  For linear time-invariant systems, a separation principle holds: stable
observer and stable state feedback can be designed for the time-invariant
system, and the combined observer and feedback will be stable. For non-linear
systems, a local separation principle holds around steady-states, as the
linearized system is time-invariant. This paper addresses the issue of a
non-linear separation principle on Lie groups. For invariant systems on Lie
groups, we prove there exists a large set of (time-varying) trajectories around
which the linearized observer-controler system is time-invariant, as soon as a
symmetry-preserving observer is used. Thus a separation principle holds around
those trajectories. The theory is illustrated by a mobile robot example, and
the developed ideas are then extended to a class of Lagrangian mechanical
systems on Lie groups described by Euler-Poincare equations.
