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Lecture 23 loop transfer function

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The document discusses open loop transfer functions and stability analysis using Nyquist plots. It begins with an outline of topics including partial fraction expansion, open loop systems, Nyquist plots, and stability criteria. It then provides examples of using partial fraction expansion to decompose transfer functions with real distinct roots, complex conjugate roots, and repeated roots. The document explains open loop and closed loop system nomenclature. It introduces the Nyquist stability criterion, which involves plotting the open loop transfer function on the Nyquist plot and checking if it encircles the critical point at -1.

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Plane of the Open Loop Transfer Function B(0) B(iw) -B(iw) ()Bi Professor Walter W. Olson -1 Real Imaginary Stable Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo -1 is called the critical point Unstable Loop Transfer Function
Outline of Today’s Lecture  Review  Partial Fraction Expansion  real distinct roots  repeated roots  complex conjugate roots  Open Loop System  Nyquist Plot  Simple Nyquist Theorem  Nyquist Gain Scaling  Conditional Stability  Full Nyquist Theorem
Partial Fraction Expansion  When using Partial Fraction Expansion, our objective is to turn the Transfer Function 2 m k i i i i ni ni     K s z s s (  ) (  2  )   1 1 2 r n q  w w  w w s s p s s (  ) (  2  ) i  1 i i  1 i ni ni G s ( )  into a sum of fractions where the denominators are the factors of the denominator of the Transfer Function: K A ( s ) A ( s ) A ( s ) B ( s ) B ( s ) n q 1 2 1 G s ( ) ... ...         2 2 s s  p s  p s  p s   w s  w s   w s  w 2 2 n n n q nq nq 1 2 1 1 1 r Then we use the linear property of Laplace Transforms and the relatively easy form to make the Inverse Transform.
Case 1: Real and Distinct Roots n i  i  Put the transfer function in the form of ( ) ... n where the are called the residue at the pole and determined by   ... 1 ( ) 0 1 2 1 2 ( ) G s ( ) 0 0 3 3 n i i s G s s s p a a a a G s s s p s p s p a p a sG s a s p                           3 1 2 1 1 2 2 ( ) ( ) ... n s p s p s p n n s p a s p G s a s p G s a s p G s        
Case 1: Real and Distinct Roots Example         s s   2 4 1 5 0 1 2 ( ) ( ) 1 5  2   4   1   5   5   1  0 1 2       2 2 2 2 6 8 6 5 5 0 1 2 0 1 2 1 2 1 0 1 2 1 2 2 1 0 0 0.6 0.75 6 5 6 5 3.6 0.15 5 8 1.6 1.6 0.75 ( ) 1 G s s s s a a a G s s s s s s a s s a s s a s s s s a s s a s s a s s a a a a a a a a a a a a a a G s s s                                                             5 0.15 5 s  ( ) 1.6 0.75 0.15 t t g t   e   e 
Case 2: Complex Conjugate Roots 2 2  1 2   1 1 1 ... ( ) ... ( 2 ) We can either solve this using the method of matching coefficients which is usually more difficult or by a method similar to that previously used as follows: 2 q i i i i G s s s s s  w w  w w      s s      w  w     w  w   1 1 1 1 1 1 1 1 1 1 A ( s ) a a then the term   s  s  s    s    2  w w  w w  1  w w  1 proceeding as before i i i         2 2 2 1 1 1 1 2 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 s 2 1 1 1 1 s 1 a s G s a s G s  w w   w w   w w   w w               
Case 3: Repeated Roots n n n n i i i G s n i s p i   n i   1 1 1 1 2 2 2 3 3 ... ( ) n ...( ) ... i Form the equation with the repeated terms expanded as ( ) ... ... ... ( ) ( ) n ( ) ( ) n ( ) ( ) s p n n i s p n s p a a a G s s p s p s p a s p G s d a s p G s ds d a s p G s ds d a ds                             n         3 1     1 1 ( ) ... i ( ) s p n n n i s p s p G s d a s p G s ds    
