L₂-Gain Analysis of Periodic Event-Triggered Control and Self-Triggered Control Using Lifting

We analyze the stability, and L2-gain properties of a class of hybrid systems that exhibit time-varying linear flow dynamics, periodic time-triggered jumps, and arbitrary nonlinear jump maps. This class of hybrid systems encompasses periodic event-triggered control, self-triggered control, and networked control systems including time-varying communication delays. 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 derived 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 (meaning the ℓ2-gain is smaller than 1) 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.