Foundations — Transfer function — Laplace domain, poles and zeros

3.5.27 · D1 · Physics › Guidance, Navigation & Control (GNC) › Transfer function — Laplace domain, poles and zeros

Is page pe kuch bhi assume nahi kiya gaya. Parent note Transfer function padhne se pehle, wahan jo bhi symbol tumhare samne aate hain unhe pehle earn karna hoga. Hum unhe ek-ek karke build karte hain, har ek pichle wale ke upar tikaa hua.

0. "Signal" aur "system" asal mein hain kya

Kisi bhi symbol se pehle, do words.

• Signal bas ek number hai jo time ke saath badalta rehta hai. Rocket ka angle, wire pe voltage, swing ki height — har cheez ek aise value ki tarah hai jise tum clock ke against ek wiggly line ki tarah plot kar sako.
• System (ya machine) ek rule hai jo ek signal ko doosre signal mein badalta hai. Tum ek input signal daalo, aur ek output signal bahar aata hai.

Poora topic ek sawaal hai: $u(t)$ diya hua ho toh $y(t)$ kya hoga?

1. Derivative $\frac{d}{dt}$ — "rate of change" wala symbol

Picture: $y(t)$ ke graph pe ek instant pe khado aur curve ke along ek chhota seedha ruler rakh do. Uska tilt hi $\dot{y}$ hai. Steep upward line → bada positive $\dot{y}$; flat line → $\dot{y}=0$; downhill line → negative $\dot{y}$.

Topic ko yeh kyun chahiye: real machines ko describe kiya jaata hai ki unki quantities kaise change hoti hain, unki raw value se nahi. "Jitna zyada push karo, utni tezi se velocity change hoti hai" — yeh sentence $\ddot{}$ aur $\dot{}$ ke baare mein hai. Yahi sentences differential equations hain, aage aata hai.

2. Differential equation — machine ka rule book

Picture: ek mass spring pe damper ke saath (ek shock absorber). $\ddot{y}$ uska acceleration hai, $3\dot{y}$ drag hai jo motion resist karta hai, $2y$ spring hai jo wapas kheenchti hai, aur $u(t)$ woh haath hai jo use push kar raha hai.

Topic ko yeh kyun chahiye: har GNC loop — autopilot, attitude control — inhi mein se ek hota hai. Har naye command $u(t)$ ke liye inhe directly solve karna slow aur painful hai. Is topic ka baaki hissa ek shortcut hai jo is calculus ko algebra mein badal deta hai.

3. Exponential $e^{something}$ — woh shape jis se sab kuch bana hai

Picture:

• Agar $a>0$: ek curve jo zyada se zyada tezi se grow karti hai (runaway).
• Agar $a<0$: ek curve jo smoothly zero ki taraf decay karti hai (calm down).
• Agar $a=0$: ek flat line (kuch nahi hota).

Topic ko yeh kyun chahiye: parent note claim karta hai ki har pole ek response "$\propto e^{pt}$" deta hai. Ab tum exactly jaante ho woh curve kaisi dikhti hai: growing agar exponent mein number positive hai, decaying agar negative. Stability bas "exponential kis direction mein jaata hai" ho jaayega.

4. Imaginary aur complex numbers — $j$, aur $\sigma +j\omega$

Picture: "spooky number" mat socho, socho flat map pe ek point. Horizontal axis real part $\sigma$ hai; vertical axis imaginary part $\omega$ hai. Toh $\sigma +j\omega$ bas ek 2-D plane pe ek address hai jise complex plane kehte hain.

Topic ko yeh kyun chahiye: parent ka pole $s=\sigma +j\omega$ wahi address hai. "Left half-plane" ka matlab sirf $\sigma <0$ hai — point vertical axis ke baayi taraf baitha hai — aur $e^{\sigma t}$ with $\sigma <0$ ki wajah se machine calm ho jaati hai. Bilkul vertical axis pe baithne wale points marginal edge hain. Ab woh sentence literal hai, magic nahi.

5. Operator $\mathcal{L}$, Laplace variable $s$, aur integral $\int$

Topic ko yeh kyun chahiye: parent define karta hai $\mathcal{L}\{f(t)\}={\int }_0^{\infty }f(t)e^{-st}dt.$ Ab tum har piece padh sakte ho: operator $\mathcal{L}$ (machine), signal $f(t)$ pe apply hota hai, ek integral (saare time pe add karo) hai us signal ka decaying-oscillating exponential $e^{-st}$ ke saath multiply karke (complex frequency $s$ se bana hua). Kernel $e^{-st}$ exactly isliye choose kiya gaya kyunki — §3 se — yeh derivative ka eigenfunction hai, jo $\frac{d}{dt}$ ko "$s$ se multiply karo" mein convert karne ki trick hai. Yeh Laplace transform poora hai.

6. Polynomials, roots, aur factoring — $N(s)$, $D(s)$

Picture: har factor $(s-r)$ ek "switch" hai jo poori product ko $0$ kar deta hai jab $s=r$ ho. Roots complex plane pe woh special addresses hain jahan polynomial vanish hoti hai.

Topic ko yeh kyun chahiye: transfer function hai $G(s)=\frac{N(s)}{D(s)}$, do polynomials ka ratio. Zeros upar wale $N(s)$ ke roots hain; poles neeche wale $D(s)$ ke roots hain. Inhe dhundhna = roots dhundhna. Parent jo behaviour ke baare mein kehta hai woh sab §4 wale plane pe in root addresses se padha jaata hai — ek edge case bhi shamil hai jahan root imaginary axis pe land karta hai (marginal ringing).

7. Ratio $\frac{Y(s)}{U(s)}$ — sab kuch jodna

Jab §1–§6 samajh aa jaayein, transfer function kuch naya nahi hai:

• Machine ki differential equation lo (§2).
• Usse Laplace-transform karo operator $\mathcal{L}$ se (§5) taaki har $\frac{d}{dt}$ ek multiply-by-$s$ ban jaaye (§3 ka eigenfunction trick), initial-condition terms zero set karke.
• Equation ek polynomial equation ban jaati hai (§6).
• Ratio $G(s)=\frac{Y(s)}{U(s)}$ = output-over-input pe rearrange karo.

$Y(s)$ aur $U(s)$, $y(t)$ aur $u(t)$ ke $s$-domain twins hain (§5 ki capital-letter convention). Yahi parent ke block-diagram ki currency hai.

Prerequisite map

Equipment checklist

Term padhte hain, answer zor se bolte hain, phir reveal karte hain.