CH. 3 Time Response

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 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 • 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. • 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. 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 Poles: Two real at 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 poles. • - 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