Intuition The one core ideaThe LQR takes a machine whose behaviour is described by a linear rule of motion and finds the single feedback recipe u = − K x
Before you can read one line of the parent note, you must own every symbol it throws at you. This page builds each from absolute zero, in the order that lets each rest on the one before it.
Everything in LQR happens to a single moving point . That point is the machine's situation right now .
x state x at this instant . For a satellite sliding along one axis, that list is (position, velocity). Stack those numbers into a column and you get one arrow from the origin to a point.
Plain words: x The picture: a dot in a flat plane (for 2 numbers). The origin is "perfect — at target, not moving". Being away from the origin is bad .Why the topic needs it: LQR's whole job is to drive this dot to the origin. If you don't picture the state as a movable point, nothing else makes sense.
R n n R − 3 0 2.7 2 n n x ∈ R n "x n . The ∈ belongs to .
Two dimensions we can draw (n = 2 n by analogy — still "a point that can be near or far from the origin".
Definition Vector — a stacked column of numbers = an arrowA vector is a column of numbers. Geometrically it is an arrow from the origin to a point.
x = [ x 1 x 2 ] e.g. [ position velocity ]
Definition Matrix — a grid that transforms arrowsA matrix A job is to eat one arrow and spit out another arrow — it stretches, rotates, or shears space. When you see A x x A
Definition Matrix shape "rows × columns"Every matrix has a shape written rows × columns . A matrix that is r × c r c R n n × 1 n 1
Definition Matrix–vector multiplication and the "shapes must match" ruleTo compute A x row of A column x [ a c b d ] [ x 1 x 2 ] = [ a x 1 + b x 2 c x 1 + d x 2 ] The one rule you must obey: the number of columns on the left must equal the number of rows on the right — otherwise the pairing has leftovers and the product is undefined. An r × c c × 1 r × 1
Definition The dimensions of every matrix in LQRWith x ∈ R n n u ∈ R m m forces every matrix's size:
Matrix
Shape
Why that shape
A n × n A x n x ˙ A n n
B n × m B u m u n x ˙
Q n × n sits between x ⊤ x x ⊤ Q x n
R m × m sits between u ⊤ u u ⊤ R u m
K m × n K x n x m
Nothing here is arbitrary — every shape is the only one that makes the multiplication legal.
A ⊤ A ⊤ r × c c × r [ a b ] [ a b ] multiply a row by a column (as in x ⊤ Q x x n × 1 x ⊤ 1 × n x ⊤ Q 1 × n
Definition Symmetric matrixM = M ⊤ symmetric if flipping it across its diagonal changes nothing: the number in row i j j i Q R
This single expression appears three times on the parent page (x ⊤ Q x u ⊤ R u x ⊤ P x
Definition Quadratic form — squared distance, weightedx ⊤ M x x ⊤ 1 × n M n × n 1 × n x n × 1 1 × 1 one single number . When M x 1 2 + x 2 2 M
Why squared (why not ∣ x ∣ A square is smooth at the bottom (a rounded bowl), so calculus can find its minimum cleanly. An absolute value has a sharp corner — the optimizer chokes on it. This smoothness is the whole reason LQR has a tidy closed-form answer.The picture: x ⊤ M x bowl sitting over the plane. The origin is the bottom (value 0 M
Definition Positive (semi)definite:M ≻ 0 M ⪰ 0 M ⪰ 0 M semi definite") means x ⊤ M x ≥ 0 every arrow x M ≻ 0 x ⊤ M x > 0 non-zero x genuine bowl with no flat valley floor.
