1. Instructor: Jongeun Choi
Time Response*, ME451
Instructor: Jongeun Choi
* This presentation is created by Jongeun Choi and Gabrial Gomes

2. Zeros and poles of a transfer function
Let G(s)=N(s)/D(s), then
Zeros of G(s) are the roots of N(s)=0
Poles of G(s) are the roots of D(s)=0
Im(s)
Re(s)

3. Theorems Initial Value Theorem Final Value Theorem
If all poles of sX(s) are in the left half plane (LHP), then

4. DC gain or static gain of a stable system
1.4
1.2 1
0.8
0.6
0.4
0.2
0.5 1
1.5 2
2.5 3

5. DC Gain of a stable transfer function
DC gain (static gain) : the ratio of the steady state output of a system to its constant input, i.e., steady state of the unit step response
Use final value theorem to compute the steady state of the unit step response

6. Pure integrator ODE : Impulse response : Step response :
If the initial condition is not zero, then :
Physical meaning of the impulse response

7. First order system ODE : Impulse response : Step response :
DC gain: (Use the final value theorem)

8. First order system If the initial condition was not zero, then
Physical meaning of the impulse response

9. Matlab Simulation
G=tf([0 5],[1 2]);
impulse(G)
step(G)
Time constant

10. First order system response
System transfer function :

11. First order system response
System transfer function :
Impulse response :

12. First order system response
System transfer function :
Impulse response :

13. First order system response
System transfer function :
Impulse response :
Step response :

14. First order system response
Im(s)
Re(s)

15. First order system response
Im(s)
Unstable
Re(s)

16. First order system response
Im(s)
Unstable -1
Re(s)

17. First order system response
Im(s)
Unstable
Re(s) -2

18. First order system response
Im(s)
Unstable
faster response slower response
Re(s)
constant

19. First order system – Time specifications.

20. First order system – Time specifications.
Time specs:
Steady state value :
Time constant :
Rise time :
Time to go from to
Settling time :

21. First order system – Simple behavior.
No overshoot
No oscillations

22. Second order system (mass-spring-damper system)
ODE :
Transfer function :

23. Polar vs. Cartesian representations.
… Imaginary part (frequency)
… Real part (rate of decay)

24. Polar vs. Cartesian representations.
… Imaginary part (frequency)
… Real part (rate of decay)
Polar representation :
… natural frequency
… damping ratio

25. Polar vs. Cartesian representations.
… Imaginary part (frequency)
… Real part (rate of decay)
Polar representation :
… natural frequency
… damping ratio

26. Polar vs. Cartesian representations.
… Imaginary part (frequency)
… Real part (rate of decay)
Polar representation :
… natural frequency
… damping ratio
Unless overdamped

27. Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio :
… Overdamped
… Critically damped
… Underdamped
… Undamped

28. Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio :
… Overdamped
… Critically damped
… Underdamped
… Undamped

29. Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio :
… Overdamped
… Critically damped
… Underdamped
… Undamped

30. Polar vs. Cartesian representations.
System transfer function :
All 4 cases
Unless overdamped
Significance of the damping ratio :
… Overdamped
… Critically damped
… Underdamped
… Undamped

31. Underdamped second order system
Two complex poles:

32. Underdamped second order system

33. Impulse response of the second order system

34. Matlab Simulation zeta = 0.3; wn=1; G=tf([wn],[1 2*zeta*wn wn^2]);
impulse(G)

35. Unit step response of undamped systems
DC gain :

36. Unit step response of undamped system

37. Matlab Simulation zeta = 0.3; wn=1; G=tf([wn],[1 2*zeta*wn wn^2]);
step(G)

38. Second order system response.
Stable 2nd order system:
2 distinct real poles
A pair of repeated real poles
A pair of complex poles
Im(s)
Unstable
Re(s)

39. Second order system response.
Stable 2nd order system:
2 distinct real poles
A pair of repeated real poles
A pair of complex poles
Im(s)
Unstable
Re(s)

