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  "body": "\n## Statement\n\nConsider the [[linear-quadratic-Gaussian control problem|DJB8MAV5BH]] on $[0, T]$ with matrices $A, B, C, Q, R, F$, noise covariances $\\Xi$ and $\\Theta$, and initial mean $m_0$ and covariance $\\Sigma_0$.\nLet $P$ solve the [[Riccati differential equation|Y4V154JWR7]] $-P' = A^{T} P + P A - P B R^{-1} B^{T} P + Q$ with $P(T) = F$.\nLet $\\Sigma$ solve the filter Riccati equation $\\Sigma' = A \\Sigma + \\Sigma A^{T} + \\Xi - \\Sigma C^{T} \\Theta^{-1} C \\Sigma$ with $\\Sigma(0) = \\Sigma_0$.\nLet $\\hat{x}$ be the Kalman–Bucy filter, the solution of\n$$d\\hat{x} = (A \\hat{x} + B u) \\, dt + \\Sigma C^{T} \\Theta^{-1} (dy - C \\hat{x} \\, dt), \\qquad \\hat{x}(0) = m_0.$$\nThen the control $u^*(t) = -R^{-1} B^{T} P(t) \\hat{x}(t)$ is admissible and optimal among all admissible controls.\nHere $\\hat{x}(t)$ is the [[conditional expectation|0TTDGTEFMN]] of $x(t)$ given the observations $y(s)$, $s \\le t$.\n\n## Remarks\n\nThe proof has two parts.\nFirst, for every admissible control, the error $x - \\hat{x}$ is a Gaussian process whose law does not depend on $u$, with covariance $\\Sigma(t)$.\nSecond, writing $x = \\hat{x} + (x - \\hat{x})$ splits the cost into a term that does not depend on $u$ and the cost of a fully observed linear-quadratic problem for $\\hat{x}$.\nThe [[linear-quadratic regulator|HR720TJZZR]] then gives the optimal feedback for $\\hat{x}$.\nThe gain $R^{-1} B^{T} P$ is the same as without noise, and the filter gain $\\Sigma C^{T} \\Theta^{-1}$ does not depend on the cost: estimation and control separate.\nA deterministic version holds for observer-based feedback of $x' = A x + B u$, $y = C x$.\nWith the observer $\\hat{x}' = A \\hat{x} + B u + L(y - C \\hat{x})$ and $u = -K \\hat{x}$, the error $e = x - \\hat{x}$ solves $e' = (A - L C) e$.\nIn the coordinates $(x, e)$ the closed-loop matrix is block triangular, so its eigenvalues are those of $A - B K$ together with those of $A - L C$.\nThe principle depends on linear dynamics, Gaussian noise and quadratic cost; for nonlinear problems the best control may need to act so as to improve future estimates.\n\n## Explanations\n\n### L1 human\n\nWhen you must control a system that you can only observe through noisy measurements, the separation principle gives a simple recipe.\nFirst build the best estimate of the current state from the measurements.\nThen apply the control that would be optimal if that estimate were exactly right.\nFor linear systems with Gaussian noise and quadratic costs, this two-step recipe is exactly optimal.\n\n### L2 human\n\nThe estimator is the Kalman–Bucy filter and the controller is the [[linear-quadratic regulator|HR720TJZZR]] gain, and each comes from its own Riccati equation.\nThe two Riccati equations are dual: the filter equation is the control equation for $(A^{T}, C^{T})$ run forward in time.\nIn practice this lets engineers design the estimator and the controller separately.\nThe deterministic version is the reason why pole placement for the controller and for the observer can be done independently.\n",
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