Foundations — Controllability matrix — rank test

Before you can read the parent note, you need a picture behind every symbol it throws at you: $x$, $u$, $A$, $B$, $\dot{x}$, ${\mathbb{R}}^n$, "span", "rank", $e^{At}$, and $A^{k}B$. We build each one from zero, in an order where every new idea leans only on the ones before it.

1. The state vector $x$ — "everything the machine is right now"

Picture a puck sliding on ice. To know its future you need two numbers: where it is ($p$) and how fast it moves ($v$). So its state is the pair $x=\left[\begin{array}{c}p\\ v\end{array}\right].$

Look at Figure s01. The left panel is the physical puck. The right panel plots the SAME information as a single point in a plane whose horizontal axis is position and vertical axis is velocity. That plane is called state space, and the point IS the state.

2. ${\mathbb{R}}^n$ — the space the state lives in

For the puck, $n=2$, so $x\in {\mathbb{R}}^2$ — a flat plane (the right panel of s01). A spinning spacecraft might need $n=6$ or more; you can't draw it, but the rules are identical.

Why the topic needs it :::: "Controllable" is defined as reach ANY point of ${\mathbb{R}}^n$. Without naming this space, "any target" has no meaning.

3. The input vector $u$ — "the dials you can turn"

For the puck, one dial: a force you apply. So $m=1$ and $u=[\text{force}]$. A quadcopter with four rotors has $m=4$.

4. The derivative $\dot{x}$ — "how the state is changing this instant"

The parent writes $\dot{x}=Ax+Bu$. That dot is doing a lot of work.

Look at Figure s02. At each point of state space we draw a small arrow: the arrow at point $x$ is the vector $\dot{x}$ — the direction and speed the state will drift if you touch no dials. Following the arrows traces the machine's natural motion.

5. The matrices $A$ and $B$ — the machine's wiring

Now we can read $\dot{x}=Ax+Bu$ term by term.

So the full sentence $\dot{x}=Ax+Bu$ reads:

"How the state changes now = (what its own current value drags it toward) + (what your dials are pushing it toward)."

6. Matrix–vector multiplication — how $Ax$ actually works

You cannot trust $Ax$ until you can compute it by hand.

7. Span — "everywhere a set of arrows can reach by mixing"

The parent note says the reachable states are the span of some columns. Here is that word.

Look at Figure s03.

• Left: two vectors pointing in genuinely different directions. Mixing them reaches every point of the plane — their span is all of ${\mathbb{R}}^2$.
• Right: two vectors lying along the same line. No matter how you scale and add, you never leave that line — their span is only a 1-D sliver, and the rest of the plane is unreachable.

8. Rank — "how many independent directions a set of columns really has"

The size of a span needs a number. That number is rank.

Reading s03 with this word:

• Left matrix (columns spread the plane): $\text{rank}=2$.
• Right matrix (columns share a line): $\text{rank}=1$.

9. Powers $A^{k}B$ — "how a push ripples through the wiring"

Why does the parent stack $B,AB,A^{2}B,\dots$? Each $A$ you apply is one step of letting the machine's own wiring pass the push along.

Look at Figure s04 for the puck ($A,B$ from §5):

• Red arrow $B=\left[\begin{array}{c}0\\ 1\end{array}\right]$: the dial pushes velocity only.
• Blue arrow $AB=\left[\begin{array}{c}1\\ 0\end{array}\right]$: after one ripple, that push has leaked into position.
• Together red + blue point in two different directions → span the whole plane → rank 2 → controllable. The dial on velocity secretly controls position too.

10. $e^{At}$ — the exponential that plays the whole motion forward

The parent's derivation uses $e^{At}$. You need only the idea; the Matrix exponential $e^{At}$ note has the machinery.

Prerequisite map

Each foundation on this page is one node; follow the arrows and you arrive exactly at the parent note's rank test.

Equipment checklist

Test yourself — reveal only after you've answered aloud.