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Misali da Dukkana Tsawo na Zama

Misali da Dukkana Tsawo na Zama?

Misali da dukkana tsawo na zama yana nufin farkon bayanai daga abubuwa da aka fi sani da abubuwan da suka samun a kan fadada masu sayarwarsa a lokacin da lokaci ya karkara (ya'ni a lokacin da tarihin sayarwarsa ta samu tsawo na zama).

Misali da dukkana tsawo na zama shi ne alamomin tarihi masu sayarwarwa. A gaba daya, sayarwarsa mai kyau ce wanda yake da misali da dukkana tsawo na zama mai kadan.

Sai dai, za a iya magana game da misali da dukkana tsawo na zama a kan fadada na farko daidai tushen tarihin sayarwarsa. Za a iya duba fadada tushen:

$\begin{equation*} \frac {C(s)}{R(s)}=\frac {1}{0.7s+1} \end{equation*}$

Wannan shi ne fadada tushen na farko, tare da kirkiro da take da wahid da kuma kashi na lokaci na 0.7 detas. Yana da suna fadada tushen na farko saboda 's' a kan kafin ya samu hanyar '1'. Idan yana cikin kafin ya samu $0.7s^2 + 1$ , yana cikin kafin ya samu fadada tushen na biyu.

Tarihin wannan fadada tushen zuwa dukkana tsawo na zama ana nuna a Figura-1. Ana iya gani cewa a lokacin da tarihi ya samu tsawo na zama, maida yana daidaita da dukkana. Saboda haka, misali da dukkana tsawo na zama ba a bani ba.

Wakar da aikin First order Transfer Function daidai step Input. — Figure-1: Yana wakar da aikin First order Transfer Function daidai step Input. Ana iya samun cewa kungiyar daidai aiki na zero

Yanayin wannan function zuwa unit ramp input ya shahara a Figure-2. Ana iya samun cewa a daidai aiki akwai farko daga bayanan da kuma aiki. Saboda haka, don unit ramp input, akwai kungiyar daidai aiki.

Wakar da aikin First order Transfer Function daidai Ramp Input. — Figure-2: Yana wakar da aikin First order Transfer Function daidai ramp Input. Ana iya samun cewa akwai kungiyar daidai aiki a wannan yanayi

Koyi cewa a cikin littattafai masu kontrol system za ku iya samun cewa, daidai ramp input, kungiyar daidai aiki na First order Transfer Function ce ta time constant. Daga neman Figure-2 a nan, ana iya samun cewa wannan ce daidai. A t=3 seconds, bayanan ya 3 amma aiki ya 2.3. Saboda haka, kungiyar daidai aiki ya 0.7, wanda yake daidai da time constant na First order Transfer Function.

Koyi abubuwan da suka fi kyau:

Kungiyar daidai aiki ya fiye daidai idan bayanan ita ce parabolic, ya fiye daidai da ramp input, amma ya fiye daidai da step input. Kamar yadda aka bayyana a nan, kungiyar daidai aiki na zero daidai step input, amma 0.7 daidai ramp input, amma ana iya samun cewa yake ∞ daidai parabolic input.
Ya kamata a tabbatar cewa kungiyar daidai aiki yana nufin bayanan, amma zama ba sa bayanan ba.

Zaɓi wani sistem na kontrol mai kuliya da ya shiga funshin na karamin hawa

$\begin{equation*} \frac{C(s)}{R(s)}= \frac{G(s)}{1+G(s)H(s)} \end{equation*}$

Daga cikin abubuwa, amfani da tushen da suka samu ma'ana. Ingantaccen sistema yana nufin da 'yan gaba na 'yan kasuwa i.e. ‘1+G(s)H(s)’. ‘1+G(s)H(s) = 0’ ita ce ta hanyar inganci. Yawan daɗi na 'yan gaba na 'yan kasuwa sun nuna ingantaccen sistema. Kasa na zamani na zamani yana nufin da R(s).

A cikin sistem na kontrol mai kuliya, za a iya hasashen siffar daɗi na zamani kamar $E(s)= \frac{R(s)}{1+G(s)H(s)}.$ Kasa na zamani na zamani za a iya hasasha kamar ess= $\lim_{s \rightarrow 0 } E(s)$ , inda kasa na zamani na zamani shine ranar da siffar daɗi ke fara a zamani. Daga wannan zan iya sanin cewa kasa na zamani na zamani yana nufin da R(s).

