1. Illustrations 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 Chapter 2: Mathematical Models of Systems Objectives

2. Illustrations 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. Illustrations Differential Equation of Physical Systems

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

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

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

7. Illustrations Differential Equation of Physical Systems

8. Illustrations Differential Equation of Physical Systems

9. Illustrations Differential Equation of Physical Systems

10. Illustrations Differential Equation of Physical Systems

11. Illustrations Linear Approximations

12. Illustrations Linear Approximations Linear Systems - Necessary condition Principle of Superposition Property of Homogeneity Taylor Series http://www.maths.abdn.ac.uk/%7Eigc/tch/ma2001/notes/node46.html

13. Illustrations Linear Approximations – Example 2.1

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

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

16. Illustrations The Laplace Transform

17. Illustrations The Laplace Transform

18. Illustrations The Laplace Transform

19. Illustrations The Laplace Transform

20. Illustrations 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. Fs() sz1  sp1  ()sp2  ()  or Fs() K1 sp1  K2 sp2   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 spi  ()sz1  ()  sp1  ()sp2  ()  s = - pi The Laplace Transform

21. Illustrations The Laplace Transform

22. Illustrations Useful Transform Pairs The Laplace Transform

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

24. Illustrations The Laplace Transform

25. Illustrations The Transfer Function of Linear Systems

26. Illustrations The Transfer Function of Linear Systems Example 2.2

27. Illustrations The Transfer Function of Linear Systems

28. Illustrations The Transfer Function of Linear Systems

29. Illustrations The Transfer Function of Linear Systems

30. Illustrations The Transfer Function of Linear Systems

31. Illustrations The Transfer Function of Linear Systems

32. Illustrations The Transfer Function of Linear Systems

33. Illustrations The Transfer Function of Linear Systems

34. Illustrations The Transfer Function of Linear Systems

35. Illustrations The Transfer Function of Linear Systems

36. Illustrations The Transfer Function of Linear Systems

37. Illustrations The Transfer Function of Linear Systems

38. Illustrations The Transfer Function of Linear Systems

39. Illustrations The Transfer Function of Linear Systems

40. Illustrations The Transfer Function of Linear Systems

41. Illustrations The Transfer Function of Linear Systems

42. Illustrations The Transfer Function of Linear Systems

43. Illustrations The Transfer Function of Linear Systems

44. Illustrations The Transfer Function of Linear Systems

45. Illustrations Block Diagram Models

46. Illustrations Block Diagram Models

47. Illustrations Block Diagram Models Original DiagramEquivalent Diagram

48. Illustrations Block Diagram Models Original DiagramEquivalent Diagram

49. Illustrations Block Diagram Models Original DiagramEquivalent Diagram

50. Illustrations Block Diagram Models

51. Illustrations Block Diagram Models Example 2.7

52. Illustrations Block Diagram Models Example 2.7

53. Illustrations 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. Illustrations Signal-Flow Graph Models

55. Illustrations Signal-Flow Graph Models

56. Illustrations Signal-Flow Graph Models Example 2.8

57. Illustrations Signal-Flow Graph Models Example 2.10

58. Illustrations Signal-Flow Graph Models

59. Illustrations Design Examples

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

61. Illustrations Design Examples

62. Illustrations Design Examples

63. Illustrations Design Examples

64. Illustrations Design Examples

65. Illustrations The Simulation of Systems Using MATLAB

66. Illustrations The Simulation of Systems Using MATLAB

67. Illustrations The Simulation of Systems Using MATLAB

68. Illustrations The Simulation of Systems Using MATLAB

69. Illustrations The Simulation of Systems Using MATLAB

70. Illustrations The Simulation of Systems Using MATLAB

71. Illustrations The Simulation of Systems Using MATLAB

72. Illustrations The Simulation of Systems Using MATLAB

73. Illustrations The Simulation of Systems Using MATLAB

74. Illustrations The Simulation of Systems Using MATLAB

75. Illustrations The Simulation of Systems Using MATLAB

76. Illustrations The Simulation of Systems Using MATLAB

77. Illustrations The Simulation of Systems Using MATLAB

78. Illustrations The Simulation of Systems Using MATLAB

79. Illustrations The Simulation of Systems Using MATLAB

80. Illustrations The Simulation of Systems Using MATLAB

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

82. Illustrations The Simulation of Systems Using MATLAB

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

84. Illustrations The Simulation of Systems Using MATLAB

85. Illustrations The Simulation of Systems Using MATLAB

86. Illustrations Sequential Design Example: Disk Drive Read System

87. Illustrations Sequential Design Example: Disk Drive Read System

88. Illustrations = Sequential Design Example: Disk Drive Read System

89. Illustrations P2.11

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

91. Illustrations

92. http://www.jhu.edu/%7Esignals/sensitivity/index.htm

93. Illustrations http://www.jhu.edu/%7Esignals/