Dept. of Agricultural & Biological EngineeringUniversity of Illinois

ABE 463 Electro-hydraulic systems

Laplace transform

Tony Grift

0

t

st

t

L f t f t e dt F s

Pierre-Simon Laplace “The French Newton” (1749-1827)

Why do we need a Laplace Transform?Definition Laplace TransformLaplace Transform of functions

Unit step functionRamp functionExponential functionCosine/SineImpulse function (dirac delta)

Laplace Transform of operationsConvolution

The Laplace transform can be used to transform a differential equation into an algebraic equation that can be solved. After transforming back to the time domain we obtain a solution of the differential equation in time.

0

stL f t f t e dt

Time domain: Differential equation

s-domain: algebraic equation Solution in s-domain

Inverse Laplace Transform

Solution in time domain

The Laplace transform is a linear operation

000

dtetgdtetfdtetgtftgtfL ststst

sGsFtgtfL

0 0

st stL Kf t Kf t e dt K f t e dt KF s

sKFtKfL

Red frame: Important result

Laplace transform of unit step function

ss

es

dtetL stst 1101110

0

s

tL 11

0

stL f t f t e dt F s

Definition Laplace Transform

The variable s is a constant under integration with respect to t

Laplace transform of a ramp function

22020

00

0

110110

11

sse

sdte

s

dtes

es

tdtetttfL

st

u

st

u

st

u

st

vu

st

v

Integration by parts

2

1s

ttfL

Blue frame: You should know this already

vuvuuv

vuvuuv

vuuvvu

Laplace transform of an exponential function

0 0

0

1 1 10

a s tat at st

a s t

L e e e dt e dt

ea s a s s a

1atL es a

Laplace transform of cosine function

22cosas

satL

cos Re e jatat

e cos sinjat at j at

0

0 0

2 2 2 2

1 1cos Re 0

1 Re

ja s t ja s tjat stL at e e dt e dt ea s ja s

s ja s ja ss ja s ja s a s a

Laplace transform of sine function

2 2sin aL ats a

sin Im e jatat

e cos sinjat at j at

0

0 0

2 2 2 2

1 1sin Im 0

1 Im

ja s t ja s tjat stL at e e dt e dt ea s ja s

s ja s ja as ja s ja s a s a

Laplace transform of an impulse ‘function’ (Dirac delta distribution)

0

0

lim 1 10

1lim0

st st

s

L t e dt es

e

s

...!3

1!2

11 32 tohxxxe x

Writing as a McLaurin series xe

...!3

1!2

11 32 tohssse s

Writing as a McLaurin series se

0

1/

Laplace transform of impulse (Dirac delta distribution)

...!3

1!2

11 32 tohssse s

...!3

1!2

111 2 tohsss

e s

2lim lim1 1 11 . . . 10 0 2! 3!

seL t s s h o ts

1L t

Check Laplace Transform of differentiation operation

022 0sincos

fsF

bsbsbtbLtfL

0fssFtfL

btbtfbttf cossin

2222 coscosbs

sbbtbLbs

sbtL

Example

Is this correct?

Laplace transform of operations

Laplace transform of differentiation operation

0000

00

0

fssFdtetfsf

dtestfetfdtetfdt

tdfL

st

stst

v

st

u

vuvuuv

vuvuuv

vuuvvu

Product rule:

Integration by parts

0fssFtfL

Laplace Transform of a function shifted in time

0

0

st

s t aas as

L f t a f t a e dt

e f t a e d t a e F s

:Note d t a dt

asL f t a e F s

Laplace transforms of common functions and operations

Initial value and final value theorems

Final value theorem proof (simplified)

lim lim0

f t sF st s

0 00

0st st st

v vu u u v

L f t f t e dt f t e f t s e dt f sF s

0 0

1

lim lim lim0 0

0 0 0

lim0

st stdf dfsF s f e dt e dt f t fs s sdt dt

sF s fs

Convolution

Convolution example: Moving average filter

1st iteration

2nd iteration

General

Continuous case

3322111 31 gfgfgfh

4332212 31 gfgfgfh

3

11 3

1n

knnk gfh

0

h t K f x g t x dx

Convolve this vector

1 -2 3

7 2 5 6 8 9 5 1 1 2 8 2 7 2 8 3

21 -8 18 10 17

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

3

11

k n n kn

h f g

MatLab: conv(a,[3 -2 1])

Correct answer 1 -2 3

7 2 5 6 8 9 5 1 1 2 8 2 7 2 8 3

21 -8 18 10 17 17 5 2 6 5 21 -8 25 -6 27 -5 2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

1 -2 3

3

11

k n n kn

h f g

Convolution in time domain = Multiplication in Laplace domain

0 0

0 0 0 0

0 0

0 0

st

st st

s v u

su sv

L h t H s f u g t u du e dt

f u g t u du e dt f u du g t u e dt

v t udv dt

H s f u d g v e dv

f u e du g v e dv

F s G s

The time equivalent of multiplication in the Laplace (and also Fourier) domain is called convolution

g t

G s

F s

f t

H s G s F s

Time domain

s-domain

0

*u t

u

h t g t f t f u g t u du

Impulse response of the systemat t - u

The total response is in fact the sum of all impulse responses over time weighted (multiplied) by the input signal

Dept. of Agricultural & Biological EngineeringUniversity of Illinois

ABE 463 Electro-hydraulic systems

Laplace transform

The End