Foundations — State-space representation — x' = Ax + Bu, y = Cx + Du

3.5.29 · D1 · Physics › Guidance, Navigation & Control (GNC) › State-space representation — x' = Ax + Bu, y = Cx + Du

Is page par kuch bhi assume nahi kiya gaya. Parent note State-space representation ko touch karne se pehle, hum har ek symbol banate hain jo woh use karta hai — ek arrow-over-a-letter ka matlab kya hota hai, yahan se lekar ek matrix ko vector se multiply karne par kya dikhta hai, tak. Upar se neeche padho; har block agale ko earn karta hai.

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1. Ek variable jo time ke saath badlta hai: $q(t)$

Sochो ek dot ek line par slide kar raha hai. Clock ki har tick par dot kahin baitha hota hai, aur $q(t)$ uska position label hai. Agar tum un saare snapshots ka flip-book banao toh tumhein ek curve milega: time neeche ki taraf, value upar ki taraf.

Is topic ko yeh kyun chahiye. GNC mein har cheez — rocket ki tilt, aircraft ki speed — ek number hai jo time ke saath badlti hai. $q(t)$ hamaari sabse pehli brick hai.

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2. Badlne ki rate: derivative $\dot{q}$

Figure s02 dekho. Orange line blue curve ko ek point par touch karti hai bina cross kiye — woh touching line tangent hai. Uski steepness (ek step right jaane par kitna upar uthti hai) us point par $\dot{q}$ hai. Steep-up ⇒ bada positive $\dot{q}$; flat ⇒ $\dot{q}=0$; neeche slope ⇒ negative $\dot{q}$.

Second derivative $\ddot{q}$ ("$q$-double-dot") rate of change of the rate of change hai — position ke liye yeh acceleration hai (speed khud kitni tezi se badal rahi hai).

Recall Dots padho

$\dot{q}$ ka ek phrase mein kya matlab hai? ::: $q$ abhi kitni tezi se badal raha hai (uski curve ka slope). $\ddot{q}$ ka kya matlab hai? ::: $\dot{q}$ kitni tezi se badal raha hai — acceleration agar $q$ position hai.

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3. Ek differential equation (ODE) ek rule hai, jawab nahi

Ise local behaviour ka ek law sochо: "kisi bhi moment par, mass, spring aur damping is tarah milte hain ki yeh combination pushing force ke barabar hota hai." Ise ek starting position aur speed do, aur rule batata hai ki time ke agale tiny slice mein kya hoga — phir agale mein — hamesha. Ise solve karne ka matlab hai woh curve $q(t)$ dhundhna jo rule ko har jagah maane.

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4. Ek saath kaafi numbers ka naam: vector $x$

Mass–spring ke liye parent $x_1=q$ (position) aur $x_2=\dot{q}$ (velocity) choose karta hai. Toh state ek single dot hai jo ek 2-D plane mein rehti hai jiske axes position aur velocity hain. Jaise time chalta hai, woh dot ek path trace karta hai — is plane ko phase plane kehte hain.

Recall Vector basics

${\mathbb{R}}^n$ words mein kya hai? ::: Sabhi $n$ real numbers ki lists ka set — ek $n$-dimensional space. Mass–spring ke liye do state entries kya hain? ::: Position $x_1=q$ aur velocity $x_2=\dot{q}$.

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5. Numbers ki ek grid: matrix $A$, aur $Ax$ kya karta hai

Rule dhheere padho. Pehli row $[ab]$ lo, entry-by-entry vector mein multiply karo aur add karo: $a{x}_1+b{x}_2$ — yeh pehli output entry ban jaati hai. Doosri entry ke liye doosri row ke saath bhi yehi karo. Ek matrix–vector product bas itna hi hai: weighted sums ki ek recipe, har row ke liye ek weighted sum.

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6. Do special matrices: identity $I$ aur inverse $A^{-1}$

Is topic ko yeh kyun chahiye. Transfer-function link $G(s)=C(sI-A{)}^{-1}B+D$ dono use karta hai: $sI$ "$s$ times the do-nothing matrix" hai, aur $(sI-A{)}^{-1}$ $(sI-A)$ ka undo hai. Yeh exactly wahan blow up karta hai jahan $(sI-A)$ singular ho jaata hai — aur woh spots eigenvalues hain, yaani poles.

