Observability — when KF can estimate state: Foundations

3.5.23 · D1 · Physics › Guidance, Navigation & Control (GNC) › Observability — when KF can estimate state

Is page par assume kiya gaya hai ki aapne kuch bhi nahi dekha. Isse pehle ki aap parent note on observability padh sakein, har woh symbol jo wahan milega uska matlab pehle clear hona chahiye. Hum unhe ek-ek karke banate hain, har ek pichle ke upar, aur har ek ko ek picture se pin karte hain.

0. Woh scene jise hum describe kar rahe hain

Ek ghoomte hue satellite ki picture socho. Andar kuch quantities hain jo sach hain par chhupe hain — uska asli pointing angle, uske gyroscope ki dheere-dheere barhti drift error. Bahar, ek star-tracker padhe jaane wale numbers ugalta hai. Poora subject "andar kya sach mein ho raha hai" aur "hum kya dekh paate hain" ke beech ki khaai mein rehta hai.

Figure — Observability — when KF can estimate state

Ab hum is picture ke har piece ko naam dete hain.

1. Ek vector aur symbol $x$

Picture: ek arrow, ya equivalently ek dot, ek aisi space mein float karta hua jahan har dial ke liye ek axis hai. Do dials → flat plane; dot ki position hi machine ki poori internal condition hai.

Topic ko kyun chahiye: observability is poori dot ko bahar se pin karne ke baare mein hai. Agar hum sabhi dials ko ek object $x$ mein bundle nahi kar sakte, toh hum pooch hi nahi sakte "kya humne poori cheez recover ki?"

2. State $x$ aur measurement $y$

Picture: state dot ek bade kamre mein rehta hai ( $n$ axes); measurement uska shadow hai ek chhoti wall par ( $m$ axes). Kai alag-alag dots ek hi shadow daal sakte hain — woh ambiguity hi observability ki chinta hai.

3. Matrices aur matrix–vector multiplication

$A$ aur $C$ ka koi matlab ho, pehle jaanna hoga ki matrix kya karti hai.

Picture: ek matrix poore dots ke space ko stretch, rotate, squish, ya project karti hai. Dekho ek dots ki grid andar jaaye aur ek warped grid bahar aaye.

Topic ko kyun chahiye: "state kaise evolve hota hai" aur "sensor state ko kaise padhta hai" dono exactly aisi mixing operations hain — isliye woh matrices hain.

4. Dynamics matrix $A$ , aur input $B$ , $u$

Picture: kamre ke har dot par, $A$ ek chhota arrow lagaata hai jo kehta hai "state agli instant is taraf slide karegi." Poora kamra flow arrows ka ek field ban jaata hai.

Topic ko kyun chahiye — crucial subtlety: agar $A_{ij} \neq = 0$ hai, toh dial $j$ dial $i$ mein leak hota hai. Ek chhupa dial jise hum measure nahi kar sakte woh bhi khud ko betray kar sakta hai ek aisa dial nudge karke jo hum dekh sakte hain. Yahi leaking woh mechanism hai jo chhupe cheezon ko observable banata hai. (Parent ke position–velocity example mein, velocity unseen hai lekin $A_{12} = 1$ use position push karne deta hai — isliye woh leak out ho jaati hai.)

Topic ko kyun chahiye: motion ka poora equation (agla section) $B u$ contain karta hai. Lekin kyunki $u$ jaana hua hai, hum hamesha uska effect subtract kar sakte hain aur sirf machine ke apne unforced drift ko study kar sakte hain — exactly isliye observability testing $u = 0$ set karti hai.

5. Measurement matrix $C$

Picture: $C$ Section 2 mein window ki shape hai. $C = [10]$ ek aisi window hai jo dial 1 dikhati hai aur dial 2 ko poori tarah chhupa deti hai.

6. Derivative — samay ke saath change padhna

Yeh tool kyun aur koi nahi? Hum measurement stream se information nikalna chahte hain. Ek akela snapshot $y (0)$ ek clue deta hai. Lekin slope $\overset{y}{˙} (0)$ , curvature $\overset{y}{¨} (0)$ , wagera extra independent clues hain — har naya derivative chhupe state ke baare mein ek naya fresh equation hai. Derivative precisely woh tool hai jo "samay ke saath dekhna" ko "zyada equations" mein convert karta hai.

Picture: ek hilti hue measurement curve $y (t)$ . $t = 0$ par uski height ek number hai; wahan uski slope ek doosra, independent number hai; uska bend teesra hai. Har ek chhupe dot mein ek naya probe hai.

7. Matrix exponential $e^{A t}$

Yeh tool kyun? Humein ek clean object chahiye jo "starting dot $x (0)$ " ko "baad ke kisi bhi samay ka dot" mein badle. $e^{A t}$ exactly woh propagator hai — $x (0)$ daalo, $x (t)$ nikalti hai.

Picture: starting dot pakdo aur $A$ ke flow arrows (Section 4) ko use uski path par sweep karne do; $e^{A t}$ woh recipe hai ki time $t$ par woh kahan land karta hai.

