Observability — when KF can estimate state: Foundations

This page assumes you have seen nothing. Before you can read the parent note on observability, every symbol it throws at you must first mean something. We build them one at a time, each on top of the last, each pinned to a picture.

0. The scene we are describing

Picture a spinning satellite. Inside it, some quantities are true but hidden — its real pointing angle, the slow drift error of its gyroscope. Outside, a star-tracker spits out numbers we can read. The whole subject lives in the gap between "what is really going on inside" and "what we get to see."

Figure — Observability — when KF can estimate state

We now name every piece of this picture.

1. A vector and the symbol $x$

Picture: an arrow, or equivalently a single dot, floating in a space with one axis per dial. Two dials → a flat plane; the dot's position is the full internal condition of the machine.

Why the topic needs it: observability is about pinning down this whole dot from the outside. If we could not bundle all the dials into one object $x$ , we could not ask "did we recover the whole thing?"

2. The state $x$ and the measurement $y$

Picture: the state dot lives in a big room ( $n$ axes); the measurement is its shadow on a small wall ( $m$ axes). Many different dots can cast the same shadow — that ambiguity is what observability worries about.

3. Matrices and matrix–vector multiplication

Before $A$ and $C$ can mean anything, we need to know what a matrix does.

Picture: a matrix stretches, rotates, squishes, or projects the whole space of dots. Watch a grid of dots go in and a warped grid come out.

Why the topic needs it: both "how the state evolves" and "how the sensor reads the state" are exactly such mixing operations — so they are matrices.

4. The dynamics matrix $A$ , and the input $B$ , $u$

Picture: at every dot in the room, $A$ plants a little arrow saying "the state will slide this way next instant." The whole room becomes a field of flow arrows.

Why the topic needs it — the crucial subtlety: if $A_{ij} \neq = 0$ , then dial $j$ leaks into dial $i$ . Even a hidden dial we cannot measure can still betray itself by nudging a dial we can see. This leaking is the entire mechanism that makes hidden things observable. (In the parent's position–velocity example, velocity is unseen but $A_{12} = 1$ lets it push position — so it leaks out.)

Why the topic needs it: the full equation of motion (next section) contains $B u$ . But because $u$ is known, we can always subtract its effect and study only the machine's own unforced drift — which is exactly why observability testing sets $u = 0$ .

5. The measurement matrix $C$

Picture: $C$ is the shape of the window in Section 2. $C = [10]$ is a window that shows dial 1 and hides dial 2 completely.

6. The derivative — reading change over time

Why this tool and not another? We want to squeeze information out of the measurement stream. A single snapshot $y (0)$ gives one clue. But the slope $\overset{y}{˙} (0)$ , the curvature $\overset{y}{¨} (0)$ , and so on, are extra independent clues — each new derivative is a fresh equation about the hidden state. The derivative is precisely the tool that converts "watching over time" into "more equations."

Picture: a wiggling measurement curve $y (t)$ . Its height at $t = 0$ is one number; its slope there is a second, independent number; its bend is a third. Each is a new probe into the hidden dot.

7. The matrix exponential $e^{A t}$

Why this tool? We need one clean object that turns "the starting dot $x (0)$ " into "the dot at any later time." $e^{A t}$ is exactly that propagator — feed in $x (0)$ , out comes $x (t)$ .

Picture: grab the starting dot and let the flow arrows of $A$ (Section 4) sweep it along its path; $e^{A t}$ is the recipe for where it lands at time $t$ .

Therefore the measured stream is $y (t) = C e^{A t} x (0)$ — the hidden start, propagated by time, then projected through the window.

8. Squeezing equations out of $y (t)$ — the derivative ladder

Now we use the derivative tool to turn the measurement stream into equations about the unknown $x (0)$ . Start from $y (t) = C e^{A t} x (0)$ and differentiate, using the single fact that $\frac{d}{d t} e^{A t} = A e^{A t}$ .

WHAT we just did: differentiated the known signal $y$ repeatedly and read off its value at $t = 0$ . WHY: at $t = 0$ the propagator $e^{A \cdot 0} = I$ disappears, leaving the clean pattern $C A^{k} x (0)$ . WHAT IT LOOKS LIKE: each derivative is a new probe of the wiggling curve — height, slope, bend — and each probe is one more row of information about the hidden dot.

When do we stop? By the Cayley–Hamilton Theorem, $A^{n}$ is just a mix of $I, A, \dots, A^{n - 1}$ , so $C A^{n}$ repeats information we already have. Thus only the first $n$ rows ( $k = 0, \dots, n - 1$ ) carry anything new.

9. Stacking the ladder — the observability matrix $O$

We now bundle those $n$ independent equations into a single matrix statement.

WHAT we just did: wrote the whole derivative ladder as one equation " $O$ times the unknown equals the measured data." WHY: whether we can solve for $x (0)$ now depends entirely on the shape of $O$ — a question we answer with rank in the next section. WHAT IT LOOKS LIKE: a tall stack of windows, each looking at the state after one more step of the flow.

10. Rank and the null space

These two ideas are the punchline of the whole topic, so we build them carefully.

Picture: think of $M$ as a projection casting shadows. If some direction lies flat against the light, it casts no shadow — that direction is in the null space. If nothing gets flattened, the null space is just ${0}$ and the matrix keeps every direction distinct (full rank).

11. The transpose $A^{⊤}$

Why the topic needs it: the parent states a duality — $(A, C)$ observable $⟺ (A^{⊤}, C^{⊤})$ controllable. The transpose is the mirror that turns a "reading-out" problem into a "driving-in" problem. See Controllability — when we can steer the state for the mirror image.

x \in R^{n}

mean in plain words?

x

is a list of

n

real numbers — the

n

hidden dials packed into one column vector.

What does a matrix do to a vector?

It transforms it — each output slot is a weighted mix of all input slots (go row by row).

What does entry

A_{ij}

of the dynamics matrix tell you?

How strongly hidden dial

j

drives the change in dial

i

— the "leaking" between dials.

What are

u

and

B

u

is the known command vector we apply;

B

is the

n \times p

matrix saying how each command pushes each dial.

What does the measurement matrix

C

represent?

The window — which mix of dials each sensor actually reports;

y = C x

Why do we use derivatives of

y

Each derivative gives

y^{(k)} (0) = C A^{k} x (0)

, a new independent equation about

x (0)

What is

e^{A t}

in one sentence?

The propagator that slides the starting state

x (0)

forward to

x (t)

along the flow of

A

Write the observability matrix in terms of

C

and

A

O = [C; C A; C A^{2}; \dots; C A^{n - 1}]

What is the null space

ker (M)

The nonzero directions

v

with

M v = 0

— inputs the matrix erases to zero (cast no shadow).

State the link between rank, null space and observability.

Observable

⟺ ker (O) = {0} ⟺ rank (O) = n

What does the transpose

A^{⊤}

let us say about observability?

(A, C)

observable

⟺ (A^{⊤}, C^{⊤})

controllable — the observability/controllability duality.

Why can a hidden dial still be observable?

A_{ij} \neq = 0

it leaks into a measured dial, betraying itself over time.

Now proceed to Observability — when KF can estimate state with every symbol earned. Related next steps: State-space representation of LTI systems, Cayley–Hamilton Theorem, Observability Gramian & degree of observability.

Foundations — Observability — when KF can estimate state