Linear Quadratic Regulator (LQR) — Riccati equation, optimal gains: Foundations

3.5.35 · D1 · Physics › Guidance, Navigation & Control (GNC) › Linear Quadratic Regulator (LQR) — Riccati equation, optimal

Parent note ki ek bhi line padhne se pehle, tumhe har woh symbol apna banana hoga jo woh tumhare saamne phenkta hai. Yeh page har ek symbol ko bilkul zero se build karta hai, us order mein jo har ek ko pichle par tikne deta hai.

0. Har cheez ke peeche ki picture: state as a point

LQR mein jo bhi hota hai woh ek akele chalte hue point par hota hai. Woh point machine ki abhi ki situation hai.

Figure — Linear Quadratic Regulator (LQR) — Riccati equation, optimal gains

Seedhe alfaaz mein: $x$ jawaab deta hai "machine abhi kya kar rahi hai?"
Picture: ek flat plane mein ek dot (2 numbers ke liye). Origin hai "perfect — target par, move nahi kar raha". Origin se door hona bura hai.
Topic ko kyun chahiye: LQR ka poora kaam is dot ko origin par laana hai. Agar tum state ko ek movable point ki tarah nahi socho, toh baaki kuch bhi samajh nahi aayega.

Do dimensions hum draw kar sakte hain ( $n = 2$ ); zyada $n$ ke liye hum analogy se sochte hain — phir bhi "ek point jo origin ke paas ya door ho sakta hai".

1. Vectors aur matrices — poore topic ki alphabet

Definition LQR mein har matrix ki dimensions

$x \in R^{n}$ (state, $n$ numbers) aur $u \in R^{m}$ (control, $m$ numbers) ke saath, "shapes must match" rule har matrix ka size force karta hai:

Matrix	Shape	Woh shape kyun
$A$	$n \times n$	$A x$ ko $\overset{x}{˙}$ mein add karne ke liye ek $n$ -vector return karna chahiye; toh $A$ $n$ khata hai, $n$ return karta hai.
$B$	$n \times m$	$B u$ $m$ -vector $u$ khaata hai aur $\overset{x}{˙}$ mein shamil hone ke liye ek $n$ -vector return karna chahiye.
$Q$	$n \times n$	$x^{⊤} Q x$ mein $x^{⊤}$ aur $x$ ke beech baithta hai; ek $n$ -vector khaana chahiye.
$R$	$m \times m$	$u^{⊤} R u$ mein $u^{⊤}$ aur $u$ ke beech baithta hai; ek $m$ -vector khaana chahiye.
$K$	$m \times n$	$K x$ $n$ -vector $x$ khaata hai aur ek $m$ -vector (ek control) return karna chahiye.

Yahan kuch bhi arbitrary nahi hai — har shape woh ek hi hai jo multiplication ko legal banati hai.

2. Quadratic form $x^{⊤} M x$ — "kitna bura hai" measure karne wala ek number

Yeh akela expression parent page par teen baar aata hai ( $x^{⊤} Q x$ , $u^{⊤} R u$ , $x^{⊤} P x$ ). Ise ek baar apna banao aur teeno free hain.

Kyun squared (kyun $∣ x ∣$ nahi)? Square bottom par smooth hota hai (ek rounded bowl), toh calculus cleanly minimum dhundh sakta hai. Absolute value mein ek sharp corner hota hai — optimizer usmein phase jaata hai. Yahi smoothness woh poori wajah hai jis se LQR ka ek clean closed-form answer hai.
Picture: $x^{⊤} M x$ ek bowl hai jo plane ke upar rakha hai. Origin bottom hai (value $0$ ); jitna bahar jaao, utna upar chadhte ho. $M$ decide karta hai bowl kitni steep hai har direction mein.

Definition Positive (semi)definite:

$M ≻ 0$ aur $M ⪰ 0$ $M ⪰ 0$ (padho " $M$ positive semidefinite hai") matlab hai $x^{⊤} M x \geq 0$ har arrow $x$ ke liye — bowl kabhi zero se neeche nahi jaata. $M ≻ 0$ ("positive definite") zyada strict hai: $x^{⊤} M x > 0$ har non-zero $x$ ke liye — ek sachcha bowl bina kisi flat valley floor ke.

Symbol	Bowl shape	Kahan use hota hai
$⪰ 0$	flat floor ho sakta hai	$Q$ (state cost) — kuch directions ko penalize na karna theek hai
$≻ 0$	har jagah strictly curved	$R$ (control cost) — curve hona zaroori hai taaki ek unique sasta control mile, aur $R^{- 1}$ exist kare

Yeh table exactly wajah batati hai ki parent kyun insist karta hai " $R ≻ 0$ strictly": control cost mein flat floor ka matlab hoga "kuch control free hai", minimum unique nahi hoga, aur $R^{- 1}$ (aage define hoga) exist nahi karega.

