Dissipativity theory and stability of feedback interconnections for hybriddynamical systems

In this paper we develop a unified dynamical systems framework for a genera l class of systems possessing left-continuous flows; that is, left-continuo us dynamical systems. These systems are shown to generalize virtually all e xisting notions of dynamical systems and include hybrid, impulsive, and swi tching dynamical systems as special cases. Furthermore, we generalize dissi pativity, passivity, and nonexpansivity theory to left-continuous dynamical systems. Specifically, the classical concepts of system storage functions and supply rates are extended to left-continuous dynamical systems providin g a generalized hybrid system energy interpretation in terms of stored ener gy, dissipated energy over the continuous-time dynamics, and dissipated ene rgy over the resetting events. Finally, the generalized dissipativity notio ns are used to develop general stability criteria for feedback interconnect ions of left-continuous dynamical systems. These results generalize the pos itivity and small gain theorems to the case of left-continuous, hybrid, and impulsive dynamical systems.