Balancing
The pole starts 15 degrees from upright. A controller that pushes the cart in proportion to the pole's angle and rate, and a little toward the origin, catches it in under two seconds and holds it. At a fifth of the gains it falls.
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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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Swinging up
An acrobot is two links with one motor, at the elbow; the shoulder turns freely. A linear controller can balance it only near upright, so from hanging down it first has to swing itself up. The swing-up controller here was not derived: it is nine weights found offline by a cross-entropy search over 1600 simulated swings, run as compiled native code, and the page contains only those nine numbers. In your browser the arm swings up in 1.188 seconds, hands over to a balancing controller, and settles to within 0.005 radians of upright.
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