Kalman gain — minimizes trace of covariance: Foundations

3.5.22 · D1 · Physics › Guidance, Navigation & Control (GNC) › Kalman gain — minimizes trace of covariance

Is page par kuch bhi assume nahi kiya gaya. Parent note Kalman gain padhne se pehle, tumhe uske har symbol ko padhna aana chahiye: $x$ , $\overset{x}{^}$ , $e$ , $P$ , $E [\cdot]$ , $tr (\cdot)$ , $H$ , $z$ , $v$ , $R$ , $K$ , aur chhote superscripts $^{-}$ , $^{+}$ , $^{⊤}$ . Hum inhe ek-ek karke build karenge, har ek ko ek picture se anchor karke.

0. Stage set karo: "state" aur "estimate"

Figure s01 (neeche): ek horizontal number line jisme ek green dot dayi taraf true state $x$ mark kar raha hai aur ek blue dot uske baayi taraf humara estimate $\overset{x}{^}$ mark kar raha hai; ek red arrow blue dot se green dot tak jaata hai, labelled $e = x - \overset{x}{^}$ — woh error jo hum abhi define karne wale hain.

Figure — Kalman gain — minimizes trace of covariance

1. Error $e$ — "kitna galat hain" yeh measure karo

2. Expectation $E [\cdot]$ — "sab luck ko average karo" machine

3. Zero-mean assumption — kyun $E [e] = 0$ matter karta hai

Abhi tak hamare paas sirf estimation error $e$ hai; sensor aur uska noise baad mein §7 mein aayega. Toh filhaal zero-mean idea sirf $e$ ke liye bataya ja raha hai (sensor noise define hone ke baad hum wahan bhi isko repeat karenge).

4. Variance aur covariance $P$ — ek guess ka "blur"

Jab state mein kai components hain (position aur velocity), toh ek component per variance kaafi nahi hoti — alag-alag components ki errors linked ho sakti hain. Hum inhe sab ek matrix mein collect karte hain. ( $^{⊤}$ "transpose" symbol jo abhi neeche use hoga, woh poori tarah §6 mein define hai; filhaal $e e^{⊤}$ ko "errors ka column times wahi errors row mein laid out" padho, jo ek square table banata hai.)

Intuition

$P$ hamesha symmetric aur positive semidefinite hoti hai $P = E [e e^{⊤}]$ se do properties seedhi nikalti hain aur yahi "ellipse" picture ko valid banati hain:

Symmetric ( $P = P^{⊤}$ ): kyunki $e e^{⊤}$ har $e$ ke liye symmetric hai (entry $i, j$ equals entry $j, i$ , dono $e_{i} e_{j}$ hain), isliye uska average bhi symmetric hai. Ek symmetric matrix ke perpendicular axes hote hain — ellipse ke axes well-defined hote hain.
Positive semidefinite ( $P ⪰ 0$ ): kisi bhi direction $u$ ke liye, $u$ ki direction mein error ki variance $u^{⊤} P u = E [(u^{⊤} e)^{2}] \geq 0$ hai — ek squared cheez negative nahi ho sakti. Iska matlab ellipse ki "negative width" kabhi nahi hogi, jo nonsense hota.

Ek matrix jo symmetric aur positive semidefinite hai woh exactly wohi hai jo ek sensible ellipse draw kar sakti hai — isliye $P$ ko is tarah picture kiya ja sakta hai.

Figure s02 (neeche): ek wide blue ellipse aur ek small green ellipse, dono ek white dot $\overset{x}{^}$ par centred, "state component 1" aur "state component 2" axes ke upar drawn. Wide ellipse labelled "big blur = big $P$ (unsure)"; tight wali "small blur (sure)" — dikhata hai ki ellipse ka size hi uncertainty $P$ ki matra hai.

5. Euclidean norm $∥ \cdot ∥$ aur trace $tr (\cdot)$

Trace se pehle, notation ka ek chhota piece jo parent note mein kaam aata hai:

6. Transpose $^{⊤}$ aur matrices $H$ — reshape karna aur mapping karna

Figure s03 (neeche): bayi taraf ek blue box labelled "state-space" jisme $x = (p, v)$ (position aur velocity) hai, ek yellow arrow labelled " $H$ (convert)" dayi taraf pointing karta hai, aur dayi taraf ek green box labelled "measurement-space" jisme " $H x$ = position only" hai — dikhata hai ki $H$ state components ko drop/mix karke waisi cheez mein le jaata hai jo sensor actually padhta hai.

