Kalman filter derivation — predict step, update step: Question bank

3.5.21 · D5 · Physics › Guidance, Navigation & Control (GNC) › Kalman filter derivation — predict step, update step

Shuru karne se pehle, ek quick vocabulary refresher taaki neeche har symbol samajh aaye:

Recall Har symbol ka matlab (agar bhool gaye ho toh unfold karo)

$x_{k}$ ::: step $k$ par true state — woh hidden quantity (position, velocity, attitude) jise hum estimate karna chahte hain.
$F$ ::: state-transition matrix — woh linear rule jo state ko ek step aage push karta hai, $x_{k} = F x_{k - 1} + \dots$ (dekho State-space representation).
$B$ ::: control-input matrix — ek known command ko state par uske effect mein map karta hai.
$u_{k}$ ::: control vector — step $k$ par apply kiya gaya known command (jaise thruster firing), isliye deterministic push hai $B u_{k}$ .
$w_{k}$ ::: process-noise vector — har step mein add hone wala ek random, zero-mean kick kyunki dynamics model imperfect hai; yeh $x_{k} = F x_{k - 1} + B u_{k} + w_{k}$ mein aata hai aur iski covariance $Q$ hai.
$z_{k}$ ::: measurement vector — step $k$ par noisy sensor reading.
$H$ ::: observation matrix — state ko woh map karta hai jo sensor dekhta hai, $z_{k} = H x_{k} + v_{k}$ .
$v_{k}$ ::: measurement-noise vector — har sensor reading par ek random, zero-mean corruption; yeh $z_{k} = H x_{k} + v_{k}$ mein aata hai aur iski covariance $R$ hai.
$\overset{x}{^}_{k}^{-}$ ::: prior (predicted) estimate — nayi measurement dekhne se pehle sirf physics kya kehti hai.
$\overset{x}{^}_{k}$ ::: posterior estimate — measurement fuse karne ke baad.
$P_{k - 1}, P_{k}^{-}, P_{k}$ ::: previous-step posterior, current prior, aur current posterior covariance — filter ki apni estimate ke baare mein uncertainty (dekho Covariance matrices and Gaussian distributions).
$Q$ ::: process-noise covariance — dynamics model $F$ kitna galat hai ( $w_{k}$ ki covariance).
$R$ ::: measurement-noise covariance — sensor kitna noisy hai ( $v_{k}$ ki covariance).
$K_{k}$ ::: Kalman gain — measurement ke surprise ko diya gaya weight.
$y_{k} = z_{k} - H \overset{x}{^}_{k}^{-}$ ::: innovation — measurement minus prediction.
$S_{k} = H P_{k}^{-} H^{⊤} + R$ ::: innovation covariance — us surprise ki uncertainty.

Poore page ko do pictures anchor karti hain. Pehli, covariance kaisi dikhti hai aur predict vs. update usse kaise reshape karte hain:

Figure — Kalman filter derivation — predict step, update step

Doosri, innovation ka geometric matlab — sensor jahan land karta hai aur physics ne predict kiya tha, unka gap:

True or false — justify

TF1. "Kalman filter ko optimal rehne ke liye past measurements ki poori history store karni padti hai."

False — yeh recursive hai:

(\overset{x}{^}_{k - 1}, P_{k - 1})

saare past data ka complete summary hai, isliye ek step aage le jaane se saari information milti hai (yahi reason hai ki yeh Recursive Least Squares se linked hai).

TF2. "

Q = 0

set karna sabse accurate filter deta hai kyunki isse noise remove ho jaati hai."

False —

Q = 0

claim karta hai ki model perfect hai, isliye

P_{k}^{-}

zero ki taraf shrink ho jaata hai,

K_{k} \to 0

, aur filter measurements sunna band kar deta hai aur diverge ho jaata hai.

TF3. "Predict step mein covariance sirf badhh sakti hai ya same reh sakti hai."

