5.3 PID Regulyatorlar — Masalalar

Jami: 35 ta | Yechim bilan: ✅

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Asosiy Masalalar (1-12)

Masala 1 ⭐⭐

P controller: $K_p = 10$. Xatolik e = 0.5. Chiqish signali u?

Yechim

$u = K_p \cdot e = 10 \times 0.5 = 5$

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Masala 2 ⭐⭐

PI controller: $K_p = 5$, $K_i = 2$. e(t) = 1 (doimiy), t = 3 s. Chiqish?

Yechim

$u_P = K_p \cdot e = 5 \times 1 = 5$

$u_I = K_i \int_0^3 1 \, dt = 2 \times 3 = 6$

$u = u_P + u_I = 5 + 6 = 11$

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Masala 3 ⭐⭐

PD controller: $K_p = 8$, $K_d = 0.5$. e = 2, $\frac{de}{dt} = -4$. Chiqish?

Yechim

$u = K_p e + K_d \frac{de}{dt} = 8(2) + 0.5(-4) = 16 - 2 = 14$

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Masala 4 ⭐⭐

Diskret PID: $K_p = 2$, $K_i = 0.5$, $K_d = 0.1$, dt = 0.01 s. e[n] = 1, e[n-1] = 1.2, integral_sum = 5. Chiqish?

Yechim

$P = K_p \cdot e = 2 \times 1 = 2$

$I = K_i \cdot (integral\_sum + e \cdot dt) = 0.5 \times (5 + 0.01) = 2.505$

$D = K_d \cdot \frac{e - e_{prev}}{dt} = 0.1 \times \frac{1 - 1.2}{0.01} = 0.1 \times (-20) = -2$

$u = P + I + D = 2 + 2.505 - 2 = 2.505$

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Masala 5 ⭐⭐

Motor tezlik kontroli: setpoint = 1000 rpm, actual = 800 rpm, $K_p = 0.5$. PWM duty cycle (0-100%)?

Yechim

$e = 1000 - 800 = 200$ rpm

$u = K_p \cdot e = 0.5 \times 200 = 100$

PWM = 100% (to'yingan)

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Masala 6 ⭐⭐

Dron altitude: setpoint = 10 m, actual = 8 m. $K_p = 5$. Thrust qo'shimchasi (N)?

Yechim

$e = 10 - 8 = 2$ m

$\Delta thrust = K_p \cdot e = 5 \times 2 = 10$ N

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Masala 7 ⭐⭐

Steady-state error: P controller $K_p = 20$, plant DC gain = 1. Step input uchun SS error?

Yechim

Yopiq sikl: $\frac{Y}{R} = \frac{K_p \cdot 1}{1 + K_p \cdot 1} = \frac{20}{21}$

$e_{ss} = 1 - \frac{20}{21} = \frac{1}{21} = 0.048 = 4.8\%$

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Masala 8 ⭐⭐

Integral action: xatolik e = 0.1 doimiy, $K_i = 2$. 5 s da integral komponenti?

Yechim

$u_I = K_i \int_0^5 0.1 \, dt = 2 \times 0.1 \times 5 = 1$

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Masala 9 ⭐⭐

Ziegler-Nichols: $K_u = 50$, $T_u = 0.5$ s. PID parametrlari?

Yechim

$K_p = 0.6 K_u = 30$

$T_i = T_u/2 = 0.25$ s → $K_i = K_p/T_i = 120$

$T_d = T_u/8 = 0.0625$ s → $K_d = K_p \cdot T_d = 1.875$

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Masala 10 ⭐⭐

Anti-windup: integral cheklovi ±100. Hozirgi integral = 95, e = 0.2, $K_i = 50$, dt = 0.1. Yangi integral?

Yechim

Yangi integral (cheklovsiz): $95 + 50 \times 0.2 \times 0.1 = 95 + 1 = 96$

96 < 100 → chegarada emas

Yangi integral = 96

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Masala 11 ⭐⭐

Robot qo'li: $\theta_{ref} = 90°$, $\theta_{actual} = 85°$. $K_p = 2$ N·m/deg. Motor moment?

