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        Modeling in the Frequency
                Domain
•   Review of the Laplace transform

•   Learn how to find a mathematical model, called a transfer
    function, for linear, time-invariant electrical, mechanical,
    and electromechanical systems

•   Learn how to linearize a nonlinear system in order to find
    the transfer function




                                                  Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                    Copyright © 2004 by John Wiley & Sons. All rights reserved.
2

                    Introduction to Modeling
•       we look for a mathematical representation where the input, output, and
        system are distinct and separate parts




    •    also, we would like to represent conveniently the interconnection of
         several subsystems. For example, we would like to represent cascaded
         interconnections, where a mathematical function, called a transfer
         function, is inside each block




                                                            Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                              Copyright © 2004 by John Wiley & Sons. All rights reserved.
3

                  Laplace Transform Review

•   a system represented by a differential equation is difficult to model
    as a block diagram
•   on the other hand, a system represented by a Laplace transformed
    differential equation is easier to model as a block diagram



    The Laplace transform is defined as




     where s = σ + jω, is a complex variable.



                                                        Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                          Copyright © 2004 by John Wiley & Sons. All rights reserved.
4

              Laplace Transform Review

The inverse Laplace transform, which allows us
to find f (t) given F(s), is




   where




                                            Control Systems Engineering, Fourth Edition by Norman S. Nise
                                              Copyright © 2004 by John Wiley & Sons. All rights reserved.
5

Laplace Transform Review
  Laplace transform table




                            Control Systems Engineering, Fourth Edition by Norman S. Nise
                              Copyright © 2004 by John Wiley & Sons. All rights reserved.
6



Problem: Find the Laplace transform of


Solution:




                                         Control Systems Engineering, Fourth Edition by Norman S. Nise
                                           Copyright © 2004 by John Wiley & Sons. All rights reserved.
7

Laplace Transform Review
 Laplace transform theorems




                              Control Systems Engineering, Fourth Edition by Norman S. Nise
                                Copyright © 2004 by John Wiley & Sons. All rights reserved.
8

Laplace Transform Review
 Laplace transform theorems




                              Control Systems Engineering, Fourth Edition by Norman S. Nise
                                Copyright © 2004 by John Wiley & Sons. All rights reserved.
9
                 Partial-Fraction Expansion

•      To find the inverse Laplace transform of a complicated function,
       we can convert the function to a sum of simpler terms for which we
       know the Laplace transform of each term.

                                       N ( s)
                            F ( s) =
                                       D( s)

    Case 1: Roots of the Denominator of F(s) Are Real and Distinct

    Case 2: Roots of the Denominator of F(s) Are Real and Repeated

    Case 3: Roots of the Denominator of F(s) Are Complex or Imaginary




                                                         Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                           Copyright © 2004 by John Wiley & Sons. All rights reserved.
10

Case 1: Roots of the Denominator of F(s) Are Real and Distinct

  If the order of N(s) is less than the order of D(s), then




 Thus, if we want to find Km, we multiply above equation by ( s + pm )




                                                         Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                           Copyright © 2004 by John Wiley & Sons. All rights reserved.
11




If we substitute s = − pm in the above equation, then we can find Km




                                                    Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                      Copyright © 2004 by John Wiley & Sons. All rights reserved.
12


Problem Given the following differential equation, solve for y(t) if
all initial conditions are zero. Use the Laplace transform.




 Solution




                                                    Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                      Copyright © 2004 by John Wiley & Sons. All rights reserved.
13




Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
14

Case 2: Roots of the Denominator of F(s) Are Real and Repeated




 First, we multiply by (s+ p )r and we can solve immediately for K1 if s= - p1
                            1




                                                          Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                            Copyright © 2004 by John Wiley & Sons. All rights reserved.
15




•   we can solve for K2 if we differentiate F1(s) with respect to s and
    then let s approach –p1

•   subsequent differentiation allows to find K3 through Kr

•   the general expression for K1 through Kr for the multiple roots is




                                                      Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                        Copyright © 2004 by John Wiley & Sons. All rights reserved.
16
Case 3: Roots of the Denominator of F(s) Are Complex or Imaginary




