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CH. 3 Time Response

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CH. 3 Time Response

Chapter3. Time Response

- 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

- 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

- 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

- 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

- 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

- Transfer function from step response.

Chapter3. Time Response

- Overdamped response
- Underdamped response

Chapter3. Time Response

- Undamped response
- Critically damped response

Chapter3. Time Response

Chapter3. Time Response

- 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

- Natural frequency, ωn : frequency of oscillation of the system without damping.
- Damping ratio, ζ : exponential decay frequency / natural frequency.

Chapter3. Time Response

Chapter3. Time Response

- Overdamped system: two first-order systems.
- Step response of under-damped second order systems:

Chapter3. Time Response

- Evaluation of peak time:
- Evaluation of %OS:
- Evaluation of settling time:

Chapter3. Time Response

- Evaluation of rise time:

Chapter3. Time Response

- Pole plot:

Chapter3. Time Response

Chapter3. Time Response

- 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

- Additional poles.

Chapter3. Time Response

- Additional zeros.

Chapter3. Time Response

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

Chapter3. Time Response

- 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