Kendali Digital 4

8/16/2019 Kendali Digital 4

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Kendali Digital

Pertemuan 4

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Menentukan periodesampling

• Apakah semakin kecil periodesampling semakin bagus??

• Semakin kecil periode samplingsemakin besar kecepatan komputasiang dibutuhkan!

• "erapakah sampling time ang pas?#ntuk sistem dg pole2 real$maka


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0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

A m p l i t u d e

k=0.36k=0.75

k=1

k=2

k=5

%

&e'ie( o) s*plane

+ocations o) polesin,uence the 'alues o)-.'ershootSettling time

+ocations o) polesare ad/usted bthe 'alue o) thegain 0k

.'ershoot and setlling time


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&elation bet(een s and

1

12

21

2

1

:sTustin'

1

1

1: bacwardsEuler'

1

1:forwardsEuler'

:ionsapproximatSome

relationOriginal

+

−

=−

+

≈=

−=

−≈=

−

=+≈=

=

z

z

T sor s

T

sT

e z

Tz

z sor

Tse z

T

z

sor sT e z

e z

sT

sT

sT

sT


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Mapping s*plane into *plane 01

( )

period).sampling:T frequenc!signal:" TondependwillanglesT#e

plane.$%of circleunitt#eintomappedare plane$sof axisimaginaron t#elocations&ole

1sincos

stable)"marginall plane$sof axisimaginarinare polest#at t#emeansit(f

ω ω

ω ω ω σ

σ ω σ

ω

ω σ ω σ ω σ

⇒

∠=+==→=

=

===

+=

= ++

T T jT e z

eeee z

j s

e z

T j

T jT T jT T j sT


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1

Mapping the lines o) constant 6

( ) ( )[ ]

( ) ( )

{ } { } z j z

T jr T r

T jT r

er reee z

j s

T T jT jT

ma*+e

sincos

sincos

'

+=

+=

+=

=←==→≠

+=

ω ω

ω ω

σ

ω σ

σ ω ω σ


11/30

Mapping s*plane into *plane 02

11

Alias

region

Aliasregion

( )

T F f

f F f F

s

s s

π π π ω =≤=

≥

2

freq.analog:freq.!Sampling:2:criterion ,quist


12/30

7ime response based on pole positions

12


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Sho( the positions o) poles (hich ha'e the same damping ratio 08-constant and ha'e di9erent undumped natural )re:! 0(n;

Sho( the positions o) poles (hich ha'e the same undumped natural)re:! 0(n - constant$ but ha'e di9erent damping ratio;

13

plane$sonandof +e-iew nω ζ

)"

sin1

tanase

)"

agnitude

1

1

2

1$

22

2

ζ

ζ θ ζ

ζ

ω

ω ω σ ζ ω ζω ω σ

f

f

j j s

n

nd

nnd

=

=⇒

−

=

=

=+=

−+=+=

−


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Ans(er

14

'=nω

'>n

ω

'=ζ

1=

ζ


15/30

15

Sho( the


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1=

"2) plane$%onand nω ζ


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1%

plane$%onand/inding nω ζ

22

2

2)")"

nn

n

s s sr s y

ω ζω ω ++=

( )

( ) 22

22

2

2

22

1(2

ln1

ln

ln

1

ln

ln

1

12

0s:&ole

θ ω

θ ζ

ζ

ζ

θ

ζω

θ

ζω

ζ ω θ

ζ ω ζω

ζω

θ

+=⇔

+

−=⇔

−

−=−

=⇔

−=⇔=

−===

−±−=−±−

=

−

r T

r

r

T r

T r er

T ree z

ja

acbb

n

n

nT

n j sT

nn

n


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>ample

1

1sTssumefrequencnaturalundampedt#eandratiodampingt#e/ind

342.'

230.'435.'

)"1

)"

r"%)

"%)

:sstemcontrolloopcloseddigitalof functiontransfera6i-en

2

=

+−

+=

+

=

z z

z

z G

z G

( )

( ) 72.'ln1

28.'ln

ln

57'.'978.'315.'8.'

'342.'%:equationsticc#aracterisstemT#e

:Solution

22

22

2(1

2

=+=

=+

−

=

±∠=±∠=±=

=+−

θ ω

θ ζ

θ

r T

r

r

r j z

z

n


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Solution b graphicalmethod

1

315.'8.''342.'%:equationsticc#aracterisstemT#e

2(1

2

j z z

±=

=+−

71.27.

28. ===T

n

π ω ζ


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Sstem Stabilit

Sstem is stable i) the roots o) the characteristic e:! areinside the unit circle!

Some methods to check )or the stabilit o) a discrete*timesstem!1! @inding the 'alue o) roots o) the characteristic e:!

directl!

2! " doing some stabilit test (ithout


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@inding the roots

• " )actoriation to


22/30

>ample 1 o) method 1

heck (hether the sstem as sho(n in the


23/30

Ans(er

23

( )

( ) ( )( ) ( )

( )

stablenotissstem870.1870.1

148.

927.116"%)1:equationsticc#aracteriT#e

1148.

927.1)"

12

1

121)"

2

1

21

2

22)1"

2

01)"

)"1)"

+"%)="%) :functiontransfersstemloop$closedT#e

2

2

2

21

2

1

→−=→=+

=−+→=+

=−

=

−−=

−−−−=

−−

−−=

+−−=

+

−=

+=

−

−

−

−−

−

−−

−

z z

z

sT z

z G

e z

e

e z z

e z z z G

e z

z

z

z z

s se Z

s s

e Z z G

z G z G

T

T

T

T

T

TsTs


24/30

>ample 2 o) method 1

@or the case o) eample 1$


25/30

Ans(er

Keep the sampling time on 7$ (e ha'e-

25

( )

( )

( )

1ln21

44

124

:if stable bewillsstemT#e

24

'12

'121eq.c#ar.T#e

12)"

2

2

2

2

22

2

2

2

2


26/30

Special Pro/ect

2=

D MotorMikrokontroler

7achometerAmpli


27/30

7hank ou

2%


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7ime domain speci time2.

o-ers#ootaximum1.

:e performanct#eof &arameters

sstemt#eof ratiodamping:

sstemt#eof frequencnatural undamped:

ζ

ω n


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&eminder- in,uence o) the dampingratio

2


30/30

•

&emind poles position on s*plane

3

222222

222

d

1

part(mag.

part(+ealfrequenc(natural?ndamped

cossinratio(amping

*s

ζ ω ω ζ ω σ ω ω

ω σ ω

ζω σ

ω

ω

σ β θ ζ

ω σ

−=−=−=

⇒+=

=

===

+=

nnnnd

d n

n

n

n

plane$sonand nω ζ

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