Stabilization of a cart with inverted pendulum

We consider the problem of stabilizing a cart moving along a straight line with an inverted pendulum installed on it. The control objective is to stabilize the cart at a given target point so that the pendulum is in the upper vertical position. The main difficulty associated with solving this problem is that the two subsystems (the cart and the pendulum) must be stabilized simultaneously using one control. A new control law is proposed based on the introduction of a second-order reference system, the trajectory of which is taken as the target one for the cart with the pendulum. By extending the reference system to the fourth order and introducing an algebraic condition coupling the two systems, the target trajectory is found in the four-dimensional phase space of the original system and a control law is constructed that ensures the trajectory of the closed-loop system asymptotically approaching the target one. The control law obtained in this paper is applicable to systems with an arbitrary ratio of pendulum and cart masses, since the closed-loop system does not depend on the mass characteristics of the system. The range of the system parameters is found for which the linearized system is stable. The presentation is illustrated with numerical examples that demonstrate the efficiency of the proposed control.