Dissipativity Theory for Switched Systems

A framework of dissipativity theory for switched systems using multiple storage functions and multiple supply rates is set up. Each subsystem of a switched system is associated with a storage function to describe the "energy" stored in the subsystem, and is associated with a supply rate that represents energy coming from outside the subsystem when the subsystem is active. The exchange of "energy" between the active subsystem and an inactive subsystem is characterized by cross-supply rates. Stability is reached when all supply rates can be made negative, as long as the total exchanged energy between the active subsystem and any inactive subsystems is finite in some sense. Two special forms of dissipativity, passivity and L ₂ -gain, are addressed. For both cases, asymptotic stability is guaranteed under certain "negative" output feedback plus asymptotic zero state detectability. Switched passivity conditions and switched L ₂-gain inequalities are, respectively, derived, which are generalizations of classical ones. Feedback invariance of passivity and a small-gain theorem are also given.