                                           has complex or pure imaginary roots

 •   the coefficients K2, K3 are found through balancing the coefficients
     of the equation after clearing fractions

      (K 2s + K3 )
                   is put in the form of                            by completing
     ( s + as + b)
        2



     the squares on ( s 2 + as + b) and adjusting the numerator


                                                          Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                            Copyright © 2004 by John Wiley & Sons. All rights reserved.
17




Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
18


                   The Transfer Function (1)
Let us consider a general nth-order, linear, time-invariant differential
equation is given as:




where c(t) is the output, r(t) is the input, and ai, bi are coefficients.


Taking the Laplace transform of both sides (assuming all initial
conditions are zero) we obtain




                                                        Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                          Copyright © 2004 by John Wiley & Sons. All rights reserved.
19


                The Transfer Function (2)
Transfer function is the ratio G(s) of the output transform, C(s),
divided by the input transform, R(s)




               C ( s)            (bm s m + bm −1s m −1 + ... + b0 )
                      = G ( s) =
               R( s)             (an s n + an −1s n −1 + ... + a0 )




                                                                  Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                    Copyright © 2004 by John Wiley & Sons. All rights reserved.
20

         Electric Network Transfer Functions

Voltage-current, voltage-charge, and impedance relationships for capacitors,
resistors, and inductors




                                                             Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                               Copyright © 2004 by John Wiley & Sons. All rights reserved.
21


Problem Find the transfer function relating the capacitor voltage,
Vc(s), to the input voltage, V(s)




 Solution:




                                                    Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                      Copyright © 2004 by John Wiley & Sons. All rights reserved.
22




Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
23

Transfer function-single loop via transform methods




                                               Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                 Copyright © 2004 by John Wiley & Sons. All rights reserved.
24

Complex Electric Circuits via Mesh Analysis

 1. Replace passive element values with their impedances.

 2. Replace all sources and time variables with their Laplace
    transform.

 3. Assume a transform current and a current direction in each
    mesh.

 4. Write Kirchhoff's voltage law around each mesh.

 5. Solve the simultaneous equations for the output.

 6. Form the transfer function



                                                 Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                   Copyright © 2004 by John Wiley & Sons. All rights reserved.
25
Problem: find the transfer function, I2(s)/V(s)




                                                  Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                    Copyright © 2004 by John Wiley & Sons. All rights reserved.
26




Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
27


Complex Electric Circuits via Nodal Analysis

1. Replace passive element values with their admittances

2. Replace all sources and time variables with their Laplace
   transform.

3. Replace transformed voltage sources with transformed current
   sources.

4. Write Kirchhoff's current law at each node.

5. Solve the simultaneous equations for the output.

6. Form the transfer function.


                                                      Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                        Copyright © 2004 by John Wiley & Sons. All rights reserved.
28


                   Operational Amplifiers




Inverting OP AMP




                                        Control Systems Engineering, Fourth Edition by Norman S. Nise
                                          Copyright © 2004 by John Wiley & Sons. All rights reserved.
29

Non-inverting OP AMP




                       Control Systems Engineering, Fourth Edition by Norman S. Nise
                         Copyright © 2004 by John Wiley & Sons. All rights reserved.
30

Example




          Control Systems Engineering, Fourth Edition by Norman S. Nise
            Copyright © 2004 by John Wiley & Sons. All rights reserved.
31

  Translational Mechanical System Transfer
                  Functions
Newton's law: The sum of forces on a body equals zero; the sum of
moments on a body equals zero.




                                                                           K… spring
                                                                           constant




                                                                           fv … coefficient of
                                                                           viscious friction



                                                                           M … mass




                                                 Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                   Copyright © 2004 by John Wiley & Sons. All rights reserved.
32




Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
33

Problem Find the transfer function, X(s)/F(s), for the system




Solution:




                                                   Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                     Copyright © 2004 by John Wiley & Sons. All rights reserved.
34


    Translational Mechanical System Transfer
                    Functions

•   The required number of equations of motion is equal to the
    number of linearly independent motions. Linear independence
    implies that a point of motion in a system can still move if all
    other points of motion are held still.

