1. Mathematical Models of Systems
Objectives
We use quantitative mathematical models of physical systems to design and analyze control systems. The dynamic behavior is generally described by ordinary differential equations. We will consider a wide range of systems, including mechanical, hydraulic, and electrical. Since most physical systems are nonlinear, we will discuss linearization approximations, which allow us to use Laplace transform methods.
We will then proceed to obtain the input–output relationship for components and subsystems in the form of transfer functions. The transfer function blocks can be organized into block diagrams or signal-flow graphs to graphically depict the interconnections. Block diagrams (and signal-flow graphs) are very convenient and natural tools for designing and analyzing complicated control systems

2. Six Step Approach to Dynamic System Problems
Introduction
Six Step Approach to Dynamic System Problems
Define the system and its components
Formulate the mathematical model and list the necessary assumptions
Write the differential equations describing the model
Solve the equations for the desired output variables
Examine the solutions and the assumptions
If necessary, reanalyze or redesign the system

3. Differential Equation of Physical Systems

4. Differential Equation of Physical Systems
Electrical Inductance
Describing Equation
Energy or Power
Translational Spring
Rotational Spring
Fluid Inertia

5. Differential Equation of Physical Systems
Electrical Capacitance
Translational Mass
Rotational Mass
Fluid Capacitance
Thermal Capacitance

6. Differential Equation of Physical Systems
Electrical Resistance
Translational Damper
Rotational Damper
Fluid Resistance
Thermal Resistance

7. Differential Equation of Physical Systems

8. Differential Equation of Physical Systems

9. Differential Equation of Physical Systems

10. Differential Equation of Physical Systems

11. Linear Approximations

12. Linear Approximations
Linear Systems - Necessary condition
Principle of Superposition
Property of Homogeneity
Taylor Series

13. Linear Approximations – Example 2.1

14. The Laplace Transform Historical Perspective - Heaviside’s Operators
Origin of Operational Calculus (1887)

15. Historical Perspective - Heaviside’s Operators
Origin of Operational Calculus (1887)
v =
H(t)
Expanded in a power series
(*) Oliver Heaviside: Sage in Solitude, Paul J. Nahin, IEEE Press 1987.

16. The Laplace Transform

17. The Laplace Transform

18. The Laplace Transform

19. The Laplace Transform

20. The Laplace Transform
The Partial-Fraction Expansion (or Heaviside expansion theorem)
Suppose that
The partial fraction expansion indicates that F(s) consists of
a sum of terms, each of which is a factor of the denominator.
The values of K1 and K2 are determined by combining the
individual fractions by means of the lowest common
denominator and comparing the resultant numerator
coefficients with those of the coefficients of the numerator
before separation in different terms. F s ( ) z1 + p1 p2 × or K1 K2
Evaluation of Ki in the manner just described requires the simultaneous solution of n equations.
An alternative method is to multiply both sides of the equation by (s + pi) then setting s= - pi, the
right-hand side is zero except for Ki so that Ki pi
s = - pi

21. The Laplace Transform

22. The Laplace Transform
Useful Transform Pairs

23. The Laplace Transform
Consider the mass-spring-damper system

24. The Laplace Transform

26. Transfer function Circuit Systems Mechanical Systems Transfer Function
Modern Control Systems

27. Circuit Systems Inductor Capacitor Resistor ．(Kirchhoff Current Law)
．(Kirchhoff Voltage Law)
．(Ohm’s Law)
Modern Control Systems

28. Example ： RC Series Circuit
Circuit Systems
Example ： RC Series Circuit
By Kirchhoff and Ohm’s Law
Fig. 2.1
Example 2.2：By Kirchhoff Current Law (Node Analysis)
Fig. 2.2
Modern Control Systems

29. Mechanical Systems (Translational Motion)
Spring
: spring constant
Damper
: Velocity
: viscous friction
Constant
: Viscous Friction Force
Modern Control Systems

30. Mechanical Systems (Translational Motion)
Mass
: Mass
: inertia force
：acceleration
Modern Control Systems

31. Mechanical Systems (Translational Motion)
Example : Mass-Spring-Damper
Form Newton’s Second Law
Under ZIC, take Laplace transform both sides
(Transfer Function)
Fig. 2.3
Note: ZIC=Zero Initial Condition
Modern Control Systems

