Lie Algebraic Criteria for Stability of Switched Systems of Differential Algebraic Equations (DAEs)

Abstract

In this letter, we prove two Lie algebraic criteria for exponential stability of switched systems of DAEs for a class of switching sequences. It is known that if the differential flows associated with the subsystems of a switched DAE commute, then the switched DAE with regular and stable DAE subsystems is asymptotically stable. Our first Lie algebraic result substitutes the condition of commutativity of the differential flows with the more general condition of solvability of the Lie algebra generated by them, and thus, clearly, generalizes the existing commutativity based result. There is, however, one caveat: the Lie algebraic criterion provides stability under switching signals that have a non-zero infimum dwell-time. This is in stark contrast with the case of conventional switched linear systems where the Lie algebraic condition works for all switching signals. We further explore this unique phenomenon of switched DAEs in this letter: we show that a switched DAE satisfying the Lie algebraic criterion is not necessarily stable for all arbitrary switching sequences. Finally, we derive a Lie algebraic stability criterion in terms of the original system matrices, as opposed to the earlier result involving differential flows.