A note on input-to-state stability and averaging of fast switching systems

Abstract

We consider the averaging method for stability of rapidly switching nonlinear and linear systems with disturbances. First, we show that the input-to-state stability (ISS) analysis results for continuous-time systems in [7] apply directly to nonlinear rapidly switched systems that we consider. We show that the notions of strong and weak averages that were proposed in [7] play an important role in the context of switched systems. A direct application of results in [7] yields conditions under which ISS of the strong average implies semi-global practical ISS (SGP-ISS) of the switched system. A similar result was shown to hold for weak averages but the conclusion is slightly weaker as we can only prove semi-global practical differential ISS (SGP-DISS) that requires the derivatives of disturbances to be bounded. Although these results follow directly from [7], to the best of our knowledge they were not known in the switched systems literature. In the second part of the paper, we use the notions of strong and weak averages, as well as stronger conditions on the linear switched system and the ISS property to obtain stronger conclusions. In particular, we give conditions under which ISS of the strong and weak average respectively imply global ISS and global DISS of the actual system. We state two auxiliary results on closeness of solutions between weak or strong averages and the actual linear switched system that were not reported in [7]. These results are of interest in their own right but also they are used to prove the global ISS or DISS results.