LQG — LQR + Kalman filter, separation principle: Visual walkthrough

3.5.36 · D2 · Physics › Guidance, Navigation & Control (GNC) › LQG — LQR + Kalman filter, separation principle

Yeh parent LQG note ka pictures-first companion hai. Hum assume karte hain ki aapne State-space Representation aur LQR — Linear Quadratic Regulator ke ideas sirf passing mein dekhe hain; jo chahiye woh hum yahan rebuild karenge.

Step 1 — "State" aur "control" actually dikhte kaise hain

Isko term by term padhte hain, bilkul wahin jahan har symbol baitha hai:

$\overset{x}{˙}$ — dot ki velocity (har second mein $x$ kitna change hota hai). Overdot matlab "rate of change".
$A x$ — drift: dot ki natural tendency. Agar $A > 0$ toh dot apne aap zero se door bhaagta hai (unstable — ise madad chahiye).
$B u$ — hamara influence: knob $u$ dot ko kitni strongly move karta hai. $B$ knob ka "leverage" hai.

WHY yahan se start karein? Neeche sab kuch is dot ko control karne aur dekhne ke baare mein hai. Agar aap dot, arrows, aur yeh dekh sako ki hum use kahan le jaana chahte hain (origin), toh aap poori derivation follow kar sakte ho.

PICTURE: dot position $x$ par, ek red drift arrow use baahir push kar raha hai, ek lavender control arrow jise hum aim kar sakte hain.

Figure — LQG — LQR + Kalman filter, separation principle

Step 2 — Hum "badness" ko bowl se kyun measure karte hain

Term by term:

$Q x^{2}$ — squared position error. Squaring se yeh hamesha positive rehta hai aur badi errors ko disproportionately punish karta hai. $Q$ yeh hai ki hum accuracy ki kitni parwah karte hain.
$R u^{2}$ — squared effort. $R$ yeh hai ki hum fuel/energy bachane ki kitni parwah karte hain.

WHY squares, aur $∣ x ∣$ nahi? Do reasons, dono visual. (1) Square ek smooth bowl hai — uska ek hi lowest point hai aur koi sharp corners nahi, isliye calculus slope ko zero set karke bottom dhundh sakta hai. (2) Quadratic bowl ka bottom ek straight-line feedback rule se reach hota hai — optimal knob simply $x$ ke proportional nikalta hai. Koi aur penalty shape woh clean linear answer tod deti hai. Tool ka yeh choice (quadratic) apne diye hue answer se earn hoti hai.

PICTURE: do parabolic bowls — coral mein position bowl, mint mein effort bowl — aur unka sum, ek wide bowl jiske bottom par hum baithna chahte hain.

Step 3 — Cost-to-go: future ki shape ka andaza

WHY guess karein ki yeh bhi bowl hai? Running penalty bowl hai ( $Q x^{2}$ ), physics linear hai, toh yeh bahut reasonable guess hai ki accumulated penalty bhi bowl hogi: $V (x) = P x^{2}, P > 0.$

$P$ — ek single positive number jo future-cost bowl ki steepness set karta hai. Zyada $P$ = off-target start karne ki zyada cost.

"Ek form guess karo, phir verify karo ki woh equation satisfy karta hai" — yahi Bellman method hai. Is bowl ka slope hai $\frac{d V}{d x} = 2 P x$ — ek line mein hume woh slope chahiye hoga.

PICTURE: cost-to-go bowl $V = P x^{2}$ , aur current dot par tangent arrow ke roop mein local slope $2 P x$ drawn hai.

Step 4 — Future ko balance karna: Bellman/HJB condition

Naye piece ko padho: $2 P x$ future-cost bowl ka slope hai (Step 3), aur $(A x + B u)$ dot ki velocity hai (Step 1). Unka product yeh hai ki dot move karte waqt aapki remaining future cost kitni tezi se gir rahi hai. Balance = dono effects optimum par cancel ho jaate hain.

