Control of a base-excited inverted pendulum with two degrees of rotational freedom

The purpose of this paper is to develop a method that can be used to study the control and stability of a base-excited inverted pendulum with multiple degrees of rotational freedom. The stabilization of such an inverted pendulum is achieved by applying appropriate control torques at the base point. The inclusion of base point motion leads to a control system with time varying parametric excitation which makes the control task challenging. The pendulum studied in this paper has two degrees of rotational freedom and the base point moves freely in the vertical direction with the only restriction being that the acceleration must be continuous. First, a piecewise continuous feedback control is designed to determine the stabilizing torques. Such a control law makes the control system non-smooth, which does not meet the requirement of all classical theories on the existence and uniqueness of solutions. Thus, the existence and uniqueness of the solution to the proposed control system are studied using the solution concept developed by Filippov. Lyapunovʹs second method and LaSalleʹs invariance principle are then employed to prove the global and asymptotic stability of the control system. The robustness of the controller with respect to the physical parameter variations and measurement errors is also investigated and the control system stability is shown to be largely insensitive to this class of uncertainties. In order to reflect the actual implementation scenario, the discontinuous terms in the control law are approximated by continuous functions. Such an approximation makes the stability analysis highly challenging. It is proven that the pendulum can be stabilized in a controlled region around the upright position using the generalized Lyapunov analysis concept in which a quasi-Lyapunov function is constructed. Simulations are performed to support the theoretical analyses presented in this study.