Foundations — Controllability matrix — rank test

3.5.32 · D1 · Physics › Guidance, Navigation & Control (GNC) › Controllability matrix — rank test

Parent note padhne se pehle, tumhe har us symbol ka ek picture chahiye jo woh throw karta hai: $x$, $u$, $A$, $B$, $\dot{x}$, ${\mathbb{R}}^n$, "span", "rank", $e^{At}$, aur $A^{k}B$. Hum inhe zero se build karenge, ek aisi order mein jahan har nayi idea sirf pehle waali ideas par lean karti hai.

1. State vector $x$ — "machine abhi kya hai"

Ek puck ko ice par slide karte socho. Uska future jaanne ke liye tumhe do numbers chahiye: kahan hai ($p$) aur kitni tez move kar raha hai ($v$). Toh uski state yeh pair hai: $x=\left[\begin{array}{c}p\\ v\end{array}\right].$

Figure s01 dekho. Left panel physical puck hai. Right panel SAME information ko ek plane mein single point ke roop mein plot karta hai, jiska horizontal axis position hai aur vertical axis velocity. Woh plane state space kehlata hai, aur woh point HI state hai.

2. ${\mathbb{R}}^n$ — woh space jahan state rehti hai

Puck ke liye $n=2$, toh $x\in {\mathbb{R}}^2$ — ek flat plane (s01 ka right panel). Ek spinning spacecraft ko $n=6$ ya zyada chahiye ho sakta hai; tum draw nahi kar sakte, lekin rules bilkul same hain.

Topic ko yeh kyun chahiye :::: "Controllable" define hota hai as ${\mathbb{R}}^n$ ke kisi bhi point tak reach karna. Iss space ka naam liye bina, "any target" ka koi meaning nahi.

3. Input vector $u$ — "woh dials jo tum ghuma sakte ho"

Puck ke liye, ek dial: ek force jo tum apply karte ho. Toh $m=1$ aur $u=[\text{force}]$. Char rotors waale quadcopter ka $m=4$ hoga.

4. Derivative $\dot{x}$ — "state abhi kaise change ho rahi hai"

Parent likhta hai $\dot{x}=Ax+Bu$. Woh dot bahut kaam kar raha hai.

Figure s02 dekho. State space ke har point par hum ek chota arrow draw karte hain: point $x$ par arrow vector $\dot{x}$ hai — woh direction aur speed jis par state drift karegi agar tum koi dial nahi chuote. Arrows follow karne se machine ki natural motion trace hoti hai.

5. Matrices $A$ aur $B$ — machine ki wiring

Ab hum $\dot{x}=Ax+Bu$ ko term by term padh sakte hain.

Toh poora sentence $\dot{x}=Ax+Bu$ padhta hai:

"State abhi kaise change ho rahi hai = (uski khud ki current value ise kahan drag karti hai) + (tumhare dials ise kahan push kar rahe hain)."

6. Matrix–vector multiplication — $Ax$ actually kaise kaam karta hai

Jab tak tum $Ax$ haath se compute nahi kar sakte, tab tak usp ek trust nahi kar sakte.

7. Span — "kuch arrows mixing karke har jagah tak pahunch sakte hain"

Parent note kehta hai reachable states kuch columns ka span hain. Yeh word yahan hai.

Figure s03 dekho.

• Left: do vectors genuinely alag directions mein point karte hain. Inhe mix karne se plane ke har point tak pahuncha ja sakta hai — inका span poora ${\mathbb{R}}^2$ hai.
• Right: do vectors ek hi line ke saath lete hain. Chahe tum kaise bhi scale aur add karo, tum kabhi woh line nahi chhodte — inका span sirf ek 1-D sliver hai, aur plane ka baaki hissa unreachable hai.

8. Rank — "columns ke ek set mein actually kitni independent directions hain"

Ek span ke size ko ek number chahiye. Woh number rank hai.

Iss word ke saath s03 padhna:

• Left matrix (columns plane spread karte hain): $\text{rank}=2$.
• Right matrix (columns ek line share karte hain): $\text{rank}=1$.

9. Powers $A^{k}B$ — "ek push wiring se kaise ripple karta hai"

Parent $B,AB,A^{2}B,\dots$ kyun stack karta hai? Har $A$ jo tum apply karte ho machine ki khud ki wiring ko push aage pass karne ka ek step hai.

Figure s04 dekho puck ke liye ($A,B$ §5 se):

• Red arrow $B=\left[\begin{array}{c}0\\ 1\end{array}\right]$: dial sirf velocity push karta hai.
• Blue arrow $AB=\left[\begin{array}{c}1\\ 0\end{array}\right]$: ek ripple ke baad, woh push position mein leak ho gaya hai.
• Saath mein red + blue do alag directions mein point karte hain → poora plane span karte hain → rank 2 → controllable. Velocity par dial secretly position bhi control karta hai.

10. $e^{At}$ — woh exponential jo poori motion aage play karta hai

Parent ki derivation $e^{At}$ use karti hai. Tumhe sirf idea chahiye; Matrix exponential $e^{At}$ note mein machinery hai.

Prerequisite map

Is page par har foundation ek node hai; arrows follow karo aur tum exactly parent note ke rank test par pahunch jaoge.

Equipment checklist

Apne aap ko test karo — sirf tab reveal karo jab tum aloud answer de chuke ho.