Topic #26 Nyquist Stability Theory Reference textbook : Control - - PowerPoint PPT Presentation

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ME 779 Control Systems

Nyquist Stability Theory

Topic #26

Reference textbook:

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

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

Feedback transfer function

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

2 1 2 1 n m

p s p s p s z s z s z s K s H s G        poles and zeros of the

• pen-loop transfer function

Control Systems: Nyquist Stability Theory

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Characteristic equation

) )...( )( ( ) )...( )( ( ) )...( )( ( ) ( ) ( 1

2 1 2 1 2 1 n m n

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

the number of closed-loop poles - that is zeros of 1+GH will be equal to the number of open-loop poles

Control Systems: Nyquist Stability Theory

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) )...( )( ( ) )( )( ( ) ( ) ( 1

2 1

2 1

n c c c

p s p s p s z s z s z s s H s G

n

       

n

c c c

z z z ... ,

2 1

Are also poles of the close-loop transfer function zeros of 1+G(s)H(s),

Control Systems: Nyquist Stability Theory

Characteristic equation

) )...( )( ( ) )...( )( ( ) )...( )( ( ) ( ) ( 1

2 1 2 1 2 1 n m n

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

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1 2

1 2

... 1 ( ) ( ) ( ) ( ) ... ( ) .

n

c c c n

s z s z s z G s H s s p s p s p        

Magnitude

1 2

1 2

1 ( ) ( ) ( ) ( ) ( )

n

c c c n

s z s z s z G s H s s p s p s p               

Angle

Control Systems: Nyquist Stability Theory

Characteristic equation

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s-plane to 1+GH plane mapping

phase angle of the 1+G(s)H(s) vector, corresponding to a point

• n the s-plane is the difference between the sum of the phase
• f all vectors drawn from zeros of 1+GH(close loop poles) and
• pen loops on the s plane. If this point s is moved along a

closed contour enclosing any or all of the above zeros and poles, only the phase of the vector of each of the enclosed zeros or open-loop poles will change by 360o. The direction will be in the same sense of the contour enclosing zeros and in the opposite sense for the contour enclosing open-loop poles.

Control Systems: Nyquist Stability Theory

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Control Systems: Nyquist Stability Theory

s-plane to 1+GH plane mapping

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Control Systems: Nyquist Stability Theory

s-plane to 1+GH plane mapping

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THEORY When a closed contour in the s-plane encloses a certain number of poles and zeros of 1+G(s)H(s) in the clockwise direction, the number of encirclements by the corresponding contour in the G(s)H(s) plane will encircle the point (-1,0) a number of times given by the difference between the number

• f its zeros of 1+G(s)H(s) and the open-loop poles it enclosed
• n the s-plane

Control Systems: Nyquist Stability Theory

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THEORY

Contour on the s-plane for checking the existence of closed-loop poles

GH from the polar plot GH from the mirror image of the polar plot Magnitude zero since n >m

Control Systems: Nyquist Stability Theory

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Modified contour on the s-plane for checking the existence of closed-loop poles

j

s e   

Magnitude of GH remains the same along the contour, phase of β changes from 270 to 90 degrees

THEORY

Control Systems: Nyquist Stability Theory

Control Systems: Nyquist Stability Theory

Conclusion

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ME 779 Control Systems

Checking Nyquist Stability

Topic #28

Reference textbook:

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

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Procedure

(1) Locate open-loop poles on the s-plane (2) Draw the closed contour and avoid open-loop poles on the imaginary axis (3) Count the number of open-loop poles enclosed in the above contour of step 2, say P (4) Plot G(j)H(j) and its reflection on the GH plane and map part of the small semi-circle detour

• n the s-plane around poles (if any) on the

imaginary axis.

Control Systems: Checking Nyquist Stability

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Control Systems: Checking Nyquist Stability

Procedure

(5) Once the entire s-plane contour is mapped on to the GH plane, count the number of encirclements

• f the point (-1,0) and its direction. Clockwise

encirclement is considered positive, say N. (6) The number of closed-loop poles in the right- half s-plane is given by Z=N+P. if Z >0, the system is unstable. (7) Determine gain margin, phase margin, and critical value of open-loop gain.

