1. Transient Response Analysis Of Control Systems
Control Systems (CS)
Lecture-5
Transient Response Analysis Of Control Systems
Dr. Imtiaz Hussain
Associate Professor
Mehran University of Engineering & Technology Jamshoro, Pakistan
URL :
6th Semester 14ES (SEC-I)
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.

2. Outline Introduction Standard Test Signals
Transient Response of 1st Order Systems
Transient Response of 2nd 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
Time response of a dynamic system response to an input expressed as a function of time.
System
The time response of any system has two components
Transient response
Steady-state response.

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)
ramp signal with slope A
r(t)
unit ramp signal

11. Standard Test Signals Parabolic signalt
p(t)
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.
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 1st 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 1st Order System
Consider the following 1st order system t
δ(t) 1

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

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

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

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

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

25. Step Response of 1st 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 1st 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 1st 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. cs(0)=0

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

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

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

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

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

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

39. 1st 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. 1st Order System with & W/O Zero
If T>α the response will be same

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

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

43. 1st Order System with a Zero

44. 1st Order System with & W/O Zero
1st 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 jω

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

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

50. Comparison

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

52. First Order System With Delays1
Unit Step
Step Response t td

53. First Order System With Delays

54. Examples of First Order Systems
Armature Controlled D.C Motor (La=0) u ia T Ra La J  B eb
Vf=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 1st 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.

58. Second Order Systems
A general second-order system is characterized by the following transfer function.
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 2nd order transfer function.

60. Second Order Systems
Two poles of the system are

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 jω

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 jω

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 jω

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 jω

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

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

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

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

69. S-Plane
For Underdamped system therefore, δ jω

70. S-Plane
For Undamped system therefore, δ jω

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

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

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

74. S-Plane

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.

76. Example-5
The natural frequency of closed loop poles of 2nd 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.

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

78. S-Plane

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

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

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

83. Time-Domain Specification
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
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

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:

87. Step Response of underdamped System

88. Step Response of underdamped System
When

89. Step Response of underdamped System

90. Step Response of underdamped System

91. Step Response of underdamped System

92. Step Response of underdamped System

93. Step Response of underdamped System

94. Step Response of underdamped System

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

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 tr, peak time tp, maximum overshoot Mp, and settling time 2% and 5% criterion ts when the system is subjected to a unit-step input.

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

98. Example#6
Rise Time

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

100. Example#6
Maximum Overshoot

101. Example#6

102. Example#7
For the system shown in Figure-(a), determine the values of gain K and velocity-feedback constant Kh 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 Kh, obtain the rise time and settling time. Assume that J=1 kg-m2 and B=1 N-m/rad/sec.

103. Example#7

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

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

106. Example#7

107. Example#7

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.

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.

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).

111. Example#10

112. Example#10

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.

114. Example # 11
Assume the roll dynamics, which relates the roll-angle output, 𝜃(𝑠), to a disturbance-torque input, TD(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.

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

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

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

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