lecture-5_transient_response_analysis_of_control_systems.pptx

xzcvxvcxvxcvcxvcxvcxvxcvcx
xcvcxvcxvcxv

1. Dr. Imtiaz Hussain Associate Professor Mehran University of Engineering & Technology Jamshoro, Pakistan email: [email protected] URL : http://imtiazhussainkalwar.weebly.com/ Lecture-5 Transient Response Analysis Of Control Systems Note: I do not claim any originality in these lectures. The contents of this presentation are mostly taken from the book of Ogatta, Norman Nise , Bishop and B C. Kuo and various other internet sources. 1 Control Systems (CS) 6 th Semester 14ES (SEC-I)

2. Outline Introduction Standard Test Signals Transient Response of 1 st Order Systems Transient Response of 2 nd Order Systems S-Plane Transient Response Specifications of 2md order System Examples

3. Introduction In time-domain analysis the response of a dynamic system to an input is expressed as a function of time. It is possible to compute the time response of a system if the nature of input and the mathematical model of the system are known. Usually, the input signals to control systems are not known fully ahead of time. For example, in a radar tracking system, the position and the speed of the target to be tracked may vary in a random fashion. It is therefore difficult to express the actual input signals mathematically by simple equations.

4. Standard Test Signals The characteristics of actual input signals are a sudden shock, a sudden change, a constant velocity, and constant acceleration. The dynamic behavior of a system is therefore judged and compared under application of standard test signals – an impulse, a step, a constant velocity, and constant acceleration. Another standard signal of great importance is a sinusoidal signal.

5. Time Response of Control Systems System The time response of any system has two components Transient response Steady-state response. Time response of a dynamic system response to an input expressed as a function of time.

6. Time Response of Control Systems When the response of the system is changed form rest or equilibrium it takes some time to settle down. Transient response is the response of a system from rest or equilibrium to steady state.  Transient Response Steady State Response The response of the system after the transient response is called steady state response.

7. Time Response of Control Systems Transient response depend upon the system poles only and not on the type of input. It is therefore sufficient to analyze the transient response using a step input. The steady-state response depends on system dynamics and the input quantity. It is then examined using different test signals by final value theorem.

8. Standard Test Signals Impulse signal The impulse signal imitate the sudden shock characteristic of actual input signal. If A=1, the impulse signal is called unit impulse signal. t δ (t) A

9. Standard Test Signals Step signal The step signal imitate the sudden change characteristic of actual input signal. If A=1, the step signal is called unit step signal t u(t) A

10. Standard Test Signals Ramp signal The ramp signal imitate the constant velocity characteristic of actual input signal. If A=1 , the ramp signal is called unit ramp signal t r(t) r(t) unit ramp signal r(t) ramp signal with slope A

11. Standard Test Signals Parabolic signal The parabolic signal imitate the constant acceleration characteristic of actual input signal. If A=1 , the parabolic signal is called unit parabolic signal. t p(t) parabolic signal with slope A p(t) Unit parabolic signal p(t)

12. Relation between standard Test Signals Impulse Step Ramp Parabolic

13. Laplace Transform of Test Signals Impulse Step

14. Laplace Transform of Test Signals Ramp Parabolic

15. First Oder System The first order system has only one pole. Where K is the D.C gain and T is the time constant of the system. Time constant is a measure of how quickly a 1 st order system responds to a unit step input. D.C Gain of the system is ratio between the input signal and the steady state value of output.

16. First Oder System The first order system given below. D.C gain is 10 and time constant is 3 seconds. And for following system D.C Gain of the system is 3/5 and time constant is 1/5 seconds.

17. Impulse Response of 1 st Order System Consider the following 1 st order system t δ (t) 1

18. Impulse Response of 1 st Order System Re-arrange above equation as In order represent the response of the system in time domain we need to compute inverse Laplace transform of the above equation.

