Lyapunov Functions to Avoid Squashed Sublevel Sets for Interconnections Containing Non-ISS Components *

Abstract This paper raises the issue of undesirable sublevel sets generated by Lyapunov functions for interconnection of integral input-to-state stable (iISS) systems. Sublevel sets are important information of forward invariant sets describing behavior of systems. This paper demonstrates for the previously available Lyapunov function that in the case of small stability margins, sublevel sets are compressed in one direction and stretched in the other direction extremely when a component system is not input-to-state stable (ISS). This paper also shows conservativeness of such sublevel sets by proposing a new Lyapunov function producing much better sublevel sets. The developments reveal practical issues to which we need to attend carefully when a component system is not ISS.