Performance of Feedback Control Systems

1. Performance of Feedback Control Systems Prof. Marian S. Stachowicz Laboratory for Intelligent Systems ECE Department, University of Minnesota Duluth February 25 – March 2, 2010

2. Outline Introduction Test Input Signals Performance of a second-order system Effects of a Third Pole and a Zero on the Second-Order System Response Estimation of the Damping Ratio The s-plane Root Location and the Transient Response 2 Control Systems Objectives The ability to adjust the transient and steady-state response of a feedback control system is a beneficial outcome of the design of control systems. One of the first steps in the design process is to specify the measures of performance. In this chapter we introduce the common time-domain specifications such as percent overshoot, settling time, time to peak, time to rise, and steady-state tracking error. We will use selected input signals such as the step and ramp to test the response of the control system. The correlation between the system performance and the location of the system transfer function poles and zeros in the s-plane is discussed. We will develop valuable relationships between the performance specifications and the natural frequency and damping ratio for second-order systems. Relying on the notion of dominant poles, we can extrapolate the ideas associated with second-order systems to those of higher order. Objectives The ability to adjust the transient and steady-state response of a feedback control system is a beneficial outcome of the design of control systems. One of the first steps in the design process is to specify the measures of performance. In this chapter we introduce the common time-domain specifications such as percent overshoot, settling time, time to peak, time to rise, and steady-state tracking error. We will use selected input signals such as the step and ramp to test the response of the control system. The correlation between the system performance and the location of the system transfer function poles and zeros in the s-plane is discussed. We will develop valuable relationships between the performance specifications and the natural frequency and damping ratio for second-order systems. Relying on the notion of dominant poles, we can extrapolate the ideas associated with second-order systems to those of higher order.

3. References for reading R.C. Dorf and R.H. Bishop, Modern Control Systems, 11th Edition, Prentice Hall, 2008, Chapter 5.1 - 5.12 2. J.J. DiStefano, A. R. Stubberud, I. J. Williams, Feeedback and Control Systems, Schaum's Outline Series, McGraw-Hill, Inc., 1990 Chapter 9 3 Control Systems

4. Test Input Signal Since the actual input signal of the system is usually unknown, a standard test input signal is normally chosen. Commonly used test signals include step input, ramp input, and the parabolic input. 4 Control Systems In a radar tracking system for antiaircraft missiles, the position and speed of the target to be tracked may vary in an unpredictable manner. From the step function to the parabolic function, they become progressively faster with respect to time. For the purposes of analysis and design, it is necessary to assume some basic types of test inputs so that the performance of a system can be evaluated. The responses due to these inputs allow the prediction of the system’s performance to other more complex inputs. In a radar tracking system for antiaircraft missiles, the position and speed of the target to be tracked may vary in an unpredictable manner. From the step function to the parabolic function, they become progressively faster with respect to time. For the purposes of analysis and design, it is necessary to assume some basic types of test inputs so that the performance of a system can be evaluated. The responses due to these inputs allow the prediction of the system’s performance to other more complex inputs.

5. General form of the standard test signals 5 Control Systems

6. Test signals r(t) = A tn 6 Control Systems The ramp signal is the integral of the step input, and the parabola is is the integral of the ramp input. The unit impulse function is also useful for test signal purposes. The responses due to these inputs allow the prediction of the system’s performance to other more complex inputs. The ramp signal is the integral of the step input, and the parabola is is the integral of the ramp input. The unit impulse function is also useful for test signal purposes. The responses due to these inputs allow the prediction of the system’s performance to other more complex inputs.

7. Table 5.1 Test Signal Inputs 7 Control Systems Step-for representation a stationary target Ramp- for track a constant angular position (first derivatives are constant) Parabolas- can be used to represent accelerating targets (second derivatives are constant) Step-for representation a stationary target Ramp- for track a constant angular position (first derivatives are constant) Parabolas- can be used to represent accelerating targets (second derivatives are constant)

8. fig_07_01 8 Control Systems 1.Step inputs represent constant position. An antenna position control is an example of a system that can be tested for accuracy using step function for control of the satellite in geostationary orbit. 2. Ramp inputs represent constant velocity inputs. A position control system that tracks a satellite that moves across the sky at constant angular velocity 3. Parabolas, whose second derivatives are constant, represent constant-acceleration inputs to position control systems and can be used to represent accelerating targets, such as missile, to determine the steady-state error performance..1.Step inputs represent constant position. An antenna position control is an example of a system that can be tested for accuracy using step function for control of the satellite in geostationary orbit. 2. Ramp inputs represent constant velocity inputs. A position control system that tracks a satellite that moves across the sky at constant angular velocity 3. Parabolas, whose second derivatives are constant, represent constant-acceleration inputs to position control systems and can be used to represent accelerating targets, such as missile, to determine the steady-state error performance..

