Input-Output-to-State Stability Tools for Hybrid Systems and Their Interconnections

We present results for the analysis of input/output properties of a general class of hybrid systems given by a flow set, a flow map, a jump set, a jump map, and an output map. For this class of systems, the notion of input-output-to-state stability is introduced in the first part of the technical note. Under mild assumptions on the functions and sets defining a hybrid system, sufficient conditions for this notion in terms of Lyapunov functions are derived. Equivalences between Lyapunov functions for input-output-to-state stability for asymptotic and exponential decay rates are established. The sufficient conditions and equivalences are linked to the existence of norm observers for hybrid systems. These results are used in the second part of the technical note to study interconnections of hybrid systems. An interconnection result in terms of a Lyapunov-based small gain theorem is also presented. Examples illustrate the results.