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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.