Linear Quadratic Regulator (LQR)

Instead of picking pole locations, LQR asks what you care about: how much each error matters and how expensive effort is. It then finds the feedback gains that minimize that total cost, which is often the best starting point for a real robot controller.

Weight on position error q Weight on effort r

The cost

LQR picks the feedback $u = -K\mathbf{x}$ that minimizes

$$J = \int_0^\infty \left(\mathbf{x}^\top Q\,\mathbf{x} + u^\top R\,u\right)dt, \qquad K = R^{-1}B^\top P$$

where $P$ solves the algebraic Riccati equation $A^\top P + PA - PBR^{-1}B^\top P + Q = 0$. For a cart with $Q = \mathrm{diag}(q, 0)$, the answer has a closed form:

$$k_1 = \sqrt{q/r} \qquad k_2 = \sqrt{2\sqrt{q/r}}$$

What to notice

Only the ratio $q/r$ matters. Make effort cheap, with a small $r$, and the cart rushes back with big forces. Make it expensive and the controller is gentle and slower. LQR always gives a well-damped response, with a damping ratio of about 0.7 here, and it guarantees good stability margins. The faint curves show the same start for ten times cheaper and ten times costlier effort, to compare.