Qualitative analysis of differential automata: existence and stability of limit cycles

Abstract

We introduce a new class of hybrid dynamical systems. We call these hybrid systems cyclic linear differential automata (CLDA). Our main results show that any CLDA can be reduced to a linear discrete-time system with periodic coefficients. Any CLDA has no singular points. Therefore, the simplest attractor in such systems is a periodic trajectory. We call a CLDA globally stable if it has a periodic trajectory which attracts all other trajectories of this 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. Furthermore, we prove global stability of a class of switched server flow networks.