Yechim

$e = 90 - 85 = 5°$

$\tau = K_p \cdot e = 2 \times 5 = 10$ N·m

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Masala 12 ⭐⭐

Line follower: sensor = 3 (o'rtadan chapda), $K_p = 20$. Motorlar tezlik farqi?

Yechim

$e = 0 - 3 = -3$ (0 = markaz)

$\Delta v = K_p \cdot e = 20 \times (-3) = -60$

Chap motor: base - 30 O'ng motor: base + 30

(O'ngga burilish)

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O'rtacha Masalalar (13-25)

Masala 13 ⭐⭐⭐

Overshoot: $\zeta = 0.5$. Foiz overshoot?

Yechim

$M_p = e^{-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}} = e^{-\frac{3.14 \times 0.5}{0.866}} = e^{-1.81} = 0.163 = 16.3\%$

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Masala 14 ⭐⭐⭐

Settling time (2%) kerak: $t_s < 2$ s. $\zeta = 0.7$ bo'lsa, $\omega_n$ minimal?

Yechim

$t_s = \frac{4}{\zeta\omega_n} < 2$

$\omega_n > \frac{4}{\zeta \cdot t_s} = \frac{4}{0.7 \times 2} = 2.86$ rad/s

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Masala 15 ⭐⭐⭐

Kaskadli PID: Tashqi (pozitsiya) $K_{p1} = 2$, ichki (tezlik) $K_{p2} = 5$. Pozitsiya xatoligi 0.5 m, tezlik xatoligi 0.1 m/s. Kuch chiqishi?

Yechim

Tashqi: $v_{ref} = K_{p1} \cdot e_{pos} = 2 \times 0.5 = 1$ m/s

Ichki: $F = K_{p2} \cdot e_{vel} = K_{p2} \cdot (v_{ref} - v_{actual})$

Agar $v_{actual}$ berilmagan, $e_{vel} = 0.1$: $F = 5 \times 0.1 = 0.5$ N

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Masala 16 ⭐⭐⭐

Derivative filter: $\tau_f = 0.01$ s, raw derivative = 100. Filtrlangan qiymat (oldingi = 50)?

Yechim

$\alpha = \frac{dt}{dt + \tau_f}$ (dt = 0.01 deb faraz qilsak)

$\alpha = \frac{0.01}{0.02} = 0.5$

$D_{filtered} = \alpha \cdot D_{raw} + (1-\alpha) \cdot D_{prev}$ $= 0.5 \times 100 + 0.5 \times 50 = 75$

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Masala 17 ⭐⭐⭐

Quadcopter roll: $K_p = 5$, $K_d = 0.5$. $\phi_{ref} = 0$, $\phi = 10°$, $\dot{\phi} = 20°/s$. Differential thrust?

Yechim

$e = 0 - 10 = -10°$ $\dot{e} = 0 - 20 = -20°/s$

$u = K_p \cdot e + K_d \cdot \dot{e} = 5(-10) + 0.5(-20) = -50 - 10 = -60$

(Chapga burish uchun thrust farqi -60 birlik)

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Masala 18 ⭐⭐⭐

Position vs Velocity form PID. Velocity form $\Delta u$ ni toping: $K_p = 2$, $K_i = 1$, $K_d = 0.1$. $e[n] = 5$, $e[n-1] = 6$, $e[n-2] = 7$.

Yechim

Velocity form: $\Delta u = K_p(e_n - e_{n-1}) + K_i \cdot e_n + K_d(e_n - 2e_{n-1} + e_{n-2})$

$= 2(5-6) + 1(5) + 0.1(5 - 12 + 7)$ $= -2 + 5 + 0 = 3$

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Masala 19 ⭐⭐⭐

Bumpless transfer: Manual mode u = 50, Auto ga o'tishda integral = ?

