On necessary and sufficient conditions for exponential and L2 stability of planar reset systems

In this paper we provide necessary and sufficient conditions for exponential stability and L2 stability of planar reset systems, i.e., systems involving a First Order Reset Element (FORE) and a linear plant having dimension one. The proof relies on Lyapunov tools developed in a recent novel representation of a class of reset systems incorporating this special planar case. Explicit Lyapunov functions are given to show both exponential and L2 stability. Based on this Lyapunov function, an explicit estimate of the L2 gain, depending on the system’s parameters, is provided. Moreover, via the same tools, it is shown that the gain estimates go to zero as certain parameters (in particular, the FORE pole) become arbitrarily large, thus allowing to establish a small gain result showing stability of certain higher order SISO linear plants under the action of a FORE.