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

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Ch 3 time response

CH. 3 Time Response

Chapter3. Time Response


3 1 introduction

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

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


Ch 3 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

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

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 systems1

3.4 First-order systems

  • Transfer function from step response.

Chapter3. Time Response


3 5 second order systems

3.5 Second-order systems

  • Overdamped response

  • Underdamped response

Chapter3. Time Response


3 5 second order systems1

3.5 Second-order systems

  • Undamped response

  • Critically damped response

Chapter3. Time Response


Ch 3 time response

Chapter3. Time Response


Classification of step responses

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

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


Ch 3 time response

Chapter3. Time Response


Transient response specifications of second order systems

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

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 systems1

Step response of under-damped second order systems.

  • Evaluation of rise time:

Chapter3. Time Response


Step response of under damped second order systems2

Step response of under-damped second order systems.

  • Pole plot:

Chapter3. Time Response


Ch 3 time response

Chapter3. Time Response


3 6 system response with additional poles and zeros

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


Ch 3 time response

  • Additional poles.

Chapter3. Time Response


Ch 3 time response

  • Additional zeros.

Chapter3. Time Response


Ch 3 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


Ch 3 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


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