•   Another name for the number of linearly independent motions
    is the number of degrees of freedom.




                                                      Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                        Copyright © 2004 by John Wiley & Sons. All rights reserved.
35

    Problem: Find the transfer function, X2(s)/F(s), of the system




Solution
•      we draw the free-body diagram for each point of motion and then use
       superposition

•      for each free-body diagram we begin by holding all other points of
       motion still and finding the forces acting on the body due only to its
       own motion.

•      then we hold the body still and activate the other points of motion one
       at a time, placing on the original body the forces created by the
       adjacent motion


                                                              Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                Copyright © 2004 by John Wiley & Sons. All rights reserved.
36




a. Forces on M1 due only
to motion of M1




b. forces on M1 due only to
motion of M2




   c. all forces on M1


                              Control Systems Engineering, Fourth Edition by Norman S. Nise
                                Copyright © 2004 by John Wiley & Sons. All rights reserved.
37




a. Forces on M2 due only
to motion of M2;




 b. forces on M2 due only
 to motion




  c. all forces on M2




                            Control Systems Engineering, Fourth Edition by Norman S. Nise
                              Copyright © 2004 by John Wiley & Sons. All rights reserved.
38




Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
39




Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
40


          Rotational Mechanical System Transfer
                        Functions
Torque-angular velocity, torque-angular displacement, and
impedance rotational relationships for springs, viscous
dampers, and inertia




                                                       •    the mass is replaced by inertia

                                                       •    the values of K, D, and J are
                                                            called spring constant, coefficient
                                                            of viscous friction, and moment of
                                                            inertia, respectively
                                                       •    the concept of degrees of freedom
                                                            is the same as for translational
                                                            movement



                                                                      Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                        Copyright © 2004 by John Wiley & Sons. All rights reserved.
41


Problem Find the transfer function




Solution
           a. Torques on J1
           due only to the
           motion of J1




                                     Control Systems Engineering, Fourth Edition by Norman S. Nise
                                       Copyright © 2004 by John Wiley & Sons. All rights reserved.
42




b. torques on J1
due only to the
motion of J2




c. final free-body
diagram for J1




                     Control Systems Engineering, Fourth Edition by Norman S. Nise
                       Copyright © 2004 by John Wiley & Sons. All rights reserved.
43




a. Torques on J1
due only to the
motion of J1
b. torques on J1
due only to the
motion of J2
c. final free-body
diagram for J1




                     Control Systems Engineering, Fourth Edition by Norman S. Nise
                       Copyright © 2004 by John Wiley & Sons. All rights reserved.
44




a. Torques on J2
due only to the
motion of J2
b. torques on J2
due only to the
motion of J1
c. final free-body
diagram for J2




                     Control Systems Engineering, Fourth Edition by Norman S. Nise
                       Copyright © 2004 by John Wiley & Sons. All rights reserved.
45




Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
46




Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
47

Transfer Functions for Systems with Gears




- the distance travelled along each gear's circumference is the same

                        θ 2 r1 N1
                           = =
                        θ1 r2 N 2

- the energy into Gear 1 equals the energy out of Gear 2

                        T2 θ1 N 2
                          =   =
                        T1 θ 2 N1




                                                  Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                    Copyright © 2004 by John Wiley & Sons. All rights reserved.
48




T2 θ1 N 2
  =   =
T1 θ 2 N1




            Control Systems Engineering, Fourth Edition by Norman S. Nise
              Copyright © 2004 by John Wiley & Sons. All rights reserved.
49




Rotational mechanical impedances can be reflected through gear
trains by multiplying the mechanical impedance by the ratio




                                                  Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                    Copyright © 2004 by John Wiley & Sons. All rights reserved.
50


A gear train is used to implement large gear ratios by cascading
smaller gear ratios.




