On equivalence of two stability criteria for continuous-time switched systems with dwell time constraint

In this note, a study on the equivalence of two stability criteria for continuous-time switched linear systems with dwell time constraint is presented. It is demonstrated that, for any dwell time satisfying the conditions in stability criterion proposed in Geromel and Colaneri (2006), the conditions in stability criterion of Allerhand and Shaked (2011) also hold as long as the number of decision variables and related LMIs is sufficiently large, which implies the two stability criteria are intrinsically equivalent. The equivalence is obtained by showing that two criteria can be derived from each other with a sufficiently large number of decision variables and LMIs. A numerical example is proposed to illustrate the theoretical results, and an extension to uncertain case is briefly presented.