## Linear Control Theory: Part 0

The purpose of this post is to introduce you to some of the basics of control theory and to introduce the Linear-Quadratic Regulator, an extremely good hammer for solving stabilization problems.

To start with, what do we mean by a control problem? We mean that we have some system with dynamics described by an equation of the form

$\dot{x} = Ax,$

where $x$ is the state of the system and $A$ is some matrix (which itself is allowed to depend on $x$). For example, we could have an object that is constrained to move in a line along a frictionless surface. In this case, the system dynamics would be

$\left[ \begin{array}{c} \dot{q} \\ \ddot{q} \end{array} \right] = \left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]\left[ \begin{array}{c} q \\ \dot{q} \end{array} \right]. $

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