Kamar yadda aka ambaci, ingantaccen sistema yana nufin da 'yan gaba na 'yan kasuwa i.e. 1 + G(s)H(s). Haka ‘1’ shine mafi yawa, saboda haka ingantaccen sistema yana nufin da G(s)H(s), wanda yake da ƙarin canza. Saboda haka, za ku iya fahimtar Bode plot, Nyquist plot an kawo da G(s)H(s), amma suke nuna ingantaccen $\frac{C(s)}{R(s)}$ .
G(s)H(s) yana ake kira funshin daɗiɗa mai sarrafa kuma $\frac{C(s)}{R(s)}$ yana ake kira funshin daɗiɗa mai sarrafa gaba. Ta hanyar tattaunawa funshin daɗiɗa mai sarrafa kadaidaita G(s)H(s), za a iya samun zafiya ta funshin daɗiɗa mai sarrafa gaba tun daga Bode plot & Nyquist plot.

Misalai Daga Ƙarfin Tsawo

Daga Ƙarfin Tsawo Don Inganci Mai Siffar Tsohuwa

Yanzu, za a bayyana, daga ƙarfin tsawo a cikin ƙungiyar daɗiɗa mai sarrafa gaba da misalai na lambar. Za mu bude da ƙungiya daɗiɗa mai sarrafa tare da inganci mai siffar tsohuwa.

Misali-1:

Sani ƙungiyar daɗiɗa mai sarrafa (system-1) kamar yadda ake nuna a Figure-3:

Closed Loop Control System — Figure-3: Ƙungiyar Daɗiɗa Mai Sarrafa Gaba

Inganci mai siffar 'Rs' yana ɗaukarsa mai siffar tsohuwa.

Duk ƙarin halayen System-1 suna nufin a Figure-4.

Steady State Value Block Diagram — Raisa-4: Muhimmanci Ma'adani na Yawancin Nuna a Tashar Sistem

Yana iya samun cewa ma'adanta mai yawa na nuna ta shirya shi ne 0.5, saboda haka ma'adanta mai yawa na nuna ta shi ne 0.5. Idan tashar da ke gaba ya zama layi da kuma abubuwan nuna suna daidai, yana iya samun muhimman ma'adanin da za su iya samun a haka:

A cikin tushen bayanai kamar $s\rightarrow 0$ , zan iya samun ma'adanta mai yawa na tushen bayanai.

Zan iya kalkulasar fayiladda hakan:

$\begin{equation*} \frac{C(s)} {R(s)}= \frac{4}{s+8} \end{equation*}$

Daga baya $R(s)$ = ci gaba na tsari = $\frac{1}{s}$ , zan iya sayyar wannan hakan:

$\begin{equation*} C(s)= \frac{4}{s+8} \times R(s)= \frac{4}{s(s+8)} \end{equation*}$

Zamani na iya aiki a cikin zama ta kasa:

$\begin{equation*} \lim_{s \rightarrow 0 } sC(s) = s\frac{4}{s+8}\frac{1}{s} =\frac{1}{2} \end{equation*}$

A zan iya amfani da haka don tafi yawan zamani na iya aiki a cikin zama ta kasa. Misali:

Na gaba $R(s)= \frac{1}{s}$ (Na gaba mai yawa)

Zamani na iya aiki a cikin zama ta kasa= $\lim_{s \rightarrow 0 }\ sR(s)=s \frac{1}{s}$ = 1.

Kuma za a iya kalkulata sabon alama a haka:

$\begin{equation*} E(s)= \frac{R(s)}{1+G(s)H(s)}= \frac{s+4}{s(s+8)} \end{equation*}$

Zama da hada na tashin alamun bayanin (yana nufin zama da hada na tashin alamun bayanin) shine:

$\begin{equation*} \lim_{s \rightarrow 0} sE(s)= s\frac{s+4}{s(s+8)}= \frac{1}{2} \end{equation*}$

Duk da cewa, zan iya samun cewa daga Raisu-4, farkon bayanin da shirin bayanin yana cikin 0.5. Saboda haka, zama da hada na tashin alamun bayanin yana cikin 0.5.

