[PPT] - Topic # 3 Second-order Systems Reference textbook : Control PowerPoint Presentation

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ME 779 Control Systems

Second-order Systems

Topic # 3

Reference textbook:

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

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Control Systems: Second-order Systems

Learning Objectives

Differential equations
Normalized form of differential equations
System transfer function
Pole-zero map
Overdamping
Critical damping
Underdamping
Undamped
Impulse, step, sinusoidal repsonse
Frequency response: magnitude and phase

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Differential equation

) (t K x k y y c y m      

y(t) response Mass, m (kg) Damping coefficient, c (N-s/m) Stiffness, k (N/m) K static sensitivity x(t) input

Control Systems: Second-order Systems

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m t Kx y m k y m c y ) (      

By dividing throughout by m

n

k m  

Undamped natural frequency, rad/s

2 c km  

Damping factor 2

( ) 2

n n

Kx t y y y m     

Control Systems: Second-order Systems Differential equation: Normalized form

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System transfer function

 

2 2

( ) ( ) 2

n n

Y s K X s m s s     

Control Systems: Second-order Systems

2

( ) 2

n n

Kx t y y y m     

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Control Systems: Second-order Systems

Classification of damping factors

Damping factor Type Property Overdamped Exponential decay Critically damped Exponential decay Underdamped Oscillatory decay Undamped Oscillatory

1   1   1    

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Pole-zero map

 

2 1,2

1

n

s       

ζ >1 overdamped

Poles

Control Systems: Second-order Systems

 

2 2

( ) ( ) 2

n n

Y s K X s m s s     

 

2 2 1,2

2 2 4 2

n n n

s       

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Pole-zero map ζ =1 critically damped

1,2 n

s   

Poles

Control Systems: Second-order Systems

 

2 2

( ) ( ) 2

n n

Y s K X s m s s     

 

2 2 1,2

2 2 4 2

n n n

s       

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Pole-zero map ζ <1 underdamped

1,2 n d

s j     

2

1

d n

    

Damped natural frequency 2

tan 1     

Control Systems: Second-order Systems

 

2 2

( ) ( ) 2

n n

Y s K X s m s s     

 

2 2 1,2

2 4 2 2

n n n

j s       

Poles

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Control Systems: Second-order Systems

Pole-zero map ζ =0 undamped

 

2 2

( ) ( )

n

Y s K X s m s   

1,2 n

s j  

Poles

s-plane

σ

jω

x x

n

j

n

j 

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Impulse response

2

2 ( )

n n i

K y y y x t m      

Differential equation

2 2 2 2 2

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

i n n i n n n n n

Kx Y s m s s Kx m s s                                      

Laplacian of the output

Overdamped case (ζ>1)

Control Systems: Second-order Systems

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 

2 2

( ) sinh 1 1

nt

i n n

Kx y t e t m



   



          

Time-domain response

Control Systems: Second-order Systems

2 2 2

1 1 ( ) 2 1 ( 1) ( 1

i n n n n n

Kx Y s m s s                           

Impulse response Overdamped case (ζ>1)

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( )

nt

i n n

Kx y t te m



 



      

Control Systems: Second-order Systems

Impulse response Critically damped case (ζ=1)

 

2 2 2

1 1 ( ) 2

i i n n n

Kx Kx Y s m s s m s                      

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12 n d

s j     

Poles

1 ( ) ( )( )

i n d n d

Kx Y s m s j s j               

( ) sin

nt

i d d

Kx y t e t m



 



      

Laplace

utput

Time-domain

utput

Control Systems: Second-order Systems

Impulse response Undamped case (ζ<1)

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Impulse response function

( ) sin

nt

d d

K h t e t m



 





 

( ) sin ( )

n

t d d

K y t e F t d m

 

    



 



Duhamel’s integral

Control Systems: Second-order Systems

Impulse response Undamped case (ζ<1)

( ) sin

nt

i d d

Kx y t e t m



 



      

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Control Systems: Second-order Systems

Impulse response

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  

2 2

1 1 ( ) 1) 1

i n n n n

Kx Y s m s s s                           

Laplace of the

utput

   

2 2 2 2

( ) 1 cosh 1 sinh 1 1

nt

i n n n

Kx y t e t t m



      



                       

Time-domain output

Control Systems: Second-order Systems

Step response Overdamped case (ζ>1)

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1 1 ( ) ( )( )

i n d n d

Kx Y s m s s j s j                    

Laplace of the

utput

2 2

( ) 1 cos sin 1

nt

i d d n

Kx y t e t t m



    



                     

Time-domain of the output

Control Systems: Second-order Systems

Step response Underdamped case (ζ<1)

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Control Systems: Second-order Systems

Step response Underdamped case (ζ<1)

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2 2

( ) 1 sin( ) 1

nt

i d n

Kx e y t t m



   



             

2 1

1 tan   



 

Control Systems: Second-order Systems

Step response Underdamped case (ζ<1)

s-plane

σ

jω x x

2

tan 1     





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2 2

( ) 1 1 sin( ) 1

n r

t r n d r i

y t m e t Kx



   



    

r d

t     

Rise time

Control Systems: Second-order Systems Control Systems: Second-order Systems

Step response Underdamped case (ζ<1)

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d p

t   

Peak time

2

1 2

( ) 1

i p n

Kx y t e m

  



 

           

Peak response

Control Systems: Second-order Systems Control Systems: Second-order Systems

Step response Underdamped case (ζ<1)

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Percentage overshoot Mp

2

1

( ) %

p p

y t y M e y

      

  

Control Systems: Second-order Systems Control Systems: Second-order Systems

Step response Underdamped case (ζ<1)

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Settling time

4

s n

t  

3

s n

t  

2% criterion 5% criterion

Control Systems: Second-order Systems Control Systems: Second-order Systems

Step response Underdamped case (ζ<1)

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Second-order systems

Control Systems: Second-order Systems

Step response Comparison of damping factors

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Sinusoidal response

2

2 sin

n n

K y y y A t m      

Differential equation

2 2 2 2

1 ( ) 2

n n

KA Y s m s s s                  

Laplacian of the

utput

Control Systems: Second-order Systems

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 

2 2

(j ) (j ) 2

n n

Y KA X m j        

Putting s=jω

2 2

( ) 1 ( ) (1 2 )

n

Y j m X j KA r jr       

n

r   

Frequency ratio

Frequency response function

Control Systems: Second-order Systems Sinusoidal response

2 2

( ) 1 ( ) 2

n n

Y s KA X s m s s           

ω forcing frequency

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 

2 2 2 2

( ) 1 20log dB ( ) 1 2

n

Y j m M X j K r r        

1 2

2 tan 1 r r  

 

       

Magnitude

Phase

Control Systems: Second-order Systems Sinusoidal response

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Magnitude Control Systems: Second-order Systems Sinusoidal response

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Phase

Control Systems: Second-order Systems Sinusoidal response