9. Steady-state error Is a difference between input and the output for a prescribed test input as 9 Control Systems

10. Application to stable systems Unstable systems represent loss of control in the steady state and are not acceptable for use at all. 10 Control Systems The expressions we derive to calculate the steady-state error can be applied erroneously to unstable system. Thus the engineer must check the system for stability while performing steady-state error analysis and design. We assume that all the systems in examples are stable.The expressions we derive to calculate the steady-state error can be applied erroneously to unstable system. Thus the engineer must check the system for stability while performing steady-state error analysis and design. We assume that all the systems in examples are stable.

11. fig_07_02 11 Control Systems Fig. a Output 1 has zero-steady-state error, and output 2 has a finite steady-state error Fig. b Output 3 steady-state error is infinite as measured vertically between input and output 3 after the transients have died down, and t approaches infinity Fig. a Output 1 has zero-steady-state error, and output 2 has a finite steady-state error Fig. b Output 3 steady-state error is infinite as measured vertically between input and output 3 after the transients have died down, and t approaches infinity

12. Time response of systems c(t) = ct(t) + css(t) The time response of a control system is divided into two parts: ct(t) - transient response css(t) - steady state response 12 Control Systems The time response of a control system is divided into two parts: the transient response and the steady-state response. The time response of a control system is divided into two parts: the transient response and the steady-state response.

13. Transient response All real control systems exhibit transient phenomena to some extend before steady state is reached. 13 Control Systems

14. Steady-state response The response that exists for a long time following any input signal initiation. 14 Control Systems

15. fig_04_01 15 Control Systems A pole of the input function generates the form of the forced response ( that is the pole at the origin generated a step function at the output. A pole of the transfer function generate the form of the exponential response A pole on the real axis generate an exponential response of the form Exp[-at] where -a is the pole location on real axis. The farther to the left a pole is on the negative real axis, the faster the exponential transit response will decay to zero. 4. The zeros and poles generate the amplitudes for both the transit and steady state responses ( see A, B in partial fraction extension)A pole of the input function generates the form of the forced response ( that is the pole at the origin generated a step function at the output. A pole of the transfer function generate the form of the exponential response A pole on the real axis generate an exponential response of the form Exp[-at] where -a is the pole location on real axis. The farther to the left a pole is on the negative real axis, the faster the exponential transit response will decay to zero. 4. The zeros and poles generate the amplitudes for both the transit and steady state responses ( see A, B in partial fraction extension)

16. Poles and zeros A pole of the input function generates the form of the forced response ( that is the pole at the origin generated a step function at the output). A pole of the transfer function generate the form of the exponential response 3. The zeros and poles generate the amplitudes for both the transit and steady state responses ( see A, B in partial fraction extension) 16 Control Systems

17. fig_04_02 17 Control Systems

18. fig_04_03 18 Control Systems Write the output, c(t), in general terms. Specify the transit and steady state parts of solutions for t > 0. By inspection, each system pole generate an exponential as part of transient response. The input’ pole generates the steady-state response.Write the output, c(t), in general terms. Specify the transit and steady state parts of solutions for t > 0. By inspection, each system pole generate an exponential as part of transient response. The input’ pole generates the steady-state response.

19. fig_04_04 19 Control Systems First order system without zeros., where the input pole at the origin generate the steady-state css(t)=1 and the system pole at - a , generate the transit response -Exp(-at)First order system without zeros., where the input pole at the origin generate the steady-state css(t)=1 and the system pole at - a , generate the transit response -Exp(-at)

20. fig_04_05 20 Control Systems 1/a = time constant of the response. From Fig., the time constant can be described as the time for 1 - Exp[-at] to rise to 63 % of initial value. 1/a = time constant of the response. From Fig., the time constant can be described as the time for 1 - Exp[-at] to rise to 63 % of initial value.

