Stabilization of stable manifold of upright position of the spherical pendulum

The stabilization problem of the upright position of the controlled spherical pendulum is treated in detail. This problem is reduced to the stabilization of the stable manifold /spl Omega//sub st/ of the upright position of the unforced spherical pendulum. It is shown that for any smooth feedback control derived by the speed-gradient algorithm with the objective to stabilize /spl Omega//sub st/ the closed loop system has a limit cycle /spl Gamma/, which does not belong to the desired attractor /spl Omega//sub st/. It is shown that /spl Gamma/ is hyperbolic.