Topic #10 Higher-order Systems Reference textbook : Control - - PowerPoint PPT Presentation

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ME 779 Control Systems

Higher-order Systems

Topic #10

Reference textbook:

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

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Learning Objectives

• General closed-loop response function
• Pole-zero map
• overdamped closed-loop poles
• critically damped closed-loop poles
• undamped closed-loop poles
• negatively underdamped closed-loop poles
• negatively overdamped closed-loop poles
• Transient response using residues
• Example

Control Systems: Higher-order Systems

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) ( ) ( 1 ) ( ) ( ) ( s H s G s G s R s C  

General closed-loop response function

1 2 1 2

( )( )...( ) ( ) ( ) ( )( )...( )

m n

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

Open-loop transfer function

1 2 1 2 1 2

( )( )( )...( ) ( ) ( ) ( )( )...( ) ( )( )...( )

n n m

G s s p s p s p C s R s s p s p s p K s z s z s z           

Control Systems: Higher-order Systems

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Pole-zero map

Pole-zero map Normalized response

(a) (b)

Overdamped closed-loop poles

Control Systems: Higher-order Systems

5 Pole-zero map Normalized response

(a) (b)

Pole-zero map Critically damped closed-loop poles

Control Systems: Higher-order Systems

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Normalized response Pole-zero map

(b) (a)

Underdamped closed-loop poles Pole-zero map

Control Systems: Higher-order Systems

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Pole-zero map Normalized response

(a) (b)

Undamped closed-loop poles Pole-zero map

Control Systems: Higher-order Systems

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Pole-zero map Normalized response

(a) (b)

Negatively underdamped closed-loop poles Pole-zero map

Control Systems: Higher-order Systems

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Pole-zero map

Negatively overdamped closed-loop poles Pole-zero map

Control Systems: Higher-order Systems

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Transient response using residues

1 2 1 2 1 2

( ) ( )( )( )...( ) ( ) ( )( )...( ) ( )( )...( )

n n m

R s G s s p s p s p C s s p s p s p K s z s z s z           

1

( )

n i c i i

a b C s s s p



  



Step response

Control Systems: Higher-order Systems

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( )

n i i

c p t i c t b a e



   

1 2 1 2 1 2

( )( )( )...( ) ( ) ( )( )...( ) ( )( )...( )

c n i i n m

G s s p s p s p a s p s s p s p s p K s z s z s z c s pi              

 

1 2 1 2 1 2

( )( )( )...( ) ( )( )...( ) ( )( )...( )

n n m

G s s p s p s p b s s s p s p s p K s z s z s z s             

Transient response using residues

Control Systems: Higher-order Systems

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1

( ) ( 5) K G s s  

Example

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( ) ( 1) K H s s  

K1=1 K2 =5

1 2

( ) ( ) ( 5)( 1) ( 5)( 1) K K K G s H s s s s s      

1 1 2

( 1) ( ) ( ) ( 5)( 1) K s C s R s s s K K     

2

( ) ( 1) 1 ( ) ( 5)( 1) 5 6 10 C s s s R s s s s s         

1

3

c

p j   

2

3

c

p j   

Control Systems: Higher-order Systems

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Example

 

2

1 ( ) 6 10 s C s s s s    

Unit step input

 

2

( 1) 0.1 6 10 s b s s s s s       

 

1 2

( 1) ( 2 ) ( 3 ) 0.05 0.35 ( 3 )(2 ) 6 10 3 s j a s j j j j s s s s j                  

 

2 2

( 1) ( 2 ) ( 3 ) 0.05 0.35 ( 3 )( 2 ) 6 10 3 s j a s j j j j s s s s j                   

Control Systems: Higher-order Systems

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0.1 ( 0.05 0.35 ) ( 0.05 0.35 ) ( ) 3 3 j j C s s s j s j           

Example

3

( ) 0.1 ( 0.1cos 0.7sin )

t

c t e t t



   

Control Systems: Higher-order Systems

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1 1 2 1 2 1

1 ( 5) ( ) 5 9 1 5 1 ( 5) K s G s K K K s s s s s                   

1

1 lim ( ) 9

p s

K G s



 

1 1 ( ) 0.9 1 1 1 9

e p

s K      

Example

Control Systems: Higher-order Systems

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( ) 0.1 ( 0.1cos 0.7sin )

t

c t e t t



   