Swinging up the spherical pendulum via stabilization of its first integrals. (Q1426265)

This paper considers stabilization of the hyperbolic upright equilibrium of the spherical pendulum. The authors' approach is to select a switching control law, restricting their efforts to finding the stable manifold of the upright equilibrium, and then to stabilize this set. The controlled spherical pendulum is a nonlinear system with two equilibrium points. At these points, in contrast to the equilibria of a planar pendulum, the vector field of the unforced spherical pendulum does not have any linear approximations. The unforced spherical pendulum has two independent first integrals -- the total energy \(H_0\) and the generalized momentum \(p_0\) that corresponds to rotation in the horizontal plane. The upright position belongs to the invariant subset \(\Omega_0\) of the phase space defined by \(H_0= 2mgl\) and \(p_0= 0\), where \(m\) and \(l\) are the mass and length of the pendulum and \(g\) is the acceleration due to gravity. The authors prove that \(\Omega_0\) is a stable manifold of the upright equilibrium. This makes it possible to construct a family of nonnegative storage functions that are zero on \(\Omega_0\) and positive outside in order to construct controllers. This ensures that the appropriate storage function decreases along the closed loop system solutions.