Visual walkthrough — PID control — proportional, integral, derivative terms

3.5.38 · D2 · Physics › Guidance, Navigation & Control (GNC) › PID control — proportional, integral, derivative terms

Step 1 — Woh gap jo hum close karna chahte hain

KYA. Hamare paas ek target hai — ek number jo hum reach karna chahte hain. Target ko $r(t)$ bolte hain, setpoint (subscript... koi subscript nahi, sirf $r$; $(t)$ ka matlab hai "yeh time ke saath change ho sakta hai"). Hamare paas actually measured value $y(t)$ hai, process variable (jo ek sensor report karta hai). Sabse important quantity unke beech ka gap hai:

$e(t)=\underset{\text{where we want to be}}{\underbrace{r(t)}}-\underset{\text{where we are}}{\underbrace{y(t)}}$

$e(t)$ error hai. Positive $e$ ka matlab hai hum target se neeche hain; negative $e$ ka matlab hai hum usse zyada upar nikal gaye.

KYUN. Duniya mein har controller ek aisi machine hai jo $e$ padhti hai aur ek push decide karti hai. Toh sabse pehli cheez jo draw karni hai woh yeh hai ki $e$ kaisi dikhti hai — "value vs time" ke graph par ek vertical gap.

PICTURE. Laal curve woh hai jahan hum actually hain, $y(t)$. Flat kaali line target $r$ hai. Laal gap $e(t)$ hai.

Step 2 — Proportional push: present pe react karo

KYA. Sabse simple possible rule: push karo us proportion mein jitna bada gap abhi hai.

$u_P(t)=\underset{\text{how hard we push per unit error}}{\underbrace{K_p}}\times \underset{\text{the present gap}}{\underbrace{e(t)}}$

Yahan $u(t)$ control signal hai — woh actual command jo actuator ko bheji jaati hai (ek thruster, ek motor). $K_p$ proportional gain hai: ek dial. Bada $K_p$ = aggressive.

Proportional kyun, maan lo constant kyun nahi? Ek constant push ko pata nahi hota kab rokna hai. Proportional push automatically fade hoti hai jaise aap approach karte ho — bada gap, badi dhakka; chhota gap, halka dhakka; zero gap, haath hat jaate hain. Yeh natural "jitna zyada drift karo utna zyada steer karo" law hai.

PICTURE. Laal arrow ki height (push) gap ki height track karti hai. Jaise gap left-to-right shrink hoti hai, arrow bhi us ke saath shrink hota hai.

Step 3 — P akela kyun hamesha miss karta hai: steady-state gap

KYA. Maano koi cheez constant humse lad rahi hai — gravity, drag, ek steady solar torque. Use disturbance $d$ bolte hain. Rest par ("steady state"), hamari push exactly $d$ ko cancel karni chahiye, warna hum ab bhi move kar rahe hote. Toh rest par:

$K_p{e}_{ss}=d⟹e_{ss}=\underset{\text{never zero if }d\ne 0}{\underbrace{\frac{d}{K_p}}}$

$e_{ss}$ steady-state error hai — sab kuch settle hone ke baad bacha hua gap.

KYUN yeh kabhi zero nahi ho sakta. Zero error zero push dega ($K_p\times 0=0$), aur phir kuch bhi $d$ ka oppose nahi karega, toh system drift karega aur error wapas badhega. Ek hi resting place hai — ek nonzero gap jo sirf itni push produce kare ki $d$ ko hold kar sake.

PICTURE. Laal curve settle ho jaata hai — lekin woh target se neeche settle hota hai, ek permanent gap $e_{ss}$ par parked. $K_p$ badhane se gap shrink hota hai lekin kabhi close nahi hota (dashed red = bada $K_p$).

Step 4 — Integral: past yaad rakh ke gap mitao

KYA. Sirf present height pe react karne ki jagah, error ka poora history add karo — error curve ke neeche shaded area — aur usi ke basis par push karo.

