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

25. The Transfer Function of Linear Systems

26. The Transfer Function of Linear Systems
Example 2.2

27. The Transfer Function of Linear Systems

28. The Transfer Function of Linear Systems

29. The Transfer Function of Linear Systems

30. The Transfer Function of Linear Systems

31. The Transfer Function of Linear Systems

32. The Transfer Function of Linear Systems

33. The Transfer Function of Linear Systems

34. The Transfer Function of Linear Systems

35. The Transfer Function of Linear Systems

36. The Transfer Function of Linear Systems

37. The Transfer Function of Linear Systems

38. The Transfer Function of Linear Systems

39. The Transfer Function of Linear Systems

40. The Transfer Function of Linear Systems

41. The Transfer Function of Linear Systems

42. The Transfer Function of Linear Systems

43. The Transfer Function of Linear Systems

44. The Transfer Function of Linear Systems

45. Block Diagram Models

46. Block Diagram Models

47. Block Diagram Models Original Diagram Equivalent Diagram

48. Block Diagram Models Original Diagram Equivalent Diagram

49. Block Diagram Models Original Diagram Equivalent Diagram

50. Block Diagram Models

51. Block Diagram Models
Example 2.7

52. Example 2.7
Block Diagram Models

53. 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.

54. Signal-Flow Graph Models

55. Signal-Flow Graph Models

56. Signal-Flow Graph Models
Example 2.8

57. Signal-Flow Graph Models
Example 2.10

58. Signal-Flow Graph Models

59. Design Examples

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

61. Design Examples

62. Design Examples

63. Design Examples

64. Design Examples

65. The Simulation of Systems Using MATLAB

66. The Simulation of Systems Using MATLAB

67. The Simulation of Systems Using MATLAB

68. The Simulation of Systems Using MATLAB

69. The Simulation of Systems Using MATLAB

70. The Simulation of Systems Using MATLAB

71. The Simulation of Systems Using MATLAB

72. The Simulation of Systems Using MATLAB

73. The Simulation of Systems Using MATLAB

74. The Simulation of Systems Using MATLAB

75. The Simulation of Systems Using MATLAB

76. The Simulation of Systems Using MATLAB

77. The Simulation of Systems Using MATLAB

78. The Simulation of Systems Using MATLAB

79. The Simulation of Systems Using MATLAB

80. The Simulation of Systems Using MATLAB

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

82. The Simulation of Systems Using MATLAB

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

84. The Simulation of Systems Using MATLAB

85. The Simulation of Systems Using MATLAB

86. Sequential Design Example: Disk Drive Read System

87. Sequential Design Example: Disk Drive Read System

88. Sequential Design Example: Disk Drive Read System=

89. P2.11

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