2-Laplace Transform Review

8/18/2019 2-Laplace Transform Review

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TRANSFER FUNCTION

DEFINITION


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TRANSFER FUNCTION

• The transfer function is the ratio ofthe Lapace transfor! of the outputof a s"ste! to the Lapace transfor!

of the input#

• As an e$a!pe% the transfer function%&'s( for a s"ste! representation)*


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TRANSFER FUNCTION

• A +enera nth or,er% inear% ti!e*in-ariant ,i.erentia e/uation is)*

0here ) c(t) is output

r(t) is inputa an, b are constants

)(...

)()(

)(...

)()(

01

1

101

1

1 t r b

dt

t r d b

dt

t r d bt ca

dt

t cd a

dt

t cd a

m

m

mm

m

mn

n

nn

n

n

+++=+++−

−

−−

−

−


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TRANSFER FUNCTION

• Ta1in+ Lapace transfor! of 2othsi,e)*

• Assu!e a initia con,itions are 3ero%

• Therefore% the transfer function is

conditioninitial)(...)()(

conditioninitial)(...)()(

0

1

1

0

1

1

++++

=++++

−

−

−

−

s Rb s R sb s R sb

sC a sC sa sC sa

m

m

m

m

n

n

n

n

)()...()()...( 01

10

1

1 s Rb sb sb sC a sa sa m

m

m

m

n

n

n

n +++=+++

−

−

−

−

0

1

1

0

1

1

...

...)(

)(

)(

a sa sa

b sb sb sG

s R

sC

n

n

n

n

m

m

m

m

+++

+++==

−

−

−

−


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TRANSFER FUNCTION

• It separate output% input an, s"ste!#

• The transfer function can 2erepresente, as a 2oc1 ,ia+ra! assho0n 2eo0)*


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TRANSFER FUNCTION FORFEED7AC8 CONTROL S9STEM

• The +enera fee,2ac1 for!'si!pi6e, !o,e( is sho0n 2eo0)*


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• Forward path transfer function,G(s) is the ratio of the Lapacetransfor! of the output 'C(s( ) of a

s"ste! to the Lapace transfor! ofthe input 'E(s)( or

• Feedback path transfer function,

H(s) is the ratio of the Lapacetransfor! of the output 'C(s)( of as"ste! to the Lapace transfor! of

the input 'C(s)( or

)()()(

s E

sC sG =

.

)(

)()(

sC

s B s H =

.


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• Open-loop transfer function, G(s)H(s) is the ratioof the Lapace transfor! of the output 'B(s( ) of as"ste! to the Lapace transfor! of the input 'E(s)( or

• For the unity feedback case 'H(s) : 4(% open*ooptransfer function 'OLTF( is e/ua to for0ar, pathtransfer function#

• Closed-loop transfer function, M(s) is the ratio ofthe Lapace transfor! of the output 'B(s( ) of a s"ste!to the Lapace transfor! of the input 'R(s)( or

)(

)()(

s E

s B sG =

.

)(

)()(

s R

sC s M =

.


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• We o2tain the e/ui-aent% or cose,*oop transfer function 'CLTF( sho0n2eo0)*


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7LOC8 DIA&RAMS

• Su2s"ste! is represente, as a 2oc10ith an input% an output an, a TF#

• Co!pose, of !utipe su2s"ste!s#

• Mutipe su2s"ste!s are interconnecte,%fe0 !ore sche!atic ee!ents !ust 2ea,,e, to the 2oc1 ,ia+ra!#

• Ne0 ee!ents are)* – Su!!in+ ;unctions

– Pic1o. points


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CASCADE FORM


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FEED7AC8 FORM


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ASSI&NMENT 4

• Pro-e the cose,*oop transferfunction 'CLTF( of ne+ati-e fee,2ac1s"ste! as sho0n 2eo0 is

)()(1

)(

s H sG

sG

+

.


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MOVIN& 7LOC8S

a( To the eft past a su!!in+ ;unction


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MOVIN& 7LOC8S

c( To the eft past a pic1o. point


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MOVIN& 7LOC8S

,( To the ri+ht past a pic1o. point


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MOVIN& 7LOC8S

• E$a!pe


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MOVIN& 7LOC8S

• E$a!pe


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2-Laplace Transform Review

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