21. Transient response specification for a first-order system Time-constant, 1/a Can be described as the time for (1 - Exp[- a t]) to rise to 63 % of initial value. Rise time, Tr = 2.2/a The time for the waveform to go from 0.1 to 0.9 of its final value. Settling time, Ts = 4/a The time for response to reach, and stay within, 2% of its final value 21 Control Systems Rise time is found by solving equation c(t) = 1- Exp[-a t] for difference in time at c(t) = 0.9 and c(t) = 0.1. Tr = 2.31/a - 0.11/a = 2.2/a Settling time, for example 2% by letting c(t) = 0,98 = 1- Exp[-a t] Rise time is found by solving equation c(t) = 1- Exp[-a t] for difference in time at c(t) = 0.9 and c(t) = 0.1. Tr = 2.31/a - 0.11/a = 2.2/a Settling time, for example 2% by letting c(t) = 0,98 = 1- Exp[-a t]

22. fig_04_06 22 Control Systems If we can identify K and a from laboratory testing, we can obtain the G(s) of the system. Assume the unit step response given in Fig. We measure the time constant, that is the time for amplitude to reach 63% of its final value: 63 x 0.72 = 0.45, or about 0.13 sec . Hence a=1/0.13=7.7 From equation, we see that the forced response reaches a steady-state value of K/a =0.72 . K= 0.72 x 7.7= 5.54 G(s) = 5.54/(s+7.7) . In reality the Fig was generated using the transfer function, G(s)= 5/(s+7)If we can identify K and a from laboratory testing, we can obtain the G(s) of the system. Assume the unit step response given in Fig. We measure the time constant, that is the time for amplitude to reach 63% of its final value: 63 x 0.72 = 0.45, or about 0.13 sec . Hence a=1/0.13=7.7 From equation, we see that the forced response reaches a steady-state value of K/a =0.72 . K= 0.72 x 7.7= 5.54 G(s) = 5.54/(s+7.7) . In reality the Fig was generated using the transfer function, G(s)= 5/(s+7)

23. Identify K and a from testing 23 Control Systems We measure the time constant, that is the time for amplitude to reach 63% of its final value: 63 x 0.72 = 0.45, or about 0.13 sec . Hence a=1/0.13=7.7 From equation, we see that the forced response reaches a steady-state value of K/a =0.72 . K= 0.72 x 7.7= 5.54 G(s) = 5.54/(s+7.7) . In reality the Fig was generated using the transfer function, G(s)= 5/(s+7)We measure the time constant, that is the time for amplitude to reach 63% of its final value: 63 x 0.72 = 0.45, or about 0.13 sec . Hence a=1/0.13=7.7 From equation, we see that the forced response reaches a steady-state value of K/a =0.72 . K= 0.72 x 7.7= 5.54 G(s) = 5.54/(s+7.7) . In reality the Fig was generated using the transfer function, G(s)= 5/(s+7)

24. Exercise A system has a transfer function G(s)= 50/(s+50). Find the transit response specifications such as Tc, Tr, Ts. 24 Control Systems

25. 25 Control Systems

26. Steady-state response If the steady-state response of the output does not agree with the steady-state of the input exactly, the system is said to have a steady-state error. It is a measure of system accuracy when a specific type of input is applied to a control system. 26 Control Systems

27. 27 Control Systems

28. Steady-state error 28 Control Systems

29. fig_07_04 29 Control Systems

30. 30 Control Systems

31. Control Systems 31

32. Performance of a second-order system 32 Control Systems

33. Numerical example of the second-order system 33 Control Systems A second order system exhibits a wide range of response. Varying a first-order system’s parameter simple change the speed of the response. Changes in the parameters of a second-order system can change the form of the response. Can display characteristic much like a first-order system or display damped or pure oscillation for its transient response. The general case: which has two finite poles and no zeros. The term in the numerator is simple scale. Changing a and b we can show all possible transient responses. The unit step response then can be found using C(s)= R(s) G(s), where R(s) = 1/s, followed by a partial-fraction expansion and inverse Laplace transform.A second order system exhibits a wide range of response. Varying a first-order system’s parameter simple change the speed of the response. Changes in the parameters of a second-order system can change the form of the response. Can display characteristic much like a first-order system or display damped or pure oscillation for its transient response. The general case: which has two finite poles and no zeros. The term in the numerator is simple scale. Changing a and b we can show all possible transient responses. The unit step response then can be found using C(s)= R(s) G(s), where R(s) = 1/s, followed by a partial-fraction expansion and inverse Laplace transform.