32. Mechanical Systems (Translational Motion)
Example ：(Suspension system)
Mathematical Model:
Fig. 2.4
Tire spring constant
Modern Control Systems

33. ：angular acceleration
Mechanical Systems (Rotational Motion)
Spring
T：torque
: angle
Damper
：angular velocity
Inertia
：angular acceleration
Modern Control Systems

34. Gear train
(1)
(2)
(3)
(4)
no energy loss
Modern Control Systems

35. Mechanical Systems (Rotational Motion)
Example : Mass－Spring－Damper (Rotational System)
Form Newton’s Second Law
Under ZIC, take Laplace transform both sides
Fig. 2.5
Modern Control Systems

36. Transfer Function Definition ZIC: Zero Initial Condition
Modern Control Systems

37. Transfer Function: Gain that depends on the frequency of input signal
When input
Under ZIC, the steady state output
(2.1)
where
Special Case:
is also called the DC gain.
Conclusion: Under ZIC, for sinusoidal input, the steady state
Output is also a sinusoidal wave.
Modern Control Systems

38. With reference to (2.1), we know
Transfer Function
Example
Find the output y(t) ?
With reference to (2.1), we know
and
Fig. 2.6
 A=0.707 <1, Attenuation!

39. Derivation of T.F. from Differential Equation
Transfer Function
Derivation of T.F. from Differential Equation
A Second-Order Example
Set I.C. =0 and Take L.T. both sides
(Transfer Function from r to y)
Modern Control Systems

40. Example : A second-order Circuit
Transfer Function
Example : A second-order Circuit
From Kirchihoff Voltage Law, we obtain
Fig. 2.7
(Transfer Function from )
Modern Control Systems

41. Example : Mass-Spring-Damper
Transfer Function
Example : Mass-Spring-Damper
Form Newton’s Second Law
Take Laplace Transfrom both sides
Fig. 2.8
Transfer Function
Modern Control Systems

42. Transfer Function
Modern Control Systems

43. Transfer Function
Example
Under ZIC, take L.T., we get the transfer function
Poles：
Zeros：
Char. Equation:

44. Transfer Function
Time Constant of a first-order system
Consider
: (Time Constant)

45. Transfer Function Example The output voltage For RC=1
is called step-function. When A=1, it is called unit step function.
The output voltage
For RC=1
Modern Control Systems

46. Time Constant: Measure of response time of a first-order system
2.3 Transfer Function
Time Constant: Measure of response time of a first-order system A 1 2 3 4 5
gain-time constant form
pole-zero form
Modern Control Systems

48. Types of Systems
Static System: If a system does not change with time, it is called a static system.
Dynamic System: If a system changes with time, it is called a dynamic system.

49. Dynamic Systems
A system is said to be dynamic if its current output may depend on the past history as well as the present values of the input variables.
Mathematically,
Example: A moving mass M y u
Model: Force=Mass x Acceleration

50. Mechanical Systems Translational Mass Spring Damper/dashpot Rotational
Inertia
Damper

51. Mechanical System Building Blocks
Basic building block: spring, dashpots, and masses.
Springs represent the stiffness of a system
Dashpots/Damper represent the forces opposing motion, for example frictional or damping effects.
Masses represent the inertia or resistance to acceleration.
Mechanical systems does not have to be really made up of springs, dashpots, and masses but have the properties of stiffness, damping, and inertia.
All these building blocks may be considered to have a force as an input and displacement as an output.

52. Mass
The mass exhibits the property that the bigger the mass the greater the force required to give it a specific acceleration.
The relationship between the force F and acceleration a is Newton’s second law as shown below.
Energy is needed to stretch the spring, accelerate the mass and move the piston in the dashpot. In the case of spring and mass we can get the energy back but with the dashpot we cannot.
Force
Acceleration
Mass

53. Translational Spring, k (N)
x(t)
Fa(t)

54. Dashpot/Damper
The dashpot block represents the types of forces experienced when pushing an object through a fluid or move an object against frictional forces. The faster the object is pushed the greater becomes the opposing forces.
The dashpot which represents these damping forces that slow down moving objects consists of a piston moving in a closed cylinder.
Movement of the piston requires the fluid on one side of the piston to flow through or past the piston. This flow produces a resistive force. The damping or resistive force is proportional to the velocity v of the piston: F = cv or F = c dv/dt.

