Foundations — LQG — LQR + Kalman filter, separation principle

This page assumes you have seen none of the notation in the parent note. We build every letter, arrow, and squiggle from the ground up, in an order where each rung of the ladder rests on the one below it. If you have read the parent LQG topic and hit a symbol you couldn't parse, this is the page that fixes it.

0 — What is "a system" and why the arrows?

Before any letters, picture the object we control.

1 — The state $x$ : "everything you'd need to know to predict the future"

Picture: in figure s01, we draw the two-number state $x=(\text{position},\text{velocity})$ as a single dot in a state plane — one axis is position, the other is velocity. The dot's location is the state; the axes are not the room, they are the two state coordinates.

Why the topic needs it: everything in LQG — steering, estimating, cost — is measured relative to the state. Our whole goal will be "drive $x$ to zero" (bring the dot to the target at the origin).

2 — Vectors and the transpose $x^{\top }$

Why we ever need $x^{\top }$: to build a single number out of a vector's size we compute $x^{\top }x=x_1^2+x_2^2+\dots$ — the sum of squares, i.e. how far the dot is from the origin, squared. A row times a column collapses a whole vector into one honest number. That single number is what we will try to make small.

3 — Matrices $A,B,C$ : "machines that reshape a vector"

Picture: a matrix is an arrow-bender. Give it the arrow $x$; it hands back a different arrow $Ax$.

The three named grids each need one more symbol first — the control $u$ (what you push with) and the noises $w$, $v$ — so we introduce those now, before writing any equation that uses them.

Now the three named grids, each answering one physical question:

Why the topic needs them: they are the rules of the room. Every equation in LQG is these three grids plus noise. See State-space Representation.

4 — $\dot{x}$ : the dot over a symbol means "rate of change"

Why this notation and not, say, "$\Delta x$"? Because the machine changes continuously, not in jumps. The overdot is the derivative — the instantaneous rate — which is the only honest way to describe smooth motion. It answers "where is the dot heading this very instant?"

Putting §3 and §4 together (with $u$, $w$, $v$ already named) gives the state and measurement equations: $\dot{x}=Ax+Bu+w,y=Cx+v$ Read the first in plain words: the velocity of the dot $(\dot{x})$ is: its natural drift $(Ax)$ plus the effect of my push $(Bu)$ plus a random shove $(w)$. The second says: what the sensor reports $(y)$ is the clean projection of the truth $(Cx)$ plus sensor error $(v)$.

5 — $y$, $\hat{x}$, $e$ : look, guess, and how wrong we are

$u$ was defined in §3; the measurement $y$ appeared just above. Here we name what we build from $y$.

6 — The size of the noise: $\mathcal{N}(0,W)$ and $\mathcal{N}(0,V)$

We named $w$ and $v$ in §3; now we say precisely how big they are.

Why "0"? If noise had an average other than zero it would be a known drift, and we'd fold it into $A$ or $B$. True noise averages out — hence centred at 0.

7 — $Q$, $R$ and "penalise = square it"

The cost we minimise is $J={\int }_0^{\infty }(\underset{\text{off-target badness}}{\underbrace{x^{\top }Qx}}+\underset{\text{effort badness}}{\underbrace{u^{\top }Ru}})dt.$

The two ordering constraints you'll meet:

• $Q⪰0$ ("positive semidefinite"): badness is never negative — you can be neutral about some directions.
• $R\succ 0$ ("positive definite"): every push costs something — you can't get free control.

8 — $P$, $\Sigma$, and $K$, $L$ : the answers the maths hands back

These four are outputs — you don't choose them, you solve for them.

Both are squeezed out of Riccati equations (see Riccati Equation) — one for the controller ($P$), one dual copy for the filter ($\Sigma$). That duality is the engine behind the separation principle.

9 — Two hidden prerequisites the parent leans on

The prerequisite map

Equipment checklist

Cover the right side and answer aloud; reveal to check.