34. Overdamped 34 Control Systems This function has a pole at the origin that comes from the unit step input and two real poles that come from the system. The input pole at the origin generates the steady-state response; each of the two system poles on the real axis generates an exponential transit response The poles tell us the form of the response without the calculation of the inverse Laplace transform. Overdamped - refers to a large amount of energy absorption in the system. As the energy absorption is reduced, an overdamped system will become underdamped with overshoot.This function has a pole at the origin that comes from the unit step input and two real poles that come from the system. The input pole at the origin generates the steady-state response; each of the two system poles on the real axis generates an exponential transit response The poles tell us the form of the response without the calculation of the inverse Laplace transform. Overdamped - refers to a large amount of energy absorption in the system. As the energy absorption is reduced, an overdamped system will become underdamped with overshoot.

35. Underdamped 35 Control Systems This function has a pole at the origin that comes from the unit step input and two complex poles that come from the system. The real part of the pole matches the exponential decay frequency of the frequency of the sinusoid’s amplitude, while the imaginary part of the pole matches the frequency of the sinusoidal oscillation. This function has a pole at the origin that comes from the unit step input and two complex poles that come from the system. The real part of the pole matches the exponential decay frequency of the frequency of the sinusoid’s amplitude, while the imaginary part of the pole matches the frequency of the sinusoidal oscillation.

36. fig_04_08 36 Control Systems

37. Undamped 37 Control Systems This function has a pole at the origin that comes from the unit step input and two imaginary poles that come from the system. This function has a pole at the origin that comes from the unit step input and two imaginary poles that come from the system.

38. Critically damped 38 Control Systems This function has a pole at the origin that comes from the unit step input and two multiple real poles that come from the system. Critically damped responses are the fastest possible without the overshoot that is characteristic of the underdamped response.This function has a pole at the origin that comes from the unit step input and two multiple real poles that come from the system. Critically damped responses are the fastest possible without the overshoot that is characteristic of the underdamped response.

39. fig_04_10 39 Control Systems The step responses for the four cases of damping.The step responses for the four cases of damping.

40. fig_04_11 40 Control Systems

41. Summary Overdamped Poles: Two real at - ?1, - ?2 Underdamped Poles: Two complex at - ?d + j?d, - ?d - j?d Undamped Poles: Two imaginary at + j?1, - j?1 Critically damped Poles: Two real at - ?1, 41 Control Systems

42. 42 Control Systems Figure: 05-04Figure: 05-04

43. Response to unit step input 43 Control Systems

44. Natural frequency ?n - the frequency of natural oscillation that would occur for two complex poles if the damping were equal to zero Damping ratio ? - a measure of damping for second-order characteristic equation 44 Control Systems For example, the frequency of oscillation of a series RLC circuit with the resistance shorted would be the natural frequency.For example, the frequency of oscillation of a series RLC circuit with the resistance shorted would be the natural frequency.

45. 45 Control Systems Figure: 02-09Figure: 02-09

46. Finding ?n and ? for a second-order system 46 Control Systems From s1, s2 we see that the various cases of second-order response are a function of ?.From s1, s2 we see that the various cases of second-order response are a function of ?.

47. fig_04_13 47 Control Systems For damping ratio values.For damping ratio values.

48. 48 Control Systems Figure: 02-12Figure: 02-12

49. Transit response 49 Control Systems

50. Unit impulse response 50 Control Systems Which is derivative of the response to a step input.Which is derivative of the response to a step input.

51. 51 Control Systems

52. fig_04_19 52 Control Systems Step responses of second-orderunderdamped systems as poles move:a. with constant real part;b. with constant imaginary part;c. with constant damping ratioStep responses of second-orderunderdamped systems as poles move:a. with constant real part;b. with constant imaginary part;c. with constant damping ratio

53. Standard performance measures 53 Control Systems

54. fig_04_14 54 Control Systems

55. Settling time The settling time is defined as the time required for a system to settle within a certain percentage of the input amplitude. 55 Control Systems

56. Settling time 56 Control Systems

57. Rise time The time it takes for a signal to go from 10% of its value to 90% of its final value 57 Control Systems The swiftness of the response is measured by the rise time Tr and the peak time TpThe swiftness of the response is measured by the rise time Tr and the peak time Tp

58. Rise time 58 Control Systems

59. Peak time Peak time is the time required by a signal to reach its maximum value. 59 Control Systems

60. Peak time 60 Control Systems

61. Percent overshoot Percent Overshoot is defined as: P.O. = [(Mpt – fv) / fv] * 100% Mpt = The peak value of the time response fv = Final value of the response 61 Control Systems

62. Percent overshoot 62 Control Systems

63. Percent overshoot and normalized peak time versus ? 63 Control Systems Percent overshoot and normalized peak time versus damping ratioPercent overshoot and normalized peak time versus damping ratio

64. Finding transient response 64 Control Systems For the system find the peak time, percent overshoot, and settling time. For the system find the peak time, percent overshoot, and settling time.