55. Rotational Damper, Bm (N-m-sec/rad)
Fa(t) Bm

56. Rotational Spring, ks (N-m-sec/rad)
Fa(t) ks

57. Mechanical Building Blocks
Equation
Energy representation
Translational
Spring
F = kx
E = 0.5 F2/k
Damper
F = c dx/dt
P = cv2
Mass
F = m d2x/dt2
E = 0.5 mv2
Rotational
T = k
E = 0.5 T2/k
T = c d/dt
P = c2
Moment of inertia
T = J d2/dt2
P = 0.5 J2

58. Building Mechanical Blocks
Output, displacement
Mass
Mathematical model of a machine mounted on the ground
Ground
Input, force

59. Building Mechanical Blocks
Moment of inertia
Mathematical model of a rotating a mass
Torque
Torsional resistance
Block model
Shaft
Physical situation

60. Example 1
Consider the following system
Free Body Diagram M

61. Example 1 Differential equation of the system is:
Taking the Laplace Transform of both sides and ignoring
Initial conditions we get

62. Example 1 if

63. Example 2
Find the transfer function of the mechanical translational system given in Figure-1.
Free Body Diagram M
Figure-1

64. Ways to Study a System System Experiment with actual System
Experiment with a model of the System
Physical Model
Mathematical Model
Analytical Solution
Simulation
Frequency Domain
Time Domain
Hybrid Domain

65. Model
A model is a simplified representation or abstraction of reality.
Reality is generally too complex to copy exactly.
Much of the complexity is actually irrelevant in problem solving.

66. What is Mathematical Model?
A set of mathematical equations (e.g., differential eqs.) that describes the input-output behavior of a system.
What is a model used for?
Simulation
Prediction/Forecasting
Prognostics/Diagnostics
Design/Performance Evaluation
Control System Design

67. Black Box Model When only input and output are known.
Internal dynamics are either too complex or unknown.
Easy to Model
Input
Output

68. Grey Box Model
When input and output and some information about the internal dynamics of the system is known.
Easier than white box Modelling.
u(t)
y(t)
y[u(t), t]

69. White Box Model
When input and output and internal dynamics of the system is known.
One should know have complete knowledge of the system to derive a white box model.
u(t)
y(t)

70. Basic Elements of Electrical Systems
The time domain expression relating voltage and current for the resistor is given by Ohm’s law i-e
The Laplace transform of the above equation is

71. Basic Elements of Electrical Systems
The time domain expression relating voltage and current for the Capacitor is given as:
The Laplace transform of the above equation (assuming there is no charge stored in the capacitor) is

72. Basic Elements of Electrical Systems
The time domain expression relating voltage and current for the inductor is given as:
The Laplace transform of the above equation (assuming there is no energy stored in inductor) is

73. V-I and I-V relations Component Symbol V-I Relation I-V Relation
Resistor
Capacitor
Inductor

74. Example#1
The two-port network shown in the following figure has vi(t) as the input voltage and vo(t) as the output voltage. Find the transfer function Vo(s)/Vi(s) of the network. C
i(t)
vi( t)
vo(t)

75. Example#1
Taking Laplace transform of both equations, considering initial conditions to zero.
Re-arrange both equations as:

76. Example#1
Substitute I(s) in equation on left

77. Example#1
The system has one pole at

78. Example#2
Design an Electrical system that would place a pole at -3 if added to another system.
System has one pole at
Therefore, C
i(t)
vi( t)
v2(t)

79. Example#3
Find the transfer function G(S) of the following two port network.
i(t)
vi(t)
vo(t) L C

80. Example#3
Simplify network by replacing multiple components with their equivalent transform impedance.
I(s)
Vi(s)
Vo(s) L C Z