3. Time $t$ , dot $\overset{x}{˙}$ — rate of change

Derivative kyun aur difference kyun nahi? Physics continuously hoti hai — machine discrete steps mein jump nahi karti, woh flow karti hai. Derivative woh tool hai jo ek instant par flow measure karta hai.
Picture: har point par, $\overset{x}{˙}$ dot ke saath attached ek chhota arrow hai jo dikhata hai woh aage kahan jayega. In arrows ka poora field machine ka "wind map" hai.

4. Control $u$ , gain $K$ , aur feedback law $u = - K x$

Minus sign kyun? Feedback ko error oppose karna chahiye, jaise ek spring ek stretch hue mass ko wapas kheenchta hai. Plus sign tumhe aur door dhakelta — control ke bilkul ulta.
$x$ ke proportional kyun (linear kyun)? Parent ka punchline yahi hai: linear plant + quadratic (bowl) cost force karta hai ki best recipe exactly yahi linear wali ho. Isi $- K x$ shape ko doosre raastoon se paane ke liye Pole Placement aur PID Controllers dekhein.

5. Cost integral $J$ — poore time ka badness add karna

Picture: har instant par do bowls; $J$ woh shaded area hai jo time flow hone par sweep hoti hai. Ek twitchy, wasteful controller ka bada shaded area hoga; LQR controller ka sabse chhota possible area hoga.

6. Value function $P$ aur Riccati equation (preview)

Foundations topic ko kaise feed karti hain

Equipment checklist

Har cue padho, zyorr se jawaab do, phir reveal karo. Agar koi rokta hai, woh section dobara padho.

x \in R^{n}

ka seedhe alfaaz mein kya matlab hai?

x

ek point (arrow) hai jo

n

real numbers se bana hai — machine ki abhi ki poori situation.

Matrix ki geometrical job kya hai?

Woh ek arrow khaati hai aur transformed arrow return karti hai (stretch/rotate/shear);

A x

hai "

x

jab

A

act kare uske baad".

A

ko vector se multiply karne ke liye "shapes must match" rule bolo.

A

ke columns ki sankhya vector ke rows ki sankhya ke barabar honi chahiye; ek

r \times c

matrix ek

c \times 1

vector khaata hai aur

r \times 1

return karta hai.

A

B

Q

R

K

ki shapes kya hain?

A

hai

n \times n

B

hai

n \times m

Q

hai

n \times n

R

hai

m \times m

K

hai

m \times n

A^{⊤}

kya karta hai aur humein iska kyun zaroorat hai?

Rows ko columns mein flip karta hai (

r \times c \to c \times r

); zaroori hai taaki shapes line up hon jab ek row ko column se multiply karo, jaise

x^{⊤} Q x

mein.

x^{⊤} M x

kaun sa akela number produce karta hai, aur uski shape kya hai?

Ek scalar jo origin se weighted squared distance measure karta hai — plane ke upar ek bowl.

Error ko square kyun karte hain

∣ x ∣

ki jagah?

Square bottom par smooth hota hai, toh calculus ek clean unique minimum dhundh leta hai;

∣ x ∣

mein ek sharp corner hai.

M ⪰ 0

aur

M ≻ 0

mein farq?

⪰ 0

ek flat valley floor allow karta hai (

x^{⊤} M x \geq 0

);

≻ 0

har jagah strictly curved hai (

x \neq = 0

ke liye

> 0

R^{- 1}

kya hai aur kab exist karta hai?

"Undo" matrix jo

R^{- 1} R = I

satisfy kare; ek square matrix ke liye exist karta hai jo kisi non-zero arrow ko zero par squash na kare — guaranteed jab

R ≻ 0

ho.

R ≻ 0

strictly kyun hona chahiye (sirf

⪰ 0

nahi)?

Taaki ek unique sasta control exist kare aur

R^{- 1}

exist kare

K = R^{- 1} B^{⊤} P

mein.

t

kya hai aur

d t

ka kya matlab hai?

t

time parameter hai (

t = 0

se seconds);

d t

time ka ek infinitesimally chhota step hai jo integral mein har instant ko weight deta hai.

\overset{x}{˙}

kya represent karta hai?

State point ki instantaneous velocity — time ke saath iski rate of change,

d x / d t

\overset{x}{˙} = A x + B u

ko alfaaz mein padho.

State ki velocity uska apna natural drift

A x

hai plus woh push

B u

jo hum control ke zariye inject karte hain.

u = - K x

mein minus sign kyun?

Feedback ko error oppose karna chahiye aur state ko origin ki taraf wapas kheenchna chahiye, jaise ek spring.

Cost ke liye

\int_{0}^{\infty} (\dots) d t

kya compute karta hai?

Running total badness (error + effort) poore future time mein continuously sum ki gayi.

V (x) = x^{⊤} P x

kya hai?

State

x

se best possible remaining total cost; symmetric

P ⪰ 0

se shaped ek bowl.

Riccati equation kya deliver karta hai?

Matrix

P

, jisse optimal gain

K = R^{- 1} B^{⊤} P

read off hota hai.