State Feedback and Linear Quadratic Regulation

Corresponds to lectures: Pole placement by state feedback; Linear quadratic regulator: discrete-time finite horizon; LQR via Lagrange multipliers; Infinite horizon linear quadratic regulator; Continuous-time linear quadratic regulator; Linear quadratic stochastic control. Both halves of this note design a feedback law $u = -Kx$ using controllability (controllability-observability-and-canonical-forms), but with different goals. Pole placement picks $K$ directly to put the closed-loop eigenvalues ($A - BK$) wherever desired. LQR instead picks $K$ to minimize a quadratic cost trading off state deviation against control effort — derived first via Lagrange multipliers for the finite-horizon discrete-time case, then extended to the infinite-horizon and continuous- time cases (where the cost-to-go satisfies a Riccati equation), and finally to the stochastic case, where the optimal feedback law turns out to be the same as the noise-free case (certainty equivalence) — the fact that makes this note’s LQR pair naturally with estimation-and-the-kalman-filter‘s state estimation.

Outline (TODO — flesh out each)

• Pole placement by state feedback
• Linear quadratic regulator: discrete-time finite horizon
• LQR via Lagrange multipliers
• Infinite horizon LQR — the algebraic Riccati equation
• Continuous-time LQR
• Linear quadratic stochastic control — certainty equivalence