Abstract
This paper develops the swinging-up control of an inverted pendulum using the Lie-Backlund approach to equivalence and flatness. In this system, the friction between the cart and the rail is considered to reach the real system. Two goals of controlling the inverted pendulum are investigated in this work. The first one is to balance the pendulum by applying differential flatness; and the second one is to keep the inverted pendulum in a position of muddled vertical balance along a trajectory, when it initially starts with zero angle of the vertical position. Our main contribution consists in applying the exact flat output for the synthesis of the differential flatness control law. A better precise trajectory tracking is then obtained, with an accurate control signal developed for a particular case of a non minimum phase inverted pendulum.