1. CH. 3 Time Response
Chapter3. Time Response

2. 3.1 Introduction
How to use poles and zeros to determine the response of a control system.
How to describe quantitatively the transient response of first- and second- order systems.
How to find the time response from the transfer function.
How to approximate higher-order systems as first and second order.
Chapter3. Time Response

3. 3.2 Poles, Zeros, and system Response
Poles: values of s that cause the transfer function to become infinite.
Zeros: values of s that cause the transfer function to become zero.
Output response of a system: sum of two responses, forced response and natural response.
Poles and zeros of a first-order system.
Chapter3. Time Response

4. A pole on the real axis generate an exponential response.
A pole of the input function generates the form of the forced response.
A pole of the transfer function generates the form of the natural response.
A pole on the real axis generate an exponential response.
The zeros and poles generate the amplitudes for both the forced and natural response.
Chapter3. Time Response

5. 3.3 Transient-response specifications
Rise time, Tr : The time required for the response to rise from 10% to 90% (or from 0% to 100%) of the final value.
Peak time, TP : The time required to reach the first, or maximum, peak.
Percent overshoot, %OS : The amount that the response curve overshoots the steady-state value at the peak time, expressed as a percentage of the steady-state value.
Settling time, TS : The time required for the transient’s damped oscillations to reach and stay within ±2% (or ± 5%) of the steady-state value.
Chapter3. Time Response

6. 3.4 First-order systems
Time constant : , The time required for the step response to rise to 63% of its final value.
Rise time, Tr
Settling time, TS
Chapter3. Time Response

7. 3.4 First-order systems Transfer function from step response.
Chapter3. Time Response

8. 3.5 Second-order systems Overdamped response Underdamped response
Chapter3. Time Response

9. 3.5 Second-order systems Undamped response Critically damped response
Chapter3. Time Response

10. Chapter3. Time Response

11. Classification of step responses
Overdamped responses
Poles: Two real at
Underdamped responses
Poles: Two complex at
Undamped responses
Poles: Two imaginary at
Critically damped responses
Chapter3. Time Response

12. The general second-order system
Natural frequency, ωn : frequency of oscillation of the system without damping.
Damping ratio, ζ : exponential decay frequency / natural frequency.
Chapter3. Time Response

13. Chapter3. Time Response

14. Transient-response specifications of second order systems.
Overdamped system: two first-order systems.
Step response of under-damped second order systems:
Chapter3. Time Response

15. Step response of under-damped second order systems.
Evaluation of peak time:
Evaluation of %OS:
Evaluation of settling time:
Chapter3. Time Response

16. Step response of under-damped second order systems.
Evaluation of rise time:
Chapter3. Time Response

17. Step response of under-damped second order systems.
Pole plot:
Chapter3. Time Response

18. Chapter3. Time Response

19. 3.6 System response with additional poles and zeros.
- Additional pole의 허수 축과의 거리가 대표극점(dominant poles)의 허수 축과의 거리에 비해 상당히 클 때 그 극점은 무시될 수 있다.
- 그 거리가 상대적으로 작을 때 (5배 이하) specs에 영향.(특히 rise time)
Chapter3. Time Response

20. Additional poles.
Chapter3. Time Response

21. Additional zeros.
Chapter3. Time Response

22. - Affect the residue, do not affect the nature of the response.
- 허수축에 가까운 zero는 specs에 영향.(특히 overshoot)
- Nonminimum-phase systems.
Chapter3. Time Response

23. Time response from the transfer function.
극점 중에서 대표극점(dominant poles)을 선택.
대표극점(dominant poles): 극점들 중에서 과도응답이 가장 오래 지속되어 시스템 응답에 지배적인 역할을 하는 극점, 허수축에 가장 가까이 있는 극점.(residue 고려)
대표극점의 단순 이차시스템으로 근사화하여 시간역 성능 예측.
추가극점이나 영점을 대표극점과의 거리를 고려하여 영향 고려.
Chapter3. Time Response