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

3.5.29 · D2 · Physics › Guidance, Navigation & Control (GNC) › State-space representation — x' = Ax + Bu, y = Cx + Du

Prerequisites jin par hum lean karte hain (sab tumhari vault mein hain): the parent topic, aur baad mein Matrix exponential, Eigenvalues and stability.

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Step 1 — $\dot{x}=ax$ aakhir keh kya raha hai?

KYA HAI. Matrices bhool jao. Sirf EK number $x(t)$ se shuru karo — ise ek tank mein paani ki height socho, ya time $t$ pe ek line pe dot ki position. Letter ke upar chhota dot, $\dot{x}$, shorthand hai "abhi is waqt $x$ kitni tezi se badal raha hai" — iska speed.

YEH EQUATION KYUN. Sabse simple possible rule jo ek system follow kar sakta hai woh hai: "mera speed meri current size ke proportional hai." Likha hua: $\dot{x}=ax$. Number $a$ growth rate constant hai. Agar $a>0$, toh jitne bade ho utni tezi se badhoge (runaway). Agar $a<0$, toh jitne bade ho utni tezi se zero ki taraf shrink karoge (decay). Yahi sab kuch ka seed hai — poora state-space solution yahi idea hai jo matrix ka costume pehne hua hai.

PICTURE. Figure dekho. Horizontal axis $x$ hai (hum kitne bade hain). Vertical arrows $\dot{x}=ax$ dikhate hain — har jagah speed. Jahan $x$ bada hai wahan arrow lamba hai; jahan $x=0$ wahan arrow ki length zero hai (ek dot jo $0$ pe baitha hai kabhi move nahi karta — yahi equilibrium hai).

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Step 2 — Answer $e^{at}$ kyun hai (aur $e$ kahan se aata hai)

KYA HAI. Hum $x(t)$ ka ek formula chahte hain jo "mera speed $a$ times khud mujhse equal hai" ko har instant pe satisfy kare.

WHY $e$. Socho: kaunsa function differentiate hone par khud hi ban jaata hai (ek factor $a$ tak)? Zyaadatar functions differentiate hone par apna shape badal lete hain. Lekin ek special number hai, $e\approx 2.718$, exactly aise bana hai ki $e^t$ differentiate hokar khud ban jaata hai. Exponent ko $a$ se multiply karo aur chain rule hume ek factor $a$ deta hai: $\frac{d}{dt}\underset{\text{our guess}}{\underbrace{e^{at}}}=\underset{\text{falls out of the exponent}}{\underbrace{a}}\underset{\text{unchanged shape}}{\underbrace{e^{at}}}.$ Yeh precisely $\dot{x}=ax$ hai. Toh $x(t)=e^{at}x(0)$, jahan $x(0)$ starting value hai (time zero pe size) — yeh sirf poori curve ko upar ya neeche scale karta hai.

PICTURE. Figure teen cases ko same axes pe plot karta hai taaki tum ek saath sab signs dekh sako: teal $a<0$ ($0$ ki taraf decay), plum $a=0$ (flat line — kabhi nahi badalta), orange $a>0$ (upar ki taraf explode). Kisi bhi point pe har curve ki slope equals $a$ times us point ki apni height — woh chhote slope arrows se padho.

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Step 3 — Upgrade: ek saath kaafi saari quantities → vector $x$

KYA HAI. Ek real vehicle ek number nahi hota. Spring pe mass ko position aur velocity chahiye; ek rocket ko attitude, rates, positions chahiye. Inhe numbers ke ek column mein stack karo — state vector $x\in {\mathbb{R}}^n$ (yahan ${\mathbb{R}}^n$ ka matlab sirf "$n$ real numbers ki ek list" hai).

MATRIX KYUN. Har entry ki speed baaki sab pe depend kar sakti hai. "Speed = rate times size" ban jaata hai "poori list ki speed = rates ki ek table times poori list." Woh table matrix $A$ hai. Toh scalar $\dot{x}=ax$ bada hokar $\dot{x}=Ax$ ban jaata hai — same sentence, bold letters.

PICTURE. Left: Step 1 ka scalar arrow-on-a-line. Right: state ab ek plane mein rehta hai ($x_1$ horizontal, $x_2$ vertical). Har point pe matrix $A$ ek chhota arrow assign karta hai — ek velocity field. Ek trajectory woh dot hai jo hamesha apne pairo ke neeche arrow follow karta hai.

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Step 4 — $e^{At}$ ka kya matlab ho sakta hai? Series se banao

KYA HAI. Scalar case mein answer $e^{at}$ tha. Hum chahte hain ki vector answer $e^{At}x(0)$ ho. Lekin tum $e$ ko ek matrix tak raise nahi kar sakte… jab tak tum decide na karo ki iska matlab kya hai.