7. Measurement $z$ , sensor noise $v$ , noise covariance $R$

Definition Sensor noise

$v$ aur uski covariance $R$ $v$ woh random error hai jo har reading ke andar hota hai. Bilkul §3 ke estimation error ki tarah, hum assume karte hain ki yeh zero-mean hai, $E [v] = 0$ (sensor true reading ke aaspaas wobble karta hai bina kisi built-in offset ke). Uska typical size tab true covariance $R = E [v v^{⊤}]$ hai — sensor ki khud ki uncertainty ellipse (symmetric aur positive semidefinite, usi argument se jo §4 mein $P$ ko woh properties deta hai).

Picture: true reading ke aaspaas ek fuzzy halo; wide halo = noisy sensor (bada $R$ ), tight halo = precise sensor (chhota $R$ ).
Kyun zaroori hai: sensor par kitna trust karna hai woh completely uski noisiness par depend karta hai. $R$ wahi noisiness hai, aur woh gain formula ke andar hi baitha hai. Dekho Innovation and Residuals ki $z - H \overset{x}{^}$ kaisa "surprise" ban ke filter react karta hai.

8. Superscripts $^{-}$ aur $^{+}$ — pehle vs baad mein

9. Gain $K$ — woh slider jo sab kuch tie karta hai

Prerequisite map

Equipment checklist

Test karo apne aap ko — tum parent note ke liye ready ho sirf tab jab bina dekhe har sawaal ka jawab de sako.

\overset{x}{^}

mein hat, matlab kya hota hai?

"Humare hisaab se best estimate / guess of" — kabhi true value nahi.

Estimation error likho aur batao yeh kahan point karta hai.

e = x - \overset{x}{^}

; humari guess se wapas truth ki taraf gap/arrow.

E [\cdot]

kya compute karta hai aur humein isko kyun chahiye?

Infinitely many random repetitions mein average; noise random hota hai isliye hum strategies ko average behaviour se compare karte hain.

Error ko average karne se pehle square kyun karte hain?

Taaki positive aur negative errors cancel na hon —

E [e]

0 ho sakta hai jabki guess phir bhi galat ho;

E [e^{2}]

true spread capture karta hai.

Filter kaun si zero-mean assumptions karta hai, aur kyun?

E [e] = 0

aur

E [v] = 0

(unbiased); sirf tab

P = E [e e^{⊤}]

aur

R = E [v v^{⊤}]

true covariances hain na ki raw second moments.

∥ w ∥

ka matlab kya hai?

Euclidean (straight-line) length

w_{1}^{2} + w_{2}^{2} + \dots

; aur

∥ w ∥^{2} = w^{⊤} w

P

words mein kya hai, aur iske saath kaun si picture jaati hai?

Error covariance

P = E [e e^{⊤}]

; guess ke aaspaas ek uncertainty ellipse — uska size humara blur hai.

P

mein hamesha kaun si do properties hoti hain, aur yeh kyun matter karti hain?

Symmetric (

P = P^{⊤}

) aur positive semidefinite (

u^{⊤} P u \geq 0

); yahi exactly woh hain jo

P

ko ek sensible uncertainty ellipse draw karne deti hain.

Dikhao kyun

tr (P) = E [∥ x - \overset{x}{^} ∥^{2}]

tr (P) = tr (E [e e^{⊤}]) = E [tr (e e^{⊤})] = E [e^{⊤} e] = E [∥ e ∥^{2}]

Matrix

H

kya karta hai?

State-space ko measurement-space mein map karta hai;

H \overset{x}{^}

hai "agar humari guess sahi ho toh sensor kya padhega."

v

aur

R

kya hain?

v

har reading ke andar random sensor noise hai;

R = E [v v^{⊤}]

uska size hai (sensor ki khud ki uncertainty).

Superscripts

^{-}

aur

^{+}

mein farq batao.

^{-}

= prior, measurement use karne se pehle;

^{+}

= posterior, measurement use karne ke baad.

Kaun si assumptions mein posterior ellipse prior se kabhi badi nahi hoti?

Linear-Gaussian setting mein optimal gain ke saath; nonlinear models, bure gains, ya non-Gaussian noise ke saath fail ho sakta hai.

Ek sentence mein, Kalman gain

K

kya hai?

Woh 0-to-1 slider setting jo decide karta hai ki sensor ki surprise ka kitna part humari guess mein fold karein, choose kiya jaata hai after-blur minimize karne ke liye.