Mathematical Problems in Engineering
Volume 7 (2001), Issue 4, Pages 299-335
Dissipativity theory and stability of feedback interconnections
for hybrid dynamical systems
1School of Aerospace Engineering, Georgia Institute of Technology, Atlanta 30332-0150, GA, USA
2Mechanical and Aerospace Engineering, University of Missouri, Columbia 65211, MO, USA
Received 19 October 2000
Copyright © 2001 Wassim M. Haddad and Vijaysekhar Chellaboina. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper we develop a unified dynamical systems framework for a general class of systems possessing left-continuous flows; that is, left-continuous dynamical systems. These systems are shown to generalize virtually all existing notions of dynamical systems and include hybrid, impulsive, and switching dynamical systems as special cases. Furthermore, we generalize dissipativity, 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 providing a generalized hybrid system energy interpretation in terms of stored energy, dissipated energy over the continuous-time dynamics, and dissipated energy over the resetting events. Finally, the generalized dissipativity notions are used to develop general stability criteria for feedback interconnections of left-continuous dynamical systems. These results generalize the positivity and small gain theorems to the case of left-continuous, hybrid, and impulsive dynamical systems.