Abstract: We consider affine switched systems as perturbations of linear ones, the equilibria playing the role of perturbation parameters. We study the stability properties of an affine switched system under arbitrary switching, assuming that the corresponding linear system is uniformly exponentially stable. It turns out that the affine system admits a minimal invariant set Ω, whose properties we investigate. In the two-dimensional bi-switched case when both subsystems have non-real eigenvalues we are able to characterize Ω completely and to prove that all trajectories of the system converge to Ω. We also explore the behavior of minimal-time trajectories in Ω by constructing optimal syntheses.