Heaviside Expansion         1 1 Heaviside Expansion Formula: L where are the distinct roots of ( ) 15( 2) 2 Example: ( ) ( 2 25) Roots of the denominator are 0, 1 4.899, and i n i b t i i i A s A b e B s d B b ds b n B s s G s s s s i                                  2 2 2    15. 73.485i 15. 73.485i 48.0     3      1 2 1 0 1 4.899 1 4.899 1 4.899 1 4.899 ( 2 25) 2 2 3 4 25 15( 2) L 3 4 25 30 ( ) 25 ( 0 9.798i 48.00 9.798i ) 1.2 0.6 1.408i i i s t i s s t i t i t i i d B s s s s s s s ds s G s e s s g t e e e g t e                                           1 4.899 0.6 1.408i t i t e     
Loop Nomenclature Reference Input R(s) +- Output y(s) Error signal E(s) Open Loop Signal B(s) Plant G(s) Disturbance/Noise Sensor H(s) Prefilter F(s) Controller C(s) +- The plant is that which is to be controlled with transfer function G(s) The prefilter and the controller define the control laws of the system. The open loop signal is the signal that results from the actions of the prefilter, the controller, the plant and the sensor and has the transfer function F(s)C(s)G(s)H(s) The closed loop signal is the output of the system and has the transfer function F ( s ) C ( s ) G ( s )  C s G s H s 1 ( ) ( ) ( )
Closed Loop System ++ Output y(s) Error signal E(s) Open Loop Signal B(s) Plant P(s) Controller C(s) Input r(s) The closed loop transfer function is             -1     n s n s y s C s P s d s d s n s n s                         ( )               ( ) ( ) 1 1 The characteristic polynomial is ( ) 1 For stability, the roots of ( ) m c p c p c p yr c p c p c p c p c p c p G s r s C s P s n s n s d s d s n s n s d s d s s C s P s d s d s n s n s s        ust have negative real parts While we can check for stability, it does not give us design guidance
Note: Your book uses L(s) rather than B(s) To avoid confusion with the Laplace transform, I will use B(s) Open Loop System ++ Output y(s) Error signal E(s) Open Loop Signal B(s) Plant P(s) Controller C(s) Input r(s) Sensor b s n s n s             ( ) The open loop transfer function is ( )    ( ) c p c p B s C s P s r s d s d s -1 If in the closed loop, the input r(s) were sinusoidal and if the signal were to continue in the same form and magnitude after the signal were disconnected, it would be necessary for n  s  n  s  ( w )  c p   1 0     c p B i d s d s
Open Loop System Nyquist Plot Error signal E(s) ++ Output y(s) Open Loop Signal B(s) Plant P(s) Controller C(s) Input r(s) Sensor -1 n  s  n  s  ( w )  c p   1 0     c p B i d s d s -1 Real Imaginary Plane of the Open Loop Transfer Function B(0) B(iw) B(i) -1 is called the critical point B(-iw)
Simple Nyquist TheoremError signal E(s) ++ Output y(s) Open Loop Signal B(s) Plant P(s) Controller C(s) Input r(s) Sensor -1 -1 Plane of the Open Loop Transfer Function Real Imaginary B(0) B(iw) -B(iw) ()Bi -1 is called the critical point Stable Unstable Simple Nyquist Theorem: For the loop transfer function, B(iw), if B(iw) has no poles in the right hand side, expect for simple poles on the imaginary axis, then the system is stable if there are no encirclements of the critical point -1.
Example 1  Plot the Nyquist plot for ( )  2  2 2 B s s s s    1    2 B i ( ) i w i w w i   2 2 w  B (0)   i B (1 i )   0.4  0.2 i B (  1 i )   0.4  0.2 i B (2 i )   0.1  0.05 i B (  2 i )   0.1  0.05 i -1 Im Re Stable
Example  Plot the Nyquist plot for 10   ( ) 2 2 B s s s s    10 20 20 10           2 4 ( ) 2 2 4 (0) (1 ) 4 2 ( 1 ) 4 2 (2 ) 1 0.5 ( 2 ) 1 0.5 (4 ) 0.077 0.135 ( 4 ) 0.077 135 i B i i i i B i B i i B i i B i i B i i B i i B i i w w w w w w                          -1 Im Re Unstable
Nyquist Gain Scaling  The form of the Nyquist plot is scaled by the system gain K   B s ( ) s s s   2 2   Show with Sisotool
Conditional Stabilty  While most system increase stability by decreasing gain, some can be stabilized by increasing gain  Show with Sisotool 2 K s s (0.25 0.12 1)    2  ( ) 1.69 1.09 1 B s s s s   
Full Nyquist Theorem  Assume that the transfer function B(iw) with P poles has been plotted as a Nyquist plot. Let N be the number of clockwise encirclements of -1 by B(iw) minus the counterclockwise encirclements of -1 by B(iw)Then the closed loop system has Z=N+P poles in the right half plane.  Show with Sisotool K s i s i ( 5 2 )( 5 2 )             ( ) .5 2 .5 2 2 6 2 6 B s s s i s i s i s i         
