Foundations — PID control — proportional, integral, derivative terms

3.5.38 · D1 · Physics › Guidance, Navigation & Control (GNC) › PID control — proportional, integral, derivative terms

Isse pehle ki tum PID law ki ek bhi line padh sako, tumhe kuch symbols aur pictures mein fluent hona padega. Yeh page unhe ek-ek karke, usi order mein build karta hai jis order mein woh ek-doosre par depend karte hain. Agar parent note ne koi symbol bina explain kiye likha tha, yahi page hai jahan usse explain kiya jaata hai.

1. Time — woh stage jis par sab kuch hota hai

Jab hum kuch aise likhte hain jaise $r(t)$ ya $y(t)$, toh chhota sa "$(t)$" ka matlab sirf yeh hai ki "is cheez ki har moment par alag value hoti hai" — yeh time ka function hai, koi fixed number nahi.

Figure dekho: wahi physical machine tumhe values ka pura curve deti hai, har instant par ek dot. Woh curve hi woh raw material hai jis par ek controller feed karta hai.

2. Teen signals: setpoint, process variable, error

Figure mein notice karo: error positive ya negative ho sakta hai, aur yeh zero se cross karta hai. Ek controller ko teeno cases mein correctly behave karna chahiye, aur yahi reason hai ki $e$ ka sign itni carefully track kiya jaata hai.

In teeno signals ke baare mein aur padhne ke liye ki yeh wires mein kaise connect hote hain, dekho Feedback control loops.

3. Control signal $u(t)$ — woh command jo tum bhejte ho

PID controller ka poora kaam yeh compute karna hai ki $e(t)$ ko $u(t)$ mein kaise badla jaaye. Hum ise $u=f(e)$ likhte hain: "command kuch aisa hai jo error par apply hoti hai."

$u$ actually spacecraft par kahan jaata hai, iske liye dekho Reaction wheels & attitude control.

4. Gains — tuning knobs $K_p,K_i,K_d$

5. Integral ${\int }_0^{t}e(\tau )d\tau$ — beete waqt ko jodna

Ab ek real piece of calculus enter hota hai. Panic mat karo — hum ise ek picture se build karte hain.

Figure mein shaded region wahi integral hai: time ko $d\tau$ width ke patle rectangles mein kato, har ek ki height $e(\tau )$, aur unke saare areas jodo. Ek persistent small error dheere-dheere ek bada area bharta hai — yahi woh "memory" hai jo eventually steady-state offset khatam karti hai.

Digital version is area ko rectangles ke running sum se replace karta hai — yahi reason hai ki flight code mein $\sum e_j\Delta t$ likha hota hai.

6. Derivative $\frac{de}{dt}$ — trend padhna

Figure mein ek error curve par do points aur unpar tangent lines dikhti hain. Ek steep downward tangent (cyan) ka matlab hai "target ki taraf tezi se dive kar raha hai" — derivative term ise padhta hai aur pehle hi ease off karta hai.

7. Equilibrium & steady state — "resting" ka idea

8. Do aur words jinhe parent lean karta hai

9. Foundations topic ko kaise feed karte hain

Diagram ko top-to-bottom padho: time tumhe signals deta hai, signals tumhe error dete hain, error teen tarike se padha jaata hai, har reading times ek gain command ka hissa ban jaati hai, aur command (disturbance se ladte hue) system ko move karti hai — jiska sensor loop ko wapas $y$ par close karta hai. Woh loop hi parent topic hai, PID control — proportional, integral, derivative terms.

Equipment checklist

Khud test karo — tum parent page ke liye tabhi ready ho jab tum bina dekhe har ek ka jawab de sako.