---
tag: DJB8MAV5BH
kind: problem
name: Linear-quadratic-Gaussian control
msc: ["93E20", 49N10, "93E11"]
uses: [HR720TJZZR, 22735FGJV6, B219G9SHJA, 01MMVW8CKC, 0KX66CZMBB, 0WQNRCNY3M, HNK1Y62G7V]
about: [HR720TJZZR]
sources:
  - url: https://en.wikipedia.org/wiki/Linear–quadratic–Gaussian_control
same_as:
  wikidata: Q1798304
  wikipedia: Linear–quadratic–Gaussian control
history:
  posed_by: "Peter D. Joseph and Julius T. Tou"
  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)."
---

## Problem

Let $A$, $B$, $C$ be real matrices of sizes $n \times n$, $n \times m$ and $k \times n$, and let $T > 0$.
Let $Q, F$ be symmetric [[positive semidefinite|01MMVW8CKC]] $n \times n$ matrices, and let $R$ be a symmetric [[positive definite|0KX66CZMBB]] $m \times m$ matrix.
Let $W$ and $V$ be independent [[Wiener processes|22735FGJV6]] in $\R^n$ and $\R^k$ with covariance matrices $\Xi t$ and $\Theta t$.
Here $\Xi$ is positive semidefinite and $\Theta$ is positive definite.
Let the initial state $x(0)$ be Gaussian with mean $m_0$ and covariance $\Sigma_0$, independent of $W$ and $V$.
The state $x$ and the observation $y$ solve the [[stochastic differential equations|B219G9SHJA]]
$$dx = (A x + B u) \, dt + dW, \qquad dy = C x \, dt + dV, \qquad y(0) = 0.$$
A 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 $y$.
The *linear-quadratic-Gaussian control problem* is to minimize the [[expected value|0WQNRCNY3M]]
$$J(u) = \mathbb{E}\Big[ \int_0^T \big( x^{T} Q x + u^{T} R u \big) \, dt + x(T)^{T} F x(T) \Big]$$
over all admissible controls $u$.

## Remarks

The problem is solved, and its solution is the separation principle.
The 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.
Without noise and with full observation of the state, the problem reduces to the linear-quadratic regulator.
The infinite-horizon version uses the average cost per unit time and the algebraic Riccati equations of the regulator and of the filter.
LQG controllers can have poor robustness margins, which was one motivation for robust control methods.

## Explanations

### L1 human

The linear-quadratic-Gaussian problem is the linear-quadratic regulator in a noisy world.
The system is pushed by random disturbances, and one cannot see its state, only noisy measurements of some of its parts.
One still wants to keep the state near zero with little control effort, on average.
The answer splits into two simple parts: estimate the state as well as possible, then act as if the estimate were the true state.