Invariant manifolds in control problems

Abstract

Invariant manifolds are useful tools for the investigation of nearly all nonlinear systems. Especially for the determination of stabilizing controls the center-stable manifold characterizes the proper feedback controls.

The method is demonstrated for the stabilization of a tethered satellite in the local vertical position by applying tension control. While in-plane perturbations can be extinguished in finite time, the tension control acts as parametric excitation for out-of-plane perturbations and is only able to cause a slow algebraic decay for both kinds of perturbations. An analytical or numerical power series expansion of the center-stable manifold at the target state provides the proper feedback controls.