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

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

The trick: any $n$ -th order linear ODE becomes $n$ coupled first-order equations by naming derivatives as new states.

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

Newton's law for a mass on a spring with damping and applied force $F$ : $m\ddot{q} + c\dot{q} + kq = F$

Step 1 — pick the state. Why? The system's memory is "where it is" and "how fast it moves": $x_1 = q \ (\text{position}), \qquad x_2 = \dot{q}\ (\text{velocity}).$

Step 2 — write each state's derivative. Why? We need $\dot{x}$ in terms of $x$ and $u=F$ . $\dot{x}_1 = \dot{q} = x_2$ $\dot{x}_2 = \ddot{q} = \frac{1}{m}\big(F - c\dot{q} - kq\big) = -\frac{k}{m}x_1 - \frac{c}{m}x_2 + \frac{1}{m}F$

Step 3 — stack into matrix form. Why? So a computer/controller sees one clean object.

= \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$$ **Step 4 — output.** *Why?* Say a sensor reads position only: $$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)

Formula Why the matrix exponential appears

Compare scalar $\dot{x}=ax \Rightarrow x=e^{at}x_0$ . For vectors, define the matrix exponential $e^{At} := \sum_{k=0}^{\infty}\frac{(At)^k}{k!},$ which satisfies $\frac{d}{dt}e^{At}=Ae^{At}$ (differentiate the series term by term).

Derivation of the full solution. Multiply $\dot{x}-Ax=Bu$ by the integrating factor $e^{-At}$ : $e^{-At}\dot{x} - e^{-At}Ax = e^{-At}Bu \;\Rightarrow\; \frac{d}{dt}\big(e^{-At}x\big) = e^{-At}Bu.$ Integrate from $0$ to $t$ and multiply back by $e^{At}$ : $\boxed{\,x(t)=e^{At}x(0)+\int_0^t e^{A(t-\tau)}Bu(\tau)\,d\tau\,}$ First term = free/zero-input response (memory). Second = forced/zero-state response (input).

Common mistakes (Steel-man them)

Forecast-then-Verify

Recall Forecast before checking

Q: For the mass–spring–damper with $m=1, k=1, c=0$ (no damping), predict the eigenvalues of $A$ and the motion. Forecast: $A=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ , $\det(\lambda I-A)=\lambda^2+1=0\Rightarrow\lambda=\pm i$ . Purely imaginary → undamped oscillation at $\omega=1$ rad/s. ✅ Matches physics: no damping ⇒ it rings forever.

Flashcards

What are the two state-space equations?

\dot{x}=Ax+Bu

(state/dynamics) and

y=Cx+Du

(output).

What is the "state" of a system?

The minimal set of variables needed with future inputs to fully predict future behavior.

What does the

A

matrix represent physically?

The internal dynamics — how the state feeds back on itself; its eigenvalues set stability.

What does

B

do?

Maps control inputs

u

into rates of change of the state.

What does

C

do?

Selects/combines states into measured outputs

y

When is

D

nonzero?

When an input instantly (algebraically) affects an output — direct feedthrough.

How do you convert an

n

-th order ODE to state-space?

Define states as the function and its first

n-1

derivatives, giving

n

coupled first-order equations.

What is the full solution of

\dot{x}=Ax+Bu

x(t)=e^{At}x(0)+\int_0^t e^{A(t-\tau)}Bu(\tau)d\tau

Condition for stability in state-space?

All eigenvalues of

A

have negative real part.

Transfer function from state-space?

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

Why is state-space not unique?

A coordinate change

\tilde x=Tx

gives

(TAT^{-1},TB,CT^{-1},D)

describing the same system.

What determines the system order

n

The dimension of the state vector = number of independent energy-storing / memory elements.

Recall Feynman: explain to a 12-year-old

Imagine a toy car with a spring. To know where it'll go next, you only need two things right now: where it is and how fast it's moving. That's its "state." A push is your input. The rule "position and speed + push → how they change in a tiny moment" is the whole physics. We just write that rule as boxes of numbers (matrices) so a computer can play the movie forward. The $C$ box says which of those two things the car's little sensor can actually read out.

Connections

Transfer functions and G(s) — $G(s)=C(sI-A)^{-1}B+D$ links the two worlds.
Eigenvalues and stability — poles = eigenvalues of $A$ .
Matrix exponential — engine of the solution $e^{At}$ .
Controllability and Observability — do $B$ and $C$ actually reach every state?
LQR optimal control — designs feedback $u=-Kx$ directly on $(A,B)$ .
Kalman filter — estimates unmeasured states from $y=Cx$ .
Linearization of nonlinear systems — where $A,B$ come from as Jacobians.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, state-space ka basic idea ekdum simple hai: kisi bhi system ka future sirf do cheezon pe depend karta hai — uska abhi ka state (yaani uski memory) aur aap jo input de rahe ho. Jaise ek spring pe laga mass — usse predict karna ho to bas position aur velocity chahiye. Yahi do variable "state" ban jaate hain, aur inhe hum ek vector $x$ me daal dete hain.

Ab magic yeh hai: koi bhi bada ODE (jaise $m\ddot q + c\dot q + kq = F$ ) ko hum kaat kar chhote-chhote first-order equations me tod dete hain. Har derivative ko naya state naam de do, aur sab kuch matrix form me aa jaata hai: $\dot x = Ax + Bu$ . Yahan $A$ system ka apna behaviour (andar ka dynamics) batata hai, $B$ batata hai input kaise push karta hai, $C$ batata hai sensor kya measure karta hai, aur $D$ direct shortcut hai (mostly zero).

GNC me yeh isliye important hai kyunki rocket ya aircraft me bahut saare inputs (thrust, fins, gimbal) aur bahut saare outputs (attitude, position) hote hain — transfer function ek-ek karke handle nahi kar sakta, par state-space sab ek saath matrices me pakadta hai. Aur sabse mast baat: stability ka pura raaz $A$ matrix ke eigenvalues me chhupa hai. Agar sab eigenvalues ka real part negative hai, system stable hai; warna wobble ya blow-up. Isi foundation pe LQR aur Kalman filter jaise sabhi advanced controllers khade hote hain, isliye ise achhe se samajhna 80/20 wala kaam hai.

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

WHY do we need state-space?

WHAT is the state?

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

The solution (derived, not dumped)

Common mistakes (Steel-man them)

Forecast-then-Verify

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Guidance, Navigation & Control (GNC)

Connections