EGGN 307 Introduction to Feedback Control Systems Transfer by rma97348

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									              EGGN 307
Introduction to Feedback Control Systems

            Transfer Functions
             (Lectures 13-15)

           Professor Kevin L. Moore
                  Spring 2010

  http://engineering.mines.edu/course/eggn307a
Previously
 3.0 Laplace Transform Modeling

   3.1 Review of Complex Numbers
   3.2 Laplace Transforms
   3.3 Inverse Laplace and LODE Solutions
   3.4 Transfer Functions
   Recall: Laplace differentiation theorem (1)
                                                    Colorado School of Mines


            The differentiation theorem



            Higher order derivatives




Due to Katie Johnson or Tyrone Vincent or someone                              3
   Differentiation Theorem (revisited)
                                                              Colorado School of Mines



                                                                      d n 1
      dn
      dt n
                                      
           f t   s n F s   s n 1 f 0   s n  2 
                                                        d
                                                        dt
                                                           f 0     n 1 f 0 
                                                                      dt
                                                                                  

            Differentiation Theorem when initial conditions are
             zero




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   Introduction to Transfer Functions
                                                    Colorado School of Mines


            Consider a system with input r(t), output x(t), and
             zero initial conditions:



            The Laplace Transform of the output is related to the
             Laplace Transform of the input by a function of s




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   Definition of a Transfer Function
                                                    Colorado School of Mines


            Definition




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•Transfer Function of a General LODE
•Impulse Response of a General LODE
                                         Colorado School of Mines




3.0 Laplace Transform Modeling

3.1 Review of Complex Numbers
3.2 Laplace Transforms
3.3 Inverse Laplace and LODE Solutions
3.4 Transfer Functions
3.5 Block Diagrams




                                                                    13
   Block Diagram Components
                                                                Colorado School of Mines


          Block diagrams are graphical representations of
           algebraic relationships.


              System                            Summing Junction

                             C (s)            R1 ( s )                     Y (s)
    R(s)
                G(s)                                                

                                                         R2 ( s )


       C ( s )  G ( s ) R( s )              Y ( s)   R1 ( s)  R2 ( s)

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   Basic Block Diagram Simplifications (1)
                                                                 Colorado School of Mines


            Cascade Systems
              R(s)                       X (s )                       C (s)
                          G1 ( s )                    G2 ( s )


                 X ( s)  G1 ( s) R( s) C ( s)  G2 ( s) X ( s)

                           C ( s)  G2 ( s)G1 ( s) R( s)

                         R(s)                         C (s)
                                     G2 ( s)G1 ( s)


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   Basic Block Diagram Simplifications (2)
                                                          Colorado School of Mines


            Systems In Parallel


                                   G1 ( s )
                                                
              R(s)
                                                
                                   G2 ( s )         C ( s)  G1 ( s) R( s)  G2 ( s) R( s)




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   Basic Block Diagram Simplifications (3)
                                                        Colorado School of Mines


            Feedback Connections

               R(s)         E (s)                     C (s)
                                        G(s)
                            


                                        H (s )



                               C ( s)  G( s) E ( s)
                               E (s)  R(s)  H (s)C (s)

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   Basic Block Diagram Simplifications (4)
                                                            Colorado School of Mines




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


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




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   Example 1
                                                         Colorado School of Mines


            Simplify the following block diagram. All variables are
             functions of s.

             R                                                           C
                        G1                          G2
                                      




                                                    H1
                                             

                                             
                                                    H2




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   Example 1 (2)
                                                           Colorado School of Mines




                                  
           R            G1                            G2                    C
                                      




                                                    H1  H 2




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   Example 1 (3)
                                                                  Colorado School of Mines




           R                                            G2                         C
                        G1                      1  G2 ( H1  H 2 )




                        R                G2G1                    C
                                   1  G2 ( H1  H 2 )




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   Other Manipulations (1)
                                                            Colorado School of Mines


            Sometimes additional manipulations are needed to
             apply basic simplifications
            Goal: move the pick-off point so that the dotted box
             has a single input and a single output.

                                                       
               R                                                   C
                                         G1
                          



                                         G2


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   Other Manipulations (2)
                                                         Colorado School of Mines


            Pick off points can be moved around blocks

                                                                        1
                                                                        G1

               G1                                   G1

                                                         G1

               G1                                        G1



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   Other Manipulations (3)
                                                                     Colorado School of Mines


                                                       
                           R                                    C
                                            G1
                                   



                                            G2


            Equivalent Diagram
                                                            1
                                                            G1
                                                                    
               R                                                                C
                                       G1
                          



                                       G2

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   Example 2
                                                    Colorado School of Mines




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   Example 2 (2)
                                                    Colorado School of Mines




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   Block diagrams just represent algebra
                                                            Colorado School of Mines