Isliye measured stream hai $y (t) = C e^{A t} x (0)$ — chhupa start, time se propagate hoke, phir window se project hokar.

8. $y (t)$ se equations nikalna — derivative ladder

Ab hum derivative tool ko use karte hain measurement stream ko unknown $x (0)$ ke baare mein equations mein badalne ke liye. $y (t) = C e^{A t} x (0)$ se shuru karke differentiate karo, ek hi fact use karke ki $\frac{d}{d t} e^{A t} = A e^{A t}$ .

WHAT humne abhi kiya: jaane hue signal $y$ ko baar-baar differentiate kiya aur $t = 0$ par uski value padhi. WHY: $t = 0$ par propagator $e^{A \cdot 0} = I$ disappear ho jaata hai, clean pattern $C A^{k} x (0)$ chhod kar. YEH KAISA DIKHTA HAI: har derivative hilti hue curve ka ek naya probe hai — height, slope, bend — aur har probe chhupe dot ke baare mein information ki ek aur row hai.

Kab rukein? Cayley–Hamilton Theorem ke anusaar, $A^{n}$ simply $I, A, \dots, A^{n - 1}$ ka ek mix hai, isliye $C A^{n}$ woh information repeat karta hai jo humारे paas already hai. Isliye sirf pehle $n$ rows ( $k = 0, \dots, n - 1$ ) mein kuch naya hai.

9. Ladder stack karna — observability matrix $O$

Ab hum un $n$ independent equations ko ek single matrix statement mein bundle karte hain.

WHAT humne abhi kiya: poori derivative ladder ko ek equation " $O$ times unknown equals measured data" ki tarah likha. WHY: kya hum $x (0)$ ke liye solve kar sakte hain yeh ab sirf $O$ ki shape par depend karta hai — ek sawaal jo hum agle section mein rank se answer karte hain. YEH KAISA DIKHTA HAI: windows ka ek tall stack, har ek flow ke ek aur step ke baad state ko dekh raha hai.

10. Rank aur null space

Ye do ideas poore topic ka punchline hain, isliye hum inhe dhyaan se banate hain.

Picture: $M$ ko ek projection socho jo shadows daalta hai. Agar koi direction light ke flat against padi hai, toh woh koi shadow nahi dalti — woh direction null space mein hai. Agar kuch bhi flat nahi hota, toh null space sirf ${0}$ hai aur matrix har direction ko alag rakhta hai (full rank).

11. Transpose $A^{⊤}$

Topic ko kyun chahiye: parent ek duality state karta hai — $(A, C)$ observable $⟺ (A^{⊤}, C^{⊤})$ controllable. Transpose woh mirror hai jo "reading-out" problem ko "driving-in" problem mein badle. Mirror image ke liye dekho Controllability — when we can steer the state.

x \in R^{n}

ka plain words mein kya matlab hai?

x

n

real numbers ki ek list hai —

n

chhupe dials ek column vector mein packed.

Ek matrix ek vector ke saath kya karta hai?

Woh use transform karta hai — har output slot saare input slots ka weighted mix hota hai (row by row jaao).

Dynamics matrix ki entry

A_{ij}

aapko kya batati hai?

Hidden dial

j

, dial

i

ki change ko kitni strongly drive karta hai — dials ke beech "leaking."

u

aur

B

kya hain?

u

known command vector hai jo hum apply karte hain;

B

n \times p

matrix hai jo batata hai ki har command har dial ko kaise push karta hai.

Measurement matrix

C

kya represent karta hai?

Window — har sensor actually dials ka kaunsa mix report karta hai;

y = C x

Hum

y

ke derivatives kyun use karte hain?

Har derivative

y^{(k)} (0) = C A^{k} x (0)

deta hai,

x (0)

ke baare mein ek naya independent equation.

e^{A t}

ek sentence mein kya hai?

Woh propagator jo starting state

x (0)

A

ke flow ke saath

x (t)

tak aage slide karta hai.

Observability matrix ko

C

aur

A

ke terms mein likho.

O = [C; C A; C A^{2}; \dots; C A^{n - 1}]

Null space

ker (M)

kya hai?

Woh nonzero directions

v

jinke liye

M v = 0

hai — woh inputs jo matrix zero kar deta hai (koi shadow nahi dalte).

Rank, null space aur observability ke beech link batao.

Observable

⟺ ker (O) = {0} ⟺ rank (O) = n

Transpose

A^{⊤}

humein observability ke baare mein kya kehne deta hai?

(A, C)

observable

⟺ (A^{⊤}, C^{⊤})

controllable — observability/controllability duality.

Ek chhupa dial phir bhi observable kyun ho sakta hai?

Agar

A_{ij} \neq = 0

hai toh woh ek measured dial mein leak ho jaata hai, samay ke saath khud ko betray karta hua.

Ab Observability — when KF can estimate state par jao jahan har symbol earn kiya hua hai. Related next steps: State-space representation of LTI systems, Cayley–Hamilton Theorem, Observability Gramian & degree of observability.

Foundations — Observability — when KF can estimate state