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  "body": "\n## Problem\n\nLet $A$, $B$, $C$ be real matrices of sizes $n \\times n$, $n \\times m$ and $k \\times n$, and let $T \u003e 0$.\nLet $Q, F$ be symmetric [[positive semidefinite|01MMVW8CKC]] $n \\times n$ matrices, and let $R$ be a symmetric [[positive definite|0KX66CZMBB]] $m \\times m$ matrix.\nLet $W$ and $V$ be independent [[Wiener processes|22735FGJV6]] in $\\R^n$ and $\\R^k$ with covariance matrices $\\Xi t$ and $\\Theta t$.\nHere $\\Xi$ is positive semidefinite and $\\Theta$ is positive definite.\nLet the initial state $x(0)$ be Gaussian with mean $m_0$ and covariance $\\Sigma_0$, independent of $W$ and $V$.\nThe state $x$ and the observation $y$ solve the [[stochastic differential equations|B219G9SHJA]]\n$$dx = (A x + B u) \\, dt + dW, \\qquad dy = C x \\, dt + dV, \\qquad y(0) = 0.$$\nA control is admissible if $u(t)$ depends only on the observations $y(s)$, $s \\le t$, that is, $u$ is adapted to the [[filtration|HNK1Y62G7V]] generated by $y$.\nThe *linear-quadratic-Gaussian control problem* is to minimize the [[expected value|0WQNRCNY3M]]\n$$J(u) = \\mathbb{E}\\Big[ \\int_0^T \\big( x^{T} Q x + u^{T} R u \\big) \\, dt + x(T)^{T} F x(T) \\Big]$$\nover all admissible controls $u$.\n\n## Remarks\n\nThe problem is solved, and its solution is the separation principle.\nThe optimal control is $u = -R^{-1} B^{T} P(t) \\hat{x}(t)$, where $P$ solves the Riccati equation of the [[linear-quadratic regulator|HR720TJZZR]] and $\\hat{x}(t)$ is the Kalman–Bucy filter estimate of $x(t)$ from the observations.\nWithout noise and with full observation of the state, the problem reduces to the linear-quadratic regulator.\nThe infinite-horizon version uses the average cost per unit time and the algebraic Riccati equations of the regulator and of the filter.\nLQG controllers can have poor robustness margins, which was one motivation for robust control methods.\n\n## Explanations\n\n### L1 human\n\nThe linear-quadratic-Gaussian problem is the linear-quadratic regulator in a noisy world.\nThe system is pushed by random disturbances, and one cannot see its state, only noisy measurements of some of its parts.\nOne still wants to keep the state near zero with little control effort, on average.\nThe answer splits into two simple parts: estimate the state as well as possible, then act as if the estimate were the true state.\n",
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      "first_stated": "1961",
      "notes": "Joseph and Tou stated the separation of estimation and control in On linear control theory, AIEE Transactions (1961), after the work of Kalman on linear filtering and regulation (1960).",
      "posed_by": "Peter D. Joseph and Julius T. Tou"
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  "markdown": "https://tamnd.github.io/math2/node/DJB8MAV5BH.md",
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  "name": "Linear-quadratic-Gaussian control",
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