Impulse response and transfer function (GNC connection): Foundations

4.6.33 · D1 · Maths › Ordinary Differential Equations › Impulse response and transfer function (GNC connection)

Ye page assume karta hai ki aapko parent note ki notation ke baare mein kuch nahi pata. Hum har symbol ground up se build karenge, har ek sirf pehle waale pe lean karega. Upar se neeche padho.

1. Time ka function: $y (t)$ , $u (t)$

Ek strip chart imagine karo: horizontal axis time $t$ (seconds) hai, vertical axis quantity hai. Jaise pen right move karta hai, ek curve trace hoti hai — woh curve hi $y (t)$ hai.

$u (t)$ = input: hum system ke saath kya karte hain (ek thruster command, ek push).
$y (t)$ = output: system kya karta hai waapis (ek angle, ek tank level).

Ye topic ko kyun chahiye: poora subject hai "input andar jaata hai, output bahar aata hai." Dono time ke functions hain, isliye pehle hume agree karna hoga ki time-function kaisa dikhta hai.

Figure — Impulse response and transfer function (GNC connection)

2. Derivative: $y^{'}$ , $\overset{y}{˙}$ , $y^{(n)}$

Hume ek tarika chahiye yeh kehne ka ki "output abhi kitni tezi se badal raha hai?"

Slope kyun aur kuch kyun nahi? Rate of change physics ki natural language hai: velocity position ka slope hai, aur Newton ke laws forces ko rates of change se relate karte hain. Ek spacecraft ko turn karte describe karne ke liye, hume baat karni hogi ki uska angle kitni tezi se move karta hai.

Teen notations jo sab ek hi cheez matlab rakhte hain:

Symbol	Padho aise	Matlab
$y^{'}$	"y prime"	first derivative (slope)
$\overset{y}{˙}$	"y dot"	time ke saath pehla derivative (physics style)
$y^{''}$ , $\overset{y}{¨}$	"y double-prime / double-dot"	derivative ka derivative = slope khud kitna badalta hai
$y^{(n)}$	"y super-n"	$n$ -th derivative (slope operation $n$ baar apply karo)

Figure dekho: peak par slope zero hai (flat top); upar jaate slope positive hai; neeche aate negative hai. Yeh sign-changing slope hi $y^{'}$ hai.

3. ODE aur uske coefficients: $a_{n} y^{(n)} + \dots + a_{0} y = u$

Parent ka central object hai: $a_{n} y^{(n)} + \dots + a_{1} y^{'} + a_{0} y = u (t) .$

$a_{0}, a_{1}, \dots, a_{n}$ constant numbers hain (yeh time ke saath nahi badalte).
Har term, $y$ ke kisi derivative ko ek weight $a_{k}$ ke saath mix karta hai.

Yeh shape kyun? Real machines balance laws follow karte hain (mass × acceleration = force, etc.), aur woh laws naturally ek quantity ko uski rates of change se relate karte hain. Ek Linear Constant-Coefficient ODE sabse simple family hai jo springs, tanks, aur spacecraft model karne ke liye kaafi rich hai — dekho Linear constant-coefficient ODEs.

4. Dirac delta: $δ (t)$ — perfect tap

Hum system ko jitna ho sake utna sharply "kick" karna chahte hain. Ek idealised tap kaisi dikhti hai?

Ise limit ke roop mein build karo: width $w$ aur height $1/ w$ ka ek rectangle lo. Uska area hamesha $1$ hai. Ab $w \to 0$ shrink karo: yeh lambi aur patli hoti jaati hai lekin area $1$ rakhi rehti hai. Woh limiting spike hi $δ (t)$ hai. Figure mein shrinking rectangles dekho.

Topic ko yeh kyun chahiye: delta mathematical "single perfect tap" hai. Iske response mein system ki fingerprint hoti hai. Aur sifting woh tool hai jo baad mein kisi bhi input ko taps ki ek row mein kaatne deta hai. Zyada detail mein Dirac delta function.

5. Zero-state ( $t = 0$ se pehle rest mein)

Ek still swing ko seedha neeche latka hua imagine karo push karne se pehle. Hum iss par isliye insist karte hain taaki jo response hum measure karein woh purely input se aaye, kisi leftover motion se nahi. Yahi "zero-state" assumption hai jo parent ke har formula ke peeche hai.

6. Impulse response: $h (t)$

Still swing, ek sharp push, phir woh swing karta hai aur dheere-dheere ruk jaata hai. Woh poori dying-down curve hi $h (t)$ hai — system ki fingerprint ya personality.

