  We give a general framework for the treatment of perturbations of types and
structures in continuous logic, allowing to specify which parts of the logic
may be perturbed. We prove that separable, elementarily equivalent structures
which are approximately $\aleph_0$-saturated up to arbitrarily small
perturbations are isomorphic up to arbitrarily small perturbations (where the
notion of perturbation is part of the data). As a corollary, we obtain a
Ryll-Nardzewski style characterisation of complete theories all of whose
separable models are isomorphic up to arbitrarily small perturbations.
