Topic #29 Nyquist plots: Closed-loop response Reference textbook : - - PowerPoint PPT Presentation

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ME 779 Control Systems

Nyquist plots: Closed-loop response

Topic #29

Reference textbook:

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

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CLOSED LOOP FREQUENCY RESPONSE

( ) ( ) ( ) 1 ( ) ( ) C j G j R j G j H j       

Nyquist plots: Closed-loop response

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( ) 20log dB ( )

r

C j M R j   

Peak Magnitude 3 dB is considered good

Nyquist plots: Closed-loop response

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Constant M-circles for unity feedback systems

) ( 1 ) ( ) (    j G j G j M  

jy x j G   ) ( 

2 2 2 2

( ) (1 ) x y M j x y     

Nyquist plots: Closed-loop response

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2 2 2 2 2 2

) 1 ( y x y M x M    

2 2 2 2 2 2

2 ) 1 ( ) 1 ( M x M y M M x     

2 2 2 2 2 2

1 1 2 M M x M M y x     

Nyquist plots: Closed-loop response

Constant M-circles for unity feedback systems

6 2 2 2 2 2 2

1 1                    M M y M M x

1          M M

Adding

.

The above equation represents a family

• f circles with its center at

         , 1

2 2

M M

and radius

2

1 M M 

.

Nyquist plots: Closed-loop response

Constant M-circles for unity feedback systems

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Family of M-circles corresponding to the close loop magnitudes (M) of a unit feedback system

Nyquist plots: Closed-loop response

Constant M-circles for unity feedback systems

M>1

M<1

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Constant N-circles

) ( 1 ) (    j G j G M      

x y x y   

 

1 tan tan

1 1



Nyquist plots: Closed-loop response

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        

 

x y x y N 1 tan tan tan

1 1

tan()=N

B A B A B A tan tan 1 tan tan ) tan(    

Nyquist plots: Closed-loop response

Constant N-circles

A B

1 1 1 y y x x N y y x x              

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2 2

y x x y N   

2 2 2

2 1 4 1 2 1 2 1                        N N y x

Nyquist plots: Closed-loop response

Constant N-circles

2 2

y x x y N    

2

1 1 4 4N 

Add

• n both sides

2 2 2 2

1 1 1 1 4 4 4 4 y x x y N N N       

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Nyquist plots: Closed-loop response

Constant N-circles

2 2 2

2 1 4 1 2 1 2 1                        N N y x

The above equation represents a family

• f circles with its center at

1 1 , 2 2N       

and radius

2

2 1 4 1        N

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Nyquist plots: Closed-loop response

Constant N-circles

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10 ( ) ( 2)( 4) G s s s s   

Example

Nyquist plots: Closed-loop response

Determine the closed-loop magnitude ratio and bandwidth of the feedback system whose forward transfer function is given by

10 ( ) ( 2)( 4) G s s s s   

and H(s)=1, by (1) direct computation and (2) using M and N circles

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Nyquist plots: Closed-loop response

Example

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Nyquist plots: Closed-loop response

Example

16 Freq., rad/s Closed-loop Magnitude ratio Closed- loop Phase angle, deg 0.1 1.0 355 0.5 1.1 335 0.8 1.2 316 0.9 1.2 308 1.0 1.2 300 1.1 1.3 290

Closed-loop magnitude and phase

Nyquist plots: Closed-loop response

Freq., rad/s Closed-loop Magnitude ratio Closed- loop Phase angle, deg 1.2 1.3 280 1.3 1.2 269 1.4 1.2 258 1.5 1.1 248 1.6 1.0 238 1.7 0.9 230 1.8 0.8 222 1.9 0.7 216 2.0 0.6 210

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Nyquist plots: Closed-loop response

Example

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Nyquist plots: Closed-loop response

Example

Nyquist plots: Closed-loop response

Conclusion

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