                                                   Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                     Copyright © 2004 by John Wiley & Sons. All rights reserved.
51

Problem Find the transfer function,




 Solution:




                                      Control Systems Engineering, Fourth Edition by Norman S. Nise
                                        Copyright © 2004 by John Wiley & Sons. All rights reserved.
52

Electromechanical System Transfer Functions
Electromechanical systems: robots, hard disk drives, …

Motor is an electromechanical component that yields a displacement output
for a voltage input, that is, a mechanical output generated by an electrical
input.




                                                         Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                           Copyright © 2004 by John Wiley & Sons. All rights reserved.
53

    Derivation of the Transfer Function of Motor




Back electromotive force (back emf):




               where Kb is a constant of proportionality called the back emf constant;

                       dθ m / dt = ωm (t ) is the angular velocity of the motor

Taking the Laplace transform, we get:


                             Torque:
                                                                Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                  Copyright © 2004 by John Wiley & Sons. All rights reserved.
54




The relationship between the armature current, ia(t), the applied armature voltage,
ea(t), and the back emf, vb(t),




                       Tm ( s )
          I a ( s) =
                        Kt



                                                               Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                 Copyright © 2004 by John Wiley & Sons. All rights reserved.
55




We assume that the armature inductance, La, is small compared to the armature
resistance, Ra,




                                                          Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                            Copyright © 2004 by John Wiley & Sons. All rights reserved.
56




   The mechanical constants Jm and Dm can be found as:




Electrical constants Kt/Ra and Kb can be found through a dynamometer test of
the motor.




                                                              Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                Copyright © 2004 by John Wiley & Sons. All rights reserved.
57




K t Tstall
   =
Ra   ea




                       ea
             Kb =
                    ωno −load




                                Control Systems Engineering, Fourth Edition by Norman S. Nise
                                  Copyright © 2004 by John Wiley & Sons. All rights reserved.
58

                         Nonlinearities

A linear system possesses two properties: superposition and homogeneity.

              Superposition - the output response of a system to the sum of
              inputs is the sum of the responses to the individual inputs.

              Homogeneity - describes the response of the system to a
              multiplication of the input by a scalar. Multiplication of an input
              by a scalar yields a response that is multiplied by the same
              scalar.




                      Linear system                     Nonlinear system
                                                              Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                Copyright © 2004 by John Wiley & Sons. All rights reserved.
59

Nonlinearities




                 Control Systems Engineering, Fourth Edition by Norman S. Nise
                   Copyright © 2004 by John Wiley & Sons. All rights reserved.
60

                        Linearization
•   if any nonlinear components are present, we must linearize the system
    before we can find the transfer function

•   we linearize nonlinear differential equation for small-signal inputs about
    the steady-state solution when the small signal input is equal to zero




                                        ma is the slope of the curve at point A




                                                            Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                              Copyright © 2004 by John Wiley & Sons. All rights reserved.
61




Solution:



  At π/2 df/dx = - 5




                       Control Systems Engineering, Fourth Edition by Norman S. Nise
                         Copyright © 2004 by John Wiley & Sons. All rights reserved.