SERIES KYUN. Yaad karo ordinary exponential ki ek honest definition — ek infinite sum: $e^s=1+s+\frac{s^2}{2!}+\frac{s^3}{3!}+\cdots$ Har term sirf powers aur counting numbers se divide karna hai — aur hum ek square matrix ki powers le sakte hain ($A^2=AA$, $A^3=AAA$, …) aur numbers se divide kar sakte hain. Toh hum simply wohi recipe reuse karte hain: $e^{At}:=I+At+\frac{(At{)}^2}{2!}+\frac{(At{)}^3}{3!}+\cdots$ jahan $I$ identity matrix hai (matrices ke liye "$1$" — kisi bhi vector ko unchanged chodta hai).

ODE solve karta hai check karo. Series ko term by term differentiate karo (har power ek se drop hoti hai, exactly jaise $\frac{d}{dt}{t}^k=k{t}^{k-1}$): $\frac{d}{dt}{e}^{At}=A+A^{2}t+\frac{A^3{t}^2}{2!}+\cdots =A(I+At+\frac{(At{)}^2}{2!}+\cdots )=A{e}^{At}.$ Toh $x(t)=e^{At}x(0)$ sach mein $\dot{x}=Ax$ satisfy karta hai.

PICTURE. Hum series ko contributions ki ek growing stack ke roop mein dikhate hain: flat $I$, phir tilt $At$, phir curve $\frac{(At{)}^2}{2}$, … har bar pehle se chhota kyunki bottom mein $k!$ tezi se badhta hai. Unka sum smooth propagator $e^{At}$ hai.

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Step 5 — Crank ghumaao: integrating-factor trick input $u$ ko handle karta hai

KYA HAI. Control add karo. Real equation hai $\dot{x}=Ax+Bu$, jahan $u$ woh input hai jo tum apply karte ho (thrust, fin angle) aur $B$ (size $n\times m$) batata hai ki har input har state ko kitna push karta hai.

INTEGRATING FACTOR KYUN. Hum $Ax$ term ko "peel off" karna chahte hain taaki right side pe sirf input bache — phir plain integration kaam khatam kar deta hai. Peeling tool hai $e^{-At}$ se multiplication (inverse slide — $t$ seconds ki dynamics undo karo). Dekho kaise do $A$-terms ek single derivative mein collapse ho jaate hain:

