Observability — when KF can estimate state

3.5.23 · Physics › Guidance, Navigation & Control (GNC)

Observability KYA hai?

KF ke liye yeh KYUN matter karta hai: Kalman Filter measurements $y$ se $\overset{x}{^}$ estimate karta hai. Agar system unobservable hai, toh state-space ke kuch directions mein koi measurement pipeline nahi hota — un directions mein KF ki error covariance $P$ kabhi shrink nahi hoti (ya toh badi rehti hai ya badhti hai). Tumhe ek "confident-looking" filter milta hai jo andar se andha hota hai.

Hum ise TEST kaise karte hain? — Observability Matrix Derive Karna

Hum jaanna chahte hain: kya $y (t)$ se $x (0)$ pin down ho sakta hai? $u = 0$ set karo (inputs known hain, isliye unka effect subtract karo; sirf unforced response observability test karta hai).

Step 1 — output likho. $\overset{x}{˙} = A x$ ka solution hai $x (t) = e^{A t} x (0)$ , isliye $y (t) = C e^{A t} x (0) .$ Yeh step kyun? $e^{A t}$ initial state ko propagate karta hai; $C$ use sensors pe project karta hai.

Step 2 — $t = 0$ par derivatives se information nikalte hain. $y$ ko baar baar differentiate karo aur $t = 0$ par evaluate karo: $y (0) = C x (0), \overset{y}{˙} (0) = C A x (0), \overset{y}{¨} (0) = C A^{2} x (0), \dots$ Yeh step kyun? Known signal $y$ ki har derivative humein unknown $x (0)$ ke baare mein ek aur linear equation deti hai.

Step 3 — $n - 1$ par ruko (Cayley–Hamilton). Cayley–Hamilton theorem ke according, $A^{n}$ , $I, A, \dots, A^{n - 1}$ ka ek linear combination hai. Isliye $C A^{n}$ se koi nayi information nahi milti — higher derivatives redundant hain. Yeh step kyun? Yeh ek finite test guarantee karta hai: sirf $n$ blocks ki zaroorat hai.

Step 4 — equations stack karo. $n$ blocks ka system yeh hai:

O C C A C A^{2} ⋮ C A^{n - 1} x (0) = y (0) \overset{y}{˙} (0) ⋮ y^{(n - 1)} (0) .

Unobservable subspace $ker (O)$ hai: iska koi bhi $x (0)$ $y (t) \equiv 0$ produce karta hai aur invisible rehta hai.

Figure — Observability — when KF can estimate state

Worked Examples

Common Mistakes

Kalman Filter se Connection

Flashcards

Kalman Filter kab ek given state direction estimate kar sakta hai?

Sirf tab jab woh direction observable ho, yaani woh

ker (O)

ke bahar ho / output mein contribute kare.

n

-state system ke liye matrices

A, C

ke saath observability matrix likho.

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

Observability rank test batao.

System observable

⟺ rank (O) = n

(full column rank).

O

mein

A^{n - 1}

par kyun ruk jaate hain?

Cayley–Hamilton:

A^{n}

lower powers ka linear combo hai, isliye higher derivatives koi nayi information nahi dete.

Physically unobservable subspace kya hai?

ker (O)

— initial states jo

y (t) \equiv 0

produce karte hain; kisi bhi estimator ke liye invisible.

Observability vs controllability — duality statement.

(A, C)

observable

⟺ (A^{⊤}, C^{⊤})

controllable.

Unobservable, unstable direction mein KF covariance ka kya hota hai?

Yeh kabhi shrink nahi hoti (measurement update blind hai) aur process noise

Q

se badhti hai — estimate diverge ho jaata hai.

Observability se weaker condition jo phir bhi stable KF deti hai?

Detectability — saare unobservable modes asymptotically stable hain.

Kya sirf position sensor ke saath position–velocity observable hai?

Haan, kyunki velocity position ko drive karti hai (

A_{12} \neq = 0

), isliye yeh output mein leak ho jaati hai.

Observability ka degree measure karne ka tool, sirf yes/no nahi?

Observability Gramian

W_{o} = \int_{0}^{T} e^{A^{⊤} t} C^{⊤} C e^{A t} d t

; chote eigenvalues = weakly observable.

Recall Feynman: 12-saal ke bacche ko samjhao

Socho ek band kamra hai jisme andar kuch gears ghoom rahe hain, aur tum sirf ek choti khidki se jhank sakte ho. Kuch gears tum seedha dekh sakte ho. Baaki tum nahi dekh sakte — lekin agar koi chupi gear ek visible gear ko dhakelta hai, uski movement us gear mein dikhti hai jo tum dekh sakte ho, toh tum use phir bhi figure out kar sakte ho! "Observable" matlab hai: teri choti khidki se, har chupi gear akhir mein khud ko zahir kar deti hai. Agar koi gear apne sealed box mein ghoom raha hai jiska tumse visible koi connection nahi, woh hamesha raaz rehega — aur koi bhi clever guessing (yahi Kalman Filter hai) use reveal nahi kar sakti. $O$ trick bas yeh hai: sirf yeh mat dekho ki visible gear kahan hai, balki yeh bhi dekho ki woh kitni tezi se chal raha hai, aur uski speed kaise change ho rahi hai — inka har ek ek naya clue hai.

Connections

Kalman Filter — prediction & update
Controllability — when we can steer the state
Cayley–Hamilton Theorem
Observability Gramian & degree of observability
Detectability & Stabilizability
Riccati Equation & steady-state covariance
Gyro bias estimation / star-tracker calibration
State-space representation of LTI systems