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

3.5.29 · Physics › Guidance, Navigation & Control (GNC)

HOW to derive it from a physical ODE (from scratch)

Trick yeh hai: koi bhi $n$ -th order linear ODE $n$ coupled first-order equations ban jaata hai derivatives ko naye states ke roop mein naam dekar.

Worked example Mass–spring–damper (Feynman-style)

Ek spring pe lage mass ke liye damping aur applied force $F$ ke saath Newton ka law: $m \overset{q}{¨} + c \overset{q}{˙} + k q = F$

Step 1 — state chunna. Kyun? System ki memory hai "woh kahan hai" aur "kitni tezi se move kar raha hai": $x_{1} = q (position), x_{2} = \overset{q}{˙} (velocity) .$

Step 2 — har state ka derivative likhna. Kyun? Hume $\overset{x}{˙}$ chahiye $x$ aur $u = F$ ke terms mein. $\overset{x}{˙}_{1} = \overset{q}{˙} = x_{2}$ $\overset{x}{˙}_{2} = \overset{q}{¨} = \frac{1}{m} (F - c \overset{q}{˙} - k q) = - \frac{k}{m} x_{1} - \frac{c}{m} x_{2} + \frac{1}{m} F$

Step 3 — matrix form mein stack karna. Kyun? Taaki computer/controller ek clean object dekhe.

= \underbrace{\begin{bmatrix}0 & 1\\ -\tfrac{k}{m} & -\tfrac{c}{m}\end{bmatrix}}_{A} \begin{bmatrix}x_1\\ x_2\end{bmatrix} + \underbrace{\begin{bmatrix}0\\ \tfrac{1}{m}\end{bmatrix}}_{B} F$$ **Step 4 — output.** *Kyun?* Maano ek sensor sirf position read karta hai: $$y = \underbrace{\begin{bmatrix}1 & 0\end{bmatrix}}_{C}\begin{bmatrix}x_1\\x_2\end{bmatrix} + \underbrace{0}_{D}\,F$$

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

The solution (derived, not dumped)

Common mistakes (Steel-man them)

Forecast-then-Verify

Recall Check karne se pehle forecast karo

Q: Mass–spring–damper ke liye jisme $m = 1, k = 1, c = 0$ (no damping) ho, $A$ ke eigenvalues aur motion predict karo. Forecast: $A = [0 - 1 10]$ , $det (λ I - A) = λ^{2} + 1 = 0 \Rightarrow λ = \pm i$ . Purely imaginary → $ω = 1$ rad/s par undamped oscillation. ✅ Physics se match karta hai: no damping ⇒ hamesha ring karta rahega.

Flashcards

Do state-space equations kya hain?

\overset{x}{˙} = A x + B u

(state/dynamics) aur

y = C x + D u

(output).

Kisi system ki "state" kya hoti hai?

Variables ka woh minimal set jo future inputs ke saath future behavior fully predict karne ke liye zaroori ho.

A

matrix physically kya represent karta hai?

Internal dynamics — state apne aap par kaise feedback karta hai; iske eigenvalues stability set karte hain.

B

kya karta hai?

Control inputs

u

ko state ke rates of change mein map karta hai.

C

kya karta hai?

States ko measured outputs

y

mein select/combine karta hai.

D

nonzero kab hota hai?

Jab koi input instantly (algebraically) output ko affect kare — direct feedthrough.

n

-th order ODE ko state-space mein kaise convert karte hain?

States ko function aur uske pehle

n - 1

derivatives ke roop mein define karo, jo

n

coupled first-order equations deta hai.

\overset{x}{˙} = A x + B u

ka full solution kya hai?

x (t) = e^{A t} x (0) + \int_{0}^{t} e^{A (t - τ)} B u (τ) d τ

State-space mein stability ki condition?

A

ke saare eigenvalues ka real part negative ho.

State-space se transfer function?

G (s) = C (s I - A)^{- 1} B + D

State-space unique kyun nahi hoti?

Coordinate change

\tilde{x} = T x

(T A T^{- 1}, T B, C T^{- 1}, D)

milta hai jo same system describe karta hai.

System order

n

kya determine karta hai?

State vector ki dimension = independent energy-storing / memory elements ki sankhya.

Recall Feynman: ek 12-saal ke bachche ko explain karo

Ek toy car imagine karo jisme spring lagi ho. Yeh jaanne ke liye ki woh aage kahan jaayegi, abhi sirf do cheezein chahiye: woh kahan hai aur kitni tezi se chal rahi hai. Yahi uski "state" hai. Ek dhakka tera input hai. "Position aur speed + dhakka → ek chhote moment mein kaise change hote hain" yahi poori physics hai. Hum bas yeh rule numbers ki boxes (matrices) ke roop mein likhte hain taaki computer movie aage chala sake. $C$ box batata hai ki un donon mein se kaun si cheez car ka chhota sensor actually read kar sakta hai.

Connections

Transfer functions and G(s) — $G (s) = C (s I - A)^{- 1} B + D$ donon worlds ko link karta hai.
Eigenvalues and stability — poles = eigenvalues of $A$ .
Matrix exponential — solution $e^{A t}$ ka engine.
Controllability and Observability — kya $B$ aur $C$ actually har state tak pahunchte hain?
LQR optimal control — directly $(A, B)$ par feedback $u = - K x$ design karta hai.
Kalman filter — $y = C x$ se unmeasured states estimate karta hai.
Linearization of nonlinear systems — jahan $A, B$ Jacobians ke roop mein aate hain.

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

WHY do we need state-space?

WHAT is the state?