1. Cauchys Principle for Function F ( z ) 1. Cauchys Principle for - PowerPoint PPT Presentation

8. Encirclement Criterion for Stability - Continued

8. Encirclement Criterion for Stability - Continued • Evaluate 1 + K b ( z ) a ( z ) along C 1 and plot it: Called C 2

8. Encirclement Criterion for Stability - Continued • Evaluate 1 + K b ( z ) a ( z ) along C 1 and plot it: Called C 2 • C 2 encircles origin N = ( n − Z ) − ( n − P )

8. Encirclement Criterion for Stability - Continued • Evaluate 1 + K b ( z ) a ( z ) along C 1 and plot it: Called C 2 • C 2 encircles origin N = ( n − Z ) − ( n − P ) = P − Z times

8. Encirclement Criterion for Stability - Continued • Evaluate 1 + K b ( z ) a ( z ) along C 1 and plot it: Called C 2 • C 2 encircles origin N = ( n − Z ) − ( n − P ) = P − Z times • Want Z = 0 for stability.

8. Encirclement Criterion for Stability - Continued • Evaluate 1 + K b ( z ) a ( z ) along C 1 and plot it: Called C 2 • C 2 encircles origin N = ( n − Z ) − ( n − P ) = P − Z times • Want Z = 0 for stability. i.e., N = P for stability

8. Encirclement Criterion for Stability - Continued • Evaluate 1 + K b ( z ) a ( z ) along C 1 and plot it: Called C 2 • C 2 encircles origin N = ( n − Z ) − ( n − P ) = P − Z times • Want Z = 0 for stability. i.e., N = P for stability P = No. of unstable poles of 1 + K b ( z ) a ( z )

8. Encirclement Criterion for Stability - Continued • Evaluate 1 + K b ( z ) a ( z ) along C 1 and plot it: Called C 2 • C 2 encircles origin N = ( n − Z ) − ( n − P ) = P − Z times • Want Z = 0 for stability. i.e., N = P for stability P = No. of unstable poles of 1 + K b ( z ) a ( z ) = No. of unstable poles of a ( z ) + Kb ( z ) a ( z )

8. Encirclement Criterion for Stability - Continued • Evaluate 1 + K b ( z ) a ( z ) along C 1 and plot it: Called C 2 • C 2 encircles origin N = ( n − Z ) − ( n − P ) = P − Z times • Want Z = 0 for stability. i.e., N = P for stability P = No. of unstable poles of 1 + K b ( z ) a ( z ) = No. of unstable poles of a ( z ) + Kb ( z ) a ( z ) = No. of unstable poles of b ( z ) a ( z )

8. Encirclement Criterion for Stability - Continued • Evaluate 1 + K b ( z ) a ( z ) along C 1 and plot it: Called C 2 • C 2 encircles origin N = ( n − Z ) − ( n − P ) = P − Z times • Want Z = 0 for stability. i.e., N = P for stability P = No. of unstable poles of 1 + K b ( z ) a ( z ) = No. of unstable poles of a ( z ) + Kb ( z ) a ( z ) = No. of unstable poles of b ( z ) a ( z ) = no. of open loop unstable poles

8. Encirclement Criterion for Stability - Continued • Evaluate 1 + K b ( z ) a ( z ) along C 1 and plot it: Called C 2 • C 2 encircles origin N = ( n − Z ) − ( n − P ) = P − Z times • Want Z = 0 for stability. i.e., N = P for stability P = No. of unstable poles of 1 + K b ( z ) a ( z ) = No. of unstable poles of a ( z ) + Kb ( z ) a ( z ) = No. of unstable poles of b ( z ) a ( z ) = no. of open loop unstable poles N should be equal to the number of open loop unstable poles 8 Digital Control Kannan M. Moudgalya, Autumn 2007

9. Procedure to Calculate K Using Nyquist Plot

9. Procedure to Calculate K Using Nyquist Plot Im( z ) Im( F ( z )) C 2 C 1 Re( F ( z )) Re( z )

9. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z )

9. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z ) • Evaluate 1+ K b ( z ) a ( z ) along the unit circle ( C 1 ) and plot C 2

9. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z ) • Evaluate 1+ K b ( z ) a ( z ) along the unit circle ( C 1 ) and plot C 2 • C 2 should encircle origin P times = open loop unstable poles

9. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z ) • Evaluate 1+ K b ( z ) a ( z ) along the unit circle ( C 1 ) and plot C 2 • C 2 should encircle origin P times = open loop unstable poles • K has to be known to do this

9. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z ) • Evaluate 1+ K b ( z ) a ( z ) along the unit circle ( C 1 ) and plot C 2 • C 2 should encircle origin P times = open loop unstable poles • K has to be known to do this • Want to convert this into a design approach to calculate K

9. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z ) • Evaluate 1+ K b ( z ) a ( z ) along the unit circle ( C 1 ) and plot C 2 • C 2 should encircle origin P times = open loop unstable poles • K has to be known to do this • Want to convert this into a design approach to calculate K • Evaluate 1 + K b ( z ) a ( z ) − 1 = K b ( z ) a ( z ) along C 1 , plot, call it C 3

9. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z ) • Evaluate 1+ K b ( z ) a ( z ) along the unit circle ( C 1 ) and plot C 2 • C 2 should encircle origin P times = open loop unstable poles • K has to be known to do this • Want to convert this into a design approach to calculate K • Evaluate 1 + K b ( z ) a ( z ) − 1 = K b ( z ) a ( z ) along C 1 , plot, call it C 3 9 Digital Control Kannan M. Moudgalya, Autumn 2007

10. Procedure to Calculate K Using Nyquist Plot

10. Procedure to Calculate K Using Nyquist Plot Im( z ) Im( F ( z )) C 2 C 1 Re( F ( z )) Re( z )

10. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z )

10. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z ) • For stability, plot of Kb ( z ) /a ( z ) , C 3 , should encircle the point ( − 1 , 0) , P times

10. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z ) • For stability, plot of Kb ( z ) /a ( z ) , C 3 , should encircle the point ( − 1 , 0) , P times • Still need to know K

10. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z ) • For stability, plot of Kb ( z ) /a ( z ) , C 3 , should encircle the point ( − 1 , 0) , P times • Still need to know K • Evaluate b ( z ) a ( z ) along C 1 and plot.

10. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z ) • For stability, plot of Kb ( z ) /a ( z ) , C 3 , should encircle the point ( − 1 , 0) , P times • Still need to know K • Evaluate b ( z ) a ( z ) along C 1 and plot. Call it C 4 .

10. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z ) • For stability, plot of Kb ( z ) /a ( z ) , C 3 , should encircle the point ( − 1 , 0) , P times • Still need to know K • Evaluate b ( z ) a ( z ) along C 1 and plot. Call it C 4 . • For stability, C 4 should encircle point ( − 1 /K, 0) , P times

10. Procedure to Calculate K Using Nyquist Plot Im( F ( z )) Im( z ) Im( F ( z )) C 2 C 3 C 2 C 1 Re( F ( z )) Re( F ( z )) Re( z ) • For stability, plot of Kb ( z ) /a ( z ) , C 3 , should encircle the point ( − 1 , 0) , P times • Still need to know K • Evaluate b ( z ) a ( z ) along C 1 and plot. Call it C 4 . • For stability, C 4 should encircle point ( − 1 /K, 0) , P times • C 4 is the Nyquist plot 10 Digital Control Kannan M. Moudgalya, Autumn 2007

11. Example of Nyquist Plot to Design Controller

11. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1)

11. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1) Im( z ) D B E A Re( z ) C 1

11. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1) Im( z ) D B E A Re( z ) C 1 • C 1 should not go through pole/zero

11. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1) Im( z ) D B E A Re( z ) C 1 • C 1 should not go through pole/zero • Indent it with a semicircle of radius → 0

11. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1) Im( z ) D B E A Re( z ) C 1 • C 1 should not go through pole/zero • Indent it with a semicircle of radius → 0 • Number of unstable poles, P = 0

11. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1) Im( z ) D B E A Re( z ) C 1 • C 1 should not go through pole/zero • Indent it with a semicircle of radius → 0 • Number of unstable poles, P = 0 • Evaluate b ( z ) a ( z ) along main C 1

11. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1) Im( z ) D B E A Re( z ) C 1 • C 1 should not go through pole/zero • Indent it with a semicircle of radius → 0 • Number of unstable poles, P = 0 • Evaluate b ( z ) a ( z ) along main C 1 11 Digital Control Kannan M. Moudgalya, Autumn 2007

12. Example of Nyquist Plot to Design Controller

12. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1)

12. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1) 1 e jω � � G = e jω ( e jω − 1)

12. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1) 1 e jω � � G = e jω ( e jω − 1) 1 = 2 ω � 2 ω � e j 3 e j 1 2 ω − e − j 1

12. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1) 1 e jω � � G = e jω ( e jω − 1) 2 ω � = − je − j 3 2 ω 1 = 2 sin ω 2 ω � e j 3 e j 1 2 ω − e − j 1 2

12. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1) 1 e jω � � G = e jω ( e jω − 1) 2 ω � = − je − j 3 2 ω 1 = 2 sin ω 2 ω � e j 3 e j 1 2 ω − e − j 1 2 cos 3 2 ω − j sin 3 � � = − j 2 ω 2 sin ω 2

12. Example of Nyquist Plot to Design Controller G ( z ) = b ( z ) 1 a ( z ) = z ( z − 1) 1 e jω � � G = e jω ( e jω − 1) 2 ω � = − je − j 3 2 ω 1 = 2 sin ω 2 ω � e j 3 e j 1 2 ω − e − j 1 2 cos 3 2 ω − j sin 3 � � = − j 2 ω 2 sin ω 2 = − sin 3 2 ω − j cos 3 2 ω 2 ω 2 sin 1 2 sin 1 2 ω 12 Digital Control Kannan M. Moudgalya, Autumn 2007

13. Example of Nyquist Plot - Continued

13. Example of Nyquist Plot - Continued Im( z ) D B E A Re( z ) C 1

13. Example of Nyquist Plot - Continued Im( z ) D B E A Re( z ) C 1 Im B D A Re E

13. Example of Nyquist Plot - Continued Im( z ) • Transfer function is given by D B E A Re( z ) C 1 Im B D A Re E

13. Example of Nyquist Plot - Continued Im( z ) • Transfer function is given by D B = − sin 3 − j cos 3 E 2 ω 2 ω A e jω � � G Re( z ) 2 sin ω 2 sin 1 2 ω 2 C 1 Im B D A Re E

13. Example of Nyquist Plot - Continued Im( z ) • Transfer function is given by D B = − sin 3 − j cos 3 E 2 ω 2 ω A e jω � � G Re( z ) 2 sin ω 2 sin 1 2 ω 2 C 1 • At point A , ω = 180 o , G = 0 . 5 Im B D A Re E

13. Example of Nyquist Plot - Continued Im( z ) • Transfer function is given by D B = − sin 3 − j cos 3 E 2 ω 2 ω A e jω � � G Re( z ) 2 sin ω 2 sin 1 2 ω 2 C 1 • At point A , ω = 180 o , G = 0 . 5 Im • At point B , ω = 120 o B D A Re E

13. Example of Nyquist Plot - Continued Im( z ) • Transfer function is given by D B = − sin 3 − j cos 3 E 2 ω 2 ω A e jω � � G Re( z ) 2 sin ω 2 sin 1 2 ω 2 C 1 • At point A , ω = 180 o , G = 0 . 5 Im • At point B , ω = 120 o G = − sin 3 2 120 − j cos 3 2 120 2 120 B D A 2 sin 1 2 sin 1 2 120 Re E