Knob pe minimize karo — sab kuch $u$ mein bowl ki tarah treat karo aur slope zero set karo: $\frac{d}{d u} [R u^{2} + 2 P x B u] = 2 R u + 2 P B x = 0 ⟹ u^{⋆} = - \frac{P B}{R} x = - K x$

$K = \frac{P B}{R}$ (scalar; general mein $R^{- 1} B^{⊤} P$ ) — feedback gain, yahan se janam leta hai.
Minus sign hi poora point hai: agar dot right drift kare, left push karo. Yeh negative feedback hai.

PICTURE: $u$ mein effort-versus-future bowl, uska minimum marked, aur resulting rule "u equals minus K times x" ek slanted feedback line ke roop mein dikhaya gaya hai.

Step 5 — Loop band karna, aur Riccati equation

Term by term (scalar CARE):

$2 P A$ — future cost mein drift ka contribution (general mein $A^{⊤} P + P A$ ).
$- P^{2} B^{2} / R$ — optimal push kitna buyback karta hai (general mein $- P B R^{- 1} B^{⊤} P$ ). Subtract isliye hai kyunki control cost reduce karta hai.
$+ Q$ — unavoidable running penalty.

$P > 0$ solve karo, $K = P B / R$ read karo, aur controlled dot ab yeh obey karta hai: $\overset{x}{˙} = (A - B K) x .$ Agar $A - B K < 0$ toh dot zero par decay karta hai — stable. Yahi poora LQR result hai.

PICTURE: unstable open-loop trajectory (bhaagta hua) vs closed-loop trajectory ( $A - B K$ , decay karta hua), same starting dot.

Step 6 — Lekin hum dot dekh nahi sakte: predict + correct

$\overset{x}{^}$ — estimate; ek doosra dot jo real wale ke peechhe bhaagta hai.
$y - C \overset{x}{^}$ — innovation, woh surprise: measured minus expected. Zero surprise = koi correction nahi chahiye.
$L$ — Kalman gain: surprise ki taraf kitna jump karna hai. Sensors par zyada trust ⇒ zyada $L$ .

PICTURE: true dot (coral) fog band ke peechhe chhupa hua; estimate dot (lavender) aage predict kar raha hai, phir ek correction arrow use noisy measurement tick ki taraf kheench raha hai.

Step 7 — Error dot aur sundar duality

Padho: gap $e$ apne aap sickta hai agar $A - L C < 0$ , lekin process noise $w$ aur filtered sensor noise $Lv$ se constantly kick milti hai.

Ab $A - L C$ ki shape dekho. Controller ke $A - B K$ se compare karo:

Regulator	Estimator
$A - B K$	$A - L C$
$A \to A^{⊤}$	$B \to C^{⊤}$ , $K \to L^{⊤}$

Yeh same algebra hai substitution $(A, B, Q, R) \to (A^{⊤}, C^{⊤}, W, V)$ ke saath. Toh $L$ ek dual Riccati Equation se aata hai, aur poora Kalman Filter design, LQR design hai jo mirror mein dekh raha hai. (Yeh observability side hai jo controllability side ko mirror kar rahi hai.)

PICTURE: error dot $e$ decay karta hua $A - L C$ ke under, side-by-side mirrored state dot ke saath jo $A - B K$ ke under decay kar raha hai.

Step 8 — Dono ko jodna: block-triangular matrix

Chaar blocks padho:

Top-left $A - B K$ — controller dynamics (Step 5).
Bottom-right $A - L C$ — estimator dynamics (Step 7).
Top-right $B K$ — leak: estimation error true state ko distort karta hai. (Noise abhi bhi cost karta hai — separation design ke baare mein hai, zero cost ke baare mein nahi.)
Bottom-left $0$ — hero. Error pure physics se evolve karta hai; controller gain $K$ use corrupt nahi kar sakta.

WHY zero important hai: neeche diagonal ke zero wali block matrix upper-triangular hoti hai, aur upper-triangular block matrix ke eigenvalues sirf diagonal blocks ke eigenvalues hote hain: $eig [A - B K 0 B K A - L C] = eig (A - B K) \cup eig (A - L C)$ Controller poles aur estimator poles kabhi mix nahi karte. Har ek akela design karo; union simply kaam karta hai. Yahi Separation Principle hai.