Yechim

$u_{auto} = K_p \cdot e + integral$

Transfer paytida $u_{auto} = u_{manual}$:

$integral = u_{manual} - K_p \cdot e = 50 - K_p \cdot e$

Agar $e = 0$ (setpoint = measured): $integral = 50$

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Masala 20 ⭐⭐⭐

Feed-forward + PID: Reference = 100, $K_{ff} = 0.8$, PID chiqishi = 5. Umumiy chiqish?

Yechim

$u = K_{ff} \cdot ref + u_{PID} = 0.8 \times 100 + 5 = 85$

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Masala 21 ⭐⭐⭐

Dead-band: |e| < 0.5 bo'lganda PID o'chiriladi. e = 0.3, $K_p = 10$. Chiqish?

Yechim

$|e| = 0.3 < 0.5$ → Dead-band ichida

$u = 0$ (yoki oxirgi qiymat saqlanadi)

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Masala 22 ⭐⭐⭐

Gain scheduling: $v < 5$ m/s → $K_p = 2$, $v \geq 5$ → $K_p = 1$. v = 4 m/s, e = 3. Chiqish?

Yechim

$v = 4 < 5$ → $K_p = 2$

$u = K_p \cdot e = 2 \times 3 = 6$

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Masala 23 ⭐⭐⭐

Rate limiter: chiqish o'zgarishi max 10 birlik/s. Oldingi u = 50, yangi u = 80, dt = 0.5s. Haqiqiy chiqish?

Yechim

Maksimal o'zgarish: $10 \times 0.5 = 5$

Kerakli o'zgarish: $80 - 50 = 30$

30 > 5 → cheklangan

$u_{actual} = 50 + 5 = 55$

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Masala 24 ⭐⭐⭐

Setpoint weighting: $b = 0.5$ (P uchun). $r = 100$, $y = 60$, $K_p = 2$. P komponenti?

Yechim

$e_P = b \cdot r - y = 0.5 \times 100 - 60 = -10$

$u_P = K_p \cdot e_P = 2 \times (-10) = -20$

(Oddiy PID: $e = r - y = 40$, $u_P = 80$)

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Masala 25 ⭐⭐⭐

Back-calculation anti-windup: $u = 120$, $u_{sat} = 100$, $T_t = 0.5$, dt = 0.01. Integral tuzatish?

Yechim

$\Delta integral = \frac{dt}{T_t}(u_{sat} - u) = \frac{0.01}{0.5}(100 - 120) = 0.02 \times (-20) = -0.4$

Integral 0.4 ga kamayadi har qadamda.

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Murakkab Masalalar (26-35)

Masala 26 ⭐⭐⭐⭐

Ziegler-Nichols reaction curve: L = 0.5 s (delay), T = 2 s (time constant). PID?

Yechim

$K_p = \frac{1.2T}{L} = \frac{1.2 \times 2}{0.5} = 4.8$

$T_i = 2L = 1$ s → $K_i = \frac{K_p}{T_i} = 4.8$

$T_d = 0.5L = 0.25$ s → $K_d = K_p \cdot T_d = 1.2$

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Masala 27 ⭐⭐⭐⭐

Auto-tuning relay: Relay amplitude d = 5, oscillation amplitude a = 2, period $T_u$ = 1 s. $K_u$ va PID?

Yechim

$K_u = \frac{4d}{\pi a} = \frac{4 \times 5}{3.14 \times 2} = \frac{20}{6.28} = 3.18$

$K_p = 0.6 K_u = 1.91$ $T_i = 0.5$ s → $K_i = 3.82$ $T_d = 0.125$ s → $K_d = 0.24$

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Masala 28 ⭐⭐⭐⭐

SIMC tuning: $K = 2$, $\tau = 5$ s, $\theta = 1$ s (FOPDT model). Closed-loop $\tau_c = 2$ s uchun PI?

Yechim

SIMC PI: $K_p = \frac{\tau}{K(\tau_c + \theta)} = \frac{5}{2(2 + 1)} = \frac{5}{6} = 0.83$

$T_i = \min(\tau, 4(\tau_c + \theta)) = \min(5, 12) = 5$ s

$K_i = \frac{K_p}{T_i} = \frac{0.83}{5} = 0.17$

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Masala 29 ⭐⭐⭐⭐

Lambda tuning: Plant = $\frac{2}{5s+1}e^{-s}$. Lambda = 3 s uchun PI?

