5.3 PID Regulyatorlar — Nazariya

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Kirish

PID (Proportional-Integral-Derivative) — eng keng tarqalgan boshqarish algoritmi. Robot, dron, raketa — hammasi PID ishlatadi.

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1. Asosiy Tushuncha

Yopiq sikl boshqarish

  r(t)    e(t)     u(t)      y(t)
───────►(+)───►[PID]───►[Plant]───┬───►
         │-                       │
         └────────────────────────┘

• $r(t)$ — reference (maqsad qiymat)
• $e(t) = r(t) - y(t)$ — xatolik
• $u(t)$ — boshqarish signali
• $y(t)$ — chiqish (o'lchangan qiymat)

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2. PID Formulasi

Vaqt sohasida

$$u(t) = K_p e(t) + K_i \int_0^t e(\tau)d\tau + K_d \frac{de(t)}{dt}$$

Komponentlar

P (Proportional):

$$u_P = K_p \cdot e(t)$$

• Joriy xatolikka proporsional
• Tezkor javob
• Faqat P → doimo xatolik (steady-state error)

I (Integral):

$$u_I = K_i \int_0^t e(\tau)d\tau$$

• Xatolik yig'indisi
• Steady-state error yo'qotadi
• Haddan tashqari → tebranish, windup

D (Derivative):

$$u_D = K_d \frac{de(t)}{dt}$$

• Xatolik o'zgarish tezligi
• Overshoot kamaytiradi
• Shovqinga sezgir

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3. Transfer Funksiyasi

Laplace sohasida:

$$C(s) = K_p + \frac{K_i}{s} + K_d s = \frac{K_d s^2 + K_p s + K_i}{s}$$

Yoki standart shakl:

$$C(s) = K_p \left(1 + \frac{1}{T_i s} + T_d s\right)$$

Bu yerda:

• $T_i = K_p/K_i$ — integral vaqt
• $T_d = K_d/K_p$ — derivative vaqt

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4. Tuning Usullari

Ziegler-Nichols (Chastota usuli)

1. $K_i = 0$, $K_d = 0$ qo'ying
2. $K_p$ ni barqaror tebranish boshlanguncha oshiring
3. Kritik $K_u$ va davr $T_u$ ni yozib oling

Controller	$K_p$	$T_i$	$T_d$
P	$0.5K_u$	—	—
PI	$0.45K_u$	$T_u/1.2$	—
PID	$0.6K_u$	$T_u/2$	$T_u/8$

Ziegler-Nichols (Javob usuli)

Step javobdan $L$ (delay) va $T$ (time constant) o'lchang:

Controller	$K_p$	$T_i$	$T_d$
P	$T/L$	—	—
PI	$0.9T/L$	$L/0.3$	—
PID	$1.2T/L$	$2L$	$0.5L$

Manual Tuning

1. $K_i = 0$, $K_d = 0$ dan boshlang
2. $K_p$ oshiring — tezkor javob, lekin tebranish
3. $K_d$ qo'shing — overshoot kamayadi
4. $K_i$ qo'shing — steady-state error yo'qoladi
5. Fine-tune qiling

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5. Diskret PID

Mikrokontrollerda:

$$u_n = K_p e_n + K_i \sum_{i=0}^{n} e_i \Delta t + K_d \frac{e_n - e_{n-1}}{\Delta t}$$

Pseudokod

class PID:
    def __init__(self, Kp, Ki, Kd, dt):
        self.Kp = Kp
        self.Ki = Ki
        self.Kd = Kd
        self.dt = dt
        self.integral = 0
        self.prev_error = 0
    
    def compute(self, setpoint, measured):
        error = setpoint - measured
        
        # P
        P = self.Kp * error
        
        # I
        self.integral += error * self.dt
        I = self.Ki * self.integral
        
        # D
        derivative = (error - self.prev_error) / self.dt
        D = self.Kd * derivative
        
        self.prev_error = error
        
        return P + I + D

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6. Anti-Windup

Integral to'planib ketishi muammo:

Clamping

self.integral = max(min(self.integral, max_integral), -max_integral)

Back-calculation

Saturatsiyada integral kamaytirish:

$$\frac{d}{dt}\int e \, dt = e + \frac{1}{T_t}(u_{sat} - u)$$

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7. Derivative Filtrlash

Shovqinni kamaytirish:

$$D_{filtered} = \frac{K_d s}{1 + \tau_f s}$$

Yoki diskretda:

self.d_filtered = alpha * derivative + (1-alpha) * self.d_filtered_prev

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8. Kaskadli PID

Ichki va tashqi sikl:

  r ──►[Outer PID]──►[Inner PID]──►[Plant]──┬──► y
        │                   │               │
        └───────────────────┴───────────────┘

Misol: Dron balandlik kontroli

• Tashqi: Balandlik → tezlik reference
• Ichki: Tezlik → thrust

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9. Robotikada Qo'llanilish

Dron Attitude Control

Roll, pitch, yaw uchun alohida PID:

Roll PID:  θ_error → Motor differential
Pitch PID: φ_error → Motor differential
Yaw PID:   ψ_error → Motor differential

Robot Qo'li

Har bir bo'g'in uchun PID:

$$\tau = K_p(\theta_{ref} - \theta) + K_d(\dot{\theta}_{ref} - \dot{\theta})$$

Motor Tezlik Kontroli

PWM chiqish:

$$PWM = K_p e + K_i \sum e + K_d \frac{de}{dt}$$

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10. Tipik Muammolar

Muammo	Sabab	Yechim
Sekin javob	$K_p$ past	$K_p$ oshiring
Overshoot	$K_p$ yuqori, $K_d$ past	$K_d$ oshiring, $K_p$ kamaytiring
Tebranish	$K_i$ yuqori	$K_i$ kamaytiring
Steady-state error	$K_i = 0$ yoki past	$K_i$ qo'shing
Shovqin kuchayishi	$K_d$ yuqori	$K_d$ kamaytiring, filtr qo'shing

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Xulosa

Parametr	Ta'sir
$K_p ↑$	Tezkor, lekin overshoot
$K_i ↑$	SS error kamayadi, lekin tebranish
$K_d ↑$	Overshoot kamayadi, lekin shovqinga sezgir

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Keyingi Qadam

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