Generally False — haalaanki

+ Q

term (jo positive-semidefinite hai, ek real covariance hone ke naate) sirf uncertainty add kar sakti hai, lekin

F P_{k - 1} F^{⊤}

term

P

ko shrink kar sakti hai jab dynamics contracting hoti hain — matlab har direction ek step mein 1 se chhote factor se multiply hoti hai, isliye errors zero ki taraf decay karti hain (formally,

F

ki largest eigenvalue-magnitude 1 se neeche hai). Yeh "sirf badhh sakti hai" sirf tab hota hai jab dynamics contracting nahi hoti.

TF4. "Update step mein (optimal gain ke saath) covariance sirf shrink ya same reh sakti hai."

True — likho

P_{k} = (I - K_{k} H) P_{k}^{-}

. Kyunki

P_{k}^{-}

ek covariance hai, yeh symmetric positive-semidefinite hai, aur dikhaaya ja sakta hai ki

P_{k}^{-} - P_{k} = K_{k} S_{k} K_{k}^{⊤}

jahan

S_{k} = H P_{k}^{-} H^{⊤} + R

bhi positive-semidefinite hai; isliye difference PSD hai, matlab

P_{k} ⪯ P_{k}^{-}

— observe karna uncertainty ko hatata hai, kabhi add nahi karta.

TF5. "Kalman gain ek constant hai jo tum ek baar tune karte ho, jaise PID gain."

False —

K_{k}

ko har step mein current

P_{k}^{-}

aur

R

se recompute kiya jaata hai; yeh shuru mein bada hota hai (unsure prior) aur steady-state value ki taraf settle hota hai.

TF6. "Agar measurement noise

R

bahut bada hai, toh filter basically sensor ko ignore karta hai."

True — bada

R

S_{k}

ko bada banata hai aur

K_{k} = P_{k}^{-} H^{⊤} S_{k}^{- 1}

chhota ho jaata hai, isliye correction

K_{k} y_{k}

tiny hoti hai aur filter apni prediction par coast karta hai.

TF7. "Posterior estimate hamesha prior aur measurement ke beech hoti hai."

Sirf 1-D mein True — wahan

\overset{x}{^} = \overset{x}{^}^{-} + K (z - \overset{x}{^}^{-})

jahan

0 \leq K \leq 1

ek genuine convex blend hai jo kisi bhi input se overshoot nahi karta. Multiple dimensions mein update ek matrix-weighted combination hai

\overset{x}{^}_{k} = \overset{x}{^}_{k}^{-} + K_{k} y_{k}

, isliye result zaruri nahi ki prior aur measurement ke "beech" ki line par ho — components un directions mein move kar sakte hain jo koi bhi input point nahi karta, kyunki

P

mein correlations correction ko redistribute karti hain.

TF8. "Kalman filter assume karta hai ki noises Gaussian hain."

Aadha sach — diye gaye covariances ke saath kisi bhi noise ke liye yeh optimal linear estimator hai, lekin overall optimal estimator tab hota hai jab noise Gaussian ho (dekho Bayesian inference).

TF9. "Do independent estimates fuse karna hamesha aapko dono akele se zyada confident banata hai."

True — precisions (inverse variances) add hote hain, isliye

1/ σ^{2} = 1/ σ_{a}^{2} + 1/ σ_{b}^{2} > max (1/ σ_{a}^{2}, 1/ σ_{b}^{2})

; fused variance dono se chhoti hoti hai.

Spot the error

SE1. "Innovation

y_{k} = z_{k} - H x_{k}

Galat — aap kabhi true

x_{k}

nahi jaante; innovation prediction use karta hai:

y_{k} = z_{k} - H \overset{x}{^}_{k}^{-}

, yahi computable version hai.

SE2. "

P_{k} = (I - K_{k} H) P_{k}^{-}

kisi bhi gain

K

ke liye kaam karta hai jo aap choose karo."

Galat — woh shortcut sirf optimal

K_{k}

ke liye valid hai; kisi aur gain ke liye Joseph form

(I - K_{k} H) P_{k}^{-} (I - K_{k} H)^{⊤} + K_{k} R K_{k}^{⊤}

use karna padega warna

P_{k}

non-symmetric ya indefinite ho sakti hai.