Yechim

$K = 2$, $\tau = 5$, $\theta = 1$

$K_p = \frac{\tau}{K(\lambda + \theta)} = \frac{5}{2(3 + 1)} = 0.625$

$T_i = \tau = 5$ s

$K_i = 0.125$

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Masala 30 ⭐⭐⭐⭐

2-DOF PID: $K_p = 2$, $b = 0.5$ (reference weight), $c = 0$ (derivative weight). Step response overshoot kamayadi, isbotlang.

Yechim

Oddiy PID: Reference step → katta P signal → overshoot

2-DOF: $e_P = br - y$, $e_D = cr - y$

$b = 0.5$: P ga reference faqat 50% ta'sir qiladi $c = 0$: D faqat output ga javob beradi

Step paytida:

• P komponenti kichikroq (b < 1)
• D smooth (c = 0, reference derivative yo'q)

Natija: Kamroq overshoot, smooth javob.

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Masala 31 ⭐⭐⭐⭐

IMC-PID: Plant $G(s) = \frac{3}{2s+1}$, filter $\lambda = 1$ s. PID parametrlari?

Yechim

IMC controller: $Q(s) = \frac{G^{-1}(s)}{1 + \lambda s} = \frac{2s+1}{3(1+s)}$

PID approximation: $K_p = \frac{\tau}{K\lambda} = \frac{2}{3 \times 1} = 0.67$

$T_i = \tau = 2$ s

$T_d = 0$ (FOPDT uchun)

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Masala 32 ⭐⭐⭐⭐

Discrete PID - Tustin transform: $K_d s$ → diskret. dt = 0.01 s.

Yechim

Tustin (bilinear): $s = \frac{2}{dt}\frac{z-1}{z+1}$

$K_d s \to K_d \frac{2}{dt}\frac{z-1}{z+1} = \frac{200K_d(z-1)}{z+1}$

Differentsiya tenglamasi: $D[n] = -D[n-1] + 200K_d(e[n] - e[n-1])$

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Masala 33 ⭐⭐⭐⭐

Model Reference Adaptive PID: Reference model $\omega_n = 10$, $\zeta = 0.7$. Actual response $\omega = 8$, $\zeta = 0.5$. Gains qaysi yo'nalishda o'zgartiriladi?

Yechim

$\omega$ past (8 < 10):

• Bandwidth kam → $K_p$ va/yoki $K_i$ oshirish

$\zeta$ past (0.5 < 0.7):

• Damping kam, overshoot ko'p → $K_d$ oshirish yoki $K_p$ kamaytirish

Optimal: $K_d ↑$, $K_p$ biroz $↑$

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Masala 34 ⭐⭐⭐⭐

Windup example: Setpoint = 100, output stuck at 50 (physical limit). $K_p = 1$, $K_i = 0.5$, dt = 0.1 s. 10 s da integral?

Yechim

$e = 100 - 50 = 50$ (doimiy)

Integralsiz: $integral = K_i \sum e \cdot dt = 0.5 \times 50 \times 10 = 250$

Bu katta windup! Anti-windup kerak.

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Masala 35 ⭐⭐⭐⭐

Parallel vs Ideal PID: Parallel $K_p = 2$, $K_i = 1$, $K_d = 0.5$. Ideal formaga o'tkazing.

Yechim

Parallel: $u = K_p e + K_i \int e + K_d \dot{e}$

Ideal: $u = K_p(e + \frac{1}{T_i}\int e + T_d \dot{e})$

$K_p = 2$ (bir xil)

$T_i = \frac{K_p}{K_i} = \frac{2}{1} = 2$ s

$T_d = \frac{K_d}{K_p} = \frac{0.5}{2} = 0.25$ s

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✅ Tekshirish Ro'yxati

• 1-12: Asosiy PID
• 13-25: Tuning va amaliy
• 26-35: Murakkab usullar

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Keyingi Qadam

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