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02 elec3114

  • 1. 1 Modeling in the Frequency Domain • Review of the Laplace transform • Learn how to find a mathematical model, called a transfer function, for linear, time-invariant electrical, mechanical, and electromechanical systems • Learn how to linearize a nonlinear system in order to find the transfer function Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 2. 2 Introduction to Modeling • we look for a mathematical representation where the input, output, and system are distinct and separate parts • also, we would like to represent conveniently the interconnection of several subsystems. For example, we would like to represent cascaded interconnections, where a mathematical function, called a transfer function, is inside each block Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 3. 3 Laplace Transform Review • a system represented by a differential equation is difficult to model as a block diagram • on the other hand, a system represented by a Laplace transformed differential equation is easier to model as a block diagram The Laplace transform is defined as where s = σ + jω, is a complex variable. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 4. 4 Laplace Transform Review The inverse Laplace transform, which allows us to find f (t) given F(s), is where Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 5. 5 Laplace Transform Review Laplace transform table Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 6. 6 Problem: Find the Laplace transform of Solution: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 7. 7 Laplace Transform Review Laplace transform theorems Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 8. 8 Laplace Transform Review Laplace transform theorems Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 9. 9 Partial-Fraction Expansion • To find the inverse Laplace transform of a complicated function, we can convert the function to a sum of simpler terms for which we know the Laplace transform of each term. N ( s) F ( s) = D( s) Case 1: Roots of the Denominator of F(s) Are Real and Distinct Case 2: Roots of the Denominator of F(s) Are Real and Repeated Case 3: Roots of the Denominator of F(s) Are Complex or Imaginary Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 10. 10 Case 1: Roots of the Denominator of F(s) Are Real and Distinct If the order of N(s) is less than the order of D(s), then Thus, if we want to find Km, we multiply above equation by ( s + pm ) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 11. 11 If we substitute s = − pm in the above equation, then we can find Km Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 12. 12 Problem Given the following differential equation, solve for y(t) if all initial conditions are zero. Use the Laplace transform. Solution Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 13. 13 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 14. 14 Case 2: Roots of the Denominator of F(s) Are Real and Repeated First, we multiply by (s+ p )r and we can solve immediately for K1 if s= - p1 1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 15. 15 • we can solve for K2 if we differentiate F1(s) with respect to s and then let s approach –p1 • subsequent differentiation allows to find K3 through Kr • the general expression for K1 through Kr for the multiple roots is Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 16. 16 Case 3: Roots of the Denominator of F(s) Are Complex or Imaginary has complex or pure imaginary roots • the coefficients K2, K3 are found through balancing the coefficients of the equation after clearing fractions (K 2s + K3 ) is put in the form of by completing ( s + as + b) 2 the squares on ( s 2 + as + b) and adjusting the numerator Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 17. 17 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 18. 18 The Transfer Function (1) Let us consider a general nth-order, linear, time-invariant differential equation is given as: where c(t) is the output, r(t) is the input, and ai, bi are coefficients. Taking the Laplace transform of both sides (assuming all initial conditions are zero) we obtain Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 19. 19 The Transfer Function (2) Transfer function is the ratio G(s) of the output transform, C(s), divided by the input transform, R(s) C ( s) (bm s m + bm −1s m −1 + ... + b0 ) = G ( s) = R( s) (an s n + an −1s n −1 + ... + a0 ) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 20. 20 Electric Network Transfer Functions Voltage-current, voltage-charge, and impedance relationships for capacitors, resistors, and inductors Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 21. 21 Problem Find the transfer function relating the capacitor voltage, Vc(s), to the input voltage, V(s) Solution: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 22. 22 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 23. 23 Transfer function-single loop via transform methods Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 24. 24 Complex Electric Circuits via Mesh Analysis 1. Replace passive element values with their impedances. 2. Replace all sources and time variables with their Laplace transform. 3. Assume a transform current and a current direction in each mesh. 4. Write Kirchhoff's voltage law around each mesh. 5. Solve the simultaneous equations for the output. 6. Form the transfer function Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 25. 25 Problem: find the transfer function, I2(s)/V(s) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 26. 26 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 27. 27 Complex Electric Circuits via Nodal Analysis 1. Replace passive element values with their admittances 2. Replace all sources and time variables with their Laplace transform. 3. Replace transformed voltage sources with transformed current sources. 4. Write Kirchhoff's current law at each node. 5. Solve the simultaneous equations for the output. 6. Form the transfer function. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 28. 28 Operational Amplifiers Inverting OP AMP Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 29. 29 Non-inverting OP AMP Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 30. 