            If you ever get stuck – just convert back to algebraic
             relationships and solve by hand
            Add variables at inputs to blocks


                                                       
               R                  A                                 C
                                         G1
                          


      C  G1 A  A                              B
                                         G2
      A  R  G2 B
      B  G1 A


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   Block diagrams just represent algebra (2)
                                                         Colorado School of Mines




          C  G1 A  A                  A  R  G2G1 A
           A  R  G2 B                        1
                                        A          R
           B  G1 A                        1  G2G1


                                   1           1
                      C  G1            R          R
                               1  G2G1    1  G2G1

                                    G1  1
                               C          R
                                  1  G2G1


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                                   Colorado School of Mines




3.6 Transfer Functions of Physical Systems


    Mesh   and Nodal Equations




                                                              29
   Electrical impedance
                                                               Colorado School of Mines


            To work with algebraic relationships, we take the Laplace
             Transform of the defining equations with zero initial conditions
              – Recall: Transfer Function  zero initial conditions


                resistor             capacitor             inductor
                                       dv                       di
                   v  iR             C i                  vL
                                        dt                      dt
              V ( s)  RI ( s)     CsV ( s )  I ( s )   V ( s )  LsI ( s )
                                           1
                                   V (s)     I ( s)
                                           Cs
            Impedance: Ratio of Laplace Transform of across variable to
             Laplace Transform of through variable



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   Finding Transfer Functions using mesh or nodal equations
                                                        Colorado School of Mines

                         Vout s 
          Example: Find
                         Vin s 
                                 R
                                                         

                            
                      Vin                   C       L    Vout
                            


                                                         
            Steps
             1. Convert elements to impedances
             2. Either
                • apply Kirchoff’s Current Law (KCL) at every node, or
                • apply Kirchoff’s Voltage Law (KVL) around every mesh
             3. Solve resulting system of equations for output variable


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   Mesh equations from KVL
                                                                             Colorado School of Mines


                                       R        
                                                                             
                                      I1 ( s )              I 2 (s)
                              
                   Vin (s )
                              
                                                              1
                                                                        sL   Vout ( s )  sLI 2 ( s )
                                                             sC
                                                      
                                                                             

                                                1
                      Vin ( s)  RI1 ( s)         ( I1 ( s)  I 2 ( s))
                                               sC
                                  1
                             0       ( I 2 ( s)  I1 ( s))  LsI 2 ( s)
                                 sC
            Two equations: need to solve for I 2 ( s )



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   Cramer’s Rule
                                                        Colorado School of Mines


             Cramer’s Rule is a tool that can be used to solve
              algebraic equations.

                   b1   a11 a12   x1 
         If
                  b    a   a22   x2 
                   2   21        
                                        determinant
         Then           b1     a12                    a11     b1
                      b2 a22                         a21 b2
                 x1                            x2 
                      a11 a12                        a11 a12
                       a21 a22                        a21 a22

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   Mesh equations in Matrix-Vector form
                                                             Colorado School of Mines


            The mesh equations can be re-written in matrix-
             vector form
                                                 1
                       Vin ( s)  RI1 ( s)         ( I1 ( s)  I 2 ( s))
                                               sC
                                   1
                              0      ( I 2 ( s )  I1 ( s ))  LsI 2 ( s )
                                  sC

                     Vin ( s ) R  1        
                                                  1   I ( s) 
                                 sC          sC    1 
                                  1              1         
                      0   sC              Ls    I 2 ( s )
                                                sC 

            Use Cramer’s Rule


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   Mesh equations in Matrix-Vector form (2)
                                                           Colorado School of Mines



                     Vin ( s ) R  1       
                                                 1   I ( s) 
                                 sC         sC    1 
                                  1             1         
                      0    sC
                               
                                             Ls    I 2 ( s )
                                                  sC 

                           1
                      R      Vin ( s )
                          sC
                          1
                                0                  1
                        sC                         sC Vin ( s )
          I 2 ( s)                     
                     R
                         1
                              
                                   1      R  sC Ls  sC    sC 
                                                1           1      1 2

                        sC       sC
                        1            1
                            Ls 
                       sC           sC

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   Mesh equations in Matrix-Vector form (3)
                                                            Colorado School of Mines




                                 Vout ( s )  sLI 2 ( s )

                                          sLVin ( s )
                           Vout ( s )  2
                                       s RLC  Ls  R


                           Vout ( s )     sL
                                       2
                           Vin ( s ) s RLC  Ls  R




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   Mesh equations from circuit diagram
                                                                 Colorado School of Mines

            Mesh equations in matrix/vector form:
         sum of impedances on mesh 1                     shared impedance


                     Vin ( s ) R  1        
                                                  1   I ( s) 
                                 sC          sC    1 
                                  1              1         
                      0   sC              Ls    I 2 ( s )
                                                sC 
                                                              sum of impedances on mesh 2
                                       R
                                                                 
