Consider the inverted pendulum system shown in Fig. 1. We desire to balance the pendulum on the vertical position by controlling the cart position and velocity. m₁ mg L Friction = bx Fig. 1 Inverted pendulum system The actual system is nonlinear and the desired equilibrium point is the zero point (0-0). If we linearize the system in the neighborhood of this point we obtain the linearized equations. Your task is to design a closed loop controller to control the position of the pendulum, whose inclination can be changed by the controller. 1. Derive the dynamic equations of the system. The nonlinear equations of motion can be determined using the Euler-Lagrange method.