This paper analyzes the asymptotic behavior of inter-event times in planar linear systems, under event-triggered control with a general 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. This viewpoint allows us to analyze the inter-event times by studying the fixed points of the , which represents the evolution of the “angle” of the state from one event to the next. We provide a sufficient condition for the convergence or non-convergence of inter-event times to a steady state value under a scale-invariant event-triggering rule. Following up on this, we further analyze the inter-event time behavior in the special case of threshold based event-triggering rule and we provide various conditions for convergence or non-convergence of inter-event times to a constant. We also analyze the asymptotic average inter-event time as a function of the angle of the initial state of the system. With the help of ergodic theory, we provide a sufficient condition for the asymptotic average inter-event time to be a constant for all non-zero initial states of the system. Then, we consider a special case where the is an orientation-preserving homeomorphism. Using rotation theory, we comment on the asymptotic behavior of the inter-event times, including on whether the inter-event times converge to a periodic sequence. We illustrate the proposed results through numerical simulations.

中文翻译：

---

事件触发控制下平面系统事件间时间的渐近行为

本文分析了在具有一般类尺度不变事件触发规则的事件触发控制下平面线性系统中事件间时间的渐近行为。在此设置中，事件间时间是事件状态“角度”的函数。这一观点使我们能够通过研究 的不动点来分析事件间时间，它代表了从一个事件到下一个事件的状态“角度”的演变。我们提供了在尺度不变事件触发规则下事件间时间收敛或不收敛到稳态值的充分条件。在此基础上，我们进一步分析了基于阈值的事件触发规则的特殊情况下的事件间时间行为，并提供了事件间时间收敛或不收敛到常数的各种条件。我们还分析了渐近平均事件间时间作为系统初始状态角度的函数。借助遍历理论，我们为系统所有非零初始状态的渐近平均事件间时间为常数提供了充分条件。然后，我们考虑一种特殊情况，其中 是方向保持同胚。使用旋转理论，我们评论事件间时间的渐近行为，包括事件间时间是否收敛到周期序列。我们通过数值模拟说明了所提出的结果。