Maqola ikki fazali qadamli dvigatel tezligini kuzatish uchun MATLAB/Simulink-OPC-PLC tizimini Smith predictor bilan birlashtiradi. MATLAB hisoblash vaqti, OPC aloqasi, PLC execution, actuator interface va sensor feedback kechikishlari yagona pure delay sifatida modellashtirilgan va nominal umumiy kechikish 0.40 s deb ishlatilgan. Delay-free nominal loop uchun gain margin 9.6 dB, phase margin 52.3°, crossover frequency 3.1 rad/s va \(\|T(j\omega)\|_\infty=1.38\) olingan.

40 rpm reference bo‘yicha uchta feedback konfiguratsiyasi besh marta takrorlangan. Model feedback + Smith bo‘lgan Case 2: 1.5 s settling, 0% overshoot, 1.83% steady-state error va 0.22 rpm RMSE bilan eng yaxshi natijani beradi. Smith ishlatilmagan Case 3 esa 4.6 s settling va 59.75% overshoot ko‘rsatadi. Natijalar aynan shu PLC-OPC-dvigatel muhitiga tegishli bo‘lib, boshqa sanoat tizimlariga avtomatik kafolat bermaydi.

Jarayon va Smith modeli

\[ G(s)=\frac{K}{Ts+1}e^{-\tau s}. \tag{1} \]

\[ Y(s)= \frac{\hat G(s)} {1+\hat G(s)C(s)} R(s). \tag{2} \]

Manbadagi (2) ifoda keyingi (12) tenglama bilan to‘liq bir xil emas; ushbu ichki tafovut saqlanadi.

\[ T_0(s)= \frac{C(s)\hat G(s)} {1+C(s)\hat G(s)}. \]

Teorema 1

\[ U(s)=C(s)E(s). \tag{3} \]

\[ Y(s)=\hat G(s)e^{-\tau s}U(s). \tag{4} \]

\[ \tilde Y(s)=\hat G(s)U(s). \tag{5} \]

\[ \hat Y(s)=\hat G(s)e^{-\tau s}U(s). \tag{6} \]

\[ Y_{\mathrm{est}} = \tilde Y+(Y-\hat Y). \tag{7} \]

\[ Y=\hat G e^{-\tau s}U=\hat Y. \tag{8} \]

\[ Y-\hat Y=0, \quad Y_{\mathrm{est}}=\hat G U. \tag{9} \]

\[ E=R-\hat G U. \tag{10} \]

\[ U=C(R-\hat G U) \Rightarrow (1+C\hat G)U=CR. \tag{11} \]

\[ \frac{Y}{R} = \frac{\hat G e^{-\tau s}C} {1+C\hat G}. \tag{12} \]

Model aniq bo‘lsa, pure delay xarakteristik maxrajni o‘zgartirmaydi.

Teorema 2: kichik-gain robustligi

\[ G= \hat G(1+W_m\Delta)e^{-\tau s}, \qquad \|\Delta\|_\infty\le1. \]

\[ Y=\hat G(1+W_m\Delta)e^{-\tau s}U. \tag{13} \]

\[ Y_{\mathrm{est}} = \hat G U+\hat G e^{-\tau s}W_m\Delta U. \tag{14} \]

\[ E=R-\hat G U-\hat G e^{-\tau s}W_m\Delta U. \tag{15} \]

\[ U=C(R-\hat G U-\hat G e^{-\tau s}W_m\Delta U). \tag{16} \]

\[ (1+C\hat G)U+C\hat G e^{-\tau s}W_m\Delta U=CR. \tag{17} \]

\[ U+ \frac{C\hat G}{1+C\hat G} e^{-\tau s}W_m\Delta U = \frac{C}{1+C\hat G}R. \tag{18} \]

\[ (I+T_0e^{-\tau s}W_m\Delta)U=U_0. \tag{19} \]

\[ \|T_0e^{-\tau s}W_m\|_\infty<1. \tag{20} \]

\[ \|T_0W_m\|_\infty<1. \tag{21} \]

