Extended Kalman Filter (EKF) — linearization, Jacobians: Question bank

3.5.24 · D5 · Physics › Guidance, Navigation & Control (GNC) › Extended Kalman Filter (EKF) — linearization, Jacobians

Shuru karne se pehle, is page mein use hone wale har symbol ko simple shabdon mein samjhate hain, taaki kuch bhi achanak na aaye:

$f$ nonlinear motion function hai: yeh tumhara current state guess leta hai aur next state return karta hai.
$h$ nonlinear measurement function hai: yeh ek state leta hai aur return karta hai ki tumhara sensor actually kya padhna chahiye.
$F$ aur $H$ Jacobians hain — $f$ aur $h$ ke slopes (partial derivatives) ki matrices. Yeh curves ke local "straight-line stand-ins" hain, sirf us point ke paas valid hain jahan inhe compute kiya gaya hai.
$P$ covariance hai: ek blob jo batata hai ki tumhara state estimate kitna uncertain (aur kitna correlated) hai.
$\hat{x}_{k}^{-}$ predicted state hai (motion ke baad, measurement se pehle); $\hat{x}_{k - 1}$ previous estimate hai.
$Q$ process-noise covariance hai: har step mein duniya kitni extra uncertainty inject karti hai kyunki tumhara motion model $f$ imperfect hai. Yeh predicted covariance mein add hota hai.
$R$ measurement-noise covariance hai: tumhara sensor kitna noisy hai. Bada $R$ matlab "is reading par zyada bharosa mat karo."
$S$ innovation covariance hai: measured aur predicted ke beech "surprise" ka total spread, state spread ko sensor space mein push karke aur sensor noise se banaya gaya, $S = H P^{-} H^{⊤} + R$ .
$K$ Kalman gain hai: us surprise ka woh fraction jo tum actually apna estimate correct karne ke liye use karte ho, $K = P^{-} H^{⊤} S^{- 1}$ . Yeh har step mein recompute hota hai aur prediction aur sensor ke beech trust balance karta hai.

Sahi ya galat — justify karo

True: mean nonlinear f se propagate hoti hai, covariance Jacobian se

True — next state ka best guess current best guess ka

f

hota hai, lekin spread local sensitivity se transform hoti hai, isliye

P^{-} = F P F^{⊤} + Q

. Mean curve use karta hai, covariance tangent use karta hai.

True: agar f already linear hai, toh EKF exactly linear Kalman filter ban jaata hai

True — ek linear map

F x

ka Jacobian constant matrix

F

hi hota hai, isliye koi approximation nahi bachti aur machinery Kalman Filter (linear) jaisi hi ho jaati hai.

True: Jacobian ek point par curve ka sabse best possible constant-matrix approximation hai

True — yeh first-order Taylor term hai, aur koi bhi constant matrix local slopes ko is se behtar match nahi karta. Isliye EKF kisi bhi doosri matrix ke bajaye ise choose karta hai.

True: do alag states ek jaisa Jacobian de sakti hain

True — ek curve ka slope repeat ho sakta hai (jaise pendulum ka

- \frac{g}{L} cos θ

θ

aur

- θ

dono par equal hota hai). Jacobian local behaviour describe karta hai, state ki identity nahi.

False: kyunki EKF linearize karta hai, yeh sirf mildly nonlinear systems hi handle kar sakta hai

Aadha-galat — yeh strong nonlinearity bhi theek se handle karta hai jab tak ek uncertainty blob par curve lagbhag straight ho. Takleef badi curvature-times-uncertainty se aati hai, nonlinearity se nahi per se.

False: process-noise covariance

Q

ko zero karne se filter zyada accurate ho jaata hai

Galat —

Q = 0

se covariance shrink hoti hai aur filter over-confident ho jaata hai, naya data ignore karta hai, aur diverge kar deta hai.

Q

jaanbujhkaar estimate ko model errors ke baare mein humble rakhti hai.

