LQR Control for an Inverted Pendulum

A LQR controller is used to stabilize a cart-pole system around its unstable equilibrium with the cart at the origin and the pole in its upright position. A region of attraction for the LQR controller on the systems linearized dynamics is empirically found. Outside of this region an energy pumping scheme is derived to drive the system to its homoclinic orbit. A hybrid system is found to stabilize the cart-pole from any initial conditions by switching to the LQR policy when the system reaches its region of attraction. The cart-pole is described by the following dynamics: