Here are the equations used:

is the length of the pendulum

is the small mass on top of the pendulum

is the mass of the sled

is the force applied to the sled to keep the pendulum in the vertical position

is the force on the small mass (here 0)

is the rotation of the pendulum

is the rotation rate of the pendulum

is the rotational acceleration of the pendulum

is the displacement of the sled

is the velocity of the sled

is the acceleration of the sled

is the acceleration due to gravity

is the angular damping of inverted pendulum shaft

is the initial angle of the pendulum

is the initial angle rate assumed as zero

is the initial displacement of the sled assumed at zero

is the initial velocity of the sled assumed at zero

is the proportional displacement gain

is the proportional angle gain

is the derivative displacement gain

is the derivative angle gain