True: EKF covariance true uncertainty ko underestimate kar sakta hai

True — linear tangent us curvature ko throw away kar deta hai jo actually ek Gaussian ko spread karta hai, isliye

P

aksar optimistically small nikalta hai. Yeh ek jaana-maana EKF failure mode hai jise UKF mitigate karta hai.

False: innovation

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

Galat — innovation mein nonlinear prediction

h (\hat{x}_{k}^{-})

use honi chahiye.

H

sirf innovation covariance

S

aur gain

K

ke andar appear karta hai;

H \hat{x}^{-}

use karna mean ko linearize kar deta, jo hum specifically nahi karte.

Galti dhundho

H

ko previous estimate

\hat{x}_{k - 1}

par compute karo taaki us point ko reuse kar sako jahan tumne

F

nikala tha."

Galat evaluation point —

h

ko predicted state

\hat{x}_{k}^{-}

par linearize karna chahiye, kyunki yahi woh state hai jiske against measurement compare ki ja rahi hai. Yeh classic #1 EKF bug hai.

"Kyunki mean

f

use karta hai, symmetry ke liye covariance

f (P)

honi chahiye."

Koi meaningful "

f (P)

" hota hi nahi —

P

ek second-moment object hai, state nahi, isliye yeh linear sensitivity

F P F^{⊤}

se transform hota hai, kabhi bhi

f

mein matrix feed karke nahi.

"Ek radar bearing residual

35 9^{\circ} - 1^{\circ} = 35 8^{\circ}

matlab filter bahut off hai."

Residual wrap hi nahi kiya gaya — sachi angular error

- 2^{\circ}

hai. Angle residuals ko

(- π, π]

mein wrap karna zaroori hai; dekho atan2 & Angle Wrapping.

"Mera initial guess rough hai lekin filter hamesha converge kar leta hai, isliye main ise trust karoonga."

Tangent-line approximation sirf

\hat{x}

ke paas valid hai; ek bura linearization point garbage

F, H

deta hai aur filter bilkul diverge kar sakta hai. Achha

\hat{x}_{0}

seed karo.

"Kalman gain

K

ek fixed tuning constant hai jo main ek baar set karta hoon."

K

har step mein

P^{-}, H

aur measurement-noise covariance

R

se recompute hota hai (

K = P^{-} H^{⊤} S^{- 1}

ke zariye); yeh automatically shrink hota hai jab sensor noisy ho aur grow karta hai jab sensor pe trust ho. Ise hard-code karna poore filter ko defeat kar deta hai.

"Main bearing measurement ke liye

arctan (y / x)

use karoonga, simple rakhne ke liye."

arctan

quadrant II aur III ko I aur IV par collapse kar deta hai, aur half plane ke liye galat angle deta hai.

atan2 (y, x)

use karo, jo dono signs padhta hai aur chaaron quadrants cover karta hai.

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

thoda non-symmetric nikla, koi baat nahi."

Numerically yeh drift karta hai aur positive-definiteness kho sakta hai, filter ko silently break kar ke. Joseph-form update use karo ya symmetry force karo — covariance zaroori hai ki symmetric aur positive-definite rahe.

Kyun wale sawaal

Mean

f

se kyun jaati hai lekin covariance

F

se kyun jaati hai?

Akela sabse likely next state literally current most-likely state ka

f

hota hai; lekin uncertainty is baare mein hai ki nearby points kaise spread hote hain, aur first order tak woh slope matrix

F

se spread hote hain. Alag sawaal, alag tools.

F

\hat{x}_{k - 1}

par lekin

H

\hat{x}_{k}^{-}

par kyun evaluate karte hain?

F

us motion ko linearize karta hai jo tum abhi apply karne wale ho, isliye yeh motion se pehle ki state use karta hai;

H

sensor ko us state ke around linearize karta hai jahan tum pahunche, isliye yeh motion ke baad ki state use karta hai. Har tangent wahan liya jaata hai jahan uska curve actually use hota hai.