19. Impulse Response of 1 st Order System If K=3 and T= 2s then

20. Step Response of 1 st Order System Consider the following 1 st order system In order to find out the inverse Laplace of the above equation, we need to break it into partial fraction expansion Forced Response Natural Response

21. Step Response of 1 st Order System Taking Inverse Laplace of above equation Where u(t)=1 When t=T

22. Step Response of 1 st Order System If K=10 and T= 1.5s then

23. Step Response of 1 st Order System If K=10 and T=1, 3, 5, 7

24. Step Response of 1 st order System System takes five time constants to reach its final value.

25. Step Response of 1 st Order System If K=1, 3, 5, 10 and T=1

26. Relation Between Step and impulse response The step response of the first order system is Differentiating c(t) with respect to t yields

27. Example#1 Impulse response of a 1 st order system is given below. Find out Time constant T D.C Gain K Transfer Function Step Response

28. Example#1 The Laplace Transform of Impulse response of a system is actually the transfer function of the system. Therefore taking Laplace Transform of the impulse response given by following equation.

29. Example#1 Impulse response of a 1 st order system is given below. Find out Time constant T=2 D.C Gain K=6 Transfer Function Step Response Also Draw the Step response on your notebook

30. Example#1 For step response integrate impulse response We can find out C if initial condition is known e.g. c s (0)=0

31. Example#1 If initial Conditions are not known then partial fraction expansion is a better choice

32. Ramp Response of 1 st Order System Consider the following 1 st order system The ramp response is given as

33. 5 10 15 2 4 6 8 10 Time c(t) Unit Ramp Response Unit Ramp Ramp Response Ramp Response of 1 st Order System If K=1 and T=1 error

34. 5 10 15 2 4 6 8 10 Time c(t) Unit Ramp Response Unit Ramp Ramp Response Ramp Response of 1 st Order System If K=1 and T=3 error

35. Parabolic Response of 1 st Order System Consider the following 1 st order system Do it yourself Therefore,

36. Practical Determination of Transfer Function of 1 st Order Systems Often it is not possible or practical to obtain a system's transfer function analytically. Perhaps the system is closed, and the component parts are not easily identifiable. The system's step response can lead to a representation even though the inner construction is not known. With a step input, we can measure the time constant and the steady-state value, from which the transfer function can be calculated.

37. Practical Determination of Transfer Function of 1 st Order Systems If we can identify T and K from laboratory testing we can obtain the transfer function of the system.

38. Practical Determination of Transfer Function of 1 st Order Systems For example, assume the unit step response given in figure. From the response, we can measure the time constant, that is, the time for the amplitude to reach 63% of its final value. Since the final value is about 0.72 the time constant is evaluated where the curve reaches 0.63 x 0.72 = 0.45, or about 0.13 second. T= 0.13s K=0.72 K is simply steady state value. Thus transfer function is obtained as:

39. 1 st Order System with a Zero Zero of the system lie at -1/ α and pole at -1/T . Step response of the system would be:

40. 1 st Order System with & W/O Zero If T> α the response will be same

41. 1 st Order System with & W/O Zero If T> α the response of the system would look like

42. 1 st Order System with & W/O Zero If T< α the response of the system would look like

43. 1 st Order System with a Zero

44. 1 st Order System with & W/O Zero 1 st Order System Without Zero

45. Home Work Find out the impulse, ramp and parabolic response of the system given below.

46. Example#2 A thermometer requires 1 min to indicate 98% of the response to a step input. Assuming the thermometer to be a first-order system, find the time constant. If the thermometer is placed in a bath, the temperature of which is changing linearly at a rate of 10°min , how much error does the thermometer show?

47. PZ-map and Step Response -1 -2 -3 δ j ω

48. PZ-map and Step Response -1 -2 -3 δ j ω

49. PZ-map and Step Response -1 -2 -3 δ j ω

50. Comparison

51. First Order System With Delays Following transfer function is the generic representation of 1 st order system with time lag. Where t d is the delay time.

