Utafiti huu unaunganisha MATLAB/Simulink, OPC Server na PLC na Smith predictor kwa udhibiti wa kasi wa stepper motor ya awamu mbili. Ucheleweshaji unaotokana na computation ya MATLAB, mawasiliano ya OPC, utekelezaji wa PLC, interface ya actuator na feedback ya sensor unawakilishwa kama equivalent pure delay; chanzo kinatumia takriban 0.40 s kama nominal total delay. Kwa nominal delay-free loop, gain margin ni 9.6 dB, phase margin 52.3°, crossover frequency 3.1 rad/s na complementary sensitivity peak \(\|T(j\omega)\|_\infty=1.38\).

Configurations tatu za feedback zilijaribiwa mara tano kwa target speed ya 40 rpm. Case 2, ambayo inatumia model feedback ndani ya Smith structure, ilitoa settling time ya 1.5 s, 0% overshoot, 1.83% steady-state error na RMSE ya 0.22 rpm. Case 3 bila Smith predictor ilikuwa na settling time ya 4.6 s na overshoot ya 59.75%. Hata hivyo matokeo yanahusu hardware, model na network architecture iliyojaribiwa; hayatoi guarantee kwa kila PLC-based industrial system.

Plant model na Smith predictor

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

Equation (2) imechapishwa kwenye source kama:

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

Hii haitoshi kabisa na closed-loop expression inayopatikana baadaye katika Equation (12); tofauti hiyo ya source haijafichwa.

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

Theorem 1: model match kamili

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

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

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

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

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

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

\[ Y-\hat Y=0, \qquad Y_{\mathrm{est}}=\hat GU. \tag{9} \]

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

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

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

Kwa model inayolingana kikamilifu, pure delay haingii kwenye characteristic denominator inayotawala poles.

Theorem 2: multiplicative model uncertainty

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

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

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

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

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

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

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

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

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

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

Theorem 3: delay mismatch

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

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

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

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

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

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

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

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

Theorem 3 ina typographical omission kwenye statement yake ya kwanza; proof ndiyo inayoonyesha wazi condition ya Equation (24).

System identification

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

MATLAB `tfest` ilitoa:

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

Inconsistency katika delay values

Arithmetic sum ya components ni 1.12 s, si 0.40 s. Inawezekana source ilikusudia 0.08 s katika component ya mwisho, lakini hakuna correction hiyo iliyofanywa kwa niaba ya waandishi.

Umuhimu unaowezekana kwa Afrika Mashariki

Utafiti haukufanyika katika mfumo wa viwanda wa Afrika Mashariki. Hata hivyo PLC, motor drives, OPC communication na delay compensation ni mada zinazoweza kuhusika katika automation ya viwanda, conveyors, processing plants na engineering laboratories katika eneo hilo. Kabla ya kuhamisha method, network latency, PLC cycle time, motor/load dynamics, sensor processing na local maintenance infrastructure lazima vipimwe upya.

Mbinu na Matokeo ya Utafiti

Hardware/software iliyotumika ni MATLAB/Simulink OPC Toolbox, KEPServerEX, FX3U PLC, Omron CWZ6C-1000 encoder, DM422C driver na two-phase stepper motor. OPC sampling period ni 0.1 s.

\[ \|T(j\omega)\|_\infty=1.38. \]

Kwa uncertainty ya juu ya 15%:

\[ 1.38\times0.15=0.207<1. \]

Phase-margin delay mismatch approximation:

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

Observed delay variation ilikuwa ±0.08 s, ambayo ni ndogo kuliko 0.29 s.

PID coefficients zimeandikwa kwenye source kama:

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

Kila configuration ilirudiwa mara tano chini ya operating conditions zilezile. Case 2 ina performance bora kwa metrics zilizoripotiwa. Source text pia inasema error integral ni 0.37 rpm²·s kwa Case 2, lakini Table 2 ina ISE=927.9 rpm²·s; inconsistency hiyo imehifadhiwa wazi.

Katika robustness test, gain \(K\) ilipunguzwa 10% na time constant \(T\) ikaongezwa 15%. Tracking ilibaki stable, lakini settling error ikaongezeka kwa takriban 1%.

Maelezo ya Chanzo na Mbinu

Waandishi: Nguyen Quang Binh, Hoang Van Quyet, Le Van Tuan; corresponding author ni Le Van Tuan.

Taasisi: CAPITI, Hanoi, Vietnam.

DOI: 10.65153/s2pkyh47.

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

Hali ya uchapishaji: peer-reviewed research article.

Open access: jarida lina Diamond Open Access policy, lakini PDF haijataja article-specific reuse licence.

Scientific limitations: matokeo ni ya semi-real-time setup iliyojaribiwa. Source inconsistencies kuhusu delay sum, Equation (2)/(12), Theorem 3 statement na Case 2 ISE hazijasahihishwa kimyakimya.