40. Second order system response.
Stable 2nd order system:
2 distinct real poles
A pair of repeated real poles
A pair of complex poles
Im(s)
Unstable
Re(s)

41. Second order system response.
Stable 2nd order system:
2 distinct real poles
A pair of repeated real poles
negative real part
A pair of complex poles
zero real part
Im(s)
Unstable
Re(s)

42. Second order system response.
Stable 2nd order system:
2 distinct real poles
A pair of repeated real poles
negative real part
A pair of complex poles
zero real part
Im(s)
Unstable
Re(s)

43. Second order system response.
Stable 2nd order system:
2 distinct real poles
A pair of repeated real poles
negative real part
A pair of complex poles
zero real part
Im(s)
Unstable
Re(s)

44. Second order system response.
Im(s)
2 distinct real poles = Overdamped
Unstable
Re(s)

45. Second order system response.
Im(s)
Repeated real poles = Critically damped
Unstable
Re(s)

46. Second order system response.
Im(s)
Complex poles negative real part = Underdamped
Unstable
Re(s)

47. Second order system response.
Im(s)
Complex poles zero real part = Undamped
Unstable
Re(s)

48. Second order system response.
Im(s)
Underdamped
Unstable
Undamped
Overdamped or Critically damped
Re(s)
Underdamped

49. Overdamped system response
System transfer function :
Impulse response :
Step response :

50. Overdamped and critically damped system response.

51. Overdamped and critically damped system response.

52. Overdamped and critically damped system response.

53. Overdamped and critically damped system response.

54. Polar vs. Cartesian representations.

55. Polar vs. Cartesian representations.
System transfer function :
All 4 cases
Unless overdamped
… Cartesian overdamped
Significance of the damping ratio :
… Overdamped
… Critically damped
… Underdamped
… Undamped

56. Polar vs. Cartesian representations.
System transfer function :
All 4 cases
Unless overdamped
… Cartesian overdamped
Significance of the damping ratio :
… Overdamped
… Critically damped
… Underdamped
… Undamped

57. Polar vs. Cartesian representations.
System transfer function :
All 4 cases
Unless overdamped
… Cartesian overdamped
Significance of the damping ratio :
… Overdamped
Overdamped case:
… Critically damped
… Underdamped
… Undamped

58. Second order impulse response – Underdamped and Undamped

59. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

60. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

61. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

62. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

63. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

64. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

65. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

66. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

67. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

68. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

69. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

70. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

71. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

72. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

73. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

74. Second order impulse response – Underdamped and Undamped
Increasing / Fixed

75. Second order step response – Underdamped and Undamped

76. Second order impulse response – Underdamped and Undamped
Higher frequency oscillations
Faster response
Slower response
Unstable
Lower frequency oscillations

77. Second order impulse response – Underdamped and Undamped
Less damping
Unstable
More damping

78. Second order step response – Time specifications.
1.4
1.2 1
0.8
0.6
0.4
0.2
0.5 1
1.5 2
2.5 3

79. Second order step response – Time specifications.
… Steady state value.
… Time to reach first peak (undamped or underdamped only).
… % of in excess of
… Time to reach and stay within 2% of
1.4
1.2 1
0.8
0.6
0.4
0.2
0.5 1
1.5 2
2.5 3

80. Second order step response – Time specifications.
… Steady state value.
More generally, if the numerator is not , but some :

81. Second order step response – Time specifications.
… Peak time.
Therefore,
is the time of the occurrence of the first peak :

82. Second order step response – Time specifications.
… Percent overshoot.
Evaluating at ,
is defined as:
Substituting our expressions for and :

83. Second order step response – Time specifications.
… Settling time.
Defining with , the previous expression for can be re-written as:
As an approximation, we find the time it takes for the exponential envelope to reach 2% of
when

84. Typical specifications for second order systems.
How many independent parameters can we specify? 3