௤^

^

The

equivalent

transfer

function

of

cascaded

blocks

is

the

product

of

the

individual

transfer

functions

K.^ Webb

MAE

Feedback

Form

^

Of

obvious

interest

to

us,

is

the

feedback

form

ݏ^

ݏ^

ܴൌ^

ݏ^

ܺെ^

ݏ^

ݏ^

ܴൌ^

ݏ^

ܻെ^

ݏ^

ܴൌ^

ݏ^

ܴൌ^

ݏ^

⋅^

^

The

closed

‐loop

transfer

function

,ܶ^

ݏ^

,^ is

ݏ^

ܻൌ^

ܴݏ ݏ^

ൌ^

K.^ Webb

MAE

Feedback

Form

^

Note

that

this

is^

negative

feedback

,^ for

positive

feedback

ݏ^

ൌ^

^

The

factor

in the

denominator

is^

the

loop

gain

or

open

‐loop

transfer

function

^

The

gain

from

input

to

output

with

the

feedback

path

broken

is^

the

forward

path

gain

• here,

^

In^

general:

ݏ^

forward path gain

1 െ loop gain

ݏ^

ൌ^

K.^ Webb

MAE

Unity

‐Feedback

Systems

^

We’re

often

interested

in

unity

‐feedback

systems

^

Feedback

path

gain

is

unity

^

Can

always

reconfigure

a^

system

to

unity

‐feedback

form

^

Closed

‐loop

transfer

function

is:

ݏ^

ൌ^

K.^ Webb

MAE

Block

Diagram

Algebra

^

Often

want

to

simplify

block

diagrams

into

simpler,

recognizable

forms

^

To

determine

the

equivalent

transfer

function

^

Simplify

to

instances

of

the

three

standard

forms,

then

simplify

those

forms

^

Move

blocks

around

relative

to

summing

junctions

and

pickoff

points

• simplify

to

a

standard

form

^

Move

blocks

forward/backward

past

summing

junctions

^

Move

blocks

forward/backward

past

pickoff

points

K.^ Webb

MAE

Moving

Blocks

Forward

Past

a

Summing

Junction

^

The

following

two

block

diagrams

are

equivalent:

ݏ^

ܷൌ^

ଵ^

ଶ^

ݏ^

ܷൌ^

ଵ^

ݏ^

ܷ൅^

ଶ^

ݏ^

K.^ Webb

MAE

Moving

Blocks

Relative

to

Pickoff

Points

^

We

can

move

blocks

backward

past

pickoff

points:

^

And,

we

can

move

them

forward

past

pickoff

points:

K.^ Webb

MAE

Block

Diagram

Simplification

• Example

2

^

Find

the

closed

‐loop

transfer

function

of

the

following

system

through

block

‐diagram

simplification

K.^ Webb

MAE

Block

Diagram

Simplification

• Example

2

^

and

ଵ^

ݏ^

are

in

feedback

form

ଵ^

ଵ^

K.^ Webb

MAE

Block

Diagram

Simplification

• Example

2

^

Simplify

the

feedback

subsystem

^

Note

that

we’ve

dropped

the

function

of

ݏ^

notation,

,^

for

clarity

ଵ

ଶ

K.^ Webb

MAE

Block

Diagram

Simplification

• Example

2

^

Simplify

the

two

parallel

subsystems