Analysis of Inter-Event Times for Planar Linear Systems Under a General Class of Event Triggering Rules

This paper analyzes the evolution of inter-event times for a planar linear system under a general class of scale-invariant event triggering rules. For scale-invariant event triggering rules, the inter-event time is a function of only the "angle" of the state at an event. We analyze the properties of this inter-event time function such as periodicity and continuity. In particular, the inter-event time function is continuous except for finitely many "angles" and we provide sufficient conditions under which the inter-event time function is continuous. Then, we analyze the evolution of the "angle" of the state from one event to the next and reduce the problem of studying the evolution of the inter-event times to that of studying the "angle" map and its fixed points. For a specific triggering rule, we provide necessary conditions for the convergence of inter-event time to a steady state value. We illustrate the proposed results through numerical simulations.