ME 647 System Dynamics #1

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ME 779 Control Systems

Laplace transform

Topic # 1

Reference textbook:

Control Systems, Dhanesh N. Manik,Cengage Publishing, 2012

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Control Systems: Laplace transform

Learning Objectives

• Laplace transform of typical time-domain functions• Partial fraction expansion of Laplace transform functions• Final value theorem• Initial value theorem• System transfer function• General transfer function: poles and zeros, block diagram• Force response• Types of excitations• Impulse response function

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System dynamics is the study of characteristic

behaviour of dynamic systems

First-order systems

Second-order systems} Differential equations

Laplace transforms convert differential equations

into algebraic equations

System transfer function can be defined

Transient response can be obtained

They can be related to frequency response


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ℒ{x(t)}=X(s)=

0

( ) stx t e dt


Basic definition

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No. Function Time-domain x(t)= ℒ-1

{X(s)} Laplace domain X(s)= ℒ{x(t)}

1 Delay δ(t-τ) e-τs

2 Unit impulse δ(t) 1

3 Unit step u(t)

s

1

4 Ramp t 2

1

s

5 Exponential decay e-αt

s

1

6 Exponential approach

te 1 )(

ss


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7 Sine sin ωt 22

s

8 Cosine cos ωt 22 s

s

9 Hyperbolic sine sinh αt 22

s

10 Hyperbolic cosine cosh αt 22 s

s

11 Exponentially

decaying sine

wave

te t sin 22)(

s

12 Exponentially

decaying cosine

wave

te t cos 2 2( )

s

s


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Partial fraction expansion of Laplace transform functions

• Unrepeated factors

• Repeated factors

• Unrepeated complex factors

Factors of the denominator


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Unrepeated factors

By equating both sides, determine A and B


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Example

2( )

( 1)( 2)

sY s

s s

Expand the following equation of Laplace

transform in terms of its partial fractions

and obtain its time-domain response.

2

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

s A B

s s s s

2 4

( )( 1) ( 2)

Y ss s

2( ) 2 4t ty t e e


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Repeated factors

2 2 2

( ) ( )

( ) ( ) ( ) ( )

N s A B A B s a

s a s a s a s a


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EXAMPLE

Expand the following Laplace transform in terms of its

partial fraction and obtain its time-domain response 2

2( )

( 1) ( 2)

sY s

s s

2 2

2

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

s A B C

s s s s s

2

2 4 4( )

( 1) ( 1) ( 2)Y s

s s s

2( ) 2 4 4t t ty t te e e


}

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Complex factors: they contain conjugate pairs

in the denominator

2 2

( )

( )( ) ( )

N s As B

s a s a s


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EXAMPLE

Express the following Laplace transform in

terms of its partial fractions and obtain its

time-domain response.

2 1( )

( 1 )( 1 )

sY s

s j s j

2 2

2 1( )

( 1) 1 ( 1) 1

sY s

s s

( ) 2 c o s s int ty t e t e t


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Final-value theorem

0

( ) lim ( )limt

s

y t sY s

EXAMPLE

2

2( )

( 1) ( 2)

sY s

s s

Determine the final value of the time-domain

function represented by 2( ) 2 4 4t t ty t te e e


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2 1( )

( 1 )( 1 )

sY s

s j s j

( ) 2 cos sint ty t e t e t

Initial-value theorem

0

( ) lim ( )limt

s

y t sY s

EXAMPLE

Determine the initial value of the time-domain

response of the following equation using

the initial-value theorem

(2 1)2

( 1 )( 1 )lim

s

s s

s j s j


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SYSTEM TRANSFER FUNCTION

Block diagram

System transfer function is the

ratio of output to input in the

Laplace domain


( )( )

( )

Y sG s

X s

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)(

)(

)())((

)())(()(

1

1

21

21

j

n

j

i

m

i

n

m

ps

zs

Kpspsps

zszszsKsG

1 2, ... mz z z

General System Transfer Function

are called zeros

1 2, ... np p p are called the poles

Number of poles n will always be greater than the number of zeros m



K is a constant

(Laplace transform is a rational polynomial)

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EXAMPLE

Obtain the pole-zero map of the following transfer function

)51)(51)(5)(4)(3(

)42)(42)(2()(

jsjssss

jsjsssG

Zeros Poles

s=2 s=3

s=-2-j4 s=4

s=-2+j4 s=5

s=-1-j5

s=-1+j5

(1)



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Zeros Poles

EXAMPLE

Zeros Poles

s=2 s=3

s=-2-j4 s=4

s=-2+j4 s=5

s=-1-j5

s=-1+j5



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Forced response

)()())((

)())(()()()(

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21 sRpspsps

zszszsKsRsGsC

n

m

R(s) input excitation


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TYPES OF EXCITATIONS

1. Impulse

2. Step

3. Ramp

4. Sinusoidal


Forced response

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Impulse input

xi

)()( atxtx i

Dirac delta

function


Forced response

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( ) ( ) ( )o ot t t dt t

Laplace transform of an impulse input

sa

ii

st exdtatxesX

)()(0

Integral property of Dirac

delta function


Forced response

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Step input

0

( ) st ii

xX s e x dt

s

Laplace transform of step input


Forced response

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Example

The following transfer function is subjected to a unit step input.

Determine the response

( 1)( )

( 4)

sG s

s

1

1 1

( )( ) ( ) ( )

( )

s z A BC s R s G s

s s p s s p

p1=-4, z1=-1

1 41 1

1 1

1 3( ) 1

4 4

p t tz zc t e e

p p

1( )R s

s


Forced response

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0.25

Step

response

Example


Forced response

41 3( )

4 4

tc t e

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Ramp input

450

2

0

1( ) stX s e tdt

s

Laplace transform of the ramp input


Forced response

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Sinusoidal input

2 2

0

( ) sinstX s e t dts


Forced response

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IMPULSE RESPONSE FUNCTION

Time-domain response of a system subjected to unit impulse excitation

h(t)= ℒ-1 {G(s))}

It is the inverse Laplace transform of the system transfer function


Forced response

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Convolution Integral

Each infinitesimal strip of force

defines an impulse response

function

dFF )(^

dthFy )(^^

Response due to each

strip of the force

t

dthFty0

)()()( Total response due to

entire force history


Forced response

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ME 647 System Dynamics #1