Stabilization of the inverted cart-pendulum system with linear friction

This study presents the design of a feedback-based smooth and continuous control action, for the movement stabilization of a mechanical system consisted of a dampen pendulum mounted on a cart. the designs assumes that the pendulum is initialized in the upper-half plane. Also assumes that the involved viscous friction force is known. The control strategy is based on some nonlinear transformation that allow to rewrite the system as if it were an simple chain of integrator of fourth degree, perturbed by a nonlinear function. This nonlinear function, which models the dynamics of the original system, vanishes at the origen. Once the system has been transformed, the simple control action, consisted of two parts, one linear and other nonlinear or linear by parts (saturation function), is applied. After that, the linear part of the control action brought the pendulum near to the origin. Subsequently, the quasilinear part slowly brought the cart to the origin, assuring the desired behaviour in closed-loop. The stability analysis, which supports the feedback control action, follows an approach known a the Lyapunov method. Convincing numerical simulations and laboratory experiments that were included.