Topic # 2 First-order Systems Reference textbook : Control Systems, - - PowerPoint PPT Presentation

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ME 779 Control Systems

First-order Systems

Topic # 2

Reference textbook:

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

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Control Systems: First-order Systems

Learning Objectives

• Differential equation
• System transfer function
• Pole-zero map
• Normalized reponse
• Impulse response
• Step response
• Ramp response
• Sinusoidal response
• Frequency response

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DIFFERENTIAL EQUATIONS

) (t K x y y    

Governing differential equation



Time constant, sec K Static sensitivity (units depend on the input and output variables) y(t) Response of the system x(t) Input excitation

Control Systems: First-order Systems

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System transfer function

( ) ( ) ( ) (1 ) Y s K G s X s s    

(1 ) K s  

Block diagram representation

• f transfer function

Control Systems: First-order Systems

) (t K x y y    

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Pole-zero map

Control Systems: First-order Systems

( ) (1 ) K G s s   

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Normalized response

• Static components are taken out leaving only

the dynamic component

• The dynamic components converge to the same

value for different physical systems of the same type or order

• Helps in recognizing typical factors of a system

Control Systems: First-order Systems

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Impulse response

) (t K x y y

i

   

1 ( ) 1 (1 )

i i

Kx Kx Y s s s                 





t i e

Kx t y



 ) (

Governing differential equation Laplacian of the response Time-domain response

Control Systems: First-order Systems

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Impulse response function

( )

t

K h t e  





By putting xi=1 in the impulse response

( ) ( )

t t

K y t e F t d



  



 



Response to any force excitation

Control Systems: First-order Systems

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( )

i

y t Kx 

/ t 

Normalized impulse response

Control Systems: First-order Systems

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Step response

( )

i

y y K x u t   

Step input of level xi

( ) 1 (1 )

i i i

Kx Kx Kx Y s s s s s       

Output in the Laplace domain

         

  t i

e Kx t y 1 ) (

Output in the time-domain

Control Systems: First-order Systems

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( )

i

y t Kx

/ t 

Control Systems: First-order Systems

Normalized step response

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Ramp response

K t y y    

x(t)=t (input)

2 2

1 ( ) 1 (1 ) K Y s s s s s s          

Laplace of the output



 

t

e t K t y



   ) (

Time-domain response

ss

y t K   

Steady-state response

Control Systems: First-order Systems

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( ) y t K

/ t 

Control Systems: First-order Systems Normalized ramp response



 

t

e t K t y



   ) (

( ) x t ( ) y t K

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Sinusoidal response

t KA y y   sin   

Sinusoidal excitation of amplitude A and frequency ω

 

2 2 2 2 2 2 2 2

1 ( ) (1 ) 1/ 1 K A s Y s s s s s s                                

Output in the Laplace domain

 

2 / 2

( ) 1 cos sin 1

t

y t e t t KA



      



         

Time-domain

• utput

Control Systems: First-order Systems

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Control Systems: First-order Systems

Sinusoidal response

 

2 / 2

( ) 1 cos sin 1

t

y t e t t KA



      



         

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2

( ) 1 sin( ) 1 ( )

ss

y t t KA      

Normalized steady-state response

1

tan  





Phase angle between input and output

Control Systems: First-order Systems

 

2

( ) 1 cos sin 1

ss

y t t t KA                

Sinusoidal response

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Steady-state response

Output frequency same as Input frequency

Control Systems: First-order Systems

Sinusoidal response

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Steady-state sinusoidal response from system transfer function

( ) 1 ( ) 1 Y s X s KA s   

Normalized system transfer function

• f a first-order system

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( ) 1 ( ) 1 ( ) Y j X j KA     

By replacing s=jω Magnitude

1

( ) tan ( ) ( ) Y j X j KA   



  

Angle

Control Systems: First-order Systems

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Frequency response (magnitude) in decibels

10 2

1 20log dB 1 ( )  

Control Systems: First-order Systems

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Frequency response: phase

1

tan ( )  





Control Systems: First-order Systems