Teorema 3: delay mismatch

\[ G(s)=\hat G(s)e^{-(\tau+\Delta\tau)s}. \]

\[ G= \hat G e^{-\tau s} [1+(e^{-\Delta\tau s}-1)]. \tag{22} \]

\[ \delta(s)=e^{-\Delta\tau s}-1. \]

\[ [I+T_0e^{-\tau s}\delta]U=U_0. \tag{23} \]

\[ \|T_0\delta\|_\infty<1. \tag{24} \]

\[ |T_0(j\omega)|\,|\delta(j\omega)|<1. \tag{25} \]

\[ |e^{-j\omega\Delta\tau}-1| = 2\left| \sin\left( \frac{\omega\Delta\tau}{2} \right) \right|. \tag{26} \]

\[ |\delta(j\omega)| \le \min\{2,\omega|\Delta\tau|\}, \qquad \omega_c|\Delta\tau|<\varphi_m. \]

Teorema 3 ning source statement qismida inequality matni to‘liq emas; proof esa (24) shartini aniq beradi.

System identification

\[ G(s)= \frac{ b_ms^m+b_{m-1}s^{m-1}+\cdots+b_1s+b_0 }{ s^n+a_{n-1}s^{n-1}+\cdots+a_1s+1 } e^{-\tau s}, \qquad n\ge m. \tag{27} \]

`tfest` bilan olingan model:

\[ G(s) = \frac{0.7255} {s^2+15.19s+77.4} e^{-0.4s}. \tag{28} \]

Delay sonlaridagi manba ziddiyati

Berilgan individual qiymatlar 1.12 s bo‘ladi. 0.8 s qiymat 0.08 s deb o‘zboshimchalik bilan tuzatilmagan.

O‘zbekiston uchun mumkin bo‘lgan ahamiyati

Bu eksperiment O‘zbekiston sanoat tizimida bajarilmagan. Shunga qaramay, PLC, OPC va motor drive delay muammolari ishlab chiqarish, konveyer tizimlari, mechatronics va avtomatlashtirish laboratoriyalarida dolzarb. Mahalliy qo‘llashdan oldin network latency, PLC scan time, actual motor-load model va sensor processing delay qayta identifikatsiya qilinishi kerak.

Tadqiqot Usuli va Natijalari

Platforma: MATLAB/Simulink OPC Toolbox, KEPServerEX, FX3U PLC, Omron CWZ6C-1000 encoder, DM422C driver va ikki fazali stepper motor. OPC sampling period = 0.1 s.

\[ \|T(j\omega)\|_\infty=1.38, \qquad 1.38\times0.15=0.207<1. \]

\[ \Delta\tau_{\max} \approx \frac{52.3^\circ\pi/180}{3.1} \approx0.29\,s. \]

Observed variation ±0.08 s bo‘lib, 0.29 s additional-delay bound ichida.

\[ (K_p;K_i;K_d) = (27,65;\;243,85;\;0). \]

Har bir case bir xil operating conditions ostida besh marta takrorlangan. Case 2 barcha asosiy jadval mezonlarida eng yaxshi javob beradi. Matndagi 0.37 rpm²·s error integral va Table 2 dagi 927.9 rpm²·s ISE bir-biriga mos kelmaydi.

Robustness testda \(K\) 10% kamaytirilgan va \(T\) 15% oshirilgan; tracking stable qolgan, settling error taxminan 1% oshgan.

Manba va Usul Bo‘yicha Izoh

Mualliflar: Nguyen Quang Binh, Hoang Van Quyet, Le Van Tuan; corresponding author — Le Van Tuan.

Muassasa: CAPITI, Hanoi, Vietnam.

DOI: 10.65153/s2pkyh47.

Jurnal: Journal of Science and Technology of East Asia University of Technology, 2(2), 2026, 64-74.

Nashr maqomi: peer-reviewed research article.

Litsenziya: PDF ichida article-specific reuse license ko‘rsatilmagan; jurnal Diamond Open Access.

Cheklovlar: delay raqamlari, Equation (2)/(12), Theorem 3 statement va ISE matn/jadval qiymatlaridagi source inconsistencies o‘zgartirilmagan.