Explaining the ‘`mystery’' of periodicity in inter-transmission times in two-dimensional event-triggered controlled system

—Motivated by scenarios where the communi- cation or the computation resources are limited, event-triggered control consists of transmitting data between the plant and the controller according to the actual system needs, and not the elapsed time since the last transmission instant as in traditional sampled-data control, so that the desired control objective is achieved. A range of tech- niques are nowadays available to design event-triggered controllers. However, we generally have only very little information about the actual behaviour of the transmission instants and thus about the amount of transmissions being actually generated, though this is a key feature of the design. In this paper, we analyse the inter-event times, i.e., the times between two successive transmission instants, when the plant is modeled as a two-dimensional linear time- invariant system. The controller is a state-feedback law and the triggering rule is the relative threshold policy, which is allowed to be time-regularized. One of the main results in this paper is the explanation of the oscillatory behaviour of the inter-event times when the constant used to deﬁne the threshold is small relative to 1 , a phenomenon commonly observed in simulations but never explained so far. More generally, the presented results help to understand the behaviour of the inter-event times, instead of solely relying on numerical simulations, and thereby can be exploited to rigorously evaluate the performance of the considered triggering condition in terms of (average) inter-transmission times.