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Example 1

Using Nyquist criterion, determine the stability of a feedback system for K=1, whose open-loop transfer function is given by

( ) ( ) ( 1) K G s H s s s  

.

Step 1 Locate open-loop poles on the s-plane Open loop poles are at s=0 and –1. Step 2 Draw the closed contour on the s-plane to check the existence

• f closed-loop poles in the right-half s-plane.

Control Systems: Checking Nyquist Stability

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Open-loop poles and s-plane contour

Control Systems: Checking Nyquist Stability

Example 1

6 2

1 ( ) ( ) 1 G j H      

1

( ) ( ) tan 2 G j H j    



   

Control Systems: Checking Nyquist Stability

Example 1

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• No. Frequency,

rad/s Magnitude Phase, degrees 1 0.2 4.9029 259 2 0.4 2.3212 248 3 0.8 0.9761 231 4 1 0.7071 225 5 4 0.0606 194 6 10 0.01 186 7 50 0.0004 181 8 100 0.0001 181 9 200 180

Control Systems: Checking Nyquist Stability

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No. Frequency, rad/s Magnitude Phase, degrees 10

• 200

180 11

• 100

0.0001 179 12

• 50

0.0004 179 13

• 10

0.01 174 14

• 4

0.0606 166 15

• 1

0.7071 135 16

• 0.8

0.9761 129 17

• 0.4

2.3212 112 18

• 0.2

4.9029 101

Control Systems: Checking Nyquist Stability

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, s-plane, deg , GH plane, deg 270 101 280 91 290 80 300 69 310 58 320 46 330 35 340 23 350 12

Control Systems: Checking Nyquist Stability

, s-plane, deg , GH plane, deg 10 348 20 337 30 325 40 314 50 302 60 291 70 280 80 269 90 259

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The system is stable

Control Systems: Checking Nyquist Stability

Example 1

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Determine the stability of the feedback system whose

• pen-loop transfer function given by

55 ( ) ( ) ( 2)( 4) G s H s s s s   

.

Locate open-loop poles on the s-plane Draw the closed loop contour on the s-plane

Control Systems: Checking Nyquist Stability

Example 2

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Control Systems: Checking Nyquist Stability

Example 2

55 ( ) ( ) ( 2)( 4) G s H s s s s   

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The number of open-loop pole enclosed, P is zero 2 2

( ) ( ) 4 16 K G j H j        

1 1

( ) ( ) tan tan 2 2 4 G j H j     

 

    

Control Systems: Checking Nyquist Stability

Example 2

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No. Frequency Magnitude Phase, degrees 1 1.5 Positive frequencies 3.4332 213 2 2 2.1741 198 3 2.5 1.4568 187 4 2.83 1.1446 180 5 3 1.017 177 6 3.5 0.7334 169 7 4.5 0.4122 156 8 5 0.319 150 9 5.5 0.2513 146 10 6 0.201 142 11 7 0.1339 136 12 8 0.0932 131 13 9 0.0673 126

Control Systems: Checking Nyquist Stability

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14

• 9

Negative frequencies 0.0673 234 15

• 8

0.0932 229 16

• 7

0.1339 224 17

• 6

0.201 218 18

• 5.5

0.2513 214 19

• 5

0.319 210 20

• 4.5

0.4122 204 21

• 3.5

0.7334 191 22

• 3

1.017 183 23

• 2.83

1.1446 180 24

• 2.5

1.4568 173 25

• 2

2.1741 162 26

• 1.5

3.4332 147

Control Systems: Checking Nyquist Stability

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, s- plane, deg , GH plane, deg 270 147 280 132 290 117 300 101 310 84 320 68 330 51 340 34 350 17 , s-plane, deg , GH plane, deg 10 343 20 326 30 309 40 292 50 276 60 259 70 243 80 228 90 213

Control Systems: Checking Nyquist Stability

Example 2

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Z=N+P=2 The system is unstable

Control Systems: Checking Nyquist Stability

Example 2

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Determine the stability of the feedback system whose

• pen-loop transfer function given by

2

( ) ( ) ( 1) K G s H s s s  

.

