# Linear Quadratic Gaussian (LQG) Control

In control theory, the linear–quadratic–Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns linear systems driven by additive white Gaussian noise. The problem is to determine an output feedback law that is optimal in the sense of minimizing the expected value of a quadratic cost criterion. Output measurements are assumed to be corrupted by Gaussian noise and the initial state, likewise, is assumed to be a Gaussian random vector.

Under these assumptions an optimal control scheme within the class of linear control laws can be derived by a completion-of-squares argument. This control law which is known as the LQG controller, is unique and it is simply a combination of a Kalman filter (a linear–quadratic state estimator (LQE)) together with a linear–quadratic regulator (LQR). The separation principle states that the state estimator and the state feedback can be designed independently. LQG control applies to both linear time-invariant systems as well as linear time-varying systems, and constitutes a linear dynamic feedback control law that is easily computed and implemented: the LQG controller itself is a dynamic system like the system it controls. Both systems have the same state dimension.

This topic includes the following resources and journeys:

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## State Space, Part 4: What is LQR control?

17 min
Beginner
Video
Theory

The Linear Quadratic Regulator (LQR)LQR is a type of optimal control that is based on state space representation. In this video, we introduce this topic at a very high level so that you walk...

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## Guaranteed Margins for LQR Regulators

10 min
Intermediate
Peer Reviewed Paper
Theory

John Doyle's famous paper! He presents a counterexample that shows that are no guaranteed margins for LQG systems.

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## MATLAB Command: lqr

Intermediate
Article / Blog
Application

MATLAB command documentation for the Linear-Quadratic Regulator (lqr) function.

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## Train Custom LQR Agent with MATLAB

Intermediate
Example
Application

This example shows how to train a custom linear quadratic regulation (LQR) agent to control a discrete-time linear system modeled in MATLAB®.

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## Control Bootcamp: LQG Example in Matlab

13 min
Beginner
Video
Theory

This video combines the LQR and Kalman filter in Matlab on the example of an inverted pendulum on a cart. We stabilize the full nonlinear system with a measurement of a single variable (the...

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## The Linear Quadratic Regulator (LQR)

Intermediate
Article / Blog
Theory

Lecture notes for ECE717 on LQR control by Laurent Lessard. There is a section that shows how the Algebraic Riccati Equation is part of the LQR solution by "completing the square".

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Beginner
Article / Blog
Theory

In these notes, we will derive the solution to the finite-horizon linear quadratic regulator (LQR) problem in several different ways. Fundamentally, LQR can be viewed as a large least...

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## Introduction to Linear Quadratic Regulator (LQR) Control

96 min
Beginner
Video
Theory

In this video we introduce the linear quadratic regulator (LQR) controller. We show that an LQR controller is a full state feedback controller where the gain matrix K is computed by solving...

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## Why the Riccati Equation Is important for LQR Control

14 min
Intermediate
Video
Theory

This Tech Talk looks at an optimal controller called linear quadratic regulator, or LQR, and shows why the Riccati equation plays such an important role in solving it efficiently. The talk...

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## Linear Quadratic Regulator (LQR) Control for the Inverted Pendulum on a Cart...

13 min
Beginner
Video
Theory

Here we design an optimal full-state feedback controller for the inverted pendulum on a cart example using the linear quadratic regulator (LQR). In Matlab, we find that this is a simple one...

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## Control Bootcamp: Linear Quadratic Gaussian (LQG)

8 min
Beginner
Video
Theory

This lecture combines the optimal full-state feedback (e.g., LQR) with the optimal full-state estimator (e.g., LQE or Kalman Filter) to obtain the sensor-based linear quadratic Gaussian (LQG...

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## Peter Ponders PID - LQR Optimizing Two Outputs

16 min
Beginner
Video
Theory
To make this LQR video different I use LQR to control two output and find the best 'compromise' between controlling the level and temperature in a mixing tan... See More

## Peter Ponders PID - Yet Another Linear Quadratic Control Video but...

22 min
Beginner
Video
Theory
This videos covers topics not found or hard to find in control books or other videos.This video converts tracking position, velocity and acceleration for a m... See More