Summary  Open Loop System  Nyquist Plot  Simple Nyquist Theorem  Nyquist Gain Scaling  Conditional Stability  Full Nyquist Theorem -1 Next Class: Stability Margins Im Re Unstable

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Lecture 23 loop transfer function

  • 1. Plane of the Open Loop Transfer Function B(0) B(iw) -B(iw) ()Bi Professor Walter W. Olson -1 Real Imaginary Stable Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo -1 is called the critical point Unstable Loop Transfer Function
  • 2. Outline of Today’s Lecture  Review  Partial Fraction Expansion  real distinct roots  repeated roots  complex conjugate roots  Open Loop System  Nyquist Plot  Simple Nyquist Theorem  Nyquist Gain Scaling  Conditional Stability  Full Nyquist Theorem
  • 3. Partial Fraction Expansion  When using Partial Fraction Expansion, our objective is to turn the Transfer Function 2 m k i i i i ni ni     K s z s s (  ) (  2  )   1 1 2 r n q  w w  w w s s p s s (  ) (  2  ) i  1 i i  1 i ni ni G s ( )  into a sum of fractions where the denominators are the factors of the denominator of the Transfer Function: K A ( s ) A ( s ) A ( s ) B ( s ) B ( s ) n q 1 2 1 G s ( ) ... ...         2 2 s s  p s  p s  p s   w s  w s   w s  w 2 2 n n n q nq nq 1 2 1 1 1 r Then we use the linear property of Laplace Transforms and the relatively easy form to make the Inverse Transform.
  • 4. Case 1: Real and Distinct Roots n i  i  Put the transfer function in the form of ( ) ... n where the are called the residue at the pole and determined by   ... 1 ( ) 0 1 2 1 2 ( ) G s ( ) 0 0 3 3 n i i s G s s s p a a a a G s s s p s p s p a p a sG s a s p                           3 1 2 1 1 2 2 ( ) ( ) ... n s p s p s p n n s p a s p G s a s p G s a s p G s        
  • 5. Case 1: Real and Distinct Roots Example         s s   2 4 1 5 0 1 2 ( ) ( ) 1 5  2   4   1   5   5   1  0 1 2       2 2 2 2 6 8 6 5 5 0 1 2 0 1 2 1 2 1 0 1 2 1 2 2 1 0 0 0.6 0.75 6 5 6 5 3.6 0.15 5 8 1.6 1.6 0.75 ( ) 1 G s s s s a a a G s s s s s s a s s a s s a s s s s a s s a s s a s s a a a a a a a a a a a a a a G s s s                                                             5 0.15 5 s  ( ) 1.6 0.75 0.15 t t g t   e   e 
  • 6. Case 2: Complex Conjugate Roots 2 2  1 2   1 1 1 ... ( ) ... ( 2 ) We can either solve this using the method of matching coefficients which is usually more difficult or by a method similar to that previously used as follows: 2 q i i i i G s s s s s  w w  w w      s s      w  w     w  w   1 1 1 1 1 1 1 1 1 1 A ( s ) a a then the term   s  s  s    s    2  w w  w w  1  w w  1 proceeding as before i i i         2 2 2 1 1 1 1 2 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 s 2 1 1 1 1 s 1 a s G s a s G s  w w   w w   w w   w w               
  • 7. Case 3: Repeated Roots n n n n i i i G s n i s p i   n i   1 1 1 1 2 2 2 3 3 ... ( ) n ...( ) ... i Form the equation with the repeated terms expanded as ( ) ... ... ... ( ) ( ) n ( ) ( ) n ( ) ( ) s p n n i s p n s p a a a G s s p s p s p a s p G s d a s p G s ds d a s p G s ds d a ds                             n         3 1     1 1 ( ) ... i ( ) s p n n n i s p s p G s d a s p G s ds    
  • 8. Heaviside Expansion         1 1 Heaviside Expansion Formula: L where are the distinct roots of ( ) 15( 2) 2 Example: ( ) ( 2 25) Roots of the denominator are 0, 1 4.899, and i n i b t i i i A s A b e B s d B b ds b n B s s G s s s s i                                  2 2 2    15. 73.485i 15. 73.485i 48.0     3      1 2 1 0 1 4.899 1 4.899 1 4.899 1 4.899 ( 2 25) 2 2 3 4 25 15( 2) L 3 4 25 30 ( ) 25 ( 0 9.798i 48.00 9.798i ) 1.2 0.6 1.408i i i s t i s s t i t i t i i d B s s s s s s s ds s G s e s s g t e e e g t e                                           1 4.899 0.6 1.408i t i t e     