PICTURE: $2 \times 2$ block matrix chaar tiles ke roop mein drawn, bottom-left tile ek glowing zero, arrows dikhate hain "error → state" (leak) lekin koi arrow nahi "state → error".

Step 9 — Edge cases jinpar kabhi mat trip karo

Degenerate limits (inhe pehle wale figures par padho):

$V \to 0$ (perfect sensor): $L \to$ large, estimate truth par snap ho jaata hai, LQG $\to$ LQR.
$V \to \infty$ (useless sensor): $L \to 0$ , filter $y$ ko ignore karta hai aur akele model par trust karta hai.
$W \to 0$ (perfect model): filter bhi model par lean karta hai; error fast decay karta hai.
$Q \to 0$ (accuracy free): $K \to 0$ , kuch mat karo — correct karne ko kuch nahi.
$R \to \infty$ (fuel priceless): $K \to 0$ phir — kabhi push mat karo.

PICTURE: ek 2-D map, horizontal axis = trust-in-sensor (small $V$ ), vertical = trust-in-model (small $W$ ), chaar corner behaviours labelled ke saath.

Ek-picture summary

Do independent bowls (ek control ke liye, ek estimation ke liye) do gains produce karte hain ( $K$ , $L$ ); block-triangular matrix apne corner zero ke saath unhe bina interference ke fuse karta hai.

Recall Feynman retelling — ek story ki tarah bolo

Ek line par ek dot hai jo apne aap door bhaagta hai, aur mere paas ek knob hai use wapas dhakeline ke liye. Main "badness" define karta hoon — far-off-squared plus push-squared — ek bowl — kyunki bowls ka ek bottom hota hai aur calculus usse dhundhta hai. Bottom tak rolling mujhe batati hai: kitna door ho usi proportion mein push karo, strength $K$ se. Yahi LQR half hai.

Lekin main dot dekh nahi sakta; main uski ek foggy, noisy shadow dekhta hoon. Toh main ek doosra dot chalata hoon, mera guess, jo physics se predict karta hai aur phir fog jo dikhata hai uski taraf jump karta hai, jump-size $L$ ke saath. $L$ choose karna ek same bowl problem hai mirror mein dekha hua — yahi Kalman half hai.

Ab trick: main apna guess knob mein feed karta hoon. Jab main real dot aur guessing-error ko side by side stack karta hoon, toh equation mein bottom-left corner mein ek zero hota hai — kyunki mera knob dot ko shove kar sakta hai lekin woh kabhi error ke heal hone ke tarike ko mess up nahi kar sakta. Iss zero wali matrix ke poles sirf controller ke plus estimator ke hote hain, kabhi tangle nahi hote. Toh dono halves akele design karo aur glue karo — optimal. Sirf ek price: error abhi bhi dot mein leak hoti hai (top-right block), toh noise abhi bhi cost karta hai, aur koi free robustness nahi mili — isliye LQG/LTR exist karta hai.

Recall Quick self-test

Optimal control state mein linear kyun hai? ::: Kyunki cost aur cost-to-go quadratic bowls hain, aur $u$ mein quadratic ka bottom $x$ ka linear function hai. Stacked matrix ki kaunsi single feature separation prove karti hai? ::: Bottom-left block mein zero (upper-triangular ⇒ eigenvalues diagonal blocks ka union hain). Kya separation ka matlab hai noise performance ko hurt nahi karta? ::: Nahi — $B K$ leak error ko state mein inject karta hai, toh LQG cost ideal LQR cost se zyada hoti hai. Jab sensor noise $V \to 0$ , LQG ka kya hota hai? ::: Kalman gain $L$ badhta hai, estimate truth par snap ho jaati hai, aur LQG reduce ho jaata hai LQR mein.

See also: LQG — LQR + Kalman filter, separation principle · LQR — Linear Quadratic Regulator · Kalman Filter · Riccati Equation · Dynamic Programming and Bellman Equation · Controllability and Observability · LQG-LTR Loop Transfer Recovery · H-infinity Control