This paper presents some stability synthesis results for the discrete-time linear switching systems whose dynamics contain reset functions and are determined by exogenous and uncontrollable events. First, for autonomous linear switching systems, conditions of stability are given in terms of a new definition of -controlled invariant set, and existence conditions to obtain such a set are presented. Then, under the assumption that the discrete state is known and the continuous state is unavailable for feedback, this new result is used to find the sufficient conditions for the existence of an observer-based stabilizing controller and dynamic output feedback controller. Such a design can be formulated in terms of linear matrix inequalities, which are numerically feasible with commercially available software. Finally, illustrative examples are given to indicate the effectiveness of the proposed design.