自動控制原理英文課件Chapter3Transient Response

Time

Domain

AnalysisofControl

SystemsThird

ClassChapter

3Review2Definition

of

time

response;Typical

test

signals;

Performance

specifications

of

steady-stateresponse

–

steady-state

error;Impact

of

disturbance

to

steady-state

error;

Impact

of

parameter

variation

to

steady-staterror.Outline3Transient

ResponsePerformance

CriteriaTransient

Response

of

1st-Order

SystemsTransient

Response

of

2nd-Order

SystemsApproximation

of

High-Order

SystemsTransient

Response4

Transient

response

is

the

response

of

a

systea

certain

input

which

dies

out

as

time

becomeslarge;

For

linear

control

systems,

the

characterizathe

transient

response

is

oftendone

by

use

ofunit-step

function

as

input;

The

response

of

a

control

system

when

the

inpuis

a

unit-step

function

is

called

the

unit-steresponse.MaximumOvershootPerformance

CriteriaCommonly

used

performance

criteria

characterizing

transient

responset1.0y(t)tstr

tp0.01.0+2%1.0-2%Maximum

OvershootRise

Time5Peak

TimeSettling

TimePerformance

CriteriaMaximum

OvershootRise

Time6Peak

TimeSettling

TimeThe

difference

between

maximum

value

andthe

steady-state

value

of

output.Rise

time

is

defined

as

the

time

required

for

the

step

respto

rise

from

zero

to

the

final

value

of

output.

(In

some

othsituation,

rise

time

is

defined

as

the

time

required

for

thresponse

to

rise

from

10

to

90

percent

of

the

final

value.

)Peak

time

is

defined

as

the

time

required

for

the

step

respto

rise

from

zero

to

the

maximum

value.Settling

time

is

defined

as

the

time

required

for

the

step

response

to

decrease

and

stay

within

a

specified

percenta

of

its

final

value.

A

frequently

used

figure

is

2%Performance

CriteriaMaximum

Overshoot

is

used

to

measure

the

relative

stability

of

a

consystem.

A

system

with

a

large

overshoot

is

usually

undesirable.Rise

time

,

peak

time ,

settling

Time

are

used

to

measure

the

quicknesthat

a

system

responds

to

the

unit-step

input.The

methods

commonly

used

to

study

the

performance

of

a

system

in

transient:Computer

simulation

–

numerical

method;

Referring

to

transient

performance

of

the

first-order

and

second

ordesystem,

estimate

those

of

higher

order

systems7Transient

Response

of

the

1st-Order

Systems8Transient

Response

of

the

2nd-Order

SystemsNatural

undamped

frequencyTime

constant,

reciprocal

of

the

natural

undamped

frequencyDamping

ratio9Transient

Response

of

the

2nd-Order

SystemsRoots

of

the

characteristic

equation:SetReal

part

of

the

rootsImaginary

part

of

the

roots10Transient

Response

of

the

2nd-Order

SystemsRelationship

between

the

characteristic-equation

roots

andand11Transient

Response

of

the

2nd-Order

SystemsWhenthe

unit-step

response

is:Undamped12Transient

Response

of

the

2nd-Order

SystemsWhenthe

unit-step

response

is:13UnderdampedTransient

Response

of

the

2nd-Order

SystemsWhenthe

unit-step

response

is:Critically

damped14Transient

Response

of

the

2nd-Order

SystemsWhenthe

unit-step

response

is:Overdamped15Step

Response

of

Systems

with

Different

Damping

Ratios16Transient

Response

of

the

2nd-Order

SystemsTwo

useful

experimental

formulas

to

estimate

the

maximum

overshoot

andsettling

time

of

the

step

response

of

2nd-order

under

damped

systems17ExampleQ:

please

find

the

maximum

overshoot

and

settling

time

of

the

following

systemwhereA:18s=tf("s");z=0.3;g=1/(s^2+2*s*z+1);step(g,20)grid

onRelationship

Between

the

Damping

Ratio

and

the

Maximum

Overshootζ↑

σ％↓19Relationship

Between

the

Damping

Ratio

and

the

Settling

TimeActualEstimated20Relationship

Between

the

Natural

Undamped

Frequency

and

Other

Criteria21ExampleQ:

please

design

a

second-order

system

with

a

damping

ratio

equal

to

0.707and

a

settling

time

equal

to

0.5s.