Wani babban hanyar da za a iya amfani da shi don kula zama da hada na tashin alamun bayanin shine gano sabbin bayanin alamun, kamar yadda ake bayyana a nan:

Kalkula da ci gaba da takwasu Kp = $\lim_{s \rightarrow 0 } G(s)H(s)$ , zaka iya samun Kp = 1, ess= $\frac{1}{1+Kp}$ . Zaka iya samun amsar da dama.

Idan alamar yadda tana cikin yadda, ya kai $R(s)=\frac{3}{s}$ (wannan shi ne alamar yadda, amma bai wani yadda), kuma ci gaba da takwasu ess= $\frac{3}{1+Kp}$

Idan alamar yadda ta yi, kuma kana son cikakken Kalkula, Velocity error coefficient Kv= $\lim_{s \rightarrow 0 }s G(s)H(s)$ , ess= $\frac{1}{Kv}$

Idan ma input ita ce unit parabolic input, kuma Bara, Acceleration error coefficient Ka= $\lim_{s \rightarrow 0 } s^2G(s)H(s)$ , ess= $\frac{1}{Ka}$ .

Daga tattalin error constants Kp, Kv da Ka, za ku iya fahimtar yadda error na steady state ta yi amfani da input.

PI Controller And Steady State Error

A PI controller (i.e. a proportional controller plus integral controller) reduces the steady state error (ess), but has a negative effect on the stability.

PI controllers have the advantage of reducing the steady-state error of a system, while having the disadvantage of reducing the system’s stability.

A PI controller reduces stability. This means that damping decreases; peak overshoot and settling time increases due to PI controller; Roots of characteristics equation (poles of closed-loop transfer function) in left-hand side will come closer to the imaginary axis. The system order also increases due to PI controller, which tends to reduce the stability.

Consider two characteristics equations, one is s3+ s2+ 3s+20=0, another is s2+3s+20=0. Just by observation, we can tell you that system related to first equation has lower stability as compared to second equation. You can verify it by finding the roots of the equation. So, you can understand higher order characteristics equations have lower stability.

Now, we will add one PI controller (Proportional Plus Integral controller) in system-1 (Figure-3) and examine the results. After inserting PI controller in system-1, various steady state values are shown in Figure-5, It can be seen that output is exactly equal to the reference input. It is the advantage of PI controller, that it minimizes the steady state error so that output tries to follow reference input.

PI Controller Block Diagram — Rikitarin-5: Zanen PI Controller yana zama a wannan diagram

Transfer function ta PI controller zai iya kula da tarihin $Kp+\frac{Ki}{s}$ ko $\frac{Kps+Ki}{s}.$ Ana iya magana wani tambayar da ya fi amfani da shi cewa idan input ta transfer function duka zero maka output yake ita ce zero. Don haka, a halin yanzu input ta PI controller ita ce zero, amma output ta PI controller ita ce balon (ya ni 1). Wannan bayani ba a gina a cikin littattafan control system bane, don haka za a bayar a kan:

(1) Steady state error ba zero bale, ya faruwa zuwa zero, haka 's' ba zero bale, ya faruwa zuwa zero, Saboda haka idan a lokacin da yawa steady state error ita ce 2x10-3, a lokacin da 's' (muna magana game da 's' a denominator ta PI controller) ita ce 2x10-3, saboda haka output ta PI controller ita ce '1'.

Za a duba wasu control system a Rikitarin-6:

Closed Loop Control System with PI Controller — Rikitarin-6: Misalai na Closed Loop Control System da PI Controller

A halin yanzu, muna iya cewa, a lokacin da yawa steady state error ita ce 2x10-3, a lokacin da 's' ita ce 4×10-3; saboda haka output ta PI controller ita ce '0.5'. Yana nufin cewa 'ess' da 's' sun faruwa zuwa zero, amma ratio su ita ce balon.

A cikin littattun da sauki masu kontrol, ba za a samu s=0 ko t=∞; zan iya samu
$s\rightarrow 0, t\rightarrow 0.$

(2) Bayanan da biyu shine ya'ayyace ta hanyar tsari yana duka, 's' yana duka a tsari. PI controller transfer function shi ne $\frac{Kps+Ki}{s}$ . A cikin littattun da matematika, za a samu $\frac{0}{0}$ ba a tabbas, saboda haka ya kamata a yi waɗanda ɗaya (tattaunawa Figure-7).

