Swinging up the spherical pendulum via stabilization of its first integrals.
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.
scientific article; zbMATH DE number 2056661