$u_I(t)=\underset{\text{gain on memory}}{\underbrace{K_i}}\underset{\text{running area = accumulated past error}}{\underbrace{{\int }_0^{t}e(\tau )d\tau }}$

${\int }_0^t\dots d\tau$ symbol ka matlab sirf hai "time $0$ se ab tak shaded area." $\tau$ ek dummy stand-in hai "har past instant" ke liye. $K_i$ integral gain hai.

KYUN yeh leftover gap ko khatam karta hai. Dekho "steady" ab kya demand karta hai. Steady state mein push constant honi chahiye. Lekin integral term tab tak change karna band nahi karti jab tak area mein kuch naya add nahi ho raha:

$\frac{d{u}_I}{dt}=K_{i}e(t)=0⟹e=0$

Toh ek hi possible resting state hai — zero error. Jab tak koi bhi gap bachi hai, area badhta rahega, push badhti rahegi, aur yeh gap ko kuch nahi tak hunt karega.

PICTURE. Error curve ke neeche shaded red area time ke saath badhta hai. Jaise accumulator bharta hai, push chadhti hai jab tak error target line par nahi aa jaata — $e_{ss}=0$.

Step 5 — Derivative: future predict karo overshoot rokne ke liye

KYA. Is baar kuch add nahi karo — balki error curve ki slope dekho: kya gap tezi se close ho raha hai?

$u_D(t)=\underset{\text{gain on anticipation}}{\underbrace{K_d}}\underset{\text{slope of the error curve}}{\underbrace{\frac{de(t)}{dt}}}$

$\frac{de}{dt}$ (bolo "dee-ee-dee-tee") error ka rate of change hai — laal curve ki steepness. $K_d$ derivative gain hai.

KYUN yeh damp karta hai. Jab hum target ki taraf rush karte hain, gap tezi se shrink ho raha hota hai, toh uski slope negative hai: $\dot{e}<0$. Tab $u_D=K_d\dot{e}<0$ — ek braking command jo approach ko pahunche se pehle ease kar deta hai, woh overshoot kill karta hai jo P+I akele karte.

PICTURE. Jahan error curve steep aur falling hai, laal tangent line neeche point karti hai — aur derivative term exactly wahan ek backward brake laagati hai.

Step 6 — Degenerate cases (reader ko kabhi stranded mat chhodho)

KYA / KYUN / PICTURE. Chaar corner situations, har ek ka jawab:

• $e=0$ lekin $d\ne 0$ (perfect aim, phir bhi disturbance se lad rahe hain): P kuch nahi deta, D kuch nahi deta — sirf I (uska stored area) push hold karta hai. I ke bina system immediately wapas target se drift kar jaata.
• $e$ constant aur nonzero (offset ke saath parked): P ek steady push deta hai, D zero deta hai (flat slope), I badhta rehta hai — proof ki yeh yahan rest nahi karega.
• $e$ change ho raha hai lekin momentarily zero (target ke through whipping): P zero hai, I steady hold karta hai, lekin D tezi se fire karta hai kyunki slope steep hai — yeh fly-through ko brake karta hai.
• Teeno zero (target par, still, koi history nahi): command zero hai. Yahi ek saccha equilibrium hai, aur har term iss par agree karta hai.

Step 7 — Teeno pushes ko ek command mein stack karo

KYA. Present, past, aur future pushes add karo. Woh sum hi PID control law hai.

$\boxed{u(t)=\underset{\text{present: speed}}{\underbrace{K_{p}e(t)}}+\underset{\text{past: kill offset}}{\underbrace{K_i{\int }_0^{t}e(\tau )d\tau }}+\underset{\text{future: damping}}{\underbrace{K_d\frac{de(t)}{dt}}}}$

KYUN yahi teen aur sirf yahi. Yeh precisely value, integral, derivative hain — teen cheezein jo calculus kisi bhi signal se nikal sakta hai. Har ek pichle term ki bimari theek karta hai: P fast hai lekin biased, I bias hataata hai lekin overshoot karta hai, D overshoot tame karta hai. Milke, ek balanced push.

PICTURE. Teen alag curves (ek per term) sum hoke ek single laal command curve banaate hain.

Ek-picture summary

Upar sab kuch, compressed: wahi error curve teen tarike se padhna — height (P), area (I), slope (D) — ek boat ko ek dot ki taraf le jaana.

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