65. 65 Control Systems

66. Gain design for transient response 66 Control Systems Design the value of gain, K, for the feedback control system so that the system will respond with a 10% overshoot. Design the value of gain, K, for the feedback control system so that the system will respond with a 10% overshoot.

67. 67 Control Systems

68. Performance Indices Elevator Control Systems 68

69. Simplified description of a control system 69 Control Systems

70. Elevator input and output 70 Control Systems When the fourth floor button is pressed on the first floor, the elevator rises to the fourth floor with a speed and floor level accuracy designed for passenger comfort. Push of the fourth-floor button is an input that represent a desired output, shown as a step function. The performance can be seen from elevator response curve in the figure.2 Two major measures of performance are apparent : the transient response and the steady-state error. Passenger comfort and passenger patience are dependent upon the transient response. If this response is too fast, passenger comfort is sacrificed; if too slow, passenger patience is sacrificed. The steady-state error is another important specification since passenger safety and convenience would be sacrificed if the elevator is not properly level. When the fourth floor button is pressed on the first floor, the elevator rises to the fourth floor with a speed and floor level accuracy designed for passenger comfort. Push of the fourth-floor button is an input that represent a desired output, shown as a step function. The performance can be seen from elevator response curve in the figure.2 Two major measures of performance are apparent : the transient response and the steady-state error. Passenger comfort and passenger patience are dependent upon the transient response. If this response is too fast, passenger comfort is sacrificed; if too slow, passenger patience is sacrificed. The steady-state error is another important specification since passenger safety and convenience would be sacrificed if the elevator is not properly level.

71. 71 Control Systems

72. Transient response 72 Control Systems Two major measures of performance : the transient response and the steady-state error. Two major measures of performance : the transient response and the steady-state error.

73. Steady-state error Passenger safety and convenience would be sacrificed if the elevator is not properly level. 73 Control Systems

74. Performance Indices A performance index is a quantitative measure of the performance of a system and is chosen so that emphasis is given to the important system specifications. 74 Control Systems

75. Response of the system 75 Control Systems

76. ISE - Integral of Square of Error 76 Control Systems The upper limit T is a finite time chosen somewhat arbitrarily so that the integral approaches a steady-state value. It is usually to apply the settling time Ts.The criterion will discriminate excessively overdamped systems and excessively underdamped systems. The minimum value of the integral occurs for a compromise value of the damping. The upper limit T is a finite time chosen somewhat arbitrarily so that the integral approaches a steady-state value. It is usually to apply the settling time Ts.The criterion will discriminate excessively overdamped systems and excessively underdamped systems. The minimum value of the integral occurs for a compromise value of the damping.

77. The Integral Squared Error 77 Control Systems

78. IAE - Integral of the Absolute Magnitude of the Error 78 Control Systems This index is particularly useful for computer simulation studies.This index is particularly useful for computer simulation studies.

79. ITAE - Integral of Time Multiplied by Absolute Error 79 Control Systems In order to reduce the contribution of large initial error to the value of the performance integral, as well as to place an emphasis on errors occurring later in the response. Provides the best selectivity of the performance indices; that is, the minimum value of the integral is readily discernible as the system parameters are varied.In order to reduce the contribution of large initial error to the value of the performance integral, as well as to place an emphasis on errors occurring later in the response. Provides the best selectivity of the performance indices; that is, the minimum value of the integral is readily discernible as the system parameters are varied.

80. ITSE - Integral of Time Multiplied by Squared Error 80 Control Systems

81. General form of the performance integral 81 Control Systems Where f [.] is a function of the error, input, output, and time. Cleary, one can obtain numerous indices based on various combinations of the system variables and time.Where f [.] is a function of the error, input, output, and time. Cleary, one can obtain numerous indices based on various combinations of the system variables and time.

82. Section 5.9 82 Control Systems

83. Performance criteria 83 Control Systems A single-loop feedback control system is shown in Fig.a , where the natural frequency is the normalized value, wn=1. Calculate three performance indices for various values of damping ratio. A single-loop feedback control system is shown in Fig.a , where the natural frequency is the normalized value, wn=1. Calculate three performance indices for various values of damping ratio.