                                   I1 ( s )    I 2 (s)
                                              1
                    Vin (s )                                sL Vout (s )
                               
                                              sC
                                                                 

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   Exercise: Find the mesh equations
                                                                                   Colorado School of Mines

                   1
                                                                      Vin  I1  I1  I 2 
                                                                                1
                 I1 ( s )                 I 2 (s)                                s
           
                                                                       0   I1  I 2   3I 2  2I 3  I 2 
     Vin                                1
                                                            3
                                                                              1
           
                                        s                                      s
                       I1  I 2                I3  I 2
                                                                      0  2I 3  I 2   Vout
                                                                 
               I 3  I1
                                            2
                                                            2s Vout
                             I 3 ( s)
                                                                 


                  Vin ( s ) 1  1
                                   s                        1
                                                             s     0   I1 ( s ) 
                   0 1                                               I ( s )
                                                          s 23  2
                                                          1
                             s                                        2 
                   0   0
                                                         2   2  2s   I 3 ( s )
                                                                                   
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   Mechanical impedance
                                                           Colorado School of Mines




        Element        Equation        Impedance           Admittance

         mass           f  M
                             x
                                         X (s)
                                         F (s)       1
                                                     Ms2
                                                            F (s)
                                                            X (s)    Ms2

       damper           f  Dx
                                        X (s)
                                         F (s)      1
                                                     Ds
                                                            F (s)
                                                            X (s)     Ds


        spring          f  Kx
                                         X (s)
                                         F (s)   K
                                                  1          F (s)
                                                             X (s)   K

           Admittance, or the inverse of impedance, is more commonly
            used in mechanical systems.
           Both admittances and impedances are commonly used in
            electrical systems.

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   Nodal equations for current input
                                                                         Colorado School of Mines




                                            V1        R   V2
                                                                     

                                                                1
                 I in (s )                       Ls
                                                               sC   Vout ( s )  V2 ( s )
                                                                     
   sum of admittances connected to node 1                             negative of shared admittance



                     I in ( s )  Ls  R
                                    1    1
                                            R  V1 ( s ) 
                                             1

                     0   1                1         
                                 R Cs  R  V2 ( s )


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   Nodal equations for mechanical systems
                                                                          Colorado School of Mines


                    K1                             K2                               u
                                   M1                                M2


                           x1                                        x2
         Newton’s Laws at each node
                   M 11   K1 x1  K 2 ( x1  x2 )
                      x
                         M 2 2   K 2 ( x2  x1 )  u
                             x
         Newton’s Laws with Admittances
                0  M 1s 2 X 1 ( s)  K1 X 1 ( s)  K 2 ( X 1 ( s)  X 2 ( s))
           U ( s)  M 2 s 2 X 2 ( s)  K 2 ( X 2 ( s)  X 1 ( s))

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   Nodal equations from mechanical diagram
                                                             Colorado School of Mines


     The same pattern as electrical nodal equations!

       K1                K2              u
                M1               M2


                x1                x2                Sum of admittances


                                                     Negative of shared admittances


       0   M 1s 2  K1  K 2                  K 2   X 1( s) 
     U ( s )                                                   
                  K2                     M 2 s  K 2   X 2 ( s )
                                                  2




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          Colorado School of Mines




•Motors




                                     43
   Electromechanical systems (1)
                                                             Colorado School of Mines


            What happens when you run current through a wire
             in a magnetic field?
                                      f    (out of screen)
               B



                                  i

            Lorentz force equation (SI units) f  i  B
            Key result: Magnitude of force is proportional to
             current and magnetic field
                                          f  Ki
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   Electromechanical Systems (2)
                                                                 Colorado School of Mines


            What happens when you move a wire through a
             magnetic field?
                                     
                                     x         (out of screen)
               B                 



                                 Vb        i
                                 
            Conservation of energy:             iVb  xf
                                                       
                                                 iVb  xKi
                                                       
            Key result: induced voltage is proportional to velocity
                                         Vb  xK
                                              
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   Recall - DC Motor
                                                           Colorado School of Mines


            DC motors are used in many control applications
             (e.g., robots, disk drives)




            Torque on load                           K f i f - magneticflux
                                                    
                                                                              
                                                                               
                 Tm  K1ia  K1 K f i f ia                 Tm - torque       
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   Field Controlled DC motor – armature current fixed
                                                                        Colorado School of Mines


            Circuit                     Transducer              Mechanical load

                                 Tm (t )  K1K f ia i f (t )     Tm

                                            K mi f (t )


         I f ( s)        1                  Tm ( s )                  (s)      1
                                                     Km                    2
         V f ( s) ( L f s  R f )           I f (s)                 Tm ( s ) Js  bs

            Overall transfer function
                           ( s)                  K m / JL f
                                     
