Periodic Trajectory Tracking for Control-Affine Driftless Systems on Compact Lie Groups

Abstract

We treat the periodic trajectory tracking problem: given a periodic trajectory of a control-affine, left-invariant driftless system in a compact and connected Lie group G and an initial condition in G, find another trajectory of the system satisfying the initial condition given and that asymptotically tracks the periodic trajectory. We solve this problem locally (for initial conditions in a neighborhood of some point of the periodic trajectory) when G is semisimple and the system is Lie-determined (i.e., controllable), and only for a class of periodic trajectories (which we call regular). Finally, we present a set of sufficient conditions to ensure the existence of such trajectories.