Transfer function — Laplace domain, poles and zeros

3.5.27 · Physics › Guidance, Navigation & Control (GNC)

YEH CHAHIYE HI KYUN?

Ek GNC system (rocket ka attitude loop, ek autopilot) linear differential equations se describe hota hai. Har nayi command ke liye unhe haath se solve karna bahut painful hai. Hum ek shortcut chahte hain jo yeh answer kare: "agar main X command karun, toh vehicle kya karega?"

Laplace transform wohi shortcut hai. Yeh $\frac{d}{d t}$ ko $s$ se multiplication mein convert kar deta hai, toh ek differential equation ek polynomial equation ban jaati hai. Output-polynomial aur input-polynomial ka ratio hi transfer function hai.

LAPLACE TRANSFORM KYA HAI (first principles)

Kernel $e^{- s t}$ KYUN? Kyunki $e^{- s t}$ differentiation ka eigenfunction hai: $\frac{d}{d t} e^{- s t} = - s e^{- s t}$ . Aise exponential se multiply karke aur integrate karke signal ko "sort" kiya jaata hai — kiti tezi se badhta/ghatata hai ( $σ$ ) aur kiti tezi se oscillate karta hai ( $ω$ ) — is hisaab se.

Woh property jo saara kaam karti hai — HOW $s$ replaces $\frac{d}{d t}$

Derivative rule ko definition se integration by parts use karke derive karo: $L {f^{'} (t)} = \int_{0}^{\infty} f^{'} (t) e^{- s t} d t$ Maano $u = e^{- s t}$ , $d v = f^{'} (t) d t \Rightarrow d u = - s e^{- s t} d t$ , $v = f (t)$ : $= [f (t) e^{- s t}]_{0}^{\infty} + s \int_{0}^{\infty} f (t) e^{- s t} d t = - f (0) + s F (s)$

Zero initial conditions KYUN? Ek transfer function system ko khud describe karta hai, kisi ek particular start state ko nahi. Hum deliberately ICs ko zero karte hain taaki answer sirf machine par depend kare.

Transfer function banana

ODE se kaise nikaalte hain — worked example

System: $\overset{y}{¨} + 3 \overset{y}{˙} + 2 y = u (t)$ (ek damped mass on a spring; $u$ = force).

Step 1 — Har term ka Laplace lo, zero ICs. Kyun? Derivatives → powers of $s$ . $s^{2} Y + 3 s Y + 2 Y = U$

Step 2 — $Y$ ko factor out karo. Kyun? Hum ratio $Y / U$ chahte hain. $(s^{2} + 3 s + 2) Y = U$

Step 3 — Ratio banao. $G (s) = \frac{Y}{U} = \frac{1}{s ^{2} + 3 s + 2} = \frac{1}{( s + 1 ) ( s + 2 )}$

Poles aur Zeros — roots jo behaviour decide karte hain

Pole $e^{pt}$ KYUN banta hai? Kyunki $L^{- 1} {\frac{1}{s - p}} = e^{pt}$ . Partial fractions $G$ ko $\frac{A _{k}}{s - p _{k}}$ ki sum mein split karta hai, aur har piece $A_{k} e^{p_{k} t}$ mein invert ho jaata hai.

Figure — Transfer function — Laplace domain, poles and zeros

Example: apne system ke poles padho

$G (s) = \frac{1}{( s + 1 ) ( s + 2 )}$ → poles $s = - 1$ aur $s = - 2$ par, koi finite zero nahi. Dono poles left half-plane mein hain → response $\sim c_{1} e^{- t} + c_{2} e^{- 2 t}$ → stable, koi oscillation nahi. Slow pole $s = - 1$ (imaginary axis ke zyada paas) dominate karta hai (time constant $τ = 1$ s vs $0.5$ s).

Common mistakes ka steel-man

Forecast-then-Verify checkpoint

Flashcards

Laplace domain mein

s

se multiply karna kya operation represent karta hai?

Time ke saath differentiation,

L {f^{'}} = s F (s) - f (0)

Transfer function define karo.

Ratio

Y (s) / U (s)

output aur input Laplace transforms ka, zero initial conditions ke saath.

Poles kya hote hain?

Denominator ke roots; frequencies jahan

G (s) \to \infty

; yeh system ke natural modes

e^{pt}

set karte hain.

Zeros kya hote hain?

Numerator ke roots; jahan

G (s) = 0

; yeh transient shape karte hain lekin stability nahi.

Poles ke terms mein stability condition?

Saare poles left half-plane mein (negative real part).

Pole ke real part ka physical meaning?

Growth/decay rate (

σ

); negative = decaying/stable.

Pole ke imaginary part ka physical meaning?

Us mode ki oscillation frequency (

ω

\frac{1}{( s + 1 ) ( s + 2 )}

ke poles aur behaviour?

s = - 1, - 2

; stable, non-oscillatory, response

c_{1} e^{- t} + c_{2} e^{- 2 t}

Laplace integral mein

e^{- s t}

kyun?

Yeh

d / d t

ka eigenfunction hai, derivatives ko

s

se multiplication mein convert karta hai.

s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}

ke poles?

s = - ζ ω_{n} \pm j ω_{n} 1 - ζ^{2}

Recall Feynman: 12-saal ke bacche ko explain karo

Ek jhula socho. Agar tum ise ek baar push karo, toh yeh thoda hilega aur ruk jaayega. Jis tarah yeh hilta hai — kitni tezi se jhulta hai aur kitni jaldi shant hota hai — yeh jhule ne khud decide kiya hai, tumhare push ne nahi. "Transfer function" ek card hai jo jhule ki personality batata hai. Us card par jo special numbers hain woh poles hain: woh kehte hain "yeh jhula 2 seconds mein shant hota hai aur ek second mein do baar jhulta hai." Agar koi pole kabhi kahe "yeh cheez hamesha ke liye badi hoti jaati hai," toh jhula toot gaya hai (unstable). Zeros kuch pushes ke liye mute button ki tarah hain — woh wiggle ki shape badlate hain lekin jhula nahi tod sakte.

Connections

Laplace transform — woh tool jo time → frequency domain map karta hai
Block diagrams and feedback loops — transfer functions signal paths par multiply hote hain
Stability - Routh-Hurwitz criterion — factor kiye bina pole locations check karta hai
Root locus — gain badalne par poles kaise move karte hain
Second-order systems - damping ratio and natural frequency
Bode plot — imaginary axis par $G (j ω)$ evaluate karo
State-space representation — equivalent time-domain description; eigenvalues = poles