  We propose a modification technique for discrete time systems for
exponentially fast convergence to compact sets. The extension technique allows
us to use tools defined on Euclidean spaces to systems evolving on manifolds by
modifying the dynamics of the system such that the manifold is an attractor
set. We show the stability properties of this technique using the simulation of
the rigid body rotation system on the unit sphere $S^3$. We also show the
improvement afforded due to this technique on a Luenberger like observer
designed for the rigid body rotation system on $S^3$.