SE3. "Gain hai

K_{k} = P_{k}^{-} H^{⊤} (H P_{k}^{-} H^{⊤})^{- 1}

Galat —

+ R

missing hai; denominator innovation covariance

S_{k} = H P_{k}^{-} H^{⊤} + R

hai, aur

R

drop karna ek noisy sensor ko blindly trust karna hoga.

SE4. "Predict mean update karta hai lekin covariance ko akela chodh deta hai."

Galat — predict covariance bhi propagate karta hai:

P_{k}^{-} = F P_{k - 1} F^{⊤} + Q

; ise bhoolna filter ko overconfident banata hai aur woh achhi measurements ko reject kar dega.

SE5. "Kyunki

S_{k}

ek covariance hai, hum isse simply ek number ki tarah divide kar sakte hain."

Galat — vector form mein

S_{k}

ek matrix hai, isliye hum uske inverse

S_{k}^{- 1}

se multiply karte hain; scalar division sirf 1-D case mein sense banata hai.

SE6. "Update ke baad

P_{k}

phenk do — sirf estimate

\overset{x}{^}_{k}

aage carry forward hoti hai."

Galat —

P_{k}

aapke confidence ki memory hai; agla predict step ise

P_{k + 1}^{-}

compute karne ke liye chahiega, aur iske bina recursion continue nahi ho sakti.

SE7. "

B u_{k}

noise ka source hai, isliye ise

Q

mein include karo."

Galat —

B u_{k}

ek known control input hai (ek deterministic push); yeh mean ko move karta hai aur koi uncertainty add nahi karta, jabki

Q

(random kick

w_{k}

ki covariance) unknown modelling error hai.

Why questions

WHY1. Position–velocity state predict karne ke baad

P

ki off-diagonal non-zero kyun ho jaati hai?

Kyunki uncertain velocity future position mein feed hoti hai, isliye unki errors correlated ho jaati hain; filter record karta hai ki aaj ka velocity error kal ek position error cause karega.

WHY2. Fused variance dono input variances mein se chhoti se bhi chhoti kyun hoti hai?

Do independent readings mein se har ek partial information carry karta hai, aur information combine karna sirf estimate ko sharpen kar sakta hai — precisions add hote hain, isliye total precision dono se zyada hoti hai.

WHY3. Hum

P_{k}

ki jagah

P_{k}

ka trace kyun minimise karte hain?

Aap "ek matrix minimise" nahi kar sakte, lekin trace saare state components mein total (summed) mean-squared error hai, ek single scalar cost jo differentiate karne par optimal gain deta hai.

WHY4. Optimal

K

ko Joseph form mein plug karne par

K R K^{⊤}

term kyun cancel ho jaata hai?

Joseph form expand karo

P_{k} = (I - K H) P^{-} (I - K H)^{⊤} + K R K^{⊤} = P^{-} - K H P^{-} - P^{-} H^{⊤} K^{⊤} + K (H P^{-} H^{⊤} + R) K^{⊤} = P^{-} - K H P^{-} - P^{-} H^{⊤} K^{⊤} + K S K^{⊤}

. Optimal

K = P^{-} H^{⊤} S^{- 1}

satisfy karta hai

K S = P^{-} H^{⊤}

, isliye

K S K^{⊤} = P^{-} H^{⊤} K^{⊤}

, jo exactly

- P^{-} H^{⊤} K^{⊤}

term ko cancel kar deta hai aur

P_{k} = P^{-} - K H P^{-} = (I - K H) P^{-}

bacha deta hai.

WHY5.

S_{k}

ko sirf

R

ki jagah innovation covariance kyun kaha jaata hai?

Surprise

y_{k}

uncertainty inherit karta hai dono noisy sensor (

R

) aur uncertain prediction ko

H

se push karne (

H P_{k}^{-} H^{⊤}

) se, isliye

S_{k}

dono sources ko sum karta hai.

