LQR computes the cart-pole's gains from the Riccati equation

Adam Zettel · September 22, 2026 · written with an agent

K = R^{- 1} B^{⊤} P, A^{⊤} P + P A - P B R^{- 1} B^{⊤} P + Q = 0

The PD-controlled cart-pole has four gains that were tuned by hand. Linearize the same dynamics, choose how much each state error and each unit of push should cost, and the Riccati equation gives the gains instead. From 25 degrees they land the pole thirteen times closer to upright than the hand-tuned ones, for a third less cost and about the same peak push.

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Linearize the cart-pole about the upright and you have $z ˙ = A z + B u$ . Pick $Q$ for how much you dislike each state and $R$ for how much you dislike pushing, and the gains that minimize $\int z^{⊤} Q z + u^{⊤} R u d t$ are $K = R^{- 1} B^{⊤} P$ , where $P$ solves the algebraic Riccati equation. The only choices made by hand are $Q$ and $R$ , and they state the tradeoff directly: angle error against cart position, and state error against force.

$A$ and $B$ are not written out here. They are computed from the library's own cartpole_rate by central differences, so the linear model matches the nonlinear one this post simulates, and the first test checks that they agree near upright.

$P$ comes from integrating the differential Riccati equation backwards until it stops changing, which takes only matrix multiplication and addition. To check it, put $P$ back into the algebraic equation: the largest entry of the residual is $7 \times 1 0^{- 13}$ .

What comes out, on this cart-pole with these weights: $K = (- 4.47, - 8.22, - 63.03, - 16.02)$ , and a closed loop whose slowest mode decays like $e^{- 1.28 t}$ . Simulating the nonlinear cart-pole for six seconds from 25 degrees, those gains end 0.00096 rad from upright against the hand-tuned gains' 0.0124, thirteen times closer, with an integrated cost of 6.13 against 8.98 and a peak force of 27.5 N against 26.2. The hand-tuned gains also balance the pole; the Riccati gains are the ones that minimize the cost set by $Q$ and $R$ .

Printed

K: [-4.47214, -8.22385, -63.0331, -16.0163]

residual: 6.53699e-13

cl_re: [-8.39442, -4.45241, -1.28419, -1.28419]

slowest: -1.28419

lqr_run: [0.000962049, 6.13006, 27.5034]

pd_run: [0.0124326, 8.9753, 26.1799]

The pole's angle, from 25 degrees

LQR, gains computedPD, gains by hand

Tests 5 of 5 pass

the linear model A z + B u matches the nonlinear rate near upright (passes)
P solves the algebraic Riccati equation: the residual is under 1e-8 (passes)
every closed-loop eigenvalue has a negative real part (passes)
from 25 degrees the pole ends upright, within a thousandth of a radian (passes)
the computed gains cost less than the hand-tuned ones on the same start (passes)