  Creating a simulation of a system enables the tuning of control systems
without the need for a physical system. In this paper, we employ Lagrangian
Mechanics to derive a set of equations to simulate an inverted pendulum on a
cart. The system consists of a freely-rotating rod attached to a cart, with the
rod's balance achieved through applying the correct forces to the cart. We
manually tune the proportional, integral, and derivative gain coefficients of a
Proportional Integral Derivative controller (PID) to balance a rod. To further
improve PID performance, we can optimize an objective function to find better
gain coefficients.
