Stabilization of a chain of integrators with nonlinear perturbations: application to the inverted pendulum

A solution to the long standing problem of the pendulum on a cart is presented. This solution involves a reformulation of the original system model in order to show that the inverted pendulum system belongs to a particular class of nonlinear systems: a class consisting of four cascaded integrators and a nonlinear "perturbation" term. Our controller first brings the pendulum close to the vertical unstable equilibrium point and then regulates the cart position around the origin. We prove that using the proposed control strategy, all state variables converge to zero for a given set of initial conditions (θ(0), θ(0)) belonging to a special domain of attraction.