Amsa lokaci na Tattalin Sistem na Tsarin Niyaye (Misali Da ake Koyar da Shi)

Tartar da tsarin kawar da gudanar da ake kula da 's' a tushen denominator ta transfer function ta.

Idan darajar da s a tushen denominator ta transfer function ta tsarin kawar da gudanar shi ita 2, wanda ya ce tsarin kawar da gudanar na ikili.

Tushen jin transfer function ta tsarin kawar da gudanar na ikili ta shi ne

A nan, ζ da ωn su ne damping ratio da natural frequency ta tsarin kawar da gudanar, har zuwa (za a bayyana waɗannan abubuwa biyu a cikin bayanan da ba).

Wadanda ake koyar da formula ta, output ta tsarin kawar da gudanar ta shi ne

Idan muna zama unit step function a matsayinta tsarin kawar da gudanar, maka output equation ta tsarin kawar da gudanar ta zai iya duba kamar yadda ake rubuta a baya

Daga baya, za a samun inverse Laplace transform ta equation ta, muna samu

Tushen output c(t) ta shi zai iya duba kamar yadda ake rubuta a baya

Error ta signal ta response ta shi ne e(t) = r (t) – c(t), saboda haka.

Daga tushen expression ta, ana iya cewa error ta signal ta shi ne oscillation type ta fi sani mai dace a lokacin da ζ < 1.

Frequency ta oscillation ta shi ne ωd da time constant ta exponential decay ta 1/ζωn.

Wanda, ωd, ana amfani da shi a cikin damped frequency ta oscillation, da ωn ne natural frequency ta oscillation. Term ta ζ yana haɗa da damping sosai saboda haka wannan term ta ce damping ratio.

Za a samun halayyar output signal, daɗi ɗaya ko kadan idan an yi damar damping ratio, saboda haka muna aiki a yi damar waɗannan cases, ɗaya da ɗaya.

Amma tun a yi damar waɗannan, muna aiki a yi analysis of the time response of a second order control system. Muna aiki a yi haka a yi analysis of the unit step response of a second order control system in the frequency domain, before converting it into the time domain.

Step Response of Second Order System

Idan damping ratio ita zero, muna iya koyar da tushen expression ta output signal ta shi kamar yadda ake rubuta a baya

Saboda a nan, ba a kan exponential term, time response ta tsarin kawar da gudanar ta shi ne un-damped don unit step input function da damping ratio ta zero.

Page 137. Figure 6.4.3. of the book automatic control system by Hasan.

Tunan muna aiki a yi damar case a cikin damping ratio ta unity.



A nan, tushen output signal ta, ba a kan oscillating part a subjective unit step function. Saboda haka time response ta tsarin kawar da gudanar ta shi ne critically damped.

Tunan muna aiki a yi damar time response ta tsarin kawar da gudanar ta subjective unit step input function idan damping ratio ta greater than one.

Taking inverse Laplace transform of both sides of the above equation we get,


In the above expression, there are two time constants.

For the value of ζ comparatively much greater than one, the effect of faster time constant on the time response can be neglected and the time response expression finally comes as

Figure 6.4.5 of the page 139 of the book automatic control system by Hasan.

Critical Damping Time Response of Control System

Tushen time response expression ta tsarin kawar da gudanar ta ikili subject to unit step input function ta shi ne a baya.

The reciprocal of constant of negative power of exponential term in the error part of the output signal is actually responsible for damping of the output response.

Here in this equation it is ζωn. The reciprocal of constant of negative power of exponential term in error signal is known as time constant.

We have already examined that when the value of ζ (also know as damping ratio) is less than unity, the oscillation of the response decays exponentially with a time constant 1/ζωn. This is called under damped response.

On the other hand. when ζ is greater than unity, the response of the unit step input given to the system, does not exhibit oscillating part in it.

This is called over damped response. We have also examined the situation when damping ratio is unity that is ζ = 1.

In that situation the damping of the response is governed by the natural frequency ωn only. The actual damping at that condition is known as critical damping of the response.

As we have already seen in the associated expressions of time response of control system subject to input step function, the oscillation part is present in the response when damping ratio (ζ) is less than one and it is not present in the response when damping ratio is equal to one.