What is the Linear Quadratic Regulator (LQR) and how is it designed?

LQR finds optimal state feedback gain K minimizing quadratic cost: J = integral(x'Qx + u'Ru)dt, where Q penalizes state deviation and R penalizes control effort. The optimal gain K = R^(-1)*B'*P where P is the solution of the algebraic Riccati equation. Q/R trade-off: larger Q/R ratio gives faster response but more control effort. LQR guarantees stability and good margins (GM >= 6 dB, PM >= 60 degrees) for single input. Extensions include LQG (adds Kalman filter for noisy measurements) and LQI (includes integral action).