Inverted Pendulum on a Cart

The inverted pendulum on cart to be controlled is shown in the attached image. Its structure consists of a cart and pendulum where the pendulum is hinged to the cart via a pivot and only the cart is actuated, where:

Lagrange's equations are applied with respect to $\theta$ and $x$ coordinates. The nonlinear state space model of inverted pendulum on cart system can then be obtained as:

$$\begin{align*} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= \dfrac{mg \sin{x_1}+ \dfrac{ m\cos{x_1}(u-f\dot{x})-m^2lx_2^2\cos{x_1}\sin{x_1}}{(M + m)}}{\dfrac{4}{3}ml - \dfrac{m^2l\cos^2{x_1}}{(M+m)}} \\ \dot{x}_3 &= x_4 \\ \dot{x}_4 &= \dfrac{u - f x_1 + \dfrac{3}{4}mg\cos{x_1}\sin{x_1} - mlx_2^2\sin{x_1}}{(M+m) - \dfrac{3}{4}m\cos^2{x_1}}, \end{align*}$$

where the state variables are consequently assigned as $x_1 = \theta$, $x_2 = \dot{\theta}$, $x_3 = x$ an $x_4=\dot{x}$, and where the input $u$ is the applied force $F$.