L₂-gain analysis of periodic event-Triggered systems with varying delays using lifting techniques

In this paper we study the stability and L₂-gain properties of periodic event-Triggered control (PETC) systems including time-varying delays. We introduce a general framework that captures these PETC systems and encompasses a class of hybrid systems that exhibit linear flow, aperiodic time-Triggered jumps (possibly with different deadlines) and arbitrary nonlinear time-varying jump maps. New notions on the stability and contractivity (L₂-gain strictly smaller than 1) from the beginning of the flow and from the end of the flow are introduced and formal relationships are deduced between these notions, revealing that some are stronger than others. Inspired by ideas from lifting, it is shown that the internal stability and contractivity in L₂-sense of a continuous-Time hybrid system in the framework is equivalent to the stability and contractivity in l₂-sense of an appropriate time-varying discrete-Time nonlinear system. These results recover existing works in the literature as special cases and indicate that analysing different discrete-Time nonlinear systems (of the same level of complexity) than in existing works yield stronger conclusions on the L₂-gain. At the end of the paper we indicate several extensions of the framework, which even include the possibility of the interjump times depending on the state, such that, for instance, self-Triggered control systems can also be included allowing their stability and contractivity analysis. A numerical example is presented showing how stability and contractivity analyses are carried out for PETC systems with delays.