Hum poore Gaussian ko exactly curve se push kyun nahi kar sakte?

Ek Gaussian jo curve se push hoti hai woh skewed aur non-Gaussian nikla aati hai, aur KF equations sirf means aur covariances ki language bolte hain. Linearize karna output ko Gaussian rakhta hai taaki math valid rahe.

Pendulum Jacobian mein

cos θ

kyun hai aur iska kyun matter karta hai?

g

gravitational acceleration aur

L

pendulum length ke saath, yeh restoring term

- \frac{g}{L} sin θ

ka derivative hai.

θ = 0

ke paas yeh

- \frac{g}{L}

hota hai (stiff, linear oscillator), lekin

θ = π /2

ke paas yeh vanish ho jaata hai — linearized stiffness gaayab ho jaati hai, jise state-dependent Jacobian sahi se report karta hai.

Jab nonlinearity severe ho toh UKF prefer kyun karein?

EKF sirf pehla Taylor term rakhta hai aur curvature discard kar deta hai; UKF iske bajaye sample points ka ek set true

f, h

se push karta hai, bina kabhi Jacobian banaye curvature capture karta hai.

Range partial

\partial r / \partial x = x / r

ek cosine ke barabar kyun hai?

x / r

target ke bearing angle ka cosine hai, isliye yeh kehta hai "range sabse tezi se badhti hai jab tum sensor ki taraf seedha move karo aur bilkul nahi jab tum sideways move karo" — Jacobian mein baka ek geometric sanity check.

Bahut chhoti measurement-noise covariance

R

filter ko kabhi kabhi jittery kyun banati hai?

Chhota

R

matlab "sensor lagbhag perfect hai," isliye gain

K

bada ho jaata hai aur estimate har noisy reading ko chase karne lagta hai. Filter noise par overtrust karta hai aur smooth karne ki jagah rattles karta hai.

Edge cases

Origin

(0, 0)

par range Jacobian kya hai?

Undefined —

r = 0

denominators mein

r

aur

r^{2}

daal deta hai, isliye bearing meaningless ho jaati hai aur covariance update blow up ho jaata hai. Sensor par baitha target ek genuine singularity hai jisse guard karna zaroori hai.

Jab target door jaata hai (bada

r

) toh bearing partials ka kya hota hai?

\partial θ / \partial x = - y / r^{2} \to 0

: door ke targets ke move karne par bearing zyada nahi badalti, isliye angle measurements long range par weak ho jaate hain aur filter range par lean karta hai.

Agar Jacobian

F

mein koi state-dependence nahi nikla, toh yeh kya batata hai?

Tumhari dynamics actually linear hain aur tumhe kabhi EKF ki zaroorat hi nahi thi — plain Kalman Filter (linear) exact aur cheaper hai.

Kya hoga agar

\hat{x}_{k}^{-}

exactly wahan land kare jahan

h

non-differentiable ho (jaise

∣ \cdot ∣

kink)?

Jacobian wahan undefined hai; tangent-line premise bilkul fail ho jaata hai. Tumhe sub-gradient use karna hoga, model smooth karna hoga, ya derivative-free filter pe switch karna hoga.

Pendulum ke inverted point

θ = π

par Jacobian kya kehta hai?

- \frac{g}{L} cos π = + \frac{g}{L}

, ek positive stiffness jo kehti hai equilibrium unstable hai — linearization sahi se sign flip karta hai aur top se door runaway growth predict karta hai.

Agar innovation covariance

S_{k}

almost singular ho toh kya?

Gain

K = P^{-} H^{⊤} S^{- 1}

S^{- 1}

chahiye; ek near-singular

S

(chhote

R

aur degenerate

H

se) gain ko explode kar deta hai.

R

ko regularize karo ya check karo ki

H

rank-deficient toh nahi.