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7. Directions jinhein ek matrix rotate nahi karta: eigenvalues $\lambda$

Zyaadatar vectors jab ek matrix act karta hai toh ek naye direction mein chale jaate hain. Kuch rare log apna direction rakhte hain aur sirf grow ya shrink karte hain. Aisi direction ke saath dynamics $\dot{x}=Ax$ scalar rule $\dot{x}=\lambda x$ mein collapse ho jaati hai — jiska solution $e^{\lambda t}$ hai. Toh $\lambda$ decide karta hai ki woh mode grows, decays, ya oscillates karta hai. Isliye parent claim karta hai "stability $A$ mein rehti hai."

Recall Eigen-intuition

Ek sentence mein, eigenvector kya hai? ::: Ek direction jise $A$ sirf stretch karta hai, kabhi rotate nahi. Eigenvalues stability kyun decide karte hain? ::: Har eigen-direction ke saath system $\dot{x}=\lambda x$ maanta hai, toh $e^{\lambda t}$ grow karta hai agar $\mathrm{Re}(\lambda )>0$ aur decay karta hai agar $\mathrm{Re}(\lambda )<0$.

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8. Woh number jo khud ko badhata hai: $e$ aur $e^{at}$

Woh self-reproducing property exactly isliye hai kyunki exponentials linear ODEs solve karte hain: ek linear rule kehta hai "rate amount ke proportional hai," aur $e^{at}$ woh shape hai jo "meri slope meri height ke proportional hai" maanti hai. Positive $a$ ⇒ runaway growth; negative $a$ ⇒ zero tak decay; imaginary exponent ⇒ oscillation. Parent yeh matrix exponential $e^{At}$ tak promote karta hai (dekho Matrix exponential) — yehi idea ek saath poore vector ke liye.

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9. Time ke saath add karna: integral $\int$

Is topic ko yeh kyun chahiye. Poora solution $x(t)=e^{At}x(0)+{\int }_0^t{e}^{A(t-\tau )}Bu(\tau )d\tau$ ke do pieces hain: start ki memory, plus har past push ka accumulated effect. Woh accumulation — har chhota input apna bit add karta hua — exactly woh hai jo ek integral sum karta hai. Dummy letter $\tau$ bas "har past instant" ko label karta hai taaki hum ise present time $t$ se confuse na karein.

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Prerequisite map

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Equipment checklist

Khud test karo — jab har reveal obvious lage tab tum parent note ke liye ready ho.

• Main $q(t)$ padh sakta/sakti hoon ::: ek number ki tarah jo time mein har moment par alag value leta hai.
• Main geometrically bata sakta/sakti hoon $\dot{q}$ kya hai ::: us instant par $q$-vs-time curve ke tangent ka slope.
• Mujhe pata hai derivatives physics mein kyun aate hain ::: kyunki laws of motion rates batate hain (force velocity ki rate of change set karta hai).
• Main ODE ka order define kar sakta/sakti hoon ::: dots (derivatives) ki highest number jo appear karti hai.
• Main mass–spring ke liye state vector imagine kar sakta/sakti hoon ::: position–velocity plane mein ek point jiska path motion hai.
• Main $Ax$ haath se compute kar sakta/sakti hoon ::: $A$ ki har row lo, $x$ ke saath weighted sum banao, results stack karo.
• Mujhe $A,B,C,D$ ki shape rule pata hai ::: rows = output ki size, columns = input ki size.
• Mujhe pata hai $A^{-1}$ aur $I$ kya karte hain ::: $I$ vector ko unchanged chhodta hai; $A^{-1}$ $A$ ko undo karta hai toh $A^{-1}A=I$.
• Main bata sakta/sakti hoon eigenvalue ka matlab kya hai ::: stretch factor $\lambda$ ek direction ke saath jise $A$ rotate nahi karta ($Av=\lambda v$).
• Main explain kar sakta/sakti hoon kyun $e^{at}$ $\dot{x}=ax$ solve karta hai ::: uski slope khud times $a$ ke barabar hai, rule se exactly match karti hai.
• Main ${\int }_0^{t}fd\tau$ padh sakta/sakti hoon ::: time $0$ se $t$ tak $f$ ke neeche accumulated area.

Jab yeh second nature ho jaayein, koodein the parent topic mein aur Matrix exponential, Eigenvalues and stability, aur Transfer functions and G(s) mein solution machinery mein.