\;\;\Longrightarrow\;\; \frac{d}{dt}\big(e^{-At}x\big)=e^{-At}Bu.$$ Left side ab ek cheez ka clean derivative hai (product rule, backward chala). $0$ se $t$ tak integrate karo, phir peeling undo karne ke liye $e^{At}$ se wapas multiply karo: $$x(t)=e^{At}x(0)+\int_0^t e^{A(t-\tau)}Bu(\tau)\,d\tau.$$ **PICTURE.** Figure answer ko visually split karta hai: ek **teal curve** = free response $e^{At}x(0)$ (hum kahan se shuru hue uski yaad, koi input nahi), aur **orange stacked slivers** = forced response, jahan time $\tau$ pe har past input kick $Bu(\tau)$ ko bacha hua time $(t-\tau)$ aage slide kiya jaata hai $e^{A(t-\tau)}$ ke zariye aur aisi sab kicks ko add kiya jaata hai (yahi $\int$ karta hai). ![[deepdives/dd-physics-3.5.29-d2-s05.png]] > [!formula] Complete solution > $$\boxed{\,x(t)=\underbrace{e^{At}x(0)}_{\text{free / zero-input}}+\underbrace{\int_0^t e^{A(t-\tau)}Bu(\tau)\,d\tau}_{\text{forced / zero-state}}\,}$$ > - $e^{At}x(0)$ ::: woh past jahan se shuru kiya tha, aage propagate kiya gaya. > - $Bu(\tau)$ ::: time $\tau$ pe deliver ki gayi ek input kick. > - $e^{A(t-\tau)}$ ::: us kick ko bacha hua $t-\tau$ seconds aage slide karta hai. > - $\int_0^t\ldots d\tau$ ::: shuruwat se abhi tak ki har kick ko add karta hai. --- ## Step 6 — Picture se seedha stability padhna (eigenvalues) **KYA HAI.** Free response kab die out hoga vs. blow up karega? Jawab poora $A$ mein hai, uske **eigenvalues** ke zariye — special numbers $\lambda$ aur directions $v$ jahan $Av=\lambda v$ (ek direction jise matrix sirf *stretch* karta hai, rotate nahi). **EIGENVALUES KYUN.** Aisi special direction $v$ ke along, vector equation *wapas scalar ban jaati hai*: $\dot{x}=Ax$ turn ho jaata hai $\dot{c}=\lambda c$ amount $c$ ke liye jo $v$ ke along hai. Aur hum woh Step 2 mein already solve kar chuke hain — agar rate negative ho toh decay karta hai. Possibly complex $\lambda=\sigma+i\omega$ ke liye: - $\sigma=\mathrm{Re}(\lambda)$ ::: badhna ($\sigma>0$) ya decay ($\sigma<0$) set karta hai — *envelope*. - $\omega=\mathrm{Im}(\lambda)$ ::: kitni tezi se oscillate karta hai yeh set karta hai — *ringing*. **PICTURE — har case dikhaya gaya.** Complex plane, vertical axis ($\mathrm{Re}=0$ line) se split. Hum chaar eigenvalue spots plot karte hain aur har ek se hone wali chhoti motion: teal (left half, decaying spiral — **stable**), plum (axis pe, pure oscillation — **marginal**), orange (right half, growing spiral — **unstable**), aur ek real negative $\lambda$ (pure decay, koi ring nahi). Rule ek sentence mein hai: **sab dots axis ke strictly left mein ⇒ stable.** ![[deepdives/dd-physics-3.5.29-d2-s06.png]] > [!definition] Eigenvalue reading > $$Av=\lambda v,\qquad \lambda=\sigma+i\omega.$$ > - $v$ ::: ek direction jise matrix sirf scale karta hai. > - $\sigma<0$ har **har** eigenvalue ke liye ::: poora state $0$ tak decay karta hai → **stable**. Dekho [[Eigenvalues and stability]]. > - $\sigma=0$ ::: hamesha ke liye ring karta hai (marginal); $\sigma>0$ ::: kam se kam ek direction explode karta hai. --- ## Ek-picture summary Sab kuch ek canvas pe: ek **scalar seed** $\dot{x}=ax\to e^{at}$ (series ke zariye) **matrix propagator** $e^{At}$ mein grow karta hai; propagator initial state ko aage carry karta hai (teal) jabki integral input kicks rake in karta hai (orange); aur **$A$ ke eigenvalues** complex plane pe, apni left/right position se, decide karte hain ki poori movie fade out hogi ya bhaag jaayegi. ![[deepdives/dd-physics-3.5.29-d2-s07.png]] > [!recall]- Feynman: ek 12-saal ke bachche ko batao > Rabbit-population rule se shuru karo: *yeh kitni tezi se badalta hai yeh ek number times abhi kitna bada hai iske equal hota hai.* Jawab hai "$e$ to the power (woh number times time)" — $e$ woh ek magic number hai jiska growth curve khud apni slope copy karta hai. Ab ek population ki jagah unki ek **poori team** ki imagine karo, jo ek doosre ko nudge karti hain. Akela number ek *box of numbers* (matrix $A$) ban jaata hai, aur magic "$e$-to-a-number" ban jaata hai "$e$-to-a-box" — jise hum same adding-up recipe se samajhte hain, bas box ko baar baar khud se multiply karte hain. Woh box, $e^{At}$, ek **time machine** hai: use batao tum kahan se shuru hue, yeh batayega tum kahan hooge. Agar tum system ko pokte bhi rehte ho (tumhara control $u$), toh tum bas saare pokes add kar lete ho, har ek utna aage slide kiya hua jitna pehle poka tha. Aur team eventually shant hogi ya control se bahar spin karegi? Box ke special "stretch numbers" (uske eigenvalues) dekho: agar woh sab number-line-with-an-up-axis ke *left* side pe hain, toh sab kuch shant ho jaata hai. Yahi poora state-space ka story hai, ek saans mein. > [!mnemonic] > **Seed → Series → Slide → Sum → Sign.** > ($a x$ ka Seed; Series se $e$; $e^{At}$ start ko Slide karta hai; integral pokes ko Sum karta hai; eigenvalue ka Sign stability decide karta hai.) ## Connections - [[Matrix exponential]] — Steps 4–5 iska poora construction hai. - [[Eigenvalues and stability]] — Step 6, sign rule. - [[Transfer functions and G(s)]] — same poles $\det(sI-A)=0$ ke zariye. - [[Controllability and Observability]] — kya $B$ ki kicks, aur $C$, har direction tak pahunchti/dekhti hain. - [[LQR optimal control]] aur [[Kalman filter]] — dono is exact solution pe run karte hain. - [[Linearization of nonlinear systems]] — jahan real $A$ aata hai.