81. Transform Impedance (Resistor)
iR(t)
vR(t) + -
IR(S)
VR(S)
ZR = R
Transformation

82. Transform Impedance (Inductor)
iL(t)
vL(t) + -
IL(S)
VL(S)
LiL(0)
ZL=LS

83. Transform Impedance (Capacitor)
ic(t)
vc(t) + -
Ic(S)
Vc(S)
ZC(S)=1/CS

84. Equivalent Transform Impedance (Series)
Consider following arrangement, find out equivalent transform impedance. L C R

85. Equivalent Transform Impedance (Parallel)

86. Equivalent Transform Impedance
Find out equivalent transform impedance of following arrangement. L2 R2 R1

87. Back to Example#3
I(s)
Vi(s)
Vo(s) L C Z

88. Example#3
I(s)
Vi(s)
Vo(s) L C Z

89. Operational Amplifiers

90. Example#4
Find out the transfer function of the following circuit.

91. Example#5
Find out the transfer function of the following circuit. v1

92. Example#6
Find out the transfer function of the following circuit. v1

93. Example#7
Find out the transfer function of the following circuit and draw the pole zero map.
10kΩ
100kΩ

94. The Transfer Function of Linear Systems

95. The Transfer Function of Linear Systems
Example 2.2

96. The Transfer Function of Linear Systems

97. The Transfer Function of Linear Systems

98. The Transfer Function of Linear Systems

99. The Transfer Function of Linear Systems

100. The Transfer Function of Linear Systems

101. The Transfer Function of Linear Systems

102. The Transfer Function of Linear Systems

103. The Transfer Function of Linear Systems

104. The Transfer Function of Linear Systems

105. The Transfer Function of Linear Systems

106. The Transfer Function of Linear Systems

107. The Transfer Function of Linear Systems

108. The Transfer Function of Linear Systems

109. The Transfer Function of Linear Systems

110. The Transfer Function of Linear Systems

111. The Transfer Function of Linear Systems

112. The Transfer Function of Linear Systems

113. The Transfer Function of Linear Systems

114. Block Diagram Models

115. Block Diagram Models

116. Block Diagram Models Original Diagram Equivalent Diagram

117. Block Diagram Models Original Diagram Equivalent Diagram

118. Block Diagram Models Original Diagram Equivalent Diagram

119. Block Diagram Models

120. Block Diagram Models
Example 2.7

121. Example 2.7
Block Diagram Models

122. Signal-Flow Graph Models
For complex systems, the block diagram method can become difficult to complete. By using the signal-flow graph model, the reduction procedure (used in the block diagram method) is not necessary to determine the relationship between system variables.

123. Signal-Flow Graph Models

124. Signal-Flow Graph Models

125. Signal-Flow Graph Models
Example 2.8

126. Signal-Flow Graph Models
Example 2.10

127. Signal-Flow Graph Models

128. Design Examples

129. Design Examples
Speed control of an electric traction motor.

130. Design Examples

131. Design Examples

132. Design Examples

133. Design Examples

136. The Simulation of Systems Using MATLAB

137. The Simulation of Systems Using MATLAB

138. The Simulation of Systems Using MATLAB

139. The Simulation of Systems Using MATLAB

140. The Simulation of Systems Using MATLAB

141. The Simulation of Systems Using MATLAB

142. The Simulation of Systems Using MATLAB

143. The Simulation of Systems Using MATLAB

144. The Simulation of Systems Using MATLAB

145. The Simulation of Systems Using MATLAB

146. The Simulation of Systems Using MATLAB

147. The Simulation of Systems Using MATLAB

148. The Simulation of Systems Using MATLAB

149. The Simulation of Systems Using MATLAB

150. The Simulation of Systems Using MATLAB

151. The Simulation of Systems Using MATLAB

152. The Simulation of Systems Using MATLAB
error
Sys1 = sysh2 / sysg4

153. The Simulation of Systems Using MATLAB

154. The Simulation of Systems Using MATLAB
error
Num4=[0.1];

155. The Simulation of Systems Using MATLAB

156. The Simulation of Systems Using MATLAB

157. Sequential Design Example: Disk Drive Read System

158. Sequential Design Example: Disk Drive Read System

159. Sequential Design Example: Disk Drive Read System=

160. P2.11

161. P2.11
+Vd Vq Id Tm K1 K2 Km Vc
-Vb K3 Ic