52. First Order System With Delays 1 Unit Step Step Response t t d

53. First Order System With Delays

54. Examples of First Order Systems Armature Controlled D.C Motor ( L a =0 ) u i a T R a L a J  B e b V f =constant

55. Examples of First Order Systems Electrical System

56. Examples of First Order Systems Mechanical System

57. Second Order Systems We have already discussed the affect of location of poles and zeros on the transient response of 1 st order systems. Compared to the simplicity of a first-order system, a second-order system exhibits a wide range of responses that must be analyzed and described. Varying a first-order system's parameter (T, K) simply changes the speed and offset of the response Whereas, changes in the parameters of a second-order system can change the form of the response. A second-order system can display characteristics much like a first-order system or, depending on component values, display damped or pure oscillations for its transient response . 57

58. Second Order Systems A general second-order system is characterized by the following transfer function. 58 un-damped natural frequency of the second order system, which is the frequency of oscillation of the system without damping. damping ratio of the second order system, which is a measure of the degree of resistance to change in the system output.

59. Example#3 Determine the un-damped natural frequency and damping ratio of the following second order system. Compare the numerator and denominator of the given transfer function with the general 2 nd order transfer function. 59

60. Second Order Systems Two poles of the system are 60

61. Second Order Systems According the value of , a second-order system can be set into one of the four categories: Overdamped - when the system has two real distinct poles ( >1). -a -b -c δ j ω 61

62. Second Order Systems According the value of , a second-order system can be set into one of the four categories: 2. Underdamped - when the system has two complex conjugate poles (0 < <1) -a -b -c δ j ω 62

63. Second Order Systems According the value of , a second-order system can be set into one of the four categories: 3. Undamped - when the system has two imaginary poles ( = 0). -a -b -c δ j ω 63

64. Second Order Systems According the value of , a second-order system can be set into one of the four categories: 4. Critically damped - when the system has two real but equal poles ( = 1). -a -b -c δ j ω 64

65. S-Plane δ j ω Natural Undamped Frequency. Distance from the origin of s-plane to pole is natural undamped frequency in rad /sec. 65

66. S-Plane δ j ω Let us draw a circle of radius 3 in s-plane. 3 -3 -3 3 If a pole is located anywhere on the circumference of the circle the natural undamped frequency would be 3 rad /sec . 66

67. S-Plane δ j ω Therefore the s-plane is divided into Constant Natural Undamped Frequency ( ω n ) Circles. 67

68. S-Plane δ j ω Damping ratio. Cosine of the angle between vector connecting origin and pole and – ve real axis yields damping ratio. 68

69. S-Plane δ j ω For Underdamped system therefore, 69

70. S-Plane δ j ω For Undamped system therefore, 70

71. S-Plane δ j ω For overdamped and critically damped systems therefore, 71

72. S-Plane δ j ω Draw a vector connecting origin of s-plane and some point P . P 72

73. S-Plane δ j ω Therefore, s-plane is divided into sections of constant damping ratio lines. 73

74. S-Plane 74

75. Example-4 Determine the natural frequency and damping ratio of the poles from the given pz -map. Also determine the transfer function of the system and state whether system is underdamped, overdamped, undamped or critically damped. 75

76. Example-5 The natural frequency of closed loop poles of 2 nd order system is 2 rad/sec and damping ratio is 0.5 . Determine the location of closed loop poles so that the damping ratio remains same but the natural undamped frequency is doubled. 76

77. Example-5 Determine the location of closed loop poles so that the damping ratio remains same but the natural undamped frequency is doubled. 77

78. S-Plane 78

79. Time-Domain Specification 79 For 0< <1 and ω n > 0 , the 2 nd order system’s response due to a unit step input looks like

80. Time-Domain Specification 80 The delay ( t d ) time is the time required for the response to reach half the final value the very first time.

81. Time-Domain Specification 81 The rise time is the time required for the response to rise from 10% to 90%, 5% to 95%, or 0% to 100% of its final value. For underdamped second order systems, the 0% to 100% rise time is normally used. For overdamped systems, the 10% to 90% rise time is commonly used. 81

82. Time-Domain Specification 82 The peak time is the time required for the response to reach the first peak of the overshoot. 82 82

83. Time-Domain Specification 83 The maximum overshoot is the maximum peak value of the response curve measured from unity. If the final steady-state value of the response differs from unity, then it is common to use the maximum percent overshoot. It is defined by The amount of the maximum (percent) overshoot directly indicates the relative stability of the system.

