Controllability matrix — rank test

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

Setup (WHAT)

Test ko scratch se derive karna (HOW + WHY)

Matrix rank is sab mein kahan se aata hai? Chalte hain ise build karte hain, memorize nahi karte.

$\dot{x}=Ax+Bu$ ka solution $x_0=0$ (rest) se shuru karke yeh hai $x(T)={\int }_0^T{e}^{A(T-\tau )}Bu(\tau )d\tau .$

Yeh step kyun? Yeh standard variation-of-parameters solution hai; matrix exponential $e^{At}$ state ko propagate karta hai, aur integral input history ka effect accumulate karta hai.

Origin se reachable states ka set $e^{A(T-\tau )}Bu(\tau )$ form ke saare vectors ka span hai. Isliye reachable subspace $e^{At}B$ ke columns se span hota hai sabhi $t$ ke liye.

Ab Cayley–Hamilton theorem use karo: har matrix $A$ apni characteristic polynomial satisfy karta hai, isliye $A^n$ (aur usse bade powers) ko $I,A,A^2,\dots ,A^{n-1}$ ke linear combination ke roop mein likha ja sakta hai. Matrix exponential expand karke $e^{At}={\sum }_{k=0}^{\infty }\frac{(At{)}^k}{k!}$ Yeh step kyun? Har power $A^k$ finite set $\{I,A,\dots ,A^{n-1}\}$ par collapse ho jaata hai. Isliye $e^{At}B$ hamesha sirf inke linear combinations produce karta hai $B,AB,A^{2}B,\dots ,A^{n-1}B.$

Isliye reachable subspace exactly inhi blocks ko side by side stack karne ka column span hai. Woh stack HI controllability matrix hai.

Rank $n$ kyun aur zyada kyun nahi? Tum kabhi $n$ se zyada dimensions reach nahi kar sakte (state $n$-dimensional hai). Rank $=n$ matlab columns poora state space span karte hain. Rank $<n$ matlab ek "dead" subspace hai jise koi bhi input touch nahi kar sakta.

Worked examples

Common mistakes (Steel-man them)

Flashcards

Connections

• Observability matrix — rank test (dual: kya hum outputs se state dekh sakte hain? swap $A\to A^T$, $B\to C^T$)
• State-space representation
• Cayley–Hamilton theorem
• Pole placement & state feedback (controllability chahiye saare poles place karne ke liye)
• Kalman decomposition (system ko controllable/uncontrollable parts mein split karta hai)
• Matrix exponential $e^{At}$
• Stabilizability (kamzor cousin: unstable modes controllable hone chahiye)

Concept Map