Control Systems: Checking Nyquist Stability

Example 3

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Locate open-loop poles

• n the s-plane

Draw the closed loop contour

• n the s-plane

Control Systems: Checking Nyquist Stability

Example 3

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Count the number of open-loop poles enclosed in the closed contour of the s-plane The number of open-loop poles enclosed, P is zero.

2 2

( ) ( ) 1 K G j H j      

1

( ) ( ) tan G j H j    



    Control Systems: Checking Nyquist Stability

Example 3

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No. Frequency Magnitude Phase, degree s 1 0.49 Positive frequencies 3.74 154 2 0.50 3.58 153 3 0.60 2.38 149 4 0.80 1.22 141 5 1.00 0.71 135 6 2.00 0.11 117 7 3.00 0.04 108 8 4.00 0.02 104 9 5.00 0.01 101 10 6.00 0.00 99 11 7.00 0.00 98 12 8.00 0.00 97 13 9.00 0.00 96 14 10.00 0.00 96

Control Systems: Checking Nyquist Stability

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15

• 10.00

Negative frequenci es 0.00 264 16

• 9.00

0.00 264 17

• 8.00

0.00 263 18

• 7.00

0.00 262 19

• 6.00

0.00 261 20

• 5.00

0.01 259 21

• 4.00

0.02 256 22

• 3.00

0.04 252 23

• 2.00

0.11 243 24

• 1.00

0.71 225 25

• 0.80

1.22 219 26

• 0.60

2.38 211 27

• 0.50

3.58 207 28

• 0.49

3.74 206

Control Systems: Checking Nyquist Stability

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, s-plane, deg , GH plane, deg 270 206 280 184 290 162 300 139 310 116 320 93 330 70 340 47 350 23 10 337 20 313 30 290 40 267 50 244 60 221 70 198 80 176 90 154

Control Systems: Checking Nyquist Stability

Example 3 Phase changes around the detour

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N=2 and Z=N+P=2 and hence the system is unstable

Control Systems: Checking Nyquist Stability

Example 3 GH Plane mapping

Control Systems: Checking Nyquist Stability

Conclusion

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1

ME 779 Control Systems

Nyquist plots: Gain and phase margin

Topic #27

Reference textbook:

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

2

Gain Margin and Phase Margin phase crossover frequency is the frequency at which the open-loop transfer function has a phase of 180o

p



The gain crossover frequency is the frequency at which the open-loop transfer function has a unit gain

g



Nyquist plots: Gain and Phase margin

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( ) ( ) ( 2)( 4) K G s H s s s s   

Nyquist plots: Gain and Phase margin

Beginning from the gain margin equation based on root-locus plots

1

20log

c

K GM K 

, where Kc is the open-loop gain corresponding to marginal stability and K1 is the open-loop gain at another arbitrary point on the root-locus, prove that

20log ( ) ( )

p p

GM G j H j    

;

p

 is the phase crossover frequency.

S-plane Root-locus Kc

j 

Kc K1 K1

p



p



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Nyquist plots: Gain and Phase margin

The open-loop transfer function in terms of open-loop poles and zeros is given by

1 2 1 2

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

m n

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

1 2 1 2

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

m n

K j z j z j z G j H j j p j p j p               

Magnitude of the Open-loop frequency response function

1

1 1 2 1 2

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

p p p m K p p p p p n

K j z j z j z G j H j j p j p j p               

1 2 1 2

( )( ) ( ) ( ) ( ) 1 ( )( ) ( )

c

c p p p m K p p p p p n

K j z j z j z G j H j j p j p j p                

The ratio of equations result in

20log ( ) ( )

p p

GM G j H j    

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Nyquist plots: Gain and Phase margin

