Foundations — LQG — LQR + Kalman filter, separation principle

3.5.36 · D1 · Physics › Guidance, Navigation & Control (GNC) › LQG — LQR + Kalman filter, separation principle

Yeh page yeh maanta hai ki tumne parent note ka koi bhi notation pehle nahi dekha. Hum har letter, arrow, aur squiggle ko ground up se build karte hain, ek aisi order mein jahan ladder ki har rung uske neeche wali par tikhi ho. Agar tumne parent LQG topic padha hai aur koi aisa symbol mila jise tum parse nahi kar sake, toh yeh woh page hai jo use fix karta hai.

0 — "Ek system" kya hota hai aur arrows kyun hote hain?

Kisi bhi letter se pehle, us object ki picture banao jise hum control karte hain.

1 — State $x$ : "woh sab jo tum bhavishya predict karne ke liye jaanna chahoge"

Picture: figure s01 mein, hum do-number state $x=(\text{position},\text{velocity})$ ko ek state plane mein single dot ke roop mein draw karte hain — ek axis position hai, doosri velocity. Dot ki location hi state hai; axes kamra nahi hain, woh do state coordinates hain.

Topic ko yeh kyun chahiye: LQG mein sab kuch — steering, estimating, cost — state ke relative measure hota hai. Hamara poora goal hoga "$x$ ko zero par drive karo" (dot ko origin par target tak laao).

2 — Vectors aur transpose $x^{\top }$

$x^{\top }$ ki zaroorat kyun kabhi padti hai: ek vector ke size se ek single number banane ke liye hum $x^{\top }x=x_1^2+x_2^2+\dots$ compute karte hain — squares ka sum, yaani dot origin se kitna door hai, squared. Ek row times column, poore vector ko ek honest number mein collapse kar deta hai. Woh single number hi woh hai jise hum chhota karne ki koshish karenge.

3 — Matrices $A,B,C$ : "machines jo ek vector ko reshape karti hain"

Picture: ek matrix ek arrow-bender hai. Use arrow $x$ do; yeh alag arrow $Ax$ wapas deta hai.

Teen named grids mein se har ek ko pehle ek aur symbol chahiye — control $u$ (jisse tum push karte ho) aur noises $w$, $v$ — isliye hum unhe abhi introduce karte hain, koi bhi equation likhne se pehle jo unhe use kare.

Ab teen named grids, har ek ek physical question ka jawab deta hua:

Topic ko yeh kyun chahiye: yeh kamre ke rules hain. LQG ki har equation yahi teen grids plus noise hain. Dekho State-space Representation.

4 — $\dot{x}$ : kisi symbol ke upar dot ka matlab "rate of change"

Yeh notation kyun, na ki "$\Delta x$"? Kyunki machine continuously badlti hai, jumps mein nahi. Overdot derivative hai — instantaneous rate — jo smooth motion describe karne ka ek maatra honest tarika hai. Yeh jawab deta hai "dot abhi is waqt kahan ja raha hai?"

§3 aur §4 ko jodne par ($u$, $w$, $v$ pehle se named hain) state aur measurement equations milti hain: $\dot{x}=Ax+Bu+w,y=Cx+v$ Pehle ko plain words mein padho: dot ki velocity $(\dot{x})$ hai: uska natural drift $(Ax)$ plus mere push ka effect $(Bu)$ plus ek random shove $(w)$. Doosra kehta hai: sensor kya report karta hai $(y)$ hai truth ka clean projection $(Cx)$ plus sensor error $(v)$.

5 — $y$, $\hat{x}$, $e$ : dekho, andazo lagao, aur hum kitne galat hain

$u$ §3 mein define hua tha; measurement $y$ abhi upar aaya. Yahan hum yeh naam dete hain jo hum $y$ se build karte hain.

6 — Noise ki size: $\mathcal{N}(0,W)$ aur $\mathcal{N}(0,V)$

Humne $w$ aur $v$ ko §3 mein naam diya; ab hum precisely batate hain ki woh kitne bade hain.

"0" kyun? Agar noise ka average zero ke alawa kuch hota toh woh ek jaani-pehchani drift hoti, aur hum use $A$ ya $B$ mein fold kar lete. Sachchi noise average out ho jaati hai — isliye 0 par centered.

7 — $Q$, $R$ aur "penalise = isko square karo"

Jo cost hum minimise karte hain woh hai $J={\int }_0^{\infty }(\underset{\text{off-target badness}}{\underbrace{x^{\top }Qx}}+\underset{\text{effort badness}}{\underbrace{u^{\top }Ru}})dt.$

Do ordering constraints jo tumhein milenge:

• $Q⪰0$ ("positive semidefinite"): badness kabhi negative nahi — tum kuch directions ke baare mein neutral ho sakte ho.
• $R\succ 0$ ("positive definite"): har push kuch toh cost karta hi hai — free control nahi milti.

8 — $P$, $\Sigma$, aur $K$, $L$ : woh answers jo maths wapas deta hai

Yeh chaar outputs hain — tum inhe choose nahi karte, tum inke liye solve karte ho.

Dono Riccati equations se nikale jaate hain (dekho Riccati Equation) — ek controller ke liye ($P$), ek dual copy filter ke liye ($\Sigma$). Woh duality hi separation principle ke peeche ka engine hai.

9 — Do hidden prerequisites jinhe parent lean karta hai

Prerequisite map

Equipment checklist

Right side cover karo aur zor se jawab do; check karne ke liye reveal karo.