30 Example Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 31. 31 Translational Mechanical System Transfer Functions Newton's law: The sum of forces on a body equals zero; the sum of moments on a body equals zero. K… spring constant fv … coefficient of viscious friction M … mass Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 32. 32 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 33. 33 Problem Find the transfer function, X(s)/F(s), for the system Solution: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 34. 34 Translational Mechanical System Transfer Functions • The required number of equations of motion is equal to the number of linearly independent motions. Linear independence implies that a point of motion in a system can still move if all other points of motion are held still. • Another name for the number of linearly independent motions is the number of degrees of freedom. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 35. 35 Problem: Find the transfer function, X2(s)/F(s), of the system Solution • we draw the free-body diagram for each point of motion and then use superposition • for each free-body diagram we begin by holding all other points of motion still and finding the forces acting on the body due only to its own motion. • then we hold the body still and activate the other points of motion one at a time, placing on the original body the forces created by the adjacent motion Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 36. 36 a. Forces on M1 due only to motion of M1 b. forces on M1 due only to motion of M2 c. all forces on M1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 37. 37 a. Forces on M2 due only to motion of M2; b. forces on M2 due only to motion c. all forces on M2 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 38. 38 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 39. 39 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 40. 40 Rotational Mechanical System Transfer Functions Torque-angular velocity, torque-angular displacement, and impedance rotational relationships for springs, viscous dampers, and inertia • the mass is replaced by inertia • the values of K, D, and J are called spring constant, coefficient of viscous friction, and moment of inertia, respectively • the concept of degrees of freedom is the same as for translational movement Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 41. 41 Problem Find the transfer function Solution a. Torques on J1 due only to the motion of J1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 42. 42 b. torques on J1 due only to the motion of J2 c. final free-body diagram for J1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 43. 43 a. Torques on J1 due only to the motion of J1 b. torques on J1 due only to the motion of J2 c. final free-body diagram for J1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 44. 44 a. Torques on J2 due only to the motion of J2 b. torques on J2 due only to the motion of J1 c. final free-body diagram for J2 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 45. 45 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 46. 46 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 47. 47 Transfer Functions for Systems with Gears - the distance travelled along each gear's circumference is the same θ 2 r1 N1 = = θ1 r2 N 2 - the energy into Gear 1 equals the energy out of Gear 2 T2 θ1 N 2 = = T1 θ 2 N1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 48. 48 T2 θ1 N 2 = = T1 θ 2 N1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 49. 49 Rotational mechanical impedances can be reflected through gear trains by multiplying the mechanical impedance by the ratio Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 50. 50 A gear train is used to implement large gear ratios by cascading smaller gear ratios. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 51. 51 Problem Find the transfer function, Solution: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 52. 52 Electromechanical System Transfer Functions Electromechanical systems: robots, hard disk drives, … Motor is an electromechanical component that yields a displacement output for a voltage input, that is, a mechanical output generated by an electrical input. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 53. 53 Derivation of the Transfer Function of Motor Back electromotive force (back emf): where Kb is a constant of proportionality called the back emf constant; dθ m / dt = ωm (t ) is the angular velocity of the motor Taking the Laplace transform, we get: Torque: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 54. 54 The relationship between the armature current, ia(t), the applied armature voltage, ea(t), and the back emf, vb(t), Tm ( s ) I a ( s) = Kt Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 55. 55 We assume that the armature inductance, La, is small compared to the armature resistance, Ra, Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 56. 56 The mechanical constants Jm and Dm can be found as: Electrical constants Kt/Ra and Kb can be found through a dynamometer test of the motor. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 57. 57 K t Tstall = Ra ea ea Kb = ωno −load Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 58. 58 Nonlinearities A linear system possesses two properties: superposition and homogeneity. Superposition - the output response of a system to the sum of inputs is the sum of the responses to the individual inputs. Homogeneity - describes the response of the system to a multiplication of the input by a scalar. Multiplication of an input by a scalar yields a response that is multiplied by the same scalar. Linear system Nonlinear system Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 59. 59 Nonlinearities Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 60. 60 Linearization • if any nonlinear components are present, we must linearize the system before we can find the transfer function • we linearize nonlinear differential equation for small-signal inputs about the steady-state solution when the small signal input is equal to zero ma is the slope of the curve at point A Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 61. 61 Solution: At π/2 df/dx = - 5 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.