84. Time-Domain Specification 84 The settling time is the time required for the response curve to reach and stay within a range about the final value of size specified by absolute percentage of the final value (usually 2% or 5%).

85. Step Response of underdamped System The partial fraction expansion of above equation is given as Step Response 85

86. Step Response of underdamped System Above equation can be written as Where , is the frequency of transient oscillations and is called damped frequency . The inverse Laplace transform of above equation can be obtained easily if C(s) is written in the following form: 86

87. Step Response of underdamped System 87

88. Step Response of underdamped System When 88

89. Step Response of underdamped System 89

90. Step Response of underdamped System 90

91. Step Response of underdamped System 91

92. Step Response of underdamped System 92

93. Step Response of underdamped System 93

94. Step Response of underdamped System 94

95. Time Domain Specifications Rise Time Peak Time Settling Time (2%) Settling Time (5%) Maximum Overshoot 95

96. Example#6 Consider the system shown in following figure, where damping ratio is 0.6 and natural undamped frequency is 5 rad/sec . Obtain the rise time t r , peak time t p , maximum overshoot M p , and settling time 2% and 5% criterion t s when the system is subjected to a unit-step input. 96

97. Example#6 Rise Time Peak Time Settling Time (2%) Maximum Overshoot Settling Time (5%) 97

98. Example#6 Rise Time 98

99. Example#6 Peak Time Settling Time (2%) Settling Time (5%) 99

100. Example#6 Maximum Overshoot 100

101. Example#6 101

102. Example#7 For the system shown in Figure-(a), determine the values of gain K and velocity-feedback constant K h so that the maximum overshoot in the unit-step response is 0.2 and the peak time is 1 sec . With these values of K and K h , obtain the rise time and settling time. Assume that J=1 kg- m 2 and B=1 N-m/rad/sec . 102

103. Example#7 103

104. Example#7 Comparing above T.F with general 2 nd order T.F 104

105. Example#7 Maximum overshoot is 0.2 . The peak time is 1 sec 105

106. Example#7 106

107. Example#7 107

108. Example#8 When the system shown in Figure(a) is subjected to a unit-step input, the system output responds as shown in Figure(b). Determine the values of a and c from the response curve. 108

109. Example#9 Figure (a) shows a mechanical vibratory system. When 2 lb of force (step input) is applied to the system, the mass oscillates, as shown in Figure (b). Determine m , b , and k of the system from this response curve. 109

110. Example#10 Given the system shown in following figure, find J and D to yield 20% overshoot and a settling time of 2 seconds for a step input of torque T(t). 110

111. Example#10 111

112. Example#10 112

113. Example # 11 Ships at sea undergo motion about their roll axis, as shown in Figure. Fins called stabilizers are used to reduce this rolling motion. The stabilizers can be positioned by a closed-loop roll control system that consists of components, such as fin actuators and sensors, as well as the ship’s roll dynamics. 113

114. Example # 11 Assume the roll dynamics, which relates the roll-angle output, , to a disturbance-torque input, T D (s), is Do the following: Find the natural frequency, damping ratio, peak time, settling time, rise time , and percent overshoot. Find the analytical expression for the output response to a unit step input.   114

115. Step Response of critically damped System ( ) The partial fraction expansion of above equation is given as Step Response 115

116. 116

117. 117 Example 12 : Describe the nature of the second-order system response via the value of the damping ratio for the systems with transfer function Second – Order System Do them as your own revision

118. Example-13 For each of the transfer functions find the locations of the poles and zeros, plot them on the s-plane , and then write an expression for the general form of the step response without solving for the inverse Laplace transform. State the nature of each response ( overdamped , underdamped , and so on). 118

119. Example-14 Solve for x(t) in the system shown in Figure if f(t ) is a unit step. 119

120. End of Lecture-5 To download this lecture visit http://imtiazhussainkalwar.weebly.com/