CH. 3 Time Response

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CH. 3 Time Response. 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.

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### CH. 3 Time Response

Chapter3. Time Response

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

3.2 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

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

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

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

3.4 First-order systems
• Transfer function from step response.

Chapter3. Time Response

3.5 Second-order systems
• Overdamped response
• Underdamped response

Chapter3. Time Response

3.5 Second-order systems
• Undamped response
• Critically damped response

Chapter3. Time Response

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

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

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

Chapter3. Time Response

Step response of under-damped second order systems.
• Evaluation of peak time:
• Evaluation of %OS:
• Evaluation of settling time:

Chapter3. Time Response

Step response of under-damped second order systems.
• Evaluation of rise time:

Chapter3. Time Response

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

Chapter3. Time Response

Chapter3. Time 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