Find

the

poles

of

the

system

and

calculate

inatural

undamped

frequency.A:

becauseandThenAccording

towe

can

getTherefores-z

wn450jw22s1s2High-order

Systems’

Time

ResponseLaplace

transform

of

the

step

response

of

a

high-order

systemTime

response

of

the

system

with

a

step-function

as

input2324High-order

Systems’

Time

Responses-

a-

5a

The

time

response

of

a

high-order

system

is

the

linear

combination

ofthose

of

1st

and

2nd

order

systems;

Poles

of

a

high-order

system

which

is

far

away

from

the

imaginary

axis

has

less

impact

on

the

system’s

transient

response

than

those

close

to

thimaginary

axis.

The

farther

the

distance

is,

the

smaller

the

impact

is.

Zeros

of

a

high-order

system

also

has

impact

on

the

system’s

timeresponse.

They

mainly

affect

the

magnitude

and

phase

of

a

dynamic

mode.

The

performance

of

a

high-order

system

is

mainly

determined

by

itscontrolling

polesControlling

PolejwControlling

PoleImpact

of

Additional

Poles

and

ZerosOriginal

system:25Impact

of

Additional

ZerosWith

additional

zero:26Impact

of

Additional

PolesWith

additional

pole:2728Conclusion

on

the

Impact

of

Additional

Poles

and

Zerosy(t)y1

(t)1

×1s2

+

0.8s

+

1

sY

(s)=s2×s

+

0.8s+

1s

+

1

1Y1(s)

=y(t)y2

(t)s

0.8s

1

s1

1×+2

+Y

(s)

=1

12×(s2

+

0.8s+

1)(s+

1)

sY

(s)=Decrease

Damping

RationIncrease

Damping

RationApproximation

of

High-order

Systems

by

Low-Order

SystemsExample:29To

keep

the

final

value

the

sameApproximation

of

High-order

Systems

by

Low-Order

SystemsExample:30Approximation

of

High-order

Systems

by

Low-Order

Systems31Approximation

CriterionThe

criterion

of

finding

the

low-order

,

given ,

is

thatthe

following

relation

should

be

satisfied

as

close

as

possible:This

condition

implies

that

the

amplitude

characteristics

of

the

two

systin

the

frequency

domain

( )

are

similar.32Wrap-up33Transient

ResponsePerformance

CriteriaTransient

Response

of

1st-Order

SystemsTransient

Response

of

2nd-Order

SystemsApproximation

of

High-Order

SystemsAssignment34Page

729，(4)10，11，12Review

questionsQ1:

The

maximum

overshoot

of

a

unit-step

response

of

the

second

order

prototypesystem

will

never

exceed

100

percent

when

the

damping

ratio

and

the

naturalfrequency

are

all

positive.

(T)

(F)35Review

questionsQ2:

For

the

second-order

prototype

system,

when

the

undamped

natural

frequencyincreases,

the

maximum

overshoot

of

the

output

stays

the

same.(T)

(F)36Review

questionsQ3:

The

maximum

overshoot

of

the

following

system

will

never

exceed

100%

when,

and

T

are

all

positive.(T)

(F)37Reference

answers:Q3:

False.Let ,

and

T

varies

from

0

to

5,

and

then

the

relationship

between

T

andthe

overshoot

can

be

shown

in

the

following

graph:Reference

answers:Let ,

and

T

varies

from

0

to

500,

and

then

the

relationship

between

T

andthe

overshoot

can

be

shown

in

the

following

graph:40Review

questionsQ4:

Increasing

the

undamped

natural

frequency

will

generally

reduce

the

rise

timeof

the

step

response(T)

(F)Reference

answers:Q4:

True.Let ,

and

varies

from

0.5

to

1.5,

and

then

the

relationship

between

andthe

rising

time

can

be

shown

in

the

following

graph:Reference

answers:Q4:

True.Let ,

and

varies

from

0.5

to

1.5,

and

then

the

relationship

between

andthe

rising

time

can

be

shown

in

the

foll