" $h$ " kyun? Yeh response hai jo ek single tap ke haath waapis diya jaata hai. Kyunki system LTI hai, yeh ek curve secretly sab kuch ka response contain karti hai — yeh aage prove hota hai.

7. Convolution: $*$ symbol

Ab punchline machinery. Hum arbitrary input ko taps mein kaatenge aur responses add karenge.

Symbols ko dhyaan se padho:

$τ$ (Greek "tau") ek dummy time hai — woh instant jab koi particular tap apply kiya gaya tha.
$u (τ)$ = time $τ$ par jo tap tha woh kitna strong tha.
$h (t - τ)$ = fingerprint, $τ$ par start karne ke liye shifted, present time $t$ par evaluate kiya gaya (time-invariance hume bas ise shift karne deti hai).
Saare $τ$ par integrate karna = har pichli tap ke after-effects ko add up karna. Dekho Convolution.

Integral kyun aur plain sum kyun nahi? Taps time mein infinitely densely packed hain, isliye "unhe add karo" ka matlab hai "integrate karo." Ek causal system ke liye ( $h = 0$ $t < 0$ ke liye) aur input $0$ se start kare, limits $0$ se $t$ tak shrink ho jaati hain: sirf past taps present ko affect kar sakte hain.

8. Laplace transform: $L$ , $s$ , $F (s)$

Convolutions painful integrals hote hain. Hum ek aisi duniya chahte hain jahan woh plain multiplication ban jaayein.

$s$ ek (possibly complex) number hai; ise ek knob ki tarah socho.
$e^{- s t}$ ek weighting hai jo $s$ badalne par alag time-behaviours par emphasis deta hai.
Capital letters ( $F$ , $Y$ , $U$ , $H$ ) matlab "lowercase function ka Laplace transform." Dekho Laplace transform.

Teen facts jin par topic lean karta hai: $L {f^{'}} = s F (s) - f (0), L {δ} = 1, L {u * h} = U (s) H (s) .$ Aakhri wala hi poori wajah hai hum kyun bother karte hain: time mein convolution, $s$ mein multiplication ban jaata hai.

9. Transfer function $H (s)$ aur poles

Bottom polynomial $a_{n} s^{n} + \dots + a_{0}$ characteristic polynomial hai — dekho Characteristic polynomial and roots.
Poles = $s$ ki woh values jo denominator ko zero karti hain (jahan $H$ blow up hota hai).

Yahi Guidance Navigation and Control (GNC) aur Feedback control systems ka core hai: poles ko left rakho, spacecraft ko calm rakho.

Prerequisite map

Equipment checklist

Khud test karo — reveal karne se pehle kya tum har ek answer de sakte ho?

y (t)

ka matlab kya hai, ek picture mein?

Ek strip chart par ek curve: time

t

ke har instant ke liye ek output value.

Derivative

y^{'}

kya measure karta hai?

y (t)

ki slope — us instant par output kitni tezi se badal raha hai.

\overset{y}{˙}

aur

\overset{y}{¨}

ka matlab kya hai?

Pehle aur doosre time-derivatives (velocity-jaisi aur acceleration-jaisi); dots operations hain, multiplication nahi.

Ek ODE ko "LTI" kya banata hai?

Linear (inputs ko scale/add karna outputs ko scale/add karta hai) aur Time-invariant (constant coefficients

a_{k}

, isliye responses sirf time mein shift hote hain).

δ (t)

ke teen defining facts kya hain?

Zero width, infinite height, area exactly

1

(plus sifting property).

Sifting property state karo.

\int f (t) δ (t - a) d t = f (a)

— spike

f

t = a

par read off karta hai.

"Zero-state / at rest" ka matlab kya hai?

Saari initial conditions zero hain:

y (0) = 0, \overset{y}{˙} (0) = 0, \dots

— tap se pehle koi leftover motion nahi.

Impulse response

h (t)

kya hai?

Zero-state output jab input

δ (t)

ho; system ki fingerprint.

u * h

mein star ka matlab kya hai?

Convolution: flipped fingerprint ko slide karo, input se multiply karo, saare past times

τ

par integrate karo.

Hum Laplace transform se kyun bother karte hain?

Yeh convolution ko multiplication mein aur differentiation ko "multiply by

s

" mein badal deta hai, ODE ko plain algebra bana deta hai.

Pole kya hai, aur kyun care karein?

s

ki woh value jo

H (s)

ko blow up karti hai (denominator root); real part ka sign decide karta hai ki response decay karega ya explode.

Connections

Parent topic
Dirac delta function
Laplace transform
Convolution
Characteristic polynomial and roots
Linear constant-coefficient ODEs
Stability and poles (left-half plane)
Feedback control systems
Guidance Navigation and Control (GNC)