This paper deals with the analysis of evolution of inter-event times in linear systems under eventtriggered control and region-based self-triggered control. First, we consider planar systems under control with a class of scale-invariant event triggering rules. In this setting, the inter-event time is a function of the "angle" of the state at an event. We analyze the properties of this inter-event time function such as periodicity and continuity. Then, we analyze the evolution of the inter-event times by studying the fixed points of the "angle" map which represents the evolution of the "angle" of the state between two successive events. For a specific triggering rule, we provide necessary conditions for the convergence of interevent times to a steady state value. We also analyze stability and region of convergence of a fixed point of the "angle" map. Next, we consider n-dimensional systems under region-based self triggered control, in which the state space is partitioned into a finite number of conic regions and each region is associated with a fixed inter-event time. We analyze the evolution of inter-event times under the proposed self-triggered control method. In this framework, studying steady state behavior of the inter-event times is equivalent to studying the existence of a conic subregion, which is a positively invariant set under the map that gives the evolution of the state from one event to the next. We provide a sufficient condition for the existence of a special kind of positively invariant subregion called positively invariant ray. We also provide necessary and sufficient conditions for a positively invariant ray to be asymptotically stable. We also explore the existence of positively invariant subregions that do not include a positively invariant ray. We illustrate the proposed method of analysis and analytical results through numerical simulations.
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