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• 1

GH Plane Real Imaginary

ω=0 ω=∞

1 GH 

g



p



stable

( ) ( )

g g

G j H j    ( ) ( )

p p

G j H j  

Nyquist plots: Gain and Phase margin

• 1

GH Plane Real Imaginary

ω=0 ω=∞

p g

  

Marginally stable

( ) ( ) 180

g g

G j H j     ( ) ( ) 1

p p

G j H j   

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Nyquist plots: Gain and Phase margin

• 1

GH Plane Real Imaginary

ω=0 ω=∞

g



p



unstable

( ) ( )

g g

G j H j    ( ) ( )

p p

G j H j  

8

Nyquist plots: Gain and Phase margin

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20log ( ) ( )

p p

M G j H j    

Gain Margin and Phase Margin

Gain margin

( ) ( ) 180o

g g

G j H j      

Phase margin

Nyquist plots: Gain and Phase margin

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Example 1

( ) ( ) ( 1) K G s H s s s  

.

Determine gain margin, phase margin and stability of the feedback system whose open-loop transfer function given by

Nyquist plots: Gain and Phase margin

11

No. Frequency, rad/s Magnitude Phase, degrees 1

∞

270 2 0.2 4.9029 259 3 0.4 2.3212 248 4 0.786 1 232 5 0.8 0.9761 231 6 1 0.7071 225 7 4 0.0606 194 8 10 0.01 186 9 50 0.0004 181 10 100 0.0001 181 11 200 ≈0 ≈180

Example 1

Nyquist plots: Gain and Phase margin

12 p



p

 20log ( ) ( )

p p

G j H j     

g



g



• Example 1

Nyquist plots: Gain and Phase margin

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55 ( ) ( ) ( 2)( 4) G s H s s s s   

.

Example 2

Determine gain margin, phase margin and stability of the feedback system whose open-loop transfer function given by

Nyquist plots: Gain and Phase margin

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Magnitude and phase

• f the open-loop frequency

transfer function (K=55)

Example 2

No. Frequency Magnitude Phase, degrees 1 1.5 3.4332 213 2 2 2.1741 198 3 2.5 1.4568 187 4 2.83 1.1446 180 5 3 1.017 177 6 3.5 0.7334 169 7 4.5 0.4122 156 8 5 0.319 150 9 5.5 0.2513 146 10 6 0.201 142 11 7 0.1339 136 12 8 0.0932 131 13 9 0.0673 126

Nyquist plots: Gain and Phase margin

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Phase crossover frequency 2.83 rad/s

*

55/1.1446 48 K  

The gain at which the system becomes marginally stable

20log ( ) ( ) 20log 1.1446 1.17dB

p p

M G j H j        

Gain margin

Example 2

Nyquist plots: Gain and Phase margin

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Gain crossover frequency =3 rad/s and the corresponding angle Of GH=177o

Phase margin=177-180=-3o The system is unstable for K=55

Example 2

Nyquist plots: Gain and Phase margin

17

Determine gain margin, phase margin and stability of the feedback system whose open-loop transfer function given by

2

( ) ( ) ( 1) K G s H s s s  

.

Example 3

Nyquist plots: Gain and Phase margin

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No. Frequency, rad/s Magnitude Phase, degrees 1 ∞ 180 2 0.4 5.803 158 4 0.5 3.5777 153 5 0.8 1.2201 141 6 0.87 1 139 7 1 0.7071 135 8 2 0.1118 117 9 3 0.0351 108

10

4 0.0152 104

11

5 0.0078 101

Example 3

Nyquist plots: Gain and Phase margin

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The phase crossover frequency is 0 rad/s and the corresponding magnitude is infinity

20log ( ) ( ) 20log dB

p p

M G j H j         

Example 3

Nyquist plots: Gain and Phase margin

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The gain crossover frequency is 0.87 rad/s and the corresponding phase is 1390 Phase margin =1390- 1800=-410 The system is unstable for K=1. Since the gain margin is negative infinity, open-loop gain K has to be decreased infinite times for the system to be stable. Hence this system is unstable for all values of K

Nyquist plots: Gain and Phase margin

Nyquist plots: Gain and Phase margin

Conclusion

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