자동제어공학 4. 과도 응답 정 우 용

제4장 과도 응답 (Transient Response) 목 표 극점, 영점과 시스템 응답 1차 시스템 2차 시스템 미흡 감쇠 특성을 갖는 2차 시스템 추가되는 극점에 대한 시스템의 응답 영점을 갖는 시스템의 응답 시간 영역에서 구한 상태 방정식의 해

목 표 전달 함수로 나타낸 시스템의 시간 응답을 구하는 방법 극점과 영점으로부터 제어 시스템의 응답을 구하는 방법 목 표 전달 함수로 나타낸 시스템의 시간 응답을 구하는 방법 극점과 영점으로부터 제어 시스템의 응답을 구하는 방법 1차와 2차 시스템들의 과도 응답을 정량적으로 나타내는 방법 고차 시스템을 1차 또는 2차 시스템으로 축소시키는 방법 시간 응답에서 시스템의 비선형 특성을 확인하는 방법 상태 공간에서 나타낸 시스템의 시간 응답을 구하는 방법

Transient Response (과도 응답)

Solving a differential Eq or Taking the I.L.T. 시스템 응답 Output response = Forced response + Natural response (출력 응답) (강제 응답) (고유 응답) Solving a differential Eq or Taking the I.L.T. -> Laborious, time consuming 결과를 직관적으로 예측할 수 있다면 정석적인 해석 방법(qualitative analysis)을 이용하여 결과 예측 극점, 영점 및 이들 관계를 이용하여 시스템의 시간 응답을 구함 -> Rapid and simple

극점, 영점과 시스템 응답 (Poles, Zeros, and System Response) 전달 함수의 극점 (Poles of a transfer function) 전달 함수가 ∞가 되도록 하는 s값 전달 함수의 분자항의 근과 공통이 되는 분모항의 근도 포함 전달 함수의 영점 (zeros of a transfer function) 전달 함수를 0이 되도록 하는 s값 전달 함수의 분모항의 근과 공통이 되는 분자항의 근도 포함 1차 시스템의 극점과 영점 전달함수 G(s)는 s=-5인 극점, s=-2인 영점을 갖는다.

Natural response Forced response

C(t)=cforcced(t)+cnatural(t) 입력함수의 극점(input pole)은 강제 응답의 형태를 결정한다. 전달함수의 극점은 고유 응답의 형태를 결정한다. 실수축에 있는 극점의 위치가 –α일 때 이 극점은 e-αt의 형태의 지수 응답을 발생시킨다. 영점과 극점은 강제 응답과 고유 응답의 크기를 결정한다.

예제. 4.1 Forced response Natural response Forced response

1차 시스템 (First-Order System) If the input is a unit step, Taking ILT Where, is forced response (here, ) is natural response (here, )

예제

예제

Time constant ( ) a: exponential frequency 63% of final value for a rising function reach 37% of its original for a decaying function a: exponential frequency

Rise Time ( ) or Where, : time constant : exponential frequency Further the pole from the imaginary axis, the faster the transient response. Rise Time ( ) Rise time is defined as the time for the waveform to go from 0.1 to 0.9 of its final value.

Settling time ( ) Settling time is defined as the time for the response to reach and stay with 2% of its final value.

2차 시스템 The parameters of a second-order system change the form of the response whereas a first-order system change the speed of the response. Basic system

1. Overdamped Response Two distant real poles Transient response : <Example>

2. Underdamped response Complex poles Transient response : <Example>

Figure 4.8 Second-order step response components generated by complex poles

3. Critically damped Doubled real poles Fastest without overshoot Transient response : <Example>

4. Oscillatory Response or Undamped Response Two imaginary poles Transient response : <Example>

Figure 4.10 step responses for second-order system damping cases

일반적인 2차 시스템 Consider the general system Let a=0, then the G(s) is similar to then then exponential decay frequency : a/2

natural frequency (고유 주파수) : 감쇠가 없는 시스템의 진동 주파수 damping ratio (감쇠비) : 포락선(envelope)이 지수 함수적으로 감소되는 주파수와 고유 주파수를 비교하는 양

distance of the roots from the origin is From angle,

Second-order systems A prototype second-order system has the following diagram The transfer function is

The response to a unit step input or is Taking the ILT yield Where,

There are 4 cases to consider 1) If , real and unequal roots → The response is said to be overdamped. 2) If , real and equal roots → The response is said to be critically damped.

