  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.
