Analysis of Inter-Event Times in Linear Systems under Region-Based Self-Triggered Control

This paper analyzes the evolution of inter-event times (IETs) in linear systems under region-based self-triggered control (RBSTC). In this control method, the state space is partitioned into a finite number of conic regions and each region is associated with a fixed IET. In this framework, studying the steady state behavior of the IETs is equivalent to studying the existence of a conic subregion that is positively invariant under the map that gives the evolution of the state from one event to the next. We provide necessary conditions and sufficient conditions for the existence of a positively invariant subregion (PIS). We also provide necessary and sufficient conditions for a PIS to be asymptotically stable. Indirectly, they provide necessary and sufficient conditions for local convergence of IETs to a constant or to a given periodic sequence. We illustrate the proposed method of analysis and results through numerical simulations.