  • 9. Loop Nomenclature Reference Input R(s) +- Output y(s) Error signal E(s) Open Loop Signal B(s) Plant G(s) Disturbance/Noise Sensor H(s) Prefilter F(s) Controller C(s) +- The plant is that which is to be controlled with transfer function G(s) The prefilter and the controller define the control laws of the system. The open loop signal is the signal that results from the actions of the prefilter, the controller, the plant and the sensor and has the transfer function F(s)C(s)G(s)H(s) The closed loop signal is the output of the system and has the transfer function F ( s ) C ( s ) G ( s )  C s G s H s 1 ( ) ( ) ( )
  • 10. Closed Loop System ++ Output y(s) Error signal E(s) Open Loop Signal B(s) Plant P(s) Controller C(s) Input r(s) The closed loop transfer function is             -1     n s n s y s C s P s d s d s n s n s                         ( )               ( ) ( ) 1 1 The characteristic polynomial is ( ) 1 For stability, the roots of ( ) m c p c p c p yr c p c p c p c p c p c p G s r s C s P s n s n s d s d s n s n s d s d s s C s P s d s d s n s n s s        ust have negative real parts While we can check for stability, it does not give us design guidance
  • 11. Note: Your book uses L(s) rather than B(s) To avoid confusion with the Laplace transform, I will use B(s) Open Loop System ++ Output y(s) Error signal E(s) Open Loop Signal B(s) Plant P(s) Controller C(s) Input r(s) Sensor b s n s n s             ( ) The open loop transfer function is ( )    ( ) c p c p B s C s P s r s d s d s -1 If in the closed loop, the input r(s) were sinusoidal and if the signal were to continue in the same form and magnitude after the signal were disconnected, it would be necessary for n  s  n  s  ( w )  c p   1 0     c p B i d s d s
  • 12. Open Loop System Nyquist Plot Error signal E(s) ++ Output y(s) Open Loop Signal B(s) Plant P(s) Controller C(s) Input r(s) Sensor -1 n  s  n  s  ( w )  c p   1 0     c p B i d s d s -1 Real Imaginary Plane of the Open Loop Transfer Function B(0) B(iw) B(i) -1 is called the critical point B(-iw)
  • 13. Simple Nyquist TheoremError signal E(s) ++ Output y(s) Open Loop Signal B(s) Plant P(s) Controller C(s) Input r(s) Sensor -1 -1 Plane of the Open Loop Transfer Function Real Imaginary B(0) B(iw) -B(iw) ()Bi -1 is called the critical point Stable Unstable Simple Nyquist Theorem: For the loop transfer function, B(iw), if B(iw) has no poles in the right hand side, expect for simple poles on the imaginary axis, then the system is stable if there are no encirclements of the critical point -1.
  • 14. Example 1  Plot the Nyquist plot for ( )  2  2 2 B s s s s    1    2 B i ( ) i w i w w i   2 2 w  B (0)   i B (1 i )   0.4  0.2 i B (  1 i )   0.4  0.2 i B (2 i )   0.1  0.05 i B (  2 i )   0.1  0.05 i -1 Im Re Stable
  • 15. Example  Plot the Nyquist plot for 10   ( ) 2 2 B s s s s    10 20 20 10           2 4 ( ) 2 2 4 (0) (1 ) 4 2 ( 1 ) 4 2 (2 ) 1 0.5 ( 2 ) 1 0.5 (4 ) 0.077 0.135 ( 4 ) 0.077 135 i B i i i i B i B i i B i i B i i B i i B i i B i i w w w w w w                          -1 Im Re Unstable
  • 16. Nyquist Gain Scaling  The form of the Nyquist plot is scaled by the system gain K   B s ( ) s s s   2 2   Show with Sisotool
  • 17. Conditional Stabilty  While most system increase stability by decreasing gain, some can be stabilized by increasing gain  Show with Sisotool 2 K s s (0.25 0.12 1)    2  ( ) 1.69 1.09 1 B s s s s   
  • 18. Full Nyquist Theorem  Assume that the transfer function B(iw) with P poles has been plotted as a Nyquist plot. Let N be the number of clockwise encirclements of -1 by B(iw) minus the counterclockwise encirclements of -1 by B(iw)Then the closed loop system has Z=N+P poles in the right half plane.  Show with Sisotool K s i s i ( 5 2 )( 5 2 )             ( ) .5 2 .5 2 2 6 2 6 B s s s i s i s i s i         
  • 19. Summary  Open Loop System  Nyquist Plot  Simple Nyquist Theorem  Nyquist Gain Scaling  Conditional Stability  Full Nyquist Theorem -1 Next Class: Stability Margins Im Re Unstable