WHY6. Kalman filter sirf ek best guess ki jagah covariance kyun track karta hai?

P

ke bina yeh nahi jaanta ki agli measurement par kitna trust karna hai; gain

K_{k}

poori tarah

P_{k}^{-}

par depend karta hai, isliye uncertainty bookkeeping hi poora engine hai.

WHY7. Real vehicles ke liye linear filter ko Extended Kalman Filter (EKF) ya Unscented Kalman Filter (UKF) kyun chahiye?

Real dynamics aur sensors (attitude, range, bearing) nonlinear hote hain, isliye

F

aur

H

constant matrices nahi hote; EKF linearise karta hai aur UKF curvature handle karne ke liye sigma points sample karta hai.

Edge cases

EC1. 1-D mein

H = 1

ke saath sensor perfect ho jaaye (

R \to 0

) toh

K_{k}

ka kya hota hai?

K_{k} \to P_{k}^{-} / P_{k}^{-} = 1

, isliye filter poori tarah measurement adopt kar leta hai aur apni prediction discard kar deta hai — ek flawless sensor ke saath, yahi bilkul sahi hai.

EC2. Prior perfect ho jaaye (

P_{k}^{-} \to 0

) toh

K_{k}

ka kya hota hai?

K_{k} \to 0

, isliye correction

K_{k} y_{k}

vanish ho jaati hai aur measurement ignore ho jaati hai — agar aap already certain hain, toh naya noisy data kuch add nahi karta.

EC3. Agar measurement prediction se exactly match kare (

z_{k} = H \overset{x}{^}_{k}^{-}

) toh innovation kya hoga?

y_{k} = 0

, isliye

\overset{x}{^}_{k} = \overset{x}{^}_{k}^{-}

(mean unchanged), phir bhi

P_{k}

shrink hoti hai — ek "no-surprise" measurement bhi estimate confirm karti hai aur confidence badhhaati hai.

EC4. Agar ek hi timestep mein do independent measurements aayein toh?

Update step do baar sequence mein apply karo, har baar latest posterior ko naya prior treat karte hue. Order result nahi badalta kyunki, information form mein, har measurement simply apni precision

H^{⊤} R^{- 1} H

running information matrix mein aur

H^{⊤} R^{- 1} z

running information vector mein add karti hai — aur addition commutative hai, isliye pehle kaunsa add karo yeh matter nahi karta.

EC5. Agar

R

bahut zyada chhota set ho (noisy sensor par over-trust) toh kya hoga?

K_{k}

bada ho jaata hai, estimate har noise spike ke peeche bhaaghne lagti hai, aur filter jittery ho jaata hai — yeh under-tuned

R

ka classic symptom hai.

EC6. Agar kisi step mein koi measurement hi na ho (sensor dropout) toh kya hoga?

Sirf predict step run karo; estimate physics par coast karti hai aur

P

+ Q

se badhh jaati hai, correctly rising uncertainty reflect karta hai jab tak measurement wapas nahi aati (yeh IMU and GPS sensor fusion ke liye central hai jahan GPS drop out hoti hai).

EC7. Steady state mein

K_{k}

kyun change karna band kar deta hai?

Predict growth (

+ Q

) aur update shrinkage

(I - K_{k} H)

ek balance reach karte hain jahan

P_{k}^{-}

har cycle mein repeat hota hai, isliye uss fixed

P_{k}^{-}

se compute kiya gaya gain bhi constant ho jaata hai.

EC8. Agar

H

sirf state ka part observe kare (jaise position, velocity nahi) toh kya hoga?

Update phir bhi velocity ko indirectly

P

ke off-diagonal correlations ke through improve karta hai; measured position correlated velocity ko saath khenchta hai chahe

H

use directly touch na kare (dekho State-space representation).

Recall Jaane se pehle ek-line self-test

Kalman gain kis ek sawaal ka jawab deta hai? ::: "Jitna main uncertain hoon aur sensor kitna noisy hai, usse milaakar surprise ka kitna hissa mujhe actually believe karna chahiye?"