(3) Bayanan da uku shine, $\frac{1}{s}$ shine integrator. Input shi zero, integration of zero ba a tabbas. Saboda haka, output of the PI controller may be any finite value.

One basic difference in open loop control system & closed loop control system

In reference to the above explanation, we will explain one basic difference in an open-loop control system & a closed-loop control system. Differences in open-loop control system & closed-loop control system, you can find in any book of control systems*, but one basic difference which is related to the above explanation is given here and we hope certainly it will be useful for the readers.

An open loop control system can be represented as follows:

Sistem kontrol loop gaskiya (sistem kontrol feedback) zai iya cewa haka:

Fankin karkashin plant yana da yawa (Fankin karkashin plant zai iya canza bata da shawarar al'amuran jiki, kusa da sauransu). A duk filayoyinka masu mafi girma ta, muna amfani da H(s)=1; Mai karatu zai iya karar da fankin karkashin mai karatu (kamar parametere na mai karatu wanda Kp, Kd, Ki) kamar haka.

Mai karatu zai iya kasance Mai karatu Proportional (P), PI, PD, PID, Fuzzy logic controller da sauransu. Akwai biyu abubuwa da mai karatu ke yi (i) Don samun inganci, ya'ni damping ya fi kan 0.7-0.9, peak overshoot da settling time suna da kyau (ii) Tushen error steady-state yana da kyau (yana da zero).

Amma idan muna tattauna damping maka tushen error steady-state zai iya zama da kyau. Saboda haka, don in kara mai karatu zai iya ba da (inganci & tushen error steady-state) suka da kyau. Ingantaccen kara mai karatu yana da littattafan buƙataccen.

An rubuta da shi, Mai karatu PI ya haɗa tushen error steady-state (ess) da kuma ya haɗa inganci.

Yanzu, zan bayyana farkon karamin duniya da ke nuna a wasu sistem kontrol loop buka da sistem kontrol loop gaskiya, wadannan suna da alaka da bayanan da aka rubuta a nan.

Amsa Rabi-10; wani sistem kontrol loop buka.

Ko da ake magana a cikin input na iya zama input mai yawa. Saboda haka, yadda ake gani wani abu a kan input ita ce '1'. Ana iya tafi shi cewa abin da ake samu a kan output ita ce '2'. Idan ana iya canza transfer function [G(s)] ta plant baki daya, yadda zai haɗa a kan input da output? Amsa ta ita ce input ta plant ba zai canza, amma output ta plant zai canza.

Tana da daga Figures-11 &12

Closed loop system — Figure-12: Closed Loop System, Plant output is same but plant input is changed due to change in Transfer Function

Duka biyu ne closed loop control systems. A Figure-11, idan ana iya canza transfer function ta plant baki daya, yadda zai haɗa a kan input da output? A nan, input ta plant zai canza, amma output ta plant ba zai canza. Output ta plant ya ƙunshi zuwa reference input.

Figure -12 ya nuna hanyoyin da aka canza, a nan parametarin plant suka canza. Zan iya tabbata cewa input ta plant an canza zuwa 0.476 daga 0.5, amma output ba zai canza. A duk waɗannan labari input ta PI controller ba zero, specs ta PI controller suna sama amma output ta PI controller shine mafi ƙaramin haske.

Saboda haka, zan iya fahimta cewa, a cikin open loop control system output ta plant zai canza, amma a cikin closed loop control system input ta plant zai canza.

A cikin littattafai na control system, za a iya samun wannan bayanin:

"A cikin haka da yake canza na parametoci na funksiya na tushen gida, kawarwari na kontrollo na doka ita ce mai kadan da kuma kawarwari na kontrollo ta fuska."

Na gane da wannan bayanin zai zama mafi yawan fahimta saboda misalai da aka baka a cikin wannan makaltaccen.

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*Abubuwan Electrical4U, ina bukata wa game da shi wanda babu shakka a nemi labarun da ke cikin littattafai; amma mutumin da na sani shine in ba da misalai masu aiki na kimiyya da inganci bayanai daga fannin Kudin Kirkiro. Na gane da wannan makaltaccen zai taimaka maka wajen fahimtar abubuwa masu karfi game da takamun jiki da kumtakawa PI.

Bayanin: Yana iya da karfin mai karatu, babban labarai ke gaba da shirya, idani akwai inganci zaka tabbatar.

Ba da kyau kuma kara mai rubutu!

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