We study an interconnection between a switched affine system and a fast LTI dynamics. Often, the classical stabilization method based on the existence of a stable convex combination is directly used for systems on two time scales while using only the slow dynamics. Here, we provide conditions for checking the Uniform Ultimate Boundedness when the fast dynamics is taken into account. In addition, criteria for verifying asymptotical stability are provided for a special case. Numerical examples illustrate the main results.