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: $x_{t+1}=A{x}_t+B{u}_t$ where

• $x_t$ is the state at time $t$
• $u_t$ is the input at time $t$

It assumes a quadratic cost function $g(x_t,u_t)=x_t^{T}Q{x}_t+u_t^{T}R{u}_t$ $Q$, $R$ are symmetric with $Q>0$, $R>0$. $Q>0⟺\forall z:z^{t}Qz>0$

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.