As an extension of standard switched systems, this paper presents a new formulation of generalized switched systems. Such a generalization allows switching among different types of systems. Hence a hybrid system can be treated as a special case of such generalized switched systems. With the design freedom gained from this formulation, it is possible to design an appropriate switching signal between a family of continuous-time systems and a family of discrete-time systems to achieve a better performance. This paper introduces fundamental definitions and stability results. The concept of output persistent excitation condition is proposed, which links closely to uniform global asymptotic stability. To illustrate the usefulness of this new formulation, stability results of the hybrid systems are revisited to show how this formulation works.