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

This page assumes nothing. Before you touch the parent note State-space representation, we build every symbol it leans on — from what an arrow-over-a-letter means, up to what a matrix times a vector looks like. Read top to bottom; each block earns the next.

---

1. A variable that changes with time: $q(t)$

Picture a dot sliding along a line. At each tick of a clock the dot sits somewhere, and $q(t)$ is its position label. If you flip-book all those snapshots you get a curve: time along the bottom, value up the side.

Why the topic needs it. Everything in GNC — a rocket's tilt, an aircraft's speed — is a number that changes with time. $q(t)$ is our very first brick.

---

2. Rate of change: the derivative $\dot{q}$

Look at figure s02. The orange line touches the blue curve at one point without crossing it — that touching line is the tangent. Its steepness (how much it rises for one step right) is $\dot{q}$ at that point. Steep-up ⇒ big positive $\dot{q}$; flat ⇒ $\dot{q}=0$; sloping down ⇒ negative $\dot{q}$.

The second derivative $\ddot{q}$ ("$q$-double-dot") is the rate of change of the rate of change — for position that is acceleration (how fast the speed itself is changing).

Recall Read the dots

What does $\dot{q}$ mean in one phrase? ::: How fast $q$ is changing right now (the slope of its curve). What does $\ddot{q}$ mean? ::: How fast $\dot{q}$ is changing — acceleration if $q$ is position.

---

3. A differential equation (ODE) is a rule, not an answer

Think of it as a law of local behaviour: "at any moment, the mass, spring and damping conspire so that this combination equals the pushing force." Give it a starting position and speed, and the rule tells you what happens in the next tiny slice of time — then the next — forever. Solving it means finding the curve $q(t)$ that obeys the rule everywhere.

---

4. Naming several numbers at once: the vector $x$

For the mass–spring the parent picks $x_1=q$ (position) and $x_2=\dot{q}$ (velocity). So the state is a single dot living in a 2-D plane whose axes are position and velocity. As time runs, that dot traces a path — this plane is called the phase plane.

Recall Vector basics

What is ${\mathbb{R}}^n$ in words? ::: The set of all lists of $n$ real numbers — an $n$-dimensional space. For the mass–spring, what are the two state entries? ::: Position $x_1=q$ and velocity $x_2=\dot{q}$.

---

5. A grid of numbers: the matrix $A$, and what $Ax$ does

Read the rule slowly. Take the first row $[ab]$, multiply entry-by-entry into the vector and add: $a{x}_1+b{x}_2$ — that becomes the first output entry. Do the same with the second row for the second entry. That is all a matrix–vector product is: a recipe of weighted sums, one weighted sum per row.

---

6. Two special matrices: identity $I$ and inverse $A^{-1}$

Why the topic needs these. The transfer-function link $G(s)=C(sI-A{)}^{-1}B+D$ uses both: $sI$ is "$s$ times the do-nothing matrix," and $(sI-A{)}^{-1}$ is the undo of $(sI-A)$. It blows up exactly where $(sI-A)$ becomes singular — and those spots are the eigenvalues, i.e. the poles.

---

7. Directions a matrix doesn't rotate: eigenvalues $\lambda$

Most vectors get knocked to a new direction when a matrix acts. A rare few keep their direction and only grow or shrink. Along such a direction the dynamics $\dot{x}=Ax$ collapse to the scalar rule $\dot{x}=\lambda x$ — whose solution is $e^{\lambda t}$. So $\lambda$ decides whether that mode grows, decays, or oscillates. This is why the parent claims "stability lives in $A$."

Recall Eigen-intuition

In one sentence, what is an eigenvector? ::: A direction that $A$ only stretches, never rotates. Why do eigenvalues decide stability? ::: Along each eigen-direction the system obeys $\dot{x}=\lambda x$, so $e^{\lambda t}$ grows if $\mathrm{Re}(\lambda )>0$ and decays if $\mathrm{Re}(\lambda )<0$.

---

8. The number that grows itself: $e$ and $e^{at}$

That self-reproducing property is exactly why exponentials solve linear ODEs: a linear rule says "rate is proportional to amount," and $e^{at}$ is the shape that obeys "my slope is proportional to my height." Positive $a$ ⇒ runaway growth; negative $a$ ⇒ decay to zero; imaginary exponent ⇒ oscillation. The parent promotes this to the matrix exponential $e^{At}$ (see Matrix exponential) — the same idea for a whole vector at once.

---

9. Adding up over time: the integral $\int$

Why the topic needs it. The full solution $x(t)=e^{At}x(0)+{\int }_0^t{e}^{A(t-\tau )}Bu(\tau )d\tau$ has two pieces: memory of the start, plus the accumulated effect of every past push. That accumulation — every little input adding its bit — is exactly what an integral sums. The dummy letter $\tau$ just labels "each past instant" so we don't confuse it with the present time $t$.

---

Prerequisite map

---

Equipment checklist

Test yourself — you're ready for the parent note when each reveal feels obvious.

• I can read $q(t)$ as ::: a number that takes a different value at each moment in time.
• I can say what $\dot{q}$ is geometrically ::: the slope of the tangent to the $q$-vs-time curve at that instant.
• I know why derivatives appear in physics ::: because laws of motion state rates (force sets the rate of change of velocity).
• I can define the order of an ODE ::: the highest number of dots (derivatives) that appears.
• I can picture the state vector for a mass–spring ::: a point in the position–velocity plane whose path is the motion.
• I can compute $Ax$ by hand ::: take each row of $A$, form the weighted sum with $x$, stack the results.
• I know the shape rule for $A,B,C,D$ ::: rows = size of the output, columns = size of the input.
• I know what $A^{-1}$ and $I$ do ::: $I$ leaves a vector unchanged; $A^{-1}$ undoes $A$ so $A^{-1}A=I$.
• I can state what an eigenvalue means ::: the stretch factor $\lambda$ along a direction $A$ doesn't rotate ($Av=\lambda v$).
• I can explain why $e^{at}$ solves $\dot{x}=ax$ ::: its slope equals itself times $a$, matching the rule exactly.
• I can read ${\int }_0^{t}fd\tau$ ::: the accumulated area under $f$ from time $0$ to $t$.

When these are second nature, jump into the parent topic and the solution machinery in Matrix exponential, Eigenvalues and stability, and Transfer functions and G(s).