3) If complex conjugates roots or → The response is said to be underdamped. 4) If , imaginary roots → The response is said to be undamped or oscillatory.

The system in (2) oscillates at frequency , so is called the natural frequency of the prototype second order system. is the frequency of oscillation of the system without damping. The system, if underdamped, oscillates at frequency is called the damped frequency. The damping of the underdamped system depends on the exponential term is called the damping factor. At critical damping, so

Underdamped Second-Order Systems

Figure 4.13 Second-order underdamped responses for damping ratio values The lower , the more oscillatory the response.

Rise time(상승시간), Tr 파형이 최종값의 0.1에서 0.9에 도달하는데 걸리는 시간 Pick time(최고값 시간), Tp 첫 번째 최고값 또는 최대 최고값에 도달하는데 걸리는 시간 Percent overshoot, %OS 최고값 시간에서 파형의 정상 상태 또는 최종 상태를 넘는 최대값을 정상 상태값에 대해 퍼센트로 나타낸 값 Settling time(정착 시간), Ts 과도 상태가 진동하면서 감소되는 값이 정상 상태값의 ±2% 이내에 도달하여 넘지 않는데 걸리는 시간

Evaluation of the peak time or Maximum overshoots The response of the prototype second-order system to a unit-step input is The derivative : when set to zero, gives the necessary condition for the relative minima and maxima. We get

For (1) to be satisfied, we have the possible solutions It gives The maxima can be found be taking the second derivative or by evaluating for various From (1) it should be easy to see that the first relative maximum occurs at

Evaluation of overshoot The percent overshoot For unit step input Figure 4.15 Percent overshoot vs . damping ratio

Evaluation of (overshoot 2%)

The relationship between ( , %OS , ) and ( ) for underdamped second-order system. Here, poles are Figure 4.17 Pole plot for an underdamped second-order system

Let Then, Real part: Exponential damping frequency Imaginary part: Damped frequency of oscillation : inversely proportional to : only a function of

Figure 4. 18 Lines of constant peak time Tp , settling time Ts , and Figure 4.18 Lines of constant peak time Tp , settling time Ts , and percent overshoot %OS Note: Ts2 < Ts1 ; Tp2< Tp1; %OS1< %OS2

Figure 4.19 Step responses of second-order underdamped systems as poles move: a. with constant real part; b. with constant imaginary part; c. with constant damping ratio

EX) Given the following circuit, find

System Response with additional poles If the real pole is five times further to the left than the dominant poles, we assume that the system is represented by its dominant second-order pair of poles Figure 4.23 Component responses of a three-pole system: a. pole plot; b. component responses: nondominant pole is near dominant second-order pair (Case I), far from the pair (Case II), and at infinity (Case III)

System Response with zeros Generally, the zeros of a response affect the amplitude but do not affect the nature of response-exponential, damped sinusoid. Add a real axis zero (-3, -5, -10) to a two pole system, Figure 4.25 Effect of adding a zero to a two-pole system

If a is negative, placing the zero in the right-half plane. Figure 4.26 Step response of a nonminimum-phase system If a motorcycle is a nonminimum-phase system, it would initially veer left when commanded to steer right.

Effect of zeros Analysis I If the zero is far from the poles, then a is large compared to b and c The zero look like a simple gain factor and does not change the relative amplitudes of the components of the response.

Analysis 2 Another way to look at the effect of zero add a zero to a transfer function If , and is very large, is a scaled version of original response. If , and is not very large, both response important. If , and is very small, derivative term dominate more overshoot in the second-order system. If , nonminimum-phase, opposite directions (aC(s),sC(s)) derivative version scaled version

Pole-Zero cancellation PFE,