Observability matrix — rank test — Hinglish notes

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

Observability KYA hai?

Sirf $A$ aur $C$ kyun? Kyunki inputs $u$ aur $D$ $y$ mein ek jaana-maana contribution dete hain; hum use subtract kar sakte hain. Jo bachta hai woh free response hai $y_{free} = C e^{A t} x (0)$ . Observability purely iss baat ke baare mein hai ki kya $C$ aur $A$ hume $x (0)$ ki har direction "dikhne" dete hain.

Test derive kaise karein (scratch se)

$u = 0$ set karo (legitimate hai, kyunki iska effect jaana hua hai). Output hai: $y (t) = C e^{A t} x (0) .$

Differentiate kyun karein? Ek akela instant $y (0) = C x (0)$ sirf $x (0)$ ka projection $C$ par batata hai. Aur directions pakdne ke liye, $t = 0$ par derivatives lo:

y (0) \overset{y}{˙} (0) \overset{y}{¨} (0) y^{(k)} (0) = C x (0) = C A x (0) = C A^{2} x (0) ⋮ = C A^{k} x (0)

$k = n - 1$ par kyun rokein? Cayley–Hamilton theorem ke anusaar, $A^{n}$ ek linear combination hai $I, A, \dots, A^{n - 1}$ ka. Isliye $C A^{n}, C A^{n + 1}, \dots$ koi nayi rows nahi dete — har higher derivative redundant hai. Derivatives $0$ se $n - 1$ tak stack karo:

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

Hum $x (0)$ ke liye uniquely solve kar sakte hain iff matrix $O$ ka full column rank $n$ ho (taaki iska null space sirf ${0}$ ho).

Figure — Observability matrix — rank test

Worked examples

Common mistakes

Flashcards

LTI system ki observability define karo

(A, C)

observable hai agar

x (0)

u (t), y (t)

se kisi bhi finite interval par uniquely determine kiya ja sake.

Observability matrix batao

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

, size

p n \times n

Rank test batao

Observable

⟺ rank (O) = n

Power

n - 1

par kyun rokein?

Cayley–Hamilton:

A^{n}

lower powers ka combination hai, isliye

C A^{k}

jab

k \geq n

ho toh koi nayi rows add nahi hoti.

Unobservable state kya hota hai?

Ek nonzero

x_{0} \in ker (O)

; yeh

C e^{A t} x_{0} \equiv 0

deta hai, sensors ko dikh hi nahi sakta.

Observability mein

B, D, u

ko kyun ignore kar sakte hain?

Unka

y

mein contribution jaana hua hai aur subtract kiya ja sakta hai; observability sirf free response

C e^{A t} x (0)

se concern rakhti hai.

Duality statement

(A, C)

observable

⟺ (A^{⊤}, C^{⊤})

controllable.

Kya ek single output do identical decoupled modes observe kar sakta hai?

Nahi — ek scalar

C

repeated eigenvalue ko 2 eigenvectors ke saath alag nahi kar sakta;

p \geq 2

chahiye.

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho ek box hai jisme hidden gears hain (states). Aap sirf bahar ek dial dekh sakte ho (output). Observability poochti hai: kya woh dial dekh ke — uski value aur woh kitni tezi se badal rahi hai — aap pata laga sakte ho ki shuruaat mein har gear kaise ghoom raha tha? Agar koi gear dial ko kabhi hilata hi nahi, toh woh "hidden" hai, aur box observable nahi hai. Rank test sirf ek tarika hai yeh count karne ka ki dial actually kitne gears reveal kar sakta hai; agar woh count total gears ki ginti ke barabar ho, toh aap sab kuch dekh sakte ho.

Connections

Controllability matrix — rank test (dual concept)
Cayley–Hamilton theorem (kyun $A^{n - 1}$ par rokein)
Kalman Filter (state estimation converge karne ke liye observability zaroori hai)
Kalman decomposition (system ko observable/unobservable subsystems mein split karta hai)
State-space representation
Rank and null space of a matrix (linear algebra core)
PBH test (rank test ka eigenvalue-based alternative)

Observability matrix — rank test