We introduce a special class of hybrid dynamical systems: cyclic linear differential automata (CLDA). We show that any CLDA can be reduced to a linear discrete-time system with periodic coefficients. Any CLDA has no equilibrium points. Therefore, the simplest attractor in such system is a periodic trajectory. We call a CLDA globally stable if it has a periodic trajectory which attracts all other trajectories of the system. A necessary and sufficient condition for global stability of CLDA is given. We apply our result to prove global stability of a flexible manufacturing system modelled as a switched server system.