                          V f ( s)       s( s  b / J )(s  R f / L f )

Due to Katie Johnson or Tyrone Vincent or someone                                                  47
   Armature controlled DC motor – field current fixed (1)
                                                           Colorado School of Mines

                                                  
                                         Vb  K b  K b
            Key relationships:
                                         Tm  K1K t i f ia  K mia
                   Ra              La
                                                    Tm        b
                                    
              
                           Ia
        Va    
                                    Vb      J
                                    
                        Armature




Due to Katie Johnson or Tyrone Vincent or someone                                     48
   Armature controlled DC motor – field current fixed (2)
                                                    Colorado School of Mines

                                                Va  Vb
            Note (from circuit diagram): I a 
                                                Ra  La s

            Thus, Tm  K m I a

                        
                            Km
                                    Va  Vb 
                          Ra  La s


            Since net torque = Tm  Td  TL , we have
                                TL        1
                                    ,  
                               Js  b     s


Due to Katie Johnson or Tyrone Vincent or someone                              49
   Armature controlled DC motor – field current fixed (2)
                                                                            Colorado School of Mines


            Closed loop transfer function
                        ( s)
                                               Km            1
                                1          ( Ra  La s ) ( Js b )
                              
                      Va ( s ) s (1             Kb K m         1
                                              ( Ra  La s ) ( Js b )   )

                   (s)                    Km
                         
                 Va ( s ) s (( Ra  La s )( Js  b)  K b K m )

            When inductance is negligible:

                             ( s)
                                                      Km

                                      
                                                                       
                                                      Ra

                           Va ( s )       s Js  b  K b         Km
                                                                 Ra




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   Example
                                                                    Colorado School of Mines

                      Ra
                                             b  8 N  m  s rad

          Va                   
                


                                    J  7 kg  m 2


                                                     m
                                             500
                     Steady state                         Va  100 V
                    motor load curve

                                                               50       m

                                            (s)
               Find the transfer function
                                           Va ( s )
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   Example (2)
                                                    Colorado School of Mines


            Motor constant from stall torque
                                           Va  Vb Va
             m  0  Vb  kbm  0  ia         
                                             Ra     Ra
                           K mVa
              m  K mia        @ m  0
                            Ra

                    K m 100
              500 
                      Ra
               Km
                   5
                Ra




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   Example (3)
                                                    Colorado School of Mines


            Motor constant from no-load velocity

              m  0  ia  0  Va  Vb

             Vb  K b  Va  K b @  m  0

             100  K b 50

             2  Kb




Due to Katie Johnson or Tyrone Vincent or someone                              53
   Example (4)
                                                                  Colorado School of Mines


            Transfer function
                                        Km
                                             Ra   5
                                         Kb  2
                                             J 7
                                             b8

                         (s)                     Km
                                   
                                                       Ra

                        Va ( s )       s Js  b  K b K m Ra 
                                          5
                                   
                                     s7 s  18


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   Hydraulic Actuator
                                                           Colorado School of Mines

            Used for large loads
                                                    load
                                          x                           y
                                         Q
                drain 
                        ps
                         
                                                               
              supply                                           P
                         
                        ps
                                             Q                 

                         
                drain
            Nonlinear flow rate Q  g ( x, P)
            Piston area A
            Pressure P
Due to Katie Johnson or Tyrone Vincent or someone                                     55
   Two dimensional nonlinear function
                                                               Colorado School of Mines


            Operating point ( x0 , P0 )

            Linearized flow
                                 g                 g           
                           Q                x                P
                                 x ( x , P )       P ( x , P ) 
                                       0 0                0 0 


                               k xx  k PP
                 where
                                            g
                                     kx 
                                            x   ( x0 , P0 )

                                            g
                                     kP  
                                            P ( x0 , P0 )

Due to Katie Johnson or Tyrone Vincent or someone                                         56
   Free Body Diagram
                                                        Colorado School of Mines


           Piston pressure/force and flow/position relationships
                                           dy Q
                    f  PA                   
                                           dt A

                    load          y
                                      Y (s)
                      f                       N ( s)
                          f           F ( s)
                                  y
            Q


                              
                                                            Y s 
                                                                     Ps 
                                                           N s  A
                              P
                Q             




Due to Katie Johnson or Tyrone Vincent or someone                                  57
   Hydraulic transfer function
                                                                Colorado School of Mines


            Small signal relationships in Laplace domain
                         Q( s)  k xX ( s)  k PP( s)
                                         1
                           P ( s )          Y ( s )
                                      N (s) A
                                     1
                           Y ( s)  Q( s)
                                    sA
                                   Q(s)                           Y (s)
       X (s)       kx                             1
                                                 sA

                                             P(s)          1
                                     kp
                                                         N ( s) A

Due to Katie Johnson or Tyrone Vincent or someone                                          58

								
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