84. 84 Control Systems

85. Optimum system A control system is optimum when the elected performance index is minimized. The optimum value of the parameters depends directly upon the definition of optimum, that is, the performance index. 85 Control Systems

86. The coefficients that will minimize the ITAE performance criterion for a step input and a ramp input have been determined for the general closed-loop transfer function. 86 Control Systems The ability to adjust the transient and steady-state performance is a distinct advantage of feedback control systems.The ability to adjust the transient and steady-state performance is a distinct advantage of feedback control systems.

87. General closed-loop T(s) 87 Control Systems This transfer function has a steady-state error equal zero for a step input The T(s) has n poles and no zeros..This transfer function has a steady-state error equal zero for a step input The T(s) has n poles and no zeros..

88. Table 5.6 The Optimum Coefficients of T(s) Based on the ITAE Criterion for a Step Input s + ?n s2 + 1.4?ns + ?n2 s3 + 1.75?n s2 + 2.15?n2s + ?n3 s4 + 2.1 ?n s3 + 3.4 ?n2s2 + 2.7 ?n3s + ?n4 s5 + 2.8 ?n s4 + 5.0 ?n2s3 + 5.5 ?n3s2 + 3.4?n4s +?n5 s6 + 3.25?ns5 + 6.60 ?n2 s4 + 8.60?n3s3 + 7.45 ?n4s2 + 3.95?n5s +?n6 88 Control Systems

89. 89 Control Systems Figure: 05-30a,bFigure: 05-30a,b

90. 90 Control Systems Figure: 05-30cFigure: 05-30c

91. Table 5.7 The Optimum Coefficients of T(s) Based on the ITAE Criterion for a Ramp Input s2 + 3.2?ns + ?n2 s3 + 1.75?n s2 + 3.25?n2s + ?n3 s4 + 2.41 ?n s3 + 4.93 ?n2s2 + 5.14 ?n3s + ?n4 s5 + 2.19 ?n s4 + 6.50 ?n2s3 + 6.30 ?n3s2 + 5.24?n4s +?n5 91 Control Systems

92. 92 Control Systems T(s) has a steady-state error equal to zero for a ramp input. T(s) has two or more pure integrations as required to provide zero steady-state error.T(s) has a steady-state error equal to zero for a ramp input. T(s) has two or more pure integrations as required to provide zero steady-state error.

93. A: Simplification of linear system 93 Control Systems Let K =10 It is important to sort out the poles that have a dominant effect on the transient response and call these the dominant poles. The poles that are close to the imaginary axis in the LH s-plan give rise to transient responses that will decay relatively slowly, the poles that are far away from thee axis ( relative to the dominant poles) correspond to fast-decaying time responses. When a system is higher than the second order, strictly we can no longer use the damping ratio and natural frequency from prototype second-order systems. ( see Kuo p. 355). We may talk about relative damping ratio.Let K =10 It is important to sort out the poles that have a dominant effect on the transient response and call these the dominant poles. The poles that are close to the imaginary axis in the LH s-plan give rise to transient responses that will decay relatively slowly, the poles that are far away from thee axis ( relative to the dominant poles) correspond to fast-decaying time responses. When a system is higher than the second order, strictly we can no longer use the damping ratio and natural frequency from prototype second-order systems. ( see Kuo p. 355). We may talk about relative damping ratio.

94. Impulse response 94 Control Systems Impulse response and K=10Impulse response and K=10

95. 95 Control Systems

96. B: Simplification of linear systems 96 Control Systems The gain constant K is the same for original and approximate.The gain constant K is the same for original and approximate.

97. 97 Control Systems

98. Simplified model 98 Control Systems Consider the third-order system H(s)Consider the third-order system H(s)

99. Example 5.9 99 Control Systems

100. Example 5.9 100 Control Systems

101. Example 5.9 101 Control Systems Because the lower-order model has two poles, we can estimate that we would obtain a slightly overdamped step response with settling time 4 secBecause the lower-order model has two poles, we can estimate that we would obtain a slightly overdamped step response with settling time 4 sec

102. Example 5.9 102 Control Systems

103. 103 Control Systems

104. Impulse response 104 Control Systems

105. 105 Control Systems

106. Dominant poles of transfer function It has been recognized in practice and in the literature that if the magnitude of the real part of a pole is at least 5 to 10 times of a dominant pole or pair of complex dominant poles, than the pole may be regarded as insignificant insofar as the transient response is concerned. 106 Control Systems

107. Control Systems 107