چه ئەمە یەکەم تاریخی کۆنترۆل سیستەم؟

Yekemîna Sisteman Destûrê ya Yekemîn Rêjeya?

Pêşnûma Yekemîna Sisteman Destûrê

Sisteman destûrê yekemîna ji bo rengkirina derketinên daxwazan û daneyan di rêjiyê de taybetmendiyek e ku li ser pirsemên yekemîn dema herêmdeh bêtirbîne.

Funksiyona berhevz (berhemîna daxwaza-danaya) ji bo vê sisteman destûrê hatiye pêşnîkirin:

• K ji bo DC Gain (DC gain of the system ratio between the input signal and the steady-state value of output)

• T ji bo time constant of the system (the time constant is a measure of how quickly a first-order system responds to a unit step input).

Funksiyona Berhevzê Sisteman Destûrê Yekemîna

Funksiyona berhevz ji bo rengkirina derketina danaya sisteman destûrê bi daxwaza, ji bo hemî nirxên daxwaza mungkin.

Poles of a Transfer Function

Poles funksiyona berhevzê nivîsên Laplace Transform variable(s) ne, ku funksiyona berhevzê bikin da heta beser dikin.Denominateora funksiyona berhevzê rasti poles funksiyonê ye.

Zeros of a Transfer Function

Zeros funksiyona berhevzê nivîsên Laplace Transform variable(s) ne, ku funksiyona berhevzê bikin da sifir bikin.Nominator funksiyona berhevzê rasti zeros funksiyonê ye.

Sisteman Destûrê Yekemîna

Lêgera ji bo vê sisteman destûrê yekemîna bi zeros. Sisteman destûrê yekemîna bizimîne çend rojan dike ku ji bo dema ku werin ên steady-state.Eger daxwaza ji bo unit step be, R(s) = 1/s dema danaya ji bo step response C(s). Têkiliya general 1st order control system , i.e ji bo funksiyona berhevzê.

Divê du pole hene, yek ji bo pole input di origin s = 0 de û yek ji bo pole system di s = -a de, vê pole di negativ axis pole plot de ye.Bi komandê MATLAB’s pzmap, divê ji bo nasnameyan poles û zeros sisteman, ku ew bêdihêjin ji bo analizkirina cayda.Niha ji bo inverse transform, total response bikin da ku sum of forced response û natural response be.

Gava pole input di origin de, forced response bike name describe by itself that giving forced to the system so it produces some response which is forced response û pole system di -a de natural response bike ku ji bo transient response of the system.

Pas hevgirtina hesabkirin, formê general of the first-order system C(s) = 1-e-at ye ku equal to forced response "1" û natural response "e-at". Gava parameter "a" dibikin.

Teknikên zaf û inverse Laplace Transform, wan heta total response bike betirbînin lê wan dem û koyi.

Karberden poles, zeros, û wata fundamental concept bizimîne ma zi qualitative information ji bo çareserkirina pirsgirêkan û bi vê anîn, divê ji bo çend rojan dike û dema ku sisteman ên steady-state point.

Ji bo şêwekirina three transient response performance specifications, time constant, rise time, û settling time ji bo sisteman destûrê yekemîna.

Time Constant of a First Order Control System

Time constant define bikin da ku ji bo step response bi up to 63% or 0.63 of its final value. We refer to this as t = 1/a. Ji bo reciprocal of time constant, unit 1/seconds or frequency.

We call the parameter “a” the exponential frequency. Because the derivative of e-at is -a at t = 0. So the time constant is considered as a transient response specification for a first-order control system.

We can control the speed of response by setting the poles. Because the farther the pole from the imaginary axis, the faster the transient response is. So, we can set poles farther from the imaginary axis to speed up the whole process.

Rise Time of a First Order Control System

The rise time is defined as the time for the waveform to go from 0.1 to 0.9 or 10% to 90% of its final value. For the equation of rising time, we put 0.1 and 0.9 in the general first-order system equation respectively.

For t = 0.1

For t = 0.9

Taking the difference between 0.9 and 0.1

Here the equation of rising time. If we know the parameter of a, we can easily find the rise time of any given system by putting “a” in the equation.

Settling Time of a First Order Control System

The settling time is defined as the time for the response to reach and stay within 2% of its final value. We can limit the percentage up to 5% of its final value. Both percentages are a consideration.

The equation of settling time is given by Ts = 4/a.

By using these three transient response specifications, we can easily compute the step response of a given system that’s why this qualitative technique is useful for order systems equations.

Conclusion of First Order Control Systems

After learning all things related to 1st order control system, we come to the following conclusions:

• A pole of the input function generates the form of the forced response. It is because of the pole at the origin which generates a step function at the output.

• A pole of the transfer function generates a natural response. It the pole of the system.

• A pole on the real axis generates an exponential frequency of the form e-at. Thus, the farther the pole to the origin, the faster the exponential transient response will decay to zero.

• Understanding poles and zeros allows us to enhance system performance and achieve faster, more accurate outputs.