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Hi good day. I am trying to find the general Inverted Pendulum on a cart nonlinear state space equations with two degrees of freedom with x, x_dot, theta, theta_dot which represents displacement, velocity, pendulum angle from vertical, angular velocity. However from research, I am seeing different state space equations. In the image attached is Matlab code I found that gives the proper functionality but I am not sure how it was derived.

y-state

m-mass of the pendulum

M- mass of the cart

L- length of pendulum line

g- gravity

d- friction

u- force on cart

The other image shows the closest set of equations I found but I am unsure. Please help thank you.

y-state

m-mass of the pendulum

M- mass of the cart

L- length of pendulum line

g- gravity

d- friction

u- force on cart

The other image shows the closest set of equations I found but I am unsure. Please help thank you.