Topic # 1 Laplace transform Reference textbook : Control Systems, - - PowerPoint PPT Presentation

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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

Control Systems: Laplace transform

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ℒ{x(t)}=X(s)=

( )

st

x t e dt

 



Control Systems: Laplace transform 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

 

t

e 



 1

) (    s s

Control Systems: Laplace transform

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7 Sine sin ωt

2 2

   s

8 Cosine cos ωt

2 2

  s s

9 Hyperbolic sine sinh αt

2 2

   s

10 Hyperbolic cosine cosh αt

2 2

  s s

11 Exponentially decaying sine wave

t e

t



 sin 

2 2

) (      s

12 Exponentially decaying cosine wave

t e

t



 cos 

2 2

( ) s s      

Control Systems: Laplace transform

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Partial fraction expansion of Laplace transform functions

• Unrepeated factors
• Repeated factors
• Unrepeated complex factors

Factors of the denominator

Control Systems: Laplace transform

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Unrepeated factors

By equating both sides, determine A and B

Control Systems: Laplace transform

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Example

2 ( ) ( 1)( 2) s Y 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 s s s     

2

( ) 2 4

t t

y t e e

 

  

Control Systems: Laplace transform

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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         

Control Systems: Laplace transform

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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) s Y 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 4

t t t

y t te e e

  

   

Control Systems: Laplace transform

}

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Complex factors: they contain conjugate pairs in the denominator

2 2

( ) ( )( ) ( ) N s As B s a s a s        

Control Systems: Laplace transform

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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 ) s Y s s j s j      

2 2

2 1 ( ) ( 1) 1 ( 1) 1 s Y s s s      

( ) 2 cos sin

t t

y t e t e t

 

 

Control Systems: Laplace transform

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Final-value theorem  

 

( )

lim ( ) lim

t s

y t

sY s

 



EXAMPLE

2

2 ( ) ( 1) ( 2) s Y s s s   

Determine the final value of the time-domain function represented by

2

( ) 2 4 4

t t t

y t te e e

  

       

Control Systems: Laplace transform

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2 1 ( ) ( 1 )( 1 ) s Y s s j s j      

( ) 2 cos sin

t t

y t e t e t

 

      Initial-value theorem

 

 

( )

lim ( ) lim

t 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



     

Control Systems: Laplace transform

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SYSTEM TRANSFER FUNCTION

Block diagram

System transfer function is the ratio of output to input in the Laplace domain

Control Systems: Laplace transform

( ) ( ) ( ) Y s G s X s 

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

1 1 2 1 2 1 j n j i m i n m

p s z s K p s p s p s z s z s z s K s G          

 

 

 

1 2

, ...

m

z z z

General System Transfer Function

are called zeros

1 2

, ...

n

p p p

are called the poles

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

Control Systems: Laplace transform

SYSTEM TRANSFER FUNCTION 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

) 5 1 )( 5 1 )( 5 )( 4 )( 3 ( ) 4 2 )( 4 2 )( 2 ( ) ( j s j s s s s j s j s s s G             

Zeros Poles s=2 s=3 s=-2-j4 s=4 s=-2+j4 s=5 s=-1-j5 s=-1+j5

(1)

Control Systems: Laplace transform

SYSTEM TRANSFER FUNCTION

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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

Control Systems: Laplace transform

SYSTEM TRANSFER FUNCTION

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Forced response

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

2 1 2 1

s R p s p s p s z s z s z s K s R s G s C

n m

         

R(s) input excitation

Control Systems: Laplace transform

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TYPES OF EXCITATIONS

• 1. Impulse
• 2. Step
• 3. Ramp
• 4. Sinusoidal

Control Systems: Laplace transform

Forced response

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Impulse input

xi

) ( ) ( a t x t x

i

  

Dirac delta function

Control Systems: Laplace transform

Forced response



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( ) ( ) ( )

• t

t t dt t   

 

 



Laplace transform of an impulse input

sa i i st

e x dt a t x e s X

  

    ) ( ) ( 

Integral property of Dirac delta function

Control Systems: Laplace transform

Forced response

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Step input

( )

st i i

x X s e x dt s

 

 



Laplace transform of step input

Control Systems: Laplace transform

Forced response

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Example

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

( 1) ( ) ( 4) s G s s   

1 1 1

( ) ( ) ( ) ( ) ( ) s z A B C s R s G s s s p s s p       

p1=-4, z1=-1

1

4 1 1 1 1

1 3 ( ) 1 4 4

p t t

z z c t e e p p



          

1 ( ) R s s 

Control Systems: Laplace transform

Forced response

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0.25

Step response Example

Control Systems: Laplace transform

Forced response

4

1 3 ( ) 4 4

t

c t e



 

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Ramp input

450

2

1 ( )

st

X s e tdt s

 

 



Laplace transform of the ramp input

Control Systems: Laplace transform

Forced response

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Sinusoidal input

2 2

( ) sin

st

X s e t dt s   

 

  



Control Systems: Laplace transform

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

Control Systems: Laplace transform

Forced response

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Convolution Integral Each infinitesimal strip of force defines an impulse response function   d F

F

) (

^



  d t h F y ) (

^ ^

 

Response due to each strip of the force



 

t

d t h F t y ) ( ) ( ) (   

Total response due to entire force history

Control Systems: Laplace transform

Forced response