Optimal Control 1.0: LQR

This post is part of a series of posts on optimal control theory. We take a detalied look on how classical LQR control is derived. A simple implementation is provided for clarity.

Introduction

LQR is an extremely popular optomal control framework. This blog closely follows (Duriez, Brunton, and Noack 2017).

Notation

Small

Linear system

Let’s consider the linear system

˙x=Ax+Buy=Cx+Du

If the system in (1) is controllable then a proportional controller can be designed as

u=−Krx

Hence the closed loop system becomes

˙x=(A−BKr)x

We can construct a quadratic cost J that balances the regulation of x with the cost of control input u,

J(t)=∫t0[xT(τ)Qx(τ)]+uT(τ)Ru(τ)]

By solving Algebraic Riccati Equation (ARE) we get the optimal control law,

Kr=R−1BTP

where the ARE is expressed as

ATP+PA−PBR−1BTP+Q=0