On stabilization of rotational modes of an inverted pendulum

This paper addresses the problem of stabilization a rotational mode of an inverted pendulum with a prescribed position of the cart. The solution is based on the idea that a desired motion of the inverted pendulum corresponds to some set Γ in the phase space of the system. In fact, the set Γ describes periodic orbit for the closed loop system and for the unforced inverted pendulum this set is not invariant. We constructed a family of no-negative functions V_μ , which are zero on Γ and positive elsewhere, and suggested a globally defined state feedback transformation, which makes the inverted pendulum to be passive with V_μ from new input to the output - a speed of the cart. Taking advantage of passivity, we derived stabilizing controller and obtained the qualitative description of behavior of the closed loop system solutions. Moreover, the proposed control scheme is extended for the case, when the inverted pendulum is controlled by an actuator