# Linear Quadratic Regulator (LQR)

LQR is a Control method used to design a controller for linear systems, which seeks to optimize a quadratic cost function based on the system’s state and control inputs.

Brian Machado & Friends used LQR on the BracketBot, so cool.

Wow this is what is used to build the autonomous Helicopter.

The LQR setting assumes a linear dynamical system: $x_{t+1}=Ax_{t}+Bu_{t}$ where

- $x_{t}$ is the state at time $t$
- $u_{t}$ is the input at time $t$

It assumes a quadratic cost function $g(x_{t},u_{t})=x_{t}Qx_{t}+u_{t}Ru_{t}$ $Q$, $R$ are symmetric with $Q>0$, $R>0$. $Q>0⟺∀z:z_{t}Qz>0$

What happens when Q and R are not symmetric? Peter abbeel derives in class for lecture 5, we only care about the symmetric part anyways.