Symbol
Bowl shape
Used for
⪰ 0 may have a flat floor
Q some directions
≻ 0 strictly curved everywhere
R unique cheapest control exists, and so R − 1
This table is exactly why the parent insists "R ≻ 0 R − 1
Definition Matrix inverseR − 1 r r − 1 = 1/ r r = 0 R − 1 undoes R R R − 1 R − 1 R = R R − 1 = I , I identity matrix (1's on the diagonal, 0's elsewhere — the "do-nothing" matrix, the matrix version of the number 1 When does it exist? Only for a square matrix that does not squash any non-zero arrow to zero. A positive-definite R ≻ 0 u u ⊤ R u > 0 R − 1 guaranteed to exist — which is exactly why the gain K = R − 1 B ⊤ P
Definition The time variablet d t t time parameter — a single real number counting seconds from a chosen start, t = 0 t x x ( t ) t d t infinitesimally small step of time — an instant so short it is "the next moment". You will see it as the multiplier inside the cost integral, tagging how long each instant's badness is paid for.
x ˙ derivative ), x ˙ = d t d x x ˙ instantaneous velocity of the state point : which way, and how fast, the dot is sliding through the plane.
Why a derivative and not a difference? Physics happens continuously — the machine doesn't jump in discrete steps, it flows. The derivative is the tool that measures flow at an instant.The picture: at each point, x ˙
x ˙ = A x + B u is the machine. It says the state's velocity x ˙ n
A x own tendency (A n × n A x n B u push we add through the control u B n × m m u n
Both terms are n x ˙ x ˙ x u A , B
u ∈ R m u m only freedom we have; A B
K u = − K x K recipe : a fixed m × n u = − K x "look at how far off you are (x K n x m . Bigger error ⇒ bigger corrective push, automatically, everywhere.
Why the minus sign? Feedback must oppose the error, like a spring pulling a stretched mass back. A plus sign would push you further off — the opposite of control.Why proportional to x The parent's punchline: linear plant + quadratic (bowl) cost forces the best recipe to be exactly this linear one. See Pole Placement and PID Controllers for the same − K x
∫ 0 ∞ ( … ) d t ∫ ∫ 0 ∞ f d t f t t = 0 ∞ d t
The picture: two bowls at every instant; J
V ( x ) = x ⊤ P x V ( x ) best remaining total cost if you start at state x V x ⊤ P x n × n P ⪰ 0 HJB equation .
Definition The Riccati equation — the puzzle that findsP A ⊤ P + P A − P B R − 1 B ⊤ P + Q = 0 one matrix equation you solve once for P K = R − 1 B ⊤ P − P B R − 1 B ⊤ P P ⪰ 0 Controllability and Observability and Lyapunov Stability .
Real numbers R and lists R to the n
Matrix shapes and multiplication rule
Transpose and symmetric matrices
Quadratic form x-top M x = a bowl
Positive definite Q and R
Derivative x-dot = velocity
Control u and feedback law u = -K x
Value function V = x-top P x
Read each cue, answer aloud, then reveal. If any stumps you, re-read that section.
What does x ∈ R n x n
What is a matrix's job geometrically? It eats an arrow and returns a transformed arrow (stretch/rotate/shear); A x x A
State the "shapes must match" rule for multiplying A The number of columns of A r × c c × 1 r × 1
What are the shapes of A B Q R K A n × n B n × m Q n × n R m × m K m × n
What does A ⊤ Flips rows into columns (r × c → c × r x ⊤ Q x
What single number does x ⊤ M x One scalar measuring weighted squared distance from the origin — a bowl over the plane.
Why square the error instead of using ∣ x ∣ A square is smooth at the bottom, so calculus finds a clean unique minimum; ∣ x ∣
Difference between M ⪰ 0 M ≻ 0 ⪰ 0 x ⊤ M x ≥ 0 ≻ 0 > 0 x = 0
What is R − 1 The "undo" matrix with R − 1 R = I R ≻ 0
Why must R ≻ 0 ⪰ 0 So a unique cheapest control exists and R − 1 K = R − 1 B ⊤ P
What is t d t t t = 0 d t
What does x ˙ The instantaneous velocity of the state point — its rate of change with time, d x / d t
Read out x ˙ = A x + B u The state's velocity is its own natural drift A x B u
Why the minus sign in u = − K x Feedback must oppose the error and pull the state back toward the origin, like a spring.
What does ∫ 0 ∞ ( ⋯ ) d t The running total badness (error + effort) summed continuously over all future time.
What is V ( x ) = x ⊤ P x The best possible remaining total cost from state x P ⪰ 0
What does the Riccati equation deliver? The matrix P K = R − 1 B ⊤ P
