Chap04 - Laplace Transforms.ppt

8/14/2019 Chap04 - Laplace Transforms.ppt

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Chapter 4

Laplace Transforms


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Overall Course Objectives

Develop the skills necessary to function as anindustrial process control engineer.

Skills

Tuning loops

Control loop design

Control loop troubleshooting

Command of the terminology

Fundamental understanding

Process dynamics

Feedback control


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Laplace Transforms

Provide valuable insight into process

dynamics and the dynamics of feedback

systems.

Provide a major portion of the terminology

of the process control profession.


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Laplace Transforms

0

)()()( sFdtetftf stL

Useful for solving linear differential equations.

Approach is to apply Laplace transform to

differential equation. Then algebraically solve for

Y(s). Finally, apply inverse Laplace transform to

directly determine y(t).

Tables of Laplace transforms are available.


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Method for Solving Linear

ODEs using Laplace Transforms

Laplace Domain

Time Domain

dy/dt = f(t,y)

sY(s) - y(0) =

F(s,Y)Y(s) = H(s)

y(t) = h(t)


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Some Commonly Used Laplace

Transforms

2222

2

2

2

1

)(

)sin()sin(

)0()0()()(1

)0()()(!

)()(/1

as

te

s

t

ffssFsdt

tfd

ase

fsFsdt

tfd

s

nt

esFtfsStepUnit

at

at

n

n

s


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Initial-Value Theorem

)(lim)(lim 0 sFstf st

Allows one to use the

Laplace transform of afunction to determine

the initial conditions

of the function.

A good consistency

check


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Apply Initial- and Final-Value

Theorems to this Example

Laplace

transform of the

function.

Apply final-value

theorem

Apply initial-

value theorem

)4()2(

2)(

ssssY

4

1

)40()20()0(

)0(2)(lim

tft

0)4()2()(

)(2)(lim 0

tft


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Partial Fraction Expansions

32)3()2(

1

s

B

s

A

ss

s Expand into a term for

each factor in the

denominator.

Recombine RHS

Equate terms in s and

constant terms. Solve. Each term is in a form so

that inverse Laplace

transforms can be applied.

)3()2(2)3(

)3()2(

1

ss

sBsA

ss

s

3

2

2

1

)3()2(

1

ssss

s

1BA 123 BA


11/16

Heaviside Method

Individual Poles

1 2

1 2

1 2

( ) ( )( )

( ) ( )( ) ( )

( )Equation 4.4.2 :( ) /( )

i

n

n

n

i

i s a

N s N sY s

D s s a s a s a

CC C

s a s a s a

N sCD s s a


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Heaviside Method

Individual Poles

1 2

1 21 2

3( )

( 1)( 2) 1 2

( )Equation 4.4.2 :( ) /( )

3 3

2; 12 1

3 2 1( )

( 1)( 2) 1 2

i

i

i s a

s s

s C CY s

s s s s

N sCD s s a

s s

C Cs s

sY s

s s s s


13/16

Heaviside Method

Repeated Poles

1 2

2

( ) ( )( )

( ) ( )

( ) ( )

1 ( ) ( )Equation 4.4.2 :( )

i

n

n

n

i n

i i

s a

N s N sY s

D s s a

CC C

s a s a s a

d s a N sCi ds D s


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Heaviside Method

Example with Repeated Poles

31 22 2

2 3 2

1 2

1 2

1 1

2

( ) 3( )( ) ( 1) ( 2) 1 ( 1) 2

3 32; 1

2 ( 1)

1 ( ) ( )Equation 4.5.3:

( )

1 3 1 31

1 2 2 ( 2)

3( )

( 1) (

i

s s

i n

i i

s a

s s

CN s s C CY sD s s s s s s

s sC C

s s

d s a N sC

i ds D s

d s sC

ds s s s

sY s

s

2

1 2 1

2) 1 ( 1) 2s s s s


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Example of Solution of an ODE

0)0(')0(2862

2

yyydt

dy

dt

yd ODE w/initial conditions

Apply Laplace transform

to each term

Solve for Y(s)

Apply partial fractionexpansions w/Heaviside

Apply inverse Laplace

transform to each term

ssYsYssYs /2)(8)(6)(2

)4()2(

2)(

ssssY

)4(4

1

)2(2

1

4

1)(

ssssY

424

1)(

42 tt eety


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Overview

Laplace transforms are an effective way tosolve linear ODEs.

Documents

Chap04 - Laplace Transforms.ppt